Perturbations In Tokamaks Research Articles

Isochronous island bifurcations driven by resonant magnetic perturbations in tokamaks.

Recent evidence shows that heteroclinic bifurcations in magnetic islands may be caused by the amplitude variation of resonant magnetic perturbations in tokamaks. To investigate the onset of these bifurcations, we consider a large aspect ratio tokamak with an ergodic limiter composed of two pairs of rings that create external primary perturbations with two sets of wave numbers. An individual pair produces hyperbolic and elliptic periodic points, and its associated islands, that are consistent with the Poincaré-Birkhoff fixed-point theorem. However, for two pairs producing external perturbations resonant on the same rational surface, we show that different configurations of isochronous island chains may appear on phase space according to the amplitude of the electric currents in each pair of the ergodic limiter. When one of the electric currents increases, isochronous bifurcations take place and new islands are created with the same winding number as the preceding islands. We present examples of bifurcation sequences displaying (a) direct transitions from the island chain configuration generated by one of the pairs to the configuration produced by the other pair, and (b) transitions with intermediate configurations produced by the limiter pairs coupling. Furthermore, we identify shearless bifurcations inside some isochronous islands, originating nonmonotonic local winding number profiles with associated shearless invariant curves.

A Bayesian formulation for perturbed current tomography in tokamaks

An initial investigation into the application of Bayesian inference to the reconstruction of the spatial distribution of current perturbations in tokamaks from diagnostic signals is presented. Previous work in Bayesian equilibrium current tomography is extended to the case of a complex phasor representation of harmonically time varying current distributions. A forward function to predict the response of magnetic diagnostics is constructed using only electrodynamics and not reduced models of ideal MHD. The extension of this forward function to incorporate a fully kinetic model of the plasma state is suggested. The response of soft x-ray diagnostics, and the motional Stark effect diagnostic to the current perturbations are also predicted and the integration of all diagnostics into a single estimate of the current perturbation is proposed. Simulations with synthetic diagnostics in simple geometry demonstrate that the perturbed current distribution can be reconstructed subject to prior assumptions regarding solution smoothness.

Iterative approach to linear ideal MHD modeling of plasma response to 3D magnetic perturbations in tokamaks

The class of plasma instabilities known as edge-localized modes (ELMs) is of special concern in tokamaks operating in high-confinement mode, such as ASDEX Upgrade and ITER. One strategy for ELM mitigation is the application of resonant magnetic perturbations (RMPs) via external coils. Kinetic modeling accurately describes the plasma response to these RMPs ab initio, particularly the parallel shielding currents at resonant surfaces. Away from resonant surfaces, ideal magnetohydrodynamics (iMHD) is expected to yield sufficiently accurate results, providing a computationally less expensive option that could complement kinetic modeling.The code MEPHIT has been developed to solve the linearized iMHD equations in a way that is compatible with iterative kinetic modeling approaches. We consider an axisymmetric iMHD equilibrium in realistic tokamak geometry under the influence of a quasi-static non-axisymmetric external perturbation from ELM mitigation coils. The plasma responds to this external magnetic perturbation with a current perturbation, which in turn produces a magnetic field perturbation. The resulting fixed-point equation can be solved in a self-consistent manner by preconditioned iterations in which Ampère’s equation and the magnetic differential equations for pressure and current are solved in alternation until convergence is reached. After expansion in toroidal Fourier harmonics, these equations are solved on a triangular mesh in the poloidal plane using finite elements. These results are then benchmarked against established codes.

Understanding of neoclassical offset rotation based on DIII-D experiments

Neoclassical offset rotation induced by non-axisymmetric magnetic perturbations in tokamaks is investigated using NTVTOK model based on plasma profiles in one DIII-D discharge. The calculated counter-Ip (Ip indicates plasma current) ion root of neoclassical offset rotation is found to be consistent with DIII-D experimental observations. The modeling results predict that this DIII-D plasma regime is close to the marginal condition for the co-Ip electron root to exist. The importance of bounce–drift resonance is highlighted in the calculation, which affects the neoclassical offset rotation, especially the electron root. The ion root usually exists for various parameter regimes, while the electron root is only possible in low collisionality (e.g., high temperature and/or low density) regimes. The magnetic perturbation spectrum is found to influence the existence of electron roots when electrons are closer to resonant superbanana plateau regime than ions. By adjusting the plasma collisionality and tuning the spectrum of magnetic perturbations, it is possible to control the plasma rotation and hence to optimize the plasma confinement.

Complex dynamics in a quasi-periodic plasma perturbations model

<p style='text-indent:20px;'>In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power <inline-formula><tex-math id="M2">\begin{document}$ h $\end{document}</tex-math></inline-formula> and the relaxation of the instability <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient <inline-formula><tex-math id="M4">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

Unified modeling of both resonant and non-resonant neoclassical transport under non-axisymmetric magnetic perturbations in tokamaks

A numerical model for neoclassical transport under nonaxisymmetric magnetic perturbations in low collisionality plasmas in tokamaks is developed. To take into account bounce-drift resonances and magnetic drift effects, a Fourier decomposition of the drift kinetic equation in new coordinates, rather than bounce average of it, is employed. A pitch angle scattering collisional operator is used to keep accuracy in the nonresonant regimes or resonant plateau regimes with resonant pitch near pitch space boundaries. Full toroidal geometry effects are also included to increase the accuracy in the modeling of neoclassical transport in the resonant regimes. Neoclassical transport in the most important collisionless regimes, including resonant super-banana plateau and bounce-drift resonances regimes, nonresonant 1/ν and ν−ν regimes, and the transitions between them, can be modeled simultaneously in this model by numerically solving the drift kinetic equation. By application to the neoclassical toroidal viscosity modeling in one discharge in the EAST tokamak, it is found that the bounce-drift resonances dominate the contributions near the plasma core where the plasma E→×B→ drift frequency is close to the bounce frequency, while the precessional resonance dominates the contribution near the edge pedestal top where the E→×B→ drift frequency is close to zero.

A new method to suppress the Rayleigh–Taylor instability in a linear device

Rayleigh–Taylor instability (RTI) is a primary hurdle for many different fusion approaches, most of which rely on external pressure to stabilize the plasma by impeding plasma displacement. In this paper, we report a novel method that utilizes a rotating magnetic field (RMF) to drive an azimuthal electron current to reduce the charge separation caused by RTI. The fluctuation measured in the central cell of the mirror device, approximately half a device length away from the RMF, is identified as the m = 1 mode and is suppressed by the RMF in the plug cell. The azimuthal electric fields of the fluctuation are found to decrease to almost zero, and the radial confinement is improved by more than a factor of ten. The separation of the RMF region from the central cell makes this stabilization method unique because the RMF, which can complicate the local magnetic field lines, has little influence on the magnetic field configuration in the central cell. This study may shed light on the use of resonant magnetic perturbations in tokamaks as well as on stabilization methods for many other fusion experiments.Rayleigh–Taylor instability (RTI) is a primary hurdle for many different fusion approaches, most of which rely on external pressure to stabilize the plasma by impeding plasma displacement. In this paper, we report a novel method that utilizes a rotating magnetic field (RMF) to drive an azimuthal electron current to reduce the charge separation caused by RTI. The fluctuation measured in the central cell of the mirror device, approximately half a device length away from the RMF, is identified as the m = 1 mode and is suppressed by the RMF in the plug cell. The azimuthal electric fields of the fluctuation are found to decrease to almost zero, and the radial confinement is improved by more than a factor of ten. The separation of the RMF region from the central cell makes this stabilization method unique because the RMF, which can complicate the local magnetic field lines, has little influence on the magnetic field configuration in the central cell. This study may shed light on the use of resonant magnetic p...

Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016

The joint Varenna-Lausanne international workshop on the theory of fusion plasmas took place in Varenna from August 29 to September 2 2016. Several issues of interest for fusion plasmas were addressed, namely MHD stability, RF heating, collisional and turbulent transport, plasma wall interaction, and physics of burning plasmas. The articles published in this special issue illustrate nicely the well balanced combination of physics, applied mathematics, and computer sciences that characterizes this workshop.Let us mention several attractive topics, which are addressed in this issue. The question of 3D MHD equilibrium in tokamaks has received a great deal of attention, in connection with external resonant magnetic perturbations in tokamaks, and also stochastic edge in stellarators. The reader will also find some recent developments related to the effect of current drive and heating on the stability of tearing modes. As usual, turbulent transport is addressed in much detail. Several papers address specific numerical aspects of fluid and gyrokinetic codes, including code optimisation. Physics issues are abundantly dealt with, such as the impact of fast particles on turbulence, and particle transport. New numerical techniques to model wave propagation are presented, which provide significant advances in the field. Refinements such as the effect of density fluctuation on wave propagation, or the interaction between particles and the electromagnetic field near antennas, have also been studied in depth. Finally, specific issues such as nonlocal transport, decay of zonal flows, and the effect of neutrals on rotation have been investigated.A striking feature of the 2016 edition was the large number of young faces among the participants. This is a great satisfaction for the organizers since a new generation of scientists is certainly needed whilst several devices come to operation, or will do so in a foreseeable future.The diversity and quality of the papers published in the present cluster prove once more that the Varenna-Lausanne meeting has a special place among the conferences on magnetic fusion energy. It is certainly an excellent forum for exchanging ideas on theory and modelling, and for advertising the best results obtained in that field.The present volume of Journal of Physics Conference Series contains 16 outstanding papers - all peer reviewed. Let us mention another set of about 20 papers to appear in Plasma Physics and Controlled Fusion. We hope the reader will appreciate this special issue and that it will be a source of inspiration for him.Xavier Garbet, Olivier Sauter November, 8 2016http://varenna-lausanne.epfl.ch

Effects of resistivity on linear plasma responses to resonant magnetic perturbations in tokamak plasmas

Numerical studies are made of the effects of resistivity on linear plasma responses to resonant magnetic perturbations (RMPs) in tokamaks based on a reduced magnetohydrodynamic model. From a local two-field model, it is suggested that the ratio of the poloidal electron advection to the resistivity diffusion rate αm can be a figure of merit parameter in linear RMP penetration physics. The shielding efficiency is governed by αm, and when αm≳1, RMPs are effectively shielded. Global simulations using a four-field model [Hazeltine and Meiss, Phys. Rep. 121, 1 (1985)] show that there exists an effective threshold of the perpendicular electron flow (Ve,⊥c) beyond which RMPs cannot penetrate. Resistivity is found to determine Ve,⊥c which increases as resistivity becomes higher, making RMP penetration easier. At low resistivity, small Ve,⊥c renders the RMP penetration sensitive to ion collisionality and the change in q95. The kink response is observed to be closely related to the residual level of RMPs at rational surfaces and can be also strongly affected by resistivity.

Evaluation of toroidal torque by non-resonant magnetic perturbations in tokamaks for resonant transport regimes using a Hamiltonian approach

Toroidal torque generated by neoclassical viscosity caused by external non-resonant, non-axisymmetric perturbations has a significant influence on toroidal plasma rotation in tokamaks. In this article, a derivation for the expressions of toroidal torque and radial transport in resonant regimes is provided within quasilinear theory in canonical action-angle variables. The proposed approach treats all low-collisional quasilinear resonant neoclassical toroidal viscosity regimes including superbanana-plateau and drift-orbit resonances in a unified way and allows for magnetic drift in all regimes. It is valid for perturbations on toroidally symmetric flux surfaces of the unperturbed equilibrium without specific assumptions on geometry or aspect ratio. The resulting expressions are shown to match the existing analytical results in the large aspect ratio limit. Numerical results from the newly developed code NEO-RT are compared to calculations by the quasilinear version of the code NEO-2 at low collisionalities. The importance of the magnetic shear term in the magnetic drift frequency and a significant effect of the magnetic drift on drift-orbit resonances are demonstrated.

Analytical model of wall forces produced by kink perturbations in tokamaks

Analytical model of the electromagnetic forces produced by kink modes on the tokamak wall [H. R. Strauss et al., Phys. Plasmas 17, 082505 (2010)] is revisited. One of the main conclusions of the mentioned paper is that the largest force occurs at γτw≈1, where γ is the kink growth rate and τw is the wall penetration time. In the present study, a similar approach is developed under less restrictive assumptions on the plasma and dynamics of perturbation, and a different result is obtained: the force increases with γ and must be maximal at γτw→∞. Additionally, the dependence of its amplitude on the plasma parameters is clarified. All distinctions and their reasons are explained in detail. The analysis is performed in the cylindrical model incorporating a resistive wall treated without traditional thin-wall constraints and covering therefore a full range in γτw. It is applicable to either locked or rotating modes. Estimates of the sideways force are presented and compared with earlier forecasts.

Modeling of non-axisymmetric magnetic perturbations in tokamaks

A numerical model to evaluate the effects of the non-axisymmetric magnetic perturbations on magnetic topology and magnetic field ripple in tokamaks is presented in this paper. It is illustrated by using an example magnetic field perturbation induced by a coil system on the EAST tokamak. The influence of the choice of the coordinates on the spectrum is presented. The amplitude of resonant components of the spectrum are found to be independent of the coordinates system, while that of the non-resonant components are not. A better way to describe the edge topology by using the Chirikov parameter profile is proposed and checked by the numerical Poincaré plot results. The contribution of the magnetic perturbation on local toroidal field ripple can be significant. One approximate method to model the helical ripple on the perturbed flux surface induced by a given non-axisymmetric magnetic field perturbation is presented. All of the spectrum analysis is applicable in case the plasma response is taken into account in the input of perturbed magnetic field.

Modelling of three dimensional equilibrium and stability of MAST plasmas with magnetic perturbations using VMEC and COBRA

It is known that magnetic perturbations can mitigate edge localized modes (ELMs) in experiments, for example, MAST [Kirk et al., Nucl. Fusion 53, 043007 (2013)]. One hypothesis is that the magnetic perturbations cause a three dimensional corrugation of the plasma and this corrugated plasma has different stability properties to peeling-ballooning modes compared to an axisymmetric plasma. It has been shown in an up-down symmetric plasma that magnetic perturbations in tokamaks will break the usual axisymmetry of the plasma causing three dimensional displacements [Chapman et al., Plasma Phys. Controlled Fusion 54, 105013 (2012)]. We produce a free boundary three-dimensional equilibrium of a lower single null MAST relevant plasma using VMEC [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)]. The safety factor and pressure profiles used for the modelling are similar to those deduced from axisymmetric analysis of experimental data with ELMs. We focus on the effect of applying n = 3 and n = 6 magnetic perturbations using the resonant magnetic perturbation (RMP) coils. A midplane displacement of over ±1 cm is seen when the full current is applied. The current in the coils is scanned and a linear relationship between coil current and midplane displacement is found. The pressure gradient in real space in different toroidal locations is shown to change when RMPs are applied. This effect should be taken into account when diagnosing plasmas with RMPs applied. The helical Pfirsch-Schlüter currents which arise as a result of the assumption of nested flux surfaces are estimated for this equilibrium. The effect of this non-axisymmetric equilibrium on infinite n ballooning stability is investigated using COBRA [Sanchez et al., J. Comput. Phys. 161, 576–588 (2000)]. The infinite n ballooning stability is analysed for two reasons; it may give an indication of the effect of non-axisymmetry on finite n peeling-ballooning modes, responsible for ELMs; and infinite n ballooning modes are correlated to kinetic ballooning modes which are thought to limit the pressure gradient of the pedestal [Snyder et al., Phys. Plasmas 16, 056118 (2009)]. The ballooning mode growth rate gains a variation in toroidal angle. The equilibria with midplane displacements due to RMP coils have a higher ballooning mode growth rate than the axisymmetric case and the possible implications are discussed.

The mechanism of particles transport induced by electrostatic perturbation in tokamak

Particle transport in tokamak devices due to wave-particle resonance induced diffusion is studied. The diffusion coefficient is derived both analytically using quasilinear theory, and numerically using a test particle code, and the obtained diffusion coefficient agrees with each other in its validity regime. Dependence of the diffusion coefficient on turbulence intensity, turbulence radial mode structures, and particle energy is investigated. It is found that the diffusion coefficient is proportional to the turbulence intensity, and the diffusion is maximized for Et≃Ti, and krΔ0≃1. Here, Et is the test particle energy, Ti is the thermal ion temperature, Δ0 is the distance between neighboring mode rational surfaces, and 1/kr is the half width of the fine radial mode structure on each rational surface.

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.

Control of Nonaxisymmetric Magnetic Field Perturbations in Tokamaks

The theory of control of nonaxisymmetric perturbations is dominated by the wide sensitivity range of a tokamak plasma to externally produced magnetic perturbations. External perturbations are characterized by their normal magnetic field on the unperturbed plasma surface. The first spatial distribution of on the unperturbed plasma surface in a sensitivity series is that distribution that at the smallest amplitude has a significant effect on plasma properties. The second distribution of in that series is the distribution to which the plasma has greatest sensitivity while being orthogonal to the first. Two distributions are orthogonal if the integral of their product over the unperturbed plasma surface is zero. Only a limited number of distributions in the sensitivity series can be driven to an unacceptable amplitude by credible construction errors in ITER. Essentially any external coil set that produces a nonaxisymmetric magnetic field of adequate strength with a controllable toroidal phase can null the drive for the distribution of highest plasma sensitivity. However, the simultaneous nulling of not only the first but also of a number of other distributions in the sensitivity series is far more difficult. It is the properties of these distributions of secondary importance that determine both the machine tolerances that are required for successful control and the adequacy of a given set of error field control coils. Nonaxisymmetric fields can also have beneficial effects such as the control of edge-localized modes. Implementation requires driving a normal field distribution to which the beneficial effect is sensitive while not driving detrimental distributions of high plasma sensitivity.

Free energy minimization approach to penetration of resonant magnetic perturbations in tokamaks

By applying the principle of minimum free energy an analytical model for the plasma response to externally applied resonant magnetic perturbations (RMPs) is proposed. The results are compared with ATTEMPT code calculations [D. Reiser et al., Phys. Plasmas 16, 0042317 (2009)] and reproduce qualitatively and quantitatively the numerical results on the collisionality dependence of RMP penetration characteristics. Strong increase in the radial electric field with reduced screening at RMPs above a certain threshold is also reproduced by the model.

Importance of plasma response to nonaxisymmetric perturbations in tokamaks

Tokamaks are sensitive to deviations from axisymmetry as small as δB/B0∼10−4. These nonaxisymmetric perturbations greatly modify plasma confinement and performance by either destroying magnetic surfaces with subsequent locking or deforming magnetic surfaces with associated nonambipolar transport. The Ideal Perturbed Equilibrium Code (IPEC) calculates ideal perturbed equilibria and provides important basis for understanding the sensitivity of tokamak plasmas to perturbations. IPEC calculations indicate that the ideal plasma response, or equivalently the effect by ideally perturbed plasma currents, is essential to explain locking experiments on National Spherical Torus eXperiment (NSTX) and DIII-D. The ideal plasma response is also important for neoclassical toroidal viscosity (NTV) in nonambipolar transport. The consistency between NTV theory and magnetic braking experiments on NSTX and DIII-D can be improved when the variation in the field strength in IPEC is coupled with generalized NTV theory. These plasma response effects will be compared with the previous vacuum superpositions to illustrate the importance. However, plasma response based on ideal perturbed equilibria is still not sufficiently accurate to predict the details of NTV transport and can be inconsistent when currents associated with a toroidal torque become comparable to ideal perturbed currents.

Plasma currents induced by resonant magnetic field perturbations in tokamaks

The plasma response on externally applied resonant magnetic field perturbations is studied by means of numerical simulations. It is shown that dependent on collisionality and perturbation strength, plasma currents build up which can compensate the external field. These plasma currents are accompanied by out-of-phase currents and poloidal flows at the resonant surfaces. With an increasing perturbation field the screening of the externally applied field decreases and at a certain level, the vacuum field approximation holds for the total magnetic field.

Control of Asymmetric Magnetic Perturbations in Tokamaks

The sensitivity of tokamak plasmas to very small deviations from the axisymmetry of the magnetic field |deltaB/B| approximately 10{-4} is well known. What was not understood until very recently is the importance of the perturbation to the plasma equilibrium in assessing the effects of externally produced asymmetries in the magnetic field, even far from a stability limit. DIII-D and NSTX experiments find that when the deleterious effects of asymmetries are mitigated, the external asymmetric field was often made stronger and had an increased interaction with the magnetic field of the unperturbed equilibrium. This Letter explains these counterintuitive results. The explanation using ideal perturbed equilibria has important implications for the control of field errors in all toroidal plasmas.