A BRIEF REVIEW OF MAGNETIC WELLS

The notion of ‘‘magnetic well’’ is useful as a figure of merit for tokamak operation. The object of this review is to collect and compare a variety of different forms that have been used as a well parameter. A particularly satisfying general formulation, based on a resistive stability criterion, is presented in Eq. (25). The application of these results to tokamaks is discussed in terms of the well-known analytic, near-axis approximation that displays the importance of the safety factor and of shaping. J.M. Greene A BRIEF REVIEW OF MAGNETIC WELLS GENERAL ATOMICS REPORT GA–A22135 2 The idea of ‘‘magnetic well’’ has had a rather convoluted history, and each phase has left some debris. This note is an effort to organize these developments, bring together a number of approximations, and find a satisfying formulation in Eq. (25). There are several possible reasons for the existence of several slightly different formulae, the most important of which is that magnetic wells have rarely been employed to explain experimental results in tokamaks. With improved diagnostics, and somewhat more venturesome choices of non circular equilibrium shapes, it may be a good time to review the various theories, more or less in historical order. The basic idea behind this effort was to find a principle analogous to the stable stratification of a fluid. For example, it is clear that if the atmosphere has an inversion, there is little stirring of the air, but if there is no inversion it is well stirred. This can be reduced to the principle that if the entropy density increases with altitude, the atmosphere is stable, otherwise it is marginal or unstable to convection cells of all scales. The demonstration of this result proceeds by considering slowly convecting cells that remain close to static force balance at all times, so that sound waves are not excited, while convecting sufficiently rapidly that entropy is fixed in the fluid. The work done against gravity maintaining such a convection determines stability. The condition that sound waves not be generated by the convection can be crudely expressed as the requirement that the motion be incompressible. However, this idea is misleading. For example, vertical convection must be accompanied by compression or expansion that is precisely determined from the scale height of the atmosphere. The first effort in this direction for magnetic confinement systems was that of Rosenbluth and Longmire.1 They considered the forces driving convection cells in a simple magnetic confinement system, following the procedure that is used in treating the atmosphere. This convection carries the magnetic field lines with it, while producing minimal changes in the field configuration. The conclusion that they came to was that magnetic field strength was the important quantity. If the field was weakest where the plasma pressure was highest, the configuration was stably stratified. In this formulation the gradient of the magnetic field strength plays the role that gravity plays in the atmosphere, and the plasma pressure is analogous to the atmospheric entropy. This analogy leads the simple identification of the magnetic field as an equivalent gravitational well, with the well parameter given by Freidberg2 in Eq. (4.39) as Ŵ = 2 V B 2 d dV B 2