Mean particle drift in gravity-modified acoustic waves over a compressible seabed

Acoustic-gravity waves (AGWs) are sound waves in the ocean slightly modified by a gravity wave boundary condition at the surface. Typical frequencies are in the range 0.1-1\,Hz 0.1 - 1 Hz . In the study of such waves, the acoustic properties of the seabed are usually not discussed, and the sea floor is taken to be insulating to sound waves. In this paper, we allow the seabed to be acoustically conducting such that a small part of the acoustic wave energy may propagate though the sea floor. To model this process in a principal way, we apply a Robin boundary condition for the velocity potential. Because of the energy loss at the bottom, the AGW motion becomes slightly damped in the fluid. Nonlinearly, this leads to a nonzero forcing term for the mean Eulerian velocity (the acoustic streaming), when averaged over the wave period. We utilize the fact that the planetary rotation is very slow compared with the AGW period and derive the Coriolis-Stokes force for the mean Eulerian motion in the cross-wave direction. We use that the mean Lagrangian (particle) motion is the sum of the Eulerian mean motion and the Stokes drift. We then find, by averaging over the inertial period, that the Stokes drift in the wave propagation direction is balanced by a negative Eulerian mean current (an anti-Stokes flow) at all depths. The mean Lagrangian drift in the cross-wave direction is a result of a balance between the Coriolis force and the gradient of the mean Reynolds stresses, constituting the Eulerian acoustic forcing in this case. It appears that the associated Lagrangian mass flux could be of importance for the circulation in the deep ocean.