Resonant triad interactions in a stably stratified uniform shear flow

We investigate resonant triad interactions (RTIs) in a two-dimensional stably stratified uniform shear flow confined between two infinite parallel walls in the absence of viscous and diffusive effects. RTIs occur when three interacting waves satisfy the spatial and temporal resonance conditions of the form ±km±kn=kr and ±ωm±ωn=ωr with km,n,r and ωm,n,r being the wave vectors and frequencies of the interacting waves, respectively. In two-dimensional flows, the interaction between two primary modes having the same frequency ω but different wave numbers km and kn produces two different secondary terms: one time-dependent (superharmonic) mode having frequency 2ω and wave number km+kn, and the other time-independent mean flow with zero frequency and wave number km-kn, where km and kn are the horizontal components of the wave vectors km and kn, respectively. The linear stability problem is solved analytically, which gives the eigenfunctions in the form of modified Bessel functions. The differential equations governing the spatial amplitudes of the superharmonic mode and secondary mean flow are solved numerically as well as analytically using the spectral collocation method and the method of variation of parameters, respectively. It turns out that the linear operator associated with the differential equation of both secondary modes 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 RTIs. We address two types of RTIs: self-resonances and different mode interactions. The self-resonating modes are identified using dispersion curves, but in different mode interactions the prediction of RTIs using dispersion curves is not so straightforward. Thus, to show the existence of RTIs for the different mode interaction case, we adopt a numerical method; in addition, we use the fact that the superharmonic mode diverges. Various cases of wave interactions in a stably stratified shear flow are analyzed in the presence of a resonant triad for various frequencies and mode numbers.