Kelvin Wave Instability: Critical Latitudes and Exponentially Small Effects

Abstract

The Kelvin wave has a critical latitude instability in a linear shear, \(U(y) = \varGamma y\). There is no neutral curve, but instead with viscosity neglected there is instability for all \(|\varGamma |\), however small. A power series in \(\varGamma \) is useless because \(\mathfrak {I}(c) \approx \frac{0.14}{\varGamma ^{3}} \, \exp \left( - \frac{1}{\varGamma ^{2} } \right) \) which goes to zero faster than any finite power of \(\varGamma \) as \(\varGamma \rightarrow \infty \). To capture “beyond-all-orders” effects, Hermite-Pade approximants, exponential asymptotics and numerical methods that loop off the real axis are applied to a hierarchy of models to illuminate this peculiar instability of this most important mode.