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Evolution of solitary waves in a two-pycnocline system

Published online by Cambridge University Press:  11 December 2009

M. NITSCHE ,
P. D. WEIDMAN ,
R. GRIMSHAW ,
M. GHRIST  and
B. FORNBERG
M. NITSCHE*
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
P. D. WEIDMAN
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA
R. GRIMSHAW
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK
M. GHRIST
Affiliation:
Department of Mathematical Sciences, USAF Academy, CO 80840-6252, USA
B. FORNBERG
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
*
†Email address for correspondence: [email protected]
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Abstract

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 235 - 277
Copyright
Copyright © Cambridge University Press 2009

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