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This is to certiry that the

thesis entitled

MELTWATER DRAINAGE FROM TEMPERATE GLACIAL ICE
BURROUGHS GLACIER
SOUTHEAST ALASKA

presented by

Ryan Jay Simmons

has been accepted towards fulfillment
of the requirements for

M.S. Geology

 

 

degree in

 

Major rofessor

Date 4 May 1989

0-7639 MSU is an Affirmative Action/Equal Opportunity Institution

PLACE IN RETURN BOX to man thh chockout from your record.
TO AVOID FINES return on or baton duo duo.

    
 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmdtvo AdloNEqunl Opportunity lm

MELTWATER DRAINAGE FROM TEMPERATE GLACIAL ICE
BURROUGHS GLACIER
SOUTHEAST ALASKA

By

Ryan Jay Simmons

ATHESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Geological Sciences

1989

ABSTRACT

MELTWATER DRAINAGE FROM TEMPERATE GLACIAL ICE
BURROUGHS GLACIER
SOUTHEAST ALASKA

By

Ryan Jay Simmons

A surface runoff model was applied to a temperate glacier
in southeast Alaska in order to determine the method of meltwater
drainage from unweathered glacial ice. Melt input for the model was
calculated, for points on a north-south grid over the glacier surface,
over a period of four sunny days in August 1973. The melt was then
applied to a Route-and-Lag runoff model to create discharge
hydrographs for the glacier for the duration of the study
period. The resultant model discharges were then compared to
actual drainage hydrographs for the same four day period. As a
result of this comparison, it was determined that meltwater drainage

from the glacier is predominantly through surface runoff.

ACKNOWLEDGEMENTS

I would like to extend thanks to my thesis advisor, Dr. Grahame
Larson, for his guidance and extreme patience. The members of my
thesis committee, Dr. David Long and Dr. Larry Segerlind, also
deserve mention for their cooperation. My parents, Robert and Rose
Simmons, perhaps deserve the greatest credit of all for their support,

and encouragement.

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

 

List of Tables v
List of Figures vi
1. Introduction 1
II. Background Information 3
Basin Hydrology 3
Climate and Physical Setting 5
Glacial Characteristics 6
III. Melt 8
IV. Runoff 11
V. Results 24
VI. Conclusions 28
Appendix A Ice Melt Calculations 31
Appendix Input Values for Melt Routine 37
Appendix Program for Calculating Hour Angles 42

 

and Melt 4 3
Program for Calculating Lumped Lagged Flow 44

B.
C.

Appendix D. Program for Calculating Radiation Receipts
E.

Appendix
Appendix F. Program for Calculating Distributed Lagged Flow__48
Appendix G. Program for Routing Flow 50
Bibliography 5 1

iv

Basin Lag Values

Daily Variable Data Values

LIST OF TAB LES

Maximum Values of t for each Basin of
Burroughs Glacier
Basin Conversion Constants for Travel Time
Total Stream Discharge during the Study

14

17
18
25
37

LIST OF FIGURES

 

 

 

 

 

 

 

 

 

 

1 Subbasins of Burroughs Glacier 4
2 Radiation Receipts and Melt through Time 10
3 Surface Runoff Flow Lines Burroughs Glacier 15
4 Absolute Travel Time Isochrone Map Burroughs Glacier ...1 9
5 Lumped Model Lagged Hydrographs 20
6 Distributed Model Lagged Hydrographs 21
7 Lumped vs. Distributed Model Outputs 22
8 Routed Hydrographs vs. Actual Hydrographs 23
9______Surface Slope Burroughs Glacier 39
10 Surface Azimuth Burroughs Glacier 40
11 Surface Albedo Burroughs Glacier 41

 

 

vi

I. Introduction

According to Drewry (1986), there are six main sources of
liquid water in temperate glaciers. These include: surface ice melt;
ice melt due to mechanical stress and strain; ice melt from
geothermal heating; groundwater flow; surface runoff; and, liquid
precipitation. This water may be found on top of, within, under, and
adjacent to the glacial ice. Surface (supraglacial) water is usually
drained via running streams (Drewry, 1986) which may discharge
into tubes, or moulins, which pierce the glacier surface and connect
to a system of tubular conduits branching through the body of the
glacier and converging at a single discharge point at the glacier
terminus. Water within the glacial ice itself (englacial water) can
drain through a system of intergranular veins and capillaries
between the ice crystals (Nye, 1976; Nye and Frank, 1973). Liquid
water at the base of a glacier may also drain as a thin film spread
uniformly between the ice and bedrock (Nye, 1973; Freeze, 1972).

In this study, meltwater production and surface runoff
associated with Burroughs Glacier, in southeast Alaska, is modeled
for a period of four days using solar radiation receipts and a time-
area lag-and-route method. The purpose for modeling the melt and

runoff is to determine if surface runoff alone can account for most of

2

the meltwater emanating from the glacier margin during periods of
high meltwater production.

Burroughs Glacier was chosen for the investigation because it is
hydrologically a very simple system to model. It lies completely
below the equilibrium line so that at the height of summer there is
no snow remaining on the glacier surface to absorb meltwater or
complicate the drainage. Also, since the glacier is temperate and at
the pressure melting point throughout, no meltwater is removed
from storage through refreezing. The glacier is drained by well
defined streams whose channels are generally stable and suitable for
monitoring of stream discharge.

Another reason for choosing Burroughs glacier is the
availability of usable data. Many meteorological, glacial, and
geological studies have been made in the general area surrounding
Burroughs glacier. One such study, conducted by G. Larson in the
summer of 1973, provides all the necessary data for testing a runoff
model such as will be used in this study. Larson's meteorological
data can be used to calculate melt - which can be applied to a surface
runoff model. Moreover, discharge data, from streams draining
Burroughs glacier for this period, can be used as a means of judging

model accuracy.

II BA K R INF RMATION

Burroughs Glacier is a small body of stagnant ice roughly 13
km long and 1.2 to 3 km wide. It is located just north of Wachusett
Inlet, about 120 km northwest of Juneau (59' 00" North Latitude,
136' 20" Longitude), and is one in a series of hundreds of small ice
bodies within the northeast comer of Glacier Bay National
Monument. Two tongues of ice, flowing in two different directions

from an ice divide, compose the body of the glacier.

My

Early records from the late 1800's show that Burroughs glacier
was once part of a much larger body of ice, the Cushing Plateau
(Mickelson, 1971). The ice at that time was around 850 m high in
the Burroughs glacier area. Over the years, however, it has melted at
a rate of as much as 8 m/year and has resulted in a gradual
reduction in glacier size until all that remains today are a number of
small individual glaciers occupying valley basins.

This study concentrates upon the eastern tongue of Burroughs
Glacier which lies within an east-west striking basin 27.2 square
kilometers in area (see figure 1). The tongue is 6.8 km long, 2.2 to 3

km wide, and 13.9 square kilometers in area, and is bounded to the

 

 

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north by Minnesota Ridge and to the east by the Curtis Hills and to
the south by the Bruce Hills. It is also cut off from all other ice
bodies by the encircling mountain ridges so that no ice is added to
the basin from outside sources. The ice divide separating the eastern
tongue from the western tongue is also at such a low elevation that
the ice is entirely below‘the equilibrium line (Larson, 1978). As
such, the yearly snow input on the glacier surface completely melts
each summer - allowing no buildup of snow to produce new glacial
ice. Moreover, not only does each years snow melt off, but some of
the glacial ice melts along with it.

Since the eastern tongue of Burroughs Glacier is surrounded by
mountains, surface water flow is blocked from entering from outside
the basin. The crystalline bedrock of the basin also does not allow
for any major exchange between groundwater and surfacewater.
Thus, the eastern tongue of Burroughs Glacier may be considered an
isolated system, such that any water exiting the glacier must be
derived from the glacier itself or from precipitation falling within the

basin.

lim e n h i l ettin
The general climate in the vicinity of Burroughs Glacier has
been described by Loewe (1966) and McKenzie (1968, 1970). Due to
the proximity of the pacific coastal waters, the climate may be
characterized as maritime. No permanent meteorological stations
have been located on the glacier itself, however, meteorological

measurements have been taken during numerous expeditions since

6
the year 1959 (Taylor, 1962; Mickelson, 1971; Larson, 1977 & 1978).

Records of these measurements show: an average temperature of
10°C 150; rainfall of 200 mm/month; and, mostly overcast
conditions.

The bedrock beneath and around Burroughs Glacier consists of
metamorphosed shales and limestones intruded by diorite and
granodiorite stocks. The metasediments are of paleozoic age and
generally occupy the eastern half of the basin. Moreover, the stocks
are of cretaceous age and occupy the western half of the basin. In
scattered areas, unconsolidated sediments mantle the bedrock and
consist of sandy till along with well-sorted sand and gravel. The till
ranges in thickness from a thin 'veneer' to more than 25 meters in
some places. The sand and gravel exist mostly in kame terraces near
valley walls.

On the basis of dye tracer experiments Larson (1978) was able
to divide the eastern tongue of Burroughs Glacier into three sub-
basins for drainage purposes (see figure 1). Each of these subbasins
is drained by its own stream: basin 1 by Burroughs River; basin 2 by

Bob Creek; and, basin 3 by Gull Creek.

1 ' h ri i
The surface texture of Burroughs Glacier varies greatly. Near
the glacier terminus, for example, ice crystals are generally coarse
and equigranular and range from five to ten centimeters in diameter
(Larson, 1978). They sometimes appear slightly melted at their

boundaries so that they are often loose, which gives the surface a

7

disintegrated appearance. Further away from the terminus,
however, the ice is fine grained, foliated, and does not have a
disintegrated appearance. Also, individual crystals are irregularly
shaped and are generally less than two centimeters in diameter.

The glacial surface is also characterized by several types of
structures (Taylor, 1962). These include foliation, banding, fractures
and crevasses. The foliation consists of parallel bands of bubble-free
and bubble-rich ice running parallel to Minnesota Ridge. The
banding consists of horizontal layering caused by a differential fine-
sediment content of the ice. Most of the fractures and crevasses on
the glacier surface are vertical, less than 30 cm wide, and as much as
10 meters deep. At higher elevations, they form a transverse system

while, in lower regions, they are more longitudinal.

III MELT

 

Larson (1978) found that short-wave global radiation is the
dominate energy source for melt on Burroughs Glacier during clear
sunny days. He also found that other sources of energy (such as net
long-wave radiation, convective heat transfer, and conductive heat
transfer) generally balance each other out, and that short-wave
radiation alone could be used to approximate melt at a single point
anywhere on the glacier surface.

For this reason, it was decided to use a cloudless four day
period to estimate melt on the glacier surface. This was done by
placing a north-south rectangular grid of 70 x 60 points over the
glacier (see figure 1) and calculating radiation receipts at each such
point. In order to account for melt variation through time energy
receipts were repeatedly calculated for each point at specified time
intervals. Melt was calculated for each point by multiplying
radiation receipts by a conversion constant (the latent heat of
fusion).

Melt was calculated, at a specified times for each grid point,
using a computer program based on radiation equations developed
by Kondratyev (1969). Inputs to the routine include: time of sunrise;

time of sunset; declination; transimissivity; radius vector; albedo;

9

surface lepe; and surface slope azimuth (see Appendix A). The
program applies the radiation equation to each grid point for each
half hour interval, for all four days of the period August 14 to
August 17, 1973, starting with the half hour immediately preceding
dawn and ending with the half hour immediately following dusk.
The total radiation received, and volume of melt, over the entire
surface for each day of the study, as well as the sum for the whole
four day period, was also calculated.

Figure 2 shows the variation in solar radiation receipts over the
basin surface, and the rate of meltwater production, through time.
On the figures, four peaked curves represent the radiation influx to
the surface during the four days. Solar radiation seems to be at a
maximum during solar noon of each day, decreasing with time on
either side of the peak until dawn or dusk is reached. Energy inputs
between dusk and dawn are zero. A comparison of radiation receipts
and melt shows that when there is a maximum of radiation at the
surface there is a maximum melt and when there is a minimum of

radiation inputted to the surface there is a minimum melt.

A 180

I

O

s 150

.5

E 120

8
3E 90
2 o

’6 60

E

3 30

2

3 0

25

20

15

10

Melt
(cubic meters/30 minutes)
Gt

0

Basin 1

10

 

 

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10000

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I I
0 1000 2000 3000 4000 5000 6000

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Basin 2

 

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1000 2000'1'3000 4000 5000 6000
Toime Since Beginning of Study (Minutes)

Basin 3

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~1600

1200

'I

800

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O

1000 .12000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)

Radiation Receipts

Radiation Receipts

(calories)

(calories)

(calories)

Figure 2. Radiation Receipts and Melt through Time

IV. RUNOFF

The lag-and-route method (Laurenson, 1964; Raudkivi, 1979)
was used to model meltwater runoff over the surface of Burroughs
Glacier. This method calculates the effects of both the translatory
(lagged) and storage (routed) movement of meltwater for each
subbasin of Burroughs Glacier.

The lag method was applied in two ways. In the first
application of the lag method, the subbasins were divided into
"lumped" zones and the sum of the runoff from each such zone was
calculated. In the second application of the lag method, flow from
specified points "distributed" over the drainage basin was calculated
- seemingly more representative of areal variations in slope and
melt. The first method, the "lumped" lagging method, was developed
to minimize the number of necessary calculations; however, the
second method, the "distributed" lagging method, seems to lend itself
more readily to computer application - a matter of applying a set of
flow equations to each grid point.

For the lumped lagging method, inflow (in the form of
meltwater production) was lagged by dividing each subbasin into a
number of zones by isochrones of runoff travel time to the subbasin
outlets. The area between isochrones was calculated and was

assigned an average travel time (taken as the average of the two

11

12

isochrones defining the zone). The contour interval between
isochrones was chosen as the duration of the inflow (meltwater
production) increment for ease of computation (Linsley, Kohler
&Paulus 1975). The inflow was then lagged by determining the
inflow value between isochrones and calculating an average

discharge for each isochrone zone by the equation:
Q = PA/T

where:

= discharge from the zone in question

Q
P = average melt for the zone
A = area of the zone
T = time increment of precipitation = time between zone
isochrones
A lagged discharge hydrograph, for each inflow increment, was
produced by calculating the discharge ordinate for each time

increment as follows:

In=PnAl +Pn_1A2+. . .+P1An

where:

1n = lagged hydrograph ordinate at time n

Pu: inflow (melt) at time n

An: area between the (n-1) and the n-th isochrone
n = number of subzones

The lagged hydrograph for each event was then added together to

produce a compound lagged discharge hydrograph. See Appendix E

13

for the computer routine written to calculate the lagged flow in this
manner.

For the distributed lagging method flow was calculated for each
point on the glacier surface. Runoff travel times were calculated for
each grid point on the glacier. Melt, calculated for each grid point,
was then added to a total compound basin hydrograph according to
the time that it took that melt to reach the basin outlet (the runoff
travel time). See Appendix F for the computer routine written to
calculate the lagged flow in this manner.

The lagged discharge hydrograph was routed through storage
to get the actual hydrograph for each subbasin. Routing was
accomplished through the use of the Muskingum storage routine, as
outlined by Laurenson (1964) and Raudkivi (1972). This routine
assumes outflow at any time depends upon both preceding outflow

and preceding inflow and is expressed as follows:

On = C0111 + Clln-l + C20n_1

 

 

where:
O = routed discharge ordinate
I = lagged discharge ordinate = inflow ordinate for route model
C _ 0.5t

0" k + 0.5t
C1=Co

k - 0.5t

C2: k + 0.5:

t = routing period

it = storage constant

14

In the above equation, the storage constant is the ratio of
storage to discharge. According to Linsley, Kohler and Paulhus
(1975), k is approximately equal to the travel time through the reach
through which the method is being applied and may be equated to
the average travel time for the basin (which is the basin lag).

The computer routine developed to perform the task of routing
flow is shown in Appendix G. Input to this routine included tabular
files containing the lagged hydrograph ordinates for each half-hour
time increment, the time increment for routing (one half—hour), and
the storage constant (basin lag) for each basin of Burroughs Glacier

(see table 1 for basin lag values).

1 i le

 

Basin Lag
Basin (hours_)__
1 3.22
2 3.05
3 3.21

W

The grid points used to calculate melt in chapter 3 were used in
the calculation of flow lines. These points were superimposed on a
topographic map of the area and flow lines were drawn between
them based on slope direction. Figure 3 shows the flow lines drawn

for subbasins 1, 2, and 3.

15

 

 

       

 

    

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16

Segment lengths and slopes were determined along each flow
line on the glacier surface. This was done from the extreme
upstream end to the subbasin outlet for each flow line. Segment
lengths and grid point elevations were read directly from the
topographic maps.

Travel times were determined by first calculating the relative
travel time of each point (t/rmax). This was done according to the

equations described by Laurenson (1964):

II II
1: = 2(L/Sc0-5)i = 2(L1-5/H).

1

i=0 i=0
where:
L = length of travel between any two adjacent points along a
flow path
Sc: slope of the surface between the two adjacent points
H = difference in elevation between the two points

A running sum was then made for each point along each flOw path.
Relative travel time for each point were then calculated by dividing

the running sums by the maximum sum value for the basin (trel =

t/rmax). Table 2 includes values for tmax for each basin of Burroughs
Glacier.

Laurenson noted that travel time and basin lag time are
related. In fact, in his study, he concluded that lag is an average
storage delay (travel) time. We know the basin lag from

measurements taken during the study (see table 1).

17

T12 MximmVl f

 

f h in f rr h I ir
Basin Relative Travel Time (11
1 26922
2 3482
3 17539

Assuming a linear relationship between relative travel time
and absolute (i.e. k(tr)=k(ta)), a basin constant was calculated by
dividing the absolute travel time by the relative travel time for the
point. Absolute travel times were then calculated for each point on
the surface by multiplying the relative travel time at each point by
the basin constant.

In his study, Laurenson equated basin lag with the centroid of
the time-area diagram. This could be taken as a weighted mean
(with the weight factor being the area corresponding to each relative
travel time value). Applying this to Burroughs Glacier , with the
weighted mean being calculated for the relative travel time at each
nodal point on the surface (with the weight factor being the area
surrounding each point), the centroid for each subbasin was located.
Because this "centroid" corresponds to basin lag, the basin conversion
constant could be calculated from travel time values at this point.
The basin conversion constants for basins 1, 2, and 3 are shown in

table 3.

18

T l B in nv ri n n nt
for Travel Time

Conversion Constant

 

Basin Mean 1: (hours)
1 0.610 5.279
2 0.520 5.865
3 0.460 6.978

The relative travel time value at each point was multiplied by
the conversion constant for each subbasin to get the absolute travel
time for each point. Figure 4 shows the absolute travel time
isochrone map developed for basins 1, 2, and 3.

Input to the runoff model was in the form of melt calculated
for each grid point on the surface of Burroughs Glacier. This data
was applied to both the lumped and distributed lagging routines.
Figure 5 represents the hydrographs produced for basins 1,2, and 3
by the lumped method. Figure 6 represents the lagged hydrographs
for basins l, 2, and 3 by the distributed method. Upon close
inspection, there does not appear to be any significant difference
between output from the two methods. This is further supported by
a near one-to-one relationship in a plot of lumped vs distributed
lagged outflow for each subbasin shown in figure 7. These lagged
hydrographs served as input to the routing routine. Figure 8
compares the lagged-and-routed hydrographs with actual stream

hydrographs for basins 1, 2, and 3.

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Discharge (Cubic Meters per Second)

Discharge (Cubic Meters per Second)

Discharge (cubic meters per second)

20

 

 

 

 

 

 

 

 

 

Basin 1
10
8
6
4
2
0 I ' I ' I ' r ' r r I '
0 1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)
Basin 2
1.5
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1.0 ‘
1
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0.0 V I I r I T I l I’ 1 n
0 1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)
Basin 3
8
6 -
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1
4 ..
2 u
0 I T I a T I I I t fir I I

 

 

O

1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)

Figure 5. Lumped Model Lagged Hydrographs

Discharge (Cubic Meters per Second)
Discharge (Cubic Meters per Second)

Discharge (Cubic Meters per Second)

21

Basin 1

 

10

1111

1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)

W

O

5

N

 

 

 

 

 

Basin 2
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1
.1
0.0
0 2000 5000 6000
Time Since Beginning of Study (Minutes)
Basin 3
8

o~

&

N

 

 

 

0 1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)

Figure 6. Distributed Model Lagged Hydrographs

Discharge

Distributed Model Discharge Distributed Model

Distributed Model Discharge

Figure 7.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0.0fi ‘f a t I a t v a I u t v u
0.0 0.5 1.0 1.5
Lumped Model Discharge
Basin3
8
6.
4.: 6%
: 8:3”
2; . .589
0H ' V I I t f r I v t r
0 4 6 8
Lumped Model Discharge

Lumped vs. Distributed Model Outputs

23

 

 

 

 

 

A Basin 1
3 8
< ~
U) 1 ’ l
3 6 .: ' i l‘ ‘
°' . \ i i i
W l l l ‘
3 j I ' l I
3 i . I
W l l l t
8 2 -I ‘ I ‘
q I
o I
s: I . ’
g 0 . I I I I I I ' I I
a 0 1000 2000 3000 4000 5000 6000

Time Since Beginning of Study (Minutes)

 

 

 

 

8 Basin 2

° 2

§ 1‘ :

m ..

a /\ A A.
‘ I

g 0 8 1 l " t 'l ‘i

g :i i ' ‘ l “ l

2 0 5 '- I \ “ I l i

u : ‘ " \ i “

3 0.4 1 i I I

a . i

U I

V 0.2 1

0 1

E ..

2 0-0 ' I I ' I I I I I '

g 0 1000 2000 3000 4000 5000 6000

D

Time Since Beginning of Study (Minutes)

Basin 3

 

a

N

§
‘llll‘Jl‘lll‘lJ
‘0‘“.
”——

 

 

 

O

I T ' ' ‘ I I 1

, .
0 1000 2000 3000 4000 5000 6000
Time Since Beginning of Study (Minutes)

Figure 8. Routed Hydrographs vs. Actual Hydrographs
(solid lines=actual flow; dashed iines=modeled flow)

Discharge (Cubic Meters per Second)

V RE TS

Figure 8 shows actual stream discharge and theoretical
discharge plotted over the duration of the study period for
each of the three subbasins of Burroughs glacier. To measure the
ability of the Lag-and-Route model to predict stream
discharge, a visual comparison of output discharge and actual
discharge was made. Such features as peak discharge, total
discharge volume, and time to peak were used as key points of
comparison - to determine the degree of correlation for the
hydrographs.

From a visual inspection of Figure 8, the model
hydrographs and actual stream discharge hydrographs for basins
1 and 3 seem very similar. The model produces peak discharges
very close to observed peaks for the two basins - except for the
first day. Likewise, total model flow volumes compare
favorably with actual flow volumes - with a difference of only
1.91% for basin 1 (see table 4). The simulated discharge for basin 1
also appears to match observed discharge in the time of peak flow.
On the other hand, the simulated discharge for basin 3 shows a time
of peak arriving slightly after the observed peak flow time (about 3
hours after). Also, on the first day, the observed discharge for basins
l and 3 appear truncated. Despite these differences,

model discharge and simulated discharge are remarkably similar

24

25
for basins 1 and 3.

Table 4. Total Stream Discharge
during the Study

 

Actual Modeled
Basin Volume Volume Absolute Percent
Number Discharge Discharge Difference Difference
1 1215626 1238862 23236 1.91
2 86665 162923 76259 87.99
3 788952 1013864 224912 28.51

Moreover, major differences between simulated and observed
stream flow, for basins 1 and 3, can be explained. For
instance, the apparent difference in basin lag (time of peak
flow) between the Lag-and-Route model and actual flow for
basin 3 can be explained in terms of a "short-circuit" of the
drainage system. Drainage paths on the surface of the glacier
were assumed to parallel surface slope (see figure 3). Marginal
channels collected drainage from surface flow directed to the side of
the glacier and ultimately discharged a total flow at each subbasin
outlet. However, it was observed, from aerial photographs, that a
submarginai chute existed about two thirds of the way down along
the glacier margin from the ice divide. This chute served to divert
water flow - so that it could flow straight to the stream outlet. This
would tend to decrease the average flow time for water draining
from the surface of basin 3 - explaining the earlier time of peak flow
for actual stream discharge.

Likewise, the difference in peak discharge between the

simulated flow and observed flow on the first day, for basins 1 and

26

3, can also be explained. The truncation of the actual discharge
hydrograph for that day is probably a result of some ephemeral
variation in atmospheric conditions. Larson (1978) noted that the
afternoon of the first day of the study was somewhat cloudy and
hazy - a temporary period of cloudiness, or fog, could account for a
sharp drop in radiation input to the glacial surface. And this
variation in radiation receipts could explain the variation in stream
discharge. The calculated discharge would not show this variation
because certain key atmospheric variables used in the melt model
were daily averages.

Unlike basin 1 and 3, simulated discharge for basin 2 differs
drastically from the observed discharge. Modeled peak flow values
for the basin are generally only two thirds the value for actual peak
flows. Also, total model flow volumes are only about one half that
for actual flow volumes for the basin. And the times of peak flow
for the model arrive around 4 hours sooner than the actual peak
flows.

Yet, if basin characteristics are taken into account, the
results for basin 2 can be explained. Basin 2 is composed
chiefly of fractured, and highly weathered, ice. The ice in
this zone consists of loose ice crystals in a slushy matrix. This is a
highly porous and permeable environment. So much of the water
generated at the surface is absorbed into the body of the glacier.
Basins 1 and 3, on the other hand, consist of clean (unweathered) ice
with no porous covering of snow or detrital material and are

characterized as impermeable environments. Relatively little of the

27

meltwater generated at the surface in such environments would be
lost to infiltration.

In summation, discharge generated from the Lag-and-Route
model appears to be a good approximation for stream discharge
draining basins 1 and 3. Model discharge for basin 1 is a
very good match of actual stream flow for the points of
comparison (peak flow, flow volume, and time of peak flow).
Model peak flow, and flow volume, were also very close to
actual values for basin 3. One point of difference between
model discharge and actual discharge, for basin 3, was time
of peak flow - and that could be explained as a result of an

irregularity in the basin.

VI nci ions

The lag-and-route method is a surface runoff model.

Stream discharge calculated, using this model, is composed
purely of runoff. Observed stream discharge, from subbasins

1 and 3, is equivalent to the discharge produced by the
lag-and-route model. Therefore, stream flow from basins 1 and
3 must be wholly a result of surface runoff. If it is further

noted that basins 1 and 3 are composed of clean (unweathered)
glacial ice, and that meltwater drainage from basins l and 3

is accomplished by surface runoff, then it should be accurate

to say that meltwater drainage from clean glacial ice must

occur through surface runoff. In other words, surface runoff is the
major mechanism driving meltwater drainage from unweathered
glacial ice.

A knowledge of this mechanism is of importance in many
different ways. Natural resources in "marginal" areas will have to be
utilized to meet the needs of growing human populations. Some
knowledge of glacial hydrologic characteristics will be necessary as
human settlement begins to invade glaciated regions. For example,
hydrologists must be able to predict the probability, and magnitude,
of flooding so that planners/developers/engineers can determine the

feasibility of site construction. In glaciated basins, a knowledge of

28

29

how water drains from glacial ice, and how to model that drainage,
would be indispensable in flood prediction.

Another area in which glaciated basins are being exploited is in
energy production. Hydroelectric projects are being implemented to
take advantage of the tremendous amount of meltwater generated
within some glaciated basins. These projects require a detailed
analysis of discharge volume, and variation through time, before a
site can even be considered. As before, only with a good model of
glacial meltwater drainage can a planner truly take the best
advantage of the resources available to him. The lag-and-route
model, as presented in this study, provides a good approximation.

Surface runoff models, such as the lag-and-route method, can
be applied to many different problems concerning glacial hydrology.
The lag-and-route model could be used to calculate total daily, and
seasonal, flow volumes - which could be used for sizing a reservoir,
turbine, and/or generator for a potential hydroelectric site. This
model could also be used to show daily, or seasonal, variation in
discharge rate, for a specified glacier, so that the manager of a
hydroelectric site could manipulate reservoir discharge in such a way
as to maintain the smallest variation in hydraulic head while
meltwater inflow to the hydroelectric reservoir varies.

The lag-and-route model could also be used to determine the
condition of glacial ice. Simulated meltwater discharge, for a glacier,
could be calculated and compared to observed discharge. If the two
hydrographs compare favorably, then it could be inferred that the
glacial ice is clean and relatively unweathered; if the hydrographs do

not match, then the glacial ice must be somewhat weathered and/or

3O

disintegrated. Similarly, from comparing model and observed
hydrographs, for a snow-covered glacier, it should be possible to
determine the effects of the snow cover upon meltwater drainage.
However, additional work needs to be done in the area of modeling
meltwater drainage through snow and firn. Some sort of porous
media flow model could be used to determine the hydraulic

characteristics of snow.

APPENDICES

APPENDIX A. ICE MELT CALCULATIONS

Appendix A

Ice Melt Calculations
1. Factors Affecting Melt.

a. Solar Constant. The sun is approximately 150 million km
from the earth. Because of its high surface temperatures (about
6000 K), radiant energy released is high (by Stefan's law). The ray
paths of solar radiation diverge as they travel away from the sun, so
radiation intensity decreases as the inverse of the square of the
distance from the sun. The earth intercepts only one two-billionth of
the sun's total energy output (Strahler & Strahler, 1979). The
average rate of incoming solar shortwave radiation, at the top of the
earth's atmosphere, is known as the solar constant and is around
1.94 calories per square centimeter per minute.

b. Atmospheric Conditions. Radiation received at the earth's
surface must first be filtered through the atmosphere. The radiation
intensity at the surface depends, a great deal, upon atmospheric
conditions. Overcast conditions may cut down on the amount of
radiation received at the surface by either reflecting it back out into
space or absorbing it.. Atmospheric moisture content, dust content,
or C02 content will also determine the relative amount of radiation
recieved at the surface. The higher the content the greater the
radiation absorption within the atmosphere. One measure of this
condition is the atmospheric transmissivity.

c. Lattitude. Shortwave radiation intensity varies inversely

with lattitude - with upper lattitudes receiving the least amount of

31

32

radiation. Assuming a planar wave front, ray paths are
perpendicular to the surface near the equator while the angle of
incidence decreases with lattitude. The area over which the
radiation between two rays is incident is greater as the angle of
incidence increases. The same amount of radiation is expended over
a larger area, with greater lattitude, so the intensity decreases.

(1. Time of Day. Radiation intensity increases from dawn to
solar noon and then decreases from solar noon to dusk. The
shortwave angle of incidence is relatively low early and late in the
day. This angle is greatest at solar noon, when the area between
rays is least and the radiation intensity is greatest. Radiation
intensity would then vary as the cosine of the hour angle ( which is
90 degrees at solar noon and 0 degrees at dawn and dusk). Times
for dawn, solar noon, and dusk varied relatively little during the
study period so one value for each was used for the entire period.
Expressed in minutes from midnight, the value for dawn, solar noon,
and dusk were: 450; 870; and, 1290. From these values, an hour
angle can be calculated for any time between dawn and dusk using a
simple linear equation.

Hourang = abs((noon-time)/(noon-dawn))*90)

e. Day of Year. Because the earth's axis of rotation is tilted, the
point where solar shortwave radiation rays intercepts the earth's
surfact at right angles will vary according to which hemisphere is
titlted towards the sun. The solar declination is the lattitude angle at
which the solar rays impinge perpendicular to the earth's surface

(Kondratyev, 1969). For the northern hemisphere, the maximum

33

solar declination occurs during the summer solstice (June 21) and the
minimum occurs during the winter solstice (December 22).
Therefore, radiation intensity increases with increasing declination
for the northern hemishpere, and it decreases with decreasing solar
declination for'the southern hemisphere. Calculations were made for
both solar declination and radius vector of the earth's orbit for the
four days of the study.

f. Surface Slope/Azimuth. Surface slope and orientation has a
great deal of influence upon the amount of shortwave radiation
received upon that surface. Obviously, a southerly slope would
receive much more solar radiation than a northerly facing slope.
Also, the angle of the slope with respect to the incoming solar rays
will also have an effect - a surface more perpendicular to these rays
will receive a greater radiation intensity. Some term must be
included into the generalized law of radiation transmission (Id = pm).
This term (Garnier and Ohmura, 1968; Williams, Barry & Andrews,
1972) may be defined as a complex function of lattitude (phi, hour
angle (H), azimuth (A), zenith angle (zx), and declination (delta) - and
is expressed as the angle between the sloping surface and the
incoming ray vectors (XS).

cos(XS)=((sin(phi)cos(H))(-cos(A)sin(zx)) -sin(H)(sin(A)sin(zx))

+ (cos(phi)cos(H))cos(zx))cos(delta)
+(cos(phi)(cos(A)sin(zx))

+ sin(phi)cos(zx))sin(delta)

34

The cosine of the angle XS is mutiplied by the radiation transmission
equation to get the radiation received on the sloping surface.

g. Surface Albedo. Albedo is a measure of the percentage of
radiation that is reflected back off of the surface upon which it is
received. Rosenberg (1974), Kondratyev (1969), and many others,
have described and measured the reflective abilities of many
different materials. Bolsenga ( 1978 ) described the variation in
reflectivity of ice throughout the day and noticed that there was less
than a ten percent variation. The amount of radiation actually
absorbed at the surface is calculated as the amount left over after

reflectance (1 - albedo).

2. Melt Equations

a. Direct Solar Radiation. Direct solar radiation is that which
falls directly on the earth's surface without being diffused through or
being reflected off of something else. This may be approximated by
an equation developed by Garnier and Ohmura (1968), transformed
by Williams, Barry, and Andrews (1972), and applied by Larson

(1978). The equation has the form:

10 n

Ih=-—2- 1(le mk f(Hk) H

Where:

10: solar constant = 1.94 cal/cm/min

r = radius vector of the earth's orbit
= mean-zenith-path transmissivity

P
m = optical air mass

3 5
H = time step
f(Hk) = C4cos(H-Y) + C3

Hk = hour angle measured from solar noon

Y = arctan(C1/C2)
C1 = sinA sin(q ) cos(delta)
= (cos(f)cos(q )-sin(f)cos(A)sin(q ))
= (sin(f)cos(q )+cos(f)cos(A)sin(q ))sin(d)

C4 = (Cl**2 + C2**2)**0.5

A = slope azimuth

q = theta = slope angle

f = lattitude

d = declination of the sun
Here, 1h stands for the hourly total radiation input to the surface.

b. Diffuse Radiation. Diffuse radiation consists of that radiation
incident to a surface originating from radiation scattered in the
atmosphere. Diffuse radiation is that portion of the radiation that
permeates a surface when it is shaded from direct sunlight.
Equations were also deve10ped (List, 1966; Garnier and Ohmura,
1968; Williams et a1, 1972) to approximate diffuse radiation inputs.

This equation has the form:

I() n
Do=0 .5 —°-2—k 21 (.091-pmk) cos(Zs) H

where:

Zs = zenith angle

10: solar constant 2 1.94 cal/cm/min

r = radius vector of the earth's orbit

36

p = mean-zenith-path transmissivity

m = optical air mass

H = time step

cos(ZS) = cos(dx)cos(fx)cos(H) + sin(dx)sin(f)
H = hour angle measured from solar noon
f = lattitude

d = declination of the sun

The variables for this equation are the same as for the previous
except for the zenith angle. Here, too, the time interval chosen was

30 minutes for the hourly diffuse radiation inputs to the surface.

c. Total Radiation at Surface. The total shortwave radiation
received at the earth's surface is the sum of the direct and diffuse
radiation inputs. The amount actually gained by the surface of the
glacial ice is this sum of shortwave radiation minus that percentage
that is reflected back up to the atmosphere. Given a reflectivity
value (r), the total radiation gain would be the total incident

radiation multiplied by the absorptivity (a=1-r).
Igain = It = (1h - Do)(l-r)

APPENDIX B. INPUT VALUES FOR MELT ROUTINE

APPENDIX B

Input Values for Melt Routine

The melt program requires the use of two different kinds of
data: data that changes value temporily; and, data that changes value
areally.

Certain input data pertain to every point on the glacial surface
but not to every day of the study period (they vary in value from
one day to the next) - these variables include: atmospheric
transmissivity; declination; and, radius vector. Table 8.1 displays

the values for these variables for the four days of the study.

Tle
i1Vri1D Vle

 

Variabl§\Da3L 1 2 3 4
atmospheric
transmissivity 0.51 0.67 0.64 0.71
solar

declination 14.6085 14.30 13.9915 13.6830

radius
vector 1.012995 1.012810 1.012625 1.012440

Referring to the grid system for the glacier, slope and slope
azimuth was calculated for each point on the grid. This was done by

statistically fitting a linear regression plane through every set of nine

37

38

points in a three by three matrix. At some points, which lay along
the margin of the glacier, the SIOpe and slope azimuth were
approximated by fitting a plane to the point in question and any two
or more adjacent points on the ice. Figures 8.1 and B.2 show the
variation of slope and slope azimuth over the surface of Burroughs
Glacier.

The reflectivity (or albedo) of the glacier surface was measured
during the early afternoon of August 15, 1973 by G. Larson. Total
incoming solar radiation was measured at each of 17 points at the ice
surface using an horizontally mounted Eppley pyranometer.
Reflected radiation was measured at each of these stations by
inverting the pyranometer and taking measurements. The albedo
was calculated for each point by dividing the amount of reflected
radiation by the amount of total incoming radiation. Albedo
variation over the glacier surface could then be determined by
assuming a linear variation between each pair of points on the

surface. Figure B.3 shows this variation in albedo for the glacier.

39

m 82m:

 

 

 

 

mm H Ufiw mmeOmmDm
mmoq m muflgm

. 8988 5.. «@on

 

 

 

 

 

 

mmH 03w mmeOmmDm
mBDZH Nd. modmem

 

 

41

2 893m

 

 

    
 

 

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8??

L
‘ ‘w.

  

MNKEKm Op Mb\va\m

MN: 0440 mmeOMEDm
Gama HUANMmDm

mos—I

down? now 33: oz

 

 

APPENDIX C. PROGRAM FOR CALCULATING HOUR ANGLES

APPENDIX C

Program to Calculate Hour Angles

data hangle;
dawn=450;
dusk=1290;
noon=870:
conrad=0.0174532;
do tim=465 to 1275 by 30;
hrang=abs(((noon-tim)/(noon-dawn))*90);
hrang=hrang*conrad;
output;
end;
proc print data=hangle;
run;

42

APPENDIX D. PROGRAM FOR CALCULATING RADIATION RECEIPTS
AND MELT

APPENDIX D

Program to Calculate Radiation Receipts and Melt

CMS FILEDEF INDATA DISK RAW] DAT A;
CMS FILEDEF OUTDAT A DISK MLlRD45 DAT A;
data melrad (keepzx y gridrad gridmel);
infile indata;
input x y slope az albedo;
file outdata;
HOURANG=1 .065 89;
conrad=0.0174532;
AR=1.012440;
PE=0.710;
DEC=13.6830*CONRAD;
lat=58.9666*conrad;
slope=slope*conrad;
az=az*conrad;
solcon=1 .94;
tint=30;
gridrad=0.0;
opairms=1.0/(cos(dec)*cos(lat)*cos(hourang)+sin(dec)*sin(1at));
if opairms 1e 0.0 then go to flag];
c1=-sin(az)*sin(slope)*cos(dec);
c2=(cos(lat)*cos(slope)-sin(lat)*cos(az)*sin(slope))*cos(dec);
c3=(sin(lat)*cos(slope)+cos(lat)*cos(az)*sin(slope))*sin(dec);
c5=cl/c2;
y2=atan(c5);
c4=sqrt(c1 **2+c2**2);
fh=c4*cos(hourang-y2)+c3;
cosz=1lopairms;
ir=(solcon/ar**2)*fh*tint*(pe**opairms);
d=0.5*(0.91pe**opairms)*cosz*tint*((cos(slope/2.0))**2)
*solcon/(ar**2);
if fh le 0.0 then go to flag];
gridrad=((100-albedo)/100)*(ir+d);
flagl: gridmel=gridrad/79.720;
put x 1-5 y 6-10 gridrad 11-20 .5 gridmel 21-30 .6;
run;

43

APPENDIX E. PROGRAM FOR CALCULATING LUMPED LAGGED FLOW

APPENDIX E

Program to Calculate Lumped Lagged Flow

CMS FILEDEF INDATA DISK ML3RD414 DAT A1;
CMS FILEDEF OUTDATA DISK FLO3414 DAT A1;
DATA ONE (KEEPzX Y RAD MEL);

RETAIN FLAG 0;

do;

infile indata;
INPUT X 1-5 Y 6-10 RAD 11-20 MEL 21-30;

IFNOT((X BO 6 AND Y EQ 24) OR (X EQ 7 AND (Y BO 22 OR Y BO 23
OR Y EQ 24)) OR (X BO 8 AND (Y EQ 20 OR Y EQ 23 OR Y BO 24))
OR(XEQ9AND(YEQ24ORYEQ25))OR(XEQ10AND
YEQ 18) OR (XEQ 11 AND(YEQ 17 ORYEQ 18)) OR
(XEQ 12AND(YEQ 15 ORYEQ l6ORYEQ 17 ORYEQ 18))
OR (X EQ13 AND (Y BO 15 ORY EQ16 OR Y EQ17 OR Y EQ18))
OR (X EQ 14 AND (Y EQ16 OR Y EQ17 OR Y EQ 18)) OR
(XEQ 15AND(YEQ l7ORYEQ 18))OR(XEQ l6AND
(Y BO 17 OR Y EQ 18)) OR (X BO 29 AND Y BO 50) OR
(X EQ 39 AND Y EQ 29)) THEN DO;

IFXEQ 37ANDYEQ38ANDFLAGEQOTHENDO;
FLAG+1;
X=34;
END;
OUTPUT;
END;
end;
run;
CMS FILEDEF INDATA DISK ABSTIMB DAT A1;
data two;
do;
infile indata;
input x 1-5 y 6-10 reltim 11-20 abstim 21-30;
end;
run;
PROC SORT DATA=ONE;
BYXK
RUN;
PROC SORT DATA=TWO;
BY X Y;

44

4 5
RUN;
data three;
do;
merge one two;
BY X Y;
end;
run;
DATA FOUR;
set three;

RETAIN AREAOS AREAIO AREA15 AREA20 AREA25 AREA30
AREA35 AREA4O AREA45 AREA50 AREA55 AREA6O AREA65
AREA7O AREA75 FLOWOS FLOWIO FLOW15 FLOW20 FLOW25
FLOW3O FLOW35 FLOW4O FLOW45 FLOWSO FLOW55 FLOW6O
FLOW65 FLOW7O FLOW75 COUNTOS COUNTIO COUNTIS COUNTZO
COUNT25 COUNT30 COUNT35 COUNT4O COUNT45 COUNTSO

COUNTSS COUNT60 COUNT65 COUNT70 COUNT75 FLAG 0.0;
do;

flag+1;

IF (ABSTIM GT 7.0) AND (ABSTIM LE 7.5) THEN DO;
COUNT75=COUNT75+1 ;

AREA75=AREA75+MEL;

end;

else

IF (ABSTIM GT 6.5) AND (ABSTIM LE 7.0) THEN DO;
COUNT70=COUNT70+1;

AREA70=AREA70+MEL;

end;

else

IF (ABSTIM GT 6.0) AND (ABSTIM LE 6.5) THEN DO;
COUNT65=COUNT65+1 ;

AREA65=AREA65+MEL;
end;
else

IF (ABSTIM GT 5.5) AND (ABSTIM LE 6.0) THEN DO;
COUNT60=COUNT60+1 ;

AREA60=AREA60+MEL;
end;
else

IF (ABSTIM GT 5.0) AND (ABSTIM LE 5.5) THEN DO;
COUNT55=COUNT55+1 ;

AREA55=AREA55+MEL;
end;
else

46

IF (ABSTIM GT 4.5) AND (ABSTIM LE 5.0) THEN DO;
COUNT50=COUNT50+1 ;
AREA50=AREA50+MEL;
end;
else
IF (ABSTIM GT 4.0) AND (ABSTIM LE 4.5) THEN DO;
COUNT45 =COUNT45 +1 ;
AREA45=AREA45+MEL;
end;
else
IF (ABSTIM GT 3.5) AND (ABSTIM LE 4.0) THEN DO;
COUNT40=COUNT40+1;
AREA40=AREA40+MEL;
END;
ELSE
IF (ABSTIM GT 3.0) AND (ABSTIM LE 3.5) THEN DO;
COUNT35=COUNT35+1;
AREA35=AREA35+MEL;
END;
ELSE
IF (ABSTIM GT 2.5) AND (ABSTIM LE 3.0) THEN DO;
COUNT30=COUNT30+1;
AREA30=AREA30+MEL;
END;
ELSE
IF (ABSTIM GT 2.0) AND (ABSTIM LE 2.5) THEN DO;
COUNT25=COUNT25+1;
AREA25=AREA25+MEL;
END;
ELSE
IF (ABSTIM GT 1.5) AND (ABSTIM LE 2.0) THEN DO;
COUNT20=COUNT20+1;
AREA20=AREA20+MEL;
END;
ELSE
IF (ABSTIM GT 1.0) AND (ABSTIM LE 1.5) THEN DO;
COUNT15=COUNT15+1;
AREA15=AREA15+MEL;
END;
ELSE ,
IF (ABSTIM GT 0.5) AND (ABSTIM LE 1.0) THEN DO;
COUNT10=COUNT10+1;
AREA10=AREA10+MEL;

4 7
END;
ELS
DO;
COUNT05=COUNTOS+ 1;
AREA05=AREA05+MEL;
END;

IF FLAG BO 489 THEN DO;
flow75=((area75/count75)/100.0)*(count75*10000.0)[1800.0;
flow70=((area70/count70)/l00.0)*(count70*10000.0)/1800.0;
flow65=((area65/count65)/100.0)*(count65*10000.0)/1800.0;
flow60=((area60/count60)/100.0)*(count60*10000.0)[1800.0;
flow55=((area55/count55)/100.0)*(count55*10000.0)[1800.0;
flow50=((area50/count50)/100.0)*(count50*10000.0)[1800.0;
flow45=((area45/count45)/100.0)*(count45*10000.0)ll800.0;
flow40=((area40/count40)/100.0)*(count40*10000.0)/1800.0;
flow35=((area35/count35)/100.0)*(count35*10000.0)[1800.0;
flow30=((area30/count30)/100.0)*(count30*10000.0)/1800.0;
flow25=((area25/count25)/100.0)*(count25*10000.0)ll800.0;

flow20=((area20/count20)/100.0)*(count20* 10000.0)ll 800.0;

flowl5=((area15/count15)]100.0)*(count15*10000.0)/1800.0;
flow10=((areal0/count10)/100.0)*(count10*10000.0)[1800.0;
flow05=((area05/count05)/100.0)*(count05*10000.0)I1800.0;
file outdata;
PUT FLOW75 1-10
PUT FLOW70 1-10
PUT FLOW65 1-10
PUT FLOW60 1-10 .
PUT FLOW55 1-10 .
PUT FLOWSO 1-10 .
PUT FLOW45 1-10 .
PUT FLOW40 1-10 .
PUT FLOW35 1-10 .
PUT FLOW30 1-10 .
PUT FLOW25 1-10 .
PUT FLOW20 1-10 .
PUT FLOW15 1-10 .
PUT FLOWIO 1-10 .
PUT FLOWOS 1-10 .

end;
end;
run;

C U U C U 0

eqeeeeeaeeeoeee

U

APPENDIX F. PROGRAM FOR CALCULATING DISTRIBUTED LAGGED
FLOW

APPENDIX F

Program to Calculate Distributed Lagged Flow

CMS FILEDEF INDATZ DISK ABSTIM2 DAT A;
CMS FILEDEF lNDATl DISK RAWZ DAT A;
CMS FILEDEF OUTDATA DISK ND30FL2 DAT A;
DATA _null_;
II), ,
ARRAY MELT(190) Ml-M 190;
RETAIN M1-M190 KOUNTl 0.0;
KOUNT1+1;
conrad=0.0174532;
INFILE INDATI;
INPUT X Y SLOPE AZ ALBEDO;
SLOPE=SLOPE*CONRAD;
AZ=AZ*CONRAD;
INFILE INDAT2;
INPUT X Y RELTIM FLOTIM;
FLOTIM=FLOTIM*2.0;
FLOTIM=ROUND(FLOTIM);
dawn=450;
dusk=1290;
noon=870;
lat=58.9666*conrad;
solcon=l .94;
TINT=30;
DO DAY=1TO 4 BY 1;
IF DAY EQ 1 THEN DO;
ar=1 .012995;
pe=0.5 10;
dec=14.6085*conrad;
END;
IF DAY BO 2 THEN DO;
AR=1.012810;
PE=0.670;
DEC=14.30*CONRAD;
END;
IF DAY BO 3 THEN DO;
AR=1.012625;
PE=0.640;

48

4 9

DEC=13.9915*CONRAD;

END;

IF DAY EQ 4 THEN DO;

AR=1.012440;
PE=O.710;
DEC=13.6830*CONRAD;
END;
DO TIM=465 TO 1275 BY 30;
HOURANG=ABS(((NOON-T1M)/(NOON-DAWN))*90);
HOURANG=HOURANG*CONRAD;
gridrad=0.0;
opairms=l.0/(cos(dec)*cos(lat)*cos(hourang)+sin(dec)*sin(lat));
if opairms le 0.0 then go to flagl:
C1=-SIN(AZ)*SIN(SLOPE)*COSCDEC);
C2=(COS(LAT)*COS(SLOPE)
-SIN(LAT)*COS(AZ)*SIN(SLOPE))*COS(DEC);
C3=(SIN(LAT)*COS(SLOPE)
+COS(LAT)*COS(AZ)*SIN(SLOPE))*SIN(DEC);
c5=c1/c2;
y2=atan(c5);
c4=sqrt(c1 **2+c2* *2);
fh=c4*cos(hourang-y2)+c3;
cosz=1lopairms;
ir=(solcon/ar**2)*fh*tint*(pe**opairms);
D=0.5*(0.91-PE**OPAIRMS)*COSZ*TINT
*((COS(SLOPE/2.0))**2)*SOLCON/(AR**2);
if fh 1e 0.0 then go to flagl;

GRIDRAD=((lOO-ALBEDO)/100)*(IR+D);

FLAGl: GRIDMEL=GRIDRADI79.720;
KOUN=(((TIM-465+30)/30)+((DAY-1)*48))+FLOTIM;
MELT(KOUN)=MELT(KOUN)+GRIDMEL*(10000/100)/( 1800);

END;
END;
IF KOUNTI BO 73 THEN DO;
FILE OUTDATA;
DO KOUN=1TO 190 BY 1;
PUT MELT(KOUN) 1-20 .6;
END;
end;
HID.
RUN;

APPENDIX G. PROGRAM FOR ROUTING FLOW

APPENDIX G

Program to Calculate Routed Flow

CMS FILEDEF INDAT DISK FLO3TOT DAT A;
CMS FILEDEF OUTDAT DISK ROUTFL3 DAT A:
DATA TEMP;
III
ARRAY OUT(199) 01-0199;
ARRAY IN(199) 11-1199;
T=0.50;
K=3.21;
CO=(0.5 *T)/(K+0.5*T);
C1=CO;
C2=(K-0.5*T)/(K+0.5*T);
INFILE INDAT;
FILE OUTDAT;
INPUT IN( 1);
OUT(1)=0.0;
DO KOUNT=2 TO 199 BY 1;
INPUT IN(KOUNT);
OUT(KOUNT)=CO*IN(KOUNT)+C1 *IN(KOUNT-1)+C2*OUT(KOUNT- 1);
PUT OUT(KOUNT) 1-20 .6;
END;
END.

50

BIBLIOGRAPHY

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Drewry, David "Glacial Geologic Processes", Edward Arnold
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