Solution of surface water flow equations using Clebsch variables

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Abstract
Conventionally, the flow equations for surface waters are obtained by integrating vertically the basic continuum conservation equations governing three‐dimensional motion, thus reducing the problem to two dimensions. However, when vertical transport is important, this model fails, and the basic equations for three‐dimensional motion must be solved directly. With the advent of super computers and attached array processors this has become feasible. In this paper a formulation based on so‐called Clebsch variables is presented. This formulation is basically equivalent to a formulation based on the vorticity. However, the formulation based on Clebsch variables does not exhibit the disadvantages of the formulation based on the vorticity. Since in the Clebsch representation an explicit expression for the pressure is available (generalized Bernoulli equation), this formulation may be considered as an extension of the well‐known potential theory for irrotational flow. A groundwater reservoir simulator solving for the pressure, temperature, and concentration can, in principle, be applied to determine the Clebsch variables describing flow in surface water, thus unifying the mathematical description of surface and subsurface hydrology. The conclusion is that a computer program based on a formulation with Clebsch variables is quite competitive, especially for three‐dimensional calculations where it should lead to a more economical use of the computer.

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