Semi-geostrophic particle motion and exponentially accurate normal forms

TL;DR

This paper develops an exponentially-accurate normal form for particle motion in semi-geostrophic shallow-water systems, demonstrating long-term near-balance behavior and extending the analysis to numerical methods like HPM.

Contribution

It introduces a new normal form that captures exponentially accurate slow manifold dynamics and applies it to numerical schemes, improving understanding of long-term behavior.

Findings

Normal form accurately describes particle motion in semi-geostrophic limit

HPM method remains close to balanced dynamics for exponentially long times

Provides bounds on fast motion growth near the slow manifold

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.