Resonant triad interactions in stratified shear flows*

We study resonant triad interactions between discrete internal wave modes m and n at a constant frequency ω and mode q at a frequency 2ω in a uniformly stratified shear flow. Using a weakly nonlinear framework, we obtain the inhomogeneous version of the Taylor-Goldstein-Haurwitz equation for the superharmonic wave vertical structure, the divergence of which indicates resonance. In the absence of shear, not all the locations (in the parameter space) where the horizontal wavenumber condition (k_m + k_n = k_q) is satisfied are the resonant locations[1,2]. It is shown using a weak shear asymptotic theory that even the presence of an arbitrarily small shear will result in all those locations, that are earlier non-resonant but satisfy k_m + k_n = k_q, will now become resonant. These include the resonances due to self-interaction and resonances close to ω ≈ 0 as well. For an ocean-like background shear flow, we identify all the possible resonant locations in the frequency(ω) - Richardson number(Ri) plane. We find that the total number of resonant triads are of the same order at small Ri as in the case of the Ri → ∞, although attaining a maximum at moderate Ri.

References:

1. Varma, D. and Mathur, M. (2017). JFM, 824, 286-311.

2. Vanneste, J. and Vial, F. (1994). GAFD, 78(1-4), 115-141.