Magnetic Flutter Research Articles

Edge‐Localized Mode Control and Transport Generated by Externally Applied Magnetic Perturbations

AbstractThis article reviews the subject of edge localized mode (ELM) control using externally applied magnetic perturbations and proposes theoretical mechanisms that may be responsible for the induced transport changes. The first question that must be addressed is: what is the structure of magnetic field within the plasma? Although initial hypotheses focused on the possibility of the creation of a region of stochastic field lines at the tokamak edge, drift magnetohydrodynamics theory predicts that magnetic reconnection is strongly suppressed over the region of the pedestal with steep gradients and fast perpendicular rotation. Reconnection can only occur near the location where the perpendicular electron velocity vanishes, and hence the electron impedance nearly vanishes, or near the foot of the pedestal, where the plasma is sufficiently cold and resistive. The next question that must be addressed is: which processes are responsible for the observed transport changes, nonlinearity, turbulence, or stochasticity? Over the pedestal region where ions and electrons rotate in opposite directions relative to the perturbation, the quasilinear Lorentz force decelerates the electron fluid and accelerates the ion fluid. The quasilinear magnetic flutter flux is proportional to the force and produces an outward convective transport that can be significant. Over the pedestal region where the E × B flow and the electrons rotate in opposite directions relative to the perturbation, magnetic islands with a width on the order of the ion gyroradius can directly radiate drift waves. In addition, the combination of quasilinear electron transport and ion viscous transport can lead to a large net particle flux. Since there are many transport mechanisms that may be active simultaneously, it is important to determine which physical mechanisms are responsible for ELM control and to predict the scaling to future devices (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simulation of microtearing turbulence in national spherical torus experiment

Thermal energy confinement times in National Spherical Torus Experiment (NSTX) dimensionless parameter scans increase with decreasing collisionality. While ion thermal transport is neoclassical, the source of anomalous electron thermal transport in these discharges remains unclear, leading to considerable uncertainty when extrapolating to future spherical tokamak (ST) devices at much lower collisionality. Linear gyrokinetic simulations find microtearing modes to be unstable in high collisionality discharges. First non-linear gyrokinetic simulations of microtearing turbulence in NSTX show they can yield experimental levels of transport. Magnetic flutter is responsible for almost all the transport (∼98%), perturbed field line trajectories are globally stochastic, and a test particle stochastic transport model agrees to within 25% of the simulated transport. Most significantly, microtearing transport is predicted to increase with electron collisionality, consistent with the observed NSTX confinement scaling. While this suggests microtearing modes may be the source of electron thermal transport, the predictions are also very sensitive to electron temperature gradient, indicating the scaling of the instability threshold is important. In addition, microtearing turbulence is susceptible to suppression via sheared E × B flows as experimental values of E × B shear (comparable to the linear growth rates) dramatically reduce the transport below experimental values. Refinements in numerical resolution and physics model assumptions are expected to minimize the apparent discrepancy. In cases where the predicted transport is strong, calculations suggest that a proposed polarimetry diagnostic may be sensitive to the magnetic perturbations associated with the unique structure of microtearing turbulence.

Gyrokinetic simulations with external resonant magnetic perturbations: Island torque and nonambipolar transport with plasma rotation

Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. 186, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow jr, and the corresponding j→×B→ plasma torque (density) R[jrBp/c], induced by magnetic islands that break the toroidal symmetry of a tokamak. This extends the previous GYRO formulation for the transport of toroidal angular momentum (TAM) [R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)]. The focus is on electrostatic full torus radial slice simulations of externally induced q=m/n=6/3 islands with widths 5% of the minor radius or about 20 ion gyroradii. Up to moderately strong E×B rotation, the island torque scales with the radial electric field at the resonant surface Er, the island width w, and the intensity I of the high-n micro-turbulence, as ErwI. The radial current inside the island is carried (entirely in the n=3 component) and almost entirely by the ion E×B flux, since the electron E×B and magnetic flutter particle fluxes are cancelled. The net island torque is null at zero Er rather than at zero toroidal rotation. This means that while the expected magnetic braking of the toroidal plasma rotation occurs at strong co- and counter-current rotation, at null toroidal rotation, there is a small co-directed magnetic acceleration up to the small diamagnetic (ion pressure gradient driven) co-rotation corresponding to the zero Er and null torque. This could be called the residual stress from an externally induced island. At zero Er, the only effect is the expected partial flattening of the electron temperature gradient within the island. Finite-beta GYRO simulations demonstrate almost complete RMP field screening and n=3 mode unlocking at strong Er.

Theory of coupled whistler-electron temperature gradient mode in high beta plasma: Application to linear plasma device

This paper presents a theory of coupled whistler (W) and electron temperature gradient (ETG) mode using two-fluid model in high beta plasma. Non-adiabatic ion response, parallel magnetic field perturbation (δBz), perpendicular magnetic flutter (δB⊥), and electron collisions are included in the treatment of theory. A linear dispersion relation for whistler-electron temperature gradient (W-ETG) mode is derived. The numerical results obtained from this relation are compared with the experimental results observed in large volume plasma device (LVPD) [Awasthi et al., Phys. Plasma 17, 42109 (2010)]. The theory predicts that the instability grows only where the temperature gradient is finite and the density gradient flat. For the parameters of the experiment, theoretically estimated frequency and wave number of W-ETG mode match with the values corresponding to the peak in the power spectrum observed in LVPD. By using simple mixing length argument, estimated level of fluctuations of W-ETG mode is in the range of fluctuation level observed in LVPD.

Electromagnetic effects on trace impurity transport in tokamak plasmas

The impact of electromagnetic effects on the transport of light and heavy impurities in tokamak plasmas is investigated by means of an extensive set of linear gyrokinetic numerical calculations with the code GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and of analytical derivations with a fluid model. The impurity transport is studied by appropriately separating diffusive and convective contributions, and conditions of background microturbulence dominated by both ion temperature gradient (ITG) and trapped electron modes (TEMs) are analyzed. The dominant contribution from magnetic flutter transport turns out to be of pure convective type. However it remains small, below 10% with respect to the E×B transport. A significant impact on the impurity transport due to an increase in the plasma normalized pressure parameter β is observed in the case of ITG modes, while for TEM the overall effect remains weak. In realistic conditions of high β plasmas in the high confinement (H-) mode with dominant ITG turbulence, the impurity diffusivity is found to decrease with increasing β in qualitative agreement with recent observations in tokamaks. In contrast, in these conditions, the ratio of the total off-diagonal convective velocity to the diagonal diffusivity is not strongly affected by an increase in β, particularly at low impurity charge, due to a compensation between the different off-diagonal contributions.

Electromagnetic fluid drift turbulence in static ergodic magnetic fields

Numerical simulations of three-dimensional nonlinear electromagnetic fluid drift turbulence in a tokamak plasma with externally applied stochastic magnetic-field perturbations are presented. The contributions to the radial particle transport due to nonlinearities arising from E×B advection and magnetic flutter are investigated for perturbation fields of varying strengths in the cases of low and high collisionalities. The perturbation strength is varied to study the physics for Chirikov parameters above 1. In all the cases considered a significant increase of E×B transport is found. A static contribution in the density and velocity perturbations contributes significantly to the total radial E×B transport. For low collisionality, the external perturbation leads to enhanced density and velocity fluctuations over a broad range in the toroidal wave-number spectrum, resulting in an enhanced turbulent flux. For high collisionality, the density fluctuations stay roughly the same and the velocity fluctuations are increased in an intermediate range of the toroidal wave number spectrum, separated from the maximum of the density fluctuations, thus leaving the turbulent flux almost unchanged.

Effects of impurity seeding and charge non-neutrality on electromagnetic electron temperature gradient modes in a tokamak

A linear theory of toroidal electromagnetic electron temperature gradient (ETG) mode is reported. The effects such as Debye shielding, impurities, magnetic flutter perturbations δB⊥ and compressible parallel magnetic field perturbations δB‖ are included in a fluid model. An eigenvalue equation is derived and solved analytically in local and semilocal limits. In the nonlocal limit, the eigenvalue equations are solved numerically. A comparison is also made of the linear thresholds obtained from this simple fluid model with previous gyrokinetic simulations. It is shown that the simple fluid theory results compare well with the thresholds obtained from gyrokinetic simulations.

Effects of electromagnetic turbulence in the neoclassical Ohm’s law

An Ohm’s law for tokamak plasmas has been derived, which includes the effect of electromagnetic turbulence as well as the neoclassical conductivity and bootstrap current. The most important current-driving effects of the turbulence have been identified, expressions for the driven (dynamo) current have been derived and these have been evaluated using the GYRO electromagnetic turbulence code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)]. The most important current drive mechanism, the divergence of the radial flux of parallel electron momentum induced by magnetic flutter, drives a current density which has positive peaks on low order rational surfaces, with compensating negative dips nearby, thus driving zero total current. Another current drive mechanism, the beating of the parallel electric field fluctuations with the electron density fluctuations, drives a current density which is much smaller than that driven by the magnetic flutter mechanism, but could drive a nonzero total current.

Study of Type III ELMs in JET

This paper presents the results of JET experiments aimed at studying the operational space of plasmas with a Type III ELMy edge, in terms of both local and global plasma parameters. In JET, the Type III ELMy regime has a wide operational space in the pedestal ne – Te diagram, and Type III ELMs are observed in standard ELMy H-modes as well as in plasmas with an internal transport barrier (ITB). The transition from an H-mode with Type III ELMs to a steady state Type I ELMy H-mode requires a minimum loss power, PTypeI. PTypeI decreases with increasing plasma triangularity. In the pedestal ne – Te diagram, the critical pedestal temperature for the transition to Type I ELMs is found to be inversely proportional to the pedestal density (Tcrit ∝ 1/n) at a low density. In contrast, at a high density, Tcrit, does not depend strongly on density. In the density range where Tcrit ∝ 1/n, the critical power required for the transition to Type I ELMs decreases with increasing density. Experimental results are presented suggesting a common mechanism for Type III ELMs at low and high collisionality. A single model for the critical temperature for the transition from Type III to Type I ELMs, based on the resistive interchange instability with magnetic flutter, fits well the density and toroidal field dependence of the JET experimental data. On the other hand, this model fails to describe the variation of the Type III ne – Te operational space with isotopic mass and q95. Other results are instead suggestive of a different physics for Type III ELMs. At low collisionality, plasma current ramp experiments indicate a role of the edge current in determining the transition from Type III to Type I ELMs, while at high collisionality, a model based on resistive ballooning instability well reproduces, in term of a critical density, the experimentally observed q95 dependence of the transition from Type I to Type III ELMs. Experimental evidence common to Type III ELMs in standard ELMy H-modes and in plasmas with ITBs indicates that they are driven by the same instability.

Electromagnetic transport components and sheared flows in drift-Alfvén turbulence

Results from three-dimensional numerical simulations of drift-Alfvén turbulence in a toroidal geometry with sheared magnetic field are presented. The simulations show a relation between self-generated poloidal shear flows and magnetic field perturbations. For large values of the plasma β we observe an increase of the transport if the viscous damping of the self-generated shear flows is absent. This behavior is in contrast to the standard argument that sheared flows suppress turbulence and transport via a decorrelation mechanism. An explanation of this behavior in terms of the transport related to magnetic flutter is proposed. The characteristics of the E×B flux are investigated using probability density distribution functions (PDFs). Although they are not Gaussian, no signs of algebraic tails in the PDFs are observed. The PDFs of the pointwise transport are found to agree well with a folded Gaussian, while the PDFs of the spatially averaged transport are in good agreement with an extreme value distribution.

Physics of the L-H Transition and Type III-ELMs Phenomena (Scaling Properties and Dimensionless Analysis)

The stability theory of Alfven drift-waves shows that with increasing plasma pressure electron drift waves become strongly coupled with Alfven waves thus damping radial perturbations. The Alfven drift turbulence suppression at the plasma edge was suggested as a triggering mechanism for the L to H transition [1]. After the transition the resistive interchange in-stability emerges as a dominant phenomenon which reveals itself in the form of Type III ELMs. Magnetic flutter provides the resistive driver for the interchange instability and, together with the stability condition caused by the electric shear stabilisation, creates the right dependence of the marginal temperature on density and other plasma parameters seen in experiments.

Performance near operational boundaries

The performance of ELMy H-mode operation in ASDEX Upgrade andJET is compared. Special attention is paid to variations (usuallyreductions) in this performance near the operational limits whichwill need to be approached in a next-step device. In JET it isfound that input powers substantially above the H-mode thresholdpower are required to obtain discharges with energy confinementenhancement factors at or above the usual ELMy H-mode scalings.Such a margin (as much as a factor of two in JET) is not observedin ASDEX Upgrade. It is proposed that this difference may be dueto the higher edge collisionality in ASDEX and the results arecompared to a recent theory based on interchange instabilitiesand magnetic flutter. In ASDEX Upgrade, the confinement in type IELMy discharges degrades as the density is raised due to astiffness of the temperature profiles which leads to adegradation of the core confinement. This type of stiffness isobserved in JET only at relatively high edge densities. In JET, theedge confinement degrades as the density is increased by externalgas fuelling, consistent with a constant edge pressure gradientand an edge barrier width which reduces in proportion to the edgeion poloidal Larmor radius. In both machines, H-mode performanceis limited at high density by a transition first to the type III ELMregime and then to the L-mode. The confinement penalty, relativeto good type I ELM discharges, of operating with type III ELMs isabout 25-30%. The maximum densities for operation withtype I or type III ELMs can be substantially increased byincreasing the plasma triangularity in both machines.

Three-dimensional computation of drift Alfvén turbulence

A transcollisional, electromagnetic fluid model, incorporating the parallel heat flux as a dependent variable, is constructed to treat electron drift turbulence in the regime of tokamak edge plasmas at the L - H transition. The resulting turbulence is very sensitive to the plasma beta throughout this regime, with the scaling with rising beta produced by the effect of magnetic induction to slow the Alfvénic parallel electron dynamics and thereby leave the turbulence in a more robust, non-adiabatic state. Magnetic flutter and curvature have a minor qualitative effect on the turbulence mode structure and on the beta scaling, even when their quantitative effect is strong. Transport by magnetic flutter is small compared to that by the flow eddies. Fluctuation statistics show that while the turbulence shows no coherent structure, it is coupled strongly enough so that neither density nor temperature fluctuations behave as passive scalars. Both profile gradients drive the turbulence, with the total thermal energy transport varying only weakly with the gradient ratio, . Scaling with magnetic shear is pronounced, with stronger shear leading to lower drive levels. Scaling with either collision frequency or magnetic curvature is weak, consistent with their weak qualitative effect. The result is that electron drift turbulence at L - H transition edge parameters is drift Alfvén turbulence, with both ballooning and resistivity in a clear secondary role. The contents of the drift Alfvén model will form a significant part of any useful first-principles computation of tokamak edge turbulence.

Effective plasma heat conductivity in 'braided' magnetic field-I. Quasi-linear limit

Anomalous cross-field electron transport in a specified magnetic field with weakly-destroyed flux surfaces is discussed. Following the approach developed previously in the present paper the regimes of effective transverse plasma transport are studied systematically, both in the collisional and collisionless limits. The present analysis incorporates the nonstationary of magnetic perturbations, which was not included in the previous works. Part I of this study deals with the quasi-linear approximation, which may be written as b0L0/ delta <<1, where b0= delta Bperpendicular to /B0 represents the relative magnitude of the transverse magnetic perturbation, while L0 and delta represent the longitudinal and transverse correlation lengths, respectively. It is found that some of the previously-described transport regimes cannot be considered as anomalous (in the sense that effective heat conduction chi eff>> chi perpendicular to ) for time-independent magnetic perturbations. However, these regimes can exist in the presence a finite-frequency magnetic flutter. A unified classification of quasi-linear regimes of anomalous transport is introduced in order to further extend the analysis to the strong magnetic turbulence limit b0L0/ delta >>1 (see Part II), which is considered as a companion paper. (Isichenko, ibid., vol.33, no.7, p.809-26 (1991)).

Theory of neoclassical pressure-gradient-driven turbulence

The nonlinear evolution and saturation of neoclassical pressure-gradient-driven turbulence (NPGDT), evolving from linearly unstable bootstrap-current modes, are investigated. The theoretical model is based on ‘‘neoclassical magnetohydrodynamic (MHD) equations’’ that are valid in the banana-plateau regimes of collisionality. Modes with poloidal wavelengths shorter than radial wavelengths are shown to be suppressed. From nonlinear saturation conditions, the turbulent pressure diffusivity is determined as an eigenvalue of the renormalized equations. Levels and radial scales of turbulence are determined from the pressure diffusivity and are shown to exceed mixing-length estimates by powers of a nonlinear enhancement factor. The problem of the electron heat transport resulting from stochastic magnetic fields driven by NPGDT is revisited. The reconsideration of the radial structure of magnetic flutter leads to estimates of the electron heat transport and magnetic fluctuation levels that differ qualitatively and quantitatively from previous calculations.

Magnetic fluctuations can contribute to plasma transport, ‘‘self-consistency constraints’’ notwithstanding

The recent conclusion of Terry, Diamond, and Hahm (TDH) [Phys. Rev. Lett. 57, 1899 (1986)] that in a turbulent, collisionless plasma ‘‘magnetic transport including quasilinear magnetic flutter transport . . . does not contribute to the relaxation of 〈f〉, and thus is not responsible for electron energy or momentum transport’’ is argued to be irrelevant or incorrect for a variety of situations of physical interest, including saturation by quasilinear plateau formation, induced scattering, and, most important, conventional mode coupling. The well-established theory of the mean infinitesimal response function and the spectral balance equation provides a unifying framework for understanding the work of TDH. In particular, the cancellations which lead to the TDH conclusion are special cases of well-known relationships between the response functionparticle propagator, and dielectric function. A more general, concise, and manifestly gauge-invariant algebraic derivation of the cancellations is given. Although the cancellations occur in a certain limit, the conclusions of TDH do not follow in general: The TDH picture of steady-state turbulence as consisting of small-scale ‘‘incoherent’’ ballistic ‘‘clumps’’ shielded by long-wavelength ‘‘coherent’’ dielectric response is misleading physically and incomplete mathematically since it does not describe correctly the often dominant process of renormalized n-wave coupling, particularly important for the ions. Although ion ballistic response is negligible, ions are important nevertheless: Their nonlinear contribution to the saturated potential can drive parallel electron currents, hence magnetic fluctuations, through linear mechanisms. Thus, when ion nonlinearities are considered, formulas for the magnetic contribution to transport emerge which are quite similar to the quasilinear formula.

Theory of trapped-particle-induced resistive fluid turbulence

A theory of anomalous electron heat transport, evolving from trapped-particle-induced resistive interchange modes, is proposed. The latter are a new branch of the resistive interchange-ballooning family of instabilities, destabilized when the pressure carried by the unfavorably drifting trapped particles is sufficiently large to overcome stabilizing contributions coming from favorable average curvature. Expressions for the turbulent heat diffusivity and anomalous electron thermal conductivity at saturation are derived for two regimes of trapped-particle energy: (i) a moderately energetic regime, which is ‘‘fluidlike’’ in the sense that the unstable mode grows faster than the time that it takes for particles in this energy range to precess once around the torus, and (ii) a highly energetic regime, where the trapped species has sufficiently high energy as to be able to interact resonantly with the mode. Unlike previous theories of anomalous transport, the estimates of diffusion and transport obtained here are self-consistent since the trapped particles do not ‘‘see’’ the magnetic flutter due to their rapid bounce motion. The theory is valid for moderate electron-temperature, high ion-temperature (auxiliary heated) plasmas and as such, is relevant for present- and future-generation experimental fusion devices.

Numerical simulation of electromagnetic turbulence in tokamaks

Nonlinear two- and three-fluid equations are written for the time evolution of the perturbed electrostatic potential, densities, vector potential, and parallel ion motion of collisional and trapped electron plasmas in tokamak geometry. The nonlinear terms arise from the Ẽ×B0 convection (d̃/dt=∂/∂t+ṽE ⋅ ∇⊥) and magnetic flutter [∇̃∥=∇∥+(B⊥/B0) ⋅ ∇⊥]. Simplified two-dimensional (k⊥) mode coupling simulations with a fixed average parallel wavenumber (k∥=1/Rq) and curvature drift [ωg=(Ln/R)ω* ] characteristic of outward ballooning are performed. Homogeneous stationary turbulent states of the dissipative drift and interchange modes from 0≤β&amp;lt;βcrit for both the collisional and trapped electron plasmas are obtained. Transport coefficients associated with E×B and magnetic motions are calculated. The problem of simulating plasmas with high viscous Reynolds number is treated with an absorbing mantle at the largest wavenumbers. The results are summarized by comparison to simple mixing length rules: ñ/n0∼1/k⊥Ln, B̃/B0 ∼(β/2) ⋅ q ⋅ ñ/n0, DE∼γ/k2⊥.

EFFECTS OF RESISTIVE INTERCHANGE INSTABILITIES ON ENERGY CONFINEMENT IN REVERSED-FIELD PINCH

Electron conduction losses due to magnetic flutter produced by resistive interchange instabilities and the resulted confinement deterioration mechanism are investigated analytically. Using approximate solutions of MHD equations for even and odd potential parities, the potential and magnetic perturbation levels at saturation are estimated. These results are used to calculate the stochastic magnetic field diffusion coefficients. An expression for the anomalous electron thermal conductivity is then derived for collision-less and collisional regimes. Scaling laws for energy confinement are infered therefrom.

Status of the Nevis Synchrocyclotron Modification

After 20 years of operation as a conventional synchrocyclotron producing 0.4 μA time average internal beam current of protons (increased to 1.6 μA in 1967) of ~ 385 MeV, we stopped operation in September 1970 for the major modification for which we have been planning since 1965. The major study program and the conversion project are supported by a $ 3.9 M grant from the NSF. The conversion retains the basic cyclotron magnet (2000 tons of Fe and ~ 300 tons of Cu coils), but adds a 10-in. thick Fe band around the outside magnet perimeter to lower the return path reluctance. A new larger vacuum chamber is used, with auxilliary excitation coils. Pole iron within 30 in. of the median plane is replaced by a new configuration which includes N=3 symmetry sector iron with a small median plane gap. <B xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> = 18 kG near the center and 20 kG near 80 in. radius. Strong azimuthal magnetic field flutter begins at &lt; 2 in. radius to give strong focusing νz, and νr∦1. We expect to obtain ~ 550 MeV proton energy, with high extraction efficiency for a 10 to 40 μA time average external beam (~ 50% duty factor), when operation starts later this year. Additional details are given in companion papers in these Proceedings. The system will still be a synchrocyclotron and will have a 300 Hz FM repetition rate, with a new RF system, etc.</B>