Abstract

The equatorial Kelvin wave is a travelling atmospheric or oceanographic wave that propagates easterly along the equator. It is known that this motion is destabilised with the inclusion of weak background shear, which may be physically generated by zonal jets. Furthermore, Natarov & Boyd (Dynam. Atmos. Oceans, 2001) numerically demonstrated that this instability is exponentially small with respect to the small shear parameter. In this talk, we demonstrate how this instability arises in connection with Stokes lines and the Stokes phenomenon. This problem includes various subtitles, such as an associated eigenvalue whose asymptotic expansion diverges, and the higher-order Stokes phenomenon which is generated by Stokes switching in the late-terms of the asymptotic expansion of the solution itself. Techniques for the derivation of both of these features are discussed, and an analytical result for the exponentially-small Kelvin wave instability is obtained. This is joint work with S. Griffiths, S. J. Chapman, and P. Trinh.