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Reflection of oscillating internal shear layers: nonlinear corrections

Published online by Cambridge University Press:  23 July 2020

Stéphane Le Dizès Open the ORCID record for Stéphane Le Dizès [Opens in a new window]
Stéphane Le Dizès*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, UMR 7342, 13384Marseille, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
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Abstract

In this work, we perform weakly nonlinear analysis of the reflection process of a thin oscillating wave beam on a non-critical surface in a fluid rotating and stratified along the same vertical axis in the limit of weak viscosity, i.e. small Ekman number $E$. We assume that the beam has the self-similar viscous structure obtained by Moore & Saffman (Phil. Trans. R. Soc. A, vol. 264, 1969, pp. 597–634) and Thomas & Stevenson (J. Fluid Mech., vol. 54, 1972, pp. 495–506). Such a solution describes the viscous internal shear layers of width $O(E^{1/3})$ generated by a localized oscillating source. We first show that the reflected beam conserves at leading order the self-similar structure of the incident beam and is modified by an $O(E^{1/6})$ correction with a different self-similar structure. We then analyse the nonlinear interaction of the reflected beam with the incident beam of amplitude $\varepsilon$ and demonstrate that a second-harmonic beam and localized meanflow correction, both of amplitude $\varepsilon ^{2} E^{-1/3}$, are created. We further show that for the purely stratified case (respectively the purely rotating case), a non-localized meanflow correction of amplitude $\varepsilon ^{2} E^{-1/6}$ is generated, except when the boundary is horizontal (respectively vertical). In this latter case, the meanflow correction remains localized but exhibits a triple-layer structure with a large $O(E^{4/9})$ viscous layer.

JFM classification

Boundary Layers: Boundary layer separation Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A21
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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