Semi-geostrophic particle motion and exponentially accurate normal forms

Abstract

We give an exponentially-accurate normal form for a Lagrangian particle moving in a rotating shallow-water system in the semi-geostrophic limit, which describes the motion in the region of an exponentially-accurate slow manifold (a region of phase space for which dynamics on the fast scale are exponentially small in the Rossby number). The result extends to numerical solutions of this problem via backward error analysis, and extends to the Hamiltonian Particle-Mesh (HPM) method for the shallow-water equations where the result shows that HPM stays close to balance for exponentially-long times in the semi-geostrophic limit. We show how this result is related to the variational asymptotics approach of [Oliver, 2005]; the difference being that on the Hamiltonian side it is possible to obtain strong bounds on the growth of fast motion away from (but near to) the slow manifold.