Shear instability in mode-2 internal Kelvin waves

Shear instabilities of stratified fluids are a classical topic with a broad literature. The classic instability takes the form of Kelvin-Helmholtz billows that initially develop in two dimensions, one of which is the vertical. Spanwise instability develops later as part of the transition to a three-dimensionalized state. We simulate mode-2 internal waves on the laboratory scale in a rotating frame of reference that, in the absence of rotation, form spanwise aligned billows on the wave flanks. Rotation breaks the symmetry of the classical shear instability because the wave amplitude decays away from the focussing wall (i.e. the waves generated are internal Kelvin waves). We document the development of the wave and the shear instabilities as the Rossby number is varied, finding that (i) even weak rotation (high Rossby number) leads to a significant modification of the billow three-dimensionalization, (ii) strong rotation (low Rossby number) leads to a strong near wall focussing of turbulence transition that is clearly evident in the second invariant of the velocity gradient, Q, of turbulence theory. For low rotation rates, or intermediate to high Rossby numbers, we identify novel instabilities with billow cores aligned in the along-tank direction, rather than the typical spanwise direction.