Author: Stuart R. Hudson
Requested Type: Poster
Submitted: 2022-03-11 12:48:59
Co-authors: S. A. Henneberg, D. Pfefferl ́e, P. Helander
PO Box 451
Princeton, NJ 08536
For free-boundary equilibrium calculations, the vacuum field must be computed.
Vacuum fields are easily calculated by solving Laplace’s equation, which can be written as a linear equation, A·φ= Btn, for the scalar potential, φ, with boundary conditions on the total normal magnetic field, Btn, on the computational domain.
The inner boundary is the plasma boundary, on which the total normal magnetic field is zero, and the outer boundary is arbitrary, required only to lie outside the plasma and inside the external sources of magnetic field.
Btn is the sum of fields produced by external sources, Ben, which is known a priori, and that produced by plasma currents, Bpn, which is only known a posteriori as part of the equilibrium calculation.
The field outside the plasma produced by currents inside the plasma can be computed using the virtual-casing principle, a surface integral over the total magnetic field immediately outside the
plasma boundary; and this after a suitable discretization is also linear, so we may write Bpn = B·φ.
This gives the virtual-casing self-consistent vacuum field in an arbitrary domain as the solution to the linear equation Lvc ·φ= Ben, where Lvc = A−B, which we might call the “Laplace-virtual-
casing matrix” .
This representation simplifies free-boundary equilibrium calculations and shows what the vacuum field must be for a given external field and a given plasma boundary.
(The plasma boundary will in general need to adjusted to achieve force balance across the plasma boundary).
An analysis of the eigenspectrum of Lvc, which is not self adjoint, might reveal interesting general properties of free-boundary equilibrium and stability.
 “Combined plasma-coil optimization algorithms” S. Henneberg et al., J. Plasma Physics 87, 905870226 (2021)