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Spin-Up of a Stratified Magnetofluid as a Model of Planetary Interiors
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| Geophys. Astrophys. Fluid Dynamics |
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We study the spin-up of a viscous electrically conducting Boussinesq fluid in a circular cylinder subject to thermal stratification and uniform axial magnetic field. The cylinder is a perfect conductor of electricity and heat. Physical parameters are chosen to be cnsistent with those of planetary interiors: E_m ~ E << 1, beta ~ E^1/2, sigma ~ E^-1, S ~ 1, where E, E_m, beta, sigma, and S are, respectively, the Ekman number, the magnetic Ekman number, the reciprocal square of the Alfvén Mach number, the Prandtl number and the stratification number. The linearized basic equations for axisymmetric fluid motion are solved by boundary layer analysis. Three kinds of end-plate boundary layers and two kinds of side-wall boundary layers are found to exist. The non-diffusive interior flow is solved by the Fourier-Bessel expansion method combined with the Laplace transformation. The Laplace transform slution is inverted numerically by the use of the residue theorem combined with Mathematica. Results show that the spin-up process is finished within the homogeneous non-magnetic spin-up time t = (L^2/v Omega)^1/2 even for cases with non-negligible stratification with those of Loper's conjecture.
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planetary interiors, rotating stratified fluids, MHD, boundary layers
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