Gyrokinetic simulations

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The gyrokinetic formalism [1] [2] [3] is based on first principles and provides a valuable tool for investigating low frequency turbulence in fusion plasmas.

Kinetic theory describes the evolution of the distribution function $ f(\vec r, \vec v) $ on the basis of the Vlasov equation:

$ \frac{\rm d f}{\rm d t} = \frac{\partial f}{\partial t} + \vec v \cdot \nabla_r f + \frac{q}{M}(\vec E + \vec v \times \vec B)\cdot \nabla_v f = 0 $

The gyro-kinetic approach introduces a simplification by decomposing the full particle orbits into a rapid gyration about the magnetic field lines and a slow drift of the gyro centre $ \vec R $:

$ \vec r = \vec R + \vec \rho(\alpha) $

where $ \alpha $ is the gyro-angle. By averaging over this gyro-angle one arrives at the gyro-kinetic equation, which describes the evolution of the gyro centre in a phase space with one less dimension than the full Vlasov equation due to the averaging over the gyro-phase angle:

$ f(\vec R, v_{||},v_\perp) $

The gyro-kinetic equation is only valid for studying phenomena on timescales longer than the inverse of the gyro-frequency, and spatial scales larger than the gyro-radius. This is appropriate for, e.g., ITG (ion temperature gradient) turbulence.

Research activities

The Theory Group at the Laboratorio Nacional de Fusión collaborates with the Barcelona Supercomputing Center and the Max Planck IPP at Greifswald for the development and exploitation of the EUTERPE global gyrokinetic code.

The code EUTERPE has recently been benchmarked against the TORB code [4][5] in both linear and non-linear simulations.[6]


  1. T.S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Phys. Fluids 31, 2670 (1988)
  2. A.J. Brizard and T.S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys. 2, 421 (2007)
  3. Felix I. Parra and Iván Calvo, Phase-space Lagrangian derivation of electrostatic gyrokinetics in general geometry, Plasma Phys. Control. Fusion 53 (2011) 045001
  4. R. Hatzky, T.M. Tran, A. Konies, R. Kleiber, S.J. Allfrey, Energy conservation in a nonlinear gyrokinetic particle-in-cell code for ion-temperature-gradient-driven modes in theta-pinch geometry, Phys. Plasmas, 9-3, 912 (2002)
  5. C. Nührenberg, R. Hatzky, S. Sorge, et al., Global ITG Turbulence in Screw-Pinch Geometry, IAEA TM on Innovative Concepts and Theory of Stellarators, Madrid (2005)
  6. Edilberto Sánchez, Ralf Kleiber, Roman Hatzky, Alejandro Soba, Xavier Sáez, Francisco Castejón and Jose M. Cela, Linear and non-linear simulations using the EUTERPE gyrokinetic code, IEEE Transactions on Plasma Science 38-1, 2119 (2010)