The non-linear evolution of magnetic flux ropes: 3. effects of dissipation

We study the evolution (expansion or oscillation) of cylindrically symmetric magnetic flux ropes when the energy dissipation is due to a drag force proportional to the product of the plasma density and the radial speed of expansion. The problem is reduced to a single, second-order, ordinary differential equation for a damped, non-linear oscillator. Motivated by recent work on the interplanetary medium and the solar corona, we consider polytropes whose index, c, may be less than unity. Numerical analysis shows that, in contrast to the small-amplitude case, large-amplitude oscillations are quasi-periodic with frequencies substantially higher than those of undamped oscillators. The asymptotic behaviour described by the momentum equation is determined by a balance between the drag force and the gradient of the gas pressure, leading to a velocity of expansion of the flux rope which may be expressed as …1=2c†r=t, where r is the radial coordinate and t is the time. In the absence of a drag force, we found in earlier work that the evolution depends both on the polytropic index and on a dimensionless parameter, j. Parameter j was found to have a critical value above which oscillations are impossible, and below which they can exist only for energies less than a certain energy threshold. In the presence of a drag force, the concept of a critical j remains valid, and when j is above critical, the oscillatory mode disappears altogether. Furthermore, critical j remains dependent only on c and is, in particular, independent of the normalized drag coefficient, m . Below critical j, however, the energy required for the flux rope to escape to infinity depends not only on j (as in the conservative force case) but also on m . This work indicates how under certain conditions a small change in the viscous drag coefficient or the initial energy may alter the evolution drastically. It is thus important to determine m and j from observations.