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Laboratory experiments and simulations for solitary internal waves with trapped cores

Published online by Cambridge University Press:  19 September 2014

Paolo Luzzatto-Fegiz  and
Karl R. Helfrich
Paolo Luzzatto-Fegiz*
Affiliation:
Churchill College, University of Cambridge, Storey’s Way, Cambridge CB3 0DS, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Karl R. Helfrich
Affiliation:
Department of Physical Oceanography, Woods Hole Oceanographic Institution, 266 Woods Hole Road, Woods Hole, MA 02543-1050, USA
*
†Email address for correspondence: [email protected]
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Abstract

We perform simultaneous coplanar measurements of velocity and density in solitary internal waves with trapped cores, as well as viscous numerical simulations. Our set-up comprises a thin stratified layer (approximately 15 % of the overall fluid depth) overlaying a deep homogeneous layer. We consider waves propagating near a free surface, as well as near a rigid no-slip lid. In the free-surface case, all trapped-core waves exhibit a strong shear instability. We propose that Marangoni effects are responsible for this instability, and use our velocity measurements to perform quantitative calculations supporting this hypothesis. These surface-tension effects appear to be difficult to avoid at the experimental scale. By contrast, our experiments with a no-slip lid yield robust waves with large cores. In order to consider larger-amplitude waves, we complement our experiments with viscous numerical simulations, employing a longer virtual tank. Where overlap exists, our experiments and simulations are in good agreement. In order to provide a robust definition of the trapped core, we propose bounding it as a Lagrangian coherent structure (instead of using a closed streamline, as has been done traditionally). This construction is less sensitive to small errors in the velocity field, and to small three-dimensional effects. In order to retain only flows near equilibrium, we introduce a steadiness criterion, based on the rate of change of the density in the core. We use this criterion to successfully select within our experiments and simulations a family of quasi-steady robust flows that exhibit good collapse in their properties. The core circulation is small (at most, around 10 % of the baroclinic wave circulation). The core density is essentially uniform; the standard deviation of the density, in the core region, is less than 4 % of the full density range. We also calculate the circulation, kinetic energy and available potential energy of these waves. We find that these results are consistent with predictions from Dubreil-Jacotin–Long theory for waves with a uniform-density irrotational core, except for an offset, which we suggest is associated with viscous effects. Finally, by computing Richardson-number fields, and performing a temporal stability analysis based on the Taylor–Goldstein equation, we show that our results are consistent with empirical stability criteria in the literature.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 354 - 380
Copyright
© 2014 Cambridge University Press 

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