skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Toroidal regularization of the guiding center Lagrangian

Abstract

In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v ∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.

Authors:
 [1];  [2]
  1. Courant Inst. of Mathematical Sciences, New York, NY (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)
OSTI Identifier:
1424084
Alternate Identifier(s):
OSTI ID: 1409875
Report Number(s):
LLNL-JRNL-737871
Journal ID: ISSN 1070-664X; TRN: US1801902
Grant/Contract Number:  
AC52-07NA27344; FG02-86ER53223; AC05-06OR23100; AC52-07NA2734
Resource Type:
Accepted Manuscript
Journal Name:
Physics of Plasmas
Additional Journal Information:
Journal Volume: 24; Journal Issue: 11; Journal ID: ISSN 1070-664X
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Hamiltonian mechanics; Energy; Probability theory; General physics; Plasma gyrokinetics; Stochastic processes; Mathematical physics; Nuclear fusion power; Plasma confinement; Tokamaks; Plasma Physics and Thermonuclear Processes

Citation Formats

Burby, J. W., and Ellison, C. L. Toroidal regularization of the guiding center Lagrangian. United States: N. p., 2017. Web. doi:10.1063/1.5004429.
Burby, J. W., & Ellison, C. L. Toroidal regularization of the guiding center Lagrangian. United States. doi:10.1063/1.5004429.
Burby, J. W., and Ellison, C. L. Wed . "Toroidal regularization of the guiding center Lagrangian". United States. doi:10.1063/1.5004429. https://www.osti.gov/servlets/purl/1424084.
@article{osti_1424084,
title = {Toroidal regularization of the guiding center Lagrangian},
author = {Burby, J. W. and Ellison, C. L.},
abstractNote = {In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v∥∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.},
doi = {10.1063/1.5004429},
journal = {Physics of Plasmas},
number = 11,
volume = 24,
place = {United States},
year = {2017},
month = {11}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 3 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Canonicalization and symplectic simulation of the gyrocenter dynamics in time-independent magnetic fields
journal, March 2014

  • Zhang, Ruili; Liu, Jian; Tang, Yifa
  • Physics of Plasmas, Vol. 21, Issue 3
  • DOI: 10.1063/1.4867669

Poisson integrators
journal, December 2004


Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry
journal, April 2009

  • Qin, Hong; Guan, Xiaoyin; Tang, William M.
  • Physics of Plasmas, Vol. 16, Issue 4
  • DOI: 10.1063/1.3099055

Variational symplectic algorithm for guiding center dynamics in the inner magnetosphere
journal, May 2011

  • Li, Jinxing; Qin, Hong; Pu, Zuyin
  • Physics of Plasmas, Vol. 18, Issue 5
  • DOI: 10.1063/1.3589275

Development of variational guiding center algorithms for parallel calculations in experimental magnetic equilibria
journal, April 2015


Variational principles of guiding centre motion
journal, February 1983


Regularization of Hamilton-Lagrangian Guiding Center Theories
journal, November 1985


Hamiltonian theory of guiding-center motion
journal, May 2009


Variational approach to low-frequency kinetic-MHD in the current coupling scheme
journal, March 2017


Hamiltonian guiding center equations in toroidal magnetic configurations
journal, March 2003

  • White, Roscoe; Zakharov, Leonid E.
  • Physics of Plasmas, Vol. 10, Issue 3
  • DOI: 10.1063/1.1544500

Nonlinear gyrokinetic theory for finite-beta plasmas
journal, January 1988

  • Hahm, T. S.; Lee, W. W.; Brizard, A.
  • Physics of Fluids, Vol. 31, Issue 7
  • DOI: 10.1063/1.866641

Foundations of nonlinear gyrokinetic theory
journal, April 2007