Geoscience Reference
In-Depth Information
5.5.4 Kinematic Wave Models for Snowpack Runoff
One of the earliest implementations of the kinematic wave equation in hydrology was for modelling
flow through a snowpack, by Colbeck (1974). The model was later used by Dunne
et al.
(1976) and
further analytical solutions have recently been outlined by Singh
et al.
(1997). All of these studies have
assumed a snowpack of constant porosity and hydraulic conductivity. The earlier studies assumed that all
the specified net melt would percolate downslope through the snowpack; the later study by Singh
et al.
explicitly incorporates the effects of infiltration into underlying soil. In the general case, the time-varying
infiltration rate may be a function of the depth of saturation in the pack. Singh
et al.
(1997) provide
analytical solutions for the case where the infiltration rate may be assumed to be constant.
5.5.5 Kinematic Shocks and Numerical Solutions
One of the problems of applying kinematic wave theory to hydrological systems is the problem of
kinematic shocks (see Singh, 1996). The kinematic wave velocity can be thought of as the speed with
which a particular storage or depth value is moving downslope. If the wave velocity increases with
storage, then waves associated with larger depths will move downslope faster than waves associated
with shallower depths. Very often, this may not be a problem. In channels, kinematic shocks are rare
(Ponce, 1991). On a hillslope plane of fixed width and slope subjected to uniform rainfall, wave velocity
will never decrease downslope and there will be no shock. If, however, a concave upwards hillslope is
represented as a cascade of planes and subjected to a uniform rainfall, then faster flow from the steeper
part of the slope will tend to accumulate as greater storage as the slope decreases, causing a steepening
wave until a kinematic shock front occurs. The paths followed by kinematic waves in a plot of distance
against time are known as “characteristic curves”. A shock front occurs when two characteristic curves
intersect on such a plot. The effect of such a shock front reaching the base of a slope would be a sudden
jump in the discharge.
This is clearly not realistic. It is a product of the mathematics of the kinematic wave approximation,
not of the physics of the system itself which would tend to disperse such sharp fronts. One example
of a kinematic shock that can be handled analytically using kinematic wave theory is the movement
and redistribution of a wetting front into an unsaturated soil or macropore system (Beven and Germann,
1982; Beven, 1984; Smith, 1983; Charbeneau, 1984; Germann, 1990). In fact, the Green and Ampt (1911)
infiltration model (outlined in Box 5.2) can be interpreted as the solution of a kinematic wave description
of infiltration, with the wetting front acting as a shock wave moving into the soil. Models have been
developed that try to take account of such shocks by solving for the position of the characteristic curves
(called the “method of characteristics” by, for example, Borah
et al.
, 1989), but most numerical solutions
rely on
numerical dispersion
to take care of such shocks.
Using a finite difference form of the kinematic wave equation to obtain a solution, the approximation
adds some artificial or numerical dispersion to the solution. If numerical dispersion is used to reduce the
problem of shocks in a solution of the kinematic wave equation (for cases where shocks might occur),
then the solution will not be a true solution of the original kinematic wave equation. There is a certain
irony to this problem in that the approximate solution might then be more “realistic” since, as already
noted, nature would tend to disperse such sharp fronts (Ponce, 1991). However, numerical dispersion is
not well controlled; the effect will depend on the space and time increments used in the solution, together
with any other solution parameters. The reader needs be aware, however, of the potential problem posed
by shocks and of the fact that the approximate solution may not be consistent with the solution of the
original equation in such cases. There is also a possibility that a large shock might lead to instability of
the approximate numerical solution.
Shocks occur where a larger wave (a depth moving through the flow system) catches up with or meets
another smaller one. This will tend to occur where flow is slowed for some reason (reduction in slope,
