A study of cell models: 2. The effect of time delay on the limiting forms of cascade and manifold cell model response functions

A simplified cell model was used to investigate the behavior of the impulse response function (IRF) of cell models incorporating time delay elements, as the number of subdivisions increases. The simplified model used in this study was a single branch model consisting of a number of equal cells arranged in sequence. Two cell models were studied, a cascade model and a manifold model. In this study the model representing each cell is modified by the addition of a delay element which is taken to be linearly proportional to the distance of its corresponding cell from the outlet. The limiting forms of the IRFs are derived for the two models using Laplace transforms; the cascade cell model IRF is found to converge to a rectangle regardless of the values of its parameters, while the manifold cell model IRF converges to a family of curves whose shapes depend quite sensitively on the values of the parameters chosen. The expression derived contains the exponential integral function, which has not been used before in surface hydrology. For some combinations of the parameters, the IRF has the classical (or intuitively expected) shape consisting of a steep rise and mild recession. For other values, the IRF is reminiscent of the kinematic wave solution to the overland flow equations, and some comparisons are made. The manifold cell model is found to hold promise as a suitably behaved alternative to the cascade cell model in the context of distributed surface runoff systems.