Stochastic dynamics from the fractional Fokker-Planck-Kolmogorov equation: large-scale behavior of the turbulent transport coefficient

Turbulence spreading by resonant wave-wave interactions: A fractional kinetics approach

This paper is concerned with the processes of spatial propagation and penetration of turbulence from the regions where it is locally excited into initially laminar regions. The phenomenon has come to be known as "turbulence spreading" and witnessed a renewed attention in the literature recently. Here, we propose a comprehensive theory of turbulence spreading based on fractional kinetics. We argue that the use of fractional-derivative equations permits a general approach focusing on fundamentals of the spreading process regardless of a specific turbulence model and/or specific instability type. The starting point is the Hamiltonian of resonant wave-wave interactions, from which a family of scaling laws for the asymptotic spreading is derived. Both three- and four-wave interactions are considered. The results span from a subdiffusive spreading in the parameter range of weak chaos to avalanche propagation in regimes with population inversion. Attention is paid to how nonergodicity introduces weak mixing, memory and intermittency into spreading dynamics, and how the properties of non-Markovianity and nonlocality emerge from the presence of islands of regular dynamics in phase space. Also we resolve an existing question concerning turbulence spillover into gap regions, where the instability growth is locally suppressed, and show that the spillover occurs through exponential (Anderson-like) localization in case of four-wave interactions and through an algebraic (weak) localization in case of triad interactions. In the latter case an inverse-cubic behavior of the spillover function is found. Wherever relevant, we contrast our findings against the available observational and numerical evidence, and we also commit ourselves to establish connections with the models of turbulence spreading proposed previously.

Topological approximation of the nonlinear Anderson model

We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrödinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance overlap in phase space, ranging from a fully developed chaos involving Lévy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that the quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on the infinite Cayley tree (Bethe lattice). It is found in the vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t→+∞. The second moment of the associated probability distribution grows with time as a power law ∝ t^{α}, with the exponent α=1/3 exactly. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to the details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of "stripes" propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.

Analytical description for field-line wandering in strong magnetic turbulence

We investigate analytically the random walk of magnetic field lines. In previous analytical treatments of field line wandering or random walk, it was assumed that the turbulent magnetic field is much weaker than the mean field. In the present paper, we develop an improved analytical method to describe the stochastic properties of turbulent magnetic fields. This approach is an extension of the standard theory of field line wandering and can be applied to weak as well as to strong magnetic turbulence.

Pseudochaos and low-frequency percolation scaling for turbulent diffusion in magnetized plasma

The basic physics properties and simplified model descriptions of the paradigmatic "percolation" transport in low-frequency electrostatic (anisotropic magnetic) turbulence are theoretically analyzed. The key problem being addressed is the scaling of the turbulent diffusion coefficient with the fluctuation strength in the limit of slow fluctuation frequencies (large Kubo numbers). In this limit, the transport is found to exhibit pseudochaotic, rather than simply chaotic, properties associated with the vanishing Kolmogorov-Sinai entropy and anomalously slow mixing of phase-space trajectories. Based on a simple random-walk model, we find the low-frequency percolation scaling of the turbulent diffusion coefficient to be given by D/omega proportional, variantQ;{2/3} (here Q1 is the Kubo number and omega is the characteristic fluctuation frequency). When the pseudochaotic property is relaxed, the percolation scaling is shown to cross over to Bohm scaling. The features of turbulent transport in the pseudochaotic regime are described statistically in terms of a time fractional diffusion equation with the fractional derivative in the Caputo sense. Additional physics effects associated with finite particle inertia are considered.

"Strange" Fermi processes and power-law nonthermal tails from a self-consistent fractional kinetic equation

This study advocates the application of fractional dynamics to the description of anomalous acceleration processes in self-organized turbulent systems. Such processes (termed "strange" accelerations) involve both the non-Markovian fractal time acceleration events associated with a generalized stochastic Fermi mechanism, and the velocity-space Levy flights identified with nonlocal violent accelerations in turbulent media far from the (quasi)equilibrium. The "strange" acceleration processes are quantified by a fractional extension of the velocity-space transport equation with fractional time and phase space derivatives. A self-consistent nonlinear fractional kinetic equation is proposed for the stochastic fractal time accelerations near the turbulent nonequilibrium saturation state. The ensuing self-consistent energy distribution reveals a power-law superthermal tail psi(epsilon) proportional to epsilon(-eta) with slope 6 or =eta< or =7 depending on the type of acceleration process (persistent or antipersistent). The results obtained are in close agreement with observational data on the Earth's magnetotail.

Kubo number and magnetic field line diffusion coefficient for anisotropic magnetic turbulence

The magnetic field line diffusion coefficients Dx and D(y) are obtained by numerical simulations in the case that all the magnetic turbulence correlation lengths l(x), l(y), and l(z) are different. We find that the variety of numerical results can be organized in terms of the Kubo number, the definition of which is extended from R=(deltaB/B(0))(l(parallel)/l(perpendicular)) to R=(deltaB/B(0))(l(z)/l(x)), for l(x) > or = l(y). Here, l(parallel) (l(perpendicular)) is the correlation length along (perpendicular to) the average field B(0)=B(0)ê(z). We have anomalous, non-Gaussian transport for R less, similar 0.1, in which case the mean square deviation scales nonlinearly with time. For R greater, similar 1 we have several Gaussian regimes: an almost quasilinear regime for 0.1 less, similar R less, similar 1, an intermediate, transition regime for 1 less, similar R less, similar 10, and a percolative regime for R greater, similar 10. An analytical form of the diffusion coefficient is proposed, D(i)=D(deltaBl(z)/B(0)l(x))(mu)(l(i)/l(x))(nu)l(2)(x)/l(z), which well describes the numerical simulation results in the quasilinear, intermediate, and percolative regimes.