Three-dimensional Magnetohydrodynamic Equilibria Research Articles

Improved radial differencing for three-dimensional magnetohydrodynamic equilibrium calculations

Several numerical schemes are described for accurately discretizing the radial dependence of the magnetohydrodynamic (MHD) energy of a toroidal plasma configuration. Compared with previous schemes, the new methods have significantly improved mesh convergence properties for the energy, magnetic axis, and other equilibrium parameters. This has favorable implications for both stability analysis, where small numerical errors in the energy may significantly affect the computation of marginal points, and transport applications, for which equilibrium computations on coarse meshes are desirable.

Three-dimensional plasma equilibrium near a separatrix

The limiting behavior of a general three-dimensional magnetohydrodynamic (MHD) equilibrium near a separatrix is calculated explicitly. No expansions in beta or assumptions about island widths are made. Implications of the results for the numerical calculation of such equilibria are discussed, as well as for issues concerning the existence of three-dimensional MHD equilibria.

MOMCON: A spectral code for obtaining three-dimensional magnetohydrodynamic equilibria

A new code, MOMCON (spectral moments code with constraints), is described that computes three-dimensional ideal magnetohydrodynamic (MHD) equilibria in a fixed toroidal domain using a Fourier expansion for the inverse coordinates ( R, Z) representing nested magnetic surfaces. A set of nonlinear coupled ordinary differential equations for the spectral coefficients of ( R, Z) is solved using an accelerated steepest descent method. A stream function, λ, is introduced to improve the mode convergence properties of the Fourier series for R and Z. The convergence rate of the R - Z spectra is optimized on each flux surface by solving nonlinear constraint equations relating the m ≥ 2 spectral coefficients of R and Z.

A convergent spectral representation for three-dimensional inverse magnetohydrodynamic equilibria

By rearranging terms in a polar representation for the cylindrical spatial coordinates (R,φ,Z), a renormalized Fourier series moment expansion is obtained that possesses superior convergence properties in mode number space. This convergent spectral representation also determines a unique poloidal angle and thus resolves the underdetermined structure of previous moment expansions. A conformal mapping technique is used to demonstrate the existence and uniqueness of the new representation.

Island formation and destruction of flux surfaces in three-dimensional MHD equilibria

The physics involved in the appearance of islands caused by resonant pressure driven currents in three-dimensional magnetohydrodynamic equilibria is described. Estimates of island widths are obtained by an expansion in β, with an auxiliary expansion about the magnetic axis. The theory is applied to Princeton’s heliac reference design.

Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria

An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest-descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive-definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self-consistent value for λ.

Three-Dimensional Magnetohydrodynamic Equilibria of Toroidal Stellarators

Fully nonaxisymmetric magnetohydrodynamic equilibria have been calculated numerically for the Proto-Cleo stellarator utilizing a fully three-dimensional magnetohydrodynamic equilibrium code. It was found that recently observed anomalous circulating currents in stellarators may be a result of the equilibrium condition.