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Reflection of oscillating internal shear layers: nonlinear corrections

Abstract : In this work, we perform the 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 a 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 264, 597-634 (1969)) and Thomas & Stevenson (J. Fluid Mech. 54, 495-506 (1972)). 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 a 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 ε and demonstrate that a second-harmonic beam and localized meanflow correction, both of amplitude ε 2 E −1/3 are created. We further show that for the purely stratified case (resp. the purely rotating case), a non-localized meanflow correction of amplitude ε 2 E −1/6 is generated, except when the boundary is horizontal (resp. 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.
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Contributor : Stéphane Le Dizès <>
Submitted on : Wednesday, June 3, 2020 - 5:18:19 PM
Last modification on : Monday, July 27, 2020 - 1:00:05 PM

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Stéphane Le Dizès. Reflection of oscillating internal shear layers: nonlinear corrections. Journal of Fluid Mechanics, Cambridge University Press (CUP), inPress. ⟨hal-02750645⟩

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