On the singular behavior of the Stokes drift in layered miscible fluids

Jan Erik H Weber,Kai H Christensen

doi:10.1016/j.wavemoti.2021.102712

Abstract

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.

Highlights

As pointed out by Weber et al [1], the Stokes flux due to internal waves obtained by integrating the Stokes drift [2] between the bounding planes of the model vanishes identically in a model with arbitrary continuous density stratification
The results reported here are obtained for inviscid miscible fluid layers, such as warm water above cold water, with a finite step in density
The appearance of an infinitely large negative Stokes drift at the interface in this case is correct from a mathematical point of view

Summary

Introduction

As pointed out by Weber et al [1], the Stokes flux due to internal waves obtained by integrating the Stokes drift [2] between the bounding planes of the model vanishes identically in a model with arbitrary continuous (stable) density stratification. The total Stokes flux should vanish in the miscible two-layer case This can be achieved in the following way: Due to an infinite vorticity at the interface in a two-layer model [1], the Stokes drift itself here must be infinitely large, and negative, in an infinitely thin layer. When integrated across this layer, the resulting Stokes flux must be finite and negative, and cancel the positive Stokes fluxes in the upper and lower layer This delta-function behavior has consequences for the Lagrangian mean position of the interface. We show that this problem can be resolved by considering the interfacial delta-function Stokes drift in the two-layer model.

The two-layer model in a Lagrangian description

Linear analysis

The Stokes drift and the Stokes flux

CA2Q 2

The Lagrangian mean position of the interface

Discussion and concluding remarks