Chandrasekhar-Kendall modes and Taylor relaxation in an axisymmetric torus

Abstract

The helicity-conserving Taylor relaxation of a plasma in a toroidal chamber to a force-free configuration, which means j=(j‖∕B)B with j‖∕B independent of position, can be generalized to include the external injection of magnetic helicity. When this is done, j‖∕B has resonant values, which can be understood using the eigenmodes of Taylor-relaxed plasmas enclosed by a perfectly conducting toroidal shell. These eigenmodes include a toroidal generalization of those found by Chandrasekhar and Kendall (CK) [Astrophys. J. 126, 457 (1957)] for a spherical chamber, which has no externally produced magnetic flux. It is shown that the CK modes in an axisymmetric torus are of three types: (1) helical modes as well as axisymmetric modes that have (2) and have no (3) net toroidal flux. Yoshida and Giga (YG) [Math. Z. 204, 235 (1990)] published a fourth class of modes: axisymmetric modes that have no net toroidal flux in the chamber due to toroidal flux produced by a net poloidal current in the shell canceling the net toroidal flux from the plasma currents. Jensen and Chu [Phys. Fluids 27, 2881 (1984)], as well as Taylor [Rev. Mod. Phys. 58, 741 (1986)], considered modes in which the vector potential was zero on the axisymmetric toroidal chamber. It is shown that these Jensen-Chu-Taylor modes include only the CK helical modes and the CK axisymmetric modes without net toroidal flux. If the toroidal chamber is perfectly conducting except for a cut that prevents a net poloidal current from flowing, resonances in j‖∕B occur at the eigenvalues of the axisymmetric CK modes. Jensen and Chu studied this type of resonance. Without the cut, so a poloidal current flows to conserve the net toroidal flux, it is shown that j‖∕B resonances occur at the eigenvalues of the CK modes that have no net toroidal flux and at the eigenvalues of the YG modes, which are upshifted from the eigenvalues of the axisymmetric CK modes that carry net toroidal flux.

Constrained Resonance in Magnetic Self-Organization

“…We wish to note that in the original Jensen-Chu [7] and Taylor [8] formulation, the vector potential Aj @ 0 is used in place of B nj @ 0 in defining the boundary condition for Chandrasekhar-Kendall modes. As the result, Jensen-Chu-Taylor theory restricts its discussion to pure flux-free CK modes [18]. There are no longer resonances at the eigenfrequencies where the eigenmodes carry finite toroidal flux, i.e., h j i Þ 0.…”

“…2 (black line). In a doubly connected torus, maintaining G 0 a constant implies the absence of a toroidal flux conserver and that the toroidal field coil circuits have infinite inductance [18].…”

Numerical computation of the helical Chandrasekhar–Kendall modes

“…These force-free eigensolutions are uniquely determined by the chamber geometry, and play an essential role in determining the relaxed states of a driven plasma [4,7], in which the spatial overlap of the CK modes and a vacuum field generated by external current provides the coupling between the external helicity source and the driven plasma [8]. In fact, resonant coupling [9][10][11][12] is the physical mechanism underlying the self-organization of system-scale magnetic fields by magnetic relaxation in the laboratory formation of spherical tokamak [13,14], spheromak [15], and reversed field pinch by helicity injection.…”

Exact axisymmetric Taylor states for shaped plasmas

“…For equilibria with a magnetic X-point, the location of the X-point must also be specified. The flexibility and simplicity of these solutions make them useful for verifying the accuracy of numerical solvers and for theoretical studies of Taylor states in laboratory experiments.Plasmas in both astrophysical and laboratory settings have a strong tendency to relax to minimum energy states known as Taylor states or Woltjer-Taylor states [1][2][3][4][5][6][7][8][9][10][11][12] in which the magnetic fields are force-free fields given by the equationwhere λ is a global constant. A well-known analytic solution to equation (1) is often used for theoretical studies and to interpret experiments [5][6][7]13 .…”

“…Plasmas in both astrophysical and laboratory settings have a strong tendency to relax to minimum energy states known as Taylor states or Woltjer-Taylor states [1][2][3][4][5][6][7][8][9][10][11][12] in which the magnetic fields are force-free fields given by the equation…”