## 4. Numerical solutions for a prototype modelIn 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).
## 4.1. Boundary conditionsFor the time-dependent simulation we consider a computational region in the plane, with lower and upper boundaries at and Mm respectively. In the radial direction the boundaries are taken at and Mm. At the base () and on the axis (), 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 ( 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 sunspotAt 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.
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 m s
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
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
6326 s is about 100 m s ## 4.3. Field line topologyFig. 3 depicts the geometry of the magnetic field at
## 4.4. Thermodynamic structure of the sunspotFigs. 5 show contours of constant log 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.
Let us now consider the temperature structure in the sunspot, which
is depicted in Fig. 6. At , we have assumed that
the pressure scale height
© European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |