Breakdown of the slow manifold in the Shallow-Water equations

Abstract

Abstract Numerical solutions are obtained by implicit multigrid solvers for initial-value problems in the rotating Shallow-Water Equations (SWE) with spatially complex initial conditions. Companion solutions are also obtained with the Shallow-Water Balance Equations (SWBE), both to determine the initial conditions for the SWE and to provide a comparison solution that lies entirely on the slow, advective manifold. We make use of a control parameter (here the Rossby number, R) to regulate the degree of slowness and balance. While there are measurable discrepancies between the evolving SWE and SWBE solutions for all R, there is a distinct, spatially local breakdown both of the slow manifold in the SWE solution and in the closeness of correspondence between the SWE and SWBE solutions. This critical value for breakdown is only slightly smaller than the R values at which, first, the SWE evolution becomes singular (i.e., the fluid depth vanishes), or second, a consistent initial condition for the SWBE cannot be ...