Kink Stability Research Articles

High performance Joint European Torus (JET) plasmas for deuterium–tritium operation with the MkII divertor

Planned experiments in the Joint European Torus [Plasma Physics and Controlled Fusion Research, Proceedings, 13th International Conference, Washington, D.C., 1990 (International Atomic Energy Agency, Vienna, 1991), Vol. 1, p. 27] (JET) with deuterium–tritium (D–T) plasmas require high fusion performance for α-particle heating studies and for investigation of isotope dependence in conditions relevant to the International Thermonuclear Experimental Reactor [Plasma Phys. Controlled Fusion 37, A19 (1995)]. In deuterium plasmas, the highest neutron rates have been obtained in the hot-ion high-confinement mode (H mode) which is ultimately limited by magnetohydrodynamic (MHD) phenomena when the pressure gradient approaches ideal ballooning and kink stability limits in the vicinity of the edge transport barrier. Results are reported confirming the MkII divertor’s increased closure and pumping in this regime, progress in understanding the MHD-related termination is discussed, and the use of ion cyclotron resonance heating (ICRH) in combination with high-power neutral beams to increase the neutron yield is described. In separate experiments internal transport barriers have been established through careful programming of the current ramp and heating waveforms, and neutron emission comparable with the best hot-ion H-modes achieved. Steady-state H-mode discharges exhibiting edge localized modes (ELMs) in reactor-like configurations and conditions have been demonstrated, including cases in which relevant dimensionless parameter values are preserved, ready also for testing in D–T.

The Helical Kink Instability of Isolated, Twisted Magnetic Flux Tubes

view Abstract Citations (99) References (37) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Helical Kink Instability of Isolated, Twisted Magnetic Flux Tubes Linton, M. G. ; Longcope, D. W. ; Fisher, G. H. Abstract To understand the dynamics of twisted active region flux tubes below the solar photosphere, we investigate the linear kink stability of isolated, twisted tubes of magnetic flux. We apply linearized equations of MHD to a cylindrical magnetic equilibrium (screw pinch), but with significant differences from earlier work. The magnetic field vanishes outside a radius r = R where it is confined by the higher pressure of the unmagnetized plasma. The outside boundary of the tube is free to move, displacing the unmagnetized plasma as it does so. We concentrate on equilibria where all field lines have the same helical pitch: Bθ/rB&zeta; = q = const. The main results are as follows. 1. These equilibria are stable, provided that the field line pitch does not exceed a threshold; q ≤qcr for stability. The threshold is qcr=(α)½, where α is the r2 coefficient in the series expansion of the equilibrium axial magnetic field (Bζ) about the tube axis (r = 0): Bζ(r) = BO(1 - αr2 + ⋯). When this criterion is violated, there are unstable eigenmodes, ξ ∝ e1(θ+kz). The most unstable of these have a helical pitch k which is near (but not equal to) the field line pitch q. 2. For weakly twisted tubes (qR ≪ 1) we derive growth rates and unstable eigenfunctions analytically. For strongly twisted tubes (qR ≤1), we find growth rates and unstable eigenfunctions numerically. 3. The maximum growth rate and range of unstable wavenumbers for a strongly twisted tube can be predicted qualitatively by using the analytical results from the weakly twisted case. The maximum growth rate in that case is given by ωmax = υAR(q2 - q2cr)/3.83, where υA is the axial Alfvén speed. The range of unstable wavenumbers is (- q - Δk/2) < k <(- q + Δk/2), where Δk = 4qR(q2 -q2cr)½/3.83. 4. The kink instability we find consists mainly of internal motions. Helical translations of the entire tube are stable. 5. We argue that an emerging, twisted magnetic flux loop will tend to have a uniform q along its length. The increase in the tube radius R as it rises results in a decreasing value of qcr. This means that the apex of the flux loop will become kink unstable before the rest of the tube. 6. Our results suggest that most twisted flux tubes rising through the convection zone will be stable to kinking. Those few tubes which are kink unstable, and which presumably become knotted or kinked active regions upon emergence, only become kink unstable some time after they have begun rising through the convection zone. Publication: The Astrophysical Journal Pub Date: October 1996 DOI: 10.1086/177842 Bibcode: 1996ApJ...469..954L Keywords: INSTABILITIES; MAGNETOHYDRODYNAMICS: MHD; SUN: ACTIVITY; SUN: INTERIOR; SUN: MAGNETIC FIELDS full text sources ADS |

Magnetohydrodynamic kink stability of high pressure (εβpol≫1) toroidal plasmas

High pressure toroidal equilibria with εβpol≫1 are shown to be violently unstable to magnetohydrodynamic (MHD) kink modes. Here, ε is the inverse aspect ratio of a torus with a square conducting wall and βpol is the ratio of the plasma pressure to the poloidal magnetic field pressure. A broad spectrum of toroidal mode numbers is unstable at large εβpol. These modes grow rapidly, with a very short exponential growth time of the order of the Alfvén time. The growth rate of the kink mode is independent of the magnitude of the resistivity of the plasma.

Ponderomotive feedback stabilization of external kinks and disruptions in tokamaks

A novel mechanism for feedback stabilization of external kink modes and disruptions is proposed in which modulated radio-frequency (RF) antennas apply a stabilizing ponderomotive force (PF) to the plasma. The RF power required for n=1 kink stabilization in the Princeton Beta Experiment-Modified (PBX-M) [Phys. Fluids B 2, 1271 (1990)] is estimated, and it is shown that the idea can be tested with the existing antenna system. A simulation of a preliminary n=0 modulation experiment on PBX-M also shows that the PF can balance applied vertical field oscillations.

Effect of toroidal plasma flow and flow shear on global magnetohydrodynamic MHD modes

The effect of a subsonic toroidal flow on the linear magnetohydrodynamic stability of a tokamak plasma surrounded by an external resistive wall is studied. A complex non-self-adjoint eigenvalue problem for the stability of general kink and tearing modes is formulated, solved numerically, and applied to high β tokamaks. Results indicate that toroidal plasma flow, in conjunction with dissipation in the plasma, can open a window of stability for the position of the external wall. In this window, stable plasma beta values can significantly exceed those predicted by the Troyon scaling law with no wall. Computations utilizing experimental data indicate good agreement with observations.

A model for ideal m=1 internal kink stabilization by minority ion cyclotron resonant heating

A generalized energy principle is used to determine the effect of ion cyclotron resonant heating (ICRH) on the stability of m=1 internal kink displacements in the low-frequency limit: such displacements are associated with sawtooth oscillations. An integral expression is obtained for the contribution to the plasma energy of an ICRH-heated minority ion population with strong temperature anisotropy, which relates the former to the ICRH power input and its deposition profile. The link is provided by a realistic, but analytically tractable, new model for the distribution function of the heated ions, which is based on the approach of Stix [Nucl. Fusion 15, 737 (1975)]. Numerical evaluation of the integral expression is carried out using parameters inferred from ICRH experiments in the Joint European Torus (JET) [Campbell et al., Phys. Rev. Lett. 60, 2148 (1988)]. It is shown that the ideal m=1 internal kink is stable at values of the poloidal plasma beta βp which typically lie in the range 0.4–1, depending on the radio-frequency power input and the radius r1 of the q=1 surface. Stability is thus possible at values of βp lying significantly above the magnetohydrodynamic instability threshold (≂0.3). If the perpendicular temperature T⊥ of the hot ions exceeds the parallel temperature by a factor of 10 or more, and r1 is greater than about one-third of the plasma minor radius, trapped ions have a greater stabilizing effect than passing ions. Stabilization is most easily achieved, however, if r1 is small. The hot-ion plasma energy depends strongly on the value of T⊥, but for fixed T⊥ is insensitive to the degree of anisotropy.

Diagnostics for real-time plasma control in PBX-M

An important issue for future tokamaks is real-time plasma control for the avoidance of magnetohydrodynamic instabilities and other applications that require detailed plasma profile and fluctuation data. Although measurements from diagnostics providing this information require significantly more processing than magnetic flux data, recent advancements could make them practical for adjusting operational settings for plasma heating and current drive systems as well as field coil currents. On the Princeton Beta Experiment-Modification (PBX-M), the lower hybrid current drive phasing can be varied during a plasma shot using digitally programmable ferrite phase shifters, and neural beam functions can be fully computer controlled. PBX-M diagnostics that may be used for control purposes include motional Stark-effect polarimetry for magnetic field pitch angle profiles, soft x-ray arrays for plasma position control and the separation of βp from li, hard x-ray detectors for energetic electron distributions, a multichannel electron cyclotron emission radiometer for ballooning mode identification, and passive plate eddy current monitors for kink stabilization. We will describe the present status of these systems on PBX-M, and discuss their suitability for feedback applications.

Effect of a static external magnetic perturbation on resistive mode stability in tokamaks

The influence of a general static external magnetic perturbation on the stability of resistive modes in a tokamak plasma is examined. There are three main parts to this investigation. First, the vacuum perturbation is expanded as a set of well-behaved toroidal ring functions, and is, thereafter, specified by the coefficients of this expansion. Second, a dispersion relation is derived for resistive plasma instabilities in the presence of a general external perturbation, and finally, this dispersion relation is solved for the amplitudes of the tearing and twisting modes driven in the plasma by a specific perturbation. It is found that the amplitudes of driven tearing and twisting modes are negligible until a certain critical perturbation strength is exceeded. Only tearing modes are driven in low-β plasmas with εβp≪1. However, twisting modes may also be driven if εβp≳1. For error-field perturbations made up of a large number of different poloidal and toroidal harmonics the critical strength to drive locked modes has a ‘‘staircase’’ variation with edge-q, characterized by strong discontinuities as coupled rational surfaces enter or leave the plasma. For single harmonic perturbations, the variation with edge-q is far smoother. Both types of behavior have been observed experimentally. The critical perturbation strength is found to decrease strongly close to an ideal external kink stability boundary. This is also in agreement with experimental observations.

Kink stability, propagation, and length-scale competition in the periodically modulated sine-Gordon equation.

We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on this and related nonlinear problems in four directions. First, we present the results of a numerical simulation program that are not compatible with the existence of a radiative threshold predicted by earlier calculations. Second, we carry out a perturbative calculation that helps interpret those previous predictions, enabling us to understand in depth our numerical results. Third, we apply the collective coordinate formalism to this system and demonstrate numerically that it reproduces accurately the observed kink dynamics. Fourth, we report on the occurrence of length-scale competition in this system and show how it can be understood by means of linear stability analysis. Finally, we conclude by summarizing the general physical framework that arises from our study.

Transition to detonation in dynamic phase changes

Subsonically propagating phase boundaries (kinks) can be modelled by material discontinuities which satisfy integral conservation laws plus an additional jump condition governing the phase-change kinetics. The necessity of an additional jump condition distinguishes kinks from the conventional shocks which satisfy the Lax criterion. We study stability of kinks with respect to the breakup (splitting) into a sequence of waves. We assume that all conventional shocks are admissible and that admissible kinks are selected by a prescribed kinetic relation. As we show, regardless of a particular choice of the kinetic relation, sufficiently fast-phase boundaries are unstable. The mode of instability includes an emission of a centered Riemann wave followed by a sonic shock (Chapman-Jouguet type phase boundary).

Stability and evolution of the quiescent and travelling solitonic bubbles

We simulate numerically the dynamics of the bubble-like solitons of the cubic-quintic nonlinear Schrödinger equation. In agreement with earlier predictions, slow moving bubbles have been observed to be unstable, in contrast to rapid bubbles which stabilize above a certain critical velocity. The empirical formula for the critical velocity has been confirmed to a high degree of accuracy. Based on the choice of some critical perturbation, we derive an integral criterion for stability of kinks and bubbles of the general nonlinear Schrödinger equation which provides an explanation of the appearance of the critical velocity. Finally, we follow numerically the nonlinear evolution of the unstable bubbles in one, two and three dimensions.

Effects of a hollow current profile on the ideal MHD stability of high beta p plasmas in a tokamak

The effects of current profiles on the n = 1 internal kink mode, the infinite n ballooning mode and the infernal mode in a tokamak are analysed numerically. The stabilities of plasmas with a circular cross-section and a peaked pressure profile are investigated for three types of current profile, that is parabolic, flat and hollow with an inverted q profile. In a low beta state (ϵβp < 0.5), the influence of the current profile on the n = 1 internal kink stability is small. However, the infernal mode appears in plasmas with a hollow current profile, although it is stable in plasmas with a parabolic profile. In a high beta state (ϵβp > 0.5), the growth rate of the n = 1 internal kink mode for the plasma with the hollow current profile becomes larger, and the unstable region spreads to the region of higher qmin (qmin > 2). When the plasma has a more peaked pressure profile in the hollow part of the current profile, the unstable regions of the n = 1 internal kink mode and ballooning mode shrink, and the infernal mode disappears

Internal kink stabilization by high-energy ions with nonstandard orbits

A generalized energy principle that takes into account the nonstandard, potato-shaped particle orbits of high-energy ions in the central region of a tokamak is derived. It is shown that, in the limit of zero orbit width, this energy principle reduces to the one formulated by Van Dam et al. [Phys. Fluids 25, 1349 (1982)]. The modification of hot particle stabilization theory when such orbit effects are important is investigated. In particular, a model distribution function is chosen to describe high-energy trapped ions produced by ion cyclotron resonant frequency (ICRF) heating applied near the axis of a tokamak. Standard banana orbit theory predicts that, for fixed total stored energy of energetic particles peaked about the magnetic axis, the stabilizing influence on internal kink modes is inversely proportional to the spatial spread of the hot particles. However, this scaling saturates when the spatial spread of the distribution function approaches the width of a typical nonstandard orbit. Hence, ICRF heating is most efficient in producing stabilization when the heating zone is comparable to the orbit width, while the tendency to stabilize does not improve if the heating zone is narrower than the orbit width. Further, it is shown that, if particle orbits can extend close to the q=1 surface, the tendency for stabilization is inhibited.

Ideal MHD stability of internal kinks in circular and shaped tokamaks

Stability limits for the internal kink mode are calculated for tokamaks with different current profiles and plasma cross-sections using ideal magnetohydrodynamics (MHD). The maximum stable poloidal beta at the q=1 surface (βp) is sensitive to the current profile, but for circular cross-sections it is typically between 0.1 and 0.2. Large aspect ratio theory gives similar predictions when the appropriate boundary conditions are applied at the plasma-vacuum surface. It is found that the internal kink is significantly destabilized by ellipticity. For JET geometry, the ideal MHD limit in βp is typically between 0.03 and 0.1, and arbitrarily low limits can result if the shear is reduced at the q=1 surface. A large aspect ratio expansion of the Mercier criterion retaining the effects of ellipticity and triangularity is used to illustrate the destabilizing influence of ellipticity

Group theory and kink stability equations

Starting from Schrodinger equations withSU(2) group-theoretic potentials, we consider a general family of kinks labeled by two (half-)integers (l, n) with ¦n¦≤l. A particular choice ofn=0,l=L (L positive integer) leads to a generalL-family, whereL=1 corresponds to sine-Gordon theory, whileL=2 represents the (λφ4)1+1 model. The (λφ6)1+1 model can also be recovered withl=3/2,n=−1/2, a particular case of theories labeled byl andn such thatl-n=2 which possess simple kink solutions. We also discuss one-loop order corrections to the kink masses in supersymmetric versions of theL-family. As a byproduct, we obtain the SUSY renormalization of the so-calledγ parameter in sine-Gordon theory.

Plasma Dynamics, Particle Transport, and Internal Activity in the IGNITEX Experiment

Alpha healing, alpha containment, and alpha stabilization effects are studied in the fusion ignition experiment IGNITEX. The IGNITEX device offers the possibility of producing fusion-ignited plasmas with ohmic heating alone. It is shown here that operating regimes with high probability for ignition and simplicity of operation are possible in IGNITEX. Time-dependent simulations showing the ohmic heating and alpha healing coupling through the discharge are presented. The characteristics for alpha transport with magnetic field perturbations are analyzed in detail using Monte-Carlo techniques. The stability of internal MHD modes and the interaction with alpha particles in the inner confinement region of ignited plasmas is studied. Specifically, the stability of resistive internal kinks, ideal internal kinks, and fishbones is presented. It is shown that a quiescent regime of operation is possible during the ignited phase in IGNITEX. Because of its ample ignition margin, its high alpha containment, and the possibility of operation far from marginal stability and the possibility of ignition operation with quiescent regimes in the inner region of the plasma, it is concluded that the IGNITEX device can produce fusion-ignited plasmas in a simple manner.

Existence and stability of travelling waves in (1+1) dimensions

The authors study the existence and stability of kinks and bell-shaped solitary travelling waves in (1+1) dimensions. They prove that kinks are stable and bells are unstable; this yields a one-to-one correspondence between Liapunov, energetic and linear stability of travelling waves. Numerical examples are presented as an illustration of the above results.

Quenching of external kink modes by localized radio-frequency fields

The quenching of external kink modes by a spatially localized, externally applied surface force is studied in a low-beta cylindrical tokamak model. It is shown that m=2 modes may be stabilized by the application of a radio frequency (rf) field in the ion cyclotron range of frequencies (ICRF) excited by either a toroidal or a poloidal antenna structure surrounding the plasma. Whereas linear mode coupling induced by toroidal localization is negligible, that caused by poloidal localization may degrade the stabilization and, thus, require increased power levels for stability. The model is directly applicable to the ponderomotive force exerted by an evanescent slow wave in the ion cyclotron range of frequencies. Quantitative estimates are made for ponderomotive stabilization of kink modes by means of ion-Bernstein-wave (IBW) antennas. This method of stabilization has the advantage that the antenna system does not have to match the spatial or temporal structure of the unstable modes. For typical IBW antenna systems, it is found that the required rf fields for kink stabilization are reasonable if the current profile is not too flat.

Propagation and stability of kinks in driven and damped nonlinear klein-gordon chains1

We consider the propagation of kinks in an elastic chain in a bistable or multistable potential under the action of a driving force [M. Buttiker and H. Thomas,Phys. Rev. A 37:235 (1988)]. Each element of the chain is subject to a damping force proportional to its velocity. We show that both the propagation velocity of the kinks as a function of the driving field, and the kink width as a function of propagation velocity, are determined by characteristic functions which depend only on the form of the potential. These functions can be found by considering a single particle moving in the upside-down potential of the chain. The general properties of these functions are studied and illustrated by several examples. The stability of these driven kinks is discussed. Interestingly, we find in addition to the expected discrete localized eigenmodes a two-dimensional continuum of oscillatory modes with a localized envelope.

Kink dynamics in hydrogen-bonded solids

A certain class of kinks responsible for the proton transport in hydrogen-bonded solids is studied. It is shown that their energies are close to those of ionic defects and of transduction for ATP formation. These kinks are obtained as exact solutions of the equation of motion for the protonic dynamics in the case of one-dimensional chains of hydrogen bonds. The stability of kinks is proved.