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Shear instability of internal solitary waves in Euler fluids with thin pycnoclines

Published online by Cambridge University Press:  29 August 2012

A. Almgren ,
R. Camassa  and
R. Tiron
A. Almgren
Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
R. Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
R. Tiron*
Affiliation:
UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
*
†Email address for correspondence: [email protected]
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Abstract

The stability with respect to initial condition perturbations of solitary travelling-wave solutions of the Euler equations for continuously, stably stratified, near two-layer fluids is examined numerically and analytically for a set of parameters of relevance for laboratory experiments. Numerical travelling-wave solutions of the Dubreil–Jacotin–Long equation are first obtained with a variant of Turkington, Eyland and Wang’s iterative code by testing convergence on the equation’s residual. In this way, stationary solutions with very thin pycnoclines (and small Richardson numbers) approaching the near two-layer configurations used in experiments can be obtained, allowing for a stability study free of non-stationary effects, introduced by lack of numerical resolution, which develop when these solutions are used as initial conditions in a time-dependent evolution code. The thin pycnoclines in this study permit analytical results to be derived from strongly nonlinear models and their predictions compared with carefully controlled numerical simulations. This brings forth shortcomings of simple criteria for shear instability manifestations based on parallel shear approximations due to subtle higher-order effects. In particular, evidence is provided that the fore–aft asymmetric growth observed in all simulations requires non-parallel shear analysis. Collectively, the results of this study reveal that while the wave-induced shear can locally reach unstable configurations and give rise to local convective instability, the global wave/self-generated shear system is in fact stable, even for extreme cases of thin pycnoclines and near-maximum-amplitude waves.

JFM classification

Instability: Absolute/convective instability Geophysical and Geological Flows: Internal waves Wakes/Jets: Shear layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 324 - 361
Copyright
Copyright © Cambridge University Press 2012

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