SLIDE 6 CLIMATIC MODEL OF RIVER DISCHARGE
The developed model is a linear reservoir model. It is composed of linear reservoirs in cells of the network. This means that the rate of discharge from a cell linearly depends on the inflow, is proportional to the slope in the cell, and is inversely proportional to the distance between cell centers. The variation rate of the discharge from a cell, or from a cascade of cells in the simplest version of the Kalinin- Milyukov model, is determined by the solution of a sequence of ordinary differential equations of the form
( ) ( ) ( ) dQ t k I t Q t dt ⋅ = −
(1) where k is the coefficient of the lag time for the cell, I(t) is the inflow into the cell, and Q(t) is the outflow from the
- cell. For a cascade of n cells, a system of n equations is solved. The equations connect the inflows and outflows
from sequential cells:
1 1
( ) ( ) ( ) 1,..., , ( ), ( )
i i i i i n
dQ t k I t Q t i n dt Q I I I t Q Q t
+
⋅ = − = = = = ¡ ¡ ¡ ¡
(2) The solution of linear differential equations of the form (1), (2) with respect to the outflow Q(t) can be found in different ways: as a convolution integral, by the finite differences method, or by integration under the condition I(t) = const during the calculation interval. The general solution of linear equations of type (1), at zero initial conditions, is the convolution integral (Duhamel’s integral):
( ) ( ) ( ) Q t I h t d τ τ τ
∞
= ⋅ − ⋅
∫
¡ ¡ ¡ ¡ ¡
(3) Here, h(t) is the kernel of the linear system (in Russian hydrological terminology, the lag curve). ¡