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Vertical scale selection in inertial instability†

Published online by Cambridge University Press:  14 December 2007

R. C. KLOOSTERZIEL  and
G. F. CARNEVALE
R. C. KLOOSTERZIEL
Affiliation:
School of Ocean & Earth Science & Technology, University of Hawaii, Honolulu, HI 96822, USA
G. F. CARNEVALE
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

The linear instability of a barotropic flow with uniform horizontal shear in a stratified rotating fluid is investigated with respect to perturbations invariant in the alongflow direction. The flow can be inertially unstable if there is sufficiently strong anticyclonic shear, but only for sufficiently high Reynolds numbers Re. We determine the critical Reynolds numbers required for amplification of the instability as a function of Prandtl number, strength of the stratification and magnitude of the shear. The vertical scales at the onset of the instability are calculated. For Prandtl number P < 1.44 instability always sets in through stationary overturning motions, for P > 1.44 it may also commence through overstable (oscillatory) motions. For Re exceeding the critical value, we determine the vertical scale of the most rapidly amplifying modes and the corresponding growth rates and how they vary with Re, P, the shear and the strength of stratification.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 594 , 10 January 2008 , pp. 249 - 269
Copyright
Copyright © Cambridge University Press 2008

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Footnotes

†

With an appendix by Stephen D. Griffiths

References

REFERENCES

Bayly, B. J. 1988 Three dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31, 56–64.CrossRefGoogle Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.CrossRefGoogle Scholar
Caton, F., Janiaud, B., & Hopfinger, E. J. 2000 Stability and bifurcations in stratified Taylor–Couette flow. J. Fluid Mech. 419, 93–114.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Charney, J. G. 1973 Lecture notes on planetary fluid dynamics. Dynamic Meteorology (ed. Morel, P.). Reidel.Google Scholar
Drazin, P. & Reid, W. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dunkerton, T. J. 1981 On the inertial instability of the equatorial middle atmosphere. J. Atmos. Sci. 38, 2354–2365.2.0.CO;2>CrossRefGoogle Scholar
Dunkerton, T. J. 1982 The double-diffusive modes of symmetric instability on an equatorial beta-plane. J. Atmos. Sci. 39, 1653–1657.2.0.CO;2>CrossRefGoogle Scholar
Emanuel, K. A. 1979 Inertial instability and mesoscale convective systems. Part I: Linear theory of inertial instability in rotating viscous fluids. J. Atmos. Sci. 36, 2425–2449.2.0.CO;2>CrossRefGoogle Scholar
Griffiths, S. D. 2003 a Nonlinear vertical scale selection in equatorial inertial instability. J. Atmos. Sci. 60, 977–990.2.0.CO;2>CrossRefGoogle Scholar
Griffiths, S. D. 2003 b The nonlinear evolution of of zonally symmetric equatorial inertial instability. J. Fluid Mech. 474, 245–273.CrossRefGoogle Scholar
Griffiths, S. D. 2007 The limiting form of inertial instability in geophysical flows. Submitted to J. Fluid Mech.CrossRefGoogle Scholar
Hoskins, B. J. 1974 The role of potential vorticity in symmetric stability and instability. Q. J. R. Met. Soc. 100, 480–482.CrossRefGoogle Scholar
Kline, M. 1972 Mathematical Thought from Ancient to Modern Times. Oxford University Press.Google Scholar
Kloosterziel, R. C. & Carnevale, G. F. 2003 Closed-form linear stability conditions for rotating Rayleigh-Bénard convection with rigid, stress-free upper and lower boundaries. J. Fluid Mech. 480, 25–42.CrossRefGoogle Scholar
Kloosterziel, R. C. & Carnevale, G. F. 2007 Generalized energetics for inertially stable parallel shear flows. J. Fluid Mech. 585, 117–126.CrossRefGoogle Scholar
Kloosterziel, R. C., Carnevale, G. F. & Orlandi, P. 2007 Inertial instability in rotating and stratified fluids: barotropic vortices. J. Fluid Mech. 583, 379–412.CrossRefGoogle Scholar
McIntyre, M. E. 1970 Diffusive destabilization of the baroclinic circular vortex. Geophys. Fluid Dyn. 1, 19–58.CrossRefGoogle Scholar
Ooyama, K. 1966 On the stability of the baroclinic circular vortex: a sufficient condition for instability. J. Atmos. Sci. 23, 43–53.2.0.CO;2>CrossRefGoogle Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Smyth, W. D. & Mcwilliams, J. C. 1998 Instability of an axisymmetric vortex in a stably stratified, rotating environment. Theor. Comput. Fluid Dyn. 11, 305–322.CrossRefGoogle Scholar
Smyth, W. D. & Peltier, W. R. 1994 Three-dimensionalization of barotropic vortices on the f-plane. J. Fluid Mech. 265, 25–64.CrossRefGoogle Scholar
Stone, P. H. 1966 On non-geostrophic stability. J. Atmos. Sci. 23, 390–400.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S. A. 1966 The stability of stratified Couette flow. In Notes on 1966 Summer Geophys. Fluid Dyn., pp. 80–107. Woods Hole Oceanographic Inst.Google Scholar