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

Title: Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics

Abstract

First-principles simulations of tokamak turbulence have proven to be of great value in recent decades. We develop a pseudo-spectral velocity formulation of the turbulence equations that smoothly interpolates between the highly efficient but lower resolution three-dimensional (3-D) gyrofluid representation and the conventional but more expensive 5-D gyrokinetic representation. Our formulation is a projection of the nonlinear gyrokinetic equation onto a Laguerre–Hermite velocity-space basis. We discuss issues related to collisions, closures and entropy. While any collision operator can be used in the formulation, we highlight a model operator that has a particularly sparse Laguerre–Hermite representation, while satisfying conservation laws and the H theorem. Free streaming, magnetic drifts and nonlinear phase mixing each give rise to closure problems, which we discuss in relation to the instabilities of interest and to free energy conservation. We show that the model is capable of reproducing gyrokinetic results for linear instabilities and zonal flow dynamics. Thus the final model is appropriate for the study of instabilities, turbulence and transport in a wide range of geometries, including tokamaks and stellarators.

Authors:
ORCiD logo; ;
Publication Date:
Research Org.:
Univ. of Maryland, College Park, MD (United States); Krell Institute, Ames, IA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1538932
DOE Contract Number:  
FC02-08ER54964; FG02-93ER54197; FG02-97ER25308
Resource Type:
Journal Article
Journal Name:
Journal of Plasma Physics
Additional Journal Information:
Journal Volume: 84; Journal Issue: 1; Journal ID: ISSN 0022-3778
Publisher:
Cambridge University Press
Country of Publication:
United States
Language:
English
Subject:
Physics

Citation Formats

Mandell, N.  R., Dorland, W., and Landreman, M. Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. United States: N. p., 2018. Web. doi:10.1017/s0022377818000041.
Mandell, N.  R., Dorland, W., & Landreman, M. Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. United States. doi:10.1017/s0022377818000041.
Mandell, N.  R., Dorland, W., and Landreman, M. Tue . "Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics". United States. doi:10.1017/s0022377818000041.
@article{osti_1538932,
title = {Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics},
author = {Mandell, N.  R. and Dorland, W. and Landreman, M.},
abstractNote = {First-principles simulations of tokamak turbulence have proven to be of great value in recent decades. We develop a pseudo-spectral velocity formulation of the turbulence equations that smoothly interpolates between the highly efficient but lower resolution three-dimensional (3-D) gyrofluid representation and the conventional but more expensive 5-D gyrokinetic representation. Our formulation is a projection of the nonlinear gyrokinetic equation onto a Laguerre–Hermite velocity-space basis. We discuss issues related to collisions, closures and entropy. While any collision operator can be used in the formulation, we highlight a model operator that has a particularly sparse Laguerre–Hermite representation, while satisfying conservation laws and the H theorem. Free streaming, magnetic drifts and nonlinear phase mixing each give rise to closure problems, which we discuss in relation to the instabilities of interest and to free energy conservation. We show that the model is capable of reproducing gyrokinetic results for linear instabilities and zonal flow dynamics. Thus the final model is appropriate for the study of instabilities, turbulence and transport in a wide range of geometries, including tokamaks and stellarators.},
doi = {10.1017/s0022377818000041},
journal = {Journal of Plasma Physics},
issn = {0022-3778},
number = 1,
volume = 84,
place = {United States},
year = {2018},
month = {1}
}

Works referenced in this record:

Kinetic simulation of steady states of ion temperature gradient driven turbulence with weak collisionality
journal, April 2004

  • Watanabe, T. -H.; Sugama, H.
  • Physics of Plasmas, Vol. 11, Issue 4
  • DOI: 10.1063/1.1669393

Electromagnetic effects on plasma microturbulence and transport
journal, March 2001

  • Snyder, P. B.; Hammett, G. W.
  • Physics of Plasmas, Vol. 8, Issue 3
  • DOI: 10.1063/1.1342029

Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence
journal, April 2016

  • Schekochihin, A. A.; Parker, J. T.; Highcock, E. G.
  • Journal of Plasma Physics, Vol. 82, Issue 2
  • DOI: 10.1017/S0022377816000374

Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests
journal, July 2009

  • Barnes, M.; Abel, I. G.; Dorland, W.
  • Physics of Plasmas, Vol. 16, Issue 7
  • DOI: 10.1063/1.3155085

AstroGK: Astrophysical gyrokinetics code
journal, December 2010

  • Numata, Ryusuke; Howes, Gregory G.; Tatsuno, Tomoya
  • Journal of Computational Physics, Vol. 229, Issue 24
  • DOI: 10.1016/j.jcp.2010.09.006

Discrete particle noise in particle-in-cell simulations of plasma microturbulence
journal, December 2005

  • Nevins, W. M.; Hammett, G. W.; Dimits, A. M.
  • Physics of Plasmas, Vol. 12, Issue 12
  • DOI: 10.1063/1.2118729

Comparison of initial value and eigenvalue codes for kinetic toroidal plasma instabilities
journal, August 1995


Viriato : A Fourier–Hermite spectral code for strongly magnetized fluid–kinetic plasma dynamics
journal, September 2016


Fluctuation-dissipation relations for a plasma-kinetic Langevin equation
journal, September 2014


A global collisionless PIC code in magnetic coordinates
journal, September 2007


Transition between Saturation Regimes of Gyrokinetic Turbulence
journal, October 2013


Linearized gyro-kinetics
journal, July 1978


Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria
journal, January 1982


Model Fokker-Planck Equation for a Plasma and Its Solution
journal, January 1964


Electron Temperature Gradient Turbulence
journal, December 2000


Plasma Oscillations with Diffusion in Velocity Space
journal, December 1958


Toroidal kinetic η i ‐mode study in high‐temperature plasmas
journal, March 1992

  • Dong, J. Q.; Horton, W.; Kim, J. Y.
  • Physics of Fluids B: Plasma Physics, Vol. 4, Issue 7
  • DOI: 10.1063/1.860040

Comparisons and physics basis of tokamak transport models and turbulence simulations
journal, March 2000

  • Dimits, A. M.; Bateman, G.; Beer, M. A.
  • Physics of Plasmas, Vol. 7, Issue 3
  • DOI: 10.1063/1.873896

General theory of Onsager symmetries for perturbations of equilibrium and nonequilibrium steady states
journal, November 1993

  • Krommes, John A.; Hu, Genze
  • Physics of Fluids B: Plasma Physics, Vol. 5, Issue 11
  • DOI: 10.1063/1.860614

Linearized gyro-kinetic equation with collisions
journal, January 1977

  • Catto, P. J.; Tsang, K. T.
  • Physics of Fluids, Vol. 20, Issue 3
  • DOI: 10.1063/1.861902

An Eulerian gyrokinetic-Maxwell solver
journal, April 2003


Field‐aligned coordinates for nonlinear simulations of tokamak turbulence
journal, August 1995

  • Beer, M. A.; Cowley, S. C.; Hammett, G. W.
  • Physics of Plasmas, Vol. 2, Issue 7
  • DOI: 10.1063/1.871232

Direct multiscale coupling of a transport code to gyrokinetic turbulence codes
journal, May 2010

  • Barnes, M.; Abel, I. G.; Dorland, W.
  • Physics of Plasmas, Vol. 17, Issue 5
  • DOI: 10.1063/1.3323082

Eigenfunctions and eigenvalues of the Dougherty collision operator
journal, May 2007

  • Anderson, M. W.; O’Neil, T. M.
  • Physics of Plasmas, Vol. 14, Issue 5
  • DOI: 10.1063/1.2727463

Free-energy conservation in local gyrofluid models
journal, October 2005


Developments in the gyrofluid approach to Tokamak turbulence simulations
journal, August 1993


Poloidal Flow Driven by Ion-Temperature-Gradient Turbulence in Tokamaks
journal, January 1998


Gyrokinetic approach in particle simulation
journal, January 1983


Astrophysical Gyrokinetics: Basic Equations and Linear Theory
journal, November 2006

  • Howes, Gregory G.; Cowley, Steven C.; Dorland, William
  • The Astrophysical Journal, Vol. 651, Issue 1
  • DOI: 10.1086/506172

Suppression of phase mixing in drift-kinetic plasma turbulence
journal, July 2016

  • Parker, J. T.; Highcock, E. G.; Schekochihin, A. A.
  • Physics of Plasmas, Vol. 23, Issue 7
  • DOI: 10.1063/1.4958954

A Note on the Polynomials of Hermite and Laguerre
journal, July 1938


On the kinetic theory of rarefied gases
journal, December 1949


A fully nonlinear characteristic method for gyrokinetic simulation
journal, January 1993

  • Parker, S. E.; Lee, W. W.
  • Physics of Fluids B: Plasma Physics, Vol. 5, Issue 1
  • DOI: 10.1063/1.860870

Gyrofluid turbulence models with kinetic effects
journal, March 1993

  • Dorland, W.; Hammett, G. W.
  • Physics of Fluids B: Plasma Physics, Vol. 5, Issue 3
  • DOI: 10.1063/1.860934

Generation and Stability of Zonal Flows in Ion-Temperature-Gradient Mode Turbulence
journal, December 2000


Nonlinear electromagnetic gyrokinetic simulations of tokamak plasmas
journal, November 2001


Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations
journal, December 1971


Global full- f gyrokinetic simulations of plasma turbulence
journal, November 2007


Scalings of Ion-Temperature-Gradient-Driven Anomalous Transport in Tokamaks
journal, July 1996


Toroidal gyro‐Landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes
journal, July 1994

  • Waltz, R. E.; Kerbel, G. D.; Milovich, J.
  • Physics of Plasmas, Vol. 1, Issue 7
  • DOI: 10.1063/1.870934

Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit
journal, February 2015


Fast Collisionless Reconnection and Electron Heating in Strongly Magnetized Plasmas
journal, July 2013


Numerical Studies of the Nonlinear Vlasov Equation
journal, January 1967


Transport processes and entropy production in toroidal plasmas with gyrokinetic electromagnetic turbulence
journal, June 1996

  • Sugama, H.; Okamoto, M.; Horton, W.
  • Physics of Plasmas, Vol. 3, Issue 6
  • DOI: 10.1063/1.871922

Wave-Particle Decorrelation and Transport of Anisotropic Turbulence in Collisionless Plasmas
journal, December 2007


Fluid moment models for Landau damping with application to the ion-temperature-gradient instability
journal, June 1990


Nonlinear Phase Mixing and Phase-Space Cascade of Entropy in Gyrokinetic Plasma Turbulence
journal, June 2009


Stability of E×B zonal flow in electron temperature gradient driven turbulence
journal, September 2000

  • Idomura, Y.; Wakatani, M.; Tokuda, S.
  • Physics of Plasmas, Vol. 7, Issue 9
  • DOI: 10.1063/1.1287418

New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions
journal, June 2013


Kinetic equations for low frequency instabilities in inhomogeneous plasmas
journal, January 1980

  • Antonsen, Thomas M.; Lane, Barton
  • Physics of Fluids, Vol. 23, Issue 6
  • DOI: 10.1063/1.863121

Electron temperature gradient driven turbulence
journal, May 2000

  • Jenko, F.; Dorland, W.; Kotschenreuther, M.
  • Physics of Plasmas, Vol. 7, Issue 5
  • DOI: 10.1063/1.874014

Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics
journal, March 1992

  • Hammett, G. W.; Dorland, W.; Perkins, F. W.
  • Physics of Fluids B: Plasma Physics, Vol. 4, Issue 7
  • DOI: 10.1063/1.860014

Nonlinear gyrofluid description of turbulent magnetized plasmas
journal, May 1992

  • Brizard, Alain
  • Physics of Fluids B: Plasma Physics, Vol. 4, Issue 5
  • DOI: 10.1063/1.860129

Gyrokinetic Simulations of Solar Wind Turbulence from Ion to Electron Scales
journal, July 2011


Two-dimensional gyrokinetic turbulence
journal, October 2010


Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas
journal, May 2009

  • Schekochihin, A. A.; Cowley, S. C.; Dorland, W.
  • The Astrophysical Journal Supplement Series, Vol. 182, Issue 1
  • DOI: 10.1088/0067-0049/182/1/310

The Magnetorotational Instability in a Collisionless Plasma
journal, September 2002

  • Quataert, Eliot; Dorland, William; Hammett, Gregory W.
  • The Astrophysical Journal, Vol. 577, Issue 1
  • DOI: 10.1086/342174

Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows
journal, October 2013


Ion and electron heating during magnetic reconnection in weakly collisional plasmas
journal, November 2014


Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas
journal, October 2011

  • Zocco, Alessandro; Schekochihin, Alexander A.
  • Physics of Plasmas, Vol. 18, Issue 10
  • DOI: 10.1063/1.3628639

Experimental Verification of Entropy Cascade in Two-Dimensional Electrostatic Turbulence in Magnetized Plasma
journal, February 2013


Interpreting Power Anisotropy Measurements in Plasma Turbulence
journal, February 2010


δf Algorithm
journal, July 1995

  • Denton, Richard E.; Kotschenreuther, M.
  • Journal of Computational Physics, Vol. 119, Issue 2
  • DOI: 10.1006/jcph.1995.1136

Partially Linearized Algorithms in Gyrokinetic Particle Simulation
journal, August 1993


Linearized model Fokker–Planck collision operators for gyrokinetic simulations. I. Theory
journal, December 2008

  • Abel, I. G.; Barnes, M.; Cowley, S. C.
  • Physics of Plasmas, Vol. 15, Issue 12
  • DOI: 10.1063/1.3046067

Equivalence of two different approaches to global δ f gyrokinetic simulations
journal, April 2015


Toroidal gyrofluid equations for simulations of tokamak turbulence
journal, November 1996

  • Beer, M. A.; Hammett, G. W.
  • Physics of Plasmas, Vol. 3, Issue 11
  • DOI: 10.1063/1.871538

Fourier-Hermite Solutions of the Vlasov Equations in the Linearized Limit
journal, January 1967