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, by considering the problem in the complex plane, we show that it contains a number of interesting features that were not remarked upon in the original studies of this equation. These include, in particular, the presence of inactive Stokes lines due to the higher-order Stokes phenomenon. Since an understanding of the behavior in the complex plane is often crucial for problems in exponential asymptotics, we hope that our results, as well as the techniques developed, will prove useful when solving more general linear (and even nonlinear) eigenvalue problems involving asymptotics beyond-all-orders.