Turbulence is a ubiquitous phenomenon, playing a role in many aspects of our daily life, and constituting an essential element in many fields of physics research. At the same time, its theoretical analysis remains one of the great challenges of physics. The advent of supercomputers has given us the new instrument of numerical modelling, and promises qualitative progress in our theoretical understanding of turbulence and ultimately also in our capability to predict quantitatively its consequences.

The presence of a magnetic field in a plasma adds a completely new element. In a one-fluid model it extends the set of ideal invariants of the Navier-Stokes equations from two (in the three-dimensional (3D) case: energy and kinetic helicity) to three (energy, cross-helicity and magnetic helicity). As in the 2D (but not the general 3D) hydrodynamic case, one of the invariants - the magnetic helicity - exhibits an inverse cascade, suggesting that the system drives towards self-organization and the formation of large-scale structures. More complexity is added to the system by the fact that for many phenomena a simple one-fluid description of the plasma is not adequate and has to be substituted by a more complex two-fluid model with kinetic corrections, or even a fully kinetic model.

Explaining the behaviour of magnetically confined laboratory plasmas has been a driving force in theoretical and experimental turbulence studies. The papers in this focus issue are concerned with configurations in which large-scale macroscopic instabilities are absent. In this situation, namely, in toroidal configurations with nested, closed magnetic surfaces, turbulence driven by smaller scale linear (or sometimes non-linear) instabilities determines the transport of energy across field lines. The turbulent structures are highly anisotropic, with scales of the order of the torus circumference along and (typically) of several ion Larmor radii perpendicular to the field lines. The large correlation length of turbulence along field lines requires that, in general, at least some aspects of the geometry of the plasma, and in particular its toroidal nature, have to be taken into account in these calculations. The driving mechanisms and the characteristics of the turbulence vary with the parameter regime of the plasma between the core of the toroidal plasma column and the edge, and also depend on whether electrons or ions are the plasma component that has been primarily heated. Plasma turbulence therefore involves, in principle, scales ranging from the electron gyro-radius to the plasma cross section. In numerical simulations it is therefore necessary to find a compromise between the complexity of the plasma model, the spatial resolution of the calculations and the geometrical extent of the simulated plasma region. Clearly, the advent of more powerful computers will allow these limits to be pushed back further.

The most striking result of magnetic confinement research during the last two decades has been the observation of so-called `transport barriers', across which the turbulent transport is strongly reduced, and which contribute, in spite of their narrow width, dominantly to the thermal insulation of the core plasma. Such barriers were first observed near the plasma edge (the `H-mode' barrier), but were later also produced in the plasma interior (`ITBs': internal transport barriers). Much theoretical and experimental evidence has accumulated that they are produced by sheared flows (or, equivalently, gradients in the electric field perpendicular to the flux surfaces). Both analytic and numerical models have shown that such flows, which are of a large scale within a flux surface, can spontaneously arise from smaller scale turbulence. The identification of the conditions under which they form a stable barrier, and the self-consistent simulation of such a transition remain, however, an important research topic. Transport barriers also arise in very different plasma parameter regimes, and can affect in different ways the particle, the electron, and the ion heat transport.

The articles included in this focus issue of New Journal of Physics highlight the state of the art in this field, which is characterized on the one hand by a rapid development of our capability to construct a virtual, magnetically confined plasma on a computer, and on the other, by a growing experimental effort to test the more `microscopic' predictions of these models by fluctuation studies and advanced data interpretation methods.

Focus on Turbulence in Magnetized Plasmas Contents

Aspects of flow generation and saturation in drift-wave turbulence
Volker Naulin

The role of radial electric fields in linear and nonlinear gyrokinetic full radius simulations
S J Allfrey, A Bottino, O Sauter and L Villard

Zonal flow generation in the improved confinement mode plasma and its role in confinement bifurcations
M G Shats and W M Solomon

Comparison of turbulence measurements and simulations of the low-temperature plasma in the torsatron TJ-K
C Lechte, S Niedner and U Stroth

Stellarator turbulence at electron gyroradius scales
F Jenko and A Kendl

The spatial structure of edge fluctuations in the Wendelstein 7-AS stellarator
J Bleuel, M Endler, H Niedermeyer, M Schubert, H Thomsen and The W7-AS Team

Fluctuations, sheared radial electric fields and transport interplay in fusion plasmas
C Hidalgo, M A Pedrosa and B Gonçalves

The nonlinear drift wave instability and its role in tokamak edge turbulence
Bruce D Scott

Dynamical simulations of boundary plasma turbulence in divertor geometry
X Q Xu, W M Nevins, R H Cohen, J R Myra and P B Snyder

Large-scale fluctuation structures in plasma turbulence
O Grulke and T Klinger

A model of nonlinear evolution and saturation of the turbulent MHD dynamo
A A Schekochihin, S C Cowley, G W Hammett, J L Maron and J C McWilliams

Slablike ion temperature gradient driven mode in reversed shear tokamaks
Y Idomura, S Tokuda and Y Kishimoto

Turbulent electron thermal transport in tokamaks
W Horton, B Hu, J Q Dong and P Zhu

Modulational instability of drift waves
K Hallatschek and P H Diamond

Sibylle Günter
Max-Planck-Institut für Plasmaphysik, Garching, Germany