GLACIER DISCHARGE SIMULATION BY GROUND-WATER ANALOGUE

By LEO DERIKX (Glaciology Subdivision, Inland Waters Branch, Department of Energy, Environment

Ottawa, Ontario, Canada)

ABSTRACT. TO simulate discharge from a glacier the drainage basin is considered analogous to a ground-water system. The daily input into the system is precipitation and water from melting snow and ice computed from meteorological data.

Outflow rates are computed by use of methods originally derived for non-steady ground-water flow. The daily discharge of a glacial stream is the output of a system operating by the linearized Boussinesq's equation with time-invariant boundary conditions. The parameter of the system is dependent on the type of glacier, time of year, climate and probably the slope of the glacier surface.

Data from Peyto Glacier, Alberta, Canada, for the ablation period 1968 have been used to test the preliminary discharge-simulation model.

RÉSUMÉ. Simulation de décharge glaciaire par eau souterraine analogue. Pour simuler la décharge d'un glacier, le bassin de draînage est considéré comme étant analogue à un système d'eau souterraine. L'apport journalier au système est la précipitation et l'eau de fonte de la neige et de la glace, calculée d'après les données météorologiques,

Les débits d'écoulement sont calculés par l'emploi de méthodes établies à l'origine pour l'écoulement permanent d'eau souterraine. La décharge quotidienne d'un courant glaciaire est le débit d'un système obéissent à l'équation linéaire de Boussinesq avec des conditions aux limites invariant dans le temps. Le paramètre du système est dépendant du type du glacier, de la période de l'année, du climat et probablement de la pente de la surface du glacier.

Des données du Peyto Glacier, Alberta, Canada, pour la période d'ablation de 1968 ont été utilisées pour tester le modèle préliminaire de simulation de décharge.

INTRODUCTION

The objective of this study is to predict glacier run-off as the integral response of a glacier basin to the variable weather conditions.

It is impractical, if not impossible, to determine in detail the effect of all the time- and space-variable geophysical and meteorological factors, but a study of the overall response of the glacier permits evaluation of the most significant relationships.

In order to be useful for the solution of practical hydrological problems, the model should use an analytical method that keeps the computations simple and the required information at a minimum. These requirements are met by a "black-box" system approach. A system is defined as a complex, real or conceptual, which through the application of a formula relates an input to an output. The operation of the system is governed by this formula and a set of boundary conditions.

Because of the porous nature of the snowpack, the firn and—at least part of—the glacial ice, run-off from a glacier has been considered analogous to discharge from a ground-water system, viz. the operation of both systems will be the same.

The non-steady unconfined one-dimensional flow of water through a homogeneous porous medium can be described by a linearized partial differential equation (Boussinesq's equation) with simple boundary conditions. This equation with time-invariant boundary conditions has been chosen for the operation of the system regardless of the deviations of the actual boundary conditions. Data from Peyto Glacier have been used for checking this preliminary simulation model because they were most suitable qualitatively and quantitatively.

30 Leo Derikx

Description ofPeyto Glacier basin

Peyto Glacier (lat. 51° 40' N, long. 116° 34' W.) lies in the Waputik Mountains, app­roximately 45 km north-west of Lake Louise, Alberta, Canada, on the eastern slope of the Canadian Rocky Mountains (Fig. 1). The glacier (area 13.72 km2) covers 62% of the drainage basin (area 22.07 km2) above the stream gauge and the basin extends in ele­vation from 2 000 to 3 300 m a.s.l. (Fig. 2).

The glacier is divided into two distinct sections by a 100 m high ice fall at the 2 400 m level. Above the ice fall lies a large catchment basin (80% of the glacier area), which is part of the Wapta Icefield. Below the ice fall the glacier is approximately 1 000 m wide and extends 2.4 km down Peyto

Fig. 1. The Peyto Glacier region

1000 0 lOOO 2000 3000 4000

Fig. 2. The Peyto Glacier basin

Glacier discharge simulation 31

Valley. The surface gradient is gentle to moderate with slope angles of 2 6% and a mean surface slope of 5.5 %.

Drainage streams on the glacier surface are poorly developed due to an extensive subglacial and englacial drainage system.

Computation of melt

The daily input to the system is precipitation and melt of snow and ice that can be com­puted from available meteorological data. In 1968 a meteorological station was installed at 2 220 m a.s.l. on a lateral moraine, approximately 70 m from the western margin of the glacier and 15 m above the ice surface. The station consisted of thermohygrograph, totalizing pyrano-meter, totalizing anemometer, sunshine recorder, barometer and precipitation gauge. For purposes of the computations, the following assumptions were made: glacier and snowpack were isothermal at 0° C throughout the period, the liquid-water holding capacity was satisfied, thermal quality of snowpack was 100% and temperature of the melt water was always 0° C.

Melt in the basin was computed from a residual in the heat balance. Four terms have been considered in the energy balance: (1) net radiation flux (shortwave and longwave), (2) sensible heat flux (convection), (3) latent heat flux (condensation or evaporation), and (4) the residual heat used for melting (Streten and Wendler, 1968).

1. Radiation melt

The total daily incident shortwave radiation (direct and diffuse components) has been mea­sured with a horizontally installed pyranometer. In the absence of synoptic albedo measure­ments, it was necessary to resort to the use of mean conditions neglecting spatial and tem­poral variations. For June, July, August and September, the following values have been assu­med for snowcovered areas: 0.75, 0.65, 0.55 and 0.55, respectively. In August and September, after a fresh snowfall, the albedo increases to 0.75 but. As soon as this fresh snow has melted the albedo drops to the original value 0.55.

For glacial ice a constant albedo of 0.35 has been assumed. Average values of this magnitude have been measured, among others, on South Cascade Glacier, Washington (Meier, 1969).

A correction has been applied to the measured radiation to obtain the radiation incident on a slope with mean dip of 5.5° and azimuth of 031.5°. The corrections are 0.96,0.95, 0.93 and 0.90 for June, July, August and September, respectively (Gamier and Ohmura, 1968). This correction has been applied on clear days only. No correction has been made in this model for the differences in direct radiation as a result of shadowing caused by peaks and ridges on rela­tively clear days.

However, from recent investigations it can be derived that the pyranometer at the meteoro­logical station overestimates the mean shortwave radiation incident on the entire glacier (Good-ison, unpublished). For Peyto Glacier, a seasonal correction factor for shadowing effects should be included in future calculations.

The melt due to shortwave radiation is calculated by:

Msw=Rsw^l~a' k g m ~ 2 d " 1 orMsw =0.0030(1 - a ) * ^ mm d"1 .*

* l k g m 2 d 1 = — m 3 m 2 d _ 1 = — m m d - 1 = 1 mm d _ 1 for Q =1000 kg m~ 3 .

32 Leo Derikx

The net heat loss as a result of longwave radiation from a melting snow and ice surface

depends on the incoming atmospheric longwave radiation, which is a function of cloudiness.

Records of hours of sunshine were available and daily values of the longwave radiation budget

have been computed by Rlw — Rolw(l —nc2) (Ambach, 1960). For conditions on Peyto Glacier,

the actual numerical values of « and Rolw are expected to be different from the values obtained

by Ambach (1960) or adopted by Keeler (1964). On days with low thick clouds and a positive

cloud-base temperature, the net longwave radiation balance will be positive; for these conditions

n must have a value larger than unity. The correct value of n as a function of height of cloud

base has to be evaluated empirically from data on cloud height and the corresponding net

longwave radiation balance. Tentatively, the values for Rolw and n were assumed constant,

6 000 k J m " 2 d ^ " 1 (%0.1 cal c m - 2 min""1) and 0.6 (for average cloud height), respectively

(Keeler, 1964). The incoming longwave radiation from surrounding rock walls has been neg­

lected. The subsequent freezing of melt water as a result of this net heat loss has been computed

by Mlw Z^^IZI^LL k g m-2 d- i o r Mjw = _ i 8 (l -0.6c2) mm d"1. Lf

On perfectly clear days, the maximum recorded hours of sunshine were less than the theo­

retical maximum possible hours of sunshine as given in the Smithsonian meteorological tables,

The mean difference was approximately 4 h 20 min in both July and August and was regarded

as the shadowing effect by surrounding mountains. The maximum hours of sunshine in June and

September have been obtained by extrapolation, subtracting the constant difference of 4 h 20 min

M —H from the theoretical value. Cloud cover (c) has then been calculated by c = , in which

M M is maximum possible hours of sunshine for meteorological station in particular month and H is hours of sunshine measured by the recorder. The daily cloudiness is assumed the same as the cloudiness during the sunshine period of the day.

2. Melt resulting from sensible and latent heat transfer

àq A basic equation for turbulent exchange is Q, = oK, —•— (cited by Haltiner and Martin,

dz 1957; Munn, 1966), where Qz is the flow of some property of the air through a unit horizontal area per unit time, dq/dz is the vertical gradient of this property (q is the conservative property of the air, e.g. mass of water vapor per unit mass of dry air, the amount of sensible heat per unit mass of dry air, momentum per unit mass of air or wind speed and z is the height above the surface) and Kz is the eddy diffusivity for the specific property at height z.

It is necessary to know ths variation of these properties with height; it appears that for

wind-speed variation with height the simple logarithmic law is applicable over a wide range of

stability conditions (Grainger and Lister, 1966). From this logarithmic law and additional

assumptions, it can be derived that the eddy diffusivity for momentum transfer Kz is defined by

Kz = kzV* where V» is the friction velocity (Haltiner and Martin, 1957, p. 214-30; Munn, 1966,

p. 56-59) or Kz =k'izVz ( In —

Glacier discharge simulation 33

For any property of the air, it follows assuming simularity in temperature, wind speed and water-vapor profiles :

In-Î-

q -q0 = — (qa-q0) and —-—• = • — and thus Qz = v— • —- . (1)

l n ^ dZ l n ^ Z l n ^ zo zo zo

Following methods in classical micro-meteorology (quoted by Haltiner and Martin (1957, p. 246-47) and by Munn (1966, p. 81 en 84) some additional assumptions have been made: the eddy diffusivities for water vapor, sensible heat and momentum transfer are equal and the vertical-flux divergence of momentum, heat and water vapor in the surface boundary layer is negligible.

pV R* The equation of state for a perfect gas is = , where V is the specific volume

T md

V = — S • K follows from this equation that n= and substitution of this expression Q J R*T

for Q in Equation (1) yields:

pmdKz(qa - <?0) Q=QZ

R*Tz In — Zn

k2zV for K, = — the flow per unit horizontal area per unit time is expressed by:

" l n - 1 zo

i?*r in —In — zo zo

3. Melt from sensible heat flux

The amount of sensible heat q\ present in dry air per unit mass is q\ = CpdBz, where 6 is the potential temperature. Because the difference between potential temperature and absolute temperature close to the surface is very small (0 ^ T), the sensible heat is given by q\ = CpiTz.

Substitution oîq\ =CpdTz in Equation (2) yields an expression for the sensible heat flux:

mdPk?Cpd(Ta-T0Wh H=~

z„ R*T In — In '•b

The melt Mc resulting from the transfer of sensible heat is

Me=J^£}^^— (3) LfR*T In— I n ^ -

for snow having a thermal quality of 100%.

3

34 Leo Derikx

4. Melt from latent heat flux

The mixing ratio q" is defined by :

1z = " mi (jP-ez) mdP

Thus the flux of water vapour is given by :

_mJc\ea-e^Vh

UE •

In addition to the condensate itself for every gram of water condensed on the surface Lv J of heat energy are released (latent heat of vaporization). This is sufficient to melt LJLf

of snow having a thermal quality of 100%. Adding this quantity to the condensate yields for the melt-plus-condensate

M ^ C l + A - ) ^^Zf^L. (4) Lf i ? * r i n ^ l n ^

zo zo

The water-vapor pressure has been calculated from the integrated Clausius-Clapeyron equation for saturation over water :

I n ' " ' ' '"•""" ' 1

6.1 y J?* \273

In this equation Lv must be replaced by Ls for saturation over ice. This has been done in the calculations, when the air temperature was below 0° C.

The actual vapor pressure is given for t ^ 0 ° C by :

e=6 .11exp l-^^L\XRH

or

19 8/ e=6 .11exp I \XRH.

! 273 + 1 '

5. Melt equation for basin-wide snow and ice melt

Replacing the symbols in Equations (3) and (4) by their numerical values (see "notation and units"), the formula for practical use change over to Mc =0.0100 tV and ME =0.0223 (e —6.11) V, respectively. An empirical index, adjusting point melt to basin values, has been obtained by comparing ablation calculated from the total heat balance and ablation measured at the stakes on the glacier during a period when the glacier was still entirely snow-covered. By reducing the sensible and latent heat transfer to about 30 %, the total quantities were in close agreement.

Glacier discharge simulation 35

Thus the formulae used to estimate basin melt as a result of sensible and latent heat transfer are:

Mc =0.0033 tVand ME =0.0075 (e —6.11) Fin mm d""1.

In the heat balance, the latent heat flux and the sensible heat flux have been utilized when the sum of the fluxes taking place at the surface layer (latent heat, sensible heat and longwave radiative heat) was positive. When the sum of these terms was negative, it has been assumed that the influence of at least the negative latent and sensible heat fluxes could be omitted, thus taking into account some reduction of losses by adjustment of the surface temperature. The re­sulting formula for basin-wide melt thus derived is:

M = 0.0030(1 -a)bRsw- 18(1 -0.6c 2) +0.0075(e - 6 . 1 \)V + 0 . 0 0 3 3 ^ mm d _ 1 .

Snow-cover depletion

A knowledge of the approximate extent of the snow cover in the basin all through the period is essential, because of change in albedo when ice is exposed and of a decrease in snow-covered areas in the non-glacierized parts of the basin.

At the beginning of the melt period, depth and density of the snowpack on the glacier surface had been measured. From these accumulation measurements the water equivalent of the snow was averaged over every 100 m elevation interval.

Information on the snow cover of moraines and mountain slopes in the basin was not available and an estimate of the accumulation has been made by extrapolating from correspond­ing elevation intervals on the glacier ice. Melt was calculated for the area immediately above the transient snow line. By subtracting the computed melt from the original accumulation, the remaining water equivalent of snow on ice and rock was obtained. When this amount became zero, the snow line was shifted to the next elevation interval. In this way the location of the transient snow line, and thus the approximate surface areas of ice, snow and rock, have been obtained.

Precipitation

Winter accumulation increases with elevation and the simplest approximation for summer rain­fall is obtained by assuming that the altitude variation is the same in summer as in the winter (Fig. 3). This accumulation figure is regarded as the integral result of precipitation and melt dur­ing the period October to June. During that per­iod, melt can be neglected and precipitation is significant, thus the accumulation figure is an indication of the long-term distribution of preci­pitation, if one neglects redistribution of snow by wind. For every daily amount of precipita­tion measured at the meteorological station, the distribution over the basin is assumed similar to the winter-accumulation pattern. A criterion to distinguish snow and rain in the basin is obtained from the mean daily air temperature using a lapse

r a t e Of 0 .65° C 100 m ^ 1 . Fig. 3. Winter accumulation

3*

36 Leo Derikx

Generally, precipitation falls as rain at surface air temperature above +1.5° C and as snow below this temperature. The elevation at which the temperature is + 1.5° C, or the separa­tion between rain and snow, was computed from mean daily temperature and adopted lapse rate.

The rainfall depth in every elevation interval is multiplied by the respective interval area, and the sum of all intervals up to the + 1.5° C isotherm represents the total amount of liquid precipitation in the basin. Liquid precipitation is treated in the same way as melt. Heat added to the glacier by rain is neglected.

Ground-water analogue

A "black-box" system has been used to transform a lumped input (basin-wide snow, ice melt and precipitation) into a discharge. Computation of outflow rates (output of the system) is based on methods derived for non-steady ground-water flow to parallel channels. According to Boussinesq's equation, the one-dimensional unconfined flow of water through a homogeneous porous medium is described by a non-linear partial differential equation.

dh dz ah 8 / ,Sh

However, a general analytical solution of this equation subjected to general initial and boundary conditions has not yet been found because of its non-linearity. By assuming a horizontal im­permeable layer at the bottom of the flow region and by assuming a constant "hydraulic diffu-sivity" independent of the horizontal distance, which is allowable if the variation of the water level h is very small, the equation may be linearized to :

dh - d2h m — =KD-

dt dx2 "

There is, however, no physical evidence that h is much smaller than D0 . The basic assumption

KJDn -\-hJ) in this linearized equation is that the "diffusivity" a = is independent of the hon-

mx

zontal distance x. If the additional assumption is made that for ice and snow KJmx is approxi­

mately constant, it follows that (D0 + hx) must be constant or hx must be negligible compared

t o D 0 .

This simplified and linearized differential equation with time-invariant boundary conditions

is used for the operation of the system. The linearized equation is correctly applicable only to

a porous medium in which the flow of water is dependent on one coordinate x only, and the

medium is bounded by a horizontal impermeable layer at the bottom, vertical seepage surfaces

at the channels and a free water table at the top (Fig. 4). Simple boundary conditions are

h(0, t) = h(L, t) = 0. Computational methods have been developed for ground-water flow in this

particular case (Krayenhoff van de Leur, 1958, 1962). The distribution graph «(1, /) of outflow

rates as the response to a unit block input of uniform percolation can be calculated according

to these methods (Fig. 5). The daily discharge is given by ô i = E u(\,i)-Mt where Qx repre-

Glacier discharge simulation 37

sents the discharge for a particular day, M{ is the melt on the /' ' day (same day and up to (« — 1)

perevious days). The values of «(1, 0 are determined by the value of the system parameter, defined as reser­

voir coefficient j = -1 mL~

—— , which incorporates all important hydrological properties of the KD

^ 5 î ^ S 5 ^ 5 5 î ^ 5 ^ ^ 2 ^ S ? î ^ 5 P ^ p ^

k = HYDRAULIC CONDUCTS

m ' SPECIFIC Y I E L D

eSERVOIft COEFFICIENT j= -

Fig. 4. Scheme for flow of ground-water to parallel channels

basin. The reservoir coefficient has been determined empirically and appeared to be equal to 2 d for a large part of the sea­son. The distribution of outflow resulting from a unit block input is presented in Tab­le I for y = 2 d. For this value of the re­servoir coefficient a period of 13 d is re­quired to account for 99.7% of the melt. To account for all the melt water, theore­tically an infinite number of days is ne­cessary, because the long tail of the unit hydrograph approaches zero asymptoti­cally.

The method is based on a linearized equation and superposition is allowed to obtain the discharge as the sum of differ­ent melt quantities, weighed by the res­pective ordinatesofthe distribution graph. The mathematical presentation is checked by comparing calculated output with ac­tual output measured in the melt stream below the glacier. Figure 6 shows the simulated discharge obtained by this preliminary model.

n =1.3,5

Fig. 5. Distribution graph for unit block input

MEASURED SIMULATED

JULY AUGUST

Fig. 6. Discharge of Peyto Glacier, 1968

SEPTEMBER

38 Leo Derikx

TABLE I. Distribution of out flow (j = 2 d)

u(\, 1) =0.3390

«(1,2) =0.2728

«(1,3) =0.1522

u(l, 4) =0.0923

«(1,5) =0.0559

«(1,6) =0.0339

«(1,7) =0.0206

«(1,8) =0.0125

«(1,

«(1,

«CI, «(1,

«(1,

i = 13

I i=l «(1,

9) =

10) =

11) = 12) =

13)

0

= 0.0076

= 0.0045

= 0.0028

= 0.0016

= 0.0010

= 0.9967

Results and discussion

Application of the linearized equation with the simple boundary conditions for the opera­tion of a glacier drainage system is a very crude approximation of the actual situation. The result shows a fair correspondence of calculated and measured discharge, and indicates that the drainage system as a whole or a significant part of it, through which practically all the melt water has to pass, has characteristics similar to those of a ground-water system. It is important to be aware that it is the operation of the two conceptual systems that is similar. This does not necessarily mean that the physical (real) characteristics of the glacier melt-water flow through a glacier are similar to the ground-water flow through the ground.

Actually on a glacier there are three distinguished zones of flow: the vertical percolation of melt water and rain from the surface of the snowpack and the firn into the saturated zone; the lateral, more or less horizontal flow in the saturated zone at the bottom of the snowpack and the permeable ice; the turbulent flow in melt-water channels on the surface or in pipes within the snowpack or the glacier. Probably the lateral saturated flow of water in the snowpack, in the firn and in the porous surface of the ice prevails and determines largely the operation of the whole system. The vertical unsaturated flow in the snowpack, the turbulent flow in surface channels and the internal network of pipes in the glacier are of minor importance on a daily basis because the changes in storage and the travel time are small.

The porous system is not time-invariant ; the reservoir coefficient varies during the season because of changing conditions in the snowpack and on the ice surface. The network of surface melt-water channels becomes more articulated and progresses further up glacier as a result of slushing out and melting. In this simulation model a constant value for the reservoir coefficient is used throughout the season, but Figure 6 shows that in June, when the glacier was still snow-covered, the response of the system is clearly slower. In July, August and part of September the response and thus the system itself is fairly constant. Apparently, after the very high discharge in July the characteristics of the system do not change significantly until September when the whole drainage pattern gradually becomes constricted by freezing of melt water.

1 ml? The reservoir coefficient is / = — • , in which the effective porosity m and the hydraulic

10 KD conductivity K are dependent on the porous medium itself. The distance L between the drainage channels and the mean depth D of the saturated zone are determined by some geometrical characteristics of the basin and indirectly by the properties of snow, ice and climate.

As a result j = cl, in which c is a constant independent of the geometry of the basin, and

Glacier discharge simulation 39

A is some characteristic length. The value of the reservoir coefficient is determined by type of glacier, the properties of snow and ice, the climate and probably by the slope of the glacier surface.

The presented value of the reservoir coefficient/ = 2 d should be considered an approximate value. In connection with the particular set of instruments, with the present quality of the mea­surements and especially with the methods and formulae used to calculate rainfall and melt of snow and ice on the entire glacier, the use of a distribution graph based on a reservoir coef­ficient of y = 2 d rather than j = 1 d or j = 3 d improved the simulation model for daily total discharge definitely. An objective parameter-optimization method has not been applied in this stage for the reasons :

(i) The system is time variant. This time invariance is of a seasonal nature and it might be feasible in the future to establish a practical objective procedure to optimize a time-varying linear-system parameter. The choice of several distinct periods during which the parameter is regarded constant is subjective; the requirement of continuity of storage at the transi­tion point to the next period complicates the computations and causes considerable discontinuities in discharge.

(ii) Optimization of an invariant parameter by objective methods tends to obscure a great deal of information. An objective parameter-dependent function to be optimized is the sum of the squares of the differences between observed outflow and the outflow computed from the corresponding input. In this procedure the larger differences, which are highly interesting in this stage, will affect the parameter value most and will therefore be reduced considerably.

The larger differences between measured and generated outflow are caused by imperfections in calculations of melt and rainfall, by possible non-linearities in the system operation or by the presence of other reservoirs with characteristic storage and travel-time properties (e.g. vertical unsaturated flow in a deep snowpack).

Before starting refinement of the flow calculations and application of parameter-optimiza­tion techniques, the determination of the input and therefore the energy balance of the glacier as accurate as possible is a prerequisite.

The preliminary result indicates that future research can be conducted along the same line. By applying the model to shorter time intervals, for instance on an hourly basis, a better judg­ment criterion is obtained for testing its physical correctness because of the high variation in input and output during the day.

As yet it is not possible to deal with spatial and temporal variability of all significant me­teorological parameters and physical properties of snow and ice, but for the reconstruction of the daily discharge hydrograph from melt and rainfall a "black-box" system will be an adequate tool in the future. However, to arrive at a general and adequate model applicable to all types of glaciers in different environments, more fundamental research on all significant physical processes is required.

ACKNOWLEDGEMENTS

Suggestions from and discussions with Dr O. H. L0ken and Dr A. D. Stanley (Glaciology Subdivision), Dr H. Ryckborst (Groundwater Subdivision) and Professor D . A. Krayenhoff van de Leur (Agricultural University of Wageningen, The Netherlands) are gratefully acknow­ledged.

40 Leo Derikx

REFERENCES

AMBACH, W. 1960. Investigations of the heat balance of the ablation area on the Greenland ice cap. Archiv fur Météorologie, Geophysik unci Bioklimatologle, Ser. B, Vol. 10, No. 3, p. 279-88.

GARNIER, B. J., and OHMURA, A. 1968. A method of calculating the direct shortwave radiation income of slopes. Journal of Applied Meteorology, Vol. 7, No. 5, p. 796-800.

GOODISON, B. Unpublished. The distribution of global radiation over Peyto Glacier, Alberta. [Glaciology Subdivision, Inland Waters Branch, Department of Energy, Mines and Resources, Canada, 1969.]

GRAINGER, M. E., and LISTER, H. 1966. Wind speed, stability and eddy viscosity over melting ice surfaces. Journal of Glaciology, Vol. 6, No. 43, p. 101-27.

HALTINER, G. J., and MARTIN, F. L. 1957. Dynamical and physical meteorology. New York, McGraw-Hill Book Co., Inc. KEELER, C. M. 1964. Relationship between climate, ablation and run-off on the Sverdrup Glacier 1963 Devon Island, N.W.T.

Arctic Institute of North America. Research Paper No. 27. KRAYENHOFF VAN DE LEUR, D. A. 1958. A study of non-steady groundwater flow with special reference to a reservoir coefficient.

De Ingénieur (The Hague), Vol. 70, No. 19, p. B87-94. KRAYENHOFF VAN DE LEUR, D. A. 1962. A study of non-steady groundwater flow. II. Computation methods for flow to drains.

De Ingénieur (The Hague), Vol. 74, No. 46, p. B205-92. MEIER, M. F. 1969. Glaciers and water supply. Journal of the American Water Works Association, Vol. 61, No. 1, p. 8-12. MUNN, R. E. 1966. Descriptive micrometeorology. New York, Academic Press. STRETEN, N. A., and WENDLER, G. 1968. The midsummer heat balance of an Alaskan maritime glacier. Journal of Glaciology,

Vol. 7, No. 51, p. 431-40.

A P P E N D I X

NOTATION AND UNITS

Rsw Daily incident global shortwave radiation (kJ t r r 2 ) .

Rlw Daily net longwave radiation loss from a melting ice surface (kJ m - 2 ) .

RolK Daily net longwave radiation loss on a clear day from a melting ice surface ( = 6 000 kJ m _ 2 ~ 0 . 1 cal c m - 2 min - 1 ) .

M Daily melt (mm d"1). Ms Melt resulting from shortwave radiation

flux. Mlw Melt resulting from longwave radiation

flux. Mc Melt resulting from sensible heat flux. Me Melt resulting from latent heat flux.

a Albedo of ice or snow. b Radiation correction for angle and azimuth

of mean slope of Peyto Glacier. n Factor dependent on height of cloud. c Amount of cloud cover.

Lf Latent heat of fusion ( = 334 kJ kg _ 1 = 79.7 cal g"1).

L Latent heat of sublimation ( = 2 834 kJ kg" 1 = 674.7 cal g"1).

L Latent heat of vaporization ( = 2 500 kJ kg- 1 = 595 cal g-1).

mn Molecular weight of water ( = 18 g mole- ' ) . md Molecular weight of dry air (28.97 g

mole - 1) . RH Relative humidity at screen level (%).

k Von Karman's constant ( = 0.38). ea Water-vapor pressure in screen (N m - 2 ) . e Water-vapor pressure in screen (mbar).

es Saturation vapor pressure in screen (N m" 2 ) .

e0 Saturation vapor pressure above melting ice surface ( = 6.11 X102 N i r r 2 = 6.11 mbar).

t Air temperature in screen (°C). Vb Run of wind (m d"1). V Run of wind (km d"1).

R* Universal gas constant ( = 8.314 J mole""1

K - ) . T Absolute temperature (°K).

za Height of thermohygrograph (=1.10 m). zb Height of anemometer ( = 3.00 m). z0 Surface roughness parameter ( = 0.25 X

10~2 m). p Atmospheric pressure at meteorological

station ( 7 8 5 X l O a N i r r 2 ) . C d Specific heat of dry air at constant pressure

(=1.003 kJ k g - 1 K- x = 0.24 cal g" 1 "C-1) . m Specific yield, effective porosity (%). h Rise of water table in porous medium above

original horizontal level (m). D 0 Height of original water table in porous

medium (m). D Average depth of seeping water between

two channels (m). K Hydraulic conductivity (m d _ 1 ) . x Abscissa, distance from origin (m). z Height of impermeable layer above hori­

zontal reference level (m). L Horizontal distance between two channels

(m). j System parameter, reservoir coefficient (d).