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The stability of quasi-geostrophic fields induced by potential vorticity sources

Published online by Cambridge University Press:  20 April 2006

Lee-Or Merkine
Lee-Or Merkine
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa
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Abstract

The stability of quasi-geostrophic barotropic fields induced by localized finite-amplitude potential vorticity sources or topographies which depend only on the zonal direction is investigated both analytically and numerically. The analytical study is for weak forcing. It demonstrates that the field induced by the topography is stable whereas the field induced by the potential vorticity source can be unstable. The growth rate is exponential and is a function of both nonlinearity and friction. In the absence of friction the flow field is always unstable. The instability takes the form of a current whose meridional wavenumber is that of a stationary Rossby wave. In the zonal direction the current exhibits long-scale oscillations and exponential decay. The numerical computations which are for strong forcing verify all the indications of the asymptotic study. They show a rapid exponential growth of a non-propagating but oscillatory wave packet whose location is fixed relative to the forcing. The zonal scale of the packet is that of a stationary Rossby wave. For weak forcing the instability can be responsible for changing the flow field from one quasi-steady state to another where the energy extraction takes place in the region of the source. It is efficient for potential vorticity sources whose length scale is comparable to the length scale of stationary Rossby waves. In agreement with the asymptotic study, fields induced by strong topographic forcing are found to be stable. The asymptotic analysis is also applied to baroclinic flows where the investigation is performed in the framework of the two-layer model. It is demonstrated that the same mechanism which operates in barotropic systems can also destabilize baroclinic flows which possess subcritical shears.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 116 , March 1982 , pp. 315 - 342
Copyright
© 1982 Cambridge University Press

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References

Barnard, B. J. S. & Pritchard, W. G. 1972 Cross-waves. Part 2. Experiments. J. Fluid Mech. 55, 245–255.Google Scholar
Charney, J. G. & Devore, J. G. 1979 Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205–1216.Google Scholar
Coaker, S. A. 1977 The stability of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 1–17.Google Scholar
Ebdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1954 Tables of Integral Transforms. McGraw-Hill.
Fjörtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for a two-dimensional, non divergent flow. Tellus 5, 225–237.Google Scholar
Gill, A. E. 1974a The stability of planetary waves on an infinite beta-plane. Geophys. Fluid Dyn. 6, 29–47.Google Scholar
Gill, A. E. 1974b A fine-scale model of ocean circulation. (Unpublished.)
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Tables of Integrals, Series and Products. Academic.
Hogg, N. G. 1973 On the stratified Taylor column. J. Fluid Mech. 58, 517–537.Google Scholar
Huppert, H. E. 1975 Some remarks on the initiation of inertial Taylor columns. J. Fluid Mech. 67, 397–412.Google Scholar
Lorenz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258–264.Google Scholar
Mahony, J. J. 1972 Cross-waves. Part 1. Theory. J. Fluid Mech. 55, 229–244.Google Scholar
Mccartney, M. S. 1975 Inertial Taylor columns on a beta plane. J. Fluid Mech. 68, 71–95.Google Scholar
Merkine, L. 1975 Steady finite-amplitude baroclinic flow over long topography in a rotating stratified atmosphere. J. Atmos. Sci. 32, 1881–1893.Google Scholar
Merkine, L. 1980 Linear and nonlinear resonance of Rossby waves. Geophys. Astrophys. Fluid Dyn. 15, 53–64.Google Scholar
Newell, A. C. 1972 The post bifurcation stage of baroclinic instability. J. Atmos. Sci. 29, 64–76.Google Scholar
Pedlosky, J. 1970 Finite amplitude baroclinic waves. J. Atmos. Sci. 37, 15–31.Google Scholar