By Birkhoff G.D.

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40) while variation with respect to v gives v = 0. Hence the final state is a static constant-a force-free magnetic field, where a is determined by the value of the helicity. In general, the relaxation process requires reconnection of field lines resulting in a simpler, less complex magnetic field structure. Since magnetic helicity is a measure of the linkedness and knottedness of the magnetic field, breaking such linkages seems to involve also a change of helicity. We will show or at least make plausible that while the individual helicities of the constituent flux ropes become meaningless, since the latter are merged and reorganized and hence lose their identity, the sum of their helicities, nevertheless, is conserved.

Finite amplitude slow modes, are more important. 4). In an inhomogeneous plasma the three types of modes are in general coupled, but they remain nevertheless useful for a qualitative classification of small perturbations. 25), sufficiently strong pressure and current density gradients give rise to instabilities. ) Finally, we briefly discuss the validity of the incompressibility assumption. Though this assumption is often made for linear modes, it is basically a consequence of the nonlinear equations since, for instance, any linear perturbation in a neutral fluid corresponds to a (compressible) sound wave.

13) can also be written in terms of the total pressure P = p + B2/8n: VP--B 4TT VB = 0. 13) it follows immediately that p is constant along magnetic field lines and along current density lines, B • Vp = 0, j • Vp = 0. Since, for a finite pressure gradient, B and j are not parallel, we can locally construct a surface \p(x9y,z) = const, where p is constant, p = p{\p). , there must be surfaces of constant pressure on which a field line runs on for ever, a condition which is rather stringent for general nonsymmetric configurations.