R. E. Waltz

A gyro-Landau-fluid transport model

A physically comprehensive and theoretically based transport model tuned to three-dimensional (3-D) ballooning mode gyrokinetic instabilities and gyrofluid nonlinear turbulence simulations is formulated with global and local magnetic shear stabilization and E×B rotational shear stabilization. Taking no fit coefficients from experiment, the model is tested against a large transport profile database with good agreement. This model is capable of describing enhanced core confinement transport barriers in negative central shear discharges based on rotational shear stabilization. The model is used to make ignition projections from relative gyroradius scaling discharges.

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 s...

Noncircular, finite aspect ratio, local equilibrium model

A tokamak equilibrium model, local to a flux surface, is introduced which is completely described in terms of nine parameters including aspect ratio, elongation, triangularity, and safety factor. By allowing controlled variation of each of these nine parameters, the model is particularly suitable for localized stability studies such as those carried out using the ballooning mode representation of the gyrokinetic equations.

Advances in the simulation of toroidal gyro‐Landau fluid model turbulence

The gyro-Landau fluid (GLF) model equations for toroidal geometry have been recently applied to the study ion temperature gradient (ITG) mode turbulence using the 3D nonlinear ballooning mode representation (BMR). The present paper extends this work by treating some unresolved issues conceming ITG turbulence with adiabatic electrons. Although eddies are highly elongated in the radial direction long time radial correlation lengths are short and comparable to poloidal lengths. Although transport at vanishing shear is not particularly large, transport at reverse global shear, is significantly less. Electrostatic transport at moderate shear is not much effected by inclusion of local shear and average favorable curvature. Transport is suppressed when critical E{times}B rotational shear is comparable to the maximum linear growth rate with only a weak dependence on magnetic shear. Self consistent turbulent transport of toroidal momentum can result in a transport bifurcation at suffciently large r/(Rq). However the main thrust of the new formulation in the paper deals with advances in the development of finite beta GLF models with trapped electron and BMR numerical methods for treating the fast parallel field motion of the untrapped electrons.

A theory-based transport model with comprehensive physics

A new theory-based transport model with comprehensive physics (trapping, general toroidal geometry, fully electromagnetic, electron-ion collisions, impurity ions) has been developed. The core of the model is the new trapped-gyro-Landau-fluid (TGLF) equations, which provide a fast and accurate approximation to the linear eigenmodes for gyrokinetic drift-wave instabilities (trapped ion and electron modes, ion and electron temperature gradient modes, and kinetic ballooning modes). The new TGLF transport model is more accurate, and has an extended range of validity, compared to its predecessor GLF23. The TGLF model unifies trapped and passing particles in a single set of gyro-Landau-fluid equations. A model for the averaging of the Landau resonance by the trapped particles makes the equations work seamlessly over the whole drift-wave wave-number range from trapped ion modes to electron temperature gradient modes. A fast eigenmode solution method enables unrestricted magnetic geometry. The transport model uses...

Ion temperature gradient turbulence simulations and plasma flux surface shape

A generalization of the circular ŝ-α local magnetohydrodynamic (MHD) equilibrium model to finite aspect ratio (A), elongation (κ), and triangularity (δ) has been added to a gyrokinetic stability code and our gyrofluid nonlinear ballooning mode code for ion temperature gradient (ITG) turbulence. This allows systematic studies of stability and transport for shaped flux surfaces with the same minor midplane radius label (r), plasma gradients, q, ŝ, and α while varying A, κ, and δ. It is shown that the (linear, nonlinear, and sheared) E×B terms in the equation of motion are unchanged from a circle at radius r with an effective field Bunit=B0ρdρ/rdr, where χ=B0ρ2/2 is the toroidal flux, r is the flux surface label, and B0 is the magnetic axis field. This leads to a “natural gyroBohm diffusivity” χnatural, which at moderate q=2 to 3 is weakly dependent on shape (κ) at fixed Bunit. Since Bunit/B0∝κ and 〈|∇r|2〉≈(1+κ2)/(2κ2), the label independent χITER=χnatural/〈|∇r|2〉 at fixed B0 scales as 2/(1+κ2) with much wea...

Measurements of core electron temperature and density fluctuations in DIII-D and comparison to nonlinear gyrokinetic simulations

For the first time, profiles (0.3<ρ<0.9) of electron temperature and density fluctuations in a tokamak have been measured simultaneously and the results compared to nonlinear gyrokinetic simulations. Electron temperature and density fluctuations measured in neutral beam-heated, sawtooth-free low confinement mode (L-mode) plasmas in DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] are found to be similar in frequency and normalized amplitude, with amplitude increasing with radius. The measured radial profile of two fluctuation fields allows for a new and rigorous comparison with gyrokinetic results. Nonlinear gyrokinetic flux-tube simulations predict that electron temperature and density fluctuations have similar normalized amplitudes in L-mode. At ρ=0.5, simulation results match experimental heat diffusivities and density fluctuation amplitude, but overestimate electron temperature fluctuation amplitude and particle diffusivity. In contrast, simulations at ρ=0.75 do not match either the experimentally de...

Gyro‐Landau fluid models for toroidal geometry

Gyro‐Landau fluid model equations provide first‐order time advancement for a limited number of moments of the gyrokinetic equation, while approximately preserving the effects of the gyroradius averaging and Landau damping. This paper extends the work of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for electrostatic motion parallel to the magnetic field and E×B motion to include the gyroaveraging linearly and the curvature drift motion. The equations are tested by comparing the ion‐temperature‐gradient mode linear growth rates for the model equations with those of the exact gyrokinetic theory over a full range of parameters.

Gyro-Landau fluid equations for trapped and passing particles

A new system of gyro-Landau fluid (GLF) equations for tokamak plasmas is presented. The new equations include both trapped particles, which can average the Landau resonance, and passing particles which do have a Landau resonance. The trap GLF (TGLF) model is unrestricted in trapped fraction or perpendicular wave number of the electrostatic perturbation. The linearly unstable eigenmodes of the TGLF equations include low-frequency trapped ion modes all the way up to high-frequency electron temperature gradient driftwaves. Extensive benchmarking of the linear TGLF eigenmodes with a large database of gyrokinetic linear stability calculations verifies that the TGLF model is accurate over the full range of plasma parameters tested.

Heat flux driven ion turbulence

This work is an analysis of an ion turbulence in a tokamak in the case where the thermal flux is fixed and the temperature profile is allowed to fluctuate. The system exhibits some features of self-organized critical systems. In particular, avalanches are observed. Also the frequency spectrum of the thermal flux exhibits a structure similar to the one of a sand pile automaton, including a 1/f behavior. However, the time average temperature profile is found to be supercritical, i.e., the temperature gradient stays above the critical value. Moreover, the heat diffusivity is not the same for a turbulence calculated at fixed flux than at fixed temperature gradient, with the same time averaged temperature. More precisely the diffusivity at fixed temperature is found to be larger in the edge and smaller close to the heat source.