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

Abstract

Quadratic Shafer approximants and their generalization to higher degree polynomials called Hermite-Pad é 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 ϵ has growth rate O(exp(−1/ ϵ 2)) in contrast to I(E)∼ O( exp(−1/ϵ)) 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 G S (ϵ)≡ lim δ→0∫ 0 ∞ exp −t 2 t−[1+ iδ]/ϵ dt on its branch cut. The imaginary part of G S( ϵ) can be shown to be I(G S )(ϵ)=π exp − 1 ϵ 2 , which is of the same order of magnitude as Kelvin-in-shear growth rate. Next, we use this technique to sum the divergent Rayleigh–Schrödinger perturbation series of the Hermite-with-Pole equation u yy+ ϵ 1 1+ϵy −λ−y 2 u=0, which is a simple model for equatorial waves in shear. The Hermite-with-Pole equation has been previously studied numerically and analytically in [J.P. Boyd, A. Nataroc, Stud. Appl. Math. 101 (1998) 43]. We compare Shafer approximants against numerical integration and find that for small ϵ, Shafer approximants are more efficient, primarily because with rational coefficients, one does not need multiple precision at the main stage of the calculation. Although the higher-order approximants are usually more accurate, the overall improvement of accuracy is not monotonic due to the appearance of nearly coincident zeros in the approximants.