Plasma Response to Resonant Perturbations at Tokamak Edge

In certain circumstances, plasma response suppresses magnetic islands expected at perturbed resonant magnetic surfaces. We investigate the plasma response to the resonant magnetic perturbations in a large aspect ratio tokamak perturbed by external resonant helical windings, considering polar toroidal coordinates for which analytical toroidal equilibrium solutions and perturbing fields are available. We apply an empirical approach to mimic the plasma screening effects by introducing presumed plasma current sheets on the resonance surfaces to cancel the RMP effects. Numerical examples show the effect of plasma response reducing magnetic islands at the plasma edge and also regularizing field lines around the resonant surface. The distribution of connection lengths along the plasma cross section indicates that the plasma response increases the connection lengths since more toroidal turns are performed until a field line reaches the tokamak wall.     

Onset of chaotic field line trajectories in tokamaks

Summary The onset of chaotic field line trajectories on tokamak plasmas perturbed by external resonant helical currents has been investigated. The spatial distribution of field lines is obtained integrating numerically the differential equations for the perturbed field lines and investigating the Poincaré maps characteristic of these fields. Several algorithms have been computed to identify the transition to large-scale stochasticity for this quasi-integrable dynamical system, which, for a given m.h.d. equilibrium, is determined by the enhancement of the coupling between the helical perturbation and the toroidicity, as the helical current increases. The authors of this paper have agreed to not receive the proofs for correction. Work partially supported by CNPq and FINEP.     

Plasma confinement in tokamaks with robust torus

We present a non-linear symplectic map that describes the alterations of the magnetic field lines inside the tokamak plasma due to the presence of a robust torus (RT) at the plasma edge. This RT prevents the magnetic field lines from reaching the tokamak wall and reduces, in its vicinity, the islands and invariant curve destruction due to resonant perturbations. The map describes the equilibrium magnetic field lines perturbed by resonances created by ergodic magnetic limiters (EMLs). We present the results obtained for twist and non-twist mappings derived for monotonic and non-monotonic plasma current density radial profiles, respectively. Our results indicate that the RT implementation would decrease the field line transport at the tokamak plasma edge.     

Influence of resonant helical windings on the mirnov oscillations in a small tokamak

Summary The influence of resonant helical perturbations on the plasma confined by the small tokamak TBR-1 was experimentally investigated. Strong attenuations of the Mirnov oscillations could be easily obtained by activating different resonant helical windings. The amplitudes of these oscillations were restored as the helical perturbation ended. Partially supported by CNPq.     

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade （HT-TU） when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode （m, n） helical perturbation can cause the formation of islands on rational surfaces with q=m/n and q=（m±1,±2,±3,...）/n due to the toroidicity and plasma shape （i.e. elongation and triangularity）, while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser （KAM） barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field Br（r） and the toroidal magnetic field amplitude Bφ0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.     

Tokamak magnetic field lines described by simple maps

The magnetic field line structure in a tokamak can be obtained by direct numerical integration of the field line equations. However, this is a lengthy procedure and the analysis of the solution may be very time-consuming. Otherwise we can use simple two-dimensional, area-preserving maps, obtained either by approximations of the magnetic field line equations, or from dynamical considerations. These maps can be quickly iterated, furnishing solutions that mirror the ones obtained from direct numerical integration, and which are useful when long-term studies of field line behavior are necessary (e.g. in diffusion calculations). In this work we focus on a set of simple tokamak maps for which these advantages are specially pronounced.     

Locating Cantori for Symmetric Tokamap and Symmetric Ergodic Magnetic Limiter Map Using Mean-Energy Error Criterion

We use a method based on the conservation of energy, the mean-energy error criterion, to approximately locate the place of a cantorus by locating the series of its convergents. The mean-energy error curve has nearly stationary parts in the vicinity of elliptic (minimax) orbits, the so-called magnetic islands. Stable minimax orbits converge to orbits homoclinic to a cantorus. By tracing the island series, we limit the cantorus to a narrow region. A near-critical perturbation parameter is used so that, while the cantorus may be destabilized, its high-order minimax orbits remain intact. As illustrations, we consider two symplectic maps, systematically derived from the Hamilton–Jacobi equation and Jacobi’s theorem, in the context of the magnetically confined plasmas in a tokamak: a symmetric tokamap realistically reproduces the main features of a tokamak, and a symmetric ergodic magnetic limiter (EML) map is defined to describe the action of EML rings on the magnetic field lines in the tokamak.     

Chaotic characteristics of electron motion in the static magnetic fields and self-fields of the electron beam

In this paper We study the properties of electron motion when a nonneutral electron beam goes through a static magnetic field. The self-electronic and magnetic fields also affects on the test electron. The Hamiltonian equations of electron motion are deduced and the Poincare surface of section and the Lyapunov exponent are used to prove that the electron motion becomes chaotic when the fields are strong enough.     

Bifurcation to 3D helical magnetic equilibrium in an axisymmetric toroidal device

We report the first direct measurement of the internal magnetic field structure associated with a 3D helical equilibrium generated spontaneously in the core of an axisymmetric toroidal plasma containment device. Magnetohydrodynamic equilibrium bifurcation occurs in a reversed-field pinch when the innermost resonant magnetic perturbation grows to a large amplitude, reaching up to 8% of the mean field strength. Magnetic topology evolution is determined by measuring the Faraday effect, revealing that, as the perturbation grows, toroidal symmetry is broken and a helical equilibrium is established.     

The strongest magnetic barrier in the DIII-D tokamak and comparison with the ASDEX UG

Magnetic perturbations in tokamaks lead to the formation of magnetic islands, chaotic field lines, and the destruction of flux surfaces. Controlling or reducing transport along chaotic field lines is a key challenge in magnetically confined fusion plasmas. A local control method was proposed by Chandre et al. [Nucl. Fusion 46, 33–45 (2006)] to build barriers to magnetic field line diffusion by addition of a small second-order control term localized in the phase space to the field line Hamiltonian. Formation and existence of such magnetic barriers in Ohmically heated tokamaks (OHT), ASDEX UG and piecewise analytic DIII-D [Luxon, J.L.; Davis, L.E., Fusion Technol. 8, 441 (1985)] plasma equilibria was predicted by the authors [Ali, H.; Punjabi, A., Plasma Phys. Control. Fusion 49, 1565–1582 (2007)]. Very recently, this prediction for the DIII-D has been corroborated [Volpe, F.A., et al., Nucl. Fusion 52, 054017 (2012)] by field-line tracing calculations, using experimentally constrained Equilibrium Fit (EFIT) [Lao, et al., Nucl. Fusion 25, 1611 (1985)] DIII-D equilibria perturbed to include the vacuum field from the internal coils utilized in the experiments. This second-order approach is applied to the DIII-D tokamak to build noble irrational magnetic barriers inside the chaos created by the locked resonant magnetic perturbations (RMPs) (m, n)=(3, 1)+(4, 1), with m and n the poloidal and toroidal mode numbers of the Fourier expansion of the magnetic perturbation with amplitude δ. A piecewise, analytic, accurate, axisymmetric generating function for the trajectories of magnetic field lines in the DIII-D is constructed in magnetic coordinates from the experimental EFIT Grad-Shafranov solver [Lao, L, et al., Fusion Sci. Technol. 48, 968 (2005)] for the shot 115,467 at 3000 ms in the DIII-D. A symplectic mathematical map is used to integrate field lines in the DIII-D. A numerical algorithm [Ali, H., et al., Radiat. Eff. Def. Solids Inc. Plasma Sc. Plasma Tech. 165, 83 (2010)] based on continued fraction decomposition of the rotational transform labeling the barriers for selecting and identifying the strongest noble irrational barrier is used. The results are compared and contrasted with our previous results on the ASDEX UG. About six times stronger a barrier can be built in the DIII-D than in the ASDEX UG. High magnetic shear near the separatrix in the DIII-D is inferred as the possible cause of this. Implications of this for the DIII-D and the International Thermonuclear Experimental Reactor (ITER) are discussed.     

Tokamak plasma equilibrium analysis based on the relaxation method with a specified magnetic axis

In this contribution, we have presented two techniques for the determination of plasma equilibrium position in IR-T1 tokamak: relaxation and optical methods. An analysis method of tokamak plasma equilibrium by a relaxation method with a specified magnetic axis is presented. The degrees of freedom due to designated positions of the magnetic axis are possible by using poloidal field coil currents. Stable steady-state tokamak plasma equilibria are calculated along with the magnetohydrodynamic potential energy. The plasma generates a plasma current which partially or fully cancels the magnetic field from the poloidal field coils. For low-temperature plasmas, the plasma current distribution is centrally peaked; for high-temperature plasmas, the plasma current has a hole. A centrally peaked current distribution in a low-temperature plasma is evolved into a current distribution with a hole by increasing the plasma pressure by Ohmic heating, radio frequency heating, or by neutral beam injection heating. In the second technique, an image-processing technique was used for the output signal of the charge coupled device camera and plasma emission intensity profile and then the plasma position was obtained. Results are compared and discussed.     

Observation of disruptions in tokamak plasma under the influence of resonant helical magnetic fields

Summary Disruptive instabilities were investigated in the small-tokamak TBR-1 during the application of resonant helical magnetic fields created by external helical windings. Indications were found that the main triggering mechanism of the disruption was the rapid increase of them=2/n=1 mode which, apparently after reaching a certain amplitude, interacts with other resistive modes: the internal 1/1 mode in the case of a major disruption, or with higherm components, as the 3/1 or 4/1, in the case of minor disruptions. After the coupling, the growth of the associated islands would create a chaotic field line distribution in the region between the corresponding rational magnetic surfaces which caused the gross particle transport and, finally, destroyed the confinement. In addition, investigations on higherZ _eff discharges in which a mixture of helium and hydrogen was used resulted in much more unstable plasmas but apparently did not alter the basic characteristics of the disruptions. The authors of this paper have agreed to not receive the proofs for correction.     

A solution of the motion of a charged particle in magnetic mirror systems

The paper solves the motion of a charged particle in an axially symmetrical, magnetostatic field, which forms magnetic mirror systems. The solution is based on the Hamiltonian for the motion of a charged particle in such a field. The whole problem is solved in orthogonal curvilinear coordinates, the coordinate curves being formed by the lines of force and the curves perpendicular to them. By applying the perturbation method of multi-periodical systems to the Hamiltonian in these coordinates, a Hamiltonian is obtained for the motion of the guiding centre which contains only the coordinate of motion along the lines of force, the others being cyclic. Thus, in addition to the energy two other first integrals of the equations of motion are obtained (since the impulses corresponding to the cyclic coordinates are constant), from which the conditions for confinement of motion in a magnetic field are immediately obtained. Since the Hamiltonian obtained in this way contains only one coordinate, the whole problem is solved by two quadratures, which define the dependence of the time and azimuthal angle on the line-of-force coordinate, while the other quantities are constant.In conclusion, the author thanks Dr. M. Seidl for valuable remarks and discussion on this work.     

HT-7 装置回旋质谱探测器边界离子诊断

介绍了回旋质谱探测器的原理和设计及其在HT-7 装置欧姆放电下对边缘等离子体中离子的诊断实验。探测器安装在限制器附近, 通过一小孔引进等离子体; 设置的前置偏压使电子和离子分离, 并使离子减速; 进入腔体内部的离子在射频电场和平行磁场的作用下发生回旋共振; 通过考察收集的离子电流信号中的共振峰可得到离子的荷质比、回旋频率等参数。实验中观察到荷质比为1、0. 5、0. 3333、0 . 1819 的离子共振峰。     

Design and construction of hot limiter biasing system for the tokamak

A hot limiter biasing system with a simplified fast switch circuit was designed, constructed, and installed on the IR-T1 tokamak, and then the negative voltage applied to a hot limiter inserted inside the tokamak fixed limiter and the plasma current, poloidal, and radial components of the magnetic fields, loop voltage, diamagnetic flux, and the ion saturation currents in the absence and presence of the biased limiter were measured. Results of measurements of biasing effects on the plasma equilibrium behavior and edge plasma rotation are compared and discussed.     

HT－7托卡马克中等离子体平衡研究

本文解决了二维轴对称近似下带铁芯的托卡马克中等离子体平衡问题，计算了ＨＴ－７托卡马克中的等离子体平衡位形以及极向场系统的非线性电感和垂直场系数。最后应用Ｋｉｒｃｈｈｏｆｆ方程组和平衡垂直场公式得到了一组等离子体、加热场和垂直场线圈的电流波形的自洽曲线。     

Diffusion in a collisional standard map

Test particle evaluation of the diffusion coefficient in the presence of magnetic field fluctuations and binary collisions is presented. Chaotic magnetic field lines originate from resonant magnetic perturbations (RMPs). To lowest order, charged particles follow magnetic field lines. Drifts and interaction (collisions) with other particles decorrelate particles from the magnetic field lines. We model the binary collision process by a constant collision frequency. The magnetic field configuration including perturbations on the integrable Hamiltonian part is such that the single particle motion can be followed by a collisional version of a Chirikov-Taylor (standard) map. Frequent collisions are allowed for. Scaling of the diffusion beyond the quasilinear and subdiffusive behaviour is investigated in dependence on the strength of the magnetic perturbations and the collision frequency. The appearance of the so called Rechester-Rosenbluth regime is verified. It is further shown that the so called Kadomtsev-Pogutse diffusion coefficient is the strong collisional limit of the Rechester-Rosenbluth formula. The theoretical estimates are supplemented by numerical simulations.