The Method of calculation of maximal rainwater discharge in Central Asia mountain conditions

DENISOV YURY M., SERGEEV ALEKSEY I.

Central Asian Research Hydrometeorological Institute (SANIGMI),

72, K.Makhsumov str., 700052, Tashkent, Uzbekistan

The method of calculation of maximal rainwater discharge for mountain rivers is proposed. As a result of conducted development work the formula of calculation of maximal rainwater discharge which is determined by area and evaluation characteristics of the catchment, filtration properties of underlying surface, probabilistic characteristics (frequency, layer, intensity) of rainfall has been derived. When the frequency of the rainfall maximum is found two-dimensional probability density of joint occurrence of assigned depth of rainfall and its maximal intensity is considered. Simplified assessment of frequency by maximal depth of rainfall in a year has shown that probability density of the rainfall is well described by the generalized two-parameter exponential distribution law. The proposed method has been approved in conditions of mountain river of CentralAsia and the Caspian Sea basin.

Introduction

Maximal rainwater discharge is a dynamic and changeable occurrence in comparison with melted one [5], and more detailed approach is necessary to describe it. Moreover for small river catchment areas (which are in the greatest number), where a regular hydrometeorological observation often is absent, the rainfall maximums is prevalent ones.

PRINCIPAL SCHEME OF CALCULATION OF MAXIMAL RAINWATER DISCHARGE

As a basis for heuristic approach of calculation of maximal rainwater discharge the following formula may be taken [1]

, (1)

where - maximal rainwater discharge, - maximal intensity of water release, -water release depth during rainfall, - stream velocity along the ground river bed, - length of the ground river bed, - river basin area. The coefficient depends on the dimension of variables occurring in the formula (1). Most often the water release intensity is expressed in millimeters per day. Ifbasinareabe done insquarekilometers, then. For the chosen dimension, length should be expressed in kilometers, and stream velocity - in kilometers per day.

For mountain rivers having a vertical extension, the formula should be defined more exactly. Point is that atmosphere precipitation can fall as both in the form of snow and rain. In the work [6] has been shown, that in average the precipitation falls in the form of rain if air temperature is more than 2-30С; otherwise it snows.

In this connection not only basin area and length of ground river should appear in the formula (1) but the area covered with rainfall and length the ground river which is situated at the area. If altitude of isothermal surface with its temperature to denote as , then .

Taking that into account formula (1) for single rainfall occurrence will be written as:

. (2)

When the length of water catchment is more than 10 km and the water release is more than 1 mm/min then unity can be neglected in the denominator and formula (2) will take a more simple form:

. (3)

Stream velocity is a function of [4]

(4)

and therefore the formulas (2) и (3) are implicit functions relative to . Under absence of a reliable data of the river bed roughness coefficient (, the value of in formula (4) must be assigned equal to 211 when the velocity has its unit km/day and that be done 2,44 when its unit is m/s.

For formula (3) the expression relative to can be obtained comparatively simply by substitution the formula (4) in it:

, (5)

where- riverbedinclination.

Doing the same procedure for formula (2) we find:

, (6)

where

. (7)

For determination of the values of and we are considering ways of general characteristics calculation of the particular rainfall occurrence.

THE GENERAL RAINFALL CHARACTERISTICS AND WAYS OF THEIR CALCULATION

Using approximation of the rainfall course ranged along a descending intensity [2, 3] in the form:

, (8)

where maximal intensity and rainfall duration,non-dimensional parameter and rainfall duration with its intensity equal to or more than given one (I). Performing integration we will find the rainfall depth during the rainfall

. (9)

As a result, formula (8) is written as

, (10)

where - mean rainfall intensity, - ratio of maximal intensity to mean one, which is approximately equal to 9 [2] for Central Asia conditions.

assessment of maximal intensity of WATER RELEASE

Infiltration intensity of rainfall is expressed as and soil filtration intensity as . Besides, potentially possible infiltration intensity is expressed as . The latter intensity depends on both the infiltration coefficient of catchment soils and their moisture. The less soil moisture the more its infiltration intensity under other conditions being equal. With increasing of underlying surface moisture the potentially infiltration intensity tends to the infiltration coefficient.

Referred above allow writing as follows:

. (11)

At the latter case all the rainfall amount is used for filtration and will not there be a surface runoff.

We assign as a simplifying assumption that potential infiltration intensity is remaining constant during rainfall although for diverse rainfall occurrences it may be different. According to [2], water release duration can be found from the following:

(12)

i.e.

. (13)

Integrating (12) from 0 to , we will derive expression for the value of water release depth

. (14)

The value in braces (14) is the coefficient of the runoff resulting from rain .

. (15)

Maximal water release intensity is equal to maximal rainfall intensity with the deduction of potential infiltration intensity

. (16)

Approximate value of potential infiltration intensity can be found by following argumentations. It is known from observations that the more time between rainfall occurrences the more infiltration intensity may exceed the value of the infiltration coefficient . Let in i-th month, having days, in the average, there are rainy days. It is resulted from here that an average number of days between rainfall occurrences is equal to . Then potential infiltration intensity will be written as follows:

, (17)

where and - non-dimensional parameters. The value of parameter may be accepted equal to 1/3 at first approximation.

If it rains every day, i.e. , the potential infiltration intensity must be equal to infiltration coefficient, but it is possible only when . Then

. (18)

When average value is , the ratio is probability of the rainfall occurrence being on the certain day of i-th month. The value of probability of rainfall occurrence on th day of a year can be found by interpolation regarding to time between months. In principle for mountain region the value depends on its altitude locality what is corresponds to .

CALCULATION OFFREQUENCY OF MAXIMAL RAINWATER DISCHARGE OF MOUNTAIN RIVERS

Taking into account that and assigning , where - non-dimensional coefficient of proportionality the formula (3) will be written as follows:

, (19)

where - generalized coefficient of proportionality, which is equal to

. (20)

As it is seen from (19), maximal rainwater discharge at the moment depends on two random variables - and . We designate joint density of their probability as and find frequency of the value of the specified maximum at the moment . For this the value area and must be determined in which maximal water discharge more than given value. At system of coordinates of and , on the basis of expression (19), this range is determined by the following values of variables:

, (21)

. (22)

Then frequency of the maximal discharge is equal to:

. (23)

Function can be presented as the following form:

(24)

Here - probability density of rainfall depth on a day under conditions that rain has fallen; - probability density of maximal rainfall intensity at the moment t when the rainfall depth is .

Probability of that the discharge will be exceeded on a day is equal to probability that it will not be exceeded is . Probability that it will not be exceeded for the whole year is equal to

, (25)

but probability that it will be exceeded at least once in a year i.e. the required probability will be determined by expression:

. (26)

Thus the formulated problem of determination of frequency maximal rainwater discharge has been solved fully.

Comparison of calculated norm and a 1% frequency of maximal rainwater discharges with measured values and values for different mountain regions is shown in the fig.1.

a)
/ b)

Figure 1. Comparison of the calculated values of maximal rainwater discharges having different frequency with measured ones for different mountain.

a – norm (West Tien-Shan); b – 1% frequency (river of Caspian Sea basin).

REFERENCES

[1]Alekseev G.A. Calculation of probability maximal water discharges and runoff volume of snow and rain floods. Proc. GGI, Vol.38 (92), (1953), pp. 3-65.

[2]Denisov V.M. On calculation floods resulting from rain of small Central Asia catchment. Journal Meteorology and Hydrology, No 7, (1975), pp 81-90.

[3]Denisov V.M. On calculation of principal characteristics of shower rain. Journal Meteorology and Hydrology, No 9, (1977), pp 57-61

[4]Denisov V.M. On Average velocity of uniform motion of free turbulent flows Proc. SANIGMI, -Vol.94 (175), (1982), pp 66-74.

[5]Denisov Yu. M., Sergeev A.I, Method of mountain rivers meltwaters maximum discharge computation. Proc. International Symposium on River Flood Defence,

Kassel Repots of Hydraulic Engineering No 9/2000, (2000), Vol.1, pp C-21-C-30

[6]Glazirin G.E., Denisov Yu. M.. Computation of mean by vertical density of snow cover. Journal Meteorology and Hydrology, No 9, (1967), pp 56-62