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 Central
Asia 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]
Qm 
k P qo
F,
qo L
1
hv
(1)
where Qm - maximal rainwater discharge, qo - maximal intensity of water release, h water release depth during rainfall, v - stream velocity along the ground river bed, L length of the ground river bed, F - river basin area. The coefficient k p depends on the
dimension of variables occurring in the formula (1). Most often the water release intensity
is expressed in millimeters per day. If basin area F be done in square kilometers, then
k p  0,0116 . For the chosen dimension, length L should be expressed in kilometers, and
stream velocity v - 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 F and length of ground river L should appear
in the formula (1) but the area covered with rainfall Fw and length the ground river Lw
which is situated at the area. If altitude of isothermal surface with its temperature
 a  2  3o C to denote as z a ( a ) , then Fw  F ( z a ( a )) .
Taking that into account formula (1) for single rainfall occurrence will be written as:
k P q ow
Fw .
q ow L w
1
hw v
Qmw 
(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:
Qmw  k P
hw v
Fw .
Lw
(3)
Stream velocity v is a function of Qmw [4]

v  bv Qmw J nv2

1/ 3
(4)
and therefore the formulas (2) и (3) are implicit functions relative to Qmw . Under absence
of a reliable data of the river bed roughness coefficient ( nv ) , the value of  v  bv / nv2 / 3 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 Qmw can be obtained comparatively
simply by substitution the formula (4) in it:
Q mw 
J 1/ 2
n

F 
 k P bv h w w 
Lw 

3/ 2
,
(5)
where J - river bed inclination.
Doing the same procedure for formula (2) we find:
/3
Qmw  Aq Q1mw
 k P qow Fw  0 ,
(6)
where
Aq 
q ow Lw
hw bv
 nv2

 J





1/ 3
.
(7)
For determination of the values of q ow and hw 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:
p
  
I ( )  S 1   ,
 T
(8)
where S, T  maximal intensity and rainfall duration, p  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 H
T
H

O
T
P
ST
  
I (t )d  S 1   dt 
.
T
p
1



(9)
O
As a result, formula (8) is written as
S
  a
I ( )  S 1  
 T
1
 
 S 1  
 T
m1
,
(10)
where a  H T - mean rainfall intensity, m  S a - 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 K a and soil filtration intensity as K  .
Besides, potentially possible infiltration intensity is expressed as K aП . The latter
intensity depends on both the infiltration coefficient of catchment soils K  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:
Ka for I (t )  Ka
.
Ka  
I (t ) for I (t )  Ka
(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 t B can be found from the following:
 t 
S 1  B 
T 

m 1
 K aП  0
(12)
i.e.
tB
1 
1 


  K aП  m1  H   K aП  m1 
 T  1 

  a  1  S 
 .
S 

 

 





(13)
Integrating (12) from 0 to t B , we will derive expression for the value of water release
depth hw
1 

 K aП   K aП K aП   K aП  m1 
hw  1 




H .
a   a
S  S 




(14)
The value in braces (14) is the coefficient of the runoff resulting from rain  .
1 
m

K aП 
 K aП  m1
 K aП   K aП K aП   K aП  m1  
.
  1 




  m  1 

  1  m
a   a
S  S 
S 
 S 

 


(15)
Maximal water release intensity q ow is equal to maximal rainfall intensity S with
the deduction of potential infiltration intensity K a
q ow  S  K a .
(16)
Approximate value of potential infiltration intensity K a 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 K  . Let in i-th month, having n i days, in the average, there are n xi rainy
days. It is resulted from here that an average number of days between rainfall occurrences
n coi is equal to n i / n xi . Then potential infiltration intensity will be written as follows:
 n
b
K ai   n CO
K     i
 n xi
b

 K  ,

(17)
where  and b - non-dimensional parameters. The value of parameter b may be accepted
equal to 1/3 at first approximation.
If it rains every day, i.e. n i  n xi , the potential infiltration intensity must be equal to
infiltration coefficient, but it is possible only when   1 . Then
K ai  K  Paib .
(18)
When average value is n xi , the ratio n xi / n i is probability Pai of the rainfall
occurrence being on the certain day of i-th month. The value of probability of rainfall
occurrence Pa (t ) on t  th day of a year can be found by interpolation regarding to time
between months. In principle for mountain region the value Pa (t ) depends on its altitude
locality what is corresponds to Pa ( z, t ) .
CALCULATION OF FREQUENCY OF MAXIMAL RAINWATER DISCHARGE
OF MOUNTAIN RIVERS
Taking into account that hw   H and assigning Lw  k L Fw , where k L - nondimensional coefficient of proportionality the formula (3) will be written as follows:
Q mw  K w
m 



 K 
 K   m 1  




J 1  m
 (m  1)
 H
 Pb S 
 Pb S 
 

 a 
 a 
 

3/ 2
34
Fw
,
(19)
where K w - generalized coefficient of proportionality, which is equal to

 
Kw  kP v 


kL 

3/ 2
.
(20)
As it is seen from (19), maximal rainwater discharge Qmw (t ) at the moment t
depends on two random variables - H and S . We designate joint density of their
probability as  (S , H ; t ) and find frequency PQ (t ) of the value of the specified maximum
at the moment t . For this the value area H and S must be determined in which maximal
water discharge more than given value. At system of coordinates of H and S , on the
basis of expression (19), this range is determined by the following values of variables:
S
K
Pab (t )
,
(21)


Qmw

H 
 K w  zw (t ) Fw (t ) 


2/3
m 

 K 
 K   m1 

  (m  1) 

1  m  b
  G (Qmw , S , t ) . (22)
 P (t ) S 
 P b (t ) S 


a
a






Then frequency of the maximal discharge is equal to:


Pw (Qmw ; t )  Pa (t )
KФ /

dS
Pab (t )
 (S , H ; t ) dH .
(23)
G (Q mД , S , t )
Function  (S , H ; t ) can be presented as the following form:
 (S , H ; t )   H (H , t )  S (S | H ; t ) Pa (t )
(24)
Here H (H , t ) - probability density of rainfall depth H on a day t under conditions
that rain has fallen;  S (S | H ; t ) - probability density of maximal rainfall intensity S at the
moment t when the rainfall depth is H .
Probability of that the discharge will be exceeded on a day t is equal to Pw (Qmw ; t );
probability that it will not be exceeded is 1  Pw (Qmw ; t )  . Probability that it will not be
exceeded for the whole year is equal to
365
[1  P (Q
w
mw ; t )] ,
(25)
t 1
but probability that it will be exceeded at least once in a year i.e. the required probability
Pwg (Qmw) will be determined by expression:
365
Pwg (Q mw )  1 
 [1  P
w (Q mw ; t )] .
(26)
t 1
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.
50
40
30
20
10
0
0
10
20
30
40
50
The measured discharges, m3/s
3
The calculated discharges, m
/s
b)
3
The calculated discharges, m
/s
a)
300
200
100
0
0
100
200
300
The measured discharges, m3/s
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]
[2]
[3]
[4]
[5]
[6]
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Denisov V.M. On calculation of principal characteristics of shower rain. Journal
Meteorology and Hydrology, No 9, (1977), pp 57-61
Denisov V.M. On Average velocity of uniform motion of free turbulent flows
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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
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