Symmetries of resistive two-fluid magnetohydrodynamics under reversals of toroidal field, current, and rotation

Abstract

The discrete symmetries of time-dependent, non-axisymmetric, two-fluid magnetohydrodynamics (MHD) equations are considered. Solutions of the time-independent, axisymmetric, ideal-MHD equations remain solutions under reversals of the toroidal field, current density, or rotation. Introducing non-axisymmetry, resistivity, and two-fluid effects into the equations each break different symmetries. Symmetry groups for solutions possessing up-down spatial symmetry, and the effect of flipping the magnetic geometry across the horizontal midplane, are also considered. It is shown that poloidal velocity may or may not be reversed under a simultaneous reversal of the toroidal field and a vertical reflection of the magnetic geometry, depending on the symmetry of the velocity. Because the symmetry groups of ideal, resistive, and two-fluid MHD are distinct, it may be possible to ascertain the dominant physical mechanism of various phenomena through empirical observations of symmetry-breaking. These results, which hold for nonlinear solutions in arbitrary geometry, should also be of use in testing numerical codes.The magnetic geometry of a solution to the GradShafranov equation is unaffected by reversing the direction of the toroidal field, the toroidal current, or the toroidal rotation. However, it is known from experiment that the relative directions of these quantities do impact both the stability and transport properties of tokamaks. 1-4 For example, it is found that tearing-mode excitation thresholds are sensitive to the direction of toroidal rotation relative to the plasma current, 3,4 and H-mode power thresholds are found to depend on the direction of the toroidal field. 2 The physical basis for these observed asymmetries is generally not well understood. In order to gain a better understanding, we consider here how various physical effects impact the symmetry properties of fluid models.The continuous Lie symmetries of the ideal and resistive single-fluid magnetohydrodynamics (MHD) equations have been obtained by Fuchs. 5 These symmetries include spatial translations and rotations, as well as dilations and Galilean transformations. Using these symmetries, it is possible to obtain special exact solutions of the MHD equations; 6 however, these solutions are not easily applicable to tokamak phenomenology. Similar analysis has also recently been performed on the Grad-Shafranov equation. [7][8][9] Perhaps more directly applicable to the interpretation of tokamak experiments is the separate issue of the discrete symmetries of model equations under reversal of equilibrium quantities. This issue has been considered by Cohen and Ryutov 10 who found symmetry properties of a general kinetic model, and of a resistive single-fluid model. Additional symmetries were found in the special case of up-down symmetric solutions. Catto and Simakov have extended Cohen and Ryutov's analysis to include the effect of flipping the magnetic geometry across the horizontal midplane. 11 Furthermore, Aydemir has described symmetries of axisymmetric solut...

Plasma response models for non-axisymmetric perturbations

“…Thus, to guarantee the correct result, all the physical processes should be included, making the calculations unrealistically expensive computationally. Experiments may be used as a guide to eliminate some processes but the information is not always easy to extract [30].…”