Resonant Three-Wave Interactions in a Shear Flow with Density Stratification

This paper examines resonant three-wave interactions between internal gravity waves in a uniform shear flow with stable stratification confined between two oppositely moving horizontal plates. In particular, we present the existence of 1: 2: 3 resonance interactions among internal waves in a uniform shear flow with stable density stratification by varying the Richardson number. Such resonances occur when a fundamental wave (km, ωm) interacts with its second harmonic (2km, 2ωm) and produce the third harmonic (3km, 3ωm), where km and ωm, respectively, denote the wave number and frequency of the fundamental mode. In this study, we develop an algorithm to classify resonating modes and critical Richardson number at which 1:2:3 resonances occur by solving the linearized Boussinesq Navier–Stokes equation. Finally, the existence of 1: 2: 3 resonance and the spatial structures of resonating modes are graphically demonstrated for a particular critical Richardson number.