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Critical layer and radiative instabilities in shallow-water shear flows

Published online by Cambridge University Press:  23 June 2014

Xavier Riedinger  and
Andrew D. Gilbert
Xavier Riedinger*
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
Andrew D. Gilbert
Affiliation:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK
*
†Email address for correspondence: xavier.riedinger@gmail.com
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Abstract

In this study a linear stability analysis of shallow-water flows is undertaken for a representative Froude number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}F=3.5$. The focus is on monotonic base flow profiles $U$ without an inflection point, in order to study critical layer instability (CLI) and its interaction with radiative instability (RI). First the dispersion relation is presented for the piecewise linear profile studied numerically by Satomura (J. Meterol. Soc. Japan, vol. 59, 1981, pp. 148–167) and using WKBJ analysis an interpretation given of mode branches, resonances and radiative instability. In particular surface gravity (SG) waves can resonate with a limit mode (LM) (or Rayleigh wave), localised near the discontinuity in shear in the flow; in this piecewise profile there is no critical layer. The piecewise linear profile is then continuously modified in a family of nonlinear profiles, to show the effect of the vorticity gradient $Q^{\prime } = - U^{\prime \prime }$ on the nature of the modes. Some modes remain as modes and others turn into quasi-modes (QM), linked to Landau damping of disturbances to the flow, depending on the sign of the vorticity gradient at the critical point. Thus an interpretation of critical layer instability for continuous profiles is given, as the remnant of the resonance with the LM. Numerical results and WKBJ analysis of critical layer instability and radiative instability for more general smooth profiles are provided. A link is made between growth rate formulae obtained by considering wave momentum and those found via the WKBJ approximation. Finally the competition between the stabilising effect of vorticity gradients in a critical layer and the destabilising effect of radiation (radiative instability) is studied.

JFM classification

Waves/Free-surface Flows: Critical layers Boundary Layers: Free shear layers Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 539 - 569
Copyright
© 2014 Cambridge University Press 

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