Stochastic magnetic field generation in MHD resistive instabilities: validity limits of linear stability analysis

The validity limits of a linear analysis for a resistive instability are determined. The effects of mode-coupling on the magnetic field structure are investigated in the reconnecting layer. Given an equilibrium magnetic field and a perturbation field, the conditions are found under which the equations for the magnetic field lines of force can be expressed in Hamiltonian form. These conditions can be fulfilled by a resistive instability. Consequently, in a simple equilibrium magnetic field the resistive eigenmodes have been analytically derived. This result is used to give an explicit expression of the Hamiltonian for field-line equations when two resistive eigenmodes are taken into account. The analytical form of the resulting Hamiltonian coincides with the so-called paradigm Hamiltonian (1.5 degrees of freedom) for which the Escande-Doveil renormalization procedure leads to an explicit expression for the global stochasticity threshold. Thus it can be shown that any pair of modes - in a suitable range of parameters - yields spatial stochasticity of magnetic field lines when the perturbation amplitude is still very low. Hence a limit of validity of the linear theory can be found. The linear phase of the resistive instability turns out to be relevant only to describe the onset of the instability itself. (author)