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

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4. Numerical solutions for a prototype model

In this paper we concentrate on a single set of parameters characterizing our initial state, which define a prototype sunspot model. These are shown in Table 1 and are taken from Pizzo (1986).


Table 1. Basic parameters for the prototype solution

4.1. Boundary conditions

For the time-dependent simulation we consider a computational region in the [FORMULA] plane, with lower and upper boundaries at [FORMULA] and [FORMULA] Mm respectively. In the radial direction the boundaries are taken at [FORMULA] and [FORMULA] Mm. At the base ([FORMULA]) and on the axis ([FORMULA]), we assume no flow through boundary conditions. The choice of a no flow boundary through condition at the base can be justified to some extent on the basis of the high density of matter, which plays the role of an impenetrable boundary. The assumption of axial symmetry precludes gas flow across the axis of the sunspot. On the right boundary, we allow for an inflow boundary condition. This is to be expected on physical grounds as a consequence of the horizontal pressure difference between the interior and exterior of the sunspot. We allow for the escape of matter along field lines from the upper boundary, where we use an outflow boundary condition.

The ZEUS code utilizes two rows of 'ghost' zones at each boundary. The boundary conditions ensure that the values stored in the ghost zones are consistent with the active zones. Boundary conditions are used to update the values of the thermodynamics variables (p, [FORMULA] and e), the velocity and the EMF (the actual values of the magnetic field components can be found from the difference equations). At the base and on the axis, we use reflecting boundary conditions. These equate the thermodynamic variables and the tangential velocity in the ghost zones to the values of these variables in their active zone images. The normal components of velocity and magnetic field are reflected, which implies that the EMF in a ghost zone is negative of the value in the active zone image. Actually, for the ghost zones parallel to the axis, we need to also reflect the azimuthal component of the velocity in view of the symmetry. In the present case, however, this is inapplicable since we assume that the flow does not possess an azimuthal component.

Having initialized all the magnetohydrodynamic variables, we numerically solve the MHD equations as a time dependent problem with the goal of achieving an equilibrium configuration for a sunspot.

4.2. Flow pattern in the sunspot

At [FORMULA] the fluid velocity over the entire mesh is set to zero. However, the initial state is clearly not in equilibrium. Thus, as soon as the simulation begins, we expect that the negative horizontal pressure gradient will lead to a radial inflow of matter from the right boundary. Let us first examine the nature of the flow pattern that is set up in the sunspot.

Fig. 2a-h, depict the velocity (flow) field in the computational domain as a vector plot at different instants of time. The length of the arrows, drawn at random points, is proportional to the magnitude of the velocity, whereas their orientation indicates the direction of the flow. The normalization in each panel is with respect to the largest value of the flow at that instant of time.

[FIGURE] Fig. 2. Series of vector fields for fluid velocity in the [FORMULA] plane at various instants of time during the simulation. The size of the computation domain selected in the simulation is 4 Mm in the r -direction and 1 Mm in z -direction. The length and direction of arrows indicate the magnitude and direction of fluid velocity at different regions over the computation domain: a gas flow towards the axis of the flux tube due to pressure and density gradients, b and c flow towards the base due to gravity, d reflection of the flow due to the impenetrable lower boundary and the instantaneous generation of high pressure and density near the base and axis, e fluid comes back due to gravity and tension forces in the magnetic field lines

Fig. 2a shows the flow field at time t = 6.3 s, which is radially inward due to the gradients in pressure and density. The length of the arrow having maximum length is [FORMULA] m s [FORMULA] and all other vectors are normalized with this length. Initially, the horizontal pressure gradient leads to a radial inflow of matter. Since this matter cannot flow out through the axis in view of our assumption of axial symmetry, the fluid density increases near the axis. The increase in density leads to a downflow of matter due to gravity towards the base of the flux tube. This can be discerned in Figs. 2b and c. Since we have assumed an impenetrable boundary at the base, there is an accumulation of fluid near the base, which results in a pressure buildup there. This leads to a reversal of the flow in the vertical direction, as can be seen in Fig. 2d for [FORMULA] 127 s. The depletion of fluid from near the base along with magnetic tension forces result in fluid again moving downward as indicted by Fig. 2e at t = 255 s. This pattern of flow reverses in time. Over each cycle the absolute value of the velocity decreases in magnitude, due to the gradual diminution of the pressure, gravitation and Lorentz forces, driving the flow. This is accompanied by a escape of matter along the field lines through the upward boundary. Fig. 2f-h display the flow field direction at subsequent times. It may be noted that the arrows lengths decrease in time. In Fig. 2h, the absolute value of the maximum velocity at [FORMULA] 6326 s is about 100 m s-1, which is very much less than the sound and Alfven speeds, typically in the range of ten to hundred km s-1.

4.3. Field line topology

Fig. 3 depicts the geometry of the magnetic field at t = 0 s (dashed lines) and at t = 6326 s (continuous lines). The curves are labeled by the constant u, along the field. The initial geometry of course corresponds to the potential field solution. We find that the field lines are compressed relative to the potential field and tends to become more vertical owing to the squeezing action due the radial inflow. Fig. 4 shows contours of constant field (in kilogauss) for the initial (dashed lines) and asymptotic (solid lines) solutions. Close to the axis of the field, we can clearly discern that the contours get shifted upwards, which as already mentioned is a consequence of field compression due to the flow. The heavy dashed and solid lines denote the [FORMULA] levels at the initial and final instants respectively, where [FORMULA]. The significance of these curves is that they delineate the boundary in the flux tube between the regions where pressure and magnetic forces dominate. The Lorentz forces are more important in controlling the momentum balance in the spot atmosphere above these curves (where [FORMULA]), whereas in the underlying layers pressure forces are more significant. Near the axis, the [FORMULA] level is raised upwards, but at about a radial distance of 1 Mm from the axis, the level has been shifted downwards as a consequence of the flow. This indicates that the region where magnetic forces are important (essentially the area above the [FORMULA] curve) has become much larger as a consequence of the field enhancement.

[FIGURE] Fig. 3. Field lines for the final (continuous lines) and initial potential field (broken lines) states. Labels indicate the constant value of u along each field line. Note that in the final state, the field lines are pushed inwards towards the axis with respect to the potential field

[FIGURE] Fig. 4. Contours of constant B in the final (continuous lines) and initial (broken lines) stages. In the final stage, the magnetic field near the axis is lifted up by the gas while near the base it is pushed down. The heavy dashed and solid lines denote the [FORMULA] levels at the initial and final instants respectively, where [FORMULA]

4.4. Thermodynamic structure of the sunspot

Figs. 5 show contours of constant log [FORMULA] in the spot at the initial and final instants (in the asymptotic time limit the pressure is practically constant with time). Close to the axis the contours are shifted up, whereas away from the axis they are moved downwards. The relaxation of the spot under the large radial pressure gradient at the initial instant leads to a diminution of this gradient in the final state. This is consistent with a decrease in pressure at large radial distances, whereas close to the axis, there is a slight increase in pressure (we rule out flow of matter across the spot axis). In the vertical direction, the state of hydrostatic equilibrium is almost restored after long enough time, when the flow has become very small.

[FIGURE] Fig. 5. Contours of constant log [FORMULA] in the final (continuous lines) and initial (broken lines) stages. Close to the axis there is an enhancement of the pressure, whereas away from the axis there is a decrease in pressure

Let us now consider the temperature structure in the sunspot, which is depicted in Fig. 6. At [FORMULA], we have assumed that the pressure scale height h and hence the temperature T vary only in the radial direction. Initially the isotherms, by assumption, are vertical (dashed lines) parallel to the axis of the tube. The curves are labeled by the constant value of temperature (in units of 1000 K). However, as the flux tube relaxes dynamically the scale height in general decreases upwards, resulting in a drop of temperature with height. This behaviour is equivalent to the statement that the temperature reduction is due to a decrease in the internal energy per unit mass ([FORMULA]). The isotherms become less steep and the temperature in the sunspot has a form which is closer to reality, viz., that the isotherms from the external atmosphere dip downwards into the spot. At equal geometric levels the temperature in the sunspot is less than that in the ambient medium.

[FIGURE] Fig. 6. Contours of constant temperature (in units of [FORMULA] K). Initially the isotherms are vertical (dashed lines). In the final state (solid lines), the isotherms are not so steep
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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998