Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks

The gyrokinetic equations predict that various drift type waves or modes can be unstable in a tokamak. For some of these modes, such as the ion temperature gradient mode and the electron temperature gradient mode, there exists a critical gradient, above which the mode is unstable. There are also drift type modes, such as the trapped electron mode, which do not have a critical ion temperature gradient, but have a threshold which depends upon a variety of parameters. Since the existence of unstable modes can cause increased transport, plasmas which are centrally heated tend to increase in temperature gradient until the modes become unstable. Here the measured ion temperature gradients are compared to the “critical” gradients as calculated by a gyrokinetic linear stability (GKS) code [M. Kotschenreuther et al., Phys. Plasmas 2, 2381 (1995)]. Also, the maximum linear growth rate as calculated by this code is compared to experimentally derived transport coefficients. The results show that for low confinement mode (L-mode) discharges, the measured ion temperature gradient is significantly greater than the GKS calculated critical gradient over a large region of the plasma. This is the same region of the plasma where the ion thermal diffusivity is large. For high confinement mode (H-mode) discharges the ion temperature gradient is closer to the critical gradient, but often still greater than the critical gradient over some region. For the best H-mode discharges, the ion temperature is less than or equal to the critical gradient over the whole plasma. In general, the position in the plasma where the ion thermal diffusivity starts to increase rapidly is where the maximum linear growth rate is greater than the E×B shearing rate.

It is shown that the toroidal ion temperature gradient (ITG) mode has an upper ηi (higher ηi) stability regime for experimentally relevant parameter values in addition to the lower regime with the stability threshold at ηi around one ( where Ln and LT1 are the characteristic lengths for the density and ion temperature gradients). The ITG mode is studied with a focus on the upper ηi stability regime and the β dependence (β = plasma pressure/magnetic pressure). The results of a gyrokinetic and a two-fluid model as well as a semilocal approximation are compared. It is shown that the upper stability threshold is very sensitive to and considerably reduced by finite-β effects. It is also sensitive to finite Larmor radius (FLR) effects and to (LB is the characteristic length for the toroidal magnetic field gradient). Predictions and comparisons are made with data from a joint European torus (JET) optimized shear discharge and a JET hot ion H-mode (high-performance mode) discharge.

In this study, global nonlinear electromagnetic gyrokinetic simulations are conducted to investigate turbulence 
in the internal transport barrier (ITB) region of the EAST discharge with weakly reversed magnetic shear. 
Linear simulations reveal two dominant ion temperature gradient (ITG) modes: 
a higher frequency mode at the $q=1$ surface, which dominates in the electrostatic limit, 
and a lower frequency mode near the $q_{\min}$ surface, which prevails under the experimental $\beta$ 
(the ratio of plasma pressure to magnetic pressure).
Therefore, electromagnetic effects play a important role in stabilizing ITG modes, 
and in causing the transition between the most unstable mode at different radial positions.
The linear growth rate of the unstable mode in the electrostatic limit is approximately 1.25 times higher than that of the dominant mode in the electromagnetic case.
However, in the electromagnetic nonlinear regime,
the thermal ion heat conductivity is reduced by at least a factor of 4.
This reduction primarily results from nonlinear electromagnetic effects enhancing the shearing effect of zonal flows, 
thereby further suppressing microturbulence.
It is emphasized that the electromagnetic effect on ITG with weak magnetic shear 
should be included to accurately calculate the turbulent transport.

(Received 7 December 2012 /Accepted 10 May 2013)To study the nonlinear saturation mechanism in turbulent plasmas, we plan to obseve ion temperature gra-dient (ITG) modes in the Plasma Assembly for Nonlinear Turbulence Analysis (PANTA). Linear growth ratesof ITG modes in the linear device are analyzed to determine their excitation conditions using a three-field fluidmodel to describe the ion motion. Parameter scans of the linear growth rate show the threshold for ITG modeexcitation, and the typical plasma parameters of the PANTA are found in the unstable domain.

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We investigated the modulational stability and instability of the ion temperature gradient (ITG) mode in electron-ion plasma. Ions are dynamic species, whereas electrons follow a Kappa distribution. We used the reduction perturbation approach to determine the linear dispersion relation for the fluid under consideration. The nonlinear Schrodinger equation describes nonlinear features such as nonlinear modulational stability or instability in the ITG mode. The product of dissipation and nonlinear coefficients, known as LM, contains both modulational stability and instability in the ion temperature gradient mode. Theoretical results are expanded numerically, showing the influence of various plasma parameters on modulational stability and instability, particularly the superthermality coefficient κ e . The current observations may be extended to space and laboratory plasma for modification.

Linear gyrokinetic simulations have been performed using the GTC code to investigate the effects of fast ions on dominant instability changes from TEM to ITG in the EAST. The effects of fractions (5%, 7.5%, and 10%) and temperatures (10 keV, 20 keV, and 30 keV) of fast ions are investigated. The dominant mode transitions from trapped electron mode (TEM) to ion temperature gradient (ITG) mode under different plasma parameters. A transition of the dominant instability is observed at 0.7 ≤ kθρs ≤ 0.8 and shifts to kθρs ≈ 0.7 when fast ions are considered. In addition, fast ions stabilize ITG modes, and the effect becomes more significant as the fraction of fast ions increases, reducing the growth rate by approximately 10% when the fraction of fast ions is 10%. The stabilizing effect increases with increasing temperature of fast ions, but varies little for Tf > 20 keV. Fast ions reduce the ion temperature gradient R/LTi threshold of the dominant instability changes from TEM to ITG. The threshold decreases with decreasing temperature of fast ions, with a reduction of about 10%-20% when the fraction reaches 10% in hydrogen plasmas, and about 3%-12% when deuterium is the main ion. Furthermore, the normalized electron temperature gradient R/LTe for the dominant instability changes from ITG to TEM is lower in hydrogen plasmas, decreasing from 5.4 to 4.1 as Tf increases from 10 keV to 30 keV, while in deuterium plasmas it decreases from 10.5 to 9.5. Moreover, electron–electron collisions lead to the dominant instability changes from TEM to ITG at 0.4 ≤ kθρs ≤ 0.8 and strongly stabilize the ITG mode, reducing its growth rate about 50%, while ion–ion collisions have little influence on TEM and ITG.

Turbulent transport and zonal flow (ZF) dynamics in the mixed kinetic ballooning mode (KBM) and ion temperature gradient (ITG) mode dominated turbulence states are analyzed by gyrokinetic simulations based on the experimental observations of KBMs in the HL-2A Internal Transport Barrier (ITB) plasmas with weak magnetic shear configuration. Results have shown that KBMs and ITGs are the main candidates for turbulent transport and the inclusion of fast ions (FIs) can destabilize both instabilities, which is resulted by the decrease of ballooning parameter αMHD hence reduced Shafranov shift stabilization due to the negative FI density gradient, making them more unstable according to the ŝ-αMHD diagram. In addition, the ITGs are suppressed by both dilution and finite-β stabilization effects caused by FIs. Nonlinear simulations excluding the effects of FIs indicate that the transport is minimum at βe=βeexp as the turbulence predominated by ITGs is strongly suppressed by the nonlinear electromagnetic (EM) stabilization and enhancement of ZFs. However, the transport is further increased with β e although the ZFs become stronger due to the transition from ITG to KBM dominated turbulence state. The presence of FIs can modify the relation between ZF shearing rate and β e. The transport level is insensitive to β e when βe⩽βeexp but increases significantly because of the KBM destabilization. Meanwhile, the ZF amplitude reaches maximum at βe=βeexp whereas it suffers an erosion at higher values, implying that the plasma might be a self-organized critical state owning to the interactions among ZFs, KBMs and FI effects hence setting the plasma β around β exp. The results have also provided a possible explanation that the destabilization of EM turbulence is responsible for the ion heat transport stiffness under weak magnetic shear configurations in the off-axis neutral beam injection heated ITB plasmas on HL-2A.

Using the averaged magnetic drift model and a first-order finite Larmor radius expansion, the eigenvalue equation for the ion temperature gradient (ITG) mode in tokamak plasmas is reduced to a Schrödinger-type differential equation. By invoking generalized translational invariance, the model is extended to reversed magnetic shear (RMS) configurations and benchmarked against global gyrokinetic simulations from GTC, showing good quantitative agreement. The analysis reveals a characteristic double-well potential unique to RMS profiles, which gives rise to the degeneracy between the lowest-order even and first-order odd eigenmodes when the two potential wells are sufficiently separated radially. The ITG instability is also found to resonate with the magnetic drift frequency, and its maximum growth occurs when the two rational surfaces are slightly separated. These results provide new physical insight into ITG mode behavior under RMS and offer a compact, accurate theoretical framework that bridges simplified analytic models and global simulations.

Global gyrokinetic particle simulations show that equilibrium radial electric field (E r ) shear reduces the linear growth rate, ion heat conductivity, and nonlinear turbulence amplitude for both the ion temperature gradient (ITG) and kinetic ballooning mode (KBM) microturbulence by tilting the poloidal mode structure. The increase in the magnetic shear enhances the stabilizing performance of the E r shear on linear growth rate for the ITG case but has no effect on that for the KBM case. The radial correlation length of the ITG turbulence is decreased by increasing the magnetic shear in a weak ion diamagnetic flow shear condition with low β, leading to a reduction in the effective E × B shearing rate, which weakens the suppression performance of the E r shear on the ITG turbulence amplitude. In contrast, under a larger ion diamagnetic shear flow for higher β, an increase in magnetic shear strengthens the suppression performance of the E r shear on the KBM turbulence amplitude due to an increase in the effective shearing rate by increasing the radial correlation length of the turbulence.

With the term microinstabilities, we indicate a class of plasma drift waves such as the ion temperature gradient (ITG) mode, the trapped electron mode (TEM), and the electromagnetic Alfvénic ion temperature gradient (AITG) mode. They are usually driven by spatial gradients in the plasma and they are held responsible for anomalous transport and enhanced heat flux in tokamaks. Their understanding is therefore crucial for the development of magnetically confined plasma fusion devices. Our approach is based on the gyrokinetic model (Hahm 1988) for particle dynamics. The gyrokinetic nonlinear Vlasov equation, in the presence of an electromagnetic perturbation (Brizard 1989, 1995), is derived using Lie perturbation theory (Cary and Littlejohn 1983) to average out the fast particle motion around the gyrocenter. We have linearized the gyrokinetic Vlasov equation and added the quasi‐neutrality equation and Ampère's law, providing closure to the system. A temporal spectral formulation is used, ensuring the absence of numerical instabilities due to high frequency modes. The equations are simplified with a large aspect ratio approximation for an axisymmetric toroidal configuration. The fields are expanded in a Fourier series in the three spatial directions (radial, poloidal, and toroidal) reducing the problem in a matrix form. The code EM‐GLOGYSTO (Brunner et al. 1998; Falchetto et al. 2003; Joint Varenna‐Lausanne International Workshop 2002) has been developed to look at its solutions. Recently, we have included in the model an electrostatic equilibrium potential in order to provide E×B sheared flow. The code has been optimized for running in a parallel environment using MPI routines, and it allows us to study frequencies, growth rates, and mode structures of electromagnetic microinstabilities. Results on an AITG mode are shown to illustrate these features.

Local gyrokinetic simulations, performed with the GENE code, are used to investigate the interaction between (negative) triangularity $\delta$ and magnetic shear $s$ , in both ion temperature gradient and trapped electron mode dominated plasmas. Magnetic shear turns out to influence in a non-trivial way the effect of negative triangularity. Also, $\delta <0$ is found to lead to a reduction of fluxes when $s>1$ but not anymore when shear is reduced. In certain parameter regimes, this effect can be inverted, up to obtaining higher fluxes when $\delta <0$ with respect to its $\delta >0$ counterpart.

Based on first-principles nonlinear gyrokinetic simulations, the electrostatic turbulence properties in the internal transport barrier (ITB) region of an Experimental Advanced Superconducting Tokamak discharge (#93890) are investigated. Specifically, ITBs with steep density and temperature gradients are located in the weakly negative magnetic shear region at the plasma center. In the linear stage, the growth rate and frequency of the ion temperature gradient (ITG) mode increase significantly due to resonant excitation by trapped electrons. That is, the resonance between trapped electrons and the ITG becomes strong due to the precession drift reversal of trapped electrons by the negative magnetic shear and Shafranov shift. Meanwhile, the trapped electron mode is stable in the ITB region due to only a very small fraction of electrons precessing in the direction of the electron diamagnetic drift. Nonlinear simulations show that, after considering the non-adiabatic effect of trapped electrons, the heat conductivity of ions and the turbulence intensity increase by at least a factor of 7 compared with the results only considering the adiabatic effect of electrons. The zonal charge density of trapped electrons can partially cancel that of ions, which weakens the intensity of the zonal flow, and consequently reduces the zonal flow regulation and enhances the turbulent transport.

One metric for comparing con nement properties of di erent magnetic fusion energy con gurations is the linear critical gradient of drift wave modes. The critical gradient scale length determines the ratio of the core to pedestal temperature when a plasma is limited to marginal stability in the plasma core. The gyrokinetic turbulence code GS2 was used to calculate critical temperature gradients for the linear, collisionless ion tem- perature gradient (ITG) mode in the National Compact Stellarator Experiment (NCSX) and a prototypical shaped tokamak, based on the pro les of a JET H-mode shot and the stronger shaping of ARIES-AT. While a concern was that the narrow cross section of NCSX at some toroidal locations would result in steep gradients that drive instabilities more easily, it is found that other stabilizing e ects of the stellarator con guration o set this so that the normalized critical gradients for NCSX are competitive with or even better than for the tokamak. For the adiabatic ITG mode, NCSX and the tokamak had similar critical gradients, though beyond marginal stability, NCSX had larger growth rates. However, for the kinetic ITG mode, NCSX had a higher critical gradient and lower growth rates until a/LT ≈ 1:5 a/LT;crit, when it surpassed the tokamak's. A discussion of the results presented with respect to a/LT vs R/LT is included.

We investigate the linear theory of the ion-temperature-gradient (ITG) mode, with the goal of developing a general understanding that may be applied to stellarators. We highlight the Wendelstein 7X (W7-X) device. Simple fluid and kinetic models that follow closely from existing literature are reviewed and two new first-principle models are presented and compared with results from direct numerical simulation. One model investigates the effect of regions of strong localized shear, which are generic to stellarator equilibria. These “shear spikes” are found to have a potentially significant stabilizing affect on the mode; however, the effect is strongest at short wavelengths perpendicular to the magnetic field, and it is found to be significant only for the fastest growing modes in W7-X. A second model investigates the long-wavelength limit for the case of negligible global magnetic shear. The analytic calculation reveals that the effect of the curvature drive enters at second order in the drift frequency, confirming conventional wisdom that the ITG mode is slab-like at long wavelengths. Using flux tube simulations of a zero-shear W7-X configuration, we observe a close relationship to an axisymmetric configuration at a similar parameter point. It is concluded that scale lengths of the equilibrium gradients constitute a good parameter space to characterize the ITG mode. Thus, to optimize the magnetic geometry for ITG mode stability, it may be fruitful to focus on local parameters, such as the magnitude of bad curvature, connection length, and local shear at locations of bad curvature (where the ITG mode amplitude peaks).