Flow driven interfacial waves: an inviscid asymptotic study

TL;DR

This paper uses asymptotic methods to analyze short wavelength interfacial waves driven by wind over water, deriving growth rates for instabilities with high accuracy for arbitrary shear flows.

Contribution

It provides a novel asymptotic framework for analyzing interfacial wave stability and growth rates in shear flows, including explicit expressions valid for general profiles.

Findings

Derived expressions for wave growth rates applicable to arbitrary shear profiles.

Showed that the imaginary part of the eigenvalue is exponentially small, indicating weak instability.

Validated results by comparing with exact solutions for exponential shear profiles.

Abstract

Motivated by wind blowing over water, we use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of these flows. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (1957) and rippling (e.g., Young & Wolfe 2013) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile.