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Ageostrophic instabilities in a horizontally uniform baroclinic flow along a slope

Published online by Cambridge University Press:  24 September 2007

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

The normal modes of a horizontally uniform, vertically sheared flow over a sloping bottom are considered in two active layers underneath a deep motionless third layer. The variations of the layer thickness are assumed to be small to analyse the sixth-order eigenvalue problem for finite-Froude-number typical for oceanic currents. The dispersion curves for the Rossby waves and the Poincaré modes of inertia–gravity waves (IGW) are investigated to identify the different types of instabilities that occur if there is a pair of wave components which have almost the same Doppler-shifted frequency related to crossover of the branches when the Froude number increases. Simple criteria for ageostrophic instabilities due to a resonance between the IGW and the Rossby wave because of the thickness gradient in either the lower or middle layer, are derived. They exactly correspond to violation of sufficient Ripa's conditions for the flow stability. In both cases the growth rate and the interval of unstable wavenumbers are shown to be proportional to the square root of the corresponding gradient of the layer thickness. These types of ageostrophic instability can coexist (and with Kelvin–Helmholtz instability). However, their role in generating unbalanced motions and mixing processes in geophysical fluids appears limited due to small growth rates and narrow intervals of the unstable wavenumbers in comparison to Kelvin–Helmholtz instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 588 , 10 October 2007 , pp. 463 - 473
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Cushman-Roisin, B. 1994 Introduction to Geophysical Fluid Dynamics. Prentice Hall.Google Scholar
Dritschel, D. G. & Viudez, A. 2007 The persistence of balance in geophysical flows. J. Fluid Mech. 570, 365–383.CrossRefGoogle Scholar
Ford, R. 1993 Gravity wave generation by vortical flows in a rotating frame. PhD Thesis, University of Cambridge.Google Scholar
Griffiths, R. W., Killworth, P. D. & Stern, M. E. 1982 Ageostrophic instability of ocean currents. J. Fluid Mech. 117, 343–377Google Scholar
Hayashi, Y. Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow water. J. Fluid Mech. 184, 477–504.Google Scholar
McIntyre, M. E. & Norton, W. A. 2000 Potential vorticity inversion on a hemisphere. J. Atmos. Sci. 57, 1214–1235, and Corrigendum 58, 949.Google 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
Meacham, S. P. & Stephens, J. C. 2001 Instabilities of gravity currents along a slope. J. Phys. Oceanogr. 31, 30–53.Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2005 Baroclinic instability and loss of balance. J. Phys. Oceanogr. 35, 1505–1517.Google Scholar
Mooney, C. J. & Swaters, G. E. 1996 Finite-amplitude baroclinic instability of a mesoscale gravity current in a channel. Geophys. Astrophys. Fluid Dyn. 82, 173–205.Google Scholar
Nakamura, K. 1988 The scale selection of baroclinic instability – effect of stratification and nongeostrophy. J. Atmos. Sci. 45, 3253–3267.2.0.CO;2>CrossRefGoogle Scholar
Orlanski, I. 1968 Instability of frontal waves. J. Atmos. Sci. 25, 178–200.Google Scholar
Pichevin, T. 1998 Baroclinic instability in a three layer flow: a wave approach. Dyn. Atmos. Oceans 28, 179–204.Google Scholar
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasigeostrophic model. Tellus 6, 273–286.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
Ripa, P. 1991 General stability conditions for a multi-layer model. J. Fluid Mech. 222, 119–137.Google 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
Stone, P. 1966 On non-geostrophic baroclinic stability. J. Atmos. Sci. 23, 390–400.Google Scholar
Stone, P. 1970 On non-geostrophic baroclinic stability: Part 11. J. Atmos. Sci. 27, 721–726.Google 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.Google 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.Google 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