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Up: Three-dimensional chromospheric magnetic

4. Implementation of the computational algorithm-results

Simulation of measured photospheric field components. From the general analytical solutions, Eqs. (8)-(13) one can obtain the ``photospheric" (i.e. at z = 0 ) field components tex2html_wrap_inline2601, tex2html_wrap_inline2603, tex2html_wrap_inline2605 and tex2html_wrap_inline2607 For illustration, these functions are represented in Fig. 1 (click here); the ``observation" domain, tex2html_wrap_inline2609 (z = 0) has the dimensions tex2html_wrap_inline2613 = tex2html_wrap_inline2615 = 5 (see Eq. (15 (click here))).

Figure 1: Simulated photospheric (z=0) observations, tex2html_wrap_inline2619 (a), tex2html_wrap_inline2621 (b), tex2html_wrap_inline2623 (c) and tex2html_wrap_inline2625 (d) as obtained from the analytical expressions evaluated at z=0. For convenience, normalized quantities are used, namely tex2html_wrap_inline2629 tex2html_wrap_inline2631, tex2html_wrap_inline2633 and tex2html_wrap_inline2635

Simulation of photospheric-field components, after removal of the tex2html_wrap_inline2637 ambiguity in the tex2html_wrap_inline2639 and tex2html_wrap_inline2641 components. When suitable techniques are used to remove the tex2html_wrap_inline2643 ambiguity (see, e.g. Cuperman et al. 1993; Li et al. 1993), one obtains the field components tex2html_wrap_inline2645 and tex2html_wrap_inline2647 shown in Figs. 2 (click here)a and b. The corresponding contours of constant values are shown in Figs. 2 (click here)d and e. For completness, the field component tex2html_wrap_inline2649 is also represented in Figs. 2 (click here)c and f. (In Figs. 2 (click here)d,e,c solid (dashed) curves indicate positive (negative) contour values; the heavy solid curves represent the contour value tex2html_wrap_inline2651

Figure 2: Simulated photospheric (z=0) observations tex2html_wrap_inline2655 (a) and tex2html_wrap_inline2657 (b) obtained after removal of the tex2html_wrap_inline2659 ambiguity; (d), (e): contours of constant tex2html_wrap_inline2661 and tex2html_wrap_inline2663-values; (c), (f) like (a), (d), for tex2html_wrap_inline2665

Computation of the FFF-function tex2html_wrap_inline2667. Using the results illustrated in Fig. 2 (click here), by the aid of Eq. (6 (click here)), one obtains the nonlinear FFF-function tex2html_wrap_inline2669 shown in Fig. 3 (click here)a and the tex2html_wrap_inline2671-function shown in Fig. 3 (click here)d (recall the definition tex2html_wrap_inline2673 the corresponding relative error (N-numerical, A-analytical)
as a function of tex2html_wrap_inline2675 is shown in Fig. 3 (click here)e. As can be seen, the computational accuracy is exceptional: tex2html_wrap_inline2677 For completness, we show in Figs. 3 (click here)b and c the spatial dependence of the functions tex2html_wrap_inline2679 and tex2html_wrap_inline2681 which enter the expression for tex2html_wrap_inline2683

Figure 3: (a), (d) Numerically computed non-linear FFF-function, tex2html_wrap_inline2685 and tex2html_wrap_inline2687, respectively; (e) relative error tex2html_wrap_inline2689 as a function of tex2html_wrap_inline2691; (b), (c) spatial dependence of the functions tex2html_wrap_inline2693 and tex2html_wrap_inline2695 used for the calculation of the function tex2html_wrap_inline2697

Electrical currents. From the tex2html_wrap_inline2699 -values shown in Fig. 3 (click here)a and the field components tex2html_wrap_inline2701, one can obtain the (normalized) FFF electrical current density components tex2html_wrap_inline2703 The computed quantities are shown in Figs. 4 (click here)a, b and c; the corresponding contours of constant values are shown in Figs. 4 (click here)d, e and c. (Here, tex2html_wrap_inline2705.

Figure 4: (a)-(c) Photospheric electrical current densities tex2html_wrap_inline2707 tex2html_wrap_inline2709 and tex2html_wrap_inline2711 based on the results shown in Fig. 2 and Fig. 3; (d)-(f) corresponding contours of constant current values

Simulation of the longitudinal component tex2html_wrap_inline2713. From the analytic expression, Eq. (10 (click here)) one obtains discrete tex2html_wrap_inline2715- values in horizontal planes parallel to the observational one, at vertical distances tex2html_wrap_inline2717 apart ( q=1,2,...); these ``simulated" values are indicated by circles in Fig. 5 (click here). Then, upon using a high-order interpolation method, from these values one obtains the much higher-density set of tex2html_wrap_inline2721-values in horizontal planes at distances tex2html_wrap_inline2723 apart from each other, as shown by the continuous curves in Fig. 5 (click here); actually, tex2html_wrap_inline2725, represents the vertical (z) integration step. The top (bottom) figure represents contours of constant tex2html_wrap_inline2729-values in the plane tex2html_wrap_inline2731

Figure 5: Circles: ``Simulated observations" of tex2html_wrap_inline2733 - contours as obtained from the analytical expression; solid curves: interpolated values obtained upon using high order interpolation techniques. Top (bottom): y=0 (x=0) planes

Reconstruction of the magnetic field components, tex2html_wrap_inline2737 & tex2html_wrap_inline2739 Upon using Eqs. (1 (click here))-(6 (click here)) with (i) ``simulated" boundary conditions (at the photosphere) represented by the functions tex2html_wrap_inline2741 shown in Fig. 2 (click here), (ii) the non-linear FFF function tex2html_wrap_inline2743 shown in Fig. 4 (click here)a, and (iii) the ``simulated" longitudinal component tex2html_wrap_inline2745 shown in Fig. 5 (click here), by Eqs. (4 (click here))-(6 (click here)) and the high order, corrective vertical extrapolation method described in Appendix B, one obtains the final result - the three-dimensional chromospheric magnetic field. Thus, Fig. 6 (click here) illustrates the computed functions tex2html_wrap_inline2747 and tex2html_wrap_inline2749 at the vertical distances tex2html_wrap_inline2751 and 20 respectively; (tex2html_wrap_inline2753 represents the photosphere). Figure 7 (click here) shows contours of equal values of the functions represented in Fig. 6 (click here). For completness, the spatial structure of the ``measured" tex2html_wrap_inline2755 - component is also indicated. The ``stretching factors" indicated on the figures (1, 1.2, 1.4 and 1.6) are used for the convenience of graphical representation.

Figure 6: Reconstructed magnetic field components tex2html_wrap_inline2761 (left column) and tex2html_wrap_inline2763 (middle column) at several height values tex2html_wrap_inline2765, as indicated; for comparison, the corresponding tex2html_wrap_inline2767 - component is shown in the right column. (Actually, to emphasize the neutral line, tex2html_wrap_inline2769 we represent the quantity tex2html_wrap_inline2771, rather than just tex2html_wrap_inline2773

Figure 7: Contours of equal values of the functions represented in Fig. 6

Finally, in Fig. 8 (click here) we show contours of constant-value magnetic field components tex2html_wrap_inline2775 and tex2html_wrap_inline2777 in the plane y=0 (top) and x = 0 (bottom); and in Fig. 9 (click here) we show the reconstructed FFF electrical current density components tex2html_wrap_inline2783 at tex2html_wrap_inline2785 ( left: tex2html_wrap_inline2787 right: corresponding contours of constant values). The average relative error tex2html_wrap_inline2789 and maximum relative error tex2html_wrap_inline2791 in the computation of tex2html_wrap_inline2793 and tex2html_wrap_inline2795 as a function of the normalized height tex2html_wrap_inline2797 are shown in Fig. 10 (click here), by solid tex2html_wrap_inline2799 and dotted tex2html_wrap_inline2801 curves. As can be seen, the computational relative accuracy is very good: tex2html_wrap_inline2803 tex2html_wrap_inline2805

Figure 8: Contours of constant magnetic field values tex2html_wrap_inline2807 and tex2html_wrap_inline2809 in the plane y=0 (top) and x=0 (bottom)

Figure 9: Left: Reconstructed FFF electrical current densities tex2html_wrap_inline2815 tex2html_wrap_inline2817 and tex2html_wrap_inline2819 at several height values tex2html_wrap_inline2821 as indicated; right: corresponding contours of constant current values

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