A self-similar solution of expanding cylindrical flux ropes for any polytropic index value

We found a new class of solutions for MHD equations that satisfies the condition that cylindrical flux ropes can expand self-similarly even when the polytropic index γ is larger than 1. We achieved this by including the effects of elongation along the symmetry axis as well as radial expansion and assuming that the radial expansion rate is the same as the elongation rate. In previous studies (Osherovich et al., 1993a, 1995), a class of self-similar solutions was described for which cylindrical flux ropes expand only in the medium where γ is less than 1. We compare the models including elongation and excluding elongation observationally by using the WIND key parameters. The difference in the fitting results of the magnetic field between these two models is slight. However the fitting of the velocity is improved when elongation is included and when new geometric parameters that are necessary to represent the elongation are introduced. The values of these parameters are almost the same scale as the radius of flux ropes, which is consistent with the assumption of the isotropic expansion. This new exact solution to a time-dependent two-dimensional MHD problem can also be used to test numerical codes.