One of the most remarkable features of turbulence is that it operates in a radically different way in two-dimensional (2D) flows than it does in three-dimensional (3D) flows. Whilst the former is characterised by an inverse energy cascade that sees larger, less dissipative structures emerge, the latter tends to very efficiently dissipate energy by transferring it to small scales where it is dissipated by viscous friction.

The question of whether turbulence obeys two or three-dimensional dynamics therefore has drastic consequences for the natural and industrial processes where it is involved. This concerns numerous classes of realistic systems under the influence of rotation, stratification or magnetic fields as well as in purely 2D geometrical configurations. The tendency to two-dimensionality in stratified flows and rotating flows is a prominent feature of planetary flows such as atmospheres and oceans. This feature has also been observed in electrically conducting flows under a strong magnetic field (MagnetoHydroDynamic flows), extending the relevance of the question of flow dimensionality to astrophysical and laboratory plasma flows (for instance, the understanding of particles and heat flux dynamics in the magnetic-field transverse directions has tremendous importance in the achievement of thermo-nuclear fusion), but also to liquid-metals engineering problems in the nuclear and metallurgical industries. The common tendency to two-dimensionality, however, hides a variety of physical mechanisms: the propagation of inertial waves along the rotation axis promotes two-dimensionality in rotating flows, while eddy currents induced play this role in MHD flows, by generating Alfven waves if magnetic advection is important (in plasmas for instance), or by diffusing momentum along the field if it isn't (as often in liquid metals). In magnetized plasmas, drift waves play a leading role in momentum transport perpendicular in the directions perpendicular to the magnetic field. Their dynamics share many characteristics with inertial (Rossby) waves, both modelled by the Charney-Hasegawa-Mima equation. The geometry of the fluid domain too can favour either 2D or 3D dynamics, in particular if it is very thin along one direction, as in Hele-Shaw cells, or soap films. Nevertheless, the question of dimensionality of turbulence is conditioned by a number of features that are common to these systems: the boundaries are part of the very definition of the concept of two-dimensionality. Consequently, experimentalists willing to investigate flows with 2D dynamics are confronted to their influence. The way in which the flow is driven leaves a non-universal signature in the flow dynamics too.

Recent work tends to show that both the walls and the forcing play a crucial role in deciding whether the dynamics of the flow is closer to a 2D or a 3D one. At least two basic forms of three-dimensionality exist too: variation of physical quantities in the third direction and appearance of a third component in the velocity field. These can interact with each other and but it also recently emerged both from theoretical and experimental work that this latter form is strongly connected to the existence of an inverse energy cascade.

The purpose of this Colloquium is to gather researchers from different communities who are in fact working on this same problem. The many recent advances, a few of which are mentioned above, provide the ideal ground to seed a concerted approach on the question of the dimensionality of turbulence. In exposing to each other the mechanisms that govern the transition between 2D and 3D dynamics in the particular type of turbulence they are studying, it is hoped that the participants will discover more common ground. This will bring progress in the general understanding not only of how three-dimensionality appears or vanishes but also, and perhaps most importantly, in how it determines whether turbulence obeys 2D or 3D dynamics. In particular, bringing together expertise from several disciplines, and from both theoretical and experimental background, will help us distinguish mechanisms that are discipline-specific from those linked to boundaries and flow forcing and to find out if universal mechanisms exist.