Ion Temperature Gradient Turbulence Research Articles

Turbulence, transport, and zonal flows in the Madison symmetric torus reversed-field pinch

The robustness and the effect of zonal flows in trapped electron mode (TEM) turbulence and Ion Temperature Gradient (ITG) turbulence in the reversed-field pinch (RFP) are investigated from numerical solutions of the gyrokinetic equations with and without magnetic external perturbations introduced to model tearing modes. For simulations without external magnetic field perturbations, zonal flows produce a much larger reduction of transport for the density-gradient-driven TEM turbulence than they do for the ITG turbulence. Zonal flows are studied in detail to understand the nature of their strong excitation in the RFP and to gain insight into the key differences between the TEM- and ITG-driven regimes. The zonal flow residuals are significantly larger in the RFP than in tokamak geometry due to the low safety factor. Collisionality is seen to play a significant role in the TEM zonal flow regulation through the different responses of the linear growth rate and the size of the Dimits shift to collisionality, while affecting the ITG only minimally. A secondary instability analysis reveals that the TEM turbulence drives zonal flows at a rate that is twice that of the ITG turbulence. In addition to interfering with zonal flows, the magnetic perturbations are found to obviate an energy scaling relation for fast particles.

First steps towards modeling of ion-driven turbulence in Wendelstein 7-X

Due to foreseen improvement of neoclassical confinement in optimised stellarators—like the newly commissioned Wendelstein 7-X (W7-X) experiment in Greifswald, Germany—it is expected that turbulence will significantly contribute to the heat and particle transport, thus posing a limit to the performance of such devices. In order to develop discharge scenarios, it is thus necessary to develop a model which could reliably capture the basic characteristics of turbulence and try to predict the levels thereof. The outcome will not only be affordable, using only a fraction of the computational cost which is normally required for repetitive direct turbulence simulations, but would also highlight important physics. In this model, we seek to describe the ion heat flux caused by ion temperature gradient (ITG) micro-turbulence, which, in certain heating scenarios, can be a strong source of free energy. With the aid of a relatively small number of state-of-the-art nonlinear gyrokinetic simulations, an initial critical gradient model (CGM) is devised, with the aim to replace an empirical model, stemming from observations in prior stellarator experiments. The novel CGM, in its present form, encapsulates all available knowledge about ion-driven 3D turbulence to date, also allowing for further important extensions, towards an accurate interpretation and prediction of the ‘anomalous’ transport. The CGM depends on the stiffness of the ITG turbulence scaling in W7-X, and implicitly includes the nonlinear zonal flow response. It is shown that the CGM is suitable for a 1D framework turbulence modeling.

Benchmarking of flux-driven full-F gyrokinetic simulations

Two full-F global gyrokinetic codes are benchmarked to compute flux-driven ion temperature gradient (ITG) turbulence in tokamak plasmas. For this purpose, the Semi-Lagrangian code GYrokinetic SEmi-LAgrangian and the Eulerian code GT5D are employed, which solve the full-F gyrokinetic equation with a realistic fixed flux condition. The equilibrium poloidal flow profile formation processes are benchmarked and compared against the local neoclassical theory. The simulations above are carried out without turbulence, which agree well with each other and with the theoretical estimates. Here, a lot of attention has been paid to the boundary conditions, which have huge impacts on the global shape of radial electric field. The behaviors of micro-instabilities are benchmarked for linear and nonlinear cases without a heat source, where we found good agreements in the linear growth rates and nonlinear critical gradient level. In the nonlinear case, initial conditions are chosen to be identical since they dominate the transient turbulence behavior. Using the appropriate settings for the boundary and initial conditions obtained in the benchmarks above, a flux-driven ITG turbulence simulation is carried out. The avalanche-like transport is assessed with a focus on spatio-temporal properties. A statistical analysis is performed to discuss this self-organized criticality (SOC) like behaviors, where we found a 1/f spectra and a transition to 1/f3 spectra at high-frequency side in both codes. Based on these benchmarks, it is verified that the SOC-like behavior is robust and not dependent on numerics.

Ion temperature gradient turbulence close to the finite heat flux threshold

The dependence of the heat flux on the temperature gradient length in collisionless ion temperature gradient turbulence has recently been revisited. It has been found that the heat flux is discontinuous at a finite heat flux threshold larger than the (Dimits) interpolated threshold. In this paper, the influence of collisions on the heat flux close to the threshold is investigated. It is found that up to relatively high collision frequencies, relevant to the modern day experiments, a discontinuous behaviour of the heat flux as a function of the gradient length persists. Collisions, however, do lead to a reduction in the gradient length at which the discontinuity is observed. Below the finite heat flux threshold, a state of low turbulence with a vanishing small heat flux persists, which can drive the zonal flow against the collisional dissipation. This state is characterised by the fully developed staircases in the radial ExB shearing profile. Increasing the collision frequency at a fixed gradient length leads to the loss of the fully developed staircase structure with the ExB shearing profile having the form of a sawtooth that allows for avalanche formation and a finite heat flux. At very high collision frequencies or gradient lengths well above the threshold the staircase structure is lost. The simulations indicate that the long wave length zonal flow saturates through a mechanism that directly involves the turbulence intensity.

Nonlinear gyrokinetic simulation of ion temperature gradient turbulence based on a numerical Lie-transform perturbation method

The electrostatic gyrokinetic nonlinear turbulence code NLT, which is based on a numerical Lie-transform perturbation method, is developed. For improving the computational efficiency and avoiding the numerical instabilities, field-aligned coordinates and a Fourier filter are adopted in the NLT code. Nonlinear tests of the ion temperature gradient driven turbulence with adiabatic electrons are performed for verifying the NLT code by comparing with other gyrokinetic codes. The time evolution of the ion heat diffusivity and the relation between the ion heat diffusivity and the ion temperature gradient are compared in the nonlinear tests. Good agreements are achieved from the nonlinear benchmarks between the NLT code and other codes. The mode structures of the perturbed electric potential representing different phases have been simulated.

Neutral recycling effects on ITG turbulence

The effects of recycled neutral atoms on tokamak ion temperature gradient (ITG) driven turbulence have been investigated in a steep edge pedestal, magnetic separatrix configuration, with the full-f edge gryokinetic code XGC1. An adiabatic electron model has been used; hence, the impacts of neutral particles and turbulence on the density gradient are not considered, nor are electromagnetic turbulence effects. The neutral atoms enhance the ITG turbulence, first, by increasing the ion temperature gradient in the pedestal via the cooling effects of charge exchange and, second, by a relative reduction in the shearing rate.

Full-f XGC1 gyrokinetic study of improved ion energy confinement from impurity stabilization of ITG turbulence

Flux-driven full-f gyrokinetic simulations are performed to study carbon impurity effects on the ion temperature gradient (ITG) turbulence and ion thermal transport in a toroidal geometry. Employing the full-f gyrokinetic code XGC1, both main ions and impurities are evolved self-consistently including turbulence and neoclassical physics. It is found that the carbon impurity profile self-organizes to form an inwardly peaked density profile, which weakens the ITG instabilities and reduces the overall fluctuations and ion thermal transport. A stronger reduction appears in the low frequency components of the fluctuations. The global structure of E × B flow also changes, resulting in the reduction of global avalanche like transport events in the impure plasma. Detailed properties of impurity transport are also studied, and it is revealed that both the inward neoclassical pinch and the outward turbulent transport are equally important in the formation of the steady state impurity profile.

Negative viscosity from negative compressibility and axial flow shear stiffness in a straight magnetic field

Negative compressibility ion temperature gradient (ITG) turbulence in a linear plasma device controlled shear de-correlation experiment can induce a negative viscosity increment. However, even with this negative increment, we show that the total axial viscosity remains positive definite, i.e., no intrinsic axial flow can be generated by pure ITG turbulence in a straight magnetic field. This differs from the case of electron drift wave turbulence, where the total viscosity can turn negative, at least transiently. When the flow gradient is steepened by any drive mechanism, so that the parallel shear flow instability (PSFI) exceeds the ITG drive, the flow profile saturates at a level close to the value above which PSFI becomes dominant. This saturated flow gradient exceeds the PSFI linear threshold, and grows with ∇Ti0 as |∇V∥|/|k∥cs|∼|∇Ti0|2/3/(k∥Ti0)2/3. This scaling trend characterizes the effective stiffness of the parallel flow gradient.

Main-Ion Intrinsic Toroidal Rotation Profile Driven by Residual Stress Torque from Ion Temperature Gradient Turbulence in the DIII-D Tokamak.

Intrinsic toroidal rotation of the deuterium main ions in the core of the DIII-D tokamak is observed to transition from flat to hollow, forming an off-axis peak, above a threshold level of direct electron heating. Nonlinear gyrokinetic simulations show that the residual stress associated with electrostatic ion temperature gradient turbulence possesses the correct radial location and stress structure to cause the observed hollow rotation profile. Residual stress momentum flux in the gyrokinetic simulations is balanced by turbulent momentum diffusion, with negligible contributions from turbulent pinch. The prediction of the velocity profile by integrating the momentum balance equation produces a rotation profile that qualitatively and quantitatively agrees with the measured main-ion profile, demonstrating that fluctuation-induced residual stress can drive the observed intrinsic velocity profile.

Intrinsic parallel rotation drive by electromagnetic ion temperature gradient turbulence

The quasilinear intrinsic parallel flow drive including parallel residual stress, kinetic stress, cross Maxwell stress and parallel turbulent acceleration by electromagnetic ion temperature gradient (ITG) turbulence is calculated analytically using electromagnetic gyrokinetic theory. Both the kinetic stress and cross Maxwell stress also enter the mean parallel flow velocity equation via their divergence, as for the usual residual stress. The turbulent acceleration driven by ion pressure gradient along the total magnetic field (including equilibrium magnetic field and fluctuating radial magnetic field) cannot be written as a divergence of stress, and so should be treated as a local source/sink. All these terms can provide intrinsic parallel rotation drive. Electromagnetic effects reduce the non-resonant electrostatic stress force and even reverse it, but enhance the resonant stress force. Both the non-resonant and resonant turbulent acceleration terms are also enhanced by electromagnetic effects. The possible implications of our results for experimental observations are discussed.

Isotopic dependence of impurity transport driven by ion temperature gradient turbulence

Hydrogenic ion mass effects, namely, the isotopic effects on impurity transport driven by ion temperature gradient (ITG) turbulence are investigated using gyrokinetic theory. For non-trace impurities, changing from hydrogen (H) to deuterium (D), and to tritium (T) plasmas, the outward flux for lower (higher) ionized impurities or for lighter (heavier) impurities is found to decrease (increase), although isotopic dependence of the ITG linear growth rate is weak. This is mainly due to the decrease of outward (inward) convection, while the isotopic dependence of diffusion is relatively weak. In addition, the isotopic effects reduce (enhance) the impurity flux of fully ionized carbon (C6+) for weaker (stronger) magnetic shear. In the trace impurity limit, the isotopic effects are found to reduce the accumulation of high-Z tungsten (W). Moreover, the isotopic effects on the peaking factor of trace high-Z W get stronger with stronger magnetic shear.

On the threshold of magnetic island width in nonlinear mutual destabilization of tearing mode and ion temperature gradient mode

Nonlinear multi-scale interactions between the tearing mode and the ion temperature gradient (ITG) mode are investigated by means of numerical simulations in a self-consistent 5-field Landau-fluid model. It is observed that there exists a threshold of magnetic island width in the nonlinear evolution of interaction, above which the ITG turbulence can enhance the island growth significantly. Dependence of the threshold on basic plasma parameters is deeply analyzed. It is found that the higher ion viscosity may raise the threshold through its effect on the E×B drift and the diamagnetic drift of electron density gradient in different ways, both of which play a synergetic role in determining the threshold. Moreover, the effects of plasma resistivity, gradient length of equilibrium current sheet as well as magnetic shear of field line on the threshold are discussed based on the analyses of the initial growth rate of islands.

In-out asymmetry of zonal flow shear and turbulence reduction

In-out asymmetry of ion temperature gradient turbulence in toroidal geometry is studied by performing nonlinear gyrokinetic simulation using the GyroKinetic Plasma Simulation Program code [Kwon et al., Nucl. Fusion 52, 013004 (2012)]. Effects of self-generated zonal flow shear on the in-out asymmetry of radial correlation length and amplitude of turbulence are addressed by varying collisionality. Both quantities exhibit strong in-out asymmetry (longer and higher, respectively, at the low field side) in the absence of zonal flows. When the zonal flow shear (which is higher at the low field side) gets stronger, the radial correlation length decreases with its in-out asymmetry also getting reduced as expected from E×B shear decorrelation theory. On the other hand, in-out asymmetry of turbulence amplitude behaves differently from that of the radial correlation length.

Gyrokinetic study of turbulent convection of heavy impurities in tokamak plasmas at comparable ion and electron heat fluxes

In tokamaks, the role of turbulent transport of heavy impurities, relative to that of neoclassical transport, increases with increasing size of the plasma, as clarified by means of general scalings, which use the ITER standard scenario parameters as reference, and by actual results from a selection of discharges from ASDEX Upgrade and JET. This motivates the theoretical investigation of the properties of the turbulent convection of heavy impurities by nonlinear gyrokinetic simulations in the experimentally relevant conditions of comparable ion and electron heat fluxes. These conditions also correspond to an intermediate regime between dominant ion temperature gradient turbulence and trapped electron mode turbulence. At moderate plasma toroidal rotation, the turbulent convection of heavy impurities, computed with nonlinear gyrokinetic simulations, is found to be directed outward, in contrast to that obtained by quasi-linear calculations based on the most unstable linear mode, which is directed inward. In this mixed turbulence regime, with comparable electron and ion heat fluxes, the nonlinear results of the impurity transport can be explained by the coexistence of both ion temperature gradient and trapped electron modes in the turbulent state, both contributing to the turbulent convection and diffusion of the impurity. The impact of toroidal rotation on the turbulent convection is also clarified.

Gradient-driven flux-tube simulations of ion temperature gradient turbulence close to the non-linear threshold

It is shown that Ion Temperature Gradient turbulence close to the threshold exhibits a long time behaviour, with smaller heat fluxes at later times. This reduction is connected with the slow growth of long wave length zonal flows, and consequently, the numerical dissipation on these flows must be sufficiently small. Close to the nonlinear threshold for turbulence generation, a relatively small dissipation can maintain a turbulent state with a sizeable heat flux, through the damping of the zonal flow. Lowering the dissipation causes the turbulence, for temperature gradients close to the threshold, to be subdued. The heat flux then does not go smoothly to zero when the threshold is approached from above. Rather, a finite minimum heat flux is obtained below which no fully developed turbulent state exists. The threshold value of the temperature gradient length at which this finite heat flux is obtained is up to 30% larger compared with the threshold value obtained by extrapolating the heat flux to zero, and the cyclone base case is found to be nonlinearly stable. Transport is subdued when a fully developed staircase structure in the E × B shearing rate forms. Just above the threshold, an incomplete staircase develops, and transport is mediated by avalanche structures which propagate through the marginally stable regions.

A 5D gyrokinetic full-[formula omitted] global semi-Lagrangian code for flux-driven ion turbulence simulations

This paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic Gysela code is one of the few international 5D gyrokinetic codes able to perform global, full-f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the Gysela code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties.

Comparison of gradient and flux driven gyro-kinetic turbulent transport

Flux and gradient driven ion temperature gradient turbulence in tokamak geometry and for Cyclone base case parameters are compared in the local limit using the same underlying gyro-kinetic turbulence model. The gradient driven turbulence described using the flux tube model with periodic boundary conditions has a finite ion heat flux Qi≈10n0T0ρ*2vth, where n0 (T0) is the background density (temperature), ρ*=ρ/R is the normalized Larmor radius, R is the major radius of the device, and vth is the ion thermal velocity at the nonlinear threshold of the temperature gradient length for turbulence generation. Consequently, the gradient driven local transport model is unable to accurately describe heat fluxes below Qi<10n0T0ρ*2vth, since no stationary fully developed turbulent state can be obtained. The turbulence in the flux driven case shows intermittent behaviour and avalanches for Qi<10n0T0ρ*2vth. Isolated avalanches disappear for Qi>10n0T0ρ*2vth, and at higher heat fluxes, the statistics of the turbulence is the same for the flux and gradient driven case. The nonlinear upshift of the temperature gradient length threshold for turbulence generation (known as the Dimits shift) is larger in the case of flux driven turbulence. This higher nonlinear upshift is attributed to the generation of structures in the radial temperature profile, known as staircases [Dif-Pradalier, Phys. Rev. E 82, 025401 (2010)]. Avalanches are initiated at specific locations and have roughly the same radial extent of 50–70 ion Larmor radii. The staircases are obtained at low heating rates, and become unstable and break up at higher heating rates. At the heat fluxes for which staircase formation is observed, no stationary gradient driven simulations can be obtained.

Enhancement of residual stress by electromagnetic fluctuations: A quasi-linear study

A study is conducted on the impact of electromagnetic (EM) fluctuations on residual Reynolds stress in the context of the quasi-linear theory. We employ a fluid formulation describing EM ion temperature gradient turbulence. Analyses show that finite plasma β (=plasma thermal energy/magnetic energy) significantly increases the residual stress, potentially leading to the strong enhancement of flow generation in high β plasmas. We identify that this strong increase of residual stress originates from the reinforcement of radial ⟨k∥⟩ (=spectrally averaged parallel wavenumber) asymmetry due to the deformation of eigenfunctions near a rational surface.

Multi-scale gyrokinetic simulations: Comparison with experiment and implications for predicting turbulence and transport

To better understand the role of cross-scale coupling in experimental conditions, a series of multi-scale gyrokinetic simulations were performed on Alcator C-Mod, L-mode plasmas. These simulations, performed using all experimental inputs and realistic ion to electron mass ratio ((mi/me)1∕2 = 60.0), simultaneously capture turbulence at the ion (kθρs∼O(1.0)) and electron-scales (kθρe∼O(1.0)). Direct comparison with experimental heat fluxes and electron profile stiffness indicates that Electron Temperature Gradient (ETG) streamers and strong cross-scale turbulence coupling likely exist in both of the experimental conditions studied. The coupling between ion and electron-scales exists in the form of energy cascades, modification of zonal flow dynamics, and the effective shearing of ETG turbulence by long wavelength, Ion Temperature Gradient (ITG) turbulence. The tightly coupled nature of ITG and ETG turbulence in these realistic plasma conditions is shown to have significant implications for the interpretation of experimental transport and fluctuations. Initial attempts are made to develop a “rule of thumb” based on linear physics, to help predict when cross-scale coupling plays an important role and to inform future modeling of experimental discharges. The details of the simulations, comparisons with experimental measurements, and implications for both modeling and experimental interpretation are discussed.

Interaction between neoclassical effects and ion temperature gradient turbulence in gradient- and flux-driven gyrokinetic simulations

Neoclassical and turbulent transport in tokamaks has been studied extensively over the past decades, but their possible interaction remains largely an open question. The two are only truly independent if the length scales governing each of them are sufficiently separate, i.e., if the ratio ρ* between ion gyroradius and the pressure gradient scale length is small. This is not the case in particularly interesting regions such as transport barriers. Global simulations of a collisional ion-temperature-gradient-driven microturbulence performed with the nonlinear global gyrokinetic code Gene are presented. In particular, comparisons are made between systems with and without neoclassical effects. In fixed-gradient simulations, the modified radial electric field is shown to alter the zonal flow pattern such that a significant increase in turbulent transport is observed for ρ*≳1/300. Furthermore, the dependency of the flux on the collisionality changes. In simulations with fixed power input, we find that the presence of neoclassical effects decreases the frequency and amplitude of intermittent turbulent transport bursts (avalanches) and thus plays an important role for the self-organisation behaviour.