  Up: Three-dimensional chromospheric magnetic

# 2. General formulation and basic equations

## 2.1. Vector magnetograph observations

Modern vector magnetographs provide the following magnetic field components : (i)   and  in a horizontal plane, , at the photosphere (z = 0); and (ii) the longitudinal (line of sight) component (x, y, z) at a (finite) number of horizontal planes (z) situated at distances ( ) above the plane (0) and parallel to it.

These observational data need some theoretical treatment before they can be used for reconstruction purposes: (a) the ambiguity in the observed components (x, y, 0) and (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 (x, y, z) - values at a relatively large number of intermediate horizontal planes parallel to (0) and situated at distances apart, where 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 (x, y, 0) and (x, y, 0), as well as the longitudinal field component (x, y, 0) in a plane (z = 0) at the photosphere and (ii) the longitudinal field component (x, y, z) at vertical distances apart, reconstruct the three-dimensional chromospheric magnetic field in half the space above the domain (z = 0).

## 2.3 Basic equations - force free field (FFF) model

The steady state FFF equations are and Thus, Eq. (1) states that the electric current density is proportional to the magnetic field . Upon taking the divergence of Eq. (1 (click here)), by (2 (click here)), one obtains which indicates the constancy of along individual field lines. In order of ascending complexity, Eqs. (1 (click here))-(3 (click here)) may describe the cases of current-free configurations , linear force-free field configurations and nonlinear force-free field configurations . The components of Eqs. (1 (click here)) and (2 (click here)) are:   and Thus, if the field components at the photosphere, (x, y; z = 0), (x, y; z = 0) and (x, y; z = 0) are given (and, therefore is determined) one can calculate numerically the horizontal derivatives and then, by Eqs. (4 (click here))-(7 (click here)) one can obtain the vertical derivatives , and proceed to the vertical integration to the height , and so on.

## 2.4. Study case

As a study case we consider a magnetic configuration generated by electric currents satisfying the FFF condition everywhere except along the line y = 0, z = -a (below the photosphere), where an infinite straight line current is flowing (Low 1982; Cuperman & Ditkowski 1990).  and where The free generating function is related to the quantity by the equation In this work we use the following generating function: Thus, in the continuation we use the values of the field components and (and consequently, ) 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.  Up: Three-dimensional chromospheric magnetic

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