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On the non-separable baroclinic parallel flow instability problem

Published online by Cambridge University Press:  29 March 2006

Michael E. McIntyre
Michael E. McIntyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Perturbation series are developed and mathematically justified, using a straightforward perturbation formalism (that is more widely applicable than those given in standard textbooks), for the case of the two-dimensional inviscid Orr-Sommerfeld-like eigenvalue problem describing quasi-geostrophic wave instabilities of parallel flows in rotating stratified fluids.

The results are first used to examine the instability properties of the perturbed Eady problem, in which the zonal velocity profile has the form u = z + μu1(y, z) where, formally, μ [Lt ] 1. The connexion between baroclinic instability theories with and without short wave cutoffs is clarified. In particular, it is established rigorously that there is instability at short wavelengths in all cases for which such instability would be expected from the ‘critical layer’ argument of Bretherton. (Therefore the apparently conflicting results obtained earlier by Pedlosky are in error.)

For the class of profiles of form u = z + μu1(y) it is then shown from an examination of the O(μ) eigenfunction correction why, under certain conditions, growing baroclinic waves will always produce a counter-gradient horizontal eddy flux of zonal momentum tending to reinforce the horizontal shear of such profiles. Finally, by computing a sufficient number of the higher corrections, this first-order result is shown to remain true, and its relationship to the actual rate of change of the mean flow is also displayed, for a particular jet-like form of profile with finite horizontal shear. The latter detailed results may help to explain at least one interesting feature of the mean flow found in a recent numerical solution for the wave régime in a heated rotating annulus.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 40 , Issue 2 , 3 February 1970 , pp. 273 - 306
Copyright
© 1970 Cambridge University Press

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References

Barcilon, V. 1964 Role of the Ekman layers in the stability of the symmetric regime obtained in a rotating annulus. J. Atmos. Sci. 21, 291–9.Google Scholar
Bretherton, F. P. 1966a Critical layer instability in baroclinic flows. Quart. J. Roy. Met. Soc. 92, 325–34.Google Scholar
Bretherton, F. P. 1966b Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Quart. J. Roy. Met. Soc. 92, 335–45.Google Scholar
Brown, J. A. 1969a A numerical investigation of hydrodynamic instability and energy conversions in the quasi-geostrophic atmosphere. Part I. J. Atmos. Sci. 26, 352–65.Google Scholar
Brown, J. A. 1969b A numerical investigation of hydrodynamic instability and energy conversions in the quasi-geostrophic atmosphere. Part II. J. Atmos. Sci. 26, 366–75.Google Scholar
Burger, A. P. 1966 Instability associated with the continuous spectrum in a baroclinic flow. J. Atmos. Sci. 23, 272–7.Google Scholar
Charney, J. G. 1947 The dynamics of long waves in a baroclinic westerly current. J. Meteor. 4, 135–62.Google Scholar
Charney, J. G. & Stern, M. 1962 On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci. 19, 159–72.Google Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, vol I. New York: Interscience.
Derome, J. & Dolph, C. L. 1969 Three-dimensional non-geostrophic disturbances in a baroclinic zonal flow. Geophysical Fluid Dynamics (in press).Google Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1, 33–52.Google Scholar
Eliasen, E. 1961 On the interactions between the long baroclinic waves and the mean zonal flow. Tellus 13, 40–55.Google Scholar
Eliassen, A. 1952 Slow thermally or frictionally controlled meridional circulations in a circular vortex. Astrophysica Norvegica, 5, 19–60.Google Scholar
Fowlis, W. W. & Hide, R. 1965 Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J. Atmos. Sci. 22, 541–58.Google Scholar
Garcia, R. V. & Norscini, R. 1969 A contribution to the baroclinic instability problem. Tellus (in press).Google Scholar
Green, J. S. A. 1960 A problem in baroclinic stability. Quart. J. Roy. Met. Soc. 86, 237–51.Google Scholar
Green, J. S. A. 1970 Transfer properties of the large-scale eddies, and the general circulation of the atmosphere. Quart. J. Roy. Met. Soc. (in press).Google Scholar
Holmboe, J. 1959 On the behaviour of baroclinic waves. The Rossby Memorial Volume (ed. B. Bolin), pp. 333–49. New York: Rockefeller.
Joseph, D. D. 1967 Parameter and domain dependence of eigenvalues of elliptic partial differential equations. Arch. Rat. Mech. Anal. 24, 325–51.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lin, C. C. 1961 Some mathematical problems in the theory of the stability of parallel flows. J. Fluid Mech. 10, 430–8.Google Scholar
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157–67.Google Scholar
Lorenz, E. N. 1967 The Nature and Theory of the General Circulation of the Atmosphere. Geneva: World Meteorological Organization.
Magaard, L. 1963 Baroclinic instability. Geophysical Fluid Dynamics Notes III, p. 63. Woods Hole Oceanographic Institution.Google Scholar
McIntyre, M. E. 1967 Convection and baroclinic instability in rotating fluids. Ph.D. thesis, University of Cambridge.
McIntyre, M. E. 1969a Diffusive destabilization of the baroclinic circular vortex. Geophysical Fluid Dynamics (in press).Google Scholar
McIntyre, M. E. 1969b Role of diffusive overturning in nonlinear axisymmetric convection in a differentially heated rotating annulus. Geophysical Fluid Dynamics (in press).Google Scholar
McIntyre, M. E. 1969c Baroclinic instability of Murray's continuous model of the polar night jet (submitted to Quart. J. Roy. Met. Soc.).Google Scholar
Miles, J. W. 1964 Baroclinic instability of the zonal wind. Rev. Geophys. 2, 155–76.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. London: McGraw-Hill.
Orr, W. M. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. Roy. Irish. Acad. 27, 9–138.Google Scholar
Pedlosky, J. 1964a The stability of currents in the atmosphere and the ocean. Part I. J. Atmos. Sci. 21, 201–19.Google Scholar
Pedlosky, J. 1964b The stability of currents in the atmosphere and the ocean. Part II. J. Atmos. Sci. 21, 342–53.Google Scholar
Pedlosky, J. 1964c An initial value problem in the theory of baroclinic instability. Tellus 16, 12–17.Google Scholar
Pedlosky, J. 1965 On the stability of baroclinic flows as a functional of the velocity profile. J. Atmos. Sci. 22, 137–45.Google Scholar
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus, 6, 273–86.Google Scholar
Phillips, N. A. 1963 Geostrophic motion. Rev. Geophys. 1, 123–76.Google Scholar
Rayleigh, Lord 1895 On the stability or instability of certain fluid motions, III. Scientific Papers 4, 203–9. Cambridge University Press.
Smagorinsky, J. 1964 Some aspects of the general circulation. Quart. J. Roy. Met. Soc. 90, 1–14.Google Scholar
Stone, P. H. 1969 The meridional structure of baroclinic waves. J. Atmos. Sci. 26, 376–89.Google Scholar
Thompson, P. D. 1959 Some statistical aspects of the dynamical processes of growth and occlusion in simple baroclinic models. The Rossby Memorial Volume (ed. B. Bolin), pp. 350–8. New York: Rockefeller.
Titchmarsh, E. C. 1958 Eigenfunction Expansions, Part II. Oxford University Press.
Walin, G. 1969 Some aspects of time-dependent motion of a rotating stratified fluid. J. Fluid Mech. 36, 289–307.Google Scholar
Williams, G. P. 1969 Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow. J. Fluid Mech. 37, 727–50.Google Scholar