On Bragg resonances and wave triad interactions in two-layered shear flows

TL;DR

This paper investigates how background shear flows, including uniform and piecewise linear velocity profiles, modify Bragg resonance and wave triad interactions, using analytical and numerical methods to extend existing theories.

Contribution

It extends the classical resonance conditions to account for background shear flows, incorporating Doppler shifts and intrinsic frequency changes, and adapts computational tools accordingly.

Findings

Background velocity alters resonance conditions via Doppler shifts.

Shear flows significantly influence wave interactions and resonance.

Extended spectral method effectively models shear effects in wave resonance.

Abstract

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline, and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the Higher Order Spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both in theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of studying these kinds of problems, to incorporate piecewise constant background shear.