Internal wave transmission in nonuniform flows

We compute transmission coefficients for internal waves propagating in a fluid with continuously varying stratification and background shear. In stationary fluid the transmission is characterized by the ratio of transmitted to incident energy. More generally, transmission across the shear is appropriately characterized by the ratio of transmitted to incident pseudoenergy flux. First, we examine the transmission and reflection of internal waves incident upon a weakly stratified layer in stationary fluid focusing upon the opposing limits of piecewise-linear theory and a heuristic application of Wentzel-Kramers-Brillouin (WKB) theory. We find the WKB prediction is reasonably accurate if the distance of transition from strong to weak stratification is as small as one sixth the vertical wavelength of the transmitted waves. In the limit of infinitesimally small transition distances the prediction of piecewise-linear theory is reproduced. Second, we consider the transmission of internal waves across a shear layer which initially is uniformly stratified. In particular, we show that significant transmission is possible across critical layers if the minimum gradient Richardson number is less than 1∕4. Finally, we show that internal waves can partially transmit across a mixed region that results from the evolution of an unstable shear layer. Transmission across critical layers occurs for waves whose horizontal phase speed matches the background flow speed at levels where the gradient Richardson number is less than 1∕4.