Three-dimensional free boundary calculations using a spectral Green's function method

Abstract

The plasma energy W p = j [\B 2-h p)dV is minimized over a toroidal domain ft p using an inverse representation for the cylindrical coordinates R = ^ R mn (s) cos(m6-nf) and Z = VJ Z mn (s) sin(m#-n<;) , where (.s,0,f) are radial, poloidal, and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when tt p is varied to make the total pressure-B 2 + p continuous at the plasma surface E p and when tie vacuum magnetic field B^ satisfies the Neumann condition B w • dS p = 0. The vacuum field is decomposed as B v = Bo I V$, where Bo is the field arising from plasma currents and external coils and $ is a single-valued potential necessary to satisfy B v • d*£, p = 0 when p == 0. A Green's function method is used to obtain an integral equation over S p for the scalar magnetic potential $ = YJ$ mrl sin(mfl-n<;). A linear matrix equation is solved for $ mn to determine ^B% on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to I-2 stellarator equilibria are presented.

Three-dimensional anisotropic pressure free boundary equilibria

“…A free boundary version of the 3D VMEC code [6] combined with the NESTOR code [5] has been modified to compute anisotropic equilibria with nested magnetic flux surfaces (called the ANIMEC code). The energetic particle pressure moments are calculated analytically from the variant of a bi-Maxwellian distri- We have concentrated the applications to a comparative study of off-axis high and low field side energetic particle deposition with large parallel and perpendicular pressure anisotropy at β 4.5% with zero net toroidal current enclosed within each flux surface in a 2-period quasiaxisymmetric stellarator reactor system.…”

“…where n p is the exterior normal vector to the plasma-vacuum interface [5,6]. The Laplace equation can be converted to the integral relation…”

“…A more detailed description of the vacuum treatment adopted can be found in Refs. [5,6]. Finally, the continuity of the total pressure across the plasma-vacuum interface must be satisfied, namely The flux surface averaged thermal pressure p, the hot particle parallel pressure p h and the hot particle perpendicular pressure p ⊥ profiles as a function of the radial variable s (roughly proportional to volume enclosed).…”

“…For comparative purposes, we show in Fig. 7 the flux surface averaged radial force balance errors for the equilibrium state (LF p ⊥ > p , β 4.5%) computed with an old version of VMEC [6] where the model described was first implemented, with ANIMEC just before the preconditioning is turned on and with the preconditioned ANIMEC result. We obtain an order of magnitude improvement between the old VMEC code and the non-preconditioned ANIMEC code with respect to the averaged radial force balance error and another 3-4 orders of magnitude reduction with preconditioning.…”

“…The three-dimensional (3D) magnetohydrodynamic equilibrium code VMEC [4] coupled with the NESTOR vacuum solver [5] putes general toroidal free boundary scalar pressure equilibria with nested magnetic flux surfaces [6]. The goal of this work is to extend this code to anisotropic pressure conditions driven by the energetic particle species.…”