Lagrangian drift in a permeable bottom layer induced by internal gravity waves

Abstract

The mean drift in a permeable homogeneous bottom layer caused by internal gravity waves in an overlying stratified fluid, is investigated theoretically. We apply a Lagrangian formulation, together with the long-wave approximation, for the motion in the fluid and the fluid-saturated permeable layer. In the stratified fluid, we assume a constant Brunt-Väisälä frequency and inviscid flow. In the permeable bed, the density is constant. Here we apply a simple macroscopic version of Darcy’s law. For internal waves with frequency ω, a fundamental small parameter in the permeable bed is R=ωK/ν2. Here K is the effective permeability, and ν2 is the overall eddy viscosity, representing the small-scale turbulence of the interstitial fluid. The Lagrangian mean flow in each layer is weakly damped, with a damping rate proportional to R, and composed of contributions from an infinite, but discrete set of eigen-modes. For each mode the mean drift in the permeable bed is an order R2 smaller than the Stokes drift in internal waves at the bottom of the fluid layer. For spatially damped waves, it is particularly interesting that the wave-induced Eulerian mean current in the permeable bed may exceed the Stokes drift if the bed thickness is smaller than the upper layer thickness. It is suggested that the explored wave-induced particle drift in the permeable bottom layer could provide a physical model for the slow net transport of bio aerosols and smoke particles in the tropical rainforest.