Computations in Plasma Physics

Computations in Plasma Physics D. Biskamp, Garching (Max-Planck-Ihstitut fur Plasmaphysik) Plasma physics plays an important role in astrophysics as well as in nuclear fusion research. It is a complex field where a great variety of different physical effects occur and interact. It is convenient to distinguish several levels of description of plasma behaviour: a) MHD theory. To describe the global pro perties of a plasma configuration, e.g. the discharge column in a fusion experiment, a fluid model is most suitable, the magnetohydrodynamic (MHD) theory. It accounts, in particular, for the intimate connection between the plasma and magnetic field resulting from the high electrical conduc tivity; b) Microscopic theory. A hot, dilute plasma may be in a state which strongly differs from local thermodynamic equilibrium, and hence is susceptible to velocity space in stabilities or micro-instabilities. This implies excitation of certain collective modes to some level of microscopic turbulence which may give rise to transport effects far exceeding those due to particle collisions. These processes are described by the Vlasov, or collisionless Botzmann equa tion; c) Transport theory. In nuclear fusion research the understanding of transport across the confining magnetic field is very important. Many different processes occur and influence each other; plasma diffusion, electron and ion thermal conduction, parti cle interactions, penetration and radiation of impurity ions. These and other effects are described by a complex system of coupled transport equations. As at each level of description one has to deal with multi-dimensional differential or integro-differential equations, which in general are strongly non-linear, and have complicated boundary conditions in com plex geometries, analytical methods such as perturbation theory can provide only a qualitative answer at best. With the advance of powerful scientific computers in the past decade, numerical computing has entered practically all fields Fig. 1 — Stellarator plasma equilibrium configu ration. 4 of plasma physics. On the one hand, it has become a powerful instrument in the eva luation as well as the planning of experi ments; on the other, our theoretical under standing of basic plasma phenomena is now achieved to a large extent through computer simulations. MHD Computations In MHD theory, the plasma is considered to be an electrically conducting fluid of mass density p, pressure p and velocity v. It couples to the magnetic field B, which is, at least in part, generated by electrical cur rents within the plasma itself. The MHD equations are: dp/dt + v.pv = 0, (1) 2 ( ) (3) (4) Equation (4) is Faraday's law, where the generalized Ohm's law has been used to eliminate the elec tric field. In many cases, the plasma resistivity η may be neglected. Equations (1) —(4) are then called ideal MHD theory. Equilibrium and stability are usually in vestigated in this framework. If dynamic processes take place, a small but finite resistivity may, however, play an important role. MHD Equilibrium A basic problem is the calculation of equilibrium configurations. In nuclear fu sion research, for instance, it is necessary to find configurations with a high plasma pressure for a given externally applied magnetic field. In equilibrium, magnetic and pressure forces balance each other: (5) The solution of this simple-looking equation, together with the condition, in general constitutes a very difficult pro blem. Fortunately, in many cases of in terest, the symmetry of the configuration simplifies the problem to a twodimensional one, reducing eq. (5) to a single equation for ψ, the component of the vector potential in the direction of the ignorable coordinate, which is of the form Lψ = G(ψ). (6) Here L is a linear differential operator (a generalization of the two-dimensional Laplacian), and G a functional of ψ Sur faces of constant ψ contain the magnetic field lines and are hence called magnetic surfaces. G contains two free functions, one being the pressure assignment to each magnetic surface. In most cases of inte rest, eq. (6) can only be treated numerical ly; efficient codes have been developed, in particular for axial and helical symmetry, both of which play an important role in fu sion research. As G is, in general, non linear in ψ, the solution of (6) requires the use of iteration methods. Special techni ques have to be applied to cope with the possibility of multiple solutions. Whereas two-dimensional equilibria are now calculated routinely, the calculation of fully three-dimensional equilibria, such as those appearing in stellarator magnetic plasma confinement, is much more involv ed, and the first numerical attempts date back only a few years. Apparently a solu tion of eq. (5) can be obtained only by a relaxation procedure, where starting from an initial guess at the configuration, the net force F is gradually reduced to zero. The procedure becomes clear from the equivalent energy formulation. The initial distribution of p and B is adjusted under the constraints implied by the MHD equa tions in such a way that the plasma energy W = d3 x [B2/2 + p/(γ—1)] (7) decreases monotonically. A state of minimum energy is thus approached which represents an MHD equilibrium, the Euler equation of the variational principle 8 W = 0 being just eq. (5). Several codes have been developed. They differ primarily in the choice of coordinates, using either a fixed Eulerian grid or a semi-Lagrangian scheme by treating the instantaneous flux surfaces as coordinate surfaces. The latter approach is inherently more accurate, but leads to special numerical problems and is restricted to configurations with only one magnetic axis. Various relaxation schemes have been developed in order to minimize the number of iterations. It is clear from the fact that eight functions defined on a threedimensional grid have to be followed over many iterations, that a numerical solution of this problem is only just marginally possible, even on the largest existing com puters. For instance, mesh grids are limited to about 303points. Fig. 1gives an example of a computed stellarator equilibrium. MHD Stability An equilibrium configuration is, how ever, physically relevant only if it is at least locally stable. It is true that, by minimizing the energy (7), the three-dimensional codes lead in principle to a stable equilibrium. In practice, however, the iteration procedure does not allow for certain modes or displa cements. These have to be excited explicit lyto check for stability. Generally speaking, the three-dimensional computations are still too coarse to allow reliable statements to be made about the stability of the resul ting equilibria. The two-dimensional equilibrium codes solving equations of type (6) do not distin guish between stable and unstable confi gurations. Their stability therefore has to be investigated separately. In recent years, numerical stability codes ha (...truncated)