Toroidal Ion Temperature Gradient Research Articles

ITG modes with finite-β effects and the upper ηi stability regime

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.

Initial Value Problem of the Toroidal Ion Temperature Gradient Mode

The initial value problem of the toroidal ion temperature gradient mode is studied based on the Laplace transform of the ion gyrokinetic equation and the electron Boltzmann relation with the charge neutrality condition. Due to the toroidal magnetic drift, the Laplace-transformed density and potential perturbations have a branch cut as well as poles on the complex-frequency plane. The inverse Laplace transform shows that the temporal evolution of the density and potential perturbations consists of the normal modes and the continuum mode, which correspond to contributions from the poles and the branch cut, respectively. The normal modes have exponential time dependence with the eigenfrequencies determined by the dispersion relation while the continuum mode shows power-law decay oscillation. For the stable case, the long-time asymptotic behavior of the potential and density perturbations is dominated by the continuum mode which decays slower than the normal modes.

Large ηi Toroidal ITG Modes in Fluid andKinetic Descriptions

A study of the toroidal ion temperature gradient (ITG) orηi mode [ηi =Ln/LT,Lj = j | dj/dr | -1(characteristic scale length)] at larger ηi values,ηi << 1, has been performed using a two-fluid model. Itis shown that the upper ηi stability threshold(ηiup) is in the range ηiup ≈ 10 orlarger and that it is sensitive to FLR (finite Larmor radius) effectsand toεn = 2Ln/LB.Comparisons are made with the results of a gyrokinetic model reportedearlier. The two-fluid description is found to reproduce the resultsof the gyrokinetic description with a good accuracy in the regimesb0 = k2ρi2 << 1and εn << 1 (k = perpendicular wavenumber,ρi = ion Larmor radius) whereas for largerb0 and εn the fluid model gives alower ηiup than the gyrokinetic model. In the lowb0/low εn regime the densitygradient independent relationεT = LT/LB ≈ b02τ/8at the upper ηi stability threshold is derived(τ = Te/Ti).

Theory of drift wave turbulence

Toroidal ion temperature gradient (ITG) driven drift mode turbulence has been analyzed analytically and numerically. By using weak nonlinearity arguments and random phase approximation, dynamic and wave kinetic equations are derived. It is found that three different nonlinearities, namely E × B, convective and diamagnetic nonlinearities, play important role in the turbulent spectral transfer. The power spectra of the weak ITG-mode turbulence are obtained analytically for |k| ≫ 1 and |k| < 1 ranges in the wave number space. It is shown that forward energy cascading due to convective and diamagnetic nonlinearities will balance the inverse energy cascading due to E × B nonlinearity at |k| ≈ 1/ρs (k is wave number, ρs = (cs/ωci), cs is the sound velocity and ωci is the ion cyclotron frequency) and results in energy condensation at |k| ≈ 1/ρs.

Toroidal ion temperature gradient mode stability at large temperature gradients

It is well known that the ion temperature gradient (ITG) or ηi mode (ηi = [(1/Ti)(dTi/dr)]/ [(1/n0)(dn0/dr)] = Ln/LT, Ln = density scale length, LT = temperature scale length) has an instability threshold (ηilow) around one for density scale lengths of the order of the magnetic field curvature radius LB. In this work the existence of an upper ηi stability threshold (ηiup) in the range ηiup ≈ 10 or larger is shown using a kinetic ion description. The lowest ηiup is found to occur at wavelengths corresponding to k⊥2ρi2 ≈ 1 (k⊥ = perpendicular wavenumber, ρi = ion Larmor radius). It is also shown that the density gradient (small Ln) is stabilizing - lowering the growth rate and the upper ηi stability threshold.

Stabilization of toroidal modes by finite- effects

The effects of finite on the toroidal ion temperature gradient mode has been investigated by a multi-pole fluid model. Experimental data from the Joint European Torus (JET) have been used to ensure the relevance to experiments. The mode is strongly stabilized by finite- effects well below the MHD ballooning mode stability threshold.

Stability of toroidal ion temperature gradient modes in the presence of inverted q-profiles

The stability of the ion temperature gradient (ITG) mode in tokamak geometries is examined analytically for a non-monotonic q-profile [negative magnetic shear, (S)] for all ranges of wavelengths, short, intermediate and long. It is shown that the shear reversal has a weak stabilising influence on the unstable mode [ηi >> ηicrit, ηi = (d ln Ti/d ln ni)]. Hence, the best way to stabilise these modes would be to optimise the equilibrium profiles so as to keep ηi < ηicrit and also S ⩽ 0 in the plasma interior as envisioned by Rewoldt et al (1993).

Basic studies of fluctuations, transport and their control

The first experimental study of a mixed slab - toroidal ITG (ion temperature gradient) driven mode in the Columbia linear machine (CLM) is reported. The effects of the modest levels of magnetic curvature are seen to be further destabilization and reduction of the real frequency of the mode. However, with a further increase in the mirror ratio, the mode amplitude decreased. The mode has the appearance of a hybrid ITG-dissipative trapped electron mode. We have measured the reduction of anomalous fluctuation driven particle transport due to feedback via cross-correlation of Langmuir probe and capacitive probe data for a centrifugal mode driven by E×B rotation of the plasma column. The particle flux is obtained from cross power spectra of density and potential fluctuation. The resulting diffusion coefficient clearly indicates that feedback can reduce the transport by up to a factor of three. To investigate the question of non-linear saturation of the mode we have performed a bispectral analysis of fluctuation data. The results clearly indicate strong mode coupling in the spectral width of the mode. Furthermore, we resolve the spectral width to reveal that it is composed of radial harmonics. These observations suggest that the nonlinear mode coupling between these radial harmonics is the saturation mechanism of the instability, as well as the underlying cause of the consequent anomalous transport.

Transport processes and entropy production in toroidally rotating plasmas with electrostatic turbulence

A new gyrokinetic equation is derived for rotating plasmas with large flow velocities on the order of the ion thermal speed. Neoclassical and anomalous transport of particles, energy, and toroidal momentum are systematically formulated from the ensemble-averaged kinetic equation with the gyrokinetic equation. As a conjugate pair of the thermodynamic force and the transport flux, the shear of the toroidal flow, which is caused by the radial electric field shear, and the toroidal viscosity enter both the neoclassical and anomalous entropy production. The interaction between the fluctuations and the sheared toroidal flow is self-consistently described by the gyrokinetic equation containing the flow shear as the thermodynamic force and by the toroidal momentum balance equation including the anomalous viscosity. Effects of the toroidal flow shear on the toroidal ion temperature gradient driven modes are investigated. Linear and quasilinear analyses of the modes show that the toroidal flow shear decreases the growth rates and reduces the anomalous toroidal viscosity.

Stationary turbulent spectra of toroidal ηi mode

Toroidal ion temperature gradient (ITG) driven drift mode turbulence has been analyzed analyti- cally and numerically. By using weak nonlinearity arguments and random phase approxima- tion, dynamic and wave kinetic equations are derived. Three different nonlinearities, namely E×B, convective, and diamagnetic nonlinearities, play important roles in the turbulent spectral transfer. The power spectra of the weak ITG-mode turbulence are obtained analytically for |k|≫1 and |k|&amp;lt;1 ranges in the wave number space. Forward energy cascading due to convective and diamagnetic nonlinearities will balance the inverse energy cascading due to E×B nonlinearity at |k|≊1/ρs (k is the wave number, ρs=cs/ωci, where cs is the sound velocity and ωci is the ion cyclotron frequency) and results in energy condensation at |k|≊1/ρs.

Poloidal shear flow effect on toroidal ion temperature gradient mode: A theory and simulation

The effect of poloidal shear flow on the global structure and stability of toroidal ion temperature gradient mode is studied theoretically and also numerically using a toroidal particle simulation code. Reasonable agreements are found between the theoretical and simulation results; both indicating that with increasing shear flow the toroidal ion temperature gradient mode is stabilized in the increase of poloidal asymmetry and in the reduction of radial mode width.

Action at distance and Bohm scaling of turbulence in tokamaks

Recent transport experiments in tokamaks have suggested the concept of ‘‘action at distance’’ in which the local turbulence depends on gradients at a distance larger than the correlation length. Furthermore, the scaling of the ion thermal diffusivity is not always consistent with local gyro-Bohm-like transport but rather scales worse than Bohm-like. This work is an attempt to reconcile these observations with simplified numerical simulations of toroidal ion temperature gradient (ITG) mode turbulence using a fast two-dimensional (2-D) inhomogeneous full radius turbulence code. It is found that action at a distance is possible, but only at weak damping rates, since the propagation range is given simply by the curvature drift group velocity divided by the average damping rate. The correlation lengths always scale linearly with the gyroradius. It is found that Bohm scaling or worse is possible when the gradients are close to the ITG threshold and the radial modes keep the turbulence level small enough to avoid destroying the slow to form global eigenmodes. In contrast to local ITG ballooning modes, the global eigenmode growth rates decrease with increasing gyroradius from the effect of larger diamagnetic rotational shear. This behavior results in an increase of the correlation time with increasing gyroradius, when the gradients are close to the thresholds. Thus, at sufficiently large relative gyroradius, the breaking of gyro-Bohm scaling can result from increased stability not mixing length.

Summary of ion temperature gradient instability studies in the Columbia linear machine

The toroidal ion temperature gradient (ITG) driven mode has been studied by starting with the slab mode and gradually turning on a mirror field in the Columbia Linear Machine (CLM). The magnetic curvature produces an extra drive for the mode and reduces its real frequency in the plasma frame, consistent with a new model of a hybrid ITG and trapped electron mode. Furthermore, the anomalous local ion thermal conductivity due to the slab mode has been measured to be 0.5 m2/sec, which is 3 orders of magnitude larger than the classical and 1 order of magnitude lower than the Bohm. Its scaling with the fluctuation amplitude, the linear growth rate and etai are all found to be linear. Lastly, flow shear is found to stabilize the ITG modes, but at levels nearly an order of magnitude larger than the theoretical predictions.

Field-aligned coordinates for nonlinear simulations of tokamak turbulence

Turbulence in tokamaks is characterized by long parallel wavelengths and short perpendicular wavelengths. A coordinate system for nonlinear fluid, gyrokinetic ‘‘Vlasov,’’ or particle simulations is presented that exploits the elongated nature of the turbulence by resolving the minimum necessary simulation volume: a long thin twisting flux tube. It is very similar to the ballooning representation, although periodicity constraints can be incorporated in a manner that allows E×B nonlinearities to be evaluated efficiently with fast Fourier transforms (FFT’s). If the parallel correlation length is very long, however, enforcing periodicity can introduce artificial correlations, so periodicity should not necessarily be enforced in the poloidal angle at θ=±π. This method is applied to high resolution three-dimensional simulations of toroidal ion temperature gradient (ITG) driven turbulence, which predict fluctuation spectra and ion heat transport similar to experimental measurements.

ITG and DTE mode driven transport in a contaminated toroidal plasma

The electrostatic quasi-linear particle and thermal diffusivities, due to the toroidal ion temperature gradient (ITG) driven modes and the dissipative trapped electron (DTE) mode, are considered in the dissipative limit, ω << vef/ϵ. Impurities are included by assuming a plasma with three particle species (electrons, main ions and impurity ions). The diffusivities are studied by computing interpretative radial transport profiles based on a TEXTOR background variation, using a saturation level approximation. The transport is found to be driven also by impurity modes, in particular when a light impurity species, such as helium, is assumed. The existence of inward fluxes of particles and heat is discussed

Negative shear effect on toroidal ion temperature gradient mode

A study of the effect of negative shear on the toroidal ion temperature gradient mode is given. It is shown that the negative shear can give a substantial stabilizing effect on the mode when the safety factor q is small.

Toroidal gyro-Landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes

The method of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] to model Landau damping has been recently applied to the moments of the gyrokinetic equation with curvature drift by Waltz, Dominguez, and Hammett [Phys. Fluids B 4, 3138 (1992)]. The higher moments are truncated in terms of the lower moments (density, parallel velocity, and parallel and perpendicular pressure) by modeling the deviation from a perturbed Maxwellian to fit the kinetic response function at all values of the kinetic parameters: k∥vth/ω, b=(k⊥ρ)2/2, and ωD/ω. Here the resulting gyro-Landau fluid equations are applied to the simulation of ion temperature gradient (ITG) mode turbulence in toroidal geometry using a novel three-dimensional (3-D) nonlinear ballooning mode representation. The representation is a Fourier transform of a field line following basis (ky′,kx′,z′) with periodicity in toroidal and poloidal angles. Particular emphasis is given to the role of nonlinearly generated n=0 (ky′ = 0, kx′ ≠ 0) ‘‘radial modes’’ in stabilizing the transport from the finite-n ITG ballooning modes. Detailing the parametric dependence of toroidal ITG turbulence is a key result.

Kinetic resonance damping rate of the toroidal ion temperature gradient mode

The linear damping rates of the toroidal ion temperature gradient (ηi) mode due to the toroidal resonance are calculated in the local kinetic limit. The well-known Landau contour method is generalized to treat the analytic continuation problem of the guiding center dispersion function in the toroidal resonance system where the resonance occurs from both the magnetic ∇B-curvature drift and the parallel ion transit drift. A detailed numerical analysis is presented for the dependence of the damping rate of the toroidal ηi mode on various parameters such as εn, ky, and the trapped electron fraction. In addition, a consideration is presented on the decay problem of the ballistic response by phase mixing in the toroidal system, which is directly related to the kinetic damping problem of the normal modes by the toroidal resonance.

Electromagnetic effect on the toroidal ion temperature gradient mode

A systematic study of the electromagnetic effects on the toroidal ion temperature gradient mode is presented using the local and nonlocal theories with the full kinetic terms. For the nonlocal study, a numerical code is developed to solve the electromagnetic gyrokinetic equation in the ballooning space. The electromagnetic coupling to the shear Alfvén mode is shown to give a stabilization of the toroidal temperature gradient mode at almost the same plasma pressure as that at which the kinetically modified magnetohydrodynamic (MHD) ballooning mode becomes destabilized. The transitional β value is shown to be lower in the full kinetic description than in the fluid theory. Possible correlations of these stability results with experimental observations are discussed.

Long-wavelength microinstabilities in toroidal plasmas*

Realistic kinetic toroidal eigenmode calculations have been carried out to support a proper assessment of the influence of long-wavelength microturbulence on transport in tokamak plasmas. In order to efficiently evaluate large-scale kinetic behavior extending over many rational surfaces, significant improvements have been made to a toroidal finite element code used to analyze the fully two-dimensional (r,θ) mode structures of trapped-ion and toroidal ion temperature gradient (ITG) instabilities. It is found that even at very long wavelengths, these eigenmodes exhibit a strong ballooning character with the associated radial structure relatively insensitive to ion Landau damping at the rational surfaces. In contrast to the long-accepted picture that the radial extent of trapped-ion instabilities is characterized by the ion-gyroradius-scale associated with strong localization between adjacent rational surfaces, present results demonstrate that under realistic conditions, the actual scale is governed by the large-scale variations in the equilibrium gradients. Applications to recent measurements of fluctuation properties in Tokamak Fusion Test Reactor (TFTR) [Plasma Phys. Controlled Nucl. Fusion Res. (International Atomic Energy Agency, Vienna, 1985), Vol. 1, p. 29] L-mode plasmas indicate that the theoretical trends appear consistent with spectral characteristics as well as rough heuristic estimates of the transport level. Benchmarking calculations in support of the development of a three-dimensional toroidal gyrokinetic code indicate reasonable agreement with respect to both the properties of the eigenfunctions and the magnitude of the eigenvalues during the linear phase of the simulations of toroidal ITG instabilities.