## 3. Modeling the penumbra of a fluted sunspot## 3.1. The Beckers-Schröter profileThe 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 () and the inclination angle of the magnetic field with the vertical at the level . 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 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 The field amplitude is given by With the coordinate transformation (15) we rewrite the Beckers-Schröter profile (36) as Modulo a phase factor this profile corresponds to the absolute value of the complex function on the real axis (). The most general complex function with the same absolute value on the real axis can be written as on the real axis (). As we will see below the
inclusion of an appropriate function with The magnetic field components are the given by the expressions Another important piece of information given by the observations is the inclination angle 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 where the last term is the phase of . 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 and the radial coordinate for . Therefore, we can write the inclination as The task now is to find a complex function
which has the property for
and which represents the radial variation of
the average inclination for . At first sight
this might seem easy because we have already presented such a function
above (). A closer inspection of the inclination
angle and the magnetic field components for values of
reveals, however, that a naive choice of
On the other hand, if increases without
bound for the sign of both
and 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 which has a
real part that tends to a constant as A function which fulfills all these conditions is where For the imaginary part vanishes as required
and for fixed and the
real part has the limit ## 3.2. An exampleIn the present paper we restrict our treatment to the function
For the example we choose the outer boundary of the penumbra to be
normalized () and the inner boundary of the
penumbra to be located at . The origin of the
transformed coordinate system is choosen as
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 and
. In this example we have chosen
. The value of
For the modulation of the inclination angle with , we choose a simple harmonic function The parameter is the amplitude of the
inclination variation which we choose to be and
The last two parameters to determine are those of the magnetic
field strength, namely and
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.
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 (or another set of parameters with the same function ) or by extending the domain beyond the penumbral boundary and shifting the origin of the transformed radial coordinate system further out. © European Southern Observatory (ESO) 1998 Online publication: March 30, 1998 |