Resonant Rational Surface Research Articles

Application of 3D MHD equilibrium calculation to RMP experiments in the J-TEXT tokamak

The application of resonant magnetic perturbation (RMP) coils could break the initial axisymmetry and change the magnetic topology in tokamak systems. To understand the plasma equilibrium response to the RMP fields, three-dimensional (3D) non-linear magnetohydrodynamics equilibrium calculations have been carried out using the HINT code for an RMP field-penetration experiment on J-TEXT. The HINT code does not assume perfectly nested flux surfaces, and is able to consider directly the change of magnetic topology due to the RMP field penetrations. Correlations between 3D equilibrium calculations and experimental observations are presented. The magnetic topologies calculated by HINT were compared with the field topologies obtained from a vacuum approximation method. It turns out that the effects of redistribution of plasma pressure and current due to the formation of magnetic islands at various resonant rational surfaces should be considered self-consistently for understanding the change of magnetic structure. Such changes include changes in the shape and size of magnetic islands, and the distribution of stochastic fields around the magnetic islands and at the plasma boundary, which plays an important role for plasma-wall interactions.

Verification of the ideal magnetohydrodynamic response at rational surfaces in the VMEC code

The VMEC nonlinear ideal MHD equilibrium code [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)] is compared against analytic linear ideal MHD theory in a screw-pinch-like configuration. The focus of such analysis is to verify the ideal MHD response at magnetic surfaces which possess magnetic transform (ι) which is resonant with spectral values of the perturbed boundary harmonics. A large aspect ratio circular cross section zero-beta equilibrium is considered. This equilibrium possess a rational surface with safety factor q = 2 at a normalized flux value of 0.5. A small resonant boundary perturbation is introduced, exciting a response at the resonant rational surface. The code is found to capture the plasma response as predicted by a newly developed analytic theory that ensures the existence of nested flux surfaces by allowing for a jump in rotational transform (ι=1/q). The VMEC code satisfactorily reproduces these theoretical results without the necessity of an explicit transform discontinuity (Δι) at the rational surface. It is found that the response across the rational surfaces depends upon both radial grid resolution and local shear (dι/dΦ, where ι is the rotational transform and Φ the enclosed toroidal flux). Calculations of an implicit Δι suggest that it does not arise due to numerical artifacts (attributed to radial finite differences in VMEC) or existence conditions for flux surfaces as predicted by linear theory (minimum values of Δι). Scans of the rotational transform profile indicate that for experimentally relevant levels of transform shear the response becomes increasing localised. Careful examination of a large experimental tokamak equilibrium, with applied resonant fields, indicates that this shielding response is present, suggesting the phenomena is not limited to this verification exercise.

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calculations. This rescues the possibility of constructing MHD equilibria with current sheets and continuous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface.

Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria

Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylor's relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac δ-function, which presumably prevents the formation of islands, and the Pfirsch-Schlüter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.

Noble magnetic barriers in the ASDEX UG tokamak

The second-order perturbation method of creating invariant tori inside chaos in Hamiltonian systems (Ali, H.; Punjabi, A. Plasma Phys. Contr. F. 2007, 49, 1565–1582) is applied to the axially symmetric divertor experiment upgrade (ASDEX UG) tokamak to build noble irrational magnetic barriers inside chaos created by resonant magnetic perturbations (m, n)=(3, 2)+(4, 3), with m and n the poloidal and toroidal mode numbers of the Fourier expansion of the magnetic perturbation. The radial dependence of the Fourier modes is ignored. The modes are considered to be locked and have the same amplitude δ. A symplectic mathematical mapping in magnetic coordinates is used to integrate magnetic field line trajectories in the ASDEX UG. Tori with noble irrational rotational transform are the last ones to be destroyed by perturbation in Hamiltonian systems. For this reason, noble irrational magnetic barriers are built inside chaos, and the strongest noble irrational barrier is identified. Three candidate locations for the strongest noble barrier in ASDEX UG are selected. All three candidate locations are chosen to be roughly midway between the resonant rational surfaces ψ32 and ψ43. ψ is the magnetic coordinate of the flux surface. The three candidate surfaces are the noble irrational surfaces close to the surface with q value that is a mediant of q=3/2 and 4/3, q value of the physical midpoint of the two resonant surfaces, and the q value of the surface where the islands of the two perturbing modes just overlap. These q values of the candidate surfaces are denoted by q MED, q MID, and q OVERLAP. The strongest noble barrier close to q MED has the continued fraction representation (CFR) [1;2,2,1∞] and exists for δ≤2.6599×10−4; the strongest noble barrier close to q MID has CFR [1;2,2,2,1∞] and exists for δ≤4.6311×10−4; and the strongest noble barrier close to q OVERLAP has CFR [1;2,2,6,2,1∞] and exists for δ≤1.367770×10−4. From these results, the strongest noble barrier is found to be close to the surface that is located physically exactly in the middle of the two resonant surfaces.

Off-axis current drive and real-time control of current profile in JT-60U

Aiming at optimization of current profile in high-β plasmas for higher confinement and stability, a real-time control system of the minimum of the safety factor (qmin) using the off-axis current drive has been developed. The off-axis current drive can raise the safety factor in the centre and help to avoid instability that limits the performance of the plasma. The system controls the injection power of lower-hybrid waves, and hence its off-axis driven current in order to control qmin. The real-time control of qmin is demonstrated in a high-β plasma, where qmin follows the temporally changing reference qmin,ref from 1.3 to 1.7. Applying the control to another high-β discharge (βN = 1.7, βp = 1.5) with m/n = 2/1 neo-classical tearing mode (NTM), qmin was raised above 2 and the NTM was suppressed. The stored energy increased by 16% with the NTM suppressed, since the resonant rational surface was eliminated. For the future use for current profile control, current density profile for off-axis neutral beam current drive (NBCD) is for the first time measured, using the motional Stark effect diagnostic. Spatially localized NBCD profile was clearly observed at the normalized minor radius ρ of about 0.6–0.8. The location was also confirmed by multi-chordal neutron emission profile measurement. The total amount of the measured beam driven current was consistent with the theoretical calculation using the ACCOME code. The CD location in the calculation was inward shifted than the measurement.