Reconstruction of the magnetic perturbation in a toroidal reversed field pinch

Abstract

A new method to obtain the radial profile of the magnetic perturbation in a toroidal force-free plasma having a circular cross section is developed. The toroidal geometry produces poloidal harmonics in the equilibrium quantities (at the leading order m = ±1, n=0), which act as mediators between perturbations with the same toroidal number and different poloidal numbers. The approach proposed here, based on the contravariant representation of the magnetic field in flux co-ordinates, is formally simple and rigorous and maintains a nice similarity with the cylindrical treatment. The method is quite general and can be applied to any circular low-beta plasma. In this work we describe its application to the Reversed Field eXperiment (RFX) plasma. It is customary in reversed field pinches to approach the analysis of MHD instabilities by using a cylindrical geometry. Nonetheless, the effect of a more realistic toroidal geometry can play an important role, and indeed we found that the toroidal effects on the magnetic perturbations are not negligible.

A 3D approach to equilibrium, stability and transport studies in RFX-mod improved regimes

“…The present RFP version of VMEC was benchmarked against a typical RFP equilibrium [ISHW09] both in the axisymmetric [ZT04] and in the helical (from SHEq) cases showing a good match between the codes when run with the same constraints and boundary conditions. Currently VMEC is being run to provide equilibria as close as possible to experimental measurements, to understand the role of the boundary in helping the formation and sustainment of the helical state.…”

“…The latter is determined through a linear approximation from the measured spectrum of fluctuations obtained using 4 poloidal arrays of 48 toroidally distributed pick-up coils for the toroidal component and saddle coils for the radial component (note that these measurements are part of the real-time control of the magnetic boundary of RFX-mod [LUC06,MAR06]). The resulting total shift is calculated taking into account the toroidal geometry as described in [ZT04] and provided to VMEC as input. Although with this coils layout we are able to measure modes up to n = 24, in VMEC we impose the machine periodicity to N fp = 7, so that the comparison can be done only with respect to this mode and its harmonics.…”

“…This is explicitly taken into account in the SHEq code as it corresponds to the eigenfunction of the dominant mode used in defining the helical flux [SHEq,ZT04]. In VMEC the equilibrium is the result of a global calculation so that the radial component is part of what we might call the zero-order field.…”

Beyond the Intelligent-shell Concept: the Clean-mode-control for Tearing Perturbations

“…In the computations shown in the next chapter the derivative gains are fixed according to this rule. The system of equations is closed providing the radial field at the resonant surface for each mode: we evolve the phases ϕ m,n , while the amplitude are imposed, for example from the experimental estimates [18]. In fact the Rutherford-like models [19], which would provide equations for the amplitude evolution at the resonant surfaces in the non-linear regime are valid in single helicity conditions.…”

“…We will show some simulations examples considering the evolution of the m = 1, n = −7 ∼ −19 modes non-linearly interacting with the m = 0, n = 1 ∼ 12 modes. The amplitudes at the resonant surfaces are imposed according to the values estimated in the reference shot from the RFX-mod edge magnetic data using a Newcomb's equation solver [18]. Since we are considering a boundary different from the RFX-mod one, these amplitudes must be taken only as indicative of a high current RFP regime.…”