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Astron. Astrophys. 361, 734-742 (2000)

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2. The uncombed penumbral model

The penumbral magnetic field organization proposed by SM has one key ingredient. The two magnetic components proposed could not be separated in the vertical direction but rather the more vertical component winds around the horizontal flux tubes (that occupy a limited range of heights). This winding is responsible for the generation of the NCP in the model because the LOS will cross the boundaries between the two components. It is the jumps of the thermodynamic, magnetic and velocity parameters at the boundaries what generates the NCP. In the SM model, the jumps occupied a fixed optical depth interval of [FORMULA] simulating the effects of the two current sheets bounding the horizontal tubes. This optical depth interval corresponds approximately to 10-15 km in the atmosphere, which is of the order of a typical resistive boundary layer (Schüssler, 1986). In the uncombed penumbral model that we use, we preserve this key property of a more vertical component with field lines that flow past the horizontal flux tubes (the vertical component will be referred as the background component in the rest of the paper).

Our realization of the uncombed penumbral model is slightly different from that used by SM. The background model temperature is taken from the mean model derived by del Toro Iniesta et al. (1994). For reasons that will become clear later, we have scaled the temperature upwards by 5040/((5040/T)-0.01), which is equivalent to heat the atmosphere by 50 oK. After this scaling, the model is brought back to hydrostatic equilibrium (as it was the case of the original model). The background magnetic field strength we use follows the law: [FORMULA] in the optical depth interval [FORMULA] 1. The inclination of the magnetic field is given by: [FORMULA] over the same interval which implies a change of 2.5o over the whole atmosphere. This gives a net inclination gradient of [FORMULA] that fits very well in the interval allowed by Solanki et al. (1993). Micro- and macroturbulence are set everywhere equal to 1 km s-1. As it will be explained later, the macroscopic velocity in the background model could not be set to zero. To reproduce the CLV of the NCP an Evershed-like outflow of 1 km s-1 was used in the background model.

Following SM, the horizontal component is supposed to be organized into flux tubes that carry a larger part of the Evershed flow. Other properties of the horizontal tubes are different:

  1. The tubes have a diameter of 100 km. Their height can be either 150 km above the [FORMULA] level (the subscript b stands for the background model) or 250 km.

  2. The horizontal tubes can be either cooler than the background or hotter (SM tubes were isothermal). The temperature profile of the tube is given by a gaussian with an amplitude of [FORMULA]K. The gaussian profile is used for practical reasons in the treatment of the radiative transfer. Cooler tubes are suggested by the observations of Rüedi et al. (1999). On the other hand, the existence of hotter tubes is suggested by the simulations of Schlichenmaier et al. (1998): as part of the convective transport of energy, the rising tubes are hotter than their surrounding and optically thick. Once they reach the upper layers they radiate efficiently, cool down and become optically thin in the continuum. Thus both types of tubes (hot and cold) may play a role.

  3. The magnetic field strength inside the tubes is always smaller than in the background model. The field strength profile in the tube follows the same gaussian as for the temperature. The amplitude of the gaussian is -500 G.

  4. The gas pressure inside the tubes was forced to be in lateral equilibrium with the background total pressure: [FORMULA]. This links the perturbation used in the tube field strength to the tube gas pressure and density (through the ideal gas law). The total optical thickness of the tube is then closely related to the perturbation used for the field strength. This was the reason to use a gaussian profile in the field strength instead of a step function. This latter case would have generated strong jumps in the optical thickness of the tube that could have been difficult to follow numerically.

  5. The field inclination was not represented by a gaussian function but by a step function instead. As mentioned in the introduction, it is the jump in the field inclination what is known to be important for the generation of the NCP. The field inclination does not couple to any thermodynamic quantity and the jumps in this magnitude were much easier to treat numerically. The tubes were considered to have always a homogeneous field inclination of [FORMULA] (the subscript t stands for the tube model). The jump in inclination was then of around [FORMULA].

  6. The macroscopic velocity inside the tube was set to 2 km s-1. Note that this is only a factor two larger than that in the background model. This is similar to the velocities used by SM but contrast seriously with the simulations of Schlichenmaier et al. (1998).

The models are build on a geometrical height scale and afterwards their own optical depth scale is computed. Fig. 1 shows the stratifications of the background model (dotted lines) and of the model with a tube at a height of 150 km above the [FORMULA] level. In the top left panel the temperature of the cold (solid) and hot (dashed) models are shown. The optical depth scale is very sensitive to the field strength perturbation (top right in the figure). Through the lateral total pressure balance condition, the density inside the tube increases by a factor 1.5 for the hot tube and by a factor 1.8 for the cold tube. The net effect, in both cases, is that the atmosphere below the tube has a higher optical depth than in the background and the local [FORMULA] is reached before. Because the [FORMULA] level is higher (at a smaller temperature), the continuum intensity is reduced for both hot and cold tubes. Thus, the temperature fluctuations that we are modeling correspond always to a darker structure compared to the background atmosphere. For models with tubes at 150 km above the [FORMULA] level, the Wilson depression is 17 km. For tubes at 250 km, the Wilson depression is 9 km. The continuum intensity (normalized to the quiet sun) of the cold tube model at 150 km is 0.74 and 0.80 if the tube is at 250 km. For the hot tubes, the same numbers are 0.86 and 0.85 respectively. Because the background model had a slightly higher temperature than the original del Toro Iniesta et al. (1994) model, the continuum intensity of the background model is 0.93. Combining all these models with different filling factors one can obtain continuum intensities well within the observed range. Once the models have their own optical depth scale, the Stokes profiles of the FeI lines at the 6302 Å region are synthesized using a Hermitian method to integrate the radiative transfer equation as proposed by Bellot Rubio et al. (1998). Unless stated otherwise, the profiles of the different models with horizontal tubes are always combined with the profiles derived from the background model using a filling factor of 50% for both atmospheres. In this way, we try to account for the unresolved nature of the penumbral fine structure.

[FIGURE] Fig. 1. From top left clockwise, temperature, field strength, magnetic field inclination and velocity of the background model (dotted lines) and of a modified model that includes a cold tube at a height of 150 km (solid lines). The dashed lines in the first plot represents the temperature run of a model with a hot tube. The numbers above this plot give the height (in km) from the [FORMULA] level in the cold model.

The two bottom panels of Fig. 1 give the inclination with respect to the solar vertical of the background and tube model (right panel) and of the velocity (left panel). Note that this last case represent the modulus of the velocity in the models, not the disk center LOS values (that would be computed by multiplying this panel times the cosine of the inclination panel). The minus sign is used because the Evershed flow is an outflow and this is the sign convention used by SIR.

The models we have built have a number of ad-hoc properties. We do not claim that they perfectly represent the complex structuring of the magnetic field in penumbrae. We only expect from them to get an understanding of some of the recent observational results that simply do not fit with a smoother magnetic field representation.

2.1. CLV of the NCP derived from the model

SM already showed that the uncombed penumbral model predicts the observed NCP and its CLV in the center- and limb-side penumbra. However, it is important that we verify that our modified uncombed model still concurs with the observed NCP. We take advantage of new data of NCP derived from observations of the ASP. The observations are described in Martínez Pillet et al. (1997). Two sunspots within active regions NOAA 7197 and NOAA 7201 were followed during their disk passage. The two round leader sunspots of these active regions are used here to measure the NCP along the line that joins the disk center and the spot center. This line first passes through the so called center-side penumbra, crosses the umbra and leaves the spot passing through the limb-side penumbra. A four degree sector centered in the umbra is averaged to increase the number of points included in the statistics. Mean values and 3 times the standard deviation are plotted in Fig. 2 for the limb- (squares) and center-side penumbra (triangles) as a function of the cosine of the heliocentric angle [FORMULA]. The upper panels of Fig. 2 correspond to FeI line at 6302.5 Å and the bottom panels to FeI line at 6301.5 Å. The predictions of the uncombed model for cold and hot tubes (left and right respectively in Fig. 2) and for tubes at 150 km and at 250 km above [FORMULA] (solid and dotted lines respectively) are shown. (In all cases a filling factor of 50% has been assigned to both the background model and the model with the horizontal tubes.) Several well known features can be recognized. The limb-side penumbra has a larger NCP, and of opposite sign, to that in the center-side penumbra. However, note that for [FORMULA] the mean values of the NCP in the center-side penumbra do have a tendency to be of the same sign as in the limb-side penumbra. This fact has not been clearly acknowledged in previous observational works, something that may be due to the lower spatial resolution of the data used (the best resolution of our ASP data is somewhere between 1-2 "). Our CLV curves predict that a penumbra observed right at disk center will show an NCP of 1.15 mÅ for FeI 6301.5 Å and of 1.49 mÅ for FeI 6302.5 Å.

[FIGURE] Fig. 2. Top left panel: CLV of the observed NCP for the FeI line in the limb direction (squares) and in the center direction (triangles). The solid line is the prediction of the model with a cold tube at 150 km. Dashed line is for a cold tube at 250 km. Bottom left panel: The same but for FeI 6301. Right panels correspond to tubes hotter than the environment.

The error bars in Fig. 2 give an idea of the range of observed values. We do not expect to generate a good fit to these data with our realizations of the uncombed penumbral model. But, rather, we expect the predictions of the models to fall somewhere within the observed range. Note that these predictions do not include an average of the penumbral conditions at different radial distances (i.e. like averaging different inclinations of the background model) but they refer to one single radial point somewhere in the outer penumbra. Taking this into account and considering that no parameter search has been made, the agreement is quite satisfactory.

The models clearly show the (observed) rapid fall towards negative values of the center-side penumbra NCP at high µ values, including an NCP=0 crossing point. This zero NCP point does not depend on the height of the tube. It is easy to understand why the uncombed penumbral model has a point in the center-side penumbra where no NCP is generated. As pointed out by SM, wherever the LOS velocities of the background and tube models are the same, no NCP is generated. This implies solving the equation [FORMULA] where the + sign applies to the limb-side and the - to the center-side penumbra. In our case, [FORMULA], [FORMULA], [FORMULA] and [FORMULA] gives a solution only for the center-side penumbra at a µ value of 0.93. These considerations also show that the CLV of the NCP makes very difficult an scenario where the background velocity would be zero ([FORMULA]). In this case, the NCP=0 point would be seen in the limb-side penumbra (as long as [FORMULA]). Thus, if the flux tubes are close to the horizontal, but not diving down within the penumbra, the Evershed outflow should also be present in the background model.

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000