Curved interplanetary flux ropes with non-linear force-free magnetic field

It is possible to construct an exact toroidal force-free non-linear solenoidal magnetic field. Beginning with a force-free condition for the magnetic field in the toroid, we should satisfy that curl of B is parallel to B. The latter condition is met only if the gradient of the ratio of magnitudes of B to that of curl B, which is a scalar function alpha, is perpendicular to the magnetic field B. This can be provided in axial symmetry once another scalar function, f, which relates the gradient of alpha to the vector which has the same magnitude of B but perpendicular to it, is properly selected. After a series of transformations, we obtain a non-linear differential equation for alpha. It can be solved by its expansion into a sum of harmonic functions. Alpha is found by applying non-linear methods. It enables then finding the magnetic field and current density. The solution is applied for the study of interplanetary magnetic clouds.