A conservation law related to kelvin's circulation theorem

Abstract

The governing equations for a moving hydrodynamic surface lead to a local conservation law for a surface velocity variable q s not in common use. When the surface is closed and applied forces are conservative, the law reduces to Kelvin's circulation theorem. When the flow is irrotational, it reduces to Bernoulli's law. Incorporation of the conservation law into a numerical water wave model cast in an Eulerian representation can result in (1) reduction of the prognostic equations from two spatial dimensions to one, and (2) realization of formal accuracy to all orders in nonlinearity. In the companion paper [1], the shallow water diagnostic equation (Poisson's equation) is also reduced to a one-dimensional problem. The prognostic equation derived here thus allows a purely one-dimensional treatment of traditionally two-dimensional shallow water waves. This yields significant resolution and execution speed benefits for the numerical integration of the overall system. Techniques also exist that reduce the deep water diagnostic equation to one dimension. Thus the new prognostic equation should be useful in modeling two-dimensional deep water waves as a one-dimensional problem.