Stellarator magnetic analysis: resolvability problems

Abstract

In all probability the only way to understand the theoretical possibilities and limitations of magnetic analysis is the use of a trial-and-error method. To demonstrate this, a virgin field of rigorous mathematical foundations of a three-dimensional (3D) equilibrium problem is briefly presented. In the first part of the paper we clarify the problem of plasma profile determination. We previously have established the limitations following from a direct 3D plasma equilibrium problem on the possibility of the moderate current density profile identification. Here we demonstrate the limitations on the pressure profile determination under zero net current regime of operation. Realistic experimental conditions that significantly complicate the use of magnetic analysis for equilibrium identification are outlined. The second part of the paper deals with the peculiarities of diamagnetic measurements in stellarators. The influence of the 3D plasma induced magnetic fields is briefly sketched. We will also treat in detail an attractive, although questionable, possibility for the estimation of the electron cyclotron resonance heating absorbed power from the jump in the time derivative of the diamagnetic signal. It is shown that the limitations of such a method are completely determined by the accuracy of the experimental system set-up, but do not depend on the mathematical properties of the equations considered, as was previously expected.

Magnetic diagnostics: General principles and the problem of reconstruction of plasma current and pressure profiles in toroidal systems

“…This can be easily done. Let us multiply the Maxwell equation j = ∇ × B by q ≡ ∇f × a (17) where a is some constant vector and f is a harmonic function (∇ 2 f = 0). Under these conditions ∇× q = ∇(a • ∇f ) and…”

“…However, the unjustified optimism about the aforementioned reconstruction of the profiles has not disappeared completely. Although little supported by published results [3][4][5], optimism about this was taken up by stellarator researchers [13][14][15][16][17], among whom attempts to gather evidence in favour of a positive answer still continue [6][7][8]. Earlier such attempts were made with a strong belief in a positive outcome, later with reinsuring reserves (leaving, however, freedom even for mutually exclusive conclusions), but always along the same lines, prescribing different profiles first and then demonstrating that they correspond to different amplitudes of measurable magnetic quantities.…”

Diagnostics for steady state plasmas

“…This is due to the nature of the equilibrium reconstruction, where the modifications by the plasma can be considered as a perturbation on the flux surface geometry. The reconstruction of the equilibrium including the current profile by the external magnetic measurements is an ill-posed problem [295,308,309]. Thus, the quality of any equilibrium reconstruction is very sensitive to inaccuracies as introduced by noise, stray fields and systematic errors due to signal integration imperfections (section 12.…”

“…The reconstruction of the equilibrium including the current profile by the external magnetic measurements is an ill-posed problem [295,308,309]. Thus, the quality of any equilibrium reconstruction is very sensitive to inaccuracies as introduced by noise, stray fields and systematic errors due to signal integration imperfections (section 12.3).…”