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Astron. Astrophys. 327, 786-794 (1997)

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3. Initial configuration

3.1. Initialization of the magnetic field

Let 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 u and vector potential [FORMULA], as [FORMULA], where r is the radial distance from the flux tube axis. The pressure can be determined in terms of u, which is initially unknown. We specify u by choosing a potential field at the initial epoch - clearly not an equilibrium solution. We evolve this configuration in time to see whether a final equilibrium state results.

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)

[EQUATION]

Let us choose the flux tube axis to be at [FORMULA] and its base at [FORMULA] The level [FORMULA] at this stage is somewhat arbitrary, since we have yet not specified its location with respect to the photosphere in the quiet atmosphere. At [FORMULA], we assume that the field [FORMULA] is vertical along the axis and the field line constant [FORMULA] The latter involves no loss of generality as it the gradient of u that matters. Since we assume axial symmetry, the field lines cannot not cross the axis of the flux tube. Along the lower boundary [FORMULA] a Dirichlet boundary condition is used by specifying the vertical magnetic field [FORMULA] On the top [FORMULA] and side [FORMULA] surfaces, we use Neumann boundary conditions (normal derivative set to zero) similar to Pizzo (1986).

Following Pizzo (1986), we assume a Gaussian variation of [FORMULA] along the base of the flux tube

[EQUATION]

where [FORMULA] is the axial field strength at [FORMULA], and [FORMULA] is the e-folding distance.

The components of the magnetic field in terms of the field line constant u are:

[EQUATION]

[EQUATION]

[EQUATION]

Using Eqs. (18) and (21), the field line constant u along [FORMULA] is generated,

[EQUATION]

At large radial distance from the axis, [FORMULA] approaches zero while u approaches a constant value [FORMULA] in accordance with Neumann boundary conditions, assumed on the outer boundaries. Along [FORMULA] i.e., along the axis of the flux tube, the axisymmetry is used to specify the magnetic field.

Using [FORMULA] kG, [FORMULA] Mm, [FORMULA] Mm and [FORMULA] 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 [FORMULA] We use this solution to calculate the magnetic field over the computational domain in the r-z plane and thereby initiate the numerical simulation.

[FIGURE] Fig. 1. Magnetic field topology for the potential solution, computed over a rectangular domain in the [FORMULA] plane, assuming that the field lines intersect the outer (r = 20 Mm) and upper (z = 12 Mm) boundaries normally. This solution is used to specify the magnetic field at the initial stage of the simulation. The labels on contours indicate the values of u in units of (kG Mm2)

3.2. Initialization of the hydrodynamic variables

From 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

[EQUATION]

where [FORMULA] is the gas pressure along the lower boundary [FORMULA] and h, the isothermal scale height along a field line, is

[EQUATION]

For a perfect gas, the temperature T can be related to h as follows

[EQUATION]

where [FORMULA] is the mean molecular weight, and [FORMULA] erg/deg-mol is the universal gas constant.

Given [FORMULA] and [FORMULA], [FORMULA] 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 [FORMULA] level in our model. Let [FORMULA] be the umbral gas pressure at [FORMULA] based on the sunspot-sunspot model. At the equivalent geometric depth, let the gas pressure in the quiet photosphere model be [FORMULA]

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 u as [FORMULA]. Using Eqs. (18) and (22), we obtain [FORMULA] The rate of increase of [FORMULA] to [FORMULA] matches with the decrease of [FORMULA] if

[EQUATION]

For [FORMULA] dyne cm-2 and [FORMULA] dyne cm-2, we find

[EQUATION]

It shows that the gas pressure increases sharply from the axis to [FORMULA], after which it becomes almost constant.

We now need to specify the variation of [FORMULA]. For simplicity, we assume that h is independent of z, and take the following dependence with respect to u,

[EQUATION]

The functional dependence of [FORMULA] and h on u in Eqs. (26)-(28) are the same as those used by Pizzo (1986). We choose [FORMULA] km and [FORMULA] km, which correspond to temperatures on the axis and in the ambient medium of 6.9 [FORMULA] K and 1.5 [FORMULA] K respectively, assuming [FORMULA].

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 straight. Clearly, the surfaces of constant u do not coincide with surfaces of constant r, but we make this assumption only to specify the starting value of p, since the final equilibrium state is unlikely to depend upon the precise values of the variables at the initial instant of time.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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