Vol. 14, No. 1, 2021

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Near-critical reflection of internal waves

Roberta Bianchini, Anne-Laure Dalibard and Laure Saint-Raymond

Vol. 14 (2021), No. 1, 205–249
Abstract

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.

Keywords
internal waves, near-critical reflection, boundary layers, Boussinesq approximation
Mathematical Subject Classification 2010
Primary: 35Q35, 35Q86, 76D10
Milestones
Received: 27 March 2019
Accepted: 7 October 2019
Published: 19 February 2021
Authors
Roberta Bianchini
IAC
Consiglio Nazionale delle Ricerche
Rome
Italy
Anne-Laure Dalibard
Sorbonne Université
Laboratoire Jacques-Louis Lions
UMR CNRS-UPMC 7598
Paris
France
Laure Saint-Raymond
École Normale Supérieure de Lyon
UMPA
UMR CNRS-ENSL 5669
Lyon
France