## 3. Initial configuration## 3.1. Initialization of the magnetic fieldLet us consider an initial configuration which is static (i.e.
without flows) and one in which the magnetic field varies smoothly in
the radial direction. In cylindrical geometry the magnetic field for
an axisymmetric configuration can be expressed, following Low (1975),
in terms of the field line constant Let us consider a cross section of the flux tube in a rectangular domain of the r-z plane, and initialize the magnetic field on all the grid points using the potential solution, i.e., the solution of the equation (Low 1975) Let us choose the flux tube axis to be at
and its base at The level
at this stage is somewhat arbitrary, since we
have yet not specified its location with respect to the photosphere in
the quiet atmosphere. At , we assume that the
field is vertical along the axis and the field
line constant The latter involves no loss of
generality as it the gradient of Following Pizzo (1986), we assume a Gaussian variation of along the base of the flux tube where is the axial field strength at , and is the e-folding distance. The components of the magnetic field in terms of the field line
constant Using Eqs. (18) and (21), the field line constant At large radial distance from the axis,
approaches zero while Using kG, Mm, Mm and Mm, the potential solution of Eq. (17) is computed using the Cyclic Reduction Algorithm (CRA) developed by Swarztrauber (1974, 1977). Fig. 1 shows the contours of the field line constant We use this solution to calculate the magnetic field over the computational domain in the r-z plane and thereby initiate the numerical simulation.
## 3.2. Initialization of the hydrodynamic variablesFrom observations it is well known that sunspots are cool and possess a pressure that is less than that of the ambient photosphere. The decrease of the pressure with height, leads to a fanning out of the magnetic field lines with height. For a axisymmetric and vertical flux tube in magnetostatic equilibrium, Low (1975) has shown that the gas pressure variation along a field line has the form where is the gas pressure along the lower
boundary and For a perfect gas, the temperature where is the mean molecular weight, and erg/deg-mol is the universal gas constant. Given and , can be specified over all the grid points in the r-z plane. The temperature can be found from Eq. (25) and the density is readily obtained using the perfect gas law. The components of the magnetic field have already been computed using the potential solution based on Eq. (17). Thus, in principle, we have the necessary information to solve the initial value MHD problem, subject to some choice of boundary conditions. It should be pointed out that the pressure and magnetic field distributions chosen in this way are not self-consistent, since the magnetic field is in general not potential. However, we follow this procedure only to specify the initial values of the variables. For our model sunspot simulations, we require a representative umbral atmosphere along the axis and a quiet photospheric atmosphere at large horizontal distances, where the field vanishes. For thick flux tubes with a distributed current, we construct a smooth transition between the quiet photospheric atmosphere and the sunspot. Following Pizzo (1986), we extract the values at the base using a smooth transition from the sunspot-sunspot model of Avrett (1981) on the axis to the convection model of Spruit (1977) for the quiet photosphere. The location of our base is chosen similar to Pizzo (1986) to lie at a depth of 120 km (i.e. below continuum optical depth unity) in the sunspot-sunspot model, which is displaced relative to the photosphere by the Wilson depression (i.e., the level corresponding to which the continuum optical depth in the vertical direction is unity). This is the level in our model. Let be the umbral gas pressure at based on the sunspot-sunspot model. At the equivalent geometric depth, let the gas pressure in the quiet photosphere model be We now consider a smooth transition of the pressure from the axis
to the exterior, chosen in such a way that it has the same functional
dependence on For dyne cm It shows that the gas pressure increases sharply from the axis to , after which it becomes almost constant. We now need to specify the variation of . For
simplicity, we assume that The functional dependence of and The pressure along a field line can now be calculated using Eq.
(23). This pressure distribution is at best an approximation to the
magnetostatic solution and as already stated is used only to initiate
the simulation. From a practical point of view, it is more convenient
to calculate the pressure from Eq. (23), assuming that the field lines
are © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |