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Astron. Astrophys. 332, 1075-1081 (1998)

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3. Modeling the penumbra of a fluted sunspot

3.1. The Beckers-Schröter profile

The solution (35) at first sight leaves a large degree of freedom to model fluted penumbrae. However, if one wants to calculate fields which reproduce the basic features of the observations the solution class is considerably restricted. We will illustrate this by considering a specific example.

We assume that the basic data we have are the absolute value of the magnetic field ([FORMULA]) and the inclination angle [FORMULA] of the magnetic field with the vertical at the level [FORMULA]. Within the framework of the theory developed so far, we can only represent spots which have little or no variation of field strength with azimuth. For reasons of mathematical convenience, we assume that

[EQUATION]

This is the radial dependence of the field strength deduced by Beckers & Schröter (1969) from their observations. Other radial profiles like that of the Schatzman field (Schatzman 1965) could also be dealt with, but the Beckers-Schröter profile has the advantage of greater mathematical simplicity.

From Eq. (35) the magnetic field is given by

[EQUATION]

The field amplitude is given by

[EQUATION]

With the coordinate transformation (15) we rewrite the Beckers-Schröter profile (36) as

[EQUATION]

Modulo a phase factor this profile corresponds to the absolute value of the complex function

[EQUATION]

on the real axis ([FORMULA]). The most general complex function [FORMULA] with the same absolute value on the real axis can be written as

[EQUATION]

with

[EQUATION]

on the real axis ([FORMULA]). As we will see below the inclusion of an appropriate function Q is indispensible for a proper representation of the penumbral magnetic field. A simple example of such a function Q is given by

[EQUATION]

with q a real number. This example also illustrates that it is most convenient to write [FORMULA] in the form

[EQUATION]

The magnetic field components are the given by the expressions

[EQUATION]

Another important piece of information given by the observations is the inclination angle [FORMULA] of the magnetic field vector, i.e. the angle between the magnetic field vector and the vertical direction. For the magnetic field with the components given by Eqs. (46) and (47), the inclination angle is given by the equation

[EQUATION]

where the last term is the phase of [FORMULA]. We remark that with the present theory, we can only represent fields that have an inclination angle which is additive in the dependence on the azimuth [FORMULA] and the radial coordinate [FORMULA] for [FORMULA]. Therefore, we can write the inclination as

[EQUATION]

with the identities

[EQUATION]

The task now is to find a complex function [FORMULA] which has the property [FORMULA] for [FORMULA] and which represents the radial variation of the average inclination for [FORMULA]. At first sight this might seem easy because we have already presented such a function above ([FORMULA]). A closer inspection of the inclination angle and the magnetic field components for values of [FORMULA] reveals, however, that a naive choice of f will cause problems. For [FORMULA] the phase of [FORMULA] vanishes, but for [FORMULA] it contributes to the inclination angle. Actually for [FORMULA] this contribution tends to [FORMULA]. Since the average inclination angle should be positive for [FORMULA] this implies that the radial magnetic field component will change its sign as z increases unless [FORMULA] also increases with z sufficiently fast. In other words, the field lines would bend backwards toward the sunspot center at a finite height z. This is of course not an acceptable solution. The real part of the linear function [FORMULA] for example does not increase with z and therefore the linear function is not a viable choice.

On the other hand, if [FORMULA] increases without bound for [FORMULA] the sign of both [FORMULA] and [FORMULA] will change an infinite number of times as the argument of sine, respectively cosine tends to infinity. This as well is not an acceptable solution. We therefore have to find a function [FORMULA] which has a real part that tends to a constant as z goes to infinity and fulfills all the conditions mentioned previously.

A function which fulfills all these conditions is

[EQUATION]

where q, [FORMULA] and [FORMULA] are real numbers. The real and imaginary parts of this function are

[EQUATION]

For [FORMULA] the imaginary part vanishes as required and for [FORMULA] fixed and [FORMULA] the real part has the limit q. Other functions may fulfill these requirements as well, but this form of f has the advantage of being relatively simple. Of course, if the radial variation of the inclination angle would be given by observations, then the observations would prescribe the real part of f for [FORMULA]. Here, however, we are merely interested to illustrate the method developed and therefore choose a simple form of f. Actually, once one has found one f with the desired properties, it is possible to add any other complex function [FORMULA] which is bounded for [FORMULA] and [FORMULA] fixed. An example of such a function is the linear function [FORMULA]. This means that even with the restrictions discussed above one still has considerable freedom to model the variation of the average inclination angle with radius across the penumbra.

3.2. An example

In the present paper we restrict our treatment to the function f given in Eq. (52). We treat the parameter q as a free parameter which we choose such that we get a satisfying field line shape. The parameters [FORMULA] and [FORMULA] are determined by imposing the (average) inclination angles at the inner and outer boundary of the penumbra. This leads to two simple linear equations for [FORMULA] and [FORMULA].

For the example we choose the outer boundary of the penumbra to be normalized ([FORMULA]) and the inner boundary of the penumbra to be located at [FORMULA]. The origin of the transformed coordinate system is choosen as [FORMULA] locating it approximately halfway between the inner and outer boundary of the penumbra. The (average) inclination angles at the inner and outer boundary are chosen as [FORMULA] and [FORMULA]. In this example we have chosen [FORMULA]. The value of q influences the inclination of the field lines for [FORMULA]. Higher values of q give a larger inclination for large z and vice versa. The resulting values for [FORMULA] and [FORMULA] are listed together with the other parameter values in Table 1.


[TABLE]

Table 1. Parameter values used in the example shown in Figs. 1-3


For the modulation of the inclination angle with [FORMULA], we choose a simple harmonic function

[EQUATION]

The parameter [FORMULA] is the amplitude of the inclination variation which we choose to be [FORMULA] and m is the azimuthal wavenumber of the variation which we choose to be 66 in accordance with Martens et al. (1996). Of course, any other modulation with azimuth would be possible as well as long as it is periodic with period [FORMULA].

The last two parameters to determine are those of the magnetic field strength, namely [FORMULA] and L. These parameters will be determined by the measured values of the magnetic field strength at the inner and outer boundary ([FORMULA] and [FORMULA]). In the present example we have taken [FORMULA] G and [FORMULA] G. Again, from the two conditions on the inner and outer penumbral boundary, we get two equations for the parameters [FORMULA] and L, which can be easily solved. For the field strengths used we also give the values of [FORMULA] and L in Table 1. The results are shown in Figs. 1 and 2. In Fig. 1 we show field line plots of the field lines in the planes [FORMULA], [FORMULA] and [FORMULA] corresponding to average, maximum and minimum inclination.

[FIGURE] Fig. 1. Field line plots in the planes [FORMULA] corresponding to average inclination (left), [FORMULA] corresponding to maximum inclination (middle) and [FORMULA] corresponding to minimum inclination (right). The field lines plotted cross the lower or left boundaries at the same location in all three plots to facilitate the comparison. Therefore the distance between field lines does not necessarily reflect the strength of the magnetic field. Though not shown here, we emphasize that it is possible to extend the model in such a way that the return of the low-lying field lines to the surface is included in accordance with the observations by Westendorp-Plaza et al. (1997).

In Fig. 2 we show a set of representative field lines having their foot points either close to the umbral-penumbral boundary or to the outer penumbral boundary. The field lines are chosen so that they outline locations of minimum and maximum inclination. The flutedness of the field is obvious. In Fig. 3 we show a slice of the full field over two wavelengths in azimuth to emphasize the details of the field structure. Especially in Fig. 3 we see that the field lines with the maximum inclination form long shallow loops across the penumbra as required for their association with the Evershed effect.


[FIGURE] Fig. 2. Three-dimensional plot of sets of field lines originating close to the inner and outer penumbral boundary. The flutedness of the field is obvious. The azimuthal wave number is [FORMULA] in this case.

[FIGURE] Fig. 3. Close-up of a part of Fig. 2.

We remark that the model could also be modified to include the return flux observed by Westendorp-Plaza et al. (1997). This could be done either by choosing different functions [FORMULA] (or another set of parameters with the same function [FORMULA]) or by extending the domain beyond the penumbral boundary and shifting the origin of the transformed radial coordinate system further out.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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