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

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4. Gradients from the model

In a recent paper, Sánchez Almeida (1998) has proposed that the existing estimates of the inclination gradient are only compatible with penumbral magnetic structures with sizes in the range of 1 to 15 km. The estimates used there correspond to the quantity:


where z indicates height in the atmosphere. Values of [FORMULA] ([FORMULA] means some kind of average over the resolution element volume) and its rms fluctuations can be obtained from different sources. For example, from a sunspot vector magnetogram one can use [FORMULA], [FORMULA] and [FORMULA] together with the [FORMULA] and infer mean values of [FORMULA], (following Sánchez Almeida 1998, we neglect in this section variations of B with height) and the rms fluctuations between different pixels ([FORMULA]). An estimate of a quadratic mean ([FORMULA]) can be obtained from the measurements of NCP (that is assumed to provide a similar volume average as before). This is a much less direct estimate than the mean of [FORMULA], but Sánchez Almeida (1998) shows that, under some assumptions, one can expect a proportionality between [FORMULA] and the measured NCP. The estimates found in this way are given in the first row of Table 1.


Table 1. Inclination gradients.
a) From Sánchez Almeida (1998)

Using the models built here, we can estimate the magnitude of these different average gradients. The [FORMULA] is easily measured from the models as the average with height of the quantity defined by Eq. (1). It is important here to point out that, in the uncombed model, [FORMULA] is a function that has two sharp peaks of opposite signs. The location of the peaks corresponds to the bottom (negative peak) and the top of the tube (positive peak). These two jumps cancel in the final [FORMULA] if they are equally weighted by the averaging process. Thus, [FORMULA] is determined from the gradient of the background model. In Table 1, we give (first column) the value of [FORMULA] for the models with tubes at 150 km and 100% filling factor (second row), 150 km and 50% filling factor (third row), 250 km and 100% filling factor (fourth row) and 250 km with 50% filling factor (fifth row). In a similar way, one trivially computes [FORMULA] as predicted by these models (second column of Table 1). But now, when one computes the quadratic mean of P, the two jumps do not cancel. Indeed, it is the magnitude of the jumps at the tube boundaries, and the fraction of heights that they occupy, that determines the value of [FORMULA]. In this case, the contribution of the background gradient is negligible. Interestingly, the linear and quadratic averages in Table 1 agree very well with the observed ones. The almost two orders of magnitude difference between them are naturally explained from the fact that, in the first case (linear mean), the contributions of the boundaries of the tubes cancel while, in the second case (quadratic mean), they do not. The almost perfect match between the models and the observations should be no mystery. It is due to selecting as inclination jump (30o) the value needed to explain the NCP (which in turn determines [FORMULA]; see Sánchez Almeida & Lites, 1992; SM; Sánchez Almeida, 1998).

We note in passing that our study also supports a proportionality between the NCP and the value of [FORMULA] (Sánchez Almeida, 1998). In particular, although the agreement between the magnetic field inclination of the average atmosphere and the atmosphere retrieved by SIR in the bottom right panel of Fig. 4 is not particularly good, both models share the same value of [FORMULA] (but a totally different value of [FORMULA]). Because both models generate Stokes V profiles with the same NCP, a proportionality seems to hold between these two magnitudes.

The estimate of [FORMULA] cannot come directly from our four uncombed penumbral models. To obtain a sensible estimate of this quantity, we need to say something about how different realizations of the model can display different values of [FORMULA]. To this end, we follow a scheme where the different realizations correspond to random values of the tube height within the line forming region (Sánchez Almeida, private communication). A penumbral flux tube centered at a height [FORMULA] will contribute with a mean gradient of:


Here, the function [FORMULA] represents a weighting function assigned to each atmospheric height z. For the order-of-magnitude estimate we pursue, a gaussian representation of g representing a typical line forming region will suffice. [FORMULA] is the gradient as estimated from Eq. 1 when the tube is at [FORMULA]. To estimate the rms fluctuations, we assume that the random variable [FORMULA] has a flat density distribution in the range [-100,400] km. That is, there will always be one tube centered somewhere in this range, all heights being equally probable. [FORMULA] has contributions from the tube and from the background atmosphere. The latter, being always the same, does not contribute to the rms values. Here, we use a constant background inclination with a null contribution to P. To compute Eq. 2, we further assume that the tubes remain the same as they travel through the atmosphere (that is, for different values of [FORMULA]). This invariance allows us to treat Eq. 2 as the convolution of the two functions g and P. For [FORMULA], we use a unit area gaussian with a FWHM of 200 km (shown in Fig. 7). [FORMULA] is taken as the combination of two gaussians of opposite sign separated by 100 km (the tube size) and, both, with a FWHM of 10 km (also shown in Fig. 7). Note that, now, the amplitude of the gaussians is given by the fact that their integral should account for the total jump in inclination: [FORMULA]. The convolution of these two functions, [FORMULA], is shown in Fig. 7 with a dashed line. This function can be used to estimate the rms values of fully resolved penumbral observations. These observations would correspond to independent realizations of the model for random choices of the tube height [FORMULA]. According to Fig. 7, the peak to peak variation of these realizations would be 0.0026 km-1 and the resulting rms would be three times smaller, 8.7 10-4 km-1. Now, if we take into account that the observations in Table 1 have a spatial resolution of around 1000 km, we should reduce this rms by an additional factor of [FORMULA]. The exact value is given in the last row of Table 1 as our rms estimate. Given all the simplifications made in this calculation, the agreement between our estimate ([FORMULA]3 10-4 km-1) and the observed value (1 10-4 km-1) is more than satisfactory. The quadratic mean predicted in this random [FORMULA] model also compares satisfactorily with the observed value.

[FIGURE] Fig. 7. Inclination gradient ([FORMULA], solid line, right ordinate), weighting function ([FORMULA], solid line, left ordinate) and the convolution of these two functions ([FORMULA], dashed line, left ordinate) as a function of height in the atmosphere ([FORMULA] and [FORMULA]) or as a function of penumbral flux tube height ([FORMULA]). The vertical line corresponds to the maximum of the weighting function.

We, then, conclude that even if the observed values of [FORMULA] and [FORMULA] differ by almost two orders of magnitude, the uncombed penumbral model proposed by SM is able to predict this difference. Building a range of [FORMULA] values that gives rise to the observed rms fluctuations seems not to be a problem for the model as well. However, one may question how can penumbrae have locally mean values of [FORMULA] as large as several 10-2 km-1 while, at the same time, preserving the [FORMULA] condition. Remember that this value is one order of magnitude larger than [FORMULA] and three orders of magnitude larger than the observed [FORMULA] (which is derived from using the [FORMULA] averaged over the resolution element). To study the implications of these large local values of [FORMULA], we write the null divergence condition in cylindrical coordinates in the following form:


where we have neglected the contribution of the radial term (this is perfectly justifiable in the mid penumbra). Eq. (3) tells us that the local jumps in [FORMULA] should be accompanied by newly generated azimuthal fields [FORMULA]. This is nothing but stating that the background field lines cannot end where they find the tubes, but have to wind around them. How large are these azimuthal fields? If we re-write Eq. (3) as:


(with d the tube diameter) we estimate the magnitude of the azimuthal field to be [FORMULA], i.e., a purely azimuthal field vector.

Already SM, noticed the need for these azimuthal fields, although they did not estimate their importance. SM pointed out that the azimuthal fields needed at the top and at the bottom of the tubes give a contribution of opposite sign to the NCP and, thus, they were neglected in the radiative transfer (as we have done here).

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Online publication: October 2, 2000