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This paper investigates the influence of turbulent dynamics on the neo-classical equilibrium in a tokamak, with an emphasis on the turbulence driven stationary electric current. The neo-classical solution is evaluated using the Hirschmann-Sigmar formalism, in which the turbulent dynamics enter as a forcing term. The latter forcing terms are evaluated through time averages of gyro-kinetic turbulence simulations and are linked with the velocity non-linearity in the gyro-kinetic equation. The time averaged turbulent forcing terms connected with the velocity non-linearity provide a non-negligible current drive, despite being a correction of second order in the normalized Larmor radius. For ITG turbulence, the force exerted due to the heat flux balance is the dominant contribution to the current. The parallel fluctuations of electron density/temperature and the electrostatic potential drive the majority of the current, which is in magnitude comparable to the bootstrap current in the kinetic cyclone base case and increases the total current by a few percent in cases with an experimentally relevant heat flux. An up-down symmetry breaking mechanism is required for turbulent current drive, which is provided in this study by a background rotation or rotation gradient. Consequently, the current is nearly linear in the plasma rotation or its gradient. [ABSTRACT FROM AUTHOR]

This paper investigates the so-called tertiary instabilities driven by the zonal flow in gyro-kinetic tokamak core turbulence. The Kelvin Helmholtz instability is first considered within a 2D fluid model and a threshold in the zonal flow wave vector k ZF > k ZF , c for instability is found. This critical scale is related to the breaking of the rotational symmetry by flux-surfaces, which is incorporated into the modified adiabatic electron response. The stability of undamped Rosenbluth-Hinton zonal flows is then investigated in gyro-kinetic simulations. Absolute instability, in the sense that the threshold zonal flow amplitude tends towards zero, is found above a zonal flow wave vector k ZF , c ρ i ≈ 1.3 ( ρ i is the ion thermal Larmor radius), which is comparable to the 2D fluid results. Large scale zonal flows with k ZF < k ZF , c are unstable for sufficiently large amplitudes with increasing trend for an increasing radial scale. However, the critical E × B-shearing rate associated with the stability boundary ω E×B , c exceeds typical values connected to the pure flow state at marginal stability by more than an order of magnitude, which therefore lies deeply in the stable parameter region. Furthermore, the impact of zonal temperature perturbations on the tertiary instability is examined. Although temperature perturbations favor instability, the realistic values of gradient-driven gyro-kinetic simulations still lie deeply in the stable parameter regime. Therefore, the relevance of the tertiary instability as a saturation mechanism to the zonal flow amplitude is questioned, as most of the zonal flow intensity is concentrated in modes satisfying k ZF ≪ k ZF , c as well as ω E×B ≪ ω E×B , c . [ABSTRACT FROM AUTHOR]

Verification is the fundamental step that any turbulence simulation code has to be submitted in order to assess the proper implementation of the underlying equations. We have carried out a cross comparison of three flux tube gyrokinetic codes, GENE [F. Jenko et al., Phys. Plasmas 7, 1904 (2000)], GKW [A. G. Peeters et al., Comput. Phys. Commun. 180, 2650 (2009)], and GS2 [W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)], focusing our attention on the effect of realistic geometries described by a series of MHD equilibria with increasing shaping complexity. To simplify the effort, the benchmark has been limited to the electrostatic collisionless linear behaviour of the system. A fully gyrokinetic model has been used to describe the dynamics of both ions and electrons. Several tests have been carried out looking at linear stability at ion and electron scales, where for the assumed profiles Ion Temperature Gradient (ITG)/Trapped Electron Modes and Electron Temperature Gradient modes are unstable. The capability of the codes to handle a non-zero ballooning angle has been successfully benchmarked in the ITG regime. Finally, the standard Rosenbluth-Hinton test has been successfully carried out looking at the effect of shaping on Zonal Flows (ZFs) and Geodesic Acoustic Modes (GAMs). Inter-code comparison as well as validation of simulation results against analytical estimates has been accomplished. All the performed tests confirm that plasma elongation strongly stabilizes plasma instabilities as well as leads to a strong increase in ZF residual and GAM damping. [ABSTRACT FROM AUTHOR]

The interplay between toroidal rotation u, parallel flow shear u', and perpendicular flow shear cE in the stabilisation of tokamak turbulence is investigated in non-linear flux-tube gyrokinetic simulations. The simulations are performed for a reference L-mode DIII-D plasma (the so-called shortfall case) at r=a = 0:8, varying the flow parameters around their nominal values. Depending on the respective signs of u, u', and λE, turbulence is found to be enhanced, reduced, or unchanged. When the coupling is favorable, the overall effect on the non-linear heat fluxes can be very large, even at moderate flow values. The ion heat flux is, for instance, decreased by a factor of 3 when the direction of the parallel flow shear is reversed with respect to its nominal value. Even more surprising, keeping u' and λE at their nominal values, the ion heat flux decreases by more than 50% when the toroidal flow is reversed. The relevance of this mechanism in the experiments which depends on the ability to decouple u, u', and cE is discussed. The interplay between u and u0 observed in the non-linear simulations qualitatively follows the linear stability results and is interpreted in the frame of a simple fluid model. [ABSTRACT FROM AUTHOR]

This paper studies the effect of rotation on microinstabilities under experimentally relevant conditions in the spherical tokamak National Spherical Torus Experiment (NSTX). The focus is specifically on the centrifugal force effects on the impurity and momentum transport in the core (r/a = 0:7) of an H-mode plasma. Due to relatively high beta, the linear simulations predict the presence of both microtearing mode (MTM) and hybrid ion temperature gradient-kinetic ballooning mode (ITG-KBM) electromagnetic instabilities. Rotation effects on both MTM and ITG-KBM growth rates and mode frequencies are found to be small for the experimental values. However, they do influence the quasi-linear particle and momentum fluxes predicted by ITG-KBM (MTM contributes only to electron heat flux). The gradient of the intrinsic carbon impurity in the sourcefree core region is predicted to be locally hollow, strengthened by centrifugal effects. This result is consistent with experimental measurements and contradicts neoclassical theory that typically provides a reasonable explanation of the impurity profiles in NSTX. The diffusive and Coriolis pinch contributions to momentum transport are found to be relatively weak. Surprisingly, the strongest contribution derives from a centrifugal effect proportional to the product of rotation and rotation shear, which predicts an inward momentum flux roughly three times bigger than the Coriolis pinch, suggesting it should be considered when interpreting previous experimental pinch measurements. [ABSTRACT FROM AUTHOR]

The effect of profile shearing on toroidal momentum transport is studied in linear and non-linear gyro-kinetic simulations. Retaining the radial dependence of both plasma and geometry parameters leads to a momentum flux that has contributions both linear in the logarithmic gradients of density and temperature, as well as contributions linear in the derivatives of the logarithmic gradients. The effect of the turbulence intensity gradient on momentum transport is found to be small for the studied parameters. Linear simulations at fixed normalized toroidal wave number predict a weak dependence of the momentum flux on the normalized Larmor radius ρ= ρ/R. Non-linear simulations, however, at sufficiently small ρ* show a linear scaling of the momentum flux with ρ*. The obtained stationary rotation gradients are in the range of, although perhaps smaller than, current experiments. For a reactor plasma, however, a rather small rotation gradient should result from profile shearing. [ABSTRACT FROM AUTHOR]

In tokamaks, turbulent particle and toroidal momentum transport are both characterized by the presence of off-diagonal contributions which play an essential role in establishing the profile shapes of the density and the toroidal rotation under most conditions. In this paper similarities and differences between the two turbulent transport channels are pointed out and, thereby, interesting physical aspects which connect the two channels are identified. The main contributions to off-diagonal particle and toroidal momentum transport are reviewed by means of a rather simplified description, which aims at providing, when possible, a direct connection between theoretical, modelling and experimental research. [ABSTRACT FROM AUTHOR]

The electrostatic potential related to a magnetic island structure with imposed width and rotation frequency is studied by means of gyrokinetic simulations, which allow its self-consistent determination via the Poisson equation. An adiabatic response of the trapped ions at the island separatrix leads to a significant smoothing of the potential with respect to analytic calculations based on a complete flattening of the pressure profile inside the island. As a consequence, the magnitude of the polarization current is drastically reduced. When the island size is comparable to the ion banana width, the adiabatic response covers the whole island region, leading to a reduced density flattening for islands rotating in the electron diamagnetic direction. This confirms previous results based on drift-kinetic simulations. [ABSTRACT FROM AUTHOR]

The effect of a finite βe = 8πneTe/B2 on the turbulent transport of toroidal momentum in tokamak plasmas is discussed. From an analytical gyrokinetic model as well as local linear gyrokinetic simulations, it is shown that the modification of the parallel mode structure due to the nonadiabatic response of passing electrons, which changes the parallel wave vector k∥ with increasing βe, leads to a decrease in size of both the diagonal momentum transport as well as the Coriolis pinch under ion temperature gradient turbulence conditions, while for trapped electron modes, practically no modification is found. The decrease is particularly strong close to the onset of the kinetic ballooning modes. There, the Coriolis pinch even reverses its direction. [ABSTRACT FROM AUTHOR]

Two symmetries of the local nonlinear δf gyrokinetic system of equations in tokamaks in the high flow regime are presented. The turbulent transport of toroidal angular momentum changes sign under an up-down reflection of the tokamak and a sign change of both the rotation and the rotation shear. Thus, the turbulent transport of toroidal angular momentum must vanish for up-down symmetric tokamaks in the absence of both rotation and rotation shear. This has important implications for the modeling of spontaneous rotation. [ABSTRACT FROM AUTHOR]

Tokamak experiments operate with a rotating plasma, with toroidal velocity which can be driven externally but can also arise spontaneously. In the frame that corotates with the plasma, the effects of the centrifugal force are felt through a centrifugal drift and an enhanced mirror force [Peeters et al., Phys. Plasmas 16, 042310 (2009)]. These inertial terms become important in the case of strong rotation, as is common in spherical devices, and are also important for heavy impurity ions even at small toroidal velocities. In this work, the first gyrokinetic simulations including the centrifugal force in a strongly rotating plasma are presented. The enhanced mirror force redistributes density over a flux surface and modifies the trapping condition, destabilizing trapped electron modes. At intermediate scales this can result in promotion of the trapped electron mode over the ion temperature gradient (ITG) mode as the dominant instability, which under marginal conditions could result in an enhanced electron heat flux. The centrifugal drift acts to damp the residual zonal flow of the geoacoustic mode, while its frequency is increased. For nonlinear ITG dominated turbulence, increased trapped electron drive and reduced zonal flow lead to an increase in ion heat diffusivity if the increased rotation is not accompanied by rotational shear stabilization. An increased fraction of slow trapped electrons enhances the convective particle pinch, leading to an increase in the steady state density gradient with strong rotation. Linear ITG mode results show an increased pinch of heavy trace impurities due to their strong centrifugal trapping. [ABSTRACT FROM AUTHOR]

Nondiffusive anomalous momentum transport in toroidal plasmas occurs through symmetry breaking mechanisms. In this paper the contribution of sheared E×B flows to parallel momentum transport [R. R. Dominguez and G. M. Staebler, Phys Fluids B 5, 3876 (1993)] is investigated with nonlinear gyrokinetic simulations in toroidal geometry. The background perpendicular shear is treated independently from the parallel velocity shear to isolate a nondiffusive, nonpinch contribution to the parallel momentum flux. It is found that the size of the term depends strongly on the magnetic shear, with the sign reversing for negative magnetic shear. Perpendicular shear flows are responsible for both symmetry breaking and suppression of turbulence, resulting in a shearing rate at which there is a maximum contribution to the momentum transport. The E×B momentum transport is shown to be quenched by increasing flow shear more strongly than the standard linear quench rule for turbulent heat diffusivity. [ABSTRACT FROM AUTHOR]

The paper derives the gyro-kinetic equation in the comoving frame of a toroidally rotating plasma, including both the Coriolis drift effect [A. G. Peeters et al., Phys. Rev. Lett. 98, 265003 (2007)] as well as the centrifugal force. The relation with the laboratory frame is discussed. A low field side gyro-fluid model is derived from the gyro-kinetic equation and applied to the description of parallel momentum transport. The model includes the effects of the Coriolis and centrifugal force as well as the parallel dynamics. The latter physics effect allows for a consistent description of both the Coriolis drift effect as well as the ExB shear effect [R. R. Dominguez and G. M. Staebler, Phys. Fluids B 5, 3876 (1993)] on the momentum transport. Strong plasma rotation as well as parallel dynamics reduce the Coriolis (inward) pinch of momentum and can lead to a sign reversal generating an outward pinch velocity. Also, the ExB shear effect is, in a similar manner, reduced by the parallel dynamics and stronger rotation. [ABSTRACT FROM AUTHOR]

Recent developments in the gyrokinetic theory have shown that, in a toroidal device, the Coriolis drift associated with the background plasma rotation significantly affects the small scale instabilities [A. G. Peeters et al., Phys. Rev. Lett. 98, 265003 (2007)]. The later study, which focuses on the effect of the Coriolis drift on toroidal momentum transport is extended in the present paper to heat and particle transport. It is shown numerically using the gyrokinetic flux-tube code GKW [A. G. Peeters and D. Strintzi, Phys. Plasmas 11, 3748 (2004)], and supported analytically, that the Coriolis drift and the parallel dynamics play a similar role in the coupling of density, temperature, and velocity perturbations. The effect on particle and heat fluxes increases with the toroidal rotation (directly) and with the toroidal rotation gradient (through the parallel mode structure), depends on the direction of propagation of the perturbation, increases with the impurity charge number and with the impurity mass to charge number ratio. The case of very high toroidal rotation, relevant to spherical tokamaks, is investigated by including the effect of the centrifugal force in a fluid model. The main effect of the centrifugal force is to decrease the local density gradient at the low field side midplane and to add an extra contribution to the fluxes. The conditions for which the inertial terms significantly affect the heat and particle fluxes are evidenced. [ABSTRACT FROM AUTHOR]

Fluid and gyrokinetic calculations of the toroidal momentum diffusivity in a tokamak are compared. The four-moment gyrofluid model predicts the Prandtl number connected with the ion temperature gradient mode reasonably well provided the drift term is kept in the momentum balance. Without the drift term in the momentum balance, some previous gyrofluid models predicted small values of the Prandtl number in the range of experimental observations. It is shown that the drift term enters in the fluid equations through the gyroviscosity. Gyrokinetic calculations of the ion temperature gradient mode with kinetic electrons, and for experimentally relevant parameters yield a Prandtl number in the range 0.7–1.2. [ABSTRACT FROM AUTHOR]

Experimental observations on core particle and impurity transport from the Axial Symmetric Divertor Experiment Upgrade [O. Gruber, H.-S. Bosch, S. Günter et al., Nucl Fusion 39, 1321 (1999)] and the Joint European Torus [J. Pamela, E. R. Solano, and JET EFDA Contributors, Nucl. Fusion 43, 1540 (2003)] tokamaks are reviewed and compared. Robust general experimental behaviors observed in both the devices and related parametric dependences are identified. The experimental observations are compared with the most recent theoretical results in the field of core particle transport. [ABSTRACT FROM AUTHOR]

Abstract: Linear and quasi-linear transport models for core transport of a tokamak plasma are reviewed. The dependence of the modelled transport on the main plasma parameters is discussed. The comparison of the models with the experiment is reviewed both directly in terms of the predicted stored energy as well as indirectly discussing key physics ingredients of the models. To cite this article: A.G. Peeters et al., C. R. Physique 7 (2006). [Copyright &y& Elsevier]

Linear gyrokinetic calculations and a quasilinear model are applied to compute the radial particle fluxes and investigate the properties of the particle transport in a tokamak plasma induced by ion temperature gradient and trapped electron mode instabilities. The paper focuses on the specific problem of the dependence of anomalous pinch and density peaking on collisionality. [ABSTRACT FROM AUTHOR]

It is shown from a symmetry in the gyrokinetic equation that for up–down symmetric tokamak equilibria and for u[lowercase_phi_synonym]>ρυthi/r (where u[lowercase_phi_synonym] is the toroidal velocity, υthi is the thermal ion velocity, ρ is the Larmor radius, and r is the radius of the flux surface), the transport of parallel momentum can be written as the sum of a diffusive and a pinch contribution with no off-diagonal terms due to temperature and pressure gradients. The measured parallel velocity gradient in ASDEX Upgrade [O. Gruber, H.-S. Bosch, S. Günter et al., Nucl. Fusion 39, 1321 (1999)] is insufficient to drive the parallel velocity shear instability. The parallel velocity is then transported by the ion temperature gradient mode. The diffusive contribution to the transport flux is investigated using a linear gyrokinetic approach, and it is found that the diffusion coefficient for parallel velocity transport divided by the ion heat conductivity coefficient is close to 1, and only weakly dependent on plasma parameters. [ABSTRACT FROM AUTHOR]

It has been shown [Maccio, Phys. Plasmas 8, 895 (2001)] that a uniform radial electric field can stabilize the toroidal ion temperature gradient mode in tokamak plasmas. In this paper, using the ballooning transform, the mechanism leading to stabilization is further clarified. However, the stabilizing effect depends on the choice of the background distribution function. It is shown that if the electric field is connected with a toroidal plasma rotation the stabilizing effect becomes ineffective. Consequently, in many current experiments this effect should not play a role. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The theoretical prediction of density profiles in tokamak plasmas plays a major role for the prediction of the plasma performance in a fusion reactor. The density peaking measured in plasmas in high confinement mode of the ASDEX Upgrade tokamak [O. Gruber, H.-S. Bosch, S. Günter et al., Nucl. Fusion 39, 1321 (1999)] is shown to decrease with increasing collisionality. This experimental behavior is explained with a theoretical fluid transport model for ion temperature gradient and trapped electron modes, GLF23, which includes a valid description of the effects of collisions on these instabilities. Collisionless reactive models, like the Weiland model, are in disagreement with the experimental observations. The difference between the predictions of the two models must be ascribed to collisionality. This has been ascertained by a detailed comparison of the physics content of the two models and by the implementation of modified, collisionless versions of the GLF23 model, which yield results analogous to those of the Weiland model and in disagreement with the experiment. It is shown that the anomalous particle pinch decreases with collisionality and the relative role of the neoclassical Ware pinch becomes important at high collisionality, that is close to the density limit in present large tokamak experiments, while it is practically negligible at low collisionality. The present results reconciliate apparently contradictory observations on the existence of an anomalous particle pinch collected so far in tokamaks. [ABSTRACT FROM AUTHOR]

The neoclassical ion transport in a tokamak plasma with circular cross section is studied with guiding center particle simulations. A Monte Carlo model of pitch-angle scattering which includes momentum conservation is employed. The model includes the whole plasma inside closed flux surfaces, but the focus is on the near-axis transport due to particles with wide orbits which lead to a different heat flux than in the standard theory. Neither of two recent theories of banana regime transport near the axis is confirmed by the simulations, nor do the numerical results for the plateau regime agree with a recent theory of wide-orbit transport in this regime. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In guiding-center simulations of a tokamak plasma edge, bifurcations of both the radial electric field and the electrode current are observed for a threshold value of the electrode bias voltage, in qualitative agreement with experiments. The simulations are neoclassical and Monte Carlo based, also containing the perpendicular viscosity effect. Above the voltage threshold, an inward moving, solitary radial electric field structure is generated with a concomitant drop in the electrode current. The position, width, and height of the soliton depend on the magnitude and radial variation of the conductivity (and, thus, on local plasma parameters), finite orbit effects, and also on the perpendicular viscosity and neutral density. The obtained E[sub r] structures are narrower and higher than the ones measured in the edge polarization experiments in the Ju¨lich tokamak TEXTOR-94 [S. Jachmich et al., Plasma Phys. Controlled Fusion 40, 1105 (1998)]. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Numerical simulations using a three-dimensional Fokker?Planck code show that for small tokamaksthe transport of electrons across the magnetic surfaces at a level consistent with anomalous transporthas a large influence on the formation of the quasilinear plateau during electron cyclotron resonantheating. The reduction of wave absorption due to the plateau formation is much smaller when the effects of radial transport are included. This is explained by the localized nature of the resonance invelocity space and the strong dependence of resonant energy on radial position. Impact of the effecton diagnostic signals is discussed. [ABSTRACT FROM AUTHOR]

A new mechanism for toroidal momentum transport in a tokamak is investigated using the gyro-kinetic model. First, an analytic model is developed through the use of the ballooning transform. The terms that generate the momentum transport are then connected with the poloidal derivative of the ballooning envelope, which are one order smaller in the normalised Larmor radius, compared with the derivative of the eikonal. The mechanism, therefore, does not introduce an inhomogeneity in the radial direction, in contrast with the effect of profile shearing. Numerical simulations of the linear ion temperature gradient mode with adiabatic electrons, retaining the finite ρ* effects in the E × B velocity, the drift, and the gyro-average, are presented. The momentum flux is found to be linear in the normalised Larmor radius (ρ*) but is, nevertheless, generating a sizeable counter-current rotation. The total momentum flux scales linear with the aspect ratio of the considered magnetic surface, and increases with increasing magnetic shear, safety factor, and density and temperature gradients. [ABSTRACT FROM AUTHOR]

Linear gyro-kinetic stability calculations predict unstable toroidal ion temperature gradient modes (ITGs) with normalised poloidal wave vectors well above one (kθρi>1) for standard tokamak parameters with adiabatic electron response. These modes have a maximum amplitude at a poloidal angle θ that is shifted away from the low field side (θ≠0). The physical mechanism is clarified through the use of a fluid model. It is shown that the shift of the mode away from the low field side reduces the effective drift frequency which allows for the instability to develop. Numerical tests using the gyro-kinetic model confirm this physical mechanism. Furthermore, it is shown that modes localized away from the low field side can be important also for kθρi<1 close to the threshold of the ITG. In fact, modes with maximum amplitude at θ≠0 can exist for normalised temperature gradient lengths below the threshold of the ITG obtained for the case with the maximum at θ=0. [ABSTRACT FROM AUTHOR]

A local gyrokinetic description of the centrifugal effects on impurity transport in tokamak plasmas is presented, which extends previous models with the inclusion of the gradient of the background toroidal angular velocity in the equilibrium distribution. The equations are implemented in a gyrokinetic code. An analytical model is derived and formulae are proposed which allow the calculation of centrifugal effects on impurity transport in the limit where centrifugal effects are large only for heavy impurities in trace concentration, but are negligible for the bulk plasma. The analytic formulae are shown to be in quantitative agreement with the numerical results and are proposed for complementing present transport models. [ABSTRACT FROM AUTHOR]

The paper discusses the impurity transport in a tokamak in the Pfirsch Schlu¨ter regime. A reduced charge state method is developed for the set of coupled equations that describe the poloidal variation of the heat flux and the third Laguerre polynomial expansion of the distribution function. A comparison is made with analytic equations in the disparate mass limit, and the influence of the cross terms of the coupled equations, which have been neglected in earlier work, on the transport coefficients of a hydrogen helium plasma is discussed. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]