Rotating Tokamak Plasmas Research Articles

Probabilistic locked mode predictor in the presence of a resistive wall and finite island saturation in tokamaks

We present a framework for estimating the probability of locking to an error field in a rotating tokamak plasma. This leverages machine learning methods trained on data from a mode-locking model, including an error field, resistive magnetohydrodynamics modeling of the plasma, a resistive wall, and an external vacuum region, leading to a fifth-order ordinary differential equation (ODE) system. It is an extension of the model without a resistive wall introduced by Akçay et al. [Phys. Plasmas 28, 082106 (2021)]. Tearing mode saturation by a finite island width is also modeled. We vary three pairs of control parameters in our studies: the momentum source plus either the error field, the tearing stability index, or the island saturation term. The order parameters are the time-asymptotic values of the five ODE variables. Normalization of them reduces the system to 2D and facilitates the classification into locked (L) or unlocked (U) states, as illustrated by Akçay et al., [Phys. Plasmas 28, 082106 (2021)]. This classification splits the control space into three regions: L̂, with only L states; Û, with only U states; and a hysteresis (hysteretic) region Ĥ, with both L and U states. In regions L̂ and Û, the cubic equation of torque balance yields one real root. Region Ĥ has three roots, allowing bifurcations between the L and U states. The classification of the ODE solutions into L/U is used to estimate the locking probability, conditional on the pair of the control parameters, using a neural network. We also explore estimating the locking probability for a sparse dataset, using a transfer learning method based on a dense model dataset.

On the interplay between interchange turbulence and sheared flows

The presence and the position of an X-point, namely, if the ion diamagnetic drift points toward it or not, strongly impact the edge plasma rotation in tokamaks. In the absence of kinetic effects such as magnetic ripple or ion orbit loss, the shape of the velocity profile results from the balance between neoclassical predictions and turbulent flow generation. In this contribution, we derive a reduced model of turbulence plunged in a shear flow. This model is based on (1) a description of the impact of a sheared flow on the interchange turbulence and (2) a prediction of the poloidal momentum generated by the turbulence. It includes the effects of both the magnetic topology and the finite shear layer width. The model is verified against 2D non-linear flux-driven simulations. Finally, the model predictions of the edge rotation resulting from the equilibrium between the neoclassical prediction and the poloidal momentum generation by the turbulence are invoked to describe the observations from experiment managed in the WEST tokamak. It points out the important role of the magnetic shear in the turbulence tilting and in the flow generation.

Parametric dependencies of resonant layer responses across linear, two-fluid, drift-MHD regimes

Non-axisymmetric magnetic fields arising in a tokamak either by external or internal perturbations can induce complex non-ideal MHD responses in their resonant surfaces while remaining ideally evolved elsewhere. This layer response can be characterized in a linear regime by a single parameter called the inner-layer Δ, which enables outer-layer matching and the prediction of torque balance to non-linear island regimes. Here, we follow strictly one of the most comprehensive analytic treatments including two-fluid and drift MHD effects and keep the fidelity of the formulation by incorporating the numerical method based on the Riccati transformation when quantifying the inner-layer Δ. The proposed scheme reproduces not only the predicted responses in essentially all asymptotic regimes but also with continuous transitions as well as improved accuracies. In particular, the Δ variations across the inertial regimes with viscous or semi-collisional effects have been further resolved, in comparison with additional analytic solutions. The results imply greater shielding of the electromagnetic torque at the layer than what would be expected by earlier work when the viscous or semi-collisional effects can compete against the inertial effects, and also due to the intermediate regulation by kinetic Alfvén wave resonances as rotation slows down. These are important features that can alter the non-axisymmetric plasma responses including the field penetration by external fields or island seeding process in rotating tokamak plasmas.

Synergy of Turbulent Momentum Drive and Magnetic Braking.

In absence of external torque, plasma rotation in tokamaks results from a balance between collisional magnetic braking and turbulent drive. The outcome of this competition and cooperation is essential to determine the plasma flow. A reduced model, supported by gyrokinetic simulations, is first used to explain and quantify the competition only. The ripple amplitude above which magnetic drag overcomes turbulent viscosity is obtained. The synergetic impact of ripple on the turbulent toroidal Reynolds stress is explored. Simulations show that the main effect comes from an enhancement of the radial electric field shear by the ripple, which in turn impacts the residual stress.

Low-frequency zonal flow eigen-structures in tokamak plasmas

The nonlocal eigenmode analysis of low-frequency zonal flows (ZFs) in toroidally rotating tokamak plasmas is performed in the framework of the reduced one-fluid ideal magnetohydrodynamic model. It is shown that for typical profiles of plasma parameters toroidal plasma rotation results in the global ZF formation on the periphery of plasma column. For some types of equilibria these ZFs are aperiodically unstable that leads to the excitation of the differential plasma rotation at the tokamak plasma edge.

Nonlinear excitation of geodesic acoustic mode by drift waves in anisotropic tokamak plasmas with toroidal rotation

Nonlinear excitation of geodesic acoustic mode (GAM) by ions drift waves is investigated in anisotropic toroidally rotating tokamak plasmas by using an ideal magnetohydrodynamic model and standard double adiabatic equations. It is found that the radial wave number kx of the effectively excited GAM and the GAM frequency increase with the ratio of the perpendicular pressure to the parallel pressure χ as well as the toroidal rotation Mach number. The equilibrium anisotropy and the ratio of the radial scale length of the equilibrium density to the tokamak major radius are shown to play a significant role in the growth rate and frequency shift of GAM. The growth rate increases with the normalized GAM frequency first and then decreases to zero as the GAM frequency increases. When the GAM frequency increases again, the growth rate continues to increase but with a relatively smaller magnitude.

Machine learning methods for probabilistic locked-mode predictors in tokamak plasmas

A rotating tokamak plasma can interact resonantly with the external helical magnetic perturbations, also known as error fields. This can lead to locking and then to disruptions. We leverage machine learning (ML) methods to predict the locking events. We use a coupled third-order nonlinear ordinary differential equation model to represent the interaction of the magnetic perturbation and the plasma rotation with the error field. This model is sufficient to describe qualitatively the locking and unlocking bifurcations. We explore using ML algorithms with the simulation data and experimental data, focusing on the methods that can be used with sparse datasets. These methods lead to the possibility of the avoidance of locking in real-time operations. We describe the operational space in terms of two control parameters: the magnitude of the error field and the rotation frequency associated with the momentum source that maintains the plasma rotation. The outcomes are quantified by order parameters that completely characterize the state, whether locked or unlocked. We use unsupervised ML methods to classify locked/unlocked states and note the usefulness of a certain normalization of the order parameters. Three supervised ML classifiers are used in suite to estimate the probability of locking in the region of control parameter space with hysteresis, i.e., the set of control parameters for which both locked and unlocked states can exist. The results show that a neural network gives the best estimate of the locking probability. An analogy of the present locking model with the van der Waals equation of state is also provided.

Nonlinear feedback algorithm for tearing-mode magnetic control in tokamak

Results of calculations and analysis of the tearing-mode evolution under the effect of resonant magnetic perturbation produced by a nonlinear feedback system in rotating tokamak plasma are presented. The TEAR-code used for calculations is based on the visco-resistive magnetohydrodynamic approximation that gives coupled diffusion-type equations for the magnetic flux perturbation and for the plasma rotation velocities in toroidal and poloidal directions. The code is supplemented by a computational unit simulating the nonlinear algorithm of the feedback system. The feedback gain and phase shift are automatically regulated in real time. The gain depends on the tearing-mode amplitude. At high feedback input signals, the gain does not exceed some finite value to reduce technical requirements for the feedback actuator and to mitigate the phase instability. In the process of the tearing-mode suppression, the gain gradually increases to provide the mode control under reduction of feedback input signal and rise of the intrinsic tearing-mode stability index. Besides that, the real-time control of the feedback phase shift is provided to additionally mitigate the phase instability. According to simulation, the used nonlinear characteristics of the feedback algorithm improve the efficiency of the tearing-mode suppression.

Ideal internal kink stability in presence of plasma flow and neoclassical toroidal viscosity due to energetic particles

The influence of energetic particles (EPs) on the ideal internal kink mode, in rotating tokamak plasmas, is numerically investigated by simultaneously solving MHD-kinetic hybrid equations together with a toroidal momentum balance equation utilizing the MARS-Q code (Liu et al 2013 Phys. Plasmas 20 042503). The neoclassical toroidal viscous (NTV) torque, induced by precessional drift resonances of trapped energetic particles, acts as the momentum sink term to damp the plasma flow. Quasi-linear initial value simulations show local reduction of the flow amplitude and enhancement of the flow shear near the q = 1 rational surface (q is the safety factor) due to EP induced NTV. Both effects in turn destabilize the internal kink mode. These numerical findings are robust against the initial linear stability of internal kink, the initial plasma flow profile, as well as the equilibrium distribution model for EPs.

Guiding-centre theory for kinetic-magnetohydrodynamic modes in strongly flowing plasmas

A kinetic-magnetohydrodynamic model with kinetic pressure closure is derived from a consistent guiding-centre framework. Higher-order (gyroviscous) corrections to the pressure tensor are derived in complex geometry from a reduced kinetic equation. The proposed model allows for flows of the order of the thermal ion velocity, taking into account important centrifugal effects due to the E × B flow, as well as the effects of diamagnetic flows associated with finite Larmor radius corrections to both ion fluid inertia and long mean-free path contributions. Wave–particle interactions, such as toroidal drift-resonance, are retained. Furthermore, the linearised model includes a quasi-neutrality equation, allowing the effects of a parallel electric field to be studied in fast rotating tokamak plasmas.

Low-Frequency Continuous MHD Spectrum of Toroidally Rotating Tokamak Plasmas with Anisotropic Pressure

Continuous-spectrum equations for low-frequency ideal magnetohydrodynamic perturbations in toroidally rotating plasmas with anisotropic pressure in axisymmetric tokamaks are derived in the framework of the Chew–Goldberger–Low model. In the rotating coordinate system, these equations describe the toroidal coupling of Alfven and slow magnetosonic modes due to the curvature of magnetic field lines, the pressure anisotropy, and the centrifugal and Coriolis effects. The derived general equations are applied to study the spectra of both the zonal flows and the general electromagnetic modes in low-pressure large-aspect-ratio tokamaks. The condition for the instability of zonal flows due to plasma stratification over the poloidal angle on a magnetic surface is obtained. It is shown that a similar instability takes place for the general modes localized in the vicinities of rational magnetic surfaces. Stabilization of this instability by the Alfven effect for the modes localized far from rational surfaces is shown.

Role of parallel compression in potential vorticity mixing and zonal flow generation: a gyrokinetic simulation study

From global gyrokinetic simulations of toroidal ion temperature gradient-driven turbulence, we show that the ion parallel compression has a strong influence on the generation and radial profile formation of zonal flows. The kinetic potential vorticity (PV) flux and its fluid expression are used to elucidate the zonal flow generation mechanism. In the absence of sheared equilibrium flow, the dominant contributions to the net PV flux are shown to come from the parallel compression and the grad-B drift, and to largely cancel out each other. With a finite parallel rotation shear, however, the parallel compression-driven flux becomes dominant over the grad-B drift-driven one, leading to a change in the radial zonal flow profile. The imbalance between the parallel compression and the grad-B drifts results in a considerable amplification of the zonal flow and a reduction of turbulence fluctuation levels as compared to the non-rotating plasma. These findings demonstrate an essential role of the parallel compression in the zonal flow generation and confinement improvement for rotating tokamak plasmas.

In–out impurity density asymmetry due to the Coriolis force in a rotating tokamak plasma

The effect of the Coriolis force due to the impurity toroidal and poloidal rotation on the in–out impurity density asymmetry in a rotating tokamak plasma is investigated. The role of the Coriolis force is figured out explicitly in the toroidally rotating plasma frame. The in–out impurity density asymmetry can be induced by the Coriolis force with , in this case, when moving along the magnetic field line from the outboard side to the inboard side in a magnetic flux surface, one sees a positive Coriolis force. The combined effects of the centrifugal force, the Coriolis force, the poloidal electric field and the ion-impurity friction force on the in–out impurity density asymmetry are discussed. The proposed theoretical model can be applied to improve the understanding of the in–out impurity density asymmetry observed in the tokamak experiments.

Gyrokinetic theory of turbulent acceleration and momentum conservation in tokamak plasmas

Understanding the generation of intrinsic rotation in tokamak plasmas is crucial for future fusion reactors such as ITER. We proposed a new mechanism named turbulent acceleration for the origin of the intrinsic parallel rotation based on gyrokinetic theory. The turbulent acceleration acts as a local source or sink of parallel rotation, i.e., volume force, which is different from the divergence of residual stress, i.e., surface force. However, the order of magnitude of turbulent acceleration can be comparable to that of the divergence of residual stress for electrostatic ion temperature gradient (ITG) turbulence. A possible theoretical explanation for the experimental observation of electron cyclotron heating induced decrease of co-current rotation was also proposed via comparison between the turbulent acceleration driven by ITG turbulence and that driven by collisionless trapped electron mode turbulence. We also extended this theory to electromagnetic ITG turbulence and investigated the electromagnetic effects on intrinsic parallel rotation drive. Finally, we demonstrated that the presence of turbulent acceleration does not conflict with momentum conservation.

Impurity effect on geodesic acoustic mode in toroidally rotating tokamak plasmas

The geodesic acoustic modes (GAMs) are analytically investigated in toroidally rotating tokamak plasmas with impurity ions such as carbon and tungsten by using the gyrokinetic equation. The non-trace and trace impurity effect on the GAM with or without toroidal rotation are studied and compared, respectively. The results show that in the non-rotation case, the non-trace impurity decreases (increases) the frequency (damping rate) of the GAM mainly due to the polarization current, while the trace impurity has little effect on the GAM. When toroidal rotation is considered, the non-trace impurity still significantly decreases (increases) the frequency (damping rate) of the GAM. Furthermore, as toroidal rotation increases, the frequency (damping rate) of the GAM with the non-trace impurity increases (decreases) more slowly than that without the non-trace impurity, especially when the non-trace impurity concentration is relatively large. Nevertheless, the trace impurity has little effect on the GAM in the weak rotation regime, while it greatly increases (decreases) the frequency (damping rate) of the GAM when toroidal rotation is sufficiently large. These results are mainly due to the additional drifts induced by toroidal rotation. In addition, it is found that the isotope effect has significant influence on the GAM and it also affects both the non-trace and trace impurity as well as toroidal rotation effect on the GAM.

Physics conditions for robust control of tearing modes in a rotating tokamak plasma

The disruptive collapse of the current sustained equilibrium of a tokamak is perhaps the single most serious obstacle on the path toward controlled thermonuclear fusion. The current disruption is generally too fast to be identified early enough and tamed efficiently, and may be associated with a variety of initial perturbing events. However, a common feature of all disruptive events is that they proceed through the onset of magnetohydrodynamic instabilities and field reconnection processes developing magnetic islands, which eventually destroy the magnetic configuration. Therefore the avoidance and control of magnetic reconnection instabilities is of foremost importance and great attention is focused on the promising stabilization techniques based on localized rf power absorption and current drive. Here a short review is proposed of the key aspects of high power rf control schemes (specifically electron cyclotron heating and current drive) for tearing modes, considering also some effects of plasma rotation. From first principles physics considerations, new conditions are presented and discussed to achieve control of the tearing perturbations by means of high power ) in regimes where strong nonlinear instabilities may be driven, such as secondary island structures, which can blur the detection and limit the control of the instabilities. Here we consider recent work that has motivated the search for the improvement of some traditional control strategies, namely the feedback schemes based on strict phase tracking of the propagating magnetic islands.

Poloidal rotation driven by electron cyclotron resonance wave in tokamak plasmas

The poloidal electric filed, which is the drive field of poloidal rotation, has been observed and increases obviously after the injection of electron cyclotron resonance wave in HL-2A experiment, and the amplitude of the poloidal electric field is in the order of 103 V/m. Through theoretical analysis using Stringer rotation model, the observed poloidal electric field is of the same order as the theoretical calculation value. In addition, the magnetic pump damping which would damp the poloidal rotation is calculated numerically and the calculation results show that the closer to the core plasmas, the stronger the magnetic pump damping will be. Meanwhile, according to the value of the calculated magnetic pump damping, the threshold of the poloidal electric field which could overcome magnetic pump damping and drive poloidal rotation in tokamak plasmas is given out. Finally, the poloidal rotation velocity over time at different minor radius is studied theoretically.

Finite-orbit-width effects on the geodesic acoustic mode in the toroidally rotating tokamak plasma

The Landau damping of geodesic acoustic mode in a toroidal rotating tokamak plasma is analytically investigated by taking into account the finite-orbit-width resonance effect to the 3rd order. The analytical result is shown to agree well with the numerical solution. The dependence of the damping rate on the toroidal Mach number M relies on krρi. For sufficiently small krρi, the damping rate monotonically decreases with M. For relatively large krρi, the damping rate increases with M until approaching the maximum and then decreases with M.

The impact of the ion-cyclotron-resonance location on the poloidal asymmetries of impurity density in an ICRF-heated rotating plasma

The recently proposed set of zeroth-order moment equations to model poloidal density asymmetries induced by temperature anisotropies in rotating tokamak plasmas (Bilato et al 2014 Nucl. Fusion 54 072003) is here extended to account for the effects of the localization of the ion-cyclotron (IC) resonance on the poloidal inhomogeneity of the density of the cyclotron-heated ion species. This additional effect has a significant impact on the poloidal density asymmetries due to radio-frequency heating, a key issue for high-Z impurity transport, leading, in particular, to a reduced ICRF heating impact on the in–out accumulation of high-Z impurities when the IC resonance is located on the HFS.

Energetic particle driven geodesic acoustic mode in a toroidally rotating tokamak plasma

Energetic particle (EP) driven geodesic acoustic modes (EGAMs) in toroidally rotating tokamak plasmas are analytically investigated using the hybrid kinetic-fluid model and gyrokinetic equations. By ignoring high-order terms and ion Landau damping, the kinetic dispersion relation is reduced to the hybrid one in the large safety factor limit. There is one high-frequency branch with a frequency larger than , the transit frequency of EPs with initial energy, which is always stable. Two low-frequency solutions with a frequency smaller than are complex conjugates in the hybrid limit. In the presence of ion Landau damping, the growth rate of the unstable branch is decreased and the damping rate of the damped branch is increased. The toroidal Mach number is shown to increase , the normalized real frequency of both branches. Although not affecting the instability critical condition, the Mach number decreases the growth rate when is larger than a critical value and enlarges the growth rate when . The ion Landau damping effect is negligible for large M. But the discrepancy between the kinetic dispersion relation and the hybrid one becomes ignorable only for .