Equilibrium of a toroidal plasma column in steady-state and high-frequency magnetic fields

The authors consider the equilibrium of a toroidal plasma column in a steady-state magnetic field on which is superimposed a multipole high-frequency field produced by currents flowing around the major circumference of the torus which are proportional to exp[i (ωt-nψ)], where ψ is the polar angle in the meridional cross-section of the torus and ω is the frequency determined by the generator.They demonstrate the possibility of achieving equilibrium when the pressure of the high-frequency field is less than the kinetic pressure of the plasma, ignoring the reaction of the circuit to the displacement of the plasma column; i.e. it is assumed that this displacement does not give rise to image currents in the circuit.An expression is derived for displacement of the column as a function of the amplitude of the high-frequency field, the kinetic pressure of the plasma, the depth of the skin layer, the multipole order (n), and of geometric parameters such as the radius of the plasma column rp, the radius of the circuit rk and the radius of the torus R. The displacement of the column is proportional to the kinetic pressure of the plasma, inversely proportional to the square of the amplitude of the current in the circuit, and independent of the steady-state magnetic field. Expressed as a function of the multipole order (n), displacement is at a minimum when n is equal to the whole number closest to 1 + [1/2 ln(rk/rp)]. When rk/rp∼2, the optimum value of n is 2 (quadrupole). When n = 1 (dipole), displacement increases in proportion to ln(c/ωrp). On the assumption of a quasi-steady state (c/ωrp ≫ 1) equilibrium containment of the plasma in the dipole case is found to be difficult to achieve in practice. When n = 0 equilibrium is quite impossible to achieve. The validity of this conclusion is dependent on the assumption of a quasi-steady state and on the further assumption that the circuit will not react to displacement of the column.