Structure of Force-Free Magnetic Fields in Equilibrium of Toroidal plasma in High β Tokamak

Abstract

For a sharp boundary equilibrium model of an axisymmetrical-toroidal dense plasma sunounded by a pressureless plasma with force-free fields, structure of force-free magnetic fields is studied by using analytical solutions of the Helmholtz equation for toroidal coordinates and Fourier decomposition. Geometrical form of the magnetic surfaces inside the separatrix surrounding the dense plasma is revealed for the poloidal beta β_p and the aspect ratio of the dense plasma R_p/r_p=coshσ_p which are characteristic of a high β tokamak.
A condition of the equilibrium is determined and the maximum β_p is given by β_p=1/4(R_p/r_p+r_p/R_p)+1/2. The magnetic field equation ∇×H=αH is reduced to the Helmholtz equation(∇∇²+αα²)(Hφe^iφ)=0 for the condition α=constant, which is solved without the approximation of the expansion about the inverse aspect ratio. The force-free magnetic fields can be evaluated in a high accuracy under the equilibrium condition of the toroidal dense plasma.
For the force-free magnetic surfaces similar to the vacuum magnetic ones, by increasing-α or Rp/rp=coshσp, the separatrix surrounding the dense plasma is expanded and the layers of the magnetic surfaces inside the separatrix are increased. Thus, by decreasing R_p/r_p for the fixed separatrix, the layers of the magnetic surfaces inside the separatrix are increased with the increase of -α.