Plasma Equilibrium Configuration of Hill’s Vortex Type with n = 3 Toroidal Asymmetry

A new plasma equilibrium configuration is found. It is a generalization of the known magnetohydrodynamic structure of Hill’s vortex type for the case of axial asymmetry with the toroidal wave number n = 3. The analysis is based on a solution to a system of coupled partial differential equations derived from exact equations of the three-dimensional equilibrium of static plasma (force balance) under the assumption of small asymmetry and in the absence of the main toroidal magnetic field. It is shown that the search for corrections to the basic axisymmetric equilibrium in the class of polynomials in powers of z (by analogy with Solov’ev solution) leads to an overspecified problem and requires satisfying compatibility conditions. The resulting solution contains two free coefficients that determine the amplitude of the toroidal inhomogeneity of a magnetic configuration. It also depends on the initial equilibrium parameters, namely, the elongation of magnetic surfaces in the basic configuration of a Hill’s vortex. The toroidal asymmetry of magnetic surfaces is accompanied by the appearance of a toroidal magnetic field, which can be oppositely directed and is absent in the basic vortex configuration. The shape of magnetic surfaces in $\varphi = {\text{const}}$ cross-section differs significantly from the “axisymmetric” one. Corrections to the magnetic axis are calculated, which to ensure the sufficiency of expansion in the weak asymmetry parameter at the small gradient of $\Psi $.