Dynamical phenomena in sunspots. I. Time dependent relaxation to equilibrium

Abstract

We have adapted a general purpose time-dependent 2-D code to study dynamical phenomena in sunspots. In the first part of our investigation, we numerically simulate the dynamical relaxation to equilibrium of a sunspot. Treating the sunspot as a thick axisymmetric flux tube in cylindrical geometry, we solve the time dependent MHD equations to examine the evolution of a sunspot towards equilibrium, starting from an arbitrary initial state. Initially, we choose a potential magnetic field configuration and assume hydrostatic equilibrium along field lines, which allows the pressure variation along the field to be determined, for a known temperature distribution. We also assume that all quantities in the tube have a smooth and continuous radial variation. In particular the pressure increases radially from the tube axis to the photospheric value. The absence of Lorentz forces to balance the radial pressure gradient, leads to an inflow of gas towards the axis accompanied by an increase in the magnetic field strength. A complex flow pattern develops in the tube, which eventually dies out due to escape of matter upwards along the field lines. In the quasi-equilibrium state it is found that the field lines near the center of a large spot assume a configuration which is almost potential while those at the periphery depart significantly from the initial state, due to being pushed inwards by the gas flow. Our method is applicable to both thin and thick flux tubes. Further it can be readily extended to any coordinate system with 2 or 3 coordinates, and to discontinuous configurations such as current sheets. Forthcoming studies will focus on an extension of the present study to an analysis of dynamical effects in sunspots associated with nonlinear waves and examine the transport of energy by these to the corona