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

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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).

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