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Spin-down of a Boussinesq fluid of small Prandtl number in a circular cylinder

Published online by Cambridge University Press:  29 March 2006

Takeo Sakurai ,
Alfred Clark  and
Patricia A. Clark
Takeo Sakurai
Affiliation:
National Center for Atmospheric Research, Boulder, Colorado Present address: Kyoto University, Kyoto, Japan.
Alfred Clark
Affiliation:
Joint Institute for Laboratory Astrophysics, Boulder, Colorado Present address: Department of Mechanical and Aerospace Sciences, University of Rochester, Rochester, New York 14627.
Patricia A. Clark
Affiliation:
Department of Astro-geophysics, University of Colorado, Boulder, Colorado
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Abstract

The impulsive linear spin-down of a stably stratified Boussinesq fluid in a circular cylinder is analyzed under the assumption that the Prandtl number P = ν/κ is small (ν is the kinematic viscosity, κ the thermal diffusivity). The nature of the spin-down process depends on the ordering of P with respect to the Ekman number E. For P < E½, the spin-down is similar to that of an unstratified fluid. For P > E½ the process is similar to that for a stratified fluid with P = O(1). The distinctive case P = O(E½) is analyzed in detail. For that case it is shown that for N [Gt ] Ω, the asymptotic state of rigid rotation is reached in a time of the order of (N/Ω)2τ, where N is the Brunt-Väisälä frequency, Ω the angular velocity and τ the thermal diffusion time for the cylinder. We calculate spin-down times for parameter values corresponding to the solar interior. For an angular velocity as large as that suggested by Dicke (5·74 × 10−5 sec −1) the spin-down time is less than the age of the sun. For an angular velocity comparable to the surface value (2middot;87 × 10−6 sec −1), the spin-down time is greater than the age of the sun. These results suggest that a uniformly and rapidly rotating solar interior is not possible, but we cannot rule out a state of non-uniform rotation producing an oblateness as large as that measured by Dicke & Goldenberg (1967).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 49 , Issue 4 , 29 October 1971 , pp. 753 - 773
Copyright
© 1971 Cambridge University Press

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References

Barcilon, V. & Pedlosky, J. 1967a Linear theory of rotating stratifled fluid motions. J. Fluid Mech., 29, 1–16.Google Scholar
Barcilon, V. & Pedlosky, J. 1967b A unified linear theory of homogeneous and stratified rotating fluids. J. Fluid Mech., 29, 609–621.Google Scholar
Clark, A., Clark, P. A., Thomas, J. H. & Lee, N. H. 1971 Spin-up of a strongly stratified fluid in a sphere. J. Fluid Mech., 45, 131–150.Google Scholar
Clark, A., Thomas, J. H. & Clark, P. A. 1969 Solar differential rotation and oblateness. Science, 164, 290–291.Google Scholar
Dicke, R. H. 1964 The sun's rotation and relativity. Nature, 202, 423–435.Google Scholar
Dicke, R. H. 1970 Internal rotation of the sun. A. Rev. Astron. & Astrophys., 8, 297–328.Google Scholar
Dicke, R. H. & Goldenberg, H. M. 1967 Solar oblateness and generel relativity. Phys. Rev. Lett. 18, 313-316.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech., 17, 385–404.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, 402–411.Google Scholar
Howard, L. N., Moore, D. W. & Spiegel, E. A. 1967 Solar spin-down problem. Nature, 214, 1297–1299.Google Scholar
Hsueh, Y. 1969 Buoyant Ekman layer. Phys. Fluids, 12, 1757–1762.Google Scholar
Roxburgh, I. W. 1964 Solar rotation and the perihelion advance of the planets. Icarus, 3, 92–97.Google Scholar
Sakurai, T. 1969ac Spin-down of Boussinesq fluid in a circular cylinder. J. Phys. Soc. Japan, 26, 840–848.Google Scholar
Sakurai, T. 1969b Spin-down problem of rotating stratified fluid in thermally insulated circular cylinders. J. Fluid Mech., 37, 689–699.Google Scholar
Sakurai, T. 1970 Spin-down of Boussinesq fluid in the circular cylinder, as a simulation of the solar spin-down procedure. In Stellar Rotation (ed. A. Slettebak), pp. 329–339. Dordrecht-Holland: D. Reidel.
Schwafzschild, M. 1958 Structure and Evolution of the Stars. Princeton University Press.
Walin, G. 1969 Some aspects of time-dependent motion of a stratrfied rotating fluid. J. Fluid Mech, 36, 289–307.Google Scholar
Watson, G. 1958 The Theory of Bessel Functions, 2nd edn. Cambridge University Press.