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

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

Feb 1997

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997angeo..15..152f&link_type=abstract

Annales Geophysicae, vol. 15, Issue 2, pp.152-164

Computer Science

Numerical Analysis

6

Scientific paper

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, , 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 (1/2)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, . Parameter 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 remains valid, and when is above critical, the oscillatory mode disappears altogether. Furthermore, critical remains dependent only on and is, in particular, independent of the normalized drag coefficient, *. Below critical , however, the energy required for the flux rope to escape to infinity depends not only on (as in the conservative force case) but also on *. 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 * and from observations. Acknowledgements.This work is supported in part by NASA Grant NAG 5-2834. Topical Editor R. Schwenn thanks M.A. Berger and V. Bothmer for their help in evaluating this paper.--> Correspondence to: C. J. Farrugia-->

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