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Regional-scale distributed modelling of glacier meteorology and melt, southern Coast Mountains, Canada Shea, Joseph Michael 2010

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Regional-Scale Distributed Modelling of Glacier Meteorology and Melt, southern Coast Mountains, Canada by Joseph Michael Shea  B.Sc., McMaster University, 2001 M.Sc., University of Calgary, 2004  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Geography)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2010 c Joseph Michael Shea 2010  Abstract Spatially distributed regional scale models of glacier melt are required to assess the potential impacts of climate change on glacier response and proglacial streamflow. The objective of this study was to address the challenges associated with regional scale modelling of glacier melt, specifically by (1) developing methods for estimating regional fields of the meteorological variables required to run melt models, and (2) testing models with a range of complexity against observed snow and ice melt at four glaciers in the southern Coast Mountains, ranging in size from a small cirque glacier to a large valley glacier. Near-surface air temperature and humidity measured over four glaciers in the southern Coast Mountains of British Columbia were compared to ambient values estimated from a regional network of off-glacier weather stations. Systematic differences between measured and ambient conditions represent the effects of katabatic flow, and were modelled as a function of flow path length calculated from glacier digital elevation models. Near-surface wind speeds (ug ) were classified as either katabatic or channelled, and were modelled based on Prandtl flow (for katabatic winds) or gradient wind speeds. Models for atmospheric transmissivity, snow and ice albedo, and incoming longwave radiation were tested and developed from observations of incident and reflected shortwave radiation (K↓ and K↑) and incoming longwave (L↓) radiation. Data from a regional climate network were used to run a degree-day model, a radiation-indexed degree-day model, a simple energy balance model (including tuned parameters for turbulent exchange) and two full energy balance models (incorporating stability corrections, with and without corrections for katabatic effects on air temperature and humidity). Modelled ii  Abstract melt was compared to mass balance measurements of seasonal snow and ice melt. Models were also compared based on their ability to predict date of snow disappearance, given an initial snowpack water equivalence. The degree-day model outperformed the simple energy balance and radiationindexed degree-day approaches, while the full energy balance model without katabatic boundary layer corrections yielded the lowest errors.  iii  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  viii  List of Figures  xii  List of Symbols  xvi  Acknowledgements 1 Introduction  xx 1  1.1  Glacier Melt Modelling . . . . . . . . . . . . . . . . . . . . . .  2  1.2  Meteorological Inputs for Melt Modelling . . . . . . . . . . .  4  1.3  Melt Model Selection . . . . . . . . . . . . . . . . . . . . . . .  5  1.4  Thesis Objectives and Outline . . . . . . . . . . . . . . . . . .  6  2 Study Area and Data Collection 2.1  2.2  9  Meteorological Data . . . . . . . . . . . . . . . . . . . . . . .  9  2.1.1  Glacier Meteorological Stations . . . . . . . . . . . . .  9  2.1.2  Ambient Meteorological Stations . . . . . . . . . . . .  17  2.1.3  Meteorological Data Post-Processing . . . . . . . . . .  17  Mass Balance and Snowline Retreat Observations . . . . . . .  18  2.2.1  Winter Balance . . . . . . . . . . . . . . . . . . . . . .  20  2.2.2  Summer Ablation . . . . . . . . . . . . . . . . . . . . .  21  2.2.3  Snowline Retreat . . . . . . . . . . . . . . . . . . . . .  21  iv  Table of Contents 3 Distributing Meteorological Fields, Part 1  28  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  28  3.2  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  3.2.1  Temperature . . . . . . . . . . . . . . . . . . . . . . .  31  3.2.2  Vapour Pressure . . . . . . . . . . . . . . . . . . . . .  33  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  35  3.3.1  Temperature . . . . . . . . . . . . . . . . . . . . . . .  35  3.3.2  Vapour Pressure . . . . . . . . . . . . . . . . . . . . .  48  3.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  3.5  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .  58  3.3  4 Distributing Meteorological Fields, Part 2  60  4.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  4.2  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  4.2.1  Data Preparation and Modelling Approach . . . . . .  62  4.2.2  Katabatic Wind Speed Modelling . . . . . . . . . . . .  64  4.2.3  Non-katabatic Wind Speed Models . . . . . . . . . . .  68  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  68  4.3.1  Glacier Wind Characterization . . . . . . . . . . . . .  68  4.3.2  Surface Winds in Katabatic Flow . . . . . . . . . . . .  73  4.3.3  Surface Winds in Non-Katabatic Flow . . . . . . . . .  79  4.3.4  Modelled Surface Wind Speeds . . . . . . . . . . . . .  79  4.3.5  Comparison of Wind Speed Models . . . . . . . . . . .  88  4.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  89  4.5  Conclusion  92  4.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 Distributing Meteorological Fields, Part 3  93  5.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  93  5.2  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  5.2.1  Temperature . . . . . . . . . . . . . . . . . . . . . . .  99  5.2.2  Vapour Pressure . . . . . . . . . . . . . . . . . . . . .  99  5.2.3  Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . 100  5.3  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101  v  Table of Contents 5.3.1  Temperature . . . . . . . . . . . . . . . . . . . . . . . 101  5.3.2  Vapour Pressure . . . . . . . . . . . . . . . . . . . . . 106  5.3.3  Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . 109  5.3.4  Application . . . . . . . . . . . . . . . . . . . . . . . . 113  5.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116  5.5  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  6 Radiation Modelling 6.1  6.2  6.3  6.4  119  Solar Radiation and Transmissivity . . . . . . . . . . . . . . . 119 6.1.1  Background and Objectives . . . . . . . . . . . . . . . 119  6.1.2  Methods and Data . . . . . . . . . . . . . . . . . . . . 121  6.1.3  Results . . . . . . . . . . . . . . . . . . . . . . . . . . 125  6.1.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 137  Albedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2.1  Background and Objectives . . . . . . . . . . . . . . . 138  6.2.2  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 140  6.2.3  Results . . . . . . . . . . . . . . . . . . . . . . . . . . 143  6.2.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 147  Longwave Radiation . . . . . . . . . . . . . . . . . . . . . . . 150 6.3.1  Background and Objectives . . . . . . . . . . . . . . . 150  6.3.2  Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 151  6.3.3  Results . . . . . . . . . . . . . . . . . . . . . . . . . . 153  6.3.4  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 169  Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 170  7 Melt Model Test Data  172  7.1  Background and Objectives . . . . . . . . . . . . . . . . . . . 172  7.2  Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . 176  7.3  7.2.1  Snow Density Modelling . . . . . . . . . . . . . . . . . 176  7.2.2  Interpolation of Initial SWE . . . . . . . . . . . . . . . 178  7.2.3  Mass Balance and Snowline Retreat Data . . . . . . . 179  7.2.4  Error Analysis . . . . . . . . . . . . . . . . . . . . . . 179  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181  vi  Table of Contents  7.4  7.3.1  Snow Density Modelling . . . . . . . . . . . . . . . . . 181  7.3.2  Initial SWE Interpolation . . . . . . . . . . . . . . . . 186  7.3.3  Surface Temperature Loggers and Snowline Retreat . 188  7.3.4  Error Analysis . . . . . . . . . . . . . . . . . . . . . . 193  Discussion and Recommendations . . . . . . . . . . . . . . . . 194  8 Melt Modelling  197  8.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197  8.2  Study Areas and Climate Data . . . . . . . . . . . . . . . . . 199  8.3  Melt Modelling Methods . . . . . . . . . . . . . . . . . . . . . 200  8.4  8.3.1  Degree-day Model . . . . . . . . . . . . . . . . . . . . 200  8.3.2  Radiation-indexed Degree-day Model . . . . . . . . . . 200  8.3.3  Simplified Energy Balance Model . . . . . . . . . . . . 203  8.3.4  Full Energy Balance Model . . . . . . . . . . . . . . . 208  Meteorological Inputs . . . . . . . . . . . . . . . . . . . . . . 213 8.4.1  Temperature . . . . . . . . . . . . . . . . . . . . . . . 214  8.4.2  Vapour Pressure . . . . . . . . . . . . . . . . . . . . . 214  8.4.3  Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . 215  8.5  Melt Model Comparisons . . . . . . . . . . . . . . . . . . . . 218  8.6  Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.6.1  Modelled Melt and Benchmark Evaluations . . . . . . 218  8.6.2  Full Energy Balance Model Analyses . . . . . . . . . . 224  8.7  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227  8.8  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229  9 Conclusion 9.1  9.2  231  Summary of Key Findings . . . . . . . . . . . . . . . . . . . . 231 9.1.1  Glacier Meteorology . . . . . . . . . . . . . . . . . . . 231  9.1.2  Radiative Fluxes . . . . . . . . . . . . . . . . . . . . . 233  9.1.3  Melt Model Test Data . . . . . . . . . . . . . . . . . . 233  9.1.4  Glacier Melt Modelling . . . . . . . . . . . . . . . . . 234  Suggestions for Future Research . . . . . . . . . . . . . . . . . 235  References Cited  238 vii  List of Tables Table 1.1  Glacier energy balance studies . . . . . . . . . . . . .  8  Table 2.1  Location and instrumentation of glacier AWS . . . . .  10  Table 2.2  Summary of on-glacier AWS periods of operation . . .  16  Table 2.3  Meteorological instrumentation specifications. . . . . .  16  Table 2.4  Ambient AWS locations and data source . . . . . . . .  17  Table 2.5  Dates of initial snow water equivalence (SWE0 ) measurements for mass balance and snow course sites . . .  19  Table 2.6  Snow course locations and ID . . . . . . . . . . . . . .  19  Table 2.7  Specifications of submersible temperature loggers used in this study . . . . . . . . . . . . . . . . . . . . . . .  Table 3.1 Table 3.2  22  Mean ambient and observed temperatures at glacier AWS . . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  Piecewise temperature model parameters . . . . . . .  41  0◦ C  Table 3.3  Elevation of  isotherm . . . . . . . . . . . . . . . .  Table 3.4  Mean ambient and observed vapour pressures at glacier  47  AWS . . . . . . . . . . . . . . . . . . . . . . . . . . . .  49  Table 3.5  Vapour pressure model parameters . . . . . . . . . . .  53  Table 4.1  Directional constancy (dc) and mean observed wind speeds (ug , in m s−1 ) at glacier AWS.  Table 4.2  . . . . . . . . .  Optimized eddy diffusivities and model errors for Prandtl wind speed model . . . . . . . . . . . . . . . . . . . .  Table 4.3  70 77  Surface wind speed models for channeled katabatic flows 78  viii  List of Tables Table 4.4  Surface wind speed model coefficients and modelled errors for channeled flows . . . . . . . . . . . . . . . .  Table 4.5  Mean (x,  m s−1 )  and standard deviation (σ,  m s−1 )  in  observed and modelled uN K , uK , and uKc . . . . . . . Table 4.6  88  Fitted coefficients for estimating Tg , eg , and ug , obtained from glacier boundary layer analyses . . . . . .  Table 5.2  87  Modelled wind speed errors using E-P methods and a representative station . . . . . . . . . . . . . . . . . .  Table 5.1  79  98  Summary of models for estimating piecewise temperature transfer functions from topographic indices . . . 103  Table 5.3  Temperature transfer function coefficients estimated from meteorological observations (obs) and from topographic indices (est) for withheld AWS sites . . . . 104  Table 5.4  Errors in modelled temperatures and vapour pressures 105  Table 5.5  Model summary for vapour pressure transfer functions and FPL . . . . . . . . . . . . . . . . . . . . . . . . . . 107  Table 5.6  Vapour pressure transfer function coefficients estimated from meteorological observations (obs) and from topographic indices (est) . . . . . . . . . . . . . . . . . . . 108  Table 5.7  Model summary for fitted eddy diffusivities KH and KM  Table 5.8  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110  Eddy diffusivity coefficients for estimating uK observed from meteorological observations (obs) and estimated from topographic indices (est) . . . . . . . . . . . . . . 111  Table 5.9  Errors in modelled wind speeds . . . . . . . . . . . . . 111  Table 6.1  Atmospheric transmissivity models . . . . . . . . . . . 125  Table 6.2  Radiation and transmissivity correlations . . . . . . . 127  Table 6.3  Errors in modelled daily transmissivity . . . . . . . . . 131  Table 6.4  Errors in modelled daily solar radiation totals . . . . . 132  Table 6.5  Fitted model parameters for daily precipitation-adjusted ∆T1 transmissivity models . . . . . . . . . . . . . . . . 135  ix  List of Tables Table 6.6  Hourly transmissivity model coefficients and errors . . 136  Table 6.7  Errors in modelled hourly radiation  Table 6.8  Snow albedo parameterizations. . . . . . . . . . . . . . 139  Table 6.9  Simple correlations (r) between snow and ice albedo  . . . . . . . . . . 137  station pairs . . . . . . . . . . . . . . . . . . . . . . . . 145 Table 6.10 Snow and ice albedo models . . . . . . . . . . . . . . . 147 Table 6.11 Clear-sky emissivity parameterizations . . . . . . . . . 158 Table 6.12 Cloud factor (F ) models . . . . . . . . . . . . . . . . . 160 Table 6.13 Summary of hourly and daily L↓ model errors . . . . . 164 Table 7.1  Snow density model summary . . . . . . . . . . . . . . 182  Table 7.2  Snow density summary by site . . . . . . . . . . . . . 185  Table 7.3  Summary of Loess and polynomial fitting of SWE0 data versus elevation . . . . . . . . . . . . . . . . . . . 187  Table 7.4  Snowline retreat sensor locations and estimated day of snowline retreat (DOY), Place Glacier  Table 7.5  . . . . . . . 190  Snowline retreat sensor locations and estimated day of snowline retreat (DOY), Helm Glacier . . . . . . . . 191  Table 7.6  Snowline retreat sensor locations and estimated day of snowline retreat (DSR), Weart Glacier . . . . . . . 192  Table 7.7  Snowline retreat sensor locations and estimated day of snowline retreat, Bridge Glacier . . . . . . . . . . . 192  Table 7.8  Winter and summer balance error statistics . . . . . . 194  Table 8.1  Melt model parameters and constants for degree-day, radiation-indexed degree-day, and simple energy balance models . . . . . . . . . . . . . . . . . . . . . . . . 201  Table 8.2  Summary of previous RIDD model parameter values . 203  Table 8.3  Melt model parameters and constants for the full energy balance model . . . . . . . . . . . . . . . . . . . . 210  Table 8.4  Coefficients for kabatatic boundary layer corrections, based on results from Chapter 5 . . . . . . . . . . . . 216  Table 8.5  Mean ablation season glacier wind speeds . . . . . . . 217  x  List of Tables Table 8.6  Melt model errors . . . . . . . . . . . . . . . . . . . . 220  xi  List of Figures Figure 1.1  Glacier melt models overview. . . . . . . . . . . . . . .  3  Figure 2.1  Southern Coast Mountains study area . . . . . . . . .  11  Figure 2.2  Place Glacier AWS locations. . . . . . . . . . . . . . .  12  Figure 2.3  Weart Glacier AWS locations. . . . . . . . . . . . . . .  13  Figure 2.4  Bridge Glacier AWS locations. . . . . . . . . . . . . .  14  Figure 2.5  Example of the on-ice meteorological stations (a) and a close-up view of the Gimbel joint used to mount pyranometers horizontal to the surface (b). . . . . . .  Figure 2.6  Helm Glacier mass balance and snowline retreat observations . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.7  24  Weart Glacier mass balance and snowline retreat observations . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.9  23  Place Glacier study area, mass balance and snowline retreat observations . . . . . . . . . . . . . . . . . . .  Figure 2.8  15  25  Bridge Glacier mass balance and snowline retreat observations . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 2.10 Method for calculating column-averaged snow density  26 27  Figure 2.11 Example of the surface temperature logger being deployed on the ice surface, Bridge Glacier 2006 . . . . .  27  Figure 3.1  Conceptual piecewise regression model . . . . . . . . .  32  Figure 3.2  Conceptual model for KBL vapour pressure analyses .  35  Figure 3.3  Hourly temperature gradients . . . . . . . . . . . . . .  37  Figure 3.4  Testing horizontal variability in ambient temperature and vapour pressure . . . . . . . . . . . . . . . . . . .  38 xii  List of Figures Figure 3.5  Temperature extrapolations for an independent station 39  Figure 3.6  Piece-wise temperature models . . . . . . . . . . . . .  Figure 3.7  Boxplots of predicted near-surface temperature residuals 45  Figure 3.8  Synoptic types . . . . . . . . . . . . . . . . . . . . . .  46  Figure 3.9  Vapour pressure gradients . . . . . . . . . . . . . . . .  51  42  Figure 3.10 Vapour pressure estimation for an independent ambient station  . . . . . . . . . . . . . . . . . . . . . . . .  52  Figure 3.11 Vapour pressure models . . . . . . . . . . . . . . . . .  54  Figure 3.12 Vapour pressure residuals by hour of day and synoptic type . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  Figure 4.1  NCEP grid used in calculation of flow indices. . . . . .  63  Figure 4.2  Logic for glacier wind speed modelling . . . . . . . . .  64  Figure 4.3  Theoretical Prandtl profiles . . . . . . . . . . . . . . .  66  Figure 4.4  Glacier AWS wind direction histograms . . . . . . . .  71  Figure 4.5  Hourly glacier wind speeds versus ambient temperatures 72  Figure 4.6  Ambient temperature - wind speed hysteresis . . . . .  74  Figure 4.7  Wind speed and temperature cross-correlations . . . .  75  Figure 4.8  Error analysis, katabatic flow optimization . . . . . . .  76  Figure 4.9  Global model for estimating uKc from southerly flow strengths V . . . . . . . . . . . . . . . . . . . . . . . . .  77  Figure 4.10 Global models for non-katabatic flows . . . . . . . . .  80  Figure 4.11 Time-series of observed and modelled wind speeds, PM2 82 Figure 4.12 Time-series of observed and modelled wind speeds, WM1 83 Figure 4.13 Time-series of observed and modelled wind speeds, BM1 84 Figure 4.14 Wind speed residuals, PM2 . . . . . . . . . . . . . . .  85  Figure 4.15 Wind speed residuals, WM1 and BM1 . . . . . . . . .  86  Figure 4.16 Predicted wind speed profiles . . . . . . . . . . . . . .  91  Figure 5.1  Morphometric parameters calculated for Place Glacier. 97  Figure 5.2  Models for estimating KBL temperature transfer functions from topographic indices . . . . . . . . . . . . . 102  Figure 5.3  Observed and modelled Tg at independent testing sites.104  xiii  List of Figures Figure 5.4  Models for estimating j1 and j2 from FPL . . . . . . . 106  Figure 5.5  Observed and modelled eg at independent testing sites. 108  Figure 5.6  Estimating eddy diffusivity coefficients KH and KM from flow path lengths . . . . . . . . . . . . . . . . . . 109  Figure 5.7  Observed and modelled uK at independent testing sites.112  Figure 5.8  Application of boundary layer development models of temperature, vapour pressure, and wind speeds . . . . 115  Figure 6.1  Coefficients of variability for hourly and daily solar radiation . . . . . . . . . . . . . . . . . . . . . . . . . 126  Figure 6.2  Scatterplot matrix of daily transmissivity observed at four mountain stations. . . . . . . . . . . . . . . . . . 128  Figure 6.3  Scatterplot matrix of hourly tranmissivity observed at four mountain stations. . . . . . . . . . . . . . . . . . 129  Figure 6.4  Transmissivity model fits with and without precipitation adjustment for S∆T1 ,ea at each site . . . . . . . . 133  Figure 6.5  Transmissivity model residuals . . . . . . . . . . . . . 134  Figure 6.6  Daily K↓ residuals . . . . . . . . . . . . . . . . . . . . 134  Figure 6.7  Modelled and observed hourly solar radiation fluxes . 136  Figure 6.8  Fisheye photo of the field of view of an inverted pyranometer . . . . . . . . . . . . . . . . . . . . . . . . . . 141  Figure 6.9  Time-series of observed albedo . . . . . . . . . . . . . 143  Figure 6.10 (a) PDD snow albedo model, (b) snow albedo residuals versus daily atmospheric transmissivity . . . . . . . . 147 Figure 6.11 Time series of observed and predicted daily albedo . . 148 Figure 6.12 Comparison of off-glacier and on-glacier L↓ . . . . . . 154 Figure 6.13 Differences in on- and off-glacier L↓ . . . . . . . . . . . 156 Figure 6.14 On- and off-glacier emissivities . . . . . . . . . . . . . 157 Figure 6.15 Predicted versus observed clear-sky emissivities at (a) hourly and (b) daily timescales . . . . . . . . . . . . . 159 Figure 6.16 Modelled and observed cloud factor F . . . . . . . . . 161 Figure 6.17 Time-series of observed and modelled incoming longwave radiation . . . . . . . . . . . . . . . . . . . . . . 165 xiv  List of Figures Figure 6.18 Boxplots of hourly L↓ residuals by hour of day . . . . 166 Figure 6.19 Modelled L↓ residuals versus ambient T . . . . . . . . 167 Figure 6.20 Modelled versus observed L↓ for test cases . . . . . . . 168 Figure 7.1  Accumulated PDD snow density model and residuals . 183  Figure 7.2  Difference maps of snow density  Figure 7.3  Comparison of polynomial regressions and Loess curve  . . . . . . . . . . . . 184  fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Figure 7.4  Tidbit temperature records . . . . . . . . . . . . . . . 190  Figure 7.5  Dates of snowline retreat versus elevation, for all sites and years. . . . . . . . . . . . . . . . . . . . . . . . . . 191  Figure 7.6  Observed specific net balance (bn ) versus date of snowline retreat obtained at Place and Helm Glaciers, 2006 - 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . 193  Figure 8.1  Modelled and observed bs for various melt models . . 221  Figure 8.2  Modelled and observed MT B for various melt models. 222  Figure 8.3  Melt model residuals versus observations. . . . . . . . 223  Figure 8.4  Observed and modelled (a) hourly and (b) daily net radiation at PM2, 2007 ablation season . . . . . . . . 225  Figure 8.5  Modelled snow pack temperatures . . . . . . . . . . . 226  Figure 8.6  Mean monthly energy fluxes for full energy balance models with (left) and without (right) KBL corrections 226  xv  List of Symbols bn  net mass balance  bs  summer mass balance  bw  winter mass balance  C  surface temperature deficit  Ca  specific heat capacity of air  CD  total drag coefficient  CE  bulk latent heat transfer coefficient  CH  bulk sensible heat transfer coefficient  d0  initial snow depth  df  final snow depth  ea  ambient vapour pressure  ef  surface vapour pressure  eg  glacier near-surface vapour pressure  es  saturation vapour pressure  F  cloud factor  Fm  modified melt factor  G  goodness-of-fit index  g  gravitational acceleration  h  surface height  k  von Karman’s constant  K↓  incoming shortwave radiation  K ↓cs  potential clear sky insolation  K↑  outgoing shortwave radiation  KH  eddy diffusivity for heat  KM  eddy diffusivity for momentum  xvi  List of Symbols ki  Degree-day melt factor for ice  ks  Degree-day melt factor for snow  L↓  incoming longwave radiation  L↑  outgoing longwave radiation  Lf  latent heat of fusion  Lv  latent heat of evaporation  Ma  molar mass of dry air  Mi  ice melt  Ms  snow melt  PDD  accumulated postive degree days  p  atmospheric pressure  p0  mean sea-level pressure  Q∗  net radiation  QA  advected turbulent heat fluxes  QE  latent heat flux  QG  ground heat flux  QH  sensible heat flux  QR  heat flux from precipitation  ∆QS  change in snowpack heat storage  R  rainfall  Rb  bulk Richardson stability number  Rg  ideal gas constant  RH  relative humidity  Rig  bulk Richardson gradient number  ri  radiation factor for ice  rs  radiation factor for snow  SWE0  initial snow water equivalence  T∗  critical ambient temperature  Ta  ambient air temperature  Tg  glacier near-surface air temperature  Td  mean daily temperature  Ts  surface temperature  TZ=0  sea-level temperature xvii  List of Symbols T0  threshold temperature for melt  U  westerly wind speed  ug  wind speed  uK  katabatic wind  uKc  channelled katabatic wind  uN K  non-katabatic wind  u(z)  wind speed profile  V  southerly wind speed  V  ∗  critical southerly wind speed  Z  elevation  z  measurement height  ∆zs  snow depth for ∆QS calculation  zx  roughness length of temperature and vapour pressure  z0  roughness length  α  surface slope  αf  fresh snow albedo  αi  ice albedo  αs  snow albedo  β  parameters for KBL modelling  a  atmospheric emissivity  cs  clear-sky atmospheric emissivity  s  surface emissivity  0  effective atmospheric emissivity  γe  vertical vapour pressure gradient  γT  vertical temperature gradient  λ  length scale  µ  momentum scale  Θ  wind vector and stability correction  θ  potential temperature  θ(z)  temperature deficit profile  θZ  solar angle of incidence  xviii  List of Symbols ρi  ice density  ρs  snow density  σ  Stefan-Boltzmann constant  τ  atmospheric transmissivity  xix  Acknowledgements First and foremost, this work would not have been possible without the guidance and support of my advisor, Dan Moore. His insights, scientific curiosity, and attention to editorial detail are hopefully manifested in this thesis, and his committment for this project was unwavering. I also thank my thesis committee, comprised of Dr. Garry Clarke, Dr. Ian McKendry, and Dr. Tim Oke, for their feedback and contributions. University examiners, Dr. Andreas Christen and Dr. Mike Novak provided numerous insights and suggestions. Special thanks go to Ellen Morgan, Roger Hodson, and Natalie Stafl, whose assistance in the field was central to obtaining both data and memorable field trips. Other people who generously volunteered their time and their backs include Faron Anslow, Chris Borstad, Rob Burrows, John Chapman, Jeff Kane, Derek van der Kamp, John Richards, Rich So, and Brett Wheler. Ivan Liu created the portable weather station tripod, which made the field loads bearable. Kerstin Stahl provided data, collaboration, and early inspiration, the British Columbia Ministry of the Environment provided the opportunity to do research at Weart and Helm Glaciers, and Blackcomb Helicopters never failed to pick us up from the field. Many thanks to a strong supporting cast in the Geography Department at the University of British Columbia for much-needed discussions and distractions, including Josh Caulkins, Emily Davis, Marwan Hassan, Derek van der Kamp, Scott Krayenhoff, Sonya Powell, Joanna Reid, Russell Smith, Amanda Stan, and Andr´e Zimmerman. Funding for this research was provided by both NSERC grants to JMS, and NSERC and CFCAS grants to RDM through the Western Canadian Cryospheric Network. The opportunity to collaborate and receive feedxx  Acknowledgements back through WCCN meetings was incredibly beneficial. Mike Demuth at the Geological Survey of Canada provided invaluable LiDAR data for the study sites, in-kind-support through shared helicopter flights, and has single-handedly maintained the Canadian glacier mass balance monitoring program. Finally, I acknowledge the patience and understanding of my wife Shelley, who has tolerated my PhD mistress and encouraged me throughout. This thesis is dedicated to Connall and Tate, who provided cheerleading when I needed it most. Any remaining errors and shortcomings in this thesis are, unfortunately, my own.  xxi  Chapter 1  Introduction Glacier fluctuations are significant at local, regional, and global scales, with impacts on the hydrology and ecology of alpine streams [Fagre et al., 2003; Fleming, 2005], geomorphic hazards and water quality [Moore et al., 2009], regional water supplies [McCarthy et al., 2001; Barnett et al., 2005] and hydroelectric power generation [Tangborn, 1984], and global sea-level changes [Oerlemans and Fortuin, 1992; Raper and Braithwaite, 2006]. Glacier mass balance is the sum of accumulation (inputs) and ablation (outputs) on an annual basis, and interannual mass balance variability is determined by climate [Moore and Demuth, 2001; Shea and Marshall, 2007]. Glacier mass balance is also the driving variable in dynamic glacier and ice sheet models, which are necessary for modelling the evolution of glacier area and volume. Glacier mass balance thus provides a critical link between climatic change and glacier responses [Marshall and Clarke, 1999], which are of global interest. Despite the significance of glacier mass balance, a maximum of only 90 sites worldwide have been monitored for mass balance changes in any given year [Dyurgerov, 2002]. Declining summer streamflows in British Columbia have been linked to glacier retreat [Stahl and Moore, 2006], yet only two glaciers in the southern Coast Mountains are currently monitored for annual mass changes. The total area of these monitoring sites (ca. 5 km2 ) represents less than 0.1% of the total glaciated area in the southern Coast Mountains (ca. 8000 km2 ). Developing glacier mass balance models that can be applied with confidence to unmonitored regions thus represents an important goal that is complicated by the fact that processes which govern glacier fluctuations (accumulation and ablation) vary in both time and space. Research presented in this thesis focuses on ablation, and on ablation 1  1.1. Glacier Melt Modelling modelling methods.  1.1  Glacier Melt Modelling  Regional analyses of glacier melt are required to identify the impacts of climate change in glacierized basins. However, distributed melt models have typically focused on a single glacier with detailed climatological measurements [Arnold et al., 1996; Klok and Oerlemans, 2002; Hock, 2005; Klok et al., 2005]. Two important issues regarding model performance are portability and complexity. The performance of different melt models has been previously compared for individual glaciers [Hock, 1999; Pellicciotti et al., 2005; Pellicciotti et al., 2008; Gudmundsson et al., 2009], but the portability of a given modelling approach between glaciers has only recently been examined [Carenzo et al., 2009]. In snow hydrology, melt model portability and the relation between complexity and performance has been addressed numerous times, most notably by the Intercomparison of Snow Models project organized by the World Meteorological Organization [WMO, 1986], and more recently by Essery et al. [1999], Strasser et al. [2002], Etchevers et al. [2004] and Franz et al. [2008]. Each of these studies attempted to address the performance of a given model in relation to its input parameters and assumptions. Approaches to glacier melt modeling range in complexity from lumped statistical models to empirical temperature-index and energy balance models (Figure 1.1), which can operate at scales ranging from point melt estimates to fully distributed models. The surface energy balance of a melting glacier can be expressed as: QM + ∆QS = Q∗ + QH + QE + QR + QA  (1.1)  where QM is the energy involved in melting (QM > 0) or refreezing (QM < 0), ∆QS is the change in energy storage within the snowpack, Q∗ is the net radiation, QH and QE are the turbulent fluxes of sensible and latent heat, respectively, QR is the sensible heat flux supplied from rain, and QA is the 2  1.1. Glacier Melt Modelling local advection of sensible heat from surrounding snow-free terrain (all in units of W m−2 ). In this formulation, fluxes directed towards the surface are positive, and fluxes away from the surface are negative. A summary of energy balance studies conducted at mid-latitude glaciers (Table 1.1) indicates that net radiative fluxes tend to dominate the surface energy balance, though turbulent fluxes represent between 5 and 50% of the net energy at the surface. At maritime sites, the contribution from turbulent fluxes tends to be greater than that observed at continental locations, and on shorter time-scales turbulent fluxes of energy can contribute substantial amounts of melt energy [Hock, 2005]. All other modelling approaches are essentially simplifications of the energy balance approach, where melt determined by the surface energy balance is parameterized through air temperature [e.g. Ohmura, 2001], or air temperature and solar radiation [e.g. Hock, 1999], or atmospheric circulation [Shea and Marshall, 2007].  Figure 1.1: Glacier melt models overview. The success of temperature-index approaches to melt modelling is due mainly to the relation between air temperature and fluxes of longwave radiation and, to a lesser degree, sensible heat [Ohmura, 2001]. High air temperatures are also correlated to clear-sky conditions during the summer, 3  1.2. Meteorological Inputs for Melt Modelling providing a further link between temperature and solar radiation. To determine which melt model is most appropriate for regional melt modelling, it is necessary to examine the meteorological drivers required for each approach.  1.2  Meteorological Inputs for Melt Modelling  Energy balance melt models require measurements or estimates of shortwave and longwave radiation, and observations of near-surface air temperature (Tg ), vapour pressure (eg ), and wind speed (ug ) [Hock, 2005]. These variables may be measured directly on the glacier surface [van de Wal et al., 1992; van de Wal and Russell, 1994; Hock and Holmgren, 1996; Brock and Arnold, 2000; M¨ olg and Hardy, 2004; Pellicciotti et al., 2005; Anslow et al., 2008], at stations adjacent to the glacier [Munro, 1991, 2004; Machguth et al., 2006], or extracted from regional climate networks [Greuell and Oerlemans, 1986; Oerlemans and Fortuin, 1992] or mesoscale atmospheric models [Cook et al., 2003; Box et al., 2004]. Distributing meteorological fields on melting glaciers, regardless of the source data, is problematic due to katabatic flows and the development of a katabatic boundary layer (KBL) during melt periods. In contrast, a temperature-indexed approach requires only regional air temperatures [Braithwaite, 1977; Shea et al., 2009], which serve as an index of the total energy available for melt. The KBL is a well-documented phenomenon of melting glaciers and ice sheets [Holmgren, 1971; Munro and Davies, 1978; van den Broeke, 1997b; Hock, 2005]. At mid-latitude glaciers in the ablation season, ambient air temperatures (Ta ) typically exceed snow- or ice-surface temperatures (Ts = 0◦ C). Cooled air near the surface and the resulting density gradient produce katabatic flows, which are typified by strong down-glacier wind speeds [Munro and Davies, 1978; Ohata, 1989; Obleitner, 1994; van den Broeke, 1997b; Greuell et al., 1997; Strasser et al., 2004]. Katabatic winds reinforce the sensible heat exchange and cooling of the near-surface layers, creating a shallow but intense temperature inversion [van den Broeke, 1997b]. As this situation arises when conditions for melt are favourable, assumptions of a constant environmental lapse rate or reference wind speed based on either 4  1.3. Melt Model Selection on- or off-glacier data may be unsuitable for parameterizing temperatures, wind speeds, and incoming longwave radiation over a melting glacier surface [Greuell and B¨ ohm, 1998; Greuell and Smeets, 2001; Klok et al., 2005]. In the snow hydrology literature, numerous studies have demonstrated methods for generating fields of T , ea , and u for distributed energy balance melt modelling [e.g. Susong et al., 1999; Garen and Marks, 2005]. However, these schemes do not account for modification of air temperature, humidity and wind speed by the development of katabatic flows, and thus are generally not applicable to glacier settings. Few glaciological studies have compared on-ice meteorological quantities to those measured off-site [Stenning et al., 1981; Lang, 1986; Oerlemans, 2000], and even fewer have directly examined relations between meteorological variables measured within the katabatic boundary layer at the same glacier [Greuell and B¨ohm, 1998; Strasser et al., 2004; Marshall et al., 2007]. To enable regional energy balance melt modelling, methods for transforming regional climate data to on-glacier fields of T , ea , and u are required.  1.3  Melt Model Selection  It is important to reconcile the complexity of a given model with both (a) portability and (b) input data requirements. While energy balance approaches theoretically offer the most portability due to the complete description of surface energy fluxes, the generation of appropriate regional climate fields has been neglected in the glacio-meteorological literature. Conversely, while temperature indexed models are the simplest approach, model portability is a concern [Shea et al., 2009]. Walter et al. [2005] demonstrated that a heavily parameterized energy balance approach achieved greater accuracy than a degree-day approach for predicting snow melt at four open and level measurement sites across the United States, whereas Franz et al. [2008] suggested that degree-day snowmelt models are preferable due to the large uncertainty in energy balance approaches. Radiation-indexed degree day model (RIDDM) coefficients are fairly stable between sites and years within a limited geographic region [Carenzo et al., 2009], but for substantially dif5  1.4. Thesis Objectives and Outline ferent climates RIDDM coefficients require local optimization [Schneeberger et al., 2003; Pellicciotti et al., 2005]. To date, there have been no intercomparisons of glacier melt models of varying complexities at a range of locations.  1.4  Thesis Objectives and Outline  The processes that govern snow and ice melt on temperate glaciers have been well studied, and both empirical and physically based melt models have been used to successfully model glacier melt at the scale of individual glaciers (Table 1.1). However, modelling skill depends to great degree on the quality of the input data, and glacierized regions are typically devoid of high-altitude climate stations. Physically based approaches for modelling melt furthermore require the input of meteorological data that represent conditions near the surface, as the development of the katabatic boundary layer precludes the use of standard meteorological extrapolation techniques. Research presented in this thesis is focused on two main themes: the estimation of meteorological quantities for glacier melt modelling, and the portability of melt models of varying complexities between sites. Analysis of observational data forms a significant portion of this research, and field data collection and instrument specifications are described in Chapter 2. Techniques for estimating near-surface meteorological variables within katabatic boundary layers are developed and tested in Chapters 3 through 5. Chapter 3 presents models for estimating near-surface temperature and vapour pressure, constructed with meteorological observations made at four glacier sites in the southern Coast Mountains of British Columbia. Chapter 4 describes the nature of near-surface wind speeds on melting glaciers, and suggests methods for estimating wind speeds. The results of Chapters 3 and 4 are re-examined in relation to the topographic characteristics of each site in Chapter 5, which provides models for estimating the effects of katabatic boundary layer development on near-surface meteorological variables based on digital elevation data. Methods for distributing radiative fluxes for energy balance modelling 6  1.4. Thesis Objectives and Outline are explored in Chapter 6, which focuses on atmospheric transmissivity, snow and ice albedo, and longwave radiation using surface observations. A description of methods used to generate datasets for testing melt models is given in Chapter 7, which includes the application of temperature loggers for monitoring snowline retreat. The final research chapter (Chapter 8) outlines four melt models of varying complexity, and examines model performance at four sites in the southern Coast Mountains of British Columbia. Individually, each chapter represents an important contribution to the fields of glaciology and climatology. Together, this research provides the foundation for future investigations into the effects of climate change on both individual glacier melt totals and regional glacier melt, as well as guidelines for regional modelling of glacier melt in hydrologic models.  7  1.4. Thesis Objectives and Outline  Table 1.1: Energy flux contributions from mid-latitude glacier studies, expressed as a percentage of total melt energy. M = maritime glacier, and QM = Q∗ + QH + QE . The quantity (QH + QE )/QM describes the contribution of turbulent fluxes to total melt energy. Elev (m)  Q∗ (%)  Snow Surfaces a 46.0N 2876 b 46.8N 2500 b 46.8N 2630 c 51.7N 2500 d 51.7N 2510 e 46.4N 2540 e 46.4N NA f 47.1N 2945 f 47.1N 3225 g 46.8N 3420 h 45.8N 3550 i (M) 54.8N 810 j (M) 43.4S 2150 k 43.1S NA  87 91 93 65 43 95 53 76 78 61 100 33 61 52  Ice Surfaces f 47.1N a 46.0N c 51.7N l 51.7N m (M) 43.4S n(M) 67.9N  73 85 51 49 21 66  41 22 55 29  Snow and Ice Surfaces o (M) 48.3N 1650 62 f 47.1N 2420 72 p 27.7N 4956 85 p 27.7N 5245 94 q (M) 69.3N 1715 71 r (M) 61.6N 1570 76 r (M) 60.6N 1450 65  29 23 10 9 25 17 25  Source  Lat (◦ )  2310 2876 2300 2280 NA 1375  QH (%)  QH + QE QM  Date (d)  0.13 0.09 0.10 0.35 0.56 0.05 0.47 0.23 0.21 0.40 0.01 0.67 0.39 0.46  Jul-Aug (20) Jul (10) Jul (18) Jun-Jul (20) Jul (14) May (18) Summer Jun-Aug (46) Jun-Aug (46) Aug-Sep (28) Jul (25 d) Aug-Sep (31) Mar (4) Jan-Feb (53)  7 29 25 5  0.27 0.15 0.48 0.51 0.80 0.34  Jun-Aug (46) Jul-Aug (13) Jun-Jul (15) Aug (6) Feb (4 d) Jul-Aug (39)  9 4 5 -1 5 8 15  0.38 0.27 0.15 0.08 0.30 0.21 0.25  Jun-Aug (92) Jun-Aug (46) May-Sep (125) May-Sep (125) May-Aug (85) Season Season  QE (%) 13  10 20 34 8 8 48 19 20  -1 -10 1 48 -3 -1 4 1 40  24 44  -23 23 39  30  16  22  5 15  a  Willis et al. [2002]; b van de Wal et al. [1992]; c Munro [1990]; d F¨ohn [1973]; e Pl¨ uss and Mazzoni [1994]; f Greuell and Smeets [2001]; g Ambach and Hoinkes [1963]; h de La Casini`ere [1974]; i Konya et al. [2004]; j Kelliher et al. [1996]; k Hay and Fitzharris [1988]; l Derickx [1975]; m Ishikawa et al. [1992]; n Hock and Holmgren [1996]; o Anslow et al. [2008]; p Kayastha et al. [1999]; q Klok et al. [2005]; r Giesen et al. [2009]  8  Chapter 2  Study Area and Data Collection Four glaciers in the southern Coast Mountains of British Columbia were chosen as observation sites (Figure 2.1). Site selection was based on the existence of ongoing or previous glaciological studies (Place, Helm, and Bridge), as well as glacier size and accessibility. This chapter details the meteorological observations, mass balance and initial snow water equivalence measurements, and the snowline retreat observations.  2.1 2.1.1  Meteorological Data Glacier Meteorological Stations  Automatic weather stations (AWS) were operated on glacier surfaces between May and October at Place Glacier (Figure 2.2), Weart Glacier (Figure 2.3), and Bridge Glacier (Figure 2.4). Handheld GPS coordinates were used to locate the station at approximately the same location each season, though in 2008 two stations at Place Glacier were relocated in order to document lateral variability in observed weather data (Figure 2.2, Table 2.1). At Weart Glacier, station WM1 was located in the same location in 2007 and 2008, and WM2 was in operation only in 2007. One climate station (BM1) was operated at Bridge Glacier in 2008 to test assumptions about the scale dependence of katabatic boundary layer development. An illustration of the floating station design is given in Figure 2.5. Station locations and elevations are given in Table 2.1, and periods of record for each station and season are given in Table 2.2. Weather stations 9  2.1. Meteorological Data used in this study were based on a portable tripod design, with wooden feet that prevented the station from melting into the snow or ice surface. With a floating station, sensors remained at a relatively constant measurement height (ca. 1.7 m) throughout the ablation season. At all stations, tensecond measurements of air temperature (Tg ), relative humidity (RHg ), wind speed (ug ), and wind direction (Θg ) were recorded as 10 minute means using a Campbell Scientific CR10X datalogger, and instrument specifications are given in Table 2.3. Hourly means were calculated from the 10-minute data. Reflected solar radiation (K↑) was measured at most glacier AWS during the periods of record indicated in Table 2.2. Kipp&Zonen pyranometers were inverted and suspended using custom-made Gimbel joints (Figure 2.5) mounted 1.7 m above the ground surface. Gimbel joints were used to maintain the (K ↑) sensor in an approximately horizontal position despite the shifting of the AWS due to surface melt. Net radiation (Q∗ ) was measured at PM2 in 2007 with a Kipp&Zonen NR-lite net radiometer. Table 2.1: Location and instrumentation of glacier AWS used in this study. T = temperature, RH = relative humidity, u = wind speed, Θ = wind direction, K ↑ = reflected shortwave radiation, L ↓ = incoming longwave radiation, Q∗ = net radiation. Incoming longwave radiation was measured at off-glacier stations. Site  Easting  Northing  Z (m)  Instrumentation  PM1 PM2 PM3 PM4 PM1.08 PM3.08 WM1 WM2 BM1  528297 528523 528404 527297 528523 528166 517126 516062 457788  5586081 5585611 5584819 5584624 5585611 5585534 5555997 5554090 5629116  1960 2012 2100 2313 2047 2043 2168 2290 1745  T , RH, u, Θ, K↑ T , RH, u, Θ, K↑, Q∗ , L↓ T , RH, u, Θ, K↑ T , RH, u, Θ, K↑ T , RH, u, Θ, K↑ T , RH, u, Θ, K↑ T , RH, u, Θ, K↑, Q∗ T , RH, u, Θ T , RH, u, Θ, K↑  10  2.1. Meteorological Data  Figure 2.1: Location of glacier mass balance sites and ambient automatic weather stations (AWS; triangles), Environment Canada AWS (circles), and snow course sites (diamonds). All map coordinates are given in UTM Zone 10N.  11  2.1. Meteorological Data  Figure 2.2: Place Glacier AWS locations.  12  2.1. Meteorological Data  Figure 2.3: Weart Glacier AWS locations.  13  2.1. Meteorological Data  Figure 2.4: Bridge Glacier AWS locations.  14  2.1. Meteorological Data  (a) Glacier AWS station  (b) Measuring reflected radiation with inverted pyranometer  Figure 2.5: Example of the on-ice meteorological stations (a) and a closeup view of the Gimbel joint used to mount pyranometers horizontal to the surface (b).  15  2.1. Meteorological Data  Table 2.2: Summary of on-glacier AWS periods of operation. Variations in station operation are due to deployment dates, collection dates, and battery life. Year  Site  Period of operation  2006  PM1 PM2 PM3 PM4  Aug Aug Aug Aug  2007  PM1 PM2 PM3 PM4 WM1 WM2  May - Sep (50 d) May - Sep (49 d) May - Sep (39 d) May - Sep (50 d) Jul - Sep (58 d) Jul - Sep (58 d)  2008  PM1.08 PM2 PM3.08 PM4 WM1 BM1  May - Aug (98 d) May - Sep (132 d) May - Sep (117 d) May - Aug (111 d) Jun - Sep (115 d) May - Aug (77 d)  -  Sep Sep Sep Sep  (50 (49 (39 (50  d) d) d) d)  Table 2.3: Meteorological instrumentation specifications. Parameter  Sensor  T RH Q∗  Rotronic HC-S3 Rotronic HC-S3 Kipp & Zonen NR-lite  u  RM Young 01503a MetOne 014ab RM Young 01503 Met One 024a Kipp & Zonen CMP6 Kipp & Zonen CGR3  Θ K L↓  Range  Accuracy ◦C  -30 to +60 0-100% 0.2 - 100 µm -30 to +70 ◦ C 0 - 100 m s−1 0 - 45 m s−1 0 - 360◦ 0 - 360◦ 0 - 2000 W · m−2 Not listed  ±0.2◦ C ±1.5% @ 23◦ C −1%/m s−1 0.12%/K ±0.3 m s−1 ±0.11 m s−1 ±3◦ ±5◦ ±5% (daily) < 5%  a: stall speed: 1.0 m s−1 b: stall speed: 0.45 m s−1 16  2.1. Meteorological Data  2.1.2  Ambient Meteorological Stations  Hourly T and RH were also obtained from six stations located at off-glacier sites (Figure 2.1), and are referred to as the ambient climate data (Ta and RHa ). Pemberton (204 m asl), Whistler Low (922 m asl), and Whistler High (1620 m asl) are long-term climate monitoring sites operated by Environment Canada [Environment Canada, 2010]. Ridge stations at Weart Glacier (2220 m asl), Helm Glacier (2192 m asl), and Bridge Glacier (1619 m asl) were installed as part of this study, and maintained year-round from 2006 to 2008. A seventh ambient site located on a ridge above Place Glacier (Figure 2.2) was in operation intermittently, and is used for independent model testing. Table 2.4: Ambient AWS locations and data source Site  Easting (m)  Northing (m)  Elevation (m)  Pembertona Whistler Low Levela Whistler High Levela Helm Ridgeb Weart Ridgeb Bridge Ridgeb Place Ridgeb  518517 501430 503577 501490 518349 463200 526534  5572019 5548637 5548004 5534354 5556981 5632482 5586943  204 933 1640 2192 2220 1640 2075  a: Environment Canada [2010] b: This study  2.1.3  Meteorological Data Post-Processing  Incoming longwave radiation (L ↓) measured with Kipp & Zonen CGR3 pyrgeometers required post-processing to separate the longwave radiation emitted by the sensor body from the net incoming longwave radiation (Lnet ) recorded by the sensor. Following the CGR3 manual [Kipp & Zonen, 2009], Steinhart-Hart equations were used to calculate the sensor temperature Tsens (C) from the voltage (v) returned by the sensor: Tsens = a + b [ln(v)] + c [ln(v)]3  −1  − 273.15  (2.1) 17  2.2. Mass Balance and Snowline Retreat Observations where a = 1.03 × 10−03 , b = 2.39 × 10−04 , and c = 1.57 × 10−07 . The sensor was assumed to have an emissivity of 1, and the incoming longwave radiation measured at the sensor was calculated as 4 L↓= Lnet − σ Tsens  (2.2)  where σ is the Stefan-Boltzmann constant (5.67×10−8 W m−2 K−4 ).  2.2  Mass Balance and Snowline Retreat Observations  For this study, winter accumulations were measured at four sites (Table 2.5) to set initial conditions for the melt models. Summer ablation was then observed through both (a) the glaciological method and (b) surface temperature measurements to provide data sets for testing melt model performance. Full mass balance measurements were conducted at Place and Helm Glaciers (Figure 2.1) with the Geological Survey of Canada, while winter accumulations were measured on Weart and Bridge Glaciers. Figures 2.6 and 2.7 show mass balance monitoring locations (accumulation and ablation), snow density pits, and surface temperature loggers on Place and Helm. Figures 2.8 and 2.9 illustrate accumulation measurement locations, snow density pits, and surface temperature loggers at Weart and Bridge Glaciers. gives the dates of mass balance measurements. Historical snow density data were also obtained for three snow courses operated by the River Forecast Centre, British Columbia Ministry of Environment [British Columbia Ministry of Environment, 2010] between 2006 and 2008. Snow course station details are given in Table 2.6, and station locations are shown in Figure 2.1. Table 2.5 provides the dates of accumulation (or initial SWE) and density measurements conducted between 2006 and 2008.  18  2.2. Mass Balance and Snowline Retreat Observations  Table 2.5: Dates of initial snow water equivalence (SWE0 ) measurements for mass balance and snow course sites Site  2006  Helm Place Weart Bridge Dog Mountain  Callaghan Creek  Orchid Lake  SWE0 Collection Dates 2007 2008  22 Apr 24 Apr 19 Jul 17 Apr 29 Dec, 1 Feb, 6 Mar, 3 Apr, 28 Apr, 15 May, 5 Jun, 20 Jun 31 Jan, 28 Feb, 1 Apr, 25 Apr, 1 Jun 30 Jan, 24 Feb, 27 Mar, 25 Apr, 15 May, 7 Jun, 20 Jun  20 Apr 21 Apr 9 July 24 May 27 Dec, 31 Jan, 1 Mar, 2 Apr, 26 Apr, 16 May, 28 May, 14 Jun 29 Jan, 27 Feb, 31 Mar, 30 Apr, 28 May 29 Jan, 27 Feb, 30 Mar, 30 Apr, 16 May, 28 May, 15 Jun  7 May 5 May 26 Jun 31 May 1 Jan, 4 Feb, 26 Feb, 2 Apr, 1 May, 15 May, 5 Jun, 16 Jun 1 Feb, 29 Feb, 31 Mar, 30 Apr, 29 May 1 Feb, 25 Feb, 1 Apr, 5 May, 15 May, 1 Jun, 16 Jun  Table 2.6: Snow course locations and ID Site  ID  Easting (m)  Northing (m)  Elevation (m)  Dog Mountain Orchid Lake Callaghan Creek  3A10 3A19 3A20  503630 496382 492853  5468219 5486748 5534603  1007 1178 1009  19  2.2. Mass Balance and Snowline Retreat Observations  2.2.1  Winter Balance  Specific winter balance (bw ) is the total winter snow accumulation (in m w.e.) at a particular point, calculated from snow depth (d, in m) and columnaveraged snow density (ρs , in kg m−3 ): bw = d  ρs ρw  (2.3)  where ρw is the density of water (1000 kg m−3 ). In this formulation, winter ablation (through melt or sublimation) is assumed to be negligible. At Bridge and Weart Glaciers, winter balances are also referred to as initial snow water equivalent, or SWE0 , as the dates of measurement occurred after melt onset (Table 2.5). Snow depths (d) reported for each site are the mean of three to five measurements spaced 1 m apart, and were measured using solid aluminum avalanche probes. At higher elevations, the height of the previous summer’s snow surface, which defines the bottom of the winter snowpack, was confirmed through several lines of evidence: a) the hardness of the layer; b) a correspondence with snow depths measured below the previous summer’s snowline; and c) by reaching the base of the snowpack with the density corer. Snow densities were measured using both density pits and density corers. Snow density pits were constructed by excavating snow to a depth of 1.5 m below the snow surface and then sampling and weighing a known volume of snow at 10 cm intervals. At Place and Helm Glacier, snow densities below 1.5 m were sampled by extracting snow cores of varying length using a Kovacs snow density corer. The length and weight of each core was measured in the field, and total sample volume and density were calculated from the core barrel diameter (9 cm). At Bridge and Weart Glacier, snow densities below 1.5 m were not sampled due to logistical constraints, however snow densities in isothermal snowpacks typically do not change with depth. As the snow density samples were not evenly distributed with depth, column-averaged snow densities ρ¯s were calculated by integrating the SWE in the entire snow column using the observed layer densities and depths,  20  2.2. Mass Balance and Snowline Retreat Observations and dividing by the total snow pit depth (Figure 2.10). Chapter 7 discusses further data reduction methods used to derive mass balance curves and to estimate initial SWE conditions for the melt models.  2.2.2  Summer Ablation  Glaciological ablation measurements rely on ablation stakes, which are aluminum poles drilled into the ice surface at the end of the summer season. A reference measurement of the height of the pole above the ice surface is made during installation. At the end of the following summer, the height above the ice surface is again recorded, providing a record of change in ice surface elevation. In high melt areas, ablation poles are measured and redrilled in mid-summer to prevent melt-out. In the ablation area, which is located below the elevation of the end-ofsummer snowline, ablation (a) is the sum of snow melt (in m w.e., determined from winter accumulation measurements) and ice melt (in m w.e), calculated from the change in ice surface height. To convert change in ice surface height to a water equivalent, an ice density of 900 kg m−3 is assumed [Paterson, 1994]. In the accumulation area, ice melt is equal to zero, and the total ablation is calculated as the difference between the winter balance and the SWE remaining at the end of the summer.  2.2.3  Snowline Retreat  Observations of snowline retreat can supplement mass balance data for testing melt models. The date of snow disappearance at a particular location represents an observation of total summer ablation that is approximately equivalent to the maximum winter accumulation. Thus, with knowledge of winter accumulation totals, observations of snowline retreat can be used to infer the timing and the quantity of total snowmelt at particular locations. Satellite imagery has been used previously to estimate snowline elevations [Knap et al., 1999; Sidjak, 1999], but this approach is limited temporally due to satellite repeat times and weather conditions. As an alternative to remotely sensed estimates of snowline retreat, this 21  2.2. Mass Balance and Snowline Retreat Observations  Table 2.7: Specifications of submersible temperature loggers used in this study Model TBI32-20-50 TBI32-05-37 UTBI-001  Range -20◦ C  50◦ C  to ◦ -4 C to 37◦ C -20◦ C to 30◦ C  Accuracy ±0.4◦ C ±0.2◦ C ±0.2◦ C  study uses temperature records collected at discrete locations on the glacier surface to infer dates of snowline retreat. Submersible temperature loggers known as Tidbits are composed of thermistors encased in a hard epoxy, and are typically used to measure stream temperatures. For this study, the loggers were placed on the glacier surface at the end of summer, and recovered the following season. During the winter, daily and seasonal temperature signals are muted due to the insulating properties of the overlying snow. When the winter snow cover is removed, the diurnal temperature signal reappears as the Tidbits absorb solar radiation, though the actual temperature value is irrelevant for this application. Only the change in diurnal temperature range is required to estimate the date of snowline retreat. Tidbit locations were matched closely with the winter balance sampling program to provide estimates of initial water equivalence, and thus of the total snowmelt to the date of snowline retreat. Further details on estimating the date of snowline retreat are given in Chapter 7 . Temperature data recorded by the Tidbits are transferred between the logger and a computer through optical cables. Table 2.7 lists the models and specifications of the sensors used. The manufacturer (Onset Corporation) lists the thermal response times as 5 minutes in water and 18 minutes in air with a wind speed of 1 m/s. At glacier mass balance sites (Figures 2.6 and 2.7), Tidbits were deployed along the stake network centerline by attaching the sensor to the ablation pole with a 1 m piece of cord during end-of-summer ablation measurements. At Bridge and Weart glaciers (Figures 2.8 and 2.9), Tidbits were deployed in 22  2.2. Mass Balance and Snowline Retreat Observations  Figure 2.6: Helm Glacier mass balance sites (circles), snow density pits (squares), automatic weather stations (black triangle), and surface temperature loggers (white triangles). a longitudinal transect following the winter accumulation measurements. If no ablation poles were available, one metre lengths of cord were used to anchor the Tidbits to rocks found on the glacier surface (Figure 2.11). Flagging tape was used to aid in the visual identification of the sensor the following season. Positional information (easting, northing, and elevation) obtained from a handheld GPS was used to locate the sensor the following summer. For more accurate estimate of the elevation of the sensors, elevations of the Tidbits were extracted from LiDAR DEMs of the study sites.  23  2.2. Mass Balance and Snowline Retreat Observations  Figure 2.7: Place Glacier mass balance sites (circles), snow density pits (squares), and surface temperature loggers (white triangles).  24  2.2. Mass Balance and Snowline Retreat Observations  Figure 2.8: Weart Glacier snow depth observations in 2006 (circles), 2007 and 2008 (diamonds), snow density pits (squares), and surface temperature loggers (white triangles).  25  2.2. Mass Balance and Snowline Retreat Observations  Figure 2.9: Bridge Glacier snow depth observations in 2007 (diamonds) and 2008 (circles), snow density pits (squares), and 2006 locations of surface temperature loggers (white triangles).  26  2.2. Mass Balance and Snowline Retreat Observations  ρs(1)  ρs(1)  d1  SWE1 = d1  (ρ )  ρs(2)  d2  SWE2 = d2  (ρ )  ρs(n)  dn  SWEn = dn  (ρ )  ρs = ρw  w  ρs(2) w  ρs(n) w  (Σ ΣSWE ) d  Figure 2.10: Method for calculating column-averaged snow density  (a) Tidbit  (b) Tidbit on ice  Figure 2.11: Example of the surface temperature logger being deployed on the ice surface, Bridge Glacier 2006 27  Chapter 3  Distributing Meteorological Fields for Modelling Glacier Melt, Part 1: Temperature and Vapour Pressure 3.1  Introduction  Distributed melt models have been employed in both glaciology and snow hydrology at a variety of scales, from individual glaciers [Hock, 2003] to mountain ranges [Machguth et al., 2006], and from large snowmelt-dominated basins [Barnett et al., 2005] to the Greenland Ice Sheet [Box et al., 2004]. Distributed melt models range in complexity from the degree-day approach, which requires only estimates of air temperature, to physically based energy balance (EB) models, which specify all incoming and outgoing energy fluxes. Melt models perform well at sites where local, on-glacier automatic weather stations (AWS) are available to provide input data [Hock, 1999; Klok and Oerlemans, 2002; Pellicciotti et al., 2005, e.g.]. However, at sites where such data are not available, generation of input data by extrapolation or downscaling can introduce substantial uncertainty [Anslow et al., 2008]. Numerous studies have examined procedures for generating meteorological fields for driving distributed snow melt models [Susong et al., 1999; Walter et al., 2005; Garen and Marks, 2005]. However, these approaches may not be suitable for modelling glacier melt unless the fields are adjusted to account for processes occurring in the katabatic boundary layer (KBL). 28  3.1. Introduction Over melting glaciers, turbulent energy fluxes (QH and QE ) are usually parameterized with the bulk aerodynamic approach, which requires measurements of temperature, vapour pressure and wind speed at only a single level. Direct measurements of turbulent energy fluxes using an eddy diffusivity approach are nearly impossible to maintain on melting and shifting glaciers. The success of the bulk aerodynamic approach for estimating turbulent fluxes over melting snow and glaciers has been established previously [Moore, 1983; Hay and Fitzharris, 1988; Munro, 1989; Klok et al., 2005], and is based in part on the modest data requirements. Hence, there is a need to focus on developing methods for reducing the errors associated with estimating meteorological input fields. These methods should furthermore be distinct from those used in snow-melt modelling, as a well-developed katabatic boundary layer (KBL) will introduce substantial departures from standard extrapolations and interpolations. Meteorological fields required for EB melt modelling include near-surface temperature (Tg ), vapour pressure (eg ), and wind speed (ug ), which are used primarily to calculate QH and QE . Secondary uses of meteorological input fields may include albedo modelling, parameterization of atmospheric transmissivity, and estimating incoming longwave radiation, depending on the approach selected. Few observational studies have attempted to describe variations in Tg , eg , or ug at the same glacier, and fewer still have developed empirical or physical models to distribute these variables over the glacier surface. Several studies have demonstrated that temperatures within the KBL are lower than those at the same elevation outside the KBL [Braithwaite, 1977; Greuell and B¨ ohm, 1998; Strasser et al., 2004] or suggested that standard atmospheric lapse rates (6.0 K km−1 ) are unsuitable to estimate Tg [Munro, 2004; Klok et al., 2005]. Greuell and Smeets [2001] used flow distance and sensible heat exchange to estimate Tg from potential temperatures, and empirical corrections to estimate Tg from ambient air temperature were suggested by Braithwaite et al. [2002]. The portability of either approach has not been tested at other sites, yet the temperature value used to estimate QH is important as EB melt models are highly sensitive to temperature 29  3.1. Introduction [Munro, 1991; Gerbaux et al., 2005]. Variations in vapour pressure within the KBL have received little attention in previous observational and modelling studies. Results from Pasterze Glacier (PASTEX) indicate a decrease in mean eg with altitude [Greuell et al., 1997], while Strasser et al. [2004] found no pattern. Distributed EB melt models have either a) assumed linear variations in relative humidity or vapour pressure with height [Arnold et al., 1996; Klok and Oerlemans, 2002] or b) have held relative humidity constant over the entire glacier surface [Hock and Noetzli, 1997; Hock and Holmgren, 2005] and calculated vapour pressure using prescribed temperature lapse rates. Susong et al. [1999] used seven-day smoothed vapour pressure differences between high and low elevation sites to generate distributed fields of ea , but this method was applied in a snow melt study. Glacier melt models are sensitive to changes in eg [Gerbaux et al., 2005], as the strength (and the sign) of the latent heat flux (Table 1.1) will have implications for modelled melt totals. At mid-latitude glaciers in the ablation season, the overlying ambient air temperature (Ta ) exceeds the surface temperature (Ts ), which cannot rise above 0◦ C. The resulting density gradient produces katabatic flows, which are typified by consistent down-glacier winds. Katabatic winds reinforce the sensible heat exchange and cooling of the near-surface layers, and glacier winds in particular demonstrate remarkably consistent speeds and directions at the lower reaches of large glaciers [Ohata, 1989; Obleitner, 1994; van den Broeke, 1997b; Greuell et al., 1997; Strasser et al., 2004]. Katabatic boundary layers (KBL) are characterized by a shallow and intense temperature inversion and consistent wind speeds [van den Broeke, 1997b]. The development of an adjusted boundary layer (where measured meteorological variables are representative of the surface over which the air is flowing) is proportional to the fetch and the surface roughness [Oke, 1987]. Large-scale eddy simulations have been used to model nocturnal katabatic flows [Skyllingstad, 2003], but there are few studies of simpler transfer functions or empirical models that relate meteorological quantities observed within katabatic boundary layers to flow distance and forcing strength. Given the consistency of katabatic winds on melting glaciers, 30  3.2. Methods KBL development and transfer functions should be predictable based on the glacier morphometry and the strength of the temperature difference driving katabatic flow, giving rise to a method for distributing near-surface meteorological variables. The objective of this research is to develop models to predict distributed fields of air temperature, vapour pressure, and wind speed over melting glaciers, using ambient meteorological fields as input. This chapter focuses on air temperature and vapour pressure fields, while Chapter 4 examines wind speed. In Chapter 5, model parameters from Chapters 3 and 4 are related to topographic indices derived from digital elevation models (DEMs) to allow prediction of meteorological fields at unmonitored glaciers.  3.2  Methods  Study areas, automatic weather station (AWS) locations, sampling methods, and instrument specifications are described in Chapter 2.  3.2.1  Temperature  Temperature observations from the six ambient AWS were regressed against elevation. Hourly vertical temperature gradients (γT ) and reference level temperatures (TZ=0 ) derived from the regression analysis were then used to estimate ambient air temperatures (Ta ) at glacier AWS elevations. An original conceptual model to estimate near-surface temperatures (Tg ) from Ta (Figure 3.1) evolves from observations of Ta and Tg . Above some critical threshold temperature (T ∗ ), the temperature difference between the surface and the overlying air mass is sufficient to induce katabatic flows, and the slope of the best-fit line between Tg and Ta > T ∗ reflects the strength of boundary layer cooling. Below T ∗ , Tg should be approximately equal to Ta , as the KBL will be poorly developed or non-existent. Near-surface temperatures (Tg ) for site x at time t were thus estimated from ambient temperatures (Ta ) using a four-term piecewise model that estimates a critical ambient temperature threshold for boundary layer effects  31  3.2. Methods  (a) Observational data  (b) Conceptual model  Figure 3.1: Ambient and observed near-surface temperatures at PM2, 20062008 (a), and the conceptual piecewise regression model for near-surface temperatures (b). Coefficient k2 represents the critical ambient temperature (T ∗ ) for the development of katabatic flow, k1 is the corresponding on-glacier temperature, and k3 and k4 are the slopes below and above T ∗ , respectively. (Figure 3.1): Tg (x, t) =  k1 + k4 · (Ta − k2 ), Ta ≥ k2 k1 − k3 · (k2 − Ta ), Ta < k2  (3.1)  where ki are fitted coefficients representing the threshold ambient temperature for boundary layer development (T ∗ = k2 ), the strength of the boundary layer cooling effect (k4 ), the equivalent threshold temperature for nearsurface temperatures (k1 ), and the strength of the boundary layer effect below T ∗ (k3 ), which is assumed be approximately 1. Separate models were fitted for each glacier AWS location using a nonlinear regression nlinfit.m function in Matlab using all available observations of Tg and Ta . Fitted coefficients were then used to predict Tg , and residuals were examined to evaluate model skill on seasonal and inter-annual timescales. Physical implications of the piecewise coefficients were also examined, with a focus on parameters k2 and k4 , which represent the katabatic ‘switch’ and the magnitude of the glacier cooling effect, respectively.  32  3.2. Methods  3.2.2  Vapour Pressure  For both ambient and on-glacier AWS, saturation vapour pressures (es ) were calculated using Teten’s formulae [Bolton, 1980]: es =  6.108 × 10(9.5 T /T +265.5) , T > 0◦ C 6.108 × 10(7.5 T /T +237.3) , T ≤ 0◦ C  (3.2)  where T is the observed air temperature. Vapour pressures were then calculated using observedRH: ea = es ·  RH . 100  (3.3)  Calculated hourly ambient vapour pressures were regressed against elevation, and the fitted elevational vapour pressure gradients (λe ) and reference level vapour pressures (eZ=0 ) were used to calculate ambient vapour pressures at glacier AWS sites. While gradients of RHa and Ta could also have been used to estimate ea , vapour pressure gradients were statistically stronger than RHa gradients, though this is primarily due to the dependence of es on T . The assumption that vapour pressures decrease linearly with elevation is in contrast to observations that atmospheric vapour pressure decreases exponentially with height over the depth of the troposphere [Barry, 1992; Visconti, 2001]. The decision to use a linear regression is supported by the fact that over the elevation range used in this study (200 to 2800 m) a linear relation appears appropriate. Furthermore, this study specifically uses surface observations of vapour pressure, and therefore calculates elevational gradients for near-surface vapour pressure. While the availability of moisture at the ambient AWS sites will exert some control on observed humidities and calculated λe , atmospheric vapour pressures are subject to vertical and horizontal mixing, and may not represent surface conditions. The relationship between free-air and surface vapour pressure lapse rates, particularly in mountainous terrain, would form an interesting avenue of future research, but is beyond the scope of the present study.  33  3.2. Methods The relation between near-surface (eg ) and ambient vapour pressure (ea ) depends on near-surface temperatures (Tg ) and surface vapour pressures (ef ), with three possible situations: 1. Tg > 0 and ea > ef : with near-surface temperatures fixed at 0◦ C, ef is constant at 6.11 hPa, the saturation vapour pressure at 0◦ C. With ea > ef the gradient will drive condensation towards the surface, removing moisture from the KBL (Figure 3.2). This is often assumed to be the typical case in glacier melt studies [Hock, 2005], and results in positive (energy directed towards the surface) latent heat fluxes (Table 1.1). 2. Tg > 0 and ea < ef : the reversed vapour pressure gradient generates evaporation and/or sublimation at the surface, adds moisture to the KBL, and results in a negative latent heat flux. 3. Tg < 0: with near-surface temperatures less than 0◦ C, the saturation vapour pressure immediately adjacent to the surface is no longer fixed at 6.11 hPa. Vapour gradients are likely negligible, as both ef and ea will be small, due to the low temperatures. Slopes of the lines of best-fit are thus expected to be close to unity. Two separate linear functions were used to describe near-surface vapour pressures: eˆg (x, t) =  j1 · ea + j2 , Tg > 0 j3 · ea + j4 , Tg < 0  (3.4)  where ji are fitted coefficients, and ea is the estimated ambient vapour pressure. With near-surface temperatures above 0◦ C, vapour pressure depletion or enhancement through condensation or evaporation occurs in response to ambient vapour pressures. This regression line should cross the 1:1 line near 6.11 hPa, the saturation vapour pressure at 0◦ C. The degree of enhancement or depletion should correspond to the level of boundary layer development, assuming minimal entrainment of ambient air from above. A separate linear fit for observations below 0◦ C is expected to have a slope near 34  3.3. Results  (a) Observational Data  (b) Conceptual Model  Figure 3.2: Ambient and observed near-surface vapour pressures at PM2, 2006-2008 (a), and conceptual model for KBL vapour pressure analysis. Observed eg are stratified by near-surface temperature. unity. Near-surface vapour pressures were estimated from ambient vapour pressure gradients and model coefficients derived from the regression analyses, and model performance was examined using seasonal and inter-annual residuals.  3.3 3.3.1  Results Temperature  Summaries of ambient and near-surface glacier temperature records used in the analysis (not shown) show that diurnal temperature variability is greater for the ambient stations, as is the range of observed temperatures. Hourly near-surface temperature gradients calculated through linear regressions of ambient temperature and station elevation range from -12 to +2 K km−1 (Figure 3.3). Reference-level (Z = 0) temperatures calculated from the regression analysis follow the patterns observed in the ambient temperature series, and range from 0 to +40◦ C. Coefficients of determination (R2 ) calculated for the temperature gradient analyses are generally high; however, lower R2 values are observed when temperature gradients are close to or 35  3.3. Results greater than zero (temperature inversions). Horizontal variations in hourly observed Ta over the study area are minimal. Near-surface temperatures at Bridge Glacier ridge station, located in a more continental setting, were successfully predicted using the other 5 ambient stations (Figure 3.4a). Near-surface temperatures predicted for the independent Place Glacier ridge station with γT and TZ=0 (Figure 3.5) have modelled absolute errors of 0.70◦ C. Observed temperatures at the ridge station tended to be greater than those predicted from the gradient analysis for most of the 2008 ablation season, and the difference appears to be greatest in the evening. Errors in predicting early evening temperatures may be a consequence of the strong diurnal temperature patterns observed at the valley stations. Nighttime radiative cooling and cold-air drainage in the valleys would lower temperature estimates made for higher elevations. Mean values of ambient and observed near-surface temperatures calculated for the periods of observation at each station (Table 3.1) illustrate KBL effects on Tg . At all sites and for all years, Tg is lower than Ta . This difference appears to be greatest for stations located at lower glacier elevations.  36  3.3. Results  (a) Temperature gradient  (b) Reference level (Z = 0) temperature  (c) Coefficient of Determination  Figure 3.3: Calculated temperature gradients (top), reference temperature (TZ=0 ), and the coefficient of determination (R2 ) for the ambient temperature regression analysis, 2007 ablation season 37  3.3. Results  (a) Temperature  (b) Vapour Pressure  Figure 3.4: Testing horizontal variability in ambient temperature and vapour pressure. Observed ambient temperatures (a) and vapour pressures (b) versus predicted for Bridge Glacier ridge station, 2007. Predicted values are based on linear regressions fitted to all ambient stations except Bridge.  38  3.3. Results  (a) Scatterplot  (b) Boxplot  (c) Time-series  Figure 3.5: Temperature extrapolation results for an independent ambient station (Place Glacier ridge station, 2008 ablation season): observed versus predicted temperatures (a), boxplots of residuals by hour (b), and time-series of residuals (c)  39  Table 3.1: Mean ambient temperature (Ta ), observed temperature (Tg ), and temperature difference (∆T = Ta −Tg ) at glacier AWS, all in units of ◦ C. Means were calculated only for periods where the glacier AWS was in operation, and Z is station elevation. 2006  2007  2008  Z(m)  Ta  Tg  ∆T  Ta  Tg  ∆T  Ta  Tg  ∆T  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08 PM Ridge  1960 2012 2100 2313 2168 2290 1745 2047 2043 2075  8.28 7.99 9.46 6.45 -  5.76 5.41 7.35 4.25 -  -2.52 -2.58 -2.11 -2.20 -  5.31 5.88 5.38 5.52 6.56 5.95 -  3.90 4.22 4.32 4.29 3.97 4.54 -  -1.41 -1.66 -1.06 -1.23 -2.59 - 1.41 -  5.29 3.70 7.09 7.11 4.82 5.21 5.27  3.59 3.11 3.94 4.29 3.89 4.29 5.66  -1.70 -0.59 -3.15 -2.82 -0.93 -0.92 0.39  3.3. Results  Site  40  3.3. Results Piecewise Temperature Models Global (all available data) scatterplots of ambient and observed temperatures and the piecewise regression curves are shown in Figure 3.6, and optimized coefficients for the global models are given in Table 3.2. Parameter k2 represents the critical ambient temperature (T ∗ ) required to initiate development of the KBL. Global model values for T ∗ range from 4.29 to 6.48◦ C. Below T ∗ , KBL development is limited, and near-surface temperatures measured on the glacier are similar to ambient temperatures estimated for the same elevation. As the differences between Ta and Ts are small below T ∗ , slopes of the best-fit lines (parameter k3 , unitless) range between 0.85 and 0.95, indicating that boundary layer development is minimal. Above T ∗ , slopes of the line of best fit are significantly different from 1, and for the global models these range from 0.32 (BM1) to 0.72 (PM4). Table 3.2: Optimized piecewise model parameters (k1 , k2 , k3 , k4 ), sample size (n), coefficient of determination (R2 ), and root mean squared error (RMSE) for global temperature models. The last column illustrates that k1 ≈ k2 · k3 (see Discussion). Site  k1 ◦ ( C)  k2 ◦ ( C)  k3  k4  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08  5.51 4.76 4.55 3.89 3.70 6.74 4.08 4.82 4.82  6.48 5.60 5.12 4.29 4.90 8.37 6.08 5.35 5.12  0.85 0.89 0.95 0.95 0.83 0.83 0.63 0.91 0.93  0.43 0.48 0.69 0.72 0.35 0.60 0.32 0.64 0.62  n  R2  RMSE  k2 · k3 (◦ C)  3429 7174 3854 5075 3418 1410 1849 2355 02799  0.90 0.91 0.94 0.93 0.68 0.87 0.82 0.96 0.96  1.04 1.10 0.97 1.25 1.54 1.17 1.01 0.86 0.80  5.51 4.98 4.86 4.08 4.07 6.94 3.83 4.87 4.76  41  3.3. Results  42  Figure 3.6: Global models of observed (Tg ; y-axis) versus ambient (Ta ; x-axis) near-surface temperatures and the piecewise fit results  3.3. Results Observed temperatures within the KBL are lower than estimated ambient air temperatures, particularly at lower elevation stations (WM1, PM1, and PM2) and at the higher range of ambient temperatures. Piecewise models explain over 90% of the variability in observed Tg , except for WM1 at Weart Glacier. Cross-glacier varability in boundary layer development was tested in 2008 at Place Glacier, when three AWS were arranged across the glacier at approximately the same elevation (Figure 2.2). All three sites had identical values for coefficient k1 (4.82◦ C), and values for T ∗ (k2 ) ranged from 5.12 to 5.61◦ C (Table 3.2, Figure 3.6). Fitted slopes below T ∗ (unitless) were between 0.88 and 0.93 at the three sites in 2008, and the main difference between the three sites was observed in coefficient k4 . The slope of the best fit line above T ∗ was found to be 0.62 and 0.64 at the outer sites, and 0.43 at the middle station, a result which may be explained by several factors. Stations near the edge may receive greater heat inputs through sensible and longwave energy from the surrounding rocky terrain, and the middle station likely has a more developed KBL. This issue is examined further in the companion chapter on generating meteorological scaling functions using morphometric parameters derived from glacier DEMs. There is no seasonal variability in the magnitude or the sign of the modelled temperature residuals, indicating that the success of the piecewise model does not depend on the evolution of the glacier surface (snow versus ice) or on other seasonally varying phenomena. Similarly, the global piecewise models work well between years, and on glaciers of vastly different spatial scales. Residuals typically range between ±4◦ C, and lag-1 (one hour) autocorrelations range between 0.77 and 0.85, indicating a high degree of autocorrelation. Boxplots of temperature residuals by hour suggest that there may be a small component of error related systematically to time of day, and also to position on the glacier (Figure 3.7a). At lower elevation stations, the piecewise model overestimates daytime temperatures, while at the higher elevation stations (PM4 in 2007 and 2008) the model underestimates daytime temperatures. While katabatic flow is assumed to be driven by Ta − Ts , 43  3.3. Results the residual pattern by hour suggests that solar heating may play a minor role. At the higher elevation stations, more daytime heating occurs near the surface of the glacier than can be accounted for by both the ambient lapse rates and the piecewise analysis. This, in turn, will generate enhanced katabatic flow which will cause further cooling at lower elevation stations. Figure 3.7b gives boxplots of temperature residuals according to synoptic type. Synoptic types were classified using a cluster analysis of mean daily surface and 500 hPa pressure heights from April to October, extracted from the National Center for Environmental Prediction and National Climate and Atmospheric Research (NCEP/NCAR) re-analysis data set (Kalnay et al. [1996]; National Center for Environmental Prediction [2009]; pers. comm. D. van der Kamp). Synoptic type composites are given in Figure 3.8. Synoptic setting appears to have no effect on the modelled near-surface temperature residuals, suggesting that KBL development is related primarily to a thermal forcing, as opposed to a mechanical forcing.  44  3.3. Results  (a) Hour  (b) Synoptic Type  Figure 3.7: Boxplots of predicted near-surface temperature residuals by (a) hour of day, and (b) synoptic type for three stations, PM2 (top), WM1 (middle), and BM1 (bottom). Some synoptic types are not represented due to differences in observation periods.  45  3.3. Results  46  Figure 3.8: Mean surface and 500 hPa surfaces for synoptic types classified using a principle components analysis (PCA) followed by cluster analysis (pers. comm., D. van der Kamp)  3.3. Results The height of a 0◦ C isotherm (ZT =0 ) associated with the onset of KBL influence was calculated using the station elevations, values obtained for k2 , and a lapse rate of -0.0057 K m−1 (Table 3.3). For all sites and seasons ZT =0 ranges from 2758 to 3063 m, with two outliers that likely result from partial datasets and sampling error (PM3 in 2006 and WM2 in 2007). ZT =0 is similar between stations with large differences in glacier size and elevation range, and one possible interpretation is that the onset of katabatic flow development depends primarily on large-scale air-mass characteristics, as opposed to local thermal conditions. Table 3.3: Estimated minimum elevation of the 0◦ C isotherm (ZT =0 , in m) required for development of katabatic boundary layers. Parameter k2 (◦ C) was obtained from annual piecewise temperature analyses, and Z is the AWS elevation (m). 2006  2007  2008  Site  Z  k2  ZT =0  k2  ZT =0  k2  ZT =0  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08  1954 2012 2095 2313 2164 2281 1745 2047 2043  5.61 4.82 10.46 3.66 -  2889 2815 3838 2923 -  6.55 5.15 4.54 4.50 4.43 8.37 -  3045 2870 2851 3063 2902 3676 -  5.61 5.12 4.38 6.08 5.35 5.12  2947 2948 2894 2758 2845 2948  47  3.3. Results  3.3.2  Vapour Pressure  Hourly ambient vapour pressure gradients (γe ), reference level vapour pressures (eZ=0 ) and the coefficients of determination (R2 ) obtained from the vapour pressure regression analyses are shown in Figure 3.9. Vapour pressures observed at PMR were estimated successfully using hourly γe and eZ=0 , though a bias towards overestimating ea was observed at this site. Time-series of residual (observed minus predicted) vapour pressures suggest that the skill of the ea models may be tied to general weather patterns (not shown). Vapour pressure models may also be biased by hour of day, as expected values at the ambient ridge station were consistently greater than observed between 10:00 and 16:00. Strong lateral climatic gradients are expected to introduce errors in estimating vapour pressures throughout the study region, given the differences in proximity to oceans and position with respect to major orographic barriers. Vapour pressures observed at BMR are less than those predicted from the remaining ambient stations, though it appears that the vertical gradients are similar and the differences are due to a systematic difference in moisture content (Figure 3.4b). Though not as strong as the temperature results, vapour pressure gradients are unaffected by the removal of the ambient Bridge station, suggesting that vertical gradients of vapour pressure are orders of magnitude stronger than those in the horizontal. Mean observed and ambient vapour pressures (Table 3.4) calculated for each station and season indicate that, on average, the vapour pressure gradient supports condensation at the surface, resulting in lower vapour pressures within the KBL. This is physically consistent with previous energy balance studies showing positive latent energy fluxes at the glacier surface (Table 1.1). Differences in ea and eg are greatest at lower elevation stations where the KBL is expected to be most developed.  48  Table 3.4: Mean ambient (ea ) and observed (eg ) vapour pressures (in hPa), and difference (∆e = eg − ea ) for glacier AWS. Ambient means were calculated only for periods where each glacier AWS was in operation. 2006  2007  2008  Z (m)  ea  eg  ∆e  ea  eg  ∆e  e  eg  ∆e  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08 PM Ridge  1960 2012 2100 2313 2168 2290 1745 2047 2043 2075  6.74 6.57 6.53 5.71 -  6.17 5.99 6.17 5.75 -  -0.57 -0.58 -0.36 0.04 -  6.22 6.32 6.05 5.72 6.67 6.24 -  5.99 6.13 6.19 5.60 6.32 6.28 -  -0.23 -0.19 0.14 -0.12 -0.35 0.04 -  6.17 5.36 6.34 7.28 5.97 6.22 6.05  5.95 5.19 5.96 6.01 5.95 6.01 5.85  -0.22 -0.17 -0.38 -1.27 -0.02 -0.21 -0.20  3.3. Results  Site  49  3.3. Results Vapour Pressure Models Scatterplots of ea and eg for the station models further illustrate the postulated behaviour of water vapour within the KBL (Figure 3.11). With ambient vapour pressures greater than 6.11 hPa and temperatures greater than 0◦ C, observed eg show moisture depletion at all sites. The expected mechansism for moisture depletion within the KBL is condensation at the surface, resulting from the gradient between ea and es . Below ea = 6.11 hPa, enhancement of vapour pressure is observed at all sites, as the gradient is reversed and evaporation or sublimation from the snow/ice surface will enhance the moisture content of the near-surface air. At all sites the relationship between observed and predicted ea appears to be linear when Tg > 0. Stratifying eg observations by Tg reveals that a single linear model is inappropriate for estimating eg (Table 3.5; Figure 3.11). Linear models for the subset with Tg < 0 show lines of best fit near unity and y-intercepts near zero, indicating that ambient and near-surface conditions are highly similar. In contrast, linear vapour pressure models for Tg > 0 exhibit moisture depletion within the KBL when ea > 6.11 hPa, and moisture enhancement when ea < 6.11 hPa, and model slopes range between 0.61 (WM1) to 0.74 (PM3). These effects appear to be greatest at lower elevation stations, where the KBL is expected to be more developed. Coefficients of determination for all vapour pressure models range between 0.39 and 0.82, and RMSE vary between 0.24 and 0.64 hPa.  50  3.3. Results  (a) ea gradient  (b) Reference level (Z = 0) ea  (c) Coefficient of determination, R2  Figure 3.9: Calculated vapour pressure gradients (top), reference-level vapour pressures (ea(Z=0) ), and coefficient of determination (R2 ) for the ambient ea gradient analysis, 2007 51  3.3. Results  (a) Scatterplot  (b) Boxplot  (c) Time-series  Figure 3.10: Observed versus predicted vapour pressures (a), boxplot of residuals by hour (b), and time-series of residuals (c), Place Glacier ridge station, 2008 ablation season  52  Table 3.5: Optimized vapour pressure model parameters (ji ), coefficient of determination (R2 ), and root mean squared error (RMSE) for vapour pressure, subset by observed near-surface temperature. Tg > 0◦ C  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08  1960 2012 2100 2313 2168 2290 1745 2047 2043  j1  j2 (hPa)  0.63 0.67 0.74 0.67 0.61 0.79 0.43 0.72 0.74  2.05 1.89 1.67 1.83 2.22 1.34 2.92 1.68 1.47  n  R2  RMSE (hPa)  3099 6236 3429 3997 3187 1337 1786 1941 2339  0.74 0.75 0.71 0.70 0.74 0.68 0.65 0.79 0.82  0.52 0.57 0.68 0.64 0.51 0.64 0.45 0.50 0.50  j3  j4 (hPa)  1.04 0.83 1.03 0.89 0.63 1.15 1.21 0.91 0.85  -0.42 0.68 0.01 0.49 1.41 -0.69 -1.69 0.45 0.60  n  R2  RMSE (hPa)  330 938 425 1078 231 73 63 414 460  0.62 0.66 0.79 0.77 0.39 0.40 0.73 0.80 0.80  0.42 0.40 0.36 0.37 0.52 0.52 0.24 0.29 0.30  3.3. Results  Site  Z (m)  Tg ≤ 0◦ C  53  3.3. Results  54  Figure 3.11: Global linear models of observed versus ambient near-surface vapour pressures, subset by observed near-surface temperatures (Tg > 0 = blue, Tg < 0 = red)  3.3. Results Time-series of vapour pressure residuals show an equal scatter of positive and negative residuals throughout the ablation seasons at all sites (not shown). Relationships between ambient vapour pressures and boundary layer vapour pressures are thus assumed to be applicable both within and between ablation seasons. As with the temperature regressions, the residuals are strongly autocorrelated, with first-order lag correlations of 0.8 or greater. However, boxplots of vapour pressure residuals by hour of day show no significant departures in residuals related to diurnal cycles (Figure 3.12). Predicted values of eg are less than observed at most sites in the late afternoon, but residuals could not be satisfactorily fitted to sinusoidal functions of time of day. Synoptic setting was again examined for its possible influence on the modelled vapour pressure residuals (Figure 3.12). For the Place Glacier sites, vapour pressure is overpredicted for synoptic types 5 and 7 (Figure 3.8), which are characterized by westerly/northwesterly flow aloft supplying cooler and drier air masses at the glacier AWS elevations. However, in a melt modelling scenario, these synoptic types would be expected to produce relatively small amounts of melt given the low mean temperatures, and the errors in estimated melt from latent heat fluxes would be negliglible.  55  3.3. Results  (a) Hour  (b) Synoptic Type  Figure 3.12: Boxplots of predicted near-surface varpour pressure residuals by (a) hour of day, and (b) synoptic type for three stations, PM2 (top), WM1 (middle), and BM1 (bottom). Some synoptic types are not represented due to differences in observation periods.  56  3.4. Discussion  3.4  Discussion  Observation of a systematic variability in fitted piecewise model coefficients supports the physical basis for estimating KBL temperature and vapour pressures from ambient conditions. Katabatic boundary layer development is a function of both the temperature forcing (Ta − Ts ) that generates katabatic flows, but also of location relative the onset of KBL development. Observed near-surface and estimated ambient temperatures are approximately equal below a critical ambient temperature threshold (T ∗ = k2 ), indicating that KBL development is minimal. Slopes of best-fit lines below T ∗ (k3 ) are near unity at all sites. Coefficient k4 gives the slope of Tg to Ta during periods of expected KBL development, and can be interpreted as a scaling function that represents changes in near-surface temperatures due to sensible heat exchange associated with down-glacier air flow. Consistent with this interpretation, k4 decreases with distance down-glacier for all sites and years, but remains constant at a given site both within and between years. Coefficient k1 is the critical near-surface temperature for KBL development, and from Table 3.2 it can be seen that k1 ≈ k2 · k3 . The robustness of this relation is interpreted as a statistical artifact of the fitting procedure. The relation between k1 , k2 , and k3 is required to estimate KBL effects for independent sites. Transfer functions for estimating near-surface vapour pressures from ambient conditions rely on near-surface temperatures. From the temperature analysis, it can be seen that below T ∗ , Ta ≈ Tg . Thus, near-surface temperatures less than zero correspond to sub-zero ambient air which has low water vapour capacity. At Tg < 0◦ C, moisture gradients between the surface and the overlying ambient air are minimal, and linear model coefficients j3 (slope) and j4 (intercept) are close to one and zero, respectively. With Tg > 0, surface vapour pressures are constrained by the saturation vapour pressure at 0◦ C (6.11 hPa). Warm and moist overlying air sets up a gradient that directs condensation towards the surface, removing vapour from the KBL, while drier air will reverse the gradient and cause evaporation 57  3.5. Conclusions from the surface. Slopes of the best-fit lines (j1 ) are interpreted as a latent heat exchange coefficient that varies between positive and negative based on the moisture content of the overlying ambient air.  3.5  Conclusions  Glacier boundary layer effects on near-surface temperature and vapour pressure are strong, and need to be considered in any attempt to develop distributed fields of these quantities. Simple empirical models that account for cooling through katabatic flow and moisture extraction or addition were developed by examining datasets of observed near-surface and estimated ambient meteorological datasets. Modelling requires only accurate estimates of ambient temperature and vapour pressure gradients, and all model results suggest that KBL development is consistent within and between seasons and also appears to be influenced by upstream flow distance. Near-surface air temperature was successfully modelled using a piecewise approach; above a critical ambient temperature (T ∗ ), buoyancy forcing causes an increase in katabatic wind speeds, resulting in enhanced cooling by sensible heat transfer as air travels over the 0◦ C snow or ice surface. Values for T ∗ calculated at all sites suggest that the onset of katabatic boundary layer development is not a local parameter, and that ambient temperatures at approximately 2900 m can be used as an index for the onset of KBL cooling. Sites with larger upstream flow distances appear to experience a greater degree of cooling, which is consistent with classical boundary layer theory. Near-surface vapour pressures within the KBL (eg ) respond to a moisture gradient which is determined by ambient vapour pressure (ea ) and nearsurface temperature (Tg ), which together controls the saturation vapour pressure at the snow/ice surface (ef ). These physical constraints give rise to a linear model for near-surface vapour pressure that will enhance vapour pressure within the boundary layer through evaporation or sublimation when ef > ea , and will remove moisture from the boundary layer through condensation when ef < ea . When Tg is below below 0◦ C, the gradient is minimal, 58  3.5. Conclusions and ef ≈ eg . The degree of moisture enhancement or removal appears to be greatest at sites with large upstream fetches, suggesting a link between position on the glacier and the level of KBL development. Approaches developed in this study require only ambient temperature and vapour pressure gradients for modelling Tg and eg . In the following chapter, physical and empirical models for estimating wind speeds within the glacier boundary layer are examined, and the third section of this analysis examines relations between calculated scaling functions for Tg , eg , and ug and morphometric parameters derived from glacier DEMs. With scaling functions dependent on glacier morphometry and elevational gradients of Ta and ea , distributed energy balance model errors associated with input climate fields can be addressed and minimized.  59  Chapter 4  Distributing Input Meteorological Fields for Glacier Melt Modelling, Part 2: Wind Speed 4.1  Introduction  Katabatic flows are common features over sloping surfaces with stable boundary layer conditions [Parmhed et al., 2004], and glacier winds are a classic example of well-developed katabatic flow. Katabatic theory and large-scale eddy simulations have demonstrated that wind speeds in thermally driven flows will increase with both the temperature deficit between the surface (Ts ) and the overlying ambient air (∆T = Ta − Ts , where Ta represents ambient conditions), and the distance down-slope [Oerlemans and Grisogono, 2002; Skyllingstad, 2003]. Surface winds may also be dynamically modified by mountainous terrain [Whiteman, 2000], creating a challenge for distributed wind speed modelling in glacierized environments. Due to the intensely stable nature of air above a melting mid-latitude glacier, katabatic flows are thought to be decoupled from ambient or synoptic gradient flows [S¨ oderberg and Parmhed, 2006]. However, both valley/mountain wind systems and channeling of synoptic flows may affect near-surface wind speeds through both enhancement (when flows are in the same direction) and turbulence (when flows are in opposition). Studies of mountain wind systems [Whiteman and Doran, 1993; Weber 60  4.1. Introduction and Kaufmann, 1998] have observed channeling of synoptic winds through (a) downward momentum transport, (b) forced channeling (gradient winds aligned with valley orientation), and (c) pressure-driven channeling (gradient winds blowing cross-valley). Several investigations have studied the interactions of thermal and mechanical winds [McKendry et al., 1986; Heinemann, 1999], while others have demonstrated that gradient winds can be scaled to surface winds in chanelled flows [Ryan, 1977; W¨orlen et al., 1999]. Mesoscale atmospheric models have been used to estimate katabatic flows in Greenland [Bromwich et al., 2001] and snow transport processes in mountainous terrain [Liston and Sturm, 1998; Bernhardt et al., 2009], and numerous glacier-based studies have used physical models to evaluate vertical profiles of katabatic flows [Grisogono and Oerlemans, 2001; Oerlemans and Grisogono, 2002; Parmhed et al., 2004]. For modelling distributed glacier melt, however, near-surface wind speeds are required for modelling turbulent energy fluxes, and representative observations of wind speed are often assumed to be constant over the entire study domain [Munro, 1991; Brock et al., 2000; Hock and Holmgren, 2005]. Anslow et al. [2008] acknowledged the issue of assuming a single wind speed over the entire glacier in a study at South Cascade Glacier, but no attempts have been made to use physical or empirical models to distribute wind speeds over melting glaciers. One particular issue related to wind speed modelling is the variety of scales that need to be considered, as any set of surface observations may include winds ranging from micro-scale eddies to synoptic-scale gusts. The objectives of this study are to use a mixture of physical and empirical methods to estimate near-surface wind speeds (ug ) at automatic weather stations located on glaciers in the southern Coast Mountains of British Columbia (Figure 2.1). Specifically, this study uses the Prandtl model to estimate thermally driven winds, and scales mechanical winds to gradient wind speeds calculated from 700 hPa pressure heights.  61  4.2. Methods  4.2 4.2.1  Methods Data Preparation and Modelling Approach  Study areas, automatic weather station (AWS) locations, and instrument specifications are described in Chapter 2. Reanalysis data from NCEP [National Center for Environmental Prediction, 2009] centred on 50◦ N, 122.5◦ W (Figure 4.1) were used to calculate gradient wind flow strengths, vorticity, and flow direction from gridded geopotential heights [Losleben et al., 2000; Shea and Marshall, 2007] at the 700 hPa level. Using the geostrophic approximation, westerly (U ) and southerly (V ) components of synoptic flow (in m s−1 ) were calculated from geopotential height gradients, and combined to determine total flow strength F and flow direction β, given in degrees. Vorticity (ζ) describes the curvature of the geostrophic wind, with positive (negative) vorticity giving cyclonic (anticyclonic) curvature. Flow indices and vorticity were calculated on a 2.5◦ grid using a centred-mean differencing algorithm given by Losleben et al. [2000] and adjusted for use at 50◦ latitude. Hourly flow indices were estimated using linear interpolations between the 4-times daily NCEP data. To examine the katabatic signal in ug the directional constancy (dc) was calculated for each site: dc =  u2 + v 2 u2 + v 2  (4.1)  where the overbars represent averages calculated for each season and station. Directional constancy ranges from 0 (random) to 1 (constant speed and direction), and it has been used as a metric for examining local wind systems on short timescales [van den Broeke, 1997a]. Histograms of observed wind direction were also constructed to identify variability in the glacier wind direction. Drawing upon the results of the previous chapter (Chapter 3), observed wind speeds at each station were stratified according to estimated Ta and calculated values for T ∗ , the threshold ambient temperature for KBL cooling. Winds speeds at Ta > T ∗ were classified as katabatic winds (uK ), and wind speeds at Ta < T ∗ were classified as non-katabatic (uN K ). Non-katabatic 62  4.2. Methods  Figure 4.1: NCEP grid used in calculation of flow indices. winds were assumed to be mechanically forced by the synoptic scale airflow, and were modelled as a function of the geostrophic wind component aligned parallel to the valley axes. One factor that may complicate the relation between wind speed and temperature is a temporal lag that has been previously observed in glacier settings [Heinemann, 1999; Pellicciotti et al., 2008]. Mean hourly temperatures and wind speeds were calculated for katabatic (Ta > T ∗ ) flows in each glacier AWS dataset, and plotted by hour of day to illustrate the observed hysteresis. Cross-correlation analyses were conducted to identify the lag associated with maximum correlation between ambient temperature and wind speed. Katabatic flows were then stratified again using a critical threshold value  63  4.2. Methods  Figure 4.2: Logic for glacier wind speed modelling. Refer to the text for symbol descriptions. for gradient flows (V ∗ ) that was determined through a global optimization procedure, where a range of values for V ∗ were applied to each site to split katabatic winds into purely thermal flows (uK ), and channelled flows under katabatic conditions (uKc ). The Prandtl model for katabatic flows (Oerlemans and Grisogono [2002]; described below) was then tuned to each station for observations where Ta > T ∗ and V < V ∗ , and a global empirical model was developed to scale observed wind speeds to gradient flows for the remaining observations where T > T ∗ and V > V ∗ . The optimal value for V ∗ was determined by examining mean absolute error (MAE), mean bias error (MBE) and root mean squared error (RMSE) calculated from modelled and observed wind speeds at each station: MAE = MBE = RMSE =  1 n 1 n 1 n  |(yˆi − yi )|  (4.2)  (yˆi − yi )  (4.3)  (yˆi − yi )2  (4.4)  where yˆi is the modelled value, yi is the observed wind speed, and n is the number of observations.  4.2.2  Katabatic Wind Speed Modelling  The Prandtl model for estimating profiles of wind speed and temperature in pure katabatic flows has been used previously over melting glaciers [Griso64  4.2. Methods gono and Oerlemans, 2001; Oerlemans and Grisogono, 2002; Parmhed et al., 2004]. The Prandtl model is based on the assumption that buoyancy forcing (from the temperature deficit) and friction are the only terms that determine the downslope momentum budget. In a z-coordinate system (z is the height perpendicular to the glacier surface) the governing equations of the Prandtl model are formulated by Oerlemans and Grisogono [2002] as d dθ KH dz dz θT d du g sin(α) + KM T0 dz dz γ u sin(α) −  = 0  (4.5)  = 0  (4.6)  where γ is a background potential temperature lapse rate, α is the surface slope, u is the downslope wind component, and θT is the temperature deficit. The surface temperature T0 is assumed to equal 273.15 K, KM and KH are the eddy diffusivities for momentum and heat, respectively, and g is the acceleration due to gravity (9.8 m s−2 ). These equations can be solved to estimate the temperature deficit θ and wind speed u using constant eddy diffusivities and simple boundary conditions, with the solutions reading: θ(z) = C e−z/λ cos(z/λ) u(z) = −C µ e  −z/λ  sin(z λ)  (4.7) (4.8)  where length scale (λ) and momentum scale (µ) are given by λ =  4 T0 KM KH γ g sin2 (α)  µ =  g KH T0 γ KM  1/4  (4.9)  1/2  .  (4.10)  The surface temperature deficit C = Ta − T0 , where Ta is the ambient temperature. From these solutions, the height of the wind speed maximum is given by zm =  π λ 4  (4.11) 65  4.2. Methods and the maximum wind speed at zm is um = −C µ e−π/4 sin(π/4) = −a1 Cµ  (4.12)  where Oerlemans [1998] found a1 = 0.322. Surface slope, lapse rate, and surface temperature deficits can all be prescribed from DEMs and ambient climate data. However, specifying diffusivities for heat and momentum is problematic, and the parameter choice holds large consequences for the results. Figure 4.3 illustrates three separate wind and temperature profiles modelled with Equations [4.7] to [4.10] and by setting C = 10, α = 6, and varying KH and KM between 0.01 and 1. It has been noted that the assumption of constant eddy diffusivities with height is incorrect, as eddy diffusivities should decrease towards the surface [Oerlemans and Grisogono, 2002]. An alternative proposed by Oerlemans [1998] and Oerlemans and Grisogono [2002] uses flow-dependent parameterizations of KH and KM to match observed temperature and wind-speed profiles.  Figure 4.3: Theoretical Prandtl profiles of wind speed (solid) and temperature (dashed) for different eddy diffusivities, with surface temperature deficit C = −10 K, surface slope α = 5◦ , and lapse rate γ = −0.006 K/m.  66  4.2. Methods In this study there is no wind speed profile information available to tune coefficients in the Prandtl model. It should be noted that an attempt was made to collect profiles of temperature, wind speed, and wind direction above the glacier surface in 2008, but the balloons and instrumentation were lost in a severe wind-storm. Instead, the assumption of constant diffusivities with height was accepted, and the Prandtl model was tuned to wind speed data observed at 2 m. Guidance for initial KH and KM estimates can be found from previous boundary layer studies in katabatic flow. Munro and Davies [1977] calculated bulk gradient Richardson numbers (Rig ) from profile data obtained at Peyto Glacier, and work on nocturnal katabatic flows by Monti et al. [2002] demonstrated that KH and KM exhibit systematic variations with Rig . Using the empirical formulas derived in Monti et al. [2002]: KH KM  (−0.045)  = 0.07 Rig =  (0.22) 0.45 Rig  (4.13) (4.14)  initial estimates of KBL KH and KM were obtained using Rig values given in Munro and Davies [1977]. Eddy diffusivities calculated for the range of Rig measured at Peyto Glacier varied between 0.18 to 0.39 m s−2 . Values of KH = KM = 0.2 m s−2 were thus used as initial estimates in the current study, and the Prandtl model was fit to observations of uK . Maximum cross-correlations between Ta and uK were found at a lag of 3 hours (see below), and wind speed observations were thus offset prior to model fitting. Note that with KH = KM = 0.2 m s−2 , C = 10 K, α = 5◦ , and γ = −0.006 K/m, the height of the maximum wind speed given by Equation 4.12 is located at 13.9 m, which is within the range of wind maximum heights observed previously at mid-latitude glaciers [Munro and Davies, 1977; van den Broeke, 1997b; Oerlemans and Grisogono, 2002]. With the fitted eddy diffusivities, uK at time i was modelled using Ta observed at time i − kmax , where kmax is the lag associated with maximum cross-correlation between Ta and uK . 67  4.3. Results  4.2.3  Non-katabatic Wind Speed Models  For channelled flows during expected KBL development (uKc ; Ta > T ∗ and V > V ∗ ) an exponential function was fit to all uKc observations and absolute southerly gradient wind speed (|V |) calculated from the 700 hPa pressure heights: uKc = u6 exp(u7 · V ) + u8  (4.15)  where ui are fitted coeffficients. Non-katabatic winds at each site (Ta < T ∗ ) were scaled to gradient winds by fitting a separate function for all uN K and absolute gradient wind speed. At Place and Weart Glaciers (north-south orientation), absolute southerly wind speeds were used, and at Bridge Glacier (east-west orientation), absolute westerly wind speeds were used:  4.3 4.3.1  Place, Weart : uN K = m1 exp(m2 |V |) + m3  (4.16)  Bridge : uN K = m4 exp(m5 |U |)  (4.17)  Results Glacier Wind Characterization  As a result of both katabatic flows and channelling, wind directions on melting glaciers are dominantly down-glacier [Ohata, 1989; Obleitner, 1994; Greuell and B¨ ohm, 1998; Strasser et al., 2004]. Histograms of wind directions observed on Place, Weart, and Bridge Glaciers illustrate the consistency of down-glacier flow direction (Figure 4.4), and support the framework of katabatic flow that this study is based upon. At all sites the dominant wind direction is more or less down-glacier, though exceptions can be seen at PM3 and PM4. Wind directions at PM3 are bi-modal as a consequence of its location near the flow divide (Figure 2.2). At PM4 and WM2 lower values of dc indicate that the katabatic boundary layer is less well developed, resulting in more frequent wind reversals as upslope or gradient winds are easily entrained in the surface flow.  68  4.3. Results Calculated seasonal values of dc and mean observed wind speeds, given in Table 4.1, highlight the differences in wind speed between sites and between glaciers. Directional constancy decreases with elevation, while wind speeds are generally greatest at lower elevations. The greatest mean wind speeds were observed at Bridge Glacier, which is the largest glacier in the study, and this station is also at the lowest elevation of all the on-ice AWS. Lateral variability in wind speed was examined at Place Glacier in 2008 (Figure 2.2), and wind speeds observed at PM108, PM2, and PM308 illustrate that position plays a role in determining wind speed. Higher wind speeds are observed at PM2 (in the middle of the glacier) and PM308, which is located at the base of the large ramp leading to the upper accumulation basin. In contrast, PM108 is located at the edge of the glacier with no major upstream areas, and observed wind speeds are low in this location. The relation between observed wind speeds and ambient temperatures (Figure 4.5) was found to be positive at all glacier AWS sites. Particularly above the critical temperature thresholds (T ∗ ) established previously for KBL development (Chapter 3), wind speeds appear to be driven by ambient temperatures. Similar relations have been demonstrated by Oerlemans [1998]; Oerlemans and Grisogono [2002] for alpine glaciers. As presented in Figure 4.5, downglacier winds dominate at sites where KBL development is expected to be greatest (PM1, PM2, WM1, BM1). Below T ∗ , observed wind speeds are uncorrelated with Ta , and measured surface wind speeds likely reflect channelling of gradient winds, as katabatic flow development will be negligible.  69  4.3. Results  Table 4.1: Directional constancy (dc) and mean observed wind speeds (ug , in m s−1 ) at glacier AWS. Site  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM108 PM308  2006  2007  2008  dc  ug  dc  ug  dc  ug  0.77 0.69 0.59 0.48 -  3.11 2.80 2.72 2.77 -  0.79 0.70 0.52 0.42 0.84 0.49 -  3.15 3.15 3.29 2.72 3.42 2.45 -  0.62 0.44 0.81 0.84 0.72 0.72  3.15 3.28 3.49 3.80 2.69 3.28  70  4.3. Results  71  Figure 4.4: Distribution of observed wind directions at glacier AWS. Full lines indicating downglacier flow directions (see study area figures).  4.3. Results  72  Figure 4.5: Scatterplots of observed hourly wind speeds versus estimated ambient temperatures, subset by downglacier (black) and upglacier (red) directions. Dashed line indicates T ∗ calculated from the temperature analyses  4.3. Results  4.3.2  Surface Winds in Katabatic Flow  Wind Speed - Temperature Hysteresis Evidence of hysteresis in the relation between wind speeds and temperature may account for the wind speed variability observed in Figure 4.5, particularly at higher ambient temperatures. As shown in Figure 4.6, observed mean wind speeds depend not only on ambient air temperature, but also on the time of day. At all sites, wind speeds demonstrate early morning minima when Ta is increasing. Likewise, maximum wind speeds at all sites occur late in the day when Ta is declining. This hysteresis could account for a large portion of the noise in the observed relation between ambient temperatures and observed wind speeds, when Ta > T ∗ . Figure 4.7 indicates that peak correlation between Ta and ug occurs at a lag of between 2 and 4 hours, and was significant at all sites during continuous periods of record. Observed maximum cross-correlations are consistent between sites, between years, and also within years, suggesting that the hysteresis is a robust feature that can be incorporated into glacier wind speed models.  73  4.3. Results  74  Figure 4.6: Mean hourly wind speeds versus mean hourly ambient temperatures by time of day, calculated for all observations where Ta > T ∗ . Blue is 01:00, and red is 24:00.  4.3. Results  (a) PM2, 19 May - 27 Jul 2007  (b) PM2, 31 Jul- 13 Sep 2007  (c) WM1, 2008  (d) BM1, 2008  Figure 4.7: Cross-correlations between ambient temperature and wind speed. Dashed lines show confidence limits for a significance level of 0.05. Katabatic Flow Optimization Uisng the optimization method, model errors (Figure 4.8) calculated for pure katabatic flow, (uK ), channelled katabatic flow (uKc ) and for the full katabatic model (uK +uKc ) indicate that the optimum threshold for gradient wind channeling during katabatic flows occurs at a southerly flow speed of approximately 13 m s−1 . Root-mean squared errors for uK are minimized at V ∗ = 11 − 13 m s−1 , mean MBE for uK remains unchanged at all values of V ∗ , and MAE for uK is optimized at V ∗ = 12 − 13 m s−1 . With V ∗ = 13 m s−1 , fitted eddy diffusivity coefficients ranged from 0.14 to 0.35 m s−2 (Table 4.2). Root mean squared errors for purely katabatic flows ranged from 0.65 to 1.61 m s−1 , and for all sites the modelled flows underpredicted katabatic wind speeds, with MBE ranging between -3 and 18% of the observed means. Mean absolute errors for katabatic flows ranged between 11 and 26% of observed means, though MAE at BM1 was 44% of the observed mean. At southerly flow strengths greater than 13 m s−1 , surface wind speeds at Place and Weart Glaciers are modelled with an exponential function (Table 75  4.3. Results 4.3, Figure 4.9). The optimized value of V ∗ = 13 m s−1 is visually consistent with the observed wind speed data, and with R2 = 0.22, surface wind speeds are modelled with a MAE of 0.92 m s−1 , or approximately 25% of the mean.  (a) RMSE  (b) MBE  (c) MAE  Figure 4.8: Weighted mean RMSE, MBE, and MAE obtained for the katabatic flow optimization, calculated for Place and Weart Glacier AWS. Errors for the full katabatic model (uK + uKc ), pure katabatic flow only (uK ) and channeling under katabatic flow (uKc ) are given for each metric. Weights were recalculated for each value of V ∗ from station observations.  76  4.3. Results  Figure 4.9: Global model for estimating uKc from southerly flow strengths V.  Table 4.2: Model errors and optimized eddy diffusivity coefficients for heat (KH ) and momentum (KM ) obtained from fitting observed wind speeds to the Prandtl model for Ta > T ∗ . All station models incorporated a lag of 3 h, γ = −0.0057 K/m, and local slopes. At Place and Weart Glacier stations, observations where V > 13 m s−1 were withheld from the model fitting. Site  KH (m s−2 )  KM (m s−2 )  RMSE (m s−1 )  MBE (m s−1 ) (%)  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08  0.26 0.19 0.15 0.18 0.23 0.24 0.33 0.14 0.22  0.17 0.20 0.22 0.21 0.19 0.19 0.13 0.22 0.19  0.65 0.81 0.90 1.32 1.14 1.09 1.51 0.75 1.01  -0.12 (-4) -0.32 (-11) -0.44 (-16) -0.47 (-16) -0.61 (-18) -0.07 (-3) -0.53 (-13) -0.29 (-12) -0.49 (-15)  MAE (m s−1 ) (%) 0.37 0.50 0.54 0.74 0.73 0.57 1.77 0.46 0.68  (11) (17) (20) (26) (21) (22) (44) (19) (21)  n 1301 2662 1616 2113 1585 443 1087 865 1027  77  4.3. Results  Table 4.3: Surface wind speed models for channeled katabatic (uKc ) flows. The model (Equation [4.15]) was developed using hourly flow indices linearly interpolated from 6-hourly NCEP data. Site  Model  R2  RMSE m s−1  MBE m s−1  MAE m s−1  Place, Weart (n = 3528)  u6 = 0.042 ± 0.026 u7 = 0.135 ± 0.019 u8 = 3.09 ± 0.146  0.22  1.18  0.00  0.92  78  4.3. Results  4.3.3  Surface Winds in Non-Katabatic Flow  For non-katabatic situations (uN K ; Ta < T ∗ ), observed wind speeds at Place and Weart Glaciers (both oriented in north-south valleys) were modelled as an exponential function of |V | (Table 4.4; Figure 4.10). At Bridge Glacier (oriented east-west), westerly flow strengths (U ) were used to model nonkatabatic flows (Table 4.4; Figure 4.10). For uN K models using southerly flow and westerly flow, the coefficients of determination (R2 ) are low (0.22 and 0.12, respectively), and the scatter is large. Table 4.4: Surface wind speed model coefficients and modelled errors for channeled non-katabatic (uN K ) flows. The models (Equations [4.16] and [4.17]) were developed using hourly flow indices linearly interpolated from 6-hourly NCEP data. Site  Coefficients  R2  RMSE (m s−1 )  MBE (m s−1 )  MAE (m s−1 )  Place, Weart (n = 14221)  m1 = 0.34 ± 0.06 m2 = 0.09 ± 0.01 m3 = 2.11 ± 0.09  0.22  1.35  0.00  1.04  Bridge (n = 762)  m4 = 2.66 ± 0.17 m5 = 0.01 ± 0.002  0.13  1.44  -0.03  1.13  4.3.4  Modelled Surface Wind Speeds  Surface wind speeds modelled at each site using a combination of empirical and physical models capture both diurnal cycles and the effects of largescale atmospheric systems. Ten-day samples of modelled and observed wind speeds are given for PM2, WM1, and BM1 in Figures 4.11, 4.12 and 4.13. Wind speeds during periods of well-developed katabatic flow (Ta > T ∗ , e.g. Figure 4.11b, Figure 4.12a) are accurately modelled in terms of both wind speed magnitude and diurnal variability at Place and Weart Glaciers. At Bridge Glacier, katabatic flows observed early in the ablation season (Figure 4.13a) are characterized by consistent nighttime winds and rapid transitions to and from daytime maximum wind speeds. Such behaviour is 79  4.3. Results  (a) Place and Weart  (b) Bridge  Figure 4.10: Global models for estimating surface wind speeds for nonkatabatic flows at (a) Place and Weart Glaciers from southerly flow speeds and at (b) Bridge Glacier from westerly flow speeds. Solid lines are model fits, dashed lines are 95% prediction bounds not captured by the Prandtl model. Later in the ablation season, a different surface wind regime is observed at BM1, and the modelled wind speeds are more accurate. Overall, katabatic wind speeds are underestimated by the Prandtl model approach (Table 4.2), though the inclusion of a 3-hour lag between Ta and uK improves the temporal fit. Empirical models for estimating surface winds using either southerly or westerly gradient flows (Tables 4.4 and 4.3) assume that direct channelling is the dominant mechanism for transmitting upper atmospheric momentum to the glacier surface, and that channelled wind speeds are uniform over the entire glacier. At Place and Weart Glaciers, broad peaks in southerly flow speeds correspond with high surface wind speeds during both katabatic and non-katabatic flows (Figure 4.11a, Figure 4.12b). Wind events of shorter duration (several hours or less) are not well-modelled, and this is presumably due to the spatial and temporal limitations of the NCEP data. At Bridge Glacier, uN K were scaled to westerly flow strengths, though a weak model fit (Table 4.4) limits the skill of this approach. Analysis of wind speed residuals (observed minus predicted) for three representative sites (Figures 4.14 and 4.15) demonstrates that all three wind  80  4.3. Results speed models are equally unbiased, both within and between seasons and sites. Autocorrelation is evident in the wind speed residuals. Statistics of mean and standard deviation for all stations, subset by uN K , uK , and uKc are provided in Table 4.5. While all three models appear to underestimate observed wind speeds, the physically modelled katabatic flows are most closely matched in both their mean and standard deviation. Modelled winds during channelling events are underestimated and show substantially lower standard deviations than observed.  81  4.3. Results  Figure 4.11: Observed and modelled wind speeds (top), estimated ambient temperatures and T ∗ (dashed line, middle panel), and absolute southerly flow speeds (bottom) for two ten-day periods at PM2 in 2007 and 2008.  82  4.3. Results  Figure 4.12: Observed and modelled wind speeds (top), estimated ambient temperatures and T ∗ (dashed line, middle panel), and absolute southerly flow speeds (bottom) for two ten-day periods, WM1.  83  4.3. Results  Figure 4.13: Observed and modelled wind speeds (top), estimated ambient temperatures and T ∗ (dashed line, middle panel), and absolute southerly flow speeds (bottom) for two ten-day periods, BM1.  84  4.3. Results  (a)  (b)  (c)  Figure 4.14: Wind speed residuals (observed - expected) for PM2 in (a) 2006 , (b) 2007, and (c) 2008.  85  4.3. Results  (a)  (b)  (c)  Figure 4.15: Wind speed residuals (observed - expected) for WM1 in (a) 2007 and (b) 2008, and for BM1 2008 (c).  86  Table 4.5: Mean (x, m s−1 ) and standard deviation (σ, m s−1 ) in observed and modelled uN K , uK , and uKc . Observed uK  uN K  uKc  Modelled uK  uN K  uKc  x  σ  x  σ  x  σ  x  σ  x  σ  x  σ  PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM1.08 PM3.08  2.92 3.00 3.23 3.17 3.17 2.51 3.52 2.72 3.10  1.30 1.52 1.87 1.55 1.56 1.31 1.53 1.30 1.48  3.28 2.97 2.73 2.88 3.46 2.55 4.00 2.47 3.24  0.86 0.79 0.83 1.28 1.08 1.19 1.34 0.69 1.06  3.92 3.87 4.03 3.02 4.17 2.36 3.66 4.51  0.83 1.20 1.53 1.10 1.10 1.08 1.18 1.27  2.83 2.82 2.94 2.75 2.88 3.06 3.50 2.69 2.73  0.74 0.70 0.84 0.70 0.89 0.99 0.58 0.64 0.71  3.05 2.49 2.02 2.04 2.59 2.40 3.47 2.00 2.56  0.94 0.91 0.72 0.82 1.04 0.99 1.24 0.76 1.05  3.58 3.74 3.63 3.74 3.75 3.68 3.93 4.00  0.26 0.66 0.48 0.67 0.69 0.55 0.65 0.90  4.3. Results  Site  87  4.3. Results  4.3.5  Comparison of Empirical-Physical and Index Station Methods for Modelling Wind Speed  Table 4.6 gives model errors calculated for the full wind speed datasets at each site using a) the empirical-physical wind speed models described above and b) the wind speed data from PM2. Wind speeds at the Place Glacier sites are better modelled using the PM2 data in nearly all cases. The use of a single index station captures wind speed variability on hourly, daily, and synoptic scales, and reflects the local wind characteristics of the glacier. What is surprising is that wind speeds at Weart Glacier, a site 30 km away, are more skillfully modelled using the PM2 data than a locally tuned model. Wind speeds at Bridge Glacier are modelled with slightly greater skill using the physical and empirical wind speed model, suggesting that the methods presented above are reasonable for large-scale distributed modelling. Table 4.6: Errors in modelled wind speed using a) approaches developed in this study, and b) wind speeds observed at a single index station (PM2). All errors given in units of m s−1 , and MAE is alse expressed as a percentage of the observed mean ug . Site PM1 PM2 PM3 PM4 WM1 WM2 BM1 PM108 PM308  Empirical-Physical Models RMSE MBE MAE (%) 0.96 1.09 1.26 1.66 1.36 1.54 1.65 1.02 1.27  -0.15 -0.29 -0.31 -0.48 -0.57 -0.53 -0.32 -0.17 -0.49  0.74 0.83 0.94 1.26 1.03 1.25 1.35 0.79 0.97  (23) (27) (30) (41) (30) (51) (36) (29) (29)  Index Station RMSE MBE MAE (%) 0.61 0.83 1.35 1.17 1.45 1.70 0.62 0.56  -0.19 -0.11 -0.02 -0.37 0.52 -0.51 0.29 -0.11  0.48 (15) 0.64 (20) 1.05 (33) 0.85 (28) 1.15 (47) 1.25 (33) 0.44 (16) 0.39 (12)  88  4.4. Discussion  4.4  Discussion  Surface wind speeds modelled at three glaciers using a combined physical and empirical model give reasonable estimates of surface wind speeds and wind speed variability, particularly during periods of katabatic flow where sensible heat fluxes are expected to be greatest. It is important to recognize, however, that application of wind speeds observed at an index station to other sites will provides better estimates of surface wind speeds than using the empirically derived model presented in this research. In general, the Prandtl model underestimates katabatic wind speeds with mean bias errors between -4 and -18% of the observed mean. Mean absolute errors for modelled uK range between 11 and 44% of the observed site mean, with the greatest errors observed at Bridge Glacier. The nature of katabatic flows at Bridge Glacier and the particularly poor performance of the Prandt model at BM1 deserves further inspection. A primary cause may be the location of the station towards the glacier margins (Figure 2.4), and at the confluence of two separate arms of the glacier. This location may be affected by turbulence created by frictional effects of katabatic flow interacting with the topography at the glacier margins and by turbulence created as the separate katabatic flows converge. Furthermore, turbulence created by return anabatic winds above the katabatic layer may be sufficient to disrupt the katabatic flow. Eddy diffusivity coefficients fitted during the model optimization procedure were used to construct typical profiles of wind speed and temperature deficit to qualitatively examine differences in boundary layer structure between and sites. Holding lapse rate and temperature deficit (C = 10 K) constant between sites, local slopes and fitted eddy diffusivity coefficients were adjusted for each site. As illustrated in Figure 4.16a, results indicate that Bridge Glacier will experience significantly greater wind speeds and a deeper katabatic layer than either Place or Weart Glacier. At Place Glacier (Figure 4.16b), the greatest wind speeds and deepest boundary layer depths are observed at PM1, where boundary layer development is expected to be greatest. Katabatic winds enable continued cooling through sensible 89  4.4. Discussion heat exchange, which further reinforces the buoyancy forcing. Greater wind speeds resulting from the continued cooling should produce stronger thermal and moisture effects at sites where flow distances are greatest. The relation between fitted eddy diffusivities and flow length will be examined in the following chapter.  90  4.4. Discussion  (a) Between Glaciers  (b) Within Glacier  Figure 4.16: Comparison of vertical profiles of wind speed (a) between glaciers, and (b) within glacier. Profiles were calculated using local slopes, γ = −0.0057 K/m, temperature deficit C = 10 K, and estimated ambient temperatures, and eddy diffusivity coefficients obtained from optimization routine.  91  4.5. Conclusion  4.5  Conclusion  A diurnal hysteresis between ambient temperatures (Ta ) and wind speeds (ug ) was observed from meteorological observations within the katabatic boundary layer, and the lag between peak Ta and ug varies from 2-4 hours. Observed wind directions and calculated directional constancies indicate that the decoupling between katabatic winds and overlying valley or gradient winds is strongest at sites where the boundary layer is most developed. Three models for estimating surface wind speeds were developed by partitioning the observed data into (a) non-katabatic flows (uN K ), where Ta < T ∗ , (b) pure katabatic flows (uK ) where Ta > T ∗ and V < V ∗ , and (c) channelled flows under katabatic flow (uKc ), where Ta > T ∗ and V > V ∗ . The value of V ∗ was determined through an optimization procedure which sought to minimize the errors in modelled wind speeds for Ta > T ∗ and distinguish between katabatic flows and channelled gradient winds. Global exponential functions were used to describe the relation between channelled flows (uN K , uKc ) and gradient wind flow speeds V and U , and the Prandtl model was tuned to uK observed at individual sites. Mean absolute errors for fully modelled wind speeds ranged between 23 and 51% of observed means, though MAE for uK ranged between 11 and 44%, indicating that fitting the Prandtl model to 2 m wind speeds gives reasonable estimates of wind speed during periods where high fluxes of turbulent energy are expected. Assuming constant wind speeds from a single representative station offers the best model for surface winds in glacial environments, but the approach developed in this study provides a reasonable method for estimating wind speeds in unmonitored locations using only ambient temperatures, local slopes, and upper-air reanalysis data. To assess the potential for estimating wind speed, temperature, and vapour pressure fields across large regions, the following chapter takes results from the boundary layer meterological analyses and examines whether model coefficients that describe boundary layer Tg , eg , and ug can be estimated from morphometric properties of the glaciers themselves.  92  Chapter 5  Distributing Input Meteorological Fields for Glacier Melt Modelling, Part 3: Topographic Indices and Transfer Functions 5.1  Introduction  The results presented in Chapters 3 and 4 indicate that near-surface air temperature (Tg ), vapour pressure (eg ) and wind speed (ug ) can be modelled using predicted ambient air temperatures and vapour pressures in conjunction with simple empirical and quasi-physical models. Because of the dominantly downglacier orientation of glacier winds during both katabatic and non-katabatic conditions (this study; see also Stenning et al. [1981]; van den Broeke [1997b]; Greuell et al. [1997]; Strasser et al. [2004]), there may be systematic down-glacier variations in the meteorological variables that are related to variables such as upslope flow path length, which can be estimated from Digital Elevation Models (DEMs). Winds observed on glaciers are predominantly downglacier as a result of both thermal (katabatic) and mechanical (gradient wind channelling) flows. In a setting where wind direction is near-constant, a model of boundary layer development that depends on flow distance can be applied. Development of the katabatic boundary layer (KBL) will thus depend on the morphometry of the glacier 93  5.1. Introduction and sustained down-glacier flow. Many previous studies rely on topographic information to distribute climatic or hydrologic variables. Climate interpolation routines such as PRISM [Daly et al., 2002], snow redistribution models [Liston and Sturm, 1998], and cold-air ponding studies [Chung et al., 2006] all incorporate topographic information, while hydrologic models make extensive use of topography to describe drainage characteristics [Quinn et al., 1991] or basin residence time [McGuire et al., 2005]. In all cases, terrain parameters derived from digital elevation data demonstrate predictive capacities for distributing various physical properties throughout a study area. This study seeks to employ a similar approach for distributing model coefficients that describe the katabatic boundary layer effects on Tg , eg , and ug over melting glaciers. Empirical model coefficients obtained from nine stations operating over one to three ablation seasons at three different glaciers appear to vary systematically with position on the glacier, suggesting that the model coefficients can be predicted from DEM-derived topographic indices. Properties of melting glaciers suggest a model of katabatic boundary layer development controlled by primarily by flow distance [Oke, 1987]. Surface conditions of glaciers are well-constrained, with surface temperatures (Ts ) and vapour pressures (ef ) equal to 0◦ C and 6.11 hPa, respectively, when ambient temperatures are greater than 0◦ C. Furthermore, glacier surfaces are generally smooth, with no obstacles or vegetation to impede flows or affect boundary layer development, and the two possible surface types (snow or ice) appear to have little effect on the skill with which boundary layer effects are estimated. The objectives of this research are to provide an approach for estimating meteorological fields of Tg , eg , and ug on melting mid-latitude glaciers using topographic indices and regional climate data. Energy balance modelling studies on melting glaciers currently rely on automated weather observations made on the glacier surface, and this study seeks to provide a method for estimating meteorological fields on the overwhelming majority of midlatitude glaciers where local data are unavailable.  94  5.2. Methods  5.2  Methods  Digital elevation models of three study sites in the southern Coast Mountains of British Columbia were obtained from airborne Light Detection and Ranging (LiDAR) surveys in 2006 (C. Hopkinson and M. Demuth, pers. comm.). Glacier boundaries were digitized from hillshaded images of the LiDAR data, and updated using an ASTER scene from 2007 (B. Menounos, pers. comm.). Glacier DEMs were extracted from the elevation data using the digitized boundaries (Figures 2.2, 2.3, and 2.4). For Place and Weart Glaciers, a DEM resolution of 25 m was used, and a 50 m resolution was used for Bridge Glacier. Hydrological processing tools in the SAGA GIS software package (http: //www.saga-gis.org) and custom functions designed in Matlab were used to calculate topographic indices from the glacier DEMs. The multiple flow direction algorithm developed by Quinn et al. [1991] was used for calculating flow path length (FPL), catchment height (CH), and catchment area (CA) in SAGA. Convergence index (CI) and slope (Sl) were also calculated in SAGA. Distance to edge (D2E) was calculated in Matlab with a function that finds the minimum distance to the edge, searched in 10◦ increments. Figure 5.1 gives examples of the six morphometric indices calculated for Place Glacier. Nine meteorological stations were operated between May and October at three different glaciers between 2006 and 2008 (Figure 2.1). For each site, all collected data were compiled to produce global datasets, which were used in the empirical analyses. Near-surface temperatures at each site were estimated from Ta using a piecewise analysis with four fitted terms (Figure 3.1). Parameters k1 and k2 describe near-surface and ambient temperature thresholds for the onset of boundary layer modification, respectively, and k3 and k4 describe the ratio of near-surface to ambient temperatures without and with boundary layer modification, respectively. Near-surface vapour pressures at each site were modelled using separate linear models defined by near-surface temperatures. Parameters j1 and j2 are slopes and intercepts for estimating eg when Tg > 0◦ C, and j3 and j4 are slope and intercepts 95  5.2. Methods for estimating eg when Tg < 0◦ C. Wind speeds were estimated by tuning the eddy diffusivity coefficients for heat (KH ) and momentum (KM ) in the Prandtl model to 2 m wind speeds observed during episodes of katabatic flow. Table 5.1 summarizes the parameters obtained for each site. Using the station coordinates, topographic indices for each site were extracted using bilinear interpolation in SAGA. For all models, full models (all stations) were constructed first to examine functional form and significance. Reduced models incorporating four stations (PM2, PM4, WM1, and BM1) were then developed, and transfer functions for the remaining independent stations were estimated. At the withheld sites (PM1, PM108, PM3, PM308, and WM2), Tg , eg , and ug were calculated from FPL, transfer functions, and ambient climate data. Mean absolute errors (MAE), mean bias errors (MBE), and root mean squared errors (RMSE) were calculated to assess model performance (Equations [4.2] to [4.4]).  96  5.2. Methods  (a) Flow path length  (b) Catchment Area  (c) Catchment Height  (d) Slope  (e) Convergence index  (f) Distance to edge  Figure 5.1: Morphometric parameters calculated for Place Glacier.  97  Table 5.1: Fitted coefficients for estimating Tg , eg , and ug , obtained from glacier boundary layer analyses. Coefficients k3 , k4 , j1 , and j3 are unitless. Tg  eg  ug  k4  j1  j2 (hPa)  j3  j4 (hPa)  KH (m s−2 )  KM (m s−2 )  PM1 PM2 PM3 PM4 PM108 PM308 WM1 WM2 BM1  5.51 4.76 4.55 3.89 4.82 4.82 3.7 6.75 4.08  6.48 5.6 5.12 4.29 5.35 5.12 4.9 8.37 6.08  0.85 0.89 0.95 0.95 0.91 0.93 0.83 0.83 0.63  0.43 0.48 0.69 0.72 0.64 0.62 0.35 0.6 0.32  0.63 0.67 0.74 0.66 0.72 0.74 0.61 0.79 0.43  2.05 1.89 1.67 1.82 1.68 1.47 2.22 1.34 2.92  1.04 0.83 1.03 0.89 0.91 0.85 0.63 1.15 1.21  -0.42 0.68 0.01 0.49 0.45 0.61 1.41 -0.69 -1.69  0.27 0.22 0.18 0.19 0.16 0.24 0.26 0.25 0.33  0.17 0.19 0.21 0.21 0.22 0.19 0.18 0.18 0.14  5.2. Methods  k2 ◦ ( C)  k3  Site  k1 ◦ ( C)  98  5.2. Methods  5.2.1  Temperature  Near-surface temperatures (Tg ) for site x at time t were predicted from ambient temperatures (Ta ) using a four-term piecewise model that hinged on the critical temperature threshold for katabatic boundary layer effects: Tˆg (x, t) =  k1 + k4 (Ta − k2 ), Ta > k2 k1 + k3 (k2 − Ta ), Ta < k2  (5.1)  Parameters k1 and k2 represent critical temperatures for KBL development, and it has been shown that that k1 ≈ k2 · k3 (Table 5.1). Coefficients k3 and k4 describe the strength of glacier cooling effect below and above k2 , respectively. Parameter k2 was modelled as a linear function of elevation Z (in m asl): k2 = β1 + β2 Z  (5.2)  where βx are fitted coefficients. The coefficients k3 and k4 were estimated as exponential functions of flow path length (FPL), weighted by the number of station-days: k3 = β3 exp (β4 FPL)  (5.3)  k4 = β5 + β6 exp (β7 FPL)  (5.4)  Hourly near-surface temperatures were estimated for the test sites from flow path lengths and ambient air temperatures estimated from six local climate stations (Chapter 3).  5.2.2  Vapour Pressure  Previous empirical analyses (Chapter 3) indicate that katabatic boundary layer vapour pressures (eg ) are a function of both ambient vapour pressures (ea ), and near-surface temperatures (Tg ), which dictate the vapour pressure at the surface (ef ). Two separate linear functions were used to describe  99  5.2. Methods near-surface vapour pressures: eˆg (x, t) =  j1 ea + j2 , Tg > 0 j3 ea + j4 , Tg < 0  (5.5)  Parameters j1 and j2 represent slopes and intercepts describing the degree of moisture enhancement or removal at a given site under katabatic flow regimes, and these coefficients were modelled as exponential and logarithmic functions of flow path length (FPL): j1 = β8 exp (β9 FPL)  (5.6)  j2 = β10 + β11 ln (FPL)  (5.7)  Parameters j3 and j4 were calculated as weighted means for both the full and the reduced datasets, as no significant relations were observed between j3 , j4 , and calculated topographic indices. As near-surface vapour pressures are expected to be equal to ambient vapour pressures, it is hypothesized that j3 ≈ 1 and j4 = 0. Near-surface vapour pressures were predicted for each test site using ambient vapour pressures, ambient temperatures, and coefficients estimated from FPL (Equations [5.2] to [5.4] and [5.6] to [5.7]).  5.2.3  Wind Speed  Eddy diffusivities for heat (KH ) and momentum (KM ) were derived from an optimization routine that identified the onset of mechanically driven flows (V ∗ ) in surface wind speed data. Using a temperature-wind speed lag of 3 hours, local slopes, and a constant lapse rate (γ) of -0.0057 K/m, the Prandtl model for katabatic flows [Oerlemans, 1998; Oerlemans and Grisogono, 2002] was fit to observed 2 m wind speeds for Ta > T ∗ and southerly flow speeds at 700 hPa (V ) less than V ∗ . The fitted diffusivities give characteristic wind speed and temperature profiles that suggest a deeper boundary layer with greater wind speeds should be observed at larger glaciers, which is consistent with previous observational studies [Obleitner, 1994]. With increasing wind  100  5.3. Results speeds, friction with both the surface and the overlying air mass will also increase, suggesting that the boundary layer depth and wind speed increases are finite. As eddy diffusivities describe a diffusion rate, the development of the katabatic boundary layer will exert an influence on KH and KM . Analyses of flow path length and temperature (below) indicate a limit to cooling due to sensible heat exchange in katabatic flows. By extension, the stability of the KBL will increase with flow path length as the difference between near-surface temperatures and ambient temperatures increases. However, stability increases will be similarly limited, and the form of the relation between KH , KM and flow path length will impose a limit. Optimized KH values were fit to the estimated flow lengths using a logarithmic growth function, while an exponential decay function was used to fit KM to FPL: KH  = β12 + β13 ln (FPL)  (5.8)  KM  = β14 + β14 exp (β15 FPL)  (5.9)  From equations 5.8 and 5.9, eddy diffusivities were estimated at the testing sites. Katabatic wind speeds (uKc ) at 2 m were then modelled using local slopes, a constant lapse rate of -0.0057 K m−1 , and Ta estimated at time i − kmax , where kmax is the lag associated with maximum cross-correlation (kmax = 3).  5.3 5.3.1  Results Temperature  Transfer function coefficients describing T ∗ (k2 ) and the strength of boundary layer development for both non-katabatic and katabatic flows (k3 , k4 ) were estimated with a high degree of confidence using elevation (Z) and flow path length (FPL), respectively (Table 5.2; Figure 5.2). For both full and reduced models of k2 and k3 , parameter estimates were significant, with p-values less than 0.05. The full model describing k4 was significant, but the  101  5.3. Results  Figure 5.2: Models for estimating KBL temperature transfer function coefficients (a) k2 , (b) k3 , and (c) k4 from topographic indices. Solid lines represent full model fits, dashed lines are reduced model fits, solid black circles are reduced model AWS sites, and open circles are withheld AWS sites. reduced model was not. At WM2, the fitted value of T ∗ = 8.37◦ C appears to be the result of inadequate sampling at lower temperature values (Figure 3.6) and the similarity between fitted slopes k3 (0.83) and k4 (0.60) (Table 3.2). The observation from WM2 was thus removed from the full model for k2 . Transfer function coefficients estimated for withheld stations using Z, FPL, and β1 to β7 are similar to those observed in the meteorological analyses (Table 5.3). Comparisons of modelled and observed near-surface temperatures at the independent sites are given in Figure 5.3. Modelled temperatures agree with those observed within the KBL across the entire range of temperatures observed, and RMSE range from 0.95 to 1.32◦ C (Table 5.4). Using topographic indices and ambient climate temperature, errors in modelled temperature at independent stations are of the same magnitude as the errors in the individual piecewise analyses conducted at these sites (Table 3.2), indicating the success with which boundary layer development can be estimated using Z and FPL.  102  Table 5.2: Summary of models for estimating piecewise temperature transfer functions from topographic indices. SE is the standard error of the estimate, and p is the p-value for significance. Function  Model  k2 = β1 + β2 Z  Full  k4 = β5 + β6 exp (β7 FPL)  SE  p  β1 = 13.35 β2 = −3.93×10−3  1.77 8.63×10−4  < 0.05 < 0.05  Reduced  β1 = 12.39 β2 = −3.49×10−3  1.29 6.18×10−4  < 0.05 < 0.05  Full  β3 = 9.77×10−1 β4 = −4.43 × 10−5  1.52×10−2 5.56 × 10−6  < 0.05 < 0.05  Reduced  β3 = 9.81 × 10−1 β4 = −4.51 × 10−5  8.68 × 10−3 2.62 × 10−6  < 0.05 < 0.05  Full  β5 = 2.91 × 10−1 β6 = 7.15 × 10−1 β7 = −5.64 × 10−4  6.46 × 10−2 1.10 × 10−1 1.86 × 10−4  < 0.05 < 0.05 < 0.05  Reduced  β5 = 3.05 × 10−1 β6 = 8.63 × 10−1 β7 = −7.61 × 10−4  2.45 × 10−2 9.02 × 10−2 1.18 × 10−4  0.05 0.07 0.1  5.3. Results  k3 = β3 exp (β4 FPL)  Parameter Estimate  103  5.3. Results  Table 5.3: Temperature transfer function coefficients estimated from meteorological observations (obs) and from topographic indices (est) for withheld AWS sites Site  PM1 PM3 PM108 PM308 WM2  k1 (◦ C)  k2 (◦ C)  k3  k4  obs  est  obs  est  obs  est  obs  est  5.51 4.55 4.82 4.82 6.74  4.87 4.84 5.08 4.82 4.02  6.48 5.12 5.35 5.12 8.37  5.64 5.17 5.42 5.39 4.43  0.85 0.95 0.91 0.93 0.84  0.86 0.94 0.94 0.89 0.91  0.43 0.69 0.64 0.62 0.60  0.41 0.70 0.71 0.48 0.54  Figure 5.3: Observed and modelled Tg at independent testing sites.  104  5.3. Results  Table 5.4: Mean absolute error (MAE), mean bias error (MBE), root mean squared error (RMSE), and sample size (n) for temperatures (◦ C) and vapour pressures (hPa) modelled at independent glacier AWS. Temperature (◦ C)  Vapour Pressure (hPa)  Site  MAE  MBE  RMSE  MAE  MBE  RMSE  n  PM1 PM3 PM108 PM308 WM2  0.83 1.06 0.76 0.77 0.95  -0.17 -0.53 0.27 -0.43 -0.02  1.07 1.32 0.95 1.05 1.27  0.42 0.62 0.45 0.45 0.56  0.02 -0.38 -0.31 -0.31 -0.25  0.54 0.78 0.58 0.58 0.71  3429 3854 2355 2799 1410  105  5.3. Results  5.3.2  Vapour Pressure  Models for estimating vapour pressure transfer function coefficients are shown in Figure 5.4 and Table 5.5. Both β8 and β9 are significant in the full and reduced models, while β10 and β11 are not significant in the reduced model. Exponential and logarithmic functions are again used to estimate the scaling parameters, demonstrating that the processes of evaporation and condensation occurring within the boundary layer reach an equilibrium after a certain flow distance. Values for j1 and j2 estimated from FPL range between +44% to -16% of the values calculated from observed meteorological data (Table 5.6), demonstrating the success of the topographic approach to describing boundary layer development and the magnitude of moisture extraction or enhancement from the KBL. Observed eg is successfully modelled at all test sites (Figure 5.5), and RMSE ranges from 0.54 to 0.71 hPa (Table 5.4). Mean bias errors range from -0.38 to 0.02 hPa, suggesting that vapour pressures tend to be underestimated using FPL. However, errors in eg modelled from topographic indices are of the same magnitude as those calculated when using the actual station data (Table 3.5).  Figure 5.4: Models for estimating KBL temperature transfer function coefficients j1 (a) and j2 (b) from FPL.  106  Table 5.5: Model summary for KBL vapour pressure transfer functions and FPL. SE is the standard error of the estimate, and p is the p-value for significance. Model  Parameter Estimate  SE  p  j1 = β8 exp(β9 FPL)  Full  β8 = 7.58 × 10−1 β9 = −5.62 × 10−5  2.62 × 10−2 1.30 × 10−5  < 0.05 < 0.05  Reduced  β8 = 7.20 × 10−1 β9 = −4.93 × 10−5  2.27 × 10−2 4.53×10−5  < 0.05 < 0.05  Full  β10 = −1.53 β11 = 4.52 × 10−1  0.96 1.30 × 10−1  0.16 < 0.05  Reduced  β10 = −1.44 β11 = 4.53 × 10−1  1.67 1.37 × 10−1  0.31 0.08  j2 = β10 + β11 log(FPL)  5.3. Results  Function  107  5.3. Results  Table 5.6: Vapour pressure transfer function coefficients observed from previous meteorological analyses (obs) and estimated from topographic indices (est). Weighted means obtained from model development sites were used for global values of e3 and e4 (0.86 and 0.06, respectively) Site  PM1 PM3 PM108 PM308 WM2  j1  j2  obs  est  obs  est  0.63 0.74 0.72 0.74 0.79  0.63 0.68 0.69 0.65 0.66  2.05 1.67 1.68 1.47 1.34  2.15 1.70 1.68 2.01 1.93  Figure 5.5: Observed and modelled eg at independent testing sites.  108  5.3. Results  5.3.3  Wind Speed  Eddy diffusivities for heat and momentum were modelled as a function of flow path length (Figure 5.6; Table 5.7) using global and reduced models. The reduced model constructed with PM2, PM4, WM1, and BM1 diffusivities could not be iterated successfully, so PM4 was replaced with PM3 and the model was re-evaluated. In the reduced KH and KM models, the parameters are significant at p = 0.05. Diffusivities estimated for the withheld stations from FPL (Table 5.8) tend to be lower than observed for KH , and higher than observed for KM , though this appears to be a result of the stations used in the model development.  Figure 5.6: Estimating eddy diffusivity coefficients (a) KH and (b) KM from flow path lengths. Legend as in Figure 5.2. Katabatic wind speeds are accurately modelled at PM1 and PM308, which have the longest FPL of the test sites. At the remaining sites, low flow path lengths suggest limited boundary layer development, and observed wind speeds vary substantially from modelled. The lack of a well-developed boundary layer may contribute to increased disruption from channelled flows or anabatic winds which lie aloft of katabatic flows, resulting in the poor model results.  109  Table 5.7: Model summary for fitted eddy diffusivities KH and KM . SE is the standard error of the estimate, and p is the p-value for significance. Function KH = β12 + β13 log(FPL)  Notes:  a  Parameter Estimate 10−1  SE  p  8.94×10−2  β12 = −3.39 × β13 = 7.21 × 10−2  1.18×10−2  < 0.05 < 0.05  Reduced  β12 = −4.42 × 10−1 β13 = 8.09 × 10−2  5.61×10−1 7.13×10−2  < 0.05 < 0.05  Full  β14 = 1.10 × 10−1 β15 = 1.24×10−1 β16 = −1.67×10−4  3.84×10−2 3.20×10−2 1.07×10−4  < 0.05 < 0.05 0.16  Reduceda  β14 = 8.96 × 10−2 β15 = 1.48×10−1 β16 = −1.23×10−4  1.95×10−2 1.63×10−2 3.13×10−5  < 0.05 0.07 0.17  Full  Reduced model used PM2, PM3, WM1, and BM1 for fitting  5.3. Results  KM = β14 + β15 exp(β16 FPL)  Model  110  5.3. Results  Table 5.8: Eddy diffusivity coefficients for estimating uK observed from meteorological observations (obs) and estimated from topographic indices (est) Site  PM1 PM4 PM108 PM308 WM2  KH (m s−2 )  KM (m s−2 )  obs  est  obs  est  0.26 0.18 0.14 0.22 0.24  0.22 0.13 0.14 0.19 0.18  0.17 0.21 0.22 0.19 0.19  0.19 0.22 0.22 0.20 0.21  Table 5.9: Mean absolute error (MAE), mean bias error (MBE), root mean squared error (RMSE), and sample size (n) for uK modelled at independent glacier AWS. uK (m s−1 ) Site  MAE  MBE  RMSE  n  PM1 PM4 PM108 PM308 WM2  0.42 0.62 0.45 0.42 0.56  0.02 -0.38 -0.31 -0.11 -0.25  0.54 0.78 0.58 0.54 0.71  1301 2113 865 1027 443  111  5.3. Results  Figure 5.7: Observed and modelled uK at independent testing sites.  112  5.3. Results  5.3.4  Application  To demonstrate the application of the katabatic boundary layer models and to examine the model suitability outside the observation sites, a hot day was selected from the observation period. Ambient temperatures and vapour pressures for June 28, 2008 were estimated for Bridge Glacier using regional climate data and glacier elevations. Using full model coefficients (Tables 5.2, 5.5 and 5.7), and flow path lengths and local slopes calculated from the DEM, Tg and eg were predicted from the ambient conditions at 14:00, and ug was predicted for three hours later (17:00). Figure 5.8 illustrates the expected behaviour of KBL temperatures, vapour pressures, and wind speeds. The lowest near-surface temperatures are expected in the upper basins of the glacier where flow path lengths and adiabatic warming are optimized. At lower elevations, adiabatic warming counters the cooling effect associated with flow distance. Temperatures at the peaks surrounding the glacier are greater than expected at the terminus. The effects of condensation or evaporation at the glacier surface depend on the ambient vapour pressure gradient and the elevation at which ea = 6.11 hPa. For the given time-step, ea = 6.11 hPa near the midpoint of the glacier, and condensation at the surface below this elevation removes moisture from the KBL, while evaporation from the surface above this elevation enhances moisture in the KBL. Near-surface vapour pressures at the terminus are expected to be nearly 2 hPa below the ambient values, indicating large amounts of condensation at the surface that will enhance melt rates. Conversely, energy is being lost at the higher elevations as evaporation removes melt energy from the surface. Figure 5.8b demonstrates that station location and measurement period will likely determine the sign of the latent heat flux, and this may explain differences in latent heat flux observed at various glacier locations (Table 1.1). Wind speeds estimated for the Bridge Glacier under strong katabatic forcing (Figure 5.8c) exhibit a surface pattern that is strongly affected by slope. The strongest winds are expected near the terminus, though the break in slope at the southern-most terminus has the effect of resetting the  113  5.3. Results flow path length and produces an unexpected drop in wind speeds. The range of expected wind speeds appears to be excessive, though the lack of instrumentation prevents verification of the model results. At BM1, wind speeds observed between 14:00 and 18:00 on 28 June 2008 ranged between 5 and 6 m s−1 , corresponding to the predicted values at that location.  114  (c) Wind speed  (d) Observations  Figure 5.8: Predicted near-surface temperatures (a), difference between boundary layer and ambient vapour pressures (b) and wind speeds (c) at Bridge Glacier, 28 June 2008 at 1400. Time-series of observed Tg , eg , and ug on June 28 2008 are given in (d).  5.3. Results  (b) Vapour pressure difference  115  (a) Temperature  5.4. Discussion  5.4  Discussion  All models relating boundary layer development to FPL followed the form of either an exponential decay or a logarithmic growth. Exponential decay or logarithmic growth functions are physically consistent with the idea that boundary layer development is finite, and the empirical models indicate that the bulk of KBL development occurs within the the first 5 km of flow (Figures 5.2, 5.4, and 5.6). Numerical models of katabatic wind speeds in tranquil flows demonstrate a similar growth pattern and scale dependency [LaLaurette and Andr´e, 1985]. A simple model of katabatic wind speeds given by Stull [1988] is compared against the results observed in this study: u=  g δT x sin α Ta  1/2  (5.10)  where g is gravity (9.8 m s−2 ), δT = Ta − Tg , x is flow distance (m), and α is slope angle. The value for u will approach an equilibrium value (ueq ) where negative buoyancy is balanced by friction: ueq =  g δT h sin α Ta CD  1/2  (5.11)  Depth of the flow is given by h, and CD is the total drag coefficient. For typical values found in this study (δT = 10 K, Ta = 283 K, α = 5◦ ) and a value of CD = 0.005, an equilibrium wind speed of 11 m s−1 is reached at a flow length of approximately 4 km. However, the models described in this study include vertical variations in δT which will affect the production rate of negative buoyancy. Transfer functions estimated from the reduced models were accurate enough to demonstrate considerable skill in modelling near-surface temperatures, vapour pressures and wind speeds with only ambient conditions and FPL. While it is desirable to have as many sites as possible to tune the models that describe transfer functions using topographic indices, it is obvious that observations need to be made at sites covering a range of scales. The observed existence of a relationship between k3 and FPL was sur116  5.4. Discussion prising, as it had been assumed that below T ∗ there would be minimal boundary layer development. Results indicate that even without a strong thermal forcing, near-surface temperatures are less than ambient temperatures, and particularly so at sites with long upwind fetches. Sampling error may be a factor in this relationship, as there were few observations at BM1 where Tg < 0, Figure 3.6. The form of the function relating FPL and k4 relates to the balance of forces created by katabatic flow (negative buoyancy and turbulence). At the scale of the glaciers observed in this study, Tg reach a minimum of approximately 30% of ambient temperatures for katabatic flows. Moisture extraction from the glacier boundary layer is driven by the gradients between the ambient vapour pressure and the surface vapour pressure. As ambient gradients of vapour pressure (γe ) are typically negative (vapour pressure decreasing with elevation; Figure 3.9) and vapour extraction is linked to flow length, condensation may contribute significant amounts of melt energy at the lower elevations of large glaciers where ea is greatest. Conversely, evaporation/sublimation will occur at high elevations where ea may be less than es , though low values of FPL will limit the amount of energy that is removed from the surface. Weighted mean values calculated for e3 and e4 are close to the assumptions of a slope of unity and a y-intercept of zero, suggesting that with Tg < 0, ambient and near-surface vapour pressure amounts are approximately equal. As this study used hourly means of observed meteorological variables, it is possible that climate data of higher temporal resolution would help resolve more transient features of katabatic flows. In particular, the assumption of a constant KM should be treated with caution, as increasing katabatic flow speeds will ultimately create turbulence that will affect the value of KM . It is recommended that the higher resolution climate data be analysed to further examine the relation between meteorological variables and flow path length. Spatially distributed fields of KBL temperatures, vapour pressures, and wind speeds (Figure 5.8 demonstrate the utility of the approach presented above. Vertical gradients of temperature and vapour pressure estimated 117  5.5. Conclusions from regional climate networks provide information regarding strength of the katabatic forcing, but FPL offers a reasonable method for distributing Tg , eg , and ug over the surface of a melting glacier. More meteorological data is required to evaluate the approach, and maps of FPL provide a means for determining the sampling strategy. Glaciers of varying sizes and climatic regimes also need to be incorporated, in order to examine the robustness of transfer functions relating KBL development to FPL.  5.5  Conclusions  The formation of katabatic winds on melting mid-latitude glaciers introduces considerable differences between the glacier boundary layer and the overlying ambient conditions. Using glacier-surface observations from nine sites over three seasons, transfer functions were developed to describe the relations between KBL and ambient climates. Results from this research have demonstrated that the transfer functions can be parameterized as a function of glacier flow path length, which can be derived from a DEM. Transfer functions for estimating KBL effects can thus be scaled to any site, and near-surface meteorological variables required for energy balance modelling can be specified with considerable confidence using only estimates of the ambient conditions and digital elevation models.  118  Chapter 6  Radiation Modelling Energy-balance (EB) models explicitly account for the incoming and outgoing fluxes of energy at the surface. Their application in glacier mass balance modelling requires the precise description of the spatial variability of solar radiation, which can be challenging in topographically complex regions. In particular, the EB approach requires estimates of atmospheric transmissivity and surface albedo, both of which can vary substantially in both space and time. The EB approach also involves modelling the longwave radiative fluxes, which can be important when considering melt on subdiurnal timescales, as they control nighttime refreezing and accumulated cold content of the snowpack. Net radiation (Q∗ ) is the sum of net shortwave (K ∗ ) and longwave (L∗ ) components, and it is typically the dominant source of melt energy at mid-latitude glaciers (Table 1.1). This chapter focuses on methods for simulating radiative fluxes for regional glacier melt modelling, with particular emphasis on atmospheric transmissivity, surface albedo, and incoming longwave radiation.  6.1 6.1.1  Solar Radiation and Transmissivity Background and Objectives  Net shortwave radiation can be expressed as K ∗ = K↓ex τ (1 − α)  (6.1)  where K↓ex is the product of potential incoming shortwave radiation at the top of the atmosphere, τ is atmospheric transmissivity, and α is the surface albedo. Incident shortwave radiation at the top of the atmosphere varies 119  6.1. Solar Radiation and Transmissivity with the angle of incidence: K↓ex = S0 cos θZ .  (6.2)  where S0 is the solar constant and θZ is the angle of incidence determined from solar declination, hour angle and the latitude of the site [Iqbal, 1983]. Transmissivity exhibits systematic diurnal variations, resulting from changes in the solar path length as well as variations caused by atmospheric constituents that absorb or reflect solar radiation, including clouds, aerosols, dust, and water vapour. Glacier melt models typically use on-site measurements of global (direct and diffuse) radiation to describe daily or hourly variations in τ . Assuming a horizontal measurement surface and a sky-view factor of 1, τ is calculated as the ratio of observed (K↓) and potential shortwave radiation: τ=  K↓ . K↓ex  (6.3)  While this approach is suitable for single glaciers, shortwave radiation measurements are sparse, and distributed EB modelling at regional scales requires a model for estimating transmissivity. Radiation studies by Hay and Suckling [1979] concluded that daily observed shortwave radiation totals in British Columbia could be extrapolated to a distance of 50 km with an error of ±15%, while Suckling and Hay [1978] demonstrated that stratifying by synoptic types did not yield more accurate predictions of K↓. Where appropriate observations exist, τ can be estimated from the atmospheric concentrations of aerosols, water vapour, dust, and air molecules [Davies et al., 1975; Dozier, 1980]. However, this approach is limited by the quality of the input data and extrapolation of such a model is generally not possible. More commonly, τ is treated as a bulk property of the atmosphere, and is modelled using a variety of empirical approaches. Diurnal temperature range (∆T ) is often used in modelling τ , and the Bristow-Campbell (BC) approach [Bristow and Campbell, 1984] capitalizes on the assumption that clear (overcast) days have high (low) transmissivities, and that high  120  6.1. Solar Radiation and Transmissivity (low) radiation loadings will result in large (small) ∆T . The BC model has been developed and tested almost exclusively on low-elevation meteorological data where this assumption generally holds true. At higher elevations and in complex topography, local climate modifications (i.e. inversions, clouds) may alter the relationship between ∆T and τ . Shortwave radiation models incorporating ∆T have been successfully adopted and modified for a variety of locations (see Liu et al. [2009] for a comprehensive review). Atmospheric transmissivity has also been modelled as a function of monthly rainfall totals in monsoon regions [Matsuda et al., 2006], and daily precipitation amounts [Winslow et al., 2001]. Altitudinal effects on solar path length have also been included [Thornton and Running, 1999]. Surprisingly, empirical τ models do not typically include some measure of atmospheric moisture. Near-surface vapour pressures show strong vertical gradients and can be extrapolated with reasonable confidence (Chapter 3), and physically based atmospheric transmittance models show a strong sensitivity to vertical profiles of atmospheric water vapour, which are related to near-surface conditions [Dozier, 1980]. Thornton and Running [1999] added a vapour pressure term after noting that the highest transmittances for any given day corresponded to low vapour pressures. Single-glacier melt models often use local observations of shortwave radiation to define τ for each model time step, but for regional-scale melt modelling, a transmissivity model that can distributed with confidence using a minimum of input data is required. The objectives of this study are (1) to examine the spatial and temporal variability of global radiation and atmospheric transmissivity between four high-elevation sites in the southern Coast Mountains of British Columbia, (2) to evaluate the performance of simple daily transmissivity models, and (3) to examine the possibility of modelling transmissivity on sub-daily timescales.  6.1.2  Methods and Data  Site locations, study areas, and instrumentation for the four high-elevation sites are given in Chapter 2. Digital elevation models (DEM) for each study  121  6.1. Solar Radiation and Transmissivity area were used to calculate horizon angles in 10◦ ground azimuth intervals for each station, and solar geometry equations from Iqbal [1983] were used to compute the local solar position at 10-minute time steps. Correlations and Coefficients of Variability Simple correlation coefficients (r) and coefficients of variability (CV) were calculated for hourly and daily radiation totals observed at the four sites. Following Hay and Suckling [1979]: CV =  σK 0.5 (K↓1 +K↓2 )  (6.4)  where σK is the standard deviation of hourly or daily radiation differences for two stations, and K↓1 and K↓2 are the mean hourly or daily values of shortwave radiation at the two sites. Hourly values of CV were calculated only for times that the stations received direct solar radiation. Hourly and daily atmospheric transmissivities (τ ; Eq. [6.3]) were calculated as the ratio of observed global radiation (K↓) and potential clear-sky radiation (K↓ex ) integrated at hourly and daily scales. Scatterplot matrices and simple correlation coefficients were used to examine variability in K↓ and τ between the four mountain stations. Transmissivity Models Three previously described transmissivity models and two developed for this research were tested to identify the optimum method for estimating transmissivity over the study region (Table 6.1). The Hargreaves and Samani (HS) approach (Eq. [6.1]) was included in this study due to its relatively simple formulation, and lack of correction factors for precipitation events. Both the Bristow-Campbell (BC) and Donatelli-Marletto (DM) transmissivity models require estimates of precipitation to apply site-specific correction factors for ∆T , though several studies [Hunt et al., 1998; Liu et al., 2009] appeared to neglect the precipitation correction in search of a generalized model. Donatelli and Marletto [1994] introduced potential incoming short-  122  6.1. Solar Radiation and Transmissivity wave radiation (K↓ex ) as a seasonal scaling factor in the BC model, though it has been argued that seasonality corrections are unnecessary [Liu et al., 2009]. In light of these considerations, the two additional models developed in this study borrow from the BC and DM approaches, but include near-surface vapour pressure to enhance the contrast between clear days and overcast days or days with precipitation. The predictor variable common to all models is daily temperature range (∆T ), defined by Bristow and Campbell [1984] as ∆T1 = Tmax(j) − (Tmin(j) + Tmin(j+1) )/2  (6.5)  where Tmax and Tmin are maximum and minimum temperatures on day j. Minimum temperatures for day j − 1 were included by Bristow and Campbell [1984] to account for the advection of large-scale moist or dry air masses through the mid-latitude study region. As precipitation was not measured at the four weather stations, daily precipitation totals at the Whistler Low station (Figure 2.1) were obtained from Environment Canada [2010] to test the empirical adjustment for the effects of precipitation events on ∆T prescribed by Bristow and Campbell [1984]: ∆T(j) = 0.75 ∆T(j)  R(j) > 0  ∆T(j−1) = 0.75 ∆T(j−1) R(j−1) = 0, ∆T(j−1) < ∆T(j−2) − 2  (6.6)  where R(j) is daily rainfall (mm) on day j. Precipitation adjustments lower the estimate of ∆T on days with rainfall, and on days prior to rainfall to account for increasing cloudiness. Observations on days missing daily precipitation data were removed for the application of all models, leaving sample sizes of n = 290, 357, 458 and 358 days for Bridge, Helm, Place, and Weart Glaciers, respectively. An alternative formulation for ∆T , ∆T2 = Tmax(j) − Tmin(j)  (6.7)  was also tested, as it was found to give better modelled results than ∆T1 , 123  6.1. Solar Radiation and Transmissivity particularly at higher elevations [Liu et al., 2009]. Air temperatures observed at the ambient stations were used to compute ∆T , and hourly ambient temperature gradients (Ch. 3) were used to estimate ambient temperatures at PM3, the on-ice station. In windy situations, the exposure of ridge stations will limit local boundary layer development, and thus these may not present a diurnal temperature range similar to those experienced by valley-bottom stations, which are typically used for transmissivity models. However, temperatures measured within the katabatic boundary layer are clearly unsuitable for estimating ∆T , particularly during strong katabatic events, as the diurnal temperature signal will be strongly muted. Atmospheric transmissivity models using ∆T are limited to daily transmissivity estimates, while EB melt models are often applied on sub-daily scales. The fourth model developed for this study seeks to further improve transmissivity estimates by incorporating near-surface vapour pressure, which can be successfully predicted from standard climate networks (Chapter 3). Solar transmittance models are sensitive to atmospheric vapour pressure [Dozier, 1980], and though the relation between surface and atmospheric vapour pressures has not been investigated thoroughly, it is assumed that near-surface vapour pressure represents a combination of both local site effects and large-scale air mass moisture characteristics. Liu et al. [2009] noted that parameters in the BC model were significantly correlated with relative humidity, and Thornton and Running [1999] also recognized a relation between observed shortwave radiation and relative humidity. Vapour pressures do not exhibit strong diurnal cycles as do temperature and relative humidity, offering the potential for modelling τ on hourly timescales. Daily transmissivity models were fit to τ and the effects of using ∆T1 or ∆T2 and precipitation corrections were examined. Mean absolute error (MAE) and root mean squared error (RMSE) were employed to determine model performance, while the model parameters were examined to characterize the robustness of any given approach across the study region. The optimization procedure was also reframed by attempting to minimize the errors in modelled K↓, and compared against K↓ estimated from modelled transmissivity and Kpot . 124  6.1. Solar Radiation and Transmissivity  Table 6.1: Transmissivity models tested at four high-elevation sites in the southern Coast Mountains, where ai , bi , and ci are fitted coefficients Source  Time Scale  Abbreviation  a  Model √ τˆ1 = a1 ∆T + b1  Daily  HS  b  τˆ2 = a2 (1 − exp (b2 ∆T c2 ))  Daily  BC  Daily  DM  Daily, Hourly  S  c  τˆ3 = a3 (1 − exp (−b3  d  τˆ4 = a4 (1 − exp(b4 ·  ∆T )) K↓ex ∆T ea  c4  ))  a: Hargreaves and Samani [1982]; b: Bristow and Campbell [1984]; c Donatelli and Marletto [1994]; d: This study  Following Eq. [6.1], an hourly transmissivity model was tested by holding ∆T constant and using hourly ambient vapour pressures. The hourly model was fitted initially for all direct sunlight observations, but was stratified to observations around solar noon (1100 to 1600 h), as solar path length strongly affects calculated transmissivities during the morning and evening. Corrections for solar path length were neglected.  6.1.3  Results  Coefficients of Variability Coefficients of variability for daily shortwave radiation totals ranged between 16 and 33%, and lower CVs were observed at stations located closer together. For distances of 40 km or more, the error involved in extrapolating daily global radiation appears to be ±25% on average (Figure 6.1). This error is somewhat higher than the ±15% error at 15 km found by Hay and Suckling [1979]. Hourly radiation totals show considerably greater variation between sites, particularly during the early morning and late evening (Figure 6.1b), when 125  6.1. Solar Radiation and Transmissivity  (a) CV by distance  (b) CV by hour  Figure 6.1: Daily and hourly (between 1100 and 1500) coefficients of variation in observed solar radiation, plotted by (a) distance between stations and by (b) hour of day. Hourly CV are only calculated for direct sunlight hours, and in (a) are given as the mean (square) and standard deviation (error bars) for CV calculated between 1100 and 1500. solar altitudes are low. Hourly CVs are stable during midday, and stationpair CVs range between 45 and 85% between 1100 and 1500. Extrapolation of hourly radiation values from a single station would thus give a midday error of approximately ±70% at a radius of 50 km (Figure 6.1a). As with daily radiation CV, extrapolation errors do not appear to change significantly between 50 and 100 km of separation, though this may be due to the limited sample size. Radiation and Transmissivity Correlations Inter-station correlation coefficients (r) for observed daily radiation totals and effective transmissivity range from 0.53 to 0.91 (Table 6.2), with similar values observed for both K ↓ and τ . In comparison, hourly K ↓ are more highly correlated (r = 0.66 to 0.87) than hourly τ (r = 0.46 to 0.74) as a result of diurnal variations in potential direct solar radiation cycles. It is likely that sub-daily variability in cloud cover at the individual sites contributed to the low hourly τ correlations. These results indicate that the averaging process involved in calculating daily transmissivities improves station-pair 126  6.1. Solar Radiation and Transmissivity correlations, and that over the study domain the conditions affecting total daily transmissivities tend to be similar. Scatterplot matrices of daily and hourly τ (Figures 6.2 and 6.3) further illustrate the spatial and temporal variabilities. Clear-sky days with high τ are evident in both hourly and daily datasets as clusters at the higher range of τ . However, large differences in hourly τ occur over the entire observed range, highlighting the difficulty in assuming that observations at a single station can represent a given region. Table 6.2: Correlation coefficients for daily and hourly observed radiation and transmissivity. All correlations are significant at p = 0.05.  Bridge Helm Place Weart  Bridge 1.00 0.79 0.58 0.83  Daily  K↓  Helm 0.79 1.00 0.70 0.91  Place 0.58 0.70 1.00 0.73  Hourly Weart 0.83 0.91 0.73 1.00  Bridge 1.00 0.83 0.66 0.81  Daily τ Bridge Helm Place Weart  Bridge 1.00 0.72 0.53 0.76  Helm 0.72 1.00 0.68 0.89  Place 0.53 0.68 1.00 0.70  Helm 0.83 1.00 0.72 0.87  K↓ Place 0.66 0.72 1.00 0.72  Weart 0.81 0.87 0.72 1.00  Hourly τ Weart 0.76 0.89 0.70 1.00  Bridge 1.00 0.63 0.46 0.59  Helm 0.63 1.00 0.58 0.74  Place 0.46 0.58 1.00 0.57  Weart 0.59 0.74 0.57 1.00  Daily Transmissivity Models Comparisons of errors in modelled daily transmissivity are given in Table 6.3. At three of the four observation sites, errors in modelled daily transmissivity were minimized using the τ model developed in this study (Table 6.1, Equation [6.1]), which incorporates a vapour pressure term. However, the improvements offered by the S model over BC, HS, and DM approaches were modest. Mean MAE for S∆T1 ,ea ,P was 0.119, versus 0.125 for BC∆T1 ,P , 0.127 for HS∆T1 ,P , and 0.126 for DM∆T1 ,P . The S∆T1 ,P model gave the lowest MAE and RMSE for both Place and Weart sites, while the S∆T1 model 127  6.1. Solar Radiation and Transmissivity  Figure 6.2: Scatterplot matrix of daily transmissivity observed at four mountain stations.  128  6.1. Solar Radiation and Transmissivity  Figure 6.3: Scatterplot matrix of hourly tranmissivity observed at four mountain stations.  129  6.1. Solar Radiation and Transmissivity gave the lowest MAE and RMSE for Bridge and Helm, suggesting that the precipitation correction may not be necessary. Contrary to the results of Liu et al. [2009], all model accuracies decreased when transmissivities were estimated with ∆T2 . Figure 6.4 illustrates model fits obtained for each site using the S∆T1 ,ea model, with and without precipitation corrections. At all sites the models are unable to reach the observed maximum transmissivity, which is approximately 0.8. Low values of observed transmissivity are overestimated. This feature is common to all the transmissivity models, and scatterplots of modelled transmissivity residuals versus observed transmissivities highlight (Figure 6.5) the extent of the problem. Another criterion for model suitability in regional applications is robustness of the fitted model parameters between sites (Table 6.5). The DM model shows the most consistent parameter values between sites, though it has the greatest prediction errors. The S model also displays robust parameter estimates but has the lowest mean MAE, and is thus recommended for estimating transmissivity across the study region. Both BC and HS models contain substantial differences in fitted parameters between sites. An attempt to reframe the problem and minimize the RMSE in modelled daily K↓ sums yielded similar biases (Figure 6.6), and greater errors (Table 6.4). Low daily solar radiation totals are overestimated, and high daily solar radiation totals are underestimated with both approaches. Alternative functions and piecewise fits were attempted for all models, but lower model errors could not be obtained. It can be seen from Figure 6.5 that the S∆T1 ,τ,ea ,P model offers the least amount of systematic error and gives the best predictions of transmissivity at higher values of τ , which recommends its use in energy balance melt modelling.  130  Table 6.3: Mean absolute error (MAE) and root mean squared error (RMSE) in modelled daily transmissivity, calculated for each station and model. Figures in bold indicate the lowest model errors for a given site. Bridge  Helm  Place  Weart  Mean  MAE  RMSE  MAE  RMSE  MAE  RMSE  MAE  RMSE  MAE  RMSE  BC∆T1 BC∆T2 BC∆T1 ,P BC∆T2 ,P  0.129 0.131 0.125 0.126  0.162 0.166 0.157 0.162  0.146 0.151 0.138 0.141  0.175 0.179 0.167 0.171  0.108 0.117 0.108 0.117  0.141 0.153 0.141 0.153  0.117* 0.123 0.117 0.123  0.145* 0.151 0.145 0.151  0.125 0.130 0.122 0.127  0.156 0.151 0.152 0.159  HS∆T1 HS∆T2 HS∆T1 ,P HS∆T2 ,P  0.134 0.138 0.131 0.133  0.167 0.172 0.163 0.168  0.145 0.155 0.138 0.145  0.175 0.183 0.168 0.175  0.109 0.119 0.109 0.119  0.141 0.154 0.141 0.154  0.118 0.125 0.117 0.125  0.146 0.154 0.145 0.154  0.127 0.131 0.124 0.131  0.157 0.164 0.145 0.164  DM∆T1 DM∆T2 DM∆T1 ,P DM∆T2 ,P  0.135 0.135 0.128 0.130  0.164 0.170 0.161 0.165  0.149 0.152 0.142 0.143  0.178 0.181 0.171 0.174  0.114 0.124 0.114 0.124  0.149 0.160 0.149 0.160  0.125 0.127 0.121 0.124  0.151 0.156 0.157 0.156  0.130 0.134 0.126 0.134  0.162 0.167 0.158 0.156  S∆T1 ,ea S∆T2 ,ea S∆T1 ,ea ,P S∆T2 ,ea ,P  0.108 0.110 0.107 0.106*  0.147 0.146 0.144 0.143*  0.146 0.150 0.138* 0.141  0.172 0.177 0.165* 0.168  0.101* 0.107 0.102 0.107  0.133* 0.141 0.133 0.141  0.119 0.123 0.119 0.123  0.145 0.151 0.145 0.151  0.119 0.123 0.119* 0.120  0.150 0.154 0.147* 0.151  6.1. Solar Radiation and Transmissivity  Model  131  6.1. Solar Radiation and Transmissivity  Table 6.4: Errors in modelled daily solar radiation totals using two different modelling approaches. For Model 1, K↓ was modelled using K↓pot and transmissivity models. In Model 2, K↓pot was included in the fitting procedure. Mean absolute and root mean squared errors (MAE, RMSE) are given in units of MJ m−2 d−1 and as a percentage of the mean daily K↓ in brackets. Model 1 Site Bridge Helm Place Weart  MAE 4.31 5.17 3.68 4.45  (18.9) (24.8) (18.8) (20.6)  Model 2  RMSE 5.80 6.21 4.64 5.42  (25.5) (29.8) (23.7) (25.2)  MAE 4.46 5.71 4.33 5.06  (19.6) (27.4) (22.1) (23.5)  RMSE 6.04 6.87 5.57 6.32  (26.6) (32.9) (28.5) (29.4)  Hourly Transmissivity Models Hourly transmissivity models calculated for observations between 1100 and 1600 at the four radiation sites yielded consistent parameter values between sites (Table 6.6) and model errors of the same magnitude as the daily models. Inclusion of a vapour pressure term provides useful information about the hourly variability in diurnal transmissivities, as the other daily models tested on hourly timescales produce substantially larger errors. As a test of the transmissivity models, hourly solar radiation fluxes were predicted from fitted model parameters, potential radiation (K ↓ex ) and observed ∆T1 and hourly ea (Figure 6.7). Modelled solar radiation fluxes agree with observations, though high radiation totals are underpredicted and low radiation totals are overpredicted. Bias in the modelled values results from the form of the transmissivity function, and is discussed further below. Transmissivity models offer a better approximation of hourly solar radiation totals than assuming radiation values from a reference station (Table 6.7). Assigning radiation totals from Place station results in mean absolute errors between 30 and 36% of the observed mean, while MAE for radiation totals modelled using ∆T , ea , and K↓ex range from 26 to 32%.  132  6.1. Solar Radiation and Transmissivity  Figure 6.4: Transmissivity model fits with and without precipitation adjustment for S∆T1 ,ea at each site  133  6.1. Solar Radiation and Transmissivity  Figure 6.5: Transmissivity residuals for all sites using Bristow-Campbell, Hargreaves-Samani, Donatelli-Maretto, and Shea transmissivity models. All models used ∆T1 corrected for precipitation.  Figure 6.6: Modelled daily K↓ residuals for all sites for the S∆T1 ,ea ,P transmissivity model.  134  6.1. Solar Radiation and Transmissivity  Table 6.5: Fitted model parameters for daily precipitation-adjusted ∆T1 transmissivity models Model  Site  a  b  c  BC∆T1 ,P  Bridge Helm Place Weart  0.73 0.85 1.12 0.89  0.48 0.52 0.27 0.45  0.97 0.61 0.67 0.63  HS∆T1 ,P  Bridge Helm Place Weart  0.16 0.18 0.27 0.21  0.26 0.21 0.01 0.16  -  DM∆T1 ,P  Bridge Helm Place Weart  0.72 0.70 0.69 0.69  0.27 0.29 0.24 0.21  -  S∆T1 ,ea ,P  Bridge Helm Place Weart  0.83 0.79 0.77 0.80  1.54 1.80 1.84 1.64  0.54 0.61 0.86 0.60  135  6.1. Solar Radiation and Transmissivity  Figure 6.7: Modelled and observed hourly solar radiation fluxes  Table 6.6: Site-specific coefficients for the S∆T1 ,ea hourly transmissivity model. Observations between 1100 and 1600 were used in the analysis, a, b, and c are fitted parameters, MAE and RMSE are mean absolute error and root mean squared error (unitless), respectively, and n is the number of model observations. Site  a  b  c  MAE  RMSE  n  Bridge Helm Place Weart  0.83 0.84 0.81 0.82  1.74 1.62 1.67 1.62  0.77 0.61 0.96 0.71  0.143 0.182 0.139 0.161  0.190 0.215 0.175 0.196  1740 2143 1555 2149  136  6.1. Solar Radiation and Transmissivity  Table 6.7: Errors in modelled hourly radiation in W m−2 and as a percentage of the observed mean in brackets. ˆ K↓= K↓P M 3 Site  MAE (%)  MBE  RMSE  Bridge Helm Weart  169.9 (36) 144.3 (35) 146.4 (30)  -79.0 -46.2 -67.5  236.6 203.2 209.3  ˆ K↓= K↓ex τˆS Bridge Helm Weart  6.1.4  125.4 (26) 132.1 (32) 136.3 (28)  24.4 17.3 25.1  170.1 170.7 171.1  Discussion  Daily and hourly correlations of observed K↓ and τ demonstrate the temporal variability of solar radiation in mountainous terrain, which can be explained primarily by variations in cloud cover. Coefficients of variability calculated for the 4 mountain AWS sites give errors in extrapolation of approximately 25% for daily radiation totals, and nearly 70% for midday hourly radiation totals. This suggests that to minimize errors in modelled K↓, a robust transmissivity parameterization is preferred over a network of K↓ observation sites. A transmissivity model that incorporates an ambient vapour pressure term modestly improves the Bristow-Campbell [Bristow and Campbell, 1984] approach, which has been widely used. Ambient near-surface vapour pressure (ea ) provides a measure of water vapour present in the atmospheric column [Barry, 1992], which affects the transmission of solar radiation. Mean daily vapour pressures will provide an indication of change in synoptic-scale air mass characteristics, whereas the hourly model will likely be influenced by boundary layer processes affecting observed ea . It is noted that transmissivity models were not constructed using vapour pressures observed near the glacier surface (eg ), as these will be biased by the KBL effects discussed previously (Chapter 3). Despite the general success of the modified Bristow137  6.2. Albedo Campbell approach, the form of the transmissivity model requires further investigation, as K↓ error terms are strongly biased at both low and high values of τ . For estimating hourly K↓, modelled transmissivity and K↓ex give lower errors versus extrapolation of observations from a reference station. Modelled errors for the two approaches are approximately the same for daily K↓.  6.2 6.2.1  Albedo Background and Objectives  Albedo controls the amount of solar radiation absorbed at the surface, which dominates the net radiation budget of melting glaciers, and is therefore an important control on snow and ice melt rates. Snow albedo is a function of snow density, snow depth, water content, grain size, and impurities [Wiscombe and Warren, 1980], but also depends on the spectral characteristics of the incident radiation and angle of incidence. In distributed applications, snow albedo is often parameterized as a function of snow age (which affects all the characteristics above) or through some combination of accumulated temperatures, snow depth, and density (Table 6.8). Of the predictor variables listed in Table 6.8, temperature and solar radiation are the only variables that can be distributed regionally with confidence. Sub-daily variations in albedo caused by solar angle have been previously recognized [Brock, 2004], and the effects of clouds have also been investigated [Jonsell et al., 2003]. Near-infrared solar radiation is absorbed preferentially by clouds, resulting in relatively higher proportions of visible radiation received at the surface during overcast conditions. As snow reflectivity is high in the visible spectrum and low in the near-infrared [Wendler and Kelley, 1988; Winther, 1993], cloudy conditions produce an increased surface albedo [Brock, 2004; Hoch, 2005]. Results from the solar radiation analyses of this study (above) suggest that atmospheric transmissivity can be successfully parameterized for use at remote sites, providing an avenue  138  6.2. Albedo for incorporating cloud effects on modelled albedo. Table 6.8: Snow albedo parameterizations. Model A  Independent variables Constant albedo  Sources Braithwaite and Olesen [1989]; Munro [1991]; Hock [1999]  B  Snow age  USACE [1956]; Tangborn [1984]; Oerlemans and Hoogendoorn [1989]; Bl¨oschl et al. [1991]; Escher-Vetter [2000]; Strasser et al. [2004]  C  Snow age and depth  Oerlemans [1993]; Oerlemans and Knap [1998]; M¨olg and Hardy [2004]; Walter et al. [2005]  D  van de Wal and Oerlemans [1994]  F  Snow age, depth, and water fraction Snow age and cloud cover Temperature  G  Temperature and depth  Brock and Arnold [2000]; Klok and Oerlemans [2004]  H  Temperature and solar radiation Depth and density  Winther [1993]  J  Density, melt rate, solar angle, and cloud cover  Greuell and Oerlemans [1986]; Ranzi and Rossi [1991]  K  Grain size, age, and solar angle  Kustas et al. [1994]  E  I  Bruland and Hagen [2002] Kondo and Yamazaki [1990]; Mittaz et al. [2002]  Klok and Oerlemans [2002]  Though spatially variable [Klok and Oerlemans, 2004], ice albedo does not apparently decay temporally as in the case of snow, and in distributed models is typically assigned a constant value. Brock and Arnold [2000] used a second order polynomial based on elevation to model the effect of increased particulate accumulations on the lower elevations of the glacier, and some authors have suggested that rainfall events can remove the particulate matter leading to an increase in ice albedo [Brock, 2004; Klok and Oerlemans, 2004]. 139  6.2. Albedo Previous albedo studies have generally been limited to observations at a single point on a glacier [Cutler and Munro, 1996; Oerlemans and Knap, 1998; M¨ olg and Hardy, 2004], and have not covered multiple glaciers. While the systematic decay of snow albedo due to increasing water and dust content can be quantified through a number of different approaches, the relation between albedos observed at different locations is pertinent for regional modelling studies. It is hypothesized that the observed variability in snow albedos results from local cloud conditions, and that atmospheric transmissivity can be used to improve albedo model estimates. The first objective of this research, therefore, is to examine correlations of daily albedos observed across the study region. Secondly, this research presents a parameterization of albedo that can be distributed regionally using observed incoming and reflected radiation and modelled potential radiation. The impacts of using either near-surface (Tg ) or estimated ambient temperatures (Ta ) for estimating the temporal decay of albedo are also examined.  6.2.2  Methods  Mean daily albedos for day i were calculated by dividing total daily reflected shortwave radiation (K↑) observed at the glacier AWS by total daily incoming shortwave radiation (K↓) observed at the reference stations: α(i)snow/ice =  K↑ (i) K↓ (i)  (6.8)  Mounted at a height of 1.7 m above the surface, the field of view for the reflected radiation sensors includes the legs of the AWS tripod and the datalogger box (Figure 6.8). Depending on the local slope, radiation from surrounding terrain and sky may also be received at the sensor. An inverted pyranometer receives approximately 90% of its radiation from a circle with a radius of three times the sensor height above surface (Reifsnyder [1967]; A. Christen, pers. comm.). For the current study, reflected radiation is therefore determined primarily by the radiation reflected from a circle with a radius of 5.1 m, . Reflections from the AWS station and radiation from  140  6.2. Albedo remaining 10% are expected to have minimal influence on observed daily albedos. Analysis of a hemispherical photo (Figure 6.8) taken from the perspective of an inverted pyranometer indicates that 95.7% of the field-of-view is occupied by the snow/ice surface. The station, including the datalogger box, tripod, and wires, occupies 3.5% of the field of view, and the remaining 0.8% represents rock or sky.  Figure 6.8: Fisheye photo of the field of view of an inverted pyranometer Using multiple lines of evidence (albedo curves, observed temperatures, atmospheric transmissivity, and precipitation at Whistler Low Level station, Figure 6.9), daily albedo observations for each site were manually separated into snow albedo (αs ), ice albedo (αi ), and new snow albedo (αns ). Intersite correlations between daily snow and ice albedos were calculated and examined. Observations of snow albedo were collated from all sites into a global dataset, and snow albedo decay was modelled as a function of cumulative  141  6.2. Albedo positive degree days (PDD) following Winther [1993]: α ˆ s = m1 exp (−m2 PDD) + m3  (6.9)  and subsequently modified by atmospheric transmissivity (τ ) to account for variations due to cloud conditions: α ˆ sτ = α ˆ s + m4 τ + m5  (6.10)  where mi are coefficients fitted to global αs observations. Snowfalls occurring after June 1 were removed from the analysis to ensure that the model was representing the albedo decay of a deep winter snowpack. For each site, PDD were calculated from mean daily temperatures. Models using both ambient (Ta ) and observed near-surface (Tg ) temperatures were developed to evaluate the influence of temperature data source on the model results, and the use of maximum daily temperatures for calculating PDD sums was also investigated. Missing observations of Tg were predicted from ambient temperatures following the piecewise analysis developed in Chapter 3. As snowfall dates prior to the start of the ablation season were unknown, albedo models were initiated on May 1 of each observation season. Ice albedos were also examined for systematic variations between sites, and several possible parameterizations were tested. Positive degree days accumulated from the dates of ice exposure, observed ambient temperatures, and atmospheric transmissivity were compared to observed ice albedos. Short-term variations in αi produced by surface water may be related to temperature, as higher temperatures would result in more melt and surface water, leading to lower albedos. Ice albedo was thus modelled as a departures from a mean value of αi , based on atmospheric transmissivity: α ˆ i = m6 τ + m7  (6.11)  Oerlemans and Knap [1998] introduced a firn albedo term to account for the transition between snow and ice surfaces. Sustained negative mass balances [Moore and Demuth, 2001] in the region have depleted firn reservoirs at the 142  6.2. Albedo AWS sites, and as a result few observations of firn albedo were retrieved. However, the inclusion of a firn albedo term will be relevant in regions with extensive firn areas.  Figure 6.9: Observed albedo (α) curves for two stations in 2007 (top), daily precipitation totals at Whistler Low Level, daily atmospheric transmissivity (τ ), and cumulative PDD estimated from ambient temperatures at PM1 (bottom). Gaps in the albedo records are from station malfunctions.  6.2.3  Results  Variability of Inter-site Correlations for Snow and Ice Albedo Observed daily snow albedos among all sites had a mean of 0.65, a maximum of 0.95, and a minimum of 0.44. Mean snow albedos calculated for each site ranged between 0.57 and 0.68, though most ranged between 0.63 and 0.67. Among all sites and observations, mean αi = 0.24, with maxima and minima of 0.47 and 0.08, respectively. Mean ice albedos calculated for each 143  6.2. Albedo site ranged between 0.18 and 0.29. Fresh snowfalls had a mean albedo of 0.79. Daily snow and ice albedos exhibit high correlations among the Place Glacier sites, and weakening correlations with distance (Table 6.9). Correlations between snow albedos observed at Place Glacier range between 0.49 and 0.94 (a negative correlation was found between PM4 and PM1, though the sample size was n = 7). Snow albedo correlations between Place and Weart Glaciers range from -0.10 to 0.51. Similarly, ice albedos at Place Glacier have r values of 0.55 to 0.89. Observed ice albedos at Bridge Glacier have correlations of 0.71 and -0.68 with PM108 and PM308, respectively, though these are statistically insignificant, and based on sample sizes of less than 5.  144  Table 6.9: Simple correlations (r) between snow (top) and ice (bottom) albedo station pairs. Data affected by snowfalls after June 1 were not included in the correlation analyses. Statistically significant correlations at p = 0.05 are indicated in bold.  145  PM1 PM2 PM3 PM4 PM108 PM308 WM1 BM1  PM2  1.00 0.94 0.92 -0.34 -0.1 -  0.94 1.00 0.90 0.49 0.38 -  0.92 0.90 1.00 0.76 0.51 -  PM1.00  PM2  Ice Albedo Correlations PM3 PM4 PM1.0008  1.00 0.62 0.60 0.55 -  0.62 1.00 0.89 0.89 -  0.60 0.89 1.00 0.80 -  -0.34 0.49 0.76 1.00 0.59 0.74 0.24 -0.11  0.55 0.89 0.80 1.00 -  0.59 1.00 0.85 -0.40  1.00 0.71  PM308  WM1  BM1  0.74 0.85 1 -0.11  -0.10 0.38 0.51 0.24 1 -  -0.11 -0.40 -0.11 1  PM308  WM1  BM1  1.00 -0.68  -  0.71 -0.68 1.00  6.2. Albedo  PM1 PM2 PM3 PM4 PM108 PM308 WM1 BM1  Snow Albedo Correlations PM3 PM4 PM108  PM1  6.2. Albedo Albedo Parameterizations Observed daily snow albedos decrease from early season highs of 0.90 to mid-summer values of 0.50, and the PDD albedo decay model captures the seasonal decline (Figure 6.10a). Parameterizing αs solely as a function of cumulative PDD gives mean absolute errors (MAE) of 0.057 (Table 6.10), regardless of the temperature data source (Ta or Tg ). Modelled snow albedo residuals (observed minus predicted) were found to be inversely related to observed daily transmissivity (Figure 6.10b). Including a linear function of τ in the model improved model fit, yielding MAE = 0.048 and RMSE = 0.063. The full albedo model tends to overestimate αs for mature snowpacks, and underestimates αs for less mature snowpacks. Mean daily ice albedo observed at all sites ranged between 0.17 and 0.27, with mean αi = 0.23. No systematic variation in αi related to position on the glacier was observed. Linear regression of all αi and τ observations yielded a weak (R2 = 0.15) negative relation, with higher atmospheric transmissivities corresponding to lower ice albedos (Table 6.10). A weak inverse relation was also observed between mean daily temperatures (Ta ) and ice albedo (not shown), suggesting that melt conditions exert some control on the reflectivity of the surface, with higher temperatures producing a wetter surface, which lowers the ice albedo. However, modelled ice albedo errors were minimized using daily transmissivity. Given the relation between high transmissivities/clear skies and warm temperatures/surface melt during the ablation season, the use of daily transmissivity values will reflect both the effects of cloud filtering and surface water on ice reflectivity. Time-series of predicted and observed albedos (Figure 6.11) demonstrate the usefulness of including atmospheric transmissivity in snow and ice albedo parameterizations. Variability in observed daily snow albedos at both PM1 and PM3 was better reproduced by the full αs model, particularly in the latter half of the melt season at PM3. Variability in daily αi at the Bridge Glacier site was not reproduced by the model (Eq. [6.11]), and it is suspected that, given the field of view of the inverted pyranometer (Figure 6.8), surface water ponding may play a large role in regulating daily ice albedos.  146  6.2. Albedo  Figure 6.10: (a) PDD snow albedo model, (b) snow albedo residuals versus daily atmospheric transmissivity  Table 6.10: PDD parameterization for snow albedo (top), corrections for snow albedo based on atmospheric transmissivity (middle), and ice parameterization (bottom). PDD were calculated from mean daily ambient temperatures, and snowfall events after June 1 were removed from the analysis. Mean absolute error (MAE) and root mean square error (RMSE) are also given as a percentage of the global means Model  n  MAE (%)  RMSE (%)  667  0.057 (8.8)  0.072 (11.1)  α ˆ sτ = α ˆ s − 0.22 τ + 0.15  667  0.048 (7.4)  0.063 (9.7)  α ˆ i = m − 0.15 τ + 0.31  208  0.049 (20.4)  0.070 (29.2)  α ˆ s = 0.17 exp (−8.41 × 10  6.2.4  −3  · PDD) + 0.60  Discussion  Correlations in daily albedos among multiple sites have not been previously reported, as most albedo studies focus on single station observations. While spatial differences in the reflective qualities of the snowpack (dust content, water content) will contribute to albedo differences between sites, the evidence presented above suggests that atmospheric transmissivity can be used to assist in modelling spatial and temporal albedo variability. It is noted that the ratio of diffuse to specular reflection will vary with slope angle and the angle of the sun relative to the slope [Duguay, 1993]. While relatively 147  6.2. Albedo  Figure 6.11: Time series of observed and predicted daily albedos at (top) an ablation zone site (PM1, 2007), (middle) an accumulation zone site (PM3, 2007 ablation season), and (bottom) BM1 in 2008. Model 1 estimates αs using PDD only, and assumes a constant value for αi . Model 2 uses PDD and τ for both αs and αi , and parameterizes fresh snowfalls.  148  6.2. Albedo flat observation sites were chosen for this study, the effects of slope angle on observed albedo were not estimated. Furthermore, it is possible that microshading due to variable surface roughness at the albedo sites will affect the results presented above. These were also not considered in the analysis. High correlations in daily albedo were observed at Place Glacier, where multiple sites were separated by a maximum distance of 2 km. At greater separations, correlations decrease in strength, but this decrease appears to be the result of regional variability in cloudiness and incoming shortwave radiation, as opposed to changes in the surface properties of the respective snowpacks. The inclusion of a transmissivity term in the albedo model helps resolve the differences in spatial variations in albedo, as daily transmissivities can vary greatly over short distances (Table 6.2). The surface albedo model presented in this study uses mean daily temperatures and daily atmospheric transmissivity calculated from measured K↓ to provide regional estimates of snow and ice albedo, and gives an overall RMSE of 0.072. Previous albedo studies, focused on single sites, give similar magnitudes of errors. The albedo parameterization of Oerlemans and Knap [1998] gave RMSE of 0.067, though their model specified albedo as a function of snow depth and days since snowfall, and included accumulation season observations. As in Winther [1993], this study modelled the longterm decay of snow albedo as a function of accumulated PDD, but found that deviations from the PDD decay could be partially explained using daily atmospheric transmissivity. Particularly for sites in the accumulation zone (Figure 6.11b), a combined PDD-τ model provides a strong fit to observed snow albedos. Brock [2004] used cloud cover estimates and cloud types to improve a sub-daily albedo model, but these variables are typically unavailable at remote sites. Ice albedo was also found to be weakly related to τ , though an alternative parameterization relating mean daily temperature (which may describe the amount of water on the ice surface) to αi offered similar modelling skill. Finally, the albedo model developed in this study incorporates variables that can be predicted with reasonable confidence at remote sites, where observations of albedo are generally unavailable.  149  6.3. Longwave Radiation  6.3 6.3.1  Longwave Radiation Background and Objectives  Longwave radiation (wavelengths greater than 3 µm) emitted towards the surface (L↓) is a function of the entire vertical profile of atmospheric air temperature and atmospheric emissivity ( ). Near-surface temperature and vapour pressure measurements are typically used to parameterize L↓ since a majority of the incoming radiation is emitted by the atmosphere near the surface [Zhao et al., 1994; Konzelmann and Ohmura, 1995; Ohmura, 2001]. Sharp boundary-layer thermoclines observed over melting glaciers [Munro and Davies, 1977] present the possibility that near-surface temperatures are inappropriate for estimating L↓, though this issue does not appear to have been addressed in previous studies. Incoming longwave radiation is expressed through the Stefan-Boltzmann law: 4 0 σ Ta  L↓= where  0  (6.12)  is the effective emissivity of the atmosphere, σ is the Stefan-  Boltzmann constant (5.67×10−8 W m−2 K−4 ), and Ta is the near-surface air temperature in K. Rearranging Eq. [6.12], effective emissivity can be computed from observations of incoming longwave radiation and temperature: 0  =  L↓ . σ Ta4  To account for the presence of clouds, function of clear-sky emissivity (  cs )  0  is generally parameterized as a  and a manually observed cloud fraction  n. Alternatively, L↓ is expressed through L↓=  (6.13)  cs F  cs  and cloudiness F :  σ T4  (6.14)  where F represents the increase of L↓ due to clouds, with F ≥ 1. Numerous parameterizations for  cs  are available [Angstr¨om, 1918; Brunt,  1932; Brutsaert, 1975; Konzelmann et al., 1994; Prata, 1996], and these have been repeatedly tested against observational data [Niemel´a et al., 2001; 150  6.3. Longwave Radiation Kjaersgaard et al., 2007; Sedlar and Hock, 2008]. The Prata [1996] model has been applied in low-latitude glacier environments [M¨olg et al., 2009], wheras the Konzelmann et al. [1994] approach has been used extensively at mid-latitude settings [Hock and Holmgren, 1996; Pl¨ uss and Ohmura, 1997; Klok and Oerlemans, 2002; Mittaz et al., 2002; Sedlar and Hock, 2008]. Equation [6.14] requires a value for F , which can be estimated using either manual observations of cloud cover (n), or a further parameterization of cloud cover using atmospheric transmissivity. The latter option is explored here as the L↓ model will be applied to remote locations. Incoming longwave radiation represents a significant component of the surface energy balance, particularly for calculations of nocturnal refreezing. In terms of regional energy balance melt modelling, a longwave radiation parameterization that does not require observations of cloud cover is required, and this portion of the study examines longwave radiation data observed both on and off-glacier. The objectives of this section are (1) to compare longwave radiation collected at sites in and out of the katabatic boundary layer, (2) to test a longwave parameterization for use in regional melt models, and (3) to examine the sensitivity of the longwave radiation models to katabatic boundary layer effects.  6.3.2  Methods  Incoming longwave radiation was measured using Kipp & Zonen CGR3 pyrgeometers (Table 2.3) at an on-glacier site (PM2, 2012 m a.s.l) and an offglacier site (PMR, 2075 m a.s.l) at Place Glacier in 2007 and 2008. Station locations are given in Figure 2.2. Sky view factors for PM2 and PMR are 0.92 and 0.97, respectively, indicating that the terrestrial portion of L↓ will be minimal. Both hourly and daily data were used in the following analyses. Mean daily L↓ were calculated from daily totals of 10-minute observations, and mean daily temperatures and vapour pressures were calculated from hourly means. Hourly and daily totals of longwave radiation observed at the two sites were compared. As both datasets contain errors of measurement, an or-  151  6.3. Longwave Radiation dinary least squares (OLS) regression approach is unsuitable [Mark and Church, 1977]. Orthogonal regressions were thus performed to examine the relation between observed L↓ at on- and off-glacier sites. Effective emissivities calculated at both stations were then compared at hourly and daily timescales. Separate longwave radiation models were developed for both sites following Equation [6.14]. First, clear-sky emissivity parameterizations were fit to observed clear-sky emissivities. Estimates of cloudiness (F ) were then obtained from observations at each site, and F was parameterized as a function of transmissivity calculated at PM3 (Figure 2.2). With  cs  and F , incoming  longwave radiation was estimated from near-surface temperatures observed at both sites. This study uses the clear-sky emissivity parameterization of Konzelmann et al. [1994]: cs  = 0.23 + a1 (ea /Ta )1/b1  (6.15)  where a and b are coefficients fitted to L↓ observations where τ > 0.75 and cs  < 0.80, ea is vapour pressure (hPa), and Ta is temperature (K). The first  term represents the emittance of a completely dry atmosphere, as calculated by LOWTRAN7 [Konzelmann et al., 1994]. Cloudiness F was estimated from observational data by rearranging Eq. [6.14]: F =  L↓ 4 cs σ Ta  (6.16)  and F was then parameterized as a function of atmospheric transmissivity, following Sedlar and Hock [2008]: F = a2 τ b2 + c2 .  (6.17)  With this approach, Sedlar and Hock [2008] found errors in modelled L↓ to be similar to those obtained using direct observations of n and a parameterized 0.  In this study,  cs  and L↓ were modelled using Equations [6.14 - 6.17],  with Ta and ea observed at PM2 and PMR. F was estimated from incoming 152  6.3. Longwave Radiation radiation observed at PM3. Mean absolute error (MAE) and RMSE were used to evaluate model performance, and fitted coefficients were compared between on-glacier and off-glacier sites to identify the effects of station location and the katabatic boundary layer on longwave radiation models. For regionally distributed EB glacier melt modelling, a consideration of the source of data used to drive L↓ parameterizations is required, particularly for large glaciers which are expected to have well-developed katabatic boundary layers during warm weather. Near-surface temperatures and vapour pressures for both ambient and KBL settings can be readily estimated from a regional climate network (Chapter 3), but it is not known if katabatic boundary layer corrections are required for longwave radiation parameterization. A final modelling test quantifies errors associated with modelling L↓ within the katabatic boundary layer by comparing observational data and three separate models. Model 1 represents the ideal case, where observations of L↓, T , and ea within the glacier boundary layer are used to fit a longwave parameterization. Model 2 represents the ambient situation, where glacier surface L↓ is assumed to equal longwave irradiance calculated for sites outside the glacier boundary layer. Model 3 incorporates the results of previous katabatic boundary layer research (Chapter 3) and uses ambient climate data corrected for boundary layer effects with longwave parameterization coefficients determined from on-glacier data. Model errors are compared to identify possible biases produced by katabatic boundary layer effects.  6.3.3  Results  Comparison of on-glacier and off-glacier observations Longwave irradiance at the on-glacier and off-glacier sites is highly correlated (r = 0.91). For hourly data, the slopes of orthogonal regression lines for allsky and clear-sky conditions are 0.97 and 0.89, respectively (Figure 6.12a). This suggests that there are no systematic differences between L↓ observed at the two sites over all sky conditions, but during clear-sky conditions L↓ received at the glacier site tends to be lower than that observed at the ridge 153  6.3. Longwave Radiation station. At daily timescales, longwave observations for both all-sky and clear-sky (τ > 0.75 and  0  < 0.8) conditions show no systematic variation  (Figure 6.12b).  (a) Hourly  (b) Daily  Figure 6.12: Off-glacier (PMR) L↓ versus on-glacier (PM2) L↓ at (a) hourly and (b) daily timesteps Differences in incoming L ↓ (∆L ↓, calculated as PM2L ↓ - PMRL ↓) showed no relation to ea measured at the ridge site (not shown), but biases become apparent when ∆L↓ is compared with PMR T and L↓ (Figure 6.13). 154  6.3. Longwave Radiation For ambient temperatures between -10 and 10◦ C, ∆L↓ are evenly scattered around zero, and the amount of scatter is large. Above 10◦ C, ∆L↓ becomes small, but appears to be biased toward lower values of L↓ at the on-glacier station. The change in amount of scatter suggests that cloud conditions are less variable between the two sites at higher temperatures, when strong high-pressure situations would be expected. Mean transmissivity calculated when off-glacier temperatures are greater than 10◦ C is 0.67, demonstrating support for this hypothesis. The small bias between the two stations at higher temperatures is likely a result of katabatic boundary layer effects, described previously in Chapter 3. The depth of katabatic flow above the surface (approximately 10 - 20 m, depending on the temperature forcing and slope) will exhibit temperatures cooler than the surrounding air, leading to a marginal decrease in L↓ observed at the glacier surface. At larger glaciers, a deeper katabatic layer may lead to an even more pronounced reduction in L↓ relative to an ambient station, though this cannot be verified without further measurements.  155  6.3. Longwave Radiation  (a) Hourly  (b) Daily  (c) Hourly  (d) Daily  Figure 6.13: Hourly and daily differences in longwave radiation measured at an off-glacier site (PMR) and an on-glacier site versus (a and b) near-surface air temperatures at the off-glacier station, and (c and d) L↓ measured at the off-glacier station. Effective atmospheric emissivities calculated from Eq. [6.13] for all-sky and clear-sky conditions are compared between the on-glacier and off-glacier sites in Figure 6.14. Atmospheric emissivities appear to be higher at the onglacier site than the off-glacier site, particularly during clear-sky conditions when  0  is low. At higher values of  0  calculated emissivities appear to be  similar at both sites. As demonstrated previously, incoming L↓ at the two sites is essentially the same, and so differences between  0  at the on-glacier  and off-glacier sites (∆ 0 ) result from the temperature used in Eq. [6.13] for calculating  0.  Figures 6.14(c) and (d) illustrate the relation between ∆  0  156  6.3. Longwave Radiation and off-glacier temperatures. Emissivities at the two sites are highly correlated (r = 0.92 for hourly  0 ),  but the generation of katabatic flows during  warm and clear conditions leads to substantially lower temperatures at the on-glacier site, which results in a higher effective atmospheric emissivity.  (a) Hourly  (b) Daily  (c) Hourly  (d) Daily  Figure 6.14: On-glacier (PM2) versus off-glacier (PMR) effective emissivities for hourly (a) and daily observations (b), and differences in effective emissivity (PM2 0 - PMR 0 ) against hourly (c) and daily (d) mean temperatures observed at the off-glacier station. All-sky conditions are given in grey, and a subset of observations for clear-sky conditions is given in black. Atmospheric Emissivity Modelling Fitted coefficients obtained using the Konzelmann et al. [1994] parameterization for hourly and daily clear-sky emissivity (Equation [6.15]) are given 157  6.3. Longwave Radiation  Table 6.11: Clear-sky emissivity parameterizations obtained from hourly and daily on-glacier and off-glacier observations of L↓, temperature (Ta ), and vapour pressure (ea ). For hourly data, clear-sky emissivities were defined by τ > 0.75 and 0 < 0.8. A threshold of mean daily τ > 0.70 was used to identify clear-sky days. For comparison, fitted coefficients obtained by other studies are given. Site  cs  Hourly PM2 PMR St¨ orglaciarena West Greenlandb  = 0.23 + a1 (ea /Ta )1/b1  MAE  RMSE  n  a1 a1 a1 a1  0.020 0.020 -  0.025 0.025 -  235 284 -  0.010 0.014  0.014 0.018  25 23  = 1.14, b1 = 0.82, b2 = 0.44, b1 = 0.44, b1  = 5.01 = 7.75 = 7.00 = 8.00  Daily PM2 a1 = 1.54, b1 = 3.63 PMR a1 = 0.88, b1 = 6.73 a Sedlar and Hock [2008] b Konzelmann et al. [1994]  in Table 6.11. With hourly temperatures, vapour pressures, and longwave radiation observed at the on-glacier and off-glacier sites, modelled hourly  cs  yielded MAE of 0.02 at both PM2 and PMR, representing between 2 and 3% of the mean clear-sky emissivity, respectively. Interestingly, coefficients obtained for the off-glacier station are more similar to those obtained for on-ice stations in Greenland [Konzelmann et al., 1994] and St¨orglaciaren [Sedlar and Hock, 2008]. Mean absolute errors (and percent of mean) for modelled daily  cs  were  0.01 (1%) for both PM2 and PMR. Coefficients obtained for the daily  cs  models were different from the hourly values. Modelled and observed clearsky emissivities for both hourly and daily models are given in Figure 6.15. At both on-glacier and off-glacier sites, the hourly  cs  both high and low values of  is unbiased across the  range of observed  cs ,  whereas the daily  cs  model is biased at  cs .  158  6.3. Longwave Radiation  (a) Hourly  (b) Daily  Figure 6.15: Predicted versus observed clear-sky emissivities at (a) hourly and (b) daily timescales Cloud Factor Modelling Following the approach of Sedlar and Hock [2008], cloud factor F was estimated at hourly and daily time steps from Equation [6.17], with  cs  calcu-  lated from Equation [6.15]. Table 6.12 gives the fitted coefficients and model MAE and RMSE, while Figure 6.16 illustrates the relation between τ and F and shows the model results. Fitted coefficients for the hourly cloud factor models are similar to those obtained by Sedlar and Hock [2008], suggesting that the parameterization is portable between locations and climates. Mean absolute errors for the hourly F models were 0.070 and 0.087 at PM2 and PMR, respectively, representing 6 and 8% of the mean. Similar errors in modelled F were obtained using daily averages.  159  6.3. Longwave Radiation  Table 6.12: Cloud factor F models as a function of τ for hourly and daily timesteps, with fitted coefficients, mean absolute error (MAE), root mean square error (RMSE), sample size (n). Site  F (τ ) = −a2 τ b2 + c2  MAE  RMSE  n  Hourly PM2 PMR St¨ orglaciarena  a2 = 0.38, b2 = 1.69, c2 = 1.29 a2 = 0.46, b2 = 1.62, c2 = 1.34 a2 = 0.49, b2 = 1.62, c2 = 1.34  0.070 0.087  0.091 0.114  1669 1524  0.053 0.064  0.070 0.084  150 137  Daily PM2 a2 = 0.61, b2 = 2.98, c3 = 1.25 PMR a2 = 0.76, b2 = 2.89, c2 = 1.31 a Sedlar and Hock [2008]  160  6.3. Longwave Radiation  (a) Hourly  (b) Daily  (c) Hourly  (d) Daily  Figure 6.16: Modelled (lines) and observed (points) cloud factor F versus atmsopheric transmissivity (τ ) for PM2 (a and b), and PMR (c and d)  161  6.3. Longwave Radiation Incoming Longwave Radiation Modelling Using observed temperature, vapour pressure, and calculated atmospheric transmissivity, incoming longwave radiation was estimated at hourly and daily time-steps for each station using the model coefficients for  cs  and  F established above. Table 6.13 presents a summary of the model errors. Mean absolute errors (in W m−2 and % of mean) for modelled hourly L↓ were 17.70 (6.09%) and 21.88 (7.58%) for PM2 and PMR, respectively. For daily mean L↓, MAE (%) were 13.41 (4.58) and 15.89 (5.86) for PM2 and PMR, respectively. Mean bias errors for all models range from 0.81 to 1.48. Time-series of observed and predicted L↓ (Figure 6.17) demonstrate that both hourly and daily models are capable of modelling variations in incoming longwave radiation. The hourly model is sensitive to variations in transmissivity, which is used to parameterize cloudiness. Boxplots of residual (observed minus predicted) longwave radiation by hour of day (Figure 6.18) demonstrate that hourly model performance at both sites is unaffected by changes in effective emissivity due to solar path length. However, the longwave model performs poorly at lower temperatures. Figure 6.19 illlustrates that at both sites and on both hourly and daily timescales, observed L↓ exceeds modelled L↓ by as much as 100 W m−2 at temperatures of 0◦ C. The fact that this bias occurs in both on- and offglacier models suggests that it is not related to katabatic flows. With higher than expected values of L↓ occurring during cooler temperatures, it is possible that strong surface inversions may be contributing to the poor model performance. Both hourly and daily longwave models formulated in this study perform well during warmer periods, a characteristic which is critical for melt modelling applications. To demonstrate different methods for modelling longwave irradiance at glacier sites, three models were compared. For the first model, L ↓ was modelled using ridge station coefficients and data, which represents the most practical modelling approach. The second model uses PM2 coefficients and ambient Ta and ea , estimated from regional climate network. The third model uses PM2 coefficients, but applies temperature and vapour pressure  162  6.3. Longwave Radiation corrections to the ambient climate data (Chapter 3). Results indicate that using on-glacier L↓ coefficients with KBL-corrected climate data (Model 3) offers the best performance, and even gives lower errors in modelled L↓ than observed on-ice climate data (Table 6.13; Figure 6.20). Mean absolute errors in daily average L↓ are approximately 4% using Model 3. Greater errors are incurred if longwave radiation is estimated at glacier sites using ambient climate data and coefficients obtained from ambient L↓ observations (Model 1), though there appears to be no bias. Without correcting for the katabatic boundary layer effects, the use of on-glacier coefficients with ambient data provides hourly and daily L↓ predictions that are biased by +11.91 and +7.83 W m−2 respectively. Integrated over the course of the ablation season, this bias would result in a substantial overestimation of melt. Results of the longwave model tests suggest that a consideration of katabatic boundary layer effects is important, even on relatively small glaciers. The optimum parameterization of longwave radiation uses coefficients estimated from on-glacier observations and ambient climate data that is corrected for katabatic boundary layer effects, indicating that it is portable between sites and glacier scales.  163  Table 6.13: Summary of hourly and daily longwave model errors for PM2 and PMR. Hourly values of L↓ were modelled only during observations of direct sunlight. MAE = mean absolute error, MBE = mean bias error, RMSE = root mean square error, n = sample size, kPM2 = on-glacier parameterization, kPMR = off-glacier parameterization Model  MAE W m−2 (%)  MBE W m−2  RMSE W m−2 (%)  n  Hourly PM2 L↓ PMR L↓  kPM2 , PM2 T, ea kPMR , PMR T, ea  17.70 (6.1) 21.88 (7.6)  0.81 1.58  22.98 (7.9) 22.10 (9.7)  1670 1527  Daily PM2 L↓ PMR L↓  kPM2 , PM2 T, ea kPMR , PMR T, ea  13.41 (4.6) 15.89 (5.5)  1.11 1.48  17.17 (5.9) 19.96 (6.9)  150 137  Testing: Hourly PM2 L↓ 1) kPMR , PMR Ta , ea PM2 L↓ 2) kPM2 , Ambient Ta , ea PM2 L↓ 3) kPM2 , KBL corrected Ta , ea  20.82 (7.2) 24.50 (8.4) 17.95 (6.2)  0.72 11.91 1.10  26.56 (9.1) 30.72 (10.6) 23.34 (8.0)  1670 1670 1670  Testing: Daily PM2 L↓ 1) kPMR , PMR T, ea PM2 L↓ 2) kPM2 , Ambient Ta , ea PM2 L↓ 3) kPM2 , KBL corrected Ta , ea  14.56 (5.0) 18.22 (6.2) 12.32 (4.2)  -1.40 7.83 1.03  18.75 (6.0) 22.22 (7.6) 15.54 (5.3)  150 150 150  6.3. Longwave Radiation  Site  164  6.3. Longwave Radiation  (a) PM2 Hourly  (b) PMR Hourly  (c) PM2 Daily  (d) PMR Daily  Figure 6.17: Time-series of hourly (a and b) and daily (c and d) observed and modelled incoming longwave radiation at on-glacier and off-glacier sites. Hourly data only shows a portion of the 2008 season for clarity, and modelled daily L↓ is incomplete due to missing transmissivity data (malfunction of PM3, where K↓ was recorded). 165  6.3. Longwave Radiation  (a) PM2  (b) PMR  Figure 6.18: Boxplots of hourly L↓ residuals by hour of day. Only hours where transmissivity could be calculated were used in the analysis.  166  6.3. Longwave Radiation  (a) Hourly, PM2  (b) Hourly, PMR  (c) Daily, PM2  (d) Daily, PMR  Figure 6.19: Modelled L↓ residuals (observed - predicted) versus observed temperatures at the ridge station, for both hourly and daily timescales  167  6.3. Longwave Radiation  Figure 6.20: Scatterplots of daily longwave radiation modelled using (a) onglacier site observations and optimized coefficients, (b) off-glacier site observations and optimized coefficients, (c) ambient climate data and on-glacier coefficients, and (d) ambient climate data corrected for katabatic boundary layer effects and on-glacier coefficients. Mean absolute errors (MAE) and root mean squared errors (RMSE) are given in W m−2 .  168  6.3. Longwave Radiation ‘  6.3.4  Discussion  Comparisons of longwave radiation observed at on-glacier and off-glacier sites suggest that katabatic boundary layer effects on L ↓ are negligible, though in clear-sky conditions the on-glacier site appears to receive decreased L↓. Ohmura [2001] concluded that 2/3 of the available incoming L↓ is derived from the first 100 m of the atmosphere. As glacier boundary layer depth varies with flow path length and thus the size of the glacier (Chapter 5), it cannot be assumed that the relations observed at Place Glacier hold true for larger glaciers. Longwave irradiance may in fact be affected by strong katabatic flows at larger glaciers. However, in the absence of detailed L↓ measurements at larger sites, the analyses presented above suggest that longwave irradiance can be modelled successfully using off-glacier parameterizations and katabatic boundary layer corrections, which will account for the size of the glacier and the site location relative to a developing katabatic boundary layer. Higher effective emissivities ( 0 ) were observed at the glacier site (Figure 6.14), particularly for clear-sky and warm conditions. However, as differences in L↓ between the two sites were small, the differences in  0  appear  to compensate for katabatic boundary layer (KBL) effects. Warm, clear-sky days will produce strong katabatic flows and strong temperature inversions near the surface, requiring an enhanced  0  at the glacier site to obtain L↓  values similar to those observed at the off-glacier site. This observation suggests that emissivity parameterizations should be treated cautiously, as local climate characteristics will alter  cs  model coefficients. For example,  the optimization procedure for the two sites produced different  cs  models  (Table 6.11) due to KBL effects, despite the similarities in incoming L↓. Incoming longwave radiation was modelled at hourly and daily resolutions following the methods of Sedlar and Hock [2008]. A clear-sky emissivity (  cs )  was estimated for each time step from near-surface temperature and  vapour pressure (Eq. [6.15]), a cloud factor (F ) was empirically determined  169  6.4. Recommendations from observed atmospheric transmissivity, and incoming longwave radiation was calculated following Eq. [6.14]. Coefficients obtained by fitting cloudiness F to τ , though weak, are nearly identical to those observed by Sedlar and Hock [2008]. As transmissivity can be modelled at remote locations using sparse climatological networks, this approach provides a means for improving estimates of longwave radiation without requiring manual observations of clouds. Given that the transmissivity model requires estimates of the diurnal temperature range (∆T ), the longwave modelling process may be simplified further by modelling the cloud fraction F direcly as a function of ∆T . Further investigations are required to identify the value of such an approach. Longwave irradiance at the glacier surface can be modelled from ambient climate data with minimal errors, provided the driving data is suited to the optimized model. Without correcting for KBL effects, predictions of L↓ made from ambient climate data and on-glacier emissivity parameterizations are biased by +11.91 (+7.83) W m−2 on hourly (daily) timescales (Table 6.13). Applying KBL corrections to ambient Ta and ea and modelling L↓ with glacier-optimized coefficients provides a prediction skill equal to using the actual station observations, with biases of -0.72 (-1.03) W m−2 at hourly (daily) timescales. However, the transferrability of emissivity parameterizations is problematic due to variations in the depth of the KBL and the strength of boundary-layer cooling that has been observed over short distances. Estimates of on-glacier L↓ made from ambient climate data and models optimized with off-glacier data provide a reasonable alternative, with mean absolute errors of 5% at daily timescales. As the coefficients optimized to off-glacier data are more transferrable than the on-glacier model, this approach is recommended for the distributed energy balance modelling approach described in Chapter 8.  6.4  Recommendations  Melt models often use a single reference station with measurements of K↓ for estimating daily or hourly transmissivity. While this is encouraged for 170  6.4. Recommendations reference stations that are proximal to the study site, results presented above suggest that at distances greater than 30 km, modelling transmissivity and potential solar radiation will reduce errors in estimating K↓. Particularly on hourly timescales, variability in cloud cover between sites creates large errors if solar radiation is extrapolated from a single station. The inclusion of a vapour pressure term improves the modelling skill of transmissivity and K↓, though the systematic errors in transmissivity are still large, and further investigation of model fitting procedures is recommended. Snow and ice albedos exhibit strong daily correlations over small (≈ 2 km) distances, and virtually no correlation at greater distances. A likely explanation for the reduction in correlation with distance involves differences in transmissivity, as clouds preferentially absorb longer-wavelength radiation, resulting in increased surface albedo. While a simple positivedegree-day model captures the general decay of snow albedo with modelled errors of 8.9%, the inclusion of an atmospheric transmissivity term reduces modelled snow albedo errors to 7.4%. Observations of longwave radiation at on- and off-glacier sites suggest that the effects of temperature suppression within the katabatic boundary layer is insufficient to cause a reduction in longwave irradiance. However, models of longwave radiation demonstrate that a potential bias of ≈ 12 Wm−2 is introduced if ambient temperatures are used to estimate L↓ with coefficients obtained from on-glacier observations. Results from this research recommend the use of on-glacier stations to optimize longwave model parameters, and the use of temperatures and vapour pressures corrected for katabatic boundary layer effects to estimate L↓. This approach may help account for changes in longwave irradiance due to deep katabatic boundary layers at large glaciers, though further observations are required to confirm this assumption.  171  Chapter 7  Developing Datasets for Testing Glacier Melt Models 7.1  Background and Objectives  Models of glacier melt are typically tested against glacier mass balance observations (e.g. Hoinkes and Steinacker [1975]; Greuell and Oerlemans [1986]; Munro [1991]; Oerlemans [1992]; J´ohanneson et al. [1995]), which provide a record of both the initial snow water equivalence (SWE0 ) and total snow and ice melt (in water equivalence) at a given location over the ablation season. Initial values of SWE0 are critical, as the transition from snow to ice melt represents nearly a doubling of the melt rate. Other options for evaluating melt model performance include pro-glacial discharge measurements [Braun and Aellen, 1990; Hock, 1999; Shea et al., 2005; de Woul et al., 2006; Zhang et al., 2007], remotely sensed snowline elevations [Østrem, 1975; Turpin et al., 1997; Demuth and Pietroniro, 1999], and manual [Marcus et al., 1985; Hay and Fitzharris, 1988] or automated [Oerlemans, 2000; M¨ olg and Hardy, 2004; Shea et al., 2004; van de Wal et al., 2005; Andreassen et al., 2008; Pellicciotti et al., 2008] measurements of snow depth or surface height changes. This chapter details approaches used for generating initial and final SWE conditions used for glacier melt modelling. Annual mass balance (bn ) measurements quantify the difference between net winter accumulation (bw ) and net summer ablation (bs ): bn = bw − bs ,  (7.1)  with bs defined as a positive quantity. Accumulation and ablation measure172  7.1. Background and Objectives ments are typically made at point locations covering the elevation range of a given glacier. Winter balance (or initial snow water equivalence, SWE0 ) is calculated as SWE0 = d0  ρs ρw  (7.2)  where d0 is snow depth (in m), ρs is the density of snow (kg m−3 ) and ρw is the density of water (1000 kg m−3 ). Density is sampled less intensively than depth [Jansson, 1999], as it exhibits less spatial variability, and density measurements require more time and effort than snow depth soundings. Winter snowpack properties, including depth and density, vary in both time and space. Large-scale spatial variations in snow depth are determined mainly by meteorological patterns and topography, though generally snow depth increases with elevation. Avalanches may also affect snow distributions [Machguth et al., 2006], though the influence of avalanching are typically neglected in mass balance calculations. Snow densities are a function of primary depositional properties and secondary in situ metamorphism [Harper and Bradford, 2003]. The density of new snow varies with initial grain size and shape, temperature, precipitation rate, and wind speed during and following deposition. Density changes through time arise through metamorphism, as a result of overburden pressure and heat and mass transport. Mass balance studies typically ignore physically based models for estimating snow density in favour of simple interpolation methods [Jansson, 1999]. Physically based approaches for estimating snow densities, developed for large-scale hydrological models, incorporate both mechanical [Kojima, 1967; Pitman et al., 1991] and thermal [Walter et al., 2004] effects that contribute to snow densification, though their implementation is data-intensive. Water percolation in a dry snowpack has been shown to cause a rapid initial densification due to grain rearrangement and structural changes [Marshall et al., 1999], and the densification rate due to water percolation is orders of magnitude faster than that observed through overburden pressures in dry snowpacks [Marshall et al., 1999]. Empirical approaches to modelling snow 173  7.1. Background and Objectives density have incorporated variables such as time [Verseghy, 1991], temperature [Hedstrom and Pomeroy, 1998], and season, snow depth, and elevation [Jonas et al., 2009]. Ablation is defined as the loss of mass from an ice-body through processes of melt and sublimation/evaporation. At a point location, summer balance (bs ) is the sum of snow melt (Ms ) and ice melt (Mi ), in m w.e.: bs ≈ Ms + Mi .  ρi  SWE0 + ∆h , Ms = SWE0 ρw ≈ ρs  df Ms < SWE0 , Mi = 0 ρw  (7.3) (7.4)  where ∆h is the change in ice surface height (m) from the previous summer’s ice surface, ρi is the density of ice (900 kg m−3 ), df is the final snow depth, and ρs is the final snow density (kg m−3 ). Where ice melt is observed, snowmelt is assumed to equal the initial SWE (Ms = SWE0 ). Elevation of the transient snowline (ZSL ) during the summer melt season also indicates the location where Ms = SWE0 , providing another avenue for testing the snow component of glacier melt models. Remotely sensed data, both optical and microwave, provide a method for estimating ZSL , as the spectral characteristics of snow and ice allow for the discrimination of snowlines [Østrem, 1975; Sidjak, 1999]. However, in a given year the number of scenes suitable for estimating snowline elevations can be severely limited due to cloud cover, particularly in mountainous regions. For seasonal snow cover in forested areas, Lundquist and Lott [2008] demonstrated an alternative approach using ground-based temperature observations to estimate the date of snowline disappearance. This study presents the first application of surface temperature loggers on glaciers for estimating the elevation of the transient snowline through the ablation season and generating data for testing glacier melt models. To estimate bw or SWE0 at locations not directly sampled and to generate glacier-wide mass balance estimates, techniques for interpolating SWE observations are required. Numerous statistical and semi-physical methods  174  7.1. Background and Objectives for distributing SWE have been explored in the snow hydrology literature [Balk and Elder, 2000; Winstral et al., 2002; Anderton et al., 2004; L´opezMoreno and Nogu´es-Bravo, 2006; Carturan et al., 2009], and kriging has been examined for mass balance studies [Hock and Jensen, 1999]. These models, however, are data-intensive, and glacier mass balance studies are typically limited by the sparse density of the observational network. While snow distributions on glaciers will contain some horizontal variability [Jansson, 1999], it is generally assumed that vertical gradients in snow accumulation are much greater [Cogley, 1999], given the relatively smooth surface and lack of forest canopy or other factors that would tend to increase horizontal variability. Ordinary least squares regressions of various forms are thus often used to describe the variations in SWE with elevation [Fountain and Vecchia, 1999], though comparisons with other methods are rarely presented. An interpolation method that has not been applied previously to mass balance studies is the locally weighted regression (Loess) technique [Cleveland, 1979]. In the absence of a physically based model for distributing SWE, the Loess approach offers a number of benefits over other interpolation techniques. First, the Loess technique is considered to be robust to outliers, which is important given both the heterogeneity of factors controlling glacier mass balance and the observation of systematic outliers in previous mass balance measurements [Adams et al., 1998]. Second, the Loess technique allows for insights into local accumulation patterns that might otherwise be lost if a polynomial function is forced through data observed at all sites. Finally, Loess functions can be generated for each site without any assumptions regarding the form of the relation between elevation and accumulation. This chapter demonstrates the application of Loess interpolation techniques for glacier mass balance observations. A number of studies have used pro-glacial discharge measurements for testing melt model results. However, suitable discharge measurements can only be obtained at sites meeting specific criteria (i.e. a single stable proglacial discharge channel, little or no groundwater contribution to streamflows, limited snow melt inputs), and repeat discharge measurements are required throughout the ablation season to generate rating curves. Dis175  7.2. Data and Methods charge measurements were not used in this study, and are mentioned here for reference only. Automated surface height measurements provide point measurements of ablation when an appropriate estimate of density is available. As noted by Andreassen et al. [2008], surface height observations should only be used for model calibration and testing when surface densities are known, which limits their use to periods of ice melt only. Assuming a constant snow density during the melt season in order to convert changes in snow depth to a water equivalent will likely introduce errors in daily melt totals due to snowpack densification over the melt season. Objectives of this chapter are to develop datasets for testing glacier melt models. Specifically, this research explores methods for distributing snow density through the development and application of a physically based snow density model, and compares ordinary least squares and locally weighted regression techniques for interpolating observed SWE0 . An error analysis for point mass balance measurements is given, and the methods and results of a new surface-based observational study of snowline retreat are presented as an complementary source of data for melt model testing.  7.2  Data and Methods  This chapter utilizes mass balance and SWE0 observations, surface-based snowline retreat observations, and climate data collected from four sites in the southern Coast Mountains between 2006-2008. Snow course data collected by the River Forecast Center of the British Columbia Ministry of Environment [British Columbia Ministry of Environment, 2010] were also used in this analysis. Descriptions of the study areas and sampling methods are given in Chapter 2.  7.2.1  Snow Density Modelling  Following the methods presented in Chapter 2, mean snow densities were calculated for each mass balance density pit. Elevations for the snow depth  176  7.2. Data and Methods soundings were extracted from LiDAR DEMs of the study glaciers using GPS coordinates. Mean snow densities were also obtained for three snow courses in the region [British Columbia Ministry of Environment, 2010]. A model for spatially distributing snow densities was developed using cumulative positive degree days (PDD) and observed snow densities, where cumulative PDD were calculated from mean daily temperatures greater than 0◦ C. Hourly vertical temperature gradients (γT ) and sea-level intercepts (TZ=0 ) from January 1 onward were calculated for each season from nearsurface temperatures observed at six ambient climate stations (Chapter 3). Mean daily γT and TZ=0 were used to calculate daily mean temperatures for the elevation of each density pit and snow course site, and cumulative positive degree days (PDD) at each site were computed as the sum of mean daily temperatures (greater than 0◦ C) to the date of observation. Observed mean densities were then related to cumulative PDD using a logistic regression: ρs (P DD) =  a 1 + b exp(−c · P DD)  (7.5)  where a, b, and c are fitted coefficients representing a theoretical maximum snow density (a), and the rate of snow densification (b and c) with respect to cumulative PDD. At each stake location, accumulated PDDs were calculated using elevations extracted from a LiDAR DEM, mean daily γT and TZ=0 , and a mean density computed using Equation [7.5]. Errors in mean pit densities (observed - predicted) were then interpolated over the glacier surface assuming a linear relation with elevation, and then added to predicted density at each site to correct for bias. Densities at sounding sites should therefore reflect both the densification associated with accumulated PDD and the individual site conditions (rain events, precipitation rates, etc.) that contribute to spatial variability in snow density. Snow densities calculated via (1) simple linear interpolation between density pits and (2) the PDD-density and bias correction approach were compared for two sites to evaluate model performance.  177  7.2. Data and Methods  7.2.2  Interpolation of Initial SWE  Using elevation (Z) as the independent variable, two methods for distributing the dependent variable SWE0 were evaluated in this study. Ordinary least squares (OLS) regressions and locally weighted (Loess) regressions were developed for each glacier and season. Quadratic forms were assumed for the OLS models, as mass balance profiles for valley glaciers tend to be quadratic [Dyurgerov, 2002; Radi´c et al., 2007, e.g.]. However, linear OLS regressions were adopted where appropriate. Initial SWE observations were weighted according to the number of snow depth soundings used to calculate the mean snow depth. Individual soundings were given a weighting of 0.2, such that the standard 5-point means were weighted at a value of 1.0. In Loess regression, a locally weighted regression is constructed around each data point [Cleveland, 1979]. The proportion of total observations that are included in the calculation at each point is defined by the span f . The form of the local regression can be either linear (degree d = 1) or quadratic (d = 2), and Jacoby [2000] recommends linear forms for monotonic data. Optimal values for f were determined by calculating corrected Aikaike’s Information Criterion (AICc) for nonparametric regressions [Hurvich et al., 1998] for f = 0.1 to f = 0.9, and both local linear and quadratic Loess forms were tested. The model with the lowest AICc was then selected. To compare the two interpolation methods, a modified F -test [Jacoby, 2000; Bekele and McFarland, 2004] was applied to determine if the increased complexity of the Loess model was justified. The F -statistic was calculated as: F =  (RSSo − RSSl )/(dfl − dfo ) RSSl /(n − dfo )  (7.6)  where RSSo is the residual sum of squares for the OLS model, RSSl is the residual sum of squares for the Loess model, dfl is the equivalent number of parameters in the Loess model (reported by the Loess function), dfo is the number of parameters in the OLS model, and n is the number of observations. p-values for each F-statistic were reported as the probability of making a Type I error, which in this case represents erroneously rejecting the null hypothesis that there are is no advantage to using the Loess approach. 178  7.2. Data and Methods  7.2.3  Mass Balance and Snowline Retreat Data  Ground-based observations of snowline retreat (Chapter 2) were collected to form a second independent data set for melt model testing. Mean daily temperatures (T d ) were extracted from hourly temperatures recorded at each submersible logger site. For each record, the date of snowline retreat was estimated using a T d threshold of 2◦ C, to ensure that the sensor was fully exposed. Transmission of solar radiation through shallow snowpacks may result in temperatures greater than zero for short periods, but it is assumed that mean daily temperatures consistently greater than 2◦ C indicate complete exposure of the sensor. The standard deviation of daily temperature was also calculated to represent the change from snow-covered (low diurnal variability) to exposed (high diurnal variability due to solar heating), and the results were similar to those given by the 2◦ C threshold. Summaries of the dates of snowline retreat versus elevation were qualitatively examined for interannual and regional variability. As the date of snowline retreat corresponds to the date at which cumulative snow melt Ms = SWE0 , an estimate of (Ms ) at each Tidbit location was derived from either direct winter balance measurements or interpolated SWE0 . Values for Ms obtained at the Tidbit sites thus form a second dataset for testing melt model performance. At sites with both mass balance and snowline retreat observations, dates of snowline retreat were compared with observed summer and net balances to assess the potential for estimating net balances at sites without full mass balance monitoring.  7.2.4  Error Analysis  Estimates of total summer ablation at a given location are subject to several sources of error. There are simple measurement errors for snow density, snow depth, stake height, though these are assumed to be small [Adams, 1966]. Unassessable errors include possible ablation stake subsidence due to melting at the base of the pole [Sharp, 1951; Munro, 2006] and internal accumulation of refrozen meltwater within the snowpack at accumulation zone sites [Trabant and March, 1999]. 179  7.2. Data and Methods This error analysis focuses on errors in the estimation of (1) initial snow water equivalence (SWE0 ; m w.e.) and (2) the final melt total (M ; m w.e.). Assuming that initial snow density (ρ0 ) and snow depth (d0 ) are uncorrelated, errors in initial SWE (σSWE(0) ) can be calculated as: 2 σSWE(0)  =  2 σd(0)  SWE20  d20  +  σρ20 ρ20  (7.7)  where σd(0) is the sample standard deviation of repeat snow depth measurements, and σρs(0) is assumed to equal the root mean squared error (RMSE) of modelled snow density. At mass balance stake sites, errors in melt measurement (σM ) are calculated differently for accumulation and ablation zones: 2 σM =  2 2 σSWE(0) + σM I  bn < 0  2 σSWE(0)  bn > 0  +  2 σSWE(f)  (7.8)  In the ablation zone (bn < 0), errors in total melt (σM ) are a function of errors in both ice melt measurement (σMI ) and initial winter equivalence (Equation [7.8]). Above the transient snowline line elevation (bn < 0), σM is a function of initial and final measurements of snow water equivalence. Errors in ice melt measurement arise from errors in stake height measurements (σh ) and errors in the densities of ice (σρi ): 2 σM = σh2 I  ρi ρw  2  + σρ2i  h ρw  2  (7.9)  Errors in stake height measurements (σh ) are assumed to equal 0.10 m, a conservative estimate that takes into account stake subsidence and measurement error. The assumed glacier ice density of 900 kg m−3 [Paterson, 1994] is below the density given for pure ice (917 kg m−3 ), but errors associated with this assumption are unknown, and the second part of Eq. [7.9] is set to zero. Final SWE measurement errors (σSWEf ) are determined by errors in the  180  7.3. Results final snow depth measurement (df ) and in the final snow density (ρf ): 2 σSWE = σd2 f  ρf ρw  2 2 + σρ(f )  df ρw  2  (7.10)  where df is the final snow depth measurement. Repeated measurements of end-of-summer snowpack depths at each sampling site were not obtained during this study, and the error in end-of-summer snow depth measurements (σd(f ) ) was assumed to equal the average standard deviation of observed winter snow depths (σ d(0) ). Maximum snow densities and errors in estimated snow density (σρ ) were obtained from the snow density model (Eq. [7.5]).  7.3 7.3.1  Results Snow Density Modelling  Mean snow densities are strongly related to cumulative positive degree days (Figure 7.1). Coefficients for the logistic regression (Table 7.1) suggest a maximum snow density of density of 552 kg m−3 during the ablation season. Comparable values can be found in the literature, with late ablation season snow densities at the Haig Glacier in the Canadian Rockies ranging between 506 and 593 kg m−3 [Shea, 2004], and between 590 and 630 kg m−3 in the North Cascades [Pelto and Riedel, 2001]. Snow density residuals do not appear to be related to either snow depth (Figure 7.1b) or elevation (Figure 7.1c). Setting cumulative PDD to zero, Equation [7.5] yields an modelled initial snowpack density of 381 kg m−3 . At PDD = 0◦ C, the spread of observed ρs is greatest (Figure 7.1d), which suggests the need for corrections based on local conditions.  181  7.3. Results  Table 7.1: Snow density (ρs ) model (Equation [7.5]) coefficients fitted to mass balance density pits and snow course data, with positive degree days (PDD) calculated from January 1. Fitted parameter estimates (and 95% confidence intervals), root mean squared error (RMSE), and sample size (n) are given. a  b  c  RMSE (kg m−3 )  n  552 (±16.1)  0.45 (±0.07)  0.026 (±0.01)  36.55  78  To examine the performance of the PDD-density model versus a standard linear interpolation of observed densities, two examples were considered. Through temperature and density profiles, it was established whether the snowpack was isothermal or polythermal. Isothermal snowpacks have a near-uniform density and temperature (0◦ C) at all depths, indicating the ocurrence of water percolation and melt onset. Polythermal snowpacks exhibit more variable densities and a range of temperatures below 0◦ C, evidence that no water percolation has occurred, and melt onset has yet to occur. Recalling that snow density is typically measured at two (or more) locations during the measurement of SWE0 , two case studies were extracted from the observational data: (1) polythermal snowpack conditions exist at both pits (Place Glacier, 2006), and (2) an isothermal pit exists at the low elevation site and a polythermal exists pit at the high elevation site (Bridge Glacier 2007). Using the PDD density model, density bias corrections were calculated for each glacier from linear regressions of observed minus predicted ρs versus elevation. At Place Glacier (Figure 7.2), there is little difference between the two densities, as accumulated PDDs to the date of observation (April 21) were essentially zero over the entire glacier surface. At Bridge Glacier, cumulative PDDs of 120◦ C were estimated for lowest elevations, while PDDs were less than 30◦ C at the highest elevations. The resulting density distributions differ by up to 30 kg m−3 between the two density interpolation approaches. A density error of this magnitude corresponds to a difference of 0.12 m w.e. based on a 4 m snowpack, demonstrating the potential utility of the PDD-density approach, particularly for glaciers with large vertical 182  7.3. Results  (a)  (b)  (c)  (d)  Figure 7.1: Accumulated PDD snow density model (a), and plots of residual ρs versus (b) snow depth, (c) elevation, and (d) PDD. Mass balance data are given by white circles, and black squares denote snow course data. relief. To estimate SWE0 at the snow depth probe sites, snow densities were estimated from Equation [7.5] and corrected with a density bias correction calculated for each site. Table 7.2 summarizes the density bias corrections used in the SWE0 calculations. Snow density residuals ranged between -100 and +69 kg m−3 , and bias corrections were both positively and negatively related to elevation.  183  7.3. Results  (a) Place  (b) Bridge  Figure 7.2: Difference maps of snow density estimated from (1) PDD-method and (2)linear interpolation of observed density with elevation for (a) Place Glacier, 2006 and (b) Bridge Glacier, 2007. All maps in units of kg m−3 .  184  7.3. Results  Table 7.2: Observed mean snow-pit density (ρs ), PDD-estimated densities ρs(PDD) ), density residuals, LiDAR elevation (Z), and the bias correction function ρs Year  ρs(PDD) −3  Site  (kg m  residual )  Z  Bias Correction  (m)  (kg m−3 )  Helm Glacier H30 2006 H70 2007 H60 2008 H70  411.5 455.7 416.1 440.1  393.6 386.9 432.2 414.7  17.8 68.8 -16.1 25.4  1846 1985 1950 1985  36.7 × 10−2 Z − 660.0  Place Glacier P30 2006 P100 P40 2007 P100 P40 2008 P100  359.6 413.2 450.7 400.2 433.4 458.8  385.6 381.4 432.1 434.2 413.9 406.2  -26.0 31.7 18.6 -34.0 19.5 52.6  1914 2289 1965 2289 1965 2289  15.4 × 10−2 · Z − 320.2  Weart Glacier W02 2006 W07 WM1 2007 W07 W03 2008 W06  a a 540.0 532.8 556.3 577.5  552.1 551.8 550.4 546.5 546.5 541.0  -10.4 -13.7 9.8 36.5  2113 2353 2144 2365 2144 2283  Bridge Glacier B04 2007 B11 BT07 2008 B11  473.3 349.3 518.0 521.3  519.8 449.6 544.6 504.0  46.5 -100.3 -26.6 17.3  1610 2432 1732  −16.1 +25.4  −16.2 × 10−2 · Z + 337.8 1.0 × 10−3 · Z − 1.8  −1.5 × 10−2 · Z + 21.1 19.2 × 10−2 · Z − 4.0  7.0 × 10−2 · Z + 70.3 6.2 × 10−4 · Z − 1.3  Notes: a: Instrument error. Density pits removed from analysis  185  7.3. Results  7.3.2  Initial SWE Interpolation  For all winter balance datasets, AICc scores (not shown) indicated that Loess interpolations using linear locally weighted regressions (d = 1) were superior to those using quadratic (d = 2) local regressions. Linear local smoothing expressions are recommended for monotonic data [Jacoby, 2000], and winter balance data generally increase with elevation. Table 7.3 summarizes comparisons of Loess and OLS regression models using the F -test. The added complexity of Loess models is generally not warranted, as in most cases the calculated F -statistic is not significant enough to reject the null hypothesis. However, there are several other reasons for adopting Loess interpolation approaches with regards to initial SWE interpolation, and specific examples are discussed here. At Place Glacier in 2006 (Figure 7.3a), the Loess model offers statistically significant improvements over the quadratic OLS model. In this case, the major difference between OLS and Loess regression curves is observed at the gap in SWE0 data between 2050 and 2300 m. The Loess model has reduced errors at the observation points on either side of the data gap. At Bridge Glacier in 2007, the Loess model gives a greater RSS than the quadratic OLS model (Table 7.3, Figure 7.3b), rendering the F -test unsuitable for model comparison. However, plots of the two regression models suggest that the quadratic model should be discarded as it incorrectly presumes a decrease in SWE0 at higher elevations. In this example, the discrepancy in RSS between the two approaches is a result of the observation weighting, as the Loess model is more sensitive to observation weights, and it does not respond to the outlier that is captured by the OLS model. As a final example, Loess and quadratic curves for the Weart Glacier 2007 SWE0 data are given in Figure 7.3c. The OLS model should also be rejected in this case, as it will produce unrealistic estimates of SWE0 at both low and high elevations.  186  Table 7.3: Summary of loess and polynomial fitting of SWE0 data versus elevation. f = optimal Loess span, RSE = residual standard error, df = effective degrees of freedom, d = polynomial model degree, F = F-statistic calculated for Loess fit (with optimal span f ) and quadratic OLS model. p-values calculated for the F-statistics give the probability of making a Type I error. NA indicates that RSS errors in the Loess model exceeded those for the OLS model, and thus no F -statistic could be calculated.  Variable  n  f  Loess Model RSE RSS df (m w.e.) (m w.e.)  d  OLS Model RSE RSS (m w.e.) (m w.e.)  ANOVA Results F (p-value)  2.49 2.38  0.167 0.316  0.352 0.653  2 2  0.138 0.310  0.256 0.653  NA NA  Helm Glacier 2006 10 2007 8 2008 10  0.9 0.9 0.9  2.36 2.61 2.61  0.121 0.132 0.055  0.103 0.079 0.078  2 2 2  0.119 0.141 0.056  0.100 0.099 0.085  F = 0.70 (p = 0.43) F = 2.53 (p = 0.16) F = 1.03 (p = 0.34)  Place Glacier 2006 27 2007 34 2008 23  0.9 0.9 0.9  2.66 2.62 2.78  0.098 0.107 0.147  0.226 0.514 0.434  2 2 2  0.102 0.108 0.152  0.251 0.061 0.464  F = 8.4∗∗ (p = 0.008) F = 0.80 (p = 0.38) F = 1.71 (p = 0.21)  Weart Glacier 2006 11 1.0 2007 9 1.0 2008 8 1.0  2.26 2.27 2.30  0.139 0.219 0.114  0.158 0.294 0.066  1 2 1  0.133 0.164 0.108  0.160 0.162 0.070  F = 0.13 (p = 0.73) NA F = 0.32 (p = 0.59)  7.3. Results  Bridge Glacier 2007 13 0.9 2008 9 0.9  187  7.3. Results  (a) Place Glacier 2006  (b) Bridge Glacier 2007  (c) Weart Glacier 2007  Figure 7.3: Comparison of polynomial regressions and Loess curve fitting at (a) Place Glacier, 2006, and (b) Bridge Glacier 2007, and (c) Weart Glacier 2008.  7.3.3  Surface Temperature Loggers and Snowline Retreat  Figure 7.4 illustrates mean daily temperature records obtained from five icesurface temperature loggers at Place Glacier between 2006 and 2007, and air temperatures estimated at an elevation of 1950 m. Burial in the fall occurs 188  7.3. Results first at the higher elevations, and coldest temperatures are observed at the low elevation sensors. Following the establishment of the winter snowpack in early November, observed temperatures at the base of the snowpack decline gradually over the winter, reaching minima between -1 and −2◦ C by the end of April. Air temperatures during this period reach -20◦ C, demonstrating the insulating properties of the snowpack. At the two lower elevation sites, a warming period in early April results in an increase in temperatures at the base of the snowpack, suggesting that meltwater percolation and the associated transition from polythermal to isothermal conditions is rapid. Basal snowpack temperatures at higher elevation sites do not reach 0◦ C until early-mid May, and remain at 0◦ C until the sensor is uncovered, at which point the diurnal temperature fluctuations increase dramatically. Dates of snowline retreat, determined as the date on which T d > 2◦ C, at the logger locations are summarized in Tables 7.4, 7.5, 7.6, and 7.7. Figure 7.5 illustrates the relationship between elevation and date of snowline retreat over multiple seasons. Despite similar amounts of winter accumulation, in 2006 and 2008 snowlines at all glaciers rise more quickly than in 2007. At a given elevation, dates of snowline retreat in 2007 are roughly one month behind those observed in 2006 or 2008. The change in elevation of the snowline over time appears to be consistent between years at the same site, but varies between sites, suggesting that glacier geometry exerts some control over the rate of transient snowline rise. Dates of snowline retreat are strongly related (correlation coefficient r = 0.84) to observed specific net mass balance, as demonstrated in Figure 7.6. Earlier exposure of ice results in greater seasonal melt totals, primarily due to the differences in albedo. This relation offers the possibility for complementing a mass balance network with observations of snowline retreat, but requires further investigation. For the purposes of this study, the surface temperature loggers are used to estimate summer melt to the date of snowline retreat, the point in time where Ms = SWE0 . These observations are used as test points in the melt model comparison.  189  7.3. Results  Figure 7.4: Mean daily temperatures recorded by the Tidbits (Ts , top panel) and estimated air temperatures at 1950 m, Place Glacier 2006-2007  Table 7.4: Snowline retreat sensor locations and estimated day of snowline retreat (DOY), Place Glacier Site  Easting (m)  Northing (m)  Z (m)  DOY 2006  DOY 2007  DOY 2008  P12 P20 P21 P25 P30 P35 P40 P45 P50 P62  527910 528027 527872 528242 528211 528298 528348 528532 528657 528411  5586600 5586584 5586463 5586514 5586334 5586109 5585831 5585620 5585230 5584913  1856 1870 1867 1896 1914 1935 1965 1997 2026 2063  188 189 186 196 199 213 207 220 215 249  212 222 226 253 241 -  196 199 204 218 216 -  190  7.3. Results  (a) Helm Glacier  (b) Place Glacier  (c) Weart Glacier  (d) Bridge Glacier  Figure 7.5: Dates of snowline retreat versus elevation, for all sites and years.  Table 7.5: Snowline retreat sensor locations and estimated day of snowline retreat (DOY), Helm Glacier Site  Easting (m)  Northing (m)  Elevation (m)  DOY 2007  DOY 2008  H30 H40 H50 H60  500636 500698 500827 500918  5534869 5534672 5534303 5534047  1847 1864 1908 1950  216 256 -  202 219 218 230  191  7.3. Results  Table 7.6: Snowline retreat sensor locations and estimated day of snowline retreat (DSR), Weart Glacier Site  Easting (m)  Northing (m)  Elevation (m)  DOY 2007  DOY 2008  WRT01 WRT02 WRT03 WRT04  517629 517330 517139 516795  5556614 5556213 5555821 5555206  2062 2113 2160 2206  226 249 -  201 219 227 228  Table 7.7: Snowline retreat sensor locations and estimated day of snowline retreat (DSR), Bridge Glacier. Separate elevations are reported for the 2007 and 2008 seasons to reflect the location where the Tidbit was located. Site  Easting (m)  Northing (m)  Z 2007 (m)  BRT01 BRT02 BRT03 BRT04 BRT05 BRT06 BRT07 BRT08 BRT09  461205 460939 460624 460092 459458 458729 457700 457128 456736  5630928 5630589 5630102 5629365 5629339 5629074 5629118 5628897 5628759  1397 1458 1513 1575 1627 1681 1736 1790 1842  DOY 2007  Z2008 (m)  DOY 2008  177 173 192 204 203 207 221 227 242  1458 1512 1574 1622 1672 1734 1784 -  160 158 158 166 180 188 198 -  192  7.3. Results  Figure 7.6: Observed specific net balance (bn ) versus date of snowline retreat obtained at Place and Helm Glaciers, 2006 - 2008  7.3.4  Error Analysis  Errors in the estimates of initial SWE range between 0.04 and 0.32 m w.e. (Table 7.8), with site averages between 0.13 and 0.19 m w.e. These values represent errors of approximately 8 - 9 % of the mean winter balance at each site. Summer balance errors at Helm Glacier range between 0.13 m w.e. and 0.22 m w.e., with an average of 0.18 m w.e. At Place Glacier, errors in bs average 0.15 m w.e., with a maximum of 0.19 m w.e. and a minimum of 0.11 m w.e. Average summer balance errors represent 5 - 6 % of the total summer melt. These error estimates fall within the range of previously reported values [Pelto, 1988; Cogley et al., 1996; Fountain and Vecchia, 1999; Krimmel, 1999], though they are an order of magnitude higher than mass balance errors estimated (subjectively) by Young [1981] and Demuth and Keller [2006].  193  7.4. Discussion and Recommendations  Table 7.8: Winter and summer balance error statistics, calculated by site. Mean errors are also given as a percentage of the mean observed SWE0 or M. σSWE(0) (m w.e.)  7.4  Site  max  min  Helm Place Weart Bridge  0.23 0.22 0.32 0.31  0.13 0.11 0.06 0.04  mean (%) 0.18 0.16 0.13 0.19  (9.1%) (9.3%) (8.2%) (9.1%)  σM (m w.e.) max  min  mean (%)  0.22 0.19 -  0.13 0.11 -  0.18 (5.8%) 0.15 (5.2%) -  Discussion and Recommendations  Methods and results presented in this chapter provide data sets for testing glacier melt models from depth and density observations, snowline retreat observations, and ablation stake measurements. Of note are new approaches to estimating snow density for calculation of SWE0 , interpolating SWE0 observations, and using surface temperature loggers for estimating snowline retreat and cumulative melt totals. Glacier mass balance observations are critical components of cryospheric investigations, yet the difficulty and expenses involved in obtaining complete measurements limits their usefulness. Results of this research offer further insights into distributing SWE over alpine glaciers and observing cryospheric change. During mass balance observations, snow density measurements are made less frequently than snow depth soundings, due to both reduced spatial variability in density as compared to snow depth, and due to the time required to make detailed density observations. The relation between snow density and accumulated positive degree days observed at mass balance and snow course sites (Figure 7.1) provides a strong empirical basis for distributing snow densities throughout the region. This approach is also particularly relevant for glaciers with large elevation ranges where lower elevations may have experienced rain or meltwater percolation, giving a much higher density than higher elevations. As this study utilized data from only four mass balance sites and three snow courses, an expanded analysis of the relation 194  7.4. Discussion and Recommendations between PDD and snow density is recommended to determine if this relation applies to other climatic regions. Glacier mass balance values reported in the literature are values distilled from point observations of accumulation and ablation. However, methods for reducing point observations to an area-averaged mass balance value are not typically reported. The comparison of OLS and Loess regressions of SWE0 versus elevation provides key insights into glacier mass balance data reduction methods. For instance, observational evidence does not statistically support the added complexity of Loess models for interpolating SWE0 . Yet OLS regressions do not provide a reliable fit to every data set, and case studies presented above suggest that Loess regressions may offer a more flexible approach to intepolating SWE0 . Further investigations on the use of Loess interpolations are recommended, particularly on datasets with greater numbers of observations than those given here. This section introduces the first reported use of submersible temperature loggers for monitoring snowline retreat on alpine glaciers. Dates of snowline retreat are used in conjunction with observations of initial snow distributions to generate observations of melt totals that can be used to test melt model performance. Advantages of the ice-surface temperature logger approach include the strength of the snowline disappearance signal (Figure 7.4), the cost efficiency versus mass balance measurements, and the possibility of greater spatial density than traditional ablation stake networks. Results presented above indicate that the date of snow disappearance is strongly related to the summer melt totals, suggesting that ground-based snowline retreat observations could complement a traditional mass balance network. Temperature records obtained between the onset of winter and melt-out the following winter also provide multiple pieces of information regarding the state of the snowpack at a given location, in addition to the date of snowline retreat. The date of burial can be observed by a greatly reduced diurnal temperature range in the fall. An estimate of the cold content of the snowpack is provided by late-winter temperatures, and the date of melt onset can be inferred from the date that basal snowpack temperatures reach 0◦ C, indicating water percolation through the entire snowpack. 195  7.4. Discussion and Recommendations Errors in mass balance estimates calculated for each stake observation were found to be on the order of previously reported mass balance errors. Winter and summer balance errors of approximately 9 and 5%, respectively, will be used in assessing melt model performance.  196  Chapter 8  Melt Modelling 8.1  Introduction  The southern Coast Mountains of British Columbia are characterized by mountainous topography and heavy precipitation during the winter, which result in a glacierized area of nearly 8000 km2 . Streams and river systems throughout the region are thus affected by glacier melt, with impacts on a broad range of ecosystem values and human activities, from fisheries [Brown et al., 2005; Moore, 2006] to hydroelectric power generation [Tangborn, 1984; J´ ohannesson, 1997] to recreation and tourism [Beniston, 2003]. Glacier mass balance, defined as the balance between accumulation (mass input) and ablation (mass output), is a vital component of cryospheric research. On long time scales, dynamic glacier and ice-sheet models are driven by mass balance time series [Paterson, 1994]. Secular trends in glacier mass balances are used as indicators of climate change [Haeberli et al., 1999], while the interannual variability of discharge from glacierized catchments is related to annual glacier mass balance [Braun and Aellen, 1990; Moore and Demuth, 2001; Stahl et al., 2008]. Daily and seasonal discharge patterns are also heavily influenced by glacier coverage [Moore and Demuth, 2001; Fleming et al., 2006]. Numerous melt models exist for resolving the ablation side of the mass balance equation, with varying degrees of complexity ranging from lumped statistical models to empirical temperature-index and full energy balance models. On sub-daily time scales, radiation-indexed or energy balance models are required to capture the spatial and temporal variability in melt due to solar radiation inputs, yet on longer time scales the simple temperatureindexed approach provides equivalent modelling skill to more data-intensive 197  8.1. Introduction approaches [WMO, 1986; Braithwaite and Olesen, 1989; Hock, 2003]. While a number of snow and glacier studies have compared the performance of different ablation models [Braun and Lang, 1986; Strasser et al., 2002; Hock, 2005; Hock et al., 2007; Pellicciotti et al., 2008], models applied in these studies are typically driven with local climate data and tested against observations of melt or discharge obtained at a single site. Walter et al. [2005] specifically addressed the issue of data requirements by comparing model results for a temperature-indexed melt model and a process-based energy balance model at four different sites, with both models driven using only maximum and minimum daily temperatures. Results from their study suggest that with minimal input data and a heavily parameterized energy balance model, a process-based approach offers greater modelling skill than the degree-day approach. However, glacier melt modelling presents a different set of environmental conditions, as near-surface meteorological properties of temperature, vapour pressure, and wind speed are affected by the katabatic boundary layer (KBL). There are few studies comparing the response of energy balance models to on-glacier and off-glacier forcing data, though recent work by Munro and Marosz-Wantuch [2009] suggests that melt models driven by off-glacier data work best when adjustments for katabatic components are suppressed. As this work was conducted on only one glacier, it remains to be seen if boundary layer effects become more important at larger sites where KBL development is greater. Given the heterogeneity of mountainous and glacial environments and the scarcity of meteorological observations in most mountain regions, the final results-based chapter of this research examines issues of melt model transferability between sites. The overarching objective of this research is to provide recommendations for regional-scale spatially distributed glacier melt modelling, particularly for regions with sparse networks of weather stations. The following specific questions are addressed: (1) do more complex models, including those based on radiation-indexed melt factors and the energy balance concept, provide improved prediction skill compared to a simpler degree-day model; and (2) for the energy balance models, does ac198  8.2. Study Areas and Climate Data counting for katabatic influences on glacier meteorology improve prediction skill. In this study, five melt models of varying complexity are tested against observations of mass balance and cumulative snowmelt obtained between 2006 and 2008 at four sites in the southern Coast Mountains of British Columbia. The five models include (1) a degree-day approach (DD), (2) a radiation-indexed degree-day model (RIDD), (3) a simplified energy balance approach (SEB), (4) a full energy balance (FEB) model with KBL corrections, and (5) a full energy balance model without KBL corrections. All models are forced by air temperature and vapour pressure estimated from a regional network of off-glacier weather stations.  8.2  Study Areas and Climate Data  Climate data and mass balance and snowline retreat observations were obtained at four glaciers in the southern Coast Mountains of British Columbia (Chapter 2). Elevation data for the study glaciers were obtained from LiDAR surveys conducted in 2006 (M. Demuth, pers. comm), and merged with Terrain Resource and Inventory Management (TRIM) elevation maps based on aerial photography from the mid-1980’s. Digital elevation model (DEM) resolution of 25 m was used at Place, Helm, and Weart Glaciers, and a 50 m resolution was used at Bridge Glacier. Climate data obtained from three Environment Canada weather stations [Environment Canada, 2010] and three ridge-top stations installed at Helm Glacier, Weart Glacier, and Bridge Glacier, formed the regional climate network (RCN) used in the study. Data from the RCN are used to calculate ambient hourly and daily temperature and vapour pressure gradients (Chapter 3). Ambient climate data provide the meteorological forcings for all models, though one version of the full energy balance model uses ambient climate data corrected for katabatic boundary layer effects.  199  8.3. Melt Modelling Methods  8.3 8.3.1  Melt Modelling Methods Degree-day Model  Temperature-indexed melt models are based on empirical relations between air temperature (Ta ) and ablation [Braithwaite, 1977]. Melt M (in mm w.e. d−1 ) is calculated as    ks (Ta − T0 ), M= ki (Ta − T0 ),   0,  (Ta > T0 , SWE > 0) (Ta > T0 , SWE = 0)  (8.1)  (Ta ≤ T0 )  where ks is a melt factor snow (mm K−1 d−1 ), ki is a melt factor for ice (mm K−1 d−1 ), Ta is the mean daily ambient air temperature (◦ C), and T0 is the threshold temperature for melt to occur, assumed here to be 0◦ C. Ice surfaces, due to their lower albedos, will absorb greater amounts of solar radiation than snow, and ks is thus typically less than ki . For an equivalent temperature, melt from ice surfaces will be greater than from snow. While surveys of melt factors show a wide range of reported values [see Hock, 2003], melt factors for snow and ice obtained from historical mass balance data in the region exhibit low spatial and temporal variability [Shea et al., 2009]. This study uses snow and ice melt factors fitted to Place Glacier mass balance data between 1965 and 1995 (ks = 2.59, ki = 4.51; Table 8.1). Mean temperatures are calculated as a function of elevation using a regional regression between hourly temperatures and elevation at the RCN stations. Regressions are fitted hourly, with daily means calculated from estimated hourly temperatures.  8.3.2  Radiation-indexed Degree-day Model  To capture daily and seasonal variability in melt rates, potential direct solar radiation has been introduced to the degree-day formulation [Cazorzi and Dalla Fontana, 1996; Hock, 1999], resulting in the radiation-indexed degreeday (RIDD) model. In the formulation given by Hock [1999], melt M (in  200  8.3. Melt Modelling Methods  Table 8.1: Melt model parameters and constants for degree-day, radiationindexed degree-day, and simple energy balance models Symbol  Description  Value and Units  Degree Day Model ks Melt factor for snow ki Melt factor for ice  2.59 mm K−1 d−1 4.51 mm K−1 d−1  Radiation Indexed Degree Day Model Fm Modified melt factor 0.85 mm K−1 d−1 rs Radiation factor for snow 8.23×10−3 mm m2 W−1 d−1 K−1 ri Radiation factor for ice 1.53×10−2 mm m2 W−1 d−1 K−1 Simple Energy Balance Model a1 fitted parameters for a2 atmospheric transmissivity a3  0.80 1.71 0.65  Ca CE  1.01×103 J K−1 kg−1 variable  CH g Lv Lf Ma m1 m2 m3 m4 m5 m6 m7 p0 Rg T0 ug αf s  ρa ρw σ  heat capacity of dry air turbulent exchange coefficient for latent heat turbulent exchange coefficient for sensible heat gravitational acceleration latent heat of vapourisation latent heat of fusion molar mass of dry air  fitted parameters for snow and ice albedo  mean sea level pressure ideal gas constant mean sea level temperature average glacier wind speed fresh snow albedo snow and ice surface emissivity density of dry air water density Stefan-Boltzmann constant  variable 9.81 m s−2 2.501×106 J kg−1 0.334×106 J kg−1 2.90×10−2 kg mol−1 0.17 8.41×10−3 0.60 -0.22 0.15 -0.15 0.31 1013.25 mbar 8.31 288 K 3.13 m s−1 0.79 0.97 1 kg m−3 1000 kg m−3 5.67×10−8 W m−2 K−1  201  8.3. Melt Modelling Methods mm w.e.) is calculated as    Ta (Fm + rs Kpot ), M= Ta (Fm + ri Kpot ),   0,  (Ta > T0 , SWE > 0) (Ta > T0 , SWE = 0)  (8.2)  (Ta ≤ T0 )  where Fm is a modified melt factor (mm K−1 d−1 ), rs and ri are radiation factors for snow and ice, respectively (mm m2 W−1 d−1 K−1 ), and Kpot is potential solar radiation at the surface. Solar radiation at the top of the atmosphere K↓ex was calculated as a function of solar path length, using equations from Iqbal [1983]: K↓ex = S0 cos θZ .  (8.3)  where S0 is the solar constant (1367 W m−2 ) and θZ is the angle of incidence determined from solar declination, hour angle and the latitude of the site. Potential direct solar radiation for each site was then calculated as: Kpot = K↓ex τ cos θ  (8.4)  where τ is estimated clear-sky atmospheric transmissivity and cos θ is a correction for the angle of incidence on a sloped surface. Following the work of Hock [1999], a constant value of τ = 0.75 is used. The value of Kpot represents an index of spatial and temporal variations in energy availability rather than the energy directly available for melt [Hock, 1998], and errors associated with assuming a constant transmissivity are thus assumed to be negligible. The angle of incidence on a sloping surface was computed following Iqbal [1983]: cos θ = sin β sin α cos(ψ − γsl ) + cos β cos α  (8.5)  where β is the slope angle (in radians), α is the solar altitude, ψ is the solar azimuth, and γsl is the slope azimuth. Between sunset and sunrise and when a site is shaded by local topography, Kpot = 0.  202  8.3. Melt Modelling Methods A review of RIDD model studies suggests that melt and radiation factors optimized for observed mass balances are highly variable between regions (Table 8.2). This may be due in part to methodological differences for optimizing radiation and temperature indexed melt factors, though differences in the relative contibution of thermal energy sources (L↓, QH ) between sites will also affect RIDD parameter values. At a single glacier, Carenzo et al. [2009] found variations in fitted RIDDM parameters between time periods and sites, though not of the orders of magnitude observed in Table 8.2. For this study, values of Fm , rs , and ri were obtained from F. Anslow (pers. comm), who fit a RIDD model to historical mass balance data compiled by Shea et al. [2009] at nine sites in the study region. Table 8.2: Summary of previous RIDD model parameter values. Z = elevation, Fm = modified melt factor, and rs and ri are radiation factors for snow and ice, respectively. Z  Fm  Site  Lat.  (m)  (mm d−1 K−1 )  St¨ orglaciarena Rhoneb S. Cascadec H. Glacier d’Arollac Keqicar Baqid S. Coast Mountainse  68 47 48 46 42 51  1425 2720 1850 2900 3321 2300  1.8 1.85 6.4 14.4 3.4, 5.0 0.85  N N N N N N  rs  ri  (mm m2 W−1 d−1 K−1 ) 6.0 ×10−4 4.0×10−4 1.87×10−5 2.7×10−3 6.0×10−4 8.23×10−3  8.0×10−3 7.0×10−4 2.17×10−5 2.6×10−3 1.0×10−3 1.53×10−2  a: Hock [1999]; b: Klok et al. [2001]; c : Schneeberger et al. [2003]; d: Zhang et al. [2007]; e: Anslow, pers. comm.  8.3.3  Simplified Energy Balance Model  A simplified energy balance model (SEB) was developed to model snow and ice melt at daily time-steps using estimates of ambient temperature (Ta ) and vapour pressure (ea ) obtained from the regional climate network. For simplification, the snowpack in the EB1 model is assumed to be isothermal, turbulent fluxes are calculated using bulk exchange coefficients tuned to observed melt totals [Oerlemans, 2000], and stability corrections are neglected.  203  8.3. Melt Modelling Methods Total daily energy for melt or refreezing (QM ) is calculated as: QM = K ∗ + L∗ + QH + QE + QR + QA  (8.6)  where K ∗ is net solar radiation, L∗ is net longwave radiation, QH and QE are the sensible and latent heat fluxes, QR is the rainfall heat flux, and QA is the advection of turbulent energy. The advective term (QA ) in Eq. [8.6] is generally neglected in glacier energy balance studies, as katabatic flows should limit the advection of air masses from the surrounding terrain over the glacier. Energy storage within the snowpack is also neglected, as changes in snowpack/ice temperature during the ablation season are expected to be minimal. Refreezing processes (QM < 0) will consume the bulk of negative energy during nighttime radiative cooling events [Anslow et al., 2008]. The simple energy balance model thus tracks negative values of QM , such that melt (M ) only occurs when cumulative values of QM are positive. Daily melt totals (in m w.e.) are computed as M=  QM ρw Lf  (8.7)  where ρw is the density of water and Lf is the latent heat of fusion (Lf = 0.334 × 106 J · kg−1 ). All parameter values can be found in Table 8.1. The following sections detail methods for calculating the radiative and turbulent fluxes of energy, and the optimization procedure. Net Solar Radiation Net solar radiation is a function of surface albedo (α), atmospheric transmissivity (τ ), and potential solar radiation (Kpot ): K ∗ = (1 − α) Kpot τ.  (8.8)  Potential solar radiation was caculated as for the RIDD model, and corrections for sloping surfaces were calculated using Equations [6.2] - [8.5]. Atmospheric transmissivity, τ , was calculated with a global parameteriza-  204  8.3. Melt Modelling Methods tion developed for the region (see Chapter 6): τ = a1 1 − exp a2  ∆T ea  a3  (8.9)  where ∆T is the diurnal temperature range corrected for precipitation [Bristow and Campbell, 1984], and ea is the mean daily vapour pressure. Identical values of a1 , a2 , and a3 were assumed for all four glaciers, and were calculated as the mean of coefficients fit to observational data at each glacier site (Chapter 6). For modelling the reflected radation, albedos for fresh snow (αf ), aged snow (αs ), and ice (αi ) were initially assumed to be constant, and average albedos observed in Chapter 6 were adopted (αf = 0.79, αs = 0.65, αi = 0.23). However, this produced poor results when compared with observed net radiation (Q∗ ). The simple energy balance model instead incorporates the snow albedo model developed in this research (Chapter 6), which requires only estimates of cumulative PDD (cPDD) and transmissivity (τ ): αs = [m1 exp(−m2 cPDD) + m3 ] + [m4 τ + m5 ]  (8.10)  where mn are globally fitted coefficients. Fresh snow albedo was set to 0.79 (the average albedo of fresh snow observations), and ice albedo was also modelled as a function of τ : αi = m6 τ + m7 .  (8.11)  Cumulative positive degree days at each site were calculated from January 1 onwards using ambient temperature gradients computed from a regional network of automated weather stations (AWS).  205  8.3. Melt Modelling Methods Net Longwave Radiation Incoming longwave radiation is a function of atmospheric emissivity ( a ) and air temperature (T ) in K. From the Stefan-Boltzmann equation, L↓=  aσT  4  (8.12)  where σ = 5.67×10−8 W m−2 K−1 . The simplified energy balance model does not account for sky-view or longwave radiation emitted from surrounding terrain. Atmospheric emissivity is computed as a function of near surface temperature and vapour presure [Marks and Dozier, 1979]:  a  = 1.24  ea Ta + 273.15  1/7  ·  p p0  (8.13)  where p is the standard site pressure (in mbar), and p0 is the standard sea level pressure (1013 mbar). Standard site pressure is computed using the standard atmosphere:  p = p0  T0 T0 + γT Z  gMa Rg γT  (8.14)  where T0 is standard sea-level temperature (288 K), γT is the standard atmospheric lapse rate (0.0065 K/m), g is gravitational acceleration (9.81 m s−2 ), Ma is the molar mass of air (2.90×10−2 kg mol−1 ), and Rg is the ideal gas constant (8.31 J K−1 mol−1 ). Longwave radiation emitted from the surface is calculated from the surface temperature (Ts ) and the surface emissivity ( s ): L ↑= where  s  4 s σTs ,  (8.15)  is assumed to be equal to 0.97, though a range of surface emissivities  are given for snow and ice surfaces [Oke, 1987]. Net longwave radiation is the difference between incoming and outgoing longwave radiation, and L reflected from the surface (L↓ [1 −  s ])  is neglected.  206  8.3. Melt Modelling Methods Turbulent Fluxes For the simplified EB model, turbulent fluxes are calculated using the bulk aerodynamic approach: QH  = ρa Ca CH u (Ta − Ts )  QE = ρa Lv CE u (0.622/p) · (ea − es ),  (8.16) (8.17)  where ρa is the density of dry air (1 kg m−3 ), Ca is the specific heat of dry air (1.01×103 J kg−1 K−1 ), CH and CE are turbulent exchange coefficients, u is the wind speed (m s−1 ), Lv is the latent heat of vapourization at 0◦ C (2.501×109 J kg−1 ), and p is the site atmospheric pressure (in Pa). A mean wind speed of 3.13 m s−1 is used here, taken from a selection of meteorological studies on temperate glaciers (Table 8.4.3). The simplified energy balance model neglects the stability corrections generally used for the bulk aerodynamic approach and involves tuning the turbulent exchange coefficients to fit observed melt totals, following the methods of Oerlemans [2000] and Konya et al. [2004]. It is assumed that CH = CE . This approach essentially folds stability corrections into the fitted turbulent exchange coefficients. Whereas Oerlemans [2000] and Konya et al. [2004] used time series of surface height observations to tune point energy balance models, mass balance data from Place Glacier were used to tune TEC in the current study. Several model optimizations were run to examine the sensitivity of fitted TEC to the input data, though these simulations are not presented here for brevity. The simple energy balance model used in this study was fit to a random sample of 22 bs observations from Place Glacier, representing a range of years and elevations. Precipitation and Ground Heat Fluxes Heat conduction from the glacier to the snowpack is assumed to be zero, as ice temperatures are assumed to equal 0◦ C. Rainfall is assumed to have the  207  8.3. Melt Modelling Methods same temperature as the air, and heat from precipitation is calculated as: QR = Cw Rp Ta  (8.18)  where Cw is the heat capacity of water (4.18×106 J m−3 K−1 ) and Rp is the depth of rainfall (in m). Daily rainfall depths are set equal to those observed at the Whistler climate station for Ta > 0 at the site of interest.  8.3.4  Full Energy Balance Model  The final melt model (FEB) is an hourly energy balance approach that builds on the simple energy balance model, but includes (1) local transmissivity models, (2) stability corrections for turbulent flux calculations, and (3) heat transfer to and from the snowpack prior to melt onset. One version of the FEB model corrects ambient meteorological data for katabatic boundary layer effects, while the other does not. These corrections apply only to data used to calculate the turbulent heat fluxes. The full energy balance model is described by: ∆QS + QM = K ∗ + L∗ + QH + QE + QR  (8.19)  where ∆QS is energy storage within the snowpack, QM is the energy for melting/refreezing, and the surface energy terms are as described for the simple energy balance model. Advection of turbulent heat energy from the glacier surroundings (QA ) was again neglected, as flows during melt situations will be mainly down-glacier (Chapter 4). The FEB model was run from 01 January to 30 September at hourly time-steps, and the surface energy balance is directed to ∆QS when snowpack temperatures are below 0◦ C. Melt (in mm w.e.) is calculated following Equation [8.7], and updated hourly from the observation of initial SWE. Daily precipitation observations at Whistler are used to estimate snowfall amounts (mm w.e) at each time step, but are not modified to account for lateral or vertical precipitation gradients. Observed total daily precipitation is spread over 24 hours, and 208  8.3. Melt Modelling Methods melt energy is directed first toward the new snowfall. Net Shortwave Radiation For the FEB model, direct and diffuse components of incoming shortwave radiation were modelled separately at each point of interest (POI). K↓ex was first calculated for each glacier and time step following Eq. [6.2]. Hourly atmospheric transmissivities (τ ) were then calculated from estimated Ta and ea using a modified Bristow-Campbell approach (Eq. [6.1]) and glacier - specific coefficients calculated previously (Table 6.6). Global open-site radiation for each glacier (G0 ) was then estimated as: G0 = K↓ex τ.  (8.20)  Using the modelled transmissivity, the ratio of diffuse to global radiation (kd ) for each time step was determined using a decomposition model developed for an alpine site in western Canada [Huo, 1991]:    1.015 − 0.208τ, kd = 0.166 + 9.570τ − 36.71τ 2 + 48.50τ 3 − 22.03τ 4 ,   0.12,  τ ≤ 0.22 0.22 < τ < 0.8 τ ≥ 0.8 (8.21)  To estimate global radiation at each POI, G0 was first split into its direct (I0 ) and diffuse (D0 ) components. A binomial terrain mask (Tm ) was calculated based on horizon angles and solar azimuth (0 = shaded, 1 = open), and global radiation for each POI (G) was calculated as G = I0 cos(θ) Tm + D0 Vf + G0 αm · (1 − Vf )  (8.22)  where Vf is the sky-view factor, and αm is the mean terrain albedo, assumed to be equal to 0.1. The terrain mask sets direct solar radiation to 0 when the POI is shaded. For calculation of reflected shortwave radiation, snow and ice surface albedo are estimated from cumulative PDD, modelled τ , and globally-fitted coefficients derived from Place, Weart, and Bridge Glacier data (Chapter 6). 209  8.3. Melt Modelling Methods Mean daily Ta is calculated for each point, and cumulative PDD are then distributed to each hourly time-step. Table 8.3: Melt model parameters and constants for the full energy balance model. Other constants are found in Table 8.1. Symbol  Description  Value and Units  f1 f2 f3  Fitted parameters for cloud factor  0.46 1.62 1.34  p1 p2  Fitted parameters for clear-sky emissivity  0.82 7.75  Ts(0) ∆zs  Initial snowpack temperature Depth for snowpack temperature calculations Snow and ice emissivity Terrain emissivity Snow density  -10.0 ◦ C 0.31 m 0.97 0.95 Variable, kg m−3  s t  ρs  Net Longwave Radiation Previous results (Chapter 6) indicate that longwave irradiances observed at on- and off-glacier sites do not exhibit systematic differences despite strong near-surface inversions over melting glaciers, at least at Place Glacier. The FEB uses a longwave radiation parameterization developed from off-glacier L↓ observations. Incoming longwave radiation is computed from both sky (L↓s ) and terrain (L↓t ) source areas: L↓ = L↓s +L↓t  (8.23)  = Vf ( c s F σ Ta4 ) + (1 − Vf ) · ( t σ Ta4 ) where L↓s is estimated following Sedlar and Hock [2008],  (8.24) cs  is clear-sky  emissivity, F is cloud factor, and ambient air-temperature (Ta ) at the elevation of the POI. For L↓t , a terrain emissivity ( t ) of 0.97 was assumed, and terrain temperatures were assumed to equal ambient temperatures. Clear-  210  8.3. Melt Modelling Methods sky emissivity is calculated following Konzelmann et al. [1994]:  cs  = 0.23 + p1  eref Tref  1/p2  (8.25)  where p1 and p2 are fitted coefficients obtained from the ridge station at Place Glacier and Tref and eref are the ambient temperature and vapour pressure estimated at 2100 m, the elevation of the ridge station. Cloud factor (F ) is expressed as a function of atmospheric transmissivity: F = f1 τ f2 + f3  (8.26)  where fi are fitted coefficients also obtained from the ridge station at Place Glacier. Turbulent Fluxes Turbulent fluxes of energy are modelled using a bulk aerodynamic approach with stability corrections described by Hock [1998]: QH QE  = ρa ca CH ug · (Tg − Ts ) 0.622(ea − es ) = ρa L CE ug · P  (8.27) (8.28)  where Tg , eg , and ug are air temperature (◦ C), vapour pressure (mbar), and wind speed (m s−1 ) measured 2 m above the glacier surface, Ts and es refer to the surface temperature and vapour pressure, respectively. Turbulent transfer coefficients CH and CE are calculated as: CH = CE =  k2 ·Θ [log(z/z0 ) log(z/zx )]  (8.29)  where k is the von Karman constant (0.41), z0 is the roughness length for momentum (m), zx is the roughness length for temperature and vapour pressure, z is the height of measurement, and Θ is the stability correction [Webb, 1970; Hock, 1998]. Stability is estimated using the bulk Richardson  211  8.3. Melt Modelling Methods number Rb , which has been used in several energy balance models over snow and ice [Moore, 1983; Hock, 1998; Sicart et al., 2005]: Rb =  (Tg − Ts )z g · Tg + 273.15 u2g  (8.30)  where g is gravitational accelaration. The magnitude of the stability correction Θ depends on the sign of Rb [Sicart et al., 2005], with positive Rb for stable cases, and negative Rb for unstable cases: Θ=  (1 − 5.2 Rb )2 , (1 − 16.0 Rb  )0.75  Rb > 0 Rb < 0  (8.31)  Choosing an appropriate value for the surface roughness lengths represents a distinct challenge, as a wide range of values have been reported in the literature. Brock et al. [2006] provided a comprehensive review of published surface roughness estimates for snow and ice derived from microtopographic measurements and near-surface wind profiles. For temperate glaciers, values for z0 for snow typically range between 0.2 and 14 mm, whereas z0 for ice typically ranges between 0.1 and 15 mm. Some studies allow z0 to vary spatially and temporally [Brock et al., 2000], and surface roughness can also be treated as a model parameter [Anslow et al., 2008] and used to minimize errors in modelled ablation. Here, values for z0 are adopted from studies on Haut Glacier D’Arolla, a temperate glacier in the Swiss Alps [Pellicciotti et al., 2005]. Over snow, z0 = 1 mm, and over ice z0 = 2 mm. Roughness lengths for temperature and vapour pressure are scaled to the momentum roughness, following Hock [1998], with zx = z0 /300. Snow Pack Surface Temperature The full energy balance model accounts for heat flux to or from the snow pack, which affects the surface temperature of the snow cover, which will in turn affect outgoing longwave radiation (Eq. [8.15]) and sensible and latent heat fluxes (Eq. [8.27]). Mechanisms of heat transfer within snow include shortwave radiation penetration, movement of water and water vapour 212  8.4. Meteorological Inputs within the snowpack, and phase changes [Hock, 2005], though in energy balance studies on glaciers these processes are typically neglected due to their complex nature. Here, the change in mean temperature (∆Ts ) of the surface snow layer snow with a depth of ∆zs is modelled using finite differences [Oke, 1987; Corripio, 2002]: ∆Ts ∆QS = ρs cs ∆zs ∆t  (8.32)  where ρs is the snow density, cs is the specific heat of snow. ∆zs is the depth of snow where diurnal temperature fluctuations are assumed to be significant. Diurnal temperature variations decrease exponentially with depth below the snow surface [Oke, 1987], and Corripio [2002] and Pellicciotti et al. [2005] used ∆zs = 0.31 m as the depth where the surface temperature signal is 95% reduced. Snow density was modelled for each time step using cumulative PDD (Chapter 7) estimated at each POI. Sensitivity analyses suggest that the initial value of Ts is not critical, as the energy balance model equilibrates the snowpack temperature rapidly.  8.4  Meteorological Inputs  Density-driven katabatic flows are commonly observed on melting glaciers [Ohata, 1989; van den Broeke, 1997b; Greuell and B¨ohm, 1998; Strasser et al., 2004] when ambient temperatures are greater than 0◦ C. As the surface temperature cannot exceed 0◦ C, highly stable near-surface boundary layers are formed. The downslope drainage of cold air further modifies the katabatic boundary layer (KBL) through latent and sensible heat exchange, with a pronounced effect on meteorological properties observed over the glacier surface. In chapters 3, 4, and 5, observations of near-surface temperature (Tg ), vapour pressure (eg ) and wind speed (ug ) at the four study glaciers were used to develop methods for adjusting ambient meterological data to compensate for KBL effects. These methods are adopted here to generate meteorological time-series of Tg , eg , and ug that will be applied in one version of the full 213  8.4. Meteorological Inputs energy balance model.  8.4.1  Temperature  The development of katabatic flows over melting glaciers increases the sensible heat exchange between the surface and the overlying air. Near-surface temperatures (Tg ) for each point of interest are computed as: Tg =  k1 + k4 · (Ta − k2 ), Ta > k2 k1 + k3 · (k2 − Ta ), Ta < k2  (8.33)  where Ta is estimated ambient temperature, and ki are coefficients determined as a function of elevation (Z) and flow path length (F P L): k2 = β1 + β2 · Z  (8.34)  k3 = β3 · exp(β4 · F P L)  (8.35)  k4 = β5 + β6 · exp(β7 · F P L)  (8.36)  Here, βi are constants fitted to the full observational dataset (see Table 8.4 for a complete list of katabatic boundary layer correction coefficients).  8.4.2  Vapour Pressure  Two separate linear functions are used to describe near-surface vapour pressures: eg =  j1 · ea + j2 , Tg > 0 j3 · ea + j4 , Tg < 0  (8.37)  with j1 and j2 determined by flow path length: j1 = β8 · exp(β9 · FPL)  (8.38)  j2 = β10 + β11 · ln(FPL)  (8.39)  Following the results of previous research (see Chapter 3), j3 = 1 and j4 = 0.  214  8.4. Meteorological Inputs  8.4.3  Wind Speed  A survey of mid-latitude glacio-meterological studies (Table 8.4.3) suggest that glacier wind speeds exhibit low variability between sites. The mean near-surface wind speed at 26 sites was found to be 3.13 m s−1 , with a standard deviation of 0.75 m s−1 . For the simple energy balance model, a mean glacier wind speed (ug ) was assigned for all sites. Wind speeds are not required for either degree-day or radiation indexed-degree day approaches. Previous research (Chapter 4) suggests that wind speeds observed onglacier offer the best method for estimating wind speeds at unmonitored locations, and most distributed energy balance models assume a spatially invariant wind speed observed at a reference AWS. Thus, for the full energy balance models, wind speeds at each time step were assumed to be constant over the entire glacier, and wind speeds measured at on-glacier sites were utilized. Measured wind speeds at PM2 were applied to model test sites at Place and Helm Glaciers. Wind speeds observed at WM1 were applied for all Weart Glacier observation sites, and wind speeds at Bridge Glacier sites were assumed equal to those observed at BM1. Assuming a reference wind speed introduces errors of between 12 and 33% within the same glacier (Table 4.6). For periods where observed wind speeds at the reference stations are unavailable, the applied wind speed value is determined by Ta . For Ta greater than k2 (the critical threshold temperature for KBL development), a katabatic wind speed is estimated from Ta , local slope, and eddy diffusivities for heat (KH ) and momentum (KM ), Equations [4.7] - [4.10]. Eddy diffusivites are modelled as a function of flow path length: KH  = β12 + β13 · log(FPL)  (8.40)  KM  = β14 + β15 · exp(β16 · F P L)  (8.41)  where βi are globally fitted coefficients (Table 8.4). For Ta < k2 , the mean wind speed of 3.13 m s−1 is applied.  215  8.4. Meteorological Inputs  Table 8.4: Coefficients for kabatatic boundary layer corrections, based on results from Chapter 5 Symbol  Description  Value and Units  Temperature β1 Coefficient β2 Coefficient β3 Coefficient β4 Coefficient β5 Coefficient β6 Coefficient β7 Coefficient  for for for for for for for  estimating estimating estimating estimating estimating estimating estimating  k2 k2 k3 k3 k4 k4 k4  13.35 -3.93×10−3 9.77×10−1 -4.43×10−5 2.91×10−1 7.15×10−1 -5.64×10−4  Vapour Pressure j3 Coefficient j4 Coefficient β8 Coefficient β9 Coefficient β10 Coefficient β11 Coefficient  for for for for for for  estimating estimating estimating estimating estimating estimating  eg eg j1 j1 j2 j2  1.00 0.00 7.58×10−1 -5.62×10−5 -1.53 4.52×10−1  Wind Speed β12 Coefficient β13 Coefficient β14 Coefficient β15 Coefficient β16 Coefficient  for for for for for  estimating estimating estimating estimating estimating  KH KH KM KM KM  -3.39×10−1 7.21×10−2 1.10×10−1 1.24×10−1 -1.67×10−4  216  8.4. Meteorological Inputs  Table 8.5: Mean on-glacier wind speeds (ug ) observed during the ablation season for a sample of glacio-meteorological studies. Mean (x) and standard deviation (σ) of wind speed for all studies are given. Site Bridge Gl.a Gl. Lenguab Haut Gl. d’Arollac Haut Gl. d’Arollac Haut Gl. d’Arollac Haut Gl. d’Arollac Haut Gl. d’Arollac Lemon Creek Gl.d Koryto Gl.e McCall Gl.f Morteratschgl.rg Pasterze Gl.h Pasterze Gl.h Pasterze Gl.h Pasterze Gl.h Pasterze Gl.h Place Gl. (PM1)a Place Gl. (PM2)a Place Gl. (PM3)a Place Gl. (PM4)a Place Gl. (PM108)a Place Gl. (PM308)a Storglaci¨ areni Worthington Gl.j Weart Gl. (WM1)a Weart Gl. (WM1)a  Latitude (◦ )  Elevation (m)  Area (km2 )  ug (m s−1 )  Sensor Height(m)  51N 53 S 46 N  1745 450 2813 4.5 2912 2911 2909 3005 1200 810 1715 2104 2205 2310 2420 2945 3225 1960 2012 2100 2313 2047 2043 1375 850 2168 2290  88 ≈5 3.50  2 2  3 8 8 8  3.80 4.10 2 2.60 3.00 2.30 2.50 1.50 2.40 3.10 3.00 4.00 4.40 4.40 3.90 4.20 3.13 3.03 3.01 2.92 2.69 3.28 2.50 2.10 3.46 2.45  x: σ:  3.13 0.75  58 55 69 46 47  N N N N N  50 N  67 N 61 N 50N 50N  12.7 7.8 6.5 17.15 19.8  4  2 2 2 2 2 3 3.5 2 2 2 2 2 2 2 2 2 2 2 2 2 2  Sources: a: This study; b: Schneider et al. [2007]; c: Strasser et al. [2004]; d: Wendler [1969] (in Giesen et al. [2009]); e: Konya et al. [2004]; f : Klok et al. [2005]; g: Oerlemans [2000]; h: Greuell et al. [1997]; i: Hock and Holmgren [1996]; j: Streten and Wendler [1968] (in Giesen et al. [2009])  217  8.5. Melt Model Comparisons  8.5  Melt Model Comparisons  For all melt models and observations, statistics of mean absolute error (MAE), mean bias error (MBE), and root mean squared error (RMSE) were calculated. At mass balance sites, total modelled summer melt (total melt M minus snowfall) was compared against observed summer balance (bs ). At locations where snowline retreat was monitored using Tidbit surface temperature loggers, total modelled melt to the day of snowline retreat (MT B ) was compared against observed (or estimated) initial snow water equivalence, SWEi . To further evaluate the performance of the melt models against one another, a goodness-of-fit index (G) with respect to a benchmark was calculated [Seibert, 2001]. Here, the degree-day model is taken as the benchmark model given its simplicity and prior application in regional melt modelling [Shea et al., 2009]. For observations of melt at site i and time t, G is calculated as: G=1−  (Mobs (i, t) − Mmod (i, t))2 (Mobs (i, t) − MDD (i, t))2  (8.42)  where Mobs (i, t) and Mmod (i, t) are the observed and modelled melt, respectively, and MDD (i, t) is the degree-day modelled melt. Negative G indicates that model performance is poorer than the benchmark, while positive G indicates improvement. For equivalent models, G = 0, and a perfect fit to the observations gives G = 1.  8.6 8.6.1  Results Modelled Melt and Benchmark Evaluations  Errors in modelled summer balance (bs ) and total melt at Tidbit sites (MT B ) are given in Table 8.6. The full energy balance model (FEB) without KBL corrections gives the lowest MAE, MBE, and RMSE at both mass balance and Tidbit sites. With KBL corrections, the full energy balance model shows greater errors, particularly with regards to underestimating melt at the mass balance sites (MBE = -0.12 m w.e.). A radiation-indexed model 218  8.6. Results (RIDD) gives modest improvements over the degree day (DD) model for MT B sites, but has greater errors at the mass balance sites. Plots of modelled and observed bs and MT B , stratified by glacier (Figures 8.1 and 8.2) provide several other notable results. For instance, bs is underpredicted at Helm Glacier by all melt models except the full energy balance approach. Using the RIDD approach, modelled bs is overpredicted at low bs values, and underpredicted at bs , suggesting a systematic bias in the RIDD coefficients. For estimates of snowmelt totals only, all models underpredict melt at low initial water equivalences, indicating that model performances improve over longer time periods. In Figure 8.3, prediction errors (observed minus predicted) for both mass balance and snowline retreat sites are compared against SWEi or observed bs . These plots are useful for illustrating the performance of the models over a range of conditions and time periods. For instance, low values of SWEi indicate sites where the models are tested over short time intervals. High values for observed bs represent significant ice-melt contribution to bs , whereas low values for bs represent accumulation zone observations. Figure 8.3 reveals the value of using surface temperature loggers for evaluating model performance, as they provide cases for testing the melt model performance in the middle of the ablation season. Plotting the residuals over a range of conditions (e.g. snow melt only or mainly ice melt) also demonstrates model skills and biases. For instance, the reduction in errors provided by the full energy balance model approach over the range of observations is evident in Figure 8.3. Also, DD, RIDD, and SEB models understimate MT B , but over-estimate melt at accumulation sites (low values of bs ).  219  8.6. Results  Table 8.6: Errors in modelled summer balance (bs ) and cumulative melt (MT B ) for dgree-day (DD), radiation-indexed degree-day (RIDD), simple energy balance (SEB), and full energy balance (FEB) models with and without katabatic boundary layer (KBL) corrections. Goodness-of-fit index G is given, and n is the sample size of the test data. All error terms are given in m w.e. Model  MAE (m w.e.)  MBE (m w.e.)  RMSE (m w.e.)  G  n  bs  DD RIDD SEB FEB (KBL) FEB (no KBL)  0.44 0.47 0.41 0.40 0.36  -0.22 -0.29 -0.05 -0.13 0.00  0.53 0.57 0.54 0.49 0.45  -0.17 -0.03 0.14 0.27  53 53 43 53 53  MT B  DD RIDD SEB FEB (KBL) FEB (no KBL)  0.30 0.27 0.30 0.26 0.26  -0.26 -0.15 0.07 -0.13 -0.03  0.37 0.35 0.44 0.33 0.32  0.08 -0.40 0.19 0.23  44 44 44 44 44  Data  220  8.6. Results  (a) DD  (b) RIDD  (c) SEB  (d) FEB (KBL)  (e) FEB (no KBL)  Figure 8.1: Modelled and observed bs for various melt models  221  8.6. Results  (a) DD  (b) RIDD  (c) SEB  (d) FEB (KBL)  (e) FEB (no KBL)  Figure 8.2: Modelled and observed MT B for various melt models.  222  8.6. Results  Figure 8.3: Melt model residuals versus observations. Goodness-of-fit indices calculated using the DD model as a benchmark demonstrate the change in model performance from simple to complex melt models (Table 8.6). The addition of radiation information for the RIDD model does not improve estimates of seasonal melt totals at mass balance sites (G = -0.17), and only provides modest improvements for mid-season melt estimates (G = 0.08). A simplified energy balance approach yields no improvement in predictive capability at bs sites, and gives a substantially poorer estimate of melt at the Tidbit sites (G = −0.40). Full energy balance modelling with KBL corrections provides improved melt estimation at both mass balance and Tidbit sites, with G equal to 0.14 and 0.19, respectively. The FEB without katabatic boundary layer corrections delivers the greatest improvement over the benchmark model, and goodness-of-fit indices with respect to the DD benchmark were 0.27 and 0.23 for bs and MT B , respectively. 223  8.6. Results  8.6.2  Full Energy Balance Model Analyses  This section briefly describes analyses concerning the full energy balance model. Observations of net radiation (Q∗ ) collected at Place Glacier in 2007 are compared against modelled Q∗ in Figure 8.4. The mean absolute errors in modelled hourly and daily Q∗ are 50 W m−2 and 7 W m−2 respectively, indicating that the radiative portion of the FEB model is generally successful. Furthermore, longwave radiative fluxes at all sites were generated with a common model, and transmissivity parameters exhibit low variability between sites (Chapter 6), suggesting a high degree of portability. Evidence for relatively unbiased modelling of both the radiative fluxes and melt totals further indicates that the approach used to model turbulent energy fluxes is appropriate, and the surface roughness parameter selection is suitable. Near-surface snowpack temperatures represent an important component in the full energy balance model, as Ts is used to calculate outgoing longwave radiation and sensible heat fluxes [Tarboton and Luce, 1996]. To assess the performance of the snowpack temperature component of the FEB, the Tidbit temperature records are employed. Mean daily temperatures at the base of the snowpack (TT B ) are highly stable (Chapter 7), as opposed to temperatures in the active surface layer. However, the date of meltwater (or possibly rainwater) percolation to the base of the snowpack can be inferred from the date that basal temperatures reach 0◦ C, providing a qualitative test of the FEB in terms of its ability to predict the onset of active meltwater generation, i.e., when temperatures throughout the snowpack have reached 0◦ C. Figure 8.5 illustrates modelled snowpack temperature records for two sites: one at 1850 m, and one at 2050 m. Modelled mean daily snowpack temperatures reach 0◦ C within 1-4 days of the observed basal snowpack temperatures reaching 0◦ C. The FEB model therefore captures the timing of melt onset, which is important for estimating seasonal melt totals. Given that KBL corrections reduce the performance of the energy balance model (Table 8.6), Figure 8.6 provides a comparison of the mean monthly energy fluxes with and without KBL corrections. The differences  224  8.6. Results  (a) Hourly  (b) Daily  Figure 8.4: Observed and modelled (a) hourly and (b) daily net radiation at PM2, 2007 ablation season are generally small. Sensible and latent heat fluxes are greater in the FEB model without KBL corrections, particularly in the latter portion of the ablation season. The average latent heat flux in June is negative when boundary layer corrections are used, but positive without. Also, the FEB model without KBL corrections has higher net solar radiation flux in August due to the enhanced snow melt rate and the earlier exposure of the ice surface.  225  8.6. Results  Figure 8.5: Modelled snow pack temperatures at two Tidbit sites, 1850 m (solid) and 2050 m (dashed), Place Glacier 2007. Arrows indicate the observed date where basal snowpack temperatures first reached 0◦ C.  Figure 8.6: Mean monthly energy fluxes for full energy balance models with (left) and without (right) KBL corrections  226  8.7. Discussion  8.7  Discussion  Methods for modelling glacier melt at regional scales are restricted by the data requirements and availability. In this study, the empirically based degree-day model is used as a benchmark for evaluating the performance of more complex models due to its simplicity. In the field of glacier melt modelling, recent shifts from degree-day approaches [Braithwaite, 1984] to simplified energy balance models [Oerlemans, 2000; Konya et al., 2004; Machguth et al., 2006] or full energy balance models [Hock, 2005; Pellicciotti et al., 2008; Anslow et al., 2008] have generally not been accompanied by comparisons with simpler modelling approaches. Thus it is difficult to assess the whether a model for regionally distributed modelling increases predictive skill over simpler approaches. For the purposes of modelling glacier melt at regional scales, this study shows that the degree-day model provides accurate melt estimates with minimal input data, and the required inputs can be readily generated (Chapter 3) from regional climate networks or downscaled reanalysis products [Shea et al., 2008]. Neither radiation-indexed nor simple energy balance models offer improved modelling skill for seasonal melt totals, despite the enhanced data requirements. Only full energy balance models, incorporating stability corrections and snowpack energy storage, provide improved modelling skill over the DD approach. It is noted that the simple energy balance model was the only model optimized to a subset of the observational data, giving a mean bias error near zero but reduced performance in terms of MBE and RMSE. The tuned turbulent exchange coefficients used in SEB models likely vary in space and time, rendering the approach unsuitable for regionally distributed melt modelling without comprehensive data sets for model tuning. The use of corrections for katabatic boundary layer effects on nearsurface temperatures (Tg ) and vapour pressures (eg ) in the full energy balance model does not improve model performance (Table 8.6). This is a surprising result, as the KBL effects on Tg and eg are significant (Chapter 3). Turbulent fluxes depend on near-surface gradients of temperature and 227  8.7. Discussion vapour pressure. Without KBL corrections, temperature and vapour pressure gradients will be larger during katabatic episodes, though this may be compensated for by an increased stability correction (Eq. [8.30]). One reason for the discrepancy between energy balance models driven by ambient KBL- corrected meteorological data may be the choice of surface roughness parameter. In practice, energy balance glacier melt models often treat surface roughness lengths as a tuning parameter, when in fact this may hide the errors associated with using temperatures and vapour pressures affected by the KBL. Furthermore, surface roughness lengths will change on daily and seasonal timescales [Smeets et al., 1998]. Diagnosing the true source of the error (meteorological quantity or surface roughness) requires actual measurement of the turbulent fluxes over melting glaciers. Though a number of turbulent flux or profile measurements exist [Munro, 1989; Smeets et al., 1998; Denby and Snellen, 2002], the challenges in obtaining turbulent flux measurements in glacier settings (e.g. surface evolution, level surface, constant measurement height, and power supply) remain prohibitive. In choosing a model for regional melt modelling, increased model complexity should be weighed against the desired accuracy and the parameter uncertainty. Full energy balance models require numerous parameterizations and estimations, and cannot be recommended for regional melt modelling despite their skill. Previous studies have demonstrated the sensitivity of energy balance models to snow albedo [Willis et al., 2002], surface roughness [Hock and Holmgren, 1996; Anslow et al., 2008], and the full energy balance model presented here is also sensitive to the choice of the snow depth used for calculating energy storage. Energy balance models may also be highly sensitive to the wind speed used for calculating turbulent energy fluxes [Cook et al., 2003]. In the current study, turbulent fluxes at Helm Glacier were estimated from observations at Place Glacier, demonstrating that an index wind station provides reasonable simulations of wind speed at a distance of tens of kilometers. For degree-day modelling, air temperature is easily distributed or extracted from reanalysis products, and degree-day factors for snow and ice are spatially and temporally consistent within a region [Shea et al., 2009]. However, given the DD model biases demonstrated 228  8.8. Conclusions above, future research efforts should be directed towards refining degree-day factor estimates. The improved modelling skill between a degree-day and a full energy balance model represents a reduction in MAE of 0.07 m w.e., or 2.4% of the mean observed summer balance. For snowmelt sites only, the model improvement is 0.04 m w.e., or 2.8% of the mean initial water equivalence. Based on these numbers and on the results and discussion given above, the degree-day model is recommended for regionally distributed glacier melt modelling. This research presents the first instance of using surface temperature loggers for monitoring snowline retreat in glacier melt studies. These data represent a valuable source of information for assessing glacier melt model performance, as it can be designed to fill both spatial and temporal gaps in test data. Snowline retreat observations are complementary to both (1) mass balance data, which provide a seasonally integrated melt total, and to (2) automated surface height observations, which provide time series of surface height changes at a single location. For example, melt coefficients in the DD and RIDD models used in this study were derived from historical mass balance data. Given the underestimation of early season snowmelt (judged by the bias errors in MT B ) but the overestimation of seasonal melt in the accumulation zone (Figure 8.3), a temporally variable melt factor [e.g. Braun and Aellen, 1990; Moore, 1993; Stahl et al., 2008] may be required to simulate temporal variability in snow melt rates.  8.8  Conclusions  This study has demonstrated the application of melt models of varying complexity at glaciers of different spatial scales. Degree-day, radiation-indexed degree day, simple energy balance, and full energy balance models were developed and tested against observed mass balance and cumulative melt observations. Benchmark statistics and calculated model errors indicate that a full energy balance model without corrections for katabatic boundary layer effects on near-surface temperatures and vapour pressures provides the most 229  8.8. Conclusions accurate predictions of snow and ice melt. However, the improved accuracy of the full energy balance model may not be warranted by the increased complexity and uncertainty given the number of parameters required for energy balance modelling. A degree-day approach is thus recommended for distributed melt modelling on regional scales, provided that spatial and temporal variability in melt factors has been examined.  230  Chapter 9  Conclusion This research addressed the challenges associated with spatially distributed modelling of glacier melt at a regional scale. It focused, in particular, on methods for specifying spatially distributed fields of the meteorological data required to run glacier melt models. It then compared the relative performance of five models representing the range of complexity of current approaches, including a simple degree-day model, a radiation-indexed degree model, and three variations on an energy balance approach. This concluding chapter provides a summary of key findings of this research, followed by suggestions for further research.  9.1 9.1.1  Summary of Key Findings Glacier Meteorology  In Chapter 3, the nature of the katabatic boundary layer and its effect on near-surface meteorological variables was examined using meteorological data from four glaciers, collected over three seasons in the southern Coast Mountains of British Columbia. Measured near-surface temperatures (Tg ) and vapour pressures (eg ) were found to be strongly affected by katabatic flow, with substantial deviations from ambient conditions estimated from a regional network of off-glacier weather stations. Simple empirical models for estimating Tg and eg from ambient conditions were able to explain the turbulent and/or advective exchange of sensible and latent heat within the katabatic boundary layer. The onset of katabatic boundary layer development corresponded to a critical threshold ambient temperature (T ∗ ) that was identified using piecewise linear regressions. Values for T ∗ ranged between  231  9.1. Summary of Key Findings 4.29 and 8.37◦ C. At ambient temperatures (Ta ) greater than T ∗ , observed Tg were lower than the estimated Ta by an amount that reflects the degree of boundary layer development at a particular site. Near-surface vapour pressures were also found to be affected by KBL development through turbulent exchange with the melting snow/ice surface. Measured above-glacier vapour pressures exceeded ambient values when ambient values were less than the equilibrium vapour pressure for melting ice due to evaporation from the surface. When ambient vapour pressures were greater than the equilibrium value for melting ice, the air in the katabatic boundary layer lost vapour by condensation, depressing eg relative to ea . In Chapter 4 measured wind speed was related to ambient air temperature. The relation between hourly mean wind speed and ambient air temperature exhibited hysteresis, with maximum wind speeds lagging maximum temperatures on the order of 2 to 3 hours. A quasi-physical wind speed model incorporating these observations was developed to reflect both drainage flows during katabatic episodes, and the channelling of synoptic gradient winds during periods of strong synoptic-scale winds or nonkatabatic epsiodes. However, for predicting wind speeds on melting glaciers, observed wind speeds at a representative glacier site provide better predictive capability than the quasi-physical model. Wind speeds were found to be highly correlated between glaciers as well, illustrating the strength and regional coherence of the katabatic signal. For each automatic weather station site, coefficients from the empirical models for Tg , eg , and ug were related to flow path lengths generated from analyses of digital elevation models. These empirical relations provided a reasonable basis for predicting the strength of boundary layer development and the response of meteorological variables at a given site. Flow path lengths can be calculated for any glacier digital elevation model, offering a means for estimating near-surface meterological variables at unmonitored locations given estimates of ambient conditions. The results of Chapters 3 to 5 represent a new approach for estimating near-surface meteorological variables required for calculating turbulent heat transfers, and the results appear robust for extension to glaciers between 4 and 80 km2 . 232  9.1. Summary of Key Findings  9.1.2  Radiative Fluxes  The focus of Chapter 6 was on developing parameterizations of atmospheric transmissivity, snow and ice albedo, and incoming longwave radiation for distributing radiative fluxes in regional melt models. Atmospheric transmissivity (τ ) was modelled at four alpine sites using a modified Bristow-Campbell approach that included vapour pressure. Errors in modelled shortwave radiation (K↓) were greater when using observations from an index station than from τ and potential K↓, demonstrating support for the (τ ) model. Snow albedos (αs ) were calculated from reflected shortwave radiation measured with inverted pyranometers. Albedos were modelled as a function of accumulated positive degree days (PDD) and τ . Relations between glacier albedo and PDD have been presented in a number of previously published studies. The inclusion of τ in this study acknowledges the important role of clouds as selective absorbers of longer wavelengths of solar radiation, and provided improved predictions of albedo relative to a model based solely on PDD. Snow albedos were also found to be highly correlated at the same glacier, with reduced correspondence between glaciers, and it was hypothesized that this represents the highly variable nature of cloud cover in mountainous terrain. Ice albedos were also found to be correlated at the same glacier, and a similar inverse relation to τ was identified. Longwave radiation at on-glacier and off-glacier sites demonstrated no significant differences, suggesting that near-surface temperature inversions did not affect longwave irradiance received at the glacier site. A longwave radiation model tuned to both glacier and off-glacier observations using a clear-sky emissivity model and a cloudiness parameterization was found to yield acceptable modelling results.  9.1.3  Melt Model Test Data  Observations of initial (SWE0 ) and final (SWEf ) snow water equivalences are required for testing the performance of glacier melt models. To generate estimates of SWE0 from snow depth soundings, estimates of density are required. Snow densities observed at mass balance and snow course sites 233  9.1. Summary of Key Findings were modelled as a function of accumulated PDD, which represents an index of densification via meltwater percolation. Locally weighted (Loess) regressions were tested as an approach for estimating SWE0 as a function of elevation, compared to standard ordinary least squares (OLS) regressions using the mass balance data collected between 2006 and 2008. Statistically, the Loess models were found to offer no significant improvements over OLS regressions, though case studies demonstrated the advantages of Loess models. In addition to traditional mass balance observations, this study used temperature records obtained from submersible temperature loggers (Tidbits) placed on the ice surface at the end of an ablation season to estimate the date of snowline retreat in the following season. At the date of snowline retreat, cumulative snow melt (Ms ) is approximately equal to the initial water equivalence (SWE0 ) measured near the beginning of the ablation season. Transects of Tidbits placed on four glaciers provided observations for testing the ability of the melt models to predict cumulative snow melt. The Tidbit temperature records also provided the date at which the snowpack became isothermal (i.e., when the temperature at the snow-ice interface reached 0 degree C). Errors in SWE0 , Ms , and Mi (icemelt) were assessed through a formal error analysis. Errors in observed SWE0 range between 0.13 m w.e. and 0.19 m w.e., or approximately 9% of the mean observed SWE0 . Errors in observed total melt are of the same magnitude.  9.1.4  Glacier Melt Modelling  Melt models of varying complexity were tested using observations of glacier mass balance and snowline retreat. A degree-day model based on historical mass balance data in the region was used as a benchmark, and goodness-of-fit indices were calculated for radiation-indexed degree-day (RIDD), simplified energy balance (SEB), and full energy balance (FEB) models. One version of the FEB model used the previously developed katabatic boundary layer corrections based on flow path lengths to estimate near-surface meteoro-  234  9.2. Suggestions for Future Research logical variables from ambient conditions, while the other FEB model used uncorrected ambient conditions. Goodness-of-fit indices indicated that the full energy balance approach without KBL corrections provides the most improvement over a simple degree day model. RIDD and SEB were found to perform worse than the degree day model, at least on the seasonal time scales being investigated. It is suspected that the ambient FEB model outperformed the KBL-corrected FEB model due to the arbitrarily selected surface roughness parameter, which was kept constant for both models. Given that the improvement in modelling skill between the FEB and the DD model (0.07 m w.e.) was within the measurement errors of SWE0 and bs , the degree-day model is recommended for glacier melt modelling at regional and seasonal scales. RIDD and FEB models are required for melt modelling at higher spatial and temporal resolutions, but the increased model complexity and parameter uncertainty does not appear to be warranted for mass balance modelling. However, the FEB model without KBL corrections generated little or no bias in modelled melt, whereas the DD and RIDD models substantially underestimated melt totals (mean bias errors of -0.22 and -0.29 m w.e., respectively). For generating decadal or century scale estimates of glacier melt and glacier dynamics, a bias error of this magnitude will have considerable cumulative effects.  9.2  Suggestions for Future Research  While this research has demonstrated new approaches to modelling glacier meteorology and melt, it has exposed a number of knowledge gaps in areas which may provide fruitful avenues of future research. Investigations of the spatial patterns of meteorological conditions on temperate glaciers have typically focused on a single glacier [Greuell and B¨ohm, 1998; Strasser et al., 2004]. This study, by considering spatial patterns on multiple glaciers, has demonstrated that sampling glaciers of different scales yields valuable information. A collaborative effort at collecting and collating meteorological data from glaciers of varying sizes and climatic regions should 235  9.2. Suggestions for Future Research be attempted to evaluate the robustness of the relation between katabatic boundary layer strength and flow path length. In this regard, it would also be beneficial to use downscaled reanalysis products for estimating ambient meteorological conditions [e.g. Shea et al., 2008], as extrapolating from regional climate networks will not be possible at more remote glacier locations, which typically have sparse climate observations with a bias to sampling low elevations. Analyses of the 10-minute meteorological observations are also recommended to evaluate transient features in the development and structure of the katabatic boundary layer. Further glacier meteorological investigations should incorporate profile measurements of wind speed and temperature at multiple locations to help refine estimates of surface roughness. Futher work is recommended to refine simple parameterizations of atmospheric transmissivity. All transmissivity models tested in this study produce biases at both low and high values of observed τ , which limits the effectiveness of RIDD and EB melt models. Alternative formulations should be explored to reduce this bias. Correlations between snow albedos observed at multiple sites support the use of atmospheric transmissivity in albedo models, but the effects of τ on ice albedo were not studied extensively due to the small sample size. It is suspected that a similar relation exists, but increased monitoring of ice albedos at multiple sites are required to confirm the relation between αi and τ . While this research did not reveal a significant effect of the katabatic boundary layer on downwelling longwave radiation at Place Glacier (3.2 km2 ), it is possible that this effect could be significant at larger glaciers, which have deeper katabatic boundary layers. Paired observations of L↓ from within the KBL and at a site outside the thermal influence of the glacier are required to evaluate this possibility. An expanded study of empirical models of snow densification is suggested as another avenue for future research. In the current study, density observations from four mass balance and three snow course sites over a 3-year period were used to develop the snow density model based on accumulated positive degree days. With more observations, this model could be developed to examine interannual or spatial variability in the PDD-density relation, or 236  9.2. Suggestions for Future Research the model could be reformulated to assess the temporal evolution of snow density with regards to accumulated PDD. In mass balance calculations, elevation is commonly used as a predictor variable to interpolate observations of accumulation or ablation, as mass balance sampling methods are often insufficient to justify more complex interpolation schemes. In these cases, loess regressions may provide a more realistic estimate of the true SWE than OLS regressions. It may be useful to extend this comparison of loess and OLS regression to other glaciers to provide a further assessment of the merits of the loess approach. Results from the melt model comparisons highlight the need for developing regionally robust turbulent flux parameterizations for inclusion in energy balance models. Simple energy balance models with tuned turbulent exchange coefficients do not adequately simulate melt totals at either (a) different time scales or (b) different sites, illustrating the perils of using such an approach for regional melt modelling. While the results of this research support the use of degree-day models for regional mass balance modelling on seasonal timescales given their simplicity and relative skill, further investigations are required to refine the degree-day factor used in mass balance modelling, as the factors used in this study underestimated seasonal and snow melt totals. In particular, research is needed to focus on improving melt model performance over shorter time scales (hourly and daily) which are necessary for hydrologic modelling.  237  References Cited Adams, W. P., 1966. Ablation and run-off on the White Glacier, Axel Heiberg Island, Canadian Arctic Achipelago. No. 1 in Glaciology. McGill University, Montreal, Quebec. Adams, W. P., Cogley, J. G., Ecclestone, M. A., Demuth, M. N., April 1998. A small glacier as an index of regional mass balance: Baby Glacier, Axel Heiberg Island, 1959-1992. Geografiska Annaler 80A (1), 37–50. Ambach, W., Hoinkes, H., 1963. The heat balance of an alpine snowfield. 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