Diego del-Castillo-Negrete

Fractional diffusion in plasma turbulence

Transport of tracer particles is studied in a model of three-dimensional, resistive, pressure-gradient-driven plasma turbulence. It is shown that in this system transport is anomalous and cannot be described in the context of the standard diffusion paradigm. In particular, the probability density function (pdf) of the radial displacements of tracers is strongly non-Gaussian with algebraic decaying tails, and the moments of the tracer displacements exhibit superdiffusive scaling. To model these results we present a transport model with fractional derivatives in space and time. The model incorporates in a unified way nonlocal effects in space (i.e., non-Fickian transport), memory effects (i.e., non-Markovian transport), and non-Gaussian scaling. There is quantitative agreement between the turbulence transport calculations and the fractional diffusion model. In particular, the model reproduces the shape and space-time scaling of the pdf, and the superdiffusive scaling of moments.

Fractional diffusion models of nonlocal transport

A class of nonlocal models based on the use of fractional derivatives (FDs) is proposed to describe nondiffusive transport in magnetically confined plasmas. FDs are integro-differential operators that incorporate in a unified framework asymmetric non-Fickian transport, non-Markovian (“memory”) effects, and nondiffusive scaling. To overcome the limitations of fractional models in unbounded domains, we use regularized FDs that allow the incorporation of finite-size domain effects, boundary conditions, and variable diffusivities. We present an α-weighted explicit/implicit numerical integration scheme based on the Grunwald-Letnikov representation of the regularized fractional diffusion operator in flux conserving form. In sharp contrast with the standard diffusive model, the strong nonlocality of fractional diffusion leads to a linear in time response for a decaying pulse at short times. In addition, an anomalous fractional pinch is observed, accompanied by the development of an uphill transport region where ...

Front propagation and segregation in a reaction–diffusion model with cross-diffusion

Abstract A study of front propagation and segregation in a system of reaction–diffusion equations with cross-diffusion is presented. The reaction models predator–prey dynamics involving two fields. The diffusive part is nonlinear in the sense that the diffusion coefficient, instead of being a constant as in the well-studied case, depends on one of the fields. A key element of the model is a cross-diffusion term according to which the flux of one of the fields is driven by gradients of the other field. The original motivation of the model was the study of the turbulence–shear flow interaction in plasmas. The model also bears some similarities with models used in the study of spatial segregation of interacting biological species. The system has three nontrivial fixed points, and a study of traveling fronts solutions joining these states is presented. Depending on the stability properties of the fixed points, the fronts are uniform or have spatial structure. In the latter case, a cross-diffusion-driven pattern-forming ( k ≢0) instability leads to segregation in the wake of the front. The segregated state consists of layered structures. A Ginzburg–Landau amplitude equation is used to describe the dynamics near marginal stability.

Fractional diffusion models of non-local perturbative transport: numerical results and application to JET experiments

Perturbative experiments in magnetically confined fusion plasmas have shown that edge cold pulses travel to the centre of the device on a time scale much faster than expected on the basis of diffusive transport. An open issue is whether the observed fast pulse propagation is due to non-local transport mechanisms or if it could be explained on the basis of local transport models. To elucidate this distinction, perturbative experiments involving ICRH power modulation in addition to cold pulses have been conducted in JET for the same plasma. Local transport models have found problematic the reconciliation of the fast propagation of cold pulses with the comparatively slower propagation of heat waves generated by power modulation. In this paper, a non-local model based on the use of fractional diffusion operators is used to describe these experiments. A numerical study of the parameter dependence of the pulse speed and the amplitude and phase of the heat wave is also presented.

Truncation effects in superdiffusive front propagation with Lévy flights.

A numerical and analytical study of the role of exponentially truncated Lévy flights in the superdiffusive propagation of fronts in reaction-diffusion systems is presented. The study is based on a variation of the Fisher-Kolmogorov equation where the diffusion operator is replaced by a lambda -truncated fractional derivative of order alpha , where 1lambda is the characteristic truncation length scale. For lambda=0 there is no truncation, and fronts exhibit exponential acceleration and algebraically decaying tails. It is shown that for lambda not equal0 this phenomenology prevails in the intermediate asymptotic regime (chit);{1alpha}x1lambda where chi is the diffusion constant. Outside the intermediate asymptotic regime, i.e., for x>1lambda , the tail of the front exhibits the tempered decay varphi approximately e;{-lambdax}x;{(1+alpha)} , the acceleration is transient, and the front velocity v_{L} approaches the terminal speed v_{*}=(gamma-lambda;{alpha}chi)lambda as t-->infinity , where it is assumed that gamma>lambda;{alpha}chi with gamma denoting the growth rate of the reaction kinetics. However, the convergence of this process is algebraic, v_{L} approximately v_{*}-alpha(lambdat) , which is very slow compared to the exponential convergence observed in the diffusive (Gaussian) case. An overtruncated regime in which the characteristic truncation length scale is shorter than the length scale of the decay of the initial condition, 1nu , is also identified. In this extreme regime, fronts exhibit exponential tails, varphi approximately e;{-nux} , and move at the constant velocity v=(gamma-lambda;{alpha}chi)nu .

Universal Probability Distribution Function for Bursty Transport in Plasma Turbulence

Bursty transport phenomena associated with convective motion present universal statistical characteristics among different physical systems. In this Letter, a stochastic univariate model and the associated probability distribution function for the description of bursty transport in plasma turbulence is presented. The proposed stochastic process recovers the universal distribution of density fluctuations observed in plasma edge of several magnetic confinement devices and the remarkable scaling between their skewness S and kurtosis K. Similar statistical characteristics of variabilities have been also observed in other physical systems that are characterized by convection such as the x-ray fluctuations emitted by the Cygnus X-1 accretion disc plasmas and the sea surface temperature fluctuations.

Local and nonlocal parallel heat transport in general magnetic fields.

A novel approach for the study of parallel transport in magnetized plasmas is presented. The method avoids numerical pollution issues of grid-based formulations and applies to integrable and chaotic magnetic fields with local or nonlocal parallel closures. In weakly chaotic fields, the method gives the fractal structure of the devils staircase radial temperature profile. In fully chaotic fields, the temperature exhibits self-similar spatiotemporal evolution with a stretched-exponential scaling function for local closures and an algebraically decaying one for nonlocal closures. It is shown that, for both closures, the effective radial heat transport is incompatible with the quasilinear diffusion model.

The origin of diffusion: the case of non-chaotic systems

Abstract We investigate the origin of diffusion in non-chaotic systems. As an example, we consider 1D map models whose slope is everywhere 1 (therefore the Lyapunov exponent is zero) but with random quenched discontinuities and quasi-periodic forcing. The models are constructed as non-chaotic approximations of chaotic maps showing deterministic diffusion, and represent one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the Ehrenfest wind-tree model). In particular, a simple construction shows that these maps define non-chaotic billiards in space–time. The models exhibit, in a wide range of the parameters, the same diffusive behavior of the corresponding chaotic versions. We present evidence of two sufficient ingredients for diffusive behavior in one-dimensional, non-chaotic systems: (i) a finite size, algebraic instability mechanism; (ii) a mechanism that suppresses periodic orbits.

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Levy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Levy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker–Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Levy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Levy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.

Compression of magnetohydrodynamic simulation data using singular value decomposition

Numerical calculations of magnetic and flow fields in magnetohydrodynamic (MHD) simulations can result in extensive data sets. Particle-based calculations in these MHD fields, needed to provide closure relations for the MHD equations, will require communication of this data to multiple processors and rapid interpolation at numerous particle orbit positions. To facilitate this analysis it is advantageous to compress the data using singular value decomposition (SVD, or principal orthogonal decomposition, POD) methods. As an example of the compression technique, SVD is applied to magnetic field data arising from a dynamic nonlinear MHD code. The performance of the SVD compression algorithm is analyzed by calculating Poincare plots for electron orbits in a three-dimensional magnetic field and comparing the results with uncompressed data.