On the nonnormal–nonlinear interaction mechanism between counter-propagating Rossby waves

Abstract

The counter-propagating Rossby wave perspective to shear flow instability is extended here to the weakly nonlinear phase. The nonlinear action at a distance interaction mechanism between a pair of waves is identified and separated from the linear one. In the former, the streamwise velocity converges the far-field vorticity anomaly of the opposed wave, whereas in the latter, the cross-stream velocity advects the far-field mean vorticity. A truncated analytical model of two vorticity interfaces shows that higher harmonics generated by the nonlinear interaction act as a forcing on the nonnormal linear dynamics. Furthermore, an intrinsic positive feedback toward small-scale enstrophy results from the fact that higher harmonic pair of waves are generated in anti-phase configuration which is favored for nonnormal growth. Near marginal stability, the waves preserve their structure and numerical simulations of the weakly nonlinear interaction show wave saturation into finite amplitudes, in good agreement both with the fixed point solution of the truncated model, as well as with its corresponding weakly nonlinear Ginzburg–Landau amplitude equation.