BREAKDOWN OF THE SLOW MANIFOLD IN THE SHALLOW-WATER EQUATIONS

Numerical solutions are obtained by implicit multigrid solvers for ini tial-value problems in the rotating Shallow-Water Equations (SWE) with spatially complex initial conditions. Companion solutions are also ob tained with the Shallow-Water Balance Equations (SWBE), both to determ ine the initial conditions for the SWE and to provide a comparison sol ution that lies entirely on the slow, advective manifold. We make use of a control parameter (here the Rossby number, R) to regulate the deg ree of slowness and balance. While there are measurable discrepancies between the evolving SWE and SWBE solutions for all R, there is a dist inct, spatially local breakdown both of the slow manifold in the SWE s olution and in the closeness of correspondence between the SWE and SWB E solutions. This critical value for breakdown is only slightly smalle r than the R values at which, first, the SWE evolution becomes singula r (i.e., the fluid depth vanishes), or second, a consistent initial co ndition for the SWBE cannot be defined. This breakdown is most clearly evident in a sudden increase in vertical velocity near the center of a strong, cyclonic vortex; its behavior is primarily associated with a n enhanced dissipation rather than an initiation of gravity-wave propa gation. The numerical performance of the multigrid solvers is satisfac tory even in the difficult circumstances near solution breakdown or si ngularity.