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This paper studies the nonlinear evolution of magnetic field turbulence inproximity of steady ideal MHD configurations characterized by a small electriccurrent, a small plasma flow, and approximate flux surfaces, a physical settingthat is relevant for plasma confinement in stellarators. The aim is to gatherinsight on magnetic field dynamics, to elucidate accessibility and stability ofthree-dimensional MHD equilibria, as well as to formulate practical methods tocompute them. Starting from the ideal MHD equations, a reduced dynamical systemof two coupled nonlinear PDEs for the flux function and the angle variableassociated with the Clebsch representation of the magnetic field is obtained.It is shown that under suitable boundary and gauge conditions such reducedsystem preserves magnetic energy, magnetic helicity, and total magnetic flux.The noncanonical Hamiltonian structure of the reduced system is identified, andused to show the nonlinear stability of steady solutions against perturbationsinvolving only one Clebsch potential. The Hamiltonian structure is also appliedto construct a dissipative dynamical system through the method of doublebrackets. This dissipative system enables the computation of MHD equilibria byminimizing energy until a critical point of the Hamiltonian is reached.Finally, an iterative scheme based on the alternate solution of the two steadyequations in the reduced system is proposed as a further method to compute MHDequilibria. A theorem is proven which states that the iterative schemeconverges to a nontrivial MHD equilbrium as long as solutions exist at eachstep of the iteration.

A Reduced Ideal MHD System for Nonlinear Magnetic Field Turbulence in Plasmas with Approximate Flux Surfaces