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

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.