Octavio Betancourt

Finite Difference Scheme

To arrive at finite difference equations modeling magnetohydrodynamic equilibrium we use a technique that is motivated by the finite element method [3]. First we develop a second order accurate numerical quadrature formula for the Hamiltonian E based on a rectangular grid of mesh points over a unit cube of the space with coordinates s, u and v. We differentiate that formula with respect to nodal values of the unknowns R, ψ r0 and z0 in order to derive difference approximations to the magnetostatic equations. This procedure yields equations in a conservation form that is automatically compatible with conditions stemming from the fact that the flux function ψ is only determined up to an arbitrary additive function of s. Thus we avoid elementary difficulties with existence of solutions of the equilibrium problem which tend to obscure the more serious issues raised by the KAM theory.

Description of the Computer Code

Here we describe how to use the computer code that is based on the theory developed in the previous chapters. The procedure for running the code is quite simple and only requires following the instructions presented in this chapter. However, an understanding of the theoretical framework is needed to determine the scope and limitations of the method, to choose the most efficient parameters, and to interpret the results correctly.

The Mercier Criterion

We have presented a method to test for instabilities with low mode numbers m and n, such as kink modes. Both theoretical analysis and calculations of axially symmetric equilibria suggest that modes with high m and n, including interchange modes, may be equally critical in the search for stable high pressure configurations [34]. They can be assessed via the Mercier criterion, which is concerned with modes localized about some rational surface [39]. The assumption is made that the plasma is covered by a family of nested flux surfaces, which we have seen to be rather tenuous when the geometry is truly three-dimensional [28]. Specialized variations δW of the energy EP are introduced that are restricted to the neighborhood of a rational surface or a closed magnetic line. Then a necessary condition for stability is derived by careful optimization of δW. A preliminary version of this condition involves integrals over the relevant closed line, but considerations about the ergodicity of magnetic lines on a surface with irrational rotational transform lead to a better version involving surface integrals that occur naturally in our model of magnetohydrodynamic equilibrium. For details we refer to the book by Mercier and Luc [39], noting only that their derivation is somewhat at odds with the resonances that occur at rational surfaces in three dimensions. More specifically, the analysis suggests that in the neighborhood of a rational surface, lower energy levels can be achieved by equilibrium solutions that have island structure.

Synopsis of the Method

We consider a formulation of the variational principle of magnetohydrodynamics that enables us to calculate equilibrium and stability without recourse to the full equations of motion [3, 37]. Denote the magnetic field by B, the fluid pressure by p and the mass density by p. We assume an equation of state of the form p = pγ where γ is the gas constant. Let

The ATF-1 and the WISTOR-U

{E_p} = \smallint \smallint \smallint \left[ {\frac{{{B^2}}}{2} + \frac{p}{{\gamma - 1}}} \right]dxdydz

The Variational Principle

stand for the potential energy of a torus of plasma and let

Magnetohydrodynamic Equilibrium and Stability of Stellarators

{E_v} = \smallint \smallint \smallint \frac{{{B^2}}}{2}dxdydz