W. W. Lee

Shearing rate of time-dependent E×B flow

Theory of E×B shear suppression of turbulence in toroidal geometry [Phys. Plasmas 2, 1648 (1995)] is extended to include fast time variations of the E×B flows often observed in nonlinear simulations of tokamak turbulence. It is shown that the quickly time varying components of the E×B flows, while they typically contribute significantly to the instantaneous E×B shearing rate, are less effective than the slowly time varying components in suppressing turbulence. This is because the shear flow pattern changes before eddies get distorted enough. The effective E×B shearing rate capturing this important physics is analytically derived and estimated from zonal flow statistics of gyrofluid simulation. This provides new insights into understanding recent gyrofluid and gyrokinetic simulations that yield a reduced, but not completely quenched, level of turbulence in the presence of turbulence-driven zonal flows.

Gyro-kinetic simulation of global turbulent transport properties in tokamak experiments

A general geometry gyro-kinetic model for particle simulation of plasma turbulence in tokamak experiments is described. It incorporates the comprehensive influence of noncircular cross section, realistic plasma profiles, plasma rotation, neoclassical (equilibrium) electric fields, and Coulomb collisions. An interesting result of global turbulence development in a shaped tokamak plasma is presented with regard to nonlinear turbulence spreading into the linearly stable region. The mutual interaction between turbulence and zonal flows in collisionless plasmas is studied with a focus on identifying possible nonlinear saturation mechanisms for zonal flows. A bursting temporal behavior with a period longer than the geodesic acoustic oscillation period is observed even in a collisionless system. Our simulation results suggest that the zonal flows can drive turbulence. However, this process is too weak to be an effective zonal flow saturation mechanism.

Shear-Alfvén waves in gyrokinetic plasmas

It is found that the thermal fluctuation level of the shear-Alfven waves in a gyrokinetic plasma is dependent on plasma β(≡cs2/vA2), where cs is the ion acoustic speed and vA is the Alfven velocity. This unique thermodynamic property based on the fluctuation–dissipation theorem is verified in this paper using a new gyrokinetic particle simulation scheme, which splits the particle distribution function into the equilibrium part as well as the adiabatic and nonadiabatic parts. The numerical implication of this property is discussed.

Comparisons of gyrofluid and gyrokinetic simulations

The gyrokinetic and gyrofluid models show the most promise for large scale simulations of tokamak microturbulence. This paper discusses detailed comparisons of these two complementary approaches. Past comparisons with linear theory have been fairly good, therefore the emphasis here is on nonlinear comparisons. Simulations include simple two‐dimensional slab test cases, turbulent three‐dimensional slab cases, and toroidal cases, each modeling the nonlinear evolution of the ion temperature gradient instability. There is good agreement in both turbulent and coherent nonlinear slab comparisons in terms of the ion heat flux, both in magnitude and scaling with magnetic shear. However, the nonlinear saturation level for ‖Φ‖ in the slab comparisons shows differences of approximately 40%. Preliminary toroidal comparisons show agreement within 50%, in terms of ion heat flux and saturation level.

Nonlocal properties of gyrokinetic turbulence and the role of E×B flow shear

The nonlocal physics associated with turbulent transport is investigated using global gyrokinetic simulations with realistic parameters in shaped tokamak plasmas. This study focuses on the turbulence spreading through a transport barrier characterized by an equilibrium E×B shear layer. It is found that an E×B shear layer with an experimentally relevant level of the shearing rate can significantly reduce, and sometimes even block, turbulence spreading by reducing the spreading extent and speed. This feature represents a new aspect of transport barrier dynamics. The key quantity in this process is identified as the local maximum shearing rate ∣ωEmax∣, rather than the amplitude of the radial electric field. These simulation studies also extend to radially local physics with respect to the saturation of the ion temperature gradient (ITG) instability, and show that the nonlinear toroidal couplings are the dominant k-space activity in the ITG dynamics, which cause energy transfer to longer wavelength damped mod...

Alfvén waves in gyrokinetic plasmas

A brief comparison of the properties of Alfven waves that are based on the gyrokinetic description with those derived from the magnetohydrodynamics (MHD) equations is presented. The critical differences between these two approaches are the treatment of the ion polarization effects. As such, the compressional Alfven waves in a gyrokinetic plasma can be eliminated through frequency ordering, whereas geometric simplifications are needed to decouple the shear Alfven waves from the compressional Alfven waves within the context of MHD. Theoretical and numerical procedures of using gyrokinetic particle simulation for studying microturbulence and kinetic-MHD physics including finite Larmor radius effects are also presented.

Nonlinear δ F simulation studies of high-intensity ion beam propagation in a periodic focusing field

This paper makes use of the nonlinear Vlasov–Poisson equations to describe the propagation of an intense, non-neutral ion beam through a periodic focusing solenoidal field with coupling coefficient κz(s+S)=κz(s) in the thin-beam approximation (rb≪S). The nonlinear δF formalism is developed for numerical simulation applications by dividing the total distribution function Fb into a zero-order part (Fb0) that propagates through the average focusing field κz=const, plus a perturbation (δFb) which evolves nonlinearly in the zero-order and perturbed field configurations. To illustrate the application of the technique to axisymmetric, matched-beam propagation, nonlinear δF-simulation results are presented for the case where Fb0 corresponds to a thermal equilibrium distribution, and the oscillatory component of the coupling coefficient, δκz(s)=κz(s)−κz, turns on adiabatically over many periods S of the focusing lattice. For adiabatic turn-on of δκz(s) over 20–100 lattice periods, the amplitude of the mismatch o...

Statistically averaged rate equations for intense non-neutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov–Maxwell equations

In this paper we present a detailed formulation and analysis of the rate equations for statistically averaged quantities for an intense non-neutral beam propagating through a periodic solenoidal focusing field Bsol(x) with axial periodicity length S=const. The analysis is based on the nonlinear Vlasov–Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius rb≪S, and small transverse momentum and axial momentum spread in comparison with the directed axial momentum pz=γbmβbc. The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average 〈χ〉=Nb−1∫dXdYdX′dY′χFb, where χ(X,Y,X′,Y′,s) is a general phase function, and Fb(X,Y,X′,Y′,s) is the distribution function of the beam particles in the transverse phase space (X,Y,X′,Y′) appropriate to the Larmor frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum 〈Pθ〉, center-of-mass motion for 〈X〉 and ...

High frequency gyrokinetic particle simulation

The gyrokinetic approach for arbitrary frequency dynamics in magnetized plasmas is explored, using the gyrocenter-gauge kinetic theory. Contrary to low-frequency gyrokinetics, which views each particle as a rigid charged ring, arbitrary frequency response of a particle is described by a quickly changing Kruskal ring. This approach allows the separation of gyrocenter and gyrophase responses and thus allows for, in many situations, larger time steps for the gyrocenter push than for the gyrophase push. The gyrophase response which determines the shape of Kruskal rings can be described by a Fourier series in gyrophase for some problems, thus allowing control over the cyclotron harmonics at which the plasma responds. A computational algorithm for particle-in-cell simulation based on this concept has been developed. An example of the ion Bernstein wave is used to illustrate its numerical properties, and comparison with a direct Lorentz-force approach is presented.

3D nonlinear perturbative particle simulations of two-stream collective processes in intense particle beams

Abstract Collective processes in intense charged particle beams described self-consistently by the Vlasov–Maxwell equations are studied using a 3D multispecies nonlinear perturbative particle simulation method. The newly-developed Beam Equilibrium Stability and Transport (BEST) code has been used to simulate the nonlinear stability properties of intense beam propagation, surface eigenmodes in a high-intensity beam, and the electron–proton (e–p) two-stream instability observed in the Proton Storage Ring (PSR).