Internal wave tunnelling in stratified shear flows

If internal waves are incident upon a slab of fluid with constant density, one expects the waves to reflect and, if the slab is sufficiently thin, to partially transmit as well. Sutherland and Yewchuk (JFM, 2004) computed the analytic formulae for the transmission coefficients of such tunnelling phenomena for waves propagating in piecewise-linear background density profiles. This work extends these results to include the presence of background shear.

For a piecewise-linear stratified shear flow in which the flow has uniform density over the extent of the shear, we compute both the stability regimes and transmission coefficients as a function of a bulk Richardson number, Ri. As Ri becomes infinitely large, the band of wavenumbers that grow due to Kelvin-Helmholtz becomes increasingly narrow. Likewise, the growth rate of the most unstable mode decreases. In this limit, we assume the flow remains parallel for sufficiently long times that we might reasonably examine tunnelling across the sheared mixed region. We compute the transmission coefficient as a function of Ri, the relative wave frequency and the shear depth relative to the horizontal wavelength of the waves. For large Ri, we recover the result of Sutherland and Yewchuk (2004). For moderately large Ri, we find parameter regimes where energy is over-transmitted indicating that tunnelling waves may draw energy from the mean flow as they cross the mixed region.

More generally, by integrating the Taylor-Goldstein equation, we compute the transmission coefficient for waves in a parallel flow with arbitrary stratification and background horizontal flow. We apply these results, in particular, to the study of wave generation by convective storms and their consequent ducting and/or leakage into the stratosphere.