Nondiffusive transport regimes for suprathermal ions in turbulent plasmas

Persistent random walks of charged particles across magnetic field lines

We investigate the time evolution of the mean location and variance of a charged particle subject to random collisions that are Poisson distributed. The particle moves on a plane and is subject to a magnetic field applied perpendicular to the plane, so it is constrained to move in circles in the absence of collisions. We develop a procedure that yields analytic expressions of the mean and variance. These results are valid for arbitrary times after the start of the walk, including early on when, on average, less than one collision is expected. As an example of their applicability, we use these expressions to model experimental results and simulations of suprathermal ions propagating in a turbulent plasma in TORPEX (the TORoidal Plasma EXperiment).

Truncated Lévy motion through path integrals and applications to nondiffusive suprathermal ion transport

Fractional Levy motion has been derived from its generalized Langevin equation via path integrals in earlier works and has since proven to be a useful model for nonlocal and non-Markovian processes, especially in the context of nondiffusive transport. Here, we generalize the approach to treat tempered Lévy distributions and derive the propagator and diffusion equation of truncated asymmetrical fractional Levy motion via path integrals. The model now recovers exponentially tempered tails above a chosen scale in the propagator, and therefore finite moments at all orders. Concise analytical expressions for its variance, skewness, and kurtosis are derived as a function of time. We then illustrate the versatility of this model by applying it to simulations of the turbulent transport of fast ions in the TORPEX basic plasma device.

Particle transport at arbitrary timescales with Poisson-distributed collisions

We develop a model to investigate the time evolution of the mean location and variance of a random walker subject to Poisson-distributed collisions at constant rate. The collisions are instantaneous velocity changes where a new value of velocity is generated from a model probability function. The walker is persistent, which means that it moves at constant velocity between collisions. We study three different cases of velocity transition functions and compute the transport properties from the evolution of the variance. We observe that transport can change character over time and that early times show features that, in general, depend on the initial conditions of the walker.

Time intermittency in nondiffusive transport regimes of suprathermal ions in turbulent plasmas

Intermittent phenomena have long been studied in the context of nondiffusive transport across a variety of fields. In the TORPEX device, the cross-field spreading of an injected suprathermal ion beam by electrostatic plasma turbulence can access different nondiffusive transport regimes. A comprehensive set of suprathermal ion time series has been acquired, and time intermittency quantified by their skewness. Values distinctly above background level are found across all observed transport regimes. Intermittency tends to increase toward quasi- and superdiffusion and for longer propagation times of the suprathermal ions. The specific prevalence of intermittency is determined by the meandering motion of the instantaneous ion beam. We demonstrate the effectiveness of an analytical model developed to predict local intermittency from the time-average beam. This model might thus be of direct interest for similar systems, e.g., in beam physics, or meandering flux-rope models for solar energetic particle propagation. More generally, it illustrates the importance of identifying the system-specific sources of time-intermittent behavior when analyzing nondiffusive transport.

On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

The problem of studying anomalous superdiffusive transport by means of fractional transport equations is considered. We concentrate on the case when an advection flow is present (since this corresponds to many actual plasma configurations), as well as on the case when a boundary is also present. We propose that the presence of a boundary can be taken into account by adopting the Caputo fractional derivatives for the side of the boundary (here, the left side), while the Riemann-Liouville derivative is used for the unbounded side (here, the right side). These derivatives are used to write the fractional diffusion–advection equation. We look for solutions in the steady-state case, as such solutions are of practical interest for comparison with observations both in laboratory and astrophysical plasmas. It is shown that the solutions in the completely asymmetric cases have the form of Mittag-Leffler functions in the case of the left fractional contribution, and the form of an exponential decay in the case of the right fractional contribution. Possible applications to space plasmas are discussed.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ-stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Nonlinear Monte Carlo model of superdiffusive shock acceleration with magnetic field amplification

Fast collisionless shocks in cosmic plasmas convert their kinetic energy flow into the hot downstream thermal plasma with a substantial fraction of energy going into a broad spectrum of superthermal charged particles and magnetic fluctuations. The superthermal particles can penetrate into the shock upstream region producing an extended shock precursor. The cold upstream plasma flow is decelerated by the force provided by the superthermal particle pressure gradient. In high Mach number collisionless shocks, efficient particle acceleration is likely coupled with turbulent magnetic field amplification (MFA) generated by the anisotropic distribution of accelerated particles. This anisotropy is determined by fast particle transport, making the problem strongly nonlinear and multiscale. Here, we present a nonlinear Monte Carlo model of collisionless shock structure with superdiffusive propagation of high-energy Fermi accelerated particles coupled to particle acceleration and MFA, which affords a consistent description of strong shocks. A distinctive feature of the Monte Carlo technique is that it includes the full angular anisotropy of the particle distribution at all precursor positions. The model reveals that the superdiffusive transport of energetic particles (i.e., Lévy-walk propagation) generates a strong quadruple anisotropy in the precursor particle distribution. The resultant pressure anisotropy of the high-energy particles produces a nonresonant mirror-type instability that amplifies compressible wave modes with wavelengths longer than the gyroradii of the highest-energy protons produced by the shock.