Self-similar evolution of the two-dimensional cylindrical magnetohydrodynamic flux rope

One of the important features of the one-dimensional cylindrical self-similar magnetohydrodynamic (MHD) model of magnetic rope is that it oscillates about a force-free solution [Osherovich et al., 1993. Nonlinear evolution of magnetic flux ropes. 1. Low-beta limit. Journal of Geophysical Research 98, 13225-13231; Osherovich et al., 1995. Nonlinear evolution of magnetic flux ropes. 2. Finite-beta plasma. Journal of Geophysical Research 100, 12307-12318.] due to the reduced dimensionality of the system, as in laboratory Z pinch plasmas [Felber, 1982. Self-similar oscillations of a Z pinch. Physics of Fluids 25, 643-645]. However, such oscillations have never been confirmed by observations. Following the approach of Low [1982a. Self-similar magnetohydrodynamics. I. The γ=4/3 Polytrope and the Coronal transient. Astrophysical Journal 254, 796-805; 1982b. Self-similar magnetohydrodynamics. II. The expansion of a Stella envelope into a surrounding vacuum. Astrophysical Journal 261, 351-369], a two-dimensional self-similar MHD model under radial expansion is analyzed in cylindrical geometry with translational symmetry in the z-axis. Non-oscillatory solutions are established with polytropic index γ=1 and 2. For γ=2, the system is linear, and the plasma pressure is balanced by the longitudinal magnetic pressure. As for γ=1, the plasma pressure is balanced by the transverse magnetic pressure, and it is also the driving force of non-linearity that stresses the system. Due to the two-dimensional structure of the magnetic field and plasma, this model allows the possibility of an energetic magnetic cloud with a southward component impinging on Earth without raising expected magnetic storms.