Astron. Astrophys. 318, 289-292 (1997)

2. Numerical model

A basic assumption in this model is that the electric current is force-free, i.e. the electric current is parallel to the local magnetic field. The currents are represented by numerical particles (6400 in our case). First some initial magnetic field of arbitrary form (potential or force-free) is prescribed. Then from the bottom surface of the numerical box, which represents the photosphere, numerical particles (no mass, no forces, no equation of motion are considered) are injected upwards along the local magnetic field line. These particles change their position with the constant "velocity" ( is the magnetic field) throughout the space above the photosphere. During the "time" step , they change their position by the propagation vector d , which simultaneously represents the force-free electric current, whose intensity is prescribed by a numerical constant. Because the particles are injected successively, one particle every time step along one current path, they are after some time distributed regularly (with the d distance between neighbouring particles) along a line above the photosphere, thus depicting the electric current path. For small currents these paths correspond to the initial magnetic field lines. If particles penetrate back below the photosphere then these particles are excluded from further computations. Prescribing to each particle some electric current and using the Biot and Savart law

where R is the distance from the current element to where dB is evaluated and dl is the element of length in the current direction (which, in our case, is equal to the particle propagation vector - see above), we compute the current magnetic field, which changes the initial magnetic field as well as the current particle trajectories. Namely, at every time step, at particle positions we compute the magnetic field directions, which are then used as directions of the particle propagation. This procedure is repeated every time step. At some specific times the magnetic field is also available at any point of the computational box, e.g. for field-line drawings. The relation (1) is singular for . Moreover, the electric current is flowing in some finite area. On the other hand, there is the finite distance between numerical particles. To solve this problem, we define as the minimum interaction distance; while for the magnetic field calculations at distances the relation (1) is valid, for the case a modified relation in the form

is used (in this relation decreases to zero for ). Thus roughly corresponds to the radius of the electric current cross-section. (In principle, Eq. (2) can be replaced by some cylindrical or other approximation. But we think that these changes cause only slight effects on global equilibria. The reason why Eq. (2) was used is the simplicity of its numerical form, which is important in the case with many numerical particles.) Simultaneously, this procedure represents some smoothing of electric current. Namely, in the force-free case, the curved, infinitesimally thin electric current always forms a helical structure. This effect is also found for smoothed electric currents, but with much larger structures. To describe this helical structure with the appropriate precision, we need the smoothing distance to be several times greater than the distance between successively injected particles. Otherwise, we have more helical circles, which are not sufficiently described by particles. This smoothing procedure is a little artificial and can be considered as appropriate only if we are interested in the global aspects of electric current and magnetic field. More precise computations can be done only if one localized electric current is represented by many close current paths. When the minimum interaction distance is shorter than the distances between numerical particles d , then the procedure with becomes irrelevant and the smoothing of currents is given by d . More precision can be obtained only by the shortening of d . Therefore, according to the type of our task, we need to select the appropriate d , , number of current paths and number of particles.

Our computations start from a low electric current which is then slowly and continuously increased to the specific value of electric current. Then it is useful to keep constant this current for some time and thus to relax the resulting structure. The method can be generalized for an arbitrary number of injection positions, i.e. for arbitrary current distribution. It enables us, for example, to simulate the current flowing through a finite cross-section by more sub-currents and thus to increase the computation precision.

Applying this method to currents in the solar atmosphere we need to consider the effect of the inertial photosphere. We included this effect using the virtual mirror current as suggested by Kuperus and Raadu (1974). This current causes that no magnetic field created by the coronal currents can penetrate the photospheric layers. This procedure is in agreement with the specification of the boundary condition at the photospheric layer (see Sakurai, 1981).

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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