Generalization of Solovev's approach to finding equilibrium solutions for axisymmetric plasmas with flow

Solovev's approach of finding equilibrium solutions was found to be extremely useful for generating a library of linear-superposable equilibria for the purpose of shaping studies.This set of solutions was subsequently expanded to include the vacuum solutions of Zheng,Wootton and Solano,resulting in a set of functions [SOLOVEV_ZWS] that were usually used for all toroidally symmetric plasmas,commonly recognized as being able to accommodate any desired plasma shapes (complete-shaping capability).The possibility of extending the Solovev approach to toroidal equilibria with a general plasma flow is examined theoretically.We found that the only meaningful extension is to plasmas with a pure toroidal rotation and with a constant Mach number.We also show that the simplification ansatz made to the current profiles,which was the basis of the Solovev approach,should be applied more systematically to include an internal boundary condition at the magnetic axis;resulting in a modified and more useful set [SOLOVEV_ZWSm].Explicit expressions of functions in this set are given for equilibria with a quasi-constant current density profile,with a toroidal flow at a constant Mach number and with specific heat capacity 1.The properties of [SOLOVEV_ZWSm] are studied analytically.Numerical examples of achievable equilibria are demonstrated.Although the shaping capability of the set [SOLOVE_ZWSm] is quite extensive,it nevertheless still does not have complete shaping capability,particularly for plasmas with negative curvature points on the plasmaboundary such as the doublets or indented bean shaped tokamaks.     

Numerical study of equilibrium solutions for axisymmetric plasmas with toroidal flow obtained using Solovev approach

Solovev approach of finding equilibrium solutions, which was extended to include the vacuum solutions provided by Zheng, Wooton, and Solano, was found extremely useful for the purpose of shaping studies. Its extension to toroidal equilibria with a general plasma flow was examined theoretically in a companion paper by Chu, Hu and Guo. The only meaningful extension was found for plasmas with a pure toroidal rotation and with a constant Mach number. A set of functions {SOLOVEV_ZWSm} was obtained which fixed location of the magnetic axis for equilibria with quasi-constant current density profile, with toroidal flow at constant Mach number and with specific heat capacity 1. The set {Solovev_ZWSm} should have complete shaping capability for plasma shapes with positive curvature at the boundary; but not for plasmas with negative curvature boundary points, i.e. the doublets or bean shaped tokamaks. We report here extensive numerical studies showing the shaping capability of {Solovev_ZWSm} for plasmas with pure toroidal rotations, including the change in topology of the solution when the rotation mach number changes. Included plasma topology are the sphere (spheromaks); and the tokamaks (including the doublets).