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The steady-state form of large-amplitude internal solitary waves

Published online by Cambridge University Press:  10 November 2010

STUART E. KING ,
MAGDA CARR  and
DAVID G. DRITSCHEL
STUART E. KING*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
MAGDA CARR
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
DAVID G. DRITSCHEL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
*
†Email address for correspondence: [email protected]
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Abstract

A new numerical scheme for obtaining the steady-state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two-dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time-dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 477 - 505
Copyright
Copyright © Cambridge University Press 2010

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References

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King supplementary material

Movie 1 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 7(i) (and corresponds to figure 10). The region of the domain shown is [-0.5,0.5]x[0.5,1.0]. The zero vorticity region at the top of the domain remains stable and stagnant (in the moving frame). It can be seen that the diffusion of vorticity and the initially sharp gradients in vorticity and buoyancy lead to some weak fringing below the main area of the wave and within the stagnant region.

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Video 1 MB

King supplementary material

Movie 2 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 7(ii) (and corresponds to figure 11). Again the region of the domain shown is [-0.5,0.5]x[0.5,1.0]. From this movie it can be seen that the region at the top of the domain in which closed streamlines were found in the steady state solution is subject to an instability. This instability mixes the region at the top of the domain as time progresses.

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Video 1.3 MB

King supplementary material

Movie 3 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 9 (and corresponds to figure 12). Again the region of the domain shown is [-0.5,0.5]x[0.5,1.0]. The rotating core solution shown is subject to an instability at the rear stagnation point. This instability gives rise to a disturbance which is advected around the core of the wave. This modifies the wave sufficiently for it to slow.

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Video 2.7 MB