# An asymptotic analysis for generation of unsteady surface waves on deep water by turbulence

## TL;DR

This paper extends the mathematical analysis of unsteady surface wave generation by turbulence on deep water, including Stokes waves, and discusses energy transfer mechanisms and the role of turbulent eddy viscosity.

## Contribution

It develops an extended mathematical framework for unsteady wave growth, incorporating Stokes waves and analyzing the effects of turbulence and eddy viscosity.

## Key findings

- Agreement with Belcher and Hunt (1993) in the limit of zero wave phase speed.
- Energy transfer to wave components can be estimated from turbulence.
- Turbulent eddy viscosity diminishes the role of Miles' critical layer.

## Abstract

The detailed mathematical study of the recent paper by Sajjadi, Hunt and Drullion (2014) is pre- sented. The mathematical developement considered by them, for unsteady growing monochro- matic waves is also extended to Stokes waves. The present contribution also demonstrates agree- ment with the pioneering work of Belcher and Hunt (1993) which is valid in the limit of the complex part of the wave phase speed \c_i \downarrow 0. It is further shown that the energy-transfer parameter and the surface shear stress for a Stokes wave reverts to a monochromatic wave when the second harmonic is excluded. Furthermore, the present theory can be used to estimate the amount of energy transferred to each component of nonlinear surface waves on deep water from a turbulent shear flow blowing over it. Finally, it is demonstrated that in the presence of turbulent eddy viscosity the Miles (1957) critical layer does not play an important role. Thus, it is concluded that in the limit of zero growth rate the effect of the wave growth arises from the elevated critical layer by finite turbulent diffusivity, so that the perturbed flow and the drag force is determined by the asymmetric and sheltering flow in the surface shear layer and its matched interaction with the upper region.