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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

A Short Introduction to General Gyrokinetic Theory

Description: Interesting plasmas in the laboratory and space are magnetized. General gyrokinetic theory is about a symmetry, gyro-symmetry, in the Vlasov-Maxwell system for magnetized plasmas. The most general gyrokinetic theory can be geometrically formulated. First, the coordinate-free, geometric Vlasov-Maxwell equations are developed in the 7-D phase space, which is defined as a fiber bundle over the space-time. The Poincar{copyright}-Cartan-Einstein 1-form pullbacked onto the 7-D phase space determines particles' worldlines in the phase space, and realizes the momentum integrals in kinetic theory as fiber integrals. The infinite small generator of the gyro-symmetry is then asymptotically constructed as the base for the gyrophase coordinate of the gyrocenter coordinate system. This is accomplished by applying the Lie coordinate perturbation method to the Poincar{copyright}-Cartan-Einstein 1-form, which also generates the most relaxed condition under which the gyro-symmetry still exists. General gyrokinetic Vlasov-Maxwell equations are then developed as the Vlasov-Maxwell equations in the gyrocenter coordinate system, rather than a set of new equations. Since the general gyrokinetic system-developed is geometrically the same as the Vlasov-Maxwell equations, all the coordinate independent properties of the Vlasov-Maxwell equations, such as energy conservation, momentum conservation, and Liouville volume conservation, are automatically carried over to the general gyrokinetic system. The pullback transformation associated with the coordinate transformation is shown to be an indispensable part of the general gyrokinetic Vlasov-Maxwell equations. 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. Three examples of applications of the general gyrokinetic theory developed in the areas of plasma equilibrium and plasma waves are given. Interesting topics, such as gyro-center gauge and gyro-gauge, are discussed as well.
Date: February 14, 2005
Creator: Qin, H.
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 Integration Of The Vlasov-Maxwell System With A Variational Particle-in-cell Scheme

Description: A fully variational, unstructured, electromagnetic particle-in-cell integrator is developed for integration of the Vlasov-Maxwell equations. Using the formalism of Discrete Exterior Calculus [1], the field solver, interpolation scheme and particle advance algorithm are derived through minimization of a single discrete field theory action. As a consequence of ensuring that the action is invariant under discrete electromagnetic gauge transformations, the integrator exactly conserves Gauss's law.
Date: March 27, 2012
Creator: Squire, J.; Qin, H. & Tang, W. M.
Partner: UNT Libraries Government Documents Department

Automation of The Guiding Center Expansion

Description: We report on the use of the recently-developed Mathematica package VEST (Vector Einstein Summation Tools) to automatically derive the guiding center transformation. Our Mathematica code employs a recursive procedure to derive the transformation order-by-order. This procedure has several novel features. (1) It is designed to allow the user to easily explore the guiding center transformation's numerous nonunique forms or representations. (2) The procedure proceeds entirely in cartesian position and velocity coordinates, thereby producing manifestly gyrogauge invariant results; the commonly-used perpendicular unit vector fields e1, e2 are never even introduced. (3) It is easy to apply in the derivation of higher-order contributions to the guiding center transformation without fear of human error. Our code therefore stands as a useful tool for exploring subtle issues related to the physics of toroidal momentum conservation in tokamaks
Date: March 19, 2013
Creator: Burby, J. W.; Squire, J. & Qin, H.
Partner: UNT Libraries Government Documents Department

VEST: Abstract Vector Calculus Simplification in Mathematica

Description: We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce scalar and vector expressions of a very general type using a systematic canonicalization procedure. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by canonicalization, subsequently applying these to simplify large expressions. In a companion paper [1], we employ VEST in the automation of the calculation of Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations. __________________________________________________
Date: March 12, 2013
Creator: Squire, J.; Burby, J. & Qin, H.
Partner: UNT Libraries Government Documents Department

Toroidal Precession as a Geometric Phase

Description: Toroidal precession is commonly understood as the orbit-averaged toroidal drift of guiding centers in axisymmetric and quasisymmetric configurations. We give a new, more natural description of precession as a geometric phase effect. In particular, we show that the precession angle arises as the holonomy of a guiding center's poloidal trajectory relative to a principal connection. The fact that this description is physically appropriate is borne out with new, manifestly coordinate-independent expressions for the precession angle that apply to all types of orbits in tokamaks and quasisymmetric stellarators alike. We then describe how these expressions may be fruitfully employed in numerical calculations of precession.
Date: September 26, 2012
Creator: Burby, J. W. & Qin, H.
Partner: UNT Libraries Government Documents Department

Geometric Gyrokinetic Theory for Edge Plasma

Description: It turns out that gyrokinetic theory can be geometrically formulated as special cases of a geometrically generalized Vlasov-Maxwell system. It is proposed that the phase space of the spacetime is a 7-dimensional fiber bundle P over the 4-dimensional spacetime M, and that a Poincare-Cartan-Einstein 1-form {gamma} on the 7-dimensional phase space determines particles worldlines in the phase space. Through Liouville 6-form {Omega} and fiber integral, the 1-form {gamma} also uniquely defines a geometrically generalized Vlasov-Maxwell system as a field theory for the collective electromagnetic field. The geometric gyrokinetic theory is then developed as a special case of the geometrically generalized Vlasov-Maxwell system. In its most general form, gyrokinetic theory is about a symmetry, called gyro-symmetry, for magnetized plasmas, and the 1-form {gamma} again uniquely defines the gyro-symmetry. The objective is to decouple the gyro-phase dynamics from the rest of particle dynamics by finding the gyro-symmetry in {gamma}. Compared with other methods of deriving the gyrokinetic equations, the advantage of the geometric approach is that it allows any approximation based on mathematical simplification or physical intuition to be made at the 1-form level, and yet the field theories still have the desirable exact conservation properties such as phase space volume conservation and energy-momentum conservation if the 1-form does not depend on the spacetime coordinate explicitly. A set of generalized gyrokinetic equations valid for the edge plasmas is then derived using this geometric method. This formalism allows large-amplitude, time-dependent background electromagnetic fields to be developed fully nonlinearly in addition to small-amplitude, short-wavelength electromagnetic perturbations. The fact that we adopted the geometric method in the present study does not necessarily imply that the major results reported here can not be achieved using classical methods. What the geometric method offers is a systematic treatment and simplified calculations.
Date: January 18, 2007
Creator: Qin, H; Cohen, R H; Nevins, W M & Xu, X Q
Partner: UNT Libraries Government Documents Department

Theoretical study of transverse-longitudinal emittance coupling

Description: The effect of a weakly coupled periodic lattice in terms of achieving emittance exchange between the transverse and longitudinal directions is investigated using the generalized Courant-Snyder theory for coupled lattices. Recently, the concept and technique of transverse-longitudinal emittance coupling have been proposed for applications in the Linac Coherent Light Source and other free-electron lasers to reduce the transverse emittance of the electron beam. Such techniques can also be applied to the driver beams for the heavy ion fusion and beam-driven high energy density physics, where the transverse emittance budget is typically tighter than the longitudinal emittance. The proposed methods consist of one or several coupling components which completely swap the emittances of one of the transverse directions and the longitudinal direction at the exit of the coupling components. The complete emittance exchange is realized in one pass through the coupling components. In the present study, we investigate the effect of a weakly coupled periodic lattice in terms of achieving emittance exchange between the transverse and longitudinal directions. A weak coupling component is introduced at every focusing lattice, and we would like to determine if such a lattice can realize the function of emittance exchange.
Date: April 14, 2011
Creator: Qin, H; Davidson, R C; Chung, M; Barnard, J J & Wang, T F
Partner: UNT Libraries Government Documents Department

Free boundary, high beta equilibrium in a large aspect ratio tokamak with nearly circular plasma boundary

Description: An analytic solution is obtained for free-boundary, high-beta equilibria in large aspect ratio tokamaks with a nearly circular plasma boundary. In the absence of surface currents at the plasma-vacuum interface, the free-boundary equilibrium solution introduces constraints arising from the need to couple to an external vacuum field which is physically realizable with a reasonable set of external field coils. This places a strong constraint on the pressure profiles that are consistent with a given boundary shape at high {epsilon}{beta}{sub p}. The equilibrium solution also provides information on the flux surface topology. The plasma is bounded by a separatrix. Increasing the plasma pressure at fixed total current causes the plasma aperture to decrease in a manner that is described.
Date: September 25, 1996
Creator: Qin, H. & Reiman, A.
Partner: UNT Libraries Government Documents Department

Gyrokinetic theory for arbitrary wavelength electromagnetic modes in tokamaks

Description: A linear gyrokinetic system for arbitrary wavelength electromagnetic modes is developed. A wide range of modes in inhomogeneous plasmas, such as the internal kink modes, the toroidal Alfven eigenmode (TAE) modes, and the drift modes, can be recovered from this system. The inclusion of most of the interesting physical factors into a single framework enables one to look at many familiar modes simultaneously and thus to study the modifications of and the interactions between them in a systematic way. Especially, the authors are able to investigate self-consistently the kinetic MHD phenomena entirely from the kinetic side. Phase space Lagrangian Lie perturbation methods and a newly developed computer algebra package for vector analysis in general coordinate system are utilized in the analytical derivation. In tokamak geometries, a 2D finite element code has been developed and tested. In this paper, they present the basic theoretical formalism and some of the preliminary results.
Date: October 15, 1997
Creator: Qin, H.; Tang, W.M. & Rewoldt, G.
Partner: UNT Libraries Government Documents Department

Gyrokinetic Theory for Arbitrary Wavelength Electromagnetic Modes in Tokamaks

Description: A linear gyrokinetic system for arbitrary wavelength electromagnetic modes is developed. A wide range of modes in inhomogeneous plasmas, such as the internal kink modes, the toroidal Alfvén eigenmode (TAE) modes, and the drift modes, can be recovered from this system. The inclusion of most of the interesting physical factors into a single framework enables us to look at many familiar modes simultaneously and thus to study the modifications of and the interactions between them in a systematic way. Especially, we are able to investigate self-consistently the kinetic MHD phenomena entirely from the kinetic side. Phase space Lagrangian Lie perturbation methods and a newly developed computer algebra package for vector analysis in general coordinate system are utilized in the analytical derivation. In tokamak geometries, a 2D finite element code has been developed and tested. In this paper, we present the basic theoretical formalism and some of the preliminary results.
Date: October 1, 1997
Creator: Qin, H.; Rewoldt, G. & Tang, W.M.
Partner: UNT Libraries Government Documents Department

Symbolic Vector Analysis in Plasma Physics

Description: Many problems in plasma physics involve substantial amounts of analytical vector calculation. The complexity usually originates from both the vector operations themselves and the choice of underlying coordinate system. A computer algebra package for symbolic vector analysis in general coordinate systems, GeneralVectorAnalysis (GVA), is developed using Mathematica. The modern viewpoint for 3D vector calculus, differential forms on 3-manifolds, is adopted to unify and systematize the vector calculus operations in general coordinate systems. This package will benefit physicists and applied mathematicians in their research where complicated vector analysis is required. It will not only save a huge amount of human brain-power and dramatically improve accuracy, but this package will also be an intelligent tool to assist researchers in finding the right approaches to their problems. Several applications of this symbolic vector analysis package to plasma physics are also given.
Date: October 1, 1997
Creator: Qin, H.; Rewoldt, G. & Tang, W.M.
Partner: UNT Libraries Government Documents Department

Symbolic Vector Analysis in Plasma Physics

Description: Many problems in plasma physics involve substantial amounts of analytical vector calculation. The complexity usually originates from both the vector operations themselves and the choice of underlying coordinate system. A computer algebra package for symbolic vector analysis in general coordinate systems, General Vector Analysis (GVA), is developed using Mathematica. The modern viewpoint for 3D vector calculus, differential forms on 3-manifolds, is adopted to unify and systematize the vector calculus operations in general coordinate systems. This package will benefit physicists and applied mathematicians in their research where complicated vector analysis is required. It will not only save a huge amount of human brain-power and dramatically improve accuracy, but this package will also be an intelligent tool to assist researchers in finding the right approaches to their problems. Several applications of this symbolic vector analysis package to plasma physics are also given.
Date: October 9, 1997
Creator: Qin, H.; Tang, W.M. & Rewoldt, G.
Partner: UNT Libraries Government Documents Department

Effects on axial momentum spread on the electron-ion two-stream instability in high-intensity ion beams

Description: Use is made of the Vlasov-Maxwell equations to describe the electron-ion two-stream instability driven by the directed axial motion of a high-intensity ion beam propagating through a stationary population of (unwanted) background electrons. The ion beam is treated as continuous in the z-direction, and the electrons are electrostatically confined in the transverse direction by the space-charge potential produced by the excession charge. The analysis is carried out for arbitrary beam intensity, consistent with transverse confinement of the beam particles, and arbitrary fractional charge neutralization by the background electrons. For the case of overlapping step-function ion and electron density profiles, corresponding to monoenergetic electrons and ions in the transverse direction, detailed stability properties are calculated, including the important effects of an axial momentum spread, over a wide range of system parameters for dipole perturbations with azimuthal mode number l=1. The two-stream instability growth rate is found to increase with increasing beam intensity, increasing fractional charge neutralization, and decreasing proximity of the conducting wall. It is shown that Landau damping associated with a modest axial momentum spread of the beam ions and background electrons has a strong stabilizing influence on the instability.
Date: June 15, 2000
Creator: Davidso, R. & Qin, H.
Partner: UNT Libraries Government Documents Department

Gyrocenter-gauge kinetic theory

Description: Gyrocenter-gauge kinetic theory is developed as an extension of the existing gyrokinetic theories. In essence, the formalism introduced here is a kinetic description of magnetized plasmas in the gyrocenter coordinates which is fully equivalent to the Vlasov-Maxwell system in the particle coordinates. In particular, provided the gyroradius is smaller than the scale-length of the magnetic field, it can treat high frequency range as well as the usual low frequency range normally associated with gyrokinetic approaches. A significant advantage of this formalism is that it enables the direct particle-in-cell simulations of compressional Alfven waves for MHD applications and of RF waves relevant to plasma heating in space and laboratory plasmas. The gyrocenter-gauge kinetic susceptibility for arbitrary wavelength and arbitrary frequency electromagnetic perturbations in a homogeneous magnetized plasma is shown to recover exactly the classical result obtained by integrating the Vlasov-Maxwell system in the particle coordinates. This demonstrates that all the waves supported by the Vlasov-Maxwell system can be studied using the gyrocenter-gauge kinetic model in the gyrocenter coordinates. This theoretical approach is so named to distinguish it from the existing gyrokinetic theory, which has been successfully developed and applied to many important low-frequency and long parallel wavelength problems, where the conventional meaning of gyrokinetic has been standardized. Besides the usual gyrokinetic distribution function, the gyrocenter-gauge kinetic theory emphasizes as well the gyrocenter-gauge distribution function, which sometimes contains all the physics of the problems being studied, and whose importance has not been realized previously. The gyrocenter-gauge distribution function enters Maxwell's equations through the pull-back transformation of the gyrocenter transformation, which depends on the perturbed fields. The efficacy of the gyrocenter-gauge kinetic approach is largely due to the fact that it directly decouples particle's gyromotion from its gyrocenter motion in the gyrocenter coordinates. As in the case of kinetic theories using guiding center ...
Date: August 7, 2000
Creator: Qin, H.; Tang, W. M. & Lee, W. W.
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

Role of Zonal Flow in Turbulent Transport Scaling

Description: Transport scalings with respect to collisionality (n*) and device size (r*) are obtained from massively parallel gyrokinetic particle simulations of toroidal ion-temperature-gradient (ITG) turbulence in the presence of zonal flows. Simulation results show that ion thermal transport from electrostatic ITG turbulence depends on ion-ion collisions due to the neo-classical damping of self-generated EXB zonal flows that regulate the turbulence. Fluctuations and heat transport levels exhibit bursting behavior with a period corresponding to the collisional damping time of poloidal flows. Results from large-scale full torus simulations with device-size scans for realistic parameters show that Bohm-like transport can be driven by microscopic scale fluctuations in the ITG turbulence with isotropic spectra. These simulation results resolve some apparent physics contradictions between experimental observations and turbulent transport theories.
Date: November 15, 2000
Creator: Lin, Z.; Hahm, T.S.; Krommes, J.A.; Lee, W.W.; Lewandowski, J.; Mynick, H. et al.
Partner: UNT Libraries Government Documents Department

Continuum Edge Gyrokinetic Theory and Simulations

Description: The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential, and mirror ratio; and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the banana regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL. (5) Our 5D gyrokinetic formulation yields a set of nonlinear electrostatic gyrokinetic equations that are for both neoclassical and turbulence simulations.
Date: January 9, 2007
Creator: Xu, X Q; Xiong, Z; Dorr, M R; Hittinger, J A; Bodi, K; Candy, J et al.
Partner: UNT Libraries Government Documents Department

Critical issues for high-brightness heavy-ion beams- prioritized

Description: This study group was initiated to consider whether there were any ''show-stopper'' issues with accelerators for heavy-ion warm-dense matter (WDM) and heavy-ion inertial fusion energy (HIF), and to prioritize them. Showstopper issues would appear as limits to beam current; that is, the beam would be well-behaved below the current limit, and significantly degraded in current or emittance if the current limit were exceeded at some region of an accelerator. We identified 14 issues: 1-6 could be addressed in the near term, 7-10 are potentially attractive solutions to performance and cost issues but are not yet fully characterized, 11-12 involve multibeam effects that cannot be more than partially studied in near-term facilities, and 13-14 involve new issues that are present in some novel driver concepts. Comparing the issues with the new experimental, simulation, and theoretical tools that we have developed, it is apparent that our new capabilities provide an opportunity to re-examine and significantly increase our understanding of the number one issue--halo growth and mitigation.
Date: February 28, 2007
Creator: Molvik, A W; Cohen, R; Davidson, R; Faltens, A; Friedman, A; Grisham, L et al.
Partner: UNT Libraries Government Documents Department