By Frances Bauer

During this ebook, we document on examine in equipment of computational magneto hydrodynamics supported by way of the U.S. division of power lower than agreement EY-76-C-02-3077 with long island college. The paintings has re sulted in a working laptop or computer code for mathematical research of the equilibrium and balance of a plasma in 3 dimensions with toroidal geometry yet no sym metry. The code is indexed within the ultimate bankruptcy. models of it were used for the layout of experiments on the Los Alamos medical Laboratory and the Max Planck Institute for Plasma Physics in Garching. we're thankful to Daniel Barnes, Jeremiah Brackbill, Harold Grad, William Grossmann, Abraham Kadish, Peter Lax, Guthrie Miller, Arnulf Schliiter, and Harold Weitzner for lots of worthy discussions of the idea. we're specially indebted to Franz Herrnegger for theoretical and pedagogical reviews. Constance Engle has supplied impressive tips with the typescript. We enjoy acknowledging assistance from the employees of the Courant arithmetic and Com puting Laboratory at manhattan college. particularly we must always wish to convey our due to Max Goldstein, Kevin McAuliffe, Terry Moore, Toshi Nagano and Tsun Tam. Frances Bauer long island Octavio Betancourt September 1978 Paul Garabedian v Contents bankruptcy 1. creation 1 1. 1 formula of the matter 1 1. 2 dialogue of effects 2 bankruptcy 2. The Variational precept four four 2. 1 The Magnetostatic Equations 6 2. 2 Flux Constraints within the Plasma . 7 2. three The Ergodic Constraint .

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04 .. 25 .. 54 .. 75. 00 CR~SS SECTI~NS MAJ~R Fig. 1 Cross sections of flux surfaces. Let the toroidal field BT = 1 and let the rotational transform be a constant P = Po in the plasma region. The poloidal field Bp is given by Bp = 2npoR/L there, which corresponds to constant current density. The condition for equilibrium gives p(R) = Po [1 - (~rJ where ra is the plasma radius and Po = (2npora/L)2. In the vacuum region we have BT = 1 and Bp = (2npor;)/(LR). In Figs. 5, and rb = 1 are shown. We initialize the run with the equilibrium solution plus a perturbation of the free boundary.

5 Iterative Scheme for the Vacuum Region 29 Since '0 and Zo are functions of valone, the Courant-Friedrichs-Lewy criterion is much less restrictive for the magnetic axis equations, given that the mesh has hu and hv of the same order of magnitude. This follows from the fact that the distances between mesh points in physical space are proportional to Rhu and Lhv in the u and v directions, respectively. It comes from the derivatives with respect to u, as we have already seen in the analysis of the equation for 1/1.

Note that they scale roughly like ab a2, and a3. 5,215): 7:p is the proportionality constant between ei and the time average ofl (Fi)t/F;I in the acceleration scheme for the equations in the plasma region. NE is the number of iterations over which the average is computed. Typical values are 7:p = 2 and NE = 50. The smoothness of the solution depends on the smoothness of the coefficients ei as functions of time. Therefore, we want the ei to vary slowly. For this, NE and 7:p must be large enough to avoid oscillations resulting from a large imaginary part in the dominant eigenvalue A..