Science, Faculty of
Earth, Ocean and Atmospheric Sciences, Department of
DSpace
UBCV
Wanket, Sean
Master of Science - MSc
University of British Columbia
The rate of change of orbitally-driven variations in continental ice mass modulates volcanism by inducing variations in the compressional (lithostatic) stress regime of the wall rock surrounding hot magma reservoirs. At West Antarctica, as crustal temperatures rise in response to the delivery and storage of melt since 36 Ma, and to the insulating effects of a thick West Antarctic Ice Sheet emplaced ∼34 Ma, the effective viscosity of wall rocks decreases. Pressures in excess of lithostatic related to the injection of magma are increasingly relieved through wall rock creep at the expense of volcanism and stress changes imparted by orbitally-forced deglaciation are consequently reduced in magnitude and delayed in time. Theoretically, maximally rapid creep and a negligible volcanic response to orbital forcing corresponds to a time lag of 1/4 of the period of the glacial forcing. Clusters of tephra layers near Mount Berlin corresponding to peaks in the rate of change of volcanism occur at eight and 110 ka, but only the eight ka cluster lags a maximum in the rate of change of deglaciation by less than 1/4 of an obliquity-driven glacial cycle (∼6 kyr), suggesting the crust underwent a rheological transition before eight ka. Using a 1D numerical model of heat diffusion we study the effects of ice sheet insulation and time-dependent magma supply on the viscoelastic response of the crust to glacial cycles. Long-term ice insulation increases the crust’s proclivity for viscous creep by increasing the temperature by ∼50 K. However, a time-dependent magma supply causes the crust to alternately cool and warm. Through the exponential dependence of crustal viscosity on temperature, thermal oscillations and long-term insulation force the crust into transient behavior in which magma is preferentially stored during warmer intervals and erupted during cooler periods. We argue that the change in the response of the crust between 110 and eight ka is a consequence of such oscillations. Our data analysis applied with inferred geothermal heat fluxes and peak magma production rates in Marie Byrd Land constrain the magnitude and period of oscillation for mantle magma production to 0.01-0.1 km³/year and ∼10⁶-10⁸ years, respectively.
Potential Ice Sheet and GlacialModulation of Volcanism in WestAntarcticaConstraints on the cadence of melt delivery into thecrustbySean WanketB.Sc., The University of British Columbia, 2020A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Geophysics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2024© Sean Wanket 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Potential Ice Sheet and Glacial Modulation of Volcanism in West Antarc-tica: Constraints on the cadence of melt delivery into the crustsubmitted by Sean Wanket in partial fulfillment of the requirements forthe degree of Master of Sciencein GeophysicsExamining Committee:Dr. A. Mark JellinekProfessor, Department of Earth, Ocean and Atmospheric Sciences, UBCCo-supervisorDr. Catherine JohnsonProfessor, Department of Earth, Ocean and Atmospheric Sciences, UBCCo-supervisorDr. Anais OrsiAssistant Professor, Department of Earth, Ocean and Atmospheric Sciences, UBCSupervisory Committee MemberDr. Jonathon FinkProfessor, Department of Geology, PSUAdjunct Professor, Department of Earth, Ocean and Atmospheric Sciences, UBCAdditional ExamineriiAbstractThe rate of change of orbitally-driven variations in continental ice massmodulates volcanism by inducing variations in the compressional (litho-static) stress regime of the wall rock surrounding hot magma reservoirs.At West Antarctica, as crustal temperatures rise in response to the deliveryand storage of melt since 36 Ma, and to the insulating effects of a thickWest Antarctic Ice Sheet emplaced ∼34 Ma, the effective viscosity of wallrocks decreases. Pressures in excess of lithostatic related to the injectionof magma are increasingly relieved through wall rock creep at the expenseof volcanism and stress changes imparted by orbitally-forced deglaciationare consequently reduced in magnitude and delayed in time. Theoretically,maximally rapid creep and a negligible volcanic response to orbital forcingcorresponds to a time lag of 1/4 of the period of the glacial forcing. Clustersof tephra layers near Mount Berlin corresponding to peaks in the rate ofchange of volcanism occur at eight and 110 ka, but only the eight ka clusterlags a maximum in the rate of change of deglaciation by less than 1/4 ofan obliquity-driven glacial cycle (∼6 kyr), suggesting the crust underwent arheological transition before eight ka. Using a 1D numerical model of heatdiffusion we study the effects of ice sheet insulation and time-dependentmagma supply on the viscoelastic response of the crust to glacial cycles.Long-term ice insulation increases the crust’s proclivity for viscous creep byincreasing the temperature by ∼50 K. However, a time-dependent magmasupply causes the crust to alternately cool and warm. Through the expo-nential dependence of crustal viscosity on temperature, thermal oscillationsand long-term insulation force the crust into transient behavior in whichmagma is preferentially stored during warmer intervals and erupted dur-ing cooler periods. We argue that the change in the response of the crustbetween 110 and eight ka is a consequence of such oscillations. Our dataanalysis applied with inferred geothermal heat fluxes and peak magma pro-duction rates in Marie Byrd Land constrain the magnitude and period ofoscillation for mantle magma production to 0.01-0.1 km3/year and ∼106-108years, respectively.iiLay SummaryAkin to controlling the release of pressurized air from a rubber balloon witha valve, the pace of periodic glacial loading and unloading can modulate vol-canism by closing or opening cracks in enveloping elastic wall rocks throughwhich hot magma rises from within Earth’s crust or mantle to the surface.At West Antarctica, calculations show that crustal warming related to pro-tracted volcanism as well as insulation by the thick West Antarctic Ice Sheethave caused host rock to become less elastic, more viscous and less respon-sive to glacial modulation overall. In a 130 kyr volcanic history at MountBerlin, West Antarctica, I identify clusters of eruptions at eight ka and atabout 110 ka and show that only the eight ka events were plausibly drivenby deglaciation. I use thermal models to show that this intermittency inglacial control is a consequence of a time-dependent mantle magma.iiiPrefaceThis thesis is original unpublished work by the author, Sean Wanket. Chap-ter 2 and Section 4.1 are based on work originally published in [35]. Thenumerical modelling is based on work done by [38]. Co-Supervisors MarkJellinek and Catherine Johnson edited the writing of the manuscript.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . 11.1.1 A Brief History of Glacial Modulation of Volcanism . 11.1.2 Ice Sheet and Glacial Modulation of Volcanism inWestAntarctica . . . . . . . . . . . . . . . . . . . . . . . . 72 Data Analysis and Conceptual Model . . . . . . . . . . . . . 92.1 Volcanism and Glacial Forcing for the last 130 ka, Marie ByrdLand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Conceptual Model: Predicted Amplitude and Phase Lag forOrbital Forcing of Volcanism . . . . . . . . . . . . . . . . . . 112.3 Glacial modulation of volcanism through a temperature-dependentcrustal filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Hypothesis and Plan . . . . . . . . . . . . . . . . . . . . . . 153 Thermal Modelling Approach . . . . . . . . . . . . . . . . . . 173.1 Heat transfer across the crust with an insulating ice sheet andperiodic mantle upwelling . . . . . . . . . . . . . . . . . . . . 173.2 1D Heat Flow Models . . . . . . . . . . . . . . . . . . . . . . 19vTable of Contents3.2.1 An analytical solution of a simplified problem . . . . 193.2.2 A numerical solution of the full problem: The Dufort-Frankel Method . . . . . . . . . . . . . . . . . . . . . 203.3 Numerical Experimental Setup . . . . . . . . . . . . . . . . . 223.3.1 Experiment One: Constant qm, Variable Hi . . . . . 223.3.2 Experiment Two: Variable qm, constant Hi . . . . . . 234 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 Data Analysis: Physical and unphysical lags between maxi-mal rates of deglaciation and changes in eruption frequency . 244.2 Thermal modelling: Time-dependent ice sheet insulation andmagma supply . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.1 Experiment One: Constant qm, Variable Hi . . . . . 244.2.2 Experiment Two: Variable qm, Constant Hi . . . . . 265 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1 Conditions for gradual and punctuated thermal evolutionsinto storage, eruption and oscillating regimes . . . . . . . . . 285.2 Transitional Regime: Thermal Variations around a De = 1 . 295.3 A West Antarctic Tr and melt supply: Time lag and geother-mal heat flux constraints . . . . . . . . . . . . . . . . . . . . 306 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38AppendicesA Benthic δ18O as a proxy for global ice volume . . . . . . . . 48B Representing a Magmatic System using Fourier Transformsand a Transfer Function . . . . . . . . . . . . . . . . . . . . . . 49B.1 The Time Domain . . . . . . . . . . . . . . . . . . . . . . . . 50B.2 Fourier Domain . . . . . . . . . . . . . . . . . . . . . . . . . 50C Derivation of Initial Conditions . . . . . . . . . . . . . . . . . 54D Radiogenic Heating: Constant vs Exponential Decrease . 56viTable of ContentsE Insulation and melting at the crust/ice boundary . . . . . 59E.1 Resistance of the Ice Sheet: The insulation boundary condi-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59E.2 Accounting for ice sheet melting at the crust/ice boundary . 59F Volumetric Magma Injection Rate calculated from MantleHeat Flux Values . . . . . . . . . . . . . . . . . . . . . . . . . . 61G Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62G.1 READ ME . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62G.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64G.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65G.4 Main Program . . . . . . . . . . . . . . . . . . . . . . . . . . 73viiList of Tables3.1 Parameter and variable definitions. Values defined wherethey are not time or temperature dependent. [1] - [48] [2]- [63] [3] - [10] [4] - [26] [5] - [76] [6] - [38]. . . . . . . . . . . . 18viiiList of Figures1.1 δ18O from the LR04 stack [47]. Glacial and Interglacial peri-ods are marked on the plot. . . . . . . . . . . . . . . . . . . . 11.2 Figure 3 in [30]. (a) Ratio of eruptive frequencies from the lastdeglacial period (18-7 ka) to the last glacial period (40-20 ka).This ratio is called d/g (b) Average d/g ratio per latitude. (c)Ratio of eruptive frequencies from the last deglacial periodto the late Holocene (5-0 ka). This ratio is called d/h (d)Average d/h ratio per latitude. . . . . . . . . . . . . . . . . . 31.3 Sketches of three conceptual mechanical models for effectsof deglaciation on volcanism. (a) At Iceland, glacial un-loading at the surface of a very thin crust enhances decom-pression melting in the upper mantle directly [37]. (b) AtEastern California, effects of ice loss at the surface of a rela-tively very thick crust are attenuated above the crust-mantleboundary and do not influence melting in the upper man-tle. However, the advance and recession of mountain glaciersalternately increase and decrease the lithostatic pressure atmagma reservoir depths. Where reservoirs support active vol-canism, this periodic change acts to increase and decreasethe ”hoop stresses” that open or close dikes to the surface tofeed volcanism, thereby modifying the cadence of eruptions[34, 35]. (c) At West Antarctic crust, the crust is comparablythick to Eastern California and, thus, glacial/interglacial cy-cles potentially modify volcanism similarly. In marked con-trast to Eastern California, these cycles are superposed ona relatively thick West Antarctic Ice Sheet that has been inplace at least since 34 Ma. [3, 4, 23, 29, 32, 36, 54, 60, 61, 73, 81]. 4ixList of Figures1.4 Map of West Antarctica: red stars are the WAIS Divide andByrd ice cores and brown trapezoids are volcanoes discussedin the paper. Contours show Geothermal Heat Flux (GHF)inferred across West Antarctica from [2], with the highest val-ues co-located with active Marie Byrd Land Volcanism. ThisGHF model is produced using a “thermal isostasy method”developed in [1, 2] and is generally consistent with inferencesdrawn from regional seismic tomography and magnetic surveystudies [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Comparison of Mount Moulton eruption frequency time seriesto the first derivative of the LR04 δ18O stack. All time seriesare differentiated using the numpy function “gradient” whichperforms the centered difference approximation at all interiorpoints and forward and backward euler method at the endpoints. Dash-dotted orange line in a, c, and d is the firstderivative of LR04 and has been bandpassed between 1/5 and1/200 kyr−1. Solid orange line in b, e, and f is the LR04 signalafter removing frequencies above 1/30 kyr−1 and below 1/55kyr−1 to isolate the obliquity period. Bars are eruptions withuncertainties indicated at their tops. Black, blue, and reddash-dotted lines are the smoothed eruption frequency timeseries with a Gaussian bin widths of 500, 1000, and 5000years, respectively. Black, blue, and red solid lines are thefirst derivative of the continuous eruption frequency curves.Red zones highlight areas where there is no response to theglacial unloading and green zones highlight areas where thereis a response within 1/4 of a 41 kyr obliquity period (see text).(a) Full LR04 time series after smoothing, interpolation, anddifferentiation and the eruption frequency time series. (b)Obliquity signal in the LR04 series found using a bandpassfilter. (c-f) a-b zoomed in where the left column of plots spans0-35 ka and the right column of plots spans 100-140 ka. . . . 10xList of Figures2.2 Schematic definition of the mechanical and thermal forcingsrelevant to the paper. Heat and melt is delivered from themantle into an active magmatic system in the crust composedof basaltic sills and intrusions at 30 km depth[44] and silicicreservoirs at ∼ 3− 8 km depth [44, 51]. These feed the erup-tions represented by the bar plot in Figure 2.1. The verticaldistribution of crustal radioactive heat production, Hrad aug-ments this magmatic heating. The geothermal heat flux atthe top of the crust is transported across the ice sheet by ther-mal conduction. To introduce thermal and mechanical orbitalmodulation, we vary the thickness H of the ice sheet. A peri-odic variation in the magma (and hence magmatic heat) inputδqm(t) generates a time-dependent heat flow into the base ofthe crust, qin(t). Temperature increases with depth with alimit on the ice/crust boundary of T0(t) = 0◦C (the meltingpoint of ice) and a limit at the crust-mantle boundary, Tm =1100◦ C (liquidus temperature of crustal rocks assuming theyare basaltic andesite [38]). Note that this sketch is not to scale. 132.3 Time Lag and Magnitude plot of the response of the crustalfilter to an obliquity driven glacial forcing (τf = 41 kyr).The max time lag is 10.25 kyr: 1/4 of τf . The time lagand magnitude are plotted against De, which is equivalent tothe normalized frequency of the transfer function (AppendixB). “1” indicates lag and magnitude in an underdevelopedmagmatic system. “2” indicates lag and magnitude in a moredeveloped, hotter magmatic system. . . . . . . . . . . . . . . 153.1 Differences between the analytic solution and DF (blue) andFTCS (orange). ∆T refers to the difference between the an-alytic and numeric solutions. For DF: ∆t = 50 years, ∆z =50 m and ∆z2/κ = 66. For FTCS: ∆t = 50 years, ∆z = 72m (divergent at the same spatial step as DF), and ∆z2/κ =137. (a-g) Differences at times specified as a fraction of τd:(a) t = 0, (b) t = 0.01τd, (c) t = 0.1τd, (d) t = 0.5τd, (e) t =τd, (f) t = 3τd, (g) t = 6τd. (h-n) focus on the top three kmof the crust for the same times as in (a-g). . . . . . . . . . . . 21xiList of Figures4.1 (a,b) Phase and magnitude plot for an obliquity forcing withthe lags related to increases in eruption frequency at ∼8 kaand ∼110 ka (see Chapter 2). Black dotted line is the lag forthe 110 ka cluster found with a 5 kyr Gaussian bin width withuncertainties (red dotted lines) based on reported eruptionages. Dash-dotted lines are the lags for the eight ka clusterfor two bin widths: 500 (black) and 1000 (blue). Uncertain-ties related to eruption ages are 100 years. (c,d) Simplifiedversions of Figures 2.1e and f. Black, blue, and red sold linesare the first derivative of eruption frequency curve and solidorange line is the first derivative of LR04 obliquity signal. . . 254.2 Experiment One: Constant mantle heat flux, variable icesheet thickness. Evolutions of (a) temperature and (b) Dewith the introduction of an ice sheet at 3τd = 6.25e6 [yrs].Once an insulating ice sheet forms, the temperature of crustalrocks hosting the magmatic system at five km depth climbsand De falls. The four simulations have progressively higheraverage mantle magma volumetric input rates: 0.003 km3/year(red), 0.005 km3/year (black), 0.006 km3/year (blue), and0.008 km3/year (green) (Table 3.1). We vary the ice sheetthickness periodically by 3000 m between 1500 and 4500 mwith a period of 100 kyr (inset). . . . . . . . . . . . . . . . . 264.3 Experiment Two: Periodic mantle heat flux with a constantice sheet thickness Hi = 3 km. (a) Crust thermal evolutionsfor different Tr at five km depth. Periods, τin, between meltinjections are: 0.0016τd (1e4 yrs) (black), 0.016τd (1e5 yrs)(orange), 0.16τd (1e6 yrs) (blue), τd (6.25e6 yrs) (green), 1.6τd(1e7 yrs) (red). Inset shows the variation in mantle meltsupply with τin = τd. (b) Corresponding De regimes. ForTr ≥ 1 the magmatic system oscillates between Eruptionand Storage regimes. Shorter periods between injections drivefavor Storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 Ratio of time steps spent in the eruptive regime to time stepsspent in the storage regime; blue circles have no ice in thesimulation and red circles have an an ice sheet. The ice sheetoscillates with the same period and magnitude as Experiment1. Tr = 1 for all simulations. The ice simulation for the meltinjection rate of 0.07 km3/year has a ratio of 0 and does notappear on a logarithmic plot. . . . . . . . . . . . . . . . . . . 30xiiList of Figures5.2 The black curve is the equivalent De at five km depth for arange of geothermal heat fluxes assuming a linear geothermalgradient. The horizontal red arrows set the range of De fromthe geothermal heat flux data summarized in [11] and [2], 90 -120 mW/m2 or 0.7 < De < 50. The horizontal black and bluearrows set the range of De from Section 4.1, 119-125 mW/m2or 0.6 < De < 0.8. The black and blue arrows are the De fromSection 4.1 for 0.5 and 1 kyr Gaussian windows, respectively. 315.3 Model iterations varying Tr along the x-axis and the max δqmalong the y-axis. (a) The maximum De the crust reaches inthe model after 20 Myr of heating.(b) The average mantleproduction rate (km3/year).(c) The minimum De the crustreaches after 20 Myr of heating. (d) Constraint One (bluecontour), constraint two (black contours), and constraint three(red contour) with the valid parameter space marked by blackdashed lines between constraints. The black contours in (d)are average magma input rate in km3/year. . . . . . . . . . . 33B.1 The transfer function in the complex domain . . . . . . . . . 52D.1 Experiment One: Constant mantle heat flux, variable icesheet thickness. (a-b) Plot of temperature and De assum-ing constant radiogenic heating. (a) shows the lower boundwhere Hrad = 1.5 µW/m3 and (b) is the upper bound whereHrad = 3 µW/m3. (c-d) Plot of temperature and De assum-ing exponential decrease in Hrad. (c) shows the lower boundon the e-fold distance hr = 7.5 km and (d) shows the upperbound hr = 10 km. In (b) De above the horizontal dottedline are in the eruption regime, and below are in the storageregime. The four simulations have progressively higher aver-age mantle heat flux: 1.27 (red), 1.91 (black), 2.54 (blue), and3.17 (green). The ice sheet is introduced in the simulationswhere the vertical black dashed line is. Ice sheet thicknessvaries periodically between 1500 and 4500 m with a period of100 kyr (shown in inset plot). . . . . . . . . . . . . . . . . . . 57xiiiList of FiguresD.2 Parameter Space for four different radiogenic heating models.All are recreations of Figure 5.3e. (a) Lower bound of con-stant radiogenic heating where Hrad = 1.5 µW/m3. (b) Upperbound where Hrad = 3 µW/m3. (c)Lower bound on the e-folddistance for exponential decrease in radiogenic heating hr =7.5 km. (d) Upper bound hr = 10 km. . . . . . . . . . . . . . 58xivAcknowledgementsNone of this would have been possible without the support of Mark andCatherine. I cannot emphasize enough how helpful and encouraging youboth were the past three years. Thank you for putting up with all of mypanic emails and late night questions. And thank you both for the support.I could not imagine having a better set of supervisors to help me throughthis exciting (and sometimes very painful) process.I would also like to thank everyone in MJCJ for being such amazingfriends and lab mates. You guys helped me with the project and also helpedme to get my mind off the project when that was necessary. Thank youYoshi for showing me the ropes when I came to work for you in the lab.Thank you Cara and Francis for afternoon beers when we all needed to getout of the office. Thank you Cole and Colin for introducing me to Frosthavenand giving me code/writing advice whenever I needed it.Thank you Mom and Dad for supporting me through my whole under-grad and Master’s. You guys always believed in me and encouraged me togo through with another few years of school and I love you guys. And thankyou Olivia for being the best part of my day every day.xvChapter 1Introduction1.1 Background and MotivationPeriodic variations in the distance of the Earth’s orbit around the sun (Ec-centricity), axial tilt (Obliquity), and the orientation of the Earth’s rota-tional axis (Precession) cause changes to insolation that result in alternatingcold (glacial) and warm (interglacial) mean climates [67]. Mountain glaciersand polar ice sheets thicken and advance to lower elevations and latitudesduring glacial periods and thin and recede during interglacial times. Ec-centricity, Obliquity and Precession modes have distinct periods of 100 kyr,41 kyr, and 23 kyr, respectively. To resolve a record of the correspondingglacial/interglacial cycles, [47] have compiled and tuned a 5.3 Myr time se-ries of benthic δ18O (Figure 1.1; see Appendix A for a description of benthicδ18O).GlacialInterglacialGlacialInterglacialGlacialInterglacialGlacialInterglacialFigure 1.1: δ18O from the LR04 stack [47]. Glacial and Interglacial periodsare marked on the plot.1.1.1 A Brief History of Glacial Modulation of VolcanismIt is now widely-recognized that the frequency of volcanic eruptions in-creased globally with Earth’s retreat from the last glacial maximum [9, 30,11.1. Background and Motivation41, 46, 64, 74]. In one of the first studies to propose glacial/interglacialcycles as a modulator of volcanism, [64] reanalyzed datasets previously usedto suggest that volcanism influenced the climate. However, they foundthat in many cases increases in global deglaciation preceded increases inthe frequency of large silicic eruptions, suggesting that climate also influ-ences volcanism. Through an exhaustive time series analysis, [30] evaluatedstatistically meaningful correlations between glacial-interglacial cycles andchanges in global eruption frequency during the last glacial period (40-20ka), the deglacial period (18-7 ka), and the late Holocene (5-0 ka). In partic-ular, they found that for mid-ocean ridge segments and continental volcanicprovinces impacted by glaciation, that eruption frequencies are a factor of2-6 higher during the deglacial period. The ratios of eruptive frequencyduring deglacial times to glacial times (Figures 1.2a and b) and the lateHolocene (Figures 1.2c and d) are largest at high latitudes where magmaticsystems and glaciers interact the most [30]. In a subsequent careful statis-tical analysis of continental volcanism, [74] showed that glacial modulationis expressed reliably only in arc systems and that glacial unloading in theSouthern Andes, Kamchatka, and the Cascades caused a 2-fold increase ineruptive frequency during deglaciation at 13-5 ka.Local studies of individual volcanic systems also show that the eruptivestyle or cadence of eruptions is modulated by periodic glacial loading andunloading [6, 20, 24, 25, 28, 35, 37, 42, 50, 53, 55, 58, 59, 62, 65, 68–70,80]. [28] found, for example, that volcanism at sub-Antarctic Marion Islandoccurs only during interglacial intervals and suggested that the faulting,which may initiate volcanism, is triggered by deglaciation. A study by [58]demonstrated a link between volcanic eruption frequency in Southern Italyand Precession-driven changes in sea level over a 190 kyr time series oferuptions.In the Dyngjufjo¨ll volcanic complex of Iceland [69] found that magmaproduction in the period 10-4.5 ka was 20-30 -fold higher than the period4.5 ka - present. Recognizing that deglaciation at Iceland will potentiallyinfluence the very shallow mantle melting in that region directly, [37] mod-eled the response of the Icelandic upper mantle melting region to glacialunloading and find that the proposed 20-30x must be preceded by an ∼30kyr period of essentially no magma production during peak glacial load-ing. They also argued that predicted deglacial bursts in melt productionwill be characterized by lavas with distinctive rare earth elements (REE)and incompatible element compositions, largely consistent with subsequentgeochemical studies [50, 71].Compared to Iceland, at continental arcs, crustal stress changes related21.1. Background and MotivationFigure 1.2: Figure 3 in [30]. (a) Ratio of eruptive frequencies from thelast deglacial period (18-7 ka) to the last glacial period (40-20 ka). Thisratio is called d/g (b) Average d/g ratio per latitude. (c) Ratio of eruptivefrequencies from the last deglacial period to the late Holocene (5-0 ka). Thisratio is called d/h (d) Average d/h ratio per latitude.31.1. Background and Motivation∆PchMantleCrustMountain GlaciersHoop stress due to ∆PchMantleCrustIce SheetSilicic ReservoirBasaltic ReservoirExcess melt generatedfrom glacial unloadingIcelandEastern California∆PchHoop stress due to ∆PchMantleCrustIce SheetSilicic ReservoirBasaltic ReservoirWest Antarcticaa)c)b)Figure 1.3: Sketches of three conceptual mechanical models for effects ofdeglaciation on volcanism. (a) At Iceland, glacial unloading at the surfaceof a very thin crust enhances decompression melting in the upper mantledirectly [37]. (b) At Eastern California, effects of ice loss at the surface ofa relatively very thick crust are attenuated above the crust-mantle bound-ary and do not influence melting in the upper mantle. However, the ad-vance and recession of mountain glaciers alternately increase and decreasethe lithostatic pressure at magma reservoir depths. Where reservoirs sup-port active volcanism, this periodic change acts to increase and decrease the”hoop stresses” that open or close dikes to the surface to feed volcanism,thereby modifying the cadence of eruptions [34, 35]. (c) At West Antarc-tic crust, the crust is comparably thick to Eastern California and, thus,glacial/interglacial cycles potentially modify volcanism similarly. In markedcontrast to Eastern California, these cycles are superposed on a relativelythick West Antarctic Ice Sheet that has been in place at least since 34 Ma.[3, 4, 23, 29, 32, 36, 54, 60, 61, 73, 81].41.1. Background and Motivationto glacial loading and unloading are negligible in the relatively much deepermantle melting region. Nevertheless, [65] found a response in the magmaticsystem feeding Mocho-Choshuenco, a large Chilean stratovolcano. Followingdeglaciation, there are marked shifts in eruption rates and magma composi-tions away from preglacial (andesitic) volcanism. In a first phase followingdeglaciation (13–8.2 ka), relatively much larger eruptions of evolved (daciticand rhyolitic) lavas occurred until a transition to relatively smaller volumeeruptions of basaltic and andesitic magmas (7.3–2.9 ka). [65] proposed thatevolved magma is stored in the crust during peak glaciation and released induring the first phase, which, in turn, draws basaltic and andesitic magmasinto the erupting reservoir. Consistent with this picture, [53] identified largevolumes of mafic ignimbrites in the Sourthern Volcanic Zone of the Andeshundreds to thousands of years after a deglaciation. Implementing an an-alytical model of the flexural deflection of an elastic disk, they show thatdecompression rates at the depths of magma chambers in the crust are con-sistent with the time lag between deglaciation and this increase in eruptedvolume.In Southwest British Columbia [80] used elastic plate flexure models tostudy the periodic loading effects of the Cordilleran Ice Sheet. Using aMonte Carlo approach they show that “glacial pumping” modifies the stressregime during glacial/interglacial cycles, enhancing magma storage in theupper crust in glacial times and increased eruption frequency and volumeduring deglaciation. A key result of their work is that rapid unloading(deglaciation) produces a higher maximum eruption rate than gradual un-loading, suggesting the rate of deglaciation is a key control on the responseof a volcanic system.Expanding on the work of [24], in an analysis of the volcanic response todeglaciation in the Sierras of eastern California, [35] (herein JMS04) foundthat increases in eruption frequency are better correlated with the rate ofdeglaciation driven by changes in Earth’s obliquity than with the extentof ice loss. Furthermore, JMS04 identified distinctive time lags betweenthe mechanical deglaciation “forcing” and the basaltic (∼11 kyr) and silicic(∼3 kyr) volcanic responses. They argued that the damping implied by thetime lags reflects the temperature-controlled effective viscosity of the crustalrocks hosting the magma reservoirs. The relatively shorter lag associatedwith silicic volcanism is related to the higher effective viscosity of colderupper crustal reservoir host rocks. We discuss this mechanism in greaterdetail in Chapter 2.51.1. Background and MotivationWAIS DivideByrdMount TakaheMount BerlinMount MoultonMount Waesche9011090100Figure 1.4: Map of West Antarctica: red stars are the WAIS Divide andByrd ice cores and brown trapezoids are volcanoes discussed in the paper.Contours show Geothermal Heat Flux (GHF) inferred across West Antarc-tica from [2], with the highest values co-located with active Marie Byrd LandVolcanism. This GHF model is produced using a “thermal isostasy method”developed in [1, 2] and is generally consistent with inferences drawn fromregional seismic tomography and magnetic survey studies [11].61.1. Background and Motivation1.1.2 Ice Sheet and Glacial Modulation of Volcanism inWest AntarcticaTheWest Antarctic Ice Sheet (WAIS) has existed since the Eocene-Oligoceneboundary, approximately 34 Ma [3, 4, 23, 29, 32, 36, 54, 60, 61, 73, 81]. Ad-ditionally, obliquity-driven glaciations appear in time series of surface tem-perature and ice thickness changes in ice cores from both the West and EastAntarctic Ice Sheets [4, 23, 29, 36, 54, 60, 61, 73]. No work has yet exploredthe extent to which the glacial-interglacial modulation of volcanism occurswhere these effects are superposed on a persistent, thick and dynamic icesheet such as the WAIS.Volcanism at Marie Byrd Land began in the late Eocene, approximately36 Ma [78, 79]. Three subaerial volcanoes (Mount Berlin, Mount Takahe,and Mount Waesche, Figure 1.4) are active today [15, 17, 18, 77, 78] andthere is also evidence for recent subglacial volcanism beneath the WAIS[8, 13, 33, 49]. Evidence of eruptions from the Marie Byrd Land VolcanicGroup (MBLVG) is preserved in the West (and East) Antarctic Ice Sheet intephra layers which have been found and studied in ice cores and blue icefields [17–19, 27, 40, 56, 57].Here, we investigate the extent to which glacial-interglacial control is ex-pressed in the ∼130 kyr volcanic record from Mount Moulton, West Antarc-tica. This area is underlain by the active West Antarctic rift system thatis marked by active volcanism and high heat flow (Figure 1.4) [2, 5, 8, 11].Our work is restricted to this short interval of the volcanic history becauseof the limited data available from exposures of older tephras, the complicat-ing effects of ice deformation and ablation on the continuity and structureof particularly older tephra sequences, and challenges inherent to producingreliable eruption ages, as well as reliable age markers in detailed strati-graphies [15, 16, 18]. Accordingly, we limit our analysis to the periodicobliquity-driven glacial modulation of volcanism expressed in eruption timeseries from the Mount Moulton Blue Ice Field in Marie Byrd Land and ben-thic δ18O data for the same time interval (Section 2.1). Our initial aim is toidentify whether peaks in volcanic activity occur in concert with glacial cy-cles and, if so, their temporal relationship to maximal rates of deglaciation.We next develop a conceptual model which extends the work of JMS04 andrelates the response of a critically over-pressurized magma reservoir to glacialunloading, assuming that the temperature-controlled visco-elastic propertiesof the crust govern the amplitude and lag of the volcanic response (Section2.2). The time lag is diagnostic of the effective viscosity and relaxation timeof crustal host rocks and is a metric for the proclivity for magma storage71.1. Background and Motivationversus eruption.Crustal rocks hosting magmatic systems warm as magmatic systems age,leading to an inevitable transition to a crust rheological regime favoringmagma storage that depends on the magma supply [34, 38]. For constantlong-term average supplies typical of arc and continental magmatic systems,a full transition to a predominantly storage regime is expected to occurwithin a few to 10 million years [38]. Applied to West Antarctica wheremagmatism has persisted for ∼ 36 million years, the occurrence of recentvolcanism is, thus, in itself curious. Because the crustal warming effects of icesheet insulation will accelerate this transition, a glacial-interglacial controlover volcanism is even more surprising if the supply of magma is approxi-mately fixed and consistent in magnitude with the high inferred geothermalheat flow (Figure 1.4 and section 5). However, volcanism is possible if the themelt supply from the mantle is intermittent or periodic, rather than steady.If melt pulses are delivered into the crust with repose times sufficiently longthat crustal cooling enhances the elastic response of magma reservoir hostrocks, the likelihood for melt storage is reduced. Accordingly, we use ourdata analysis and conceptual model to explore a conjecture that the recordof Mount Berlin volcanism provides a novel constraint on the timing of meltinjections into the crust from the mantle. Indeed, time-dependence in thedelivery of mantle melts is expected from models of mantle melt extrac-tion [39], is evident in observational data from well-studied continental riftsystems [21], and has entered into recent studies of the consequent crustalrheological evolution [38, 45].To explore this idea quantitatively, we extend [38] to develop a 1D numer-ical model for the thermal and rheological evolution of the crust beneath theWAIS that includes the combined thermal effects of a periodic melt supplyfrom the mantle and overlying ice sheet insulation (section 3.1 - numericalmodel set up). Applying additional constraints from crustal heat flow stud-ies at West Antarctica, we use the model to map a parameter space thatidentifies how evolving ice insulation and periodic melt injections modify aproclivity for eruptions with a frequency modulated by glacial cycles relatedto periodic changes in Earth’s obliquity. We then use this parameter spaceto identify bounds for the cadence of melt delivery from the mantle into theWest Antarctic magmatic system.8Chapter 2Data Analysis andConceptual Model2.1 Volcanism and Glacial Forcing for the last130 ka, Marie Byrd LandThis study uses a West Antarctic volcanic history (mostly Mount Berlin[18, 77]) reconstructed initially from 48 tephra layers identified in the MountMoulton Blue Ice Field in the Flood Range of Marie Byrd Land (Figures1.4 and 2.1) [15, 18, 78]. Eight of these tephra layers have been datedwith high-precision 39Ar/40Ar methods [18, 77]. Using these ages as markerhorizons [61] constructed a time-to-depth scaling in the stratigraphy thatwas subsequently calibrated to the AICC2012 timescale [4, 72, 73]. Applyingthese results, we compile a history of discreet eruption events since 130 ka(Figure 2.1, gray bars). To produce a continuous volcanic history, followingJMS04, we interpolate the record of discrete events with a moving Gaussianwindow with a width that we vary, depending on the reported uncertaintiesin the reconstructed eruption ages.How to define clusters of eruptions in time in such a short volcanic his-tory is not straightforward. Unequivocal evidence of increases in eruptionfrequency relative to a well-understood background cadence and pace of vol-canism is not possible, which is in marked contrast to the much longer recordanalyzed in JMS04 and other studies [41]. Nevertheless, two features in theeruption time series are of interest. First, the six events between 103-123ka reflect a much higher average eruption rate than that over the precedingfour 20 kyr intervals (one event over 100-80 ka, two over 80-60 ka and 60-40ka, and one over 40-20 ka). Second, from 20-0 ka five eruptions occur withfour events over only 200 years at ∼8 ka. We will take these youngest best-age-resolved events to be “cluster 1” in our analysis. Although uncertaintiesin event timings cause the six older events to span about half of a 41 kyrobliquity cycle, we propose these events as “cluster 2”.To explore the glacial modulation of volcanism in West Antarctica we92.1. Volcanism and Glacial Forcing for the last 130 ka, Marie Byrd LandRaw DataObliquity8 ka Cluster 110 ka Clustera)b)f)e)d)c)8 ka cluster110 ka clusterFigure 2.1: Comparison of Mount Moulton eruption frequency time series tothe first derivative of the LR04 δ18O stack. All time series are differentiatedusing the numpy function “gradient” which performs the centered differenceapproximation at all interior points and forward and backward euler methodat the end points. Dash-dotted orange line in a, c, and d is the first derivativeof LR04 and has been bandpassed between 1/5 and 1/200 kyr−1. Solidorange line in b, e, and f is the LR04 signal after removing frequenciesabove 1/30 kyr−1 and below 1/55 kyr−1 to isolate the obliquity period.Bars are eruptions with uncertainties indicated at their tops. Black, blue,and red dash-dotted lines are the smoothed eruption frequency time serieswith a Gaussian bin widths of 500, 1000, and 5000 years, respectively. Black,blue, and red solid lines are the first derivative of the continuous eruptionfrequency curves. Red zones highlight areas where there is no response to theglacial unloading and green zones highlight areas where there is a responsewithin 1/4 of a 41 kyr obliquity period (see text). (a) Full LR04 timeseries after smoothing, interpolation, and differentiation and the eruptionfrequency time series. (b) Obliquity signal in the LR04 series found usinga bandpass filter. (c-f) a-b zoomed in where the left column of plots spans0-35 ka and the right column of plots spans 100-140 ka.102.2. Conceptual Model: Predicted Amplitude and Phase Lag for Orbital Forcing of Volcanismrequire a complementary time series of land ice volume changes that is notbiased by local climatic conditions. [47] compiled benthic δ18O data fromacross the globe to create a 5.3 Myr time series (LR04). To illustrate therate of change of ice loading we show the first derivative of the LR04 seriesafter bandpassing the data to isolate the obliquity signal (Figure 2.1b).An initial visual comparison of the time series for the rates of changeof deglaciation and volcanism suggests that although clusters one and twomay be related to deglaciation, there is little indication of such modulationover the intervening period from 100-20 ka. In particular, cluster one lagsthe peak rate of deglaciation in LR04 by about six kyr and is well withinthe maximum lag of 1/4 of an obliquity cycle required by the JMS04 model,which we discuss in detail below. By contrast, cluster two lags the peakrate of deglaciation by approximately 14 kyr. Significantly, this lag is about40% longer than the permitted 1/4 obliquity cycle limit and may suggest adifferent underlying mechanism. However, this picture must be viewed cau-tiously: the uncertainty in the ages of these older events limits the Gaussianbin width used to smooth the data to about five kyr and so this lag is pos-sibly long simply as a result of the uncertainties in reconstructed eruptionages. We return to potential implications of the absence of glacial controlbetween clusters one and two and the long lag associated with cluster twofor the dynamics of the Western Antarctic magmatic system in section 5.2.2 Conceptual Model: Predicted Amplitude andPhase Lag for Orbital Forcing of VolcanismA critical assumption in JMS04 and in our treatment is that crustal magmareservoirs support volcanism throughout the 130 kyr history of glacial/in-terglacial cycles. This condition is met where the magma supply from themantle is sufficiently large, on average, that the rate of pressurization ofcrustal reservoirs exceeds the rate at which resultant elastic stresses in hostwall rocks are relaxed through effectively viscous deformation [34]. A keycontrol, the Deborah number (De) conventionally expresses the ratio of thetimescale for thermally-activated viscous relaxation τm = µ(T )/E to thetimescale τf over which elastic stresses accumulate (or dissipate) (Table 3.1)[38]. Whereas eruptions can occur where De > 1 (eruptive regime), magmahas a high likelihood to remain stored in the crust where De < 1 (storageregime).At West Antarctica, it is unclear whether it is reasonable to simplyassume that De > 1 continuously over the 36 Myr life of the volcanic system112.3. Glacial modulation of volcanism through a temperature-dependent crustal filteras was done in JMS04. In particular, glacial/interglacial cycles since 130 kaare superposed on a thick ice sheet that has inhibited crustal heat loss for 34million years (Figure 2.2). Compared to ice-free situations, the addition of athree km-thick layer of insulating ice can cause De to decline significantly andwe discuss this effect quantitatively in section 3.1. A second considerationis that the extraction of mantle melts through compaction processes andtheir injection into the actively rifting lithosphere and crust beneath theWAIS will be highly time-dependent [21, 39]. Depending on the periodbetween melt injections τin, the thermal effect of this time-dependence inmelt supply can exert a basic control over the tendencies for volcanism andcrustal magma storage [38] (Figure 2.2).A useful metric for this thermal effect is the Transport number Tr =τin/τd, where τd is the time scale for thermal diffusion across the crustthickness hosting the erupting magma reservoirs [38]. In particular, whereTr≫ 1, heat inputs are diffused over the crust half thickness and a periodicmantle supply delivers a perfectly periodic variation in crust temperature.By contrast, where Tr ≪ 1 variations in the mantle melt supply are suffi-ciently short period that their effects build up over time akin to an effectivelyconstant mantle heat flux. How the combined effects of obliquity-driven pe-riodic changes ice sheet thickness and a time-dependent melt supply enterinto the glacial modulation of volcanism is unclear. In particular, whetherthese time-varying crustal heating effects conspire to inevitably drive themagmatic system into a storage regime or, alternatively into oscillating stor-age/eruption regimes over the 34 Myr age of the WAIS is unknown and acentral aim of this study.2.3 Glacial modulation of volcanism through atemperature-dependent crustal filterFollowing JMS04, the response of a pressurized crustal magma reservoirthat is supporting volcanism (“critically overpressured”) to glacial unload-ing (forcing) d∆Pf/dt is approximately given by the evolution equation forreservoir excess pressure ∆Pch:d∆Pchdt+∆Pchτm= −d∆Pfdt. (2.1)where τm = µ(T )/E is the temperature dependent Maxwell relaxation timeof the crust (Table 3.1).122.3. Glacial modulation of volcanism through a temperature-dependent crustal filterDepth = 3-8 kmDepth (L) = 30 kmSilicic ReservoirBasaltic Reservoirsmean ice heightperiodic partH = 1-3 kmHradhr = 8 kmT0TsurfT0(t)H = Hi(t) + Hi,0qin(t) = qm,0 + δqm(t)kiHi(t) + Hi,0qout(t) = (T0(t)+ Tsurf)TmFigure 2.2: Schematic definition of the mechanical and thermal forcingsrelevant to the paper. Heat and melt is delivered from the mantle into anactive magmatic system in the crust composed of basaltic sills and intrusionsat 30 km depth[44] and silicic reservoirs at ∼ 3 − 8 km depth [44, 51].These feed the eruptions represented by the bar plot in Figure 2.1. Thevertical distribution of crustal radioactive heat production, Hrad augmentsthis magmatic heating. The geothermal heat flux at the top of the crustis transported across the ice sheet by thermal conduction. To introducethermal and mechanical orbital modulation, we vary the thickness H ofthe ice sheet. A periodic variation in the magma (and hence magmaticheat) input δqm(t) generates a time-dependent heat flow into the base ofthe crust, qin(t). Temperature increases with depth with a limit on theice/crust boundary of T0(t) = 0◦C (the melting point of ice) and a limit atthe crust-mantle boundary, Tm = 1100◦ C (liquidus temperature of crustalrocks assuming they are basaltic andesite [38]). Note that this sketch is notto scale.132.3. Glacial modulation of volcanism through a temperature-dependent crustal filterTreating the crust as a mechanical filter that modulates the magnitudeand timing of any response to surface deglaciation (Figure 2.3), the Fouriertransform of Equation 2.1 gives the solution in the spectral domain in termsof the magnitude of a transfer function that expresses the ratio of the rateof change of reservoir pressure to the rate of change of glacial unloading:|H(iω)| =(∆Pchτm)/(∆Pfτf). (2.2)Here, H(iω) is the transfer function and τf is the timescale of glacial un-loading (glacial period). Assuming the reservoir supports active volcanism,∆Pch = ∆Pcrit, which is the critical overpressure required to drive magmato the surface in dikes without freezing [34, 66]. The time lag between theglacial forcing and the volcanic response is obtained from the phase:Θ =2πΓτf. (2.3)where Γ is the time lag, which is maximized at 1/4 τf (Figure 2.3). SeeAppendix B for a more detailed derivation and description of the TransferFunction.To understand the thermal control over the lag time Γ and the mag-nitude of the crustal mechanical response it is instructive to carry out athought experiment (Figure 2.3). In the special case discussed briefly in sec-tion 1 of an approximately fixed magma supply (Tr→ 0), the establishingof an insulating ice sheet will cause the crust to warm and De to declinemonotonically over the thermal history of the crust hosting the magmaticsystem. The time lag between glacial unloading and the volcanic responseshould increase, in turn, at a rate that depends on the thermal resistancesupplied by the ice layer. To build intuition, consider the evolving crustalthermal response to a rapidly-emplaced three km-thick ice layer 2.3. Thethermal affect of the ice sheet should be felt at five km depth after approxi-mately τd. At early time (t << τd, “1” in Figure 2.3), the magnitude of themechanical response of the crust is maximized and the time lag is about fivekyr. By contrast, five Myr later in the evolution (t ∼ τd, “2” in Figure 2.3)De declines by almost an order of magnitude, the magnitude of the responseis only 10% of the input deglacial forcing and the time lag is now almost 10kyr.142.4. Hypothesis and PlanTemperatureViscosityHighHighLowLow12Figure 2.3: Time Lag and Magnitude plot of the response of the crustal filterto an obliquity driven glacial forcing (τf = 41 kyr). The max time lag is 10.25kyr: 1/4 of τf . The time lag and magnitude are plotted against De, which isequivalent to the normalized frequency of the transfer function (AppendixB). “1” indicates lag and magnitude in an underdeveloped magmatic system.“2” indicates lag and magnitude in a more developed, hotter magmaticsystem.2.4 Hypothesis and PlanA magma supply sustained for time scales exceeding τd will drive De to beless than one and thus drive the magmatic system into a storage regime onaverage. Depending on the magma supply, variations in the ice thicknessand, thus the thermal resistance of the ice layer, may temper this evolutionbut will not alter its course. However, if the magma supply is quasi-periodicwith a recurrence interval that is long in comparison to τd (Tr > 1), this timedependence may cause the magmatic system to enter and exit a rheologicalregime favoring melt storage. We hypothesise that the physical lag at eightka and the absence of any physical lags before eight ka record a transitiondiagnostic of this oscillating behaviour. If this picture is reasonable, it re-152.4. Hypothesis and Planinforces a basic control of the Transport number Tr suggested in [38]. Italso suggests more generally that it may be possible to recover the grossbehavior of Tr from data analysis and constrain average mantle volume fluxto the crust which are challenging or impossible to capture otherwise. Anobvious limitation of our study is that the short volcanic record extends overonly about three obliquity cycles. Bearing in mind this practical reality, wedevelop and test the transfer function method nevertheless.16Chapter 3Thermal ModellingApproach3.1 Heat transfer across the crust with aninsulating ice sheet and periodic mantleupwellingThe heat transfer problem is defined in Figure 2.3. The heat diffusion equa-tion, boundary conditions, and initial condition at the mantle/crust andcrust/ice interfaces are:∂T∂t= κ∂2T∂z2+ S (3.1a)−kc∂T∂z(0, t) =kiHi(t) +Hi,0(T0(t)− Tsurf ) (3.1b)−kc∂T∂z(L, t) = qm + δqm(t) (3.1c)T (z, 0) = −A0h2rkce−zhr +(qmkc− A0hrkce−Lhr)z + TsurfA0h2rkc.(3.1d)Here, the volumetric heat source S = (Hrad +Hmelt) /ρcp (units in Table3.1). In addition to the radiogenic heat source Hrad we add volumetriccontributions from crustal melting Hmelt. A background mantle heat fluxqm is delivered across the crust/mantle boundary with an additional periodicfluctuation δqm(t) = cqm sin(2πt/τin) that we take to be a proxy for time-dependent melt injections. In the equation for δqm(t), cqm is the maximumvalue that δqm(t) will reach and c is the factor above the background heatflux. For the boundary condition at the crust/ice interface (equation 3.1b)ki, Hi(t), T0(t), and Tsurf are the average conductivity, time dependent icethickness, ice sheet base temperature, and surface temperature of the icesheet, respectively.Before introducing magmatism and an ice sheet at t = 0 Myr we establishan initial crustal temperature profile, unaffected by the two thermal forcings,173.1. Heat transfer across the crust with an insulating ice sheet and periodic mantle upwellingParameter Definition Value and Unitsτm Maxwell viscoelastic relaxation timescale [yrs]τf glacial forcing timescale [yrs]τin Timescale of magma injection into the crust 1e4-1e8 [yrs]τd Timescale of heat diffusion across the crust (L/2)2/κ [yrs]De Deborah Number τm/τfTr Transport Number τin/τd∆Pch pressure of a magma reservoir in excess of litho-static[Pa]∆Pcrit critical overpressure to drive magma to the sur-face and erupt[Pa]∆Pf pressure changes related to glacial unloading [Pa]E Young’s modulus of the crust 1010 [Pa]µ Dynamic viscosity [Pa-s]Γ time lag of eruption frequency behind glacial un-loading[kyr]qm background mantle heat flux 0.035 [W/m2] [1]δqm periodic part of mantle heat flux [W/m2]Tsurf temperature at the surface of the ice sheet 238 [K] [3]T0 temperature at crust/ice interface [K]Tliq emplacement temperature of magma from themantle1100 [◦ C] [6]ki average thermal conductivity of the ice sheet 2.1 [W/(m K)] [4]kc thermal conductivity of the crust 2.5 [W/(m K)] [1]κ thermal diffusivity of the crust 1.2e-6 [m2/s] [5]hr e-fold distance for radiogenic heating 7.5-10 [km] [1]L Average crust thickness 30 [km] [2]Hi,0 average ice sheet thickness 3000 [m]Hi(t) periodic variations in ice sheet thickness [m]Hrad Radiogenic Heat Source [W/m3]Hmelt Melt Heat Source [W/m3]A0 surface radiogenic heating 4 [µW/m3] [1]Table 3.1: Parameter and variable definitions. Values defined where theyare not time or temperature dependent. [1] - [48] [2] - [63] [3] - [10] [4] - [26][5] - [76] [6] - [38].183.2. 1D Heat Flow Modelsthat is given by Equation 3.1d. We find this profile by solving the steadystate form of equation 3.1a with a constant temperature surface boundarycondition and fixed mantle heat flux:−kc∂2T0∂z2= Hrad(z) = A0e−z/hr (3.2a)T0(0) = Tsurf (3.2b)−kcdT0dz(L) = qm (3.2c)Here, the exponential form of Hrad is consistent with [43] who analyzedplutons in the United States and created a physical model which shows thata longer-lived, hotter magmatic province would be enriched in radioactiveelements in the shallow crust because of magma differentiation. From thismodel, the most accurate distribution of radiogenic heat production is anexponential decrease from the top of the crust over an e-fold depth whichis distinct for each volcanic province. Applying these results, [48] find thatthe max heat production A0 ≈ 4 µW/m3 and [51] suggest an e-fold depthof eight km for West Antarctica.3.2 1D Heat Flow Models3.2.1 An analytical solution of a simplified problemTo establish a benchmark for the numerical scheme we develop next, wefirst solve a simplified version of Equation 3.1 analytically, with a perfectinsulation boundary condition at the top of the crust (qout = 0), no radio-genic heating (Hrad = 0), and fixed heat flux at the base (qin = qm). Thissimplified system of equations is:∂T∂t= κ∂2T∂z2(3.3a)∂T∂z(0, t) = 0 (3.3b)−kc∂T∂z(L, t) = qm (3.3c)T (z, 0) =qmkcz + Tsurf . (3.3d)193.2. 1D Heat Flow ModelsThe particular solution is:T (z, t) = u1(z, t) + u2(z, t) (3.4a)u1(z, t) =qm2Lkcz2 +κ(qm/kc)Lt (3.4b)u2(z, t) =∞∑n=0Bnexp(−κλ2t) cosλt (3.4c)Here, the temperature T (z, t) is separated into inhomogenous (u1) and ho-mogenous (u2) components where Bn and λ = nπ/L are the Fourier coeffi-cients and eigenvalues of the homogenous solution, respectively.3.2.2 A numerical solution of the full problem: TheDufort-Frankel MethodThe boundary conditions, time- and spatially-varying internal heating (Fig-ure 2.2), are challenging to implement with standard finite difference schemes.Most finite difference solvers in python step through time within the func-tion, however the temperature and time dependence of the boundary con-ditions and limits require checks on the temperatures at the boundaries ateach timestep. These checks cannot be implemented in standard pythonsolvers so we are required to write our own finite difference functions. Con-sequently, we implement a DuFort-Frankel (DF) scheme to solve the fullproblem numerically. DF uses a centered difference method in space andtime. To ensure stability, it includes a temporal average of the central spacestep in the spatial derivative [52]. The discretized form of Equation 3.1a is:uj+1i − uj−1i2∆t= κuji+1 − 2uj+1i −uj−1i2 + uji−1∆z2+ S, (3.5)where i is the space coordinate and j is the time coordinate. This centereddifference scheme is second order accurate, and the temporal average of thecentral time step in the spatial derivative makes the DF scheme uncondi-tionally stable. We check the accuracy of the time-stepping by applying theDF scheme to the simplified problem above (Equation 3.3) and comparingresults to the analytical solution (Figure 3.1). Unsurprisingly, solutions areboth stable and accurate where ∆t ≤ ∆z2/2κ [52]. Figures 3.1b-e show thatboth solutions depart from the analytic solution at the top of the crust, sowe have zoomed in to the top three km in Figures 3.1i-l. The DF solutionreaches the maximum error at the top of the crust after 0.1 τd, while the203.2. 1D Heat Flow Modelsa) g)f)e)d)c)b)n)m)l)k)j)i)h)Figure 3.1: Differences between the analytic solution and DF (blue) andFTCS (orange). ∆T refers to the difference between the analytic and nu-meric solutions. For DF: ∆t = 50 years, ∆z = 50 m and ∆z2/κ = 66. ForFTCS: ∆t = 50 years, ∆z = 72 m (divergent at the same spatial step asDF), and ∆z2/κ = 137. (a-g) Differences at times specified as a fraction ofτd: (a) t = 0, (b) t = 0.01τd, (c) t = 0.1τd, (d) t = 0.5τd, (e) t = τd, (f) t =3τd, (g) t = 6τd. (h-n) focus on the top three km of the crust for the sametimes as in (a-g).213.3. Numerical Experimental SetupFTCS continues to diverge from the solution at all time steps shown. TheDF scheme is at most 0.2 K hotter than the analytic solution while theFTCS scheme continues to diverge from the true solution. For this study, adifference of 0.2 K has no effect on the interpretation of the results.The final discretized form of Equation 3.5 is:uj+1i = ηuj−1i + χ(uji+1 + uji−1)+ ζ (3.6)whereα =2kc∆tρcp∆z2η =1− α1 + αχ =α1 + αζ =2∆t(1 + α)ρcpS.There is no temperature for j = −1, therefore the FTCS scheme is usedat t = 0 because this node is not required. The same is true for the boundarynodes because the i− 1 and i+ 1 nodes do not exist at the edges.3.3 Numerical experimental setup: Effects ofinsulation and periodic mantle heat deliveryon crust thermal evolution and DeFor both experiments, radiogenic heating is assumed to have an exponentialdecrease with depth over an e-fold distance of eight km [51]. Other modelsfor the distribution of radiogenic heating and their effects on the experimentresults are discussed in Appendix D.3.3.1 Experiment One: Constant qm, Variable HiTo isolate and understand the potential maximal insulation effects of anorbitally-modulated ice sheet thickness on a steady-state magmatic system,we let Tr → 0 and vary the ice sheet thickness between 1500 and 4500 mwith a period of 100 kyr [14, 22]. Following the establishing of a steady-state, we introduce the ice sheet and track crust temperature profiles andDe at five km depth.223.3. Numerical Experimental Setup3.3.2 Experiment Two: Variable qm, constant HiTo isolate and understand effects of periodic magma injections from the man-tle we carry out five simulations where the peak-to-peak periods of mantleheat flux are varied τin = 1e4, 1e5,1e6, τd, and 1e7 years corresponding toTr of 0.002, 0.017, 0.169, 1.000, and 1.685 respectively. We maintain theice sheet thickness at three km, similar to present day. We again tracktemperature and De at five km depth.23Chapter 4Results4.1 Data Analysis: Physical and unphysical lagsbetween maximal rates of deglaciation andchanges in eruption frequencyFigure 4.1 shows time lag and magnitude plots with the lags of the 110 andeight ka eruption clusters mapped on to them so that De and magnitudecan be inferred. The cluster of eruptions at 110 ka lags the obliquity drivenglacial forcing by 13.9 ± 1.7 kyr. This lag is about 3.5 kyr longer than themaximum physical lag of 10.25 kyr that is possible with the mechanismoutlined in Figure 2.2a. Therefore the data constrain neither a transferfunction magnitude nor a value for De at 110 ka. By contrast, eruptionsclustered at eight ka lag this glacial forcing by 6.0 − 6.5 kyr, depending onthe width of the gaussian window used on the discrete eruption series. Theinferred lags restrict De to ≈ 0.6 and 0.8 and magnitudes of approximately50% and 65% that of the input.4.2 Thermal modelling: Time-dependent icesheet insulation and magma supply4.2.1 Experiment One: Constant qm, Variable HiThe introduction of an insulating ice sheet causes magma reservoir wall rocksto rise 35 K, driving a gradual one order of magnitude decrease in De (Figure4.2). No periodic variations in this De related to temporal variations in icesheet thickness occur because the small amplitude of these thermal effectsare attenuated at five km depth. The crust reaches a new thermal steadystate after approximately 1.5 τd. Whereas the evolution with the lowestmantle heat flux (red lines in Figure 4.2) is consistent with active volcanism(De < 1), ice sheet insulation with the highest mantle heat fluxes drives themagmatic system into a regime favoring magma storage (De > 1).244.2. Thermal modelling: Time-dependent ice sheet insulation and magma supplyc)8 ka eruption frequency cluster110 ka eruption frequency clustera)d)b)Figure 4.1: (a,b) Phase and magnitude plot for an obliquity forcing with thelags related to increases in eruption frequency at ∼8 ka and ∼110 ka (seeChapter 2). Black dotted line is the lag for the 110 ka cluster found witha 5 kyr Gaussian bin width with uncertainties (red dotted lines) based onreported eruption ages. Dash-dotted lines are the lags for the eight ka clusterfor two bin widths: 500 (black) and 1000 (blue). Uncertainties related toeruption ages are 100 years. (c,d) Simplified versions of Figures 2.1e and f.Black, blue, and red sold lines are the first derivative of eruption frequencycurve and solid orange line is the first derivative of LR04 obliquity signal.254.2. Thermal modelling: Time-dependent ice sheet insulation and magma supplyEruptionStorageSteady StateIce SheetFormsa)b)Figure 4.2: Experiment One: Constant mantle heat flux, variable ice sheetthickness. Evolutions of (a) temperature and (b) De with the introductionof an ice sheet at 3τd = 6.25e6 [yrs]. Once an insulating ice sheet forms,the temperature of crustal rocks hosting the magmatic system at five kmdepth climbs and De falls. The four simulations have progressively higheraverage mantle magma volumetric input rates: 0.003 km3/year (red), 0.005km3/year (black), 0.006 km3/year (blue), and 0.008 km3/year (green) (Table3.1). We vary the ice sheet thickness periodically by 3000 m between 1500and 4500 m with a period of 100 kyr (inset).4.2.2 Experiment Two: Variable qm, Constant HiA periodic delivery of mantle melts with a recurrence interval comparableto or longer than the diffusion time τd (Tr ≥ 1) drives > 50 K temperaturevariations that cause De to oscillate between regimes favoring eruption ormagma storage (Figure 4.3). By contrast, the steady buildup of heat and <10 K variations for Tr < 1 maintains the crust in a magma storage regime.264.2. Thermal modelling: Time-dependent ice sheet insulation and magma supplyEruptionStoragea)b)Figure 4.3: Experiment Two: Periodic mantle heat flux with a constant icesheet thickness Hi = 3 km. (a) Crust thermal evolutions for different Trat five km depth. Periods, τin, between melt injections are: 0.0016τd (1e4yrs) (black), 0.016τd (1e5 yrs) (orange), 0.16τd (1e6 yrs) (blue), τd (6.25e6yrs) (green), 1.6τd (1e7 yrs) (red). Inset shows the variation in mantle meltsupply with τin = τd. (b) Corresponding De regimes. For Tr ≥ 1 themagmatic system oscillates between Eruption and Storage regimes. Shorterperiods between injections drive favor Storage.27Chapter 5DiscussionIn this section we will first discuss the results from Experiments One andTwo and the implications of those results on interpreting the time lags fromMount Berlin. Insulation and mantle magma supply are the only thermalforcings in our model and they determine whether the crust predominantlyerupts or stores magma. We then discuss a special case where periodicmantle magma supply combined with insulation causes the crust to enter atransitional regime, switching between storage and eruptive behavior. Thetime lags from Mount Berlin suggest that both eruptive and storage be-haviors occur, which would indicate that the magmatic system is in thistransitional regime. Finally we combine all the results along with obser-vational studies in Marie Byrd land and use them to constrain Tr and themagma production rate for this magmatic system.5.1 Conditions for gradual and punctuatedthermal evolutions into storage, eruption andoscillating regimesApproximately steady magma supplies (Tr≪ 1) in the range 0.003-0.007km3/year, combined with a fixed insulating ice sheet will drive De belowone and the magmatic system inevitably into a storage regime (Figures 4.2and 4.3). This result is insensitive to orbitally-driven changes in the icesheet thickness and thermal resistance. Because the crustal thermal effectsof the variations in ice sheet thickness over time scales of 100 kyr are rapidlyattenuated in the upper crust (Figure 4.2), they induce no shifts betweenthe eruption and storage regimes.In contrast, variations in the Transport number exert a powerful con-trol over the eruption and storage behaviour of the model system. Whereas∼ 10−3 < Tr <∼ 10−1 give rise to temperature variations < 10K and neg-ligible shifts in De, Tr ≥ 1 drives temperature variations of order 50-100 Kand associated periodic shifts between eruption and storage regimes. Ap-plied to the time lags in Figure 4.1, and assuming that the absence of glacial285.2. Transitional Regime: Thermal Variations around a De = 1modulation before eight ka is a reasonable hypothesis, our results suggestthat the combination of ice sheet insulation and magma input in a Tr ≥ 1regime is an underlying mechanism. Indeed, such behavior is not possibleunless Tr ≥ 1.5.2 Transitional Regime: Thermal Variationsaround a De = 1Magmatic systems can exhibit both storage and eruptive behaviours. Wepropose that a system that oscillates around De = 1 with variations in Degreater than one order of magnitude is transitional, exhibiting both types ofbehaviour depending on the ice insulation and magma input. A proclivityfor storage or eruptive behavior would be reflected in unequal time spent inone or the other regime. To quantify this tendency in our simulations, weplot the ratio of the time that a simulation spends in the eruptive regimeto the time it spends in the storage regime for seven melt injection rates(Figure 5.1). This property is measured from 2τd to 4τd (two crustal heatdiffusion periods) for a simulation with Tr = 1. For each melt injectionrate the simulation is run with and without an ice sheet to show the effectof insulation on the dominant behavior of a magmatic system. If the ratiois greater than one the dominant behaviour is eruptive; if the ratio is lessthan one the dominant behaviour is storage. The first blue dot on the lefthand side marked “1” has a ratio of approximately 14:1, meaning for every14 time steps spent in the eruptive regime, one time step is spent in thestorage regime. In other words that simulation displays eruptive behaviorapproximately 93% of the time. Conversely, the last red dot on the rightmarked “2” has a ratio of 1:40. This magmatic system will display storagebehavior approximately 98% of the time. In between these end memberscenarios lies the blue dot marked “3” which has a ratio pf approximately1:1. This system will behave in equal parts storage and eruptive throughoutits lifetime. Therefore, instead of viewing these magmatic systems as abinary storage vs eruptive system, the reality will be a spectrum of storageand eruptive behaviors with, more often than not, one behavior dominating.How, then, does insulation affect the dominant behaviour of a magmaticsystem? When Tr ≥ 1 insulation will increase the percentage of total timethat the system spends in the storage regime by decreasing the average Deof the crust. For magma influx with Tr = 1 and an average input rateof 0.009 km3/year (max rate of 0.03 km3/year), the eruptive/storage ratiois 1.93 when no ice sheet exists and 0.45 when insulation occurs (Figure295.3. A West Antarctic Tr and melt supply: Time lag and geothermal heat flux constraints5.1). In this case insulation has changed the system from eruptive dominantbehavior to storage dominant behavior.Equal time in both regimes123Figure 5.1: Ratio of time steps spent in the eruptive regime to time stepsspent in the storage regime; blue circles have no ice in the simulation andred circles have an an ice sheet. The ice sheet oscillates with the sameperiod and magnitude as Experiment 1. Tr = 1 for all simulations. The icesimulation for the melt injection rate of 0.07 km3/year has a ratio of 0 anddoes not appear on a logarithmic plot.5.3 A West Antarctic Tr and melt supply: Timelag and geothermal heat flux constraintsThe time lags for clusters one and two place bounds on the possible Trand volume flux conditions for the magmatic system feeding Mount Berlin(Section 4.1). Crucially, the cadence and magnitude of the magma supplyfrom the mantle must allow for glacially-modulated volcanism at eight ka,after 34 million years of ice insulation. To constrain the possible Tr-volumeflux parameter space, we will use three observational constraints:1. Geothermal heat fluxes (GHF) in the region around Mount Berlinconstrain the temperatures, and hence the range of De values at five kmdepth in the crust that must occur during the intervals between peaks inthe magma supply. These can be compared with the De values inferredfrom Figure 4.1. Accurate estimates of the GHF in West Antarctica are305.3. A West Antarctic Tr and melt supply: Time lag and geothermal heat flux constraintschallenging but intensely studied because of their implications for the sta-bility of the WAIS with climate change. Estimates of the GHF derived fromCurie point depths in the crust on the basis of magnetic surveys, from man-tle heat fluxes inferred from short wavelength lateral variations in uppermantle seismic velocity, from local radar measurements and from availablebedrock borehole data restrict GHF to be approximately in the range 90 -120 mW/m2, which is consistent with high heat flow in volcanically activeregions in other continental rift systems (see recent review by [2, 11] andreferences therein). Assuming a linear geothermal gradient, this range inGHF corresponds to 0.7 < De < 50 at five km depth and our estimate of Defrom the time lags is at the low end of this range (Figure 5.2).0.5 kyr sd1kyr sdGHF range from literature (Artemieva 2022,Burton-Johnson et al. 2020)Figure 5.2: The black curve is the equivalent De at five km depth for arange of geothermal heat fluxes assuming a linear geothermal gradient. Thehorizontal red arrows set the range of De from the geothermal heat fluxdata summarized in [11] and [2], 90 - 120 mW/m2 or 0.7 < De < 50. Thehorizontal black and blue arrows set the range of De from Section 4.1, 119-125 mW/m2 or 0.6 < De < 0.8. The black and blue arrows are the De fromSection 4.1 for 0.5 and 1 kyr Gaussian windows, respectively.2. The long-term average eruption rate at continental rifts globally andan estimate at Mount Berlin derived from the volume and age of the edi-fice can help to restrict the magnitude of the heat flux excess δqm in thesimulations. Some assumption about the degree to which mantle melts are315.3. A West Antarctic Tr and melt supply: Time lag and geothermal heat flux constraintsstored under potentially De∼1 conditions must first be made. Intrusive/ex-trusive ratios determined for continental rift systems are commonly around5:1 [75]. Given that ice insulation might enhance storage, consistent with thediscussion above, this ratio is probably a lower bound for the Mount Berlinmagmatic system. Nevertheless, taking the largest possible extrusive outputfor continental rifts of 0.06 km3/year then a 5:1 ratio implies a maximummantle production rate of approximately 0.3 km3/year. If instead we takethe median extrusive rate in the compilation of [75] to be 0.008 km3/year,the production is 0.04 km3/year.Turning to West Antarctica, [44] estimate the volume of Mount Berlinto be 200 km3. [78] give an edifice construction age of 0.01 ± 0.0053 Myrfor Mount Berlin. We use this age and the estimated volume to calculatean eruption rate of 0.02 km3/year, about twice the median value in [75] butsignificantly lower than the highest eruption rates. Whether volcanism atthe nearby Merram Peak, which formed by 0.578 ± 0.009 Myr, contributedto the volcanic record at Mount Berlin is unclear.Taking the intrusive to extrusive ratio of 5:1 and the eruption rate of0.02 km3/year, the mantle production rate is about 0.1 km3/year. We rec-ognize this value to be an upper bound (Appendix F). The contribution ofMerram peak to the volume of Mount Berlin affects the extrusive rate ofvolcanism and therefore will lower the estimated average production rate.See Appendix F for a more in depth discussion on these calculations.3. The maximum time lag found by JMS04 for basaltic volcanism fedfrom deep reservoirs hosted in warm rocks gives the minimum De foundusing the transfer function method developed in JMS04 and should be closeto the threshold at which the crust no longer responds elastically to glacialunloading. Even though this time lag is for a lower crust magmatic reservoir,analyzing the time lag with a transfer function will give results comparableto a shallow reservoir [35]. There is no response to glacial unloading beforeeight ka which suggests the De of the crust must be lower than this thresholdDe and the best estimate is De < 0.055, the lowest De found by JMS04 tostill produce a response.These constraints set the parameter space for both Tr and maximummagma injection rate and make direct estimates on mantle-crust heat ex-change. To quantify the effects of Tr and melt injection rate on the thermalregime of the crust we have run 1200 simulations of our numerical model,varying Tr and δqm in each simulation. Figure 5.3 shows the results of thesesimulations. We will use this figure and the constraints discussed above toset bounds on Tr and melt injection rate.Constraint One sets a De that the crust must cool to in periods of low325.3. A West Antarctic Tr and melt supply: Time lag and geothermal heat flux constraintsa)c)b)d)Figure 5.3: Model iterations varying Tr along the x-axis and the max δqmalong the y-axis. (a) The maximum De the crust reaches in the model after20 Myr of heating.(b) The average mantle production rate (km3/year).(c)The minimum De the crust reaches after 20 Myr of heating. (d) ConstraintOne (blue contour), constraint two (black contours), and constraint three(red contour) with the valid parameter space marked by black dashed linesbetween constraints. The black contours in (d) are average magma inputrate in km3/year.magma injection. Two contours from Figure 5.3a characterize this con-straint: simulations with a maximum De below 0.7 do not allow sufficientcooling. Constraint Two sets the maximum average magma injection rate,and therefore the maximum δqm, possible for this magmatic system. Theblack contour on Figure 5.3b characterizes this constraint: simulations withTr-δqm combinations which have an average magma injection rate higherthan 0.1 represent unrealistic melt production for this continental rift. Con-straint Three sets a De that the crust must warm to in periods of peakmagma injection. One contour from Figure 5.3c characterize this constraint:simulations with a minimum De above 0.055 do not sufficiently heat belowthis De. We combine all of these contours on one plot, Figure 5.3d, so thatwe may define a set of Tr and δqm for our model that explain the time lagsand observational behaviors of volcanism in Marie Byrd Land.The minimum average melt injection to the crust is approximately 0.01km3/year (from constraint 3), and the maximum is 0.1 km3/year (from con-335.3. A West Antarctic Tr and melt supply: Time lag and geothermal heat flux constraintsstraint 2). The crossover points between the blue and red contours constrainTr to 0.5 - 10. Tr < 1 do not oscillate over more than one order of magnitude,and therefore do not align with the data from Section 4.1. Our simulationsare restricted to tens of millions of years and Tr > 10 (τin ≥ 1e8) will notoscillate over more than one order of magnitude on this timescale. ThereforeTr should fall in the range of 1-10 but could be as low as 0.5.34Chapter 6ConclusionThis is the first study of glacial modulation of volcanism where a periodicglacial forcing is superposed on a long lived, insulating ice sheet. The pur-pose of this work is two-fold. First, we analyze a time series of eruptions fromMount Berlin to determine whether and when orbitally-driven glacial mod-ulation of volcanism occurs at Mount Berlin. Second, we exploit the timingof such modulation and the phasing between the glacial forcing and volcanicresponse to constrain the rheological structure of the crust hosting the mag-matic system and the cadence of melt input from the mantle. In markedcontrast to existing studies that identify consistent modulation over manyglacial cycles [41], we find that glacial modulation potentially occurs onlysince eight ka in a record extending to 130 ka. A key difference comparedwith all existing analyses of such effects on arc volcanoes [12, 41, 74], say, isthat periodic glaciation has been superposed on a long-lived, thick WesternAntarctic ice sheet (WAIS). We hypothesise that the insulating effect of theWAIS will act to warm the crust hosting the magma system, potentiallyenhancing a proclivity for magma storage (Deborah number De→ 1), de-pending on the timing of the magma supply. To test this hypothesis andunderstand conditions that permit glacial modulation at eight ka and notbefore, we model 1D heat flow to study how the combined thermal effects oflong-term insulation and a potentially time-dependent magma supply mod-ify the evolutions of the thermal and rheological structures of the crust. Inparticular, one goal is to determine whether either or both of these ther-mal forcings will cause large (> 50 K) oscillations in the temperature ofthe crust through time, which correspond to rheological changes sufficientto transition between eruption and storage regimes. Finally, consistent withobservational constraints on the geothermal heat flow and mass eruptionrate inferred at West Antarctica, we identify a set of conditions under whicha transition to glacial-control of volcanism by eight Ka is possible.Bearing in mind the limitations of the Mount Berlin eruption time seriesand the limitations inherent to a 1D modelling study, our results supportthe following six conclusions:35Chapter 6. Conclusion1. If the tephrachronological data from the Mount Moulton blue ice sec-tion are an accurate timeline of Mount Berlin eruptions, then there isno evidence of a response to glacial unloading at Mount Berlin untileight ka.2. At eight ka there is possible evidence for glacial unloading via thephase lag of 6.0 - 6.5 kyr3. Thermal modelling along with the time lag at eight ka constrains De atthat time to be in the range 0.6-0.8, approximately order 1. Thus, thetime scale for stress changes imparted by glacial unloading is compara-ble to the time scale for the thermally-controlled viscoelastic relaxationof these changes and the magmatic system has comparable proclivitiesfor volcanism or magma storage.4. For all magma supply conditions, insulation will cause a decrease inDe of about one order of magnitude over about 2 Myr, enhancing thelikelihood of magma storage.5. Periodic mantle magma supply consistent with current West Antarc-tic geothermal heat flow (a maximum long-term supply of about 0.1km3/yr) and with 0.5 < Tr < 10 (peak-to-peak magma injection pe-riods of order 106− 108 years) explains the occurrence of glacial mod-ulation at eight ka and not before.6. Our results suggest that the combined effects of long-term ice sheetinsulation and a periodic magma supply forced the Mount Berlin mag-matic system into a transitional regime where both storage and erup-tive behaviors have occurred since 130 Ka. Our results constrain theconditions that signal this transition.Our conclusions have important implications for future studies that seekto identify and use volcanism induced by glacial unloading to constraininformation not just about the evolving crust thermal or rheological regimebut also the time-dependence of the magma supply from the mantle. In theiranalysis of glacial modulation of volcanism in eastern California, JMSO4assumed a constant long-term magma supply to ultimately constrain thecrust rheological structure consistent with distinct time lags for silicic anddeeper basaltic magmatic responses. Because ice sheet insulation acceleratesthe thermal evolution of the magmatic system towards a transitional regime,its inclusion into our analysis adds an additional sensitivity to the temporalcharacter of the magma supply. Our results show that warming effects36Chapter 6. Conclusionof new injections that enhance storage are retained or lost from the crustdepending on the period between injections. An approximately constantaverage magma supply from the mantle over the majority of the 35 Myrlifetime of WAIS–a typical assumption in models– would, for example, leadinevitably to a transition to a storage regime well before 130 ka such that notonly is glacial modulation at eight ka not possible, but that any volcanismsince 110 ka would be surprising [34, 38]. Introducing the effects of iceinsulation have enabled the glacial modulation of volcanism to be a novelprobe of the dynamics of melt injection.Our work motivates specific future studies of West Antarctica. To beconsistent with glacial modulation at eight ka, our model predicts that themagmatic system may oscillate into and out of storage and eruption regimesseveral times over the age of the WAIS. From a modeling perspective, ex-ploring the threshold values around the transitional De=1 condition thatseparate these regimes is critical to understanding how the mechanics of thisrheological transition are expressed in the volcanic response to deglaciation.Observationally, a key scientific challenge is to extend the volcanic historyat Mount Berlin. A technical hurdle with the current eruption time series isthe need to obtain better age resolution particularly for the events clusteredat 110 ka. More broadly, an ambitious future goal is to capture volcanismacross West Antarctica over time scales spanning at least 100 ka and ide-ally longer. To this end, a useful exercise with existing data would be tocompile known eruptions to create a timeline of eruption frequency for WestAntarctica. This would require cross analyzing all ice cores, blue ice sec-tions, and volcanic deposits to verify that all known eruptions are accountedfor and each eruption is counted only once. This compilation and reanalysisof known data would not only make studies of West Antarctic eruption fre-quency possible, but would also allow analysis of individual volcanic systemseasier because many known tephra layers are already attributed to certainvolcanoes [15, 17].A provocative feature of our results is the implication that the WestAntarctic magmatic system is in a transitional regime. Such behavior isnot limited to where ice insulation occurs and is an expected feature ofthe thermal evolution of magmatic systems [38]. 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To observe this trend many studies measure the ratio of 18O to16O in the fossils of ocean-floor dwelling (benthic) organisms [7, 31, 47, 82].Comparing the fossil ratio (the sample) to a standard ratio gives the nondimensional value δ18O:δ18O =(18O16O)sample(18O16O)standard− 1 ∗ 1000 (A.1)A high δ18O means the sample ratio is larger than the standard ratio.This occurs when a larger volume of 16O enriched ice existed during thetime of the sample ratio compared to the time of the standard ratio. Byplotting δ18O as a function of time (Figure 1.1) we can create a time serieswhich tracks the relative volume of ice compared to the modern day [31, 47].Glacial periods are marked by high δ18O while interglacials have low δ18O.48Appendix BRepresenting a MagmaticSystem using FourierTransforms and a TransferFunctionGlacial cycles affect eruption frequency during the deglacial part of the cycleby changing the stresses in the upper crust, allowing for magma to reachthe surface more easily. We can describe this process through the equation:d∆Pchdt+Eµ∆Pch = −d∆Pfdt(B.1)where ∆Pch is the pressure in a magma chamber above the backgroundlithostatic pressure (over pressure), E is Young’s Modulus, µ is the viscosityof the crust around the magma chamber, and ∆Pf is the pressure causedby an ice sheet above the crust. Equation B.1 states that the change inover pressure of a magma chamber in time plus a damping term is equalto the negative change in glacial pressure in time. In term two on the LHSof equation B.1 the fraction Eµ is the inverse of the viscoelastic relaxationtimescale of the crust, τm.τm =µE(B.2)The timescale τm is the time it takes the crust to relax to an equilibriumstate of stress. A longer timescale means that the crust will take longer torelax and therefore will behave more elastically, while a shorter timescalemeans the crust is behaving more viscously. The characteristic timescale ofthe forcing, τf , will also be important. This is the timescale over which theforcing is happening.Non-dimensionalizing the equation will make this a bit easier. So to do49B.1. The Time Domainthis we say:Pch =∆Pch∆PcrPf =∆Pf∆Pcrτ =tτmInputing these new variables into equation B.1 we get:dPchdτ+ Pch = −dPfdτ(B.3)B.1 The Time DomainEquation B.1 can be separated into three parts:Input => −d∆Pfdt(B.4a)Output =>d∆Pchdt(B.4b)Dampening =>Eµ∆Pch (B.4c)The magmatic system (magma chamber and crust near the magmachamber) can be thought of as a system that filters inputs and producesoutputs. In this case the input is a periodic change in the pressure inducedby an ice sheet and the output is a change in the over pressure of a magmachamber. The right hand side (RHS) of equation B.1 is the input, the firstterm on the left hand side (LHS) is the output and the second term on theLHS describes how the filtering system dampens the input.B.2 Fourier DomainRepresenting this system in the frequency domain will make understandingthis system easier through the use of transfer functions. The transfer func-tion of a system is defined as the Fourier Transform of the output dividedby the fourier transform of the input:50B.2. Fourier DomainInput => X(iω)Output => Y (iω)Transfer Function = H(iω) =>Y (iω)X(iω)(B.5)and describes how a system responds to any input. The normalized fre-quency is ω = τmτf . For now we will leave ω in the equations, but at theend it will be useful to define it in this way. This allows us to calculate therelative magnitude and phase difference of the output of the system to anyinput.Now using the Fourier Transform and its time derivative property:F [Pch] = Pch (B.6a)F[Pchdτ]= Pch ∗ (iω) (B.6b)F[Pfdτ]= Pf ∗ (iω) (B.6c)we transform equation B.3 to the frequency domain.Pch ∗ (iω) + Pch = −Pf ∗ (iω) (B.7)In Equation B.4 we define the input to the system as the rate of changeof ice sheet height and the output as the rate of change of the overpres-sure of the magma chamber. Therefore our transfer function would be thepressurization rate of the magma chamber over the forcing rate:H(iω) =Pch ∗ (iω)Pf ∗ (iω) =Pressurization RateForcing Rate(B.8)We want to isolate H(iω) from equation B.8 in equation B.7. In thelast step below we muliply by the complex conjugate of the denominator toseparate the real and imaginary parts of the transfer function.Pch ∗ (iω)(1 +1iω)= −Pf ∗ (iω)Pch ∗ (iω)Pf ∗ (iω) = −11 + 1iω51B.2. Fourier Domain|H(iω)|Real PartImaginary PartFigure B.1: The transfer function in the complex domainH(iω) = − iω1 + iω∗ 1− iω1− iωReal ImaginaryPch ∗ (iω)Pf ∗ (iω) =[− ω21 + ω2]+i[− ω1 + ω2](B.9)Now that we have the Real and Imaginary parts of the transfer functionseparated, we can calculate the magnitude and phase. The magnitude of acomplex number is√Re2 + Im2. From figure B.1 we can see that the phase,Θ, can be represented using the definition of the cosine, sine or tangent.|H(iω)| = ω(ω2 + 1)(1/2)(B.10)Θ(iω) = tan−1(1ω)= cos−1(R[H(iω]|H(iω|) (B.11)Here we only use tangent and cosine because they are the two most usefulfor looking at end members of our equation. The two end members we are52B.2. Fourier Domaininterested in are a perfectly elastic crust, µ → ∞, and a perfectly viscouscrust, µ→ 0.Case 1 : µ→ 0 Case 2 : µ→∞|H(iω)| = 0 |H(iω)| = 1Θ(iω) =π2Θ(iω) = 0In case one we have a completely viscous material. Term two in equationB.1 goes to infinity, which means that there is infinite damping of the input.The magnitude of the transfer function goes to 0 and the phase is π/2.Referring back to equation B.8 we can see that a zero magnitude transferfunction means either the input goes to infinity or the output goes to zero.An infinitely large input is nonphysical, so we can say that the magnitude ofthe output has gone to zero. Figure B.1 tells us that a phase of π/2 meansthat the real part of the transfer function has gone to zero. So the transferfunction is purely imaginary. What these two values mean physically is thatthe crust is no longer responding to the forcing.In case two we have a perfectly elastic material. Term two in equationB.1 goes to zero, which means that there is no damping of the input. Themagnitude of the transfer function is maximized at one and the phase iszero. From equation B.8 we can see that this magnitude means that themagnitude of the output matches the input. Figure B.1 tells us that a phaseof zero means the transfer function has no imaginary part. Physically thismeans that the crust responds perfectly and in phase with the input withno lag.Another important concept that falls out of case one is the maximum lagthat is physically possible for this system. In case one we get the maximumphase of π/2. The phase of the transfer function is related to the lag of theoutput behind the input by:Γ =Θ2πτf (B.12)If we plug the maximum phase from case one into equation B.12 then weget the maximum lag of the output behind the input isτf4 . So if the outputtime series lags behind the input time series by longer thanτf4 then the lagis nonphysical.53Appendix CDerivation of InitialConditionsPreviously stated in Section 3.1 the system of equations used to calculatethe initial conditions is−kc∂2T0∂z2= Hrad = −qnethre−z/hr (C.1a)T0(0) = Tsurf (C.1b)−kcdT0dz(L) = −qm (C.1c)If radiogenic heating Hrad is assumed to be constant we can isolate∂2T0∂z2as:∂2T0∂z2=Hrad−kc . (C.2a)(C.2b)Integrating twice gives:∂T0∂z= −Hradkcz + c1 (C.3)T0(z) = −Hradkcz22+ c1z + c0 (C.4)To solve for c1 we use the neumann boundary condition:c1 =HradkcL+qmkc(C.5)To solve for c0 we use the dirichlet boundary condition:c0 = Tsurf (C.6)54Appendix C. Derivation of Initial ConditionsThis leaves use with the final result:To(z) = −Hradkcz22+(HradkcL+qmkc)z + Tsurf (C.7)If radiogenic heating is assumed to be exponential in space in the formsuggested by [38] we isolate ∂2T0∂z2like this:∂2T0∂z2=qnetkchre−z/hr (C.8)Then we integrate twice:∂T0∂z= −qnetkce−z/hr + c1 (C.9)T0(z) =qnethrkce−z/hr + c1z + c0 (C.10)To solve for c1 we use the neumann boundary condition:c1 =qnetkce−L/hr +qmkc(C.11)To solve for c0 we use the dirichlet boundary condition:c0 = Tsurf − qnethrkc(C.12)Finally we are left with:T0(z) =qnethrkce−zhr −(−qnetkce−Lhr − qmkc)z + Tsurf − qnethrkc(C.13)55Appendix DRadiogenic Heating:Constant vs ExponentialDecreaseIn this study we have made the assumption that radiogenic heat productiondecreases expoentially with depth and that the e-fold distance is eight km.Previous studies of heat flow have also used a constant heat productionmodel for the crust as well as different e-fold distances [48]. The decisionto use a constant heat production or a different e-fold distances will changethe results of Section 4.2.1 and we explore these changes here.For a constant radiogenic heat production value Hrad = 1.5 µW/m3(lower bound from [48]) mantle heat flux must be between 0.85 and 2.5W/m2 for insulation to cause the system to move from the eruptive tostorage regime. In contrast, for Hrad = 3 µW/m3 (upper bound from [48])mantle heat flux must be between 0.035 and 0.41 W/m2.Both exponential radiogenic models have a maximum heat productionof 4 µW/m3 [48]. Mantle heat fluxes of 1.23-3.0 W/m2 cause the magmaticsystem in the eruptive regime to move into the storage regime for exponentialheat production with an e-fold distance of 7.5 km (Figure D.1c). For a heatproduction with an e-fold distance of 10 km, heat fluxes of 0.79-2.45 W/m2force the magmatic system into the storage regime (Figure D.1d).56Appendix D. Radiogenic Heating: Constant vs Exponential Decreasea)d)c)b)Figure D.1: Experiment One: Constant mantle heat flux, variable ice sheetthickness. (a-b) Plot of temperature and De assuming constant radiogenicheating. (a) shows the lower bound where Hrad = 1.5 µW/m3 and (b) is theupper bound where Hrad = 3 µW/m3. (c-d) Plot of temperature and Deassuming exponential decrease in Hrad. (c) shows the lower bound on the e-fold distance hr = 7.5 km and (d) shows the upper bound hr = 10 km. In (b)De above the horizontal dotted line are in the eruption regime, and beloware in the storage regime. The four simulations have progressively higheraverage mantle heat flux: 1.27 (red), 1.91 (black), 2.54 (blue), and 3.17(green). The ice sheet is introduced in the simulations where the verticalblack dashed line is. Ice sheet thickness varies periodically between 1500and 4500 m with a period of 100 kyr (shown in inset plot).57Appendix D. Radiogenic Heating: Constant vs Exponential DecreaseConstant Hrad= 3.5 μW/m3a)d)c)b)Exponential Hrad = 4*exp(-z/7500)Exponential Hrad = 4*exp(-z/10000)Constant Hrad= 1.5 μW/m3Figure D.2: Parameter Space for four different radiogenic heating mod-els. All are recreations of Figure 5.3e. (a) Lower bound of constant radio-genic heating where Hrad = 1.5 µW/m3. (b) Upper bound where Hrad = 3µW/m3. (c)Lower bound on the e-fold distance for exponential decrease inradiogenic heating hr = 7.5 km. (d) Upper bound hr = 10 km.58Appendix EInsulation and melting at thecrust/ice boundaryE.1 Resistance of the Ice Sheet: The insulationboundary conditionThe top boundary condition for Equation 3.1 is:−kc∂T∂z(0, t) =kiHi(t) +Hi,0(T0(t)− Tsurf ) (E.1)To derive this equation we assume that the temperature gradient throughthe ice sheet is linear, and that no heat is generated within the ice sheet.This allows us to state that the heat flux into the ice sheet is equivalent tothe heat flux through the ice sheet: qin = qthrough. The flux through the icesheet is:kidTdz≈ ki∆T∆z=kiHi(t) +Hi,0(T0(t)− Tsurf ) (E.2)Where ∆T = (T0(t)− Tsurf ) is the difference in temperature from the bot-tom and top of the ice sheet and ∆z = Hi(t) +Hi,0 is the height of the icesheet. The flux into the ice sheet is −kc ∂T∂z (0, t). Combining these two weget equation 3.1b.E.2 Accounting for ice sheet melting at thecrust/ice boundaryWithout accounting for ice melting at the crust/ice interface the insulationof the ice sheet causes the top boundary to heat continuously throughoutthe simulation. This boundary can reach a maximum temperature of 0◦Cbecause anything above the melting point of ice would cause the ice tomelt. To account for this, after calculating the next temperature at the top59E.2. Accounting for ice sheet melting at the crust/ice boundaryboundary we check to see if the new temperature is higher than 0◦C. If it iswe calculate a new heat flux qmelt:qmelt = ρcp∆z (Tnext − Tmelt) /∆t (E.3)where Tnext is the temperature at the crust/ice boundary at the next timestepwithout melting and Tmelt is the melting temperature of ice. We then sub-tract this flux from the top node and recalculate the next temperature. Herewe assume that any melt water that is generated is removed from the systemimmediately.60Appendix FVolumetric Magma InjectionRate calculated from MantleHeat Flux ValuesWe need to calculate the mantle production rate Qvol using the prescribedmantle heat flux qm(t) = qm,0 + δqm(t). To start:qm = Fρ(cpT + Lf)V (F.1)where F is the percent of the total heat from the magma that actually heatsthe crust (F = 0.01 [38]), T is the difference between the emplacementtemperature of magma from the mantle Tliq(Table 3.1) and the averagetemperature of the crust from 5 - 30 km depth, and V is the flow rate perarea (units of m/s). In this case the area is the average cross sectional area ofdykes feeding the magma reservoir times the number of dykes. We multiplyby area on both sides and are left with:qmLcrustWdykeN = Fρ(cpT + Lf)Qvol (F.2)where N (best fit = 6) is the number of dykes,Wdyke is the width of the dyke(assumed to be 10 m) and Qvol is the flow rate of magma into the crust. Weisolate Qvol:Qvol =qm ∗ Lcrust ∗Wdyke ∗NF ∗ ρ ∗ (cp ∗ T + Lf) (F.3)At every timestep while the model is running melt production rate Qvol iscalculated and we save these estimations in an (Mx1) array where M is thenumber of time steps.Note that this approximation is an upper bound on the magma produc-tion rate. If we would instead assume that most dykes stop in the bottomhalf of the crust then Lcrust/2 would be a better approximation for thelength of dykes.61Appendix GCodeG.1 READ MENumerical Model For Thes i s Defense T i t l ed :” Po t en t i a l I c e Sheet and G la c i a l Modulation o fvolcanism in West Antarct i ca : Constra int on thecadence o f melt d e l i v e r y in to the c ru s t ”by Sean Wanket−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Pro j e c t d e s c r i p t i o nThis p r o j e c t models 1D heat f low through the c ru s tus ing the Dufort−Frankel method . Boundarycond i t i on s are both Nuemann BCs . The goa l i s toquant i fythe e f f e c t s o f two thermal f o r c i n g s ( I n s u l a t i o n andTime−dependent mantle magma i n j e c t i o n ) on therheo logy o f wa l l rock surrounding a magma chamber .The outputs from se tup and in t e g r a t i on . py are :T DF : temperature at each t imestep and s p a t i a l node ;NxM arrayHmagma: exce s s heat ing that occurs at the base o f thec ru s t when s p a t i a l nodes reach temperatures h igherthan the l i q u i du s ; NxM arrayUn: Deborah Number ; NxM arrayv i s c : v i s c o s i t y ; NxM arrayQvol : volume o f magma produced at the base o f thec ru s t by Hmagma; Mx1 arrayQvo l t o ta l : volume o f magma that would causeequ iva l en t heat ing to delta qm ; Mx1 array62G.1. READ ME−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−F i l e Desc r ip t i onmain melt ing . py −−> This i s the main python f i l e whichc a l l s to params . yaml to import parameters ,per forms i n i t i a l c a l c u l a t i o n s , d e f i n e s time−dependentparameters , execute s f unc t i on s fromse tup and in t e g r a t i on . py , and p l o t s the outputsa f t e r func t i on execut ion .params . yaml −−> i n i t i a l i z e a l l cons tant s andparameters that do not r e qu i r e python func t i on s toc r e a t e .s e t up and in t e g r a t i on . py −−> ” i n i t i a l ” func t i on toc r e a t e i n i t i a l temperature cond i t i on s (T( z , t ) = T( z, 0 ) ) . ”dT dt” func t i on to c a l c u l a t e FTCStemperature s tep .” integrat ion DF ” func t i on to exec ture Dufort−Frankelmethod to s o l v e the 1D pa r t i a l d i f f e r e n t i a lequat ion us ing ”dT dt” f o r the f i r s t time step .−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Necessary Packagespandasnumpymatp lo t l i bs c ipysympytqdmyaml−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Cred i t sWritten by Sean Wanket . Edited by Mark J e l l i n e k andCather ine Johnson .The model i s based on the work o f Karlstrom et a l . ,63G.2. Parameters2017 .Parameter sour c e s :q m , kc , hr , A 0 −−> Losing e t a l . , 2020L −−> Ramirez et a l . , 2017Tsurf −−> Buizer t e t a l . , 2021k i −−> A. Gow & Engelhardt , 2000kappa −−> Whittington et a l . , 2009Fu l l c i t a t i o n s in Wanket et a l . ,G.2 Parametersqm : 0 .035 # background mantle heat f l u x [W/m2]hr : 8000 # e−f o l d depth f o r r ad i og en i c heat ing [m]rho c ru s t : 2600 # dens i ty o f the c ru s t [ kg/m3]kc : 2 . 5 # thermal conduc t i v i ty o f the c ru s t [W/(m K) ]k i : 2 . 1 # average thermal conduc t i v i ty o f the i c eshee t [W/(m K) ]cp c ru s t : 800 # s p e c i f i c heat capac i ty o f the c ru s t [J /( kg K) ]Lf : 310000 # Latent heat o f f u s i on f o r c r u s t a l rocks[ J/kg ]Tl iq : 1373 # Liquidus temperature o f c r u s t a l rocks [K]Tmelt : 273 # Melt ing temperature o f i c e [K]A : 42500000 # constant f o r Arrhenius law f o rv i s c o s i t yBmu : 8 .31 # constant f o r Arrhenius law f o r v i s c o s i t yG : 141000 # constant f o r Arrhenius law f o r v i s c o s i t yEe f f : 10000000000 # Young ’ s ModulusTbase : 273 # I n i t i a l temperature at the base o f thei c e shee t [K]Tsurf : 238 # Temperature at the su r f a c e o f the i c eshee t [K]t f : 41000 # Timescale o f Deg l a c i a t i one f f i c i e n c y : 0 .01 # e f f i c i e n c y o f heat ing from magma .heat l o s t to water advect ion and volcanism64G.3. FunctionsG.3 Functionsimport numpy as npfrom tqdm import tqdmdef i n i t i a l ( params ) :# Create i n i t i a l s t a t e o f the problem#OUTPUTS:# u = temperature array with f i r s t columni n i t i a l i z e d and a l l other columns 0u = np . z e ro s ( ( params [ ’N’ ] , params [ ’M’ ] ) ) #Temperature# Set i n i t i a l c ond i t i on si f params [ ’ cons tant rad heat ing ’ ] == True : # i fr ad i og en i c heat ing i s constantu [ : , 0 ] = −params [ ’ Hrad ’ ] / params [ ’ kc ’ ] ∗ (params [ ’ z ’ ] ∗ ∗ 2 / 2 ) + ( ( params [ ’ Hrad ’ ] / params[ ’ kc ’ ] ) ∗params [ ’ L crust ’ ] + params [ ’qm’ ] /params [ ’ kc ’ ] ) ∗params [ ’ z ’ ] + params [ ’ Tsurf ’ ]# i n i t i a l c ru s t temperaturee l i f params [ ’ cons tant rad heat ing ’ ] == False : # i fr ad i og en i c heat ing i s exponent i a lf o r i in range (0 , params [ ’N’ ] ) : #steady s t a t egeothermc1 = params [ ’qm’ ] / params [ ’ kc ’ ] − params [ ’Hrad0 ’ ] ∗ params [ ’ hr ’ ] / params [ ’ kc ’ ] ∗ np .exp(−params [ ’ L crust ’ ] / params [ ’ hr ’ ] ) #constant o f i n t e g r a t i o nc0 = params [ ’ Tsurf ’ ] + params [ ’ Hrad0 ’ ] ∗params [ ’ hr ’ ] ∗ ∗ 2/ params [ ’ kc ’ ] # constanto f i n t e g r a t i o nu [ i , 0 ] = −params [ ’ Hrad0 ’ ] ∗ params [ ’ hr ’ ] ∗ ∗2/params [ ’ kc ’ ] ∗ np . exp(−params [ ’ z ’ ] [ i ] /params [ ’ hr ’ ] ) + c1∗params [ ’ z ’ ] [ i ] + c0# i n i t i a l c ru s t temperaturere turn u65G.3. Functionsde f dT dt (T, index , qi , params ) :# Use the method o f l i n e s to c a l c u l a t e thed e r i v a t i v e (dT/dt ) by c a l c u l a t i n g f l u x e s in andout o f each s p a t i a l node## INPUTS:# T: Temperature at cur rent time step# index : index at which t h i s t imestep i s happeningin the i n t e g r a t i o n func t i on# q i : heat f l u x out due to i c e shee t r e s i s t i v i t y# params : d i c t i ona ry o f a l l cons tant s used inc a l c u l a t i o n## OUTPUTS:# dT: the change in temperature from th i s t imestepto the next t imestepdT = np . empty (np . shape (T) )# q on staggered g r id order ( d e l t a x ) ˆ4 accurateq s = np . z e r o s ( l en (T)−1)q s [1 : −1 ] = −params [ ’ kc ’ ]∗ ( −0 . 5∗ (T [ 3 : ] ) + 7 .5∗T[2 : −1 ] − 7 .5∗T[1 : −2 ] + 0 . 5∗ (T[ : −3 ] ) ) /(6∗params[ ’SPACE STEP ’ ] ) # four th order cente redd i f f e r e n c eq s [ 0 ] = −params [ ’ kc ’ ] ∗ (T[1]−T[ 0 ] ) /( params [ ’SPACE STEP ’ ] ) # second order cente redd i f f e r e n c e f o r i = 1/2q s [−1] = −params [ ’ kc ’ ] ∗ (T[−1]−T[−2]) /( params [ ’SPACE STEP ’ ] ) # second order cente redd i f f e r e n c e f o r i = M − 1/2i f params [ ’ i c e ’ ] == True : # i f the re i s an i c eshee tdT [ 0 ] = (−q s [ 0 ] + q i ) /( params [ ’ rho cp ’ ] ∗params [ ’SPACE STEP ’ ] ) + params [ ’ Hrad ’ ] [ 0 ] / (params [ ’ rho cp ’ ] ) # temperature change attop boundary66G.3. FunctionsdT[1 : −1 ] = −( q s [ 1 : : ] − q s [ 0 : −1 ] ) /( params [ ’rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] ) + params [ ’Hrad ’ ] [ 1 : − 1 ] / params [ ’ rho cp ’ ] # temperaturechange at c e n t r a l nodesdT[−1] = ( params [ ’ delta qm ’ ] [ index ] + q s [−1])/( params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] )+ params [ ’ Hrad ’ ] [ −1 ] / params [ ’ rho cp ’ ] #temperature change at bottom boundaryi f T[0 ]+ params [ ’TIME STEP’ ] ∗dT [ 0 ] > params [ ’Tmelt ’ ] : # i f T at the top boundary exceedsmelt ing po int o f i c e then remove exce s sheat due to melt ingdT [ 0 ] = (−q s [ 0 ] + q i − params [ ’ rho cp ’ ] ∗params [ ’SPACE STEP’ ] ∗ (T[ 0 ] − 273) /params [ ’TIME STEP ’ ] ) /( params [ ’ rho cp ’ ] ∗params [ ’SPACE STEP ’ ] ) + params [ ’ Hrad’ ] [ 0 ] / ( params [ ’ rho cp ’ ] )e l i f params [ ’ i c e ’ ] == False : # i f the re i s not ani c e shee tdT [ 0 ] = 0dT[1 : −1 ] = −( q s [ 1 : : ] − q s [ 0 : −1 ] ) /( params [ ’rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] ) + params [ ’Hrad ’ ] [ 1 : − 1 ] / params [ ’ rho cp ’ ]dT[−1] = ( params [ ’ delta qm ’ ] [ index ] + q s [−1])/( params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] )+ params [ ’ Hrad ’ ] [ −1 ] / params [ ’ rho cp ’ ]r e turn dTdef integrat ion DF ( t ,T, params ) :# In t e g r a t e T through time us ing the Dufort−Frankel (DF) Method . The f i r s t t imestep and thetop and bottom o f each i t e r a t i o n use ForwardTime Centered Space (FTCS) to step forward .# This i s because DF r e qu i r e s the prev ioust imestep to c a l c u l a t e the next time step and67G.3. Functionst h i s i s no p o s s i b l e to s t a r t . FTCS i s u s e f u l atthe top and bottom of the domain to# both because DF uses the s p a t i a l s t ep above andbelow (which isn ’ t p o s s i b l e at the top andbottom of the domain ) and t h i s problem app l i e sf i x e d f l u x e s at the boundar ies and# FTCS makes that much e a s i e r .## INPUTS:# t : time array with l ength M where M i s thenumber o f time s t ep s# T: Temperature with shape (N,M) where N i s thenumber o f s p a t i a l s t ep s# params : d i c t i ona ry with a l l constant used in theproblem## OUTPUTS:# T: Temperature array now f i l l e d with shape (N,M)# Hmagma: exce s s heat from melt ing o f the base o fthe c ru s t and spread throughout the c ru s t (N,M)# Un: Unloading number array with s i z e (N,M)# v i s c : V i s c o s i t y array with s i z e (N,M)# Qvol : magma volume generated to c r e a t e Hmagma (M, 1 )# Qvo l t o ta l : magma volume ca l c u l a t ed fromdelta qm which would c r e a t e the equ iva l en tamount o f heat ing (M, 1 )# constant s used f o r the DF c a l c u l a t i o n salpha = 2∗params [ ’ kc ’ ] ∗ params [ ’TIME STEP’ ] / ( params[ ’ rho cp ’ ] ∗ params [ ’SPACE STEP’ ] ∗ ∗ 2 )de l t a = (1−alpha ) /(1+alpha )ch i = alpha/(1+alpha )xeta = 2∗params [ ’TIME STEP’ ] ∗ params [ ’ Hrad ’ ] [ 1 : − 1 ]/ ((1+ alpha ) ∗params [ ’ rho cp ’ ] )# Flux through the i c e shee t which w i l l be s e t asthe top boundary cond i t i on . Only f i r s t f l u x i s68G.3. Functionsc a l c u l a t ed because q i i s dependent on thetemperatureq i = np . z e ro s ( l en ( t ) )q i [ 0 ] = −(params [ ’ ki ’ ] / params [ ’ L ice ’ ] [ 0 ] ) ∗(T[0 ,0 ]− params [ ’ Tsurf ’ ] )# F i r s t time step us ing FTCST[ : , 1 ] = T[ : , 0 ] + params [ ’TIME STEP’ ] ∗ dT dt (T[ : , 0 ] , 0 , q i [ 0 ] , params )# c r ea t e array to s t o r e melt ing f l u x e s which arec a l c u l a t ed at each t imestepqmelt = np . z e ro s ( l en ( t ) )Hmagma = np . z e ro s (np . shape (T) )# c r ea t e ar rays to s t o r e magma volumes f o r exc e s smelt ing and t o t a l volumeQvol = np . z e r o s ( l en ( t ) ) # exce s s melt ing volumeQvo l t o ta l = np . z e r o s ( l en ( t ) ) # t o t a l magma volumera t e f o r a l l delta qmN = 6 # Number o f dykes r equ i r ed to ach i evec o r r e c t amount o f melt ing . Used f o r Qvo l t o ta lc a l c u l a t i o n s .f o r i i in tqdm(np . arange (1 , params [ ’M’ ] ) ) :i f i i +1 >= params [ ’M’ ] : # dont execute pastl a s t time stepbreak# ca l c u l a t e t o t a l volume ra t e f o r cur rentdelta qm valueQvo l t o ta l [ i i ] = params [ ’ delta qm ’ ] [ i i ]∗ params[ ’ L crust ’ ] ∗10 ∗N / ( params [ ’ e f f i c i e n c y ’ ] ∗params [ ’ rho c rus t ’ ] ∗ ( params [ ’ cp crus t ’ ] ∗np .mean( params [ ’ Tliq ’ ] − T[ np . argwhere (params [ ’ z ’ ] == 5000) [ 0 ] [ 0 ] : − 1 , i i ] ) + params[ ’ Lf ’ ] ) ) # ra t e o f magma i n j e c t i o n in to thec ru s t c a l c u l a t ed us ing delta qm69G.3. Functions# ca l c u l a t e f l u x through the i c e shee t at eacht imestepq i [ i i ] = ( params [ ’ ki ’ ] / params [ ’ L ice ’ ] [ i i ] ) ∗(T[ 0 , i i ]− params [ ’ Tsurf ’ ] ) # in su l a t ed heatf l u x through the i c e shee t# c a l c u l a t e the next temperature at the top o fthe s p a t i a l domain us ing FTCSi f params [ ’ i c e ’ ] == True :Tguess = T[ 0 , i i ] + params [ ’TIME STEP ’ ] ∗ ( (params [ ’ kc ’ ] ∗ (T[ 1 , i i ]−T[0 , i i ] ) /params [ ’SPACE STEP ’ ] − q i [ i i ] ) /( params [ ’ rho cp’ ] ∗ params [ ’SPACE STEP ’ ] ) + params [ ’ Hrad’ ] [ 0 ] / ( params [ ’ rho cp ’ ] ) ) # guess atemperature f o r the next time step# Check to see i f the next temperaturew i l l be above 273 K. I f yes , add in themelt f l u x to t h i s t imestep andr e c a l c u l a t e next temperature . I f no ,s e t next temperature to Tguessi f Tguess > params [ ’ Tmelt ’ ] :qmelt [ i i ] = params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP’ ] ∗ ( Tguess − params [ ’Tmelt ’ ] ) /params [ ’TIME STEP ’ ]T[ 0 , i i +1] = T[ 0 , i i ] + params [ ’TIME STEP ’ ] ∗ ( ( params [ ’ kc ’ ] ∗ (T[ 1 , i i]−T[0 , i i ] ) /params [ ’SPACE STEP ’ ] −q i [ i i ] − qmelt [ i i ] ) /( params [ ’ rho cp’ ] ∗ params [ ’SPACE STEP ’ ] ) + params [ ’Hrad ’ ] [ 0 ] / ( params [ ’ rho cp ’ ] ) )e l s e :T[ 0 , i i +1] = Tguesse l i f params [ ’ i c e ’ ] == False :T[ 0 , i i +1] = params [ ’ Tsurf ’ ]####################################################################################################################################### This part i s f o r p l o t t i n g experiment 1and I w i l l d e l e t e i t once the paper / t h e s i s70G.3. Functionsi s done j u s t in case I need to r ep l o t ###### i f t [ i i ] < 3∗params [ ’ tau d ’ ] :# T[ 0 , i i +1] = params [ ’ Tsurf ’ ]# e l i f t [ i i ] >= 3∗params [ ’ tau d ’ ] :# Tguess = T[ 0 , i i ] + params [ ’TIME STEP’ ] ∗ ( ( params [ ’ kc ’ ] ∗ (T[ 1 , i i ]−T[0 , i i ] ) /params[ ’SPACE STEP ’ ] − q i [ i i ] ) /( params [ ’ rho cp ’ ] ∗params [ ’SPACE STEP ’ ] ) + params [ ’ Hrad ’ ] [ 0 ] / (params [ ’ rho cp ’ ] ) )# # Check to see i f the next temperaturew i l l be ab le 273 K. I f yes , add in the meltf l u x to t h i s t imestep and r e c a l c u l a t e nexttemperature . I f no , s e t next temperatureto Tguess# i f Tguess > params [ ’ Tmelt ’ ] :# qmelt [ i i ] = params [ ’ rho cp ’ ] ∗ params[ ’SPACE STEP’ ] ∗ ( Tguess − params [ ’ Tmelt ’ ] ) /params [ ’TIME STEP ’ ]# T[ 0 , i i +1] = T[ 0 , i i ] + params [ ’TIME STEP ’ ] ∗ ( ( params [ ’ kc ’ ] ∗ (T[ 1 , i i ]−T[0 , i i] ) /params [ ’SPACE STEP ’ ] − q i [ i i ] − qmelt [ i i] ) /( params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] )+ params [ ’ Hrad ’ ] [ 0 ] / ( params [ ’ rho cp ’ ] ) )# e l s e :# T[ 0 , i i +1] = Tguess################################################################################################################################### ca l c u l a t e next tempreature at the bottom ofthe s p a t i a l domain us ing FTCSTguess2 = T[−1 , i i ] + params [ ’TIME STEP ’ ] ∗ ( (params [ ’ delta qm ’ ] [ i i ] − params [ ’ kc ’ ] ∗ (T[−1 , i i ]−T[−2 , i i ] ) /params [ ’SPACE STEP ’ ] ) /(params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] )+params [ ’ Hrad ’ ] [ − 1 ] / ( params [ ’ rho cp ’ ] ) )# Check to see i f the next temperatuer w i l l be71G.3. Functionsabove l i q u i du s temp . I f yes , add in themelt f l u x to t h i s t imestep and r e c a l c u l a t etemp . I f no , s e t next temperature toTguess2i f Tguess2 > params [ ’ Tliq ’ ] :qmelt magma = params [ ’ rho cp ’ ] ∗ params [ ’SPACE STEP’ ] ∗ ( Tguess2 − params [ ’ Tliq’ ] ) /params [ ’TIME STEP ’ ]T[−1 , i i +1] = T[−1 , i i ] + params [ ’TIME STEP’ ] ∗ ( ( params [ ’ delta qm ’ ] [ i i ] − params [ ’kc ’ ] ∗ (T[−1 , i i ]−T[−2 , i i ] ) /params [ ’SPACE STEP’]−qmelt magma ) /( params [ ’rho cp ’ ] ∗ params [ ’SPACE STEP ’ ] )+ params[ ’ Hrad ’ ] [ − 1 ] / ( params [ ’ rho cp ’ ] ) )Qvol [ i i ] = qmelt magma/( params [ ’ rho c rus t’ ] ∗ params [ ’ Lf ’ ] )Hmagma [ : , i i ] = params [ ’ e f f i c i e n c y ’ ] ∗params [ ’ rho c rus t ’ ] ∗ ( params [ ’ cp crus t’ ] ∗ ( params [ ’ Tliq ’]−T[ : , i i ] )+params [ ’ Lf’ ] ) ∗Qvol [ i i ]∗ params [ ’ dz ’ ] / ( params [ ’dyke length ’ ] )xeta = 2∗params [ ’TIME STEP’ ] ∗ ( params [ ’Hrad ’ ] [ 1 : −1 ]+Hmagma[1:−1 , i i ] ) / ((1+alpha ) ∗params [ ’ rho cp ’ ] )Tguess3 = de l t a ∗T[1:−1 , i i −1] + ch i ∗(T [ 2 : : ,i i ] + T[0:−2 , i i ] ) + xeta# Check to see i f a dd i t i ona l nodes abovethe bottom boundary exceed Tl iq . I f yesremove exce s s heat f l u x .i f np . any ( Tguess3>params [ ’ Tliq ’ ] ) :i n d i c e s a b o v e l i q u i d u s = np . argwhere (Tguess3>params [ ’ Tliq ’ ] )f o r kk in np . f l i p (i n d i c e s a b o v e l i q u i d u s ) :Hmagma[1:−1 , i i ] [ kk [ 0 ] ] = 0xeta = 2∗params [ ’TIME STEP’ ] ∗ ( params[ ’ Hrad ’ ] [ 1 : −1 ]+Hmagma[1:−1 , i i ] ) /((1+ alpha ) ∗params [ ’ rho cp ’ ] )T[1 :−1 , i i +1] = de l t a ∗T[1:−1 , i i −1] +72G.4. Main Programch i ∗(T [ 2 : : , i i ] + T[0:−2 , i i ] ) + xetae l s e :T[1 :−1 , i i +1] = Tguess3e l s e :T[−1 , i i +1] = Tguess2xeta = 2∗params [ ’TIME STEP’ ] ∗ params [ ’ Hrad’ ] [ 1 : − 1 ] / ((1+ alpha ) ∗params [ ’ rho cp ’ ] )# c a l c u l a t e next temperatures with in thes p a t i a l domain us ing DFT[1:−1 , i i +1] = de l t a ∗T[1:−1 , i i −1] + ch i ∗(T[ 2 : : , i i ] + T[0:−2 , i i ] ) + xeta# # ca l c u l a t e v i s c o s i t y us ing Arrhenius V i s c o s i t yLawv i s c = params [ ’A’ ] ∗ np . exp ( params [ ’G’ ] / ( params [ ’Bmu’ ] ∗T) ) # Vi s c o s i t y# #ca l c u l a t e Unloading numberUn = ( v i s c /( params [ ’ year ’ ] ∗ params [ ’ Eef f ’ ] ) ) /params[ ’ t f ’ ] # Unloading Numberre turn T,Hmagma,Un, v i s c , Qvol , Qvo l t o ta lG.4 Main Program#%%import pandas as pdimport numpy as npimport matp lo t l i b . pyplot as p l tfrom sc ipy import spar s eimport sympy as syfrom sc ipy import i n t e g r a t efrom tqdm import tqdmimport yamlimport s e tup and in t e g r a t i on me l t i n g as fn73G.4. Main Programfrom sc ipy . s i g n a l import detrendfrom matp lo t l i b import t i c k e r , cm# import parameter f i l ewith open (” params . yaml ” , ’ r ’ ) as stream :parameters = yaml . s a f e l o a d ( stream )#DOMAIN GEOMETRY AND TIME STEPPINGyear=3600∗24∗365 # one year in secondsSPACE STEP=0.05 ∗ 1e3 # gr id spacing , mTIME STEP= 50 ∗ year # time step , yearsL crus t = 30 ∗ 1e3 # t o t a l l ength o f the c ru s t in mH ice = 3 ∗ 1e3 # t o t a l l ength o f i c e in mN = round ( L crus t /SPACE STEP) # number o f nodes inc ru s tM = round (36 e6 ∗ year / TIME STEP) # number o ft imes teps = t o t a l time / length t imestept = np . arange (0 ,M∗TIME STEP,TIME STEP) # time vec torz = np . arange (0 , L crust ,SPACE STEP) # d i s t anc e vec to rc ru s tchamber depth = 5e3 # depth to the magma chamberdyke length = L crust−chamber depth # length o f thedyke from the base o f the c ru s t updz = 0.5∗(1+np . tanh (10∗ ( z−chamber depth ) / L crus t ) ) #shape o f magmatic heat ing f o r exc e s s melt i n j e c t i o n#Create Array Var iab l e srho cp = parameters [ ’ rho c rus t ’ ] ∗ parameters [ ’ cp crus t’ ] # dens i ty and heat capac i ty mu l t i p l i e d toge the rto make fu tu r e math f a s t e r during i n t e g r a t i o nkappa = parameters [ ’ kc ’ ] / ( parameters [ ’ rho c rus t ’ ] ∗parameters [ ’ cp crus t ’ ] ) # heat d i f f u s i o n in thec ru s ttau d = ( L crus t /2) ∗∗2/kappa # d i f f u s i o n t ime s ca l ethrough ha l f o f the c ru s t74G.4. Main Program#Percent ext ra mantle heat ingmagnitude above = 1000 # f a c t o r o f mantle heat f l u xabove the background that dqm w i l l reach at maximumi n j e c t p e r i o d = 3e7 ∗ year # per iod o f magma i n j e c t i o nin secondsdelta qm = parameters [ ’qm’ ] + parameters [ ’qm’ ] ∗magnitude above ∗ (np . s i n (2 ∗ np . p i ∗ t /i n j e c t p e r i o d ) )# heat f l u x cannot be negat ive so make a l l negat iveva lue s 0f o r i in range ( l en ( delta qm ) ) :i f delta qm [ i ] < 0 :delta qm [ i ] = 0# Ice shee t s u r f a c e f l u x modulation# The he ight o f the i c e shee t through time . I fconstant , s e t constant = True , i f time var iant ,s e t constant = Falsei c e = False # i s the re an i c e shee t ? i f True :i n s u l a t i o n boundary cond i t i on ; i f Fa l se : constanttemp boundary cond i t i ono r b i t a l p e r i o d = 100000∗ year # milankovi tch per iod ,can be changed to be 100 , 41 , or 23 kyri f i c e == True :constant = False # I s the i c e shee t he ightconstant or p e r i o d i c ?i f constant == True :L i c e = H ice ∗np . ones (M)e l i f constant == False :L i c e = 3000 − 1500∗np . cos (2∗np . p i ∗ t /(o r b i t a l p e r i o d ) )e l i f i c e == False :L i c e = H ice ∗np . ones (M)# Radiogenic Heatingcon s tan t rad hea t ing = False # I s r ad i og en i c heat ing75G.4. Main Programconstant throughout the c ru s t or exponent i a ldec r ea se with depthi f c on s t an t rad hea t ing == True :Hrad0 = 1 .5 e−6Hrad = Hrad0∗np . ones ( l en ( z ) )e l i f c on s tan t rad hea t ing == False :Hrad0 = 4e−6Hrad = Hrad0 ∗ np . exp(−z/ parameters [ ’ hr ’ ] )# F i l l parameter d i c t i ona ry with new paramtersparameters . update ({ ’ year ’ : year , ’SPACE STEP ’ : SPACE STEP, ’TIME STEP ’ : TIME STEP, ’ L crust ’ : L crust , ’ L ice ’ :L ice , ’N’ : N, ’M’ :M, ’ z ’ : z , ’ rho cp ’ : rho cp , ’ delta qm ’ :delta qm , ’ Hrad ’ : Hrad , ’ chamber depth ’ : chamber depth, ’ dz ’ : dz , ’ dyke length ’ : dyke length , ’ i c e ’ : i c e , ’ tau d’ : tau d , ’ cons tant rad heat ing ’ : cons tant rad heat ing, ’ Hrad0 ’ : Hrad0 })# In t e g r a t i onp r in t ( ’ i n t e g r a t i n g . . . ’ )T DF = fn . i n i t i a l ( parameters ) # i n i t i a l setupT DF,Hmagma,Un, v i s c , Qvol , Qvo l t o ta l = fn .integrat ion DF ( t ,T DF, parameters ) # i n t e g r a t e# Save ar rays i f nece s sa ryp r in t ( ’ sav ing . . . ’ )# np . save txt ( ’ exper iment2 exp 8hr 1e7 temp . csv ’ ,T DF [np . argwhere ( z == 5000) [ 0 ] [ 0 ] , : ] , d e l im i t e r = ’ , ’ )# np . save txt ( ’ exper iment2 exp 8hr 1e7 deb . csv ’ ,Un [ np .argwhere ( z == 5000) [ 0 ] [ 0 ] , : ] , d e l im i t e r = ’ , ’ )# c a l c u l a t e r a t i o o f e rupt i v e : s t o rage time along withproduct ion rate , min De , and max Depr in t ( ’ e rupt i v e / s to rage ra t i o ’ )e rupt i v e = 0s to rage = 0t o t a l = 0f o r i in Un[ np . argwhere ( z == 5000) [ 0 ] [ 0 ] , np . argmin (np .76G.4. Main Programabs ( t − 4∗ tau d ) ) : np . argmin (np . abs ( t−(6∗ tau d ) ) ) ] :t o t a l += 1 # to t a l time s t ep s in the cy c l edes ignated in the prev ious l i n ei f i > 1 :e rupt i v e += 1 # time s t ep s f o r one heat ingcy c l e spent in the e rupt i v e regimee l i f i <1:s t o rage += 1 # time s t ep s f o r one heat ingcy c l e spent in the s to rage regimepr in t ( f ’ r a t i o = { e rupt i v e / s to rage } ’ ) # r a t i o o fe rupt i v e vs s to ragep r i n t ( f ’max mass ra t e = {np .max( Qvo l t o ta l ∗( year /1 e9 ) )} ’ ) # max ra t e o f magma i n f l u x in to the c ru s tp r i n t ( f ’ average mass ra t e = {np .mean( Qvo l t o ta l ∗( year/1 e9 ) ) } ’ ) # average ra t e o f magma i n f l u x in to thec ru s tp r i n t ( f ’Min De = {np . min (Un[ np . argwhere ( z == 5000)[ 0 ] [ 0 ] , np . argmin (np . abs ( t − 4∗ tau d ) ) : np . argmin (np .abs ( t−(6∗ tau d ) ) ) ] ) } ’ )p r i n t ( f ’Max De = {np .max(Un[ np . argwhere ( z == 5000)[ 0 ] [ 0 ] , np . argmin (np . abs ( t − 4∗ tau d ) ) : np . argmin (np .abs ( t−(6∗ tau d ) ) ) ] ) } ’ )p r i n t ( ’ p l o t t i n g . . . ’ )f i g , ax = p l t . subp lo t s ( nrows = 3 , nco l s = 1 , f i g s i z e =(10 ,7 ) )ax [ 0 ] . p l o t ( t / tau d ,T DF [ np . argwhere ( z == 5000)[ 0 ] [ 0 ] , : ] , c o l o r = ’ black ’ )ax [ 1 ] . semi logy ( t / tau d , v i s c [ np . argwhere ( z == 5000)[ 0 ] [ 0 ] , : ] , c o l o r = ’ black ’ )ax [ 2 ] . semi logy ( t / tau d ,Un[ np . argwhere ( z == 5000)[ 0 ] [ 0 ] , : ] , c o l o r = ’ black ’ )f i g . s u p t i t l e ( ’ Var i a t i on s in I n su l a t i o n at the top o fthe Crust ’ )ax [ 0 ] . s e t t i t l e ( ’ Temperature ’ )ax [ 1 ] . s e t t i t l e ( ’ V i s co s i ty ’ )77G.4. Main Programax [ 2 ] . s e t t i t l e ( ’ Deborah Number ’ )ax [ 2 ] . s e t x l a b e l ( ’Time ( t /$T d$ ) ’ )ax [ 2 ] . h l i n e s (1 e0 , 1 , 6 )f i g . t i g h t l a y ou t ( )p l t . show ( )p r i n t ( ’ done . . . ’ )78
Thesis/Dissertation
eng
Geophysics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Jellinek, A. Mark
Johnson, Catherine
Potential ice sheet and glacial modulation of volcanism in West Antarctica : constraints on the cadence of melt delivery into the crust
Text