Resonant triad interactions in stably-stratified uniform shear flow

Abstract

We investigate exact and near resonant triad interactions (RTI) in a two-dimensional stably stratified uniform shear flow confined between two infinite parallel walls in the absence of viscous and diffusive effects. RTI occur when three interacting waves satisfy the resonance conditions of the form $k_1 \pm k_2 = k_3$ and $\omega_1 \pm \omega_2 = \omega_3$ with $k_i$ and $\omega_i$ being the wavenumber and frequency of the ith wave ($i \in {1,2,3}$), respectively. The linear stability problem is solved analytically, which gives the eigenfunctions in the form of the modified Bessel functions. It is identified that an interaction between two primary modes having the same frequency $\omega$ but different wavenumbers $k_m$ and $k_n$ produces two different secondary modes: one time-dependent (superharmonic) mode having frequency $2\omega$ and wavenumber $k_m +k_n$, and the other time-independent (subharmonic) mode with $\omega = 0$ and wavenumber $k_m - k_n$. The differential equation governing the spatial amplitude of the superharmonic mode is solved numerically as well as analytically using the method of variation of parameters. It turns out that the linear operator associated with the differential equation of the superharmonic mode is the same as the linear stability operator and that the solvability condition of the differential equation is found to be associated with the existence of RTI. The existence of resonant triad interactions predicted by the dispersion relation, are justified by showing the divergence of the spatial amplitude of superharmonic mode. Various cases of wave interactions in a stably stratified shear flow are analysed in the presence of a resonant triad for various frequencies and linear stratifications.