Stability Of Tokamak Plasmas Research Articles

Unveiling the Significance of Correlations in K-Space and Configuration Space for Drift Wave Turbulence in Tokamaks

Turbulence and transport phenomena play a crucial role in the confinement and stability of tokamak plasmas. Turbulent fluctuations in certain physical quantities, such as density or temperature fluctuations, can have a wide range of spatial scales, and understanding their correlation length is important for predicting and controlling the behavior of the plasma. The correlation length in the radial direction is identified as the critical length in real space. The dynamics in real space are of significant interest because transport in configuration space is primarily focused on them. When investigating transport caused by the E×B drift, the correlation length in real space represents the size of E×B whirls. It was numerically discovered that in drift wave turbulence, this length is inversely proportional to the normalized mode number of the fastest growing mode relative to the drift frequency. Considerable time was required before a proper analytical derivation of this condition was accomplished. Therefore, a connection has been established between phenomena occurring in real space and those occurring in k-space. Although accompanied by a turbulent spectrum in k-space with a substantial width, transport in real space is uniquely determined by the correlation length, allowing for accurate transport calculations through the dynamics of a single mode. Naturally, the dynamics are subject to nonlinear effects, with resonance broadening in frequency being the most significant nonlinear effect. Thus, mode number space is once again involved. Resonance broadening leads to the detuning of waves from particles, permitting a fluid treatment. It should be emphasized that the consideration here involves the total electric field, including the induction part, which becomes particularly important at higher beta plasmas.

Effects of the parallel flow shear on the ITG-driven turbulent transport in tokamak plasmas

The impact of the parallel flow shear on the tokamak plasma stability and turbulent transport driven by the ion temperature gradient (ITG) modes is analyzed by means of local gyrokinetic numerical analyses. It is shown that the parallel flow shear increases the ITG growth rate in the linear regime, and induces a broadening and shift of the radial spectrum. Then, the different effects of the finite parallel shear on the ITG turbulence characteristics are deeply analyzed in the nonlinear regime. These studies highlight that a reduction of the thermal-ion turbulent heat flux is induced by a complex mechanism involving the nonlinear generation of an enhanced zonal flow activity. Indeed, the turbulent sources of the zonal flows are increased by the introduction of the finite parallel flow shear in the system, beneficially acting on the saturation level of the ITG turbulence. The study has been carried out for the Waltz standard case below the critical threshold of the destabilization of the parallel velocity gradient instability, and then generalized to a selected pulse of a recent JET scenario with substantial toroidal rotation in the edge plasma region. It is, thus, suggested that the investigated complex mechanism triggered by the finite parallel flow shear reducing the ITG turbulent heat fluxes could be complementary to the well-established perpendicular flow shear in a region with sufficiently large plasma toroidal rotation.

A generalised formulation of G-continuous Bezier elements applied to non-linear MHD simulations

The international tokamak ITER is progressing towards assembly completion and first-plasma operation, which will be a physics and engineering challenge for the fusion community. In the preparation for ITER experimental scenarios, non-linear MHD simulations are playing an essential role to actively understand and predict the behaviour and stability of tokamak plasmas in future fusion power plant. The development of MHD codes like JOREK is a key aspect of this research effort, and provides invaluable insight into the plasma stability and the control of global and localised plasma events, like Edge-Localised-Mode and disruptions. In this paper, we present an operational implementation of a new, generalised formulation of Bezier finite-elements applied to the JOREK code, a significant advancement from the previously G1-continuous bi-cubic Bezier elements. This new mathematical method enables any polynomial order of Bezier elements, with a guarantee of G-continuity at the level of (n−1)/2, for any odd n, where n is the order of the Bezier polynomials. The generalised method is defined, and a rigorous mathematical proof is provided for the G-continuity requirement. Key details on the code implementation are mentioned, together with a suite of tests to demonstrate the mathematical reliability of the finite-element method, as well as the practical usability for typical non-linear tokamak MHD simulations. A demonstration for a state-of-the-art simulation of an Edge-Localised-Mode instability in the future ITER tokamak, with realistic grid geometry, finalises the study.

Impact of magnetic X-points on the vertical stability of tokamak plasmas

The ideal-magneto-hydro-dynamic (MHD) theory of axisymmetric modes with toroidal mode number n = 0 in tokamak plasmas is developed. These modes are resonant at the magnetic X-points of the tokamak divertor separatrix. As a consequence, current sheets form along the separatrix, which profoundly affect the stability of vertical plasma displacements. In particular, current sheets at the magnetic separatrix lead to stabilization of n = 0 modes, at least on the ideal-MHD time scale, adding an important ingredient to the mechanism of passive feedback stabilization.

Stabilization of plasma vertical position of elongated tokamak using upper and lower triangular coils

Improvement of the vertical stability of tokamak plasma with only local coils above and below the plasma was observed for the first time. The local coils in triangular shape, named upper and lower triangular (ULT) coils, were installed in TOKASTAR-2, a tokamak-stellarator hybrid device at Nagoya University. The ULT coils generate the effective radial magnetic field BReff, which is expected to stabilize the plasma vertical position. The region of the plasma position where vertical displacement events (VDEs) did not occur was extended by applying the ULT magnetic field. Stabilization of the plasma vertical position was related to the ratio of the vertical-direction derivative of the effective radial field generated by the ULT coils, ∂BReff/∂Z, to that of the axisymmetric radial field, ∂BR/∂Z. Moreover, even when VDEs occurred in discharges with the ULT magnetic field, the VDE speed was mitigated by the effect of the ULT magnetic field; the VDE speed increased with ∂BR/∂Z, while decreased with ∂BReff/∂Z.

Peeling-ballooning stability of tokamak plasmas with applied 3D magnetic fields

The poloidal harmonics of the toroidal normal modes of an unstable axisymmetric tokamak plasma are employed as basis functions for the minimisation of the 3D energy functional. This approach presents a natural extension of the perturbative method considered in Anastopoulos Tzanis et al (2019 Nucl. Fusion 59 126028). This variational formulation is applied to the stability of tokamak plasmas subject to external non-axisymmetric magnetic fields. A comparison of the variational and perturbative methods shows that for D-shaped, high βN plasmas, the coupling of normal modes becomes strong at experimentally relevant applied 3D fields, leading to violation of the assumptions that justify a perturbative analysis. The variational analysis employed here addresses strong coupling, minimising energy with respect to both toroidal and poloidal Fourier coefficients. In general, it is observed that ballooning unstable modes are further destabilised by the applied 3D fields and field-aligned localisation of the perturbation takes place, as local ballooning theory suggests. For D-shaped high βN plasmas, relevant to experimental cases, it is observed that the existence of intermediate n unstable peeling-ballooning modes, where a maximum in the growth rate spectrum typically occurs, leads to a destabilising synergistic coupling that strongly degrades the stability of the 3D system.

Projected global stability of high beta MAST-U spherical tokamak plasmas

Assessment of the limits of stability of tokamak plasmas is key to operation in high fusion performance ranges without disruption of the plasma current. Projected equilibria have been generated for the MAST-U spherical tokamak experiment, an upgrade of the previous MAST device, in order to prepare for operation. These equilibria are scanned in pressure and current profiles, and assessed with the DCON and MARS-F stability codes to find the so-called ‘no-wall’ beta limit, above which resistive wall mode instabilities can be expected in the absence of other stabilising effects. The no-wall limit was generally found to increase as plasma internal inductance increased. The equilibria are also assessed for the ‘with-wall’ limit, theoretically the highest achievable performance point, again with the DCON and MARS-F codes, including different approximate axisymmetric walls, and with the VALEN code which includes a 3D model of the surrounding conducting structure. Similar limits were found, despite the difference between the 2D and 3D codes in the treatment of the wall. Conducting passive stabilisation plates, which were newly installed in MAST-U, are in a region of significant mode perturbation when the plasma βN is sufficiently high and eddy currents are driven in these structures. Due to the increased stabilising effect of the wall in MAST-U vs. MAST, a significant gap exists between the approximate no-wall limits of βN/li = 6.71 and 7.13, found from DCON and MARS-F respectively, and the with-wall limits of βN/li = 8.23 and 8.53 for the equilibrium profiles analysed in this study. This opens a region of high beta operating space in MAST-U for potentially stable operation if non-ideal effects or active control can stabilise the resistive wall mode.

Overview of spherical tokamak research in Japan

Nationally coordinated research on spherical tokamak is being conducted in Japan. Recent achievements include: (i) plasma current start-up and ramp-up without the use of the central solenoid by RF waves (in electron cyclotron and lower hybrid frequency ranges), (ii) plasma current start-up by AC Ohmic operation and by coaxial helicity injection, (iii) development of an advanced fuelling technique by compact toroid injection, (iv) ultra-long-pulse operation and particle control using a high temperature metal wall, (v) access to the ultra-high-β regime by high-power reconnection heating, and (vi) improvement of spherical tokamak plasma stability by externally applied helical field.

Linear stability of ideal MHD configurations. II. Results for stationary equilibrium configurations

In this paper, we continue exploring the consequences of the general equation of motion (EOM) governing all Lagrangian perturbations ξ about a time-dependent, ideal magnetohydrodynamic (MHD) configuration, which includes self-gravity, external gravity, pressure gradients, compressibility, inertial effects, and anisotropic Lorentz force. We here address the specific case of MHD stability for 3D stationary equilibria, where the perturbed EOM features a symmetric operator F and an antisymmetric Doppler-Coriolis operator v·∇. For this case, we state and prove the general properties for the solutions ξ of the governing dynamical system. For axisymmetric perturbations about axisymmetric equilibria with purely toroidal, or purely poloidal magnetic fields, specific stability theorems can be formulated. We derive a useful integral expression for the quadratic quantity given by the inner product ⟨ξ,F[ξ]⟩. For deriving stability statements on MHD states where self-gravity is involved as well, we provide an upper bound on the perturbed self-gravitational energy associated with the displacement ξ. The resulting expression elucidates the role of potentially stabilizing versus destabilizing contributions and shows the role of gravity, entropy gradients, velocity shear, currents, Lorentz forces, inertia, and pressure gradients in offering many routes to unstable behavior in flowing gases and plasmas. These have historically mostly been studied for static v=0 configurations, looking at stability of exactly force-balanced states, or by assuming stationarity similar to our approach here (i.e., ∂t≡0 for the state we perturb), but typically in combination with some reduced dimensionality on the configuration of interest (translational or axisymmetry). We show that in these limits, we find and generalize expressions well-known from, e.g., the study of ideal MHD stability of tokamak plasmas or from Schwarzschild's criteria controlling convection in hydrodynamic, (external) gravitating systems. When applied to stationary, axisymmetric configurations in motion, we can use our upper bound to derive a sufficient stability criterion for perturbations of arbitrary azimuthal mode number m used in ξ(r)=η(r,z) exp(imφ).

Calculation of Neutral Beam Injected Torque and Its Effective Tangency Major Radius for EAST**supported by National Natural Science Foundation of China (Nos. 11247302, 11175211, 11175208), the National Magnetic Confinement Fusion Science Program of China (Nos. 2013GB101001 and 2013GB112004) and International Science and Technology Cooperation Program of China (No. 2014DFG61950)

Toroidal rotation has been recognized to have significant effects on the transport and magnetohydrodynamic (MHD) stability of tokamak plasmas. Neutral beam injection (NBI) is the most effective rotation generation method on current tokamak devices. To estimate the effective injected torque of the first neutral beam injection system on EAST, a simplified analytic method was derived. Calculated beam torque values were validated by those obtained from the NUBEAM code simulation. According to the results, for the collisional torque, the effective tangential radius for torque deposition is close to the beam tangency major radius. However, due to the dielectric property of tokamak plasma, the equivalent tangency major radius of the torque is equal to the average major radius of the magnetic flux surface. The results will be useful for the research of toroidal momentum confinement and the experimental analysis of momentum transport related with NBI on EAST.

Numerical Simulations of the Radio-Frequency–driven Toroidal Current in Tokamaks

Radio-frequency (rf) waves are a powerful tool for improving the performance and stability of tokamak plasmas through heating and current drive mechanisms, allowing current density profile control and steady-state operation. From first principles, and taking advantage from the ordering between the various time and space scales, fast and powerful numerical tools have been developed to calculate the rf-driven current. The current drive problem in tokamaks is first introduced with the purpose of maintaining a steady-state self-organized toroidal magnetohydrodynamic equilibrium, such that a minimal amount of the fusion power has to be recycled to control the plasma current. The strict criterion that characterizes a steady-state discharge is derived from the response of the tokamak, considered as a transformer, and of the plasma, when an external source of current is applied. The calculation of a rf-driven source of current requires solving self-consistently a set of equations describing the dynamics of wave fields and charged particles in an inhomogeneous magnetized plasma. The range of applicability of these equations is discussed, as well as numerical methods developed to solve them, such as the ray-tracing code C3PO and the three-dimensional linearized relativistic bounce-averaged electron Fokker-Planck solver LUKE. Simulations of current drive by lower-hybrid waves are presented to illustrate the applications of our numerical tools. Current drive modeling includes the effect of electron density fluctuations at the plasma edge, and the case of electron cyclotron waves used for stabilization of the 3/2 neoclassical tearing modes in ITER is studied in detail. Finally, ongoing developments, including cross effects between momentum and configuration spaces, aiming at improving current drive calculations are discussed.

A fast, magnetics-free flux surface estimation and q-profile reconstruction algorithm for feedback control of plasma profiles

The flux surfaces' layout and the magnetic winding number q are important quantities for the performance and stability of tokamak plasmas. Normally, these quantities are iteratively derived by solving the plasma equilibrium for the poloidal and toroidal flux.In this work, a fast, non-iterative and magnetics-free numerical method is proposed to estimate the shape of the flux surfaces by an inward propagation of the plasma boundary shape, as can be determined for example by optical boundary reconstruction described in Hommen (2010 Rev. Sci. Instrum. 81 113504), toward the magnetic axis, as can be determined independently with the motional Stark effect (MSE) diagnostic. Flux surfaces are estimated for various plasma regimes in the ITER, JET and MAST tokamaks and are compared with results of CRONOS reconstructions and simulations, showing agreement to within 1% of the minor radius for almost all treated plasmas.The availability of the flux surface shapes combined with the pitch angles measured using MSE allow the reconstruction of the plasma q-profile, by evaluating the contour-integral over the flux surfaces of the magnetic field pitch angle. This method provides a direct and exact measure of the q-profile for arbitrary flux surface shapes, which does not rely on magnetic measurements. Results based on estimated flux surface shapes show agreement with CRONOS q-profiles of better than 10%. The impact of the shape of the flux surfaces on the q-profile, particularly the profiles of elongation and Shafranov shift, and offsets in plasma boundary and the magnetic axis are assessed.OFIT+ was conceived for real-time plasma profile control experiments and advanced tokamak operation, and provides quickly and reliably the mapping of actuators and sensors to the minor radius as well as the plasma q-profile, independent of magnetic measurements.

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Plasma inductance and stability metrics on Alcator C-Mod

The growth rate of the vertical instability of elongated plasmas increases with the internal inductance. An upper limit to the value of the internal inductance is found, based on analytic approximations, and validated through the analysis of a large database of Alcator C-Mod plasmas. The extrapolation to ITER suggests that low-current high-q95 plasmas could exceed the original design value of li and thus be more difficult to stabilize. Metrics to characterize the vertical stability of tokamak plasmas are discussed and calculated for typical and ITER-like C-Mod plasmas. These figures contribute to the assessment of the ITER vertical stabilization loop and the establishment of what performance margin is required.

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Stability of tokamak plasmas with internal transport barriers against high n ideal magnetohydrodynamic ballooning mode

A ballooning mode equation for tokamak plasma, with the toroidicity and the Shafranov shift effects included, is derived for a shift circular flux tokamak configuration. Using this equation, the stability of the plasma configuration with an internal transport barrier (ITB) against the high n (the toroidal mode number) ideal magnetohydrodynamic (MHD) ballooning mode is analysed. It is shown that both the toroidicity and the Shafranov shift effects are stabilizing. In the ITB region, these effects give rise to a low shear stable channel between the first and the second stability regions. Out of the ITB region towards the plasma edge, the stabilizing effect of the Shafranov shift causes the unstable zone to be significantly narrowed.

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Real-time measurement of toroidal rotation (abstract)

One of the important goals in Columbia’s HBT-EP tokamak program is the improvement in the stability of tokamak plasmas by controlling the bulk plasma flow relative to the conducting wall. The method for active plasma flow control in HBT-EP is the application of oscillating resonant magnetic perturbations to oppose the velocity of magnetic islands at the q=2 surface. Real time (10 kHz) feedback control without inserting a material probe necessitates the use of an optical toroidal rotation measurement whose data is available during the shot. This is being accomplished in a novel way by seeding the deuterium plasma with 5%–10% helium and measuring the Doppler shift of the chord-integrated emission of the He II (n=4→3) line at 4686 Å. Since the electron temperature is expected to be about 30 eV at the q=2 surface, helium is not fully stripped. The shift in wavelength is calculated by measuring the change in intensity as the line moves across the passband of an interference filter that varies linearly. Filters with less than 0.2% variation from a perfectly linear slope have been obtained. Fluctuations in the plasma emission are removed by having two detectors observe the same volume of plasma. This is achieved by splitting the optical view with a ‘Y’ composed of randomized optical fibers. One detector views the plasma through a filter whose passband has a negative slope and the other channel views it through a positive-slope filter. Systematic differences such as detector sensitivity, amplifier gain, fiber losses, etc. are compensated by normalizing each signal to the signal at a particular reference time. The ratio of signals at two different times does not depend on any detector or circuit characteristic that remains constant. The Doppler shift, relative to the reference time, is a function only of the slope of the filter’s transmission. The Doppler shift at the He II impurity emisson line is 0.25 Å for a toroidal rotation of 3 km/s, and the slope of the filter passband is 8%–10% per Å, resulting in a 4% variation in signal level relative to the other channel.

Observation of floating potential asymmetry in the edge plasma of the SINP tokamak

Edge plasma properties in a tokamak is an interesting subject of study from the view point of confinement and stability of tokamak plasma. The edge plasma of SINP-tokamak has been investigated using specially designed Langmuir probes. We have observed a poloidal asymmetry of floating potentials, particularly the top-bottom floating potential differences are quite noticeable, which in turn produces a vertical electric field (Ev). This Ev remains throughout the discharge but changes its direction at certain point of time which seems to depend on applied vertical magnetic field (Bv).

Chapter 3: MHD stability, operational limits and disruptions

The present physics understandings of magnetohydrodynamic (MHD)stability of tokamak plasmas, the threshold conditions for onset of MHD instability,and the resulting operational limits on attainable plasma pressure (beta limit) anddensity (density limit), and the consequences of plasma disruption and disruptionrelated effects are reviewed and assessed in the context of their application to afuture DT burning reactor prototype tokamak experiment such as ITER. The principalconsiderations covered within the MHD stability and beta limit assessments are (i)magnetostatic equilibrium, ideal MHD stability and the resulting ideal MHD beta limit;(ii) sawtooth oscillations and the coupling of sawtooth activity to other types of MHDinstability; (iii) neoclassical island resistive tearing modes and the correspondinglimits on beta and energy confinement; (iv) wall stabilization of ideal MHDinstabilities and resistive wall instabilities; (v) mode locking effects ofnon-axisymmetric error fields; (vi) edge localized MHD instabilities (ELMs, etc.); and(vii) MHD instabilities and beta/pressure gradient limits in plasmas with activelymodified current and magnetic shear profiles. The principal considerations coveredwithin the density limit assessments are (i) empirical density limits; (ii) edge powerbalance/radiative density limits in ohmic and L-mode plasmas; and (iii) edge parameterrelated density limits in H-mode plasmas. The principal considerations covered in thedisruption assessments are (i) disruption causes, frequency and MHD instability onset;(ii) disruption thermal and current quench characteristics; (iii) verticalinstabilities (VDEs), both before and after disruption, and plasma and in-vessel halocurrents; (iv) after disruption runaway electron formation, confinement and loss; (v)fast plasma shutdown (rapid externally initiated dissipation of plasma thermal andmagnetic energies); (vi) means for disruption avoidance and disruption effectmitigation; and (vii) `integrated' modelling of disruptions and fast shutdown and ofthe ensuing effects. In each instance, the presentation within a given topical areaprogresses from a summary of present experimental and theoretical understanding to howthis understanding projects or extrapolates to an ITER class reactor regime tokamak.Examples of extrapolations to the specific ITER design concept developed during thecourse of the ITER EDA are given, and assessments of the degree of adequacy of presentunderstanding are also provided. In areas where present understanding is identified tobe less than fully adequate, areas in which continuing or new research is needed areidentified.