Toroidal Electron Temperature Gradient Research Articles

Electron temperature gradient driven transport model for tokamak plasmas

A new model for electron temperature gradient (ETG) modes is developed as a component of the multi-mode anomalous transport module [Rafiq et al., Phys Plasmas 20, 032506 (2013)] to predict a time-dependent electron temperature profile in conventional and low aspect ratio tokamaks. This model is based on two-fluid equations that govern the dynamics of low-frequency short- and long-wavelength electromagnetic toroidal ETG driven drift modes. A low collisionality NSTX discharge is used to scan the plasma parameter dependence on the ETG real frequency, growth rate, and electron thermal diffusivity. Electron thermal transport is discovered in the deep core region where modes are more electromagnetic in nature. Several previously reported gyrokinetic trends are reproduced, including the dependencies of density gradients, magnetic shear, β and gradient of β (β′), collisionality, safety factor, and toroidicity, where β is the ratio of the plasma pressure to the magnetic pressure. The electron heat diffusivity associated with the ETG mode is discovered to be on a scale consistent with the experimental diffusivity determined by power balance analysis.

Excitation of zonal flow by intermediate-scale toroidal electron temperature gradient turbulence

On the basis of gyrokinetic theory, we derive nonlinear equations for the zonal flow (ZF) generation in intermediate-scale electron temperature gradient (ETG) turbulence (with wavelength much shorter than the ion Larmor radius but much longer than the electron Larmor radius) in nonuniform tokamak plasmas. Both the spontaneous and forced generation of ZFs are kept on the same footing. The resultant Schrödinger equation for the ETG amplitude is characterized by a Navier–Stokes type nonlinearity, which is typically stronger than the Hasegawa–Mima type nonlinearity resulting from the fluid approximation. The physics underlying the three stages of ZF generation process is clarified, and the role of parallel mode structure decoupling is discussed. It is found that ZFs can be more easily excited in the intermediate-scale ETG turbulence than in the short wavelength regime.

Toroidal and slab ETG instability dominance in the linear spectrum of JET-ILW pedestals

Local linear gyrokinetic simulations show that electron temperature gradient (ETG) instabilities are the fastest growing modes for in the steep gradient region for a JET pedestal discharge (92174) where the electron temperature gradient is steeper than the ion temperature gradient. Here, ky is the wavenumber in the direction perpendicular to both the magnetic field and the radial direction, and ρi is the ion gyroradius. At , the fastest growing mode is often a novel type of toroidal ETG instability. This toroidal ETG mode is driven at scales as large as and at a sufficiently large radial wavenumber that electron finite Larmor radius effects become important; that is, , where Kx is the effective radial wavenumber. Here, ρe is the electron gyroradius, R0 is the major radius of the last closed flux surface, and 1/LTe is an inverse length proportional to the logarithmic gradient of the equilibrium electron temperature. The fastest growing toroidal ETG modes are often driven far away from the outboard midplane. In this equilibrium, ion temperature gradient instability is subdominant at all scales and kinetic ballooning modes are shown to be suppressed by shear. ETG modes are very resilient to shear. Heuristic quasilinear arguments suggest that the novel toroidal ETG instability is important for transport.

Self-organization of zonal flows and isotropic eddies in toroidal electron temperature gradient driven turbulence

Self-organization in the toroidal electron temperature gradient driven (ETG) turbulence is investigated based on a global gyrokinetic model in a weak magnetic shear configuration. Because of global profile effects, toroidal ETG modes with a higher toroidal mode number n are excited at the outer magnetic surfaces, leading to strong linear wave dispersion. The resulting anisotropic wave turbulence boundary and the inverse energy cascade generate the self-organization of zonal flows, which is the unique mechanism in the global gyrokinetic model. The self-organization is confirmed both in the decaying turbulence initialized by random noises and in the toroidal ETG turbulence. It is also shown that the self-organization process generates zonal flows and isotropic eddies depending on a criterion parameter, which is determined by the ion to electron temperature ratio and the turbulence intensity.

Interchange destabilization of collisionless tearing modes by temperature gradient

Using a fluid theory, the stability of collisionless tearing modes in plasmas is analyzed in the presence of an inhomogeneous magnetic field, electron temperature and density gradients. It is shown that small scale modes, characterized by a negative stability parameter (Δ′<0), can be driven unstable due to a combination of the magnetic field and electron temperature gradients. The destabilization mechanism is identified as of the interchange type similar to that for toroidal electron temperature gradient modes.

The study of the transition regime between slab and mixed slab-toroidal electron temperature gradient modes in a basic experiment

Electron temperature gradient (ETG) modes are suspected sources of anomalous electron thermal transport in magnetically confined plasmas as in tokamaks. Prior work in the Columbia Linear Machine (CLM) has been able to produce and identify slab ETG modes in a slab geometry [Wei et al., Phys. Plasmas 17, 042108 (2010)]. Now by modifying CLM to introduce curvature to the confining axial magnetic field, we have excited mixed slab-toroidal modes. Linear theory predicts a transition between slab and toroidal ETG modes when k∥Rckyρ∼1 [J. Kim and W. Horton, Phys. Fluids B 3, 1167 (1991)]. We observe changes in the mode amplitude for levels of curvature Rc−1≪k∥,slabk⊥ρ, which may be explained by reductions in k∥ in the transition from slab to mixed slab-toroidal modes, as also predicted by theory. We present mode amplitude scaling as a function of magnetic field curvature. Over the range of curvature available in CLM experimentally we find a modest increase in saturated ETG potential fluctuations (∼1.5×), and a substantial increase in the power density of individual mode peaks (∼4–5×).

Turbulent electron transport in edge pedestal by electron temperature gradient turbulence

We present a model for turbulent electron thermal transport at the edge pedestal in high (H)-mode plasmas based on electron temperature gradient (ETG) turbulence. A quasi-linear analysis of electrostatic toroidal ETG modes shows that both turbulent electron thermal diffusivity and hyper-resistivity exhibits the Ohkawa scaling in which the radial correlation length of turbulence becomes the order of electron skin depth. Combination of the Ohkawa scales and the plasma current dependence results in a novel confinement scaling inside the pedestal region. It is also shown that ETG turbulence induces a thermoelectric pinch, which may accelerate the density pedestal formation.

Experimental study of parametric dependence of electron-scale turbulence in a spherical tokamak

Electron-scale turbulence is predicted to drive anomalous electron thermal transport. However, experimental study of its relation with transport is still in its early stage. On the National Spherical Tokamak Experiment (NSTX), electron-scale density fluctuations are studied with a novel tangential microwave scattering system with high radial resolution of ±2 cm. Here, we report a study of parametric dependence of electron-scale turbulence in NSTX H-mode plasmas. The dependence on density gradient is studied through the observation of a large density gradient variation in the core induced by an edge localized mode (ELM) event, where we found the first clear experimental evidence of density gradient stabilization of electron-gyro scale turbulence in a fusion plasma. This observation, coupled with linear gyro-kinetic calculations, leads to the identification of the observed instability as toroidal electron temperature gradient (ETG) modes. It is observed that longer wavelength ETG modes, k⊥ρs≲10 (ρs is the ion gyroradius at electron temperature and k⊥ is the wavenumber perpendicular to local equilibrium magnetic field), are most stabilized by density gradient, and the stabilization is accompanied by about a factor of two decrease in electron thermal diffusivity. Comparisons with nonlinear ETG gyrokinetic simulations show ETG turbulence may be able to explain the experimental electron heat flux observed before the ELM event. The collisionality dependence of electron-scale turbulence is also studied by systematically varying plasma current and toroidal field, so that electron gyroradius (ρe), electron beta (βe), and safety factor (q95) are kept approximately constant. More than a factor of two change in electron collisionality, νe*, was achieved, and we found that the spectral power of electron-scale turbulence appears to increase as νe* is decreased in this collisonality scan. However, both linear and nonlinear simulations show no or weak dependence with the electron-ion collision frequency, νe/i. Instead, other equilibrium parameters (safety factor, electron density gradient, for example) affect ETG linear growth rate and electron thermal transport more than νe/i does. Furthermore, electron heat flux predicted by the simulations is found to have an order-of-magnitude spatial variation in the experimental measurement region and is also found to be much smaller than experimental levels except at one radial location we evaluated. The predicted electron heat flux is shown to be strongly anti-correlated with density gradient, which varies for a factor of three in the measurement region, which is in agreement with the density gradient dependence study reported in this paper.

Role of secondary long wavelength structures in the saturation of electron temperature gradient driven turbulence

The dynamics of secondary long wavelength structures (LWSs) in electron temperature gradient (ETG) driven turbulence are investigated by performing gyrofluid simulations and modeling analyses in a slab geometry with an emphasis of the underlying nonlinear interaction processes. It is shown that the back-reaction of the secondary LWS on the ambient fluctuations essentially contributes to saturating ETG instability and limiting the electron transport. The LWS is nonlinearly generated mainly through the beating of the most unstable ETG modes, even a weak modulation instability. The back-reaction is identified as the enhanced stabilization of the ETG modes due to the streamer-type feature of the LWS, which dominantly produces a local poloidal mode coupling among unstable and highly damped spectral components to form a global mode, besides the suppression effect of the LWS due to the radial shearing decorrelation and/or the radial mode coupling. Finally, the correspondence between the LWS in the slab model and the quasimode observed in toroidal ETG simulation [Z. Lin et al., Phys. Plasmas 12, 056125 (2005)] and the importance of the nonlinear mode coupling in the multiscale turbulence interaction are discussed.

Global profile effects and structure formations in toroidal electron temperature gradient driven turbulence

Using a gyrokinetic toroidal particle code with global profile effects, the toroidal electron temperature gradient driven (ETG) turbulence in positive and reversed shear tokamaks is studied. In the simulation, initial saturation levels of the ETG mode are consistent with the mixing length theory, which shows a Bohm (gyro-Bohm) like ρ*-scaling for a ballooning type (slab like) ETG mode in a positive (reversed) shear configuration, where ρ* is the electron Larmor radius ρte divided by the minor radius a. In a realistic small ρ* positive shear configuration, the ETG mode has a higher saturation level than the large ρ* positive shear configuration and the reversed shear configuration. In the nonlinear turbulent state, the ETG turbulence in the positive and reversed shear configurations shows quite different structure formations. In the positive shear configuration, the ETG turbulence is dominated by streamers which have a ballooning type structure, and the electron temperature Te profile is quickly relaxed by enhanced heat transport in a turbulent time scale. In the reversed shear configuration, quasi-steady zonal flows are produced in the negative shear region, while the positive shear region is characterized by streamers. Accordingly, the electron thermal diffusivity χe has a gap structure across the qmin surface, and the Te gradient is sustained above the critical value for a long time. The results suggest a stiffness of the Te profile in positive shear tokamaks and a possibility of the Te transport barrier in reversed shear tokamaks.

Dynamics of large-scale structure and electron transport in tokamak microturbulence simulations

The important issue of whether zonal flows or streamers are preferentially formed in plasma turbulence with electron gyroradius scale is studied based on a gyrofluid model of electron temperature gradient (ETG) driven turbulence. Results from three approaches are presented. It is analytically derived first that the secondary generation of different large-scale structures is determined by the spectral anisotropy of turbulent fluctuation in two-dimensional Charney–Hasegawa–Mima turbulence. This is verified subsequently using three-dimensional simulations of sheared slab ETG turbulence, which show that the magnetic shear governs the pattern selection. It is found that a weak shear favours the enhancement of zonal flows so that the electron transport is strongly suppressed. In contrast, radially elongated streamers are formed nonlinearly in stronger-shear ETG turbulence. Finally, three-dimensional toroidal ETG simulations show that streamers are excited in the linearly stable region along the field (i.e. good curvature region) through a modulation instability after initial saturation of ETG modes. Although the electron transport at the quasi-steady state becomes higher than the initial saturation level, which is dominated by fluctuations with a peaked spectrum, the averaged value is still low at around the gyro-Bohm level. Furthermore, it is shown that the enhanced zonal flows in weak shear ETG turbulence may be limited by a Kelvin–Helmholtz instability. Also, it is found that the electromagnetic effects reduce the generation of zonal flows and reverse the so-called Okawa-scaling of electron transport on the β dependence.

Electron thermal diffusivity due to the electron temperature gradient mode.

Charge neutrality breaks down in the short wavelength toroidal electron temperature gradient mode. In contrast to the ion temperature gradient mode, the wave number normalized by the Debye wave number, k/k(De), appears as a natural scale parameter, rather than the finite Larmor radius parameter k( perpendicular )rho(e). The growth rate and consequent mixing length estimate yields an electron thermal diffusivity large enough to be relevant to tokamaks.

Toroidal electron temperature gradient mode structure in the presence of nonuniform background flows

The influence of nonuniform poloidal and toroidal background plasma flows on the spatial structure and growth rate of the electrostatic electron temperature gradient (ETG) mode is investigated in the linear approximation. This derivation includes the ballooning mode formalism and a more recently developed version of the direct method by Taylor and Wilson [Plasma Phys. Controlled Fusion 38, 1999 (1996)]. It is shown that the growth rate of the ETG mode is not changed significantly by flow shear. However, it is found that the spatial structure of the ETG mode depends crucially on the derivative of the flow shear rate with respect to the minor radius of the tokamak cross section and also depends crucially on the magnetic shear. For moderate magnetic shear, the unstable ETG mode is strongly localized in the poloidal direction and is elongated along the radial direction, with a characteristic radial width much larger than the electron Larmor radius. This may explain the formation of streamer structures above the threshold of ETG mode instability. Streamers are believed to enhance electron thermal transport beyond the values provided by simple mixing length estimates. For very low values of magnetic shear, the ETG mode structure becomes extended in the poloidal direction, and the ballooning formalism does not apply. In this case, the direct method is used and it is shown that the ETG mode is strongly localized in the radial direction. The small radial extent of these modes may considerably reduce electron heat transport, which would enhance the formation of an electron thermal transport barrier.

Collisional damping of ETG-mode-driven zonal flows.

We study collisional damping of electron zonal flows in toroidal electron temperature gradient (ETG) turbulence due to the friction between trapped and untrapped electrons. With the assumption of adiabatic ions, the collisional damping is shown to occur on fast time scales approximately 0.24epsilon(1/2)tau(e). The comparison with the growth rate of electron zonal flows indicates that the shearing by electron zonal flows is unlikely to be a robust mechanism for regulating ETG turbulence. This finding vitiates the claims of several simulation studies that have ignored the effects of collisional damping of electron zonal flows and offers a possible partial explanation of the high levels of electron thermal transport observed in the National Spherical Torus Experiment.

Linear aspects of streamer formation in electron temperature gradient turbulence

The dependence of radially elongated streamer formation in toroidal electron temperature gradient driven turbulence on the magnetic field geometry (specified by shear s and shift α) is well described by means of a linear gyrokinetic analysis of the radial mode number spectrum.

Critical gradient formula for toroidal electron temperature gradient modes

Under certain conditions, the electron heat transport induced by electron temperature gradient (ETG) streamers is sufficiently large and sensitive with respect to the normalized electron temperature gradient to represent a possible cause for electron temperature profile consistency (“stiffness”). Here, linear gyrokinetic simulations of toroidal ETG modes in tokamak core and edge plasmas are presented. An algebraic formula for the threshold of the linear instability is derived from the numerical solutions of the linear gyrokinetic equations which recovers previous analytical results in the appropriate limits.

Electromagnetic effect on the toroidal electron temperature gradient mode

The electromagnetic effect on the toroidal electron temperature gradient mode is investigated using local kinetic and fluid theories. It is found that the electromagnetic effect on the toroidal mode differs from that for the slab mode. For the slab mode, the electromagnetic effect gives a strong stabilization, while for the toroidal mode there is a destabilization. In particular, the modes with perpendicular wavelength comparable to the electron skin depth are destabilized by the finite electron inertia. It is shown that this contrasting result for the toroidal mode occurs mainly because the stabilizing effect of the parallel transit drift is significantly reduced by the finite electromagnetic term.

Electromagnetic semi-local kinetic analysis of the electron temperature gradient mode in tokamaks

Electromagnetic (finite beta) effects on the toroidal electron temperature gradient mode have been investigated in terms of an electromagnetic semi-local kinetic dispersion relation. For this mode, finite beta is destabilizing, in contrast to the ion temperature gradient mode. However, a relatively steep temperature gradient, ϵT = LT/R < 0.15, is required for the instability. The most unstable mode satisfies ω ≃ ωTe (the electron transit mode).