Nonlinear force-free magnetic field and field distribution over a sunspot

The force-free magnetic flux rope with the uniform twisting has been considered. The cross-section radius of any magnetic surface varies along the symmetry axis of the system. The field distribution derived as a solution of the nonlinear differential equation for the force-free field satisfies the well known similarity condition. The basic properties of the model are: a) The twisting of the field in and over the support is weak enough. Every line of force makes only half of the turn on the distance between the sunspots. b) The total flux of the longitudinal field is a finite value; the total current along the axis is zero, because the current density changes the sign on a magnetic surface (α=0). c) The surface α=0, being stable in the MHD approximation, could play the role of the natural sunspot boundary. d) The similarity condition could be derived from the two conditions: uniform twisting (the pitch of the helix is constant) and force-free approximation. The vertical and radial field distribution obtained are in good agreement with the observational data