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Critical layer and radiative instabilities in shallow-water shear flows

  • Xavier Riedinger (a1) and Andrew D. Gilbert (a1)

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.

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Corresponding author

Email address for correspondence: xavier.riedinger@gmail.com

References

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Balmforth, N. J. 1999 Shear instability in shallow water. J. Fluid Mech. 387, 97127.
Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 2001 Disturbing vortices. J. Fluid Mech. 426, 95133.
Bassom, A. P. & Gilbert, A. D. 1999 The spiral wind-up and dissipation of vorticity and a passive scalar in a strained planar vortex. J. Fluid Mech. 398, 245270.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Berry, M. V. 1989 Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. R. Soc. Lond. A 422, 721.
Billant, P. & Le Dizès, S. 2009 Waves on a columnar vortex in a strongly stratified fluid. Phys. Fluids 21, 106602.
Bretherton, F. P. 1966 Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc. 92, 325334.
Briggs, R. J., Daugherty, J. D. & Levy, R. H. 1970 Role of Landau damping in cross-field electron beams and inviscid shear flow. Phys. Fluids 13, 421432.
Broadbent, E. G. & Moore, D. W. 1979 Acoustic destabilization of vortices. Phil. Trans. R. Soc. Lond. A 290, 353371.
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria of stability of laminar flow and for the baroclinic circular vortex. Geophys. Publ. 17, 152.
Ford, R. 1994 The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech. 280, 303334.
Glatzel, W. 1985 Sonic instabilities in supersonic shear flows. Mon. Not. R. Astron. Soc. 281, 795821.
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow waters. J. Fluid Mech. 184, 477504.
van Heijst, G. J. F. 1991 Laboratory experiments on the tripolar vortex in a rotating fluid. J. Fluid Mech. 225, 301331.
Iga, K. 1999 Critical layer instability as a resonance between a non-singular mode and continuous modes. Fluid Dyn. Res. 25, 6386.
Knessl, C. & Keller, J. B. 1995 Stability of linear shear flows in shallow water. J. Fluid Mech. 303, 203214.
Kubokawa, A. 1985 Instability of a geostrophic front and its energetics. Geophys. Astrophys. Fluid Dyn. 33, 223257.
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Q. Appl. Maths 3, 117142.
Lindzen, R. S. & Barker, J. W. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189217.
Lindzen, R. S. & Tung, K. K. 1978 Wave over-reflection and shear instability. J. Atmos. Sci. 35, 16261632.
Mak, J., Griffiths, S. D. & Hughes, D. W. 2014 Shear instabilities in shallow water MHD. J. Fluid Mech. (in preparation).
Olver, F. W. J. 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.
Papaloizou, J. C. B. & Pringle, J. E. 1987 The dynamical stability of differentially rotating disks III. Mon. Not. R. Astron. Soc. 225, 267283.
Park, J. & Billant, P. 2012 Radiative instability of an anticyclonic vortex in a stratified rotating fluid. J. Fluid Mech. 707, 381392.
Park, J. & Billant, P. 2013 The stably stratified Taylor–Couette flow is always unstable except for solid-body rotation. J. Fluid Mech. 725, 262280.
Parras, L. & Le Dizès, S. 2010 Temporal instability modes of supersonic round jets. J. Fluid Mech. 662, 173196.
Perkins, J. & Renardy, M. 1997 Stability of equatorial currents with nonzero potential vorticity. Geophys. Astrophys. Fluid Dyn. 85, 3164.
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11, 5770.
Riedinger, X., Le Dizès, S. & Meunier, P. 2010a Viscous stability properties of a Lamb–Oseen vortex in a stratified fluid. J. Fluid Mech. 645, 255278.
Riedinger, X., Le Dizès, S. & Meunier, P. 2011 Radiative instability of the flow around a rotating cylinder in a stratified fluid. J. Fluid Mech. 672, 130146.
Riedinger, X., Meunier, P. & Le Dizès, S. 2010b Instability of a vertical columnar vortex in a stratified fluid. Exp. Fluids 49, 673681.
Rossi, L. F., Lingevitch, J. F. & Bernoff, A. J. 1997 Quasi-steady monopole and tripole attractors for relaxing vortices. Phys. Fluids 9, 23292338.
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Meterol. Soc. Japan 59, 148167.
Schecter, D. A., Dubin, D. H. E., Cass, A. C., Driscoll, C. F., Lansky, I. M. & O’Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12, 23972412.
Schecter, D. A. & Montgomery, M. T. 2004 Damping and pumping of a vortex Rossby wave in a monotonic cyclone: critical layer stirring versus inertial–buoyancy wave emission. Phys. Fluids 16, 13341348.
Shepard, H. K. 1983 Decay widths for metastable states. Improved WKB approximation. Phys. Rev. D 27, 12881298.
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.
Takehiro, S.-I. & Hayashi, Y.-Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 259279.
Turner, M. R., Gilbert, A. D. & Bassom, A. P. 2008 Neutral modes of a two-dimensional vortex and their link to persistent cat’s eyes. Phys. Fluids 20, 027101.
Vanneste, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid Mech. 45, 147172.
Yim, E. & Billant, P. 2013 Spontaneous bending of a columnar vortex in stratified-rotating fluid. Bull. Am. Phys. Soc. 58 (18), 367.
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Critical layer and radiative instabilities in shallow-water shear flows

  • Xavier Riedinger (a1) and Andrew D. Gilbert (a1)

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