Analytic test cases for three‐dimensional hydrodynamic models

Exact periodic solutions are generated for the 3‐D hydrodynamic equations in linearized form. A linear slip condition is enforced at the bottom, based on the velocity at the bottom. It is shown that the bottom stress can be equivalently expressed in terms of the vertically averaged velocity, and expressions for this bottom stress coefficient are derived in terms of the primary parameters of the problem. As a result, the three‐dimensional structure may be assembled from conventional solutions to (a) the 1‐D vertical diffusion equation; and (b) the 2‐D vertically averaged shallow water equations. In the latter, the bottom stress effects are shown to be complex and frequency‐dependent, and an additional rotational term is required for their representation.