Nonlinear reflection of internal waves

The reflection of internal waves at a horizontal interface between layers of constant buoyancy frequency (a density gradient interface) is treated with weakly nonlinear theory. The interface is a simple model of the atmospheric tropopause. The waves are horizontally periodic, vertically confined within a long packet, and obey the nonlinear Schrodinger equation away from the interface. The interfacial conditions are formally taken to third order, and include higher-order linear as well as nonlinear effects. The higher-order linear affects show a modulational instability at the interface for steep waves. Numerical results for shorter wave packets show that the higher order linear terms result in non-physical wave generation, and are ultimately neglected for such cases. Non-linear effects result in a decrease in the reflected and an increase in the transmitted wave amplitudes when compared to linear theory. Overall, the nonlinear effects make the interface more transparent to upwardly propagating internal waves.