Resolution of the paradox discovered by Freidberg in the Shafranov's theory on plasma equilibrium

Abstract

Predictions of the Shafranov's analytical theory are so highly respected that some of them are reprinted without any check for consistency. One example of such a kind is considered here: the equation for the plasma shift in a tokamak first introduced in Shafranov, J. Nucl. Energy C 5, 521 (1963). Since then, it has entered a number of review papers and textbooks, though, when tested by Freidberg [Freidberg, Rev. Mod. Phys. 54, 801 (1982)], it revealed some peculiar features. The main (long unnoticed) discovery was that the external vertical magnetic field required by that equation at zero shift was significantly different from the value given by the other famous Shafranov's formula for the same field in the same configuration. It is the tremendous scale of disagreement in the very core of the otherwise perfect theory that attracts attention. Here, a wrong element in the construction is found and repaired. It is established that it must be the incorrectly imposed boundary conditions for the magnetic field at the vacuum vessel wall treated as an ideal conductor. The proposed replacement eliminates the contradiction and extends the model by incorporating the resistive-wall effects.