Pathological exponential asymptotics for a model problem of an equatorially trapped Rossby wave

TL;DR

This paper investigates a simplified eigenvalue problem modeling equatorially trapped Rossby waves, revealing complex exponential asymptotics and pathologies that impact the understanding of wave behavior in geophysical fluid dynamics.

Contribution

It uncovers previously unnoticed pathological elements in exponential asymptotics of the model, including divergent eigenvalues and non-standard eigenfunction divergence, and develops generalizable techniques.

Findings

Identification of dominant divergent eigenvalues

Discovery of non-standard divergence of eigenfunctions

Analysis of inactive Stokes lines due to higher-order Stokes phenomenon

Abstract

We examine a misleadingly simple linear second-order eigenvalue problem (the Hermite-with-pole equation) that was previously proposed as a model problem of an equatorially-trapped Rossby wave. In the singularly perturbed limit representing small latitudinal shear, the eigenvalue contains an exponentially-small imaginary part; the derivation of this component requires exponential asymptotics. In this work, we demonstrate that the problem contains a number of pathological elements in exponential asymptotics that were not remarked upon in the original studies. This includes the presence of dominant divergent eigenvalues, non-standard divergence of the eigenfunctions, and inactive Stokes lines due to the higher-order Stokes phenomenon. The techniques developed in this work can be generalised to other linear or nonlinear eigenvalue problems involving asymptotics beyond-all-orders where such pathologies are present.