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Comment on Turbulent Equipartition Theory of Toroidal Momentum Pinch

Description: This response demonstrates that the comment by Peeters et al. contains an incorrect and misleading interpretation of our paper [Hahm et al., Phys. Plasmas 15, 055902 (2008)] regarding the density gradient dependence of momentum pinch and the turbulent equipartition (TEP) theory.
Date: March 12, 2009
Creator: T.S. Hahm, P.H. Diamond, O.D. Gurcan, and G. Rewoldt
Partner: UNT Libraries Government Documents Department

Gyrokinetic Simulations of Microinstabilities in Stellarator Geometry

Description: A computational study of microinstabilities in general geometry is presented. The ion gyrokinetic is solved as an initial value problem. The advantage of this approach is the accurate treatment of some important kinetic effects. The magnetohydrodynamic equilibrium is obtained from a three-dimensional local equilibrium model. The use of a local magnetohydrodynamic equilibrium model allows for a computationally-efficient systematic study of the impact of the magnetic structure on microinstabilities.
Date: August 29, 2003
Creator: Lewandowski, J.L.V.
Partner: UNT Libraries Government Documents Department

Response to Comment on "On Higher-Order Corrections to Gyrokinetic Vlasov-Poisson Equations in the Long Wavelength Limit [Phys. Plasmas 16, 044506 (2009)]"

Description: We show in this Response that the nonlinear Poisson's equation in our original paper derived from the drift kinetic approach can be verified by using the nonlinear gyrokinetic Poisson's equation of Dubin et al. [Phys. Fluids 26, 3524 (1983)]. This nonlinear contribution in φ2 is indeed of the order of k4⊥ in the long wavelength limit and remains finite for zero ion temperature, in contrast to the nonlinear term by Parra and Catto [Plasma Phys. Control. Fusion 50, 065014 (2008)], which is of the order of k2⊥ and diverges for Ti → 0. For comparison, the leading term for the gyrokinetic Poisson's equation in this limit is of the order of k2⊥φ,
Date: November 20, 2009
Creator: Lee, W. W. & Kolensnikov, R. A.
Partner: UNT Libraries Government Documents Department

On Higher-order Corrections to Gyrokinetic Vlasov-Poisson Equations in the Long Wavelength Limit

Description: In this paper, we present a simple iterative procedure for obtaining the higher order E x B and dE/dt (polarization) drifts associated with the gyrokinetic Vlasov-Poisson equations in the long wavelength limit of k⊥ρi ~ o(ε) and k⊥L ~ o(1), where ρi is the ion gyroradius, L is the scale length of the background inhomogeneity and ε is a smallness parameter. It can be shown that these new higher order k⊥ρi terms, which are also related to the higher order perturbations of the electrostatic potential Φ, should have negligible effects on turbulent and neoclassical transport in tokamaks, regardless of the form of the background distribution and the amplitude of the perturbation. To address further the issue of a non-Maxwellian plasma, higher order finite Larmor radius terms in the gyrokinetic Poisson's equation have been studied and shown to be unimportant as well. On the other hand, the terms of o(k2⊥ρi2) ~ o(ε) and k⊥L ~ o(1) can indeed have impact on microturbulence, especially in the linear stage, such as those arising from the difference between the guiding center and the gyrocenter densities due to the presence of the background gradients. These results will be compared with a recent study questioning the validity of the commonly used gyrokinetic equations for long time simulations.
Date: February 17, 2009
Creator: Lee, W. W. & Kolesnikov, R. A.
Partner: UNT Libraries Government Documents Department

Turbulent Equipartition Theory of Toroidal Momentum Pinch

Description: The mode-independet part of magnetic curvature driven turbulent convective (TuroCo) pinch of the angular momentum density [Hahm et al., Phys. Plasmas 14,072302 (2007)] which was originally derived from the gyrokinetic equation, can be interpreted in terms of the turbulent equipartition (TEP) theory. It is shown that the previous results can be obtained from the local conservation of "magnetically weighted angular momentum density," nmi U|| R⁄B2, and its homogenization due to turbulent flows. It is also demonstrated that the magnetic curvature modification of the parallel acceleration in the nonlinear gyrokinetic equation in the laboratory frame, which was shown to be responsible for the TEP part of the TurCo pinch of angular momentum density in the previous work, is closely related to the Coriolis drift coupling to the perturbed electric field. In addition, the origin of the diffusive flux in the rotating frame is highlighted. Finally, it is illustratd that there should be a difference in scalings between the momentum pinch originated from inherently toroidal effects and that coming from other mechanisms which exist in a simpler geometry.
Date: January 31, 2008
Creator: Hahm, T. S.; Diamond, P. H.; Gurcan, O. D. & Rewaldt, G.
Partner: UNT Libraries Government Documents Department

Statistical Plasma Physics in a Strong Magnetic Field: Paradigms and Problems

Description: An overview is given of certain aspects of fundamental statistical theories as applied to strongly magnetized plasmas. Emphasis is given to the gyrokinetic formalism, the historical development of realizable Markovian closures, and recent results in the statistical theory of turbulent generation of long-wavelength flows that generalize and provide further physical insight to classic calculations of eddy viscosity. A Hamiltonian formulation of turbulent flow generation is described and argued to be very useful.
Date: March 19, 2004
Creator: Krommes, J.A.
Partner: UNT Libraries Government Documents Department

Pullback Transformations in Gyrokinetic Theory

Description: The Pullback transformation of the distribution function is a key component of the gyrokinetic theory. In this paper, a systematic treatment of this subject is presented, and results from applications of the uniform framework developed are reviewed. The focus is on providing a clear exposition of the basic formalism which arises from the existence of three distinct coordinate systems in gyrokinetic theory. The familiar gyrocenter coordinate system, where the gyromotion is decoupled from the rest of particle's dynamics, is non-canonical and non-fabric. On the other hand, Maxwell's equations, which are needed to complete a kinetic system, are initially only defined in the fabric laboratory phase space coordinate system. The pullback transformations provide a rigorous connection between the distribution functions in gyrocenter coordinates and Maxwell's equations in laboratory phase space coordinates. This involves the generalization of the usual moment integrals originally defined on the cotangent fiber of the phase space to the moment integrals on a general 6D symplectic manifold, is shown to be an important step in the proper formulation of gyrokinetic theory. The resultant systematic treatment of the moment integrals enabled by the pullback transformation. Without this vital element, a number of prominent physics features, such as the presence of the compressional Alfven wave and a proper description of the gyrokinetic equilibrium, cannot be readily recovered.
Date: January 21, 2003
Creator: Qin, H. & Tang, W.M.
Partner: UNT Libraries Government Documents Department

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

Description: We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. __________________________________________________
Date: September 25, 2012
Creator: Squire, J.; Qin, H. & Tang, W. M.
Partner: UNT Libraries Government Documents Department

Geometric Phase of the Gyromotion for Charged Particles in a Time-dependent Magnetic Field

Description: We study the dynamics of the gyrophase of a charged particle in a magnetic field which is uniform in space but changes slowly with time. As the magnetic field evolves slowly with time, the changing of the gyrophase is composed of two parts. The rst part is the dynamical phase, which is the time integral of the instantaneous gyrofrequency. The second part, called geometric gyrophase, is more interesting, and it is an example of the geometric phase which has found many important applications in different branches of physics. If the magnetic field returns to the initial value after a loop in the parameter space, then the geometric gyrophase equals the solid angle spanned by the loop in the parameter space. This classical geometric gyrophase is compared with the geometric phase (the Berry phase) of the spin wave function of an electron placed in the same adiabatically changing magnetic field. Even though gyromotion is not the classical counterpart of the quantum spin, the similarities between the geometric phases of the two cases nevertheless reveal the similar geometric nature of the different physics laws governing these two physics phenomena.
Date: July 18, 2011
Creator: Liu, Jian & Qin, Hong
Partner: UNT Libraries Government Documents Department

Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence

Description: An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E Χ B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρ[sub ]i [\sub]<< ρϑ¡ ~ LE ~ Lp << R (here ρ[sub ]i [\sub] is the thermal ion Larmor radius and ρϑ¡ = [B over Bϑ] ρ[sub ]i [\sub]), as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. We take κ[sub ] perpendicular to[/sub] ρ[sub ]i [\sub] ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ ⁄ Τ[sub ]i [\sub]~ δΒ ⁄ Β up to the second order. Extending the electrostatic theory in the presence of high E Χ B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.
Date: August 27, 2008
Creator: Hahm, T.S.; Wang, Lu, & Madsen, J.
Partner: UNT Libraries Government Documents Department

Variational Symplectic Integrator for Long-Time Simulations of the Guiding-Center Motion of Charged Particles in General Magnetic Fields

Description: A variational symplectic integrator for the guiding-center motion of charged particles in general magnetic fields is developed for long-time simulation studies of magnetized plasmas. Instead of discretizing the differential equations of the guiding-center motion, the action of the guiding-center motion is discretized and minimized to obtain the iteration rules for advancing the dynamics. The variational symplectic integrator conserves exactly a discrete Lagrangian symplectic structure, and has better numerical properties over long integration time, compared with standard integrators, such as the standard and variable time-step fourth order Runge-Kutta methods.
Date: February 11, 2008
Creator: Qin, H. & Guan, X.
Partner: UNT Libraries Government Documents Department

Nonlinear Gyrokinetic Theory With Polarization Drift

Description: A set of the electrostatic toroidal gyrokinetic Vlasov equation and the Poisson equation, which explicitly includes the polarization drift, is derived systematically by using Lie-transform method. The polarization drift is introduced in the gyrocenter equations of motion, and the corresponding polarization density is derived. Contrary to the wide-spread expectation, the inclusion of the polarization drift in the gyrocenter equations of motion does not affect the expression for the polarization density significantly. This is due to modification of the gyrocenter phase-space volume caused by the electrostatic potential [T. S. Hahm, Phys. Plasmas 3, 4658 (1996)] .
Date: March 25, 2010
Creator: Wang, L. & Hahm, T. S.
Partner: UNT Libraries Government Documents Department

Gyrokinetic Studies of Turbulence in Steep Gradient Region: Role of Turbulence Spreading and E x B Shear

Description: An integrated program of gyrokinetic particle simulation and theory has been developed to investigate several outstanding issues in both turbulence and neoclassical physics. Gyrokinetic particle simulations of toroidal ion temperature gradient (ITG) turbulence spreading using the GTC code and its related dynamical model have been extended to the case with radially increasing ion temperature gradient, to study the inward spreading of edge turbulence toward the core. Due to turbulence spreading from the edge, the turbulence intensity in the core region is significantly enhanced over the value obtained from simulations of the core region only. Even when the core gradient is within the Dimits shift regime (i.e., self-generated zonal flows reduce the transport to a negligible value), a significant level of turbulence and transport is observed in the core due to spreading from the edge. The scaling of the turbulent front propagation speed is closer to the prediction from our nonlinear diffusion model than one based on linear toroidal coupling. A calculation of ion poloidal rotation in the presence of sharp density and toroidal angular rotation frequency gradients from the GTC-Neo particle simulation code shows that the results are significantly different from the conventional neoclassical theory predictions. An energy conserving set of a fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to edge turbulence, is being derived via the phase-space action variational Lie perturbation method. Our generalized ordering takes the ion poloidal gyroradius to be on the order of the radial electric field gradient length.
Date: December 21, 2004
Creator: Hahm, T.S.; Lin, Z.; Diamond, P.H.; Rewoldt, G.; Wang, W.X.; Ethier, S. et al.
Partner: UNT Libraries Government Documents Department

Improved Conservation Properties for Particle-in-cell Simulations with Kinetic Electrons

Description: It is shown that a simple algorithm which exactly segregates between adiabatic and non-adiabatic electrons in particle-in-cell simulations of drift modes yields excellent conservation properties (e.g. particle number, energy) compared to the conventional df scheme. The removal of the free streaming term in the evolution of the marker weight is shown to be responsible for the improved linear and nonlinear properties of the simulated plasma.
Date: June 19, 2003
Creator: Lewandowski, J.L.V.
Partner: UNT Libraries Government Documents Department

Monte Carlo Sampling of Negative-temperature Plasma States

Description: A Monte Carlo procedure is used to generate N-particle configurations compatible with two-temperature canonical equilibria in two dimensions, with particular attention to nonlinear plasma gyrokinetics. An unusual feature of the problem is the importance of a nontrivial probability density function R0(PHI), the probability of realizing a set {Phi} of Fourier amplitudes associated with an ensemble of uniformly distributed, independent particles. This quantity arises because the equilibrium distribution is specified in terms of {Phi}, whereas the sampling procedure naturally produces particles states gamma; {Phi} and gamma are related via a gyrokinetic Poisson equation, highly nonlinear in its dependence on gamma. Expansion and asymptotic methods are used to calculate R0(PHI) analytically; excellent agreement is found between the large-N asymptotic result and a direct numerical calculation. The algorithm is tested by successfully generating a variety of states of both positive and negative temperature, including ones in which either the longest- or shortest-wavelength modes are excited to relatively very large amplitudes.
Date: July 19, 2002
Creator: Krommes, John A. & Rath, Sharadini
Partner: UNT Libraries Government Documents Department

Optimized Loading for Particle-in-cell Gyrokinetic Simulations

Description: The problem of particle loading in particle-in-cell gyrokinetic simulations is addressed using a quadratic optimization algorithm. Optimized loading in configuration space dramatically reduces the short wavelength modes in the electrostatic potential that are partly responsible for the non-conservation of total energy; further, the long wavelength modes are resolved with good accuracy. As a result, the conservation of energy for the optimized loading is much better that the conservation of energy for the random loading. The method is valid for any geometry and can be coupled to optimization algorithms in velocity space.
Date: May 13, 2004
Creator: Lewandowski, J.L.V.
Partner: UNT Libraries Government Documents Department

Introduction to Gyrokinetic Theory with Applications in Magnetic Confinement Research in Plasma Physics

Description: The present lecture provides an introduction to the subject of gyrokinetic theory with applications in the area of magnetic confinement research in plasma physics--the research arena from which this formalism was originally developed. It was presented as a component of the ''Short Course in Kinetic Theory within the Thematic Program in Partial Differential Equations'' held at the Fields Institute for Research in Mathematical Science (24 March 2004). This lecture also discusses the connection between the gyrokinetic formalism and powerful modern numerical simulations. Indeed, simulation, which provides a natural bridge between theory and experiment, is an essential modern tool for understanding complex plasma behavior. Progress has been stimulated in particular by the exponential growth of computer speed along with significant improvements in computer technology. The advances in both particle and fluid simulations of fine-scale turbulence and large-scale dynamics have produced increasingly good agreement between experimental observations and computational modeling. This was enabled by two key factors: (i) innovative advances in analytic and computational methods for developing reduced descriptions of physics phenomena spanning widely disparate temporal and spatial scales and (ii) access to powerful new computational resources.
Date: January 3, 2005
Creator: Tang, W.M.
Partner: UNT Libraries Government Documents Department

Comparison of Microinstability Properties for Stellarator Magnetic Geometries

Description: The microinstability properties of seven distinct magnetic geometries corresponding to different operating and planned stellarators with differing symmetry properties are compared. Specifically, the kinetic stability properties (linear growth rates and real frequencies) of toroidal microinstabilities (driven by ion temperature gradients and trapped-electron dynamics) are compared, as parameters are varied. The familiar ballooning representation is used to enable efficient treatment of the spatial variations along the equilibrium magnetic field lines. These studies provide useful insights for understanding the differences in the relative strengths of the instabilities caused by the differing localizations of good and bad magnetic curvature and of the presence of trapped particles. The associated differences in growth rates due to magnetic geometry are large for small values of the temperature gradient parameter n identical to d ln T/d ln n, whereas for large values of n, the mode is strongly unstable for all of the different magnetic geometries.
Date: June 16, 2005
Creator: Rewoldt, G.; Ku, L.-P. & Tang, W.M.
Partner: UNT Libraries Government Documents Department

A Numerical Instability in an ADI Algorithm for Gyrokinetics

Description: We explore the implementation of an Alternating Direction Implicit (ADI) algorithm for a gyrokinetic plasma problem and its resulting numerical stability properties. This algorithm, which uses a standard ADI scheme to divide the field solve from the particle distribution function advance, has previously been found to work well for certain plasma kinetic problems involving one spatial and two velocity dimensions, including collisions and an electric field. However, for the gyrokinetic problem we find a severe stability restriction on the time step. Furthermore, we find that this numerical instability limitation also affects some other algorithms, such as a partially implicit Adams-Bashforth algorithm, where the parallel motion operator v{sub {parallel}} {partial_derivative}/{partial_derivative}z is treated implicitly and the field terms are treated with an Adams-Bashforth explicit scheme. Fully explicit algorithms applied to all terms can be better at long wavelengths than these ADI or partially implicit algorithms.
Date: December 17, 2004
Creator: Belli, E.A. & Hammett, G.W.
Partner: UNT Libraries Government Documents Department

Theoretical and Numerical Properties of a Gyrokinetic Plasma: Issues Related to Transport Time Scale Simulation

Description: Particle simulation has played an important role for the recent investigations on turbulence in magnetically confined plasmas. In this paper, theoretical and numerical properties of a gyrokinetic plasma as well as its relationship with magnetohydrodynamics (MHD) are discussed with the ultimate aim of simulating microturbulence in transport time scale using massively parallel computers.
Date: September 17, 2003
Creator: Lee, W.W.
Partner: UNT Libraries Government Documents Department

Comparison of Linear Microinstability Calculations of Varying Input Realism

Description: The effect of varying ''input realism'' or varying completeness of the input data for linear microinstability calculations, in particular on the critical value of the ion temperature gradient for the ion temperature gradient mode, is investigated using gyrokinetic and gyrofluid approaches. The calculations show that varying input realism can have a substantial quantitative effect on the results.
Date: September 8, 2003
Creator: Rewoldt, G.
Partner: UNT Libraries Government Documents Department

Nonlinear Gyrokinetics: A Powerful Tool for the Description of Microturbulence in Magnetized Plasmas

Description: Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained. The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a "pull-back" (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and diffculties) of deriving nonlinear gyro fluid equations suitable for rapid numerical solution -- although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.
Date: September 27, 2010
Creator: Krommes, John E.
Partner: UNT Libraries Government Documents Department

Multispecies Density and Temperature Gradient Dependence of Quasilinear Particle and Energy Fluxes

Description: The variations of the normalized quasilinear particle and energy fluxes with artificial changes in the density and temperature gradients, as well as the variations of the linear growth rates and real frequencies, for ion temperature gradient and trapped-electron modes, are calculated. The quasilinear fluxes are normalized to the total energy flux, summed over all species. Here, realistic cases for tokamaks and spherical torii are considered which have two impurity species. For situations where there are substantial changes in the normalized fluxes, the ''diffusive approximation,'' in which the normalized fluxes are taken to be linear in the gradients, is seen to be inaccurate. Even in the case of small artificial changes in density or temperature gradients, changes in the fluxes of different species (''off-diagonal'') generally are significant, or even dominant, compared to those for the same species (''diagonal'').
Date: August 9, 2004
Creator: Rewoldt, G.; Budny, R.V. & Tang, W.M.
Partner: UNT Libraries Government Documents Department