The authors consider the equilibrium of a toroidal plasma pinch in a constant magnetic field on which is superposed a helical high-frequency field produced by helical currents flowing on a toroidal surface with major radius R and minor radius rk. The helical currents are proportional to exp[i(ωt – nψ – k||ζ)], where ω is the frequency imparted by the generator, ψ the polar angle in the meridional cross-section of the torus, and ζ is the length of arc along the major circumference of the torus.The authors demonstrate that it is possible to achieve equilibrium with high-frequency field pressures less than the kinetic pressure of the plasma.An expression is derived for the displacement of the pinch in the equilibrium state as a function of high-frequency field amplitude, plasma kinetic pressure, the multipolarity n, and such geometric parameters as the radius of the plasma pinch rp, the radius of the circuit rk and the radius of the torus R.In the helical high-frequency field, the equilibrium of the plasma pinch depends to a great extent on the strength of the constant magnetic field H0, the plasma concentration N, the frequency of the variable field f = 2π/ω, and the wavelength of the helical current λ|| = 2π/k||.The helical high-frequency electromagnetic field within the plasma is a superposition of two wave types. When H0 < Hcrit = the first type of wave transfers to the skin, and when H0 > Hcrit. it propagates. When k||rp ≪ 1 and k||rp ≪ H0/Hcrit. the field of the second type of wave differs only slightly from the vacuum field.In the region where the first type of wave transfers to the skin, the displacement Δ of the plasma pinch decreases with increasing H0 and becomes zero when H0 = Hres = : with further increases in the magnetic field strength Δ becomes negative. H0 = Hres is the condition for resonance excitation of the helical high-frequency field within the plasma. The most effective use of high-frequency power for equilibrium containment of the plasma pinch within the torus is to be expected near this resonance on the side where Δ > 0.The change in the sign of Δ is a consequence of a change in the direction of the force acting on the plasma. If there are two types of wave in the plasma, there appear – in addition to the forces associated with the pressure of each individual wave -forces produced by the interaction of the currents excited by one wave type with the field of the other wave type, and vice versa.For certain parameter values, the phase relations of the fields and currents may be such that the additional forces associated with the cross interaction may work in a direction opposite to that of the forces produced by the pressure of each individual wave.In the propagation region of the first wave, i.e. where H0 > Hcrit. positive and negative values of the plasma pinch displacement alternate (i.e. there are alternating regions of stable and unstable equilibrium respectively).
Toroidal equilibrium in the reversed field pinch corrected by a vertical field