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On the response of large-amplitude internal waves to upstream disturbances

Published online by Cambridge University Press:  22 May 2012

Roberto Camassa  and
Claudio Viotti
Roberto Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
Claudio Viotti*
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: [email protected]
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Abstract

Large-amplitude internal solitary waves generate shear flows that intensify from the wings of the waves to their maxima. Upstream perturbations of the hydrostatic equilibrium in the form of wave packets along the path of wave propagation are expected to trigger shear instability and ultimately generate Kelvin–Helmholtz roll-ups. In contrast, as shown here with accurate simulations of incompressible stratified Euler equations, large internal waves can act as suppressors of perturbations. The precise understanding of the mechanisms leading to different outcomes, including whether instability is excited, is the focus of this work. Under the action of shear flows, small-amplitude wave packets undergo stretching and filamentation, which lead to significant absorption of perturbation energy into the background shear. It is found that this typical behaviour is present in the self-induced shear by internal waves, regardless of whether the shear is stable or unstable, and can leave a quieter state in the wave’s wake for a wide range of perturbation parameters. In the unstable case, even once perturbations are selected to excite the instability, our results show that this absorption can act to reduce growth in the strong-shear region, effectively making roll-up development observable only downstream of the wave crest. Our approach is both analytical and numerical; a model valid for relatively thin pycnoclines and suitable for local spectral analysis is devised and used. Energy diagnostics on the simulations are implemented to validate the numerics and illustrate the energy exchanges between background wave flow and its shear. A link between the absorption mechanism and the clustering of local eigenvalues along the wave is proposed. This promotes an energetic coupling among neutral modes stronger than what may be expected to occur in slowly varying flows, and gives rise to multi-modal transient dynamics of the kind often referred to as non-normality effects. For those cases in which the wave-induced shear meets the conditions for local instability, it is found that the growth of disturbances is selective with respect to the sign of the mode excited upstream. Elements of this phenomenon are interpreted by asymptotic analysis for spatial growth in time-independent slowly varying media.

JFM classification

Instability: Instability Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 702 , 10 July 2012 , pp. 59 - 88
Copyright
Copyright © Cambridge University Press 2012

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