Scale-free vortex cascade emerging from random forcing in a strongly coupled system

The notions of self-organised criticality (SOC) and turbulence are traditionally considered to be applicable to disjoint classes of phenomena. Nevertheless, scale-free burst statistics is a feature shared by turbulent as well as self-organised critical dynamics. It has also been suggested that another shared feature is universal non-gaussian probability density functions (PDFs) of global fluctuations. Here, we elucidate the unifying aspects through analysis of data from a laboratory dusty plasma monolayer. We compare analysis of experimental data with simulations of a two-dimensional (2D) many-body system, of 2D fluid turbulence, and a 2D SOC model, all subject to random forcing at small scales. The scale-free vortex cascade is apparent from structure functions as well as spatio-temporal avalanche analysis, the latter giving similar results for the experimental and all model systems studied. The experiment exhibits global fluctuation statistics consistent with a non-gaussian universal PDF, but the model systems yield this result only in a restricted range of forcing conditions.

In the experimental setup 600 micron-size charged dust grains immersed in a weakly ionized plasma levitate above an electrode in a radio-frequency plasma discharge and form states ranging from hexagonal 2D crystals to 2D liquid or gaseous states. The most interesting states, however, are those that are semi-crystalline, but show strong grain transport and even vortical viscoelastic flows and turbulence due to development of defects in the crystal structure 7 . The particles are illuminated by a laser beam, their positions are recorded by a CCD camera, and their trajectories can be tracked to yield the full space-time history of every particle in the horizontal plane.
The experimental setup and methods are described in Ref. 7.
A natural unit of collective motion is a hexagonal vortex as seen in Fig. 1a, but topological restrictions do not allow a system of hexagonal cells, since such a system requires counter-streaming flows along some edges of neighbouring vortices. This velocity shear creates stress leading to break-up and merging of vortices. The velocity field of the resulting turbulent dynamics can be visualised (Fig. 1a) and animated by plotting the particle positions in time in grey-scale plots with decaying light intensity backwards in time. Vortical structures lasting shorter than the chosen decay time will not be visible in this animation. However, vortex dynamics on all scales down to the sampling time and the mean inter-particle distance can be visualised by creating a mean velocity field   In the many-body simulation one follows the trajectories of charged point masses subject to self-consistent, mutually repulsive electrostatic forces, in addition to the confining force from a parabolic confining potential. The effect of collisions with the particles of the neutral background gas has been modelled by a stochastic force with zero mean and a friction force proportional to the velocity of the point mass.
Data from the experiment has been used to model these forces as realistically as possible, but there are limitations in this respect. The main difference between the experimental and simulated dynamics is that the ratio between "thermal" particle velocities and large-scale fluid velocities are higher in the simulation than in the experiment. However, the most interesting features of the scale-free vortex cascade in the experimental flow are reproduced by the simulation, as will be shown below.
Spatial characteristics of the flow have some commonalities with 2D Navier-Stokes turbulence. A standard method of characterizing scaling in turbulence is to  but has scaling similar to the experiment for ! d less than 10 inter-particle distances.
Dust grains are partially trapped in their potential wells in the quasi-crystal, and their motion over longer distances than the mean inter-grain separation occurs through an avalanche of defects in the crystal structure. Avalanches (or bursts) can be identified as a connected cluster of particles in 3D space-time whose speed exceeds a prescribed threshold. This definition associates avalanching particles with those in the edge region of vortices. In space-time such clusters may be cylinder-shaped.
However, since vortices split and merge, avalanches generally constitute tree-like structures in space-time (Fig. 3a). Let the avalanches in a given observation series (or a simulation) be enumerated by the index In Fig. 3b we plot the mean area strongly driven sandpile models show that this form is only well described by the BHP form for a moderately strong drive. For a weak drive the PDF is a stretched exponential, and for strong drive it approaches a more symmetric form. We also find that the kinetic energy PDF for the fluid simulations depends on the details of the forcing. These rather severe limitations of the universality of the BHP distribution need to be considered in future theoretical efforts to explain its appearance.
2D turbulence is very different from its 3D counterpart in that it exhibits an inverse energy cascade to larger scales, giving rise to merging of vortices and generation of large-scale flows. This is because the phenomenon of vortex stretching is impossible in 2D. Emergence of system-size vortices has its counterpart in systemsize avalanches in SOC models. The many-body simulations of the dusty monolayer demonstrate that a scale-free distribution of vortices may result from stochastic, independent forcing of each individual dust grain. This is the essence of selforganised criticality: a scale-free hierarchy of dynamical entities emerging in an open, non-equilibrium system subject to a random drive on the smallest possible spatial scales. These apparent commonalities between 2D turbulence and sandpile avalanching, and their physical realisation in the dust monolayer, call for more general approaches to turbulence which go beyond the Navier-Stokes equation and invoke concepts from the theory of critical phenomena in non-equilibrium systems 11 .