Two-Dimensional Stellar Evolution Code Including Arbitrary Magnetic   Fields. II. Precision Improvement and Inclusion of Turbulence and Rotation
  In the second paper of this series we pursue two objectives. First, in order to make the code more sensitive to small effects, we remove many approximations made in Paper I. Second, we include turbulence and rotation in the two-dimensional framework. The stellar equilibrium is described by means of a set of five differential equations, with the introduction of a new dependent variable, namely the perturbation to the radial gravity, that is found when the non-radial effects are considered in the solution of the Poisson equation; following the scheme of the first paper, we write the equations in such a way that the two-dimensional effects can be easily disentangled. The key concept introduced in this series is the equipotential surface. We use the underlying cause-effect relation to develop a recurrence relation to calculate the equipotential surface functions for uniform rotation, differential rotation, rotation-like toroidal magnetic fields and turbulence. We also develop a more precise code to numerically solve the two-dimensional stellar structure and evolution equations based on the equipotential surface calculations. We have shown that with this formulation we can achieve the precision required by observations by appropriately selecting the convergence criterion. Several examples are presented to show that the method works well. Since we are interested in modeling the effects of a dynamo-type field on the detailed envelope structure and global properties of the Sun, the code has been optimized for short timescales phenomena (down to 1 yr). The time dependence of the code has so far been tested exclusively to address such problems. 
Linghuai Li (Yale), Sabatino Sofia (Yale), Paolo Ventura (INAF),   Valentina Penza (Universita' Di Roma "Tor Vergata"), Shao...