Ion temperature gradient modes modulational stability with kappa-distribution

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.

(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.

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.

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.

The effects of dissipative trapped electrons on ion temperature gradient-driven instability (ηi mode) in tokamak plasmas are considered. A sheared slab geometry is adopted and a linearized fluid model of ion temperature gradient (ITG) mode including dissipative trapped electrons, which are described by means of the well-known formula of the nonadiabatic electron response [P. L. Similon and P. H. Diamond, Phys. Fluids 27, 916 (1984)], is presented. Results show that in tokamak plasmas not only is there a modification of the dissipative trapped electrons on the ηi mode, but also there may exist an intrinsic oscillation mode, namely, a hybrid dissipative trapped electron ion temperature gradient mode. The higher the dissipative trapped electron fraction, the more it drives the ηi mode, that is, when the trapped electron fraction is sufficient high and the trapped electrons are dissipated strongly, the mode is dominated by the trapped electron dynamics and propagates in the electron diamagnetic direction. These analytical results can be reduced to the usual predictions of the ion temperature gradient-driven instability in the absence of the dissipative trapped electron. Numerical results further show that (a) there may be a hybrid dissipative trapped electron ITG mode and the dissipative trapped electron effect is a destabilizing effect on this mode; and (b) when the trapped electron fraction is sufficiently high and the trapped electrons are dissipated strongly, the mode is determined by the trapped electron dynamics. These conclusions are in agreement with the experimental observations in the latest simulated tokamak experiment on the Columbia Linear Machine [J. Chen and A. K. Sen, Phys. Rev. Lett. 72, 3997 (1994)].

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.

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.

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.

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.

The linearized gyrokinetic equation governing electrostatic microinstabilities in the presence of sheared equilibrium flow in both the {cflx z} and {cflx y} directions has been systematically derived for a sheared slab geometry, where in the large aspect ratio limit {cflx z} and {cflx y} directions correspond to the toroidal and poloidal directions respectively. In the familiar long perpendicular wavelength regime ({kappa}{perpendicular}{rho}i > 1), the analysis leads to a comprehensive kinetic differential eigenmode equation which is solved numerically. The numerical results have been successfully cross-checked against analytic estimates in the fluid limit. For typical conditions, the Ion Temperature Gradient ({eta}i) modes are found to be stabilized for {cflx y}-direction flows with a velocity shear scale comparable to that of the ion temperature gradient and velocities of a few percent of the sound speed. Sheared flows in the {cflx z}-direction taken along are usually destabilizing, with the effect being independent of the sign of the flow. However, when both types are simultaneously considered, it is found that in the presence of shared {cflx z}-direction flow, sheared {cflx y}-direction flow can be either stabilizing or destabilizing depending on the relative sign of these flows. However, for sufficiently large values of {upsilon}{prime}{sub y} the mode is completely stabilized regardless of the sign of {upsilon}{prime}{sub z} {upsilon}{prime}{sub y}. The importance of a proper kinetic treatment of this problem is supported by comparisons with fluid estimates. In particular, when such effects are favorable, significantly smaller values of sheared {cflx y}-direction flow are required for stability than fluid estimates would indicate.

The turbulent transport properties and dynamics of zonal flows (ZFs) in the presence of fast ions (FIs) are investigated for a typical internal transport barrier (ITB) plasma based on the gyrokinetic approach, focusing on the role of FI temperature and the effects of the toroidal rotation, including the E× B rotational shear, parallel velocity gradient (PVG) as well as the rotation velocity itself. Linear GENE simulations have shown that the core ITB plasma on HL-2A is dominated by ion temperature gradient (ITG) modes and trapped electron modes (TEMs), where the former is stabilized by FIs whereas destabilized by the PVG. Neither of the FIs or the PVG has observable effect on TEMs. The ion heat transport generally decreases at large FI temperature due to the nonlinear electromagnetic stabilization of turbulence with increased total plasma β until electromagnetic modes are excited. The transport fluxes peak around a certain FI temperature and the ZF shearing rate is significantly higher at such value compared with that in the absence of FIs, and the heat flux reduction is a result of the synergistic interaction between turbulence, ZFs and the external rotational shear. The E× B shear stabilizing and PVG destabilizing is not obvious at low normalized ITG R/L Ti, indicating they are less important in determining the stiffness level in the relatively low density and rotation scenarios regarding the HL-2A ITB discharges. The turbulence suppression is predominated by the nonlinear stabilization of ITG turbulence as well as enhanced ZFs simultaneously in the presence of FIs. These results have also provided the possible way to reduce the turbulence transport through increasing the FI temperature in the off-axis neutral beam heated plasmas such as in HL-2A.

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.

Hot ion mode has been achieved on HL-2A tokamak with a low electron density and a moderate neutral beam injection power. It is characterized by a high ratio of ion temperature (T_i) and electron temperature (T_e). The T_i/T_e is higher than 3 and ion temperature reaches to 3.5keV in the experiments with heating power no more than 1.3MW. Strong internal transport barrier (ITB), fishbone modes, kinetic ballooning modes (KBM) and ion temperature gradient (ITG) modes are also observed in the hot ion plasma. Power balance analysis given by TRANSP code suggests that electron heating is dominant but thermal ion heat diffusivity is lower than the neoclassical diffusivity in ITB region. Further analysis based on GENE code shows the ITG modes induced heat transport is dominant in experiment, but KBM induced transport will play a more critical role when the electron beta reaches to 0.48\%. More interesting, the heat flux shows a great drop when the fast ions are taken into account. Those results may indicate that the fast ions have a stabilization effect on plasma turbulence, then reduce the thermal ion transport and finally contribute to achievement of high ion temperature and formation of hot ion mode.

The stability properties of the toroidal ion temperature gradient (ITG) mode change dramatically for plasmas with flat density profiles, as found in H-mode plasmas. Even though ηi ≫ 1, the relevant stability parameter is is the ion temperature gradient scale length and R is the major radius). It is proposed that a significant element in the improved confinement observed in the L- to H-mode transient is the diminution of ITG mode turbulence, mediated by broader H-mode ion temperature profiles. Confinement results of L- and H-mode simulations of DIII-D, based on a simplified energy transport equation, are presented to support this hypothesis.

The pivotal role played by radial electric fields in thedevelopment of turbulence associated with anomalous transportis examined by means of global gyrokinetic simulations. It isshown that the stabilizing effect of E×B flows on iontemperature gradient (ITG) modes is quadratic in the shearing rate amplitude. For a given shearing rate it leads to anincrease in the critical gradient. The electric fields (zonalflows) self-generated by ITG modes interact in a nonlinear wayand it is shown that a saturated level of both the zonal flowand ITG turbulence is reached in the absence of any collisionalmechanism being included in the model. The quality of theglobal nonlinear simulations is verified by the energyconservation which is allowed by the inclusion of nonlinearparallel dynamics. This demonstrates the absence of spurious damping of numerical origin and thus confirms the nonlinear character of zonal flow saturation mechanism.