Internal Kink Research Articles

Equilibrium properties at very low aspect ratio in the Pegasus toroidal experiment

Equilibrium reconstructions of low-aspect ratio (A < 1.3) discharges in the Pegasus toroidal experiment have been performed. Magnetic diagnostics are used for equilibrium constraint and a filament code is used to estimate the significant currents flowing in the vacuum vessel walls. This technique is able to fit the global plasma parameters of plasma current and major radius to within 5%, internal inductance to within 10% and plasma pressure to within 15% as determined by Monte Carlo estimation of the uncertainties in the fit parameters. Determination of the equilibrium properties of the plasma allows an understanding of the dynamics of internal tearing modes and external ideal kink modes that limit plasma performance. Internal tearing modes were found to degrade plasma confinement when rational surfaces are located in regions of low magnetic shear early in the discharge when temperature is lower and resistivity is higher. This confinement degradation limits the maximum achievable plasma current and pressure. Disruptions with precursors growing with a time of ∼90 µs have been found to be consistent with ideal external kink modes with a hybrid growth time.

Impact of velocity space distribution on hybrid kinetic-magnetohydrodynamic simulation of the (1,1) mode

Numeric studies of the impact of the velocity space distribution on the stabilization of (1,1) internal kink mode and excitation of the fishbone mode are performed with a hybrid kinetic-magnetohydrodynamic model. These simulations demonstrate an extension of the physics capabilities of NIMROD [C. R. Sovinec et al., J. Comput. Phys. 195, 355 (2004)], a three-dimensional extended magnetohydrodynamic (MHD) code, to include the kinetic effects of an energetic minority ion species. Kinetic effects are captured by a modification of the usual MHD momentum equation to include a pressure tensor calculated from the δf particle-in-cell method [S. E. Parker and W. W. Lee, Phys. Fluids B 5, 77 (1993)]. The particles are advanced in the self-consistent NIMROD fields. We outline the implementation and present simulation results of energetic minority ion stabilization of the (1,1) internal kink mode and excitation of the fishbone mode. A benchmark of the linear growth rate and real frequency is shown to agree well with another code. The impact of the details of the velocity space distribution is examined; particularly extending the velocity space cutoff of the simulation particles. Modestly increasing the cutoff strongly impacts the (1,1) mode. Numeric experiments are performed to study the impact of passing versus trapped particles. Observations of these numeric experiments suggest that assumptions of energetic particle effects should be re-examined.

Ideal internal kink modes in a differentially rotating cylindrical plasma

The Velikhov effect leading to magnetorotational instability (MRI) is incorporated into the theory of ideal internal kink modes in a differentially rotating cylindrical plasma column. It is shown that this effect can play a stabilizing role for suitably organized plasma rotation profiles, leading to suppression of MHD (magnetohydrodynamic) instabilities in magnetic confinement systems. The role of this effect in the problem of the Suydam and the m = 1 internal kink modes is elucidated, where m is the poloidal mode number.

The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas

The latest version of the XTOR code which solves a set of the extended magnetohydrodynamic (MHD) equations in toroidal geometry is presented. The numerical method is discussed with particular emphasis on critical issues leading to numerical stability and robustness. This includes the time advance algorithm, the choice of variables and the boundary conditions. The physics in the model includes resistive MHD, anisotropic thermal diffusion and some neoclassical effects. The time advance method used in XTOR is unconditionally stable for linear MHD. First, both the ideal and the resistive MHD parts of the equations are advanced semi-implicitly and then the thermal transport part full-implicitly, using sub-stepping [H. Lütjens, Comp. Phys. Commun. 164 (2004) 301]. The time steps are only weakly limited by the departure of the nonlinear MHD dynamics from the linear one and are automatically defined by a set of nonlinear stability criteria. The robustness of the method is illustrated by some numerically difficult simulations, i.e. sawtooth simulations, the nonlinear destabilization of ballooning instabilities by an internal kink, and the dynamics of a neoclassical tearing mode in International Thermonuclear Experimental Reactor (ITER) [R. Aymar, V.A. Chuyanov, M. Huguet, et al., Nucl. Fusion 41 (2001) 1301] like geometry about its nonlinear stability threshold.

Sawtooth control using off-axis NBI

It has previously been observed that neo-classical tearing modes can be triggered by long period sawteeth even at low values of β. Since fusion-born α particles are predicted to invoke long period sawteeth, mechanisms to control sawteeth are important for ITER. Here, sawtooth behaviour in off-axis neutral beam injection (NBI) heated plasmas in JET is presented. It is found that the off-axis NBI can be used to destabilize sawteeth even in discharges with concurrent stabilizing on-axis neutral beams. Drift kinetic modelling of such discharges qualitatively shows that the passing fast ions born due to the NBI determine the stability of the n = 1 internal kink mode, thought to be related to sawtooth oscillations. In particular, when the beam ions are deposited outside the q = 1 surface, they can result in more frequent sawteeth. It is proposed that off-axis neutral beams could be used as an actuator for sawtooth control in ITER. Provided that the q = 1 surface is sufficiently core localized, the strongly passing population induced by the negative-ion NBI planned for ITER may be able to significantly destabilize the sawteeth.

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.

Sawtooth stability in neutral beam heated plasmas in TEXTOR

The experimental sawtooth behaviour in neutral beam injection (NBI) heated plasmas in TEXTOR is described. It is found that the sawtooth period is minimized with a low NBI power oriented in the same direction as the plasma current. As the beam power is increased in the opposite direction to the plasma current, the sawtooth period increases to a maximum before it begins to shorten once more. Results from both magnetohydrodynamic stability modelling including toroidal flows and modelling of the kinetic effects of the fast ions resulting from NBI heating are also presented. This model combining the gyroscopic and kinetic effects upon the stability of the n = 1 internal kink mode—thought to be associated with sawtooth oscillations—qualitatively recovers the sawtooth behaviour exhibited in the experiment. It is proposed that the sawtooth period is minimized in the co-NBI direction at the point at which the stabilization of the kink mode due to rotation is weakest. This occurs when the plasma rotation induced by the NBI balances the intrinsic rotation of the plasma. The sawtooth behaviour in the counter-NBI regime is attributed to a subtle balance of the competing stabilization from the toroidal rotation and destabilization from the presence of energetic ions.

The physics of sawtooth stabilization

Long period sawteeth have been observed to result in low-β triggering of neo-classical tearing modes, which can significantly degrade plasma confinement. Consequently, a detailed physical understanding of sawtooth behaviour is critical, especially for ITER where fusion-born α particles are likely to lead to very long sawtooth periods. Many techniques have been developed to control, and in particular to destabilize the sawteeth. The application of counter-current neutral beam injection (NBI) in JET has resulted in shorter sawtooth periods than in Ohmic plasmas. This result has been explained because, firstly, the counter-passing fast ions give a destabilizing contribution to the n = 1 internal kink mode—which is accepted to be related to sawtooth oscillations—and secondly, the flow shear strongly influences the stabilizing trapped particles. A similar experimental result has been observed in counter-NBI heated plasmas in MAST. However, the strong toroidal flows in spherical tokamaks mean that the sawtooth behaviour is determined by the gyroscopic flow stabilization of the kink mode rather than kinetic effects. In NBI heated plasmas in smaller conventional aspect-ratio tokamaks, such as TEXTOR, the flow and kinetic effects compete to give different sawtooth behaviour. Other techniques applied to destabilize sawteeth are the application of electron cyclotron current drive (ECCD) or ion cyclotron resonance heating (ICRH). In JET, it has been observed that localized ICRH is able to destabilize sawteeth which were otherwise stabilized by a co-existing population of energetic trapped ions in the core. This is explained through the dual rôle of the ICRH in reducing the critical magnetic shear required to trigger a sawtooth crash, and the increase in the local magnetic shear which results from driving current near the q = 1 rational surface. Sawtooth control in ITER could be provided by a combination of ECCD and co-passing off-axis negative-NBI fast ions.

Resistive internal kink modes in a differentially rotating cylindrical plasma

The resistive internal kink modes in a differentially rotating cylindrical plasma column are studied. It is shown that the Velikhov effect, which causes the magnetorotational instability in astrophysics, contributes to the magnetic hill/well and thereby enhances or suppresses these modes, depending on the character of radial profile of the rotation frequency. It is pointed out that, in the case of unfavorable rotation frequency profile, such a rotation-induced magnetic hill can exceed the hill effect due to the plasma pressure gradient. Under this condition, there appears a new variety of resistive-interchange modes, which are referred to as rotational modes. On the other hand, for a favorable rotation frequency profile, the Velikhov effect suppresses the resistive-interchange modes. These results concern the m&amp;gt;1 modes, where m is the poloidal mode number. In the case of perturbations with m=1, the favorable rotation frequency profile leads to decreasing the growth rate of the reconnecting mode.

Electron fishbones: theory and experimental evidence

We discuss the processes underlying the excitation of fishbone-like internal kink instabilities driven by supra-thermal electrons generated experimentally by different means: electron cyclotron resonance heating (ECRH) and by lower hybrid (LH) power injection. The peculiarity and interest of exciting these electron fishbones by ECRH only or by LH only is also analysed. Not only is the mode stability explained, but also the transition between steady state nonlinear oscillations to bursting (almost regular) pulsations, as observed in FTU, is interpreted in terms of the LH power input. These results are directly relevant to the investigation of trapped alpha particle interactions with low-frequency MHD modes in burning plasmas: in fact, alpha particles in reactor relevant conditions are characterized by small dimensionless orbits, similarly to electrons; the trapped particle bounce averaged dynamics, meanwhile, depends on energy and not mass.

Safety factor corrections to the magnetohydrodynamic internal kink mode in a tokamak

It has long been acknowledged that the well known and frequently used stability criterion βp&amp;lt;0.3 for the toroidal ideal magnetohydrodynamic internal kink mode in a tokamak is inaccurate for an empirically relevant safety factor. The present paper outlines the severity of the usual approximation, and presents improved analytical approximations of the general solution in M. N. Bussac, R. Pellat, D. Edery, and J. L. Soulé, Phys. Rev. Lett. 35, 1638 (1975), thus providing new insights into the nature of the instability, together with simple formulas that can be incorporated into transport codes with sawtooth cycle algorithms.

Modeling sawtooth stabilization by energetic ions from neutral beam injection

Recent advances in modeling the effects of anisotropic energetic ion distributions have enabled the development of a complete coherent physics explanation of sawtooth stabilization in both conventional and spherical tokamaks. As an example, a complete model has been developed to explain the asymmetric stabilization of sawteeth with respect to neutral beam injection direction in the Joint European Torus. This asymmetric sawtooth stabilization [M. Nave et al., Phys. Plasmas 13, 014503 (2006)] arises because of both the destabilizing contribution from the counterpassing ions and the strong modification of the stabilizing contribution of the nonadiabatic trapped ions due to flow shear. The fast particle effects including pressure anisotropy, sheared flows, and the adiabatic response to the internal kink mode have been modeled in general toroidal geometry for the first time.

PHOENIX: MHD spectral code for rotating laboratory and gravitating astrophysical plasmas

The new PHOENIX code is discussed together with a sample of many new results that are obtained concerning magnetohydrodynamic (MHD) spectra of axisymmetric plasmas where flow and gravity are consistently taken into account. PHOENIX, developed from the CASTOR code [W. Kerner, J.P. Goedbloed, G.T.A. Huysmans, S. Poedts, E. Schwarz, J. Comput. Phys. 142 (1998) 271], incorporates purely toroidal, or both toroidal and poloidal flow and external gravitational fields to compute the entire ideal or resistive MHD spectrum for general tokamak or accretion disk configurations. These equilibria are computed by means of FINESSE [A.J.C. Beliën, M.A. Botchev, J.P. Goedbloed, B. van der Holst, R. Keppens, J. Comp. Physics 182 (2002) 91], which discriminates between the different elliptic flow regimes that may occur. PHOENIX makes use of a finite element method in combination with a spectral method for the discretization. This leads to a large generalized eigenvalue problem, which is solved by means of Jacobi–Davidson algorithm [G.L.G. Sleijpen, H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17 (1996) 401]. PHOENIX is compared with CASTOR, PEST-1 and ERATO for an internal mode of Soloviev equilibria. Furthermore, the resistive internal kink mode has been computed to demonstrate that the code can accurately handle small values for the resistivity. A new reference test case for a Soloviev-like equilibrium with toroidal flow shows that, on a particular unstable mode, the flow has a quantifiable stabilizing effect regardless of the direction of the flow. PHOENIX reproduces the Toroidal Flow induced Alfvén Eigenmode (TFAE, [B. van der Holst, A.J.C. Beliën, J.P. Goedbloed, Phys. Rev. Lett. 84 (2000) 2865]) where finite resistivity in combination with equilibrium flow effects causes resonant damping. Localized ideal gap modes are presented for tokamak plasmas with toroidal and poloidal flow. Finally, we demonstrate the ability to spectrally diagnose magnetized accretion disk equilibria where gravity acts together with either purely toroidal flow or both toroidal and poloidal flow. These cases show that the MHD continua can be unstable or overstable due to the presence of a gravitational field together with equilibrium flow-driven dynamics [J.P. Goedbloed, A.J.C. Beliën, B. van der Holst, R. Keppens, Phys. Plasmas 11 (2004) 28].

Effect of a noncircular plasma cross section on the ideal (1, 1) mode in low-shear tokamaks

The effect of a noncircular shape of the plasma cross section on the ideal magnetohydrodynamic (MHD) stability of the (m,n)=(1,1) internal kink, or quasi-interchange (QI), mode in toroidal plasmas, where the safety factor q is close to unity in a wide area in the plasma core, is analyzed. The analysis is based on previously derived perturbation expansions of the ideal MHD equations for the (1, 1) mode, in the inverse aspect ratio, and in the ellipticity and the triangularity of the plasma cross section. For equilibria having a parabolic pressure profile and a constant central q=q0, with 0&amp;lt;q0−1⪡1, a simple analytical formulation of the stability problem is obtained, and an expression for the eigenfunction in terms of a Bessel function is found. Furthermore, noncircularity of the cross section is found to induce a whole spectrum of (1, 1) ideal modes with eigenvalues approaching the continuum (ω∕ωA)2=(q0−1)2∕3 as the number of nodes in the radial direction goes to infinity. The QI mode is the most unstable mode in this class. If the pressure is low and/or the radius of the low-shear region is small, noncircularity has a small and, usually, destabilizing effect on the QI mode, whereas in plasmas with high pressure and a large radius of the low-shear region, the combined effects of vertical elongation and positive triangularity lead to a substantial improvement of the stability of this mode compared with the situation in plasmas with circular cross section.

Sawtooth oscillations in shaped plasmas

The role of interchange and internal kink modes in the sawtooth oscillations is explored by comparing bean- and oval-shaped plasmas. The n=1 instability that results in the collapse of the sawtooth has been identified as a quasi-interchange in the oval cases and the internal kink in the bean shape. The ion and electron temperature profiles are followed in detail through the sawtooth ramp. It is found that electron energy transport rates are very high in the oval and quite low in the bean shape. Ion energy confinement in the oval is excellent and the sawtooth amplitude (δT∕T) in the ion temperature is much larger than that of the electrons. The sawtooth amplitudes for ions and electrons are comparable in the bean shape. The measured q profiles in the bean and oval shapes are found to be consistent with neoclassical current diffusion of the toroidal current, and the observed differences in q largely result from the severe differences in electron energy transport. For both shapes the collapse flattens the q profile and after the collapse return to q0≳1. Recent results on intermediate shapes are reported. These shapes show that the electron energy transport improves gradually as the plasma triangularity is increased.

MHD stability of (2,1) tearing mode: an issue for the preforming phase of Tore Supra non-inductive discharges

The early phase of a tokamak plasma discharge can have a dramatic impact on the main heating phase. This has been a persistent problem for the development of the steady state, fully non-inductive scenario using lower hybrid current drive (LHCD) on Tore Supra. The present paper reports on recent experimental and numerical investigations showing that a tearing mode coupled to the internal kink grows on q = 2 in the ohmic phase when the total current is too low, due to the weakening of field line curvature stabilization. Then, the application of LHCD drives the island to a larger size and undermines the development of the non-inductive phase. Decreasing the edge safety factor or increasing the Lundquist number S is found to be beneficial in both the linear and non-linear MHD analyses. The experimental database, which allows covering the edge safety factor dependence, supports this interpretation.

Stability Properties of Magnetic Tower Jets

Stability properties of jets propagating in a gravitationally stratified background have been examined by performing three-dimensional magnetohydrodynamic simulations. The current-carrying, Poynting flux-dominated magnetic tower jet, which possesses a highly wound helical magnetic field, is subject to the current-driven instability (CDI). We find that, under general physical conditions including small perturbations in the initial background profiles, the propagating magnetic tower jets develop the nonaxisymmetric, m = 1 kink mode of the CDI. The kink mode grows on the local Alfven crossing timescale. In addition, two types of kink modes appear in the system. At the central region where external thermal pressure confinement is strong, only the internal kink mode is excited and will grow. A large distance away from the central region where the external thermal pressure becomes low, the external kink mode is observed. As a result, the exterior of magnetic tower jets will be deformed into a large-scale wiggled structure. We also discuss extensively the different physical processes that contribute to the overall stability properties of the magnetic tower jets. Specifically, when the jet propagates in an initially unperturbed background, we find that it can survive the kink mode beyond the point predicted by the well-known Kruskal-Shafranov (K-S) criterion. The stabilization in this case comes mainly from the dynamical relaxation of magnetic twists during the propagation of magnetic towers; the magnetic pitch is reduced, and the corresponding K-S critical wavelength becomes longer as the tower jet proceeds. Furthermore, we show that the pressure-driven and Kelvin-Helmholtz instabilities do not occur in the magnetic tower jets. This strongly suggests that the CDI is the primary reason for the wiggling structures in jets.

Dynamics of resistive double tearing modes with broad linear spectra

The nonlinear evolution of resistive double tearing modes (DTMs) with safety factor values q=1 and q=3 is studied with a reduced cylindrical model of a tokamak plasma. We focus on cases where the resonant surfaces are a small distance apart. Recent numerical studies have shown that in such configurations high-m modes are strongly unstable and may peak around m=mpeak∼10. In this paper, it is first demonstrated that this result agrees with existing linear theory for DTMs. Based on this theory, a semiempirical formula for the dependence of mpeak on the system parameters is proposed. Second, with the use of nonlinear simulations, it is shown that the presence of fast growing high-m modes leads to a rapid turbulent collapse in an annular region, where small magnetic island structures form. Furthermore, consideration is given to the evolution of low-m modes, in particular the global m=1 internal kink, which can undergo nonlinear driving through coupling to fast growing linear high-m DTMs. Factors influencing the details of the dynamics are discussed. These results may be relevant to the understanding of the magnetohydrodynamic activity near the minimum of q and may thus be of interest for studies on stability and confinement of advanced tokamaks.

Kinetic stability of the internal kink mode in ITER

The kinetic stability of the n=1, m=1 internal kink mode is analyzed for realistic equilibria typical of the standard operation scenario of ITER (the International Thermonuclear Experimental Reactor) [ITER Physics Basis Editors, Nucl. Fusion 39, 2137 (1999)]. The kinetic effects modify the inertia and the perturbed potential energy δW of the mode, the two key elements determining the mode stability. Numerical results are obtained for ITER-like equilibria with different q profiles. For moderate magnetic shear within the q=1 surface, the low frequency magnetohydrodynamic (MHD) branch is fully suppressed by the kinetic effects for the expected profiles and parameters up to twice the expected plasma β while the high frequency fishbone branch is found to be destabilized as the plasma β and the radius of the q=1 surface increase. The MHD branch can be destabilized at higher plasma β or larger radii of the q=1 surface only for q profiles with a low magnetic shear within the q=1 surface.

The effect of toroidal plasma rotation on sawteeth in MAST

Results from MAST concerning the effect of toroidal rotation driven by neutral beam injection (NBI) on sawteeth are presented. The sawtooth period is shown to increase as the co-NBI power, and thus the toroidal plasma rotation, is increased. As the counter-NBI power is increased, the sawtooth period decreases to a minimum that is shorter than in Ohmically heated plasmas, before lengthening at high toroidal flows. The sawtooth period reaches a minimum when the sawtooth precursor changes the direction of rotation. This can be interpreted as the counter-NBI inducing a toroidal flow which balances the intrinsic ion diamagnetic rotation of the plasma, at which point the mode precursor changes direction. Magnetohydrodynamic stability analyses of the ideal n = 1 internal kink mode with respect to toroidal rotation at finite ion diamagnetic frequency show good accordance with the experimental results. When MAST discharges are modelled at the experimental ion diamagnetic frequency, , the ideal n = 1 internal kink mode is stabilized completely by co-rotation approaching the ion sound speed. When the rotation is oriented in the counter- direction, the mode is initially driven more unstable, before being stabilized by high toroidal flows. Experimentally, the radial extent of the sawtooth crash, and hence the q = 1 surface, varies with respect to the sawtooth period. Stability analyses show that the marginally stable radial location of the q = 1 surface reaches a minimum at approximately the same counter-toroidal rotation as that which minimizes the sawtooth period experimentally. The magnitude of the toroidal rotation at the q = 1 surface determines the stabilizing effect of the toroidal flow upon the ideal n = 1 internal kink mode.