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Lack of balance in continuously stratified rotating flows

Published online by Cambridge University Press:  25 November 2008

GEORGI G. SUTYRIN
GEORGI G. SUTYRIN*
Affiliation:
Graduate School of Oceanography, University of Rhode Island, Narragansett, RI 02882, USA
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Abstract

Periodic linear waves in a vertically sheared flow are considered in a continuously stratified layer of rotating fluid between homogeneous layers along a sloping bottom. This generalized Phillips' configuration has cyclonic horizontal shear and supports the Rossby modes related to the thickness variations of the homogeneous layers and inertia–gravity waves (IGW). While long Rossby modes with streamwise wavenumber κ < f/V (f is the Coriolis parameter, V is the maximum velocity) can be approximated by a neutral balanced solution, short waves with κ > f/V are found to have an inertial critical level and unbalanced gravity-wave-like structure beyond this level. Such ageostrophic unstable normal modes are shown explicitly to couple short Rossby waves with Doppler-shifted gravity waves. They exist even for small Froude number, although the growth rate of ageostrophic unstable modes is exponentially small in Froude number as in the Eady model. This lack of balance in continuously stratified flows agrees with the ultraviolet problem for Ripa's sufficient conditions of stability in a multi-layer model when the number of layers tends to infinity.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 615 , 25 November 2008 , pp. 93 - 100
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abramowitz, M. & Stegun, I. (Eds.) 1964 Handbook of Mathematical Functions. National Bureau of Standards, US Government Printing Office, 1046 pp.Google Scholar
Heifetz, E. & Farrell, B. F. 2007 Generalized stability of nongeostrophic baroclinic shear flow. Part II: Intermediate Richardson number regime. J. Atmos. Sci. 64, 4366–4382.CrossRefGoogle Scholar
McWilliams, J. C. 2003 Diagnostic force balance and its limits. In Nonlinear Processes in Geophysical Fluid Dynamics (ed. Fuentes, O. U. Velasco, Sheinbaum, J. & Ochoa, J.), pp. 287–304. Kluwer.CrossRefGoogle Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2005 Baroclinic instability and loss of balance. J. Phys. Oceanogr. 35, 1505–1517.CrossRefGoogle Scholar
Nakamura, N. 1988 The scale selection of baroclinic instability – effect of stratification and nongeostrophy. J. Atmos. Sci. 45, 3253–3267.2.0.CO;2>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
Poulin, F. J. & Swaters, G. E. 1999 Subinertial dynamics of density-driven flows in a continuously stratified fluid on a sloping bottom. I. Model derivation and stability characteristics. Proc. R. Soc. Lond. A 455, 2281–2304.CrossRefGoogle Scholar
Ripa, P. 1991 General stability conditions for a multi-layer model. J. Fluid Mech. 222, 119–137.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
Sutyrin, G. G. 2004 Agradient velocity, vortical motion and gravity waves in rotating shallow water model. Q. J. R. Met. Soc. 130, 1977–1989.CrossRefGoogle Scholar
Sutyrin, G. G. 2007 Ageostrophic instabilities in a horizontally uniform baroclinic flow along a slope. J. Fluid Mech. 588, 463–473.CrossRefGoogle Scholar
Vanneste, J. & Yavneh, I., 2007 Unbalanced instabilities of rapidly rotating stratified sheared flows. J. Fluid Mech. 584, 373–396.CrossRefGoogle Scholar
Yamazaki, Y. H. & Peltier, W. R. 2001 The existence of subsynoptic-scale baroclinic instability and the nonlinear evolution of shallow disturbances. J. Atmos. Sci. 58, 657–683.2.0.CO;2>CrossRefGoogle Scholar
Zeitlin, V., Reznik, G. M. & Ben Jelloul, M. 2003 Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations. J. Fluid Mech. 491, 207–228.CrossRefGoogle Scholar