In this second paper on the evolution of magnetic flux ropes we study
the effects of gas pressure. We assume that the energy transport is de
scribed by a polytropic relationship and reduce the set of ideal MHD e
quations to a single, second-order, nonlinear, ordinary differential e
quation for the evolution function. For this conservative system we ob
tain a first integral of motion. To analyze the possible motions, we u
se a mechanical analogue-a one-dimensional, nonlinear oscillator. We f
ind that the effective potential for such an oscillator depends on two
parameters: the polytropic index gamma and a dimensionless quantity k
appa the latter being a function of the plasma beta, the strength of t
he azimuthal magnetic field relative to the axial field of the flux ro
pe, and gamma. Through a study of this effective potential we classify
all possible modes of evolution of the system. In the main body of th
e paper, we focus on magnetic flux ropes whose field and gas pressure
increase steadily towards the symmetry axis. In this case, for gamma >
1 and all values of kappa, only oscillations are possible. For gamma
< 1, however, both oscillations and expansion are allowed. For gamma <
1 and kappa below a critical value, the energy of the nonlinear oscil
lator determines whether the flux rope will oscillate or expand to inf
inity. For gamma < 1 and kappa above critical, however, only expansion
occurs. Thus by increasing kappa while keeping gamma fixed (<1), a ph
ase transition occurs at kappa = kappa(critical) and the oscillatory m
ode disappears. We illustrate the above theoretical considerations by
the example of a flux rope of constant field line twist evolving self-
similarly. For this example, we present the full numerical MHD solutio
n. In an appendix to the paper we catalogue all possible evolutions wh
en (1) either the magnetic field or (2) the gas pressure decreases mon
otonically toward the axis. We find that in these cases critical condi
tions can occur for gamma > 1. While in most cases the flux rope colla
pses, there are notable exceptions when, for certain ranges of kappa a
nd gamma, collapse may be averted.