MHD spin-down of a G-type star

We study MHD time evolution of initially rigidly rotating radiative zone of a G-type star subject to the effect of the magnetic field and of the deceleration of the surface angular velocity. We solve MHD equations of an unsteady axisymmetric Eddington-Vogt-Sweet (EVS)-type flow in the radiative zone, especially with respect to the angular velocity and the toroidal magnetic field, by using the Spectral method of solution. Three kinds of poloidal magnetic configurations (see Figure 1) are examined. We assume (i) that the toroidal magnetic field is induced by the coupling of the poloidal magnetic field with the non-uniform rotation caused by the spin-down process, (ii) that surface angular velocity conforms to the Skumanich law (Skumanich, 1972) of stellar angular velocity and (iii) that the time scale is the Keivin-Helmholtz (KH)-time and not the EVS-time. Similar to Charbonneau and MacGregor (1992), we can reduce this MHD spin-down problem to the solution of the phi-components [in spherical coordinates (1, theta, phi)] of the equations of motion and the induction equations. We perform numerical calculations by combining the Spectral-Tau method with Crank-Nicolson's time-marching procedure. Our study shows (i) the existence of a law of dynamical similarity, (ii) the effectiveness of the magnetic field in agreement with Mestel et al. (1988) and (iii) the existence of instability in agreement with Mestel and Weiss (1987). Our numerical results show the importance of phase mixing in agreement with Charbonneau and MacGregor (1992).