On the Singular Behavior of the Stokes Drift in Layered Miscible Fluids

Gravity waves of the Stokes-type in a system of two horizontal layers of inviscid, stratified and miscible fluids are studied by applying a Lagrangian description of fluid motion. It is shown that for the total vertically-integrated Stokes drift (the Stokes flux) to become zero, the density must be continuous at the interface. This is not the case in a system of two immiscible homogeneous layers of different densities. For miscible fluids, however, one can extend the density of the lower layer continuously to that of the upper layer through an infinitely thin interface. This introduces a negative Stokes drift with a delta-function behavior. The corresponding finite negative Stokes flux obtained by integrating the Stokes drift across this infinitely thin layer adds to the original positive Stokes fluxes in the upper and lower layer, yielding the required vanishing total Stokes flux between the bounding planes of the model. The delta-function behavior of the Stokes drift at the interface also induces additional Lagrangian mean displacements in the vertical so that the position of the Lagrangian mean interfacial level is the same as in the absence of waves.