USSR case studies: water supply

A.N. KRENKE & V.M. KOTLYÂKOV Institute of Geography Academy of Sciences Starmonetny per 29 Moscow 109017, USSR

COMPUTING LONG-TERM AVERAGES OF ANNUAL VALUES OF RUN-OFF IN GLACIER SYSTEMS FROM AIR TEMPERATURE AT THE HEIGHT OF THE EQUILIBRIUM LINE (Krenke, 1973, 1982) (Model A)

Information on the height of the equilibrium line on glaciers zequ *s selected from the glacier inventory. We assemble the data on the firn line, on the limit of superimposed ice, on the mean perennial heights of snow line, or on the maximum height of the latter in the years with mass-balance indices, b, close to the mean perennial values:

b! = Ps - (9.5 + T s )3 (1)

or

= p - p T - T

£>2 _A__s _ s s (2)

s s

where Ps is the solid precipitation in the year in question, Ps

its long-term mean value± Ts is mean summer air-temperature in the particular year and Ts its long-term mean.

The altitude of the equilibrium line can be calculated from the glacier morphology by the Kurovsky method, or simply as the average height of its terminus and upper reaches.

Glaciers are subdivided into groups, including those of different morphology and exposure and the area-weighted average height of the equilibrium line Z e q u is computed for each group. The number of glaciers in each group is: 5 to 6 glaciers in Siberia and the Arctic and 10 to 15 glaciers in the mountains in the southern USSR. The map of contour lines of the height of the equilibrium line on glaciers is obtained from such data for the groups of glaciers.

The long-term normal summer temperatures of the air at all the stations, reduced to a certain fixed altitude, Ts r e cj, are calculated from the data on the vertical temperature gradient y averaged for a summer. The map of contour lines Ts recj is compiled from these data.

The mean summer temperature at the height of the equilibrium line Ts eqU is calculated with the help of the vertical temperature gradient for each group of glaciers on the map. The value Ts e q u may also be determined from the regular network of

81

82 A.N. Kcenke & V.M. Kotlyakov

points by way of superposition of maps of the equilibrium line altitude and adjusted temperature of the air Ts r e cj:

T = T , + Y (Z - Z ,) - AT + AT (3) s equ s red ' equ red s - ex

where Zre(j is the altitude to which air temperature is reduced and ATS is the correction for the cooling effect of glaciers on the air. It increases with increasing glacier area and increase in air temperature:

AT = f(T , F) (4) s s equ

The type and graph (nomograph) of the function (4) are determined by analogy from the field data in the region or ATS is simply assumed to be 1-1.5° for the groups with small glaciers or 2-2.5° for the groups with large glaciers. The meaning of ATex is given below; F is glacier area.

The long-time averaged seasonal melting at the height of the equilibrium line (MeqU, mm) is calculated by the empirical formulae:

M = (T + 9.5)3 (5) equ s equ

or specifically for the height ZeqU

- - 7 AS M = 1.33 (T + 9.66) (6) equ s equ T is the June to August average.

When deriving the formulae (5) and (6) we used observations on glaciers, among which 60% have N, NE and E exposures favourable for their existence, 10% have unfavourable S, SW and W exposures and 30% have intermediate MW and SE exposures. A correction ATex

for the discrepancy between the distribution of glaciers with different exposures in each group and the above-mentioned distribution of glaciers is introduced into the computations by formula (3). This correction is made with due regard to the difference in the height of the equilibrium line of glaciers of various exposures, which is transformed into ATex with the help of the vertical gradient of air temperature.

Under the assumption of the stationarity of glaciers and linearity of the distribution of the values M(Z) • F(Z) together with height Z, the total average ablation on the glacier A is taken, equal to the total ablation at the altitude of the equilibrium line:

« " Mequ (7>

and the volume of the total ablation or the glacial run-off Q is determined from:

Q = M • F (8) * equ

USSR case studies: water supply 83

The equality (8) is confirmed by the measurements of ablation distribution on glaciers.

The map of isolines of the layer of glacial run-off is plotted from the values of M for the groups of glaciers. Proceeding from this map, the glacial run-off for each river basin Qj-> can be determined from

% - Fb * "b (9)

where Fj-, is the glacierized area of the basin,

- * Si • fi

M. = 1 1 (10) b Fb

Mj is the layer of glacial run-off between contour lines, and fj is the area of glacierization between them.

The model described has been implemented for the USSR and at present is being implemented for the majority of glacierized regions of the world in the World Atlas of Snow and Ice Resources.

Modification of this model is achieved by computation of melting by the two-factor formula (Khodakov, 1978), replacing equation (6) with:

M = (T tl.3vf t 4.0)3 (11) equ s equ s

where Rs is the short-wave balance for June to August in kcal/cm^. On the basis of field data it is assumed that

Rs = 0.32 R (12)

where R is the total radiation for a summer at the stations situated in non-glacierized areas, varying over the territory of the USSR from 32 to 62 kcal/cm^. The empirical coefficient of 0.32 accounts for the albedo of melting firn (0.6), the increase of cloudiness and the increase of the total radiation of the clear sky with height as well as exposure averaged for the glaciers. This modification has been carried out for 50 groups of glaciers over the territory of the USSR.

Another modification of computation without special determination of y and ATS, presupposes the use of empirical equations, proposed by Lebedeva (1963)

r _ = <P [(Z M | - Z , ) , To .] or <p [(Zonil - Z . ) , Ï e.] (13) s equ equ red s red equ st s st

where Zst and Ts st are the altitude of the nearest weather station and the mean summer temperature there. The type of function (p is determined for large climatic regions. This modification has been performed for the glaciers of Pamir-Alai and Central Asia.

For the computations of the glacial runoff of a particular year the mean annual altitude of the equilibrium line is replaced by its altitude in the year in question; this can be determined from satellite images, and the long-term mean temperature of the air is replaced by the air temperature in this year. The other operations

84 A.N. Kcenke & V.M. KotJyakov

remain the same. This modified procedure using information from space is now undertaken for the Pamirs and Karakoram.

COMPUTING GLACIAL RUNOFF HYDROGRAPHS IN A LARGE RIVER (Abalyan, Mukhin, Polunin, 1980) (Model B)

The mean altitude of the equilibrium line ZeqU is determined for the basins of separate tributaries of the second and third order according to the Glacier Inventory.

The summation of annual positive air temperatures £(+T°), averaged for a period of about ten years, is found for this height of the equilibrium line from the data of the nearest weather stations and with regard to the vertical temperature gradient y.

The mean perennial value of ablation at the height of the equilibrium line MeqU is determined from the snow melting coefficient MRS, assumed to be 5 mm/°C:

*L™. = MR-£(+T°) (14) equ s u

According to the definition of the equilibrium line

"equ = *equ ( 1 5 )

where ÂeqU is the long-term averaged accumulation of snow (in water equivalent) at the altitude of the equilibrium line.

For a particular year accumulation at the height of the equilibrium line Aequ is determined from the total precipitation for October to May at 8 points, according to:

P8 A = A . — (16) equ equ -

*8 where Pg is the long-term normal amount of precipitation, and Pg - the amount of precipitation in this particular year for the above-mentioned months at eight weather stations.

It is assumed that the value of accumulation over the whole glacier equals accumulation at the height of the equilibrium line

A = Aequ (17)

The sums of positive temperatures for time intervals (one day, five days, ten days, one month) are calculated from the vertical gradient of the air temperature for all the altitudinal zones. On the basis of these sums and with regard to the altitudinal distribution of glacier area in the basin F(Z) snow melting M(T,Z) is calculated.

The calculation is carried out on condition that M < A. Then the calculation is continued with the ice melting coefficient MRj taken to be 8 mm/°C:

l+l h q(t) = MR • J J F(z) • [X(+T°)(z)]'dz-dt +

S t. Z , i ssl

USSR case studies: water supply 85

MR. J Jssl

J Z

F(z)'[£(+T°)(z)]«dz«dt (18)

where Zss\ is the altitude of the seasonal snow line moving up the glacier, Zj, and Zt are altitudes of the glaciers' upper heads and termini, q is the income of water to the glacier and to the runoff at the time interval from tj to t}+i, and t is time.

The model has been applied to the Vakhsh River run-off (Tutkaul gauge) with a basin area of 31 200 km*S nourishing the Nurek water reservoir for 23 years. An example of the model application for 1956 is shown in Fig. 1, representing separately the total glacia] runoff and the runoff due to melting of the perennial ice. Good agreement between the calculated runoff and the runoff estimated by typical dividing of the hydrograph testifies to the applicability of the model despite essential assumptions, some of which are counterbalanced. Overestimation of snow storage on the glacier tongue, reducing the runoff, is compensated by the sum of positive temperatures because of the omission of the air cooling caused by the glacier.

E 800-

d

VI MONTHS

FIG. 1 Hydrograph of the Vakhsh River runoff in 1956 (Q m^lsec), including the share of glacial runoff (shaded) (According to Abafyan, Mukhin, Polunin, 1980)

A COMPUTATION MODEL FOR GLACIAL RUNOFF HYDROGRAPHS (Borovikova, Denisov, Trofimova, Shentsis, 1972) (Model C)

Time variations of precipitation at different heights on the glacier are calculated from the precipitation-height curve typical of the region and plotted from the data of weather stations,

86 A.N. Krenke & V.M. Kotlgakov

totalizers, expeditions and a periglacial weather station:

P(Z,t) = P(Z0,t).[l + K2 (Z-Z0) + K3 (Z-Z0)2] (19)

where t is time, P is precipitation, K2 and K3 are empirical coefficients. Precipitation is considered solid when the air temperature is below +2°C. The corresponding level is determined from the data of the same weather station, corrected for the difference in height and multiplied by the vertical temperature gradient. The annual variation of coefficients K 2 and K3 is taken into account.

Melting is calculated by equation (14), MRS and MRj are from experimental data (Ks = 3 - 5 mm/°C, Kj = 8 mm/°C), and temperature of the air is calculated from the gradient y, varying throughout a year.

Snow storage on the glacier is determined from the balance

Xps " ms X(+T°) (20)

where P s is solid precipitation. After snow melts out at the altitudinal belt under consideration, melting is computed by the ice coefficient Kj. The total input of water from the glacier is calculated by equation (18) adding liquid precipitation, when temperatures are above +2°C.

A,

L,..

L2"

-Cv

/

FIG. 2 Linear models of runoff transformation. The q denotes water income to the glacier, Q - glacial runoff, L - losses from the runoff on internal feeding, infiltration or evaporation on Aj - the transforming capacity (according to Borovikova et. al., 197Z).

Transformation of the water input to the glacier into the runoff is calculated according to one-element or two-element models (Fig. 2 ) . In case of the one-element linear model, the equation of water balance in the glacier, provided that the glacier runoff varies proportionally with the losses of liquid water in the glacier, has the form:

. <?2 dt + 0 = 3-q

(21)

where Q is the glacial run-off, k and p are empirical parameters reflecting the time taken for water to move through the glacier

USSR case studies: water supply 87

(the averaged lag time), and runoff coefficient (ratio of the glacial run-off to precipitation as water equivalent on the glacier).

For the two-element model with the serial-parallel operation the balance equation takes the form

kl ' k2 * T2 + (kl + k2 ) * ft + » " Pik2 * I + « W W ' « dt

where indices 1 and 2 denote emptying and loss parameters for the first and second capacities.

For the non-linear single-element model, the general form of relationship between the discharge Q and losses L with the storage of liquid waters in the glacier W is as follows:

Q = CjL . wn ancj L = CjL2 . w (23)

With n=2 the balance equation takes the form

jj£ + 2-/C1-Q3/2 + 2C12-Q - 2-/C1-q.Q

1/2 = 0 (24)

Solution of the linear differential equation (21) for the single-capacity linear model is as follows:

t Q(t) = 3 • J Pj • (t-x) • q(t) • dT (25)

T=0

where Pj is the influence function, t is the time from 0 to t

t~T

Pj'Ct-T) = |- • e Te (26)

t-T

0(t) = P • J • £- • e k • q(i).dT (27) t J

T = 0

For the two-element linear model:

t , 0(t) = J P2-(t-T) • [i^- k2- g (T) + (pi+p2-pi.p2).q(T)].dT (28)

t-T t-T

A.N. Krenke & V.H. Kotlyakov

300

240

• v — < ~

1

i

hi jr

1 A

••• ' V l

V

(a)

III IV MONTHS

E O 120

''"'""

(b)

III IV MONTHS

E

d 120

4/ N

(c)

III IV MONTHS

FIG. 3 Results of runoff computations from the Pskem river basin with different modifications of runoff transformation models: (a) linear one-element model with mean parameters ; (b) non-linear two-element model with mean parameters ; (c) non-linear two-element model with mean parameters corrected for the interval of ten days. 1. Measured discharge. 2. Computed discharge (from Borovikova et al., 1972).

USSR case studies: water supply 89

To determine the empirical parameters k, k|, k2> p, Pj, P2 the least-squares technique is used under conditions of balance equality:

| - V = - k.-Cft.-Q ) (30) P q 1 T o

where V is the volume of glacial run-off from the zero moment to the t moment, Vg is the volume of water income for the same time, Q0 and Qt are discharges of glacial run-off at the initial and final moments. The least-squares method is applied for the determination of k from the equation:

J (0-^ -q)2-dt = P-k . J q.f dt \ (Q2-Q2) (31) o q o

k and p are determined from the system of equations (30) and (31). Similarly empirical parameters for the more complicated non-linear and two-element models are determined.

All the model options were implemented for a number of river basins in Central Asia. Fig. 3 demonstrates such implementation for the Pskem River, the Karangitugai gauge with a basin area of 1584 km^ (110 km^ glacier covered). The agreement between the calculated and measured values is quite satisfactory. Excellent agreement is achieved only in case of every-ten-day or monthly adjustment of computations to the actual measurements of water discharge (Fig. 3c).

The modification of this model with the refinement of water income to the glacier has been obtained from the following relation:

M = a-£(+T°) + b-Rs + c (32)

where a, b, c are empirical coefficients, and Rs is a short-wave balance. For the computations of Rs the values of total radiation in case of clear sky are pre-assigned for the given latitude, altitude and time. Cloud cover is estimated from the nearest weather station; albedo is calculated from the ratio of melted snow to the initial snow storage and from its present values for the ice; consideration is also taken of areal distribution within the basin with regard to altitude, orientation, exposure and albedo.

GLACIAL RUNOFF MODEL INCORPORATING SPECIAL FEATURES OF GLACIER-GENERATED RUNOFF (Konovalov, 1979) (Model D)

The above-mentioned special features of glacial runoff formation, referring to water input - concentration of snow and cooling of the air - are integrally reflected in the height of the seasonal snow line (ssl) and the maximum height of snow line (equilibrium line) on the glacier. They are the computation entry in models A and B and are nearly neglected in model C. In model D, incorporating the computations of the water-discharge transformation by the glacier from model C, the computation of water input to the glacier is

90 A.N. Krenke & V.M. KotJyakov

improved by introducing the elevation of the highest and lowest points on the glacier and the elevation of the equilibrium line.

The position of ZeqU in the current year is determined from equation (2) at the nearest weather station and proceeds from the assumption that ZeqU is distributed in a long-term series as a normal Gaussian function and that the maximum values of this distribution coincide with the highest and lowest elevations on the glacier. In this case the computation pattern takes the form:

z equ = f ( b 2 )

Pj(Zequ) = f [P j (b 2 >]

Fj[P;j<Zequ> = f {Fj[Pjtt>2>]}

Zequ = z l + AZ.Pj(b2)

Zequ = Zh - AZ[1 - P j (b 2 >]

AZ = Zh - Z t

(33)

(34)

(35)

(36)

(37)

(38)

where Z^ is the height of the highest, and Zt is the height of lowest point of the glacier, Pj are sample probabilities, and Fj is a quartile of the valuated normal distribution function. Precipitation in expression for b2 (2) is taken for October to April, October to May or October to June, and air temperatures for May to September or June to September.

Snow accumulation and melting are calculated up to the moment of the exposure of the lower end of the glacier in the same manner as in the previous model. A certain coefficient of snow concentration on the glacier should have been introduced, however this was not done in the implemented models. Consequently, snow storage at the lower end of the glacier, as on non-glacierized slopes, is det2rmined by the balance method from the smoothed altitudinal relationships, as in model C. Further on, the data on the glacier morphology are used, introduced into the computations ZeqU and the model of the snow line motion is plotted, indicating the real snow storage on the glacier.

The value ZeqU - Zt = AZegU on every glacier presents in any particular year the final phase of successive processes of accumulation and ablation of solid atmospheric precipitation. It is assumed that if dz/dt is the variation of the height of the seasonal snow line, at the same time intervals, At, there exists the following relationship between the relative velocities of the seasonal snow line's approach to the limit Z e q u and successive accumulation of the total layer of melted winter snow at the same level Zequ:

1 m dz(x) fck-l " d T

/ dz (T)

dM(-r) dt

= U>(T) (39)

•dM <T1 equ

USSR case studies: water supply 91

where

k-1 k-1 i J dz(-r) = J dz(T) - J dz(i) = Z(tu_,) - Z(tv> (40) t. t« fc~ i o o

k-1' v k'

presents the part remaining from AZeqU after the seasonal snow limit reached the level Z(tj) on the glacier during the period from initial t0 and intermediate moment tj. Similarly,

fck-l J dM(x) = M-(tk_1) - M(tj) (41)

t i

indirectly characterizes the remnant of winter snow unmelted by the moment tj at the altitudinal interval Z e q u - Z(t^) = AZ(tj), where M(tj._2) is the total melting layer of winter snow at the height ZeqU: M(tj) is the cumulative layer of winter snow melting by the moment tj at the same level. The rise of the snow line for the time interval fit = tj+j - t-j is determined similarly. To find wj (At) beginning with the moment tj it is possible to write:

M (At) (ût) = r3 (42)

1 Vl t

I -M. (At)

Then the appropriate formula is substituted for M^ (At) for the computation of melting.

In the formulae (37) to (42) the k is an index which denotes the final date of snow melting. It lies within the time interval between the date when ice melting begins and 31 October. To determine summer precipitation and melting the following scheme is used :

Zequ, when Pss (t) = 0

Z*(T) = Zss, when Pss (T) > 0 (43)

N N min Zss, when X Mss(AZ,i) < I Pss(AZ,t)

This is analyzed in daily reverse succession from the 31 October to the beginning of the period. In formula (43) Z s s is the altitude of the lowest elevation of summer snow-fall, Mss is the melting of summer snow, N, the number of days from the beginning of summer snow-fall, and Pss is the amount of summer snow-fall.

The first day, when Z*(T) = ZeqU, is the date of the highest position of the snow line on the glacier, and the following day is the final date of the ice melting period.

The value of summer snow accumulation and the position of its lower limit is determined the same way as in model C. Fig. 4 provides an example of the implementation of the model of snow line motion for Tsentralniy Tuyuksu Glacier in Tienshan.

92 A.N. Krenke & V.M. Kotlyakov

Z (t)km

III f\ /JL -, Pi n Pi r, r-H M n LJiJL^ 10 20 30 10 20 31 10 20 31 10 20 30

VI VII VIII IX TIME

FIG. 4 Example of computations of the seasonal snow line motion on the surface of Tsentralniy Tuyuksu Glacier during the ablation period in 1961 (from V.G. Konovalov, 1979). 1. The motion of the seasonal snow line. 2. Interpolation of the height of the lower limit of winter-spring snow. 3. Lower limit of summer snow-fal1. 4. Summer precipitation.

Melting is calculated as in model C, but separately for the surface of ice, snow, old firn and debris-covered ice. The area of these types of surface is regarded as a function of altitude. Melting under the debris-cover is calculated with the help of relations developed by Khodakov (1978):

Ï . _JLL_ (44 Mi 1+0.2 h K

where Mp is the ice melting under the moraine cover, Mj is the melting of base ice and h is the thickness of moraine in cm; k can be estimated from an empirical relationship: if s is the ratio of the debris-covered area to total glacier area then for central Asia h = 80s + 2.9.

The computation of melting for a number of glaciers in a basin is conducted for an average glacier, whose altitudinal characteristics are calculated as the mean-weight properties of glaciers in the basin, and characteristics of the area - as the simple mean properties. However median (with 50% - probability) characteristics of the area would be preferable. Resultant function of the water income q(t) is in this case multiplied by the relation of the glacierized area of the basin to the area of an averaged glacier or of a median glacier. This model has been implemented for the Tuyuksu Glacier and for some other river basins in Central Asia.

USSR case studies: water supply 93

MODIFICATION OF THE GLACIAL RUNOFF TRANSFORMATION METHOD

The C and D models incorporate the flattening unit hydrograph in a glacier. However with Iq and ft parameters constant in time the redistribution of the runoff by the glacier during the ablation season is not taken into account. Water is accumulated in the glacier in the first part of the ablation season, k is high and ft is small. In the second part of the season, due to the well-pronounced drainage, glacier cavities and the firn sequence are emptied of the water, kj falls, and ft grows, sometimes exceeding 1. By the end of the ablation period, k grows again.

Examples of the efforts made in the USSR to take into account the above-mentioned effect are presented below.

The revealed time variation of kj made the following improvement of the one-element linear model possible (Glazyrina & Glazyrin, 1978). The course of k is given in the form of cosine curve:

2ir k = kQ + AT • cos [ggjr • (n - kT)], (45)

where k0, AT and kT are parameters, n is the number of days, beginning with January. From the data of observations, kj is determined by the method, shown in model C for every ten days on the Abramov Glacier. The obtained course of kj does resemble a cosine curve. Its parameters have been determined from these data and the expression (46) is obtained:

k = 37 + 33 • cos [0.0172 (n - 30)] (46)

With (21) it gives:

{37 + 33 • cos [0.0172 (n-30)]} + Q = ft-q (47)

Therefore ft can be found by the same least-squares technique. The model has been implemented for the seven years of measurements on the Abramov Glacier (Alai Range). Due regard to changes in k increased the accuracy of hydrograph computations by 15%. The S/o index decreased from 0.38 to 0.33. Here S is the mean square deviation of the actual and computed hydrographs, and o - the mean square deviation of the actual hydrograph.

Freidlin (1980) advanced the model with several parallel entries - the income to the glacier tongue qj, the income to the firn sequence q2 and the income from the non-glacierized part of the basin q3. The run-off from these areas Qj and Q2 was determined by the genetic division of the hydrograph, and Q3 as Q - Q^ - Q2. The k], k2 and k3 parameters and nj, n2, n3 in the influence function were determined by optimization methods. Further on, the k parameter for equation (21) is determined from

k = Jq-aj + k 2 -a 2 f k 3 -a 3 (48)

where a^, a2, a3 are the weight coefficients.

94 A.N. Krenke & V.M. Kotlyakov

For the account of non-stationarity of k(t) Freidlin considers k(t) in the saturation phase as

k - * sat q

where V is the regulating volume, varying in time

(49)

V = V + J (vY - My)«dt o

o

(50)

where V0 is the surface regulation by the glacier, V is the rate of filtration, y is porosity, M is melting on the surface. As Y»A (200 to 300 mm/day versus 20 to 30 mm/day),

V k = _k Ksat

r

o + J o t J o

Y"

H-

(v-H)

p*dt

•dt

(51)

where p is the density, H is thickness of firn. k^ at the phase of water drainage from the glacier is described similarly:

max M - "-W i' i f M constancy (52)

The values of Ts(t) and k,j(t) are substituted for k = constancy into the influence function in (26) and (29).

The model has been applied to the Dzhankuat Glacier in the Caucasus and produced good results (Fig. 5) when the measured income was used.

JUNE 1969

FIG. 5 Observed (1) and calculated (2) hydrographs of the runoff from the Dzhankuat Glacier (from the model with non-stationary ^sat parameter at the saturation phase). The input was measured (according to Freidlin, 1980).

Simplification into the one-volume transformation model for the glacier proper was proposed by Golubev (1976), who considered it possible to assume the runoff coefficient p in the equation (13)

USSR case studies: water supply 95

to be 1. Then k acquires the sense of time lag and can be determined from recession curves after the water input stops (at the beginning of summer and autumn snow-falls). Such estimations undertaken for a number of glaciers demonstrated the dependence of k, measured in days, on the area of a glacier FCkm^) (Fig. 6).

k = 3.8 fi.g (F + 1) (53)

The other modifications of the computation models of the run-off tested in the USSR are to be found in the reference list published by the Working Group earlier.

kday

10 -

8 -

6 -

4 -

2 -

0.5 1.0 1.5 2-0 Ig (F + 1) _ I l I I I I I I I

0.5 1 3 5 10 30 50 100 F km2

FIG. 6 Dependence of the mean lag time of melt waters from the glacier on its area. 1. Kacabatkak (Tien-Shan). 2. Dzhankuat (Caucasus). 3. Igan (Urals). 4. Garabashi (Caucasus). 5. Bol'shoi Aktru (Altai). 6. Maliy Aktru (Altai). 7. Zeravshanskiy (Pamir-Altai). 8. Fedchenko (Pamirs).

POSSIBILITIES OF PREDICTING RUNOFF FROM GLACIERIZED AREAS

As demonstrated above, glacial runoff is determined by precipitation, known in advance, and thermal conditions, which have not yet been predicted. Therefore prediction of runoff for the whole growing season in areas of considerable glacierization still presents great difficulties.

A graphic example is furnished by the gauge of Nurek power-station situated on the vakhsh River. Its catchment area is 31 200 km^, with more than 10% occupied by glaciers. The dependence of the runoff generated by the melting of perennial snow storage on spring precipitation (Px-V) and summer thermal resources J(+T°) is presented in Fig. 7 (Abal'yan, Mukhin & Polunin, 1980). Summer thermal resources are not known in advance, at the same time winter and spring precipitation at the representative stations (the Fedchenko Glacier and Sarytash), which are correlated with snow storages in the basin, are known several months in advance. Glacier runoff bears an inverse relationship

96 A.N. Krenke & V.H. Kotlyakov

with snow storage. The mean error of forecasts, taking into account only winter-spring accumulation and considering the sums of positive temperatures as long-term mean values, constitutes 35%, while the long-term variations of the glacial runoff (in its hydrological sense) constitute 50%. Consequently, the S/cr index equals 0.70 and cannot be used in practice, though it is better than "climatic" forecasts.

0I I l I l l l l I 600 1000 1400 1800 2200

CUMULATIVE PRECIPITATION OCTOBER TO MAY

Px-v. m m

FIG. 7 Dependence of the runoff generated by the melting of perennial ice q± in the Vakhsh river basin on winter-spring precipitation and the sum of positive temperatures at the stations Fedchenko Glacier and Sarytash (Abaïyan et al., 1980).

The total runoff of the Vakhsh River is predicted very accurately; snow melt runoff is important as well as glacial runoff. Four snow-survey points have been selected among 35 points over the territory of the basin and serve as predictors of the total runoff. They have the greatest correlation with the total runoff. The index of snow thickness, averaged for these four snow survey points is connected with the total runoff by a correlation coefficient of 0.76 (Fig. 8).

It is noteworthy in Fig. 8, that the runoff in the year with the greatest snow storage is not maximum, and in the year with the least snow storage it is not minimum. This is caused by the long-term regulation of the runoff by the glaciers.

USSR case studies: water supply 97

E 1000

d

• 62

- 2 - 1 0 1 2 SNOW COVER DEVIATION FROM NORMAL

FIG. 8. Relations between the total runoff of the Vakhsh River Q with the index of snow cover

4 H. - H J = £ —z—-— where H. and H is the thickness of snow

s . , o .JH i i=l i

cover at i-gauge in a particular year and the mean long-term value. Standard deviation of snow cover thickness at snow gauge i is &j#.

The mean ten-day and monthly discharges of the glacial runoff depend first of all on the discharges of preceding periods and on the income of the month (or ten days) for which the prediction is prepared. Attempts were made (Glazyrina & Glazyrin, 1978) to replace the last argument by the income of the preceding period, in order to make the forecast of the runoff independent of temperature forecasts. Fig. 9 presents the mean ten-day values of water discharges from the Abramov Glacier for seven years and also the two predicted hydrographs.

The first hydrograph was calculated in the following way. If from equation (21) one passes to the simplest modification of finite differences and assumes At = 1 then:

k-(Qi-Qi-l) + 0i - P-qi (54)

98 A.N. Kcenke & V.M. Kotlyakov

FIG. 9 Hydrographs of the mean runoff for ten-day periods for Abramov Glacier (Alai Range); q denotes measured, qj - computed from Qj_j and qj_j, ?2 ~ computed from q\_l and q^ (according to Glazyrina & Glazyrin, 1980).

To obtain the prediction scheme, we should replace qj by q^_j in the right-hand part of this expression. Then from (54):

° i " P'qi-1 + k' Q i-1

1 + k (55)

where ~ symbol denotes the predicted value. The parameters |î and k were found by the least-squares technique, common for the seven years (S = 1.55 m^/sec, S/o =0.44). We took as the income for the second hydrograph the mean seven-year values of the corresponding ten-day periods qj and then the algorithm was transformed into

qi

3-qj _ k'^j-i (56) 1 + k

(S = 1.27 m3/sec, S/a = 0.36). The higher accuracy of the second modification testifies to the small influence of the input during the previous ten-day period on the discharge of the next ten-day period. To put it more accurately, this influence is pronounced already in the discharge of the preceding ten days.

REFERENCES

Abal'yan, T.S., Mukhin, V.M. & Polunin, A.Ya. (1980) O lednikovom pitanii i vozomzhnostyakh dolgosrochnogo prognoza stoka gornykh rek s bol'shim oledeneniem vodosbora (On glacial nourishment and

USSR case studies: water supply 99

possibilities of long-term prediction of the mountain rivers' runoff with vast glacierization of the catchment areas). In: The Data of Glaciological Studies. Chronicle. Discussion, N 39, Moscow, 42-49.

Borovikova, L.N., Denisov, Yu.M., Trofimova, E.B. & I.D. Shentsis. (1972) Matematicheskoye modelirovanie protsessa stoka gornykh rek. (On mathematical modelling of the mountain rivers' runoff). Leningrad, Hydrometeoizdat, 152 p.

Freidlin, V.S. (1980) Vozmozhnosti lineynykh modelei dlya rascheta hydrographa lednikovogo stoka (The prospects of linear modelling for the computation of hydrograph of the glacial runoff). In: The Data of Glaciological Studies. Chronicle. Discussion, 37, Moscow, 149-155.

Glazyrina, E.L. & Glazyrin, G.E. (1978) Primeneniye lineynykh modeley dlya rasheta stoka s lednikov (On the application of linear models for computations of the runoff from glaciers). In: The Data of Glaciological Studies. Chronicle. Discussion, 32, Moscow, 27-34.

Golubev, G.N. (1976) Hidrologia lednikov (Hydrology of glaciers). Leningrad, Hydrometeoizdat, 247 p.

Khodakov, V.G. (1978) Vodno-ledoviy balans raionov sovremennogo i drevinego oledeneniya SSSR (Water-ice balance of the regions of present-day and former glaciers in the USSR). Moscow, Nauka, 194 p.

Konovalov, V.K. (1979) Raschet i prognoz tayaniya lednikov Srednei Azii (Computations and predictions of the melting of glaciers in Central Asia), Leningrad, Hydrometeoizdat, 232 p.

Krenke, A.N. (1973) Climaticheskiye uslovia uschestvovaniya oledeneniya Srednei Asii (Climatic conditions of the existence of glaciers in Central Asia). Izvestia Akademii Nauk SSR, Seria geoyraph. 1, 19-23.

Krenke, A.N. (1982) Massoobmen v lédnikovykh sistemakh na terri torii SSSR (Mass exchange in regional systems over the territory of the USSR). Leningrad, Hydrometeoizdat, 288 p.

Lebedeva, I.M. (1963) Melting processes on the MSU Glacier. In: The Data of Glaciological Studies. Chronicle. Discussion. 8, 64-72.