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On a theory of amplitude vacillation in baroclinic waves

Published online by Cambridge University Press:  11 April 2006

R. K. Smith
R. K. Smith
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia 3168
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Abstract

This study contributes to the theory of amplitude vacillation for finite amplitude baroclinic waves in a two-layer, quasi-geostrophic, zonal flow as worked out by Pedlosky. In a recent paper the author has shown that Pedlosky omitted a certain side-wall boundary condition on the mean zonal flow. The neglect of this boundary condition results in an unspecified energy source at the side-wall boundaries and the physical problem is incorrectly posed.

In this paper, Pedlosky's analysis is repeated but with the side-wall boundary condition included. It is shown that the side-wall energy source is negligible only when the zonal wavenumber of the disturbance is large compared with the meridional wavenumber, and not otherwise. Moreover, the energy conversions to and from mean zonal kinetic energy corresponding to Pedlosky's calculations and those given here have essential differences, although for fixed meridional wavenumber, these differences become less pronounced as the zonal wavenumber increases.

It is also shown that, when the side-wall condition is included, the mean flow distortion associated with the wave is different in structure to that which occurs when the condition is omitted. However, as the total disturbance wavenumber a increases, the influence of the side wall on the mean flow structure is confined to a boundary layer of width comparable with the internal deformation radius 2½a−1.

Even when the deformation radius is comparable with the channel width, the conclusions of Pedlosky (1972) concerning the existence of stable periodic solutions are correct, providing viscous effects are vanishingly small, and the criteria for stability of the steady solutions obtained herein are not significantly different from those given by Pedlosky. In this viscous regime, we have also studied the evolution to limit-cycle solutions.

Evolution in the case where viscous effects are small on the time scale for the initial growth of an incipient wave, but not vanishingly small, will be discussed in a subsequent paper.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 79 , Issue 2 , 22 February 1977 , pp. 289 - 306
Copyright
© 1977 Cambridge University Press

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References

Byrd, P. F. & Friedman, M. D. 1971 Handbook of Elliptic Integrals for Engineers and Physicists. Springer.
Cole, J. 1968 Perturbation Methods in Applied Mathematics, chap. 3. Blaisdell.
Fowlis, W. W. & Pfeffer, R. L. 1969 Characteristics of amplitude vacillation in a rotating, differentially-heated fluid determined by a multi-probe technique. J. Atmos. Sci. 26, 100–108.Google Scholar
Hart, J. E. 1972 A laboratory study of baroclinic instability. Geophys. Fluid Dyn. 3, 181–209.Google Scholar
Hart, J. E. 1973 On the behaviour of large-amplitude baroclinic waves. J. Atmos. Sci. 30, 1017–1034.Google Scholar
Hide, R. 1969 Some laboratory experiments on free thermal convection in a rotating fluid subject to a horizontal temperature gradient and their relation to the theory of the global atmospheric circulation. In The Global Circulation of the Atmosphere (ed. G. A. Corby), pp. 196–221. London: Roy. Met. Soc.
Kuzmak, G. E. 1959 Asymptotic solutions of nonlinear differential equations of second order with variable coefficients. J. Appl. Math. Mech. 23, 730–744.Google Scholar
Pedlosky, J. 1970 Finite amplitude baroclinic waves. J. Atmos. Sci. 27, 15–30.Google Scholar
Pedlosky, J. 1971 Finite amplitude baroclinic waves with small dissipation. J. Atmos. Sci. 28, 587–597.Google Scholar
Pedlosky, J. 1972 Limit cycles and unstable baroclinic waves. J. Atmos. Sci. 29, 53–63.Google Scholar
Pedlosky, J. 1975 Comments on the note of R. K. Smith. J. Atmos. Sci. 32, 2027.Google Scholar
Pfeffer, R. L. & Chiang, Y. 1967 Two kinds of vacillation in rotating annulus experiments. Mon. Weather Rev. 95, 75–82.Google Scholar
Smith, R. K. 1974 On limit cycles and vacillating baroclinic waves. J. Atmos. Sci. 31, 2008–2011.Google Scholar