Semigeostrophic Particle Motion and Exponentially Accurate Normal forms

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). We show how this result is related to the variational asymptotics approach of [Oli05]; 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. Our normal form approach extends to numerical approximations via backward error analysis, and extends to particle methods for the shallow-water equations, where the result shows that particles stay close to balance over long times in the semi-geostrophic limit.