Nonexistence of magnetohydrodynamic equilibria with poloidally closed field lines in the case of violated axisymmetry

The existence of nonaxisymmetric toroidal magnetohydrodynamic (MHD) equilibria, whose magnetic field lines are closed after one poloidal turn around the magnetic axis C is investigated analytically. Up--down symmetry of the configuration with respect to the equatorial plane that contains the axis is assumed. In principle, nonaxisymmetry is manifested in the form of a noncircular axis or a variation of the geometry and/or magnetic field along a circular axis. It is proved that no equilibrium with a noncircular axis exists. For a circular axis, nonexistence is proved if the ellipticity of the cross section varies along C. Nor is variation of the triangularity, etc., up to the seventh Fourier mode with respect to the poloidal angle, allowed. For variations with still higher mode numbers, nonexistence is made plausible. For the magnetic field the situation is analogous. Nonaxisymmetric poloidal equilibria with equatorial mirror symmetry are thus practically ruled out. The method of investigation is an expansion in the distance from the magnetic axis, supported by an algebraic computer language. With growing order the number of constraints on the configuration increases until the quoted results are obtained in the sixth and higher orders. copyright 1995 American Institute of Physics