## 4. Gradients from the modelIn 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
Using the models built here, we can estimate the magnitude of these
different average gradients. The 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, 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
if they are equally weighted by the
averaging process. Thus, is
determined from the gradient of the background model. In Table 1,
we give (first column) the value of
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 as predicted by these
models (second column of Table 1). But now, when one computes the
quadratic mean of We note in passing that our study also supports a proportionality
between the NCP and the value of
(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
The estimate of 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 . 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 will contribute with a mean gradient of: Here, the function represents a
weighting function assigned to each atmospheric height
We, then, conclude that even if the observed values of
and
differ by almost two orders of
magnitude, the uncombed penumbral model proposed by SM is able to
predict this difference. Building a range of
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 as large as several
10 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 should be accompanied by newly generated azimuthal fields . 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 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). © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |