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Astron. Astrophys. 357, 1056-1062 (2000)

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3. Results

For all 18 magnetograms (see Table 1) we calculated the total flux and the error bars using Eqs. (1) and (2), respectively; and calculated the total positive and negative currents and the error bars using Eqs. (3) and (6), respectively. Along with this we estimated the value of [FORMULA] in four ways, i.e. using Eqs. (7), (8), (9) and (10). In the latter two cases we calculated the linear force-free field by the Fourier Transform Method (Alissandrakis 1981). Such algorithms were performed both for the entire active region and for the three main spots separately.

Fig. 1 shows the daily vector magnetograms of the active region. The emergence of the magnetic flux is well pronounced. Three main poles can be revealed: N, S and S1. The corresponding daily maps of [FORMULA] density ([FORMULA]) are presented in Fig. 2. In Figs. 3 and 4 the time variations of the total fluxes [FORMULA] and total currents I are shown. The monotonous growth of the total flux both in the entire AR and in the N, S and S1 poles is obvious. The same is valid also for the total currents, except for the total positive currents in S and S1: they are weak and practically constant. The growth rates of the flux and current (estimated from the least-square linear fit in Figs. 3 and 4) are presented in Table 2. One can see that the growth rate of the total positive flux (current) is very close to that of total negative flux (current) for the entire AR. Such a situation can be physically interpreted by the solenoidal nature of [FORMULA] and [FORMULA] vectors, and this confirms the reliability of our field measurement and current calculations.

[FIGURE] Fig. 1. The photospheric vector magnetograms (contours) superposed on the white light images showing the spots. The white contours in all figures demonstrate the intensity of the longitudinal field (levels = [FORMULA]500, 1500, 2500, and 3500 Gauss); solid lines show the positive field and dashed lines show the negative field. The short arrows indicate the transverse field with the intensity above 2[FORMULA] noise level. The field of view (FOV) is 157[FORMULA]157".

[FIGURE] Fig. 2. Maps of [FORMULA] density (gray-scale) with the photospheric magnetograms superposed on. The white solid (dashed) contours show positive (negative) fields with the same levels as in Fig. 1. The [FORMULA] scale (units of 10-7m-1) is shown below. The grey maps only show [FORMULA] distribution in such areas where [FORMULA] and [FORMULA]. The field of view is the same as in Fig. 1.


[TABLE]

Table 2. Growth rates of the total flux and the total current.


The plot of the total current versus the total flux is shown in Fig. 5. It is shown that for the entire AR there are two linear relationships: for the positive branch (with the coefficient [FORMULA]) and for the negative branch (with the coefficient [FORMULA]). Moreover, in both cases, the least-square linear fit does not cross the zero. When [FORMULA], [FORMULA]A, and for [FORMULA] we have [FORMULA]A. In each of the poles studied, there are both positive and negative currents, but the positive (negative) current dominates in the N pole (S, S1 poles). This results in the overall positive (right-handed) twist in the entire AR, which can be estimated from [FORMULA] by the least-square linear fit in Fig. 5. The results of this estimation are shown in Columns (2) and (4) in Table 3. Columns (3) and (5) represent the coefficient of the Spearman rank correlation (Press et al. 1994). One can see that the correlations are high enough.

[FIGURE] Fig. 3. Time variations of the total flux ([FORMULA]) for the entire region and for the main poles (N, S and S1). The bars for [FORMULA] do not exceed the size of symbols. The dashed lines represent a least-square linear fit.

[FIGURE] Fig. 4. Time variations of the total positive and negative currents (I) for the entire region and for the main poles (N, S and S1). The dashed lines represent a least-square linear fit.

[FIGURE] Fig. 5. The total current (I) versus the total flux ([FORMULA]) for the entire region and for the main poles (N, S and S1). Squares refer to positive [FORMULA]; Triangles refer to negative [FORMULA]. The dashed lines represent a least-square linear fit.


[TABLE]

Table 3. Slope coefficients of the least-square linear fit for the total flux and the total current, and coefficients of the Spearman rank correlation.


The results of direct calculations of [FORMULA] by Eqs. (7), (8), (9) and (10) are given in Fig. 6 and Table 4. The comparison of [FORMULA] calculated by different approaches allows us to conclude that the [FORMULA] is systematically larger when estimated as [FORMULA] (Column (2) in Table 4). Moreover, in the pole N the [FORMULA] is negative that disagrees with other forms of [FORMULA] estimation in this pole ([FORMULA], [FORMULA] and [FORMULA]). This may be caused by the presence of large negative [FORMULA] in places of weak field. These places should not weigh a lot in the total twist. So this case (pole N) shows that the [FORMULA] method has some intrinsic disadvantage as compared with the [FORMULA] method. In addition, [FORMULA] shows the better coincidence in values with [FORMULA] and [FORMULA] as well as the estimate of [FORMULA] from [FORMULA] (see Tables 3 and 4). This also suggests that the [FORMULA] method seems more reasonable than the [FORMULA] method to measure the twist of the flux.

[FIGURE] Fig. 6. Time variations of four forms of the parameter [FORMULA] characterizing the twist of the flux for the entire AR and for the main poles (N, S and S1). Triangles : [FORMULA], the average of [FORMULA]. Squares : [FORMULA], the average of [FORMULA] weighted by the flux element [FORMULA]. Diamonds : [FORMULA], the best-fit [FORMULA] of a constant force-free field which minimizes the difference between the x and y components of the computed and observed fields. Asterisks : [FORMULA], the best-fit [FORMULA] which minimizes the angles between the transverse components of the computed and observed fields. In the former two cases, [FORMULA] is averaged only over pixels with [FORMULA] and [FORMULA], while in the latter two cases, the regions of average satisfy the condition [FORMULA].


[TABLE]

Table 4. Four estimates of [FORMULA] parameter averaged over 3 days. All forms have the same unit of 10-8m-1.


So, except [FORMULA], all other estimated [FORMULA] are in a good agreement to show that the changes in [FORMULA] from day to day were rather small. This result indicates that the twist, or, the magnetic complexity, changes very slowly during the flux emergence.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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