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Spin-up of a strongly stratified fluid in a sphere

Published online by Cambridge University Press:  29 March 2006

Alfred Clark ,
Patricia A. Clark ,
John H. Thomas  and
Nien-Hon Lee
Alfred Clark
Affiliation:
Department of Mechanical and Aerospace Sciences University of Rochester, Rochester, New York
Patricia A. Clark
Affiliation:
Department of Mechanical and Aerospace Sciences University of Rochester, Rochester, New York
John H. Thomas
Affiliation:
Department of Mechanical and Aerospace Sciences University of Rochester, Rochester, New York
Nien-Hon Lee
Affiliation:
Department of Mechanical and Aerospace Sciences University of Rochester, Rochester, New York
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Abstract

A linear theory is developed for the spin-up of a compressible fluid, stratified by a spherical gravity field. Numerical results are obtained for the case of strong stratification (BruntVisl frequency N much greater than the rotation frequency 0). The interior flow is solved in terms of a set of angular eigenfunctions which have been obtained numerically. The principal result is that the spin-up is limited to a layer adjacent to the spherical boundary, the thickness of the layer being of the order of L(0/N), where L is the radius of the boundary. The solution is qualitatively similar to that found by Holton (1965), Walin (1969), and Sakurai (1969a, b) for a stratified fluid in a cylinder. The thickness of the spin-up layer diminishes with latitude , the variation being described roughly by the formula L0| sin |/N. For the case of slow continuous spin-up, the Ekman suction velocity has been calculated, and the results show that || = 24 is the dividing angle between suction (|| > 24) and blowing (|| < 24).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 1 , 15 January 1971 , pp. 131 - 149
Copyright
Copyright © Cambridge University Press 1971

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References

Abramowitz, N. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards, Washington.Google Scholar
Carrier, G. F. 1965 Some effects of stratification and geometry in rotating fluids J. Fluid Mech. 23, 145172.Google Scholar
Clark, A., Thomas, J. H. & Clark, P. A. 1969 Solar differential rotation and oblateness Science, 164, 290291.Google Scholar
Dicke, R. H. 1964 The sun's rotation and relativity Nature, Lond. 202, 432435.Google Scholar
Dicke, R. H. 1967a The solar spin-down problem. Astrophys. J. Lett. 149, L 121L 127.Google Scholar
Dicke, R. H. 1967b Solar models Science, 157, 960.Google Scholar
Dicke, R. H. 1970 The solar oblateness and the gravitational quadrupole moment Astrophys. J. 159, 124.Google Scholar
Dicke, R. H. & Goldenberg, H. M. 1967 Solar oblateness and general relativity Phys. Rev. Lett. 18, 313316.Google Scholar
Goldreich, P. & Schubert, G. 1967 Differential rotation in stars Astrophys. J. 150, 571587.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid J. Fluid Mech. 17, 385404.Google Scholar
Holton, J. R. 1965 The influence of viscous boundary layers on transient motions in a stratified rotating fluid: part I. J. Atmos. Sci. 22, 402411.Google Scholar
Howard, L. N., Moore, D. W. & Spiegel, E. A. 1967 Solar spin-down problem Nature, Lond. 214, 12971299.Google Scholar
Hsueh, Y. 1969 Buoyant Ekman layer Phys. Fluids, 12, 17571762.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 2. New York: McGraw-Hill.Google Scholar
Roxburgh, I. W. 1964 Solar rotation and the perihelion advance of the planets Icarus, 3, 9297.Google Scholar
Sakurai, T. 1969a Spin down of Boussinesq fluid in a circular cylinder J. Phys. Soc. Japan, 26, 840848.Google Scholar
Sakurai, T. 1969b Spin down problem of rotating stratified fluid in thermally insulated circular cylinders J. Fluid Mech. 37, 689699.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2 J. Fluid Mech. 26, 131144.Google Scholar
Walin, G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid J. Fluid Mech. 36, 289307.Google Scholar
Weinberger, H. F. 1965 Partial Differential Equations. New York: Blaisdell.Google Scholar
Wilkinson, J. H. 1965 The Algebraic Eigenvalue Problem. Oxford University Press.Google Scholar