On the inertial effects of density variation in stratified shear flows

Abstract

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i)is the ratio of density variation to the mean density and F r c is the ratio of the phase speed to the long wave speed. The second clause implies, contrary to the commonly accepted notion, that a flow with large density variations can also be Boussinesq.Indeed, we show that deep water surface gravity waves are Boussinesq while shallow water surface gravity waves arent. However, in the presence of moderate/strong shear, Boussinesq approximation implies the conventionally accepted A t O(1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces -neutral, stable and unstable -embedded in a background shear layer, are investigated. Instabilities ensuing from these configurations, which includes Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g. the destabilizing role of density stratification, stabilizing role of shear, etc. are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interactions perspective.

On the opposing roles of the Boussinesq and non‐Boussinesq baroclinic torques in surface gravity wave propagation

“…Even for infinite depth, the non‐Boussinesq torque should still be taken into account when multiple density interfaces exist (Heifetz and Mak, ; Guha and Raj, ). This is because the vortex sheet gravity waves induce far‐field horizontal fields on each other and thereby activate the non‐Boussinesq dynamics.…”

“…In other words, this implies that the Boussinesq effects are not confined to “small density variations,” as is conventionally understood. A similar conclusion has been obtained by Guha and Raj () using detailed scaling arguments.…”

Nonlinear travelling periodic waves for the Euler equations in three-layer flows

“…While reasonable on physical grounds, both the Boussinesq and rigid-lid assumptions do have clear structural mathematical consequences which are not often emphasised in the literature, for example the aforementioned breaking of symmetry, changes in the stability characteristics (Barros & Choi 2011; Boonkasame & Milewski 2012; Heifetz & Mak 2015; Guha & Raj 2018) of certain flows, and a momentum paradox (Camassa et al. 2012).…”

Presence of Holmboe waves in particle-laden intrusive gravity current

“…He established that both of these mechanisms are of equal importance for a better understanding of the instability’s features 22 . In order to gain insight into the inertial effects of variations in density, Guha and Raj examined a range of non-Boussinesq shear flows and Holmboe waves, which are a type of steadily propagating wave that can occur at the boundary between two fluids with different background densities and vorticities 23 . Some of the results, such as the destabilizing nature of density stratification and the stabilizing effects of shear, appeared to be counterintuitive; however, these instabilities could be explained in terms of wave-interaction if viewed from a different perspective 23 .…”