Shafer (Hermite-Padé) approximants for functions with exponentially small imaginary part with application to equatorial waves with critical latitude

Quadratic Shafer approximants and their generalization to higher degree polynomials called Hermite-Padé approximants have been successfully used in quantum mechanics for calculation of exponentially small escape rates. In this paper we test quadratic Shafer approximants in representing growth rates typical for equatorial atmosphere. One of the characteristic features of the equatorial Kelvin wave — the dominant mode in equatorial dynamics — is its exceptionally small linear growth rate. For example, Kelvin wave evolving on the zonal basic state with small linear shear 2 has growth rate O(exp(−1/22)) in contrast to =(E) ∼ O(exp(−1/2)) common to similar problems in quantum mechanics. It is interesting to know how well Hermite-Padé approximants handle this more computationally expensive problem. First we apply the quadratic Shafer approximants to calculate the imaginary part of the Gauss-Stieltjes function defined as