Page 5 - Positional stability of field-reversed-configurations in the presence of resistive walls

system of equations can be used for time dependent simu- lations as well as eigenvalue analysis.
To analyze positional stability of an FRC, we retain the VALEN functions for calculating surface inductances, mutu- als, and resistances, but replace the calls to DCON/IPEC. In- stead, the control surface is chosen without respect to a spe- cific flux surface and in such a way that it encloses the bulk of the plasma for the expected range of equilibria. Instead of using eigenfunctions of the force operator, the functions on the control surface are expanded in finite elements just like for the resistive wall. Both plasma and wall meshes are im- ported directly from an ABAQUS file that can be generated by most CAD programs. The calculation of Lw , Rw , Mpw and Mwp is performed as usual. The FEniCS software suite10,11 is used to compute the reluctance matrix R. We import the equilibrium plasma current profiles and interpolate it on a volumetric mesh inside the control surface. The basis func- tions defined on the control surface are then interpreted as boundary conditions for the normal magnetic field, and we calculate the resulting ⃗j × B⃗ force on the plasma for each basis function. Since the force must be a linear operator on the normal field at the boundary, this results in a system of linear equations for a force matrix Fw:
plasma
where B⃗x is the (unique) curl-free field that satisfies B⃗x · nˆ = f j (⃗x) on the boundary. The force matrix can be used to cal- culate the force for any external field B⃗x (defined through its expansion coefficients ⃗Φx on the control surface):
D. Rigid Volume - a free parameter
The linear model presented so far has a free parameter that may not be immediately obvious. The degree of free- dom arises because there is no clear, unique separation be- tween the plasma and the surrounding vacuum region. In- tuitively, we do not expect that rigid motion of “the FRC” will result in e.g. a displacement of the density profile in the divertor region – which is separated by the separatrix by both distance and the magnetic mirror. When calculating the flux that is induced in the wall by the moving plasma currents and the force that is exerted on the plasma cur- rents, we therefore have to define a boundary for the volume integrals. This boundary will be called the “rigid volume”, because it defines what part of the plasma we consider as rigidly moving, and what part we assume will remain sta- tionary. Theoretically, we expect that the rigid volume en- closes at least the separatrix, and that it excludes at least part of the SOL (both radially and axially). Numerically, we also require the rigid volume to be contained in the control surface.
R=
(F−1.F )·∇)B⃗ (⃗x)·nˆ
(32)
(w) ⃗ ⃗
IV. A.
NON-LINEAR SIMULATIONS
Outline of the Q2D code
Fij =
(jp×Bx)idV (30)

plasma
The vacuum force matrix Fv is calculated from equations (5) to (10). Since the basis functions for the control surface cor- respond to the canonical basis of the finite element repre- sentation, the integral in equation (24) vanishes, and the re- luctance matrix R can be identified with the integrand,
Q2D12 is hybrid, FRC simulation code that combines a 2- D, extended MHD code (“LamyRidge”) with a 3-D, Monte- Carlo “fast-particle” tracker (“MC”). LamyRidge and MC time steps are executed interleaved. MC follows individual ion and neutral trajectories while the field components pro- duced by the fluid plasma are held constant. LamyRidge evolves separate ion, electron and neutron fluids that ex- change energy and momentum with the fast particles. The total number of particles, energy and momentum among particles and fluids is conserved. Modeled interactions between fluid and particles include collisions, ionization and charge exchange. Particles whose energy falls under a threshold become part of the respective fluid. The compu- tational domain is bounded by an arbitrary shaped, axisym- metric resistive wall with Dirichlet boundary conditions for the temperature. Impacting particles can absorbed and re- cycled.
Mathematically, Q2D solves a continuity equation
∂n +∇·(n⃗u)= f1 (33)
∂t
for both the ion and the neutral fluid. Here n is the fluid density, ⃗u the fluid velocity, and f1 a source term due to charge exchange and fast particles falling under the energy threshold. The equation for the ion fluid velocity is
∂u ⃗⃗ ⃗
n ∂t +⃗u·∇u =jp×B−nfρfE−∇P+∇·ν∇⃗u+f2 (34)
Here⃗jp =∇×B⃗/μ0−⃗jf isthefluidplasmacurrent,⃗jf isthe plasma current from fast particles, nf is the fast ion density,
⃗⃗⃗ jp×BxdV=Fw.Φx (31)
−1
μ0vw p
5
The meaning of this equation becomes clearer if we con-
sider how it acts on a specific external field perturbation ⃗Φx.
To evaluate R .⃗Φx , we first calculate F⃗w = Fw .⃗Φx , which is the
force exerted by the perturbation on the plasma. The result-
ing plasma displacement ⃗ξ is given by the constraint that the
force from the perturbed external field is balanced by the
force exerted by the equilibrium field on on the displaced
plasma currents, ⃗ξ = F−1.F⃗ . If the plasma is displaced in vw
the ⃗ξ direction, this goes along with a change δB⃗p = (⃗ξ · ∇)B⃗p of the plasma-generated magnetic field. The normal com- ponent of this field, which is identical to the current po- tential on the control surface that would produce the same field, is δB⃗p ·nˆ.