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2. General formulation and basic equations

2.1. Vector magnetograph observations

Modern vector magnetographs provide the following magnetic field componentsgif: (i) tex2html_wrap_inline2359 tex2html_wrap_inline2361 tex2html_wrap_inline2363 and tex2html_wrap_inline2365 tex2html_wrap_inline2367 in a horizontal plane, tex2html_wrap_inline2369, at the photosphere (z = 0); and (ii) the longitudinal (line of sight) component tex2html_wrap_inline2373 (x, y, z) at a (finite) number of horizontal planes tex2html_wrap_inline2381 (z) situated at distances tex2html_wrap_inline2385 (tex2html_wrap_inline2387) above the plane tex2html_wrap_inline2389 (0) and parallel to it.

These observational data need some theoretical treatment before they can be used for reconstruction purposes: (a) the tex2html_wrap_inline2391 ambiguity in the observed components tex2html_wrap_inline2393 (x, y, 0) and tex2html_wrap_inline2399 (x, y, 0) has to be removed by the aid of some theoretical method (see, e.g., Cuperman et al. 1991b and references therein); (b) high order interpolation schemes (see, e.g., next section) have to be applied in order to obtain tex2html_wrap_inline2405 (x, y, z) - values at a relatively large number of intermediate horizontal planes parallel to tex2html_wrap_inline2413 (0) and situated at distances tex2html_wrap_inline2415 apart, where tex2html_wrap_inline2417 represents the vertical integration step.

2.2. Formulation of the problem

With the magnetograph observations corrected and improved as indicated above, the problem can be formulated as follows: given (i) the transverse field components tex2html_wrap_inline2421 (x, y, 0) and tex2html_wrap_inline2427 (x, y, 0), as well as the longitudinal field component tex2html_wrap_inline2433 (x, y, 0) in a plane tex2html_wrap_inline2439 (z = 0) at the photosphere and (ii) the longitudinal field component tex2html_wrap_inline2443 (x, y, z) at vertical distances tex2html_wrap_inline2451 apart, reconstruct the three-dimensional chromospheric magnetic field in half the space above the domain tex2html_wrap_inline2453 (z = 0).

2.3 Basic equations - force free field (FFF) model

The steady state FFF equations are

Thus, Eq. (1) states that the electric current density tex2html_wrap_inline2457 is proportional to the magnetic field tex2html_wrap_inline2459. Upon taking the divergence of Eq. (1 (click here)), by (2 (click here)), one obtains
which indicates the constancy of tex2html_wrap_inline2461 along individual field lines. In order of ascending complexity, Eqs. (1 (click here))-(3 (click here)) may describe the cases of current-free configurations tex2html_wrap_inline2463, linear force-free field configurations tex2html_wrap_inline2465 and nonlinear force-free field configurations tex2html_wrap_inline2467. The components of Eqs. (1 (click here)) and (2 (click here)) are:



Thus, if the field components at the photosphere, tex2html_wrap_inline2469 (x, y; z = 0), tex2html_wrap_inline2477 (x, y; z = 0) and tex2html_wrap_inline2485 (x, y; z = 0) are given (and, therefore tex2html_wrap_inline2493 is determined) one can calculate numerically the horizontal derivatives tex2html_wrap_inline2495 and then, by Eqs. (4 (click here))-(7 (click here)) one can obtain the vertical derivatives tex2html_wrap_inline2497, and proceed to the vertical integration to the height tex2html_wrap_inline2499, and so on.

2.4. Study case

As a study case we consider a magnetic configuration generated by electric currents satisfying the FFF condition tex2html_wrap_inline2501 everywhere except along the line y = 0, z = -a (below the photosphere), where an infinite straight line current tex2html_wrap_inline2507 is flowing (Low 1982; Cuperman & Ditkowski 1990).

The solutions of Eqs. (1 (click here)) and (2 (click here)) for this case are





The free generating function tex2html_wrap_inline2509 is related to the quantity tex2html_wrap_inline2511 by the equation


In this work we use the following generating function:


Thus, in the continuation we use the values of the field components tex2html_wrap_inline2513 and tex2html_wrap_inline2515 (and consequently, tex2html_wrap_inline2517) at the photospheric surface (z = 0) given by Eqs. (8 (click here))-(13 (click here)) as boundary conditions and integrate the system of Eqs. (4 (click here))-(7 (click here)) to determine the magnetic field components in the half space z > 0 . Comparison of the results with the values given by Eqs. (8 (click here))-(12 (click here)) for z > 0 will indicate the reliability of the numerical integration procedure proposed and implemented here.

For convenience, we use the normalizations (i = x, y, z):


The integration limits in the x, y-plane are:

b being a free parameter.

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