Geometric considerations of the evolution of magnetic flux ropes

We use flux conservation and magnetohydrodynamics (MHD) theory to discuss essential differences in the nature of the evolution of two analytical solutions describing magnetic flux tubes evolving in time. The first of these maintains the elongation of the tube, while the second maintains a constant angular extension with respect to a possible pointlike source. In the first case, free expansion of the plasma (density N) occurs only in a direction perpendicular to the flux-tube x axis. In the second case, isotropic evolution is considered. In both cases it is assumed that at initial time t(0) the flux-tube B field is the force-free magnetostatic Lundquist solution, which energetically corresponds to the most stable state for any flux-tube structure. We show that for each case conservation of magnetic flux is enough to establish the scaling with time of the B field. While both expansions may correspond to the evolution of observed flux tubes in the heliosphere, the isotropic expansion appears to capture consistently essential features associated with the actual observations of expanding coronal mass ejections within 30 solar radii. For isotropic expansion of the plasma the force-free nature of the B field is preserved for all time. As an example the MHD solutions are applied to an interplanetary magnetic cloud observed with the spacecraft Wind, which passed Earth's vicinity on June 2, 1998.