Beyond-All-Orders Instability in the Equatorial Kelvin Wave

The equatorial Kelvin wave is destabilized by cross-equatorial shear in a rather bizarre fashion. There is no well-defined inviscid neutral curve. Instead, using an exemplar the linear shear U(y) = 2y where U is the mean zonal flow and y is nondimensional latitude, the Kelvin wave is unstable for all 2, however, small. We show through multiple precision numerical calculations that the imaginary part of the phase speed is proportional to the square of the unperturbed eigenfunction at the critical latitude, that is, =(c) ∝ exp(−1/2) for small 2. Such exponential dependence on a small parameter implies that result lies “beyond-all-orders” in a perturbation series in powers of 2. We connect our numerical work here with our previous study of a model problem through “beyond-all-orders” perturbation theory to argue that the Kelvin instability is a critical latitude phenomenon.