Thesis Approval

Home , Asgard Range, Lake Chad (Antarctica), Suess Glacier

THESIS APPROVAL

The abstract and thesis of Amy Francis Ebnet for the Master of Science in Geology were presented October 31, 2005, and accepted by the thesis committee and the department.

COMMITTEE APPROVALS: ______Andrew G. Fountain, Chair

______Christina L. Hulbe

______Scott F. Burns

______Roy W. Koch Representative of the Office of Graduate Studies

DEPARTMENT APPROVAL: ______Michael L. Cummings, Chair Department of Geology

ABSTRACT

An abstract of the thesis of Amy Francis Ebnet for the Master of Science in Geology presented October 31, 2005.

Title: A Temperature-Index Model of Stream Flow in Taylor Valley, Antarctica.

The McMurdo Dry Valleys of Antarctica are a polar desert with alpine glaciers, ephemeral streams, and perennially ice-covered lakes. Essentially all water in the lakes and streams originates as glacier melt during the austral summer. Because summer temperatures hover near the melting point, small variations in the energy balance cause the melt water flux to vary from season to season by as much as an order of magnitude. I developed a model for stream flow volume in Taylor Valley, one of the dry valleys, for the purpose of estimating flow over decades to millennia.

Three simple statistical models were developed using measured stream discharge and glacier ablation, a model based solely on air temperature, one including the additional

variable spatial variations in solar radiation and one including the additional variable summer average wind-speed. Results showed that melt-water flow increased exponentially as summer average air temperatures warmed to 0°C. A temperature- based model was sufficient to predict stream flow volume from the north-facing or south-facing glaciers but not the two together. The inclusion of spatial variations in solar radiation allowed both north and south facing glaciers to be modeled. In the

Fryxell and Hoare basins, model results showed r2 values of 0.60 and 0.67 respectively from the temperature-based model and 0.73 and 0.42 from the model including solar radiation. Modeling results for the Bonney Basin were poor partly due to large differences in energy balance controls on the glaciers. Glaciers in the Bonney Basin are characterized by steep slopes and rough surfaces which are important contributors towards melt and are not accounted for in the models. The third model attempted to account for the variation in wind speed throughout the valley with little improvement.

The models predict melt at summer average temperatures below 0°C with threshold temperatures ranging from -4.9°C to -5.5°C.

A TEMPERATURE-INDEX MODEL OF STREAM FLOW IN TAYLOR VALLEY,

ANTARCTICA

AMY FRANCIS EBNET

A thesis submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE in GEOLOGY

Portland State University 2005 ACKNOWLEDGEMENTS

I would like to thank Andrew G. Fountain for giving me the opportunity to do this research and the guidance to see it through. I also was to thank Thomas Nylen for his advice and help with fieldwork, my fellow graduate students for making graduate school fun and interesting and my husband, Jon and my parents, Ken and Harriet for their love and support. Funding for this work came from NSF OPP 0096250.

i TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS ...... i

LIST OF TABLES ...... iv

LIST OF FIGURES...... v

CHAPTER

I INTRODUCTION...... 1

Climate and Setting ...... 1

Melt Models...... 5

II DATA AND PRELIMINARY ANALYSIS ...... 9

Streams ...... 9

Meteorological Data ...... 13

Mass Balance Data ...... 14

Correlations of Temperature with Stream Flow...... 16

III STREAM FLOW VOLUME MODEL ...... 26

IV MODEL APPLICATION...... 33

Temperature-Index Model...... 33

Threshold Temperature and Melt Limit ...... 39

Cliff Melt...... 40

Stream Evaporation ...... 41

Model Calibration...... 43

ii V RESULTS...... 48

Melt/Ablation Results...... 48

Stream Flow Volume Results...... 50

Threshold Temperature and Melt Limit Results ...... 56

VI ERROR ANALYSIS...... 60

VII DISCUSSION AND CONCLUSIONS...... 75

REFERENCES...... 79

iii LIST OF TABLES

Table 1: Total yearly stream flow volume (m3)...... 12

Table 2: Regression r2 and p-values for monthly stream volume and air temperature...... 20

Table 3: Coefficients of determination and p-values for exponential regressions between monthly stream volume and Lake Hoare air temperature excluding November and February...... 23

Table 4: Coefficients of determination and p-values for exponential regressions between seasonal stream volume and Lake Hoare air temperature...... 24

Table 5: Solar radiation and wind speed for glaciers...... 32

Table 6: Coefficients for the temperature index model (T), the temperature-solar radiation model (TI) and the temperature-solar radiation-wind speed model (TIU)...... 48

Table 7: Mean and standard deviation of variables used for normalization...... 48

Table 9: Threshold temperatures in °C and average melt limits and ELA in m above seal level for each glacier and model...... 57

Table 10: Lake Hoare summer average temperature and degree-days...... 66

Table 11: Summary of seasons where degree-days and meteorological data explain the scatter between air temperature and measured stream volume where stream volume is high and temperature low...... 73

Table 12: Summary of seasons where degree-days and meteorological data explain the scatter between air temperature and measured stream volume where stream volume is low and temperature high...... 74

iv LIST OF FIGURES

Figure 1: Site map of Taylor Valley, Antarctica showing locations of glaciers, lakes, stream gauges and meteorological stations...... 2

Figure 2: Taylor Glacier...... 4

Figure 3: Stream gauge on Andersen Creek...... 10

Figure 4: Linear regressions between Canada Stream and streams with incomplete flow records in Taylor Valley...... 11

Figure 5: Canada Glacier meteorological station...... 14

Figure 6: Mass balance stake locations for (a) Canada, (b) Commonwealth, (c) Howard and (d) Taylor glaciers (Fountain, accepted)...... 15

Figure 7: Ablation stake measurement on Canada Glacier...... 16

Figure 8 Polynomial regressions between monthly total stream volume from Canada Glacier and monthly average air temperature at Lake Hoare...... 17

Figure 9: Exponential regressions between monthly total stream volume from the glaciers and monthly average air temperature at Lake Hoare. Note the difference in scale...... 19

Figure 10: Stream hydrographs showing total stream volume from all thirteen streams...... 21

Figure 11: Exponential regressions through monthly stream volume and Lake Hoare air temperature for streams flowing from Canada, Commonwealth, Howard and Rhone glaciers excluding November and February...... 23

Figure 12: Exponential regressions through summer stream volume (totaled for the months of December and January) and Lake Hoare air temperature (averaged over the months of December and January) for streams flowing from Canada, Commonwealth, Howard and Rhone glaciers...... 24

Figure 13: Area-elevation distribution of selected glaciers in Taylor Valley...... 25

Figure 14: Residuals between modeled temperature and measured summer (December – January) temperature at the Taylor Valley meteorological stations...... 28

Figure 15: The sun’s zenith and azimuth angles. Adapted from Iqbal, (1983)...... 30

v Figure 16: Zenith angle as a function of hour for December 1, December 21 and January 31...... 31

Figure 17: Hourly solar radiation calculated on the summer solstice (December 21) and averaged over the ablation area of selected glacier...... 31

Figure 18: Glacier contributing areas and model calculation points for Canada and Commonwealth glaciers...... 35

Figure 19: Glacier contributing areas and model calculation points for Howard and Crescent glaciers and VonGuerard and Aiken streams...... 36

Figure 20: Glacier contributing areas and model calculation points for Suess and LaCroix glaciers...... 37

Figure 21: Glacier contributing areas and model calculation points for Sollas, Marr West, Taylor and Rhone glaciers...... 38

Figure 22: Model flow chart...... 42

Figure 23: Curve fit to the measured melt (ablation reduced by half) on Canada Glacier (a) and the resulting curve for stream flow volume from Canada Glacier compared to the total measured stream flow volume from Andersen, Green and Canada streams (b)...... 44

Figure 24: Sensitivity of T-model to (a) variations in the A coefficient compared to measured stream flow volume from Canada Glacier and sensitivity to (b) variation in the B coefficient compared to measured stream flow volume from Canada Glacier...... 45

Figure 25: Sensitivity of incorporating the solar radiation index in the TI-model and of changes in the C coefficient for (A) Howard Glacier A=0.004 and for (B) Canada Glacier A=0.004...... 46

Figure 26: Effect of incorporating wind speed into the model for melt on Howard and Taylor glaciers. A=0.03, B=1.06, C=0 ...... 47

Figure 27: Comparison of modeled melt and measured ablation for Canada, Commonwealth, Howard and Taylor glaciers...... 49

Figure 28: Modeled and measured stream volume into Lakes Fryxell, Hoare and Bonney...... 52

Figure 29: Modeled and measured stream volume for individual streams...... 52

vi Figure 30: Modeled and measured stream volume for Canada, Commonwealth, Howard, and Crescent glaciers...... 53

Figure 31: Melt limit and ELA for Canada, Commonwealth, Howard, Crescent, VonGuerard and Rhone glaciers...... 59

Figure 32: Effect of errors in temperature for melt (A) and stream flow volume (B) from Canada Glacier...... 61

Figure 33: Lake Fryxell, Hoare and Bonney basin residuals...... 62

Figure 34 Canada, Commonwealth and Howard glacier residuals ...... 63

Figure 35: Replot of Figure 27 and 29 to highlight the model underestimate in the colder, low flow seasons...... 64

Figure 36: 1990-1991 and 1991-1992 Lake Hoare daily average temperatures...... 65

Figure 37: Lake Hoare normalized average temperature and degree-days...... 669

Figure 38: Normalized Lake Hoare air temperature minus normalized stream volume from Lake Fryxell, Lake Hoare, Canada Glacier, Commonwealth Glacier and Howard Glacier versus the difference between Lake Hoare normalized air temperature and Lake Hoare normalized degree-days...... 67

Figure 39: Summer average meteorological data from Lake Hoare, Lake Fryxell, Commonwealth Glacier, Canada Glacier and Howard Glacier...... 68

Figure 40: Normalized air temperature minus normalized stream volume versus normalized relative humidity for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers...... 69

Figure 41: Normalized air temperature minus normalized stream volume versus normalized wind-speed for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers...... 70

Figure 42: Normalized air temperature minus normalized stream volume versus normalized incoming shortwave radiation for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers...... 71

Figure 43: Normalized air temperature minus normalized stream volume versus normalized albedo for Canada, Commonwealth and Howard Glaciers...... 72

vii Figure 44: Normalized air temperature minus normalized stream volume versus normalized net shortwave radiation for Canada, Commonwealth and Howard Glaciers...... 72

viii I INTRODUCTION

The McMurdo Dry Valleys (MDV) of Antarctica are a polar desert with alpine glaciers, ephemeral streams, and perennially ice-covered lakes. The ecosystem consists of microorganisms, mosses, lichens and a few invertebrates which inhabit the streams, lakes and soils of the MDV. The region receives little precipitation (snow) and essentially all water in the streams and lakes originates as glacier melt (Fountain and others, 1999a). Because summer temperatures hover near and often below the melting temperature, small variations in the available energy for melt cause the seasonal melt-water flux to vary by as much as an order of magnitude. Consequently, the ecosystem in the MDV is extremely sensitive to small changes in the energy balance of the glaciers. The study of melt-water production from the glaciers is therefore important for understanding the controls on the structure and function of the ecosystem. Melt-water production in response to climate change is also important for understanding the paleo-hydrology of the MDV.

The purpose of this thesis is to develop a simple model of glacial melt-water production to the lakes in Taylor Valley, one of the MDV, for the eventual purpose of estimating the paleo-hydrology over millennia. This work will aid our understanding of long-term hydrologic changes and contribute to understanding the ecosystem response.

Climate and Setting The MDV are located on the west coast of McMurdo Sound at 77°00’S

162°52’E (Figure 1). They cover an area of approximately 4800 km2, the largest

1 relatively ice-free area in Antarctica (Fountain and others, 1999a). The MDV are relatively ice-free because the Transantarctic Mountains block much of the ice-flow off the east Antarctic Ice Sheet toward McMurdo Sound (Chinn, 1990). Taylor

Valley, approximately 36 km long, with an area of about 400 km2 (Lewis and others,

1999), is bounded by the Asgard Range to the north and the Kukri Hills to the south.

Taylor Valley

Antarctica

Nussbaum Riegel

Figure 1: Site map of Taylor Valley, Antarctica showing locations of glaciers, lakes, stream gauges and meteorological stations.

Taylor Valley contains about 26 alpine glaciers flowing from the adjacent mountain ranges and more than 24 ephemeral streams (Fountain and others, 1999a) that drain melt-water from the glaciers into three large perennially ice-covered lakes

(Bonney, Hoare and Fryxell) located on the valley bottom. The mean annual

2 precipitation in the valley bottom is less than 10 cm water equivalent (weq) (Fountain and others, 1999a) making the valley a polar desert. Mean annual temperature varies between –14.8 and –23.0°C (Doran and others, 2002) with mean summer temperatures of about -2°C. Monthly average wind speeds range from 2-4 m s-1 and tend to increase from the coast inland (Nylen and others, 2004; Doran and others, 2002; Fountain and others, 1999a). Katabatic winds flow off the ice sheet and warm adiabatically as they descend into the valley creating a warmer and drier atmosphere (Nylen and others,

2004; Doran and others, 2002). This favors sublimation and limits the energy available for melt. An up-valley (away from the coast) gradient in climate exists with increasing air temperatures, decreasing humidity, decreasing precipitation, and higher wind speeds (Doran and others, 2002; Fountain and others 1999b). The climatic differences along the valley are enhanced by the Nussbaum Riegel, a 700 m high hill that bisects the valley 20 km from the coast between the Hoare and Bonney basins

(Fountain and others, 1999b).

Glacial melt streams flow to the ice-covered lakes for 4 to 10 weeks during the austral summer (McKnight and others, 1999). Some water is lost from the streams directly through evaporation to the atmosphere or indirectly through the hyporheic zone (Bomblies, 1998). None of the three lakes have an outlet and are surrounded by permafrost so the only loss of volume from the lakes is through sublimation of the ice cover or evaporation from the surface water (Bomblies, 1998).

Taylor Valley glaciers flow from the Asgard Range and the Kukri Hills. Taylor

Glacier, an outlet glacier of the East Antarctic Ice Sheet, flows into the valley from the

3 west (Figure 2). All of the larger glaciers (e.g. Taylor, Canada, and Commonwealth) terminate in vertical cliffs. The north-facing glaciers in the Kukri Hills have higher equilibrium line altitudes (ELA) than the south-facing glaciers in the Asgard Range due to differences in solar radiation (Fountain and others, 1999b). Because of topographic differences, the glaciers in the Asgard Range have larger accumulation basins making them larger than the glaciers in the Kukri Hills. Cool, dry, and windy conditions during summer favor sublimation over melt (Chinn, 1987) and account for

40% - 80% of the ablation on Canada Glacier (Lewis and others, 1998) and 56% ablation on Taylor in 1994-1995 (Johnston, 2004). Melt can occur at air temperatures below 0°C (Chinn, 1987; Lewis and others, 1998), some of which occurs in the subsurface through a solid-state greenhouse effect (Fountain and others, 2004).

Differences in the energy balance allow cliff faces to melt while the upper surface of the glacier remains frozen (Chinn, 1987). Cliffs can contribute up to 20% of the total melt-water runoff (Lewis and others, 1998).

Figure 2: Taylor Glacier.

4 Melt Models Surface melt on glaciers is determined by energy exchange to and from the glacier surface. Energy is primarily exchanged through net radiation (R), sensible heat

(H), latent heat (L), heat conduction through the ice (G) and melt (M),

R + H + L + G + M = 0 (1)

Each term of (1) (except M) can be determined from measurements of radiation, air and ice temperatures, wind speed and vapor pressure. Energy for melt is calculated as a residual from (1) (Paterson, 1994). Complete energy balance models require detailed meteorological measurements, which are difficult to collect and are typically unavailable for long periods of time. We wish to estimate melt for times that predate detailed meteorological measurements, so we adopt an alternative approach, a temperature-index model, in which melt is related to air temperature (T) alone. The simplest form of temperature-index model is,

M = FT (2) where F is a factor of proportionality and T is air temperature. Typically M and T are daily values. Another form is to use only temperatures above 0 °C (positive degree- day) to calculate daily melt,

n n ∑ M = F∑T + ⋅ ∆t (3) i=1 i=1

The summation on the right-hand side of equation (3) is the sum of positive air temperatures over a day where T+ is a temperature >0°C measured over a time interval

5 ∆t (Hock, 2003). The factor F in both equations varies according to the relative importance of terms in (1) (Braithwaite, 1995; Hock, 2003). For example, F is greater for low albedo snow/ice and high elevation because increased net solar radiation produces more energy for melt. F is smaller in conditions of high sublimation because sublimation reduces the energy available for melt. To estimate melt for conditions colder than 0°C, which occur in the MDV, a melt threshold temperature can be defined,

⎧F(Td − T0 ):Td > T0 M = ⎨ (4) ⎩0 :Td ≤ T0

where T0 is a threshold temperature below 0°C and F is a factor of proportionality

(Hock, 2003).

Temperature-index models generally perform well despite their great simplicity because air temperature is a factor in all terms in the energy balance equation, except for shortwave radiation (Braithwaite, 1981; Ohmura, 2001).

Although net radiation is usually the largest heat source to the surface (Braithwaite,

1981; Ohmura, 2001), it does not correlate particularly well with melt because unlike temperature, daily and monthly net radiation are not as variable as melt (Braithwaite,

1981). Temperature-index models are widely used because air temperature data are widely available and easy to interpolate and forecast (Hock, 2003). However, they fail to account for spatial variations in melt due to spatial changes in the energy balance

(Hock, 1999). The accuracy of index models also decreases with shorter time intervals because they ignore the temporal changes in the energy terms, which may

6 overwhelm the effects of temperature (Hock, 2003). Attempts to improve melt predictions usually involve temporal changes in some of these terms (e.g. Hock, 1999;

Kane and Greck, 1997; Kustas and Rango, 1994).

Previous works on melt-water modeling in the MDV have used both energy balance (Chinn, 1987; Lewis and others, 1998; Lewis and others, 1999) and temperature-index models (Dana and others, 2002; Jaros, 2003). The model of Dana and others (2002) used advanced very high resolution radiometer (AVHRR) satellite brightness temperatures of the dry valleys to estimate diurnal and seasonal discharge.

Using 1.1 km resolution pixels, they found a good correlation between the number of pixels with temperature greater than -14oC and discharge. Given the frequent AVHRR coverage of the valley, Dana and others (2002) were able to predict flow biweekly.

Jaros (2003) modeled the spatial variations in melt water production based on the temperature lapse rate but not the seasonal variation. Bomblies (1998) examined spatial variations of stream flow based on patterns of solar shading on the glaciers.

Here, I build the work of Jaros (2003) and Bomblies (1998) and model both spatial and temporal variations in melt water production to estimate stream flow volume. I present three simple models to predict total summer (December-January) melt-water runoff from all of the glaciers contributing to the three major lakes in

Taylor Valley. In addition to a basic temperature-index model, I examine a temperature and solar-radiation index model, and a model that includes wind speed.

The models calculate specific melt (m3m-2 water equivalent) over the glacier surface, and stream flow volume is estimated by the product of melt and surface area of the

7 glacier. Independent field measurements of glacier mass balance are used to constrain the magnitude of the estimated melt while measured stream flow is used to constrain the estimated stream flow volume.

8 II DATA AND PRELIMINARY ANALYSIS

Streams Stream flow measurements in the MDV are important to the monitoring of lake level change because the streams are the only major source of water to the lakes

(McKnight and others, 1994). Stream flow is also important for nutrient budgets in the lakes, a critical factor for lake ecosystems (McKnight and others, 1994).

Measurements of stream flow began in Taylor Valley in the 1990-1991 summer season and have continued to the present (except for the 1992-1993 season). Eight streams are monitored continuously in the Fryxell Basin, two in the Hoare Basin, four in the Bonney Basin, and one that flows to the coast (Figure 1). Stream stage is generally recorded at 15-minute intervals by a pressure sensor and converted to discharge (liters per second) via rating curves developed from discharge measurements (McKnight and others, 1994). Accurate stage measurements are difficult because of stream icing at low flow and the unstable nature of the stream channels during high flow (McKnight and others, 1994). The data quality is assessed during each visit to the measurement sites and a quality level of good, fair or poor is assigned to each measurement (Cozzetto, K., personal communication, 2004). Good stream flow measurements are considered accurate to within 10% whereas poor measurements may be inaccurate by more than 25% (http://huey.colorado.edu/, 2004).

Some stream-flow records are missing in the eleven-year period and were replaced by estimates based on flows from Canada Stream flowing from Canada

Glacier. Canada Stream has the longest and most complete record of all monitored

9 streams in MDV. The monthly stream flow volume for Canada Stream correlates well with most of the streams in the valley (Jaros, 2003). Linear correlations of Canada

Stream volume with eight other streams (Andersen, Green, Commonwealth, Lost Seal,

Delta, House, Priscu, Lawson, Lyons, Santa Fe) were significant at the 1x10-9 level of significance (all p-values were less than 4x10-10) and r2 values ranged from 0.77 to

0.97 (Figure 4). The data quality of Lyons and Santa Fe streams draining from Taylor

Glacier (Bonney Basin), are poor with frequent data gaps. These streams are subject to unstable stream channels and large sediment loads (Jaros, C., personal communication, 2004). Consequently, no significant correlation between these two streams and Canada Stream exists with the available data (Figure 4). For the rest of the stations, only nineteen of 143 seasons of record were estimated for the thirteen streams (Table 1).

Figure 3: Stream gauge on Andersen Creek. Photo by Jonathan Ebnet, 2005.

10 30 30

Ande r se n Green y ) ) 25

25 3 3 y hl m m hl nt 4 20 4 20 o 10 ont m 15 n

(x10 15 (x m e e e y = 0.54x + 0.08 n y = 0.51x + 0.78 e rs m m

10 e 10 u 2 u 2 r l R = 0.91 l R = 0.87 G nde vo -13 vo -11 A 5 p-value 1x10 5 p-value 8x10 0 0 0 1020304050 0 1020304050 Canada monthly volume (x104 m3) Canada monthly volume (x104 m3)

60 40

y l 35 h

) Lost Seal y

t Commonwealth l 50 3 ) n 3

h 30 t m n m 4

4 25

mo 40 o 0 h t 1 20 m x x10 al

30 ( e al 15 e w y = 0.98x + 0.54 y = 1.37x - 1.85 me ( n me

u 20 10 2 l 2

lu R = 0.90 st S

o R = 0.99 o o v 5 -13 v

mmo -27

L p-value 5x10 o 10 p-value 4x10 0 C 0 -5 01020304050 0 5 10 15 20 25 30 4 3 Canada monthly volume (x104 m3) Canada monthly volume (x10 m )

30 8

) 7

3 25

Delta ) House

3 y y

m 6 hl 4 hl 20 m 4 nt

10 5 ont o x 15 (

m 4 e m (x10 a e y = 0.55x - 1.19 e y = 0.14x + 0.18

t 10 m l 3 2 m 2 u e us l R = 0.97 R = 0.87 u D o 5 l -15 2 -13 vo p-value 3x10 H p-value 6x10 0 vo 1 -5 0 0 1020304050 0 1020304050 Canada monthly volume (x104 m3) Canada monthly volume (x104 m3)

40 25 35

)

) Lawson Priscu y 3 3 20 y 30 hl m hl m 4 4 25 0 ont 15 ont 10

20 x (

m y = 0.70x - 0.08 (x1

e y = 0.36x + 1.99 u e 15

2 on m 10 c m R = 0.97 s 2 m s R = 0.77 i 10 u u w l r

l -23 p-value 7x10 -10 P

5 La 5 p-value 4x10 vo vo 0 -5 0 0 1020304050 0 1020304050 4 3 Canada monthly volume (x104 m3) Canada monthly volume (x10 m )

9 120 8 Santa Fe

Lyons ) )

3 100 3 y = 4.83x + 6.09 y 7 hl

m 2 m hl 4 4 6 80 R = 0.14 ont 10 ont 5 p-value 0.10 m

(x 60 (x10 m e e 4 Fe m m y = 0.51x + 1.77 a ons 3 40 u u l l 2 nt Ly 2 R = 0.17 a vo vo

S 20 1 p-value 0.21 0 0 01234567 01234567 Canada monthly volume (x104 m3) Canada monthly volume (x104 m3)

Figure 4: Linear regressions between Canada Stream and streams with incomplete flow records in Taylor Valley. Data obtained from http://huey.colorado.edu/, 2004.

11 Table 1: Total yearly stream flow volume (m3). – indicates missing data and * indicates seasons where at least one month of data was estimated from Canada Stream. †The estimated stream flow volume for Huey Creek in 94-95 was negative. Original data obtained from http://huey.colorado.edu/. Basin Stream 90-91 91-92 93-94 94-95 95-96 96-97 Canada 501,609 299,657 130,138 41,704 109,654 93,011 Green 330,315 223,956 58,251 27,232 68,037 66,060 Lost Seal 525,858 429,060 185,525 19,536 64,959 46,376 Fryxell Aiken 249,898 278,849 57,047 9,447 25,048 33,298 Basin Delta 186,860 135,312 63,312 2,468 *24,075 *25,387 Crescent 208,009 80,447 38,569 1,920 7,651 7,844 VonGuerard 159,018 140,807 67,708 5,864 6,527 4,593 Huey 151,278 34,807 48,056 --† *9,712 4,221 Hoare House *71,629 *48,730 19,865 14,487 13,912 9,003 Basin Andersen *268,856 *161,155 47,954 14,808 41,479 44,780 Lyons ------53,478 -- Bonney Santa Fe ------118,687 97,278 978,154 Basin Lawson *262,008 *168,649 *126,829 100,042 135,372 109,869 Priscu *318,090 *207,209 63,531 41,214 75,282 70,996 New Harbor Commonwealth *512,712 *309,561 114,917 61,622 134,610 115,031

Basin Stream 97-98 98-99 99-00 00-01 01-02 Canada 76,859 131,843 85,334 93,662 760,594 Green 57,072 *79,091 48,331 *71,197 *360,019 Lost Seal 57,517 138,099 73,019 27,646 *687,734 Fryxell Aiken 36,975 107,819 19,557 8,588 742,847 Basin Delta 6,120 25,665 767 69 407,255 Crescent 5,380 10,236 180 0 158,737 VonGuerard 720 7,248 0 0 275,987 Huey 3,218 1,094 1,743 0 *221,856 Hoare House 30,750 19,508 31,747 7,069 117,980 Basin Andersen 25,729 88,550 73,027 *59,700 425,739 Lyons 130,544 124,010 73,340 -- -- Bonney Santa Fe 657,477 470,223 1,458,988 -- -- Basin Lawson 76,221 114,311 119,107 102,406 324,791 Priscu 111,992 78,083 46,344 25,255 518,172 New Harbor Commonwealth 94,859 156,712 101,052 99,351 760,594

12 Meteorological Data Eight meteorological stations collect data year-round in Taylor Valley (Figure

1). Four stations are located on glaciers – one each on Commonwealth, Canada,

Taylor, and Howard glaciers. The other four stations are located on the valley floor – three on the shores of each of the lakes and one at Explorer’s Cove near the coast.

Each meteorological station (Figure 5) measures air temperature, relative humidity, wind speed and direction and incoming shortwave radiation (Doran and others, 2002).

Some stations also measure outgoing shortwave radiation, incoming and outgoing longwave radiation, net radiation, ice/soil temperature, and ice ablation. Data sampling intervals over the period of record ranged from 30 to 60 seconds, and averaging intervals ranged from 10 minutes to 6 hours (Doran and others, 2002). The averaging interval was set to 15 minutes for all stations starting in November 1995.

The meteorological station at Lake Hoare, starting in December of 1985, contains the longest record (Clow and others, 1988). Most of the other stations were installed in

1993-1994 with a few added in the following years.

Figure 5: Canada Glacier meteorological station. Photo by Jonathan Ebnet, 2005.

Surface Mass Balance Data Mass balance measurements are collected on Canada, Commonwealth,

Howard (since 1993-1994), and Taylor glaciers (since 1994-1995). The height change of the snow or ice surface measured against stakes drilled vertically into the ice or snow is used to determine mass balance (Figure 6 and 7). The density of the surface material is measured if it is snow. The stakes are measured twice a year in early

November and late January to determine the winter and summer mass balance

(Fountain and others, 1994). The summer balance measurements in the ablation zone

14 were used to constrain the melt calculation, such that local estimated melt could not exceed measured ablation.

(a) (b)

Figure 6: Mass balance stake locations for (a) Canada, (b) Commonwealth, (c) Howard and (d) Taylor glaciers (Fountain, accepted). Note the difference is scale. Only the ablation zone is measured on Canada and Taylor glaciers.

Figure 7: Ablation stake measurement on Canada Glacier. Photo by Thomas Nylen 2003-2004 season.

Correlations of Temperature with Stream Flow To examine the relationship between stream flow and air temperature, monthly values of stream volume (November, December, January and February) from five glaciers in Taylor Valley (Canada, Commonwealth, Howard, Rhone and Taylor) were plotted against monthly air temperatures from the Lake Hoare meteorological station.

For all cases except Taylor Glacier, the trend between temperature and stream flow shows a gradual increase in stream flow with increase in temperature at low temperatures. Once the average temperature reaches about -1.5 °C, the slope of the trend increases sharply. Polynomial and exponential equations were fit to the monthly total stream volume and air temperature data by least squares and Marquardt-

16 Levenberg methods to determine the best form of equation to use for the model. The

Marquardt-Levenberg method is an iterative technique for finding the best-fit parameters for a nonlinear equation. The method works by making an initial guess of the parameter values and calculating the sum of squared differences between the observed and predicted data. The parameters are then adjusted in each iteration to decrease the sum of squared differences until they no longer decrease significantly

(Bates and Watts, 1988). Good fits could not be found using first, second and third order polynomials (Figure 8).

90 Canada Glacier Streams 80

e 70 m u l

o 60 First order ) V 3

m 50 m a 4

re Second order t

10 40 S (x y 30 hl Third order

ont 20 M 10 Measured 0 -14 -12 -10 -8 -6 -4 -2 0 Lake Hoare Monthly Average Temperature (°C)

Figure 8 Polynomial regressions between monthly total stream volume from Canada Glacier and monthly average air temperature at Lake Hoare.

The quality of fit was assessed with the coefficient of determination and the p- value. The coefficient of determination,

2 2 2 ∑∑(Qm − Qm ) − (Qm − Qc ) r = 2 ()Q − Q ∑ m m (5)

where, Qm and Qc are the measured and calculated stream flow volume respectively

and Qm is the mean measured stream flow volume (McClave and Sincich, 2000) increases while the p-value decreases with increase in order of the polynomial

17 equation (Table 2). The first order equation fails to depict the steep increase in stream volume at temperatures above -1.5°C. The second and third order polynomials perform better but still do not fit the highest three or four data points. The second and third order equations also causes the regressions to have a negative slope, which represents a decrease in stream flow volume with increase in temperature for a small range in temperatures, which is not representative of the trend in the data.

To capture the steep increase in stream volume at warmer temperatures an exponential equation of the form

Runoff = ae (bTemp) + c (6) was tried (Figure 9). The coefficients a and b in (6) control the curvature of the function and c controls the horizontal asymptote. This allows the regression to fit the rapid rise in stream volume much better than the polynomials (Figure 9). The exponential equation yielded higher r2 values and lower p-values than the polynomial equations (Table 2). The poor fit for Taylor Glacier may be due to poor stream data.

18 90 140 80 Canada Glacier Streams Commonwealth Glacier Streams

120 5 (0.73Temp) 3 6 (0.82Temp) 3 e e Runoff = 7x10 e + 8x10 Runoff=1.06x10 e + 2.3x10 70 m m u

l 100 lu

60 o ) ) V Vo 3 3 80 m m 50 m a m a 4 4 e e r r t 10 40 60 x10 S St ( (x y y l 30 hl h

t 40 n ont o 20 M M 20 10 0 0 -14 -12 -10 -8 -6 -4 -2 0 -14 -12 -10 -8 -6 -4 -2 0 Lake Hoare Monthly Average Temperature (°C) Lake Hoare Monthly Average Temperature (°C)

30 18

Howard Glacier Streams 16 Lawson Stream (Draining from Rhone Glacier) 25 5 (0.94Temp) 3 e Runoff=2x10 e - 2.6x10 e 14 5 (0.45Temp) 3 m m Runoff=1.3x10 e + 7.6x10 u u l l o 20 o 12 ) ) V V 3 3 m m 10 m m a a 4 4 e 15 e r r t t 8 x10 x10 S S ( ( y 10 y 6 hl hl ont ont 4 M 5 M 2 0 0 -14-12-10-8-6-4-2 0 -14 -12 -10 -8 -6 -4 -2 0 Lake Hoare Monthly Average Temperature (°C) Lake Hoare Monthly Average Temperature (°C)

140

120 Taylor Glacier Streams e 6 (0.006Temp) 6 m Runoff=6.1x10 e - 5.7x10 u

l 100 o ) V 3 80 m m a 4 re t 60 S (x10 y

hl 40

Mont 20

0 -10-8-6-4-2 0 Lake Hoare Monthly Average Temperature (°C)

Figure 9: Exponential regressions between monthly total stream volume from the glaciers and monthly average air temperature at Lake Hoare. Note the difference in scale.

19 Table 2: Regression r2 and p-values for monthly stream volume and air temperature. Glacier Regression r2 p-value 1st Order 0.37 1.7x10-05 2nd Order 0.50 1.0x10-06 Canada 3rd Order 0.57 2.2x10-07 Exponential 0.58 2.7x10-09 1st Order 0.33 5.2x10-05 2nd Order 0.47 3.1x10-06 Commonwealth 3rd Order 0.55 5.6x10-07 Exponential 0.57 6.0x10-09 1st Order 0.24 8.3x10-04 2nd Order 0.38 7.4x10-05 Howard 3rd Order 0.48 1.1x10-05 Exponential 0.48 2.2x10-07 1st Order 0.55 1.6x10-08 2nd Order 0.68 1.4x10-10 Rhone 3rd Order 0.70 2.9x10-10 Exponential 0.70 2.3x10-12 1st Order 0.10 2.9x10-01 2nd Order 0.21 3.1x10-01 Taylor 3rd Order 0.29 3.5x10-01 Exponential 0.10 3.0x10-01

The seasonal stream volume is dominated (94%) by flow in December and

January (Figure 10). Therefore, the months of November and February may be dropped from the model with little error (except for November 1997) in the estimate of total annual stream volume.

20 0.9 0.9

l 0.8 1990-1991 l 0.8 0.73 1991-1992 nua 0.7 nua 0.7 n n A 0.6 A 0.6

e e l 0.48 0.46 l a a m m 0.5 0.5 lu lu 0.4 0.4 Vo Vo 0.26 0.3 0.3 on Tot on Tot i i t t c 0.2 c 0.2 a 0.1 0.05 0.02 a 0.1 Fr Fr 0.01 0.00 0.0 0.0 Nov-90 Dec-90 Jan-91 Feb-91 Nov-91 Dec-91 Jan-92 Feb-92 Month Month

0.9 0.82 0.9

l 0.8 l 0.8 1993-1994 1994-1995 0.7 0.7 0.61 nnua 0.6 nnua 0.6 A A l l a a me me 0.5 0.5 t t o o lu lu 0.4 0.4 0.32 Vo Vo 0.3 0.3 on T on T i i t 0.13 t c 0.2 c 0.2 a a 0.07 0.1 0.02 0.03 0.1

Fr Fr 0.00 0.0 0.0 Nov-93 Dec-93 Jan-94 Feb-94 Nov-94 Dec-94 Jan-95 Feb-95 Month Month

0.9 0.9

l l 0.8 1995-1996 0.8 1996-1997 0.65

0.7 nua 0.7 n nnua 0.56

0.6 A 0.6

A e l l a a m me 0.5 0.5 t 0.40 u o l lu 0.4 0.4 Vo Vo 0.3 0.3 on Tot

on T 0.21 i i t t 0.13 c

c 0.2 0.2 a a 0.1 0.1 0.04 Fr Fr 0.01 0.00 0.0 0.0 Nov-95 Dec-95 Jan-96 Feb-96 Nov-96 Dec-96 Jan-97 Feb-97 Month Month

0.9 0.9

l 0.8 1997-1998 l 0.8 1998-1999 0.7 0.7 nnua 0.6 0.52 nnua 0.6 0.51 A A l l 0.45 a a

me 0.5 me 0.5 t t o o lu 0.4 lu 0.4

Vo 0.3 0.23 Vo 0.3

on T 0.21 on T i i t t

c 0.2 c 0.2 a 0.1 0.04 a 0.1 0.04

Fr Fr 0.01 0.0 0.0 Nov-97 Dec-97 Jan-98 Feb-98 Nov-98 Dec-98 Jan-99 Feb-99 Month Month

0.9 0.9

l l 0.8 1999-2000 0.8 2000-2001

0.7 nua 0.7 n nnua 0.57

0.6 A 0.6

A e l l 0.46 a

a 0.44 m me 0.5 0.5 t o lu lu 0.4 0.4 0.31 Vo Vo 0.3 0.3 on Tot on T i i t t c c 0.2 0.2 0.11 a a 0.1 0.05 0.05 0.1

Fr 0.01 Fr 0.0 0.0 Nov-99 Dec-99 Jan-00 Feb-00 Nov-00 Dec-00 Jan-01 Feb-01 Month Month

0.9 0.9

Total for all Seasons l l 0.8 2001-2002 0.8 0.7 0.59 0.7 nnua nnua 0.6 0.6 A A e

l 0.48 l 0.47 a a m me 0.5 0.5 t

0.39 u o l lu 0.4 0.4 Vo Vo 0.3 0.3 on Tot on T i i t t

c 0.2

c 0.2 a 0.1 0.02 0.1 0.03 0.03 Fra Fr 0.00 0.0 0.0 Nov-01 Dec-01 Jan-02 Feb-02 Nov Dec Jan Feb Month Month

Figure 10: Stream hydrographs showing total stream volume from all thirteen streams.

21 The average air temperature at Lake Hoare for the months of November and

February are -7.4°C and -8.2°C respectively, whereas the average temperature for the months of December and January are -2.0°C and -2.3°C respectively. Discarding the

November and February values reduces the range of temperature over which stream flow is compared. An exponential function fit to the December and January data resulted in curve fits (Figure 11) that had lower r2 values and higher p-values (Table 3) compared to the curve fit based on November through February values. The statistics for December and January data are worse than for November through February because the range of temperatures is smaller while the scatter at the warmer temperatures remains constant. The scatter in the December and January data is probably a result of non-temperature-related variations such as snowfall, cloud cover and wind. These influences will be discussed in Chapter 6.

22 90 140 80

Canada Glacier Streams 120 e e Commonwealth Glacier Streams m 70 m Runoff=6.9x105exp(0.67Temp) - 9x103 6 (0.72Temp) 4 lu lu 100 Runoff=1.05x10 exp - 3.7x10 60 ) ) Vo Vo 3 3 m 50 m 80 m m a a 4 4 e e r r 10 40 10 60 St St (x (x y y l 30 l h h t t 40 n 20 n o o M 10 M 20 0 0 -6 -5 -4 -3 -2 -1 0 -6 -5 -4 -3 -2 -1 0 Lake Hoare Monthly Average Temperature (°C) Lake Hoare Monthly Average Temperature (°C)

30 18 16 Lawson Stream (Draining from Rhone Glacier) 25 Howard Glacier Streams e

e 5 (0.64T) 4 5 (0.71T) 4 Runoff=1.2x10 exp + 2.6x10 m Runoff=1.9x10 exp - 1.8x10 m 14 u u l 20 l o o 12 ) V ) V

3 15 3 m 10 m m a 4 4 ream re t t

10 10 8 x10 ( S (x y S y l l 6 h h

t 5 n

o 4 ont M

0 M 2 -5 0 -6 -5 -4 -3 -2 -1 0 -6 -5 -4 -3 -2 -1 0 Lake Hoare Monthly Average Temperature (°C) Lake Hoare Monthly Average Temperature (°C)

Table 3: Coefficients of determination and p-values for exponential regressions between monthly stream volume and Lake Hoare air temperature excluding November and February. Glacier r2 p-value Canada 0.43 9.3x10-4 Commonwealth 0.43 8.6x10-4 Howard 0.40 1.7x10-3 Rhone 0.46 4.9x10-4

To determine a seasonal trend between air temperature and stream volume, I totaled the stream volume over December and January and averaged the December and January air temperatures (Figure 12). The exponential regression through the seasonal data yielded much higher coefficients of determination and lower p-values

(Table 4). Because my goal is to estimate total seasonal stream flow and because

23 summer values correlate better than monthly values, I use the summer values in the model. From this point on, summer refers to December through January.

16 25 Canada Glacier Streams 14 Commonwealth Glacier Streams

Runoff=7.04x106e(2.2Temp) + 1.3x105 20 me 12 me Runoff=1.29x107e(2.4Temp) + 1.3x105 lu lu ) ) Vo 10 Vo 3 3 15 m m m m a a 5 5 e 8 e r r x10 x10 St St ( ( 10 r 6 r e e mm 4 mm 5 Su 2 Su

0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

5 5 Howard Glacier Streams Lawson Stream (Draining from Rhone Glacier)

6 (3.3Temp) 4 4 Runoff=5.18x10 e + 1.5x10 4 5 (1.7Temp) 4 me me Runoff=8.2x10 e + 9.3x10 lu lu ) ) Vo Vo 3 3 3 3 m m m m a a 5 5 e e r r x10 x10 St St ( 2 ( 2 r r e e mm 1 mm 1 Su Su

0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

Table 4: Coefficients of determination and p-values for exponential regressions between seasonal stream volume and Lake Hoare air temperature. Glacier r2 p-value Canada 0.77 5.4x10-4 Commonwealth 0.82 1.2x10-4 Howard 0.79 2.5x10-4 Rhone 0.74 6.6x10-4

To estimate the total melt-water flux into the lakes I need to estimate stream flow in ungauged streams and, where glaciers contact the lakes, flux directly from the

24 glaciers to the lakes. Therefore, I need to estimate ice melt rather than stream flow directly. This approach is useful for estimating stream flow in past glacial conditions, where the distribution of glacier area may be different from today. This is also useful for estimating stream flow for glaciers with different area – elevation distributions

(Figure 13). The results from this chapter provided important information on the temperature-melt (expressed as stream flow) relation for the stream flow model.

LaCroix 2000 Suess Rhone

Canada 1500 )

m How ard ( n o

i 1000 Commonw ealth evat l E 500 Taylor

0 0 1020304050 Cummulative Area (x106 m 2)

Figure 13: Area-elevation distribution of selected glaciers in Taylor Valley.

25 III STREAM FLOW VOLUME MODEL

Stream flow volume from each glacier is based on the total melt and contributing area by,

Q = MdA (7) ∫A where Q is the total seasonal stream flow volume and M is the total seasonal specific melt for each stream (m3m-2) over the glacier area A. The integral over A may be approximated by,

Q = ∑ M (Tx, y,z , I,U )A(z) (8).

I choose M to be a function of three variables, temperature (T), which varies from summer to summer and is itself a function of location (x, y, z), solar radiation (I), which is constant in time but varies spatially, and wind speed (U), which also only varies spatially. As discussed earlier, M is most accurately calculated using a complete energy balance at the ice surface, but my intent is to predict melt, using as few variables as possible, during times prior to meteorological measurements in

Taylor Valley. Temperature was chosen because it is a key variable in the energy balance and is highly correlated with melt. Solar radiation was chosen because of the large spatial differences in solar radiation received by the glaciers in the valley (Dana and others, 2002; Fountain and others, 1999). Wind is included because of its fundamental importance in the turbulent terms of the energy balance equation and its demonstrated influence on partitioning ice ablation between melt and sublimation

(Johnston, 2004; Fountain and others 1999b; Lewis and others, 1998).

26 To account for temperature control on melting, the differences in temperature across the valley are first defined. The seasonal air temperature within Taylor Valley increases with distance from the coast (Doran and others, 2002) and from the center of the valley bottom toward the side (Delaney, Chad, written communication, 2003) as a result of warming over the exposed soil. Air temperature also decreases with elevation according to the adiabatic lapse rate (Doran and others, 2002). The spatial variation in summer average temperature is approximated by

Tx, y,z = 0.09x + 0.24y − 9.8z − T0 (9)

(Delany, Chad, written communication, 2003) where x is the distance from the coast, y is the distance from the valley bottom and z is the elevation (all in km) and To is the summer average temperature at the coast. The vertical gradient, -9.8 °C km-1, is the dry adiabatic lapse rate, which is typical for Taylor Valley (Lewis and others 1998).

The other gradients were determined by a least squares polynomial fit to the average summer air temperature measurements across the valleys from the 1997-1998 season.

Equation (8) was modified to predict valley temperatures from the meteorological station at Lake Hoare, which has the longest running record,

Tx, y,z = 0.09(x − xH )+ 0.24(y − yH )− 9.8(z − z H )+ TH (10) where the subscript H indicates the value at Lake Hoare. Summer average air temperatures were calculated for seven other meteorological stations in the valley and compared to the measured summer average temperatures (Figure 14). All deviations

27 are less than 1°C, and many are less than 0.5°C. The coefficient of determination between measured and calculated air temperature is r2 = 0.95.

1.0 C ce °

n 0.5 e r e f f i 0.0 e d r u at

er -0.5 p Tem -1.0 1993- 1994- 1995- 1996- 1997- 1998- 1999- 2000- 2001- 1994 1995 1996 1997 1998 1999 2000 2001 2002 Season

Canada Bonney Commonwealth Howard Taylor Fryxell Explorer's Cove

Figure 14: Residuals between modeled temperature and measured summer (December – January) temperature at the Taylor Valley meteorological stations. A positive difference indicates that the modeled temperature is greater than the measured.

Potential direct solar radiation, I, is calculated

⎛ P ⎞ 2 ⎜ ⎟ ⎜ P cos θ ⎟ ⎛ Rm ⎞ ⎝ 0 Z ⎠ I = I 0 ⎜ ⎟ ψ a cosθ I (11) ⎝ R ⎠

2 -2 ⎛ Rm ⎞ (Hock, 1999) where I0 is the solar constant (1368 W m , Fröhlich (1993)), ⎜ ⎟ is ⎝ R ⎠ the eccentricity correction factor where R is the instantaneous sun-Earth distance, and

Rm is the mean sun-Earth distance, ψa is the mean atmospheric clear-sky transmissivity, P is the atmospheric pressure, P0 is the mean atmospheric pressure at

28 sea level, θZ is the zenith angle (angle between vertical and the solar ray) and θI is the incident angle between the solar ray and the normal to the surface. Following Iqbal

(1983), the eccentricity correction factor is

2 ⎛ R ⎞ ⎜ m ⎟ = 1.000110 + 0.034221cosΓ + 0.001280sin Γ ⎝ R ⎠ (12) + 0.000719cos 2Γ + 0.000077sin 2Γ

2π (J −1) Γ = (13) 365 where Γ is the day angle (in radians) and J is the Julian Day. The incident angle of the solar ray and earth surface is calculated

cosθ I = cos β cosθ Z + sin β sinθ Z cos(ϕ sun −ϕ slope ) (14)

where β is the angle of the surface slope from horizontal, θZ is the zenith angle, φsun is the solar azimuth (the angle at local zenith between the plane of the observer’s meridian and the plane of a great circle passing through the zenith and the sun), and

φslope is the aspect of the surface (Figure 15).

29 Zenith

Sun E Path of sun θz

φsun N S

Projection of sun’s path W

Figure 15: The sun’s zenith and azimuth angles. Adapted from Iqbal, (1983).

The zenith angle is a function of latitude, solar declination and hour angle.

The hour angle is the angle measured at the celestial pole between the observer’s meridian and the solar meridian. The solar declination (δ) is the angle between a line joining the centers of the sun and earth at solar noon and the equatorial plane. The solar declination for a given day is

δ = 0.006918 − 0.399912cosΓ + 0.070257sin Γ − 0.006758cos 2Γ + 0.000907sin 2Γ (15) − 0.002697 cos3Γ + 0.00148sin 3Γ with a maximum error of 0.0006 radians. The zenith angle can be calculated

cosθ Z = sinδ sinφ + cosδ cosφ cosω (16)

30 where φ is the geographic latitude (-77°40’) and ω is the hour angle where noon is

0°, morning is positive and a one hour increment is 15° (Figure 16). Finally the solar azimuth is calculated

(sin(90 −θ z )sinφ)− sinδ cosϕ sun = (17) cos()90 −θ z cosφ

1-Dec 21-Dec 31-Jan

) 85

ees 80 r

eg 75 e (d

l 70 g n 65 A h t i 60 n e

Z 55 50

0 0 0 0 0 0 0 0 0 :0 00 00 :0 :0 :0 :0 :0 :0 :0 :0 00 2 4: 6: 8 2 8 0: 10 1 14 16 1 20 22 Hour

Figure 16: Zenith angle as a function of hour for December 1, December 21 and January 31.

Crescent 800 Howard 700 Taylor ) -2 600 Canada m

W Commonwealth ( 500 Rhone on i t 400 a di a 300 R r a l

o 200 S 100

0 0 0 0 00 00 00 00 :0 :0 :0 :0 :00 :00 :00 :00 2: 4: 6: 8: 8 0 2 0 10 12 14 16 1 2 2 Hour

Figure 17: Hourly solar radiation calculated on the summer solstice (December 21) and averaged over the ablation area of selected glacier.

31 Slope and aspect were measured at points 50 m in elevation apart along the center of the ablation zone of each glacier. The potential solar radiation was calculated at each point at hourly intervals from December 1 to January 31. For the hours when the sun drops below the plane of the glacier surface, potential solar radiation was set to zero. No effect of shadowing by surrounding terrain was included. The results were averaged over time then over the surface of the glacier, yielding a single solar radiation value for each glacier (Table 5).

The summer average wind speed (U) was calculated by averaging the summer wind speed over all years of record for each meteorological station in the valley (Table

5). The wind regime was estimated by means of an ordinary kriging method in which an ellipse defined the kriging domain for Taylor Valley, and wind speeds were estimated using the nearest two to four meteorological stations (Delany, Chad, personal communication, 2004).

Table 5: Solar radiation and wind speed for glaciers. Solar Average Radiation Wind Speed Glacier (W m-2) (m s-1) Canada 242 3.2 Commonwealth 243 3.5 Howard 266 2.7 Crescent 275 3.0 VonGuerard 279 3.2 Taylor 257 4.5 Rhone 210 4.3 LaCroix 221 3.8 Suess 214 2.7

32 IV MODEL APPLICATION

Temperature-Index Model To estimate total melt-water contribution to the lakes, I start with the temperature-index model then examine the temperature-solar radiation model and finally the temperature-solar radiation-wind model. Many temperature-index models

(hereafter, T-model) are based on a linear relationship between air temperature and melt M = f(Tx,y,z) (Hock, 1999; Kane and Greck, 1997; Braithwaite, 1995; Kustas and

Rango, 1994). As discussed previously, plots of summer stream flow volume and summer average temperature in Taylor Valley clearly show a non-linear relationship that is well approximated by an exponential function.

M = Ae(BT*) + E (18) where A, B and E are empirically-determined parameters. The constant E effectively prevents the calculation of melt below a threshold temperature (the lowest temperature at which melt occurs). T* is the normalized air temperature (discussed below). Solar radiation is added to the temperature-index model to improve the melt prediction spatially. The temperature-solar radiation (TI) model is of the form

* * M = Ae B(T +CI ) + E (19) where I* is normalized potential direct solar radiation on a glacier (discussed below) and C is an empirically-determined parameter. Finally a temperature-solar radiation- wind (TIU) model is given by the relation

* * * M = Ae B(T +CI +DU ) + E (20) where U* is normalized average wind speed for each glacier in Taylor Valley

(discussed below) and D is an empirically-determined parameter.

33 Each glacier was divided into drainages bounded by the water flow divides to different valley streams, as defined by surface topography and expressed by elevation contours on topographic maps. The drainages were further subdivided by the 50 m elevation contours to account for the elevation change on the glacier in the calculation of air temperature (Figure 18, 19, 20, and 21). Summer average temperature was calculated at the midpoint of each section for the 11 seasons using equation (9). Melt was then calculated at the midpoints (equations 18 – 20) and multiplied by the surface area of the corresponding section yielding a melt volume. Stream volume for each drainage is the sum of the melt volumes over the surface of the drainage.

Normalized values of the independent variable were used in the TI and TIU models, (equations 19 and 20) to assess the relative importance of each variable and in the T-model for consistency with the other two models. The temperatures were normalized

T − T T* = x,y,z x,y,z (21) ST where Tx, y,z is a mean and ST is a standard deviation of temperatures in the ablation zones of the glaciers in Taylor Valley. To calculate the mean and standard deviation of air temperature, air temperatures were calculated at the midpoint of the 50 m elevation intervals in the ablation zones from equation (9) of every glacier contributing melt to the three lakes in the valley for each of the eleven seasons of the model. The mean and standard deviation is calculated from all midpoints over the eleven seasons.

34 Canada Glacier

500 Canada Stream drainage area

250 Lake Fryxell drainage area

Andersen Creek drainage area

Lake Hoare 100 drainage area Green Creek drainage area

Commonwealth Glacier 500

Commonwealth Stream drainage area

250 Aiken Creek Lost Seal Creek drainage area drainage area

100 Huey Creek drainage area McKnight Creek drainage area

Figure 18: Glacier contributing areas and model calculation points for Canada and Commonwealth glaciers.

35 Howard and Crescent glaciers

500

Delta Stream 500 750 drainage area

Crescent Stream 750 drainage area

VonGuerard and Aiken Stream Contributing Areas

Aiken Stream drainage area

500 VonGuerard Stream drainage area

750

Figure 19: Glacier contributing areas and model calculation points for Howard and Crescent glaciers and VonGuerard and Aiken streams.

36 Suess Glacier

1000

750 House Stream drainage area 500

Lake Chad 250

Lake Chad drainage area Mummy Pond drainage area

LaCroix Glacier

1250

750 1000

Priscu Stream drainage area 500

250

Figure 20: Glacier contributing areas and model calculation points for Suess and LaCroix glaciers.

37 Sollas and Marr West Glaciers

500 1000

Priscu Stream drainage area

750

1250

1000

Taylor and Rhone Glaciers 1250 Lawson Stream drainage area

Lake Bonney drainage area Santa Fe Stream drainage area

250

Lyons Stream drainage area

Figure 21: Glacier contributing areas and model calculation points for Sollas, Marr West, Taylor and Rhone glaciers.

38 For the TI model, the potential direct solar radiation was calculated at the midpoint of each elevation interval on the glacier and averaged over the ablation zone of each glacier. The average solar radiation value for each glacier ablation zone was normalized using the average and standard deviation of all glaciers in the valley

I − I I* = (22) S I

where I and SI are the mean and standard deviation of the solar radiation values of all the glaciers, respectively.

For the TIU model, the summer average wind speed over all seasons and over each glacier is normalized over all glaciers in the valley

U −U U* = . (23) SU

The mean (U ) and standard deviation (SU) of the wind speed for all glaciers was calculated from summer average wind speed over all glaciers in the model.

Threshold Temperature and Melt Limit As temperature decreases with increasing elevation, calculated melt decreases to zero when the temperature drops below a threshold value. The threshold temperature for melt (Tm) is the summer average temperature below which no melt occurs. This temperature is derived from the models by setting M in equation (19) to zero and solving for T,

ln(E / A) T = − CI − DU (24). m B

39 The coefficients D and C are set to zero for the T-model and D is set to zero for the

TI-model. The melt limit is the elevation at which the threshold temperature occurs according to the lapse rate. Tm is constant and is the same for every glacier in the T- model. The melt limit, however, varies from glacier to glacier because temperatures increase with distance from the coast and the valley bottom. For the TI and TIU- models, Tm remains constant seasonally but varies spatially among glaciers due to the local effects of solar radiation and wind-speed. This also increases the variability in melt limit among glaciers.

Cliff Melt Vertical cliffs commonly form the lower margin of the glaciers in Taylor

Valley. Attempts were made to incorporate cliff aspect and shading into the models with little success because of the complexities of the enhanced melt on the cliffs.

Instead, cliff melt (Mc) was calculated separately from surface melt using a separate temperature-index model,

(bTx, y , z ) M c = ae + c (25) where a = 0.48 m, b = 0.3 °C-1, and c = -0.026 m, as I determined from the regression fit to measured ablation on the cliffs of Canada Glacier (data obtained from http://huey.colorado.edu). I applied equation (25) to all glacier cliffs in the valley.

The total volume of cliff melt was then calculated as the product of cliff melt and cliff area. The area was determined by measuring the cliff length between the 50 m contours in ArcMap and multiplying it by the cliff height measured on a LIDAR high-

40 resolution digital elevation model. Where measurements were not available, cliff height was assumed to be 20 m. The total melt from the cliffs was then added to the total runoff over the glacier surface (equation 8).

Stream Evaporation Few of the stream gauges are located near the point where water runs off the glacier. The gauges on some streams (e.g. Delta) may be several kilometers from the source. In these conditions evaporative stream losses can be important. Because our goal is to predict water flow to the lakes, it is important to define evaporative stream losses. Evaporation loss is estimated as a function of stream length and width. That is, a relatively longer and wider stream suffers relatively larger losses than narrower shorter streams. Evaporation pan experiments yield summer evaporative rate of

7.14x10-8 m3m-2s-1 of water (Gooseff and others, 2003). To obtain volume loss, the specific loss rate is multiplied by the stream length and width and the length of the melt season (two months).

41 Phase I

Distance from Coast

Equation (9) Equation (17) MELT Distance from Floor Air Temperature T-model

Elevation

Slope

Equation (10) Equation (18) MELT Aspect Solar Radiation TI-model

Elevation

Equation (19) MELT Average Wind-speed TIU-model

Phase II

Glacier Surface Area Equation (6) Runoff

MELT from Phase I

Cliff Distance from Coast

Equation (9) Cliff Air Cliff Distance from Floor Temperature

Cliff Elevation Equation (22)

Cliff Melt

Equation (6) Add Cliff Area Cliff Runoff Total Runoff

Evaporation Stream Surface Area Evaporative Loss Rate Total Input to Lake Subtract

Figure 22: Model flow chart.

42 Model Calibration The models cannot be fit directly to observed stream flow volume because of the summation in equation (8). Melt limits and thus the contributing areas of the glaciers depend on the empirically-derived coefficients in the melt equations. So the models are fit to measured ablation and then compared to measured stream flow volume. The coefficients for each melt model were determined by fitting equations

(17-19) to the measured ablation from Canada, Commonwealth, Howard, and Taylor glaciers using the Marquardt-Levenberg method in PSI-PLOT software. Sublimation makes up a significant proportion of ablation (40-80%) from the glaciers (Lewis and others 1998). To account for the fraction of ablation attributed to sublimation, the ablation data were divided in half before fitting. There is much uncertainty in the magnitude of sublimation and attributing half of the ablation to sublimation is only a rough value.

The parameter values which provide a best fit for the melt data might not produce the best fit for the stream volume. For instance, stream volume was calculated by equation (6) using the melt calculated from the T-model, equation (18), with the best fit parameter values for Canada Glacier and compared to the measured stream volume from Canada Glacier. While the results of the curve fit worked well for the melt calculation, stream volume was significantly underestimated for the warm years (Figure 23).

43 0.20 16 ) 3 measured 14 measured m 5 0.15 predicted 12 predicted ) q e

(x10 10 w e

m 0.10 8 u ( l lt o 6 e V M 0.05 a 4 b eam r

t 2 S 0.00 0 -7 -6 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

The coefficients A and B were adjusted to improve the calculated stream volume curve. The solar and wind terms represent the spatial and not temporal variations in melt and stream flow so changing C and D will not improve the stream flow results for the warm years. The estimated melt with the new coefficients are compared to the measured ablation to ensure that modeled melt does not exceed measured ablation.

The coefficients for the models were adjusted to best fit the stream flow volume data from as many streams as possible. Small changes in the coefficients can produce drastically different results. Changes in A and B, in the T-model have the largest effect at warmer temperatures (Figure 24).

44 20 20 measured measured ) ) 3 3 A=0.002 16 B=2.8 m m 16 5 5 A=0.003 B=2.9 x10 x10 ( ( A=0.004 12 B=3.0

12 e e

A=0.005 m B=3.1 m u u l l

A=0.006 o B=3.2 o 8 8 V V eam eam 4 r 4 r t t S S 0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

Figure 24: Sensitivity of T-model to (a) variations in the A coefficient compared to measured stream flow volume from Canada Glacier. The coefficient B=3.0. And sensitivity to (b) variation in the B coefficient compared to measured stream flow volume from Canada Glacier. The coefficient A=0.004.

Model accuracy is evaluated using the coefficient of determination (r2 value)

(McClave and Sincich, 2000), which indicates what fraction of the variability in stream flow volume is accounted for by the model. Negative r2 values indicate the variability in the difference between measured and calculated stream flow volume is larger than the variability in the measured stream flow volume. The total error in volume summed over the eleven summer seasons was calculated by dividing the difference between total calculated and measured stream volume by the total measured stream volume.

The parameter values which provide a best fit for stream volume from Canada

Glacier may not produce a good fit for other glaciers. The best fit T-model for Canada

Glacier (A=0.004, B=3.0) results in a poor fit for Howard Glacier (Figure 25A).

Howard Glacier is a north-facing glacier and receives more solar radiation than the south facing Canada Glacier. Applying the TI model, using the same A and B values and including a non-zero value for C in equation (19) shifts the stream flow volume

(and melt) curves along the temperature axis. The normalized solar radiation index

45 term is 0.49 for Howard Glacier and -0.40 for Canada Glacier. With a positive coefficient C the stream flow volume curve shifts towards colder air temperature for

Howard Glacier resulting in an increase in the melt at colder air temperatures relative to Canada Glacier. As shown in Figure 25, increasing C from 0 to 0.5 shifts the curve from a to b. Increasing C even further to C=1 shifts the results to curve c making a close fit to the measured data for Howard Glacier, but increases the error for Canada

Glacier. By adjusting both A and B, a reasonable fit to the stream flow volumes of both glaciers is possible (d).

cd b a 5 ad b c ) measured 20 3 )

3 measured

m B=3.0 C=0

5 4 m B=3.0 C=0 B=3.0 C=0.5 5 16

0 B=3.0 C=0.5 1 (x10 B=3.0 C=1

3 x ( e B=3.5 C=0.5 12 B=3.0 C=1 e m u m

l B=3.5 C=0.5 u

o 2 l 8 V Vo

1 m eam a r 4 t

(A) e r

S (B)

0 St 0 -5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 1 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

Figure 25: Sensitivity of incorporating the solar radiation index in the TI-model and of changes in the C coefficient for (A) Howard Glacier A=0.004 and for (B) Canada Glacier A=0.004. The arrows show how the curve shifts from a to b to c when C is changed.

The inclusion of wind-speed (TIU-model) has a similar but opposite effect on predicted melt and stream flow volume as solar radiation in the TI-model. An increase in wind speed shifts the curve towards positive temperature reducing the modeled melt/stream flow with respect to temperature. When comparing the measured melt

(half the total ablation) with respect to the estimated temperature at the stake locations for Howard and Taylor glaciers, melt on Howard Glacier occurs at cooler temperatures

46 than on Taylor (Figure 26). The average wind-speed is 4.5 ms-1 over Taylor Glacier and only 2.7 ms-1 over Howard Glacier. The same curve with D set to zero would overestimate melt for Taylor Glacier and underestimate melt for Howard Glacier.

0.35 Taylor Measured Melt 0.30 How ard Measured Melt D=0 0.25 D=-0.23 (Taylor)

) D=-0.23 (How ard) q e 0.20 w m

( 0.15 lt e M 0.10

0.05

0.00 -12 -10 -8 -6 -4 -2 0 Summer Average Temperature (°C)

Figure 26: Effect of incorporating wind speed into the model for melt on Howard and Taylor glaciers. A=0.03, B=1.06, C=0

47 V RESULTS

The T-model was optimized using data from Canada and Commonwealth glaciers and applied to the rest of the glaciers. The TI-model was optimized using data from Howard, Canada and Commonwealth glaciers and the TIU-model was optimized using data from Taylor Glacier in addition to Howard, Canada and

Commonwealth glaciers. The coefficients of the final models are presented in Table 6 and the values used to normalize the variables are presented in Table 7.

Table 6: Optimal coefficients for the temperature index model (T), the temperature- solar radiation model (TI) and the temperature-solar radiation-wind speed model (TIU). Coefficients T TI TIU A 0.004 0.004 0.006 B 3.000 3.600 2.700 C 0.000 0.520 0.330 D 0.000 0.000 -0.176 E -0.010 -0.010 -0.010

Table 7: Mean and standard deviation of variables used for normalization. Standard Variable Mean Deviation T 5.76 2.81 I 253.05 27.17 U 3.62 0.61

Melt/Ablation Results To compare estimated melt with the ablation measurements of the stake networks on Canada, Commonwealth, Howard and Taylor Glaciers, summer average temperature (equation 9) and then melt (equations 17-19) was calculated at each stake for each season (Figure 27). For Canada and Commonwealth glaciers, the T-model closely matches the half-ablation and is in most cases below the measured ablation.

48 The predicted melt for Howard Glacier is below the half-ablation group, and the predicted melt for Taylor Glacier is well above the total ablation.

0.4 0.7

Canada Glacier 0.6 Commonwealth Glacier

q) 0.3 q) e e 0.5 w w

m m 0.4

on ( 0.2 on ( i i t t

a a 0.3 bl bl A A / / t t

l l 0.2

e 0.1 e M M 0.1

0.0 0 -7 -6 -5 -4 -3 -2 -1 -9 -7 -5 -3 -1 Summer Average Temperature (°C) Summer Average Temperature (°C)

0.5 1 0.9 Howard Glacier Taylor Glacier

0.4 ) 0.8 q q) e e 0.7 w w

m m 0.3 0.6 ( n o on ( 0.5 i i t t a a 0.2 l 0.4 b bl A /A /

t 0.3 lt l e e M M 0.1 0.2 0.1 0.0 0 -12 -10 -8 -6 -4 -2 -7 -6 -5 -4 -3 -2 -1 0 Summer Average Temperature (°C) Summer Average Temperature (°C)

T-Model TI-Model TIU-Model Half-ablation Measured * ablation

Figure 27: Comparison of modeled melt and measured ablation for Canada, Commonwealth, Howard and Taylor glaciers. Summer average temperatures were calculated at each stake from equation (9).

The large underestimate of melt on Howard Glacier in the Kukri Hills is due, in part, to the fact that Howard Glacier slopes towards the north. The sun is higher in the northern sky (36° above horizontal at solstice) compared to the southern sky (10.5° above horizontal at solstice) resulting in a significant difference in potential solar radiation (roughly 266 and 242 Wm-2) on the glacier surfaces in the Kukri Hills and the Asgard Range, respectively.

To better account for spatial differences in solar radiation, the TI-model was applied. Melt estimates for Canada and Commonwealth were similar to those from

49 the T-model (Figure 27). The results for Howard Glacier, however, significantly improved and closely followed the half-ablation values. Unfortunately, the melt at

Taylor Glacier is overestimated, approximately 6 m (weq) of melt at a summer average temperature of -1 °C compared to measured ablation of less than 0.7 m (weq).

High average summer wind speeds and low humidity on Taylor Glacier may account for the large difference in predicted versus measured. Comparing wind speed and relative humidity on Taylor and Canada glaciers, for example, shows 4.5 ms-1 versus 3.2 ms-1 and 59% versus 65%, respectively. As previously mentioned, wind plays an important role in the energy balance. For air temperatures below freezing, common to summer air temperatures, higher winds enhance energy loss through latent and sensible heat exchange. Consequently, little energy is left over for melt. The

TIU-model reduced the predicted melt for Taylor Glacier so that the melt curve at least lies within the measured ablation data. The modeled melt for Canada,

Commonwealth and Howard glaciers again closely matches the half ablation and is in most cases below the measured ablation (Figure 27).

Stream Flow Volume Results

The accuracy of modeled stream flow is assessed by comparison with observed flows (Table 8). The statistical comparison is made both for individual streams and entire basins. Taylor Glacier is excluded from this analysis due to the poor quality of the observational data. These results demonstrate the ability of the model to predict stream flow volume from unmeasured glaciers and streams. The regression plots of

50 stream flow versus Lake Hoare summer average temperature for the basins, glaciers, and individual streams are shown in Figures 28, 29, and 30 respectively.

Table 8: Coefficients of determination and total volume difference for the temperature-index model (T), the temperature-solar radiation model (TI) and the temperature-solar radiation-wind speed model (TIU). The glacier name in italics follows the name of the stream that drains from it. Negative values of the coefficient of determination indicate the difference between the measured and calculated stream volume is greater than the variability of the measured stream flow volume. Negative values of total volume difference represent over predicted volume. Coefficient of Determination Total Volume Difference (%)

T- TI- TIU- T- TI- TIU- Contributing Area Model Model Model Model Model Model

Lake Fryxell Basin 0.60 0.73 0.67 27 15 13

Green Creek -0.06 -0.19 -0.28 -57 -54 -70 Canada Canada Stream 0.62 0.61 0.67 22 30 4 Canada Lost Seal Creek 0.73 0.64 0.59 8 30 28 Commonwealth Aiken Creek 0.55 0.57 0.42 34 39 46 Commonwealth Commonwealth Stream 0.60 0.68 0.66 28 -2 12 Commonwealth Delta Stream -0.03 0.71 0.66 81 -25 -21 Howard Crescent Stream -0.43 0.24 0.51 99 -46 15 Crescent VonGuerard Stream -0.48 0.15 -0.20 100 73 90 Unnamed

Lake Hoare Basin 0.67 0.42 0.62 1 44 11 Andersen Creek 0.66 0.59 0.65 -9 29 -5 Canada House Stream 0.48 -0.75 -0.01 35 95 65 Suess

Lake Bonney Basin -0.19 -0.14 -0.97 63 63 84

Lawson Creek -1.44 -3.60 -3.59 69 94 94 Rhone Priscu Stream 0.17 0.59 -0.15 57 29 72 LaCroix, Sollas and Marr West Canada Glacier 0.69 0.73 0.64 -9 6 -19 Commonwealth Glacier 0.67 0.73 0.61 22 21 27

51 2 2 0 40 6 0 - - ) ) 01 01 3 3 35 Lake Fryxell Lake Hoare 5 m m 5 5 30 1 0 9 0 1 - 1 1 9 -

x 4 x 25 90 2 ( ( 90 9 e - 2 9 m me

20 3 - 91 u l lu 91 9 o o 9

15 9 4 0 - 9 V - 9 V 0 2 4 7 - 1 7 6 9 9 98 - 0 6 m 98 0 9 9 8 5 1 -

10 8 - - 5 93- 99 9 a 0 9 9 0 - 9 - 9 - 93 96- - e ream - 00 r 1 96 95 t 95 99 97 94-

5 00 97- S 94 St 0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

10 2 0 - ) 3 9 Lake Bonney 01 m 5 8 T-Model 0 1

1 7 9 - x (

6 90 e 2 TI-Model 9 - m 5 lu 91 o

4 6 7 9

9 TIU-Model 9 5 9 8 - 0 4 V - 1 9 9 0

3 9 0 - - m 95 96 98- a 94- 97 99- 93 00- e 2 r Measured

St 1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C)

Figure 28: Modeled and measured stream volume into Lakes Fryxell, Hoare and Bonney.

4 4 2 0 ) ) 3 3 Lawson Creek 02

VonGuerard Stream 01- m m 5 5 3 01- 3 0 0 1 1 1 9 - x x ( ( 1 90 2 e 9 9 - m 6 me 2 92 2 9 90 u 7 - l 91- 1 5 9 lu 9 91- 0 4 - 0 9 9 0 9 95 - - 8 96 Vo Vo 94 00- 94- 98- 9 99 93 - m m

1 93- 1 a a 97 e e r r 99 96 95 97 01 98 00 t St S 98- 95- 94- 96- 00- 97- 99- 0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

2 0

6 1.5 2 - -0 ) ) 01 1 3 3 5 Priscu Stream House Stream 0 m m 5 5 1 0 0 9 1 1 - 4 1.0 1 x x -9 ( ( 90 0 2 9 9 2 me me

3 - -9 lu lu 91 1 9 0 8 4 9 9 Vo Vo -0 2 7 0.5 6 9 5 4 9 9 9 9 0 6 - -9 9 5 8 m m 9 9 7 1 9 0 1 3 9 9 a a - -9 - - 0 97 9 -9 98- 98- e -0 e 96- 5 - 95- 94- 93- r r 6

1 0 99 9 t t 94 97 9 0 00 S S 0 0.0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

T-Model TI-Model TIU-Model Measured Figure 29: Modeled and measured stream volume for individual streams.

52 20 25 2 -0 2 1 ) ) 3 0

18 -0 3 Canada Glacier

1 Commonwealth Glacier m 0 m

16 5 20 5 0 0 1 1 14 1 1 x x -9 -9 0 2 ( 0 12 ( 15 9 e 9 -9 2 1 m 10 me -9 9 1 lu lu 9 o 8 10 9 9 4 V Vo -9 4 -9 7 1 0 6 6 8 -9 0 6 9 7 8 m m 8 8 0 0 9 1 9 3 0 9 -9 5 5 - - - 9 a a 9 9 - - -9 9 6 - - e -0

e 5 93- 6 4 -9 -9 9 r r 00 99 95 0 99 95 9 4 t 4 97 97 0 9 9 St 2 S 0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

5 2 3

-0 Crescent Glacier (Crescent Stream) 1 ) Howard Glacier (Delta Stream) ) 1 3 3 0 -9 m m 4 0 5 5 9 2 0 0 1 1 2 -0 x x 1 ( (

3 0 1 -9 me me 0 2 2 lu lu 9 -9 2 -9 1 1 9 9 Vo Vo 4

1 4 9 -9 m m 7 9 6 a a 9 3 7 6 8 8 5 5 1 0 1 0 9 e e -9 -9 93- 1 -9 -9 r r -9 -9 6 8 -9 5 -9 t t -9 -9 -0 -0 -0 -0 8 6 5 9 9 7 9 7 4 4 0 9 0 9 9 S S 9 9 9 9 9 9 0 9 0 9 0 0 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Lake Hoare Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

T-Model TI-Model TIU-Model Measured Figure 30: Modeled and measured stream volume for Canada, Commonwealth, Howard, and Crescent glaciers.

The T-Model performed well in the Lake Fryxell and Lake Hoare basins with r2 values of 0.60 and 0.67 respectively. The model underestimates the total stream flow volume for the eleven seasons by 27% for the Fryxell Basin and by only 1% for the Hoare Basin. The model performs poorly in the Bonney Basin, underestimating total stream flow volume for Lawson and Priscu streams over the eleven seasons by

63%. In the Lake Fryxell Basin, the model performs well for four of the five streams draining from south-facing Canada and Commonwealth glaciers (Table 8). Total stream flow from Green Creek, where the model performs poorly, is overestimated by

57%. This may be a result of overestimation of the contributing area for Green Creek

(Figure 18) since stream flow volume is underestimated for the other three streams. In contrast, stream flow volume is underestimated by 80% to 100% for the streams,

53 Delta, Crescent and VonGuerard, all of which drain from the north-facing glaciers of the Kukri Hills. Two streams flow to Lake Hoare, Andersen Creek from Canada

Glacier and House Stream from Suess Glacier. The T-Model performed well for these streams, overestimating total stream flow for Andersen Creek by 9% and underestimating stream flow volume for House Stream by 35%. Stream flows into

Lake Bonney—Lawson Creek from Rhone Glacier and Priscu Stream from LaCroix,

Sollas and Marr West glaciers—are underestimated by 69% and 57% respectively.

The TI-Model improved the results for the north-facing glaciers in the Kukri

Hills with little change in the results from the south-facing Canada and

Commonwealth glaciers. The r2 value for the Fryxell Basin increased to 0.73 and the total stream flow volume error was reduced to 15%. The reduction in error is largely from the increased calculated stream flow volume for Delta, Crescent and VonGuerard streams from the solar radiation term. The r2 value for Delta Stream jumped to 0.71 and the values for Crescent and VonGuerard streams increased to 0.24 and 0.15 respectively. The total stream flow volume for Delta and Crescent streams are overestimated by 25% and 46% respectively and that for VonGuerard Stream is underestimated by 73%. Some of the water in VonGuerard Stream comes from a tributary stream from Crescent Glacier (Figure 1). It is unknown how much of

VonGuerard’s total stream flow volume comes from this tributary stream but this may account for some of the error. Statistics for the stream flow from Crescent and

VonGuerard Streams combined show a smaller 21% error in total volume.

54 The TI-Model did not improve results in the Lake Hoare Basin. The r2 value for Lake Hoare decreased from 0.67 to 0.42 and the total volume is underestimated by

44%. Most of this error is from decreased estimates of House Stream flow (95%).

Decreased flow is caused by below average solar radiation, 214 Wm-2 compared to

253 Wm-2. Considering that most of the stream flow into Lake Hoare comes directly from Canada Glacier, the effect of the under-predicting Suess on the total water input to the lake is small. Severe under prediction also occurs for Lawson Creek in the

Bonney Basin. The total volume difference for Lawson increased from a 69% underestimate from the T-model to a 94% underestimate from the TI-model. Solar radiation is relatively low for Rhone, Suess and La Croix glaciers (Table 5) because they have steep south facing slopes. For example, the solar radiation value for Rhone is only 210 Wm-2. The low solar radiation values on these glaciers cause the TI-model to under predict their melt and stream volume.

The TIU-model performed well in the Fryxell and Hoare basins. Results for the TIU model were similar to those of the TI-model in the Fryxell Basin with an r2 of

0.67 and 13% underestimate in total stream volume. The results changed little for the individual streams in the Fryxell Basin except for Crescent Stream where r2 jumped from 0.24 with the TI-model to 0.51 with the TIU-model. Results in the Hoare Basin improved over the TI-model with an r2 of 0.62 due to an increased stream flow volume for both streams. In the Lake Bonney Basin, the total stream volume for Lawson and

Priscu streams are severely underestimated (94% and 72% respectively) because the higher wind speeds in the Bonney Basin reduce the modeled stream volume for these

55 streams. Unfortunately, the ability of these models to predict total stream volume into the Bonney Basin cannot be sufficiently examined in any detail because of the lack of good quality stream data from the largest contributor to the lake, Taylor Glacier.

The streams in the valley are unequal in their importance to the total input into the lakes. For example, Canada Stream is generally the largest contributor to Lake

Fryxell, Lost Seal is the second largest, and Crescent, VonGuerard and Huey streams are generally the smallest contributors to Lake Fryxell. The total volume from the three small contributors combined is almost always smaller than the volume of one of the large contributors alone. Therefore it is more important to get good results from the large contributors than from the small. Fortunately, the models performed their best for the large magnitude streams of Canada, Lost Seal, Aiken and Commonwealth in the Fryxell basin and Andersen in the Hoare basin. The models performed poorly for the large streams, Green Creek in the Fryxell Basin and Lawson and Priscu streams in the Bonney Basin.

Threshold Temperature and Melt Limit Results The melt threshold temperature, the summer average temperature below which no melt is predicted (equation 24), for the T-model is -4.9°C throughout the valley

(Table 9). This is comparable to Johnston’s (2004) threshold (-4.4°C) for the onset of rapid channel growth on Taylor Glacier. The melt threshold temperature represents the minimum summer average temperature required for seasonal melt to occur. This does not mean that melt actually occurs at the threshold temperature. Figure 31 shows the melt limits and estimated equilibrium line altitudes (ELAs) for six glaciers in

56 Taylor Valley. The ELAs were estimated from the elevation where the glacier surface transitions from a convex (ablation zone) to a concave (accumulation zone) surface morphology (Leonard and Fountain, 2003), which were estimated on USGS topographic maps (Fountain and others, 1999a). The melt limits for Canada and

Commonwealth glaciers hover around their ELAs (375 m), with averages of 390 m and 370 m respectively. However, the melt limits for Howard, Crescent and

VonGuerard glaciers barely reach their ELAs during the warm seasons and are well below their ELAs during the cold seasons. The melt limit for Rhone Glacier is well below the ELA for all seasons.

Table 9: Threshold temperatures in °C and average melt limits and ELA in m above seal level for each glacier and model. Threshold Temp Melt Limit ELA Glacier T TI TIU T TI TIU Canada -4.9 -4.5 -5.4 394 341 449 375 Commonwealth -4.9 -4.5 -5.0 372 313 390 375 Howard -4.9 -5.7 -6.4 437 540 621 500 Crescent -4.9 -6.2 -6.5 440 588 617 500 VonGuerard -4.9 -6.4 -6.4 440 610 613 450 Rhone -4.9 -2.7 -3.2 550 313 365 1000 Average -4.9 -5.0 -5.5 439 451 509 533

The solar radiation term in the TI-model lowered the threshold temperatures for Howard, Crescent, and VonGuerard glaciers. Lower threshold temperatures increase the melt water flux at any point previously contributing, increase the elevations of the melt limits and therefore increase the contributing area. The melt limits increased for all three glaciers in the Kukri Hills by ~150m from the T-model to the TI-model. The threshold temperature for both Canada and Commonwealth

57 glaciers increased to -4.5°C with the TI-model producing melt limits ~60 m lower than those from the T-model. The low solar radiation of Rhone Glacier increased the threshold temperature by +2.2°C to -2.7°C from the T-model which requires warmer temperatures for melt to occur. The melt limits of Rhone for the TI-model are ~240 m lower than those from the T-model leaving even less contributing area for melt. For the TIU-model, Taylor and Rhone have the warmest threshold temperatures and

Howard, Crescent and VonGuerard have the coldest. As a result, Taylor and Rhone have the lowest estimated melt rate where Howard, Crescent and VonGuerard have the highest. This improved the melt prediction for Taylor Glacier but made no improvement for Rhone.

58 700 700 Canada Glacier Commonwe alth Glacier 600 600 500 ) 500 ) 400 (m t t (m i 400 i m m 300 i i 200 ff L 300 ff L o o n n 100 200 u Ru R 0 100 -100 0 -200 1 2 4 5 6 7 8 9 0 1 2 9 9 9 9 9 9 9 9 0 0 0 ------91 -92 -94 -95 -96 -97 -98 -99 -00 -01 -02 90 91 93 94 95 96 97 98 99 00 01 90 91 93 94 95 96 97 98 99 00 01 Season Season

900 900 Howard Glacier Crescent Glacier 800 800 700 700 ) )

(m 600 600 t (m t i i m m 500 500 i i

400 ff L 400 ff L o o n n 300 300 u R Ru 200 200 100 100 0 0 1 2 4 5 6 7 8 9 0 1 2 -91 -92 -94 -95 -96 -97 -98 -99 -00 -01 -02 90 91 93 94 95 96 97 98 99 00 01 90-9 91-9 93-9 94-9 95-9 96-9 97-9 98-9 99-0 00-0 01-0 Season Season

900 1000 VonGuerard Glacier 800 900 Rhone Glacier 800 700 ) ) 700

600 t (m i t (m

i 600 m i

m 500 i 500 ff L

ff L 400 o 400 o n n 300 u u

R 300 R 200 200 100 100 0 0 1 2 4 5 6 7 8 9 0 1 2 -9 -9 -9 -9 -9 -9 -9 -9 -0 -0 -0 -91 -92 -94 -95 -96 -97 -98 -99 -00 -01 -02 90 91 93 94 95 96 97 98 99 00 01 90 91 93 94 95 96 97 98 99 00 01 Season Season

T-Model TI-Model TIU-Model ELA Figure 31: Melt limit and ELA for Canada, Commonwealth, Howard, Crescent, VonGuerard and Rhone glaciers.

59 VI ERROR ANALYSIS

There are two types of errors inherent in these models—spatial, in which the models will yield poor results for some glaciers while yielding good results for others and temporal in which the models perform well for some years and poorly for others.

For the spatial errors, the largest occurred at Rhone, Suess, and LaCroix glaciers where all three models underestimate stream volume and at Taylor Glacier where the

T and TI-models overestimate melt. Rhone, Suess, and LaCroix glaciers are characterized by steep rough surfaces with many vertical walls, which enhance surface melt. Vertical walls enhance solar intensity when the sun is low in the sky, and reduce wind speed, which reduces the turbulent heat losses and leaves more energy available for melt (Lewis and others, 1999; Johnston and others, in press). On Suess Glacier, rock debris spread over the surface also enhances melt. I believe that the different surface roughnesses of these glaciers are responsible for their different melt behavior.

Taylor Glacier is characterized by a smooth flat surface with melt channels incised into the surface near the terminus. High wind speeds and low humidity over Taylor

Glacier favor sublimation over melt. The T-model and TI-model overestimate melt for this glacier, whereas the TIU-model improved the result for only Taylor.

A second source of spatial error is calculated temperatures from equation (10)

(Figure 14). The melt models depend on these calculated temperatures and thus inherit their errors. The standard deviation of the residuals between measured and calculated temperatures at the seven meteorological stations in the valley is 0.36°C.

To test the effect of errors in calculated temperature on modeled melt, three melt

60 curves were plotted together one with the estimated temperatures, one with a + 0.36°C shift and one with a – 0.36°C shift (Figure 32). This provides a maximum range of melt and stream volume values. Because of the nature of the exponential equation, the modeled melt and stream volume is most sensitive to errors in estimated temperatures at warmer temperatures.

B 2 )

A 0 3 0.25 25 - 01 m 5 0

0.20 1 20 x ) ( 1 e q 9 - e m 0.15 15 2 w 9 90 u

- l o m 91 ( V t

l 0.10 10 9 4 ow 9 9 - Me - 0 6 7 8 1 0 9 9 98 Fl 5 93 9 - - - 0 - - 0.05 5 9 - m 99 95 96 97 00 a 94 re 0.00 t 0 S -7 -6 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 Modeled Summer Average Temperature (°C) Lake Hoare Summer Average Temperature (°C)

Figure 32: Effect of errors in temperature for melt (A) and stream flow volume (B) from Canada Glacier. The points are measured melt (half ablation) in A and measured stream flow volume in B.

The temporal errors are, in part, a result of a poor fit to the data for some temperature ranges and a result of scatter in the data (Figure 33 and 34). The residuals were normalized by dividing them by the measured stream flow volume to better compare differences between modeled and measured. The residuals in the Bonney

Basin show large underestimations of stream flow for all seasons because of the poor performance of the models for Lawson and Priscu streams. Many of the residuals result because some relatively cool seasons (1993-1994, 1998-1999) produced more stream flow volume than some warmer seasons (1996-1997, 1995-1996). Also the model underpredicts flow at low temperatures (Figure 35).

61 10 300 Lake Fryxell 250 Lake Fryxell 5 )

3 200 al u m 150 5

0 d

0 100 1 esi x

( 50

-5 R l t 0 dua cen

i -10 r -50 s e e P

R -100 -15 -150 -20 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

10 300 Lake Hoare 250 Lake Hoare

) 5 3 l 200 a m u

5 150

0 d 0 si 1 100 e x

( -5 R l 50 t a

u 0 en d -10

rc -50 e P Resi -15 -100 -150 -20 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

10 300 Lake Bonney 250 Lake Bonney

) 5

l 200 3 a u

m 150 5 0 d 100 10 esi x 50 ( -5 R l t

a 0 u cen d -10 r -50 e esi P -100 R -15 -150 -20 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

T-model TI-model TIU-model

Figure 33: Lake Fryxell, Hoare and Bonney basin residuals. The left hand column is the residual (modeled minus measured stream flow volume) and the right hand column is the normalized residual (residual divided by measured stream flow volume) shown here as a percentage.

62 6 Canada Glacier 500 Canada Glacier 4

) 400 3 2 s l a m 300 5 0 u d 0 1 si

x -2 200 e ( l a -4 t R 100 u en d -6 c

r 0 -8 e Resi P -10 -100 -12 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

500 6 Commonwealth Glacier Commonwealth Glacier 4 400 ) s 3 2 l a

m 300 u 5

0 d 10 si 200 x -2 e ( l t R a -4 100 u en d -6 c

r 0 e esi -8 P R -10 -100 -12 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

6 500 Howard Glacier Howard Glacier 964 2670 4 400 ) 3 2 l a m u 300 5

0 d 0 si 1

e 200

x -2 ( R l t a -4 100 u en d -6 rc 0 e

-8 P Resi -10 -100 -12 -200 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

T-model TI-model TIU-model

Figure 34 Canada, Commonwealth and Howard glacier residuals. The left hand column is the actual residual (modeled minus measured stream flow volume) and the right hand column is the normalized residual (residual divided by measured stream flow volume) shown here as a percentage.

63 5 4 Commonwealth Glacier 99 9 94

98- Canada Glacier 9

e 4 e 93- 98- m 4 m

3 9 u - l u l 7 o ) 93 0 1 9 6 o ) 3 0 0 V 3 3 9 - -

V -

96- 7 m 00 96 99 00 m 95 9 5 - 5 ow 8 0 ow 8 2 99- 95- 9 96 1 9 - 1 - Fl

2 0 Fl (x 97 5 (x10 97 m 9 m 00- a 95 a 94- e re r t 1 94- 1 t S S

0 0 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 Lake Hoare Summer Average Temperature Lake Hoare Summer Average Temperature 4

0.8 9 - 4 7 9 93 Howard Glacier - Fryxell Basin 93 9 9

6 e - e m 0.6 98 m u l u 5 l o ) o 3 ) 3 V V 7 6

m 4 9 9 5 m

0.4 7 9 5 ow 0 9 9 6 ow 96- 95- - - 0 9 8 - - 9 Fl x10 3 96 98 ( 01 Fl 99 95 (x10 m 97- m 5 a 00- a 9 e

2 - r e

0.2 8 t r 9 94 t 5 S 0 9 1 S 0 0 - 97- 1 94- 99 00- 0.0 0 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 Lake Hoare Summer Average Temperature Lake Hoare Summer Average Temperature 6 9 7 - 9 9

Bonney Basin 9

2 2 95 96- 98- Hoare Basin 4 9 5 8 0 9 9 - e e 0 - 93- 1 94 0 m m 97 - 99- 9 u u l l 9 00 0 o o ) ) 0 3 3 98- - V V 99 m m 4 5 5

1 9 1 ow ow 7 6 01 9 9 93- - - Fl Fl (x10 (x10 00- 96 95 m m 5 9 a a - 98 e e r r 94 t t 97- S S

0 0 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 Lake Hoare Summer Average Temperature Lake Hoare Summer Average Temperature

Figure 35: Replot of Figure 28 and 30 to highlight the model underestimate in the colder, low flow seasons.

To examine why some relatively cool seasons produced more stream flow volume than some warmer seasons, I examine air temperature and the other meteorological values that affect the energy balance. I first compare degree-days because the models use summer average temperature to calculate melt rather than the span of time the temperatures were above the melting temperature. This difference could lead to an over or underestimation of modeled melt. For example, the 1990-

1991 season had a colder temperature (-1.33°C) than the 1991-1992 season (-0.94°C)

64 but yielded more stream flow. The 1990-1991 season had an intense cold period early in the season but warmer temperatures in mid season than the 1991-1992 season.

Most of the melt is generated in mid season.

4 2

C) 0 g e -2 d (

p -4 m e -6 T r

Ai -8 -10 -12 1-Dec 8-Dec 15-Dec 22-Dec 29-Dec 5-Jan 12-Jan 19-Jan 26-Jan Month

90-91 91-92 90-91 Average 91-92 Average

Figure 36: 1990-1991 and 1991-1992 Lake Hoare daily average temperatures.

The calculation of degree-days in equation (4) sums the temperatures above a threshold temperature and may provide a more sensitive indicator of melt than average temperature. The number of degree-days for each season at Lake Hoare was calculated using 0ºC and -4.9ºC threshold temperatures (the latter being the threshold temperature for the T-model (Table 9)) and the 10-minute to 6-hour temperature data, depending on the data collection intervals, from December 1 to January 31. Lake

Hoare summer average temperatures and degree-days are presented in Table 10, then normalized and plotted in Figure 37. Degree-days with -4.9ºC threshold temperature follow the average temperature more closely than degree-days with -0ºC threshold temperature.

65 Table 10: Lake Hoare summer average temperature and degree-days. Average Degree-Days Degree-Days Stream Flow Volume Temperature (°C-day) (°C-day) (x105 m3) Season (°C) (T0=0ºC) (T0=-4.9ºC) Fryxell Hoare Bonney 90-91 -1.33 39.06 243.43 22.4 3.1 5.2 91-92 -0.94 36.52 246.66 16.2 2.1 3.6 93-94 -1.96 8.36 126.68 6.4 0.7 1.5 94-95 -3.12 2.39 97.13 1.0 0.3 1.4 95-96 -1.49 24.47 213.76 2.7 0.5 1.9 96-97 -1.85 15.57 196.04 2.8 0.5 1.7 97-98 -3.30 10.57 133.02 1.9 0.2 1.4 98-99 -1.63 24.52 212.11 4.9 1.1 1.7 99-00 -2.70 2.82 150.04 2.2 0.9 1.3 00-01 -4.58 0.26 67.27 1.7 0.6 1.2 01-02 -0.84 57.23 259.16 33.4 5.3 8.2 Average -2.14 20.16 176.66 8.7 1.4 2.6 Standard Deviation 1.15 18.7 62.22 10.7 1.6 2.2

2 ays d 1 ee- r eg

d 0 / p m e -1 t m r o

N -2 nor m temp norm degree-day (0°C) norm degree-day (-4.9°C) norm stream volume -3 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season

Figure 37: Lake Hoare normalized average temperature and degree-days.

I expect the seasons where the normalized degree-day is larger than the normalized average temperature to be those with relatively high stream volumes during relatively cold temperatures and vice-versa. Comparisons of the difference

66 between normalized air temperature and normalized stream flow volume and the difference between the normalized air temperatures and the normalized degree-days

(Figure 38) shows that this is the case for most seasons.

2 Lake Fryxell 2 Lake Hoare

m m r 1 93-94 r 93-94

o 1 o y y

a 96-97 96-97 n

d 99-00 99-00 - n da -

- 95-96 95-96 - p e 94-95 91-92 0 e 0 94-95 e e mp 90-91 91-92 98-99 m 90-91 98-99 e gr gr t te 01-02 97-98 01-02 97-98 de m de m

r -1 00-01 r -1 00-01 o N No -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Canada Glacier 2Commonwealth Glacier m m r 1 r 1 o 93-94 o 93-94 y y

a 96-97 a 96-97

d 99-00 d 99-00 - n - n

- 95-96 - 94-95 95-96

e 0 94-95 98-99 e 0 98-99 e e mp 90-91 91-92 mp 90-91 91-92 gr gr te te 97-98 de de m 01-02 97-98 m 01-02 r -1 r -1 00-01

o 00-01 o N N -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Howard Glacier m r 1 o 93-94 y

a 96-97

d 99-00 - n

- 95-96

e 0 94-95 e

mp 91-92 98-99 90-91 gr te 01-02 97-98 de m r -1 00-01 o N -2 -2 -1 0 1 2 Norm temp - norm stream volume

For the case of the 1990-1991 and 1991-1992 seasons mentioned earlier, the

1990-1991 season, which had 1.5 times more stream flow than 1991-1992, had more degree-days (39) than the 1991-1992 season (36.5). An extreme case is the 2001-2002 season, which experienced more than double the stream flow than the 1991-1992

67 season, which was only 0.1°C cooler. The difference in stream flow here can also be explained by degree-days, 2001-2002 had 57 and 1991-1992 had 36.5. As shown in

Figure 38, degree-days/average temperature help explain the scatter in the stream volume-temperature relationship for all seasons except 1994-1995.

The residuals may also be examined by comparison with meteorological data

(summer average relative humidity, wind speed, shortwave radiation, and albedo)

(Figure 39). I normalized the meteorological data using equation (21) and compared them with the difference between normalized air temperature and normalized measured stream volume (Figure 40-44). ) n

360 -2

o 180 i m

at 340

W 160

adi 320 140 on ( i e R

at 120

) 300 di av -2 a

w 100

m 280 R

e 80 (W hort 260 av 60 w t

ng S 240 i 40 or m h

o 220 20 S c t n e I 200 0 N 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

100 90 95 85 90 80 75 ) 85 ) % 80 70

75 (% 65 edo ( RH b 70 60 l

A 65 55 60 50 55 45 50 40 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Season Season

5 Canada

) 5 -1 s

m 4 Commonwealth (

ed 4 pe s - 3 Lake Hoare nd i W 3 Howard 2 90-91 91-92 93-94 94-95 95-96 96-97 97-98 98-99 99-00 00-01 01-02 Lake Fryxell Season

Figure 39: Summer average meteorological data from Lake Hoare, Lake Fryxell, Commonwealth Glacier, Canada Glacier and Howard Glacier.

68 High relative humidity favors melt compared to sublimation via the latent heat term in the energy balance. Figure 40 shows that high relative humidity in 1997-1998 and 2000-2001, and low relative humidity in 1991-1992 may account for the difference between temperature and stream volume.

3 Lake Fryxell 3 Lake Hoare 2 97-98 2

e 00-01 e 97-98 v v i i y 1 y 1 00-01 t t t t 93-94 a a 94-95 l l di 93-94 98-99 di 96-97 0 0 01-02 99-00 95-96 re re mi mi 94-95 95-96 98-99 01-02 rm rm hu -1 99-00 96-97 hu -1 o o 90-91 N N -2 -2 91-92 -3 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Canada Glacier 3 Commonwealth Glacier 00-01 97-98 97-98 2

e 1 e v v i i 1 t t ty ty i i a a l l 00-01 93-94 d d i 0 96-97 i 0 94-95 98-99 re re m 94-95 98-9995-96 m 96-9795-96 rm rm 99-00 hu hu -1 01-02 o 01-02 o N -1 N 99-00 -2 -2 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Howard Glacier 97-98 00-01

e 1 v i y t t 94-95 a

l 95-96 di 0 re mi 93-9496-97 rm hu o 01-02 98-99

N -1 99-00

-2 -2 -1 0 1 2 Norm temp - norm stream volume

Figure 40: Normalized air temperature minus normalized stream volume versus normalized relative humidity for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers.

High wind-speed favors sublimation through the latent heat term in the energy balance leaving less energy for melt. Figure 41 shows that low wind-speed may account for relatively high stream flow in 1990-1991, 1994-1995, 1997-1998, and

2000-2001 and high wind-speed may account for relatively low stream flow in 1995-

1996, 1998-1999 and 1999-2000.

69 3 Lake Fryxell 3 Lake Hoare

2 d 2 d 01-02 e

e 01-02

98-99 pe pe 1 98-99 99-00 -s 1 -s 00-01 99-00

nd 95-96 i

nd 0 95-96 i 97-98 0 90-91 91-92 w

w 94-95 00-01 m 97-98

-1 96.97 r

m 96-97 r 93-94 o -1 94-95 o

N 93-94

N -2 -2 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

3 Canada Glacier 2 Commonwealth Glacier 97-98 d 2 d 1 98-99 e 01-02 e 99-00 pe pe 01-02 95-96 -s 1 -s 0 98-99 93-94 nd 99-00 nd i 0 95-96 i -1 00-01 94-95 w w 96-97 m 96-97 m r 00-01 94-95 r o -1 97-98 o -2 N N -2 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Howard Glacier 01-02 d

e 99-00 1 95-96 pe 98-99 -s 0 nd i 94-95 w 00-01 97-98

m 93-94

r -1

o 96-97 N -2 -2 -1 0 1 2 Norm temp - norm stream volume

Figure 41: Normalized air temperature minus normalized stream volume versus normalized wind-speed for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers.

Incoming short-wave radiation depends on the atmospheric conditions including cloud cover and atmosphere transparency. Relatively high incoming shortwave radiation indicates clear days, enhancing heating of the ice and melting. The plots in Figure 42 show that relatively high stream flow in 2000-2001 and 2001-2002 can be accounted for by relatively high incoming shortwave radiation and low stream flow in 1991-1992, 1995-1996, and 1996-1997 can be accounted for by low incoming shortwave radiation.

70 2 Lake Fryxell 3 Lake Hoare 93-94

2 1 01-02 g

n 93-94 i e ng 99-00 98-99 01-02 i 00-01 1 e m 00-01 99-00 av v m o

a 0 98-99 c o w

t 0 c w 94-95 95-96 r t 90-91 95-96 in n

o 96-97 i 97-98 -1 97-98 96.97 m -1 r sh hor m

r 94-95 o s o N -2 N -2 91-92 -3 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume 2 Canada Glacier 3 Commonwealth Glacier

99-00 2 93-94 g g 1 n n 00-01 i i e e 01-02 96-97 1 m m 94-95 98-99 01-02 av av 0 o o 00-01 98-99 c c w w 94-95 t t 0 95-96 r r 95-96 in in o o -1 m m -1 99-00 96-97 r r sh sh 97-98 o o N N -2 97-98 -2 -3 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Howard Glacier

g 1 01-02 99-00 n

i 94-95

e 00-01

m 98-99

av 0 93-94 95-96 co w

t 96-97 r in

o -1 m r sh o

N -2 97-98 -3 -2 -1 0 1 2 Norm temp - norm stream volume

Figure 42: Normalized air temperature minus normalized stream volume versus normalized incoming shortwave radiation for Lakes Fryxell and Hoare, and Canada, Commonwealth and Howard Glaciers.

Glaciers with high albedo will melt less than glaciers with low albedo.

Albedo, the fraction of incoming shortwave-radiation reflected by the surface, increases with the presence of snow on the glacier surface and the presence of ice or sediment decreases the albedo. The plots in Figure 43 shows that low albedo in 2001-

2002 and high albedo in 1995-1996, 1996-1997, and 1998-1999 may account for the difference between temperature and stream volume.

71 3 Canada Glacier 2 Commonwealth Glacier

2 97-98 1 97-98 96-97 94-95 95-96 do do 98-99 1 0 00-01 be be l 96-97 l 99-00 a 00-01 a 93-94 m 0 m -1 r 94-95 98-99 r o 01-02 95-96 o 01-02 N -1 N -2 99-00 -2 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 Howard Glacier 97-98 1 96-97 do 00-01 94-95 95-96 be l 0 a m r 98-99

o 99-00

N -1 93-94 01-02 -2 -2 -1 0 1 2 Norm temp - norm stream volume

Figure 43: Normalized air temperature minus normalized stream volume versus normalized albedo for Canada, Commonwealth and Howard Glaciers.

2 Canada Glacier 2 Commonwealth Glacier 93-94 99-00 01-02 e e

v 1 v 1 a a

w 01-02 98-99 w t 94-95 95-96 t 99-00 0 00-01 0 00-01

hor hor 98-99

s 96-97 s 94-95 95-96

t -1 t -1 97-98 96-97 ne ne m m

r -2 97-98 r -2 o o N N -3 -3 -2 -1 0 1 2 -2 -1 0 1 2 Norm temp - norm stream volume Norm temp - norm stream volume

2 01-02 Howard Glacier e

av 1 99-00 w

t 93-94

r 98-99 o

h 0 s t

e 95-96 00-01 94-95

n 96-97 -1 m r 97-98 No -2 -2 -1 0 1 2 Norm temp - norm stream volume

Figure 44: Normalized air temperature minus normalized stream volume versus normalized net shortwave radiation for Canada, Commonwealth and Howard Glaciers.

High net shortwave radiation enhances melt. The net shortwave radiation is the magnitude of radiation that is absorbed by the surface (incoming minus outgoing).

This is dependent on both the incoming short-wave radiation and the albedo. As with albedo, high net shortwave radiation in 2001-2002 and low net shortwave radiation in

72 1995-1996, 1996-1997 and 1998-1999 may explain the difference between temperature and stream volume (Figure 44).

In summary, Table 11 and 12 show the seasons in which degree-days and the meteorological data may qualitatively explain the scatter between air temperature and measured stream volume. Of all of the variables involved in the stream flow estimation, degree-days can explain most of the scatter followed by net radiation and relative humidity.

Table 11: Summary of seasons where degree-days and meteorological data explain the scatter between air temperature and measured stream volume where stream volume is high and temperature low. Season High High Low Wind- High Low High D-days RH speed SW In Albedo Net SW 90-91 X X n/a n/a 94-95 X X n/a n/a 97-98 X X X n/a n/a Lake Hoare 99-00 X n/a n/a 00-01 X X X X n/a n/a 01-02 X X n/a n/a 90-91 X n/a n/a 94-95 X n/a n/a Lake Fryxell 97-98 X X X n/a n/a 00-01 X X X n/a n/a 01-02 X X n/a n/a 90-91 X 94-95 X X X X Canada Glacier 97-98 X X X 00-01 X X X X 01-02 X X X X 90-91 X 94-95 X X Commonwealth 97-98 X X Glacier 00-01 X X X X X X 01-02 X X X X 90-91 X 94-95 X X X Howard 97-98 X X X Glacier 00-01 X X X X 01-02 X X X X

73 Table 12: Summary of seasons where degree-days and meteorological data explain the scatter between air temperature and measured stream volume where stream volume is low and temperature high. Season Low Low High Wind- Low High Low D-days RH speed SW In Albedo Net SW 91-92 X X X X n/a n/a 93-94 X n/a n/a Lake Hoare 95-96 X X n/a n/a 96-97 X X n/a n/a 98-99 X X n/a n/a 91-92 X n/a n/a 93-94 X X n/a n/a 95-96 X X X X n/a n/a Lake Fryxell 96-97 X X X n/a n/a 98-99 X X n/a n/a 99-00 X X X n/a n/a 91-92 X 93-94 X 95-96 X X X X Canada Glacier 96-97 X X X 98-99 X X X 99-00 X X X 91-92 X 93-94 X Commonwealth 95-96 X X X X X Glacier 96-97 X X X X X 98-99 X X X X X 99-00 X X X X 91-92 X 93-94 X X X Howard 95-96 X X X X X Glacier 96-97 X X X X X 98-99 X X X 99-00 X X X

74 VII DISCUSSION AND CONCLUSIONS

For this thesis, I developed a temperature-index model of stream flow across

Taylor Valley that can be used to predict past stream flow when air temperature is the only available data. This work follows from Hock’s (1999) work on Storglaciaren.

By including spatial variations in potential direct solar radiation I found that incorporating potential direct solar radiation improved the melt prediction in the dry valleys. Many temperature index models are developed for hourly to daily time scales and use a degree-day model (Hock, 1999; Kane and Greck, 1997; Braithwaite, 1995;

Kustas and Rango, 1994). My project differs by examining a seasonal scale, using average temperature.

In the MDV, this work has added to the work of Bomblies (1998) and Jaros

(2003) who related glacier contributing area to stream flow. Bomblies (1998) weighted the contributing areas by topographic shading and found that the correlations between contributing areas and stream flow were barely significant. Jaros (2003) weighted the contributing areas by the air temperature decrease from the adiabatic lapse rate and found a significant correlation between stream flow and the contributing areas. I took this a step further and used this relationship to predict stream flow throughout the valley. Rather than correcting for topographic shading, I corrected for solar intensity determined by the glacier slope and aspect. The studies of Bomblies

(1998) and Jaros (2003) were limited to the Fryxell Basin where I tried to extend the model into the rest of the valley. The problems I found with modeling stream flow in the Bonney Basin underscore the results of Fountain (1999b) and Johnston (2004) that

75 increased wind-speed in the Bonney Basin has a profound effect on the turbulent heat transfer favoring sublimation over melt (Fountain, 1999b) and conversely, surface roughness and debris cover on some of the glaciers in the Bonney Basin can increase their melt (Johnston, 2004).

The goal of this thesis was to develop a valley-wide temperature-index model of stream volume for Taylor Valley, Antarctica. I developed three models of summer stream flow volume for Taylor Valley, Antarctica based on summer average air temperature with spatial variations in potential solar radiation included in the second and third models and spatial variations in average wind speed included in the third model. Air temperature was interpolated across the valley with respect to elevation, distance from the coast and distance from the valley bottom by equation (10). The T- model worked well on Canada and Commonwealth glaciers because they have similar characteristics. It failed, however, to model the stream flow of glaciers with differing spatial characteristics simultaneously (e.g. Canada and Howard) or valley location

(e.g. Canada and Taylor). The TI-model included spatial variations in potential solar radiation and was successful in modeling stream flow from the north and south facing glaciers of the Hoare and Fryxell basins. The TIU-model included spatial variations in summer average wind speed as well as potential solar radiation and was fairly successful in modeling the melt from Taylor Glacier as well as the stream flow from the glaciers of the Hoare and Fryxell basins. All three models performed poorly in modeling the stream flow from the steep, rough, south-facing glaciers Rhone,

LaCroix, and Suess. Modeling results for these glaciers were poor partly due to large

76 differences in energy balance controls on the glaciers. In the Bonney Basin, it is difficult to develop a good model with little usable stream flow data. Overall, the best results come from the TI-model (equation 20) in the Fryxell Basin where A=0.004,

B=3.6, C=0.52 and E=-0.01 (Table 6). These models also predict reasonable melt values compared to ablation measurements for Canada, Commonwealth and Howard glaciers. The good results for Canada, Commonwealth and Howard glaciers give me confidence in applying these models elsewhere. The melt limits for Canada,

Commonwealth, Howard, Crescent and VonGuerard glaciers closely hover around their respective ELAs. For these glaciers, the TI-model can simulate spatial and temporal variations in the ELA. Finally, the TI-model predicts a spatial variation in melt threshold temperatures via differences in potential solar radiation, yielding melt at lower temperatures on the glaciers in the Kukri Hills (more radiation) than the

Asgard Range (less radiation) (Table 8). The average seasonal threshold temperatures for the T, TI and TIU-models are below 0ºC (-4.9ºC, -5.0ºC and -5.5ºC respectively).

These are comparable to Johnston’s (2004) threshold (-4.4°C) for the onset of rapid channel growth on Taylor Glacier.

Errors in predicted stream flow volume result from errors in the spatial extrapolation or temporal variations in other climatic variables (e.g. net radiation, humidity) and surface morphologies that is not well resolved by the index models.

Most temporal errors can be explained by degree-days and many can also be explained by a combination of meteorological variables including shortwave radiation, humidity, wind speed, and albedo.

77 There is room for much improvement on this model in the Bonney Basin. One possibility for improvement would be to add a factor for surface roughness and debris cover to improve the melt prediction for Rhone, LaCroix and Suess glaciers.

Improvement in cliff melt may be important and attempts were made to incorporate cliff melt in the TI-model and TIU-model with no success. Although the cliffs receive very intense solar radiation for part of the day, resulting in melt, the rest of the day they are shadowed, so calculated daily averages yield solar radiation values lower than the surface. Perhaps if melt were calculated on a finer time scale (hourly) the high intensity of solar radiation when the cliffs are in the sun could be applied towards a larger flux of melt. Other factors that may be included to improve the prediction of cliff melt are increased long wave radiation from rock surfaces adjacent to the cliffs and reduced wind-speeds.

The TI-model can be applied to all of the glaciers in the three basins to predict the total water input to the lakes for any season in which air temperature data exists.

This includes all seasons with air temperature data from McMurdo Station and Scott

Base as the summer average air temperatures here correlate well with Lake Hoare.

The TI-model could also be applied to the other dry valleys, Wright and Victoria.

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