Numerical solutions for the spin-up of a stratified fluid

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Abstract
Numerical solutions for the impulsively started spin-up of a thermally stratified fluid in a cylinder with an insulating side wall are presented. Previous experimental and numerical work on stratified spin-up had not provided a comprehensive and accurate set of flow-field data. Further, comparisons of this work with theory showed, in general, a substantial discrepancy. The theory was scaled using the homogeneous meridional-flow spin-up time scale and thus viscous-diffusion effects were excluded from the interior. It was anticipated that these effects could only be significant on the larger viscous-diffusion time scale. However, the comparisons with theory showed a faster rate of decay for the measurements even over the shorter meridional-flow spin-up time scale. Previous workers had suggested a number of explanations but the cause of the discrepancy was still unresolved. To provide data to extend the previous work, a numerical model was used. The model was first checked against accurate experimental measurements of stratified spin-up made using a laser-Doppler velocimeter. New accurate results which cover ranges of Ekman number (5·92 × 10−4 ≤ E ≤ 7·24 × 10−4), Rossby number (0·019 ≤ ε ≤ 0·220), stratification parameter (0·0 ≤ Sa−1 ≤ 1·03), and Prandtl number (5·68 ≤ σ ≤ 7·10) are presented. These results show the radial and vertical structure of the decaying azimuthal and meridional flows. The inertial–internal gravity oscillations excited by the impulsive spin-up are clearly seen. By making use of conclusions from the previous work and the results presented in this paper, it is established that viscous diffusion in the interior is the cause of the discrepancy with theory. Stratification causes the meridional spin-up flow to be confined closer to the boundary disks. This results in non-uniform spin-up of the interior and hence flow gradients in the interior. These gradients introduce viscous diffusion into the interior sooner than anticipated by the theory. A previous suggestion that the faster decay rate is due to angular momentum being injected into the interior from an oscillation of the meridional corner-jet flow is shown to be untenable.

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