Theoretical and numerical studies of wave-packet propagation are presented to analyze the time varying 2D mode structures of electrostatic fluctuations in tokamak plasmas, using general flux coordinates. Instead of solving the 2D wave equations directly, the solution of the initial value problem is used to obtain the 2D mode structure, following the propagation of wave-packets generated by a source and reconstructing the time varying field. As application, the 2D WKB method is applied to investigate the shaping effects (elongation and triangularity) of tokamak geometry on the lower hybrid wave propagation and absorbtion. Meanwhile, the Mode Structure Decomposition (MSD) method is used to handle the boundary conditions and simplify the 2D problem to two nested 1D problems. The MSD method is related to that discussed earlier by Zonca and Chen [Phys. Fluids B 5, 3668 (1993)], and reduces to the well-known "ballooning formalism" [J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, 396 (1978)], when spatial scale separation applies. This method is used to investigate the time varying 2D electrostatic ITG mode structure with a mixed WKB-full-wave technique. The time varying field pattern is reconstructed and the time asymptotic structure of the wave-packet propagation gives the 2D eigenmode and the corresponding eigenvalue. As a general approach to investigate 2D mode structures in tokamak plasmas, our method also applies for electromagnetic waves with general source/sink terms, either by an internal/external antenna or nonlinear wave interaction with zonal structures.
The paper presents a relativistic mechanism that can create a poloidal non-uniformity of a radial electric field on a magnetic surface in a tokamak plasma. This mechanism is associated with a significant current velocity of electrons in plasma for many experimental modes of tokamak operations. It is important that electric fields arise without the traditional separation of charges in plasma. The presented mechanism can explain the previously experimentally measured poloidal non-uniformity of the C+6 carbon ion toroidal rotation velocity in the TORE-SUPRA tokamak plasma
Spontaneous spin-up are widely observed in tokamak plasmas, which is crucially important for plasma confinement. A kinetic theory is proposed to show that a toroidal rotation of core plasma is induced by the topological transition of orbits driven by turbulent diffusion in a collisionless tokamak plasma. The theoretical prediction agrees well with the well-known Rice-scaling of intrinsic core plasma flow. This new theory predicts an intrinsic core parallel flow of $\sim 100km/s$ for the International Thermonuclear Experimental Reactor.
Turbulence is a major factor limiting the achievement of better tokamak performance as it enhances the transport of particles, momentum and heat which hinders the foremost objective of tokamaks. Hence, understanding and possibly being able to control turbulence in tokamaks is of paramount importance, not to mention our intellectual curiosity of it.
A relation between physical consequences of the so-called Ehrenfest's Paradox and the radial electric field in the classical quasi-neutral tokamak plasma is shown. Basic author's approach to the relativistic nature of the tokamak has been described in [1]. The experiment which can resolve the Ehrenfest's Paradox is presented.
A comprehensive two-fluid nonlinear theory of magnetic reconnection driven at a single, tearing-stable, rational surface embedded in an H-mode tokamak plasma is presented. The surface is assumed to be resonant with one of the dominant helical harmonics of an applied resonant magnetic perturbation (RMP). The theory described in this paper is highly relevant to the problem of understanding the physics of RMP-induced edge localized mode (ELM) suppression in tokamak plasmas.
A simplified form and some possible theoretical resolutions so-called Ehrenfest's Paradox is described. A relation between physical consequences of this relativistic Paradox and the charge density of a tokamak plasma is shown. Plasma experiments which can resolve the Ehrenfest's Paradox is presented.
The Hall term has often been neglected in MHD codes as it is difficult to compute. Nevertheless setting it aside for numerical reasons led to ignoring it altogether. This is especially problematic when dealing with tokamak physics as the Hall term cannot be neglected as this paper shows.
The Hall effect, defined as the separation of electric charges of opposite sign when they move in a magnetic field, is suggested to contribute substantially to the observed negative radial electric field Er in the core plasma in tokamaks and, respectively, to the spontaneous/intrinsic rotation of plasma. A simple way to evaluate the Hall effect contribution to the Er value, using the independently measured space distributions of magnetic field and plasma rotation velocity, is suggested. The estimates of the effect for experimental data from the TM-4 and T-10 tokamaks suggest that the above phe…
Drift-reduced plasma fluid models are commonly used in plasma physics for analytics and simulations; however, the validity of such models must be verified for the regions of parameter space in which tokamak plasmas exist. By looking at the linear behaviour of drift-reduced and full-velocity models one can determine that the physics lost through the simplification that the drift-reduction provides is important in the core region of the tokamak. It is more acceptable for the edge-region but one must determine specifically for a given simulation if such a model is appropriate.
The SMARDDA software library is used to model plasma interaction with complex engineered surfaces. A simple flux-tube model of power deposition necessitates the following of magnetic fieldlines until they meet geometry taken from a CAD (Computer Aided Design) database. Application is made to 1) models of ITER tokamak limiter geometry and 2) MASTU tokamak divertor designs, illustrating the accuracy and effectiveness of SMARDDA, even in the presence of significant nonaxisymmetric ripple field. SMARDDA's ability to exchange data with CAD databases and its speed of execution also give it the poten…
The first linear global electromagnetic gyrokinetic particle simulation on the excitation of toroidicity induced Alfven eigenmode (TAE) by energetic particles is reported. With an increase in the energetic particle pressure, the TAE frequency moves down into the lower continuum.
Decay times of plasma flows and plasma profiles have been measured after a sudden biasing switch-off in experiments on the CASTOR tokamak. A biased electrode has been used to polarize the edge plasma. The edge plasma potential and flows have been characterized by means of Langmuir and Mach probes, the radiation was measured using an array of bolometers. Potential profiles and poloidal flows can be well fitted by an exponential decay time in the range of 10 - 30 microseconds when the electrode biasing is turn off in the CASTOR tokamak. The radiation shows a slower time scale (about 1 ms), which…
Plasma terminating disruptions in tokamaks may result in relativistic runaway electron beams with potentially serious consequences for future devices with large plasma currents. In this paper we investigate the effect of plasma elongation on the coupled dynamics of runaway generation and resistive diffusion of the electric field. We find that elongated plasmas are less likely to produce large runaway currents, partly due to the lower induced electric fields associated with larger plasmas, and partly due to direct shaping effects, which mainly lead to a reduction in the runaway avalanche gain.
An analytic solution to a generalized Grad-Shafranov equation with flow of arbitrary direction is obtained upon adopting the generic linearizing ansatz for the free functions related to the poloidal current, the static pressure and the electric field. Subsequently, a D-shaped tokamak pertinent equilibrium with sheared flow is constructed using the aforementioned solution.
A local magnetic equilibrium model is presented, with finite aspect ratio and up-down asymmetrically shaped cross section, that depends on eight free parameters. In contrast with other local equilibria, which provide simple magnetic-surface parametrizations at the cost of complex poloidal-field flux descriptions, the proposed model is intentionally built to afford analytically tractable magnetic-field components. Therefore, it is particularly suitable for analytical assessments of equilibrium-shaping effects on a variety of tokamak-plasma phenomena.
It is pointed that there is a current jump between the edge plasma inside the last closed magnetic surface and the scrape-off layer and the current jump can lead the external kink modes to convert to the tearing modes, due to the current interchange effects [L. J. Zheng and M. Furukawa, Phys. Plasmas 17, 052508 (2010)]. The magnetic reconnection in the presence of tearing modes subsequently causes the tokamak edge plasma to be peeled off to link to the diverters. In particular, the peeling or peeling-ballooning modes can become the "peeling-off" modes in this sense. This phenomenon indicates t…
Flux-surface-averaged momentum loss and parallel rotation of the bulk ions at the edge of a tokamak plasma due to the ion orbit loss are calculated by computing the minimum loss energy of both the trapped and the passing thermal ions. The flux-surface-averaged parallel rotation of the bulk ions is in the co-current direction. The peak of the co-current rotation speed locates inside the last closed flux surface due to the orbit loss of the co-current thermal ions at the very edge of a tokamak plasma. The peaking position moves inward when the ion temperature increases.
The plasma beta, ratio of kinetic to magnetic pressure, inside a tokamak should stay below the Troyon limit to avoid major plasma instabilities. However, this paper argues that Troyon limit occurs only when current profiles cannot sustain high beta equilibria. This paper shows that plasmas with peak beta on the order of unity are stable to ideal MHD modes. While the theoretical (ideal-MHD) stability of unity beta equilibrium has been resolved, the experimental existence of high beta equilibria in tokamaks remains an unsolved problem of plasma physics.
This comment clarifies the relation of the research in a recently published article [Phys. Plasmas 14, 042503 (2007)] to other prior publications addressing the inclusion of electromagnetic and drift-kinetic electron physics in gyrokinetic simulation, raises a concern related to the inclusion of kinetic electrons in a system with magnetic shear, and discusses alternatives in the face of an important limitation on the general applicability of the algorithm described therein.
The effect of small deviations from a Maxwellian equilibrium on turbulent momentum transport in tokamak plasmas is considered. These non-Maxwellian features, arising from diamagnetic effects, introduce a strong dependence of the radial flux of co-current toroidal angular momentum on collisionality: As the plasma goes from nearly collisionless to weakly collisional, the flux reverses direction from radially inward to outward. This indicates a collisionality-dependent transition from peaked to hollow rotation profiles, consistent with experimental observations of intrinsic rotation.
In this paper we propose a vertical stabilization~(VS) control system for tokamak plasmas based on the extremum seeking (ES) algorithm. The gist of the proposed strategy is to inject an oscillating term in the control action and exploit a modified ES algorithm in order to bring to zero the average motion of the plasma along the unstable mode. In this way, the stabilization of the unstable vertical dynamic of the plasma is achieved. The approach is validated by means of both linear and nonlinear simulations of the overall~ITER tokamak magnetic control system, with the aim of demonstrating robus…
Solutions to a model 2D eigenmode equation describing micro-instabilities in tokamak plasmas are presented that demonstrate a sensitivity of the mode structure and stability to plasma profiles. In narrow regions of parameter space, with special plasma profiles, a maximally unstable mode is found that balloons on the outboard side of the tokamak. This corresponds to the conventional picture of a ballooning mode. However, for most profiles this mode cannot exist and instead a more stable mode is found that balloons closer to the top or bottom of the plasma. Good quantitative agreement with a 1D …
Steering tokamak plasmas is commonly viewed as a way to avoid disruptions and runaway electrons. Plasma steering sounds as safe as driving to work but will be shown to more closely resemble driving at high speed through a dense fog on an icy road. The long time required to terminate an ITER discharge compared to time over which dangers can be foreseen is analogous to driving in a dense fog. The difficulty of regaining plasma control if it is lost resembles driving on an icy road. Disruptions and runaways are associated with three issues -- a solution to one tends to complicate the solution to …
Magnetic islands (MIs), resulting from a magnetic field reconnection, are ubiquitous structures in magnetized plasmas. In tokamak plasmas, recent researches suggested that the interaction between the MI and ambient turbulence can be important for the nonlinear MI evolution, but a lack of detailed experimental observations and analyses has prevented further understanding. Here, we provide comprehensive two-dimensional observations that indicate various effects of the ambient turbulence on the nonlinear MI evolution. It is shown that the modified plasma turbulence around the MI can lead to eithe…
The first experimental measurements of vorticity and vorticity flux in a fusion device were performed in tokamak ISTTOK. This is an important achievement since vorticity plays a key role in the transport of energy and particles in plasmas and fluids. The measurements were performed with a specifically designed array of Langmuir probes in the plasma edge of the small tokamak ISTTOK. The experimental results presented in this paper, allowing for the first time a direct comparison with theoretical models, show that the vorticity flux feeds into the shear flow in the tokamak plasma edge region. Th…
A model system of equations has been derived to describe a toroidally rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM model and a nonlinear NTM model. If no wall is present, the NTM growth shows the typical threshold/saturation island widths, whereas a linearly unstable kink mode grows exponentially in this model plasma system. When a resistive wall is present, the growth of the linearly unstable RWM is accelerated by an unstable island: a form of coupled RWM-NTM mode. Crucially, this coupled system has no…
Radio frequency waves are routinely used in tokamak fusion plasmas for plasma heating, current control, and as well as in diagnostics. These waves are excited by antenna structures placed near the tokamak's wall and they have to propagate through a turbulent layer known as scrape-off layer, before reaching the core plasma (which is their target). This layer exhibits coherent density fluctuations in the form of blobs and filaments. The scattering processes of RF plane waves by single blob is studied, with the assumption that the blob has cylindrical shape and infinite length. The axis of the bl…
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 identified. The in-out impurity density asymmetry can be induced by the Coriolis force with q*v_theta_z*omega_z, 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 proposed theory is consistent with the ASDEX Upgrade experimental observations.
Using theoretical arguments, a simple scaling law for the size of the intrinsic rotation observed in tokamaks in the absence of momentum injection is found: the velocity generated in the core of a tokamak must be proportional to the ion temperature difference in the core divided by the plasma current, independent of the size of the device. The constant of proportionality is of the order of $10\,\mathrm{km \cdot s^{-1} \cdot MA \cdot keV^{-1}}$. When the intrinsic rotation profile is hollow, i.e. it is counter-current in the core of the tokamak and co-current in the edge, the scaling law presen…
The effect of momentum injection on the temperature gradient in tokamak plasmas is studied. A plausible scenario for transitions to reduced transport regimes is proposed. The transition happens when there is sufficient momentum input so that the velocity shear can suppress or reduce the turbulence. However, it is possible to drive too much velocity shear and rekindle the turbulent transport. The optimal level of momentum injection is determined. The reduction in transport is maximized in the regions of low or zero magnetic shear.
Ion cyclotron emission (ICE) is detected during edge localised modes (ELMs) in the KSTAR tokamak at harmonics of the proton cyclotron frequency in the outer plasma edge. The emission typically chirps downward (occasionally upward) during ELM crashes, and is driven by confined 3MeV fusion-born protons that have large drift excursions from the plasma core. We exploit fully kinetic simulations at multiple plasma densities to match the time-evolving features of the chirping ICE. This yields a unique, very high time resolution diagnostic of the collapsing edge pedestal density.
The effect of magnetic perturbations (MPs) on the helical self-organization of shaped tokamak plasmas is discussed in the framework of the nonlinear 3D MHD model. Numerical simulations performed in toroidal geometry with the \textsc{pixie3d} code [L. Chacón, Phys. Plasmas {\bf 15}, 056103 (2008)] show that $n=1$ MPs significantly affect the spontaneous quasi-periodic sawtoothing activity of such plasmas. In particular, the mitigation of sawtooth oscillations is induced by $m/n=1/1$ and $2/1$ MPs. These numerical findings provide a confirmation of previous circular tokamak simulations, and are …
We analyze an axisymmetric equilibrium of a plasma endowed with toroidal and poloidal velocity fields, with the aim to characterize the influence of the global motion on the morphology of the magnetic confinement. We construct our configuration assuming that the poloidal velocity field is aligned with the poloidal magnetic field lines and, furthermore, we require that the plasma mass density depend on the magnetic flux function (or equivalently, that the plasma fluid be incompressible). We then derive a sort of Grad-Shafranov equation for such an equilibrium and implement it to tokamak relevan…
Tokamak plasma current start-up assisted by Electron Bernstein waves (EBW) has been demonstrated successfully in a number of experiments. The dynamic start-up phase involves a change in field topology, as the initially open magnetic field lines form closed flux surfaces (CFS) under the initiation of a plasma current. This change in field topology will bring about a change in the current drive (CD) mechanism, and, although various mechanisms have been proposed to explain the formation of CFS, no detailed theoretical studies have previously been undertaken. Here, we report on the development of …
In this paper we have studied the influence of reversed shear on the improvement of the confinement's quality in the plasma of tokamak and especially in reducing the anomalous transport. For that, we have used a special model for the drift wave fields. Comparison between particles trajectories for normal and reversed shear is carried out in 2 D and 3 D presentation. Also, the diffusion coefficient of particles for the two cases, normal and negative shear, is evaluated.
The narrow power decay-length ($λ_q$), recently found in the scrape-off layer (SOL) of inner-wall limited (IWL) discharges in tokamaks, is studied using 3D, flux-driven, global two-fluid turbulence simulations. The formation of the steep plasma profiles measured is found to arise due to radially sheared $\vec{E}\times\vec{B}$ poloidal flows. A complex interaction between sheared flows and outflowing plasma currents regulates the turbulent saturation, determining the transport levels. We quantify the effects of sheared flows, obtaining theoretical estimates in agreement with our non-linear simu…
The Landau damping of geodesic acoustic mode (GAM) in a torodial rotating tokamak plasma is analytically investigated by taking into account the finite-orbit-width (FOW) 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 $k_r ρ_i$. For sufficiently small $k_r ρ_i$, the damping rate monotonically decreases with $M$. For relatively large $k_r ρ_i$, the damping rate increases with $M$ until approaching the maximum and then decreases with $M$.
Neutral atoms can strongly influence the intrinsic rotation and radial electric field at the tokamak edge. Here, we present a framework to investigate these effects when the neutrals dominate the momentum transport. We explore the parameter space numerically, using highly flexible model geometries and a state of the art kinetic solver. We find that the most important parameters controlling the toroidal rotation and electric field are the major radius where the neutrals are localized and the plasma collisionality. This offers a means to influence the rotation and electric field by, for example,…
Boundary plasma physics plays an important role in tokamak confinement, but is difficult to simulate in a gyrokinetic code due to the scale-inseparable nonlocal multi-physics in magnetic separatrix and open magnetic field geometry. Neutral particles are also an important part of the boundary plasma physics. In the present paper, noble electrostatic gyrokinetic techniques to simulate the flux-driven, low-beta electrostatic boundary plasma is reported. Gyrokinetic ions and drift-kinetic electrons are utilized without scale-separation between the neoclassical and turbulence dynamics. It is found …
We present recent results and technical breakthroughs for the Bayesian inference of tokamak equilibria using force-balance as a prior constraint. Issues surrounding model parameter representation and posterior analysis are discussed and addressed. These points motivate the recent advancements embodied in the Bayesian Equilibrium Analysis and Simulation Tool (BEAST) software being presently utilised to study equilibria on the Mega-Ampere Spherical Tokamak (MAST) experiment in the UK (von Nessi \emph{et. al.} 2012 \emph{J. Phys. A} \textbf{46} 185501). State-of-the-art results of using BEAST to …
Recently, theoretical analysis has identified plasma microinstabilities as the primary mechanism responsible for anomalous heat transport in tokamaks. In particular, the microtearing mode (MTM) has been credited with the production of intense electron heat fluxes, most notably through a thin self-organized boundary layer called the pedestal. Here we exploit a novel, time-dependent analysis to compile explicit experimental evidence that MTMs are active in the pedestal region. The expected frequency of pedestal MTMs, calculated as a function of time from plasma profile measurements, is shown in …
Rotation of tokamak-plasmas, not at the mechanical equilibrium, is investigated utilizing a theorem of thermodynamics, established by Prigogine. This theorem, suitably applied to toroidally confined plasmas, suggests that the global barycentric rotations of the plasma, in the toroidal and poloidal directions, are pure reversible processes. In case of negligible viscosity and by supposing the validity of the balance equation for the internal forces, we show that the plasma, even not in the mechanical equilibrium, may freely rotate as a rigid body in the toroidal direction with an angular freque…
A spherical tokamak (ST) with a plasma center column (PCC) can be formed via driven magnetic relaxation of a screw pinch plasma. An ST-PCC could in principle eliminate many problems associated with a material center column, a key weakness of the ST reactor concept. This work summarizes the design space for an ST-PCC in terms of flux amplification, aspect ratio, and elongation, based on the zero-beta Taylor-relaxed analysis of Tang & Boozer [Phys. Plasmas 13, 042514 (2006)]. The paper will discuss (1) equilibrium and stability properties of the ST-PCC, (2) issues for an engineering design, and …
Fusion-graded plasmas are one of the physically complex systems, resulting in continuous establishment of plasma theories for unclarified physical phenomena in order to thoroughly control nuclear fusion reactors. Deep learning has drawn vast attention to this field of controlled fusion plasma to link physical phenomena with control-relevant parameters without a deepened understanding about plasma theories. Albeit, quantifying the uncertainty of deep learning models has been constantly requested due to their fundamental shortage of physical understanding. Thus, a concept of a reliable deep lear…
In this work, we use a passive advection model for ion temperature fluctuations, in order to investigate their effects on Doppler Spectral line shapes. The relevance of the model is discussed in the framework of the Braginskii equations, and the subsequent Probability Density Function evaluation relies on results obtained in neutral fluids. The resulting Doppler line profiles are shown to exhibit characteristic exponential tails.
A process specific to the dynamics of the heavy impurities in turbulent tokamak plasmas is found and analysed. We show that the parallel stochastic acceleration is strongly coupled to the perpendicular transport and generates a radial pinch velocity. The interaction is produced with the hidden drifts, a quasi-coherent component of the motion that consists of a pair of average radial velocities in opposite directions. The parallel acceleration breaks this symmetry and yields a radial average velocity that can be in the inward or outward direction. The transport of the tungsten ions in three-dim…
Externally applied non-axisymmetric magnetic fields such as error field and resonant magnetic perturbation (RMP) are known to influence the plasma momentum transport and flow evolution through plasma response in a tokamak, whereas the evolution of plasma response itself strongly depends on the plasma flow as well. The nonlinear interaction between the two have been captured in the conventional error field theory with a ``no-slip'' condition, which has been recently extended to allow the ``free-slip'' condition. For comparison with simulations, we solve for the nonlinear plasma response and flo…
We present particle-in-cell (PIC) simulations of minority energetic protons in deuterium plasmas, which demonstrate a collective instability responsible for emission near the lower hybrid frequency and its harmonics. The simulations capture the lower hybrid drift instability in a regime relevant to tokamak fusion plasmas, and show further that the excited electromagnetic fields collectively and collisionlessly couple free energy from the protons to directed electron motion. This results in an asymmetric tail antiparallel to the magnetic field. We focus on obliquely propagating modes under cond…
In tokamaks, internal transport barriers, produced by modifications of the plasma current profile, reduce particle transport and improve plasma confinement. The triggering of the internal transport barriers and their dependence on the plasma profiles is a key nonlinear dynamics problem still under investigation. We consider the onset of shearless invariant curves inside the plasma which create internal transport barriers. A non-integrable drift-kinetic model is used to describe particle transport driven by drift waves and to investigate these shearless barriers onset in tokamaks. We show that …
