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Unbalanced instabilities of rapidly rotating stratified shear flows

Published online by Cambridge University Press:  25 July 2007

J. VANNESTE  and
I. YAVNEH
J. VANNESTE
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK
I. YAVNEH
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
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Abstract

The linear stability of a rotating stratified inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. Two dimensionless parameters characterize the flow: the Rossby number ε, defined as the ratio of the shear to the Coriolis frequency and assumed small, and the ratio s of the Coriolis frequency to the buoyancy frequency, assumed to satisfy s ≤ 1. An energy argument is used to show that unstable perturbations must have large, O(ε−1), wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia–gravity waves with or without turning points.

Although the waves have real phase speeds to all algebraic orders in ε, we establish that the flow is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. Three modes of instabilities are identified, corresponding to the resonance between (i) a pair of Kelvin waves, (ii) a Kelvin wave and an inertia–gravity wave, and (iii) a pair of inertia–gravity waves. Whilst all three modes of instability are active when the Couette flow is anticyclonic, mode (iii) is the only possible instability mechanism when the flow is cyclonic.

We derive asymptotic estimates for the instability growth rates. These are exponentially small in ε, i.e. of the form Im ω = a exp(-Ψ/ε) for some positive constants a and Ψ. For the Kelvin-wave instabilities (i), we obtain analytic expressions for a and Ψ; the maximum growth rate, in particular, corresponds to Ψ = 2. For the other types of instabilities, we make the simplifying assumption s ≪ 1 and find that the maximum growth rates correspond to Ψ=2.80 for (ii) and Ψ= π for (iii). The asymptotic results are confirmed by numerical computations. These reveal, in particular, that the instabilities (iii) have much smaller growth rates in cyclonic flows than in anticyclonic flows, even though Ψ = π in both cases.

Our results highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia–gravity waves and hence predict the stability of Couette flow. They are also relevant to the stability of Taylor–Couette flows and of astrophysical accretion disks.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 373 - 396
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Allen, J. S., Barth, J. A. & Newberger, P. A. 1990 On intermediate models for barotropic continental shelf and slope flow fields. Part I: Formulation and comparison of exact solutions. J. Phys. Oceanogr. 20, 1017–1042.2.0.CO;2>CrossRefGoogle Scholar
Balmforth, N. J. 1996 Shear instability in shallow water. J. Fluid Mech. 387, 97–127.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.CrossRefGoogle Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flow. J. Fluid Mech. 92, 1–14.CrossRefGoogle Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Dritschel, D. G. & Vanneste, J. 2006 Instability of a shallow-water potential-vorticity front. J. Fluid Mech. 561, 237–254.CrossRefGoogle Scholar
Dubrulle, B., Marié,, L., Normand, C., Richards, D., Hersant, F. & Zahn, J. 2005 A hydrodynamic shear instability in stratified disks. Astron. Astrophys. 429, 1–13.CrossRefGoogle Scholar
Ford, R. 1994 The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech. 280, 303–334.CrossRefGoogle Scholar
Howard, J. E. & MacKay, R. S. 1987 Linear stability of symplectic maps. J. Math. Phys. 28, 1036–1051.CrossRefGoogle Scholar
Knessl, C. & Keller, J. B. 1992 Stability of rotating shear flow in shallow water. J. Fluid Mech. 244, 605–614.CrossRefGoogle Scholar
Kushner, P. J., McIntyre, M. E. & Shepherd, T. G. 1998 Coupled Kelvin wave and mirage-wave instabilities in semi-geostrophic dynamics. J. Phys. Oceanogr. 28, 513–518.2.0.CO;2>CrossRefGoogle Scholar
McWilliams, J. C., Molemaker, M. J. & Yavneh, I. 2004 Ageostrophic, anticyclonic instability of a geostrophic, barotropic boundary current. Phys. Fluids 16, 3720–3725.CrossRefGoogle Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally-stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270–5273.CrossRefGoogle ScholarPubMed
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2005 Baroclinic instability and loss of balance. J. Phys. Oceanogr. 35, 1505–1517.CrossRefGoogle Scholar
Narayan, R., Goldreich, P. & Goodman, J. 1987 Physics of modes in a differentially rotating system – analysis of shearing sheet. Mon. Not. R. Astr. Soc. 228, 1–41.CrossRefGoogle Scholar
Papaloizou, J. C. B. & Pringle, J. E. 1987 The dynamical stability of differentially rotating discs – III. Mon. Not. R. Astr. Soc. 225, 267–283.CrossRefGoogle Scholar
Plougonven, R., Muraki, D. J. & Snyder, C. 2005 A baroclinic instability that couples balanced motions and gravity waves. J. Atmos. Sci. 62, 1545–1559.CrossRefGoogle Scholar
Ren, S. & Shepherd, T. G. 1997 Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics. J. Fluid Mech. 345, 287–305.CrossRefGoogle Scholar
Ripa, P. 1983 General stability conditions for zonal flow in a one-layer model on the β-plane or the sphere. J. Fluid Mech. 126, 463–489.CrossRefGoogle Scholar
Ripa, P. 1990 Positive, negative and zero wave energy and the flow stability problem in the Eulerian and Lagrangian–Eulerian descriptions, Pure Appl. Geophys. 133, 713–732.CrossRefGoogle Scholar
Rüdiger, G., Arlt, R. & Shalybkov, D. 2002 Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification. Astron. Astrophys. 391, 781–787.CrossRefGoogle Scholar
Sakai, S. 1989 Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves, J. Fluid Mech. 202, 149–176.CrossRefGoogle Scholar
Satomura, T. 1981 a An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148–167.CrossRefGoogle Scholar
Satomura, T. 1981 b Supplementary notes on shear instability in a shallow water. J. Met. Soc. Japan 59, 168–171.CrossRefGoogle Scholar
Satomura, T. 1982 An investigation of shear instability in a shallow water, part II: numerical experiment. J. Met. Soc. Japan 60, 227–244.CrossRefGoogle Scholar
Vanneste, J. & Yavneh, I. 2004 Exponentially small inertia–gravity waves and the breakdown of quasi-geostrophic balance. J. Atmos. Sci. 61, 211–223.2.0.CO;2>CrossRefGoogle Scholar
Warn, T. 1997 Nonlinear balance and quasi-geostrophic sets. Atmos. Ocean 35, 135–145.CrossRefGoogle Scholar
Warn, T., Bokhove, O., Shepherd, T. G. & Vallis, G. K. 1995 Rossby number expansions, slaving principles, and balance dynamics. Q. J. R. Met. Soc. 121, 723–739.CrossRefGoogle Scholar
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetric instability of centrifugally-stable stratified Taylor–Couette flow. J. Fluid Mech. 448, 1–21.CrossRefGoogle Scholar