Elementary stratified flows with stability at low Richardson number

Abstract

We revisit the stability analysis for three classical configurations of multiple fluid layers proposed by Goldstein ["On the stability of superposed streams of fluids of different densities," Proc. R. Soc. A. 132, 524 (1931)], Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], and Holmboe ["On the behaviour of symmetric waves in stratified shear layers," Geophys. Publ. 24, 67 (1962)] as simple prototypes to understand stability characteristics of stratified shear flows with sharp density transitions. When such flows are confined in a finite domain, it is shown that a large shear across the layers that is often considered a source of instability plays a stabilizing role. Presented are simple analytical criteria for stability of these low Richardson number flows. C 2014 AIP Publishing LLC. [http://dx.

Stratified shear flow instabilities in the non-Boussinesq regime

“…These results are in general agreement with the previous works on shear instability in non-Boussinesq systems [12][13][14] even if their precise set up is not identical to ours. In particular, the observation that increasing the shear (i.e., the value of F ) stabilises the instability 14 is in agreement of our results here, with the reason being that the magnitude of the shear increases the degree of asymmetry for wave propagation, which in turn affects phase-locking properties.…”

“…This suggests a physical interpretation to the work of Barros & Choi 14 , who find that a large shear across the interfaces plays a stabilising role, which is perhaps somewhat counter-intuitive as the shear is normally seen as a source of instability. To test this hypothesis, we consider a simplified form of the the Taylor-Caulfield problem 10,15,16,18,24 , where the basic state is essentially given by…”

“…In particular, the observation that increasing the shear (i.e., the value of F ) stabilises the instability 14 is in agreement of our results here, with the reason being that the magnitude of the shear increases the degree of asymmetry for wave propagation, which in turn affects phase-locking properties. Furthermore, the work of Barros & Choi 14 analysed the non-Boussinesq effect when it is combined with the effect of confinement by boundaries. As shown in some previous works 28,29 , the reduction in growth owing to confinement can also be explained in terms of wave interaction, since mirror image waves that are in anti-phase with the counter-propagating waves may be placed on the other side of the boundaries to enforce the boundary conditions accordingly, as in the method of images.…”

“…Substituting this form of solution into equation (14), the imaginary parts cancel out exactly, and the resulting dispersion relation is given by…”

“…Since the use of defects is as a model for sharp gradients in the basic state, one can ask whether it might be more appropriate to study flow instabilities in the presence of sharp density gradients without the Boussinesq approximation, since the assumption of small density variation may no longer hold. To this end, there have been several works studying shear instabilities beyond the Boussinesq approximation over the years, using smooth profiles but with a density that has a small scale height 11 , and classic profiles with defects in [12][13][14] . However, these aforementioned works in the non-Boussinesq setting focuses on solving the modified Taylor-Goldstein equation to investigate the property of growth rates with increasing deviation from the Boussinesq regime (which will be seen to be measured by a Froude number), without necessarily providing a physical reason of what causes the modifications to the instability characteristics.…”