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We examine a misleadingly simple linear second-order eigenvalue problem (theHermite-with-pole equation) that was previously proposed as a model problem ofan equatorially-trapped Rossby wave. In the singularly perturbed limitrepresenting small latitudinal shear, the eigenvalue contains anexponentially-small imaginary part; the derivation of this component requiresexponential asymptotics. In this work, we demonstrate that the problem containsa number of pathological elements in exponential asymptotics that were notremarked upon in the original studies. This includes the presence of dominantdivergent eigenvalues, non-standard divergence of the eigenfunctions, andinactive Stokes lines due to the higher-order Stokes phenomenon. The techniquesdeveloped in this work can be generalised to other linear or nonlineareigenvalue problems involving asymptotics beyond-all-orders where suchpathologies are present.

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