Numerical Method for the Structure of Rotating Axisymmetric Stars and its Thermally Driven Circulation

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We have computed equilibrium models for self-gravitating and stationary rotating toroids with viscous beating. We have chosen two kinds of rotation laws: linear velocity constant case and specific angular momentum constant case. As for the heat source we mimic heat generation due to viscosity as follows: epsilon = nuR^2 (d OMEGA/dR)^2, where nu is the kinematic viscosity whose value is uncertain and so we treat as a parameter.When we choose ring source model, its beat generation is so large that the energy cannot be transferred by radiation diffusion. Thus convective motion is excited in the central toroidal region. However, for the viscous source models, the main structure is determined by the balance between the self-gravity and centrifugal force. Thus the large energy which is mainly generated in the region near the rotation axis must be transferred by the meridional circulation because the structure itself is convectively stable. The excited meridional circulation is symmetric about the equatorial plane and its flow is clock wise direction in the upper half meridional plane. If we choose configuration whose innermost surface is very near the rotation axis or the value of kinematic viscosity to be large, the radiation pressure becomes important and toroidal convective region appears.Since the meridional circulation transfer the angular momentum, the angular velocity distribution will be changed. If we suppose that the state will settle down to a stationary state in which no circulation is allowed, the temperature constant surfaces should be shifted inward to the rotation axis much more than the pressure constant surfaces. Therefore the angular velocity distribution comes to satisfy the condition (d OMEGA/dz) < 0. Since such a distribution is unstable to Goldreich-Schubert instability, configuration will be oscillating between barotropic state and baroclinic state.