Relationship between the Horizontal Gradient of the Vertical Magnetic Field and the Horizontal Electric Current on the Photosphere in a Model Active Region of the Sun

Abstract: Based on data obtained from numerical modeling of an active region (AR) of the Sun with the codes of radiation magnetohydrodynamics (MURaM), the horizontal gradient of the vertical component of the magnetic field on three layers in the vicinity of the photosphere ∇_hB_z and the horizontal component of the electric current density j_h for the AR of the flare were calculated. It was found that ≈90% of the calculated pixels contain j_h and ∇_hB_z of the same order, but these quantities are weakly correlated. The correlation coefficient is calculated for several groups of pixels that are distinguished by the magnitude of the horizontal current and the magnetic-field configuration. The correlation coefficients for these cases did not exceed 0.34 for the lower layer and 0.54 for the upper one. The contours of the strong j_h and ∇_hB_z (in units of electric current density) are also weakly correlated. The result may indicate that caution must be taken when the horizontal-field gradient is used for approximation of the horizontal current in the photosphere (in particular, in the development of methods for the forecasting of flares). In additional, the probability-density functions PDF(|in AR photosphere are constructed. They show that the median value of the ratio of the horizontal current to the vertical current is 3.5. This is consistent with the few estimates of various components of currents in real solar ARs and indicates the importance of the further study of horizontal currents.