Understanding and predicting transport of particles, momentum and energy across the magnetic field in magnetically confined toroidal plasmas is of the utmost importance in the fusion energy program for an obvious reason: the highest the transport level, the lower the confinement time. Hence, relevant progress in the comprehension of transport phenomena will surely have an impact on future reactor design and cost.

Assume that our toroidal plasma is stable with respect to long-wavelength perturbations (comparable to the system size). Then, there are three essential mechanisms of transport: collisions (the theory of collisional plasma transport in a constant magnetic field constitutes the classical theory), magnetic field geometry (the theory of collisional plasma transport in a toroidal plasma constitutes the neoclassical theory), and microturbulence. Collisional transport theory in a magnetized plasma (gyrofrequency much larger than transit frequency, or equivalently, Larmor radius ρ much smaller than the typical length of variation of the magnetic field, L) with constant magnetic field [Braginskii- 65] closely resembles to Chapman-Enskog theories [Chapman-52] although it is not identical: parallel and perpendicular components are different in a magnetized plasma. Much more dramatic is the difference between Chapman-Enskog theories and neoclassical theory (the seminal reference for neoclassical transport is [Galeev-68]; see also the review [Hinton-76]). In the long mean-free-path regime, appropriate to typical temperatures of fusion-grade plasmas, the magnetic field geometry has enormous influence on transport to the point that transport coefficients are larger by orders of magnitude than those predicted by the classical theory.

In general, neoclassical theory sets only a lower bound on transport in toroidal plasmas. Fusion plasmas are usually unstable to perturbations with wavelengths perpendicular to the magnetic field on the order of the ion Larmor radius or smaller. We call them microinstabilities [Horton-99] and we are forced to live with them. They are not catastrophic but are responsible for the generation of microturbulence, that enhances transport (with respect to the neoclassical level) and therefore degrades confinement. Their typical frequencies are on the order of the sound speed over L. Gyrokinetic theory [Catto-78, Brizard-07] is the appropriate framework to describe microturbulence. It is a reduced kinetic theory, based on an asymptotic expansion of the Vlasov-Poisson or Vlasov-Maxwell set of equations in powers of ε = ρ / L, in which the degree of freedom associated to the gyration of particles around field lines (considered irrelevant because microturbulence time scales are much longer than the inverse of the gyrofrequency) has been rigorously averaged out.

On the theoretical side, we will mostly deal with fundamental problems in gyrokinetic theory, but not only. Some questions arising in neoclassical theory will also be addressed.



[Braginskii-65] S. I. Braginskii, in Reviews of Plasma Physics, ed. M. A. Leontovich (Consultants Bureau, New York, 1965), vol. 1, p. 205.

[Chapman-52] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University, London, 1952).

[Galeev-68] A. A. Galeev and R. Z. Sagdeev, Sov. Phys.-JETP 26, 233 (1968).

[Hinton-76] F.L. Hinton and R.D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976).

[Horton-99] W. Horton, Rev. Mod. Phys. 71, 735 (1999).

[Catto-78] P.J. Catto, Plasma Phys. 20, 719 (1978).

[Brizard-07] A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. 79, 421 (2007).





Gyrokinetic simulations

Gyrokinetic theory represents a significant step towards achieving accurate simulations of microturbulence in magnetized plasmas [Garbet-10]. The periodic motion of a charged particle around the magnetic field line is averaged out rigorously, keeping the effect of the small Larmor radius.  In this way the computing resources required are largely reduced. This allows accurate simulations of microturbulence of spatial scales larger than Larmor radius, for which the typical timescales are much larger that the gyrofrequency.

The computer code used at the Laboratorio Nacional de Fusión (LNF) is called EUTERPE [Jost-00, Kornilov-04, Kleiber-12], a global non-linear electromagnetic gyrokinetic code developed and maintained at the Max-Planck-Institut für Plasmaphysik at Greifswald with the distinctive feature that it has been designed to
allow any magnetic configuration whose MHD equilibrium can be computed by VMEC [Hirshman-98]. Thus, it is the appropriate tool for microturbulence simulations in TJ-II, which has a complicated magnetic geometry.

Within the framework of this project we study the following topics by means of gyrokinetic simulations:

Collisionless damping of zonal flows. The importance of zonal flows for the self-regulation of turbulence in fusion plasmas and the concomitant transport reduction is generally recognized. The understanding of the regulating effect of sheared flows gave rise to the shear decorrelation paradigm [Biglari-90]. Zonal flows can be generated by the non-linear interaction of unstable modes and are expected to be eventually damped by collisional (neoclassical) processes. For times short in comparison to the collision time or in hot, low-collisionality plasmas, the non-collisional limit is relevant.  We aim to understand the collisionless damping of zonal flows in TJ-II and its possible connection with recent experimental measurements of long-range correlations [Pedrosa-08, Alonso-12].

Microinstabilities in TJ-II. The density and temperature gradients in fusion plasmas constitute sources of free energy that trigger the development of different instabilities: Ion Temperature Gradient (ITG), Electron Temperature Gradient (ETG), Trapped Electron Modes (TEM), among them. Each instability possesses different signatures and effects [Horton-99]. One of the objectives of this research project is to clarify the applicability of gyrokinetic simulations to each region and type of TJ-II plasmas and characterize the existing instabilities.

Numerical investigation of basic statistical features of microturbulence. Recent analytical results on gyrokinetic absolute statistical equilibria [Zhu-10] and analytically derived phenomenological laws à la Kolmogorov [Barnes-11] might give new insights on the statistical features of the gyrokinetic turbulent state. Also the results of [R. Sánchez-08] deserve to be mentioned, where the transport of tracer particles across a sheared zonal flow in certain gyrokinetic simulations was found to be subdiffusive and Lévy. We will try to gain deeper understanding of these new findings in the course of the project.


[Alonso-12]J. A. Alonso, et. al., Nucl. Fusion, 52, 6, 63010 (2012).

[Barnes-11] M. Barnes et al., Phys. Rev. Lett. 107, 115003 (2011).

[Biglari-90] H. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids B 2, 1 (1990).

[Garbet-10] X. Garbet, Y. Idomura, L. Villard, and T. H. Watanabe, Nucl. Fusion 50, 043002 (2010).

[Hirshman-98] S. P. Hirshman and J. Breslau, Phys. Plasmas 5, 2664 (1998).

[Horton-99] W. Horton, Rev. Mod. Phys. 71, 735 (1999).

[Jost-00] G. Jost, Simulation particulaires d’ondes de dérive dans des configurations magnétiques 3D, PhD dissertation, École Polytechnique Fédérale de Lausanne, 2000.

[Kornilov-04] V. Kornilov et al., Phys. Plasmas 11, 3196 (2004).

[Kleiber-10] R. Kleiber, R. Hatzky, and A. Mishchenko, Contrib. Plasma Phys. 50, 766 (2010).

[Kleiber-12] R. Kleiber and R. Hatzky. Comput. Phys. Commun., 183, 2, 305–308 (2012).

[Pedrosa-08] M. A. Pedrosa et al., Phys. Rev. Lett. 100, 215003 (2008).

[R. Sánchez-08] R. Sánchez et al., Phys. Rev. Lett. 101, 205002 (2008).

[Zhu-10] J. Z. Zhu and G. W. Hammett. Phys. Plasmas 17, 122307 (2010).

Neoclassical simulations

Neoclassical theory [Galeev-68, Hinton-76] sets a lower bound on transport in toroidal plasmas. In the case of three-dimensional devices, neoclassical transport may become even dominant as the temperature of the plasma increases, due to the unfavorable temperature scaling of the transport coefficients. Hence, the comprehensive study of neoclassical transport is a necessary condition for both the understanding of reactor-relevant regimes and the design of future reactors.

Calculating neoclassical transport implies solving the Drift Kinetic Equation (DKE) [Hinton-76]. Apart from some particular limits in the parameter space, the DKE cannot be solved analytically, and one has to resort to numerical computations. Within this project, we will address the problems with the following tools:

DKES [HIrshman-86], a neoclassical transport code based on the radially local and monoenergetic approach. It is complemented with numerical algorithms that guarantee momentum conservation [Maassberg-09].

FORTEC-3D [Satake-06], a delta-f global Monte Carlo neoclassical code developed at NIFS, which allows to remove the local and monoenergetic approximations.

EUTERPE [Kleiber-12], a delta-f global gyrokinetic code developed at IPP (see Gyrokinetic Simulations).

When possible, these simulations will be complemented with semianalytical calculations of simplified versions of the DKE for the exact geometry of TJ-II.


[Galeev-68] A. A. Galeev and R. Z. Sagdeev, Sov. Phys.-JETP 26, 233 (1968).

[Hinton-76] F.L. Hinton and R.D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976).

[HIrshman-86] S.P. Hirshman, K.C. Shaing, W.I. van Rij, C.O. Beasley and E.C. Crume, Phys. Fluid 29, 2951 (1986).

[Maasberg-09] H. Maassberg, C.D. Beidler and Y. Turkin, Phys. Plasmas 16, 072504 (2009).

[Satake-06] S. Satake M. Okamoto, N. Nakajima, H. Sugama and M. Yokoyama, Plasma and Fusion Res. 1, 002 (2006).

[Kleiber-12] R. Kleiber and R. Hatzky, Comput. Phys. Commun. 183, 305 (2012).


The ultimate goal of our Theory and Simulation activities is to understand what is observed in experiments, specifically in the TJ-II stellarator device. Some of the measurements we would like to improve our understanding of are

  • Decay of Zonal flow-like structures detected with double Langmuir probe systems. These turbulence-generated, global ExB rotation patterns are thought to be beneficial for confinement and have been proposed to trigger Low-to-High confinement transitions. The figure shows a clear anti-correlation between an H-alpha signal (blue, related to the outward flux of particles) and the amplitude of the Zonal flow (green) measured in TJ-II.


  • Radiation asymmetries caused by impurity density re-distribution within flux surfaces. Impurities with a high charge are prone to develop density inhomogeneities on flux surfaces, which should be manifest in radiation signals. The re-distribution of impurities is mainly caused by ion-impurity friction which is also a main drive of radial impurity transport. Therefore, understanding those asymmetries is a way to test our understanding of impurity radial transport, which is one of the most important issues in future reactor viability and performance.