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Near-critical
reflection of internal waves
Roberta Bianchini, Anne-Laure Dalibard and Laure Saint-Raymond |
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Vol. 14 (2021), No. 1, 205–249
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Abstract |
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Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes’ laws, and it is expected to be singular if the slope has the same inclination as the group velocity. We prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation. |
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Keywords
internal waves, near-critical reflection, boundary layers,
Boussinesq approximation
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Mathematical Subject Classification 2010
Primary: 35Q35, 35Q86, 76D10
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Milestones
Received: 27 March 2019
Accepted: 7 October 2019
Published: 19 February 2021
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Authors |
| Roberta Bianchini | |
| IAC Consiglio Nazionale delle Ricerche Rome Italy |
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| Anne-Laure Dalibard | |
| Sorbonne Université Laboratoire Jacques-Louis Lions UMR CNRS-UPMC 7598 Paris France |
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| Laure Saint-Raymond | |
| École Normale Supérieure de
Lyon UMPA UMR CNRS-ENSL 5669 Lyon France |
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