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Energy balance analysis of the temperature dynamics of a steep proglacial stream Dufficy, Anna L. 2019

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Energy Balance Analysis of theTemperature Dynamics of a SteepProglacial StreambyAnna L. DucyB.Sc., Indiana University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Geography)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019©Anna L. Ducy, 2019The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, a thesis/dissertation entitled:Energy Balance Analysis of the Temperature Dynamics of a Steep Proglacial Streamsubmitted by: Anna L. Ducy in partial fulfillment of the requirementsfor the degree of Master of Sciencein GeographyExamining Committee:Dan R. Moore, GeographySupervisorBrett Eaton, GeographySupervisory Committee MemberAndre Zimmermann, Northwest Hydraulics Consultants, GeographySupervisory Committee MemberSara Knox, GeographyAdditional ExamineriiAbstractThe application of physically based models to assess the e↵ect of glacier retreat is complicatedby the heterogeneity of proglacial stream morphology, the diurnal fluctuations of stream flow,and the e↵ects of aeration on turbulent exchange and stream albedo. Current energy balanceparameter values derived in shallow streams provide unreliable predictions by incorporatingincorrect representations of the underlying physical processes. In proglacial streams, diurnalflow produces transient channel beds from daily drying and re-wetting, which can e↵ect tran-sient heat storage from pre-exposed boulders and hyporheic exchange. However, no availablemodel predicts the dependence of stream aeration and complex hydraulic geometry on bothturbulent exchange and residual heating inputs of steep streams. This study quantified thedependence of steep stream morphology and turbulence on the wind function and residualheating parameters in order to improve stream temperature predictions in proglacial alpineand headwater streams.The study focused on a 1-km-long reach of South Creek, a steep glacier-fed stream inthe Bridge Glacier valley of British Columbia. Energy balance input variables were acquiredfrom measured and calculated field data using above-stream and above-land weather stations,salt dilution gauging, UAV photogrammetry (SfM), and water temperature monitoring fromthe observed two-month-long melt season. Following model optimization, derived wind func-tion coecient ranges were an order of magnitude greater than the literature values for lowgradient streams (c = 20-975 W·m2·kPa1, d = 25-200 W·s·m3·kPa1). An added sta-bility term to the wind function further improved results. Nighttime optimization yieldedsmaller wind function coecient values and greater residual warming to the stream, reveal-ing a diurnal thermal regime shift. Adjustments to albedo and net radiation improved thepredictive power relative to models with the standard literature inputs. The results fromthis study contribute to our understanding of steep stream heat dynamics from a morpho-dynamic standpoint, which is of increasing importance for headwater streams in the contextof continued glacial retreat.iiiLay SummaryThe current equations available to predict stream temperature are not applicable in mountainstreams. These streams are di↵erent from the low gradient rivers in which the equations werederived due to steep profiles and high concentrations of boulders, which produce aeration,a↵ecting the surface area of the stream and the degree to which evaporation/condensationoccurs. This study collected data from South Creek in British Columbia to apply to an energybalance model where the e↵ects of aeration were calculated using a wind function equation.Results from the model show that during the day, evaporation is enhanced compared tolow gradient streams, but at night the e↵ects decrease and warming happens instead fromoutside sources such as groundwater or hyporheic exchange. Better knowledge and predictivepower of mountain stream temperature is especially important in British Columbia streamssourced by continually declining glacier meltwater.ivPrefaceThis thesis is original work completed by the author. Guidance was given by the supervisorycommittee (Dan Moore, Brett Eaton, and Andre Zimmermann) and field assistance was pro-vided by Jordyn Carss, Haley Williams, Stefan Gronsdahl, Johannes Exler, Martin Cermakand Greg Wellage.A version of this work has been published as an oral presentation (Ducy, AL. Moore,RD. and Eaton, BC. Physically based stream temperature modeling of a steep proglacialstream) on which the author acted as lead investigator and presented at the 2018 CanadianGeophysical Union (CGU) Joint Meeting.vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Review of thermal processes in steep pro-glacial streams . . . . . . . . . . . 21.2.1 Net radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Sensible and latent heat fluxes . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Non-steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.4 Specifying hydraulic geometry . . . . . . . . . . . . . . . . . . . . . . 61.2.5 Transient storage and bed heat conduction . . . . . . . . . . . . . . . 71.3 Research objectives and thesis structure . . . . . . . . . . . . . . . . . . . . 72 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Streamflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Stream gauging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Stage measurements and rating curves . . . . . . . . . . . . . . . . . 142.3 Discharge-dependent reach characteristics . . . . . . . . . . . . . . . . . . . . 142.3.1 Travel time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Stream width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.3 Fraction of aeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Stream temperature and meteorological data . . . . . . . . . . . . . . . . . . 172.5 Processing of meteorological data . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Calculation of vapour pressure . . . . . . . . . . . . . . . . . . . . . . 182.5.2 Spatial distribution of air temperature, vapour pressure and wind speed 202.5.3 Spatial distribution of solar radiation . . . . . . . . . . . . . . . . . . 202.5.4 Spatial distribution of longwave radiation . . . . . . . . . . . . . . . . 212.5.5 Net radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.6 Latent and sensible heat . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.7 Frictional heat dissipation . . . . . . . . . . . . . . . . . . . . . . . . 222.5.8 Surface-subsurface interactions . . . . . . . . . . . . . . . . . . . . . . 222.6 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.1 Model calibration and testing . . . . . . . . . . . . . . . . . . . . . . 233 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Overview of the study period . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Flow routing and the potential for groundwater flux . . . . . . . . . . . . . . 253.3 Inter-station comparison of meteorological variables . . . . . . . . . . . . . . 273.4 Hydraulic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Comparing modelled and measured solar radiation and albedo . . . . . . . . 313.6 Model optimization and sensitivity . . . . . . . . . . . . . . . . . . . . . . . 333.7 Energy balance comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Comparison of optimized and standard model . . . . . . . . . . . . . . . . . 403.9 Error trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1 Model optimization and sensitivity . . . . . . . . . . . . . . . . . . . . . . . 444.2 Hydraulic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Channel controls on the thermal regime . . . . . . . . . . . . . . . . . . . . . 484.4 Hypotheses for residual heat inputs . . . . . . . . . . . . . . . . . . . . . . . 494.5 Comparison of temperature warming in similar streams . . . . . . . . . . . . 504.6 Limitations associated with measurent resolution for stream temperature . . 504.7 Predicting stream temperature under glacier retreat . . . . . . . . . . . . . . 51vii5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1 Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A Rating Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B Salt Dilution Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64C Above-stream meteorological conditions . . . . . . . . . . . . . . . . . . . 65viiiList of Tables1.1 Input variables and parameter definitions for studies that have applied anenergy balance approach to proglacial streams, where l is the study reachlength, S is the slope, w is the stream width, ↵ is albedo, and NS is non-specified information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Summary of reach characteristics for South Creek from the most downstreamReach 1 to the most upstream Reach 4, where l:w is the ratio of the length toaverage wetted width used in the salt injection methods. Black dots representsurveyed points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Hydrometerological variables measured and instruments used. The number inbrackets indicates the number of instruments deployed. . . . . . . . . . . . . 183.1 Power law relations for width (coecients ar and br), as well as travel timeand velocity predicted from discharge of 2.5 m3s1 and average slope for eachreach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Ranges of fitted wind function coecient values associated with random vari-ations in the boundary conditions during optimization. Optimization split-sample tests include the perturbations to all three of radiation, width andtravel time for hourly data, daytime data and nighttime data, and three split-sample tests for perturbations to only one boundary condition. . . . . . . . . 393.3 Calculated changes in downstream temperature associated with each energyflux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Comparisons of the input parameters for the optimized and standard literaturemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Comparison of wind function coecients determined in previous studies . . . 45List of Figures2.1 Study area location. (a) Satellite image of part of the upper Bridge River val-ley showing the ice-contact proglacial lake and South Creek tributary stream,which flows NNW into the proglacial lake. (b) Satellite image of the portion ofimage (a) within the red box, highlighting locations of the three weather sta-tions, pressure transducers for stage height and 10 water temperature sensorsdeployed throughout the reach. . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 (a) Looking south across Bridge Lake at South Creek (star) and surroundingterrain. (b) Zoomed-in photo of Reaches 1 and 2 of South Creek taken froma floatplane looking upstream (south). Photo: A Szeitz (2016) . . . . . . . . 112.3 Longitudinal profile of South Creek, where coloured points represent individ-ual surveyed locations along the stream for each reach. . . . . . . . . . . . . 122.4 A section of one of 11 constructed orthomosaic images used in order to quan-tify the stream width-discharge relationship by measuring widths along 10-mspaced transects. Orthomosaics were also used to estimate proportion of aer-ated water surface area. Blue, red, and yellow lines indicate the downstreamextents of reaches 3, 2 and 1 respectively. Streamflow is from left to right. . 162.5 Photographs of two of the three weather stations. (Top) Open-site weatherstation located on a stream terrace. (Bottom) Above-stream weather stationlocated at the upper end of the 1-km study reach, looking downstream. . . . 193.1 Plot of meteorological conditions observed at the base weather station for thefull duration of the study period. . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Plot of discharge for Reach 4 (black, upstream) and Reach 1 (blue, down-stream) and the change in storage between both reaches. (See Appendix Afor rating curves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Plots of longitudinal water chemistry observations for electrical conductivity(top) and temperature (bottom) on four separate days. . . . . . . . . . . . . 283.4 Di↵erences in recorded wind speed (top), air temperature (center), and vapourpressure (bottom) between the two above-stream weather station locations.Doted red line indicates more similar conditions between weather stations,occurring during a low pressure storm. . . . . . . . . . . . . . . . . . . . . . 293.5 Comparison of air temperature, vapour pressure and wind speed among theupstream (US), downstream (DS) and open-site weather stations. . . . . . . 303.6 Width-discharge relations for all reaches with 95% confidence intervals out-lined in blue. Measured widths for Reaches 1 to 3 were derived from ortho-mosaic photogrammetry; measured widths for Reach 4 were estimated usinga laser range finder performed on 4 separate days. The W-Q regression linefor Reach 3 is plotted on the data for Reach 4. . . . . . . . . . . . . . . . . 323.7 Relations between aerated portion of the stream derived from UAV pho-togrammetry and discharge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Relations between velocity and discharge for each reach, based on Eqs. (2.4)and (2.5), with 95% confidence intervals outlined in blue. . . . . . . . . . . 343.9 Comparison of time series of albedo computed from the model of McMahonand Moore (2017) and values based on the downward-facing pyranometers. . 353.10 Comparison of modelled direct solar radiation and raw pyranometer valuesmeasured from the stream terrace. Grey lines represent the adjusted valuesfor each of the 37 segments of the stream when taking into account topographicshading and solar angle/position. . . . . . . . . . . . . . . . . . . . . . . . . 353.11 Distributions of parameter values for c, d, e, and  from split-sample opti-mization tests (night vs day) and the hourly optimization test. . . . . . . . . 363.12 Sensitivity of optimized parameters to the combined e↵ect of variations inthe boundary conditions for the hourly dataset. All three boundary condi-tions were randomly varied for each optimization run. Parameters c to are presented by row and boundary conditions by column (from left to right,radiation, travel time and width). . . . . . . . . . . . . . . . . . . . . . . . . 373.13 Calibrated parameter values obtained from three optimization trials with per-turbations to net radiation (left), travel time (centre) and width (right) by anerror term multiplier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.14 In each plot, the ensemble of predicted fluxes and value of Tw for all hourlyparameter sets are shown. For Tw, the ensemble of predictions is shown inblue and observed values are shown in burgundy. . . . . . . . . . . . . . . . 40xi3.15 Comparison of predicted and observed temperatures based on (top) calibratedmodel and (bottom) model based on standard parameterizations from previousstudies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.16 Boxplots illustrating the distribution of model error (modelled - observed) anddischarge at 10-minute intervals through the day for four two-week periods.Predictions were made using the best single parameter set from the hourlyoptimization test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.1 Rating curves for all four reaches including their residual standard mean error. 63B.1 Example salt dilution curves for all four reaches. . . . . . . . . . . . . . . . . 64C.1 Plot of stream temperature and meteorological conditions observed at theupstream weather station for the full study period. . . . . . . . . . . . . . . 66C.2 Plot of stream temperature and meteorological conditions observed at thedownstream weather station for full the study period. . . . . . . . . . . . . . 67xiiList of SymbolsSymbol Definition UnitsA area under the salt dilution breakthrough curve (s · µS · cm1)ar fitted coecient for width-dischargea azimuth angle ()br fitted rate coecient for width-discharger Bowen Ratio slope (rad)c fitted coecient for free convection in fw (W·m2·kPa1)cpa specific heat of air at constant pressure (J·kg1·C1)CFT calibration factor for salt dilution (g · cm · µS1 ·m3)d fitted coecient for wind speed in fw (W·s·m3·kPa1)Tf temperature change from friction (°C)✏t emissivity of surrounding terraine fitted stability coecient in fw (W·m2·kPa1·°C1)ea vapour pressure of overlying air (kPa)eNR adjustments to NR used in optimizationer linear regression coecient for travel timesat(T ) saturation vapour pressure for temperature T (kPa)etau adjustments to travel time used in optimizationew vapour pressure at water surface (kPa)ewidth adjustments to width used in optimizationECT (i) the electrical conductivity observed at time increment i (µS · cm1)ECbg the background conductivity of the stream (µS · cm1)faer fraction of aerated stream (%)fr linear regression coecient for travel timefw wind functiong acceleration due to gravity (m·s2)xiiiH stream stage (m)Ha horizon angle at a specified azimuth (a) (rad)H0 cease-to-flow reference level (m)i(t, s) angle of incidence (rad)j fitted stage-discharge coecientk fitted stage-discharge rate coecient(K#) incident solar radiation (W · m2)K#dir direct incident solar radiation (W · m2)K#diff di↵use incident solar radiation (W · m2)Lr the length of the reach (m)Lv latent heat of vaporization (J/kg)L#(t) incident longwave radiation (W · m2)M mass (g)P atmospheric pressure (kPa) residual warming (°C)Q discharge (m3s1)Qe latent heat flux (W · m2)Qh sensible heat flux (W · m2)Qf friction flux (W · m2)Qr net radiation flux (W · m2)RH measured relative humidity (%)⇢w density of water (kg·m3)S slope (rad)s sun location above horizon (rad)Ta temperature of air (C)Tw temperature of water (C)⌧r(Q) travel time per reach (s)✓ the zenith angle ()u wind speed (ms1)Vr(Q) velocity of reach (ms1)wr(Q) stream width of reach v(m)w stream width (m)x(t, z) the interpolated meteorological valuesz elevation (m)xivAcknowledgementsMany people have been influential throughout the process of completing this research project.First and foremost, I would like to thank my supervisor, Dan Moore for his unending enthu-siasm both in the field and in the oce. His technical guidance throughout this project hastaught me invaluable lessons from how to carry out fieldwork in remote places to complexmodeling problems. I am grateful to Dan, Brett Eaton, Andre Zimmerman, and Sara Knoxfor their constructive comments during the thesis writing process.Funding was provided by operating grants to Professor Dan Moore from the NaturalSciences and Engineering Research Council (NSERC) of Canada. The author was supportedby a Graduate Award from the UBC Faculty of Arts. Special thanks go to my fieldwork as-sistants who provided me with support, muscle, bear defenses, laughter, and troubleshootingassistance during our field campaigns. Thank you to Jordyn Carss, Haley Williams, StefanGronsdahl, Johannes Exler, Martin Cermak, and Greg Wellage; without them I would nothave a thesis!I would also like to thank the Earth Science Department at Indiana University and theCenter for Geospatial Data Analysis at the Indiana Geological Survey. In particular I wouldlike to thank Shawn Naylor and Dr. Darren Ficklin for their mentorship in hydrologicalresearch and field techniques, and for allowing me to be a part of their team.Additional thanks to my family for providing me access to a wonderful education and tomy father, a retired earth scientist, for instilling in me a fascination and appreciation for thenatural world.xvChapter 1Introduction1.1 MotivationStream temperature is an important aspect of aquatic habitat and a key water quality vari-able (Webb, Hannah, Moore, Brown, & Nobilis, 2008). There has been ongoing concernfor several decades about stream temperature response to land cover changes such as theremoval of riparian forest (G. Brown, 1969; Leach et al., 2012), urban development (Klein,1979; Pluhowski, 1972), and changes in flow regime associated with withdrawals, diversionsand impoundments (Gu et al., 1998; Hockey et al., 1982; Meier et al., 2003; Webb & Zhang,1997). In the last decade, there has been increasing concern that ongoing glacier retreat andthe associated decrease in late summer streamflow will result in higher stream temperaturesand changes in aquatic ecosystems, particularly in alpine environments that lack shadingby riparian forest (L. E. Brown et al., 2007; Cowie et al., 2014; Finn et al., 2013). Indeed,landscape-scale analyses have shown that summer stream temperatures are negatively asso-ciated with the fractional glacier coverage in a catchment (Fellman et al., 2014; Moore, 2006;Moore et al., 2013), supporting the hypothesis that glacier retreat should eventually lead tohigher summer stream temperatures. The use of thermal sensitivities derived from such em-pirical models to make projections for future scenarios rests on the assumption that spatialvariability across a landscape is a valid surrogate for the e↵ects of the temporal evolution ofcatchment glacier cover. However, this assumption may not be appropriate, especially underchanging climatic conditions.The most rigorous approach to diagnosing and predicting the sensitivity of stream thermalregimes to environmental changes and human activity is through the application of physi-cally based models that solve energy and water budgets along the stream network. Thesemodels have been developed using Lagrangian (Devkota & Imberger, 2012; Gu et al., 1998;Hockey et al., 1982; Leach & Moore, 2010a; Vugts, 1974), semi-Lagrangian (Yearsley, 2009)1and Eulerian frames of reference (Sinokrot & Stefan, 1993; Sridhar et al., 2004). Methodsof solution have ranged from simple Euler integration for Lagrangian models (Hockey et al.,1982) to sophisticated numerical solutions of partial di↵erential equations in an Eulerianframework that are capable of handling unsteady flow (Kim & Chapra, 1997; Younus et al.,2000) and multiple-component transient storage processes (Bingham et al., 2012; Meier etal., 2003; Schmadel et al., 2015; Westho↵ et al., 2010). All types of models have performedwell for reaches for which surface energy exchanges and the reach-scale water budget canbe characterized using on-site field measurements or calibration. However, even the mostnumerically sophisticated model may not provide reliable predictions if it incorporates in-correct representations of the underlying physical processes, particularly the surface energyexchanges. Furthermore, for models that incorporate transient storage processes, incorrectspecification of surface energy exchanges could be compensated for by incorrect calibrationof the parameters controlling transient storage processes.Table 1.1 indicates stream characteristics and parameterizations used in previous studiesof the energy balance of proglacial streams. The following sections provide a more detailedreview.1.2 Review of thermal processes in steep pro-glacialstreams1.2.1 Net radiationNet radiation includes both solar and longwave radiation. Incident solar radiation varieswith latitude, date, time of day, cloud cover, presence of atmospheric particulate matter,and the slope and aspect of the water surface. In addition, a stream can be partially or fullyshaded by the stream banks, riparian vegetation or surrounding topography. Models existto account for all of these e↵ects. For example, topographic shading can be modelled usinga digital elevation model (Johnson & Wilby, 2015) or field measurements of horizon angles(Tovar-Pescador et al., 2006). Shading by riparian vegetation can be modelled based on fieldmeasurements of vegetation height and channel geometry (Li et al., 2012; Rutherford, 1994),hemispherical photographs (Leach & Moore, 2010a; Moore, 2005), or Lidar data (Greenberget al., 2012).Incident longwave radiation depends on atmospheric temperature and humidity, cloudcover (which influence atmospheric emissivity and temperature) and the view factors asso-ciated with the stream banks, riparian vegetation, surrounding terrain and the atmosphere.Relatively accurate models exist for predicting the atmospheric emissivity (Baigorria et al.,2Table 1.1Input variables and parameter definitions for studies that have applied an energy balance approach to proglacial streams, wherel is the study reach length, S is the slope, w is the stream width, ↵ is albedo, and NS is non-specified information.Authors Location l (m) S w (m) ↵Adjustments forNet RadiationWind functionsourceChikita et al. (2010)Phelan Creek,AK, USA650 and 370 0.02 NS 0.1 None Kondo (1994)Cardenas et al. (2014)Urbach River,Switzerland340 0.02 - 0.3 6-12 NS None Chapra (1997)Magnusson et al. (2012)Damma Glacier,Switzerland520 NS 6-20 0.05 NoneWebb andZhang (1997)Khamis et al. (2015)Taillon–Gabie´touscatchment,France900 NS NS 0.1 NoneWebb andZhang (1997)Meier et al., (2003)Brenno delLucomagno -Valley Blenio,Switzerland250 0.14 6-12Calculated fromsolar angleAccounted fortopographicshadingCalibratedparameters fromMeier (1996)32004), and the view factors can be computed based on radiation geometry using similar dataas required for computing stream shading (Moore et al., 2014). Outgoing longwave radiationis a function of water temperature and emissivity, the latter of which typically ranges from0.95 to 0.97 (Anderson, 1954; Oke, 1987).Steep streams are distinct from low-gradient streams in that low-gradient streams cansafely be assumed to have horizontal water surfaces, whereas the slope and aspect of thestream surface need to be accounted for when calculating incident solar radiation and viewfactors for steep streams. Equations exist for accounting for slope and aspect on radiationreceipt (Iqbal, 1983), although they do not appear to have been applied in previous modellingstudies focused on steep streams (Cardenas et al., 2014; Magnusson et al., 2012; Meier et al.,2003).Another way in which steep mountain streams di↵er from low-gradient streams is thatthey often have high turbidity and aeration, both of which can influence albedo. Chikita etal. (2010) measured albedo near 0.1 for a moderate-gradient, highly turbid proglacial stream.Richards and Moore (2011) found that albedo downstream of Place Glacier ranged from 0.1to over 0.4, substantially greater than the values used or measured for low-gradient streams(typically 0.03 to 0.07) (Benyahya et al., 2012; Evans et al., 1998; Leach & Moore, 2010a).Measured albedo had a positive relation with discharge as a result of the increasing coverageof aerated water at higher flows. McMahon and Moore (2017) developed an empirical modelfor predicting the albedo of mountain streams, which accounts for incidence angle for directradiation, the fraction of di↵use radiation and turbidity. However, no stream temperaturemodelling studies appear to have accounted for the e↵ect of aeration on albedo. For example,Magnusson et al. (2012) assumed a value of 0.05, and Cardenas et al. (2014) followed Chikitaet al. (2010) and set albedo to 0.10.1.2.2 Sensible and latent heat fluxesMost stream temperature models use empirical wind functions to compute the sensible andlatent heat fluxes (Qh and Qe, respectively):Qe = (a+ b · u) · (ew  ea) (1.1)andQh = r ·Qe (1.2)where u is wind speed, ew and ea are the vapour pressures (kPa) at the water surface andthe overlying air, r is the Bowen ratio, and a and b are empirical coecients.4Wind functionsTo the author’s knowledge, only seven studies have derived wind functions for channelizedflow (as opposed to lakes, ponds or wetlands). Four studies derived the wind function byregression using in-stream evaporation pans in low-gradient streams (Benner & Beschta,2000; Caissie, 2006; Guenther et al., 2012; Maheu et al., 2014). Three studies derived thewind function by calibration of a heat budget model. Of these studies, Jobson (1980) andFulford and Sturm (1984) focused on relatively low-gradient artificial channels. Meier et al.(2003) used values derived from Meier (1996), but did not provide any specific informationon the slope or morphology of the channel, noting only that it was similar but with smallerslope than the streams studied in the 2003 paper (3.8% and 14.4%). Webb and Zhang (1997)presented a wind function that provided values consistent with near-stream pan evaporationat a site in England.Previous studies of the energy balance of steep streams incorporated wind function coe-cients from low-gradient stream studies rather than fitting the parameters to their respectivemodels. Magnusson et al. (2012) and Khamis et al. (2015) utilized the Webb and Zhang(1997) values, which generated latent and sensible heat fluxes that were of second-orderimportance in the energy balance, relative to net radiation. Cardenas et al. (2014) incorpo-rated the wind function suggested by Chapra (1997), which was originally developed for lakeevaporation. Chikita et al. (2010) used a dimensionless bulk transfer coecient adjusted byusing the exchange speed to correct for the usual underestimation of the coecient in smallstreams.Sensible and latent heat exchanges for a high-gradient stream with cascading flow andaeration should occur at a higher rate than from a low-gradient stream due to the greatere↵ective surface area of water. However, it does not appear that any previous studies haveattempted to derive a wind function for cascading flow. A central objective of the currentstudy is, therefore, to use an energy-budget-modelling approach to derive a wind functionunder conditions of aerated flow.Above-stream meteorologyMeasurements of above-stream air temperature, humidity and wind speed are required tocompute the sensible and latent heat fluxes. Most studies have used weather stations locatedover land to specify these values (Chikita et al., 2010; Magnusson et al., 2012; Meier etal., 2003). Studies at forested mid-latitude sites have demonstrated that conditions overthe stream can di↵er substantially from land-based measurements (Caissie, 2006; Leach &Moore, 2010a; Maheu et al., 2014). In addition, Leach and Moore (2010a) showed that there5were di↵erences in conditions recorded by two above-stream weather stations about 1 kmapart over a low-gradient stream. Considering the complexity of wind patterns in mountainsettings, in addition to the e↵ect of elevation di↵erences along a steep channel, above-streammeteorology could be even more variable than for low-gradient streams.1.2.3 Non-steady flowMany studies have focused on stream temperature during summer due to concerns about highseasonal temperatures on biota, especially cold- and cool-water fish (Eaton & Scheller, 1996;Mayer, 2012). During summer periods without rain, non-glacial streams would achieve base-flow conditions, for which streamflow can be reasonably assumed to be steady. In glacier-fedstreams, however, late-summer streamflow typically exhibits marked diel variations associ-ated with the diel cycle of glacier melt. The presence of unsteady flow has implications forspecification of hydraulic geometry as well as for transient storage processes, as outlined inthe next two sections.1.2.4 Specifying hydraulic geometryStream temperature models require knowledge of the width, depth and velocity of flow as afunction of discharge – i.e., the hydraulic geometry. For low-gradient streams with relativelysimple cross sections, hydraulic geometry can be determined by measuring depth and velocityat one or more cross sections within the reach (Leach & Moore, 2010a).Steep streams with irregular channel cross sections are more challenging to characterizethan low-gradient streams. However, Lee and Ferguson (2002) successfully used a combi-nation of salt tracer injections and width measurements to determine velocity and widthat di↵erent discharges for non-glacial step-pool streams in England, with gradients up toalmost 20%, for which discharge variations through time would have been relatively minorduring the field campaigns. For proglacial channels, it should be feasible to use salt in-jections to determine velocity-discharge relations; however, it would be dicult to samplesucient transects in a time period in which discharge remained roughly constant in orderto determine width-discharge relations, except for relatively short reaches.Magnusson et al. (2012) determined width-discharge relations for a proglacial stream byapplying an energy-balance framework based on measured incident solar radiation (with analbedo of 0.05), modelled longwave radiation, and sensible and latent heat fluxes computedusing the Webb and Zhang (1997) wind function. However, this approach is based on theassumption that all the surface energy exchanges are accurately specified. As reviewed inearlier sections, the albedo and sensible and latent heat fluxes based on “standard” literature6values may not appropriate for steep proglacial streams.Over the last decade, mapping has been revolutionized by the development of relativelylow-cost drones and software for developing digital elevation models using Structure fromMotion (SfM). As demonstrated by Tamminga et al. (2014), these technologies support themapping and interpretation of channel geometry, gradient and other characteristics overmoderate reach lengths (e.g. hundreds of m) over relatively short time intervals. Thus,the combination of salt injection trials and drone-based photogrammetry provides a viableapproach for characterizing the width-discharge relation for steep proglacial channels.1.2.5 Transient storage and bed heat conductionTransient storage processes commonly include advective heat exchanges between the mainchannel and (a) the hyporheic zone and (b) in-channel transient storage zones such as recir-culating pools (Meier et al., 2003). In streams with beds comprising large clasts, conductiveheat fluxes into and out of the clasts have also been characterized using a transient storageframework (Westho↵ et al., 2010).For a steep proglacial channel, heat exchange between the flowing water and in-streamboulders could be facilitated by the diel cycle of discharge. Proglacial streams typically de-cline in discharge through the morning, with daily minimum flows occurring around noon(Jansson et al., 2003). During morning, exposed boulders would be heated by direct expo-sure to solar radiation. As streamflow increased through the afternoon, this heat would betransferred to the flowing water, thus representing an additional source of energy. Cardenaset al. (2014) documented heating of hillslope water flowing over cli↵s before it entered astream using time-lapse thermal imaging of the stream and adjacent hillslopes.1.3 Research objectives and thesis structureThe goal of this study was to better understand the controls on temperature dynamics ofsteep proglacial streams. The primary hypothesis is that the sensible and latent heat fluxesshould be enhanced by aerated flow, such that fitted wind function parameters should di↵erfrom those established for low-gradient streams. A secondary hypothesis is that, duringthe period of rising flow each day, conduction of heat stored in in-channel boulders shouldrepresent an additional source of energy for warming the stream.To address these hypotheses within an energy-balance framework, a novel combination offield methods was required. Drone-based imagery and SfM were used, in combination withsalt injections, to define the width-discharge and velocity-discharge relations, as well as to7quantify the relation between the fraction of the stream surface that is aerated as a functionof stream discharge and channel gradient.The remainder of this thesis comprises four chapters: methods, results, discussion andconclusion.8Chapter 2Methods2.1 Study areaThis study was conducted on a glacier-fed tributary of the upper Bridge River, located inthe southern Coast Mountains of British Columbia (Figure 2.2a). The stream is unociallyknown as ”South Creek” (Ryder, 1991). South Creek is currently incising into a lateralmoraine that once impounded a glacial lake within the creek’s headwater valley (Figure2.1b). Radiocarbon dating of wood within the strandlines of the lake basin constrain thetiming of multiple drainage events of the lake to be between years 1935 and 1970, whenthe final outburst flood occurred (Ryder, 1991). Based on comparison of historic aerialphotographs, an avulsion took place in the 1980’s diverting the lower 500 m of the stream inwhat is its current channel path today. The breached lateral moraine marks Bridge Glacier’smaximum width and thickness dating back to the Little Ice Age. The river basin thereforelies within a glacially carved area where depth to bedrock is unknown. The basin existswithin the Bridge River Cones area of the larger Garibaldi Volcanic Belt, yielding bedrocklithologies of mostly granite to granodiorite. The river bed and banks of South Creek arecomposed almost entirely of granitic glacial till and reworked glaciolacustrine material andthe basin is almost completely unvegetated. Granitic cobbles and boulders dominate theclast-size portion of the channel bed with a fine to medium sand-sized matrix (MacKenzie,2012). At the upper end of the study reach, the stream’s catchment area is 18 km2, of which20% remains glacierized.The study focused on a 965-m-long section of South Creek, with the upper boundarylocated at the breached moraine. The elevation ranges from 1500 m to 1400 m above sea level(m.a.s.l.). South Creek has slopes ranging from 0.05-0.33 and wetted channel widths between5 and 20 m, with step-pool and cascade morphologies. There are no tributary confluencesalong the reach. The hydrologic regime is dominated by spring-summer meltwater, initially9(a)(b)Figure 2.1: Study area location. (a) Satellite image of part of the upper Bridge River val-ley showing the ice-contact proglacial lake and South Creek tributary stream,which flows NNW into the proglacial lake. (b) Satellite image of the portion ofimage (a) within the red box, highlighting locations of the three weather sta-tions, pressure transducers for stage height and 10 water temperature sensorsdeployed throughout the reach.10(a)(b)Figure 2.2: (a) Looking south across Bridge Lake at South Creek (star) and surroundingterrain. (b) Zoomed-in photo of Reaches 1 and 2 of South Creek taken from afloatplane looking upstream (south). Photo: A Szeitz (2016)11dominated by seasonal snowmelt and transitioning to glacier melt in the summer. Summerdischarges generally range from about 0.5 - 4 m3s1 (Moyer et al., 2016).To account for the downstream variation in stream morphology, the 965-m-long sectionof stream was divided into four reaches (Table 2.1). The upper-most reaches, R3 and R4,exhibit the highest slopes as they continue to incise into the lateral moraine. These reachesare characterized by mainly step-pool and cascading morphologies, and exhibit a higherdegree of aeration than the lower reaches. The gradient gradually decreases downstream inR2 followed by a more abrupt break in slope at a noticeable knickpoint in R1, below whicha more gentle gradient continues before its outlet into Bridge Lake. Due to this contrast inreach morphology, width and aeration analysis is divided into lower and upper R1 sections.Access to the stream was from the east, and it was not safe to cross the stream except fora location upstream of the breached moraine. Therefore, manual field measurements weremade from the right bank of the stream.Table 2.1Summary of reach characteristics for South Creek from the most downstream Reach 1 to themost upstream Reach 4, where l:w is the ratio of the length to average wetted width used inthe salt injection methods. Black dots represent surveyed points.Reach slope length (m) l:w ratio (m) morphology1 0.05 250 20.8:1 step-pool2 0.1 240 23.5:1 step-pool3 0.23 325 28.7:1 cascade and step-pool4 0.33 150 15:1 cascade and step-pool0 200 400 600 8001400144014801520R4R3R2R1Distance (m)Elevation (m)Figure 2.3: Longitudinal profile of South Creek, where coloured points represent individualsurveyed locations along the stream for each reach.122.2 Streamflow2.2.1 Stream gaugingStreamflow was measured at the lower end of each study reach by dilution gauging using themass balance method, which involves injection of known masses of salt (Richardson et al.,2017). Injections were repeated 10 to 13 times throughout the summer in each of the fourreaches in order to capture the full range of flows over the field season. Timing of the saltwave relative to the time of injection was also recorded to allow calculation of travel times.A known mass of NaCl (usually a 2 kg box) was injected at the upstream end of thereach. The boxes were weighed to the nearest 0.1 kg before and after injection to reduce theuncertainty in the injected mass to under 0.1%.In order to promote rapid dissolution and lateral mixing of the tracer, salt was injectedupstream of flow constrictions or in highly turbulent areas. It was not possible to record thesalt wave on both sides of the creek to ensure complete mixing of the salt across the channel.However, Richardson et al. (2017) made repeated measurements at three locations alongSouth Creek using di↵erent injection points, and concluded that complete mixing would beachieved at South Creek with mixing lengths from 6.5 to 24.5 times the mean wetted width.The length:width ratios used in this study are shown in Table 2.1.Discharge was computed as:Q =MCFT · A (2.1)where M is the mass of salt (g), CFT is the calibration factor (g · cm · µS1 ·m3), and A isthe area under the breakthrough curve (s · µS · cm1), computed as:A = tnXi=1[ECT (i) ECbg] (2.2)where ECT (i) is the electrical conductivity observed at time increment i (µS · cm1), ECbgis the background conductivity of the stream, n is the number of ECT observations recordedduring the passage of the salt wave, and t is the time interval between ECT observations(s).The calibration factor was derived in the field following procedures described by Richard-son et al. (2017). First, a secondary solution was made by mixing 1 g NaCl in 1 L ofstream water. The salt was pre-weighed on a laboratory scale to ± 0.001 g, and volumesof streamwater were measured with a glass volumetric flask to minimize uncertainty. Thesecondary solution was added in increments of 5 or 10 mL (measured using glass pipettes)13to 1 L of streamwater and the ECT was recorded. Increments of the secondary solution wereadded until ECT of the calibration solution exceeded the peak ECT observed during thesalt wave. At least four increments of secondary solution were added in each calibration toensure that any blunders (e.g., mis-recorded values) could be detected and corrected. CFTwas determined as the slope of a linear regression between the concentration of NaCl andECT of the secondary solution.2.2.2 Stage measurements and rating curvesContinuous water level measurements were recorded in three locations using Onset U20pressure transducers (Figure 2.1b). Stage measurements were processed by subtracting thebarometric pressure, measured with an aerially exposed transducer located at the upstreamend of the study section, from the raw pressure data. Shifts in the recorded stage occurredwhen the transducers were removed from the stream for downloading and then replacedin the stream. These shifts were computed based on the recorded stage before and afterdownloading and were corrected to ensure a consistent time series.Rating curves were constructed for each of the four reaches by relating the measureddischarges (10 to 13 per reach) to the corresponding stage measurement using a power-lawof the following form: :Q = j · (H H0)k (2.3)where Q is measured discharge, H is the stage extracted from the closest pressure transducercorresponding to the time of the discharge measurement, and j, H0 and k are parametersdetermined by nonlinear least squares regression. Data from the downstream-most pressuretransducer were not used due to intractable noise from being placed in what turned out tobe a hydraulic jump at higher flows. See Appendix A for rating curves and Appendix B forexample salt dilution curves.2.3 Discharge-dependent reach characteristics2.3.1 Travel timeRelations between travel time and discharge were established for each of the four reaches.The harmonic mean, centroid and peak travel times were first determined from the saltwaves. Following Zimmermann (2010), the harmonic mean travel time was chosen as themost appropriate measure of mean travel time for each dilution gauging in Reaches 1-3.14Relations between travel time (⌧) and discharge did not conform to the expected power-lawrelation and were instead modelled using a linear relation of the form:⌧r(Q) = er + fr · log(Q) (2.4)where er and fr are coecients fitted by linear regression for reach r. The best relationbetween ⌧ and Q for reach 4 was achieved using the lag time to peak ECT due to its roughlysymmetrical salt waves. Velocity V r(Q) was then modelled as a function of discharge foreach reach:Vr(Q) =Lr⌧r(Q)(2.5)where ⌧r(Q) is the travel time for a given discharge and Lr is the length of the reach (m).2.3.2 Stream widthWidth was measured in two ways. For the lower three reaches, width was measured remotelyusing drone-based orthomosaic images from a quadcopter UAC. South Creek poses an idealstudy area for Structure from Motion (SfM) photogrammetry due to the lack of vegetationnear the stream. Channel dimensions were determined by extracting surface area informationobtained from aerial imagery analysis using methods and suggestions from a study of braidedstreams by Javernick et al. (2014).Prior to the aerial surveys, a total of 25 ground control points were established on bothsides of the stream and surveyed using a total station. Three total station set-ups were usedto complete the longitudinal survey, with each station location surveyed from subsequentset-ups in order to ensure accuracy in the results and account for the compounded erroracquired when moving the instrument. Flights were operated at a constant speed 50 mabove the stream and flown at a range of flow conditions to account for diurnal variabilityin discharge. Overlapping images and alternating camera angles reduced artificial surfacecurvature and optimized the 3D accuracy of a given feature. Unfortunately, the highest flowswere not captured due to drone charging diculties in the early season.Following completion of the field campaigns, orthomosaic images and DEMs were createdin AgiSoft Photoscan for each complete flight. Vertical accuracy of the DEMs ranged between5 and 10 cm after comparing the coordinates of the control points from the total stationto DEM derived elevations. These orthomosaics were then analyzed to extract width andaeration data using transects spaced every 10 m, and manually measuring the wetted surfacewidth and proportion of the stream width experiencing partial, full or no aeration (Figure15Figure 2.4: A section of one of 11 constructed orthomosaic images used in order to quan-tify the stream width-discharge relationship by measuring widths along 10-mspaced transects. Orthomosaics were also used to estimate proportion of aer-ated water surface area. Blue, red, and yellow lines indicate the downstreamextents of reaches 3, 2 and 1 respectively. Streamflow is from left to right.2.4). The mean reach width within each image was computed. The corresponding dischargewas extracted from the discharge time series for the reach generated using the stage recordand the rating curve.The second approach was based on measuring widths at 25-m intervals along all fourreaches using a laser rangefinder, using the opposite bank for a target. At locations wheresome of the flow was carried by small side channels, these widths were measured using a 30-m tape and added to the width of the main channel. In addition to the total width, visualestimates were made of the fractions of each transect width that were partially or fullyaerated; however, the results were not comparable to aeration estimated using orthomosaicimages. This is likely due to the oblique perspective of the stream surface on the banks asopposed to plan-view measurements from UAV photographs.A power-law function of the following form was fit for each reach:wr(Q) = ar ·Qbr (2.6)where ar and br are fitted coecients for reach r. The power-law function was estimatedby performing a linear regression for log-transformed variables, then back-transforming therelation. The SfM-derived widths were used for the lower three reaches and the field mea-surements for the upper-most.UAV photogrammetry was restricted to reaches 1-3 due to topographic constraints.Width estimates of reach 4 were made with a laser range finder at varying flows on four16separate days but due to the lack of available data points, a power-law relationship was notmade. Instead, considering the apparent similarity in relations for Reaches 3 and 4, therelation for Reach 3 was assumed to apply to Reach 4 in the simulations.2.3.3 Fraction of aerationFor the three lower reaches, the degree of aeration was determined manually from the ortho-mosaic images using the width transects described in the preceding section. The width of fullaeration (completely white sections) and non-aerated flat water (wna) were first measuredalong the transects, leaving the remainder of the transect represented by partial aeration.Because these are only visual estimates, only the measured non-aerated section of streamwas used to eliminate biases in visually designating water as either fully or partially aerated,as non-aerated is the most obvious. Therefore, the fraction of the stream surface exposed toaeration was computed as:faer =⌃ni=1xi⌃ni=1wi(2.7)where n is the total number of transects measured in the orthomosaic for each reach, irepresents the individual transects measured, xi is the length of the complete and partialaerated section of transect i (m) and wi is the width of transect i (m).2.4 Stream temperature and meteorological dataStream temperature was monitored at ten locations along the 965-m study section usingTidbit sensors that were programmed to record at 10-minute intervals (green dots on Figure2.1b). The sensors were enclosed within short lengths of white PVC pipe with holes drilledalong the pipe’s length, and attached to concrete weights. This assembly was attached byclothesline to an anchor above the stream bank. Laboratory calibration showed that sensorswere accurate within ± 0.2°C.Three weather stations were deployed, one on a terrace located 50 m from the stream(hereafter referred to as the open-site station) and two mounted above the stream (Figure2.5a and Figure 2.5b). The above-stream stations were located at a steep, highly aeratedsection upstream (1480 m.a.s.l.) as well as a more moderately graded section downstream(1420 m.a.s.l.). Air temperature, relative humidity, wind speed and net radiation weremeasured above the stream surface on stations bolted into granite boulders of the banks. Inaddition, reflected solar radiation was recorded at the upstream and downstream stations17using a pyranometer suspended from a gimbal support to ensure it faced vertically downwards(McMahon & Moore, 2017; Richards & Moore, 2011).The open-site station monitored incident solar and longwave radiation, air temperature,relative humidity, rainfall, barometric pressure and wind speed at a height of 1.5 m above acleared ground surface, with minor topographic shading. Observations at all locations wererecorded every 10 s and averaged over 10-minute intervals using Campbell Scientific CR10Xdata loggers.Table 2.2Hydrometerological variables measured and instruments used. The number in brackets indi-cates the number of instruments deployed.Variable Instrument Sensor NameWater temperature (Tw) Temperature sensor (10) Onset Tidbit v2Stage height (H) Pressure transducer (4) Onset U20 Level LoggerAir temperature (Ta) Shielded temperature sensor (3) Rotronic HC-S3Rel. humidity (RH) Shielded humidity sensor (3) Rotronic HC-S3Wind speed (u) Anemometer (3) R.M. Young 05108Baro pressure (P ) Barometric pressure logger Onset U20 Level LoggerRainfall (R) Rain gauge TX Electronics TR-525MIncident Solar (K #) Pyranometer (3) Kipp and Zonen CM3Incident Longwave (LW #) Pyrgeometer Kipp and Zonen CGR3Stream surface area (w) UAV DGI Phantom 4Electrical conductivity (ECT ) EC probe and meter (2) WTW TetraCon 3252.5 Processing of meteorological data2.5.1 Calculation of vapour pressureVapour pressure of the water surface and overlying air (ew and ea, respectively) (kPa) werecomputed asew = esat(Tw) (2.8)andea =RH100esat(Ta) (2.9)where Ta and Tw are air and water temperatures, respectively (C), esat(T ) is the saturationvapour pressure for temperature T , and RH is the measured relative humidity (%).18(a)(b)Figure 2.5: Photographs of two of the three weather stations. (Top) Open-site weatherstation located on a stream terrace. (Bottom) Above-stream weather stationlocated at the upper end of the 1-km study reach, looking downstream.192.5.2 Spatial distribution of air temperature, vapour pressure andwind speedAir temperature, vapour pressure and wind speed (u) were assumed to vary linearly withelevation. Accordingly, the value of a variable x (i.e., Ta, ea or u) was computed for a giventime t and elevation z as follows:x(t, z) = xlwr(t) + (z  zlwr) · xupr(t) xlwr(t)zupr  zlwr (2.10)where x(t, z) is the interpolated value, xlwr(t) and xupr(t) are the recorded values of x at thelower and upper above-stream weather stations at time t, zlwr and zupr are the elevations ofthe lower and upper weather stations (m), and z is the elevation of a specific location alongthe channel (m).2.5.3 Spatial distribution of solar radiationThe first step in estimating incident solar radiation (K#) at points along the study reach wasto separate the total measured K# into its direct and di↵use components (K#dir and K#diff ,respectively) following the approach described by Erbs et al. (1982) and as applied by Leachand Moore (2010). Incident solar radiation at a specific location along the stream channelat time t (K# (t, s)) was then calculated as:K # (t, s) = K #dir (t) · (t, s) · cos[i(t, s)] +K #diff (t) · fv(s) (2.11)where (t, s) equals 1 at times when the sun is above the horizon at point s and 0 otherwise,i(t, s) is the angle of incidence for direct radiation at the stream surface, and fv(s) is the skyview factor at point s.Using a 25-m resolution DEM, sky view factors (fv) were calculated for each point alongthe channel for ground azimuth horizon angle intervals of 10° (36 points) as:fv(s) =1⇡Z 2⇡0Z Ha0cos[i(✓, a)]sin(✓)d✓da (2.12)where fv(s) is the sky view factor for point s along the stream, Ha is the horizon angle ata specified azimuth (a), ✓ is the zenith angle, i(✓, a) is the angle of incidence on the streamsurface receiving radiation from location (✓, a) in the sky, taking into account the stream’sslope and aspect at that point. The integral over zenith angle was performed over 10°azimuth intervals and then multiplied by da (10° ⇡ 0.1745 rad) and summed to approximatethe integral over azimuth angle. The local azimuth and zenith angles were calculated using20equations from Iqbal (1983).To determine (t, s), the horizon angle associated with the solar azimuth angle at timet was interpolated linearly from the values of Ha determined from the DEM. The value of(t, s) was then assigned based on the comparison of the solar elevation angle at time t tothe interpolated horizon angle.2.5.4 Spatial distribution of longwave radiationIncident longwave radiation at points along the channel was calculated as:L # (t, s) = L # (t) · fv(s) + (1fv(s)) · ✏t ·  · (Tt + 273.2)4 (2.13)where L # (t) is the measured incident longwave radiation at time t, ✏t is the emissivityof surrounding terrain, assumed equal to 0.95, and Tt is the e↵ective temperature of thesurrounding terrain (°C), assumed to equal air temperature at the open-site weather station.2.5.5 Net radiationNet radiation (Qr) was calculated as:Qr = K #(t,s) (1 ↵) + ✏wL #(t,s)  ✏wTw4 (2.14)where ↵ is stream albedo calculated using equations from McMahon et al. (2017) for white-water streams, ✏w is the emissivity of water, (L#) is the incident longwave radiation,  isthe Stefan-Boltzmann constant (5.67·108W m2 K4) and Tw is the temperature of thewater (K). The emissivity of the water was set to a constant value of 0.97 following therecommendations of Oke (1987).2.5.6 Latent and sensible heatLatent heat was calculated using a Dalton-type equation as:Qe = fw · (ea  ew) (2.15)where fw is a wind function which depends on the wind speed over the stream surface and thestability of the atmosphere directly over the stream, expressed as the temperature contrastbetween the water and overlying air. The function becomesfw = (c+ d · u) + e · (Tw  Ta) (2.16)21where c, d, and e are the fitted empirical coecients (W·m2·kPa1, W·s·m3·kPa1 andW·m2·kPa1·°C1, respectively).Sensible heat flux, Qh, was calculated from the latent heat flux and the Bowen ratio:Qh =✓cpaP0.622Lv◆✓Ta  Twea  ew◆Qe (2.17)where cpa is the specific heat of air at constant pressure (J·kg1·C1), P is atmosphericpressure (kPa) and Lv is the latent heat of vaporization (2.47·106 J/kg).2.5.7 Frictional heat dissipationAs water flows downhill, the potential energy associated with gravity is dissipated by friction.Assuming that all of this energy is consumed by heating water, this heat input can becomputed as:Qf =⇢w · g · S ·Qw(2.18)where ⇢w is the density of water (1000 kg·m3), g is the acceleration due to gravity (9.8m·m2),S is the slope at a point, Q is discharge and w is width (Meier et al., 2003). Integratingthis heat input along a stream reach, the associated change in water temperature can becomputed as:Tf =gzcw(2.19)where z is the total change in elevation.2.5.8 Surface-subsurface interactionsThe potential for influences of surface-subsurface interactions was investigated in two ways.First, the total input of water at the top of the reach was compared to the total output atthe bottom over daily and longer time frames. The di↵erence provides an indication of netgains or losses along the channel, subject to measurement error. Second, following Moore etal. (2008), longitudinal surveys of electrical conductivity and temperature were conductedon several days during late morning, when discharge was close to steady. Sharp changes inECT and/or water temperature would signal potential areas of inflow to the channel.222.6 Numerical implementationThe reach was delineated into segments based on the 25-m DEM. The length of each reachthus depended on the local gradient and was slightly greater than 25 m. The model followeda parcel i released at time ti with discharge Qi (determined from the upstream gauge) as itflowed downstream. At the upstream end of each segment, the parcel’s velocity was computedas a function of Qi and the reach containing the segment, using Eq. (2.5). Using this velocityand the length of the reach, the times at which the parcel passed the segment’s midpoint(tmid) and lower end (tl) were computed. Energy inputs were computed based on the parcel’stemperature at the upstream end of the segment and meteorological data interpolated to thesegment’s midpoint for time tmid, as described in sections 2.5.2, 2.5.3 and 2.5.4.The temperature at the lower end of the segment, Tp(l), was then calculated asTp(l) = Tp(u) +w · ds · (Qr +Qh +Qe)⇢w · cw ·Qi +g · tan  ·xcw(2.20)where Tp(u) is the temperature of the parcel at the upstream end of the segment (C), dsis the segment’s slope distance (m), Qr + Qh + Qe is the net energy exchange across thesegment’s water surface, and x is the DEM grid spacing (25 m).Once the parcel reached the bottom of the 965-m study reach, the final temperature wascomputed asTds(i) = Tp(f) +  (2.21)where Tds(i) is the final predicted temperature for parcel i (C), Tp(f) is the predictedtemperature of the parcel at the downstream end of the lowest segment (C), and  isa parameter that accounts for the constant e↵ect of any energy exchanges that are notaccounted for, along with any bias between the upstream and downstream temperatureloggers (C).In addition to tracking the temperature of each parcel as it flowed downstream, the modelalso calculated the mean energy fluxes experienced by the parcel between the upstream anddownstream ends of the reach, and the time at which the parcel arrived at the downstreamend of the study reach (tds(i)).2.6.1 Model calibration and testingThe model was calibrated based on the Lagrangian temperature change for each parcel,computed as23Tmod(i) = Tds(i) Tus(i) (2.22)where Tus(i) is the observed temperature at the upstream end of the study reach for parceli (C). These simulated di↵erences were compared to the observed change in temperature,computed asTobs(i) = Tod(ti + ⌧) Tus(i) (2.23)where Tod(ti + ⌧) is the observed temperature at the lower-most logger at time ti + ⌧ (C),and ⌧ is the time taken for a parcel with discharge Qi to travel down the entire study reach(s).Upstream boundary conditions (discharge and temperature) were recorded every 10 min.Calibration was performed on three separate datasets of released upstream parcels, wherethe total number of parcels represent all ten-minute intervals where complete boundaryconditions were recorded for that parcel’s travel time downstream. The first set includedone-sixth of the total parcels, sampling one complete set of boundary condition observations(one ten-minute interval) per hour for the entire length of the study period. This calibrationdataset is further referred to as the ”hourly” set. In addition, split-sample calibration fornighttime (21:00 - 5:00) and daytime data (7:00 - 18:00) subsets were conducted to assessthe variability of fitted model parameters under conditions with di↵erent heat exchanges.The calibration was based on finding the values of the wind function parameters c, d and eand the o↵set parameter  that minimized the root-mean-square error for the Lagrangiantemperature change:RMSE =vuut 1nnXk=1[Tobs(i)Tmod(i)]2 (2.24)Calibration was performed using the “pseudoOptim()” function in the FME package in theR programming language (Soetaert & Petzoldt, 2010).One source of uncertainty in the calibrated parameters is uncertainty in the boundaryconditions, particularly net radiation, travel time and width. To characterize the e↵ectof uncertainty in these boundary conditions, the model was optimized 100 times. In eachoptimization, net radiation, travel time and width were multiplied by normally distributedrandom variables with a mean of unity and standard deviation of 0.05.24Chapter 3Results3.1 Overview of the study periodA two-month record of discharge and meteorological data was obtained from 7 July to 7September 2017. Lake ice prevented both an earlier and later extension of the study perioddue to restricted floatplane access. Streamflow varied throughout the season with the highestflows early in the study period, up to 4.15 m3s1, and lower flows later into September, downto 0.69 m3s1 (Figure 3.2). Streamflow variability was dominated by the diurnal melt cycle.Beginning in mid-August (August 23), there was a decline in daily minimum flows but anincrease in the diel range (Figure 3.2). Air temperatures ranged between 0 and 26 °C alongthe study reach. The observational period was mainly dry with an average relative humidityof 50% and only 7 days experiencing measurable precipitation (Figure 3.1).3.2 Flow routing and the potential for groundwaterfluxComparisons of downstream and upstream water storage suggest both daily and seasonalstreamflow trends over the study reach. Overall, the study reach lost more water downstreamduring the day, corresponding to the rising limb of the hydrograph, than during the nighttimefalling limb of the hydrograph (Figure 3.2). Downstream gains occurred over night in thebeginning of the season, but as general streamflow decreased over time, the stream beganconsistently losing water downstream at all times. A shift occurred on August 23 after apeak flow event that once again restored the diel loses and gains over the reach.Paired electrical conductivity and water temperature surveys highlight slight variationsin water chemistry along the reach. As seen in Figure 3.3, there is a definitive drop in EC2504001000K↓ (Wm−2)260320380L↓ (Wm−2)0246u (ms−1 )0.00.40.8P (mm)0102030T (°C)Air Water01234Q (m3 s−1)Jul−07 Jul−15 Jul−24 Aug−02 Aug−11 Aug−20 Aug−29 Sep−06Figure 3.1: Plot of meteorological conditions observed at the base weather station for thefull duration of the study period.and corresponding spike in temperature at locations 390 m and 840 m downstream. Based onthe diurnal trends in downstream water storage, the stream would typically be losing waterduring all EC/Tw measurements as they all occured on the rising limb of the hydrograph.Therefore, the magnitude of the EC/Tw contrasts measured might underestimate the waterchemistry variations along the reach on the falling limb of the hydrograph, when the storagecalculations predict gaining streamflow downstream. In addition, as the stream gains waterat night, more locations of EC/Tw contrasts might exist than were detected during the day.26Figure 3.2: Plot of discharge for Reach 4 (black, upstream) and Reach 1 (blue, downstream)and the change in storage between both reaches. (See Appendix A for ratingcurves).3.3 Inter-station comparison of meteorological variablesFigure 3.4 shows the di↵erences in wind speed, air temperature and saturation vapor pres-sure between the two above-stream locations for the entire two month study period. Airtemperature at the upstream station was generally warmer than the downstream stationduring the day and cooler at night, with maximum di↵erences of 7.7 deg C during the dayand -6.9 deg C during the night. This diurnal signal is also present in the relative humidityvariations between both stations, with higher relative humidities upstream at night and lowerduring the afternoon, made apparent in the vapour pressure di↵erences. Wind speed di↵eredbetween the two above-stream sites by approximately ± 3 ms1. The upstream station expe-rienced consistently higher winds during periods of low pressure and stormy weather, whichcoincided with small temperature and vapour pressure gradients between the two sites (redline, Figure 3.4).The greatest variation was observed between the two above-stream weather stations andthe open-site station. Plots of observed meteorological conditions at both locations can befound in Appendix C for the full study period. Temperature was more comparable betweenthe stations than vapour pressure and wind speed. The above-stream stations exhibitedconsistently higher vapour pressures and overall lower wind speed than the open-site station.The above-stream wind speeds were limited to 5 ms1 whereas the open site experiencedwinds up to 7.5 ms1. The downstream weather station is more comparable to the open-sitestation than the upstream station. The plotted di↵erences in each variable for the upstreamand downstream values can be seen in Figure 3.5 where the discrepancies are attributed todiurnal variations between stations.South Creek exhibited a consistent downstream warming trend between the upstream2712141618Distance (m)EC (µ S/cm)0 200 400 600 80068101214Distance downstream (m)T w (°C)7/07 09:40 7/24 09:40 7/29 16:00 8/17 16:00Figure 3.3: Plots of longitudinal water chemistry observations for electrical conductivity(top) and temperature (bottom) on four separate days.and downstream water sensors for the majority of the study period, according to calculatedobserved downstream temperature changes. Negative temperature gradients (cooling) onlyoccurred for 0.1% of all ten-minute recorded observations. Downstream temperature changesranged from -0.3 to 1.7 deg C, with an average of 0.6 °C over the 965-m-long reach. In-streamtemperatures varied between a minimum of 2 °C overnight to a maximum of 15°C in theafternoon, with the greatest diel temperature swings observed in channel pools.3.4 Hydraulic geometryFigures 3.6a and 3.6b display the change in channel geometry and degree of aeration withchanging flow conditions for a reference section of the stream. The fitted power-law relationsfor width-discharge had R2 values between 0.8 and 0.9, with corresponding 95% confidence28−3−2−10123∆ u  (ms−1 )−505∆ Ta  (°C)Aug Sep−0.20.00.20.4∆ ea  (kPa)Figure 3.4: Di↵erences in recorded wind speed (top), air temperature (center), and vapourpressure (bottom) between the two above-stream weather station locations.Doted red line indicates more similar conditions between weather stations,occurring during a low pressure storm.intervals plotted in Figure 3.6 (Table 3.1). Reach 3 exhibited the highest degree of scatter.The ephemeral side channels made this reach the most challenging to measure due to di-culties in assessing flowing or stagnant water, in which case the presence of aeration was usedto di↵erentiate between the two. Reach 3 and the lower portion of Reach 1 (downstream ofthe knickpoint) had the greatest observed widths while Reach 2 was the most narrow (Ta-290 5 10 20051020Open Ta  (°C)DS Ta  (°C)best fit1:10 5 10 20051020Open Ta  (°C)US Ta  (°C)0 5 10 20051020US Ta  (°C)DS Ta  (°C)0.2 0.6 1.0 1.40.20.61.01.4Open ea  (kPa)DS ea  (kPa)0.2 0.6 1.0 1.40.20.61.01.4Open ea  (kPa)US ea  (kPa)0.2 0.6 1.0 1.40.20.61.01.4US ea  (kPa)DS ea  (kPa)0 2 4 60246Open u  (m s−1)DS u  (m s−1 )0 2 4 60246Open u  (m s−1)US u  (m s−1 )0 2 4 60246US u  (m s−1)DS u  (m s−1 )Figure 3.5: Comparison of air temperature, vapour pressure and wind speed among theupstream (US), downstream (DS) and open-site weather stations.ble 3.1). Although the ground-based width observations for Reach 4 were less reliable thanthe widths determined by photogrammetry, the values generally plot around the relation forReach 3, which is broadly consistent with the similarity of channel morphology between thereaches.There was no clear relationship between width and slope. The b exponent coecient waslarger downstream (lower and upper Reach 1) than upstream (Reach 3), but actual widthswere more similar between the steep upstream reaches and the lowest gradient sections30downstream, as exhibited by the reference width for the average discharge in Table 3.1. Thisis due to a large a coecient in Reach 3, where multi-threaded channels keep the streamwide at low discharges. Stream gradients ranged from 4 to 33% with the steepest sectionsin Reaches 3 and 4 located within or just downstream of the incised lateral moraine and theshallowest gradients at the most downstream extent.Table 3.1Power law relations for width (coecients ar and br), as well as travel time and velocitypredicted from discharge of 2.5 m3s1 and average slope for each reach.Reach ar (m) br wref (m) R2 (w) S ⌧ref (s) R2 (⌧) vref (ms1)1 lower 9.39 0.39 13.4 0.93 0.05 261 0.94 0.761 upper 8.45 0.33 11.4 0.90 0.07 261 0.94 0.762 8.82 0.24 10.9 0.87 0.07 298 0.73 0.803 9.41 0.27 12.2 0.81 0.13 438 0.9 0.624 NA NA 11.5 NA 0.14 222 0.78 0.69Aeration increased with discharge for all reaches, with the stream nearing almost fullaeration for high flows (greater than 3.5 m3s1; Figure 3.7). Although all reaches approachedfull aeration at the highest discharges, the decline in aeration with decreasing dischargetended to relate to channel gradient, with the shallowest (5%), the portion of Reach 1 belowthe knickpoint, showing the greatest decline. Although not measured, the stream containedhigh suspended sediment concentrations that increased with discharge, as observed in thefield and captured by UAV photographs.Observed velocities ranged from about 0.4 to 1.0 m·s1 and increased in a nonlinear fashionwith discharge (Figure 3.8). The steeper reaches (3 and 4) generally had lower velocities thanthe lower-gradient reaches (1 and 2), especially for discharges below 1 ms1. The amount ofscatter generally increased with discharge.3.5 Comparing modelled and measured solar radiationand albedoThe average albedo of the stream surface measured above two points upstream and down-stream was 0.3 while the average modelled albedo (of 37 continuous segments) was 0.17.Despite this discrepancy, the modelled albedo follows the same daily trend as the measuredalbedo, as seen in Figure 3.9. The pyranometers recorded higher reflected solar radiationthan was predicted in the models, particularly when incident solar radiation was low, butthe two experience greater similarities during times of greater solar elevation angles in the310.5 2.0 3.581014Lower Reach 10.5 2.0 3.5Upper Reach 10.5 2.0 3.5Reach 20.5 2.0 3.581014Reach 30.5 2.0 3.5Reach 4Width (m)Q (m3s−1)Figure 3.6: Width-discharge relations for all reaches with 95% confidence intervals outlinedin blue. Measured widths for Reaches 1 to 3 were derived from orthomosaicphotogrammetry; measured widths for Reach 4 were estimated using a laserrange finder performed on 4 separate days. The W-Q regression line for Reach3 is plotted on the data for Reach 4.afternoon hours. The model adjustments and raw measured solar radiation exhibited smallvarying di↵erences along the reach. In Reaches 3 and 4, modelled solar radiation was onaverage 20-30 W·m2 lower than the solar radiation measured on the stream terrace, witha maximum di↵erence of 500 W·m2 measured on one of these segments. In contrast, the321.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0Lower Reach 1Discharge (m3/s)Portion aerated1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0Upper Reach 1Discharge (m3/s)Portion aerated1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0Reach 2Discharge (m3/s)Portion aerated1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0Reach 3Discharge (m3/s)Portion aeratedProportion of surface aeratedQ (m3s−1)Figure 3.7: Relations between aerated portion of the stream derived from UAV photogram-metry and discharge.modelled solar receipt for lower reaches agreed within 5 to 10 W·m2. The degree of vari-ation between the modelled segments, shown by the grey lines in Figure 3.10, captures theheterogeneity in stream aspect, shading and slope along the study reach.3.6 Model optimization and sensitivityAccounting for uncertainty in the boundary conditions, the optimization yielded values ofc, d, e and  as shown in Table 3.2 and Figure 3.11. None of the hourly day-night modelranges includes 0, so it can be concluded that all fitted coecients are significantly di↵erentfrom zero. Coecient e was 0 for the nighttime split-sample model runs. Root-mean-squareerror associated with the optimizations ranged from 0.03 C at night to 0.14 C during theday, with a best-fit RMSE of 0.08 C for the hourly test, which is well within the 0.2 C331 2 3 40.30.50.70.9Reach 11 2 3 40.30.50.70.9Reach 21 2 3 40.30.50.70.9Reach 31 2 3 40.30.50.70.9Reach 4Velocity (ms−1 )Q (m3s−1)Figure 3.8: Relations between velocity and discharge for each reach, based on Eqs. (2.4)and (2.5), with 95% confidence intervals outlined in blue.accuracy of the temperature sensors.As shown in Figure 3.12, the wind function parameters (c, d and e) all have positive lin-ear relations with variations in net radiation; that is, as net radiation is perturbed upward,the optimized model increases the magnitudes of the sensible and latent heat fluxes to com-pensate. The o↵set parameter () is negatively related to perturbations in net radiation,although the variation is relatively small (± 0.02 C) compared to downstream tempera-ture changes. The o↵set has no clear relation with perturbations in travel time, and thewind function parameters have a varying response: c increases while d and e decrease withtravel time. The wind function parameters had the weakest response to perturbations insurface width, whereas the sensitivity of the o↵set parameter was similar to its sensitivity tovariations in net radiation.The combined model optimization tests assume that inputs of radiation, travel time, and340.00.20.40.60.81.0αAug−04 Aug−06 Aug−08 Aug−11Measured ModelledFigure 3.9: Comparison of time series of albedo computed from the model of McMahonand Moore (2017) and values based on the downward-facing pyranometers.02006001000K↓ (Wm−2)Aug−04 Aug−06 Aug−08 Aug−11Adjusted Open−site measurementFigure 3.10: Comparison of modelled direct solar radiation and raw pyranometer valuesmeasured from the stream terrace. Grey lines represent the adjusted values foreach of the 37 segments of the stream when taking into account topographicshading and solar angle/position.width are changing simultaneously and randomly with respect to some degree of measure-ment error (defined by the error term multiplier). In the combined error analysis for thehourly dataset, the parameter values conform to more clear relationships with adjustmentsin radiation, but the interactions between the three boundary conditions is unknown. Inorder to separate the e↵ects of each, additional optimization tests were conducted for eachboundary condition individually. When adjusting for just one input, the individual e↵ects35Figure 3.11: Distributions of parameter values for c, d, e, and  from split-sample opti-mization tests (night vs day) and the hourly optimization test.of not only radiation, but also width and travel time, become more clear, as seen in Figure3.13 and Table 3.2. When adjusting for width alone, there are strong positive correlationswith the stability parameter, coecient e, and strong negative correlations with all otherparameters. However, the range of predicted parameter values is much more narrow whenadjusting for width than for travel time or radiation (Table 3.2). When travel times areperturbed upward, the coecient c increases, while all other coecients decrease. The re-lationships with net radiation remain the same for the individual tests as they were in thecombined analysis and contain the least amount of scatter.Split-sample testing between nighttime and daytime datasets yielded very di↵erent pa-rameter ranges. During daylight hours, the free convection coecient c is an order of magni-tude higher and the stability parameter e goes from zero at night to 23-45 W·m2·kPa1°C136during the day. The estimated residual warming component () increases at night as theother coecients decrease, and is on average four times greater at night than during the day.The residual warming at night maintains a nearly constant 0.23 C, which for the dischargesobserved throughout the study season equates to 100 to 300 Wm2 of heat input over thereach.ce_NRce_tauce_width450500550600450500550600450500550600de_NRde_taude_width125150175125150175125150175ee_NRee_tauee_width20.025.030.035.020.025.030.035.020.025.030.035.0phie_NRphie_tauphie_width0.8 0.9 1.0 1.1 0.8 0.9 1.0 1.1 0.8 0.9 1.0 1.10.100.120.140.100.120.140.100.120.14Error term multiplerRange of parameter valuesFigure 3.12: Sensitivity of optimized parameters to the combined e↵ect of variations inthe boundary conditions for the hourly dataset. All three boundary condi-tions were randomly varied for each optimization run. Parameters c to are presented by row and boundary conditions by column (from left to right,radiation, travel time and width).Once the coecients are applied to the energy balance model a range of probable down-stream temperature changes can be estimated using data from each parameter set. Therange distribution of the surface flux components and derived temperature changes are pre-sented in Figure 3.14. After parameter optimization, the observed downstream temperature37ce_NRce_tauce_width460500540de_NRde_taude_width140150160170180190ee_NRee_tauee_width25.030.0phie_NRphie_tauphie_width0.8 0.9 1.0 1.1 0.8 0.9 1.0 1.1 0.8 0.9 1.0 1.10.120.140.16Error term multiplerRange of parameter valuesFigure 3.13: Calibrated parameter values obtained from three optimization trials with per-turbations to net radiation (left), travel time (centre) and width (right) byan error term multiplier.changes fell within the range of predicted temperature changes for most time-steps (bottom,Figure 3.14). Exceptions to this observation include underestimation of peak warming oncertain sunny days and underprediction of warming overnight in some instances when theobserved rate of temperature change dropped steeply just before sunrise on certain days.These nighttime discrepancies coincided with dampened sensible heat fluxes, two of suchcases represented heat losses.3.7 Energy balance comparisonsThroughout the two-month study period, the largest heat sources to the stream were fric-tional warming, net radiation, and sensible heat, while the largest heat sink was latent heat38Table 3.2Ranges of fitted wind function coecient values associated with random variations in theboundary conditions during optimization. Optimization split-sample tests include the per-turbations to all three of radiation, width and travel time for hourly data, daytime dataand nighttime data, and three split-sample tests for perturbations to only one boundarycondition.BoundaryConditionc(W·m2·kPa1)d(W·s·m3·kPa1)e(W·m2·kPa1°C1) (°C) RMSE(°C)Hourly 438-600 112-194 18-35 0.08-0.15 0.09-0.11Night 19-84 59-72 0 0.21-0.24 0.03-0.04Day 787-975 25-63 23-45 0-0.13 0.10-0.14Radiation 428-567 134-183 21-31 0.10-0.16 0.09-0.10Width 503-520 162-164 26-27 0.11-0.14 0.09-0.10Travel time 490-538 142-187 23-33 0.12 0.09-0.10Table 3.3Calculated changes in downstream temperature associated with each energy flux.Input: min (°C) max (°C) mean (°C)Net radiation (Qr) -0.1 1.5 0.2Sensible heat (Qh) -0.02 0.6 0.25Latent heat (Qe) -1.5 0.1 -0.2Friction (Qf ) 0.23 0.23 0.23Residual () day 0 0.13 0.06Residual () night 0.21 0.24 0.23(Table 3.3). Frictional warming contributed a constant 0.23 °C to the stream, while netradiation controlled the diurnal variability in Tw with daytime temperature increases upto 1.5 °C and nighttime cooling of -0.1 °C. Turbulent energy exchanges of sensible and latentheat resulted in diurnal temperature signals with multi-day trends associated with air massvariations, from acting as a heat sink (Aug 25-26) to an energy source (Sept 1-7) when wild-fire smoke penetrated the valley. Latent heat cooled the water temperature by up to -1.5 °Cand warmed it up to 0.1 °C. Sensible heat warmed the water up to 0.6 °C and only becamean energy sink on two days throughout the two month period. Based on the split-sampleoptimization tests, residual heating () warmed the stream an average of 0.06 °C duringthe day but increased to 0.23 °C during the night (Table 3.2). With an average nighttimewarming of 0.45 °C, warming can be almost entirely explained by inputs of residual heat andfriction. Daytime energy fluxes were dominated by solar radiation, sensible heat and latentheat flux.39−5000500Qr (Wm−2)−5000500Qh (Wm−2)−5000500Qe (Wm−2)−5000500Qnet (Wm−2)Jul 15 Aug 01 Aug 15 Sep 010.00.51.01.5∆ Tw (°C)Figure 3.14: In each plot, the ensemble of predicted fluxes and value of Tw for all hourlyparameter sets are shown. For Tw, the ensemble of predictions is shown inblue and observed values are shown in burgundy.3.8 Comparison of optimized and standard modelComparisons between the optimized model and the standard model from the literature high-light di↵erences in the ability to predict stream temperature. Relative to the standardliterature model, the optimized model had a lower residual standard error (as low as 0.03for split-sample tests compared to 0.2 for literature parameter values). Figure 3.15 shows400.00.51.01.52.0∆ Tw (°C)Jul 15 Jul 17 Jul 19 Jul 210.00.51.01.52.0∆ Tw (°C)∆ Tw (°C)Optimized Literature ObservedFigure 3.15: Comparison of predicted and observed temperatures based on (top) calibratedmodel and (bottom) model based on standard parameterizations from previ-ous studies.Table 3.4Comparisons of the input parameters for the optimized and standard literature modelModel Optimized LiteratureWind function calibrated (c,d,e) Webb and Zhang (1997)Albedo McMahon et al. (2017) 0.05 (constant)RMSE (°C) 0.1 0.2a week of observed and modelled downstream temperature changes including both mostlysunny and partly cloudy days. As depicted, the standard model over-predicts the degree ofwarming during the day due to unmodified solar radiation and consistently under-predictsthe warming that occurs at night. The optimized model often under-predicts peak warmingduring the day, but the underprediction is of a smaller magnitude than the literature model’s41over-estimate of peak warming.3.9 Error trends−0.20.4 error (°C) 7/7 − 7/202.03.04.0Q (m3 s−1)−0.60.0 error (°C) 7/20 − 8/102.03.0Q (m3 s−1)−0.40.00.4 error (°C) 8/10 − 8/241.02.5Q (m3 s−1)−0.20.2 error (°C) 8/24 − 9/71.02.5Q (m3 s−1)00:00 07:00 14:00 21:00Figure 3.16: Boxplots illustrating the distribution of model error (modelled - observed) anddischarge at 10-minute intervals through the day for four two-week periods.Predictions were made using the best single parameter set from the hourlyoptimization test.Overall, there is a greater tendency for the predictions to conform to the observed tem-42peratures at night than during sunlight hours for the hourly optimized parameter set. Thistrend of increasing model performance at night exists for most days in the study period.The daily onset of model underprediction occurs around the same time as the daily shiftof increasing discharge on the base of hydrograph rising limb. Figure 3.16 shows the dailytrend of the residuals with a shift towards more positive residuals around 11:00. Residualsovernight remained equally varied. The relationship between model residuals and dischargepersists throughout the season and is best portrayed by subsetting the data into four timeperiods to reflect the change in timing of the hydrograph (Figure 3.16). The model thereforehas a tendency to under-predict the peak warming that occurs in the late morning/earlyafternoon near the onset of the hydrograph’s rising limb and over-predict warming later inthe afternoon/evening. The observed drop in temperature gradient after peak warming ismore abrupt while the model typically extends the degree of intense warming late into theafternoon with a more gradual decline into the evening hours.Nighttime model error such as the under-prediction on July 17 and 18 (Figure 3.15)coincide with relatively cold water and air temperatures and low ew as a result. Split-sampletests estimated smaller wind function coecients in the night than the hourly optimizationproduced. Therefore, nightly latent and sensible heat fluxes may be over-estimated whenusing the hourly parameter terms.43Chapter 4DiscussionThe primary objective of this study was to build an energy balance model constrained bymeasured variables and physically based principles in order to both limit the physical im-plausibility of parameter values and better understand the thermal characteristics of steepstreams.4.1 Model optimization and sensitivityInter-station comparisons of meteorological data and inter-segment variations in radiationhighlight the need for high spatial resolution data in stream temperature modelling in com-plex terrain. While meteorological data are more easily recorded at an open-site locationthan a bouldery stream bank, the di↵erence in observed conditions between each location canbe significant (Figure 3.5). Wind speed was much greater at the open site due to katabaticwinds from Bridge Glacier, which South Creek did not experience to the same degree dueto topographic barriers. Even within the channel, along-stream changes in air temperature,wind, relative humidity, shading, width, velocity, and morphology a↵ect local temperaturesand are therefore important to include in physically based models. Water storage informationat the upstream and downstream extents provide insight into the gaining or losing nature ofthe system, which will a↵ect temperatures as well.Results from the model fitting analyses of the wind function coecients suggest thatturbulent exchanges are enhanced in steep streams relative to those predicted in other studiessuch as the commonly used Webb-Zhang characterization (Table 4.1). Like the Webb andZhang (1997) parameterization based on stream-side pan evaporation, Jobson (1980) andFulford and Sturm (1984) focused on low-gradient channels. The gradient and morphologyof the Meier et al. (2003) streams are unknown, but Meier et al. (2003) indicated that theyare similar to those studied in Meier (1996) with slopes ranging from 3.8% to 14.4%. Only44Table 4.1Comparison of wind function coecients determined in previous studiesStudy c(W·m2·kPa1)d(W·s·m3·kPa1)e(W·m2·kPa1°C1)Channeltype/approachThis study:(hourly, day,night)20-975 25-200 0-45 proglacial/energybalanceRichards (2008) 350-750 50-200 - proglacial/energybalanceMaheu et al.(2014)87.7 23.8 - low-gradientstream/in-streampansMeier (1996) 130 8.6 1.7 mountainstream/energybalanceFulford andSturm (1984)91.3 22.9 - mountainstream/energybalanceJobson (1980) 86.4 32.3 - aqueduct/energybalanceWebb andZhang (1997)37.7 40.9 - low gradientstreams/streamsidepansRichards (2008) arrived at similar parameter estimates in Place Creek, a steep proglacialstream in British Columbia. Model fitting for combined night and day datasets derivedlower bound estimates of 350 and 50 for c and d, respectively, and upper bounds of 750and 200 for Place Creek. The parameter values of South Creek fall within or exceed thisrange of expected wind function coecients, further supporting the hypothesis that turbulentcascading flow in steep channels enhances turbulent exchange by increasing aeration andtherefore also increasing the amount of water exposed to the atmosphere. Whereas otherstudies have classified turbulent exchanges as a second order process in low gradient streamsbelow net radiation, it is evident from the heating inputs in Table 3.3 that in steep highlyaerated streams the e↵ects of turbulent exchange at the surface are of similar importance tothe warming e↵ects of net radiation.45The addition of an optimized stability parameter and warming o↵set coecient improvedthe results compared to model output using just two wind function coecients, c and d. Meieret al. (2003) is the only study to have used the air stability parameter in an energy balanceapproach, added to better characterize the stratification of the air mass directly above thestream. The need for a stability correction may not be so apparent for low-gradient streams,for which the sensible and latent heat fluxes are of secondary importance to net radiations.However, given that saturation vapor pressure increases sharply with temperature, an in-crease in the water temperature would increase the humidity contrast between the watersurface and the overlying air, resulting in latent heat losses. In steep streams, the resultsof this study indicate that turbulent heat exchange is enhanced, making the stability of theoverlying air particularly important in governing the evaporative tendencies of this turbulentmixture of air and water.The warming coecient, , accounts for external factors contributing to stream warm-ing not defined in the energy balance model such as hyporheic exchange, hillslope runo↵,advective heat from groundwater, and bed heat or boulder heat conduction. The residualheating coecient ranged from 0 to 0.24 °C during the daytime and nighttime optimizationruns, suggesting that a minimum of 12% of the warming was not attributed to surface heatfluxes. This does not account for the standard error of 0.09 °C further unaccounted for inthe model.Based on the range of parameter values displayed in Figure 3.13 and Table 3.2, the windfunction parameters are most sensitive to inputs of net radiation and travel time. This is notonly apparent in the individual sensitivity analysis tests run on each input variable separately,but is also present in the results of the combined sensitivity analysis of Figure 3.12. Increasedwind function parameters (c, d, e) for increased net radiation is tractable given latent andsensible heat fluxes are a function of the temperature and vapour pressure gradients at andabove the stream, of which solar radiation has a large e↵ect. The contrast in parameter valuesbetween the day and night-time split sample calibrations reflects fundamental di↵erences inthe roles of the various energy fluxes. During the day, warming of the stream increaseswater temperature which, in turn, increases the surface vapour pressure and thus the vapourpressure gradient over the stream. This increased vapour pressure gradient coincides withthe increase in wind speed during the day (Figure 3.3/3.2) to promote increased evaporationduring the day, which acts to o↵set downstream warming associated with net radiation. Forthe daytime calibration, the o↵set parameter  is relatively small (less than 0.1 o C), reflectingthe dominant e↵ect of surface energy exchanges on downstream temperature changes. Indeed,the  parameter could be accounting for a small bias between the upstream and downstreamtemperature loggers.46At night, net radiation becomes negative, and stream cooling reduces the vapour pressuregradient, resulting in the latent heat flux becoming a relatively small heat loss term. Overall,the surface energy fluxes become relatively small, while the calibrated value of  increasesto values in excess of 0.2 o C, which is greater than the accuracy of the loggers. For thenight-time calibration, the value of  is likely accounting for the warming e↵ect of heatconduction from the bed and boulders, and possibly hyporheic exchange. These processeswill be discussed in more detail in section 4.4.4.2 Hydraulic geometryThe width exponent values of this study are on the lower end of those values typically found inthe literature for proglacial streams (Ashmore & Sauks, 2006; Leduc et al., 2018; Magnussonet al., 2012; Park, 1977) (Table 3.1). Most studies focus on lower gradient gravel-bed braidedstreams, apart from Leduc et al. (2018), where the width-discharge relation was calculatedusing oblique time-lapse imagery for a steep bouldery proglacial alpine stream similar toSouth Creek in the Dome Glacier forefield in Alberta, Canada. The width exponent of 0.6,as derived by Leduc et al. (2018), is higher than the range derived for South Creek. However,Leduc et al. (2018) indicated that the regression analysis produced a large amount of scatteraround the power-law curve, highlighting the challenges of measuring hydraulic geometry inhighly irregular channels with strong diurnal streamflow signals.The wide range of exponent values between reaches at South Creek can be explained bythe topographic restrictions of the study site. Narrow sections of the stream in Reaches 4and 2 are continuously incising into the lateral moraine or undercutting the adjacent ridge,which therefore restricts their ability to widen with higher flows. Relic channel banks frompast episodic outburst floods could constrain widths downstream for high flows. Althoughthe frequency and magnitude of such e↵ective flows is unknown, Ryder (1991) hypothesizedthat damming of the channel by snow and ice at the most upstream section at the breachedlateral moraine is likely to occur, causing episodic downstream spring flooding.The fitted relationships between velocity and discharge illustrate the impacts of streambed morphology and slope on channel flow. The deviation from a simple power law functionin all of the segments is consistent with other studies in steep natural streams and flumeexperiments (Milzow et al., 2006; Zimmermann, 2010)(Milzow et al., 2006; Zimmermann,2010)(Milzow et al., 2006; Zimmermann, 2010). As observed in a similar steep proglacialcreek in the Swiss Alps, Schneider et al. (2015) also noted a pattern of decreasing velocities insteeper reaches where lower relative flow depths (depth/sediment diameter) generated morespill resistance. Indeed, Reaches 3 and 4 exhibited the largest concentration of boulders47and large width exponents, and therefore lowest relative flow depth. The steepest sectionof the stream, located in these reaches, had pronounced instances of cascading flow withthe largest step-pool jumps. The organization of the bed and cascading flows increase thebed roughness and thus flow resistance, which would decrease the flow velocity as a result.Given that boulders of various sizes protruded out of the flow for the entire study reach, thestream’s ability to e↵ectively drown out the roughness elements at even high flows is limited,and thus the range of possible velocities is constrained. Cascade and step-pool morphologygenerate tumbling flow, which in turn produces strong pressure head gradients that drivehyporheic exchange in coarse porous media (Bungton & Tonina, 2009). This, in additionto channel bed roughness, can lead to longer residence times (Kasahara & Wondzell, 2003),such as those observed in South Creek.4.3 Channel controls on the thermal regimeHydraulic controls such as width, depth and travel time impact the extent of warming inproglacial streams. As observed in the highly glacerised Swiss Alps, stream surface area wasthe best predictor of downstream temperature warming (Williamson et al., 2019). Thosestreams, which were confined by steep valley walls, had smaller temperature gradients down-stream than the more shallow and wider streams that have more surface area for atmosphericenergy fluxes such as solar radiation to take place (L. E. Brown et al., 2007; Williamson etal., 2019). Conversely, temperature response to a given energy input is inversely proportionalto depth. Deeper streams are inherently faster and therefore cannot warm to the same de-gree as a stream with a longer travel time. In this study, a combination of steep gradientsand complex step-pool morphologies limited faster velocities even at high flows. This limi-tation on velocity, in conjunction with increasing aeration and width at higher discharges,enhanced the temperature response to surface energy exchanges relative to the situation forfaster moving, low-gradient streams. Schmadel et al. (2015) and Link et al. (2012) also foundthat stream temperature predictions were highly sensitive to residence time calculations andthat increasing travel times increased the thermal influence of channel hydraulics. This is inagreement with the parameter sensitivity analysis results for travel time (Figure 3.12, 3.13).Measured albedo over the stream was generally consistent with the model developedby McMahon and Moore (2017) for aerated flow, with values of 0.15 to 0.2 for mid-dayperiods. The model under-predicted albedo at low sun angles, when incident solar radiationis low. The observed increase in the extent of aeration with increasing discharge wouldbe expected to result in an increase in mean albedo over the stream, consistent with thedischarge dependence of albedo found by Richards and Moore (2011). Overall, the results48of this study provide further evidence that values of albedo typically assumed for proglacialstreams (0.05 to 0.1) are not appropriate for steep streams with extensive aeration.Stream-subsurface interactions, including hyporheic exchange, can be important influ-ences on stream temperature (King & Neilson, 2019; Leach & Moore, 2010b; Moore, 2005),especially in steep streams with step-pool morphology. For example, a flume study demon-strated that, for step-pool morphologies, the depth of hyporheic water penetration increasedwith stream gradient and the hydraulic conductivity of the bed (Hassan et al. 2014). Notonly is the bed material coarse at South Creek, which would be associated with a high hy-draulic conductivity, but the depth to bedrock is much greater than typical alpine channelsas Bridge Glacier carved out the surrounding area that is now infilled with fluvial deposits ofreworked glacial till (Ryder, 1991). Therefore, bedrock elevation would not be a constrainingfactor in determining hyporheic depths, as is usually the case in alpine channels (Bungton& Tonina, 2009). In addition, the channel bank boundaries of South Creek are neither well-defined nor vegetated, leaving lateral fluxes of groundwater less constrained throughout thechannel bank and bed material. Furthermore, the expansion of flow into side channels athigher discharge would be another mechanism driving hyporheic exchange.Unfortunately, it is dicult to quantify hyporheic exchange based on field measurementsin real channels, especially complex step-pool units (Scordo & Moore, 2009). One alternativeapproach is to determine hyporheic exchange through calibration of a stream temperaturemodel with a transient storage component (e.g. King and Neilson (2019)). However, thisapproach requires that all other energy fluxes be specified accurately. As shown by this study,the standard parameterizations for modelling surface energy exchange are not applicableto steep mountain channels (Figure 3.15), which would result in misleading estimates ofhyporheic heat exchange.4.4 Hypotheses for residual heat inputsThe tight range of the  parameter for nighttime model calibration (0.22-0.24 °C) and smallassociated RMSE (0.03 °C) suggest that the residual warming is a consistent process thatcould result from a number of energy exchanges not included in the model, including ground-water discharge, hyporheic exchange, or heat conduction from the bed or in-stream boulders.The discharge measurements suggest that the stream was dominantly losing until Aug.23, and then exhibited an alternating pattern of losing during the day and gaining at night(Figure 3.1). However, the longitudinal stream temperature and electrical conductivity pro-files suggest that there were one or two locations of relatively focused discharge into thestream, one about 370 m below the upstream boundary and the other near the bottom of49the study reach. It is entirely plausible that a stream could be overall losing along a reachbut still experience localized inputs via groundwater discharge or hyporheic exchange.As seen in Figure 3.2, the localized discharge resulted in at least a local influence ontemperature of a magnitude that could explain the calibrated value of . However, the e↵ectof hyporheic exchange and heat conduction cannot be excluded as alternative explanations.For example, Westho↵ et al. (2010) documented the e↵ect of heat exchange with in-streamclasts, and incorporated this e↵ect as a transient storage term in an advection-dispersionmodel.Further research is required to gain a better understanding of the roles of advective andconductive heat exchange in steep streams, especially streams subject to diel variations indischarge.4.5 Comparison of temperature warming in similar streamsThe rate of stream warming observed in South Creek is within the range of published valuesobserved by other proglacial stream temperature studies. The mean temperature change of0.6 °C·km1 in this study is the same as the warming observed by Cadbury et al. (2008)for a proglacial stream in New Zealand and Uehlinger et al. (2003) in the Swiss Alps, mea-suring an average warming rate of 0.6 °C·km1. Cadbury et al. (2008) cited valley/channelgeomorphology, hydroclimatological conditions and seasonal streamflow as the key drivers ofthe thermal regime. The range of warming at South Creek (-0.28 to 1.7°C·km1) is similarto those reported by Cardenas et al. (2014) of -0.23 to 1.45°C·km1 in the Urbach River inSwitzerland. The observed warming at South Creek is less than the mean observed 80 kmto the southwest at Place Creek by Richards (2008) (1.1°C·km1). While South Creek istopographically shaded in upstream areas, both streams have a northern aspect with littleto no vegetative shading. One factor attributing to the warming di↵erence is the change inelevation at Place Creek (200+ m compared to 100 m in this study) and therefore twice thecontribution of frictional warming. It is important to note that the mean warming gradientof South Creek is only three times the accuracy of the temperature sensors.4.6 Limitations associated with measurent resolutionfor stream temperatureOne important concern in the optimized modelling approach is the resolution of the streamtemperature data. Given that most downstream temperature gradients were between 0-501°C/km, many of the observed temperature changes are within the accuracy of the tem-perature loggers. Temperature was recorded at several locations along the reach with theintent of resolving warming and cooling as a function of the di↵erent channel morphologies.However, these comparisons were deemed not feasible given the low signal-to-noise ratio.Therefore, model calibration and testing focused on the full study reach.Another issue relates to the fact that the data were recorded every 10 minutes, and thedownstream temperatures were interpolated to generate an observed temperature for thearrival time of each parcel. In an extreme case, the observed stream temperature increasedby 1.6 oC over a 10-minute interval. Therefore, the use of linear interpolation introducesadditional error into the calculation of downstream temperature changes.4.7 Predicting stream temperature under glacier re-treatThe fitted model can provide insight into the e↵ects of glacier retreat on stream temperature,at least at the reach scale. As glacier surface area decreases so too will the average streamflow.For low-gradient streams, there is generally a negative relation between stream temperatureand discharge associated with the combined e↵ects of longer in-reach travel times and reducedwater depth. However, for steep streams, the situation is complicated by the e↵ect of aerationon surface energy exchanges.For the South Creek study reach, a decrease in discharge would result in a reductionin aeration, which would have two reinforcing e↵ects: it would result in an increase in theamount of absorbed solar radiation due to the decreased albedo, and would also result ina reduced heat loss by evaporation. These influences would augment the e↵ects of reduceddepth and increased travel time.As seen Figure 3.7, the relation between aeration and discharge varied among the reaches,with the steeper reaches having less sensitivity. Therefore, the e↵ect of changing dischargeon surface energy exchange is likely to depend on the magnitude of discharge change andthe channel morphology. Lower-gradient reaches would experience a more marked reductionin aeration, which would make them more sensitive to changes in discharge.The influence of decreased aeration could vary with climatic regime. The Bridge Rivervalley generally experiences low humidities, partly due to its location in the lee of the CoastMountains and also, quite likely, due to the katabatic flow from Bridge Glacier, which canextract moisture from the boundary layer through condensation (Shea & Moore, 2010).This situation contrasts with that studied by Richards (2008), where stream evaporation51calculated from a calibrated wind function were up to an order of magnitude lower thancalculated for South Creek.Other complicating factors include the e↵ects of advective exchanges associated withgroundwater discharge and hyporheic exchange, which can become more pronounced at lowerdischarges (Cadbury et al., 2008). Finally, one must consider the entire stream length upto the snout of the glacier. As glaciers retreat, the length of the stream subject to warmingwould increase, further contributing to warming (Williamson et al., 2019).52Chapter 5Conclusion5.1 Key FindingsFindings from the energy balance approximation of downstream warming at South Creekcan be summarized in the following statements.• Using a three-parameter wind function in the calculations of latent and sensible heatbest captures the degree of turbulent exchange occurring in steep, highly aeratedstreams. Optimization yielded much higher wind function coecients than those usedpreviously in the literature, especially in the daytime. Calculations of turbulent fluxesin steep streams using coecients from the literature therefore underestimate thesecomponents in the energy balance, which in this study were comparable to surfaceheat fluxes from net radiation. The larger range of parameter coecients derived fromthe optimization suggests that the wind function parameters cannot be displayed asone number, as they exist in the literature, but instead be o↵ered as a broad range thataccounts for the uncertainty in the energy balance inputs as well as the daily cycle ofchanging atmospheric demand for turbulent exchange.• Split-sample results highlight the diurnal trends of the relative thermal sinks andsources of the South Creek basin. The energy balance model was most sensitive toinputs of net radiation and travel time for the optimized parameters. Heating of theair and water from net radiation enhanced turbulent exchanges during the daytime asobserved from daytime model optimization, while nighttime optimization limited theinput of latent and sensible heating to the stream. Instead, friction (0.22 °C) and resid-ual heat (0.22 - 0.24 °C) dominated, which agrees with the average nighttime warmingof 0.45 °C. Possible sources of residual heat include streambed heat conduction andadvective warming from groundwater inflows or hyporheic exchange.53• Using predictive models and adjustments for solar radiation, albedo and net radia-tion better captures the variability of surface radiation fluxes in steep, aerated alpinestreams. Stream aspect, slope, and shading a↵ected the incoming solar radiation andreduced the overall net radiation inputs to the stream. The simulated values estimatedmost of the observed calculated albedo at the two above-stream pyranometer locationsat South Creek, whereas using a constant common literature value of 0.05 substantiallyunderestimated the range of albedo observed throughout the study.• UAV SfM photogrammetry proved to be an e↵ective and ecient means of assessingthe spatial heterogeneity of stream morphology and geometry in a bouldery channelnot accessible to standard in-stream surveying. The average width calculations werelimited to 10-m spaced transects in this study, but with the advances in raster-basedmachine learning, the resolution could be increased to the lower-most limit of theorthomosaic’s pixel length if and when such a technique is designed.• This study has highlighted the need for careful field measurements to constrain thehydrologic geometry and hydraulic conditions of a channel, along with above-streammeteorological measurements, for analysing stream energy budgets in steep proglacialchannels.5.2 Recommendations for Future WorkTo the knowledge of the author, this is only one of two studies that tested the e↵ects oftumbling and aerated flow on the surface heat flux components. Future work should focus onstudying similar steep proglacial streams using this parameterized energy balance approachin order to verify the conclusions of this study.One of the major questions remaining from this approach are the unresolved sources andcauses of residual heating along the reach. Future studies should focus on the impacts ofboulder heat conduction and hyporheic exchange in a proglacial steep stream environment.The diurnal flow provides daily inundation of boulders and drowning of steps, creating anideal experimental set up to test the presence of these e↵ects in the stream. To test heatconduction from boulders, aerial thermal imaging can be used, either using a fixed set up ona ridge (Cardenas et al., 2014) or attaching the camera to a drone. Hyporheic exchange maybe investigated using tracer methods. 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WaterResources Research, 46(9). doi: 10.1029/2009wr00791362Appendix ARating Curves0.0 0.1 0.2 0.31.01.52.02.53.0Reach 1Stage (m)Q (m3 s−1)RMSE = 0.11 m3s−10.0 0.1 0.2 0.31.01.52.02.53.0Reach 2Stage (m)Q (m3 s−1)RMSE = 0.16 m3s−10.10 0.20 0.30 0.401.52.02.53.0Reach 3Stage (m)Q (m3 s−1)RMSE = 0.15 m3s−10.10 0.15 0.20 0.25 0.301.52.02.5Reach 4Stage (m)Q (m3 s−1)RMSE = 0.11 m3s−1Figure A.1: Rating curves for all four reaches including their residual standard mean error.63Appendix BSalt Dilution Curves0 100 300 500 70020253035Reach 1Time (s)Electrical Conductivity (µ S/cm)0 100 300 500 70015202530Reach 2 Discharge m3/sVelocity (m/s)0 100 300 500 7001012141618Reach 3  Discharge m3/sVelocity (m/s)0 100 300 500 7001214161820222426Reach 4  Discharge m3/sVelocity (m/s)Figure B.1: Example salt dilution curves for all four reaches.64Appendix CAbove-stream meteorologicalconditions65012345u (ms−1 )0.00.51.01.5e (kPa)0510152025T (°C)01234Q (m3 s−1)Jul−07 Jul−15 Jul−24 Aug−02 Aug−11 Aug−20 Aug−29 Sep−06ea ew Ta TwFigure C.1: Plot of stream temperature and meteorological conditions observed at theupstream weather station for the full study period.660123456u (ms−1 )0.00.51.01.5e (kPa)510152025T (°C)01234Q (m3 s−1)Jul−07 Jul−15 Jul−24 Aug−02 Aug−11 Aug−20 Aug−29 Sep−06ea ew Ta TwFigure C.2: Plot of stream temperature and meteorological conditions observed at thedownstream weather station for full the study period.67

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