Anisotropic pressure bi-Maxwellian distribution function model for three-dimensional equilibria

Abstract

A formulation of the anisotropic pressure magnetohydrodynamic equilibrium problem for three-dimensional plasmas with imposed nested magnetic surfaces is developed based on a bi-Maxwellian model of the distribution function for the energetic particle species. The hot particle distribution function satisfies the constraint . Large parallel and perpendicular anisotropy factors can be explored within the model through the choice of the hot particle perpendicular to parallel temperature ratio T⊥/T‖. A fixed boundary version of the VMEC code has been adapted to numerically compute three-dimensional anisotropic pressure equilibria. Applications to a 10-field period Heliotron device and a 2-field quasiaxisymmetric stellarator demonstrate that the pressures do not vary significantly around the magnetic surfaces when the total parallel pressure p‖ is larger than its perpendicular counterpart p⊥. For off-axis hot particle deposition with p⊥ > p‖, p⊥ concentrates in the region where the energetic particles are generated. On the other hand, p‖ is distributed roughly uniformly around the flux surfaces in the Heliotron but is localized on the low field side in the quasiaxisymmetric machine. The hot particle density structure correlates more closely with the corresponding perpendicular rather than with parallel pressure. The specific form for the definition of β that best correlates with the Shafranov shift is identified.

Three-dimensional anisotropic pressure free boundary equilibria

“…We specifically concentrate on off-axis hot particle deposition on the high magnetic field (HF) with B C = 4.9T and the low field (LF) side with B C = 4.2T . These conditions provide a very meaningful evaluation of the versatility of the code as we can anticipate significant poloidal localisation of the energetic particle pressure distributions particularly with large perpendicular pressure anisotropy as has been previously demonstrated in fixed boundary computations [9]. We define the volume av- …”

“…Rather than determine the perpendicular pressure from the corresponding moment of the distribution function, we invoke the magnetohydrodynamic force balance parallel to the equilibrium magnetic field lines [9] to obtain…”

“…This model is very convenient because the pressure moments of the distribution function can be determined analytically and it allows for the deposition of hot particles in any region of the plasma. It is particularly useful to describe hot populations generated with ion cyclotron resonance heating (ICRH) [7,8] and has been previously implemented and applied with success in a fixed boundary version of the 3D VMEC code [9].…”

“…For 3D stellarator configurations, an expansion method was applied in which the pressures are imposed to be only a function of the radial variable [15]. More recently, fixed boundary equilibria based on the VMEC code have been devised, in addition to the bi-Maxwellian model [9], using slowing down distribution functions multiplied by a factor (μB min /E) to obtain large perpendicular anisotropy concentrated on the low field (LF) …”

Full-field drift Hamiltonian particle orbits in axisymmetric tokamak geometry

“…11 This last work also allowed for anisotropic pressure which can be very relevant for plasmas that are heated with auxiliary methods. [12][13][14][15][16][17] In particular, we extend the formalism adopted in Ref. 11 to include the finite dA ?…”