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Interactions between climate and the rise of explosive volcanic plumes : a new feedback in the Earth… Aubry, Thomas J. 2018

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Interactions between climate and the rise of explosive volcanicplumes: A new feedback in the Earth system.byThomas J. AubryBSc, Physics, Ecole Normale Supérieure de Cachan & Université Pierre and Marie Curie, 2011MSc, Ocean, Atmosphere, Climate and Space Observations, Université Pierre and Marie Curie& Ecole Normale Supérieure de Paris, 2013Diploma of the Ecole Normale Supérieure, Physics, Ecole Normale Supérieure de Cachan, 2014a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Geophysics)The University of British Columbia(Vancouver)May 2018c© Thomas J. Aubry, 2018The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:“Interactions between climate and the rise of explosive volcanic plumes: A new feedback in theEarth system.”submitted by Thomas J. Aubry in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Geophysics.Examining Committee:Prof. A. Mark Jellinek, Department of Earth, Ocean and Atmospheric Sciences, University ofBritish ColumbiaSupervisorProf. Valentina Radić, Department of Earth, Ocean and Atmospheric Sciences, University ofBritish ColumbiaSupervisory Committee MemberProf. Stephanie Waterman, Department of Earth, Ocean and Atmospheric Sciences, Universityof British ColumbiaSupervisory Committee MemberProf. Kelly Russel, Department of Earth, Ocean and Atmospheric Sciences, University ofBritish ColumbiaUniversity ExaminerProf. Gregory A. Lawrence, Department of Civil Engineering, University of British ColumbiaUniversity ExaminerProf. Alan Robock, Department of Environmental Sciences, Rutgers UniversityExternal ExaminerAdditional Supervisory Committee Members:Prof. Phil Austin, Department of Earth, Ocean and Atmospheric Sciences, University of BritishColumbiaSupervisory Committee MemberiiAbstractVolcanic plumes rising above the tropopause inject SO2 directly into the stratosphere, where itforms sulfate aerosols that modulate Earth’s radiative balance. Stratospheric volcanic sulfateaerosol forcing reduces Earth’s surface temperature and is a predominant driver of climatevariability. The processes that govern the volcanic injection of SO2 into the stratosphere arecontrolled to a large extent by climate. Thus, climate changes may affect stratospheric volcanicSO2 inputs, volcanic forcing and climate, in turn. The assessment of this potential feedbackis hindered by difficulties in understanding and constraining observationally the key processesgoverning plume rise.To address this challenge, we compile a new exhaustive database of eruption source param-eters, along with their uncertainties (Aubry et al., 2017b). We apply these data along withthe results of laboratory experiments to compare the performances of our newly proposed andpublished scalings for predicting volcanic plume heights. We demonstrate that plume heightsare captured better by scalings accounting for atmospheric conditions (Aubry et al., 2017b).Furthermore, we evaluate 1D models of volcanic plume using the experimental and naturaleruption datasets. We show that these new datasets enable reliable constraints on processescritical to plume rise including the rate of entrainment of atmosphere as well as the role con-densation of water vapor (Aubry et al. (2017a) and Chapter 4). Significant limitations in thecompiled data remain and we identify future improvements required to improve plume modelsevaluation.Next, we explore the impacts of climate projections for ongoing global warming on therise height of volcanic plumes and SO2 injection into the stratosphere. Our results reveal anovel feedback where global warming will reduce stratospheric injections of SO2 by explosiveeruptions (Aubry et al., 2016). This would lead to reduced volcanic forcing and surface cooling,and enhance global warming, in turn. To test this feedback, we develop a new idealized model ofvolcanic aerosol forcing and show that the proposed feedback may have important implicationsif greenhouse gas concentrations continue to increase at currents rates (Chapter 6). An excitingfuture direction is to assess interactions among the proposed feedback with other publishedclimate-volcano feedbacks.iiiLay SummaryExplosive volcanic eruptions that inject sulfur into the stratosphere, at altitudes higher than'10-16 km, induce a cooling of Earth’s surface. This volcanic “forcing” is one of the mostimportant drivers of Earth’s climate variability. The rise of eruptive volcanic plumes governswhich eruptions will inject gases directly into the stratosphere and impact climate. We gathernew datasets to calibrate and evaluate physical models that predict the height reached by avolcanic plume. Atmospheric conditions control this rise height to a large extent. Accordingly,we apply these models to investigate whether ongoing anthropogenic global warming will affectthe ability of explosive eruptions to inject gases into the stratosphere and cool climate. We findthat global warming will reduce stratospheric injections by explosive eruptions and, in turn,slightly reduce volcanic surface cooling and enhance global warming. Such a positive feedbackhas implications for predicting future climate and understanding past climate changes.ivPrefaceThis thesis is original work completed by Thomas J. Aubry. Guidance was given by the super-visory committee and the collaborators listed below for each chapter. This thesis includes fourpublished manuscripts and one complementary Chapter that will be submitted for publicationas one or two manuscripts.Chapters 2, 3 and 4 use a dataset of laboratory experiments published in Carazzo et al. (2014)and that is not part of this thesis. Guillaume Carazzo is responsible for initiating these expe-rience. I contributed to designing and setting up the experimental design. I am responsiblefor calibrating the experimental apparatus and performing all experiments. I conducted thelatter work as a research assistant at the Institut de Physique du Globe de Paris over the threemonths preceding my PhD. I am responsible for all the analyzes of the experimental datasetpresented in this dissertation, and I performed these analyzes exclusively during my PhD.Chapter 2 is published in Journal of Volcanology and Geothermal Research (Aubry et al., 2017b).The co-authors are A. Mark Jellinek, Guillaume Carazzo, Rose Gallo, Kim Hatcher and JackDunning. I am responsible for initiating this study, developing its design, proposing the newlydeveloped scaling, compiling the new database of eruption source parameters, analyzing all datapresented and writing the majority of the manuscript. A. Mark Jellinek and Guillaume Carazzoprovided feedback on the derivation of the scaling and the analysis of the data. Rose Gallo, KimHatcher and Jack Dunning contributed to compile ca. 10% of the database of eruption sourceparameters under my supervision. All co-authors provided editorial review of the manuscriptprior to publication.Aubry, T. J., A. M. Jellinek, G. Carazzo, R. Gallo, K. Hatcher and J. Dunning (2017), Anew analytical scaling for turbulent wind-bent plumes: Comparison of scaling laws with ana-log experiments and a new database of eruptive conditions for predicting the height of volcanicplumes , Journal of Volcanology and Geothermal Research, doi:10.1016/j.jvolgeores.2017.07.006.Chapter 3 is published in Geophysical Research Letters (Aubry et al., 2017a). The co-authorsvare Guillaume Carazzo and A. Mark Jellinek. I am responsible for initiating this study, devel-oping its design, developing algorithms to analyze laboratory experiments, analyzing all datapresented and writing the majority of the manuscript. Guillaume Carazzo performed the plumemodel runs. All co-authors provided editorial review of the manuscript prior to publication.Aubry, T. J., G. Carazzo and A. M. Jellinek (2017), Turbulent entrainment into volcanic plumes:New constraints from laboratory experiments on buoyant jets rising in a stratified crossflow,Geophysical Research Letters, doi:10.1002/2017GL075069.Chapter 4 is published in Earth and Planetary Science Letters (Aubry and Jellinek, 2018). Myco-author is A. Mark Jellinek. I am responsible for initiating this study, developing its design,running the plume models and analyzing the results. I wrote the majority of the manuscriptwith extensive feedback by A. Mark Jellinek prior submissions.Aubry, T. J. and A. M. Jellinek (2018), New insights on entrainment and condensation involcanic plumes: Constraints from independent observations of explosive eruptions and implica-tions for assessing their impacts, Earth and Planetary Science Letters, doi:10.1016/j.epsl.2018.03.028.Chapter 5 is published in Journal of Geophysical Research: Atmospheres (Aubry et al., 2016).The co-authors are A. Mark Jellinek, Wim Degruyter, Costanza Bonadonna, Valentina Radić,Margot Clyne and Adjoa Quainoo. I am responsible for initiating this study, developing itsdesign, running the plume model, analyzing all data presented and writing the majority of themanuscript. Wim Degruyter provided me with a version of the plume model which I adaptedto the study needs. A. Mark Jellinek, Wim Degruyter, Costanza Bonadonna and ValentinaRadić provided feedback on the analyzes performed for this study. Margot Clyne and Ad-joa Quainoo contributed to compiling global climate model and climate reanalyses data undermy supervision. All co-authors provided editorial review of the manuscript prior to publication.Aubry, T. J., A. M. Jellinek, W. Degruyter, C. Bonadonna, V. Radić, M. Clyne and A. Quainoo(2016), Impact of global warming on the rise of volcanic plumes and implications for future vol-canic aerosol forcing, Journal of Geophysical Research: Atmospheres, doi:10.1002/2016JD025405.Chapter 6 is a project started in collaboration with A. Mark Jellinek, Matthew Toohey andAnja Schmidt. I am responsible for initiating this project, developing its design, and for con-ducting all the work presented including writing the chapter. Mark Jellinek and MatthewToohey provided feedback on the model design, the analyzes conducted and the writing of theChapter. I will present the main results of this chapter at the American Geophysical UnionviChapman conference “Stratospheric aerosol in the post-Pinatubo era: Processes, Interactions,and Importance” in March 2018, and we expect that this work will lead to two publications.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Supplementary Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A key driver of climate variability: Volcanic aerosol-radiation interactions. . . . . 11.2 From the vent to the atmosphere: Processes governing volcanic plume rise andsubsequent aerosol forcing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Research questions and overview of the thesis. . . . . . . . . . . . . . . . . . . . . 71.4 Major contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A new analytical scaling for turbulent wind-bent plumes: Comparison ofscaling laws with analog experiments and a new database of eruptive con-ditions for predicting the height of volcanic plumes . . . . . . . . . . . . . . 102.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Overview: volcanic plume height and the problem of wind stresses . . . . 112.2.2 Plume rising in quiescent conditions: the Morton et al. (1956) scaling . . 122.2.3 Plume rising under windy conditions: the Hewett et al. (1971) scaling . . 14viii2.2.4 Plume rising under arbitrary wind stress: overview of existing ”func-tional" scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.5 Summary and motivation for a new analytical scaling law . . . . . . . . . 182.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Derivation of a new analytical scaling . . . . . . . . . . . . . . . . . . . . 192.3.2 Discussion of additional assumptions . . . . . . . . . . . . . . . . . . . . . 222.4 Data and method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 Data from laboratory experiments . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Data from explosive eruptions . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.3 Dynamical similarity of laboratory experiments . . . . . . . . . . . . . . . 252.4.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Plume height predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.2 Entrainment coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.3 Best set of parameters to evaluate mass eruption rate . . . . . . . . . . . 302.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.1 New constraint on entrainment coefficients: Implications . . . . . . . . . . 322.6.2 Plume shape and the Morton et al. (1956) similarity theory. . . . . . . . . 332.6.3 Future challenges for testing scaling and integral plume models . . . . . . 362.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Turbulent entrainment into volcanic plumes: New constraints from labo-ratory experiments on buoyant jets rising in a stratified crossflow . . . . . 413.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Data and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5.1 Best set of entrainment parameters for each model . . . . . . . . . . . . . 463.5.2 Best model for entrainment coefficients . . . . . . . . . . . . . . . . . . . 483.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6.1 New constraints on entrainment parameters and implications for volcanicplume modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6.2 Parameterization of entrainment coefficients and future work . . . . . . . 513.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 New insights on entrainment and condensation in volcanic plumes: Con-straints from independent observations of explosive eruptions and implica-tions for assessing their impacts. . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54ix4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Overview of the tested models . . . . . . . . . . . . . . . . . . . . . . . . 584.3.2 Condensation of entrained water vapor . . . . . . . . . . . . . . . . . . . . 594.3.3 Models for entrainment coefficients . . . . . . . . . . . . . . . . . . . . . . 604.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.1 Metric for model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.2 Uncertainties and Monte Carlo simulations . . . . . . . . . . . . . . . . . 634.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6.1 Best set of parameters for each model . . . . . . . . . . . . . . . . . . . . 654.6.2 Best models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.7.1 Best parameterization of condensation and implications for predictingclimate impact of explosive eruptions . . . . . . . . . . . . . . . . . . . . . 684.7.2 Best entrainment model: Implication for a plume collapse-buoyant plumetransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.7.3 What is plume height? Toward improving future databases of eruptiveparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Impact of global warming on the rise of volcanic plumes and implicationsfor future volcanic aerosol forcing . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Data and plume model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.1 Source conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2 Atmospheric conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.3 Integral volcanic plume model . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4.1 Impact of temperature and geopotential height changes under RCP8.5 . . 885.4.2 Impact of horizontal wind speed changes under RCP8.5 . . . . . . . . . . 915.4.3 Impact of combined changes of temperature, geopotential height and hor-izontal wind speed under RCP8.5 . . . . . . . . . . . . . . . . . . . . . . . 925.4.4 Summary: Results for all investigated regions, periods and scenarios . . . 935.4.5 Height projections for past eruptions . . . . . . . . . . . . . . . . . . . . . 955.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5.1 Mechanisms driving changes in plume and tropopause heights . . . . . . . 975.5.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.5.3 Implications for future volcanic forcing . . . . . . . . . . . . . . . . . . . . 102x5.5.4 Limitations and potential improvements: beyond a binary view of vol-canic aerosol forcing sensitivity to plume height . . . . . . . . . . . . . . . 1065.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 A new idealized model of stratospheric volcanic sulfate aerosol forcing:application to quantify the proposed climate-volcano feedback. . . . . . . . 1106.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 Data and model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.3 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.3.4 SO2 injection in the model . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Calibration of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4.1 Scaling for global mean stratospheric aerosol optical depth . . . . . . . . 1226.4.2 Scaling for global mean aerosol effective radius . . . . . . . . . . . . . . . 1236.4.3 Model timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.4 Model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.5 Understanding the impact of plume height on stratospheric volcanic sulfateaerosol forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5.1 Sensitivity of volcanic forcing to injection height and latitude . . . . . . . 1336.5.2 Scenarios for future stratospheric volcanic sulfate aerosol forcing . . . . . 1356.5.3 Quantification of the climate-volcano feedback proposed by Aubry et al.(2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.2.1 Better datasets for understanding volcanic plume dynamics. . . . . . . . . 1457.2.2 3D vs 1D plume models: performance and assessment of climate-volcanofeedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2.3 Climate-volcano interactions . . . . . . . . . . . . . . . . . . . . . . . . . 146Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A Supporting Information published along Chapter 2 (JVGR 2017) . . . . . . 180A.1 Degruyter and Bonadonna (2012) scaling for plume height . . . . . . . . . . . . . 180A.2 Natural eruptions parameters uncertainties . . . . . . . . . . . . . . . . . . . . . 181A.2.1 Erupted mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181A.2.2 Eruption duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182xiA.2.3 Plume height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182A.2.4 Differences with Mastin (2014) and concluding remarks on compiling adatabase of eruptive parameters . . . . . . . . . . . . . . . . . . . . . . . 183B Supporting Information published along Chapter 3 (GRL 2017) . . . . . . 189C Supporting Information submitted along Chapter 4 (EPSL, under review) 190D Supporting Information published along Chapter 5 (JGRA 2016) . . . . . . 196D.1 Evaluation of Global Climate Models . . . . . . . . . . . . . . . . . . . . . . . . . 196D.2 Significance test for changes in median H∗ and median critical M∗0 . . . . . . . . 199D.3 Sensitivity to Relative Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 200D.4 Sensitivity to source conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202E Supporting Information for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . 204E.1 Complementary figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204E.2 Ongoing and future model development . . . . . . . . . . . . . . . . . . . . . . . 205E.2.1 Datasets used for model calibration . . . . . . . . . . . . . . . . . . . . . 205E.2.2 Scalings for stratospheric aerosol optical depth and effective radius . . . . 206E.2.3 SO2 injection and transport . . . . . . . . . . . . . . . . . . . . . . . . . . 206E.2.4 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207xiiList of TablesTable 2.1 Symbols used in this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Table 2.2 Optimal exponents associated to each scaling for predicting mass eruption rates 32Table 5.1 Values of model parameters and eruption source conditions . . . . . . . . . . . 80Table 5.2 Subset of the volcanic eruptions chosen to test the impact of climate changeon plume height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Table 5.3 Relative changes in the ratio of plume to tropopause height for eruptive con-ditions for which this ratio is equal to 1 for the reference period . . . . . . . . 93Table 5.4 Relative changes in the critical mass eruption rate required to reach thetropopause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Table 6.1 Optimal values of the model parameters . . . . . . . . . . . . . . . . . . . . . 128Table 6.2 Projected changes in stratospheric volcanic sulfate aerosol forcing driven bygreenhouse gases emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Table C.1 Symbols used in conservation equations of the 1D model . . . . . . . . . . . . 191Table D.1 Global climate models evaluated and availibility of output variables for selectCMIP5 experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Table D.2 Latitudinal and longitudinal boundaries of the regions used in this study. . . . 199xiiiList of FiguresFigure 1.1 Overview of the processes governing the climate response to volcanic sulfateaerosol forcing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 2.1 Overview of entrainment parameterizations and scalings used across variouswind stress regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Cartoon of a rising plume and control volume used to derive the scaling . . . 15Figure 2.3 Predictions for volcanic plume height of published scalings . . . . . . . . . . . 18Figure 2.4 Plume shape regime diagram for laboratory experiments and natural eruptions 25Figure 2.5 Comparison of predicted vs observed plume height for laboratory experiments 27Figure 2.6 Comparison of predicted vs observed plume height for explosive eruptions . . 28Figure 2.7 New constraints on entrainment coefficients . . . . . . . . . . . . . . . . . . . 29Figure 2.8 Comparison of predicted vs observed mass eruption rate . . . . . . . . . . . . 31Figure 2.9 Plume trajectories of select laboratory experiments . . . . . . . . . . . . . . . 33Figure 2.10 Evolution of the plume asymmetry as a function of the wind Richardsonnumber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.11 Comparison of predicted vs observed plume height for the laboratory exper-iments of Contini et al. (2011) . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.12 Residual height for each scaling as a function of the atmosphere relativehumidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.1 Plume centerline for a select experiment and predictions from a 1D plumemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.2 Probability distributions of entrainment parameters (laboratory experiments) 47Figure 3.3 Probability distributions of pairwise differences between model error (labo-ratory experiments) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.4 Implications of our results for predicting plume collapse and climate impacts 50Figure 3.5 Dependence of entrainment rates on the source Richardson number and ve-locity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.1 Uncertainties on plume height-mass eruption rate relationship in 1D volcanicplume models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55xivFigure 4.2 Cartoon of a volcanic plume and parameterizations of entrainment and con-densation used in 1D models of volcanic plume . . . . . . . . . . . . . . . . . 58Figure 4.3 Map of the eruptive events for which plume height and mass eruption ratesare independently constrained in Aubry et al. (2017b) . . . . . . . . . . . . . 62Figure 4.4 Probability distributions for entrainment and condensation parameters of allmodels (natural eruptions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.5 Probability distributions of pairwise differences between model error (naturaleruptions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.6 Critical mass eruption rate required for direct stratospheric injections underdifferent climate conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.7 Regime diagram for the collapse and rise of volcanic column . . . . . . . . . . 70Figure 4.8 Sensibility of constraints on model parameters to the definition of plume height 72Figure 5.1 Map of the 12 regions for which we investigate the impact of global warmingon volcanic plume rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.2 Work-flow of our methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.3 Evaluation of the global climate models used in our study . . . . . . . . . . . 85Figure 5.4 Cartoon of a rising volcanic plume and key concepts for 1D plume models . . 86Figure 5.5 Impacts of projected changes in temperature and geopotential height on vol-canic plume height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 5.6 Impact of projected changes in wind speed on volcanic plume height . . . . . 91Figure 5.7 Projected changes for the ratio of plume height to tropopause height . . . . . 92Figure 5.8 Projected height (relative to tropopause height) of select historical eruptionsunder future climate conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 5.9 Projected heights (relative to tropopause height) of 1979-2015 eruptions un-der future climate conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 5.10 Sensitivity of plume height projections to the choice of global climate model . 99Figure 5.11 Sensitivity of plume height projections to the choice of entrainment rates inthe 1D plume model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 5.12 Projections for the flux of volcanic SO2 into the stratosphere . . . . . . . . . 103Figure 6.1 Datasets used to calibrate the new idealized model of volcanic forcing . . . . 114Figure 6.2 Structure of the new idealized model . . . . . . . . . . . . . . . . . . . . . . . 115Figure 6.3 Shape functions for predicting observed extinction from the observed SAODin the eight boxes of the model . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 6.4 Extinction in GLOSSAC compared to the extinction reconstructed from theobserved SAOD in the eight boxes of the model . . . . . . . . . . . . . . . . . 117Figure 6.5 Extinction increase following the Pinatubo 1991 and Calbuco 2015 eruptionsand resulting fit for associated “SO2 clouds” . . . . . . . . . . . . . . . . . . . 120Figure 6.6 “SO2 clouds” height, latitude and extent estimated from GLOSSAC . . . . . 121xvFigure 6.7 Global mean SAOD vs. sulfur burden in GLOSSAC and WACCM . . . . . . 122Figure 6.8 Aerosol effective radius vs global mean SAOD in SAGE4λ and WACCM . . . 123Figure 6.9 SO4 fluxes in the new idealized model of volcanic forcing . . . . . . . . . . . 126Figure 6.10 Model performance in reconstructing SAOD as a function of additional fluxesand parameterization used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 6.11 Model prediction vs. observations for the SAOD in the eight boxes of themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure 6.12 Model prediction vs. observations for the global mean SAOD . . . . . . . . . 131Figure 6.13 Shape function used to transform the SAOD predictions in the eight boxesinto latitude/altitude dependent extinction . . . . . . . . . . . . . . . . . . . 131Figure 6.14 Model prediction vs. observations for extinction at 525 nm . . . . . . . . . . 132Figure 6.15 Sensitivity of forcing amplitude and decay timescale to the latitude and alti-tude of volcanic SO2 injections . . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 6.16 Sensitivity of the time-latitude evolution of SAOD to SO2 injection altitudeand latitude for three select examples . . . . . . . . . . . . . . . . . . . . . . 134Figure 6.17 Estimated volume flow rate vs. mass of SO2 for eruptive events of the Carnet al. (2016) dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Figure 6.18 Example of scenario for future vent-level eruptive conditions, atmosphericinjections of SO2, SAOD and radiative forcing . . . . . . . . . . . . . . . . . 137Figure 6.19 Impact of the resampling strategy on scenarios for future volcanic forcing . . 139Figure 6.20 Impact of global warming on volcanic forcing for a select scenario of futureeruptive conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Figure A.1 Exact solution of conservation equations vs. proposed scaling . . . . . . . . . 184Figure A.2 Observed vs. predicted heights of laboratory plumes using two differentregimes parameters for wind stress in our proposed scaling . . . . . . . . . . 185Figure A.3 Comparison of laboratory experiment and natural eruption conditions to theassumptions used to derive the new scaling . . . . . . . . . . . . . . . . . . . 186Figure A.4 Mean angle of rise and wind speed to buoyancy-scaled velocity ratio for lab-oratory experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Figure A.5 Select cross-sections of tracer intensity in laboratory experiments, and gaus-sian fits used to derive plume radius . . . . . . . . . . . . . . . . . . . . . . . 188Figure C.1 Metric for convergence of model parameter distributions as a function of thenumber of Monte Carlo simulations performed . . . . . . . . . . . . . . . . . 192Figure C.2 Sensitivity of model error to plume height definition for model 2 . . . . . . . 193Figure C.3 Sensitivity of the condensation rate to plume height definitions for tropicaleruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194xviFigure D.1 Results sensitivity to parameterization of condensation in the 1D model ofvolcanic plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Figure D.2 Sensitivity of our results to eruption source conditions other than the masseruption rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Figure D.3 Eruption frequency as a function of erupted mass of SO2 in the Sigl et al.(2015) and Carn et al. (2016) datasets . . . . . . . . . . . . . . . . . . . . . . 203Figure E.1 Global mean SAOD in GLOSSAC compared to the global mean SAOD re-constructed from the observed SAOD in the eight boxes of the model . . . . 204Figure E.2 Distribution of parameter values maximizing the percentage of explainedSAOD variability when using a simple parameterization of the loss term . . . 205Figure E.3 Distribution of parameter values maximizing the percentage of explainedSAOD variability when using a Stokes parameterization of the loss term . . . 205xviiList of Supplementary MaterialsThe following tables are spreadsheets provided as supplementary materials. When necessary,captions are provided in the appendix of the chapter corresponding to a supplementary table.Table A.1 provides measured parameters for the laboratory experiments used in Chapter 2.Table A.2 provides parameters of the natural explosive eruptions for which plume height andmass eruption rate are independently constrained in Chapter 2.Table B.1 provides uncertainties on experimental parameters used to perform Monte Carlosimulations in Chapter 3.Table C.2 provides the regime parameters and fraction of tephra originating from pyroclasticdensity current for eruptions used in Figure 4.7 of Chapter 4.xviiiAcknowledgmentsIn fall 2011, I was looking for a lab where to visit for the international research internship of myfirst year of MSc. I would then do a joint masters in climate science and political science, andfind a job at the interface of science and politics in France. There was no question about thecity where I would go for my internship. I would head to Vancouver, work for a few months,and spend the summer accomplishing my childhood dream and paddling with the orcas of theJohnstone straight. I only had to find a research supervisor and Mark Jellinek ended up beingmy victim. At that time, I didn’t know that my supposedly short encounter with Mark andBritish Columbia would completely change my well-defined plan and lead to this incredibleadventure of more than four years. I would like to express my heartfelt thanks to all of youwho shared this journey.Some of my achievements during my research internship with Mark include: i) arrivingwithout a visa and almost not passing the custom; ii) finishing half-naked in the 1m3 tankwith Yoshi pouring water on me, after I spilled on myself half of the acidic “ocean” in whichI was dissolving pyrites; and iii) wasting Kathi’s research summer with the noise of the stirrermixing this ocean 24/7. Considering this productive internship, it remains a mystery to mewhy Mark put his trust and faith in me. I will be forever thankful for this. You opened myeyes on the obvious truth that becoming a scientist was a necessary step before thinking aboutinteracting with policy makers on science, which ultimately led me on the PhD path. Overthe last few years, you gave me the freedom to explore any idea I had and made of me anindependent scientist. You have always been enthusiastic about what I was doing, but alsoalways pushed me further. A single sentence from you, repeated multiple times to overcomemy stubbornness, has this amazing ability to change the outcome of our publications. “Maybeyou should non-dimensionalize this equation”... “Maybe you should define a metric for SO2injections”... In order to solve my English problems, I was helped by you to improve my Englishby avoiding repetition (and parenthetical comments), passive voice and endless sentences thatcan be followed by nobody but me due to the fact that I am tending to write in English basedon the French. Indeed, because communication is key in science, your outstanding efforts hasbeen allowing me to make very key improvements and these last two sentences are a tributeto your teaching and your patience. Have faith! Above everything else, you are a deeply goodhuman being. You help us to enjoy life and find what matters which is critical in an adventurewhere it is so easy to focus on the challenges, pressure, and deceptions inevitably encountered.xixMerci!I have grown surrounded by the love and support from my mother, Béatrice, father, Gabrieland my two brothers, Mathieu and Thibaut. You have all my love and gratitude. Who I amtoday begins with your love, with your encouragement to pursue my passions, with the storiesof Ulysses’ travels and endless chess games. You pushed me further and gave me the meansto grow while placing my well-being above all. I will always remember the day when, in themiddle of a frequent storm during my “classe preparatoire”, you called me to explain the planyou made so that I could study marine biology in Vancouver if I wanted to change my lifetrajectory. My two brothers are my models. Mathieu is my greatest source of inspiration fortrying to be a good scientist. The time you took with me in difficult moments of my life, York or in Vienne, has been so important to me. Thibaut is five years younger than me but,in many aspects, he is more mature than I am. You do not hesitate to radically change yourtrajectory when you know it is the right thing to do. You chose one of the most importantjob - I sure know it after four years in grad school - and you will bring much love in thisworld. My passion for science and nature also started with my grandmother Francoise and myGrandfather Jacques. I can’t thank you enough for your patience when taking me fishing andsolve the impossible knots I was constantly making with the line. I am so proud to write thefirst name of my grandfather on each of my attempt to contribute to science, and both of youare on board with me in this adventure.To the world-known “MJCJ” group and all my lab-mates. You have been the most preciousfellows during this adventure. Kathi (I hope we will soon share many more teas in the UK),Catherine (what could possibly be better than attending InSight’s launch to celebrate mydefense?), Yoshi (thanks for one of the most hilarious moment of my life in a bathroom ofTarquinia), David (and of course Séverine, Elise, and -1 month old Alec who hosted me inClermont and shared so many valuable insights on being a researcher), Kirsten, Reka, theAnnas (know that you are the best and also that one day, a french team will end the spanish-germain supremacy in champion’s league), young Colin (you better improve at TerraformingMars and deal with all this water in volcanic plumes), Gabriela (you are annoyingly good atTerraforming Mars but you bake amazing cakes), Manar (the most classy woman of UBC),Megan (rrrrr), Vahid (you better be back on your feet by the time this is submitted!), Georgia,Tasha, Geneviève, Tim and Alex (why this sudden craziness for chess and darts? still a mystery),Mike, British (see you tomorrow at Coffee Boulevard), Hannah, Lydia, Lindsay, Keelin (and herwhale jokes...). A special thank to Margot, Adjoa, Kim, Rose, Jack and Meghan, our researchgroup would not be the same without you who preferred to come to our group meetings insteadof working for mid-terms. I learned so much working with you! And to all of you who helpedme through these months of writing for papers, proposals, thesis dissertation and other funstuffs, you have an unlimited credit to solicit my feedback on whatever, with no expiry date.To Wim Degruyter and Costanza Bonadonna, who took the time to listen and help thisexcited grad student who was chasing them to talk about volcanic plumes and climate in thexxmiddle of AGU posters. To Guillaume Carazzo, who let me play for three months with hisbeautifully scaled experimental design. To Valentina Radić, who introduced me to the highlynon-chaotic world of self-organizing maps. I have learned a lot from you all and hope we willendeavor in more research adventures together!To my supervisory committee, Mark Jellinek, Valentina Radić, Phil Austin and StephanieWaterman. Together, you had a perfectly balanced mix of personalities and skills, and yourguidance was instrumental in going through this PhD journey.To all the inspiring research supervisors and professors who led me to this PhD: MyriamKhodri (I made good use of the resampling methods you taught me), Pierre-Yves Lagree andLydie Staron (who let me use Lego to build a lab experiment...), Dominique Guest and JulienCubizolles, Mr Bertrand and Bertrand and, of course, Béatrice Aubry!To Michael Lipsen and Lucy Porrit who gave me the opportunity to learn the art of teachingin which they are masters, and Yulia, Manuel and Nathalie who shared this adventure.To all the friends who supported me over these last four years: Méliné, Camille, Laure,Louis, Simon, Chloé, Baptiste, Audrey, Sonia, Vincent, Céline.To Lori Glaze, Larry Mastin, Arnau Folch, Tamsin Mather and the anonymous reviewerswho provided insightful reviews on Chapters 2-5 of this thesis, with a particular thank to LoriGlaze who took time to extensively review two of them with challenging and rigorous but alwayskind and encouraging comments. To all the enthusiastic students and researchers who took timeto discuss at conferences and answer my emails, with a particular thank to all people involvedin the Volcanic Impact on Climate and Society working group and Matthew Toohey. To AlanRobock who took the time to review this thesis and to make a comment to Earth Magazineon our publication in Journal of Geophysical Research, and JoAnna Wendel who drew thisamazing cartoon on the same paper.To Anja Schmidt, David Pyle, Costanza Bonadonna, Wim Degruyter, Davide Zanchettin,Myriam Khodri and Guillaume Carazzo who trust that I can be useful to science and supportedmy postdoc applications and the opening of a new chapter of my life as a researcher.My PhD research was generously funded by the University of British Columbia througha four-year fellowship, by the Ecole Normale Superieure de Cachan for the first few monthsof my thesis, by Mark Jellinek via his funding from the National Science and EngineeringResearch Council of Canada, and by the Earth, Ocean and Atmosphere Department via theW.H. Mathews scholarship.To all the volcanoes I hiked, and to the orcas of the Johnstone strait. This adventure startswith you, and I will never get used to see such magnificent and breathtaking creatures emergingnext to my kayak.xxiDedicationThis dissertation is dedicated to Camille Defrenne. When I proposed you to move to Vancouver,I was proposing to move in a place I knew, to work with a supervisor I knew, in an english-speaking environment where I spent six months already, and to do research which is what myentire post-secondary education prepared me to do. You, on the other hand, were proposed tomove in a place far away from your friends and family where you would knew nobody but thisguy you had been dating for barely a year; an exclusively english-speaking environment whileyou had little faith in your english skills; a place where you had no professional contact and whereyou would try to do a PhD despite your post-secondary was focused on agronomy/engineering.And yet, you gave us a chance to do this journey together. You faced and overcame someof your greatest fears to give us this opportunity, from leaving four years in a foreign place tohiking through bison herds in the Yellowstone. The Camille I met hates mathematics and shestill does, but she is doing mixed-effects models and write thousands of lines of codes in R. Youare a rigorous and demanding scientist, and you will become a brilliant researcher, or anythingelse you wish. I hope to be the next one to follow you were your career will take you, and Iwill never forget what you did for me. We have grown together through these years made ofchallenges and discoveries, and I don’t see it stop anytime soon. What’s our next journey?With all my love.xxiiChapter 1Introduction and motivationThis doctoral dissertation includes four published papers and one chapter in preparation forsubmission(s). Consequently, this introduction only gives an overview of the motivation, ra-tionale and organization of the dissertation. As a full paper, each chapter includes a detailedintroduction to the specific question(s) it addresses. The organization of this dissertation fol-lows a logical order which does not always reflect the chronological order of my PhD and/or thechronological order of the associated publications.1.1 A key driver of climate variability: Volcanicaerosol-radiation interactions.Explosive volcanic eruptions inject gas and particles into the atmosphere, which impact Earth’sclimate. At seasonal to millennial timescales, the dominant driver of the climate responseto explosive volcanism is the injection of sulfur gases, which form sulfate aerosols that mod-ify Earth’s radiative balance, in turn (Robock (2000) and Figure 1.1, green boxes). In thetroposphere, sulfate aerosols increase the concentration of cloud condensation nuclei and en-hance cloud formation (aerosol-cloud interactions), which may lead to a local net cooling ofEarth’s surface (e.g. Schmidt et al. (2012); McCoy and Hartmann (2015)) . Troposphericsulfate aerosols are washed out by precipitation and have a residence time of order one week.In the stratosphere, sulfate aerosols (Robock (2000); Timmreck (2012); Kremser et al. (2016)):1) backscatter shortwave radiation from the Sun, leading to a net cooling of Earth’s surfacethrough an increased albedo; and 2) absorb infrared radiation from the Earth and the Sun,leading to a net warming in the stratosphere (aerosol-radiation interactions). For example, fol-lowing the 1991 eruption of Mount Pinatubo, the net flux of short-wave radiation at top of theatmosphere decreased by 3-4 Wm−2 for over a year, with the global mean surface temperaturedecreasing by as much as 0.5 K and stratospheric temperatures increasing by as much as 3 K(Labitzke and McCormick, 1992; Soden et al., 2002). The residence time of sulfate aerosols inthe stratosphere is on the order of one year and is controlled by their rate of sedimentation,troposphere-stratosphere mixing and horizontal transport in the stratosphere.11.1. A key driver of climate variability: Volcanic aerosol-radiation interactions.The large difference in lifetime of sulfate aerosols in the troposphere and in the stratosphereimplies that for the same erupted mass of sulfur, an eruption plume that reaches the strato-sphere will have a much stronger and longer-lived radiative forcing compared to one that isconfined in the troposphere. The vertical distribution of volcanic sulfur gases (SO2) into the at-mosphere following a volcanic eruption is thus a critical control on stratospheric volcanic sulfateaerosol (SVSA) forcing. From ice core reconstructions and satellite observations, the injectionof volcanic sulfur into the stratosphere is one of the predominant natural climate forcings overthe past 2500 years. In particular, the coldest decades of the Common Era coincide with thelargest eruptions (Sigl et al., 2015). SVSA forcing is one of the predominant causes of the LittleIce Age (ca. 1300-1800) (Crowley et al., 2008; Miller et al., 2012; Slawinska and Robock, 2018).The 1815 Tambora eruption was followed by the infamous “year without summer” (Luterbacherand Pfister , 2015). The cooling associated with volcanic eruptions has well-documented con-sequences for human societies, in particular crop failure, starvation, and increased mortality(Ludlow et al., 2013; Toohey et al., 2016a; Manning et al., 2017).In addition to imparting an average surface cooling response, SVSA forcing interacts withmultiple major modes of variability of the climate system (Robock, 2000; Swingedouw et al.,2017). Such interactions extend the longevity of the climate response to SVSA forcing beyondthe residence time of sulfate aerosols in the stratosphere. Large tropical eruptions that reachthe stratosphere, for example, trigger a positive phase of the North Atlantic Oscillation (NAO)during the first and sometimes second winter following the eruption (Robock and Mao, 1992;Robock, 2000; Fischer et al., 2007; Driscoll et al., 2012; Zambri et al., 2017), although observa-tions and models also suggest an additional delayed decadal response (Zanchettin et al., 2013;Slawinska and Robock, 2018). This NAO response results in the so-called “winter warming”pattern with a surface warming over Eurasia and cooling over Greenland and the Middle East.Large stratospheric eruptions - at least tropical events - tend also to trigger positive phasesof the El Niño Southern Oscillation (ENSO) (Robock and Free, 1995; Robock, 2000; Li et al.,2013; Khodri et al., 2017). SVSA forcing is also a major driver of the Atlantic MultidecadalOscillation (AMO), in which cold phases are associated with large volcanic eruptions (Otterået al., 2010; Knudsen et al., 2014).Until the early 21st century, research has focused on the climate response to SVSA forcing oflarge volcanic eruptions, i.e. injecting order 10 Mt of SO2 or more such as the 1982 El Chichóneruption or the 1991 Pinatubo eruption. However, the last 20 years of satellite observations inthe absence of large eruptions are revealing an unexpectedly significant contribution to SVSAforcing from relatively small eruptions (Solomon et al., 2011; Vernier et al., 2011). In particular,multiple studies argue that such eruptions are, in part, responsible for a lower than expectedrate of global warming since 2000 (Haywood et al., 2014; Santer et al., 2014, 2015;Monerie et al.,2017). Santer et al. (2014) have investigated the correlation between lower tropical tropospherictemperatures and the stratospheric aerosol optical depth (SAOD) over 1979-2014. They findcomparable correlations over the 1990-1995 Pinatubo period and the 2005-2014 period (Figure21.1. A key driver of climate variability: Volcanic aerosol-radiation interactions.3b in Santer et al. (2014)) despite SAOD levels being at least an order of magnitude smaller forthe latter period. These and other studies demonstrate that SVSA forcing by small eruptionsexert a significant and under-appreciated forcing on climate.Overall, SVSA forcing driven by impulsive volcanic events is one of the most importantforcings of Earth’s climate. In addition, its very strong interactions with major modes ofvariability of Earth’s climate (e.g., the NAO, ENSO or AMO) make SVSA an ideal naturalexperiment to test the skills of climate models in reproducing the response of key modes ofclimate variability to transient forcing (Stenchikov et al., 2006; Driscoll et al., 2012; Khodriet al., 2017). Accordingly, international projects gathering and coordinating research groupsworking on climate-volcano interactions have been recently initiated, such as the Model Inter-comparison Project on the climatic response to Volcanic forcing (Zanchettin et al., 2016), theVolcanic Impacts on Climate and Society working group (LeGrande and Anchukaitis, 2015) andthe Interactive Stratospheric Aerosol Model Intercomparison Project (Timmreck et al., underreview).However, these initiatives generally focus on investigating processes: i) governing the emer-gence of a radiative forcing from a specified injection of SO2 in the atmosphere (green boxeson Figure 1.1), generally defined by a latitude, a vertical mass distribution, and sometimes alongitude; or ii) the climate response to this forcing (blue boxes on figure 1.1). In this thesis,I propose a new perspective on SVSA forcing by investigating the processes that govern thevertical distribution of SO2 into the atmosphere, given only knowledge of source conditions atthe eruptive vent (e.g. mass eruption rate) and atmospheric conditions during an eruption.Such a perspective requires me to address a number of grand challenges in the understand-ing and modeling of volcanic plumes, which I detail in the following section. My results leadto a novel feedback between climate and the rise of volcanic plumes. Ongoing anthropogenicglobal warming will reduce stratospheric injections of SO2 by explosive eruptions and, in turn,reduce volcanic surface cooling and enhance global warming. Such a feedback has importantimplications for predicting future climate as well as understanding climate changes in Earth’sgeological past.3AltitudeAtmospheretemperatureStratospheric warmingSurface coolingAnd modulation of major modes of climate variability, e.g., NAO, ENSO and AMO(depends on forcing spatio-temporal structure and syn-eruptive climate conditions)Surface/subsurface hydrologyMass eruption rateWindEntrainmentTropopause heightRemoval via precipitationEnhanced condensation nuclei concentrationSurface coolingAerosol-cloud interactions (1-3 weeks)Aerosol-radiation interactions (1-2 years)Removal via settling and cross-tropopause mixingSulfate aerosolsSulfate aerosolsSO2SO2 →→SO2 – sulfate aerosols conversion and dispersionXXX XXX XXXProcess or factor governing the delivery of sulfur from the vent into the atmosphereProcess governing spatio-temporal evolution of volcanic sulfate aerosol radiative forcingSW backscattering and LW absorptionClimate responseWater phasechangeFigure 1.1: Cartoon illustrating the mechanisms governing: 1) The delivery of sulfur from a volcanic vent into the atmosphereduring an explosive eruption (red boxes); 2) The spatio-temporal development of a radiative forcing from the sulfur injectioninto the atmosphere (green boxes); and 3) The climate response to this forcing (blue boxes).41.2. From the vent to the atmosphere: Processes governing volcanic plume rise and subsequentaerosol forcing.1.2 From the vent to the atmosphere: Processes governingvolcanic plume rise and subsequent aerosol forcing.In Figure 1.1 (red boxes), I sketch the main processes that govern the delivery of SO2 into theatmosphere from a volcanic vent. During typical explosive eruptions, the erupted mixture ofgas and solid pyroclastic material is initially denser than the surrounding atmosphere. However,air entrained into the column is heated by the particles and expands, causing the column toultimately rise as a buoyant plume. The rise height of a volcanic plume is the primary controlon the injection of SO2 into the stratosphere and more generally on the vertical distributionof SO2 into the atmosphere. SVSA forcing is consequently governed by plume height. Atorder zero, in a linearly stratified atmosphere without wind, the plume height H (m) scales as(Morton et al., 1956):H ∝ −1/2F 1/40 N−3/4 , (1.1)where F0 (m4s−3) is the source buoyancy flux of the plume, N (s−1) is the atmospheric strat-ification (Brunt-Väisälä frequency), and  is the rate of entrainment of atmosphere into theplume.The strength of the atmospheric stratification exerts the strongest control on plume height inthis model and is in part controlled by the atmospheric temperature profile during the eruption.The same temperature profile also governs the height of the tropopause, and thus the verticaldistribution of volcanic SO2 relative to the tropopause height.The second strongest control on the plume height in Equation 1.1 is the entrainment rate (dimensionless). In a quiescent environment, estimates for the entrainment rate vary by afactor of 2, depending on the balance between the source momentum and buoyancy fluxes forthe plume. In a windy atmosphere, wind stresses enhance velocity gradients between the risingplume and the atmosphere, which increases the entrainment rate and decreases the plume height(e.g., Hoult et al. (1969); Bursik (2001)). Published estimates of entrainment rate in a windyatmosphere are uncertain by an order of magnitude, which affects plume height by a factorof '3, with critical implications for estimating whether an explosive eruption will inject gasesdirectly into the stratosphere for specified source and environmental conditions.The third control on plume height is the plume source buoyancy flux which is mostly gov-erned by the mass eruption rate. For a given eruption duration, the mass eruption rate alsocontrols the total mass of sulfur gases injected into the atmosphere. The mass fraction of solidsand the temperatures of the plume and the atmosphere at the vent level also affect the plumebuoyancy flux and, in turn, the plume height (e.g. Woods (1993, 1995)).Depending on the balance between the momentum and buoyancy fluxes of the eruptedmixture and the rate of entrainment of atmosphere into the column, volcanic eruptions maycollapse instead of forming buoyant plumes (Woods and Wohletz, 1991; Carazzo et al., 2008a;Degruyter and Bonadonna, 2013; Jessop et al., 2016a). Buoyant co-ignimbrite columns mayform from the resulting pyroclastic flows as the largest clasts settle and air is entrained and51.2. From the vent to the atmosphere: Processes governing volcanic plume rise and subsequentaerosol forcing.heated in the upper part of the flow. Buoyancy flux and entrainment mechanics remain criticalcontrols on the rise height of co-ignimbrite columns which rise to lower altitudes than buoyantplumes rising directly from the vent for the same eruption magnitude (Woods and Wohletz, 1991;Herzog and Graf , 2010). Consequently, only very powerful (e.g. Tambora 1815 or Pinatubo1991) and thus less frequent eruptions may inject SO2 directly into the stratosphere when theycollapse.Additional factors modulate the plume buoyancy flux. Before entering the atmosphere,magmatic material may pass through layers of water or ice (e.g. lake or glacier). The meltingand evaporation of such a layer may have a large impact on the thermal and kinetic energy, watervapor content and particle size distribution of the mixture entering the atmosphere (Wilsonet al., 2013). Atmospheric water vapor entrained into a plume cools as it rises, and condensesto release latent heat. Some one-dimensional (1D) and three-dimensional (3D) models of plumerise suggest that such latent heat flux would increase the plume buoyancy flux sufficientlyto increase plume height by up to '5-10 km, depending on atmospheric humidity (Woods,1993; Glaze et al., 1997; Tupper et al., 2009). Such predictions have critical implications forpredicting the vertical distribution of SO2 following an eruption, in particular how much entersthe stratosphere, but is yet to be evaluated against observations.In addition to the challenges involved in understanding and predicting the physical controlson plume rise and stratospheric SO2 injections, restrictive observations enabling a rigorous eval-uation of models for these processes are difficult to gather. For natural eruptions, independentestimates of mass eruption rate (model input) and plume height (model output) are required.However, both parameters are subject to large uncertainties (Tupper and Wunderman, 2009;Bonadonna and Houghton, 2005; Bonadonna et al., 2015a). Furthermore, for the vast majorityof volcanic eruptions, the mass eruption rate is estimated from the plume height using models ofvolcanic plumes (e.g., Mastin et al. (2009)). The year I started my thesis (2014), Larry Mastincompiled the largest database at that time of independently constrained plume heights, masseruption rates and atmospheric conditions (Mastin, 2014). It includes 25 events but providesno uncertainty associated with eruptive parameter estimations. Despite the availibility of thisnew database, observations of natural plumes were not used to evaluate volcanic plume modelsin the recent “intercomparison” study (Costa et al., 2016). To overcome difficulties in compil-ing data from explosive eruptions, many studies use carefully-scaled laboratory experiments toevaluate models of plume rise and, in particular, to calibrate entrainment rates into buoyantjets (e.g. Morton et al. (1956); Hoult et al. (1969); Yang and Hwang (2001); Contini et al.(2011)). A critical challenge for laboratory experiments is, however, to span the large range ofsource and atmospheric conditions in which natural eruptions occur (Carazzo et al., 2014).A striking feature of the mechanics that govern plume rise and ultimately the vertical distri-bution of volcanic SO2 into the atmosphere is that most of the underlying process are controlledby weather and, in turn, climate: the surface/subsurface hydrology, the atmospheric tempera-ture profile that controls the stratification and tropopause height, the wind profile that controls61.3. Research questions and overview of the thesis.turbulent entrainment into the plume, the humidity that may impact the plume buoyancy fluxvia entrainment and condensation. Even the processes that govern the development of SVSAforcing from an “atmospheric source” of SO2 are governed by climate, in particular stratosphericcirculation and chemistry and stratosphere-troposphere exchanges (Kremser et al., 2016). Atthe much longer timescale of glacial-interglacial cycles, the growth and retreat of ice sheetsaffect the frequency of volcanic eruptions (e.g. Jellinek et al. (2004)), and thus potentially thetime-averaged rate of stratospheric SO2 injections. A natural question that emerges from theseconsiderations, and forms the major question I have addressed during my PhD, is to what ex-tent does climate change affect SVSA forcing and SVSA forcing affect climate, in turn. I choseto focus my thesis on feedbacks that may emerge from processes governing the rise of volcanicplumes.1.3 Research questions and overview of the thesis.Understanding and modeling potential feedbacks between climate and the rise of volcanicplumes requires an answer to the first research question of my PhD:1 - Can we reliably model the impact of atmospheric conditions on the rise of a volcanic plumeat syneruptive timescales?After gaining insights on this first question, I return to the major question motivating my PhDand use ongoing anthropogenic climate change as a test case for feedbacks between climate andthe rise of volcanic plumes. The second research question I address in this thesis is:2- Will ongoing anthropogenic climate change predictably alter the rise of volcanic plumes, thevertical distribution of volcanic SO2 in the atmosphere and the resulting SVSA forcing and, inturn, enhance or mitigate global warming?Chapters 2-4 of my thesis address the first question, with chapter 4 providing insights onthe second one. In chapter 2 (published in Journal of Volcanology and Geothermal Research,Aubry et al. (2017b)), I use two datasets to evaluate a newly proposed scaling and publishedscalings that predict the height of volcanic plumes from source and atmospheric conditions.The first dataset consists of analog laboratory experiments which I performed during the threemonths preceding the beginning of my PhD (Carazzo et al., 2014). The second dataset is anew and unprecedented database of eruption parameters compiled during my PhD. I use thesedatasets to evaluate 1D models of volcanic plumes in chapters 3 (published in GeophysicalResearch Letters, Aubry et al. (2017a)) and 4 (published in Earth and Planetary Letters, Aubryand Jellinek (2018)). Although my initial motivation was to understand the climate impactsof volcanic eruptions, chapters 2-4 also make fundamental contributions to understanding andpredicting risks related to devastating pyroclastic flows and ash dispersion and settling.Chapter 5-6 of my thesis address the second research question. In chapter 5 (published inJournal of Geophysical Research, Aubry et al. (2016)), I explore the potential impact of ongoingglobal warming on the rise height of volcanic plumes and on the volcanic injection of sulfur intothe stratosphere. In Chapter 6, I introduce my initial efforts to develop an idealized model of71.4. Major contributions.SVSA radiative forcing and its application to quantify the feedback suggested by the results ofchapter 5. In the conclusion section, I briefly discuss ongoing or submitted projects that willadvance further understanding of potential feedbacks between climate and SVSA forcing.1.4 Major contributions.My PhD journey connects the mechanics of plume rise to vertical SO2 distribution and SVSAforcing and, ultimately, climate impacts. To this end, I produced datasets, methodologies andfuture directions that are critical for communities working on volcanic plumes and their impacts,including climate forcing.Over the three months preceding the start of my doctoral thesis, I performed a set ofanalog laboratory experiments on wind-forced volcanic plumes at the Institut de Physique duGlobe de Paris under the direction of Guillaume Carazzo. The resulting dataset spans anunprecedented range of dynamical regimes in terms of source and environmental conditions,and covers conditions observed for most explosive eruptions. Although the production of thisdataset is not part of this thesis (Carazzo et al., 2014), I extensively analyzed this new datasetas part of my PhD and provided new constraints on scalings and 1D model of volcanic plumes(Aubry et al. (2017b,a); Aubry and Jellinek (2018)).Unsatisfied with addressing my first research question using laboratory experiments only,I tried to update the database of eruptive parameters of Mastin (2014) with a few eruptionsthat occurred shortly before or during my PhD (e.g. Cordón Caulle 2011 or Calbuco 2015). Inthe end, I produced a database with four times more events than in Mastin (2014), and withestimates of uncertainties associated to all eruptive parameters while only mean estimates werepreviously provided Aubry et al. (2017b).I applied both the experimental and natural eruption datasets to extensively evaluate 1Dmodels of volcanic plumes, and in particular their treatment of entrainment and condensationprocesses which are the leading source of uncertainties for predicting plume height (Aubry et al.(2017a); Aubry and Jellinek (2018)). The datasets I produced and my systematic assessing ofhow observational uncertainties affect model evaluation reduces uncertainties on model predic-tions for plume height, with implications ranging from the assessment of climate impacts ofvolcanic eruptions to the likelihood of the production of pyroclastic flows. More importantly, Iidentify the most critical limitations of these datasets that must be addressed to make progressbeyond my work.My application of volcanic plume dynamics to understand SVSA forcing in a self-consistentway led to a novel view of climate-volcano interactions. My work identifies a new positiveclimate-volcano feedback where anthropogenic CO2 emissions lead to increased stratificationand tropopause height, decreased volcanic plume height and SO2 injections into the strato-sphere, decreased SVSA forcing and, in turn, reduced volcanic surface cooling which mayenhance global warming (Aubry et al., 2016). Beyond revealing this feedback, my doctoralthesis will increase focus on processes that govern the vertical distribution of volcanic SO2 into81.4. Major contributions.the atmosphere from eruptive conditions at vent level, and how these processes contribute toclimate-volcano interactions.My journey through investigating this feedback motivated my development of a new ideal-ized model of volcanic forcing (Chapter 6). In contrast with previous models: 1) I use 37 yearsof satellite observations of volcanic SO2 injection and SVSA forcing to calibrate the model,instead of the only the 1991 Pinatubo eruption; and 2) The model can predict the verticalstructure of the forcing instead of prescribing it. This model is still under development, butthe very promising results obtained so far will make it a state of the art and computationallycheap tool for the large community working on stratospheric aerosols.9Chapter 2A new analytical scaling forturbulent wind-bent plumes:Comparison of scaling laws withanalog experiments and a newdatabase of eruptive conditions forpredicting the height of volcanicplumesThis chapter was published in Journal of Volcanology and Geothermal Research in September2017 ( SummaryVarious scaling relationships relate the height of volcanic plumes to eruptive source conditions,atmospheric density stratification, turbulent entrainment, and wind stresses. However, obser-vational, analog, and numerical studies used to test these scalings capture only a narrow rangeof natural eruptive conditions. In particular, existing analytical scalings are not appropriate forthe wind stress conditions typical of the majority of volcanic eruptions. Accordingly, we developa new analytical scaling for the height of buoyant plumes rising in density-stratified uniformcrossflows. We compare this scaling to existing analytical scalings (Morton et al. (1956), Hewettet al. (1971)) as well as “functional" scalings (i.e., parameterizations expressed as a functionof the Morton et al. (1956) scaling and a regime parameter related to wind stress, Degruyterand Bonadonna (2012),Woodhouse et al. (2013),Carazzo et al. (2014)) using the extensive ex-102.2. Introductionperimental dataset from Carazzo et al. (2014) along with natural events from a new databaseincluding 94 eruptive phases. Our proposed scaling best predicts the height of experimentalplumes, which enables us to constrain the ratio of the wind to radial entrainment coefficients.For natural eruptions, the Woodhouse et al. (2013) and Carazzo et al. (2014) scalings, whichaccount explicitly for wind gradient, best predict plume heights. We show that accounting foratmospheric stratification and wind improves empirical relationships between mass eruptionrates and plume heights. For tested scalings, analysis of residual heights for natural eruptionssupports the hypothesis that volcanic plumes rise higher in a wetter atmosphere. Finally, foranalog plumes rising under moderate to high wind stresses, we show that plume shapes evolveover the plume height, violating the self-similarity assumption on which all scalings and integralmodel results rely. We discuss consequences for relaxing the self-similarity hypothesis as wellas potential improvements for standard integral plume models, in turn.2.2 Introduction2.2.1 Overview: volcanic plume height and the problem of wind stressesAnalytical scaling relationships provide important insight into the dynamics of volcanic plumesby relating the plume heights to (i) eruption source conditions, (ii) atmospheric conditions and(iii) the rate of turbulent entrainment of air into rising plumes. Scalings enable a better un-derstanding of the main controls on plume height which, in turn, enables rigorous assessmentsof societal impacts of explosive eruptions related to, e.g., falling ash (e.g. Carey and Sparks(1986)), climate change on timescales of weeks to centuries (e.g. Robock (2000); Timmreck(2012)), and the delivery of nutrients to oceans, such as iron (e.g. Langmann et al. (2010);Browning et al. (2015)). Furthermore, when tested against analog laboratory experiments, an-alytical scalings enable the calibration of turbulent entrainment rates (e.g. Morton et al. (1956);Hoult and Weil (1972); Fischer et al. (1979)) which must be specified in more sophisticatedplume models (e.g. Degruyter and Bonadonna (2012); Woodhouse et al. (2013)).A number of recent eruptions strongly impacted by winds (e.g. Eyjafjallajökull in 2010,Cordón Caulle in 2011, Calbuco in 2015) have reanimated debates over how best to parameterizewind effects on turbulent entrainment in buoyant plume models, from which scalings are derived(e.g., Degruyter and Bonadonna (2012); Woodhouse et al. (2013); Mastin (2014); Bonadonnaet al. (2015b); Woodhouse et al. (2015); Girault et al. (2016); Folch et al. (2016)). Accordingly,a practical aim is to provide a self-consistent intercomparison of scalings relating the height ofvolcanic plumes to wind stresses through their effect on entrainment, as determined from bothlaboratory experiments and observations of natural eruptions.112.2. IntroductionHewett, Fay and Hoult scalingβ = 0.1-1 from lab and numerical experimentsW∗𝟎 =𝐰𝐢𝐧𝐝 𝐬𝐩𝐞𝐞𝐝𝐬𝐨𝐮𝐫𝐜𝐞 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲Morton, Taylor and Turner scalingα = 0.05-0.16 from lab and numerical experiments0N O  W I N D I N T E R M E D I A T E  W I N D S(most explosive eruptions occur in these regimes)S T R O N G  W I N D SPuyehue, 2011 Chaiten, 2008Shinmoedake, 2011Best scaling?lab experimental studies cover sparse regimes100Hmax ̴ HMTTuε= α.uEddiesuuε= α|u-w.sin(φ)|+β|w.cos(φ)|Hmax ?uφHmax ̴ HHFHw(z)uε= β.wuFigure 2.1: Entrainment hypothesis and scalings for plume height across different windregimes: quiescent atmosphere (left), intermediate winds (center), and high winds(right). The different notations are defined in the text and in table 2.1. Left andright pictures show laboratory experiment performed under W∗0 = windspeedinjectionspeed '10−3 and W∗0 ' 100, respectively. Pictures in the middle show eruptions occuringunder W∗0 ranging from 10−3 to 10− Plume rising in quiescent conditions: the Morton et al. (1956) scalingMost scalings for the rise of volcanic plumes follow the classical approach ofMorton et al. (1956).For a fully developed turbulent and incompressible plume with steady source conditions, Mortonet al. (1956) develop a similarity theory by making the following assumptions:1. The turbulent entrainment of ambient air into the plume is driven by lateral pressuregradients related to velocity differences between the plume and the atmosphere. Conse-quently, the inflow velocity u of ambient fluid into the plume is assumed to be propor-tional to the centerline velocity u of the plume at a height z such thatu = αu , (2.1)where α is the radial entrainment coefficient.2. The profiles of velocity and density across the plume are of similar form at all heightsalong the plume centerline (e.g., “top-hat” or gaussian functions).122.2. IntroductionFor a dry, quiescent and linearly stratified atmosphere, and under the Boussinesq assump-tion, a plume rising from a point source of buoyancy flux with top-hat velocity and densityprofiles will ascend to a maximum height (Figure 2.1, left) given by:HMTT = 2.8× (2− 58pi− 14α− 12F140 N− 34 ) , (2.2)where the prefactor 2.8 is the non-dimensional maximum plume height, F0 is the source buoy-ancy flux of the plume, N =√− gρa0dρadz is the atmospheric Brunt-Väisälä frequency with g theEarth’s gravity acceleration, ρa the ambient density, z the altitude, and 0 subscript denotingproperties taken at source altitude (cf Table 2.1 for a summary of symbols used). Equation2.2 is, thus, an analytical expression that explicitly relates the height of a plume to its sourcecondition, the environmental stratification and the radial entrainment rate.Explosive volcanic plumes erupt as negatively buoyant jets with dense pyroclasts in the flow.The initial momentum flux drives entrainment of atmosphere, which is heated by hot pyroclasts,in turn reducing the mean density of the jet. If the efficiency of turbulent entrainment andheating from pyroclasts is sufficiently high, the buoyancy of the column reverses before theinitial momentum flux is exhausted and the mixture rises by natural convection. The height ofthe momentum-dominated region is generally small compared to the total rise height, so thatEquation 2.2 is commonly applied to study the rise of volcanic plumes, where the buoyancyflux is specified in terms of an enthalpy flux (Wilson et al., 1978; Woods, 1995).The radial entrainment coefficient α is historically calibrated using small-scale laboratoryexperiments (e.g. Morton et al. (1956); Fischer et al. (1979); Chen and Rodi (1980); Kaminskiet al. (2005)), large-scale experiments (e.g. Dellino et al. (2014)), or numerical simulations(e.g. Suzuki and Koyaguchi (2010, 2015)). Laboratory experiments show that for radial top-hat-shaped profiles, α lies between 0.065 and 0.07 in pure jets (driven by a source of momentumonly, e.g., Fischer et al. (1979)), and between 0.1 and 0.16 in pure plumes (driven by a source ofbuoyancy only, e.g., Morton et al. (1956); George et al. (1977)). Kaminski et al. (2005), however,show that instead of being constant, α is a function of the shapes of the velocity, density andturbulent shear stress profiles, and of the vertically varying (i.e., ”local") Richardson number,which expresses the ratio of the the momentum and buoyancy fluxes:Ri = rg′u2. (2.3)Here, r is the jet radius, g′ = ρ−ρaρa g is the reduced gravity of the jet, where ρ is the jet density.132.2. IntroductionTable 2.1: Symbols used in this study. 0-subscript refers to source conditions. For plumeproperties, symbols with hat refer to properties non-dimensionalized using Equation2.16.Parameter Symbol UnitPlumeBulk velocity u m.s−1Bulk density ρ kg.m−3Radius r mDownwind or upwind radius rdown/up mPlume asymmetry factor As −Entrainment velocity u m.s−1Radial entrainment parameter α −Wind entrainment parameter β −Equivalent entrainment parameter  −Plume deflection relative to horizontal θ radPlume deflection relative to vertical φ radPlume mean deflection relative to vertical φ¯ radBulk Buoyancy g′ m.s−2Plume maximum height Hmax mScaling height from ref Href mJet/plume transition length lm mVertical distance from source z mHorizontal distance from source x mDistance from source along centerline s mBuoyancy flux f m4.s−3Momentum flux m m4.s−2Volume flux q m3.s−1Source temperature T0 KSource gas content n0 wt.%AtmosphereHorizontal wind speed w m.s−1Density ρa kg.m−3Radius r mEntrainment velocity u m.s−1Horizontal wind speed averaged over plume height W m.s−1Brunt Väisälä frequency N s−1Tropopause altitude Htp mRelative humidity RH −Regimeparameters Velocity ratio W∗ −Richardson number Ri −Gradient Richardson number Rig −Wind Richardson number Riwind −Wind Gradient Richardson number Riwindg −Wind Velocity to Pure Plume Velocity Scale Ratio V∗0 −Wind Gradient to Plume Velocity Gradient Ratio T ∗ −Constants Gravity Acceleration of Earth g m.s−2Specific gas constant R−1.K−1Empirical parameter for the Carazzo et al. (2014) scaling ac −Height of the tank (for laboratory experiment) Htank m2.2.3 Plume rising under windy conditions: the Hewett et al. (1971)scalingMorton et al. (1956) neglect the impact of atmospheric winds, which have long been recognizedto strongly affect entrainment mechanics (Hoult et al., 1969; Bursik, 2001). Qualitatively,atmospheric crosswinds bend the plume and decrease the radial gradient of axial (relative to142.2. Introductionplume axis) velocity between the plume and the atmosphere, but they create a radial gradientof normal velocity (normal to the local plume centerline, see Figure 2.2). Assuming a linearsuperposition, both velocity gradients contribute to an entrainment velocity Hoult et al. (1969):u = α|u−w sin(φ)|+ β|w cos(φ)| , (2.4)where w is the horizontal wind speed, φ is the local deflection of the plume with respect to thevertical (cf Figure 2.1, center) and β is the wind entrainment parameter.φzxsρa(z)Wφρa(z)uε(s)u(s)u(s+ds)ss+dsr(s)r(s+ds)ρ(s)ρ(s+ds)W.sin(φ)W.cos(φ)Figure 2.2: Cartoon of a rising plume (top) and of the control volume (bottom) used toderive conservation Equations. φ¯ = 1Hmax∫Hmax0 φ dz is the mean angle of rise.Whether the mechanics of entraining eddies is controlled predominantly by wind or plumevelocity gradients is expressed by the local ratio of the wind velocity to the plume velocity:W∗ = wu. (2.5)Laboratory experiments (e.g. Hoult and Weil (1972); Huq (1997); Contini et al. (2011)), fieldobservations (e.g. Hoult et al. (1969); Briggs (1972)) and numerical studies (e.g. Suzuki andKoyaguchi (2015)) suggest that β lies between 0.1 and 1. Most studies assume that β is aconstant (e.g. Hewett et al. (1971); Degruyter and Bonadonna (2012); Woodhouse et al. (2013);Bonadonna et al. (2015b)). Using the Morton et al. (1956) pure plume model and Equation2.4, and assuming a uniform wind of magnitude W such that W∗ >> 1, Hewett et al. (1971)152.2. Introductionshowed that the maximum plume height is given by (Figure 2.1, right):HHFH = 613pi−13β−23F130 N− 23W−13 . (2.6)As with Equation 2.2, this analytical scaling relates the height of a pure plume to source con-ditions and stratification, but also to the crossflow speed and the wind entrainment coefficient(instead of the radial entrainment coefficient for Equation 2.2).2.2.4 Plume rising under arbitrary wind stress: overview of existing”functional" scalingsNeither the Morton et al. (1956) (W∗ = 0) nor the Hewett et al. (1971) (W∗ >> 1) assumptionsare appropriate for the majority of explosive volcanic eruptions, which commonly occur underintermediate wind stress conditions, i.e., W∗ ' 10−2− 1 (Figure 2.1 and Carazzo et al. (2014)).Consequently, various “functional" scalings are commonly applied, which are adjustments toEquations 2.2 and 2.6 using different wind regime parameters. In this study, we will considerthe three scalings recently proposed by Degruyter and Bonadonna (2012), Woodhouse et al.(2013) and Carazzo et al. (2014). These scalings have been developed specifically for applicationto the rise of volcanic plumes and enable predictions where the Morton et al. (1956) and Hewettet al. (1971) scalings are not applicable, but are not fully analytical.Degruyter and Bonadonna (2012) scale the source buoyancy flux of an eruption as the sumof the buoyancy fluxes obtained from the Morton et al. (1956) and Hewett et al. (1971) scalings.Rearranging their scaling for the source buoyancy flux (cf Appendix A), the plume height isgiven by:HDB = HMTT1 + 0.17 β2α32V∗0 + 0.00061( β2α32V∗0 )21 + 0.48 β2α32V∗0 + 0.0072( β2α32V∗0 )2, (2.7)where the main control parameter is nowV∗0 =W(F0N)14, (2.8)where W is the horizontal wind speed averaged over the plume height. Here, V∗0 can be inter-preted the same way as W∗0 (W∗ taken at source), but with the plume velocity being scaledusing the source buoyancy flux instead of momentum flux. This scaling has been tested using1D integral models and the 1980 Mt St Helens and 2010 Eyjafjallajökull eruptions, but it hasnever been tested against laboratory experiments.For linear wind gradients (dwdz constant and w = 0 at the vent altitude), Woodhouseet al. (2013) demonstrate that plume height depends only on HMTT , the atmospheric gradientRichardson numberRiwindg =N2(dwdz )2 , (2.9)162.2. Introductionand the ratio of the entrainment coefficients βα . No analytical solution has been found for thissituation but they show that the numerical solution is well approximated by:HWHPS = HMTT1 + (0.87 + 0.05βα )(Riwindg )121 + (1.09 + 0.32βα )(Riwindg )12 + (0.06 + 0.03βα )(Riwindg ). (2.10)Using a 1D integral model of volcanic plumes, Woodhouse et al. (2013) test Equation 2.10 withsource and atmospheric conditions corresponding to the Eyjafjallajökull 2010 eruptions. Theyalso compare Equation 2.10 to a larger set of eruptions (Sparks et al., 1997; Mastin et al., 2009)but use US Standard atmosphere and idealized wind profile instead of the observed atmosphericconditions. This scaling has not been tested against experimental datasets.More recently, Carazzo et al. (2014) propose a scaling based only on laboratory experimentsand a 1D model of turbulent buoyant jet with variable entrainment (Kaminski et al., 2005).In this treatment, the height reached in a quiescent environment is corrected using the regimeparameter T ∗ to account for wind effects, whereT ∗ = (HMTTU0)(w(Htp)Htp) (2.11)expresses the ratio of the wind gradient in the troposphere (where Htp is the tropopause height)and the vertical velocity gradient in the plume. Simplified expressions relating mass eruptionrates to plume height are proposed by these authors, together with 9 empirical parametersthat depend on the eruption latitudes (cf equations 7-10 in Carazzo et al. (2014)). To builda straightforward scaling, we replace the volume flow rate calculated from their 1D model,without wind and with α dependent on the local Richardson number, by the volume flow rateobtained with the Morton et al. (1956) scaling, with a constant α. The resulting expression forthe plume height is:HCGABK = HMTT e−acT∗ (2.12)where ac is an empirical parameter constrained by laboratory experiments. The original Carazzoet al. (2014) scaling has been shown to predict accurately the source conditions of a selectionof 10 explosive volcanic eruptions, but as presented in Equation 2.12 it has not been tested yet.172.2. Introduction2.2.5 Summary and motivation for a new analytical scaling lawWind speed at 10km (m.s-1)0 20 40 60HHFH (normalized)00.511.5HFH1971(a)Wind speed at 10km (m.s-1)0 20 40 60HWHPS (normalized)00.511.5WHPS2013(b)Wind speed at 10km (m.s-1)0 20 40 60HDB (normalized)00.511.5DB2012(c)Wind speed at 10km (m.s-1)0 20 40 60HCGABK  (normalized)00.511.5CGABK2014(d)Figure 2.3: Standardized plume height as a function of the wind speed at 10km. Panels(a), (b), (c) and (d) show height predicted by the Hewett et al. (1971), Degruyterand Bonadonna (2012), Woodhouse et al. (2013) and Carazzo et al. (2014) scalingsrespectively. All height are normalized using the Morton et al. (1956) scaling withα = 0.1. We used F0 = 107 m4s−3, N = 0.015 s−1, U0 = 300 m.s−1, andassumed that the wind was linearly increasing with altitude. We use a Monte Carlomethod to randomly sample entrainment coefficients α and β and estimate the 95%confidence interval on predicted plume height (shadings). For α (respectively β),we assume a gaussian distribution of mean 0.1 (respectively 0.55) and standarddeviation 0.025 (respectively 0.225).Analytical scalings (Equations 2.2 and 2.6) give important insights on the physics of volcanicplumes, and can be used to calibrate entrainment coefficients α and β required in 1D modelsof volcanic plumes by using laboratory experiments. However, these scalings are often notapplicable to volcanic plumes because they require either unrealistically low or high wind speeds.On the other hand, functional scalings (Equations 2.7, 2.10 and 2.12) enable predictions wherethe analytical scalings of Morton et al. (1956) and Hewett et al. (1971) fail. Nonetheless,they have mostly been tested using 1D models of volcanic plumes, and a restricted number oferuptions for which eruption source conditions and heights can be inferred, albeit with largeuncertainties. Consequently, their generic predictive power is uncertain. The scalings discussedhere also do not provide consistent predictions, even in the windy conditions for which they weredesigned (Figure 2.3). In addition, the values of α and β are subject to large uncertainties (0.05182.3. Theoryto 0.16 for α and 0.1 to 1 for β). Where they are calibrated, the entrainment parameters α and βare understood over a narrow subset of the full range of wind regimes in which volcanic eruptionscommonly occur (Carazzo et al., 2014). Figure 2.3 shows, for each scaling, the uncertainty inpredicted height associated with uncertainty on entrainment coefficients. The height predictedby a given scaling can vary by almost a factor of 2 depending on the chosen values for α andβ. This uncertainty affects not only scalings, but also 1D integral volcanic plume models (e.g.Mastin (2014); Woodhouse et al. (2015)) and is recognized as the key factor to improve volcanicplume models based on entrainment parameterization (e.g. Costa et al. (2016); Devenish (2016);Macedonio et al. (2016)).To make progress, in Section 2.3, we develop a new analytical scaling law for buoyant plumesrising in density-stratified, uniform crossflows that is applicable to wind regimes where explosiveeruption most commonly occur. To help future studies in making a choice among all existingscalings, we also provide a benchmark comparison of the newly derived scaling with the fivescalings introduced in the present section. The scalings are compared using a recent set oflaboratory experiments (Carazzo et al., 2014) as well as a new dataset of observations fromnatural eruptions presented in section 2.4. In section 2.5, we compare height predictions fromeach scaling, using both experimental and natural eruptions datasets. We will show that thenewly proposed scaling places new constraints on the entrainment coefficients α and β. Wealso assess the best empirical relationship to predict mass eruption rates on the basis of theset of parameters used in each scaling. In section 2.6, we discuss the implication of the newconstraint on entrainment coefficients, and the extent to which the use of self-similarity theoryis adequate under variable wind conditions. In section 2.7, we summarize our conclusions.2.3 Theory2.3.1 Derivation of a new analytical scalingWe build a new scaling for plume height based on the buoyant plume model of Morton et al.(1956) modified by Hoult et al. (1969) to account for wind (Section 2.2). In particular, we assumethat cross-plume velocity and density profiles have a top-hat shape, and that the plume rises intoa uniform crossflow (i.e. w(z) = W ) in an environment with a linear density stratification witha strength N . The key assumptions that enable us to develop a new scaling are: i) the plumedeflection relative to the vertical φ (Figure 2.2, top) is small and ii) the entrainment rate can beapproximated by a constant. The strength of this assumption is that the equations governingthe plume rise can be recast in form to the Morton et al. (1956) equations, where the equivalententrainment coefficient  depends on both the radial and wind entrainment coefficients (α andβ) and the regime parameter W∗0 = WU0 (W∗ taken at source).Using the control volume defined in Figure 2.2 (bottom), the plume rise is governed bythe conservation of volume flux q = pir2u, momentum flux m = pir2u2, and buoyancy fluxf = pir2ug′ along the plume centerline. Let z and x be the vertical and horizontal distances192.3. Theoryfrom the source such that they are related to s, the distance from the source along the plumecenterline, and the local deflection angle φ:sin(φ) = dxds ; cos(φ) =dzds . (2.13)We note for clarification that most authors (e.g. Bursik (2001)) instead use the inclination ofthe plume centerline to the horizontal θ = pi2 − φ to relate s to x and z.Next, we parametrize the entrainment rate  following Hoult et al. (1969): =uu= α|1− Wqmsin(φ)|+ β|Wqmcos(φ)| . (2.14)Projected in the [x,z] coordinate system, conservation of volume, x- and z- momentum, andbuoyancy fluxes are (e.g. Hewett et al. (1971); Bursik (2001); Woodhouse et al. (2013)):dqds = 2 pi12 m12 (2.15a)dm sin(φ)ds = Wdqds (2.15b)dm cos(φ)ds =fqm(2.15c)dfds = − qN2 cos(φ) . (2.15d)To build physical insight, we non-dimensionalize equations 2.13 and 2.15a-2.15d using thefollowing scales (Morton et al., 1956):q = 258pi14F340 N− 54 qˆ , m = 212F0N−1mˆ , f = F0fˆ , x, z, s = 2−58pi−14F140 N− 34 xˆ, zˆ, sˆ ,(2.16)where the hats denote dimensionless quantities. Note that the entrainment rate is not includedin this non-dimensionalization and that the prefactors used are different than those of Mortonet al. (1956) because we use top-hat instead of gaussian profiles. Using Equation 2.13 and thechain rule, the dimensionless governing equations become:cos(φ)dqˆdzˆ = mˆ12 (2.17a)dmˆ sin(φ)dzˆ = 218pi14V∗0dqˆdzˆ (2.17b)cos(φ)dmˆ cos(φ)dzˆ =12fˆ qˆmˆ(2.17c)dfˆdzˆ = − qˆ . (2.17d)Equations 2.17b and 2.17c can be combined as 2mˆ sin(φ)× (2.17b) + 2mˆ× (2.17c) to give:202.3. Theorycos(φ)dqˆdzˆ = mˆ12 (2.18a)dmˆ sin(φ)dzˆ = 218pi14V∗0dqˆdzˆ (2.18b)dmˆ2dzˆ = fˆ qˆ+ 298pi14V∗0 sin(φ)mˆdqˆdzˆ (2.18c)dfˆdzˆ = − qˆ . (2.18d)To simplify further Equations 2.18a-2.18d, we use a small angle approximation such thatφ  1, and assume that the momentum flux carried by the wind is smaller than the plumebuoyancy flux such that 2 98pi 14V∗0 < 1. To 1st order in φ, the governing equations 2.18a-2.18dthen become:dqˆdzˆ = mˆ12 (2.19a)dmˆφdzˆ = 218pi14V∗0dqˆdzˆ (2.19b)dmˆ2dzˆ = fˆ qˆ (2.19c)dfˆdzˆ = − qˆ . (2.19d)We note that with a constant V∗0 (uniform wind and linear stratification), Equation 2.19bcan be written d(mˆφ−218 pi14 V∗0 qˆ)dzˆ = 0.With initial condition mˆ = qˆ = 0, the small angle approximation thus implies φ =2 18pi 14V∗0 qˆmˆ  1. Equations 2.19a, 2.19c and 2.19d are identical to Equations 11 and 12 inMorton et al. (1956), but with a variable entrainment rate: = α(1 + βα|2 18pi 14V∗0qˆmˆ|) , (2.20)using Equations 2.20, 2.16, φ 1, and 2 98pi 14V∗0 < 1. To obtain a system of governing Equationssimilar to Morton et al. (1956), we approximate the entrainment rate with a constant value ¯.From Equation 2.20, a natural scale for ¯ for pure plumes is:¯ ∝ α(1 + βα218pi14V∗0 ) . (2.21)With source conditions fˆ = 1, mˆ = qˆ = 0 specified at zˆ = 0, Equation 2.19a, 2.19c and2.19d are equivalent to Equations 11 and 12 in Morton et al. (1956) with the only differencebeing the radial entrainment rate α replaced by an effective entrainment rate  defined byEquation 2.21. Consequently, the maximum plume height scales as:212.3. TheoryHmax ∝ (1 + βα218pi14V∗0 )−12HMTT . (2.22)Equation 2.22 scales well with the numerical solution of Equations 2.15 for V∗0 6 O(1) andβα 6 10 (Figure A.1). For buoyant jets where the source momentum flux is not null, the scalefor the wind to plume velocity ratio can be based on the source momentum flux, i.e., W∗0 = WU0instead of V∗0 so that Equation 2.21 becomes:¯ ∝ α(1 + βα218pi14W∗0 ) , (2.23)and the plume height scales withHAJC = (1 + βαW∗0 )−12HMTT . (2.24)A choice between these two velocity ratios W∗0 and V∗0 to scale the entrainment rate ¯ forbuoyant jets should be based on the ratio of the jet/plume transition length lm = M340 F− 120(where M0 is the source momentum flux, e.g. Fischer et al. (1979)) and the buoyant plumeheight HMTT . The ratio lmHMTTvaries between 0.1 and 1 for the Carazzo et al. (2014) exper-iments, with the same range estimated for natural eruptions. Experiments ultimately showthat W∗0 = WU0 is the most appropriate velocity ratio scale for the W∗0 −Ri0 parameter spaceconsidered (Section 2.5 and Figure A.2).2.3.2 Discussion of additional assumptionsIn summary, in developing this correction to the Morton et al. (1956) scaling, we make fourassumptions in addition to those articulated by Morton et al. (1956):• A1 : The plume rises in a uniform wind field of speed W.• A2 : φ 1: the angle of rise is very small• A3 : V∗0 < 1: the momentum flux carried by wind is smaller than the buoyancy fluxcarried by the plume.• A4 :  ' ¯: the entrainment rate can be approximated by a mean value. We propose thatthis mean value scales with V∗0 for pure plumes (¯ ∝ α(1 + βαV∗0 )), and with W∗0 for forbuoyant jets with low Richardson number (¯ ∝ α(1 + βαW∗0 )).A uniform wind profile (A1) is not generally realistic, and depends in particular on theregion and season considered (e.g. Mastin (2014); Aubry et al. (2016)). Although the averagewind speed is expected to be the main control on the plume height, the exact shape of thewind profile is also important (Girault et al., 2016). However, it is complex to incorporate theseeffects into a scaling relationship. Ideal shapes such as uniform wind (e.g., Hewett et al. (1971))or linear wind gradient (e.g., Woodhouse et al. (2013)) are consequently commonly used.222.4. Data and methodWe expect φ 1 (A2) to hold when V∗0  1 for a pure plume and W∗0  1 for a pure jet.For buoyant jets, measurements of the mean angle of rise φ¯ = 1Hmax∫Hmax0 φdz in the Carazzoet al. (2014) experiments shows that A2 holds when V∗0 ≤ O(1) (A3) andW∗0  1 (Figure A.3),with A2 and A3 both holding for '12 experiments and being both violated for '7 experiments(out of 27, cf Figure A.4). For pure plumes and V∗0 6 1, Figure A.1 shows that the small anglehypothesis enables good prediction of the maximum plume height.The same figure also comforts that the entrainment rate can be approximated by a constantmean value (A4), scaling with V∗0 for pure plumes with V∗0 6 1. Whether the mean entrainmentrate scales with W∗0 for buoyant jets with low Richardson number is more challenging to testdirectly on the basis of the data used in this paper. However, the performance of the proposedscaling in predicting the height of the Carazzo et al. (2014) experiments (Section 2.5) alsoreflects the validity of assumptions made in deriving this scaling, including A4, for the regimeparameter space used by Carazzo et al. (2014).2.4 Data and method2.4.1 Data from laboratory experimentsWe use two sets of laboratory experiments presented in detail in Contini et al. (2011) andCarazzo et al. (2014) . In both sets of experiments, turbulent salt water plumes are injectedat a fixed rate into a tank containing salt water with a specified linear density gradient. Tosimulate the effect of wind, the nozzle through which plumes are injected was translated relativeto the tank at a fixed speed. The evolution of the plume was recorded using a digital videocamera (25 images/s). For each experiment, source and environment parameters were carefullymeasured in order to compute the different regime parameters (e.g. equations 2.3, 2.8, 2.11)and scaling heights (e.g. equations 2.2, 2.6, 2.7, 2.10, 2.12 and 2.24). In addition, the averagemaximum height Hmax was measured from the stack of all images recorded for an experimentas the maximum height where the tracer concentration becomes indistinguishable from thebackground. The duration of an experiment is much longer than the plume rise timescale, sothat transient turbulent features are removed when stacking all images together (Figure 2.9).For the Carazzo et al. (2014) dataset, we measure the average tracer concentration along aseries of profiles normal to the centerline with digital image analysis (Figure A.5). The mainsource parameters and maximum height of the 27 experiments from Carazzo et al. (2014) areindicated in Table A. Data from explosive eruptionsWe extend the database compiled by Mastin (2014) and use 94 eruption phases for whichindependent estimates of the erupted mass, the eruption duration, and the average plume heightare available. Uncertainties in estimation of these three parameters can be large and vary a lot232.4. Data and methodamong eruptions. Thus, we also strive to quantify these uncertainties for each eruption. Weaccount for various contributions to these uncertainties such as:• Erupted mass: the choice of an empirical function used to fit and integrate the square rootof isopach/isomass area vs the thickness of the deposit, which can lead to differences ofup to a factor 2 on the final erupted mass (e.g. Biass and Bonadonna (2011); Bonadonnaet al. (2015a))• Eruption duration: differences in the interpretation of the timing of deposition of tephralayers as well as the interpretation of the start and end of an eruptive phase, which canbe based on seismological, remote sensing or visual observations (e.g. Paladio-Melosantoset al. (1996); Castruccio et al. (2016))• Plume height: discrepancies between different estimates (e.g. visual, ground-based radaror satellite), differences between the height reached by the tephra plume and the gasplume, and temporal variability in the plume height (e.g. Tupper and Wunderman (2009))In Appendix B, we detail a well-constrained and poorly constrained case for each of these threeparameters, and explain important differences with estimates made by Mastin (2014) for twoeruptions.For each eruptive phase, the average wind speed, vertical shear and Brunt Väisälä frequencyare retrieved from the NCEP-NCAR reanalysis (Kalnay et al., 1996) using the same method asMastin (2014). For the few eruptive phases occuring before 1948, the 20th Century Reanalysis(Compo et al., 2011) is used to retrieve atmospheric parameters. We assumed a relative uncer-tainty of 10% for all atmospheric parameters because differences between these parameters wassmall when calculated using different reanalysis or atmospheric soundings, for one high-latitudeand one tropical eruptive phase.The calculation of the regime parameters W∗0 and T ∗ requires an independent estimationof eruptions exit velocity U0. We follow Woods and Bower (1995) to compute the exit velocityas:U0 = 1.85√Rn0T0 . (2.25)R is the specific gas constant, n0 the magma volatile content and T0 the magma temperature.We assume that the water content of melt inclusions is representative of the gas content at thevent. For the few eruptions for which we were not able to find n0 in the literature, we assignedn0 = 3.5± 3wt.% (95% confidence level) based on the distribution of n0 found for the othereruptions. We assume an exit temperature of T0 = 1200± 200K. Equation 2.25 assumes thatthere is a free decompression at the vent exit, which does not apply to all explosive eruptions(e.g., Koyaguchi et al. (2010)). To account for uncertainty related to the use of this scalingfor the exit velocity, we assume a 50% relative uncertainty, despite uncertainties on n0 and T0found in the literature would typically result in relative uncertainties of 20− 40% on the exitvelocity, using Equation Data and methodTable A.2 contains the newly compiled database. It is important to stress that these datareflect average eruptive and atmospheric parameters of each eruption phase. Many of theeruptive phases we consider do not have a steady behaviour through their entire duration.However, it is difficult to estimate time-dependent mass eruption rate independently from theplume height for the vast majority of eruptions for which other parameters are well constrained.Although the large number of eruptions in our database and our treatment of uncertainties(Section 2.4.4) make our results more robust, unsteadiness in source conditions and plumeheight remain a limitation for the compiled database.2.4.3 Dynamical similarity of laboratory experimentsExperiment (CGABK 2014)Eruption Experiments (CDCR 2011)Picture shownStrong plumeDistorsed plumeWeak plumeUnclassifiedplumeFigure 2.4: Regime diagram for the Carazzo et al. (2014) experiments (CGABK 2014,squares), volcanic eruptions (Table A.2, circles), and Contini et al. (2011) experi-ments (CDCR 2011, shaded in light blue). W∗0 and Ri0 are the x-axis and y-axisrespectively (logarithmic scale). Error bars are not shown for clarity. Typical rel-ative error for experiments is ' 10%, while errors for natural eruptions are abouthalf an order of magnitude. Red, green, blue and open symbols respectively indi-cate strong (the umbrella spreads both up- and downwind), distorted (the plumerises above the umbrella, which spreads only downwind), weak (the plume does notrise above the umbrella) and undetermined plume regimes. Two plumes picturesare shown for each regime: one is issued from analog experiments, the other onecorresponds to a real eruption: Puyehue-Cordón Caulle (5th June 2011), Eyjafjal-lajökull (10th May 2010) and Etna (24th November 2006) for the strong, distortedand weak plume regimes respectively. Eruptions or experiments for which picturesare shown are highlighted by a diamond symbol. Dashed lines, defined by empir-ical critical values of Riwind0 , show estimated regimes boundaries. Photo of MtEtna by Luigi Lodato. Photo of Eyjafjallajökull courtesy of Dr. Arason. Photo ofPuyehue-Cordón Caulle modified from a picture by air force of Chile/Reuters.252.4. Data and methodGiven large uncertainties in the measurements of erupted mass, eruption duration and averageeruption heights, analog laboratory experiments are crucial to test the performance of modelsrelating plume height to source conditions. However, to test models in dynamical regimesrepresentative of volcanic eruptions, it is necessary that experimental studies are performedin the region of the Ri0-W∗0 -V∗0 parameter space that are appropriate for explosive eruptions.Contrary to previous experimental studies (e.g. Hewett et al. (1971), Hoult and Weil (1972),Huq (1997), Yang and Hwang (2001) or Contini et al. (2011)) Carazzo et al. (2014) data spanvirtually the full range of Ri0-W∗0 -V∗0 in which explosive eruptions apparently occur (Figure 2.4and A.3). W∗0 ranges from 5.10−3 to 5.10−1, V∗0 from 10−2 to 101 and Ri0 from 10−4 to 10−1,although some explosive eruptions from Table A.2 are characterized by larger Ri0 ranging from10−1 to 100.Qualitatively, the plume style (defined hereafter) has similar transitions for experiments anderuptions, which further corroborates that the analog experiments capture the basic mechanicsof natural explosive eruptions. For Riwind0 = Ri0W∗20 > 5, most natural and experimental plumesare characterized by an umbrella spreading both upwind and downwind (strong plumes). Onthe other hand, for Riwind0 < 0.5, most plumes do not rise above their neutral buoyancy leveland only spread downwind (weak plumes). Distorted plumes rise above their neutral buoyancylevel but spread downwind only, and mostly correspond to 0.5 < Riwind0 < 5 for natural andexperimental plumes. Given large uncertainties in eruption source parameters, as well as inattributing a plume style to an eruption from a few photographs, the uncertainty on criticalRiwind0 values at which the plume transition are likely of half an order of magnitude.In contrast, the Contini et al. (2011) dataset spans only two narrow regions of the Ri0-W∗0space (Figure 2.4). Consequently, we will restrict our comparison of scalings to the Carazzoet al. (2014) experiments along with eruptive parameter database. We will discuss resultsobtained when using the Contini et al. (2011) data in Section MethodA major aim of this study is to compare the performance of our proposed scaling (Equation2.24) with the performances of published scalings (Equations 2.2, 2.6, 2.7 2.10 and 2.12) forpredicting the height of explosive volcanic eruptions from source and atmospheric conditions.Our goal is, in part, to provide a benchmark comparison of scalings for volcanic plumes. Tothis end, we fit the plume height predicted by each scaling to the observed plume height, andcompare model fits and coefficient of determination R2. For the Morton et al. (1956) andHewett et al. (1971) scalings, we use a linear fit. For the Degruyter and Bonadonna (2012),Woodhouse et al. (2013), Carazzo et al. (2014) and our proposed scaling, there is an additionalfitting parameter, which is related to the ratio of entrainment coefficients for the Degruyter andBonadonna (2012), Woodhouse et al. (2013) and our proposed scaling, while it is an empiricalparameter (ac) for the Carazzo et al. (2014) scaling.To account for uncertainties in the experimental and observational datasets, we use a Monte262.4. Data and methodCarlo method. For each variable (plume height, source buoyancy flux or mass eruption rate, andenvironment conditions), we assume a gaussian distribution centered on the estimated mean andof standard deviation chosen to match the estimated 95% confidence interval. We create 1000samples of each variable. For each scaling, we thus obtain 1000 values of the fitting parameters(e.g. βα for our proposed scaling) and the determination coefficient of the fit (R2), from whichwe estimate their mean values and the 95% confidence interval, which are taken as the 2.5thand 97.5th quantiles of their distributions. To assess whether one scaling better performs thananother, we test whether the 5th quantile of the distribution of pairwise difference of associateddetermination coefficients is larger than 0.272.5. Results2.5 Results2.5.1 Plume height predictionsHMTT(cm)0 20 40Hmax (cm)010203040R2=0.510.370.64MTT1956(a)HHFH(cm)0 20 40Hmax (cm)010203040R2= 20 40Hmax (cm)010203040R2=0.720.560.83WHPS2013(e)HDB(cm)0 20 40Hmax (cm)010203040R2=0.740.60.84DB2012(d)HCGABK(cm)0 20 40Hmax (cm)010203040R2=0.870.780.93CGABK2014Scalings builtfor volcanic plumes(f)HAJC(cm)0 20 40Hmax (cm)010203040R2=0.880.80.94This studyAnalytical scalingsfor pure plumes(c)Figure 2.5: Measured height Hmax (cm) as a function of scaling heights for the set of ana-logue experiments from Carazzo et al. (2014). Panels (a)-(f) show the Morton et al.(1956) (MTT1956), Hewett et al. (1971) (HFH1971), our proposed, Degruyter andBonadonna (2012) (DB2012), Woodhouse et al. (2013) (WHPS2013) and Carazzoet al. (2014) (CGABK2014) scalings respectively. In panels (a)-(c), scalings havebeen derived analytically for a pure plume model, with different assumptions onthe wind speed. In panels (d)-(f), scalings are functional corrections to the Mor-ton et al. (1956) intended to be applied to volcanic plumes. Error bars show theestimated 95% confidence intervals. Black dashed lines show the linear functionof slope 1. Determination coefficients are annotated on each graph, with the 95%confidence bounds indicated in subscript/superscript.For each scaling, Figure 2.5 shows the measured maximum plume height as a function ofpredicted height for laboratory experiments. The top panels show the three analytical scalings(Morton et al. (1956), Hewett et al. (1971) and this study) while the bottom panels show thescalings from functional approximations of the plume height (Degruyter and Bonadonna (2012),Woodhouse et al. (2013) and Carazzo et al. (2014)).For analytical scalings, the classical HMTT , which does not account for wind effects, and282.5. Resultsthe Hewett et al. (1971) scaling, derived assuming W∗ >> 1 (panels (a) and (b) of Figure 2.5,respectively), poorly explain the plume height variability resulting from various environmentaland source conditions with R2 = 0.510.640.37 and R2 =, respectively (subscript/superscriptindicate the 95% confidence interval). In contrast, the proposed analytical scaling outperformsall scalings, analytical and functional, for the plume height (panel (c), R2 = 0.880.940.8 ). Thebest fit is found for βα = 7.710.95.3 . For functional scalings, the Degruyter and Bonadonna (2012)and Woodhouse et al. (2013) (panels (d) and (e) of Figure 2.5 respectively) scalings capturethe plume height variations resulting from different wind forcings, although they leave about20− 25% of height variance unexplained (R2 = 0.740.840.6 and R2 = 0.720.830.56 respectively). Bestfit for these scalings are found for β2α32= 0.540.740.37 and βα = respectively. Note that here,we define Riwindg = ( WHtankN )2 (required in the Woodhouse et al. (2013) scaling) due to theabsence of a wind gradient in the Carazzo et al. (2014) experiments (where Htank is the depthof the tank used). Despite our modification its original formulation, the Carazzo et al. (2014)scaling outperforms other functional scalings with R2 = 0.870.930.78 (ac = panel (f)). Thedistribution of pairwise differences of R2 indicates that R2 for this scaling is not significantlydifferent (at the 95% level) to that for our proposed analytical scaling.HMTT(km)0 20 40Hmax (km)010203040R2=0.580.460.67MTT1956(a)HHFH(km)0 20 40Hmax (km)010203040R2=0.490.350.63HFH1971(b)HWHPS(km)0 20 40Hmax (km)010203040R2=0.630.510.74WHPS2013(e)HDB(km)0 20 40Hmax (km)010203040R2=0.580.460.68DB2012(d)HCGABK(km)0 20 40Hmax (km)010203040R2=0.620.50.71CGABK2014Scalings builtfor volcanic plumes(f)HAJC(km)0 20 40Hmax (km)010203040R2=0.580.460.67This studyAnalytical scalingsfor pure plumes(c)Figure 2.6: Same as Figure 2.5, but using data from explosive eruptions instead of analogexperiments.We repeat the same analysis using data from explosive eruptions instead of laboratory292.5. Resultsexperiments in Figure 2.6. With natural data, the differences among the performances ofthe scalings are less pronounced. The least successful prediction is obtained with the Hewettet al. (1971) scaling which assume a very strong wind, with R2 = 0.50.620.36. Predictions usingthe Morton et al. (1956), Equation 2.24 and Degruyter and Bonadonna (2012) scalings aresignificantly better than the Hewett et al. (1971) scalings with R2 = 0.58. The Woodhouseet al. (2013) and Carazzo et al. (2014) scalings outperform other scalings with R2 ' 0.63.R2 of these two scalings are not significantly different at a 95% confidence level, but they aresignificantly larger than R2 of all other scalings. In general, the relatively weak performanceswith natural data give poor constraints on fit parameters, and in particular, the entrainmentcoefficients ratio which values are however compatible with the one constrained experimentallywithin the 95% uncertainty.2.5.2 Entrainment coefficientsApplying the Carazzo et al. (2014) experimental data to test plume height scalings (Figure 2.5),we obtain β2α32= 0.540.740.37, βα = ± 2, and βα = 7.710.95.3 for the Degruyter and Bonadonna(2012), Woodhouse et al. (2013) and proposed scalings respectively. However, the expressionfrom Woodhouse et al. (2013) is valid for plumes rising in linear wind gradient. Thus, βα cannot be rigorously constrained using this scaling and the Carazzo et al. (2014) dataset (uniformwind). The constraint obtained from Degruyter and Bonadonna (2012) relies on the arbitraryassumption that the Morton et al. (1956) and Hewett et al. (1971) scalings can be linearlysuperposed to predict the buoyancy flux of a plume rising under intermediate wind forcing(0 <W∗0 < 1). R2 value for this scaling suggests that their assumption is not fully adequate. Incontrast, our scaling is analytical, and applicable to buoyant jets rising in uniform intermediatecrossflow, which corresponds to experimental conditions of Carazzo et al. (2014). Figure A.4shows that a majority of experiments fulfill the assumptions made in deriving the proposedscaling (cf Section 2.3.2), which is further corroborated by the good prediction of plume heightfor these experiments.302.5. Results,0.04 0.06 0.08 0.1 0.12 0.14 0.16- 2015 F 1979HFF 1969SK 2015 H 1997MTT 1956Possible pairs at 95% (2<) significance level)Possible pairs at 68% (1<) significance level)Figure 2.7: Summary of current constraints on entrainment coefficients. Arrows alongthe x and y axis show values of α and β calibrated from previous studies. HFF1969, H 1997, SK 2015, F 1979 and MTT 1956 refer to Hoult et al. (1969), Huq(1997), Suzuki and Koyaguchi (2015), Fischer et al. (1979) andMorton et al. (1956)respectively. Shaded area bounds the values of βα estimated using the Carazzo et al.(2014) dataset and proposed scaling (Equation 2.24), for 2 levels of significance(68%, blue, and 95%, cyan).Figure 2.7 combines existing constraints on entrainment coefficients from previous studieswith the new constraint βα = 7.710.95.3 . The large range of Ri0−W∗0 used in Carazzo et al. (2014)suggests that this constraint holds across the large variety of regimes in which α and β have beenexperimentally constrained in previous studies (Figure 2.7). The plausible space for the choiceof a pair (α,β) is considerably reduced. This new constraint may help reduce uncertaintiesin scaling and integral plume model arising from uncertainties in entrainment coefficient. Wefurther explore this result in the discussion section.2.5.3 Best set of parameters to evaluate mass eruption rateFor natural eruptions, it is common to regress the logarithm of the mass eruption rate as afunction of the logarithm of the plume height to obtain an empirical relationship between masseruption rate and plume height (e.g. Mastin (2014)). Here, to get the best possible predictionsfor mass eruption rates, we perform a multiple linear regression where the dependent variableis the logarithm of the mass eruption rate log(M0) and the predictors are log(Hmax), log(N)and Pwscaling, where Pwscaling is related to the parameter used to account for wind forcing in thedifferent scalings:• log(W ) for the Hewett et al. (1971) scaling312.5. Results• Ri−12gw for the Woodhouse et al. (2013) scaling• V∗0 for the Degruyter and Bonadonna (2012) scaling• T ∗0 for the Carazzo et al. (2014) scaling• W∗0 for our proposed scaling (section 2.3).For the latter four scalings, we developed the expression of the logarithm of the scaled masseruption rate at order 1 in the parameter used to account for wind forcing in each scalingto simplify the expression of Pwscaling. Large differences between the original scalings and oursimplified linear regression model are thus expected when Ri−12gw , V∗0 , T ∗0 or W∗0 are not smallwith respect to 1, but the linearization enables to obtain simple, compact empirical relationshipsbetween plume height, mass eruption rate, and environmental conditions.log(M0)4 6 8log(MregMTT)468 R2=0.720.650.77MTT1956(a)log(M0)4 6 8log(MregHFH)468 R2=0.730.660.78HFH1971(b)log(M0)4 6 8log(MregWHPS)3456789R2=0.730.660.79WHPS2013(e)log(M0)4 6 8log(MregDB)3456789R2=0.770.720.81DB2012(d)log(M0)4 6 8log(MregCGABK )3456789R2=0.760.690.81CGABK2014(f)log(M0)4 6 8log(MregAJC)468 R2=0.720.650.77This study(c)Figure 2.8: Same as Figure 2.6, but showing the logarithm of the regressed mass eruptionrate using the set of parameters employed in each scaling as independent regressionparameters (logarithm of the plume height, Brunt-Väisälä frequency, and wind pa-rameter associated to each scaling), as a function of the logarithm of the measuredmass eruption rate. Regression coefficients for each scaling are indicated in Table2.2. Error bars are not shown for clarity but are typically about 0.5 (half an orderof magnitude).Figure 2.8 shows the regressed mass eruption rate as a function of log(M0). Associated322.6. Discussionregression coefficients are indicated in Table 2.2. Differences among the different predictionsare small, with the mean R2 varying between 0.72 and 0.77. The largest R2 is found whenusing the set of parameters of the Degruyter and Bonadonna (2012) and Carazzo et al. (2014)scaling (no significant R2 difference between these scalings). When only using plume height asa predictor (like Mastin (2014)), we find R2 = 0.700.760.64. R2 is always significantly larger whenusing the Brunt Väisälä frequency or a parameter accounting for wind forcing as a dependentvariable. We note that using scaling relationships accounting for environmental parametersinstead of empirical relationships between plume height and mass eruption rate may provecritical in the future as global warming is expected to impact the height reached by volcanicplumes (Aubry et al., 2016).Table 2.2: Coefficients of the multiple linear regression of the logarithm of the mass erup-tion rate (in kg.s−1) associated, from 1st to 4th row respectively, to the constantterm, the logarithm of the plume height (in km), the logarithm of the Brunt-Väisäläfrequency (in s−1), and the wind parameter (the logarithm of wind speed in m.s−1for Hewett et al. (1971), dimensionless parameter otherwise). The last row gives theassociated coefficient of determination. Coefficients using only the plume height asindependent variables (like Mastin (2014)) are indicated in the 8th column (M2014).Subscripts/superscripts indicate the 95% confidence interval.Coef. MTT1956 HFH1971 this study DB2012 WHPS2013 CGABK2014 M2014C −CW − 1.13.4−1.2 −0.3−0.2−0.4 −R2 0.720.770.66 0.730.780.66 0.720.770.65 0.770.810.72 0.730.790.66 0.760.810.69 0.700.760.642.6 Discussion2.6.1 New constraint on entrainment coefficients: ImplicationsThe application of the proposed analytical scaling to the Carazzo et al. (2014) experimentsconstrains the ratio of entrainment coefficients βα = 7.710.95.3 . Critically, the large range of Ri0used in Carazzo et al. (2014) suggests that the proposed constraint holds across regimes wherethe radial entrainment coefficient α is known to vary significantly. Our result thus suggeststhat variations in β are approximately proportional to those in α. For models accounting forvariable α (e.g. Carazzo et al. (2008a)), the proposed constraint of βα ' 8 consequently providesan empirical framework to take into account corresponding variations of β.To illustrate this application, we calculate plume trajectories for experiments 23 (W ∗0 =4.3.10−2) and 27 (W ∗0 = 3.5.10−3) of the Carazzo et al. (2014) dataset using the Carazzo et al.(2008a) integral model. Figure 2.9 shows (i) the observed plume trajectories (on each panel), (ii)332.6. Discussionthe modeled plume trajectories with constant β (panels (a) and (c)) and (iii) modeled plumetrajectories with constant βα (panels (b) and (d)). We calculate trajectories for values of βvarying from 0.1 to 1 with an increment of 0.1, and for values of βα varying from 5 to 11 with anincrement of 1. For each modeled trajectory, we calculate the relative root-mean squared erroron the plume trajectory as E =√∫ xmax0 (zmod−zobs)2dx∫ xmax0 z2obsdxwhere zmod is the modeled height of thecenterline, zobs the observed height, and xmax is the horizontal distance from the source. Forexperiments 23 and 27, when using a constant β in the integral plume model, optimal valuesto best match the observed plume trajectories are different for experiment 23 (β = 0.5− 0.7)and 27 (β = 0.2− 0.4). In addition, for experiment 27, trajectories with the lowest error Eoverestimate the maximum plume height. These problems disappear when using a constant βαin the integral plume model. We find a best match to plume trajectories with βα = 6− 8 forboth experiments. These values of the ratio βα are close to the mean value of 7.7 estimated usingour proposed scaling. Plume height is also well reproduced for experiment 27. Further testingusing the integral model, which is not the purpose of this study, will be required but theseexploratory results show that the new constraint may not only help to reduce uncertainties onthe values of entrainment coefficients, but may also help improve entrainment parameterization.Note that the recent study of Folch et al. (2016) proposes to account for local variations ofβ, but assumes that β decreases with the local Richardson number on the basis of laboratoryexperiments, while α increases with it. In contrast, the good prediction of our proposed scalingfor the plume height suggests that both parameters vary similarly.342.6. Discussion2.6.2 Plume shape and the Morton et al. (1956) similarity theory.x (cm)0 2 4 6 8 10 12 14 16z (cm)0123456Run 23, constant -(a)Measured-=0.5, E=0.04-=0.6, E=0.08-=0.7, E=0.15x (cm)0 2 4 6 8 10 12 14 16z (cm)0123456Run 23, constant - /,(b)Measured-/,=7, E=0.04-/,=8, E=0.09-/,=6, E=0.1x (cm)0 1 2z (cm)02468101214161820Run 27, constant -(c)Measured-=0.3, E=0.1-=0.4, E=0.11-=0.2, E=0.11x (cm)0 1 2z (cm)02468101214161820Run 27, constant - /,(d)Measured-/,=6, E=0.13-/,=7, E=0.14-/,=8, E=0.16Figure 2.9: Plume trajectories for experiments 23 (panels (a) and (b), W ∗0 = 4.3.10−2)and 27 (panels (c) and (d), W ∗0 = 3.5.10−3) of the Carazzo et al. (2014) dataset.The thick black line shows measured trajectories. Colored discontinuous lines showtrajectories predicted by the Carazzo et al. (2008a) buoyant jet model using aconstant β (panels (a) and (c)) or a constant βα (panels (b) and (d)). In each case,we show the three values of β or βα for which the measured plume trajectory is bestreproduced. The associated relative root mean squared error (E, cf Section 2.6) isindicated for each modeled trajectory.The Morton et al. (1956) model, on which most scalings and integral models of volcanic plumerely, assumes that the plume shape is self-similar. In the presence of wind, the flow is notvertically symmetric and the self-similarity hypothesis is not obviously justified and needs tobe verified. To this end, for each of the Carazzo et al. (2014) experiments, we measure theaverage tracer concentration along plume profiles perpendicular to the centerline using thestack of all images recorded for the experiment. Here, the centerline is defined as the peak ofthe tracer concentration. For Riwind0 = Ri0W∗20 ' 103, the rise is approximately symmetric withrespect to the vertical and plume profiles are well approximated by a gaussian function at allheights. However, for smaller Riwind0 , plume profiles are skewed to the downwind side. Toquantify this asymmetry, we separate each tracer profile into a downwind and upwind half oneither side of the peak of the profile. Each half is well approximated by a gaussian fit (FigureA.5) from which we obtain the characteristic downwind width (rdown) and upwind width (rup).From these widths, we define the “plume asymmetry factor”:352.6. DiscussionAs = 2× rdown − ruprdown + rup. (2.26)Figure 2.10 shows measured As as a function of Riwind0 . We use only profiles for which R2of the gaussian fit of each half are larger than 0.9. In addition, we use profiles measured atheights below 50% of the maximum height due to partial overlapping of the rising plume anddescending or spreading umbrella cloud in the upper plume half for some experiments (FigureA.5). For the largest Riwind0 (' 103), the plume is symmetric around the vertical axis and Asdoes not exceed 0.23 in absolute value. As Riwind0 decreases, maximum values of As increases,indicating that the plume is skewed to the downwind side. For a fixed Riwind0 , As tends toincrease with height. For Riwind0 < 10, As reaches values that exceed the range found for thelargest Riwind0 experiment indicating a significant skewness. For Riwind0 < 1, As often reachesvalues above 0.6, indicating that the downwind plume width is about twice the upwind width.Note that Riwind0 seems to be a better control regime parameter on As thanW ∗0 , and is also theparameter that seems to control the transition in plume style (Figure 2.4). Riwind0 thus seemsto control both the large scale features (plume style) and the internal detail (profile shape) ofthe plume shape.Changes in shape imply that the similarity hypothesis of Morton et al. (1956) is locallyviolated and that the rate of entrainment is different at the downwind and upwind edges of theplume. In numerical simulations, Suzuki and Koyaguchi (2015) also observed velocity profilesskewed to the downwind side. They deduce that entrainment rate is larger to the upwindside because of the sharper upwind velocity gradient. We do not measure local velocity inthe Carazzo et al. (2014) experiments, but a plume skewed to the downwind side requiresa larger mass flux downwind and thus a larger downwind entrainment. A large variety oftransient turbulent features, depending on the dynamical regimes, can explain asymmetricplume shape and entrainment rate (Mahesh, 2013). In particular, based on 3D numericalsimulations, Muppidi and Mahesh (2008) show that downwind entrainment contributes to upto 90% of total entrainment for a jet in a crossflow with W∗0 ' 0.18. They find that thecontribution of downwind entrainment to the total entrainment increases as the plume rises,which is consistent with Figure 2.10. Enhanced downwind entrainment occurs because the jetacts as an obstacle for the crossflow. The pressure field is characterized by a low pressureinside the jet, and two high pressure zones at the upwind edge of the jet where the crossflowis decelerated and downstream of the jet where it is accelerated. Upwind, pressure gradientsare confined very close to the jet edge and little fluid from the crossflows is thus entrained.Downwind, however, pressure gradients extend far behind the jet edge and entrainment ofcrossflow fluid into the jet is thus enhanced (Muppidi and Mahesh, 2008).362.6. DiscussionRi0wind=Ri0/W*2010-2 10-1 100 101 102 103A s (Plume asymmetry factor)-1-0.8-0.6-0.4- downwind widthLarger upwind widthz 5 0:1Hmax0:1Hmax < z 5 0:3Hmax0:3Hmax < z 5 0:5HmaxFigure 2.10: Plume asymmetry factor (As, cf Section 2.6) as a function of the windRichardson number (Riwind0 = Ri0W∗20 ) for the Carazzo et al. (2014) experiments.Black, red and blue dots indicate measurements for z ≤ 0.1Hmax, 0.1Hmax <z ≤ 0.3Hmax and 0.3Hmax < z ≤ 0.5Hmax, respectively. Dashed lines showthe maximum absolute value of As for the experiment with the lowest Riwind0 .Positive As, above the thick dashed line, indicates that the plume upwind widthis significantly larger than the plume downwind width, and vice-versa.Evidence for violation of the self-similarity hypothesis leaves space for improvement of in-tegral models. In particular, laboratory and numerical experiments from which entrainment isdirectly measured are mostly restricted to jets, and to a few values of the regime parameterW∗0 . How the entrainment rate, and thus α and β, vary across the Ri0-W∗0 in which explosivevolcanic eruptions occur remains an open question. In addition, integral plume models usea prescribed shape for buoyancy and velocity profiles, generally parameterized using a distri-bution with a single shape parameter (the plume width). To account for changes in plumeshape, distributions with two shape parameters could be used to describe the plume profiles(e.g. a width and a skewness). Changes in the second shape parameter could be described byan empirical equation, based on laboratory or numerical experiments. Further work to bettercharacterize variations of entrainment rates and changes in plume shapes is required beforeincluding such effects in integral plume models, but would lead to an improved prediction ofplume height and an additional prediction of particle and gas distribution in the plume whichare a critical outputs for dispersion models (Folch and Felpeto, 2005).372.6. Discussion2.6.3 Future challenges for testing scaling and integral plume modelsLaboratory experimentsHMTT(cm)10 20 30 40Hmax (cm)10203040 R2=0.440.380.5MTT1956(a)HHFH(cm)10 20 30 40Hmax (cm)10203040 R2=0.950.920.97HFH1971(b)HWHPS(cm)10 20 30 40Hmax (cm)10203040 R2=0.950.930.97WHPS2013(e)HDB(cm)10 20 30 40Hmax (cm)10203040 R2=0.960.930.98DB2012(d)HCGABK(cm)10 20 30 40Hmax (cm)10203040 R2=0.910.870.94CGABK2014Scalings builtfor volcanic plumes(f)HAJC(cm)10 20 30 40Hmax (cm)10203040 R2=0.920.890.95This studyAnalytical scalingsfor pure plumes(c)Figure 2.11: Same as Figure 2.5, but using laboratory experiments from Contini et al.(2011). Red dashed lines highlight the three main values of HMTT used in theseexperiments.Ideally, validations of volcanic plume models on the basis of analog laboratory experimentsshould rest on a dataset spanning the same range of source Richardson numbers (Ri0) and windto source velocity ratios (W∗0 ) as natural explosive eruptions. However, Carazzo et al. (2014)show that most existing datasets do not fulfill this condition. To test whether datasets withrestricted Ri0 −W∗0 ranges allow a correct assessment of plume models, we use the Continiet al. (2011) experiments to test height predictions of the six scalings (the Ri0 −W∗0 spacecovered by this study is shown in Figure 2.4 by blue shadings).Predicted plume heights from Contini et al. (2011) are shown in Figure 2.11. In contrastto Figure 2.5, all scalings accounting for wind forcing show excellent performance in capturingplume height variability (R2 > 0.9). Beyond the restricted Ri0 −W∗0 used by Contini et al.(2011), the few values used for F0 and N (source buoyancy flux and stratification), and thusHMTT (red dashed lines on panel (a), Figure 2.11), facilitate fitting by scalings using an ad-ditional parameter related to wind forcing. While a restricted Ri0-W∗0 parameter space may382.6. Discussionbe justified for some applications, such as engineering oriented applications (e.g., Huq (1997);Contini et al. (2011)), the comparison of Figures 2.5 and 2.11 demonstrates that it can lead toan equivocal model validation for applications to volcanic plumes.As a final remark to this section, experiments with more complex wind profile would enablea quantitative comparison of scalings that are based on the average wind speed (e.g. Degruyterand Bonadonna (2012)) to those developed with an average wind vertical gradient (e.g. Wood-house et al. (2013)). In particular, that the Woodhouse et al. (2013) and Carazzo et al. (2014)scalings best predict the height of natural eruptions may highlight the important role of windgradients for natural eruption dynamics. However, these scalings reasonably capture the plumeheight variability from the Carazzo et al. (2014) experiments which also suggests that effectsof wind gradient and constant wind may be comparable over certain W∗0 ranges. Appropriatelaboratory experiments with wind gradients will resolve this issue.Explosive eruptionsFor explosive eruptions, our initial study used only the 25 eruptions from the Mastin (2014)dataset. Significant differences - in terms of the performance of each scaling as well as bestfit parameters - appear when we use the newly compiled dataset of 94 eruptions (not shown).These differences highlight the need to keep improving datasets with plume height and sourceconditions estimated independently, in particular by including eruptions spanning a large rangeof source and atmospheric conditions. Our study shows that despite large uncertainties innatural eruption parameters, statistically significant differences between scalings performanceexist. An improved assessment of uncertainties in eruption parameters will further improve thecomparison of scalings, and plume models in general.Last, it is commonly thought that atmospheric water vapor entrained inside a volcanic plumemay eventually condense as the plume rises and cools. The release of latent heat with conden-sation would increase the plume buoyancy flux which, in turn, will increase the plume height(Morton, 1957; Woods, 1993). Numerical modeling results suggest that for M0  106kg.s−1,this additional flux of buoyancy dominates the eruption source buoyancy flux related to ashand clast heat, and depends mostly on atmospheric humidity (e.g., Woods (1993); Degruyterand Bonadonna (2012); Woodhouse et al. (2013); Mastin (2014)).392.6. DiscussionRelative humidity at vent altitude  (%)0 50 100Hmax-HMTT (km)-10-50510r= (all)r=0.360.270.45 (M0<106)MTT1956(a)Relative humidity at vent altitude  (%)0 50 100Hmax-HHFH (km)-10-50510r= humidity at vent altitude  (%)0 50 100Hmax-HAJC (km)-10-50510r= study(c)Relative humidity at vent altitude  (%)0 50 100Hmax-HDB  (km)-10-50510r= humidity at vent altitude  (%)0 50 100Hmax-HWHPS (km)-10-50510r= humidity at vent altitude  (%)0 50 100Hmax-HCGABK  (km)-10-50510r= 2.12: Residual height for each scaling as a function of the atmosphere relativehumidity at vent altitude. Red dots show eruptions with M0 < 106kg.s−1 whichare expected to be more impacted by condensation of entrained water vapor. Thecorrelation coefficient between the residual height and relative humidity acrossall eruptions (respectively eruptions with M0 < 106kg.s−1) is indicated on eachpanel in black (respectively red).To verify the impact of humidity on plume height inferred from models, we test how residualheights from Figure 2.6 depend on relative humidity at vent altitude (Figure 2.12). Positiveresiduals are expected for eruptions impacted by condensation of entrained atmospheric watervapor, because the tested scalings do not account for condensation of entrained water vapor.Figure 2.12 shows that relative humidity at vent is indeed positively correlated with the residualplume height. The correlation is significant for all scaling used to calculate a residual, and isstronger when calculated with the only eruptions for which M0 6 106kg.s−1. Our resultsthus suggest that condensation of entrained water vapor increase the plume height. Furtheranalysis of humidity impact is beyond the scope of this paper, but it would be valuable to testif accounting for the condensation of entrained atmospheric water vapor in integral models ofvolcanic plumes improves plume height prediction.402.7. Conclusions2.7 ConclusionsIn this study, we derive a new analytical scaling for buoyant plumes rising into a density-stratified uniform crossflow of intermediate strength (W∗0 ,V∗0  1). This scaling, along withothers commonly used in volcanology (cf section 2.2), is tested using a dataset of 27 analogexperiments on buoyant jets rising in density stratified unifrom crossflow (Carazzo et al., 2014)and a new dataset of 94 explosive eruption phases. In contrast to previous studies, our exper-imental data span similar dynamical conditions as explosive eruptions. Our work provides abenchmark comparison between scalings for volcanic plume heights and supports eight mainconclusions:1. Our proposed scaling best predicts heights of plumes from analog experiments with uni-form crossflow and highlights the key control of the wind speed over buoyant jet sourcevelocity ratio (W∗0 ) on the flow.2. The application of the scaling to predict the height of laboratory experiments suggeststhat the ratio of the wind entrainment parameter β to the radial entrainment parameterα is constant through a large span of dynamical regimes, with βα = 7.710.95.3 ± 3 (95%confidence interval). This constraint is applied in an integral model of buoyant jet withα depending on the local Richardson number and seems to improve plume trajectoriespredictions.3. The Woodhouse et al. (2013) and Carazzo et al. (2014) scalings, which includes effects ofwind gradients, best captures the heights of natural explosive eruptions.4. Empirical multilinear regression using the plume height, stratification, and a wind regimeparameter slightly but significantly improve mass eruption rate predictions compared toregression using plume height only.5. Experimental datasets spanning a narrow range Ri0-W∗0 parameter space do not distin-guish plume model performance.6. The regime parameterRiwind seems to control transition in plume shape (strong, distortedand weak plumes), i.e., if a plume overshoots its level of neutral buoyancy and if it spreadsup- and downwind, or only downwind.7. Residual plume height is positively correlated with relative humidity at vent, which sug-gests that condensation of entrained water vapor increases plume height.8. Despite the successful prediction of our scaling for analog plume heights, the plume shapeis not self-similar in the presence of a crossflow and the Morton et al. (1956) similarityhypothesis is thus violated in the presence of wind. This conclusion is not new but israrely discussed for volcanic plumes. It leaves space for entrainment and plume shapeparameterization can be improved in integral model.412.7. ConclusionsOne main contribution of this work is to provide a more accurate constraint on the entrainmentcoefficients applied in volcanic plume models generally. This work will improve consistency inentrainment parameter values used by different authors, which is the main cause of uncertaintyin volcanic plume height predictions by integral plume models (Costa et al., 2016). A secondmajor contribution is the newly compiled database of eruption parameters will also contributeto improved evaluation of plume models.42Chapter 3Turbulent entrainment into volcanicplumes: New constraints fromlaboratory experiments on buoyantjets rising in a stratified crossflowThis chapter was published in Geophysical Research Letters in October 2017( SummaryPredictions for the heights and downwind trajectories of volcanic plumes using integral mod-els are critical for the assessment of risks and climate impacts of explosive eruptions, but arestrongly influenced by parameterizations for turbulent entrainment. We compare four pop-ular parameterizations using small scale laboratory experiments spanning the large range ofdynamical regimes in which volcanic eruptions occur. We reduce uncertainties on the windentrainment coefficient β which quantifies the contribution of wind-driven radial velocity shearto entrainment and is a major source of uncertainty for predicting plume height. We showthat models better predict plume trajectories if: i) β is constant or increases with the plumebuoyancy to momentum flux ratio; ii) the superposition of the axial and radial velocity shearcontributions to the turbulent entrainment is quadratic rather than linear. Our results haveimportant implications for predicting the heights and likelihood of collapse of volcanic columns.3.2 IntroductionA major control on explosive volcanic plume behavior is the turbulent entrainment of ambientatmosphere into the plume. This process is parameterized in one-dimensional integral mod-els of volcanic plumes. Important parameters, such as the wind entrainment coefficient, are433.2. Introductionsubject to uncertainties of up to an order of magnitude (e.g. Hewett et al. (1971); Suzuki andKoyaguchi (2015)) and values used in the latest volcanic plume model intercomparison studyvary by a factor of 2 among modeling groups (Costa et al., 2016). Plume height predictionsare consequently subject to uncertainties of up to a factor 3 (Figure 3.1, left), with criticalimplications for how to evaluate the climate impacts of explosive eruption (e.g. Robock (2000)).Uncertainties on entrainment parameters also affect the predictions for the collapse of volcanicplumes and the production of devastating pyroclastic flows (Degruyter and Bonadonna, 2013;Jessop et al., 2016a).In the classical entrainment parameterization developed for a buoyant jet rising into aquiescent environment, the radial entrainment velocity u is proportional to the upflow velocityu:u = α|u| , (3.1)where α is the radial entrainment coefficient. To extend this parameterization to an en-vironment with a horizontal crossflow (e.g. wind), Hoult et al. (1969) propose that the axialand radial velocity differences between the jet and the ambient fluid contribute linearly to theentrainment rate, such that:u = α|u−w cos(θ)|+ β|w sin(θ)| . (3.2)Here, θ is the local inclination of the plume with respect to the horizontal (Figure 3.1, right),w the local speed of the crossflow, and β is the wind entrainment coefficient. Devenish et al.(2010) build on Hoult et al. (1969) and propose a less restrictive power law parameterization ofthe form:u = ((α|u−w cos(θ)|)m + |βw sin(θ)|m) 1m . (3.3)In this treatment, m > 1 is a constant. As m increases the contribution to u of the largestterm among α|u−w cos(θ)| and |βw sin(θ)| increases. In terms of predictions for plume height,Devenish et al. (2010) find a better agreement with large eddy simulations for m = 1.5.Although Equation 3.3 is the most general entrainment parameterization, the values for m,α and β vary widely in the literature. This variability is a major cause of uncertainty in integralmodels of volcanic plumes (Costa et al., 2016). In the majority of models, with the exception ofDevenish et al. (2010), the exponent m is taken to be 1. The radial entrainment coefficient α iseither imposed constant with values varying between 0.05 and 0.17 (Morton et al., 1956; Chenand Rodi, 1980), or variable with the distance from the source due to local buoyancy effects(Fischer et al., 1979; Kaminski et al., 2005). The wind entrainment coefficient β is commonlytaken constant with values varying between 0.1 and 1 (Hewett et al., 1971; Bursik, 2001; Suzukiand Koyaguchi, 2015), but it can also vary with the distance from the source (Folch et al., 2016).Other parameterizations include a variable coefficient α but a constant ratio β/α (Aubry et al.,443.3. Model2017b). These large uncertainties in the values of the parameters used in Equation 3.3 leadto important differences in the predictions of jet maximum height and trajectory (Figure 3.1,left).The major aim of this study is to evaluate, calibrate, and compare the different entrainmentcoefficient models available in the literature (Section 3.3) to reduce uncertainties in modeling ofvolcanic plumes, and more generally buoyant jets. To this end, we use an experimental datasetspanning the full range of dynamical conditions in which explosive volcanic eruptions occur, andMonte Carlo simulations to account for experimental uncertainties (Section 3.4). In Section 3.5,we show the constraints obtained on the parameters (m, α, β or βα) used in the tested models,and demonstrate that important differences exist among the performance of these models. Webriefly discuss our results in Section 3.6 and summarize our main conclusions in Section ModelFollowing Morton et al. (1956) and Hoult et al. (1969), we use a plume-centered system, wheres denotes the curvilinear abscissa along the plume centerline. For “Top-Hat” radial dependenceof the plume properties, the conservation equations of mass, axial and radial momentum, andbuoyancy fluxes for the control volume in Figure 3.1 (right) are,dxds = cos(θ) ,dzds = sin(θ), (3.4)dds(ρur2)= 2ρaru, (3.5)dds(ρu2r2)= (ρa − ρ) gr2 sin(θ) +w cos(θ) dds(ρur2), (3.6)(ρu2r2) dθds = (ρa − ρ) gr2 cos(θ)−w sin(θ) dds(ρur2), (3.7)dds(g′ur2)= −N2ur2 sin(θ), (3.8)where x and z are the horizontal and vertical distance from the source, r is the jet radius(orthogonal to s), g is the Earth’s gravity acceleration, ρ is the jet density, ρa is the density ofthe atmosphere, g′ = g (ρa − ρ) /ρa is the jet reduced gravity, and N is the atmospheric Brunt-Väisälä frequency. The turbulent entrainment rate u is parameterized by using Equation 3.3together with four entrainment closures commonly used in the literature:• Model 1 (Hewett et al., 1971) uses constant values for α and β. We perform the calcula-tions for α = 0.05− 0.17 with increments of 0.01, and β = 0.1− 1 with increments of 0.1,consistent with values found in the literature.• Model 2 (Kaminski et al., 2005; Girault et al., 2016) uses a constant value for β and avariable coefficient α:453.4. Data and methodsα = 0.0675 + (1− 1A)Ri+ r2d ln(A)dz, (3.9)where A is a dimensionless shape function that depends on the flow structure (see e.g.Carazzo et al. (2006, 2008a)), z is the vertical distance from the source, and Ri = rg′/u2is the local Richardson number.• Model 3 (Folch et al., 2016) uses variable coefficients α and β. This model combines theparameterization proposed by Kaminski et al. (2005) for the radial entrainment coeffi-cient α (Equation 3.9) and the parameterization proposed by Tate (2002) for the windentrainment coefficient β based on laboratory experiments from previous studies:β = 0.34(√2|Ri|W∗)−0.125 . (3.10)Here W∗ = w/u0 and the subscript 0 denotes a value at the source.• Model 4 (Aubry et al., 2017b) uses a variable coefficient α and a constant ratio β/α atany distance from the source. This model combines the parameterization proposed byKaminski et al. (2005) for the radial entrainment coefficient α (Equation 3.9) and keepsβ/α constant because it enables good predictions of plume height by a scaling law througha large range of Ri, where α is expected to vary. However, the scaling on which theirresults are based uses several simplifying assumptions and the proposed parameterizationhas not been tested in an integral model for buoyant jets. We perform the calculationsfor β/α = 4− 12 with increments of 1.In the four models, we performed calculations for m = 1− 2 with increments of Data and methodsThe trajectory (x,z) predicted by Equations 3.3 to 3.8 together with the four models for en-trainment coefficient will be compared to measured trajectories (maximum tracer intensity) oflaboratory buoyant jets rising in a density-stratified crossflow. These experiments are presentedin detail in Carazzo et al. (2014). The full experimental dataset spans almost the entire rangeof Ri0 and W ∗0 in which explosive eruptions occur, and cover a much larger parameter spacethan previous studies (Carazzo et al., 2014).The source and environment parameters as well as the associated uncertainties were care-fully measured for the 27 experiments and are reported in Aubry et al. (2017b). We obtainedthe centerline of the jet and the associated uncertainty for each experiment from the stackof all images recorded with digital video camera using a semi-automatized script looking formaximum tracer intensity. The duration of an experiment is much longer than the plume risetimescale, so that transient turbulent features (e.g. eddies) are removed when stacking all im-463.4. Data and methodsages together. Figure 3.1 (left) shows an example of experimental buoyant jet and obtainedcenterline trajectory.CenterlineUpper/lower limit of centerlineCenterline from model runθ(s)zxsρa(z)Wρa(z)uε(s)u(s)u(s+ds)ss+dsr(s)r(s+ds)ρ(s)ρ(s+ds)W cos(θ)W sin(θ)xθ(s)E x p e r i m e n t M o d e l5 cmFigure 3.1: Left: Buoyant jet, observed centerline (cyan lines), and model centerlines(black lines) for one experiment from the Carazzo et al. (2014) dataset. The pictureshows the average of > 300 frames taken during the experiment. The thick andthin cyan lines show the plume centerline and the 95% confidence envelope of thecenterline as found by a semi-automatic script. Black lines show model trajectoriesfor model 2 for one sample of source and environmental conditions (i.e., one MonteCarlo simulation, cf Section 3.4), for all tested values of β and m = 1 or m = 2.Right: Cartoon of a rising plume (top) and of the control volume (bottom) usedto derive conservation Equations 3.4-3.8. Modified from Aubry et al. (2017b).To test how well a given model with specified entrainment parameters values reproduce theplume trajectory of an experiment, we first calculate the standardized root mean squared erroron the plume trajectory for experiment i as:ei =∫ xmax0 (zmod − zobs)2 dx∫ xmaxobs0 z2obsdx1/2 , (3.11)where zmod is the modeled height of the centerline, zobs the observed height, xmaxobs is thehorizontal distance from the source where the maximum plume height is reached for the observedcenterline, and xmax is the largest value between xmaxobs and xmaxmod (same as xmaxobs , but for the473.5. Resultsmodel trajectory).Next, we find the total weighted root mean squared error of a model prediction for plumetrajectories with specified entrainment parameter values as:E =(1nn∑i=1e2i(∆ei)2)1/2, (3.12)where n = 27 is the number of experiments, and the weights ∆ei are the 95% observationaluncertainty on ei (cf Supplementary Information S1 for the detailed calculation of ∆ei). Theseweights give more importance to experiments with smaller observational error. The error E isthen calculated for each entrainment model (1-4) and each possible combination of entrainmentparameters value (α, β, m and/or β/α). Note that E is non-dimensional and expressed asa fraction of the observational uncertainty. For example, if a model has an error of 1.3, itmeans that the model error is 30% larger than the error on plume trajectory attributable toexperimental uncertainties alone.Last, to account for experimental uncertainties, we use a Monte Carlo method to generate100 sets of experimental parameters. For each experiment and each parameter (e.g. exit velocityor plume trajectory), we assume a Gaussian distribution centered on the measured value withstandard deviation chosen to match the estimated 95% confidence interval. These distributionparameters are provided in Table B.1 for source (e.g. exit velocity U0) and environment (e.g.wind speed w) parameters for all experiments. We then draw 100 samples of each variable andrun the four models across all combination of entrainment parameters to obtain 100 values ofthe error E for each entrainment model and set of entrainment parameter values. For a givenmodel, we consequently find the best (i.e., by minimizing error E) set of entrainment parametersfor each of the 100 Monte Carlo simulations and obtain the probability distributions of theentrainment parameters of each model. In addition, the distribution of pairwise differences inerror between two models quantifies the probability for a model to outperform another.3.5 Results3.5.1 Best set of entrainment parameters for each modelFor each entrainment model (columns) and each parameter (rows), Figure 3.2 shows (i) thefrequency at which a value of the parameter minimizes the error E across all Monte Carlosimulations (red histograms) and (ii) the minimum error E obtained for each tested value ofthe parameter (black dots). Thus, red histograms show the likelihood that a given parametervalue minimizes the error E and black dots show the sensitivity of E to the value of the sameparameter, for each model.The first row of plots in Figure 3.2 shows results for the exponent m in Equation 3.3. Formodels 1 and 4, and models 2 and 3, the error on trajectory E is minimized in more than 95%of the Monte Carlo simulations when m > 1.5 and m > 1.75, respectively (as shown by red483.5. Resultshistograms). The error is sensitive to the value of m: E varies by as much as 0.5 (i.e., 50% ofexperimental uncertainty on plume trajectories) for the tested values of m.m1   1.25 1.5 1.75 2   Exponent m0   0.250.5 0.751   Model 1: constant ,  and -1  1.52  2.53  3.54  m1   1.25 1.5 1.75 2   0   0.250.5 0.751   Model 2: ,(Ri) and constant  -1  1.52  2.53  3.54  m1   1.25 1.5 1.75 2   0   0.250.5 0.751   Model 3: ,(Ri) and - (Ri,W*)1  1.52  2.53  3.54  m1   1.25 1.5 1.75 2   0   0.250.5 0.751   Model 4: ,(Ri) and constant - /,1  1.52  2.53  3.54  ,0.05 0.08 0.11 0.14 0.17Radial entrainmentcoefficient ,0   0.250.5 0.751   1  1.52  2.53  3.54  -0.1 0.3 0.5 0.7 0.9Wind entrainmentcoefficient - or -/,0   0.250.5 0.751   1  1.52  2.53  3.54  -0.1 0.3 0.5 0.7 0.90   0.250.5 0.751   1  1.52  2.53  3.54  -/,4 6 8 10 120   0.250.5 0.751   1  1.52  2.53  3.54  Probability of minimizing error EMinimum errorFigure 3.2: Values of plume model parameters minimizing the error on centerline trajec-tory (Equation 3.12). Each column of plots shows results related to one entrainmentmodel, while each row of plots shows results related to one type of model parameter.On each plot, red bars (left red axis) show the distribution (across all Monte Carlosimulations) of the parameter values for which the error E on centerline trajectoryis minimized. Black dots (right black axis) show the minimum error on centerlinetrajectory for a fixed value of the model parameter considered. Dots show the meanerror and error bars are the 95% confidence interval.In the second row of Figure 3.2, we show results for the wind entrainment parameter β formodels 1 and 2, and the ratio βα for model 4. Model 3 parameterize β using Equation 3.10and its value is thus not specified. The error E is systematically minimized for β = 0.5− 0.6with model 1 and for β = 0.4− 0.5 with model 2, with β = 0.5 being the best value in morethan 90% of the Monte Carlo simulations for both models. For model 4, the values βα = 6− 8minimize E. Models 1 and 2 are very sensitive to β with the minimum error E varying between1.2 and 4 depending on the value of β. Model 4 is less sensitive to the value of βα with the errorE varying between 1.5-2.5.493.5. ResultsIn the third row of Figure 3.2, we show results for the radial entrainment parameter α formodel 1. Models 2-4 use Equation 3.9 for α and its value is thus not specified. The valuesα = 0.05− 0.07 minimize E for model 1, with E ranging from ' 1.4 for α = 0.05 to ' 2.5 forα = Best model for entrainment coefficientsE1-E2-0.5 0 0.500. betterthan model1 (22%)Model1 betterthan model2 (78%)E1-E3-0.5 0 0.500. betterthan model1 (4%)Model1 betterthan model3 (96%)E1-E4-0.5 0 0.500. betterthan model1 (10%)Model1 betterthan model4 (90%)E2-E3-0.5 0 0.500. betterthan model2 (10%)Model2 betterthan model3 (90%)E2-E4-0.5 0 0.500. betterthan model2 (25%)Model2 betterthan model4 (75%)E3-E4-0.5 0 0.500. betterthan model3 (68%)Model3 betterthan model4 (32%)Figure 3.3: Distribution of pairwise differences of error on centerline trajectory E, foreach pair of model. For a given model i, we sampled 1000 sets of parameters fromthe distributions shown in Figure 3.2. We thus obtain 100 000 values of Ei, theerror for model i, across all Monte Carlo simulations. The resulting distributionof pairwise differences are shown in this figure, and account for both observationaland model parameter uncertainty. For two models i and j, Ei −Ej is positive ifmodel j predicts centerline trajectories better than model i, and vice-versa. Theestimated probability for a model to outperform the other one is annotated on eachplot.Figure 3.3 shows the distribution of pairwise error differences for the six possible pairs ofmodels. These distributions account for both observational uncertainties, and uncertainties onentrainment parameters constrained from Figure 3.2 (cf Figure 3.3’s caption).The only difference significant at the 95% level is between models 1 and 3, with model 1having a smaller error in 96% of the samples. However, Figure 3.3 shows some clear trends in503.6. Discussionmodel performance. Model 1 performs better than other models (with a confidence level of 78to 96%), while model 2 outperforms models 3 and 4 (with a confidence level of 75 to 90%) andmodel 4 outperforms model 3 in 68% of the samples. The magnitude of pairwise error differencesalso shows that the error is moderately sensitive to the choice of entrainment models, comparedto the choice of entrainment parameter values for each model. Indeed, most error differencesdo not exceed 0.3 (30% of the observational uncertainties) with highest differences of ' 0.5. Incontrast, the choice of β in models 1 or 2 was resulting in error differences of up to 2.5, forexample.3.6 Discussion3.6.1 New constraints on entrainment parameters and implications forvolcanic plume modelingThe constraints obtained on entrainment parameters (m, α, and β) in Figure 3.2 show notabledifferences with the values most commonly used in models of volcanic plume. First, mostmodelers use a value of m = 1 (e.g. Degruyter and Bonadonna (2012); Woodhouse et al.(2013); Mastin (2014)), except the Devenish model which uses m = 1.5 (Devenish et al., 2010;Devenish, 2016). Our results are more consistent with Devenish et al. (2010) and show thatvalues of m between 1.5 and 2 decrease the error on plume trajectories by up to 0.5.Second, Figure 3.2 suggests that β = 0.4−0.6 minimize errors in plume trajectories, for bothmodels 1 and 2. Although several models of volcanic plume use β = 0.5, the range commonlyused is 0.5− 0.9 (Costa et al., 2016). Our results show that higher values in this range mayresult in large error on plume trajectories. The compatibility of our results for β with previousstudies using a value of m = 1 suggests that the value of β is only weakly sensitive to the valueof m.Third, for model 3, the range of 6− 8 found for βα in Figure 3.2 is in remarkable agreementwith the results of Aubry et al. (2016), even though this previous study used an analyticalscaling (relating plume height to source conditions and environmental parameters) and plumeheight measurement to constrain this ratio.Last, for model 1, we find values of α corresponding to values usually found for momentum-driven jets, consistent with |Ri0| 6 10−1. These optimal values of α are for a plume risingunder a crossflow, and with m = 1.5− 2 in Equation 3.3. Previous studies mostly constrainedα using m = 1, and in the absence of crossflow.513.6. DiscussionWind entrainment coefficient -0.1 0.25 0.4 0.55 0.7 0.85 1   Critical Richardson number forcollapse-buoyant plume transition-102-101-100-10-1Buoyant PlumeCollapseExperimentallyconstrainedvalues of -(this study)Wind entrainment coefficient -0.1 0.25 0.4 0.55 0.7 0.85 1   Critical mass eruption rate (kg/s) for reaching the tropopause106107108Stratospheric injection, signi-ficant impacts on climateNo direct stratosphericinjectionFigure 3.4: Left: critical Richardson number at which the transition from buoyant plumeto collapsing plume occur as a function of the wind entrainment coefficient β. Right:critical mass eruption rate required to cross the tropopause as a function of β.For both plots, we used the model of volcanic plume of Degruyter and Bonadonna(2012) with entrainment coefficients parameterized as in Model 2, m=2, and ignoredthe effects of condensation of water vapor. Atmospheric conditions correspond tothose during the second phase of the 2011 Cordón Caulle eruption, which occurredunder strong winds, and were retrieved from the NCEP-NCAR reanalysis (Kalnayet al., 1996). Calculations were performed for a range of values of β correspondingto values previously published, and the green shading highlight the values for model2 constrained in this study. Dots and error bars show the mean and 95% confidenceintervals obtained with 300 Monte Carlo simulations where we randomly sampledthe source temperature (800-1500 K) and gas content (0.01-0.07 wt.%). For the leftplot (critical Richardson number), the vent radius was also randomly sampled (10-100 m) while for the right plot (critical mass eruption rate), the source Richardsonnumber was randomly sampled (−10−4to− 10−2).These improved constraints on entrainment parameterizations have important implicationsfor the modeling of volcanic plumes. For example, the main source of uncertainty on volcanicplume height, which governs the lifetime of ash and aerosols in the atmosphere, is the windentrainment coefficient β (e.g. Woodhouse et al. (2015)). In particular, Figure 3.4 (right) showsthat the minimum mass eruption rate required for a volcanic plume to reach the tropopauseis uncertain by more than one order of magnitude for values of β ranging from 0.1 to 1. Thenew constraints found in this study result in an uncertainty of a factor of 2. The stability ofvolcanic plumes, which controls the production of pyroclastic flows, is also determined by thevalues of entrainment coefficients (Degruyter and Bonadonna, 2013). Figure 3.4 (left) showsthat the critical Richardson number at the vent above which an eruptive column collapses issubject to uncertainties of one order of magnitude as a consequence of uncertainties on β. This523.6. Discussionuncertainty is again significantly reduced thanks to the constraints on β provided in this study.3.6.2 Parameterization of entrainment coefficients and future workAlthough our study gives some insights into which models of entrainment parameters best re-produce experimental trajectories, only one pairwise comparison exhibits differences significantat the 95% confidence level (3 pairs at the 90% level). In general, it appears that model 3,which assumes that β decreases with Ri, has larger errors than models that assume a constantβ (models 1 and 2) or β increasing with Ri.To understand how entrainment coefficients α and β depend on regime parameters Riand W∗, we assess which values of α and β minimize the error e (Equation 3.11) for indi-vidual experiments. We then assume relationships of the form α = C1(Ri0)r1(W∗0 )w1 andβ = C2(Ri0)r2(W∗0 )w2 and use multilinear regression to find the values of fitting coefficients(C1, C2, r1, r2, w1 and w2) minimizing the least square error, on a log scale. The chosen formfor these relationships has no physical basis although it is similar to parameterizations usedin other models (e.g. β for model 3, Folch et al. (2016)). However, the signs of the exponentprovide a simple test on whether entrainment coefficients decrease or increase with the regimeparameters W∗0 and Ri0.The first two panels in Figure 3.5 show the estimated values of α and β as a function ofvalues predicted by the regression against W∗0 and Ri0. We perform regressions using onlydata from plumes for which the umbrella spreads only downwind, because the dependence ofentrainment coefficients on both Ri0 and W∗0 in this regime is evident. We find the best fitrelationshipsα = 0.11Ri0.020 W∗0.120 (3.13)β = 0.76Ri0.070 W∗0.130 . (3.14)(3.15)Figure 3.5 thus suggests that both α and β increase with W∗0 and Ri0 although the de-pendence on these two parameters is weak. In addition, when constraining values of α and βfor individual experiments, uncertainties on source and environmental parameters as well asthe plume trajectory result in very large uncertainties. Consequently, there is only a > 90%confidence level that coefficients w1, w2 and r2 in the regression are positive (75% for r1).The third panel in Figure 3.5 shows the estimated values of β as a function of the param-eterization of model 3 (Tate, 2002; Folch et al., 2016) and source values W∗. The predictedvalues and the estimates are anti-correlated at the 95% confidence level despite experimentaluncertainties.Overall, Figure 3.5 explains the results of Figure 3.3. Model 3 is outperformed by othermodels because α and β seem to slightly increase, not decrease, with increasing values of W∗0533.7. Conclusionsand Ri0. The lack of significance of these trends causes differences between models 1, 2 and 4to be not significant at the 95% level. The most significant trend is the increase of α and β withW∗0 for plumes spreading downwind only and it can be interpreted using the results of Aubryet al. (2017b). They find that as crossflow speed increases, tracer profiles in the plume do notremain similar downwind, becoming skewed to the downwind side of the plume which suggestsa larger downwind entrainment. This increased downwind entrainment may explain why themodel better fits trajectories with larger values of α and β asW∗0 increases. However, the trendsfrom Figure 3.5 remain challenging to quantitatively interpret because of their low (6 90%)significance caused by uncertainties on individual values of α and β. In addition, local plumeproperties (velocity, density) were not measured in the Carazzo et al. (2014) experiments sothat we can only infer how mean entrainment coefficients depend on source and environmentalregime parameters, while parameterizations of models 2, 3 and 4 express relationships betweenlocal entrainment rates and local regime parameters. Laboratory experiments covering a largerange of parameter space with measurements of local plume properties would thus help improveour understanding of the impact of crossflows on turbulent entrainment in buoyant jets.0.11Ri00.02W0*0.120.02 0.04 0.06 0.08 0.1 0.12, downwind onlyUmbrella downwind and upwind0.76Ri00.07W0*0.130.1 0.2 0.3 0.4 0.5 0.6- $0 )!0:1250.2 0.3 0.4 0.5 0.6 0.7 0.8- 3.5: Estimated value of α or β compared to functions of the form C1Rir10 W∗w10 .For the left and center plots, fit parameters C1, r1 and w1 are estimated using amultilinear regression between α or β, and Ri0 and W ∗w0 , on a log scale. Onlyplumes for which the umbrella spreads only downwind (blue) are used to performmultilinear regressions. The right plot shows the estimated values of β as a functionof the paramaterization used in model 3. Dashed lines show the 1:1 ratio. Theestimates of α and β are subject to large uncertainties which are not shown forclarity.3.7 ConclusionsIn this study, we constrain and compare four popular parameterizations of entrainment coef-ficients involved in Equation 3.3 using small-scale laboratory experiments. We demonstratethat values of the wind entrainment coefficient β between 0.4 and 0.6 and that values of mbetween 1.5 and 2 minimize error on downwind plume trajectories for these experiments. The543.7. Conclusionslatter constraint implies that a quadratic superposition of the axial and radial velocity gradientcontributions is better than a linear superposition. In addition, we show that models where βis constant or increases with the plume Richardson number are better than the tested modelwhere β decreases with the Richardson number. Our results also suggest that entrainment co-efficients depend on the ratio of the wind velocity to the ascent velocity of the plume, althoughexperiments with local measurement of the plume properties will be required to confirm thetrends we exhibit (e.g. 3D measurements of the average velocity and density, or at least 2Dmeasurement in the plan defined by the exit velocity and the wind field). Our improved con-straints on entrainment parameterizations in volcanic plume models have implications for keypredictions for explosive eruptions. These include the height and stability of volcanic columnsand the likelihood for the production of pyroclastic flows, as well as the magnitude required foran explosive eruption to reach the stratosphere and have a long-lasting impact on climate.55Chapter 4New insights on entrainment andcondensation in volcanic plumes:Constraints from independentobservations of explosive eruptionsand implications for assessing theirimpacts.This chapter was published in Earth and Planetary Science Letters in May 2018( SummaryThe turbulent entrainment of atmosphere and the condensation of water vapor govern theheights of explosive volcanic plumes. These processes thus determine the delivery and thelifetime of volcanic ash and aerosols into the atmosphere. Predictions of plume heights usingone-dimensional “integral” models of volcanic plumes, however, suffer from very large uncer-tainties, related to parameterizations for entrainment and condensation. In particular, the windentrainment coefficient β, which governs the contribution of crosswinds to turbulent entrain-ment, is subject to uncertainties of one order of magnitude, leading to relative uncertainties ofthe order of 50% on plume height. In this study, we use a database of 94 eruptive phases toconstrain and evaluate four popular 1D models. We employ re-sampling methods to account forobservational uncertainties. We show that plume height predictions are significantly improvedwhen: i) the contribution of water vapor condensation to the plume buoyancy flux is excluded;and ii) the wind entrainment coefficient β is held constant between 0.1 and 0.4. We explore564.2. Introductionimplications of these results for predicting the climate impacts of explosive eruptions and thelikelihood that eruptions will form stable umbrella clouds or devastating pyroclastic flows. Last,we discuss the sensitivity of our results to the definition of plume height in the model in lightof a recent set of laboratory experiments and draw conclusions for improving future databasesof eruption parameters.4.2 IntroductionUncertainty dominated by condensation parameterizationUncertainty dominated by entrainment parameterizationTropopauseFigure 4.1: Uncertainties on plume height related to entrainment and condensation pa-rameterizations in integral models of volcanic plumes. Left and right panels showthe predicted plume height (km) as a function of the mass eruption rate (kg.s−1)for atmospheric conditions similar to the ones during the Tungurahua 2013 and theJune 7th 2011 morning phase of the Cordón Caulle eruption respectively. Black,blue, red and green dots correspond to models 1 to 4 respectively (cf Section4.3). Black dashed lines show the tropopause. We run models with T0 = 1200K,n0 = 4wt.% and Ri0 ' 10−2 (where Ri0 is Ri at vent altitude), and for eachvalue of the model parameters λ, m, α, β and βα specified in Section 4.5. Uncer-tainties for plume height prediction are mostly related to uncertainties on λ forM0 < 105 kg.s−1 and to uncertainties on m, α, β and/or βα for M0 > 107 kg.s−1.How explosive does an eruption have to be to produce a plume that reaches the stratosphereto affect climate? Depending on the volcanic plume model used to address this question,the answer in terms of mass eruption rate can vary by two orders of magnitude (Figure 4.1).Nevertheless, such models of volcanic plumes remain the main tool to understand and forecastthe impacts of explosive volcanic eruptions. For example, predicting the stability of a volcaniccolumn determines whether it will collapse to produce devastating pyroclastic density currents,574.2. Introductionor rise as a buoyant plume and potentially shut down major airline routes (e.g., Degruyter andBonadonna (2013); Jessop et al. (2016b)). For buoyant plumes, predicting the mass and heightof gas and particles injected into the atmosphere enables assessments of climate impacts of aneruption (e.g., Robock (2000)) as well as of risk related to ash fallout (e.g. Carey and Sparks(1986)).One-dimensionnal integral models of volcanic plumes are popular because they are simpleto implement and they successfully capture the main dynamics governing stability and heightof eruptive columns. More sophisticated 3D volcanic plume models capture the complexity ofturbulent multiphase volcanic plumes in more details, but they are so expensive computationallythat mapping an appropriately large regime parameter space is challenging. Although powerfulfor their simplicity, the success of simple 1D models depends on accurate parameterizations ofthree processes that affect the maximum height of a volcanic plume:1. the turbulent entrainment of atmosphere into the plume2. the condensation of source and entrained atmospheric water vapor within the plume3. the sedimentation and re-entrainment of particles settling from the column and the spread-ing cloud.Turbulent entrainment is particularly important because it exerts a basic control on thestability of a volcanic column as well as on the amount of entrained atmospheric water vaporand settling particles, which, in turn, affect condensation and particle re-entrainment pro-cesses. Figure 4.1 shows that uncertainties related to entrainment parameterization affect theprediction of plume height by up to a factor of 2-3, in particular at high latitudes where thecontribution of wind to entrainment is large. Furthermore, models assuming that condensedwater vapor contributes to the plume buoyancy flux predict higher plume heights by up to10 km for mass eruption rate smaller than 106 − 107kg.s−1, in particular in tropical latitudeswhere the atmosphere is relatively wetter (cf Figure 4.1 and, e.g., Woods (1993); Tupper et al.(2009)). Such predictions have important implication for potential climate effects of tropicaleruptions and requires rigorous testing against well-observed eruptions. Last, integral volcanicplume models accounting for the fallout and re-entrainment of particles predict, for a givenmass eruption rate, a decrease of the plume height for larger particles (Girault et al., 2016),although such an effect largely depends on particle size distribution (Macedonio et al., 2016).Despite careful calibrations of integral plume models using analog laboratory experiments(e.g. Morton et al. (1956); Hoult and Weil (1972); Dellino et al. (2014); Aubry et al. (2017b)),large uncertainties remain particularly for the rate of turbulent entrainment with and withouteffects of typical crosswinds. Consequently, it is critical to evaluate these models against naturaleruption data. However, such a comparison is challenging and a thorough evaluation mustinclude:• A dataset of natural eruptions with source conditions (at least mass eruption rate), plume584.2. Introductionheight and atmospheric conditions that are independently constrained and independentof any class of plume model• A large number of eruptions spanning the full range of mass eruption rate and atmosphericconditions characteristic of the majority of events in the historical records• An assessment of how the large observational uncertainties on mass eruption rate andplume height affect the evaluation of the model(s)• A test of the full range of plausible parameter values given constraints from previousexperimental or observational studies, in particular for entrainment parameterization,and ideally compare multiple models.Although studies carried out over the last decade have made progress on the above crite-ria, no existing study fulfills the full set. For example, recent studies use timeseries of observedplume height from the Eyjafjallajökull eruption in 2010 to test integral plume models with spec-ified values of entrainment parameters (e.g. Bursik et al. (2012); Degruyter and Bonadonna(2012); Woodhouse et al. (2013)). However, if a specific parameterization is able to predictthe plume height of one eruption using the constrained mass eruption rate (or vice versa), itdoes not guarantee that the same parameterization can be applied generally to other erup-tions. Other recent studies use several eruptions to evaluate integral plume models, but employsimplified atmospheric conditions (e.g. Woodhouse et al. (2013),Girault et al. (2016)). Mastin(2014) uses 25 eruptive phases with atmospheric conditions retrieved from the NCEP-NCARreanalysis (Kalnay et al., 1996) to test a 1D plume model using a single set of entrainmentparameters. This work does not explore how uncertainties in estimated plume height and masseruption rate impact the confidence on the model accuracy. Woodhouse et al. (2015) use historymatching to rigorously account for uncertainties arising from both model parameterization andnatural observations for two eruptions. However, their study focuses on showing how model andobservational uncertainties impact mass eruption rate estimates obtained from plume heightobservations using inverse modeling. In a recent “intercomparison” exercise, Costa et al. (2016)aim to compare plume height predictions of integral models of volcanic plumes differing in theirparameterizations of entrainment/condensation/particle fallout and re-entrainment. However,the study does not provide an assessment of model performance in reproducing observed plumeheights given constrained mass eruption rates.Building on previous work, our study aims to rigorously evaluate existing parameterizationsfor entrainment and condensation in integral models of volcanic plumes, which we introducein Section 4.3. We fulfill all evaluation criteria listed above. We use a database of 94 eruptivephases (Aubry et al., 2017b) with independent estimates of mean values and uncertainties forthe erupted mass of solid magmatic material (i.e., ash), eruption duration, and mean plumeheight (Section 4.4). Section 4.5 explains the methods used to account for observational andmodel uncertainties, assess the “best” set of parameter values for each tested model, and to594.3. Modelinter-compare these models. We present our main results in Section 4.6 and use them to discussthe following two outstanding questions in Section 4.7: What mass eruption rate is required fora plume to reach the tropopause and have a significant impact on climate? What conditionslead to the collapse or buoyant rise of a volcanic jet? Finally, to make modeling studies in thefuture more robust, we discuss critical improvements to the eruptive parameter database.4.3 Model4.3.1 Overview of the tested modelsTurbulent entrainment:uε= ((α|u-wcos(ϕ)|)m+(β|wsin(ϕ)|)m)1/mModels 1-4  four parameterizations of α and βuϕswCondensation at fixed rate λuεH+RHFigure 4.2: Cartoon of a volcanic plume rising in the atmosphere and parameterizationsof entrainment and condensation used in the one-dimensional integral models ofvolcanic plume 1-4. The curvilinear coordinate s follows the plume centerline.Plume property profiles, such as the plume velocity u, are top-hat in shape (i.e,are zero beyond the plume radius, r, and within the plume have constant valuesfrom the centerline to r). The inflow of atmospheric air into the plume u dependson the radial gradient of axial velocity between the plume and the atmosphere(|u − wa cos(ϕ)|) and to the radial gradient of ortho-axial velocity (|wa sin(ϕ)|)where ϕ, the local plume deflection with respect to the horizontal, defines thedirection of the local axial direction. Water vapor inside the plume is condensed ata rate λ. Figure adapted from Aubry et al. (2016). Orange dashed lines show theplume heights H and H+R which are discussed in Section 4.7.3.The one-dimensional integral plume models that we evaluate are based on the turbulent buoyantplume model of Morton et al. (1956) which has been adapted for volcanic plumes by Houltet al. (1969); Woods (1988); Glaze et al. (1997); Bursik (2001). In particular, it assumes thatatmosphere is entrained into the plume with an entrainment velocity:604.3. Modelu = ((α|u−w cos(ϕ)|)m + (β|w sin(ϕ)|)m) 1m , (4.1)where u is the rise velocity of the plume, w is the horizontal wind speed, and ϕ is the localdeflection of the plume with respect to the horizontal (Figure 4.2). The three entrainmentparameters that are responsible for most differences between integral models of volcanic plumes,are:• the radial entrainment coefficient α, which sets the contribution of the axial velocity dif-ference between the plume and the atmosphere (u−w cos(ϕ)) to the entrainment velocityu• the wind entrainment coefficient β, which sets the contribution of the radial velocitydifference between the plume and the atmosphere (w sin(ϕ)) to u• the exponent m, which governs how the axial and radial velocity differences are combinedto calculate u.In addition, the tested models assume that the radial profiles of plume properties (e.g. velocity)are top-hat in shape (i.e., are zero beyond the plume radius, r, and within the plume haveconstant values from the centerline to r), and that source and atmospheric conditions aresteady. The governing equations of the models are:dds (ρdur2φd) = 2urρadφad (4.2a)dds (ρvur2φv) = 2urρavφav − λρvr2φv (4.2b)dds (ρlur2φl) = λρvr2φv (4.2c)dds (ρsur2φs) = 0 (4.2d)dds (ρu2r2) = g(ρa − ρ)r2 sin(ϕ) +w cos(ϕ)d(ρur2)ds (4.2e)ρu2r2dϕds = g(ρa − ρ)r2 cos(ϕ)−w sin(ϕ)d(ρur2)ds (4.2f)dds (gur2 ρcT − ρcaTaρa0ca0Ta0) = − ρρa0ur2N2 sin(ϕ) + gLρa0ca0Ta0d(ρlφlur2)ds . (4.2g)Equations 4.2a-4.2d give the conservation of the mass fluxes of dry air, water vapor, liquidvapor and solid pyroclasts respectively. Equations 4.2e-4.2f are the conservation of verticaland horizontal momentum fluxes written in terms of the variables s and ϕ, defined in Figure4.2. Equation 4.2g gives the conservation of heat flux. Table S1 defines all symbols used inEquations 4.2 and more details on their derivation are provided in previous studies on integralmodel of volcanic plumes (e.g., Degruyter and Bonadonna (2012)). In this study, we focus on614.3. Modelhighlighting differences in integral plume models that emerge from the parameterization of theentrainment velocity (Equation 4.1) or the condensation of entrained water vapor. We do notevaluate parameterizations for the sedimentation and re-entrainment of particles.4.3.2 Condensation of entrained water vaporWater vapor can condense inside a volcanic plume, release latent heat and increase the plumebuoyancy flux. Two models are commonly used to account for this process. Woods (1993)assumes that as soon as water vapor partial pressure exceeds the saturation pressure, the excesswater vapor is condensed. Glaze et al. (1997) assume that water vapor condenses at a constantrate λ. For rapid condensation rates (λ ' 10−2s−1), plume height predictions are very similarfor both models (e.g. Glaze et al. (1997); Degruyter and Bonadonna (2012); Woodhouse et al.(2013)). When λ 6' 10−4s−1, however, the Glaze et al. (1997) model produces results similarto integral models that do not account for condensation of water vapor. Thus, we decided touse the Glaze et al. (1997) formulation for condensation of water vapor in Equations 4.2 as it issimilar to other models for end-member values of the condensation rate λ, but also allows fora moderate impact of water vapor condensation on plume height for intermediate values of λ(i.e., λ ∼ 10−3s−1).4.3.3 Models for entrainment coefficientsIntegral models of volcanic plumes use Equation 4.1 to parametrize the entrainment velocity.Most models assume m=1 (e.g. Degruyter and Bonadonna (2012); Woodhouse et al. (2013)).Devenish et al. (2010) use a value ofm=1.5 calibrated against large-eddy simulations of buoyantplumes. The major source of uncertainties among integral models of volcanic plumes is theparameterization of the entrainment coefficients α and β (e.g. Costa et al. (2016)). In thisstudy, we test the following four entrainment coefficient models.Model 1: constant α and constant β.In model 1, entrainment coefficients α and β are constant (Morton et al., 1956; Hoult et al.,1969). Small-scale and large-scale laboratory experiments or numerical simulations constrainα between 0.05 and 0.16 and β between 0.1 and 1 (e.g. Morton et al. (1956); Fischer et al.(1979); Chen and Rodi (1980); Kaminski et al. (2005); Contini et al. (2011); Dellino et al.(2014); Suzuki and Koyaguchi (2015)). Smaller values of α are found for pure jets (driven by asource of momentum only, e.g., Fischer et al. (1979)), while higher values correspond to pureplumes (driven by a source of buoyancy only, e.g., Morton et al. (1956)). α = 0.09− 0.1 andβ = 0.5− 0.9 are the ranges of values most commonly used in integral models of volcanic plumes(Costa et al., 2016).Model 2: variable α and constant β.In model 2, β is constant and we parameterize α using Carazzo et al. (2008b). Kaminskiet al. (2005) show that α is a function of the local Richardson number, which expresses the624.4. Dataratio of the the momentum and stabilizing buoyancy fluxes:Ri = rg′u2, (4.3)where r is the jet radius and g′ its reduced gravity. Carazzo et al. (2008b) use:α = 0.0675 + (1− 1A)Ri+ r2d ln(A)dz, (4.4)where the shape function A depends on the source diameter, the vertical distance from thesource z, and the source Richardson number.Model 3: variable α and variable β.In model 3, α is given by Equation 4.4 and β is a function of Ri and W∗ = wU0 , the ratio ofwind speed (w) to plume velocity at vent altitude (U0) (Folch et al., 2016):β = 0.34 ∗ (√2|Ri|W∗)−0.125 . (4.5)Model 4: variable α and constant βα .In model 4, we again use Equation 4.4 for α and assume that the ratio βα is constant, on thebasis of laboratory experiments showing that βα seems constant through a large range of Ri,where α is expected to vary (Aubry et al., 2017b).In models 2-4 above, for α and β, we impose minimum values of 0.05 and 0.1 and maximumvalues of 0.16 and 1 respectively, because these models parameterizations can lead to valuesof α and β largely outside the range in which they were constrained from observations. Tosummarize this section, we will test four popular physical models for entrainment coefficients.The chosen parameterization for condensation of water vapor inside the plume enables us toproduce results similar to other parameterizations used in integral models depending on thevalue of the condensation rate λ. Each model thus has one parameter related to the condensationof entrained water vapor (λ) and between one and three parameters related to the entrainmentEquation 4.1 (m, α and β or βα).4.4 DataThe inputs for the models presented in Section 4.3 are the vent elevation, the vent radius (R0),the velocity at vent (U0), the magmatic gas content at the vent (n0, which we assume to be100% water vapor), the temperature at the vent (T0), and atmospheric conditions (pressure,temperature, relative humidity and horizontal wind speed). One output of the model whichis often constrained for explosive eruptions is the maximum plume height. We calculate thisproperty as the plume altitude when the vertical component of the plume velocity becomessmaller than 0.01m.s−1. Among source conditions, the maximum plume height depends mostlyon the mass eruption rate:M0 = piρ0R20U0 , (4.6)634.4. Datawhere ρ0 is the density of the plume at vent elevation. Thus, in order to evaluate integralplume models, we follow the classic approach to compare modeled and observed plume heightfor explosive eruptions for which the maximum plume height and the mass eruption rate areindependently constrained from observations.Figure 4.3: Location, mass eruption rate, and season of the eruptive phases from theAubry et al. (2017b) dataset used to evaluate volcanic plume models 1-4. The sizeof the symbols is proportional to the mass eruption rate. Colors show the seasonof the start date of the eruptive phases. DJF, MAM, JJA and SON indicate astart date between December and January, March and May, June and August, andSeptember and November respectively. We apply an offset of 0.5◦ latitude andlongitude to eruptions from the same volcano for clarity. Dotted lines show theEquator and the tropics of Cancer and Capricorn.We use the database of Aubry et al. (2017b) (Figure 4.3) which provides mean estimates ofthe erupted mass of solid material, eruption duration, and mean plume height of 94 eruptivephases with buoyant plumes as well as estimates of 95% confidence intervals for these threeparameters. For most eruptions, data availability only enables mean, minimum and maximumestimates of these parameters. In this case, Aubry et al. (2017b) assume that a confidence levelof 95% can be associated to the range defined by the minimum and maximum estimates. Wewill assume n0 = 4± 3wt.% and T0 = 1200± 300K on the basis of values compiled from theliterature (Aubry et al. (2017b)). Vent altitudes are also compiled, and atmospheric conditionsare retrieved from the NCEP-NCAR reanalysis (Kalnay et al., 1996), or the 20th CenturyReanalysis (Compo et al., 2011) for the few eruptive phases occuring before 1948. The masseruption rate is calculated from the erupted mass of solid, the eruption duration and the gascontent, and U0 and R0 are chosen using Equation 4.6 (cf Section 4.5). The database includesboth “weak” (bent-over by wind and spreading downwind only) and “strong” (with an umbrella644.5. Methodcloud spreading both downwind and upwind) plumes.One of the main limitations of testing steady integral plume models using a database of av-erage eruptive parameters lies in the unsteadiness of explosive eruptions. However, it is difficultto estimate time-dependent mass eruption rate independently from the plume height for thevast majority of eruptions for which other parameters are well constrained. In addition, Figure4.3 shows that the dataset contains a large number of eruptions, spanning varied latitudes/sea-sons (and thus, atmospheric conditions) and a large range of mass eruption rates. Althougheach eruptive phase is represented by average parameters, we thus expect that the resultingvariability in maximum plume height should be captured by integral models of volcanic plume.4.5 Method4.5.1 Metric for model evaluationFor a given entrainment model e and set of model parameters p, we calculate the standardizedroot mean squared error E on plume height as:Ee,p =√√√√ 1NN∑i=1(H i −H ie,p)2(∆H i)2, (4.7)where H i is the observed plume height of eruption i, H ie,p is the plume height predicted bymodel e with the set of parameters p for eruption i, ∆H i is the estimated 95% confidenceinterval onH i−H ie,p calculated on the basis of observational uncertainties, and N is the numberof eruptions. The normalization by ∆H i (detailed later in this section) allows us to give lessweight to eruptions for which plume height or source conditions are poorly constrained. Onaverage, for a model e with set of parameters p, the model does not reproduce observed plumeheight within the 95% confidence interval if Ee,p > 1.In this study, we calculate H ie,p as the maximum height of the plume centerline. We discussthe sensitivity of our results to the definition of maximum plume height in the model in Section4. Uncertainties and Monte Carlo simulationsThe dataset compiled by Aubry et al. (2017b) has the advantage that it provides estimates ofuncertainties on each of the parameters. In this study, we use Monte Carlo simulations (re-sampling method) to account for these uncertainties. For each variable (plume height, eruptedsolid mass, eruption duration, gas content and temperature at the vent), we assume a gaussiandistribution centered on the estimated mean with standard deviation chosen to match theestimated 95% confidence interval. Once these variables are sampled, the value of R20U0 mustmatch M0piρ0 (Equation 4.6). We randomly sample U0 within values that fulfill three conditions:• 5 m≤ R0 ≤150 m654.5. Method• 10 m.s−1 ≤ U0 ≤ 500 m.s−1• The plume does not collapse for any of the models and parameters sets tested.For 3 eruptions out of 94 (two events with the smallest mass eruption rate and one with thelargest), we extended the above ranges to 0.5 m ≤ R0 ≤ 700 m and 0.5 m.s−1 ≤ U0 ≤ 700m.s−1 to match the extreme mass eruption rates. The third condition insures that we do notarbitrarily disadvantage models/parameters sets with lower entrainment rates and thus morelikely to produce collapse, because Ee,p would likely be much larger for models for which collapseoccurs for some eruptions. We use the Degruyter and Bonadonna (2013) regime diagram forcollapse/buoyant plume regime to calculate ranges of U0 that fulfill the third condition.We perform 230 Monte Carlo simulations. In each simulation, we: i) sample source con-ditions and plume height for all eruptions; and ii) calculate the error Ee,p for each model andeach combination of parameter values among:• λ (all models): 10−5, 10−4, 10−3 and 10−2 s−1• m (all models): from 1 to 2 with increments of 0.25• α (model 1): from 0.05 to 0.16 with increment of '0.016• β (models 1 and 2): from 0.1 to 1 with increment of '0.07• βα (model 4): 1 to 13 with increment of 1.Next, for each Monte Carlo simulation, we find the “best” sets of parameters and model(s) thatminimizes the error E. We thus obtain a probability distribution constructed with 230 samplesfor best model parameters and minimum error. Half the number of Monte Carlo simulationsused is sufficient to obtain the convergence of the probability distributions of parameters ofmodel 1, which has the most parameters to constrain (Figure C.1).We estimate the weights ∆H i in Equation 4.7 as ∆H i = 1.97σi with:(σi)2 =1Ns,e,p∑s∑e∑p((H is −H is,e,p)− µ(H is −H is,e,p))2 , (4.8)where H is is the observed height of eruption i for the Monte Carlo simulation s, H is,e,p the heightpredicted for eruption i by model e with model parameter set p and source parameters fromthe same Monte Carlo simulation as the observed height, and µ(H is −H is,e,p) is the average ofH is−H is,e,p for a given model e and set of parameters p. As defined, σi is similar to the standarderror on the difference between the observed height and predicted height, but is independent ofthe model/set of parameters and only accounts for observational uncertainties.664.6. Results4.6 Results4.6.1 Best set of parameters for each model6 (s-1)10-5 10-4 10-3 10-2Condensation rate 60   0.250.5 0.751   Model 1: constant ,  and - (s-1)10-5 10-4 10-3 10-20   0.250.5 0.751   Model 2: ,(Ri) and constant  - (s-1)10-5 10-4 10-3 10-20   0.250.5 0.751   Model 3: ,(Ri) and - (Ri,W*) (s-1)10-5 10-4 10-3 10-20   0.250.5 0.751   Model 4: ,(Ri) and constant - /,   1.25 1.5 1.75 2   Exponent m0   0.250.5 0.751   1.25 1.5 1.75 2   0   0.250.5 0.751   1.25 1.5 1.75 2   0   0.250.5 0.751   1.25 1.5 1.75 2   0   0.250.5 0.751,0.05 0.1 0.15Radial entrainmentcoefficient ,0   0.250.5 0.751 0.3 0.5 0.7 0.9Wind entrainmentcoefficient - or -/,0   0.250.5 0.751 0.3 0.5 0.7 0.90   0.250.5 0.751,1 3 5 7 9 11 130   0.250.5 0.751 value minimizing the errorMinimum errorFigure 4.4: Values of integral plume model parameters minimizing the error on observedplume height. Each column of plots shows results related to one entrainment model,while each row of plots show results related to one type of model parameter. Oneach plot, red bars are associated to the left red axis and show the probability dis-tribution (across all Monte Carlo simulations) for the parameter value minimizingthe error on plume height. Black dots, associated to the right black axis, show theminimum error on plume height for a fixed value of the model parameter consid-ered. Dots show the mean error and error bars are the 95% confidence interval.Across all models, sets of parameters, and Monte Carlo simulations, the error E on the 94eruptive phases plume height (Equation 4.7) varies between 1.2 and 3.1. The minimum error674.6. Resultsis larger than 1, meaning that the discrepancies between observed and modeled plume heightscannot be explained only by the reported observational uncertainties. Figure 4.4 shows, foreach model (columns) and each parameter (row), (i) the frequency at which a value of theparameter minimizes the error E across all Monte Carlo simulations (red histograms) and (ii)the minimum error E obtained for each tested value of the parameter (black dots).The first row of plots in Figure 4.4 show results for the condensation rate λ. Values of λ 610−4 s−1 result in better plume height predictions in 90− 95% of the Monte Carlo simulationsfor models 1, 2 and 4, and ' 56% for entrainment model 3 (red histograms). The effect ofcondensation of water vapor on the plume buoyancy flux is negligible for such values of λ(Glaze et al., 1997; Degruyter and Bonadonna, 2012). For all models except model 3, the errorE (black dots) decreases by ' 0.1-0.2 (i.e., 10 to 20% of the observational uncertainty) forλ 6 10−4 s−1 relative to λ > 10−3 s−1, for which condensation impacts the height of smallplumes. Entrainment model 3 is less sensitive to λ with E varying only by ' 0.05 for the testedvalues of λ.The second row of plots in Figure 4.4 shows results for the exponent m in the entrainmentEquation 4.1. For entrainment models 1, 2 and 4, the error is relatively insensitive to the valueof m, with the 5 values tested being roughly equally probable. For entrainment model 3 wherem is the only adjustable parameter in the entrainment equation 4.1, values of m > 1.5 result ina smaller error on plume height in more than 85% of the simulations, with an error 0.05− 0.1smaller relative to values of m 6 1.25.The third row of plots in Figure 4.4 shows results for the wind entrainment coefficient β formodels 1 and 2, or the wind to radial entrainment coefficients ratio βα for model 4. The error isvery sensitive to these parameters, with differences of up to 0.25 for the tested values of β andβα . 0.1 6 β 6 0.4 and 1 6βα 6 4 minimize the error on plume height in more than 95% of theMonte Carlo simulations for models 1, 2 and 4. These values are in the lower range of previousconstraints on these parameters derived from laboratory experiments and field observations (cfreferences listes in Section 2.3). In addition, these values of β orβα are weakly consistent withresults obtained with a similar methodology, but using analogue buoyant jets from laboratoryexperiments instead of natural eruptions, and plume trajectories instead of plume height (Aubryet al., 2017a).Last, the fourth row of the plot in Figure 4.4 shows results for the radial entrainmentcoefficient α for model 1. Values of α between 0.08 and 0.16 minimize the error on plumeheight in more than 95% of the Monte Carlo simulations, and the sensitivity of the error on thevalue of α is relatively small with differences of 0.05-0.15 depending on the chosen value of α.684.6. Results4.6.2 Best modelsE1-E2-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model1 (63%)Model1 betterthan model2 (37%)E1-E3-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model1 (1%)Model1 betterthan model3 (99%)E1-E4-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model1 (37%)Model1 betterthan model4 (63%)E2-E3-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model2 (1%)Model2 betterthan model3 (99%)E2-E4-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model2 (23%)Model2 betterthan model4 (77%)E3-E4-0.3 -0.2 -0.1 0 0.1 0.2 0.300. betterthan model3 (94%)Model3 betterthan model4 (6%)Figure 4.5: Distribution of pairwise differences of model error E , for each pair of model.Y-axis show the frequency and x-axis show the error difference. For a given modeli, we sampled 1000 set of parameters from the distributions shown on Figure 4.4.We thus obtain 230 000 values of Ei, the error for model i, across all Monte Carlosimulations. The resulting distribution of pairwise differences are shown on thisfigure, and account for both observational and model parameter uncertainty. Fortwo models i and j, Ei−Ej is positive if model j predicts plume height better thanmodel i, and vice versa. The probability for one model to better than the other isannotated on each plot.Figure 4.5 shows the distribution of pairwise differences between model error, for each possiblepair of models. These distributions account for both observational errors, and errors on en-trainment parameters constrained from Figure 4.4 (cf legend of Figure 4.5). Model 3 has errorslarger than any other model at the 95% confidence level (94% for model 4). The differencesbetween other models are not significant at the 95% level. Model 2 outperforms models 1 and4, but with confidence levels of only 63 and 77 %, respectively. Model 1 outperforms model 4with a confidence level of 63%. The error E depends moderately on the chosen model, withmean differences of up to 0.1, and maximum differences of up to 0.3. Our conclusions on model“ranking” are in excellent agreement with those found with laboratory experiments for whichmodels 1 and 2 better predict plume trajectories, and model 3 performs worst to a high (> 90%)694.7. Discussionsignificance (Aubry et al., 2017a).4.7 Discussion4.7.1 Best parameterization of condensation and implications forpredicting climate impact of explosive eruptionsLGM  185 ppm  22.2 CHolocene  280 ppm  24.9 CMWP  280 ppm  25.3 CLIA  280 ppm  25.0 CPresent  340 ppm  26.0 CRCP8.5  1950 ppm  36.5 CCritical MER to reach the tropopause (kg.s-1 )1061076=10-5s-1 (slow condensation)6=10-2s-1 (rapid condensation)Figure 4.6: Critical mass eruption rate (MER) required to reach the tropopause underdifferent climate conditions, as predicted by model 2 with slow (blue) and fast (red)condensation. We assumed β = 0.4, m = 1.5, a vent altitude of 2000 m, and sourcetemperature and gas content of 1200K and 5wt.% respectively. Atmospheric profilesare retrieved from 50 years of monthly output of global climate model (GCM)simulations, at 0.93◦S 78.75◦W, close to the Cotopaxi volcano (Ecuador). Thecritical MER is calculated for each month, and dots and error bars show the meanand 95% confidence interval, respectively. Last glacial maximum (LGM), Holocene,medieval warm period (MWP, 1050-1100), little ice age (LIA, 1650-1700), present(1955-2005) and RCP8.5 (2250-2300) conditions are all obtained from simulationof the Max Planck Institute MPI-ESM model, from experiments of the CoupledModel Intercomparison Project Phase 5 (Giorgetta et al., 2013). The approximateglobal atmospheric CO2 concentration (ppm) and the 50-year average temperature(◦C) at 1000 hPa at the chosen location is labeled for each period.Figure 4.4 shows that models 1, 2 and 4 predict plume height better when the condensation rateis small or negligible (λ 6 10−4 s−1). Although this result is less clear for model 3, this modelalso has significantly larger errors than the other 3 models (Figure 4.5). Such small values of λhave implications for predicting the critical mass eruption rate required for a tropical eruption704.7. Discussionto cross the tropopause and impact climate through long-lived (1-2 years) sulfate aerosol forcing(Robock, 2000). Figure 4.6 shows, for example, that over a large range of climate conditions forλ = 10−5 s−1, the critical mass eruption rate to reach the tropopause is 50 to 250% larger thanit is for λ = 10−2 s−1. This effect on the critical mass eruption rate is magnified with globalwarming for 23rd century climate projections with the high greenhouse gas emission scenarioRCP8.5. As a consequence, the projected increase of the critical mass eruption rate as Earthwarms, which is caused by changes in the temperature lapse rate in the tropical troposphere,is more significant for λ = 10−5 s−1 than for λ = 10−2 s−1. This conclusion strengthens thefeedback proposed in Aubry et al. (2016), with ongoing global warming leading to less frequentstratospheric injections and, in turn, less surface cooling resulting from volcanic aerosol forcing.The condition that λ is very small is surprising with respect to predictions of 3D plume models(e.g. Tupper et al. (2009)) and results in supersaturation in the 1D model. This result alsoleads to the question whether condensation of entrained water vapor is important for volcanicplume height, or if additional latent heat flux could be compensated by a process ignored inthe 1D model.4.7.2 Best entrainment model: Implication for a plume collapse-buoyantplume transitionParameterization of turbulent entrainment has a significant impact on predictions for bothplume height and the conditions governing the gravitational stability of volcanic plumes. Inparticular, using model 1 with m = 1, Degruyter and Bonadonna (2013) show that the collapse-buoyant plume transition can be understood in terms of the two regime parameters Ri0 andW ∗0 , and the two entrainment coefficients α and β. Additional factors such as particle sizedistribution or vent shape are expected to impact collapse conditions (e.g. Jessop et al. (2016b))but will not be discussed here. Building on the work of Degruyter and Bonadonna (2013), Figure4.7 shows the mean transition predicted by the four entrainment models tested in this studyas a function of Ri0 and W ∗0 , and with entrainment and condensation parameters sampledfrom distributions of Figure 4.4. Qualitatively, all models show a similar behaviour alreadyhighlighted in Degruyter and Bonadonna (2013): In the absence of wind, plumes collapse wherethe Richardson number is sufficiently high, i.e., where the jet momentum flux is too smallrelative to the initial negative buoyancy flux. However, as W ∗0 increases, the critical Ri0 abovewhich collapse occurs increases because of the additional momentum flux carried by the wind.714.7. DiscussionRange of Ri0 for total collapse in Carazzo et al (2008)Buoyant PlumesPlume CollapseFigure 4.7: Ratio of the mass of tephra originating from PDC or co-PDC plumes and thetotal erupted mass of tephra (shading) as a function of the Richardson number (Ri0)and the wind to plume velocity ratio (W ∗0 ). Dark and light disks show eruptionsdominated by PDC and buoyant plume respectively. The thick black arrow showsthe range of Ri0 in which Carazzo et al. (2008a) found total collapse, but for whichwe could not estimate W ∗0 due to lack of constraint on wind. Continuous linesshow mean buoyant plume-collapse transitions calculated using the four modelsdiscussed in this study, with entrainment and condensation parameters constrainedfrom Figure 4.4. The dashed line shows the prediction of model 1 with α = 0.15,β = 1 and m = 1. Uncertainties are not shown to keep the figure readable but aretypically on the order of a quarter to half an order of magnitude for observationand model transitions. Table C.2 compiles the data and references used in thisfigure.Furthermore, Figure 4.7 shows novel quantitative differences among the models in terms ofthe critical Ri0 at which the collapse occurs, and how this condition is modified asW ∗0 increasesfrom 0. First, using the Carazzo et al. (2008b) parameterization for α, models 2, 3 and 4 predictcollapse at a Richardson number that is 50% lower than model 1 in the absence of wind. Second,the value ofW ∗0 above which wind stresses will strongly affect the critical Ri0 varies significantlyamong the models. For example, for model 1, the critical Ri0 for W ∗0 = 10−0.1 is twice thecritical Ri0 for W ∗0 = 0. The same doubling of the critical Ri0 in the absence of wind onlyrequires W ∗0 = 10−1.1 for model 3. Consequently, relatively small wind stresses are requiredto affect the predicted transition Ri0 in model 3 relative to model 1. Third, the dependenceof the critical Ri0 on W ∗0 varies among models. For large W ∗0 , the value of Ri0 at which thetransition between collapsing jets and buoyant plumes occurs increases linearly for models 1,724.7. Discussion2 and 4 (consistent with Degruyter and Bonadonna (2013)). However, a power-law with anexponent of 0.75 fits better the Ri0-W ∗0 transition for model 3. Thus, the critical Ri0 increasesless with W ∗0 for model 3. This behavior is expected as for this model, the wind entrainmentcoefficient β decreases with W ∗ so that the additional momentum flux at higher W ∗0 is partiallycompensated by a smaller entrainment rate.In Figure 4.7, we also show how predicted collapsing jet/buoyant plume transitions comparewith the mass fraction of pyroclastic density current (PDC) deposits of natural eruptions forwhich we can infer Ri0 (Table C.2 and Carazzo et al. (2008a)). The PDC mass fraction isindicative of the gravitational stability of a volcanic plume. We also estimate W ∗0 for eruptionsfor which meteorological observations or climate reanalysis are available.In general, relatively lower values of Ri0 and higher values of W ∗0 correspond with a lowerproportion of PDC deposits which is consistent with expectations from all models. Model andobservational uncertainties do not permit a conclusion about the model that best captures theplume collapse-buoyant plume transition. However, models in which α is parameterized usingCarazzo et al. (2008a) are generally in better agreement with the data. In addition, Figure4.7 shows clearly that a choice of high entrainment coefficients (α = 0.15, β = 1) results ina large overestimation of the critical Richardson number at which the collapse-buoyant plumetransition occur. Such values are used, for example, in the Puffin model (Pouget et al., 2016)in the plume model intercomparison exercise (Costa et al., 2016).4.7.3 What is plume height? Toward improving future databases oferuptive parameters.In this study, we demonstrate that despite large uncertainties, observations now available formany eruptions enable constraints on optimal parameters (α, β, m and λ) of integral 1D modelsof volcanic plume. An additional source of uncertainty lies simply in the definition of plumeheight in a 1D model. In Section 4.6, we use the maximum height of the plume centerline(H herafter, cf Figure 4.2). Other authors (e.g. Mastin (2014)) use the sum of the maximumcenterline height and plume radius at this height, or the maximum centerline height calculatedin the absence of wind when it is smaller than the sum of plume height and radius calculatedwith observed wind conditions (H +R hereafter, cf Figure 4.2 and Figure 1 in Mastin (2014)).734.7. Discussion6 (s-1)10-5 10-4 10-3 10-20   0.250.5 0.751   Condensation rate-0.1 0.3 0.5 0.7 0.9Probability0   0.250.5 0.751   Wind entrainment coefficientEruptions andRMSE based on HEruptions andRMSE based on H+RLaboratoryexperimentsFigure 4.8: Same as Figure 4.4, but showing results obtained when calculating the errorE using the maximum height of the plume centerline (H, red), the sum of themaximum height of plume centerline and plume radius, or the maximum plumeheight in the absence of wind if it is smaller than the former (H+R, blue), andresults obtained from laboratory experiments (Aubry et al., 2017a), with similarmethod and an error metric based on plume trajectory instead of plume height(green). For simplicity, we only show results for model 2, and the parameters β(left) and λ (right). The latter is not constrained from lab experiments.Figure 4.8 (left) shows the probability distribution for the wind entrainment coefficient βwhen the model plume height is H (red) and H +R (blue) for model 2. For the same set ofmodel parameters, H +R is larger than H. Consequently, to match the observed plume height,the distribution of β is shifted toward relatively larger values (corresponding to smaller plumeheight) when using H +R. We have greater confidence in the results using the maximum heightof the centerline H because:• In 75% of the runs using H +R, β = 0.9− 1 minimizes the error on plume height. Theactual optimal values may thus be larger than 1 as we only test values up to 1. Such valuesare at the higher end of published values of β from a large number of studies spanning a0.1-1 range (cf Section 2.3).• Optimal values of β from analysis of laboratory experiments spanning a large range Ri0-W ∗0 values (Aubry et al., 2017a) restrict β to be 0.4-0.5 for this model (green histogram onFigure 4.8). This experimental study used a methodology similar to the one we use, butused an error metric built on the root mean square error on plume trajectory instead ofplume height, so that the definition of plume height does not impact experimental results.• In 90% of the Monte Carlo simulations, the minimum error E obtained across the entiremodel parameter space (m, β and λ) is smaller when using H instead H + R (FigureC.2).744.7. DiscussionFor the condensation rate λ (Figure 4.8, right), we find higher values when using H + Rinstead of H. This result is not straightforward to interpret because plume height increaseswith λ, so that to match the same observed height, relatively smaller values of λ are expectedwith H +R relative to H. The Monte Carlo simulations in which we find optimal values of thecondensation rate λ larger than 10−3s−1 are also found to correspond to the simulations whereβ > 0.9. For strongly bent-over plumes, mostly occurring at high latitude under windy but dryconditions, a high value of the wind entrainment coefficient is then the only way to obtain valuesof H +R close to the observed plume height. For eruptions with smaller difference between Hand H +R and impacted by wind and humidity, the high wind entrainment coefficient decreasesplume height too much which can be compensated by a larger condensation rate. This explainswhy Monte Carlo simulations for which β > 0.9 is the optimal value of the wind entrainmentcoefficient are also the ones where λ > 10−3s−1. In Figure C.3, we use only tropical troposphericeruptions, which are expected to be the ones most impacted by condensation of water vapor,to calculate E in Equation 4.7. We then recover values of λ 6 10−4s−1 in more than 95% ofthe Monte Carlo simulation when using H +R, showing that the higher values of λ found whenusing all eruptions is an artifact caused by the high value of β imposed by eruptions stronglyimpacted by wind.Overall, Figures 4.8 and C.3 show that constraints on entrainment parameters are sensitiveto the definition of plume height in 1D models, while constraints on the condensation rate λare robust to this definition. Our results suggest that the maximum centerline height (H) isthe best definition for the dataset of Aubry et al. (2017b). The distribution obtained with labexperiments, using an error metric on plume trajectories, most likely falls in-between the onesobtained with natural eruptions using an error metric on plume height with H and H + Rbecause one definition is more appropriate for some eruptions and the other one for othereruptions.The question of how to define plume height is closely tied to the plume shape regime. H orH +R may be appropriate for weak plumes while H is the single possibility for strong plumes.In addition, 1D model are expected to better capture the dynamics of weak plumes becausethe self-similarity and entrainment hypotheses on which they are built are violated in/abovethe umbrella cloud of a strong plume. A natural question is thus whether model performancesare different for strong and weak plumes, and whether optimal parameter values also differ forthese two asymptotic regimes. Answering this question with data from natural eruptions ischallenging. Aubry et al. (2017b) attempt to attribute a plume shape regime to each eruptivephase of the database we use but could do so for only ' 30 phases out of 94. In addition, evenfor phases where they indicate a regime, attributing a single regime to eruptive phases thatcommonly transition between weak and strong plumes is challenging (e.g. Bonadonna et al.(2015b)) unless the precise time history of plume style evolution is known (e.g. Eyjafjallajökull2010 or Calbuco 2015). Our categorization of plume shape regime is thus only qualitative.Regime parameters controlling the shape of the plume (e.g. Degruyter and Bonadonna (2012);754.8. ConclusionsCarazzo et al. (2014); Bonadonna et al. (2015b); Aubry et al. (2017b)) can be used in theabsence of direct observations, but their inference depends on the plume height or the masseruption rate, and potentially on entrainment rates.While the database used in this study thus does not enable a comparative analysis of ourresults for weak vs. strong plumes, such analyses can be conducted for laboratory experiments.For example, Aubry et al. (2017a) show that optimal entrainment coefficient values exhibit anapparent dependence on the ratio of wind to plume source velocity (W ∗0 ) for weak plumes, butnot for strong plumes.Building on the limitations of our study, we can draw two key directions for improvingdatabases of eruptive parameters. First, systematically include qualitative information accom-panying quantitative information on plume height to choose the plume height definition (Hor H +R) best adapted to each eruption. Second, improve the characterization of the plumestyle (weak vs. strong) of each eruptive phase to explore how model performance and optimalparameterization varies between these two regimes. We suggest other improvements, which wedid not discuss on the basis of our results:• Compile atmospheric data from multiple source, as plume height predictions are as sen-sitive to wind strength as entrainment coefficient values (Pouget et al., 2016)• Use recent improvements in the estimation erupted volume or mass and their uncertaintiesfrom a deposit (e.g. Bonadonna et al. (2015a)) to revisit estimates of mass eruption ratein the database.• Compile data (e.g. grain size distribution) necessary to the evaluation of parameteriza-tions of particle settling and re-entrainment, which we could not perform in this study.Until these limitations are addressed, in particular the first two, laboratory experiments willremain an important tool to evaluate models of volcanic plume. Nonetheless, our study demon-strates that the inclusion of a large number of eruptive phases and observational uncertaintiesenable significant progresses in putting tight constraints on optimal plume model parameters,and in discriminating which model make the best predictions for plume height.4.8 ConclusionsWe have provided detailed probability distributions of entrainment and condensation parame-ters for four different models of volcanic plumes on the basis of observations from 94 eruptivephases, and compared the performance of these models. Our four main conclusions are:1. Models predict plume height more accurately when the condensation rate is small, i.e.,when the contribution of condensation of water vapor in the plume to the plume buoyancyflux is negligible.2. Models predict plume height more accurately for values of the wind entrainment coefficientβ of 0.1-0.5.764.8. Conclusions3. Models in which β is constant (models 1 and 2) or increases with the local Richardsonnumber Ri (model 4) better predict plume height than the model where β decreases withRi (model 3).4. None of the tested models provide satisfactory predictions for plume height given obser-vational constraints on mass eruption rate and atmospheric conditions yet.Among these four conclusions, the second one is sensitive to the definition of “plume height”in the model. Accordingly, one major step in improving databases of eruptive parameters is tosystematically include qualitative information on plume height measurements. Improving andaugmenting such databases accordingly will also enable to investigate the extent to which unsat-isfactory model predictions (4th conclusion) are due to limitations of the 1D model themselvesor biases in the observational data. Finally, we have discussed the application of our resultsfor understanding the impacts of explosive volcanism. First, the critical mass eruption ratefor which a plume reaches the stratosphere and has long-term impacts on climate is 2-3 timeslarger for the constrained values of condensation rate. One outcome is also a stronger increaseof this critical mass eruption rate as Earth is warming, and thus less frequent stratosphericinjections by volcanic eruptions and volcanically induced surface cooling. Second, we showedthat the buoyant plume-collapse transition is sensitive to the parameterization of entrainmentcoefficients α and β, and that models 2-4 with parameters as constrained in this study areconsistent with observations of collapsed and buoyant plumes.77Chapter 5Impact of global warming on the riseof volcanic plumes and implicationsfor future volcanic aerosol forcingThis chapter was published in Journal of Geophysical Research: Atmospheres in November 2016( SummaryVolcanic eruptions have a significant impact on climate when they inject sulfur gases into thestratosphere. The dynamics of eruption plumes is also affected by climate itself, as atmosphericstratification impacts plumes height. We use an integral plume model to assess changes involcanic plume maximum rise heights as a consequence of global warming, with atmosphericconditions from an ensemble of global climate models (GCM), using three representative con-centration pathways (RCP) scenarios. Predicted changes in atmospheric temperature profilesdecrease the heights of tropospheric and lowermost stratospheric volcanic plumes and increasethe tropopause height, for the RCP4.5 and RCP8.5 scenarios in the coming three centuries.Consequently, the critical mass eruption rate required to cross the tropopause increases by upto a factor 3 for tropical regions, and up to 2 for high-latitude regions. A number of recentlower stratospheric plumes, mostly in the tropics (e.g., Merapi, 2010), would be expected to notcross the tropopause starting from the late 21st century, under RCP4.5 and RCP8.5 scenario.This effect could result in a ' 5− 25% decrease in the average SO2 flux into the stratospherecarried by small plumes, which frequency is larger than the rate of decay of volcanic strato-spheric aerosol, and a ' 2− 12% decrease of the total flux. Our results suggest the existenceof a positive feedback between climate and volcanic aerosol forcing. Such feedback may haveminor implications for global warming rate but can prove to be important to understand thelong-term evolution of volcanic atmospheric inputs.785.2. Introduction5.2 IntroductionExplosive volcanic eruptions eject gases and ash into the atmosphere, which act to modifyEarth’s global radiative energy balance. At annual to centennial timescales, the injection ofsulfur gases, resulting in the formation of sulfur aerosols, has the largest impact on Earth’sradiative balance via scattering of Sun radiation and absorption of Sun and Earth radiation(aerosol-radiation interactions) (Robock, 2000; Timmreck, 2012). Tropospheric volcanic aerosolsare washed out within a few weeks. It is therefore commonly assumed that tropospheric aerosol-radiation interactions from individual eruptions are negligible at a global scale, although aerosolparticles enhance cloud condensation nuclei and, thus, have an indirect impact via aerosol-cloudinteractions on Earth’s radiative balance (Schmidt et al., 2012). Stratospheric volcanic aerosols,by comparison, have a typical e-folding time of one year and exert a significant influence onclimate over these timescales. These relatively long-lived particles scatter shortwave radiationand absorb longwave radiation, resulting in a net cooling of the troposphere and a net warmingof the stratosphere (Robock, 2000; Timmreck, 2012). In addition to these global effects on airtemperature, stratospheric volcanic aerosol-radiation interactions can cause significant changesin atmospheric and oceanic circulation, sea ice dynamics (e.g., Robock (2000); Shindell et al.(2004); Mignot et al. (2011); McGregor and Timmermann (2010); Driscoll et al. (2012); Stoffelet al. (2015); Toohey et al. (2016a)), and precipitation patterns (e.g., Iles and Hegerl (2015)).Whether an eruptive plume reaches the stratosphere also controls ozone depletion by halogenspecies injected by a volcano, although this forcing is small relative to aerosol-radiation inter-actions and largely depends on halogen scavenging in the plume (Tabazadeh and Turco, 1993;Textor et al., 2003; Timmreck, 2012; Carn et al., 2016).In the context of present day global warming, which is mostly driven by anthropogenic green-house gas emissions, volcanic aerosols are of particular importance because their atmospherictemperature fingerprint is opposed to the one of CO2, i.e., a net warming of the troposphereand a net cooling of the stratosphere (Hartmann et al., 2013). In particular, climate modelsneglecting aerosol-radiation interactions of stratospheric volcanic eruptions since 1998 are over-estimating global warming, even though no major volcanic eruption occurred during this period(Solomon et al., 2011; Haywood et al., 2014; Ridley et al., 2014; Santer et al., 2014).Critically, most projections from global climate models (GCMs) impose a constant volcanicradiative forcing (Collins et al., 2013a). Only some decadal projections experiments assumethat a Pinatubo-like eruption will occur at one given year to test sensitivity of short-termprojections to volcanic eruptions (Taylor et al., 2012). Thus, GCMs are unable to predicttemperature changes resulting from future eruptions, although their ability to simulate theclimate response to past volcanic eruptions is continuously improved (Timmreck, 2012; Flatoet al., 2013). Prediction of changes in future volcanic aerosol-radiation interaction would allowimproved prediction of future climate.There are two key controls on volcanic aerosol-radiation interactions resulting from a par-ticular eruption:795.2. Introduction1. How much sulfur gas is expelled.2. Whether this sulfur gas reaches the stratosphere.Both controls partly depend on eruption source conditions, and, in particular, on the masseruption rate of the eruptive plume. The exact timing, global location, and source conditionsof future eruptions are impossible to predict, which is a reason why most climate projectionsassume a constant volcanic radiative forcing. In addition, the height of a given volcanic plumeH depends strongly on atmospheric stratification (Morton et al., 1956; Wilson et al., 1978;Woods, 2010):H ∝ N−κ1Mκ20 , (5.1)where N is the Brunt-Väisälä frequency, M0 is the mass eruption rate, κ1= 34 and κ2=14 inthe absence of wind (Morton et al., 1956) and κ1= 23 and κ2=13 under strong wind conditions(Hewett et al., 1971). The Brunt-Väisälä frequency mostly depends on the temperature lapserate :N2 =gT(gcp− Γ), (5.2)where g is the Earth’s gravitational acceleration, T is the atmospheric temperature, cp is theair specific heat capacity, Γ=− dTdz is the lapse rate and z is the altitude.A major effect of present day global warming is the decrease of the temperature lapse rateΓ in the tropical troposphere (e.g., Simmons et al. (2014); Sherwood and Nishant (2015)),and hence an increase in the strength of the stratification which could result in a decreaseof tropospheric plume height, in the tropics (Equation 5.1). The key question we ask in thispaper is, thus: how will global warming impact the heights of plumes of future eruptions? Inparticular, will more or fewer eruptive plumes reach the stratosphere than at present, and howwill it impact future volcanic aerosol-radiation interactions? Some of these questions are raisedby Glaze et al. (2015) in the context of past climate change, but have never been investigatedinto detail in the context of the present day climate change. Understanding the climate change-driven controls on variations in volcanic plume height has fundamental implications also on thedistribution of hazards associated with the dispersal and sedimentation of both lapilli-sized andash-sized particles, e.g., from proximal damage to buildings and infrastructures to far-field riskto aviation and human health (Rymer , 2015).Our paper is structured in the following way. Our methodology is described in detail insection 5.3: we use an integral volcanic plume model to predict changes in volcanic plume heightdriven by changes in atmospheric temperature, geopotential height and wind fields inferred fromGCM projections. In section 5.4, we show the impact of predicted changes of these fields onthe plume height, as well as the impact of their combined effects. In section 5.5, we testthe sensitivity of our results regarding the plume model parameterization and choice of GCM.805.3. Data and plume modelLastly, we estimate changes in the flux of volcanic SO2 into the stratosphere driven by changesin plume height, and discuss the implications of our results for future volcanic forcing.5.3 Data and plume model  ChileNew ZealandEcuadorIndonesiaPhilippinesCentral America Rift ValleyJapanCascade ArcKamchatkaAleutiansIceland1800−present eruptions of:VEI7 VEI6 VEI5 VEI4 VEI3Figure 5.1: Global map with the 12 volcanically active regions selected for this study(black rectangles). Orange dots show large explosive eruptions (VEI of 3 to 7) forthe last 2 centuries (from Global Volcanism Program database).We apply an integral volcanic plume model to compute the height of explosive volcanic plumes.In each model run, we specify eruption source conditions and atmospheric conditions. Weuse atmospheric conditions associated with 12 active volcanic regions (Figure 5.1) over fourdifferent time intervals. The sample of 12 regions is chosen based on its large scatter bothlatitudinally and longitudinally, which facilitates the sensitivity test of our results to regionalclimate variability. The projections for atmospheric conditions are based on three differentgreenhouse-gas emission scenarios from an ensemble of three GCMs. Our overall methodologyis summarized by the flow chart presented in Figure 5.2.a and the following sections providemore details on the data and integral volcanic plume model that are used.815.3. Data and plume modelAtmospheric conditionsFrom an ensemble of three general circulation models: temperature, geopotential height, wind, and relative humidity profilesPlume altitude probability distributionEruption Source Conditionse.g., mass eruption rate, gas contentTime Period20 years starting 1981, 2081, 2181 or 2281RegionAmong 12 volcanic active areasForcing ScenarioGreenhouse gas representative concentration pathwayIntegral Volcanic Plume Model (a) (b)Figure 5.2: (a) Flow chart summarizing the methodology used. To compute the plumealtitude probability distribution, we use an integral volcanic plume model. Erup-tion source conditions are sampled from a fixed parameter space. Atmosphericconditions depends on the chosen region, period, and greenhouse gas forcing (Rep-resentative Concentration Pathway). (b) Example of plume altitude probabilitydistribution obtained for M0=3.7 106 kg s−1 in the Philippines, for the 1981-2000period. The spread of the distribution is due to variability in temperature, geopo-tential height and horizontal wind within the 20 year period.5.3.1 Source conditionsTable 5.1: Values of parameters used in the integral volcanic plume model (greek symbols)and of eruption source conditions (symbols with 0-subscript).Parameter Symbol Unit Value RangeRadial entrainment coefficient α - 0.1 0.07− 0.13Wind entrainment coefficient β - 0.7 0.35− 1Condensation rate λ s−1 0 0− 0.098Temperature T0 K 1200 1000− 1400Gas mass fraction n0 - 0.04 0.01− 0.07Velocity U0 m s−1 75− 300 75− 300Vent radius R0 m 10− 150 10− 150Vent height H0 m 1500 local topographyaaVent height is sampled from a distribution representative of the altitude of volcanoes in theregion considered (cf. Section D.4) or from the Carn et al. (2016) datasetSource conditions that must be specified for each run of the integral volcanic plume model arethe vent altitude and radius, and the gas-ash mixture exit velocity, gas content and temperature.We use two approaches to specify the source conditions of the model. First, we sample source825.3. Data and plume modelconditions in a fixed parameter space (Table 5.1). A key source parameter controlling the heightreached by a volcanic plume (Equation 5.1) is the mass eruption rate M0 :M0 = piρ0R20U0 , (5.3)which is controlled by the vent radius R0, the exit velocity U0, and the bulk density of the ejectedmixture ρ0 which depends on the magma temperature and gas content. We will initially varyM0 by considering variations in R0 and U0 only (section 5.4). The range in which we sampleR0 and U0 is chosen to obtain mass eruption rates of ' 106 − 108 kg s−1, which ensures thatplume heights are between ' 50− 150% of the present day tropopause height. We return to thesensitivity of our results to natural variability in other source parameters, including the ventaltitude, in section 5.5.Next, we use the dataset of Carn et al. (2016) to test the model using source conditionsinferred for historical eruptions. We use this dataset because it covers a longer period andincludes more eruptions than, for example, Brühl et al. (2015) or Mills et al. (2016). The Carnet al. (2016) dataset includes the mass of SO2, height of SO2 injection, Volcanic ExplosivityIndex (VEI, Newhall and Self (1982)), vent altitude, latitude and longitude of eruptions ob-served by satellites since 1979. Estimates of SO2 loading into the atmosphere are based onsatellite measurements in the ultraviolet (UV), infrared (IR) and microwave spectral bands.We only use explosive eruptions between 1980 and 2015, of VEI larger than 3 and for which theestimated SO2 injection altitude is higher than 50% of the tropopause altitude. In addition, weuse three basaltic eruptions: an eruptive event at Mt Etna (2011, Italy), and the large fissureeruptions of Laki (1783-1784) and Bárðarbunga (2014-2015) in Iceland. We estimate the masseruption rate of all historical eruptions used on the basis of the observed height reached bytheir plumes using the integral volcanic plume model described in Section 5.3.3. To do this,we specify atmospheric conditions retrieved from the National Centers for Environmental Pre-diction (NCEP)/National Center for Atmospheric Research (NCAR) reanalysis (Kalnay et al.,1996), and all other parameters as in Table 5.1 except the vent altitude, and the gas contenttaken equal to 0.9 for the Bárðarbunga plume which contained little ash (Schmidt et al., 2015).Table 5.2 summarizes the date, location, mass, altitude, and altitude range of injected SO2,and the estimated mass eruption rate of 10 explosive eruptions from the Carn et al. (2016)dataset as well as the three basaltic eruptions used. For the Laki (1783-1784) eruption, weuse a mean plume altitude of 11 km corresponding to the range of plume altitudes of 9-13 kmestimated by Thordarson and Self (2003) for explosive plumes during the first three monthsof the eruption, during which most of the SO2 was released. Uncertainties in the altitudereached by volcanic SO2 plumes are large, including when they are estimated using satellitemeasurements. For example, estimates from Carn et al. (2016) are often in the higher rangeof values found in Brühl et al. (2015) or Mills et al. (2016). Another example is the Nabro(2011) eruption, for which Bourassa et al. (2013) report tropospheric plume altitudes of 13-16km while Vernier et al. (2013) and Fromm et al. (2013) reports stratospheric altitudes of 16-19835.3. Data and plume modelkm.Table 5.2: Subset of the volcanic eruptions chosen to test the impact of climate change onplume height. The top group consists of eruptions with relatively large stratosphericinjections in the late 20th century. The middle group consists of eruptions withrelatively small stratospheric injections in the early 21st century with a distinctfootprint on climate (Santer et al., 2015). The bottom group consists of basalticeruptions, either stratospheric or tropospheric. SO2 mass and plume altitudes aretaken from Carn et al. (2016), except for the Laki eruptions (Thordarson and Self ,2003), and the range indicated for plume altitude corresponds to estimated rangefrom other studies, when available. We also indicate the stratospheric aerosol opticaldepth peak after the eruption, defined as the stratospheric aerosol optical depthof the month preceding the eruption subtracted from the first peak in the globalmonthly mean stratospheric aerosol optical depth in the 12 months following aneruption.Volcano Date Country Latitude Vent Altitude (km) SO2 Plume Altitude (km) Estimated M0 (kg s−1) SO2 (Mt) ∆τEl Chichón, A Mar.29, 1982 Mexico 17.4◦N 1.2 17a 1.3 107 0.75a 9.2 10−2 bEl Chichón, B Apr.4, 1982 Mexico 17.4◦N 1.2 28a 3.0 108 7a 9.2 10−2 bMt Pinatubo Jun.15, 1991 Philippines 15.0◦N 1.7 25a (17-28)c,d,e 1.7 108 18a 1.4 10−1 bManam Jan.27, 2005 Papua New Guinea 4.1◦S 1.8 24a (18-24)c,d,f 8.1 107 0.14a 8.0 10−4 bSoufrière Hills May 20, 2006 Montserrat (UK) 16.7◦N 0.2 20a (17-21)c,d,g 4.1 107 0.2a 2.2 10−3 bKasatochi Aug.7, 2008 Russia 52.2◦N 0.3 15a (10-18)c,d,h 3.4 107 2a 1.5 10−3 bSarychev Jun.16, 2009 Russia 48.1◦N 1.5 17a (11-17)c,d 3.8 107 1.2a 2.6 10−3 bMerapi Nov.4, 2010 Indonesia 7.5◦S 3 17a (14-18)c,d 5.5 106 0.3a 1.0 10−3 bNabro Jun.13, 2011 Eritrea 13.4◦N 2.2 18a (10-19)c,d,i,j,k,l 1.8 107 0.68a 3.4 10−3 bKelut Feb.13, 2014 Indonesia 8.0◦S 1.7 19a (17-26)d 2.9 107 0.2a 2.5 10−3 kLaki Jun.8, 1783 - Feb.7 1784 Iceland 64◦N 1.7 11 (9-13)n 3.7 106 122n -Etna Aug.20, 2011 Italy 37.7◦N 3.4 9a 5.6 105 0.004a -Bárðarbunga Sep. 2014 - Dec. 2014 Iceland 64.6◦N 2 5a (3-5)o 7.1 104 4.3a -aCarn et al. (2016), bSato et al. (1993), cBrühl et al. (2015), dMills et al. (2016), e Guo et al. (2004), f Tupper et al. (2007), g Prata et al. (2007), h Waythomas et al. (2010),iFromm et al. (2013), jVernier et al. (2013), kBourassa et al. (2013), lClarisse et al. (2014), m Rieger et al. (2015),n Thordarson and Self (2003), o Schmidt et al. (2015)Last, in Section 5.5, we use the Sigl et al. (2015) dataset in addition to the Carn et al.(2016) dataset to estimate SO2 flux into the stratosphere. Sigl et al. (2015) use Greenland andAntarctic ice-cores to reconstruct the mass of volcanic aerosols produced in the stratosphereby eruptions over the past 2500 years. Figure D.3 shows the distribution of erupted mass ofSO2 using these two datasets. The Carn et al. (2016) dataset enables to characterize smallstratospheric injections (≤3 Mt of SO2), which occur with a frequency that is larger than therate of decay of stratospheric sulfate aerosol and contribute strongly to the “stratospheric aerosolbackground” (Solomon et al., 2011). The Sigl et al. (2015) dataset, on the other hand, enablesto characterize large stratospheric injections (≥3 Mt of SO2) which occur with a frequency thatis much smaller than the rate of decay of stratospheric sulfate aerosol, and thus act as impulsiveforcings.845.3. Data and plume model5.3.2 Atmospheric conditionsChoice of GCM, period and RCP scenarioWe retrieve the temperature (T ), pressure (P ), horizontal wind speed (V ), and relative humidity(RH) profiles required for each run of the integral volcanic plume model. These fields areretrieved from an ensemble of three Coupled Model Intercomparison Project Phase 5 (CMIP5)GCMs:• BCC-CSM1.1 is the coarse resolution version of the Earth System Model (ESM, coupledclimate-carbon cycle model) of the Beijing Climate Center Climate System Model (BCC-CSM,Wu et al. (2014)). The horizontal resolution is approximately 2.8125◦× 2.8125◦ with26 levels for the atmospheric component .• CanESM2 is the Earth system model of the Canadian Centre for Climate Modelingand Analysis (Chylek et al., 2011). The horizontal resolution is approximately 1.875◦×1.875◦ with 35 levels for the atmospheric component.• MPI-ESM-LR is the Earth system model of the Max Planck Institute (MPI, Giorgettaet al. (2013)). The horizontal resolution is approximately 1.875◦× 1.875◦ with 47 levelsfor the atmospheric component.We choose these GCMs because of the availibility of long-term (2005-2300) climate projectionsoutputs with a daily resolution (Table D.1). Profiles of fields are drawn from GCM outputover 8 to 15 pressure levels. Because the integral volcanic plume model uses height levels andis integrated with a vertical resolution of a few tens of meters, we also retrieve geopotentialheight (Z) profiles and interpolate the field profiles drawn from GCM results using a cubicinterpolation scheme (after testing several interpolation methods). Because the duration oflarge explosive eruptions is typically of the order one day (e.g., Mastin et al. (2009)), we usedaily atmospheric variables, retrieved from 12 regions in which explosive eruptions potentiallyreaching the stratosphere (Volcanic Explosivity Index > 3, Newhall and Self (1982)) mostfrequently occur (Figure 5.1, Table D.2). For each region we derive the spatially-averaged dailyatmospheric profiles. All GCM outputs are obtained from the Climate and EnvironmentalRetrieval and Archive database ( We use (Taylor et al., 2012):• Historical experiments where GCMs were run for the 1850-2005 period with imposedatmospheric composition (e.g., CO2), solar forcing, aerosols, and land use changes inferredfrom observations.• Representative Concentration Pathways (RCP) experiments where GCMs were run withdifferent forcing scenarios, in particular in terms of CO2 concentrations, but also in termsof other greenhouse gases, aerosols and land use change. We use the RCP2.6, RCP4.5and RCP8.5 experiments, and the periods 2081-2100, 2181-2200 and 2281-2300.855.3. Data and plume modelWe take our reference period to be 1981-2000, for which data are retrieved from the historicalexperiments. Our choice of RCP scenarios and periods allows us to explore the impact of alarge range of greenhouse gas forcings (Van Vuuren et al., 2011):• For the RCP2.6 scenario, Earth radiative forcing peaks at +3 W m−2 (relative to pre-industrial period) in the mid 21st century before decreasing (+2.6 W m−2 in 2100, '+2 W m−2 in 2300). In the fifth assessment report (AR5), the International Panel onClimate Change (IPCC) project global mean surface air temperature anomalies in 2081-2100 relative to 1986-2005 of 1.0± 0.4, and 0.6± 0.3 in 2281-2300 (CMIP5 multi-modelmean ± 1 standard deviation across individual models, Collins et al. (2013b)).• For the RCP4.5 scenario, the radiative forcing peaks at +4.5 W m−2 in 2100 and is stablein the following centuries. Projected temperature anomalies for this scenario are 1.8± 0.5in 2081-2100 and 2.5± 0.6 in 2281-2300.• For the RCP8.5 scenario, the radiative forcing peaks at +8.5 W m−2 in 2100, +12 W m−2in the mid 23rd century and is steady afterwards. Projected temperature anomalies forthis scenario are 3.7± 0.7 in 2081-2100 and 7.8± 2.9 in 2281-2300.Current CO2 emissions slightly exceeded the RCP8.5 scenario over 2010-2014 (Sanford et al.,2014)). For each period and RCP experiment, we use only one run for the GCMs with multipleruns available. We make this choice because for the 22nd and 23rd century, most GCMs onlyhave outputs for the last 20 years of these centuries from a single run available (Table D.1).For consistency, we used the same period duration and number of runs for the 20th and 21stcenturies.Performance of chosen GCMsThere is large variability in the capabilities of GCMs for reproducing past climate, as well asin their predictions of future climate. The performance of given GCMs also strongly depend onregion, field variable (e.g., temperature) and altitude range (Gleckler et al., 2008; Flato et al.,2013). The three GCMs (BCC-CSM1.1, CanESM2 and MPI-ESM-LR) we select for this studymust perform well for all four fields (T , V , RH and Z) and in each of the 12 regions chosen.Following Gleckler et al. (2008) , we compare how GCM historical runs reproduce climate overthe 1960-2000 period, our reference period for GCM ranking. In addition to our selected threeGCMs we use 13 other GCM for this evaluation analysis since we are interested in the relativeperformance of the selected GCMs within a model ensemble. The 16 GCMs (Table D.1) areselected following previous GCM evaluation studies (e.g., Flato et al. (2013)). We choose theNCEP-NCAR reanalysis (Kalnay et al., 1996) as a reference dataset, but obtain very similarresults using the ERA40 reanalysis (Uppala et al., 2005). This section provides a brief overviewof our evaluation procedure and main results. The reader is referred to the Section D.1 forfurther details.865.3. Data and plume modelGCMs are compared to the reference dataset on the basis of their root mean squared errors(RMSE) assessed on (i) the monthly average of a field (T , V , RH or Z) (ii) the monthly standarddeviations, over time, in a field (iii) the frequency of occurrence of one field characteristicprofiles. For the latter metric, for a given region, month and field, we demean and normalizedaily profiles by substracting the monthly mean and dividing by the monthly standard deviation,at each altitude. We then identify characteristic profiles and their frequency of occurrence inthe reference dataset using a Self Organizing Map algorithm (SOM, Kohonen (1982)). Next, foreach demeaned and standardized profile of a GCM, we find the best matching profile among thecharacteristic profiles of the reference dataset. We can then compare the frequency of occurrenceof a characteristic profile in a GCM and in the reference dataset (Radić et al., 2015). Moredetails on this metric are given in Supporting Information. Since we are interested in therelative model performance, we define the relative RMSE as the error relative to the medianerror of the 16 GCMs. In this way, a relative model error of, for example, 0.5 means thatthe GCM has a 50% larger error than the median model error. Figure 5.3 shows the relativeRMSE for the three GCMs used for this study and their ensemble across all evaluation metrics.For simplicity, we grouped the 12 regions into three groups of regions: northern extra-tropical,tropical and southern extra-tropical region.-1 -0.5 0 0.5 1 1.5avgstdfrqavgstdfrqavgstdfrqavgstdfrqTVZRHSouthern Extra-Tropical RegionsBCC-CSM1.1CanESM2MPI-ESM-LRother tested GCMsELT3 ensembleRelative root mean squared error (unitless)-1 -0.5 0 0.5 1 1.5Tropical Regions-1 -0.5 0 0.5 1 1.5Northern Extra-Tropical RegionsFigure 5.3: Relative root mean square error (unitless, relative to the GCM median error)for the T , V , Z, and RH fields, the three evaluation metrics (average, standarddeviation and frequency of characteristics patterns noted “avg", “std" and “frq",respectively on the figure) and the three groups of regions. Small black dots showthe relative error for the 16 GCMs tested (Table D.1). Diamonds symbols showthe three GCMs selected to be used in this study (BCC-CSM1.1, CanESM2 andMPI-ESM-LR) and their ensemble (ELT3) A negative error (left of the dashed line)indicates that a GCM performs better than the median GCM. More details on theGCM evaluation procedure are given in Section D.1.875.3. Data and plume modelFor all metrics, two of our selected GCMs (MPI-ESM-LR and Can-ESM2) perform betterthan the median model, especially for the tropical and northern high-latitudes regions. MPI-ESM-LR outperforms most GCMs for temperature related metrics. For BCC-CSM1.1, errorsare generally close to or larger than the GCM median error. The error of the ensemble ofthe chosen three GCMs (ELT3) is always below the GCM median error, for errors on averagefields. In particular, ELT3 outperforms most GCMs in reproducing the mean temperature andhorizontal wind speed profile (except for wind over the southern extra-tropical regions). ELT3is sometimes outperformed by CanESM2 or MPI-ESM-LR. However, using this ensemble forour study will allow us to better account for uncertainties related to spread in GCMs projectionsof future climate. Sensitivity of our results to the choice of GCMs will be further discussed insection Integral volcanic plume modelTurbulent entrainment:uε= α|u-Vsin(φ)|+β|Vcos(φ)|uφ sVCondensation at fixed rate λuεHHbFigure 5.4: Cartoon of a volcanic plume rising in the atmosphere and problem definitionfor the integral volcanic plume model developed in Section 2.3. Plume properties,such as the plume velocity u, depend only on the distance along the plume centerlines and plume properties profiles are top-hat (constant inside the plume and null out-side). The inflow of atmospheric air into the plume u is proportional to the radialgradient of axial velocity between the plume and the atmosphere (|u− V sin(φ)|)and to the radial gradient of ortho-axial velocity (|V cos(φ)|) where φ, the localplume deflection with respect to the vertical, defines the local axial direction. Thegreen dashed lines shows the maximum plume altitudeH and the altitude of neutralbuoyancy Hb.885.3. Data and plume modelTo compute the height reached by a volcanic plume, we use an integral volcanic plume modeldescribed in Degruyter and Bonadonna (2012), which is based on the 1D buoyant plume modelof Morton et al. (1956) adapted by Woods (1988) for explosive eruptions. The model alsoincludes the effects of atmospheric wind and humidity on the plume rise (Bursik, 2001; Glazeet al., 1997). We use the maximum height reached by the plume H (also called overshootheight, Figure 5.4), but we verified that using the height of the neutral buoyancy level Hbinstead does not impact our results. Plume properties (e.g., temperature, velocity or relativehumidity) profiles across the plume are assumed to be top-hat in shape and thus depend only onthe position along the plume centerline s (Figure 5.4). Plume rise is governed by conservationequations for mass, momentum and energy rates (Degruyter and Bonadonna, 2012).Turbulent motions mix surrounding atmosphere into a rising plume. To characterize thiscritical phenomenon we, employ the entrainment hypothesis (Morton et al., 1956), modified toaccount for wind effect (Hewett et al., 1971), to specify the inflow entrainment velocity normalto the centerline u as:u = α|u− V sin(φ)|+ β|V cos(φ)| . (5.4)Here u is the average axial velocity of the plume and φ is the plume deflection with respect to thevertical direction (Figure 5.4). α is the radial entrainment coefficient (Morton et al., 1956) andrelates u to the radial gradient of axial velocity. β is the wind entrainment coefficient (Hewettet al., 1971) and relates u to the radial gradient of normal velocity. The major effect of wind isto enhance entrainment rates. On the basis of the experiments of Carazzo et al. (2014), we takeα=0.1 and β=0.7 unless otherwise specified. These values are within the range commonly usedin buoyant plume models (e.g., Costa et al. (2016)). Integral volcanic plume models capturethe first-order effects of atmospheric temperature and wind stresses variations on the rise of theplume (e.g., Degruyter and Bonadonna (2012); Woodhouse et al. (2013); Mastin (2014); Folchet al. (2016)). Uncertainties on the entrainment coefficients (Table 5.1) are the main sourcesof uncertainty on the plume height (e.g., Mastin (2014), Woodhouse et al. (2015), Bonadonnaet al. (2015b), Costa et al. (2016)) and will be discussed in section 5.5.In addition to temperature and wind, atmospheric humidity can impact the plume rise.Entrained water vapor can condense inside a plume, leading to an additional buoyancy fluxrelated to release of latent heat (Morton, 1957; Woods, 1993). To include these effects, wefollow Glaze et al. (1997) and assume that water vapor condensation inside the plume occursat a specified constant rate λ when water vapor pressure is above the saturation pressure. Thereader is referred to Degruyter and Bonadonna (2012) for further details on the integral volcanicplume model. How to most accurately capture the effects of humidity on plume rise in integralmodels is a challenge that is largely unexplored (in Aubry and Jellinek (2018), i.e. Chapter 4,which was published after this chapter, we made a posteriori contributions to improve this aspectof the model). Furthermore, simulation of humidity and cloud formation is one of the mainchallenges for GCMs (Flato et al., 2013). Consequently, in this study, the impact of projectedchanges in relative humidity will be discussed in section 5.5 but is not considered (i.e., λ=0) in895.4. Resultsour main results (section 5.4).For given eruption source conditions, region, period, and RCP scenario, the volcanic plumemaximum height depends on the exact weather conditions during the eruption. As futureweather conditions are projected with a large range of uncertainty, we apply a method thatallows us to assess the probability of occurrence of most prevailing (characteristic) weatherconditions in terms of temperature, wind speed, relative humidity and geopotential height. Tothis end, we use a SOM algorithm to cluster the GCMs daily profiles from each 20-year periodinto ' 60 representative profiles, each of those having an associated frequency of occurrenceover 20 years. We then run the integral volcanic plume model for each representative profileto obtain a probability distribution of the plume altitude using the frequency of occurrenceof each profile (Figure 5.2 b). This distribution accounts for both variability in atmosphericconditions as simulated by one GCM within a 20-year period (e.g., due to seasonal cycle) andthe inter-GCM variability as we use a three-model ensemble.In addition to plume height, for each characteristic profile identified by the SOM algorithm,we estimate the tropopause height by interpolating the temperature profile and finding thelowest altitude at which the temperature lapse rate is less than 2 K km−1, for at least 2 km(following the World Meteorological Organization definition). Although the vertical resolutionof GCM datasets used is coarser than the multidecadal changes in tropopause height, previousstudies demonstrate that estimates on the basis of interpolation of coarse temperature profilesare reliable to assess multidecadal changes in tropopause height (e.g., Santer et al. (2003)).5.4 ResultsTo understand how global warming might impact the height reached by volcanic plumes, wefirst analyze distinct effects of projected changes in temperature and geopotential height profiles(which control the lapse rate), and horizontal wind speed profiles for 2 regions (one high-latitude,Chile, and one tropical, Philippines) under strong greenhouse gas forcing (scenario RCP8.5).We then assess the combined impacts of changes in temperature, geopotential height and windfor the same forcing and regions, and summarize results for all regions (Figure 5.1), periods(1981-2000, 2081-2100, 2181-2200 and 2281-2300) and forcing scenarios (RCP2.6, RCP4.5 andRCP8.5). Finally, we illustrate our results by projecting changes in the height of historicaleruptions if they were to occur under future climate conditions.5.4.1 Impact of temperature and geopotential height changes underRCP8.5In this section, we fix the horizontal wind speed to the average of the reference period (1981-2000) for each region. Figure 5.5 shows the temperature as a function of geopotential heightin Chile (a) and in the Philippines (b), for the reference (1981-2000), 2081-2100 and 2281-2300periods. For both regions, the temperature increases with time in the troposphere, decreasesin the stratosphere, and the tropopause height increases.905.4. ResultsGeopotential height (km)051015202530Temperature ( K)180 200 220 240 260 280 300 320Tropo-pauseTropospheric lapse rate:6.4 K km-1 (6.2 6.7)6.1 K km-1 (6 6.4)6 K km-1 (5.7 6.2)Chile(a)M0 (kg s-1)106 107 108Plume altitude (km)510152025(c)1981-20002081-2100, RCP852281-2300, RCP85M0 (kg s-1)106 107 108H* (Plume/Tropopause)0.511.5Tropopause(e)051015202530Temperature ( K)180 200 220 240 260 280 300 3206.7 K km-1 (6.4 6.9)6.4 K km-1 (6.1 6.5)5.8 K km-1 (5.6 6.2)Philippines(b)M0 (kg s-1)106 107 108510152025(d)M0 (kg s-1)106 107 1080.511.5(f)Figure 5.5: Impacts of projected changes in temperature and geopotential height on vol-canic plume height for RCP8.5 (wind fixed to reference period average):Left (a, c, e) and right (b, d, f) columns show results for the Chile and Philippinesregions, respectively. Top row (a, b) shows the temperature as a function of thegeopotential height. Bold lines show the median temperature, and shaded areasshow the interval between the 5th and 95th quantiles. The median tropopauseheight is shown by a square on the top row, with an error bar showing the intervalbetween the 5th and 95th quantiles. Blue, orange and red correspond to the ref-erence period (1981-2100), 2081-2100 RCP8.5, and 2281-2300 RCP8.5 projections,respectively. The values of the median tropospheric lapse rate are indicated onpanels (a) and (b) with 5th and 95th quantiles indicated in parenthesis. Centerrow (c, d) shows the maximum plume altitude (H) as a function of the mass erup-tion rate. Bottom row (e, f) shows H∗, the ratio of the maximum plume height totropopause height, as a function of the mass eruption rate.In the tropical region (Philippines), changes in median temperature and tropopause heightfrom one period to another are large compared to the seasonal and inter-annual variability over915.4. Resultseach period. In contrast, the changes are smaller compared to variability in the high-latituderegion (Chile), mostly because of the higher seasonality. Between the late 23rd century andthe reference period, the tropospheric lapse rate is projected to decrease by 0.9 K km−1 in thePhilippines and by 0.4 K km−1 in Chile. The stratospheric lapse rate is projected to increaseby ' 1K km−1 on average between the tropopause and ' 30 km altitude, which results in aslightly positive lapse rate in the lower stratosphere in Chile, for the 2281-2300 period (wherethe lapse rate is defined as Γ = −dTdz ).Volcanic plume heights vary with projected temperature and geopotential height changes(Figure 5.5, panels c and d). In particular, where the lapse rate decreases, plume heightdecreases and vice-versa. In the Philippines, for mass eruption rates of order of magnitude107 kg s−1, plume heights are projected to decrease by 2-3 km in the upper troposphere.Decrease in tropospheric plume height is weaker (< 1 km) and less significant in Chile. Forboth regions, stratospheric plume (M0  107 kg s−1) heights are predicted to increase by' 2 km, with a more significant increase in the tropical region. The uncertainty in plumeheight due to temperature variability over one period is small ('1-2 km for both regions).The ratio of the maximum plume altitude to the tropopause altitude (H∗) declines forboth regions and all M0, as greenhouse gas forcing increases (Figure 5.5, panels e and f). Inthe Philippines, for an eruption whose median H∗ was equal to 1 in the reference period,H∗ decreases by 0.2-0.3 in 2281-2300. Similar changes are predicted for Chilean plumes, butare smaller and less significant due to relatively small decreases in tropospheric plume heightand larger temperature variability. In the stratosphere, although plume heights increase, H∗decreases by ' 0.2− 0.3 for both regions because the tropopause height increases over the sameperiod.925.4. Results5.4.2 Impact of horizontal wind speed changes under RCP8.5Altitude (km)051015202530Horizontal wind speed (m s-1)0 10 20 30 40 50 60 70Chile(a)M0 (kg s-1)106 107 108H* (Plume/Tropopause), RCP852281-2300, RCP85051015202530Horizontal wind speed (m s-1)0 10 20 30 40 50 60 70TropopausePhilippines(b)M0 (kg s-1)106 107 1080.40.60.811. 5.6: Impact of projected changes in wind speed on volcanic plume height forRCP8.5 (temperature and geopotential height fixed to reference period average):The temperature and geopotential height profiles are fixed to their averages for thereference period. Top row (a, b) shows the horizontal wind speed as a function ofthe geopotential height. Bottom row (c, d) shows H∗ as a function of the masseruption rate, with a fixed tropopause altitude. Regions, color and shading are thesame as for Figure 5.5.We now fix the temperature and geopotential height to their average values for the referenceperiod for each region while we apply daily wind profiles from GCM runs in the plume model.Overall, we observe no significant change in projected wind profiles in either region (Figure 5.6,panels a and b). For example, in Chile, there is a decrease in median tropospheric wind speedand an increase in median stratospheric wind speed. However, these changes are small relativeto the wind variability over one period. Similar conclusions apply to the Philippines, where thewinds are weaker and changes are smaller relative to Chile. For both regions, the wind speedvariability in time increases with greenhouse gas forcing.Variations of H∗ (Figure 5.6, panels c and d) only reflect variations in plume height sincethe temperature profiles, and thus the tropopause height, are constant. For a given M0 andover one period, wind variability causes H∗ to vary by 0.1 to 0.4 around its median, which935.4. Resultsmakes the changes in H∗ driven by long-term wind speed changes in response to increasinggreenhouse forcing negligible compared to these uncertainties.5.4.3 Impact of combined changes of temperature, geopotential height andhorizontal wind speed under RCP8.5Median 𝑀0∗ for which H*=1Median H* for which 𝑀0∗ = 1Figure 5.7: Changes in H∗ as a function of the dimensionless mass eruption rate M∗0(normalized to the median mass eruption rate required to reach the tropopause in1981-2000) for RCP8.5:(a) and (b) show result for Chile and Philippines, respectively. Bold lines showthe median, and shadings show the interval between the 5th and 95th quantiles.Blue, orange and red correspond to the reference period (1981-2100), 2081-2100RCP8.5, and 2281-2300 RCP8.5 projections respectively. Dotted lines of corre-sponding colors show the median value of H∗ reached in M∗0=1 (i.e., the medianmass eruption rate for which the tropopause is reached in 1981-2000). Dashed linesof corresponding colors show the median value of M∗0 required to reach H∗=1 (i.e.,the tropopause).We now analyze the effect of combined changes in temperature, geopotential height and windspeed. To facilitate the discussion, we define a normalized mass eruption rate M∗0= M0Mtp,ref0,where M tp,ref0 is the median critical mass eruption rate for which H∗=1 for the referenceperiod (1981-2000). Thus, our normalization for M0 is dependent on the region, but indicatesvariations in M0 required to reach the tropopause.Figure 5.7 shows H∗ as a function of M∗0 . Evolution of H∗ as the greenhouse gas forcingincreases is the same as when varying the temperature and geopotential height only (Figure 5.5).For a given M∗0 and period, uncertainties on H∗ originating from variability of temperature,geopotential height and wind speed are comparable to those obtained when varying the windspeed only (Figure 5.6). For example, in the Philippines, the median H∗ decreases by up to' 0.15 in the upper troposphere, for the late 21st century, and up to ' 0.25 for the 23rd century(RCP8.5). Decrease of plume height and increase of tropopause height contribute equally to945.4. Resultschanges in H∗, and result in the increase of the critical mass eruption rate required to cross thetropopause. It is increased by a factor 1.65 for the late 21st century compared to the referenceperiod, and a factor 2.8 for the 23rd century. We observe similar trends for Chile (Figure 5.7,left), although the magnitude of changes in H∗ or critical M∗0 to reach the tropopause aresmaller.5.4.4 Summary: Results for all investigated regions, periods and scenariosTable 5.3: H∗ reached for M∗0=1, i.e., the median plume altitude, relative to thetropopause height, reached for a mass eruption rate equal to the one required toreach the tropopause in 1981-2000. The table provides the values for each region(rows), and period and scenario (columns) considered in this study. Bold valuesindicate 99% significant changes relative to the reference period (cf. SupportingInformation D.2 for details on the significance test).2081-2100 2181-2200 2281-2300RCP2.6 RCP4.5 RCP8.5 RCP2.6 RCP4.5 RCP8.5 RCP2.6 RCP4.5 RCP8.5Chile 1 0.99 0.98 1 1 0.92 0.99 0.99 0.93New Zealand 0.99 0.98 0.97 0.99 0.97 0.92 1.01 0.97 0.88Ecuador 0.96 0.94 0.89 0.97 0.94 0.82 0.97 0.93 0.78Indonesia 0.98 0.96 0.92 0.99 0.96 0.83 0.99 0.95 0.8Phillippines 0.95 0.96 0.89 0.94 0.93 0.79 0.96 0.94 0.75Central America 0.96 0.94 0.89 0.96 0.92 0.82 0.96 0.94 0.79African Ridge 0.97 0.96 0.91 0.98 0.95 0.83 0.99 0.93 0.8Japan 0.99 0.98 0.96 0.99 0.96 0.94 1 0.98 0.92Cascade 1 0.99 0.96 0.99 0.98 0.93 1.02 0.99 0.92Kamchatka 1.01 0.99 0.94 1.01 1 0.87 1.02 0.98 0.88Aleutians 1 1 0.96 1.03 1.03 0.9 1.01 1 0.87Iceland 1 0.99 0.94 0.99 0.99 0.9 1 0.98 0.88We summarize our results with two key values. The first is the median value of H∗ for whichM∗0=1 (horizontal dotted lines in Figure 5.7; Table 5.3). The second is the median value ofM∗0 for which H∗=1 (vertical dashed lines in Figure 5.7; Table 5.4). For the reference period,we estimate the 99% confidence interval on the median H∗ for which M∗0=1 or median M∗0 forwhich H∗=1 by using a bootstrap method (cf. Section D.2).For M∗0=1, H∗ mostly decreases by 0 to 0.25 relative to the 1981-2000 reference period(Table 5.3). For the RCP2.6 scenario, H∗ increases by 0 to 0.03 in some extratropical regions,and always decreases for tropical regions. Decreases in H∗ are stronger and more statisticallysignificant for tropical regions, higher RCP scenarios, and more distant future for RCP4.5 andRCP8.5, for which the radiative forcing does not stabilize before 2300 (cf. Section 5.3.2 andVan Vuuren et al. (2011)). For RCP8.5, the median H∗ reached with M∗0=1 decreases by955.4. Results' 0.2 in tropical regions and ' 0.1 in extra-tropical regions, compared to the reference period.Changes are statistically significant for all tropical regions and most extratropical regions forRCP8.5 and for tropical regions for RCP4.5.Table 5.4 shows the median M∗0 for which H∗=1. The median critical mass eruption raterequired to reach the tropopause generally increases by a factor up to 2.8 depending on theregion, period and scenario. As for Table 5.3, changes are more significant for tropical regions,stronger radiative forcing, and time periods further away in the future. In particular, for theRCP8.5 scenario, the critical mass eruption rate is increased by a factor 2 to 2.8 in tropicalregions for the 22nd and 23rd centuries, and 1.25 to 2 in extra-tropical regions. Again, forthis scenario, changes are statistically significant in all tropical regions and most extratropicalregions. Values in Tables 5.3 and 5.4 are unchanged if we use the plume neutral buoyancyheight Hb instead of the maximum plume height H (Figure 5.4) to define H∗.Table 5.4: Same as Table 5.3, but showing the median M∗0 required to reach H∗=1.2081-2100 2181-2200 2281-2300RCP2.6 RCP4.5 RCP8.5 RCP2.6 RCP4.5 RCP8.5 RCP2.6 RCP4.5 RCP8.5Chile 0.98 1.04 1.1 0.99 1.01 1.53 1.05 1.08 1.42New Zealand 1.09 1.15 1.18 1.03 1.19 1.57 0.95 1.19 1.85Ecuador 1.24 1.32 1.66 1.14 1.33 2.13 1.17 1.37 2.46Indonesia 1.12 1.23 1.5 1.09 1.24 2.11 1.06 1.29 2.52Phillippines 1.32 1.25 1.65 1.34 1.39 2.29 1.21 1.3 2.8Central America 1.24 1.38 1.79 1.25 1.52 2.41 1.21 1.37 2.75African Ridge 1.14 1.21 1.59 1.11 1.27 2.27 1.08 1.41 2.44Japan 1.04 1.1 1.23 1.04 1.2 1.25 0.98 1.11 1.42Cascade 0.99 1.05 1.29 1.04 1.1 1.44 0.9 1.08 1.53Kamchatka 0.95 1.06 1.36 0.93 0.99 1.86 0.88 1.09 1.92Aleutians 1.01 1 1.2 0.86 0.87 1.69 0.97 1.02 1.84Iceland 0.99 1.05 1.37 1.04 1.04 1.67 0.99 1.11 1.94965.4. Results5.4.5 Height projections for past eruptionsEl Chichón 1982, AEl Chichón 1982, BPinatubo 1991Manam 2005Soufrière Hills 2006Kasatochi 2008Sarychev 2009Merapi 2010Nabro 2011Kelut 2014Laki 1783-1784Etna 2011Bárðarbunga 2014-2015H* 20th centurylarge stratosphericeruptionsLargest 21st century stratospheric eruptionsBasaltic eruptionsObserved1981-2000 possible range2081-2100 projected range for RCP852281-2300 projected range for RCP85Figure 5.8: Observed and projected H∗ for past volcanic eruptions (Table 5.2):Parameters for eruptions shown are listed in Table 5.2. The observed H∗, takenfrom Carn et al. (2016), is shown in black, with vertical bars showing the estimateduncertainty based on height estimates from different studies. We assume a relativeuncertainty in plume height of ±20% where we could not find estimates differentfrom Carn et al. (2016). Blue, orange and red dots show the predicted median H∗for the 1980-2000, 2081-2100 (RCP8.5) and 2281-2300 (RCP8.5) periods, with ver-tical bars showing the 5th and 95th quantiles. The horizontal dashed line indicatesthe tropopause, which corresponds to H∗=1.To illustrate the effects of changes in volcanic plume and tropopause height, we first test how theheight of 13 historical eruptions (Table 5.2) would change relative to the tropopause height as aconsequence of greenhouse gas emissions. For each eruption, Figure 5.8 shows H∗ inferred fromCarn et al. (2016) and predicted values for the 1981-2000 reference period, 2081-2100 (RCP8.5)and 2281-2300 (RCP8.5). Atmospheric conditions used to predict H∗ are associated with theregion closest to the volcano considered except for the Etna eruption for which we retrievedreanalysis and GCM atmospheric profiles over Sicily (Figure 5.1, Table D.2). Eruptions withH∗975.4. Resultsabove 1 cross the tropopause. The observed H∗ generally lies within the range predicted usingGCM historical runs for the 1981-2000 period. Predicted H∗ for the late 21st century for theRCP8.5 scenario is lower than that which is predicted for the reference period. For 2 eruptions(El Chichón 1982 A and Merapi 2010), the predicted median H∗ is below 1, indicating thatthe probability that the eruption will cross the tropopause is less than 50%. For the late 23rdcentury and a RCP8.5 scenario, the median H∗ for 4 eruptions is below 1, with a probability tocross the tropopause of less than 5% for El Chichón 1982 A and Merapi 2010. The El Chichón1982 B and Pinatubo eruptions remain largely above the tropopause although H∗ decreasesfor these eruptions as well. The value of H∗ for analyzed basaltic eruptions also decreases. Inparticular, our results suggest that a Laki-type eruption would have less than 50% chance ofcrossing the tropopause in between 2100 and 2300, under the RCP8.5 scenario.Figure 5.8 illustrates the impact of global warming on different size and type of plumes,but does not reflect that smaller eruptive plumes (e.g., Merapi 2010) are more frequent thanlarger eruptive plumes (e.g., Pinatubo 1991). Accordingly, we project H∗ for the subset oferuptions from the Carn et al. (2016) dataset described in Section 5.3.1 (i.e., in particular,VEI>3 and observed H∗ > 0.5). Figure 5.9 (panel (a)) shows the observed H∗ and mass ofinjected SO2 as a function of latitude and time. Panels (b)-(f) shows median H∗ predictionunder a 1981-2000, 2081-2100 and 2281-2300 climate (RCP4.5 and RCP8.5 for future periods).We show only stratospheric plumes (i.e., for which H∗ > 1) and indicate on each panel thecorresponding estimate for the global and tropical volcanic fluxes of SO2 into the stratosphere.There is again a good agreement between H∗ calculated from the Carn et al. (2016) dataset(Figure 5.9.a) and the values calculated for the reference period climate, using GCM historicalruns (Figure 5.9.b). For the reference period, the total flux of volcanic SO2 into the stratosphereis 1.26 Mt/yr, about 0.9 Mt/yr of which are injected in the tropics. Under a 2081-2100 climateevolving under a RCP4.5 or RCP8.5 scenario, or 2281-2300 climate under RCP4.5, we find thatthere would be '15-20 fewer eruptions reaching the stratosphere, on average, with most of theeruptions shifted below the tropopause being in the tropics. However, the flux of volcanic SO2into the stratosphere would only decrease by 0.04-0.06 Mt/yr (or 3 to 5%) for the total flux and0.03-0.04 Mt/yr (or 3 to 4%) for the tropics. For a 2281-2300 climate under a RCP8.5 scenario,'40 eruptions out of '200 in this dataset would be tropospheric rather than stratospheric.The corresponding reduction in the SO2 injected into the stratosphere is 0.22 Mt of SO2/yr(17%), 0.16 Mt of SO2/yr (18%) of which occurring in the tropics. Last, for eruptions thatremain in the stratosphere, H∗ decreases by 0.1-0.4 depending on the time period and scenarioconsidered.985.5. Discussion1980 1985 1990 1995 2000 2005 2010 2015Latitude-50050Stratospheric SO2:Total:1.23 Mt/yrTropics:0.88 Mt/yr(a) 1980-2015 SO2 loading and H*1980 1985 1990 1995 2000 2005 2010 2015-50050Stratospheric SO2:Total:1.26 Mt/yrTropics:0.9 Mt/yrMass ofSO2 (Mt):(b) Projection under 1981-2000 climate10110-110-210-311.21.41980 1985 1990 1995 2000 2005 2010 2015Latitude-50050Stratospheric SO2:Total:1.22 Mt/yrTropics:0.87 Mt/yr(c) Projection under 2081-2100 RCP4.5 climate1980 1985 1990 1995 2000 2005 2010 2015-50050Stratospheric SO2:Total:1.2 Mt/yrTropics:0.86 Mt/yrH*(d) Projection under 2081-2100 RCP8.5 climateYear1980 1985 1990 1995 2000 2005 2010 2015Latitude-50050Stratospheric SO2:Total:1.21 Mt/yrTropics:0.86 Mt/yr(e) Projection under 2281-2300 RCP4.5 climateYear1980 1985 1990 1995 2000 2005 2010 2015-50050Stratospheric SO2:Total:1.04 Mt/yrTropics:0.74 Mt/yr(f) Projection under 2281-2300 RCP8.5 climateFigure 5.9: Same as Figure 5.8, but showing observed and projected H∗ (color scale) asa function of time and latitude for all eruptions retained in the Carn et al. (2016)dataset (dashed lines show the tropics). The size of the circles is proportional tothe logarithm of the mass of SO2 injected. Only stratospheric injections (H∗ >1)are shown. Panel (a) shows the original Carn et al. (2016) dataset. In panels(b)-(f), we assume that the same sequence of eruptions occur (i.e., same sourceparameters), but use climate conditions representative of the labeled period andRCP scenario. For panels (b)-(f), we used the median H∗ for each eruption. Thetotal and tropical volcanic flux of SO2 into the stratosphere are indicated on eachpanel.5.5 Discussion5.5.1 Mechanisms driving changes in plume and tropopause heightsUnder a RCP4.5 or RCP8.5 scenario, GCM projections imply that eruptions must have a largermass eruption rate to reach the tropopause. This result is a consequence of: i) a decrease oftropospheric volcanic plume height and ii) an increase of the tropopause height. The decrease intropospheric plume height is a consequence of the decrease in tropospheric temperature lapserate (Figure 5.5). Indeed, there is a remarkable agreement between the decrease in plume995.5. Discussionheight predicted by applying change in tropospheric temperature lapse rate in Equations 5.1and 5.2, and decrease in plume height predicted by our volcanic plume model using dailyprofiles of temperature, wind speed and relative humidity. When fixing temperature profilesbut varying horizontal wind speed (Figure 5.6), we observe no large change in the medianplume height but an increased difference between the 5th and 95th quantile of plume heightprobability distribution. Horizontal wind speed is thus a source of uncertainty on plume heightfor a particular eruption, but multidecadal changes in wind speed in response to greenhouse gasemissions do not drive any significant shift of the plume height probability distribution. Ourresults apply to both explosive silicic eruptions plumes and thermal plumes related to basalticeruptions (Figure 5.8).Although our results rely on GCM predictions, they require only a decrease of troposphericlapse rate and an increase of the tropopause height. Both CMIP5 GCMs and observations ex-hibit a decrease of the tropospheric temperature lapse rate in the tropics, over the 1960-2010 pe-riod (Fu et al., 2011; Simmons et al., 2014; Sherwood and Nishant, 2015). In particular, CMIP5GCMs simulate well the shape of warming rate profiles in the tropical troposphere, which con-trols the change in lapse rate (Mitchell et al., 2013). Also, an increase of the tropopause heightis found consistently in GCMs and observations (e.g., (Santer et al., 2003)).A key question is to assess how past changes in temperature lapse rate and tropopause heighthave impacted the rise of volcanic plumes. Glaze et al. (2015) discuss how the height of a plumeproduced by a flood basalt eruption would change in an atmosphere typical of the Miocene.They suggest that a warmer atmosphere would cause a decrease in plume height. The near-vent atmospheric temperature controls the temperature difference between the erupted ash-gasmixture and the atmosphere, and thus the plume source buoyancy flux. However, near thevent, the plume is hundreds of degrees Kelvin warmer than the atmosphere and the sourcebuoyancy flux would thus not be significantly affected by a few-degrees Kelvin change of theatmospheric temperature. In addition, the plume height only weakly depends on the plumesource buoyancy flux relative to the atmospheric stratification (Equation 5.1). A change inthe mean tropospheric temperature without a change in the lapse rate would also affect thestratification (Equation 5.2) but again it would be negligible as atmospheric temperature is oforder hundreds of degrees Kelvin.5.5.2 Sensitivity analysisIn this section we test the sensitivity of our results to the choice of GCM (section 5.3.2) and tothe entrainment coefficient values applied in our volcanic plume model (section 5.3.3). We alsobriefly discuss the sensitivity of our results to the parameterization of water droplet condensa-tion in the model and the sensitivity to variability in eruption source conditions other than themass eruption rate.1005.5. DiscussionChoice of GCMsM0*10 -1 100 101H*, RCP852281-2300, RCP85M0*10 -1 100 101H**10 -1 100 101H**10 -1 100 101H* 5.10: Same as Figure 5.7, but with only the Philippines region shown. Panels (a),(b), (c) and (d) show the result obtained when using projection from BCC-CSM-LR , CanESM2, MPI-ESM-LR, and ELT3, respectively. Daily RCP runs for the23rd century were not available for CanESM2.We analyze how our results differ when using an individual GCM of the ELT3 ensemble (BCC-CSM-LR, CanESM2 and MPI-ESM-LR) relative to the results when their ensemble was used.Figure 5.10 shows H∗ as a function of M∗0 for the Philippines, for the 3 individual GCMsand the ensemble ELT3, for the reference period and the late 21st and 23rd century for theRCP8.5 scenario. First, on the basis of our volcanic plume model, all GCM projections resultin a decrease of H∗ and an increase of the critical mass eruption rate required to reach thetropopause. For the 2081-2100 period, BCC-CSM-LR, CanESM2 and MPI-ESM-LR predictan increase by a factor 1.35, 1.34 and 1.55 of the critical mass eruption rate required to reachthe tropopause, all significant at the 99% confidence level. For the 2281-2300 period, BCC-CSM-LR and MPI-ESM-LR predict an increase by a factor 1.99 and 3.16, respectively, both1015.5. Discussionbeing significant again. An extended (2100-2300) RCP8.5 run of the CanESM2 model was notavailable.All three GCMs we use and their ensemble (ELT3) thus show similar trends and differencesin the results do not change our conclusions. Although using an ensemble with more GCMswould make our analysis more complete statistically, we are limited by the availability of ex-tended RCP runs with daily outputs (Table D.1). For similar reasons, we also use a single runfrom each model. However, when comparing results using 1 or 3 runs for historical experimentsfor the CanESM2 and MPI-ESM-LR, we did not find any significant difference. Finally, it isimportant to stress that 2 out of the 3 GCMs used (MPI-ESM-LR and CanESM2) are amongthe better performing GCMs according to the evaluation metrics tested in section 5.3.2, whichgives greater confidence in our results.Volcanic plume model parametersIn integral volcanic plume models, the values of the entrainment coefficients α and β (Equa-tion 5.4), which govern the mixing of atmosphere into the volcanic plume, must be assigned.Entrainment coefficients are identified as the main source of uncertainties on the plume height(e.g., Costa et al. (2016)). To test the sensitivity of our results to entrainment coefficients, weshow H∗ as a function of M∗0 for the Philippines and for a RCP8.5 scenario, for the 6 differentcases for entrainment coefficients (Figure 5.11). We obtain similar results when the ratio of en-trainment coefficients βα is kept constant (“Standard", “Weak" and “Strong" entrainment ratescases corresponding to panels (a), (b) and (c) of Figure 5.11). When the ratio βα is increased(“Weak radial entrainment rates", panel (d) of Figure 5.11), uncertainties on H∗ induced bywind are larger and changes of H∗ are slightly less statistically significant. This behavior isexpected as the dependence of the plume height on wind is increased when increasing βα .In contrast, when the ratio βα is decreased (“Strong radial entrainment rate", panel (e) ofFigure 5.11), the significance of the changes slightly increases. Finally, we test the sensitivityof the results to the random choice of values for the entrainment coefficients, because entrain-ment coefficients depend on the plume dynamics and might vary between eruptions (“Variableentrainment rates", panel (f) of Figure 5.11). In this case, despite the increase by ' 50% of theupper bound uncertainty in H∗, the median H∗ undergoes negligible change. The increase inthe median critical mass eruption rate required to reach the tropopause is thus not sensitive tothe value of entrainment coefficients used in the integral volcanic plume model; e.g., it variesbetween 2.71 and 3.02 for the 6 cases investigated and is always significant at the 99% level for2281-2300.1025.5. DiscussionM0*10 -1 100 101H* entrainment (, = 0:1, - = 0:7)(a)1981-20002081-2100, RCP852281-2300, RCP85M0*10 -1 100 101H* entrainment (, = 0:07, - = 0:49)(b)M0*10 -1 100 101H* entrainment (, = 0:12, - = 0:84)(c)M0*10 -1 100 101H* radial entrainment (, = 0:07, - = 0:7)(d)M0*10 -1 100 101H* radial entrainment (, = 0:12, - = 0:48)(e)M0*10 -1 100 101H* sampled entrainment(f)Figure 5.11: Same as Figure 5.10, but showing sensitivity of the results to entrainmentrates α and β (Equation 5.4). Results are shown for the Philippines region usingthe ensemble ELT3. For panel (a) to (e), we run the integral volcanic plume modelwith fixed values of α and β, labelled in each panel. The ratio βα is equal to 7, 10and 4 for panels (a)-(c), panel (d) and panel (e), respectively. For panel (f), werandomly sample values of α and β using a Monte-Carlo simulation; we assumethat α and βα have normal distributions of mean 0.1 and 7 and width 0.015 and1.5 respectively (based on a refined calibration of entrainment coefficients usingthe experiments of Carazzo et al. (2014)).Additional factors affecting the height of volcanic plumesThe release of latent heat caused by condensation of entrained water vapor can increase volcanicplume heights, which is discussed in Supporting Information (Figure D.1). The impact ofchanges in atmospheric humidity projected by GCMs largely depends on the condensation rateλ used in the integral volcanic plume model (see Section 5.3.3 and Table 5.1). For the end-1035.5. Discussionmember case λ=0.098 s−1 (large condensation rate), the median plume height of troposphericplume and uncertainties on plume height increase, especially in tropical regions. However, fortropical regions, the increase of the median mass eruption rate required to reach the tropopausediffers negligibly from the model results that do not consider the condensation effect (λ=0) andremain significant at the 99% level for a RCP8.5 scenario. In addition to the mass eruption rate,plume height is influenced by other source conditions. We test how the source temperature andgas content as well as the vent altitude impact our results in Supporting Information (FigureD.2). Among these three factors, uncertainty in the vent altitude is the main factor increasinguncertainty on plume height, but this does not affect our conclusions.5.5.3 Implications for future volcanic forcingChanges in volcanic plume height and tropopause height could have significant implicationsfor future volcanic forcing as the longevity of volcanic aerosol-radiation interactions dependsstrongly on whether volcanic SO2 is injected directly into the stratosphere. The dispersal ofvolcanic particles also depends on plume height and wind speed and direction (e.g., Burdenet al. (2011)). A combined variation of all these parameters could have a significant effect onthe distributions of the associated hazards. In addition, atmospheric conditions also have asignificant effect on plume dynamics and, therefore, on the occurrence of associated hazardousprocesses (e.g., buoyant plume versus pyroclastic density currents, Degruyter and Bonadonna(2013)). Although we acknowledge that these are key issues that should be explored in detailin the future, we only discuss the implications of our results for future volcanic forcing in thisstudy.Volcanic SO2 injection efficiency metricThe potential decrease of H∗ by ' 5− 25% relative to 1981-2000 (for a RCP4.5 or RCP8.5scenario, in the coming three centuries) has significant implications for plumes ascending toan altitude just a few kilometers above the tropopause. Although eruptions associated withthese small plumes generally inject relatively moderate quantities of SO2 into the stratosphere(Table 5.2), they have a significant footprint on climate (Solomon et al., 2011; Santer et al.,2014, 2015) and are more frequent than the eruptions associated with very tall plumes (Brownet al., 2014). A generic SO2 injection efficiency metric accounting for both the quantity of SO2injected and the height of injection is a useful tool to further parameterize or characterize theimpact of climate change on volcanic aerosol-radiation interactions. We propose this injectionefficiency to be of the form:ηSO2 =∫ ∞M∗cn¯SO2M∗0 f(M∗0 )dM∗0 . (5.5)1045.5. DiscussionM $c / N 4H3tp10-1 100 1012 SO2 (Mt/yr)00.511.522.52SO2 estimate: Scaling-based approach2SO2 estimate: Monte Carlo approach(a)20thCRCP4.5, 21stCRCP4.5, 23rdCRCP8.5, 21stCRCP8.5, 23rdC20th CRCP4.5, 21st CRCP4.5, 23rd CRCP8.5, 21st CRCP8.5, 23rd C2 SO2 (Mt/yr) p=0.71p=1"Small" eruptions,tropical(b)20th CRCP4.5, 21st CRCP4.5, 23rd CRCP8.5, 21st CRCP8.5, 23rd C00."Small" eruptions,global(c)20th CRCP4.5, 21st CRCP4.5, 23rd CRCP8.5, 21st CRCP8.5, 23rd C00. eruptions,global(d)Figure 5.12: Projections of the volcanic SO2 flux into the stratosphere SO2 , over a cen-tury, in Mt/yr, for 1981-2000, 2081-2100 (RCP4.5 and RCP8.5) and 2281-2300(RCP4.5 and RCP8.5). Panel (a) shows SO2 as a function of the critical masseruption rate M∗c and the values of SO2 for the different scenario estimated us-ing the scaling-based approach of Section (Htp is the tropopause height).Panels (b)-(d) show the median SO2 estimated using the Monte-Carlo approachof Section Panel (b) shows the contribution of small (injecting less than 3Mt of SO2) tropical eruptions, panel (c) the contribution of small eruptions, andpanel (d) the total flux. In panels (b)-(d), for future periods, the probability pfor SO2 to be lower than under the reference climate is indicated.Here n¯SO2 is the ratio of the mass of SO2 injected by an eruption and its normalized masseruption rate M∗0 , which is assumed to be a constant, f(M∗0 ) is the time-averaged frequencyof an eruption of mass eruption rate M∗0 , and M∗c is the critical normalized mass eruption raterequired to reach the tropopause and is equal to 1 for the reference period by definition of M∗0 .Climate controls ηSO2 by governing M∗c , whereas crustal magmatic processes might control1055.5. Discussionf(M∗0 ) over time scales of 102 to 104 years, and magmatism related to mantle dynamics andplate tectonics enter at time scales of order > 106 years.To estimate ηSO2 for the reference period, we take n¯SO2 to be the average value of the ratioof the mass of SO2 injected by an eruption to its normalized mass eruption rate M∗0 in theCarn et al. (2016) dataset. To estimate f(M∗0 ), we use the Carn et al. (2016) dataset for thefrequent eruptions injecting less than ' 3 Mt of SO2 that contribute to aerosol background.We use the Sigl et al. (2015) dataset for intermittent events injecting more than ' 3 Mt ofSO2. Figure D.3 shows the distribution of erupted mass of SO2 from both datasets, to whichwe fit f(M∗0 ) using a Kernel distribution. Figure 5.12 (a) shows the estimated values of ηSO2as a function of M∗c . Using M∗c=1, we find ηSO2=1.45 Mt/yr for the reference period, which isclose to the value of 1.23 Mt/yr estimated in Figure 5.9 using the Carn et al. (2016) datasetonly. To estimate ηSO2 for an arbitrary period, we use Equation 5.1. Let rT be the ratioof the tropopause height of the period considered to the tropopause height of the referenceperiod. Let rN be the ratio of the Brunt-Väisälä frequencies for the same periods. Then, usingEquations 5.1, M∗c = r4T r3N . Figure 5.12 shows this scaling-based estimate of ηSO2 for a RCP4.5and RCP8.5 scenario. Using average changes in tropical tropopause height and tropospherictemperature lapse rate to calculate M∗c , we find ηSO2=1.34 Mt/yr and ηSO2=1.31 Mt/yr forthe late 21st century for the RCP4.5 and RCP8.5 scenarios respectively, and ηSO2=1.23 Mt/yrand ηSO2=1.0 Mt/yr for the late 23rd century for the same scenarios (Figure 5.12 (a)). Relativedecreases in the volcanic injection of SO2 into the stratosphere using this simple, scaling basedapproach are thus remarkably close to the ones estimated in Figure 5.9.Magnitude and likelihood of projected changes in volcanic SO2 fluxes into thestratosphereEstimates of ηSO2 on the basis of either the scaling-based approach of Section 4.3.1 or fromFigure 5.9 rely on several assumptions. In particular, estimates from Figure 5.9 assume that:(i) the 1980-2015 sequence of eruptions will be repeated in the future; (ii) all volcanic SO2 isinjected at the maximum plume altitude; and (iii) the plume altitude is the median altitudefor the considered period and RCP scenario. In addition, we use a steady-state plume model,which can not account for the potential additional transport of SO2 across the tropopause byatmospheric circulation (e.g., Bourassa et al. (2012)) or by natural convection after absorptionand warming (e.g., de Laat et al. (2012)).In a preliminary effort to relax some of these assumptions, we use a Monte Carlo methodto estimate future stratospheric injection of volcanic SO2 over a century, for a specified timeperiod and forcing scenario. For one simulation, we randomly sample 36525 days (100 years)in the 1980-2015 period, which is the longest period with available plume height and SO2loading for most eruptions. For each day corresponding to an eruption in the Carn et al. (2016)dataset injecting less than 3 Mt of SO2, we assume that an eruption occurs with the followingcharacteristics:1065.5. Discussion• The region and vent altitude is the same as for the original eruption.• The mass eruption rate is 10ψ ×MCarn where MCarn is the mass eruption rate of theoriginal eruption and ψ is a random number between -0.3 and 0.3. Since 100.3 ' 2, theresulting mass eruption rate is within a factor 2 of the one of the original eruption. Thisapproach enables us to randomize the mass eruption rate, while preserving its order ofmagnitude such that the distribution of mass eruption rates is similar to the one inferredfor the 1980-2015 period.• The mass of SO2 is 10φ×MSO2Carn where MSO2Carn is the mass of SO2 of the originaleruption and φ is a random number between -0.3 and 0.3, where the choice of randomnumber range is based on the same argument as for the mass eruption rate.• Atmospheric conditions correspond to a day randomly sampled from the GCM ensemble,for the specified period and scenario.• The SO2 is uniformly distributed between Hb and 2H− Hb where H is the maximumplume altitude and Hb the altitude of neutral buoyancy of the plume. This approachis approximately equivalent to distributing the SO2 over a layer of height 30-50% of themaximum height. For a steady plume and in the absence of additional vertical transportby atmospheric winds or thermal convection, we would have distributed the SO2 in a layerof thickness H−Hb. Here we arbitrarily double this thickness to explore a larger verticalspread of the SO2 due to unsteadiness and spreading mechanisms mentioned above. Thechosen layer thickness is coherent with uncertainties on observed plume height shown onFigure 5.8, which are due to a large extent to unsteadiness of the eruption, or uncertaintiesrelated to vertical transport of the plume.Last, we randomly sample a 100-year period in the Sigl et al. (2015) dataset from which weexcluded eruptions injecting less than 3 Mt of SO2. We assume that corresponding samplederuptions inject SO2 directly into the stratosphere, regardless of atmospheric conditions.We perform 300 Monte Carlo simulations of 100 years of volcanic eruptions for the late21st and late 23rd centuries for RCP4.5 and RCP8.5 scenarios, as well as for the referenceperiod. Results are not sensible to the number of simulations performed for more than '100simulations. Figure 5.12 shows the median flux of SO2 into the stratosphere ηSO2 (panel (d))as well as the median global (panel(c)) and tropical (panel(b)) flux of volcanic SO2 into thestratosphere due to small eruptions only (i.e., the ones injecting less than ' 3 Mt of SO2 thatare sampled from the Carn et al. (2016) dataset). The probability for projected stratosphericfluxes of future time periods to be smaller than fluxes for the reference period is also reportedon each panel.Panel (c) (Figure 5.12) shows that the flux of SO2 into the stratosphere related to smalleruptions may decrease by ' 5-25% for a RCP4.5 or RCP8.5 scenario depending on the periodconsidered. A decrease is “likely” (66 to 90% probability, using the IPCC AR5 likelihood1075.5. Discussionscale, Mastrandrea et al. (2010)) by the 23rd century but “about as likely as not” (33 to 66%probability) for the 21st century due to large uncertainties related to future eruptive conditions.Projected decreases of the tropical flux of SO2 carried by small eruptions (panel (b)) are larger(' 10-50%), and “likely” (66 to 90% probability) to “very likely” (≥90%) . However, panel(d) shows that the total flux, including the contribution from large eruptions, would undergo asmaller decrease (' 2-12%) that would be “about as likely as not” due to the large simulatedvariability in volcanic SO2 fluxes when including contribution from all eruptions. Reductionsare even smaller and less likely for a RCP2.6 scenario (not shown).To summarize, our results suggest that global warming may significantly decrease the back-ground volcanic flux of SO2 into the stratosphere sustained by small (≤ 3 Mt of SO2) andfrequent (compared to the rate of decay of stratospheric sulfate aerosols) stratospheric injec-tions. However, the effect on the total flux of SO2 into the stratosphere is small because of thecontributions of large (≥ 3 Mt of SO2) and infrequent (compared to the rate of decay of strato-spheric sulfate aerosols) events. As a final remark on this result, our view may be conservativebecause we assume that large eruptions inject SO2 into the stratosphere regardless of climate,which is not the case at least for basaltic eruptions such as the 1783-1784 eruptions of Laki(Thordarson and Self , 2003) as shown in Figure 5.8.Critically, our estimates of a decrease of the flux of volcanic SO2 into the stratosphere chal-lenges the use of steady volcanic forcing for climate projections in two ways. First, our resultssuggest a new positive feedback between climate and volcanic aerosol-radiation interaction:(i) global warming decreases the frequency of eruptions with stratospheric injections; (ii) lessfrequent stratospheric volcanic injections result in a decrease of the long-term average sulfateaerosol concentration in the stratosphere and thus of the albedo of the atmosphere; and (iii) areduced atmospheric albedo will enhance global warming. Assuming a long-term average vol-canic forcing of small eruptions (V EI 6 5) of order of magnitude -0.1 W.m−2 (Solomon et al.,2011), and that the relative variations in this average would be of the same order of magnitudeas change in the average volcanic SO2 flux into the stratosphere, the order of magnitude ofthis feedback would be 10−2 W.m−2/◦C. It may thus make a negligible contribution to globalwarming rate, although we note that the order of magnitude of projected changes in strato-spheric SO2 flux is comparable to the increase in volcanic stratospheric SO2 since 2002 whichhas been argued to contribute to overestimates of global warming rate by GCMs (e.g., Solomonet al. (2011); Santer et al. (2014)). The proposed feedback may also prove important for under-standing the evolution of volcanic aerosol forcing in the future, as well as the overall impact ofEarth’s climate on the distribution of volcanic inputs in the atmosphere. Second, our statisticalanalyses suggest that for a given climate, the average flux of volcanic SO2 into the stratosphereover a century may vary by a factor ' 5− 10, which would likely have important consequencesfor forcing related to volcanic aerosol-radiation interactions and may increase uncertainties infuture climate projections.1085.5. Discussion5.5.4 Limitations and potential improvements: beyond a binary view ofvolcanic aerosol forcing sensitivity to plume heightThe discussion of our results is grounded in the assumption that only stratospheric aerosolsexert a significant influence on global climate. Although this is a good first approximation, theshift in impact between a tropospheric and stratospheric injection of SO2 is not as abrupt. Thefollowing considerations enter the full picture of volcanic forcing:1. For stratospheric plumes, aerosol-radiation interactions are sensitive to the plume height,although most sensitivity studies focus on the impact of the eruption season and lati-tude. Stoffel et al. (2015) test the sensitivity of climate response to plume height for theSamalas 1257 eruption, and report larger aerosol optical depth and 40◦N-90◦N land tem-perature anomalies for an upper stratospheric injection (36-43 km) compared to a lowerstratospheric injection (22-26 km), with differences by up to a factor '2 depending onthe season. A sensitivity study for high latitude eruptions using a GCM coupled with astratospheric chemistry/aerosols microphysics module suggests similar effects (MatthewToohey, personal communication). For high latitude eruptions, aerosol clouds issued fromstratospheric plumes smaller than the tropical tropopause spread along constant poten-tial temperature surface and may thus cross the tropopause and be scavenged at midlatitudes (Holton et al., 1995). Carn et al. (2016), on the basis of satellite measurements,also shows that the e-folding time for SO2 removal increases with the plume height, andsuggests thatH∗ is the main parameter controlling the longevity of SO2. Greater longevityfor SO2 may lead a slower aerosol production and to a reduced but longer lasting peakof volcanic aerosol-radiation interactions (Timmreck, 2012). Thus, the decrease of H∗for large stratospheric plumes (Figures 5.7, 5.9) might have important consequences forfuture radiative forcing even if they are not shifted below the tropopause.2. Tropospheric eruptive plumes also impact climate by increasing cloud condensation nu-clei concentrations and, in turn, cloud reflectivity (aerosol-cloud interactions). For exam-ple, during the Bárðarbunga 2014-2015 eruption (Iceland), McCoy and Hartmann (2015)report increases of up to 2 W m−2 in the reflected solar radiation, over the North At-lantic. Schmidt et al. (2012) estimate that the long term average volcanic aerosol-cloudinteractions forcing is ' -0.3 to -1.6 W m−2, depending on the background aerosol con-centrations). As aerosol and nucleated cloud radiative properties depend on the heightof injection of volcanic SO2 in the troposphere (Schmidt et al., 2016), volcanic aerosol-cloud interactions may also depend on the height of volcanic plumes. As a result, alarger injection of volcanic SO2 into the troposphere and the decrease of the height oftropospheric plumes (Figure 5.8) may increase future volcanic aerosol-cloud interactionsforcing, although the projected increase in volcanic SO2 flux into the troposphere is small(' 0− 5%, estimated from panel (c) of Figure 5.12 and tropospheric flux estimates fromHalmer et al. (2002) and Carn et al. (2016)).1095.6. Conclusions3. An injection of SO2 directly into the stratosphere may not be necessary for the SO2 orsulfur aerosol to reach the stratosphere and result in significant aerosol-radiation interac-tions. Upper tropospheric volcanic sulfur gases or aerosols may be transported to someextent through the tropopause by atmospheric circulation (Bourassa et al., 2012, 2013;Clarisse et al., 2014) or by convection driven as a result of absorption of Earth and Sunradiation, which has been suggested for the Black Sunday fire (de Laat et al., 2012).4. Even when a volcanic eruption produces a stable plume, part of the erupted materialmay collapse to form pyroclastic flows (Carazzo and Jellinek, 2012). Part of the SO2 lostto pyroclastic flows may however be entrained into co-ignimbrite columns (Woods andWohletz, 1991). Although the height reached by co-ignimbrite plumes are typically lowerthan the main plinian column with which they are associated, they may transport SO2into the stratosphere for very large eruptions such as Tambora in 1815 or Pinatubo in1991 (Herzog and Graf , 2010). Such effects would not be captured by the model used inthis study.Different modeling approaches can be applied to tackle some of these four limitations. Forexample, in order to estimate changes in volcanic aerosol-radiation interactions, our plumemodel can provide SO2 altitude and loading to an idealized volcanic aerosol model, such asEasy Volcanic Aerosol (Toohey et al., 2016b), or to a GCM coupled with a stratospheric chem-istry/aerosols microphysics module, such as MAECHAM5-HAM (e.g., Toohey et al. (2011)).The use of a 3-dimensional plume model instead of an integral volcanic plume model may enableto better account for the complexity of the flows resulting from a volcanic eruptions, such asco-ignimbrite plumes.As a final note to this discussion, global warming may impact volcanic aerosol forcingvia mechanisms different than the one proposed here. For example, the gradual melting ofcontinental snow and ice cover implies that future eruptions are less likely to melt and entrainsurface water into the eruption plume, which may affect both the probability of collapse of aplume (Koyaguchi and Woods, 1996) and the radiative forcing of the eruption (LeGrande et al.,2016). Changes in atmospheric circulation may affect the distribution and e-folding time ofstratospheric aerosols (e.g., McLandress and Shepherd (2009); Jones et al. (2016)) and changesin water vapor may affect the aerosol size, and thus their radiative properties and e-folding time(e.g., Gettelman et al. (2010)). Finally, a number of studies show that eruption frequency isimpacted by continental ice-sheets, alpine glacier or sea-level change (e.g. Hall (1982); McGuireet al. (1997); Jellinek et al. (2004)). The response of volcanic aerosol forcing to these combinedeffects may improve our understanding of the evolution of volcanic aerosol forcing.5.6 ConclusionsIn this study, we investigate whether the ongoing global warming, driven by anthropogenicgreenhouse gas emissions, will shift volcanic eruption plume height relative to the tropopause1105.6. Conclusionsheight. We compute volcanic plume heights using an integral volcanic plume model. At-mospheric conditions are obtained from an ensemble of GCM runs for historical and RCPexperiments.We find that the critical mass eruption rate required to reach the tropopause will increaseas a consequence of: (i) a decrease in the heights of tropospheric plumes driven by a decreaseof the tropospheric temperature lapse rate; and (ii) an increase of the tropopause height. Thisresult is independent of the choice of GCMs and insensitive to parameterizations for the volcanicplume model. Depending on the latitudinal zone, RCP scenario and time period considered, thecritical mass eruption rate increases by up to a factor of 2.8 relative to the late 20th century.This increase is significant in tropical regions for a RCP4.5 scenario and all tested regionsfor a RCP8.5 scenario. This result implies that eruptions rising a few kilometers above thetropopause under current climate conditions may be shifted to the stratosphere in the future.As a consequence, we estimate that the flux of SO2 into the stratosphere associated to small(≤ 3 Mt of SO2) frequent (compared to the rate of decay of stratospheric sulfate aerosols)eruptions would likely decrease by ' 5 − 25% over the next three centuries, for a RCP4.5or RCP8.5 scenario. The amplitude and likelihood of such decrease is more pronounced fortropical injections. Due to the contribution of large (≥ 3 Mt of SO2) infrequent (compared tothe rate of decay of stratospheric volcanic aerosol) eruptions, and to large uncertainties in futureeruptive source conditions, the total flux of volcanic SO2 into the stratosphere is projected todecrease by ' 2− 12%, with the likelihood of such decrease being weak. Finally, our resultschallenge the popular use of steady volcanic radiative forcing in climate projections for thecoming centuries. Instead, our work suggests that greenhouse gas driven climate change willresult in less cooling from volcanic eruptions, potentially resulting in a positive feedback. Theexpected amplitude for this feedback is small, although it has been argued that the increasein stratospheric SO2 injections since 2002, which amplitude are comparable to the decreaseprojected in our study, has contributed to the overestimation of global warming rate by GCMs(e.g., Solomon et al. (2011); Santer et al. (2014)). While processes linking eruptive sourceconditions to the distribution of volcanic SO2 are neglected in past GCMs experiments onvolcanic forcing (e.g., (Stenchikov et al., 2006; Driscoll et al., 2012)) and in the next ModelIntercomparison Project on the climatic response to Volcanic forcing (Zanchettin et al., 2016),we demonstrate that such processes may prove critical to the understanding of past and futurevolcanic forcing.111Chapter 6A new idealized model ofstratospheric volcanic sulfate aerosolforcing: application to quantify theproposed climate-volcano feedback.This chapter summarizes ongoing work on an idealized model of volcanic forcing. We expect itwill lead to one publication on the model and its performance in reproducing volcanic forcingobserved during the satellite era, and another publication on the applications of the model todesign scenarios for future volcanic forcing and quantifying the feedback proposed in Aubry et al.(2016).6.1 SummaryAubry et al. (2016) show that ongoing global warming will reduce the height of volcanic plumesrelative to the tropopause height and, in turn, volcanic injections of SO2 into the stratosphere.They propose it will result in a positive feedback with reduced stratospheric volcanic aerosolforcing, reduced volcanic surface cooling, and, in turn, enhanced global warming. However, thestrength of this feedback is not quantified yet. Idealized scaling-based box models of volcanicaerosol forcing are potential tools to estimate the strength of this feedback but are subject tocritical limitations. First, they do not account for plume height and prescribe a vertical structurefor the forcing. Second, they are calibrated against only a few eruptions, most commonly the1991 eruption of Pinatubo.In this chapter, we design a new idealized model of volcanic forcing that overcomes theselimitations. Given an inventory of volcanic SO2 emissions and injection heights, the modelmakes excellent predictions for the global mean stratospheric aerosol optical depth over thecomplete 1979-2015 period, but also for the evolution of the latitudinal and vertical structure1126.2. Introduction and Motivationof extinction with time.First, we apply this model to extensively test the sensitivity of stratospheric volcanic sulfateaerosol forcing to the latitude and altitude of injection of SO2. Such a sensitivity study: i)cannot be performed in previous idealized models that prescribe the vertical structure of theforcing; and ii) are computationally expensive to run with coupled chemestry-climate models.Second, we apply this model to design scenarios for future volcanic forcing by resampling aninventory of past SO2 injections. We show that scenarios resampling only from ice-core recordsof volcanic sulfate injections neglect up to 90% of volcanic stratospheric SO2 inputs and upto 70% of the average stratospheric volcanic sulfate aerosol forcing. Third, we show that thefeedback proposed in Aubry et al. (2016) may reduce the 100-year average stratospheric volcanicsulfate aerosol forcing (on the order of 0.1 W .m−2) by up to 17%. We estimate the feedbackstrength to be 1.7 × 10−3W .m−2/W .m−2, which suggests minor implications for moderateclimate changes, although how this feedback might interact with other modes in the climatesystem is unclear. Exciting future research directions to explore with coupled chemistry-climatemodels include: i) quantifying the impact of this feedback for basaltic eruptions for which weexpect more significant impacts; ii) exploring the climate impact of this feedback in the contextof major climate change, such as a RCP8.5 scenario or in extreme climate variabilities includingSnowball Earth; iii) investigate the combined effect of this feedback with other proposed climate-volcano feedbacks.6.2 Introduction and MotivationAubry et al. (2016) suggest the existence of a positive feedback where CO2-driven global warmingleads to reduced plume height relative to tropopause height and, in turn, reduced injections ofvolcanic SO2 into the stratosphere. The expected consequence is reduced stratospheric volcanicsulfate aerosol forcing. However, they do not directly assess this forcing change. In addition,beyond a change in the flux of volcanic SO2 into the stratosphere, the potential feedback involvesdistinct responses in the tropics and high-latitude regions as well as changes in height of eruptiveplumes that would still reach the stratosphere. Consequently, to quantify the feedback theypropose, it is critical to use projected changes in the vertical and latitudinal distributions ofvolcanic SO2 in the atmosphere as input to a model of volcanic forcing.Fully dynamic coupled chemistry-climate models (e.g. Niemeier et al. (2009); Dhomseet al. (2014); Sheng et al. (2015); Mills et al. (2016)) can be used to predict the radiativeforcing response to change in volcanic SO2 distribution. They would also enable assessmentof the corresponding climate response. However, the forcing predicted by these models for aspecified volcanic SO2 injection is subject to large uncertainties (Figure 3a in Zanchettin et al.(2016)). Such uncertainties would add to uncertainties related to future eruptive history orvolcanic plume model (Aubry et al., 2016). Given the computational cost of coupled chemistry-climate models, exploring how the propagation of these uncertainties affect the assessment ofthe proposed feedback and associated climate response is not realistic within the framework of1136.2. Introduction and Motivationmy PhD. A simpler strategy is thus required to get a first estimation of the proposed feedbackand perform extensive sensitivity analyses.Idealized models or “emulators” of volcanic aerosol forcing are a class of model well suited toobtaining a first-order estimate of the feedback proposed by Aubry et al. (2016). These modelshave been developed to reconstruct the time-latitude evolution of volcanic forcing before thesatellite era using constraints from ice-core on the timing and mass of sulfur injected by ancienteruptions (e.g. Gao et al. (2008); Crowley and Unterman (2013); Toohey and Sigl (2017)).They are box models, where each box corresponds to a latitudinal region of the stratosphere.For a prescribed sulfur injection in one of the boxes, the evolution of the mass of sulfate aerosolis governed by timescale(s) for: i) the production of sulfate from SO2; ii) the mixing betweenthe boxes; and iii) the loss of aerosol to the troposphere. Global mean stratospheric aerosoloptical depth (SAOD) and effective radius are scaled from the total mass of sulfate in theboxes. The simplicity of such models imply that their results must be interpreted carefullyand depend on the parameterization employed. However, these models generally rely on only afew parameters and are computationally inexpensive so that conducting sensitivity studies toexplore uncertainty propagation is straightforward.The state of the art idealized model of volcanic forcing is the Easy Volcanic Aerosols model(EVA, Toohey et al. (2016b)). Like all idealized models of volcanic forcing, it has two importantlimitations with regard to quantifying the feedback proposed in Aubry et al. (2016):1. The vertical structure of the forcing produced by the model does not depend on char-acteristics of volcanic sulfur injections, in particular plume height. This is an obviousproblem for assessing changes in volcanic forcing related to changes in plume height.2. EVA is calibrated using data from the 1991 Pinatubo eruption. Thus, one should becareful when applying this model to other eruptions. In particular, the feedback proposedin Aubry et al. (2016) would mostly affect eruptions largely different from Pinatubo,i.e. basaltic eruptions and silicic eruptions injecting order(s) of magnitude less sulfurthan Pinatubo and with lowermost stratospheric injections (instead of middle to highstratosphere for Pinatubo).Consequently, the two major objectives of this chapter are:1. Build on EVA to develop an idealized model of volcanic aerosol forcing: i) accounting forplume height to determine the forcing resulting from a sulfur injection; ii) including thevertical structure of aerosol extinction; and iii) calibrated against eruptions spanning alarge range of mass of erupted sulfur, plume height and latitude.2. Apply this model to quantify the feedback proposed in Aubry et al. (2016).The first objective will be of interest to all modelers working on stratospheric aerosols as wellas to the various communities using idealized forcing models. The second objective aims to1146.3. Data and model structureanswer the major question of my PhD and has fundamental implications for understandingpast climate changes and predicting future climate.Section 6.3 introduces the datasets used to calibrate the new idealized model as well asthe structure of the model. Section 6.4 details how the model was calibrated and evaluated.In Section 6.5, we propose three applications of the model including quantifying the feedbackproposed by Aubry et al. (2016). This project was only started a few months before the sub-mission of this doctoral dissertation. It is thus evolving rapidly and Appendix E provides anon-exhaustive overview of the directions and improvements that will be tested in the comingmonths. We summarize our conclusions as of February 2018 in Section Data and model structure6.3.1 DataOur strategy is to calibrate the model so that it best reproduces observations for atmosphericoptical properties (model output) given an inventory of volcanic sulfur emissions (model input).For optical properties, we use GLOSSAC (Thomason et al., 2018), which is the National Aero-nautics and Space Administration state of the art reconstruction of extinction (the fraction ofradiation not transmitted per unit of thickness) from satelitte data. It contains latitude-altitudeextinction at 525nm from 1979 to 2016. The treatment of data gaps or missing values is detailedin Thomason et al. (2018). As volcanic sulfur emission inventory, we use Carn et al. (2016)who report the date, location, mass of SO2 and altitude of volcanic emissions over 1978-2015.In addition to these two datasets, we use:• SAGE4λ Arfeuille et al. (2013), which is a previous reconstruction of atmospheric opticalproperties using mostly satellite data. It contains extinction at 550nm as a function oftime, latitude and altitude from 1960 to 2010.• Aerosols properties derived from the coupled chemistry-climate model WACCM (Millset al. (2016)) using an inventory of volcanic sulfur emission.These two datasets enable further testing of the parameterizations of the idealized forcingmodel. Satellite observations are subject to large uncertainties, in particular for parametersderived from direct observations (e.g. aerosol size distribution). Using data from a prognosticaerosol model consequently enables me to strengthen confidence in parameterizations derivedfrom observational data. Figure 6.1 shows the global mean SAOD time series of the GLOSSAC,SAGE4λ and WACCM datasets as well as stratospheric SO2 injections from the Carn et al.(2016) inventory.1156.3. Data and model structure1980 1985 1990 1995 2000 2005 2010 20153 x 10-13 x 1003 x 1013 x 1023 x 1033 x 104Stratospheric SO2 loading (kt)3 x 10-310-23 x 10-210-13 x 10-1Global mean SAOD (~525nm)WACCM (model, Mills et al. 2016)GLOSSAC (satellite, Thomason et al., under review)SAGE4  (satellite, Arfeuille et al. 2013)Stratospheric volcanic sulfur loading (Carn et al., 2016)Figure 6.1: The left red y-axis shows the mass of SO2 injected into the stratosphere (redbars, data from Carn et al. (2016)). The right black axis shows global mean SAODat ' 525nm from GLOSSAC, SAGE4λ and WACCM.6.3.2 Model structureEVA separates the stratosphere into three latitudinal bands (southern high latitudes, tropicsand northern high latitudes) which is consistent with respect to the structure of the Brewer-Dobson circulation (e.g. Plumb (1996); Neu and Plumb (1999); Butchart (2014)). A logicalproposition for the vertical structure of the model could use three bands:• The lowermost extratropical stratosphere/uppermost tropical troposphere (' 10-16km),where cross-tropopause mixing and transport at mid-latitudes is an important control onthe transport of aerosols from the stratosphere to the troposphere.• The lowermost tropical stratosphere (' 16-21km) where aerosols in the tropics may betransported directly into the lowermost extratropical stratosphere due to the latitudinaldependence of isoline of potential temperature.• The “overworld” (≥ 21km).The proposed structure including three latitudinal and three vertical bands results in a “8-box” model if we keep only stratospheric boxes and exclude the uppermost tropical troposphere.Figure 6.2 shows a cartoon of the proposed model structure. Boxes are indexed from 1 to 8from top to bottom and South to North.1166.3. Data and model structure10km16km21km22.5oS 22.5oNSH extra-tropics NH extra-tropicsTropicsOverworldLowermost Tropical StratosphereLowermost Extra-tropical StratosphereTropopauseB1 B2 B3B4 B5 B6B7 B8Figure 6.2: Cartoon showing the proposed 8 boxes for the idealized model, and theirapproximate boundaries and positions relative to the tropopause. The boxes areindexed from bottom to top and left to right.In EVA, gaussian shape functions (in latitude and height) are used to produce latitude-altitude distribution of extinction given mass of aerosols in the three latitudinal boxes. Here, weuse a multilinear regression approach to produce extinction distribution. At each latitude λ andaltitude z, we perform a multilinear regression where the extinction timeserie EXT525(λ, z, t)is the dependent variable and the mean SAOD timeseries of the eight boxes SAODi(t) are theindependent variables:EXT525(λ, z, t) =8∑i=1ci(λ, z)× SAODi(t) + (λ, z, t) (6.1)where i = 1..8 is the number of the boxes, (λ, z, t) is the error, and ci(λ, z) are the regressioncoefficients of box i for latitude λ and altitude z. We perform this multilinear regression usingthe GLOSSAC dataset. The 8 panels of Figure 6.3 show the values of the regression coefficientsfor the 8 boxes as a function of altitude and latitude.1176.3. Data and model structureBox 1Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 2Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 3Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 4Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 5Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 6Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 7Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 8Latitude (oN)-50 0 50Altitude (km)102030Reg. Coef. (km-1 ) 6.3: Regression coefficients ci in Equation 6.1 as a function of latitude and alti-tude. Each panel corresponds to one of the model’s boxes, organized in the sameway as Figure 6.2.Figure E.1 shows that this 8-box model enables to reproduce well the global mean SAODtimeserie of GLOSSAC, for both El Chichón/Pinatubo and the relatively smaller 21st centuryeruptions. Furthermore, we demonstrate that the spatio-temporal structure of extinction inGLOSSAC is also reproduced given knowledge of SAOD in the 8 boxes (Figure 6.4). In par-ticular, the evolution of the vertical and latitudinal structure of the extinction following the ElChichón and Pinatubo eruptions are well reproduced. Many of the patterns associated withsmall 21st century eruptions are broadly reproduced. For these eruptions, some of the detailsof the 3D evolution of the extinction do not perfectly match the original data. However, givenlarge relative uncertainties extinction measurement from satelites, the multilinear regressionapproach does a realistic prediction and justifies further the choice of a 8-box model structure.1186.3. Data and model structureTime TimeSAOD (log)Latitude (oN) “Overwor ld ”  (>21km)Lowermost  t ropica l  st ratosphere  (15.5 -21km)Lowermost  extra -tropica l  st ratosphere  (<15.5km)Northern  Hemisphere  (>22.5 oN)Tropics  (22.5 oS-22.5 oN)Southern Hemisphere  (<22.5 oS)GLOSSAC Reconstructed (8 boxes)SAOD (log)SAOD (log)Ext (km-1, log)Latitude (oN)Latitude (oN)Altitude (km)Altitude (km)Altitude (km)Ext (km-1, log)Ext (km-1, log)Figure 6.4: Comparison of the extinction or SAOD field from GLOSSAC (left column)and reconstructed from the 8-box multilinear regression (Equation 6.1, right col-umn) for different regions of the atmosphere. The top three rows show extinctionas a function of altitude and time for the three latitudinal bands of the model. Thebottom three rows show SAOD as a function of latitude and time for the threevertical bands of the model.1196.3. Data and model structure6.3.3 Model equationsThe model will follow the approach of EVA: the global mean SAOD at 525nm and effectiveradius will be scaled from the total mass of SO4 in the boxes. Transport equations will governthe production, transfer and loss of SO4 particles among the boxes and the latitudinal andvertical distribution of extinction will be derived from the distribution of SO4 mass in the 8boxes. Wavelength-dependent extinction, single scattering albedo, and scattering asymmetryfactor will be deduced from the effective radius and the extinction at 525nm using the Mietheory and assuming a log-normal aerosol size distribution with an effective standard deviationof 1.2.Scalings for SAOD and effective radius will be discussed in Sections 6.4.1 and 6.4.2, respec-tively. As in EVA, we will assume that the evolution of the mass of SO2 in a box i M iSO2 isgoverned by the equation:dM iSO2dt= Si −M iSO2τ iprod, (6.2)where Si is a source term determined from a dataset containing inventory of volcanic SO2emissions (c.f. Section 6.3.4) and τ iprod is an effective timescale for the conversion of SO2 intosulfate aerosols. Accordingly, the production of SO4 in a box i will be of the form:PROD =M iSO2τ iprod. (6.3)As in EVA, we will assume that two-way mixing can occur between two boxes and is pro-portional to the SO4 mass difference between them. The two-way mixing flux from a box i toa box j is:MIXING =M iSO4 −MjSO4τ ijmix, (6.4)where τ ijmix is a mixing timescale.We will also assume that one-way mixing can happen between two boxes, with the flux froma box i to a box j being proportional to the mass of SO4 in box i:OWM =M iSO4τ ijowm, (6.5)where τ ijowm is a one-way mixing timescale. In EVA, one-way mixing terms are used to repre-sent the residual Brewer-Dobson circulation from the tropics to the extra-tropics not accountedfor in the two-way mixing terms. In our “2D” box model, one-way mixing term could addition-ally be used to model the meridional cross-tropopause transport of aerosols at mid-latitudes.Last, for the loss term, a first approach follows EVA’s approach where the loss of aerosol in1206.3. Data and model structurebox i is proportional to the mass of SO4 in the same box:LOSSEV A = −M iSO4τ iloss, (6.6)where τ iloss is a loss timescale. In our model, we will assume that the SO4 loss flux from a boxthat is not in contact with the tropopause (i.e., all boxes except boxes 5, 7 and 8) correspondsto a positive flux for the box located directly below. For example, the loss term in box 1,−M1SO4τ1loss, corresponds to a flux +M1SO4τ1lossin box 4.A second approach for the loss term consists in accounting for a dependence of the losstimescale on the aerosol size distribution. If we assume that an aerosol particle settles accordingto Stokes’s law, then the time required for a particle to settle through a box of specified heightis inversely proportional to the square of the radius of the aerosol particle. If we further assumea log-normal size distribution of the aerosol population, the aerosol size is proportional to theeffective radius. Thus, we can use a loss term of the form:LOSSStokes = −M iSO4 ×R2effτ iloss ×R20, (6.7)where R0 is a non-dimensionalization factor chosen as the minimum value of effective radiuswe impose in the model (cf Section 6.4.2).The general equation governing the evolution of aerosol massM iSO4 in one of the eight boxesi will then be:dM iSO4dt= PROD+MIXING+OWM + LOSS , (6.8)where the production term PROD is governed by Equation 6.3, two-way and one-way mixingterm(s) MIXING and OWM are governed by Equation 6.4 and 6.5 respectively, and the lossterm LOSS is governed by equation 6.6 (EVA-like) or 6.7 (Stokes scaling) and may includepositive terms related to the loss of aerosols in the box located above box i, if any. Note thattimescales τloss, τmix and τowm are not physical timescales and depends on the configuration ofthe 8 boxes of the model.The final configuration of the model depends on the following choices:1. Among which boxes to include contributions for two-way and one-way mixing2. The dependence of the timescales τprod, τloss, τmix and τowm on latitude, altitude andseason (e.g. like mixing timescales in EVA).We further discuss these choices in Section SO2 injection in the modelThe Carn et al. (2016) dataset provides the latitude, date, estimated mass of SO2 and estimatedheight for each reported volcanic SO2 injection into the atmosphere. A simple method to include1216.3. Data and model structureSO2 in the 8-box model is to inject the entire mass into the box which corresponds to the pointdefined by the eruption latitude and estimated injection height. However, in the absence ofa transport equation for SO2 in the model, a more realistic approach is to distribute the SO2spatially instead of injecting 100% of the mass in a single box.To decide for the spatial distribution of injected SO2, we calculate the increase of theextinction at 525nm in GLOSSAC for the 5 months following each eruption in the Carn et al.(2016) dataset. The concentration of sulfate aerosol, and thus extinction, should increase inthe presence of SO2 so that one can assume that the extinction derivative is representative ofSO2 location at the peak of production of aerosols. This assumption ignores the role of sometransport processes, e.g. the radiative lifting (de Laat et al., 2012) or atmospheric circulation(Bourassa et al., 2012), which would shift the height at which extinction increase is observedrelated to the height of SO2 injection. We visually inspected the obtained extinction derivativespatial patterns for all eruptions and various averaging time-windows. An average between0 and 5 months after the eruption produces the clearest SO2 clouds, inferred from regionof extinction increase, at expected latitude and altitude and is consistent with the effectiveproduction timescale of 6 months found in Toohey et al. (2016b) for the 1991 Pinatubo eruption.Some eruptions were discarded due to the absence of a clear pattern, or due to the spatial andtemporal proximity of an eruption with similar or larger SO2 injection. At this stage, theeruption selection is based on visual inspection of individual pattern but we will use a cut-offvalue of the coefficient of determination of the fit in the future.Latitude(oN)-60 -40 -20 0 20 40 60Altitude (km)5101520253035Pinatubo 1991 (GLOSSAC) #10-3246810121416(d Ext)/(dt)Carn et al (2016) eruption locationRegion used for fitLatitude(oN)-60 -40 -20 0 20 40 60Altitude (km)5101520253035Pinatubo 1991 (gaussian fit) #10-3246810121416Latitude(oN)-60 -40 -20 0 20 40 60Altitude (km)5101520253035Calbuco 2015 (GLOSSAC) #10-30.511.52Latitude(oN)-60 -40 -20 0 20 40 60Altitude (km)5101520253035Calbuco 2015 (gaussian fit) #10-30.511.52Figure 6.5: Left: average derivative of extinction at 525nm in GLOSSAC between 0 and5 months after the Pinatubo 1991 (top) and Calbuco 2015 (bottom) eruptions. Thered triangle shows the eruption latitude and estimated height from the Carn et al.(2016) dataset. Right: 2-dimensional gaussian fit (Equation 6.9) of the left plots.The black dashed lines highlight the region used to perform each fit.1226.3. Data and model structureFigure 6.5 (left) shows examples of obtained patterns for the extinction derivative after thePinatubo 1991 and Calbuco 2015 eruptions. Two relatively well-defined regions of extinctionincrease can be identified nearby the estimated injection latitude/altitude from the Carn et al.(2016) dataset. For the Calbuco 2015 eruption, some of the higher values at latitudes ≥ 70oSseem to be unrelated to the eruptions although a more careful analysis would be requiredto confirm this interpretation. For each identified eruption cloud, we choose a region (blackdashed rectangle on Figure 6.5) that contains most of the cloud and excludes signals that maybe associated with other eruptions. This region is determined on the basis of a case by casevisual inspection. We then fit the extinction derivative with a 2D gaussian model:dExtdt= k1 + k2 × e−(λ−λ0)2(∆λ)2 × e−(h−h0)2(∆h)2 . (6.9)Figure 6.5 (right) shows this fit for the Pinatubo and Calbuco eruptions. The chosen fitimplies that the main axis of the 2D gaussian function are horizontal and vertical respectively,which is not perfectly realistic for these two and other eruptions. However, Figure 6.5 showsthat the fitting function chosen captures the location, vertical and latitudinal extent of theregion where extinction increases following the eruption.Figure 6.6: Left scatter plots: estimate of cloud central latitude (oN) and altitude (kmabove sea level, asl) from a gaussian fit of extinction derivative (Figure 6.5 andEquation 6.9) as a function of estimates from the Carn et al. (2016) dataset. Righthistograms: distribution of cloud thickness and latitudinal extent, determined asthe gaussian widths ∆h and ∆λ in Equation 6.9 used to fit the GLOSSAC extinctionderivative.Figure 6.6 (left scatter plots) shows that the central latitude (λ0) and altitude (h0) ofthe cloud are strongly correlated to the latitude and altitude reported in Carn et al. (2016).Although this is not surprising, it is encouraging given the large uncertainties in both datasetsand the simplicity of our approach to identify a SO2 cloud. For the eruption altitude, the centralheight of the GLOSSAC gaussian fit is ' 85% smaller than the altitude estimate provided inCarn et al. (2016). This may be related to systematic errors in one of the two datasets and/or1236.4. Calibration of the modelto issues related to vertical transport of the SO2 (and intermediate species in the reaction chainleading to the production of sulfate aerosols) during the first few months after the eruption.We also show distributions of the gaussian widths ∆h and ∆λ in Equation 6.9, which reflectthe vertical and latitudinal extent of the cloud (Figure 6.6, right). They are not significantlycorrelated to the eruption latitude or height. Thus, when running the box model using theCarn et al. (2016) dataset for SO2 inputs, we distribute the SO2 mass among the boxes usinga 2D gaussian distribution. The center of the gaussian is determined by latitude and altitudeestimate from Carn et al. (2016), and gaussian widths are the average cloud thickness of 2.8km and average latitudinal extent of 13o calculated from Figure Calibration of the model6.4.1 Scaling for global mean stratospheric aerosol optical depthStratospheric sulfur injection (TgS)10-3 10-2 10-1 100 101 102SAOD 525nm increase10-510-410-310-210-1100GLOSSAC and Carn et al 2016 (SO2)GLOSSAC +Carn et al (2016) data 1979-2016power law fit exponent=1 +/-0.2Stratospheric SO4 burden (normalized by maximum value)10-2 10-1 100SAOD 550nm10-310-210-1100WACCMWACCM data 1990-2014power law fit exponent=0.944+/-0.002Figure 6.7: Left: global mean SAOD peak (GLOSSAC) as a function of correspondingstratospheric loadings (Carn et al. (2016)). The SAOD peak is the difference be-tween the peak value of SAOD and the pre-eruptive value. Right: global meanSAOD as a function of the global stratospheric SO4 burden in WACCM. Red linesshow best power law fit in both panels.EVA assumes a linear scaling between the mass of SO4 and SAOD at 550nm, which is calibratedusing data from the 1991 Pinatubo eruption (Toohey et al., 2016b). Here we further test thisscaling using three datasets (Figure 6.7). First, we identify all clear SAOD peaks in the 1979-2016 GLOSSAC SAOD timeseries (we found 20 of them) and stratospheric loading(s) in theCarn et al. (2016) dataset that likely correspond to these peaks. We fit the difference betweenthe SAOD peak and the SAOD before the associated eruption(s) as a function of the total SO2loading using a power law (Figure 6.7, left). We find an exponent of 1± 0.2 which supportsa linear scaling, at least for eruptions up to Pinatubo size (' 18 Mt of SO2). Secondly, we1246.4. Calibration of the modelcalculate the total SO4 burden and global mean SAOD at 550 nm timeseries in the 1990-2014WACCM run. We also fit them using a power-law fit and find an exponent of 0.944± 0.02which also supports a linear scaling (Figure 6.7, right).A linear scaling for SAOD thus appears reasonable, even when tested against eruptions otherthan Pinatubo and with WACCM. It also considerably simplifies the model and the calibrationof its parameters (cf Section 6.4). Consequently, we assume the following scaling between theglobal mean SAOD at 525nm SAOD525 and the total SO4 burden MSO4:SAOD525 = A×MSO4 , (6.10)where A is a constant in TgS− Scaling for global mean aerosol effective radiuslag (month, >0 when SAOD leads)-5 0 5 10Correlation0. SAOD-REff lagentire period1996-20141990-1996SAOD10-2 10-1Reff (7 m, lagged by 2 months) SAOD-REff scaling1990-19961996-2014Power-law fit exponent = 0.31 +/- 0.01 (all data)lag (month, >0 when SAOD leads)-5 0 5 10Correlation0.70.750.80.850.90.95SAGE4 6  (1985-2005) SAOD-REff laglog(SAOD)10-2 10-1Reff (7 m, lagged by one month) 6  (1984-2005) SAOD-REff scalingDataPower-law fit exponent = 0.28 +/- 0.01Figure 6.8: Top panels show analyses with WACCM and bottom panels with SAGE4λ.Left panels show the correlation between SAOD and effective radius (global mean,weighted by the number of aerosol particles) as a function of the time lag betweenthe two timeseries. Right panels show a scatter plot of the effective radius vs.SAOD when applying the lag for which the maximum correlation is found. Thecontinuous line show the best power-law fit. For the top panels (WACCM), greencrosses and red triangles show data for the Pinatubo and post-Pinatubo periodrespectively.EVA uses a 1/3 power law between the effective radius Reff and the SO4 burden, which hasa physical foundation detailed in Toohey et al. (2016b). With the linear scaling between SO41256.4. Calibration of the modelburden and SAOD (Equation 6.10), this yields a 1/3 power law between SAOD and Reff . Thereare two potential reasons for modifying the scaling used in Toohey et al. (2016b): i) there seemsto be a time lag between SAOD and Reff peak for the Pinatubo eruption; and ii) the e-foldingdecay timescale is different for Reff and SAOD for Pinatubo.For both the SAGE4λ and WACCM datasets, we investigate the time lag and optimalscaling between SAOD and the global mean (weighted by number of SO4 particles) effectiveradius (Figure 6.8). Toohey et al. (2016b) use the maximum value of Reff which explains somesmall differences in effective radius values among our results and their study. We use SAGE4λinstead of GLOSSAC because Thomason et al. (2018) do not provide estimates for the aerosolsize distribution or effective radius because these parameters are not directly measured andcan only be derived from extinction observations. We only use the period of the SAGE2instrument (1985-2005) because other periods use scalings calibrated on SAGE2 to derive aneffective radius.For WACCM, there is an optimal lag of 2 months between the SAOD and effective radiustimeseries when SAOD leads(top left panel of Figure 6.8). The same optimal lag is found whenanalyzing only the Pinatubo (1990-1996) or post-Pinatubo (1996-2014) period. For SAGE4λ,the optimal lag is one month (bottom left panel of Figure 6.8) although the correlation is closeto maximum value for 0 and 2 months. When fitting a power law to the data (to which weapply the optimal lag), we find an optimal exponent of 0.31± 0.01 and 0.28± 0.01 for WACCMand SAGE4λ respectively (bottom panels of Figure 6.8). A 1/3 power law is thus appropriateand we use the following scaling for the effective radius (µm):Reff (t) = 0.8× SAOD525(t− 2)1/3 , (6.11)where t is the time in months and the constant 0.8µm is found from fits with WACCM andSAGE4λ in Figure 6.8. With minimum values of Reff of 0.11 and 0.14 µm in the WACCMand SAGE4λ timeseries, we set the minimum value of the effective radius to 0.15 µm (Tooheyet al. (2016b) use 0.2µm) .Overall, for both SAOD and effective radius, the scalings we use do not differ much fromEVA. However, we gained confidence that:• these scalings are supported by state-of-the-art datasets• these scalings are supported by coupled chemistry-climate model results• these scalings are applicable to “small” eruptions (and El Chichón 1982), whereas thecalibration of EVA is based on the 1991 Pinatubo eruption.6.4.3 Model timescalesAll time constants potentially involved in the right-hand side of Equation 6.8 as well as thefactor A involved in the SAOD scaling (Equation 6.10) must be calibrated. For example, for1266.4. Calibration of the modelbox 1, we have to find the parameters A, τ1loss, τ1prod , τ12mix and τ12owm best reproducing theSAOD timeserie in box 1 given the equation:dM1SO4dt=M1SO2τ1prod+M2SO4 −M1SO4τ12mix+M2SO4τ12owm− M1SO4 ×R2effτ1loss ×R20, (6.12)and assuming that: i) box 1 only mixes with box 2; ii) there is a one-way mixing term beingfrom box 2 to box 1 (similar to the residual Brewer-Dobson circulation term in EVA); and iii)the loss term follows a Stokes scaling (i.e., Equation 6.7 instead of 6.6) .Given the linear scaling for the global mean SAOD, each box i follows the scaling wi ×SAODi = A×M iSO4 where A is the same constant as in Equation 6.10 and wi are knownweights that can be calculated from the latitudinal extent of each box. Furthermore, if weassume that τprod is the same in each box and does not depend on season, the mass of M iSO2 ina box i at any time t is given by:M iSO2(t) =∑k,tk≤tM ike− t−tkτprod , (6.13)where k is an index representing eruptions in the Carn et al. (2016) dataset, tk the dateof the k-th eruption, and M ik the mass of SO2 injected by eruption k in box i calculated asdescribed in Section 6.3.4. Let wSAODi = wi × SAODi be the weighted SAOD in each box.Equation 6.12 can then be written:dwSAOD1dt=A×M1SO2τprod+wSAOD2 −wSAOD1τ12mix+wSAOD2τ12owm− wSAOD1 ×R2effτ1loss ×R20, (6.14)where τprod is specified and wSAOD andMSO2 are obtained from GLOSSAC and Carn et al.(2016). Consequently, calculating the time derivative with the Euler method, the calibration ofmodel parameters is a simple least-squared linear problem with 8 boxes × (N -1) observations(with N=456 the number of months in the GLOSSAC dataset) and as many independentvariables as model parameters. For example, the value of the independent variable associatedwith the model parameter 1τ1lossis −wSAOD1×R2effR20for the dependent variable values associatedwith box 1 and +wSAOD1×R2effR20for those associated with box 2. As the value of the independentvariables is dependent on the specified value of τprod, we solve the linear least-square problemfor values of τprod between 2 and 12 months with increments of 0.5 months. We then choose thevalue of τprod and associated solution minimizing the coefficient of determination of the multiplelinear regression. Potential seasonal variations of each model parameter can be investigated byusing observations belonging only to specified months in the multiple linear regression, andlooking for seasonal trends after solving optimal coefficient values in each box.We initially tested relatively complex models where each timescale depends on latitude andaltitude, i.e. with generally 20 or more model parameters and as many independent variables1276.4. Calibration of the modelin the multilinear regression. With the exception of a few coefficients (A and τloss for the threebottom boxes), the parameters were poorly constrained, with 95% confidence interval admittingboth positive and negative values for time constants. Thus, we instead started from a muchsimpler model. We assume that: i) there are no one-way mixing terms; ii) loss timescalesdepend on altitude only (i.e., three τloss); and iii) mixing timescales depend on latitude onlywith no mixing between the extra-tropical lowermost stratosphere (boxes 7 and 8) and thetropical lowermost stratosphere (box 5) (i.e., two τmix). SO4 fluxes in this configuration areillustrated in Figure 6.9.NH extra-tropicsSH extra-tropics TropicsOverworldTrop. LTExtra-Trop. LTProduction:   𝑀𝑆𝑂2𝑖𝜏𝑝𝑟𝑜𝑑Mixing:𝑀𝑆𝑂4𝑖 −𝑀𝑆𝑂4𝑗𝜏𝑚𝑖𝑥Loss:𝑀𝑆𝑂4𝑖 𝑅𝑒𝑓𝑓2𝜏𝑙𝑜𝑠𝑠𝑅02 or 𝑀𝑆𝑂4𝑖𝜏𝑙𝑜𝑠𝑠𝐺𝑙𝑜𝑏𝑎𝑙 𝑚𝑒𝑎𝑛 𝑆𝐴𝑂𝐷 = 𝐴 × 𝑀𝑆𝑂4𝑡𝑜𝑡𝑅𝑒𝑓𝑓 = 0.8 × 𝑆𝐴𝑂𝐷(𝑡 − 2)1/3 𝝉𝒎𝒊𝒙𝟏𝝉𝒎𝒊𝒙𝟏𝝉𝒎𝒊𝒙𝟐𝝉𝒎𝒊𝒙𝟐𝝉𝒍𝒐𝒔𝒔𝟏 𝝉𝒍𝒐𝒔𝒔𝟏 𝝉𝒍𝒐𝒔𝒔𝟏𝝉𝒍𝒐𝒔𝒔𝟐𝝉𝒍𝒐𝒔𝒔𝟐𝝉𝒍𝒐𝒔𝒔𝟐𝝉𝒍𝒐𝒔𝒔𝟑𝝉𝒍𝒐𝒔𝒔𝟑Figure 6.9: Same as Figure 6.2, but showing the SO4 fluxes between the boxes for a simplemodel where the mixing timescale τmix depends only on latitude with horizontalmixing only, the loss timescales τloss depends only on altitude, the productiontimescale τprod is independent on latitude/altitude, and without one-way mixing.Starting from this simplified model, and trying to increase its complexity, we came to thefollowing conclusions based on the multilinear regression approach (Figure 6.10)• The coefficient of determination of the multilinear regression (R2) is not significantlyimproved when adding one-way mixing terms to represent the residual Brewer-Dobsoncirculation.• R2 is not significantly improved when adding diagonal mixing terms between the lower-most stratospheric boxes (7, 5 and 8) (Figure 6.10).• When using a settling timescales scaling following Stokes law, R2 is not significantlyimproved when adding one-way mixing term between the extra-tropical lowermost strato-spheric boxes (7 and 8) and the tropical troposphere.• Weighting or not the mixing term by the volumes of the boxes (i.e., having a mixing1286.4. Calibration of the modelproportional to concentration difference instead of mass difference) does not significantlychange R2 (Figure 6.10).• R2 is slightly improved when including a Stokes scaling (i.e., loss term governed by Equa-tion 6.7) instead of a simple EVA-like scaling (i.e. loss term governed by Equation 6.6)(Figure 6.10).• None of the coefficients involved in the multilinear regression (i.e. all except τprod) exhibita clear seasonal cycle; in addition, mixing timescales are poorly calibrated and sometimenegative when using one or a few selected months to investigate a potential seasonal cycle.=prod (month)0 2 4 6 8 10 12Coefficient of determination R20.250.30.350.40.450.5stokes=0, weight=0, diag=0stokes=0, weight=1, diag=0stokes=1, weight=0, diag=0stokes=1, weight=1, diag=0stokes=0, weight=0, diag=1stokes=0, weight=1, diag=1stokes=1, weight=0, diag=1stokes=1, weight=1, diag=1Figure 6.10: R2 of the multilinear regression for predicting the derivative of weightedSAOD in all boxes, as described earlier in this section, as a function of the value ofthe production timescale τprod (in months). The model corresponds to the simpleversion shown in Figure 6.9, with different symbols corresponding to differentadditional complexity: (i) Equation 6.7 (Stokes=1) or Equation 6.6 (Stokes=0) forthe loss term; (ii) with (weight=1) or without (weight=0) weighting mixing termsby box volumes to get a mixing flux proportional to difference in concentrationinstead of mass; and (iii) with (diag=1) or without (diag=0) non-horizontal mixingbetween box 5 (lowermost tropical stratosphere) and boxes 7 and 8 (lowermostextra-tropical stratosphere).Consequently, until we can demonstrate that increased model complexity leads to significantimprovements of SAOD predictions, we will use the model defined by the cartoon of Figure 6.10with Equation 6.7 (Stokes scaling) or Equation 6.6 (EVA-like) governing the loss terms.1296.4. Calibration of the modelColumns 2 and 3 of Table 6.1 show the best estimates and 95% confidence intervals of modelparameters obtained from the multilinear regression approach described previously. Overall,all parameters but the mixing timescales are relatively well constrained. The values of mixingtimescales are surprisingly low (' twice smaller) compared to optimal values found by Tooheyet al. (2016b). Loss timescales are ' 5 times larger when using a Stokes-like scaling for theloss term, which is expected as the factor Reff2R20in this scaling varies between 1 and 10 withR0 = 0.15µm. The value of the coefficient of determination R2 is larger by 0.02 when using theStokes-like scaling for the loss terms.Parameter MLR (no Stokes) MLR (Stokes) MCS (no Stokes) MCS (Stokes)τprod(months) 4 (3-7) 4.5 (3-7) 7.38 6.74A(×102)(TgS−1) 1.06 (1.02-1.11) 1.09 (1.04-1.14) 2.07 1.91τ1loss(months) 7.5 (7-8) 38 (36-41) 4.58 25.07τ2loss(months) 10 (9-11) 50 (46-54) 8.14 31.15τ3loss(months) 10 (9-11.5) 48 (43-54) 2.10 13.63τ1mix(months) 26 (21-37) 24 (19-33) 3.89 8.77τ2mix(months) 40 (28-70) 41 (28-74) 4.48 6.24R2 0.44 0.46 0.7022 0.5863Table 6.1: Optimal parameter values for the model of Figure 6.9 with or without a Stokes-like scaling in the loss term. For time constants, all values are given in months. Thevalue of A is in TgS−1. Columns 2 and 3 show optimal values and 95% intervalsobtained from multilinear regression (MLR) predicting the derivative of SAOD ineach box from SAOD and the mass of SO2. Columns 4 and 5 show refined valuesfrom Monte Carlo simulations (MCS) where we minimize the percentage of explainedSAOD variance to determine optimal model parameters and only use SO2 inputs topredict SAOD.The multilinear regression predicts the derivative of SAOD in each box using knowledge ofSAOD and the mass of SO2 in all boxes at the same time step. In contrast, we seek to minimizethe error on predicted SAOD. Consequently, in a second phase, we use Monte Carlo simulationsto obtain optimal model parameters.One Monte Carlo simulation consists in randomly sampling the value of all model parametersin a range corresponding to 95% confidence intervals from the multilinear regression, adapted toinclude values used in Toohey et al. (2016b) or if the range explored appeared insufficient afterprelimnary tests. We run 320000 simulations and find the parameter combination maximizingthe mean percentage of explained variance of the weighted SAOD (R2) in the 8 boxes (weightedby the variance of each box).Optimal parameter values are reported in columns 4 and 5 of Table 6.1. Figures E.2 andE.3 show that parameter values for which the highest values of R2 are obtained fall in the rangetested for each parameter. They also confirm that timescales governing the production and loss1306.4. Calibration of the modelof aerosols are relatively well constrained, with a clear peak in the distributions. There are alsopeaks in the distribution of mixing timescales, but values differing by one order of magnitudeenable to obtain a similar R2.There are significant differences in parameter values constrained from the multilinear regres-sion approach vs. the Monte Carlo approach (Table 6.1). In addition, up to 10% more SAODvariability is explained when using an EVA-like (Equation 6.6) instead of Stokes (Equation 6.7)parameterization for loss timescales, which contradicts result of the multilinear regressions.These contradictions are not necessarely surprising as the two approaches are not directly com-parable:• We maximize the percentage of explained variance on SAOD in the Monte Carlo simula-tions instead of the time derivative of SAOD in the multilinear regression approach.• In the multilinear regression, we use SO2 inputs and observed SAOD at a given timestept to predict the SAOD derivative at the same timestep. The Monte Carlo simulationsonly use SO2 inputs.• When using the idealized model of aerosol with the Stokes scaling for the loss term inthe Monte Carlo approach, errors on SAOD predicted from the only SO2 inputs result inerrors on predicted Reff which will affect the loss term via Equation 6.7 and propagateto the next time step.Given the better SAOD prediction using the EVA-like parameterization, we will thus use Equa-tion 6.6 for the loss term in the rest of this chapter, and until we are able to improve the model.Note that the preliminary calibration using a multilinear regression approach is important todetermine appropriate model complexity and ranges for model parameters with a computation-ally cheap method. As an example, starting with a complex model with 20 different timescales,' 1010 Monte Carlo simulations would be necessary to explore 3 values for each parameters.6.4.4 Model performanceFigure 6.11 shows the weighted SAOD in GLOSSAC and as predicted by the new idealized modelof aerosol forcing in the 8 boxes of the model. In most boxes, at least 80% of the SAOD varianceis explained by the model, with excellent predictions for the vertical band corresponding to thelowermost tropical stratosphere (R2 ≥ 0.90). The model explains only 56% of the variance inthe lowermost stratosphere of northern high-latitudes (box 8). In particular, it underestimatesthe SAOD peak associated with the El Chichón 1982 eruption and overestimate the SAODpeaks associated with the Kasatochi 2008 and Sarychev 2009 eruptions.1316.4. Calibration of the model1980 1990 2000 2010Weighted SAOD10-310-2Box 1, R2=0.81GLOSSACNew idealized model1980 1990 2000 2010Weighted SAOD10-310-2Box 2, R2=0.91980 1990 2000 2010Weighted SAOD10-310-2Box 3, R2=0.791980 1990 2000 2010Weighted SAOD10-310-2Box 4, R2=0.941980 1990 2000 2010Weighted SAOD10-310-2Box 5, R2=0.941980 1990 2000 2010Weighted SAOD10-310-2Box 6, R2=0.91980 1990 2000 2010Weighted SAOD10-310-2Box 7, R2=0.891980 1990 2000 2010Weighted SAOD10-310-2Box 8, R2=0.56Figure 6.11: SAOD in the 8 boxes of the model (Figure 6.9). The black lines show ob-served SAOD (GLOSSAC) and the red lines show the model prediction using theCarn et al. (2016) volcanic SO2 injection inventory. The coefficient of determina-tion (R2) is reported for each box.Figure 6.12 shows the global mean SAOD for GLOSSAC and as predicted by the new model,which explains 97% of the variance. The most notable discrepencies between the model andGLOSSAC are:• The peaks associated with the 1982 El Chichón and 2011 Nabro eruptions, which areslightly underestimated.• The peak associated with the Kasatochi 2008 eruption which is largely overestimated.• For several periods, e.g. 1985-1991, 2000-2008 and 2014-2016, the global mean SAODis underestimated by the model but the amplitude of the peaks associated with majoreruptions is in good agreement with GLOSSAC.These problems may be related to: i) errors in GLOSSAC, e.g. underestimation of SAODin the lowermost stratosphere, Mills et al. (2016); ii) errors in the Carn et al. (2016) volcanicSO2 injection inventory, e.g. error on erupted mass and plume height or missing events; and iii)to the model itself (e.g. physical prameterizations or absence of non-volcanic source of SO2).However, given the simplicity of the model and the large uncertainties in the input dataset(Carn et al., 2016) and observations for the output (GLOSSAC, Thomason et al. (2018)), thepercentage of explained variance for the global mean SAOD is satisfying.1326.4. Calibration of the modelTime1975 1980 1985 1990 1995 2000 2005 2010 2015 2020Global mean SAOD (525nm)10-210-1GLOSSACNew model, R2=0.97Figure 6.12: Global mean SAOD from GLOSSAC (blue) and predicted by the new model(red).Figure 6.13 shows the shape function used to convert SAOD predictions in the 8 boxesinto 3D (time, latitude, altitude) extinction predictions. As in Section 6.3.2 (Equation 6.1 andFigure 6.3), we use a multilinear regression approach to determine these shape functions. Wenormalize the shape functions by their integral over the entire latitude-altitude domain, so thatthe global mean SAOD as calculated from the sum of weighted SAOD in the 8 boxes is equalto the integral of extinction.Latitude (oN)-50 0 50Altitude (km)102030Box 1Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 2Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 3Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 4Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 5Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 6Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 7Reg. Coef. (km-1 ) (oN)-50 0 50Altitude (km)102030Box 8Reg. Coef. (km-1 ) 6.13: Shape functions used to transform the timeseries of weighted SAOD in the8 boxes into a 3D extinction field. These shape functions are determined using amultilinear regression approach (Equation 6.1) as done for Figure 6.4.1336.4. Calibration of the modelTime TimeSAOD (log)Latitude (oN) ͞Overwor ld ͟  ;>ϮϭkmͿLowermost  t ropica l  st ratosphere  (15.5 -21km)Lowermost  extra -tropica l  st ratosphere  (<15.5km)Northern  Hemisphere  (>22.5 oN)Tropics  (22.5 oS-22.5 oN)Southern Hemisphere  (<22.5 oS)GLOSSAC New modelSAOD (log)SAOD (log)Ext (km-1, log)Latitude (oN)Latitude (oN)Altitude (km)Altitude (km)Altitude (km)Ext (km-1, log)Ext (km-1, log)Figure 6.14: Same as Figure 6.4, but with the right column showing the prediction ofthe new model using the Carn et al. (2016) volcanic SO2 injection inventory.1346.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingLast, Figure 6.14 shows the time-altitude extinction in the three latitudinal bands of themodel or the time-latitude SAOD in the three vertical bands of the model for GLOSSAC andfor the model prediction. Whereas previous idealized model of volcanic forcing would have thesame vertical structure at all time, our new model is able to capture the broad evolution of thevertical structure of extinction following the El Chichón and Pinatubo eruptions, but also forsmaller eruptions of the 21st century. It is also able to broadly capture the time evolution of thelatitudinal structure of SAOD as previous models. Overall, most of the discrepencies betweenGLOSSAC and the model predictions in Figure 6.14 are related to problems already noted forthe global mean SAOD, e.g. an underestimation of the SAOD peak following the 1982 eruptionof El Chichón or an overestimation of the one associated with the Kasatochi 2008 eruption.6.5 Understanding the impact of plume height onstratospheric volcanic sulfate aerosol forcing6.5.1 Sensitivity of volcanic forcing to injection height and latitudeBefore applying the new idealized model of volcanic forcing to design scenarios for future vol-canic forcing and quantify the feedback proposed in Aubry et al. (2016), we apply the modelto test the sensitivity of the forcing to the height and latitude of a volcanic injection of SO2into the atmosphere. Such sensitivity studies to eruption latitude and altitude with coupledchemistry-climate models are computationally expensive and, consequently, nearly absent fromthe literature to my knowledge.Injection latitude (oN)-50 0 50Injection altitude (km asl)10152025SAOD peakNormalized SAOD peak0. latitude (oN)-50 0 50Injection altitude (km asl)10152025SAOD e-folding timeSAOD e-folding time (months)1012141618Figure 6.15: Global mean SAOD peak (left) and e-folding time (right) as a function ofthe latitude and height of a 1000 kt injection of SO2 by a volcanic eruption inMarch. SAOD peak values are normalized by the maximum value obtained, andthe normalized value does not depend on the injected mass of SO2. The red lineshows the mean tropopause height as a function of latitude.Not surprisingly, we find that SAOD peak and e-folding timescales are smaller for lowermost-stratospheric eruptions. More surprisingly, for eruptions which inject SO2 above the height of1356.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingthe tropical tropopause, the forcing is roughly independent of the latitude of injection. Tofurther illustrate this important result, we show the time-latitude evolution of SAOD for aneruption injecting 10000 kt of SO2 in the stratosphere at: i) 0oN and 20 km above sea level(a.s.l) (Figure 6.16, top); ii) 60oN and 20 km a.s.l (Figure 6.16, center); and iii) 60oN and13 km a.s.l (Figure 6.16, bottom). Both injections at an altitude of 20 km result in a globalincrease of extinction and have a comparable global mean forcing (Figure 6.15).Time (month)5 10 15 20 25 30 35 40 45 50 55 60Latitude(oN)-5005010000kt of SO2 at 0oN and 20km a.s.lSAOD-2-1.5-1Time (month)5 10 15 20 25 30 35 40 45 50 55 60Latitude(oN)-5005010000kt of SO2 at 60oN and 20km a.s.lSAOD-2-1.5-1Time (month)5 10 15 20 25 30 35 40 45 50 55 60Latitude(oN)-5005010000kt of SO2 at 60oN and 13km a.s.lSAOD-2-1.5-1Figure 6.16: SAOD as a function of time and latitude for three injections of 10000kt ofSO2 at different altitude and latitude labeled above each panel.This result is in agreement with a recent set of simulations with the coupled chemistry-climate model MAECHAM (Matthew Toohey, personal communication) but contradicts resultsobtained with global climate models in which a forcing is prescribed from previous idealizedvolcanic aerosol forcing model. For example, Oman et al. (2005) tested the climate response tothe 1912 Katmai eruption, but the forcing implemented in the model is derived from an idealizedmodel of volcanic forcing (Amman et al., 2003) that does not include a vertical structure andconfines aerosols of high-latitude eruptions to high latitudes. On the basis of Figures 6.15 and6.16, the results of their study would likely be different if they use our new idealized modelof volcanic forcing given the constrained plume height of 22-26 km a.s.l. for Katmai 1912(Bonadonna and Costa, 2013). In contrast, in EVA, SAOD latitudinal spread for high-latitudeeruptions is not confined to high-latitudes but the forcing is also independent of plume height.Thus, EVA would predict the same forcing for the high-latitude injections at 13km and 18km1366.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingshown on Figure 6.16, while our new model clearly predicts that SAOD increase is confinedto high-latitudes when the injection altitude is smaller than the tropical tropopause. Our newmodel thus has critical implications for climate models that do not include a prognostic aerosolsscheme and must use prescribed volcanic forcing.6.5.2 Scenarios for future stratospheric volcanic sulfate aerosol forcingWe apply our new idealized model to generate scenarios for future volcanic forcing. Our method-ology mostly follows Aubry et al. (2016). We first generate scenarios for eruptive source condi-tions at the vent (i.e., not including plume height). We assume that future eruptive conditionswill be statistically similar to those observed for the past 2500 years and resample these con-ditions from the Sigl et al. (2015) dataset (stratospheric sulfur injections from ice-core for thepast 2500 years) and Carn et al. (2016) (satellite observations of sulfur injections since 1979).Aubry et al. (2016) (Figure D.3) show that eruptions injecting ≤ 100.5 Mt of SO2 and ≥ 100.5Mt of SO2 seem better represented in the Carn et al. (2016) and Sigl et al. (2015) datasets,respectively. Thus, we filter eruptions above or below this mass threshold in these datasetsaccordingly. In addition, for the Carn et al. (2016) dataset, we calculate the volume flow rateat the vent (which is the product of the mass eruption rate over the density of the columnat vent level) for which a 1D model of volcanic plume predicts the same plume height as theinjection height reported by Carn et al. (2016) using observed atmospheric conditions duringthe eruption. The 1D model uses a wind entrainment coefficient of β = 0.4 and neglect con-densations following the results of Chapters 3 and 4. In Carn et al. (2016), we filter out alleruptions for which the ratio of the plume altitude to tropopause height H∗ is smaller than0.5, because such eruptions can likely not have significant stratospheric injections regardless ofclimate conditions.To generate scenarios for eruptive condition at the vent for 100 years:• We randomly re-sample an equivalent of 100 years of eruptive events, from both the Carnet al. (2016) and Sigl et al. (2015) datasets.• For the re-sampled erupted mass of SO2 and volume flow rate, we further multiply themby a random (on logarithmic scale) factor between 10−0.3 and 100.3. This enables to notreproduce the exact same eruptive conditions as the dataset we resample from, but topreserve the order of magnitude of mass and volume flow rates re-sampled, and thus themagnitude-frequency distribution for these two parameters.• The location of each event (i.e. vent height and latitude) is the one corresponding tothe event originally re-sampled. For events sampled from the ice-core dataset, Sigl et al.(2015) estimate whether the eruption occured in the tropics, southern high-latitude ornorthern high-latitudes. However, they cannot attribute a specficic volcano to each eventdetected in the ice-core. We then re-sample a random location in the Carn et al. (2016)1376.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingamong locations belonging to the altitude range attributed to the event sampled from theSigl et al. (2015) dataset.• We attribute a random date in the 100 year period to each eruptive event resampled.Mass of SO2 (kg)106 107 108 109 1010 1011Volume flow rate (m3 .s-1 )105106107108109Carn et al 2016 (1<H*<1.2)Carn et al 2016 (H*>1.2)Power-law fit and 95% confidence bound for Carn et al 2016Applied relationship with upper/lower bound for Sigl et al (2015)Figure 6.17: Volume flow rate as a function of the erupted mass of SO2 for eruptionswith H∗ ≥ 1 in the Carn et al. (2016) dataset. Blue dots and black diamondsshow events for which H∗ ≤ 1 and H∗ ≥ 1.2 respectively. The black continuousand dashed lines show the best power law fit and 95% confidence interval. Redlines show the same applied to the range of erupted mass of SO2 for events in theSigl et al. (2015) dataset, and with lower and upper limits for the 95% bounds asdiscussed in the text.Injection height cannot yet be estimated from ice cores and we thus do not have volumeflow rate estimate for the Sigl et al. (2015) dataset. Consequently, we find the best power-lawfit between the erupted mass of SO2 and the volume flow rate for stratospheric eruptions ofthe Carn et al. (2016) datasets and the 95% confidence bound for the fit prediction. To obtaina volume flow rate for the events resampled from the Sigl et al. (2015) dataset, we samplea value within the 95% prediction of this fit for the resampled value of the erupted mass ofSO2. We impose a higher bound of 2× 108 m3.s−1 for the obtained volume flow rates becauseeruptions with such rate will reach the higher stratosphere regardless of climate conditions.Higher volume flow rates can be a problem when applied in the plume model, e.g. because thesimulated plume will rise higher than the maximum altitude of the input atmospheric data.We also impose a lower bound of 106.5 m3.s−1 for the volume flow rate of events resampledfrom the Sigl et al. (2015) dataset. For such values, the majority of injection heights in theCarn et al. (2016) are 20% higher than the tropopause height. Erupted mass of SO2 of the Siglet al. (2015) datasets likely correspond to the mass erupted in the stratosphere as the e-foldingtime of tropospheric aerosols would not allow them to be deposited at the poles except foreruptions occuring nearby the poles. Thus, the chosen lower bound for the volume flow rate1386.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingensures that injections from the Sigl et al. (2015) datasets are stratospheric, for atmosphericCO2 concentrations equivalent or smaller than those during 1979-2015.0 10 20 30 40 50 60 70 80 90 100Latitude(oN)-50050100Example of scenario for eruptive conditions at vent levelVFR (m3.s-1, color):105106107108SO2 (kg,size):105.5107108.51010H* (color):0.751.251.752.250 10 20 30 40 50 60 70 80 90 100Latitude(oN)-50050100Corresponding scenario for atmospheric injections under 1981-2000 climate0 10 20 30 40 50 60 70 80 90 100Latitude(oN)-50050Corresponding scenario for SAODSAOD-3-2.5-2-1.5-1-0.5Time (year)0 10 20 30 40 50 60 70 80 90 100SW forcing (W.m-2 )10-2100Corresponding scenario for radiative forcingFigure 6.18: Example of scenario generated for volcanic forcing. The top panel shows thescenario obtained for vent-level eruptive conditions using the resampling strategydescribed in this section. The mass of SO2 (marker size) and volume flow rate(marker color) of each event are shown as a function of time and latitude. Thesecond panel shows the corresponding scenario for atmospheric injections, withmarker color indicating the ratio of injection height to tropopause height H∗.This scenario is obtained by using the scenario for vent-level conditions as inputof a 1D plume model, with atmospheric conditions from the MPI-ESM-LR modelfor the 1981-2000 period in CMIP5 historical experiment. The third panel showsthe time-latitude evolution of SAOD obtained by inputing the scenario for at-mospheric injections in the new idealized model of stratospheric volcanic aerosolforcing. The last panel shows the corresponding short-wave radiative forcing (ab-solute value) as a function of time.Figure 6.18 (top) shows the time, latitude, volume flow rate and erupted mass of SO2 for allevents of one of the eruptive scenario generated. Next, we proceed as Aubry et al. (2016) anduse a 1D model of volcanic plume to predict the height of atmospheric injections correspondingto scenarios:• The volume flow rates are taken from scenarios generated for eruption source conditions;1396.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingwe assume that the erupted mixture has a gas content of 4w.t% and a temperature of1200K.• Atmospheric conditions are obtained from the MPI-ESM-LR global climate model (Gior-getta et al., 2013), which performance relative to other CMIP5 models are discussed inAubry et al. (2016), and which has the best vertical resolution for outputs among themodels used in Aubry et al. (2016).• We specify a 20-year period (starting in 1981, 2081 or 2281) and CMIP5 experiment(historical, RCP2.6, RCP4.5 or RCP8.5) as done in Aubry et al. (2016). For each re-sampled eruptive event, we then randomly choose a day of the 20-year period and useatmospheric conditions of this day and the eruptive region closest to the re-sampledlocation among the 13 regions defined in Aubry et al. (2016).• We run the plume model with these source and atmospheric conditions for each re-sampledevent, with model parameterization being the same as the one used to infer the volumeflow rate of eruptions in the Carn et al. (2016) dataset from observed plume height andatmospheric conditions (Figure 6.17).Figure 6.18 (second panel) shows an example of obtained scenario for “atmospheric injec-tions”, i.e. the time, latitude, erupted mass of SO2, and height of injection relative to tropopauseH∗ for each eruptive event re-sampled.Next, we use the scenarios for atmospheric injections as inputs to the new idealized modelof volcanic forcing. For each eruptive event, SO2 is spread among the 8 boxes as describedin Section 6.3.4 but adapting the lower vertical boundaries of boxes 7, 5 and 8 to match thetropopause height corresponding to the same event. This enables to account for potentialchanges in plume height and tropopause height. Figure 6.18 (third panel) shows the obtainedSAOD as a function of time and latitude.Last, we calculate the monthly global mean stratospheric aerosol optical depth at 525nm.The corresponding short-wave radiative forcing F at the top of the atmosphere can be roughlyestimated from a linear scaling:F = k× SAOD525nm . (6.15)Estimates for the coefficient k varies between -15 and -30Wm−2 for SAOD calculated atwavelengths between 500 and 550nm (e.g. Lacis et al. (1992); Sato et al. (1993); Ramachandranet al. (2000); Hansen et al. (2002)). Here we will use the recent estimate of -23Wm−2 byAndersson et al. (2015) which is also consistent with data published in Santer et al. (2014, 2015).The last panel of Figure 6.18 shows the radiative forcing timeseries derived from Equation 6.15.In this project, we choose to design 100-year volcanic forcing scenario by resampling eruptiveconditions from:1406.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcing• Ice-core data (Sigl et al., 2015), in which large eruptions are well-preserved but smallerinjections of sulfur are not detected.• Satellite observations, which include small volcanic injections of sulfur into the strato-sphere, but for which large eruptions are poorly represented because only 39 years ofobservations are available.The short period of satellite observations is an important limitation to design 100-yearlong scenario as we have little to no understanding of variability in atmospheric injections(magnitude, frequency and location) by relatively small volcanic eruption at such timescale.However, this approach is reasonable considering that the return period of eruptions injecting≤ 100.5 Mt of SO2 into the atmosphere is much smaller than the duration of satellite observationsavailable.Stratospheric SO2 injections (Mt.yr-1)0 0.5 1 1.5 2 2.5 3Probability00.10.20.3Resampling from ice-core and satelitte dataResampling from ice-core data onlyMean forcing (W.m-2)-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0Probability00. of SO2 mass neglected when resampling from ice-core data only0 20 40 60 80 100Probability00. of forcing neglected when resampling from ice-core data only0 20 40 60 80 100Probability00. 6.19: Comparison of the probability distribution for the flux of volcanic SO2 intothe stratosphere (left) and mean volcanic forcing (right) obtained when designing100-year volcanic scenario with resampling from: i) ice-core and satelitte data(blue histograms on top panels); and ii) ice-core data only (red histograms ontop panels). The bottom panels show the probability distributions for the relativeerror on volcanic SO2 into the stratosphere (left) and mean volcanic forcing (right)when resampling from the ice-core dataset only.In contrast to Aubry et al. (2016) and this study, Bethke et al. (2017) choose to use onlythe Sigl et al. (2015) ice-core dataset to design scenarios of volcanic forcing. Their pioneeringwork leads the way to the inclusion of realistic forcing in global climate model projections andto quantifying uncertainties related to this forcing. However, an important question is how theinclusion of relatively smaller eruptions would impact their results. Figure 6.19 investigates howthe inclusion of satelitte observations in resampling strategy affects the probability distribution1416.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingfor the flux of volcanic SO2 into the stratosphere (top left) and mean volcanic forcing (top right)obtained for 100-year volcanic scenario. Scenarios constructed with both satelitte observationsand ice-core records have injections of sulfur into the stratosphere higher by up to 0.5Mt.yr−1,and corresponding 100-year average forcing higher by up to 0.15W .m2. Bottom panels ofFigure 6.19 show that when neglecting small eruptions and resampling from ice-core only, thestratospheric injections of SO2 are underestimated by 20 to 90% and the corresponding mean100-year forcing is underestimated by 15 to 70%. We conclude that accounting for relativelysmall injections of SO2 into the stratosphere is critical when designing volcanic forcing scenario.6.5.3 Quantification of the climate-volcano feedback proposed by Aubryet al. (2016)Aubry et al. (2016) show that increases of the tropopause height and tropospheric stratificationcaused by global warming would lead to decreased SO2 injections into the stratosphere. Toquantify the impact on volcanic forcing, we can use the methodology of Figure 6.18. Fora given scenario of eruptive conditions at the vent, we calculate the corresponding injectionheight and tropopause height using the plume model, and for different greenhouse gas emissionstrajectories. For each obtained scenario of atmospheric injection (corresponding to differentgreenhouse gas emissions), we then calculate the corresponding forcing timeserie by runningthe plume model.Time (year)0 10 20 30 40 50 60 70 80 90 100-F (W.m2 )10-1100Late 20thC,F=-0.23W.m-2Late 21stC, RCP2.6,F=-0.22W.m-2Late 23rdC, RCP2.6,F=-0.23W.m-2Late 21stC, RCP4.5,F=-0.22W.m-2Late 23rdC, RCP4.5,F=-0.23W.m-2Late 21stC, RCP8.5,F=-0.22W.m-2Late 23rdC, RCP8.5,F=-0.2W.m-2Figure 6.20: Short-wave forcing timeseries obtained for one scenario of eruptive condi-tions at vent level (Figure 6.18, top), but using different greenhouse gas emissionstrajectory to resample climate conditions used to obtain a scenario for volcanicforcing (Figure 6.18). The 100-year mean forcing is labeled for each scenario ofgreenhouse gas emissions.1426.5. Understanding the impact of plume height on stratospheric volcanic sulfate aerosol forcingFigure 6.20 shows the predicted forcing timeseries for one scenario of eruptive conditionsat vent level, and for different climate conditions corresponding to the 1981-2000, 2081-2100,2281-2300 periods and historical, RCP2.6, RCP4.5 and RCP8.5 experiments. The 100-yearmean forcing is labeled in legend. Differences in forcing timeseries are caused by differencein atmospheric conditions during each eruptive event in the scenario. For example, the meanforcing for the RCP8.5 pathway and 2281-2300 period is ' 10% smaller than the one for the1981-2000 period, which is consistent with the feedback proposed in Aubry et al. (2016). Becausewe randomly sample atmospheric conditions in an experiment corresponding to a particularperiod and scenario, the forcing associated with an eruption may be larger in warmer climatebecause of variability at daily timescale of local atmospheric conditions. For example, the largeforcing peak at year ' 17 of the scenario shown on Figure 6.20 is larger for the RCP8.5 pathwaycompared to the historical period.To quantify the mean change in forcing and uncertainty associated with the feedback pro-posed by Aubry et al. (2016), we generate 240 scenarios of eruptive conditions at vent level.We then calculate the difference in mean forcing caused by different atmospheric conditions asin Figure 6.20. Table 6.2 shows the mean and 95% confidence intervals of the volcanic forcingchange driven by global warming. Our results suggest that:• Changes in volcanic forcing driven by anthropogenic greenhouse gases emission would besignificant only for a RCP8.5 scenario because of the large uncertainty on future eruptiveconditions.• For a RCP 8.5 scenario, we find an average forcing change of up to 2× 10−2W .m2 for the23rd century, with a 97.5th quantile of 5.45× 10−2W .m2 across all scenarios generated.These corresponds to relative changes of -7 and -17% of the 100-year mean volcanic forcing.• We estimate the feedback strength to be 1.7 × 10−3W .m−2/W .m−2 (on the order of10−3W .m−2.K−1) which is two to three order of magnitude smaller than estimates ofmajor feedbacks of the climate system such as the water vapor or albedo feedbacks (Sodenand Held, 2006).Overall, our results thus suggest that the feedback proposed by Aubry et al. (2016) will beimportant to understand volcanic inputs into the stratosphere if we maintain a high greenhousegas emission trajectory. In particular, relative changes in the volcanic aerosol “background”forcing (Solomon et al., 2011) would be much higher than the values reported in Table 6.2,especially for the tropics. However, such changes would have minor impacts on climate. Ourassumptions on eruptive gas content and temperature means that our modeling work is mostlyapplicable to silicic eruptions. As discussed in Aubry et al. (2016), the climatic relevance ofthe feedback proposed may be much stronger for basaltic eruptions (e.g. Laki 1783-1784) asthese eruptions can inject significant quantities of sulfur into the atmosphere, but there plumesdo not rise as high as those of silicic eruptions. Consequently, relative changes in the ratio of1436.6. Conclusionsplume to tropopause height have more important implications for the mass of SO2 injected intothe stratosphere for these plumes.RCP 2.6,21stCRCP 4.5,21stCRCP 8.5,21stCRCP 2.6,23rdCRCP 4.5,23rdCRCP 8.5,23rdCGreenhouse gasforcing (W .m2)0.85 2.75 6.75 0.05 2.75 10.80Volcanic forcingchange (10−2W .m2)0.132.35−1.54 0.152.16−1.69 0.512.82−1.35 0.091.92−1.56 0.372.36−1.36 2.035.450.05Relative volcanicforcing change (%)−0.44.6−6.9 −0.65.3−6.9 −1.84.7−9.0 −0.34.9−7.3 −1.34.2−8.7 −7.0−0.2−17.0Table 6.2: Absolute (3rd row) and relative (4th row) changes in 100-year average strato-spheric volcanic sulfate aerosol forcing. We report the mean values across the 240scenarios for eruptive conditions generated, and the 2.5th and 97.5th quantiles are in-dicated as subscript and superscript respectively. All values refer to change relativeto the 1981-2000 historical experiment. Each column correspond to a RCP scenarioand period, with the second row indicating the anthropogenic forcing relative to1981-2000 inferred from Figure 11 in Van Vuuren et al. (2011).6.6 ConclusionsWe are designing a new idealized model of volcanic forcing which, in contrast to previousmodels: i) accounts for the height of volcanic plumes; ii) does not prescribe a vertical structurefor the forcing; and iii) is calibrated using comprehensive satellite observations from 1979 to2015, instead of the only 1991 Pinatubo eruption. Several aspects of the model are yet to beimproved (Appendix E). Nonetheless, the model predictions for the full 3D (time, latitude andaltitude) evolution of extinction at 525nm from an inventory of volcanic SO2 emissions arepromising. Our three main conclusions suggested by the new idealized model at this time are:1. High-latitude eruptions which inject SO2 above the altitude of the tropical tropopausehave a global climate impact comparable to tropical eruptions.2. Small eruptions (injecting ≤ 100.5 Mt of SO2) make a significant contribution to volcanicforcing. In particular, scenarios for future volcanic forcing constructed on the basis of onlyice-core records of volcanic sulfate injections neglect up to 90% of volcanic stratosphericSO2 inputs and up to 70% of the average stratospheric volcanic sulfate aerosol forcing.3. We estimate the feedback proposed in Aubry et al. (2016) (global warming leads to de-creased volcanic forcing via plume height decrease and tropopause height increase) to1.7× 10−3W.m−2/W.m−2.The model will be improved and its main predictions will be compared to experiments fromcoupled chemistry-climate models. However, it shows promising improvements over previous1446.6. Conclusionsidealized model and may become a key tool for global climate model that do not includeprognostic capabilities for stratospheric volcanic sulfate aerosol forcing.On its own, the proposed feedback likely has minor implications for moderate climatechanges. However, it may prove important to understand volcanic forcing and its climateimpact in the context of major climate change, e.g. a RCP8.5 scenario or past climate changesincluding Snowball Earth and the Cretaceous Thermal Maximum. This feedback may also havemore significant implications for basaltic eruptions which we have not explored quantitativelyyet. Last, an exciting research direction is to understand how this feedback may combine withknown (e.g. Jellinek et al. (2004)) and recently revealed (Fasullo et al., 2017; Hopcroft et al.,2017) climate-volcano feedbacks to modulate stratospheric volcanic sulfate aerosol forcing andthe climate response to this forcing.145Chapter 7Conclusions and outlook7.1 Concluding remarksMy thesis is motivated by two simple considerations (Chapter 1):• Stratospheric volcanic sulfate aerosol forcing is a key driver of climate variability.• Most processes governing the rise of volcanic plumes and injection of volcanic gases intothe stratosphere are affected by climate.Consequently, we asked the question: Is there a significant feedback between climate andthe rise of volcanic plumes? We demonstrated that the projected climate response to ongoinganthropogenic greenhouse gas emissions will lead to a decrease of the height of volcanic plumesrelative to the tropopause height. Thus, we proposed a new positive feedback where globalwarming leads to reduced stratospheric volcanic sulfate aerosol, reduced surface cooling byvolcanoes, and, in turn, enhanced global warming (Aubry et al., 2016). Our results suggestthat this feedback is important to understand volcanic inputs into the stratosphere, but thatit will have a weak impact for moderate climate changes (Chapter 6), at least applied to siliciceruptions (Aubry et al., 2016).Starting to draw an answer to our initial question led us to face and overcome multiplechallenges related to volcanic plume dynamics and volcanic aerosol forcing. Our contributionsto making progresses in these two major topics of Earth science are at least as important as thefeedback newly proposed. We compiled a new and exhaustive observational dataset of eruptionsource parameters, unprecedented for its size and by the systematic inclusion of uncertaintieson plume height and mass eruption rate (Aubry et al., 2017b). Applied together with theresults of an extensive series of analogue experiments I completed before starting my PhD,we demonstrated that this dataset enables critical improvements in the evaluation of volcanicplume models. We identify clear directions to overcome significant and specific limitations ofobservational data (Chapter 4).1467.2. Future directionsWe used the observational and experimental datasets to provide new constraints on scalings(Aubry et al., 2017b) and 1D models for volcanic plume rise (Aubry et al., 2017a), including ascaling proposed in this thesis. In particular, we improved the understanding and calibrationof turbulent entrainment which is the leading source of uncertainty in volcanic plume models.we explored the implications of our results for predicting the collapse of volcanic columns andproduction of devastating pyroclastic flows, as well as for predicting the climate impacts ofexplosive eruptions (Aubry et al., 2017b,a).Last, to quantify the proposed feedback, we are developing a new idealized model of vol-canic forcing (Chapter 6), which is at an early stage. This model already shows promisingperformance and overcomes several limitations of previous idealized models, including theirabsence of vertical structure and any dependence on plume height, as well as their calibrationagainst only a few select eruptions or the 1991 Pinatubo eruption. Our model suggests thathigh-latitude eruptions injecting sulfate above the altitude of the tropical tropopause may havea global impact, and that accounting for the role of relatively small eruptions is critical whendesigning scenarios for future volcanic aerosol forcing. Our contributions to understandingthe rise of volcanic plumes and subsequent volcanic forcing have been published in four peer-reviewed publication (Aubry et al., 2016, 2017b,a; Aubry and Jellinek, 2018), with Chapter 6to be submitted soon.7.2 Future directions7.2.1 Better datasets for understanding volcanic plume dynamics.When I started my PhD, I wanted to design a new laboratory experiment enabling to explorethe dynamics of particle-laden buoyant jets rising in a stratified ambient with a non-uniformcrossflow. Since I built a prototype in the first summer of my PhD, I was never able to returndespite my passion for science being driven by laboratory experiments. Experiments on buoyantjets combining a density stratification and non-uniform crossflow in regimes relevant to volcanicplumes are nearly absent in the literature, let alone experiments with particle-laden jets andwind. In particular, I want to explore how the combined effect of wind gradients and particlesnon fully coupled to the flow affect the turbulent entrainment rate in buoyant jets. Woulda “classic” formulation of the entrainment hypothesis enable accurate model predictions forplume height by 1D plume models for such experiments? I will work on a new experimentalsetup as soon as I get the opportunity.A year after compiling an unprecedented database of eruption source parameters, we havehighlighted the limitations of this database, the overcoming of which will be one of my majorresearch objectives for the coming years. The new database will maintain the advancementsmade during my thesis – including a large number of eruptions and quantifying uncertaintiesin addition to mean estimates of source parameters. In addition, the following improvementsare required:1477.2. Future directions1. Distinguishing estimates of “plume height” that correspond to different layers of the plume(e.g. gas or ash, maximum height or intrusion) or to different metrics for these layers (e.g.upper boundary or center height).2. Revisiting estimates of erupted mass from field deposit and their uncertainties usingrecent improvements in quantifying uncertainties emerging from the determination of anisopach mass from field measurement (e.g.Engwell et al. (2015)) or from the integrationof thinning trend (e.g.Bonadonna et al. (2015a)).3. Attributing a plume style (weak or strong) to all events of the database independently ofany other parameters (plume height or mass eruption rate) and model.4. Compiling atmospheric conditions from multiple sources.To make this future database a reference dataset, I will develop it in collaboration withexperts in volcanic plume modeling and field observations. This will ensure that the databaseis state of the art and answer major modeler needs. A key aspect will also be to develop adedicated website where researchers can comment suggest revisions of the particular events orparameters, advertise the availability of new data for an eruptive event, and give feedback onthe database.7.2.2 3D vs 1D plume models: performance and assessment ofclimate-volcano feedback.My thesis uses scalings and 1D models of volcanic plume rise. The use of such models wasinstrumental in analyzing the sensitivity of my results. For example, the assessment of theproposed feedback using Monte Carlo simulations to account for uncertainties in future eruptionsequence would be impossible with a 3D model due to its computational cost.However, the robustness of the feedback proposed in my thesis would be strengthened if Ican show that 3D models predictions compare well to 1D model predictions for a few selectexperiments. I have discussed an experimental design with Matteo Cerminara (INGV, Italy)where we would compare 1D and 3D models predictions for: i) three climate scenarios; ii) onehigh-latitude and one tropical case; and iii) a dozen of mass eruption rates for which eruptiveplumes rise in the uppermost troposphere to lowermost stratosphere under present-day climateconditions.In addition to testing the proposed feedback in 3D vs 1D models, I would like to assess theperformance of 3D vs 1D models in predicting observed plume heights from field-constrainedmass eruption rates. The recent plume model intercomparison study (Costa et al., 2016) givesinteresting insights on differences between 3D and 1D models but does not evaluate their per-formance. It would be of interest to the entire community to know whether the computationalcost of 3D models enables significantly better predictions for plume height compared to 1Dmodels. The database I propose to develop is the ideal tool to address this question.1487.2. Future directions7.2.3 Climate-volcano interactionsMy PhD work shows that the proposed feedback may have important climatic implications forlarge changes of atmospheric CO2 concentration and/or basaltic eruptions. I would like to ex-plore these implications in a global climate model. One first possibility is to directly implementpredicted changes in the vertical distribution of volcanic SO2 into a coupled chemistry-climatemodel. One second possibility is to implement forcing scenarios derived from the new idealizedvolcanic aerosol forcing model into a climate model without a prognostic aerosol scheme. Thesecond solution is computationally cheaper, but I would like to test both solutions as impor-tant processes, e.g. aerosols grow and coagulation, are absent from the idealized model we aredeveloping. An exciting research direction is also to assess changes in volcanic forcing and theassociated climate response arising from the feedback we propose and recently revealed mech-anisms that would amplify (Fasullo et al., 2017) or reduce (Hopcroft et al., 2017) the climateresponse to volcanic eruptions.Independently from the proposed feedback, our applications of the new idealized volcanicaerosol forcing model has several major results such as the importance of relatively small erup-tions for designing scenarios for volcanic forcing, or the potentially global impact of high-latitudeeruptions. 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