Appears in collection : 2026 - T2 - WS1 - Vortices and vorticity in geophysical flows

We consider the rotating shallow water equations with small (planetary) Rossby and Froude numbers on a surface of revolution with variable Coriolis parameter having opposite signs at the poles. The large variation of the linear operator in the PDE is a possible mechanism of short-time instability as the small parameters tend to zero. However, we prove that such instability does not happen in this case: classical solutions satisfy uniform estimates on a time interval independent of the small parameters.

The most novel part of our approach is to find the explicit formula of a modified Laplacian which commutes with the large linear operator of the system. Further, upon a unitary transformation, the solution converges strongly to a limit for which the governing system is identified.