Aris' moment transformation was used to convert the advective-diffusion equation for two-dimensional (longitudinal and transverse) open channel flow into temporal moment equations. The equation for the zeroth temporal moment of the concentration distribution includes the initial concentration distribution which is a Dirac delta function for instantaneous injections. This equation can be solved analytically for some specific velocity distributions but cannot be solved numerically because of the δ function. Using an implicit finite difference scheme, numerical solutions for both spatial and temporal moments were obtained for plane and centered line source initial conditions and for three velocity distributions. The results were used to examine the relationship between the special and temporal moments during both the initial period and the dispersive period.
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Tsai et al. (1978) studied this question.