Abstract

The Kelvin wave is perhaps the most dynamically important of the equatorially trapped waves in the terrestrial atmosphere and ocean. Theoretically, it can be understood from the linear dynamics of a rotating stratified fluid, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear, and the growth rate is exponentially small as the shear tends to zero.  This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause. Here we show how the growth rate and full spatial structure of the instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with confluent hypergeometric functions in the exponentially-decaying tails of the Kelvin wave, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging: it turns out that the growth rate scales as p3 exp(-1/p2) in the limit of small nondimensional shear p, meaning that special high-precision calculations are required even when p is not that small (e.g., 0.2).