Spin-up of a stratified magnetofluid as a model of planetary interiors

摘要

We study 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 of heat. Physical parameters are chosen to be consistent with those of planetary interiors:Em#x223C; E #x2264; 1, #x3B2; #x223C; E#xBD;, #xF2; #x223C; E#x2212;1, S #x223C; 1, whereE, Em'#x3B2;, #xF2;andSare, respectively, the Ekman number, the magnetic Ekman number, the reciprocal square of the Alfv#xE9;n Mach number, the Prandtl number and the stratification number see definitions just after (2.22). 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 Fourier-Bessel expansion method combined with Laplace transformation. The Laplace transform solution is inverted numerically by the use of residue theorem combined withMathematica(Wolfram Res., 1992). Results show that the spin-up process is finished within the homogeneous non-magnetic spin-up timet = (L2/ #x3BD;#x3A9;)#xBD;even for cases with non-negligible stratification in accordance with those of Loper's conjecture (Loper, 1976b,c).