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Asymptotic theory of thermal convection in rapidly rotating systems

Published online by Cambridge University Press:  26 April 2006

Jun-Ichi Yano
Affiliation:
Theoretische Physik IV. Physikalisches Institut, Universität Bayreuth, Postfach 101251, W-8580 Bayreuth, Germany Current affiliation: NCAR, PO Box 3000, Boulder, Colorado 80307-3000, USA.

Abstract

An asymptotic theory of marginal thermal convection in rotating systems is constructed for the limit of rapid rotation. Many self-gravitating astronomical bodies, including the major planets, the Sun, and the Earth's liquid core, correspond to this limit. In the laboratory, an analogous system can be constructed with a very rapidly rotating apparatus, in which the centrifugal force plays the role of self-gravitation. The formulation is offered in such a way that both these geophysical systems and laboratory analogues are included as special cases. When the inclination of the outer boundaries relative to the equatorial plane is considered weak, the two types of system are identical at leading order. In this limit, the asymptotic analysis is profoundly simplified, because the system satisfies the Taylor-Proudman theorem to leading order. Nevertheless the system contains a very peculiar property: the mode defined by a conventional WKBJ theory implicitly assuming a locality of convection in the radial direction perpendicular to the axis of rotation cannot be accepted as a correct marginal mode, because a modulation equation gives an exponential growth in the radial direction, which contradicts an implicit initial assumption. The erroneous behaviour is traced to a spatial dispersion of thermal Rossby waves, which governs the marginal mode. The difficulty is resolved by extending the analysis to a complex plane of the radial coordinate of the point where convection amplitude attains its maximum. Such a complex radial distance is defined as the point where the wave dispersion disappears locally. The projection of the solution onto the real axis results in an inclination of the Taylor columns with respect to the radial direction. This is in good agreement with the most recent numerical studies. The isolation of convective Taylor columns in the radial direction weakens and the spiralling gets stronger as the Prandtl number decreases, as a result of the need to displace the critical radial distance further from the real axis.

Type
Research Article
Copyright
© 1992 Cambridge University Press

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References

Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441460.Google Scholar
Busse, F. H. 1983 A model of mean zonal flows in the major planets. Geophys. Astrophys. Fluid Dyn. 23, 153174.Google Scholar
Busse, F. H. 1986 Asymptotic theory of convection in a rotating cylindrical annulus. J. Fluid Mech. 173, 545556.Google Scholar
Busse, F. H. & Carrigan, C. R. 1976 Laboratory simulation of thermal convection in rotating planets and stars. Science 191, 8183.Google Scholar
Busse, F. H. & Hood, L. L. 1982 Differential rotation driven by convection in a rapidly rotating annulus. Geophys. Astrophys. Fluid Dyn. 21, 5974.Google Scholar
Carrigan, C. R. & Busse, F. H. 1983 An experimental and theoretical investigation of the onset of convection in rotating spherical shells. J. Fluid Mech. 126, 287305.Google Scholar
Chandrasekahr, S. 1961 Hydrodynamic and Hydromagnetic Instability. Clarendon.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hirsching, W. R. & Yano, J.-I. 1992 Metamorphosis of marginal thermal convection in rapidly rotating self-gravitating spherical shells. Geophys. Astrophys. Fluid Dyn. (submitted).Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mecch. 22. 473537.Google Scholar
Ingersoll, A. P. & Pollard, D. 1982 Motion in the interior and atmospheres of Jupiter and Saturn: Scale analysis, anelastic equations, barotropic stability criterion. Icarus 52. 6280.Google Scholar
Kawahara, T. 1973 The derivative-expansion method and non-linear dispersive waves. J. Phys. Soc. Japan 35. 15371544.Google Scholar
Miller, J. C. P. 1950 On the choice of standard solutions for a homogeneous linear differential equation of the second order. Q. J. Mech. Appl. Maths, 3, 225235.Google Scholar
Olver, F. W. J. 1974 Asymptotics and Special Functions. Academic.
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.
Roberts, P. H. 1968 On the thermal instability of a rotating fluid sphere containing heat sources.. Phil. Trans. R. Soc. Lond. A 263, 93117.Google Scholar
Soward, A. M. 1977 On the finite amplitude instability of a rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9. 1974.Google Scholar
Soward, A. M. & Jones, C. A. 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Maths 36, 1942.Google Scholar
Zhang, K. K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535556.Google Scholar
Zhang, K. K. & Busse, F. H. 1987 On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 39, 119147.Google Scholar