Virtual wave stress and transient mean drift in spatially damped long interfacial waves

Abstract

The mean drift in spatially damped long gravity waves at the boundary between two layers of immiscible viscous fluids is investigated theoretically by applying a Lagrangian description of motion. The focus of the paper is on the development of the drift near the interface. The initial drift (inviscid Stokes drift + viscous boundary-layer terms) associated with the instantaneously imposed wave field does not generally fulfill the conditions at the common boundary between the layers. Hence, transient Eulerian mean currents develop on both sides of the interface to ensure continuity of velocities and viscous stresses. The development of strong jet-like Eulerian currents increasing with time in this problem is related to the action of the virtual wave stress (VWS). Very soon (after a few wave periods) the transient Eulerian part dominates in the Lagrangian mean current. This effect is similar to that found for the drift in short gravity waves with a film-covered surface. A new relation is derived showing that the difference between the VWS’s at the interface is given by the divergence of the total horizontal wave momentum flux in a two-layer system. Our analysis with spatially damped waves also yields the Lagrangian change of the mean surface level and mean interfacial level (the divergence effect) due to periodic baroclinic wave motion.