gyro-Bohm Transport Research Articles

Required toroidal confinement for fusion and omnigeneity

Deuterium–tritium (DT) burning requires a long energy confinement times compared to collision times, so the particle distribution functions must approximate local Maxwellians. Non-equilibrium thermodynamics is applicable, which gives relations among transport, entropy production, the collision frequency, and the deviation from a Maxwellian. The distribution functions are given by the Fokker–Planck equation, which is an advection–diffusion equation. A large hyperbolic operator, the Vlasov operator with the particle trajectories as its characteristics, equals a small diffusive operator, the collision operator. The collisionless particle trajectories would be chaotic in stellarators without careful optimization. This would lead to rapid entropy production and transport—far beyond what is consistent with a self-sustaining DT burn. Omnigeneity is the weakest general condition that is consistent with a sufficiently small entropy production associated with the thermal particle trajectories. Omnigeneity requires that the contours of constant magnetic field strength be unbounded in at least one of the two angular coordinates in magnetic surfaces and that there be a symmetry in the field-strength wells along the field lines. Even in omnigenous plasmas, fluctuations due to microturbulence can produce chaotic particle trajectories and the gyro-Bohm transport is seen in many stellarator and tokamak experiments. The higher the plasma temperature above 10 keV, the smaller the transport must be compared to gyro-Bohm for a self-sustaining DT burn. The hot alphas of DT fusion heat the electrons. When the ion–electron equilibration time is long compared to the ion energy confinement time, a self-sustaining DT burn is not possible, which sets a limit on the electron temperature.

Mesoscopic Transport Events and the Breakdown of Fick’s Law for Turbulent Fluxes

This paper presents a pedagogical review of the physics of mesoscopic transport events and their role in the breakdown of Fick’s Law for turbulent transport in magnetically confined plasma. It is now clear that the conventional picture of localized turbulence and quasi-linear calculation of fluxes fails to address and account for the phenomenology of tokamak transport. One key issue is the observed departure from the expected gyro-Bohm transport scaling. The causes of this breakdown of Fickian thinking include turbulent avalanching and pulse propagation (turbulence spreading). Both are mesoscopic transport events, and both tend to de-localize the flux–gradient relation. Turbulence spreading is the process of self-scattering and expansion of a slug or other local exciton of turbulence. Spreading is described by theoretically-motivated, phenomenological reaction–diffusion models for the turbulence activity (intensity) field, much in the spirit of Ginzburg–Landau theory. Such models imply that spreading will occur by propagation of intensity fronts. After discussing the basic theory, this paper presents several critical tests of turbulence spreading models using gyrokinetic simulation. Applications include rho-star scaling, penetration of transport barriers and core-edge coupling. Relevant experiment–theory comparisons are addressed, as well. Avalanching refers to a process whereby correlated topplings of nearby localized cells overturn sequentially and drive a burst of transport. Avalanching is a process intrinsic to systems that support a broad range of scales l between a cell size Δ and system size L, i.e. Δ < l < L. Avalanching is also a natural way to produce transport events on scales that exceed the cell size or correlation length. Therefore, the PDF (probability distribution function) of avalanches as a function of l is a crucial quantity, necessary for predicting confinement in a system like ITER, with a very large-scale separation between L and Δ. Avalanching emerged from the theory of selforganized criticality but is a more general phenomenon. The paper traces the intellectual prehistory of avalanching through the advent of self-organized criticality. Special focus is devoted to reduced continuum models of avalanching. The physics of avalanching in confined plasma is discussed in detail, via several multi-faceted comparisons to flux-driven fluid and gyrokinetic simulations. The dominance of bursty, large transport events in the flux is identified. Evidence for avalanching in basic and confinement experiments is summarized. The paper concludes with sections on selected special topics, a discussion of the relation between turbulence spreading and avalanching, and a list of possible future directions. Throughout the paper, an effort is made to set fusion theory and phenomenology in the context of ideas discussed in the broader scientific community.

Transport of radial heat flux and second sound in fusion plasmas

Simple flux-gradient relations that involve time delay and radial coupling are discussed. Such a formulation leads to a rather simple description of avalanches and may explain breaking of gyroBohm transport scaling. The generalization of the flux-gradient relation (i.e., constitutive relation), which involve both time delay and spatial coupling, is derived from drift-kinetic equation, leading to kinetic definitions of constitutive elements such as the flux of radial heat flux. This allows numerical simulations to compute these cubic quantities directly. The formulation introduced here can be viewed as an extension of turbulence spreading to include the effect of spreading of cross-phase as well as turbulence intensity, combined in such a way to give the flux. The link between turbulence spreading and entropy production is highlighted. An extension of this formulation to general quasi-linear theory for the distribution function in the phase space of radial position and parallel velocity is also discussed.

JET and the Physics Basis of ITER

JET has made unique contributions to the physics basis of ITER by virtue of its ITER-like geometry, large plasma size and D-T capability. The paper discusses recent JET results and their implications for ITER in the areas of standard ELMy H-mode, D-T operation and advanced tokamak modes. In ELMy H-mode the separation of plasma energy into core and pedestal contributions shows that core confinement scales like gyroBohm transport. High triangularity has a beneficial effect on confinement and leads to an integrated plasma performance exceeding the ITER Q = 10 reference case. A revised type I ELM scaling predicts acceptable ELM energy losses for ITER, while progress in physics understanding of NTMs shows how to control them in ITER. The D-T experiments of 1997 have validated ICRF scenarios for heating ITER/a reactor and identified ion minority schemes (e.g. (3He)DT) with strong ion heating. They also show that the slowing down of alpha particles is classical so that the self-heating by fusion alphas should cause no unexpected problems. With the Pellet Enhanced Performance mode of 1988, JET has produced the first advanced tokamak mode, with peaked pressure profiles sustained by reversed magnetic shear and strongly reduced transport. More recently, LHCD has provided easy tuning of reversed shear and reliable access to ITBs. Improved physics understanding shows that rational q-surfaces play a key role in the formation and development of ITBs. The demonstration of real time feedback control of plasma current and pressure profiles opens the path towards fully controlled steady-state tokamak plasmas.

Effect of isotope mass on transport simulations of Joint European Torus high-mode plasmas with Edge Localized Modes

The effect of isotopic mass on heat and particle transport in Joint European Torus (JET) [P.-H. Rebut et al., Nucl. Fusion 25, 1011 (1985)] plasma discharges is studied using the Multi-Mode model in the BALDUR predictive transport code [Bateman et al., Phys. Plasmas 5, 1793 (1998)]. Temperature and density profiles from these simulations generally agree with the experimentally measured profiles for high-mode JET discharges with Edge Localized Modes in hydrogen, deuterium, and tritium discharges. It is surprising that a purely gyro-Bohm transport model, used in these simulations, correctly predicts the experimentally observed improvement in confinement as the isotope mass is increased—given the fact that gyro-Bohm diffusion coefficients increase with isotope mass when the shapes of all the plasma profiles are held fixed. However, in the JET experiment, it was found that the electron and ion temperature at the top of the edge pedestal increases systematically as the isotope mass in increased (J. G. Cordey et al., Report No. JET-P (98)53, 1998). The numerical simulations reported here show that this increase in the edge temperatures and subsequent broadening of the temperature profiles account for the improvement in confinement as the isotope mass is increased.

Time dependent transport simulations of JET H mode plasmas

A drift wave based transport model is used to self-consistently predict the time evolution of temperature and density profiles in JET H mode tokamak discharges. It is found that the same theoretically derived gyro-Bohm transport model previously used to simulate systematic scans of L mode discharges is equally successful in modelling JET ELMy H mode plasmas, implying that core transport is not intrinsically different from L mode confinement. The only difference between the L mode and H mode simulations results from the boundary conditions (i.e. density and temperature pedestals), which are taken from experimental data in both cases. Here, standardized experimental data from 16 JET H mode discharges in the ITER Profile Database are used, including dimensionless parameter scans in relative gyro-radius ρ*, collisionality ν and plasma β. Imperfections in dimensionless similarity for three pairs of scans in relative gyro-radius cause a purely gyro-Bohm transport model to exhibit worse than gyro-Bohm confinement. For the β scan, the model indicates a somewhat stronger β dependence than that observed with a thermal energy confinement scaling of Bτ ∝ β-0.7. More than half of the predicted β scaling is found to result from finite-β effects in the model. The model demonstrates a collisionality scaling of Bτ ∝ ν*-0.3 with some unfavourable dependence arising from neoclassical transport in the plasma core region. The overall goodness of fit obtained when comparing the global and local predictions results in a root mean square error averaging less than 13% for the total stored energy and averaging less than 9% for the density and temperature profiles relative to the maximum experimental values. When E × B shear effects are added to the model, the resulting change in the mean RMS error for the temperature profiles is less than 4%.

Predictive tests of ITG based models of tokamak heat transport on ITER database discharges

Three gyro-Bohm transport models are applied to the L mode discharges of the ITER profile database for predictive tests of plasma confinement. All three models are based on ion temperature gradient driven turbulence but emphasize different aspects of the resulting turbulent transport. It is found that the model using the long wavelength component of the turbulence for the radial correlation length yields the predictions with the lowest overall residual error for a given database. Extrapolations for the performance of ITER are given for the models used.

Overview of ITER physics deuterium-tritium experiments in JET

An overview of JET experimental results in DT plasmas directly relevant to ITER modes of operation is presented. Experiments in D:T mixtures varying from 100:0 to 10:90 and those carried out in hydrogen plasmas show that the H mode threshold power has an approximately inverse isotope mass dependence. Matching some of the key dimensionless parameters to the ITER values, the ITER similarity experiments with ITER shape and safety factor q show that the global energy confinement time is practically independent of isotopic mass (∼A0.03±0.08), where A is the atomic mass of the hydrogenic species. Subtracting the edge pedestal energy (which scales as ∼A0.57±0.2) from the total stored energy leads to a ∼A-0.17±0.1 dependence of confinement in the plasma core, very similar to that expected from the gyro-Bohm transport (∼ A-0.2) model. The observed scaling of the edge pedestal energy is consistent with a model in which the edge pressure gradient saturates at the ballooning limit over a region of width that scales as the ion poloidal Larmor radius governed by the average energy of the fast ions in the edge. The steady state total stored energy for a given input power in both ICRH and NBI discharges is the same despite the lower edge pedestal in the ICRH case, which is compensated for by more peaked power deposition profiles in ICRH. The ELM frequency is smaller with NBI; it decreases with isotopic mass in both NBI and ICRH discharges. A steady state, type I ELMy H mode discharge with ITER shape and q at 3.8 T/3.8 MA with an input power of 22 MW produced a Q ≈ 0.18 for 3.5 s and extrapolates well to ignition with ITER parameters. Here, Q is the ratio of fusion output power to input power. The thermal ELMy H mode confinement in both deuterium and tritium gas fuelled plasmas decreases significantly when the plasma density exceeds 0.75 of the Greenwald (nGW) limit, and the maximum density achieved is 0.85nGW. In L mode, the density limit decreases with increasing isotope mass roughly in accordance with code predictions. ITER reference ICRH scenarios have been evaluated. Second harmonic heating of tritium at the densities available in JET produces strong tails and heats electrons predominantly as expected. The 3He minority in 50:50 D:T and tritium dominated plasmas showed strong bulk ion heating leading to ion temperatures up to 13 keV with ICRH alone. Deuterium minority ion cyclotron heating in tritium plasmas at a power level of 6 MW produced steady state record values of Q ≈ 0.22 for more than 2.5 s.

Deuterium-tritium operation in magnetic confinement experiments: results and underlying physics

A review of experimental results obtained in JET D-T plasmas is presented. In discussing the underlying physics, results previously obtained on tokamak fusion test reactor (TFTR) are also taken into account. In JET, the maximum fusion power output of 16.1 MW has been obtained in an edge localized mode (ELM)-free hot-ion H-mode featuring an edge confinement barrier in a single-null divertor plasma with a where is the total input power to torus. A steady-state H-mode discharge, with plasma shape and safety factor q similar to that of ITER, produced 4 MW for 5 s (22 MJ). The steady-state results extrapolate well to ignition with ITER parameters using the normalized plasma pressure achieved on JET. Also, the advanced tokamak regime using optimized magnetic shear configuration featuring an internal transport barrier produced 8.2 MW of fusion power. With regards to reactor physics issues, a clear identification of electron heating by fusion born -particles has been made both in JET and TFTR. The JET experiments show that the H-mode threshold power has approximately an inverse isotopic mass dependence and that it does not depend on the method of auxiliary heating. The global energy confinement time in the TFTR D-T supershot regime scales as but in the JET H-modes, it is found to be practically independent of isotopic mass where A is the atomic mass of the hydrogenic species. In JET, the plasma core and the edge appear to have different underlying confinement physics, the former follows the gyro-Bohm transport model whereas the edge pedestal energy scales as . The maximum edge pressure in H-modes is analysed in relation to the ion poloidal Larmor radius at the edge. The fast ions driven by neutral beam injections (NBI) or ion cyclotron resonance heating (ICRH) could play an important role in setting the width of the edge pedestal. The thermal ELMy H-mode confinement both in D or T gas fuelled plasmas decreases significantly when the plasma density exceeds 0.75 of the Greenwald limit and the maximum density achieved is . The ICRH scenarios for a reactor have been evaluated. For example, minority in 50:50 D:T and tritium-dominated plasmas showed strong bulk ion heating leading to ion temperatures up to 13 keV with ICRH alone. Deuterium-minority ion cyclotron heating in tritium plasmas at a power level of 6 MW produced steady-state record values of for more than 2.5 s. Finally, the on-site closed-cycle tritium reprocessing plant and remote handling tools at JET have been used routinely and provided an integrated demonstration of safe and reliable operations of a tokamak device in reactor-relevant conditions.

Predictive transport simulations of JET L and H mode gyro-radius scaling experiments

Predictive transport code simulations of gyro-radius scaling experiments have been performed using experimental data from JET profile databases. Both L mode and H mode discharges are considered. Self-consistently derived steady state profiles of electron and impurity density, temperatures and heat diffusivities are compared with the experimental profiles. The simulations are based on a first principle gyro-Bohm transport model for ion temperature gradient (ITG) modes, impurity modes and collisionless trapped electron (CTE) modes. In the good confinement region, the experimental profiles as well as the transport scalings are well reproduced by the simulations. In particular, apparent Bohm- and Goldston-like scalings of the local diffusivities are obtained in the L mode simulations, whereas gyro-Bohm scalings are obtained in the H mode scans. The scalings obtained in the L mode simulations are a result of small systematic deviations in the similarity conditions between the low field and high field discharges. In the H mode simulations, the profile effects are generally smaller, resulting in a gyro-Bohm scaling of the heat diffusivities. Since profile effects arise through the edge boundary conditions, the different scalings obtained in the simulations are attributed to differences in edge confinement between L and H mode plasmas.

Thermal electron transport in regimes with low and negative magnetic shear in Tore Supra

The magnetic shear effect on thermal electron transport is studied in a large variety of non-inductive plasmas in Tore Supra. An improved confinement in the region of low and negative shear was observed and quantified with an exponential dependence on the magnetic shear (Litaudon, et al., Fusion Energy 1996 (Proc. 16th Int; Conf. Montreal, 1996), vol. 1, IAEA, Vienna (1997) 669). This is interpreted as a consequence of a decoupling of the global modes (Romanelli and Zonca, Phys. Fluids B 5 (1993) 4081) that are thought to be responsible for anomalous transport. This dependence is proposed in order to complete the Bohm-like L mode local electron thermal diffusivity so as to describe the transition from Bohm-like to gyroBohm transport in the plasma core. The good agreement between the predictive simulations of the different Tore Supra regimes (hot core lower hybrid enhanced performance, reversed shear plasmas and combined lower hybrid current drive and fast wave electron heating) and experimental data provides a basis for extrapolation of this magnetic shear dependence in the local transport coefficients to future machines. As an example, a scenario for non-inductive current profile optimization and control in ITER is presented

Drift wave transport simulations of dimensionally similar JET discharges

Predictive transport simulations of dimensionally similar low confinement mode discharges have been performed using experimental data from Joint European Torus (JET) databases. The simulations, based on a gyro-Bohm transport model for temperature gradient driven drift waves, are found to reproduce the experimental profiles of density and temperatures well. In particular, apparent Bohm- and Goldston-like scalings of the local diffusivities are obtained as the gyroradius is varied. The results are due to small systematic deviations in the similarity conditions combined with marginal stability effects

Coherent drift-wave structures in toroidal plasmas

Using the ion-temperature-gradient-driven drift waves as a paradigm for drift-wave anomalous transport, we explore the structure of the linear and nonlinear modes. Two phases of transport are shown to exist: (i) Bohm-like transport for parameters close to marginal stability; (ii) gyro-Bohm transport for turbulent convection cells in systems driven away from marginal stability. Nonlinear relaxation to large-scale coherent convective structures is observed in three-dimensional toroidal particle simulations.

Theory-based transport modeling of the gyro-radius experiments

Self-consistent predictive transport simulations of temperature and density profiles have been carried out for ten dimensionally similar low (L) mode discharges from the Tokamak Fusion Test Reactor (TFTR) [D. Grove and D. M. Meade, Nucl. Fusion 25, 1167 (1985)], Doublet III-D Tokamak [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)], and the Joint European Torus [P. H. Rebut, R. J. Bickerton, and B. E. Keen, Nucl. Fusion 25, 1011 (1985)], where only the normalized Larmor radius was allowed to vary. It is found that a purely gyro-Bohm transport model predicts temperature and density profiles that match the experimental data from these ρ* scans very well. In particular, a combination of theoretically derived transport models is used in these simulations, including the Weiland model for transport due to drift waves (ion temperature gradient and trapped electron modes) and the Guzdar–Drake model for transport due to resistive ballooning modes. These gyro-Bohm transport models depend very sensitively on the shapes as well as the magnitudes of the profiles. As the magnetic field, density and temperature are changed in each dimensionally similar series of discharges, the penetration length of neutrals from the edge varies considerably. This effect causes the shape of the density profiles to change near the edge of the plasma, which causes the scaling of our transport model diffusivities to differ significantly from their fundamentally gyro-Bohm scaling.

Transport inferred from consideration of particle orbits in drift turbulence

Particle orbits in the 2D potential hills/valley ('blobs') of low-frequency drift turbulence are analysed. For most of a given blob's lifetime a particle initially on the blob circulates on an orbit locked to that blob even though the blob shape and amplitude are changing. This locking corresponds to an adiabatic invariant being preserved. However, when the blob amplitude decreases to a critical value, the adiabatic invariant is destroyed, the locking ceases, and the particle detaches from the dying blob and attaches to a newly forming blob. This gives a detailed physical picture of the well known gyro-Bohm transport scaling D approximately gamma /k2 where gamma is the turbulence autocorrelation time (corresponding to blob lifetime) and k is the perpendicular wavenumber (corresponding to blob diameter). It is further shown that gyro-Bohm transport is consistent with global measurements in a wide variety of experiments, in addition to the large tokamaks to which it has traditionally been applied. Finally, it is shown that ion orbit behavior becomes qualitatively different if the blobs become sufficiently sharp and steep.