Constructing criteria to diagnose the likelihood of extreme events in the case of the electric power grid.

A set of new criteria for the diagnosis of the possible occurrence of a large blackout are constructed, using the output from a model for the dynamics of the electric power grid (the OPA model). The approach used is to look for maximum values of the Transfer Entropy between time series of the system variables and the time series of large blackouts. The best criterion is found by looking at the number of overloaded lines at the beginning of the day.

Does size matter?

Failures of the complex infrastructures society depends on having enormous human and economic cost that poses the question: Are there ways to optimize these systems to reduce the risks of failure? A dynamic model of one such system, the power transmission grid, is used to investigate the risk from failure as a function of the system size. It is found that there appears to be optimal sizes for such networks where the risk of failure is balanced by the benefit given by the size.

Dynamical coupling between gradients and transport in fusion plasmas.

The dynamical coupling between density gradients and particle transport has been investigated using similar experimental tools in the plasma boundary of different tokamak (JET, ISTTOK) and stellarator (TJ-II) devices, showing that the size of turbulent events is minimum in the proximity of the most probable density gradient. Experimental results were found to be consistent with results from two very different models of plasma turbulence and transport. The present findings, common to several plasma devices, suggest the importance of self-regulation mechanisms between plasma transport and gradients in fusion devices.

Topological characterization of the transition from laminar regime to fully developed turbulence in the resistive pressure-gradient-driven turbulence model.

For the resistive pressure-gradient-driven turbulence model, the transition from laminar regime to fully developed turbulence is not simple and goes through several phases. For low values of the plasma parameter beta, a single quasicoherent structure forms. As beta increases, several of these structures may emerge and in turn take the dominant role. Finally, at high beta, fully developed turbulence with a broad spectrum is established. A suitable characterization of this transition can be given in terms of topological properties of the flow. Here, we analyze these properties that provide an understanding of the turbulence-induced transport and give a measure of the breaking of the homogeneity of the turbulence. To this end, an approach is developed that allows discriminating between topological properties of plasma turbulence flows that are relevant to the transport dynamics and the ones that are not. This is done using computational homology tools and leads to a faster convergence of numerical results for a fixed level of resolution than previously presented in Phys. Rev. E 78, 066402 (2008).

The impact of risk-averse operation on the likelihood of extreme events in a simple model of infrastructure.

A simple dynamic model of agent operation of an infrastructure system is presented. This system evolves over a long time scale by a daily increase in consumer demand that raises the overall load on the system and an engineering response to failures that involves upgrading of the components. The system is controlled by adjusting the upgrading rate of the components and the replacement time of the components. Two agents operate the system. Their behavior is characterized by their risk-averse and risk-taking attitudes while operating the system, their response to large events, and the effect of learning time on adapting to new conditions. A risk-averse operation causes a reduction in the frequency of failures and in the number of failures per unit time. However, risk aversion brings an increase in the probability of extreme events.

Nature of transport across sheared zonal flows in electrostatic ion-temperature-gradient gyrokinetic plasma turbulence.

It is shown that the usual picture for the suppression of turbulent transport across a stable sheared flow based on a reduction of diffusive transport coefficients is, by itself, incomplete. By means of toroidal gyrokinetic simulations of electrostatic, collisionless ion-temperature-gradient turbulence, it is found that the nature of the transport is altered fundamentally, changing from diffusive to anticorrelated and subdiffusive. Additionally, whenever the flows are self-consistently driven by turbulence, the transport gains an additional non-Gaussian character. These results suggest that a description of transport across sheared flows using effective diffusivities is oversimplified.

Characterization of nondiffusive transport in plasma turbulence via a novel Lagrangian method.

A novel method to probe and characterize the nature of the transport of passive scalars carried out by a turbulent flow is introduced. It requires the determination of two exponents which encapsulate the statistical and correlation properties of the component of interest of the Lagrangian velocities of the flow. Numerical simulations of a magnetically confined, near-critical turbulent plasma, known to exhibit superdiffusive radial transport, are used to illustrate the method. It is shown that the method can easily detect the change in the dynamics of the radial transport that takes place after adding to the simulations a (subdominant) diffusive channel of tunable strength.

Evidence of long-distance correlation of fluctuations during edge transitions to improved-confinement regimes in the TJ-II stellarator.

Long-distance coupling between edge parameters' fluctuations has been investigated in the TJ-II stellarator. Results show long-range correlations in potential fluctuations, which are amplified by the development of radial electric fields during transitions to improved-confinement regimes, whereas there is no correlation between ion saturation current signals. These experimental findings suggest the importance of long-range correlations as a new fingerprint of the plasma behavior during the development of edge shear flows and the key role of electric fields to amplify them.

Topological characterization of flow structures in resistive pressure-gradient-driven turbulence.

Visualization of turbulent flows is a powerful tool to help understand the turbulence dynamics and induced transport. However, it does not provide a quantitative description of the observed structures. In this paper, an approach to characterize quantitatively the topology of the flows is given. The technique, which can be applied to any type of turbulence dynamics, is illustrated through the example of resistive ballooning instabilities.

Fractional generalization of Fick's law: a microscopic approach.

In the study of transport in inhomogeneous systems it is common to construct transport equations invoking the inhomogeneous Fick law. The validity of this approach requires that at least two ingredients be present in the system. First, finite characteristic length and time scales associated with the dominant transport process must exist. Second, the transport mechanism must satisfy a microscopic symmetry: global reversibility. Global reversibility is often satisfied in nature. However, many complex systems exhibit a lack of finite characteristic scales. In this Letter we show how to construct a generalization of the inhomogeneous Fick law that does not require the existence of characteristic scales while still satisfying global reversibility.

Renormalization of tracer turbulence leading to fractional differential equations.

For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides a first step toward establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.

Topological instability along invariant surfaces and pseudochaotic transport.

The paper describes the complex topological structure of invariant surfaces that appears in a quasi-stationary regime of the tokamak plasma, and it considers in detail anomalous transport of particles along the invariant surfaces (isosurfaces) that have topological genus greater than 1. Such dynamics is pseudochaotic; i.e. it has a zero Lyapunov exponent. Simulations discover such surfaces in confined plasmas under a fairly low ratio of pressure to the magnetic field energy (beta). The isosurfaces correspond to quasi-coherent structures called "streamers" and the streamers are connected by filaments. We study distribution of time of particle separation, Poincaré; recurrences of trajectories, and first time arrival to the system's edge. A model of a multibar-in-square billiard, introduced by Carreras et al. [Chaos 13, 1175 (2003)] is studied with renormalization group method to obtain a distribution of the first time of particles arrival to the edge as a function of the number of bars, which appears to be power-like. The characteristic exponent of this distribution is discussed with respect to its dependence on the number of filaments that connect adjacent streamers.

Nondiffusive transport in plasma turbulence: a fractional diffusion approach.

Numerical evidence of nondiffusive transport in three-dimensional, resistive pressure-gradient-driven plasma turbulence is presented. It is shown that the probability density function (pdf) of tracer particles' radial displacements is strongly non-Gaussian and exhibits algebraic decaying tails. To model these results we propose a macroscopic transport model for the pdf based on the use of fractional derivatives in space and time that incorporate in a unified way space-time nonlocality (non-Fickian transport), non-Gaussianity, and nondiffusive scaling. The fractional diffusion model reproduces the shape and space-time scaling of the non-Gaussian pdf of turbulent transport calculations. The model also reproduces the observed superdiffusive scaling.

Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations.

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

Complex dynamics of blackouts in power transmission systems.

In order to study the complex global dynamics of a series of blackouts in power transmission systems a dynamical model of such a system has been developed. This model includes a simple representation of the dynamical evolution by incorporating the growth of power demand, the engineering response to system failures, and the upgrade of generator capacity. Two types of blackouts have been identified, each having different dynamical properties. One type of blackout involves the loss of load due to transmission lines reaching their load limits but no line outages. The second type of blackout is associated with multiple line outages. The dominance of one type of blackout over the other depends on operational conditions and the proximity of the system to one of its two critical points. The model displays characteristics such as a probability distribution of blackout sizes with power tails similar to that observed in real blackout data from North America.

Topological instability along filamented invariant surfaces.

In dynamical systems with a zero Lyapunov exponent, weak mixing can be governed by a specific topological structure of some surfaces that are invariant with respect to particle dynamics. In particular, when the genus of the invariant surfaces is more than one, they may have weak mixing and the corresponding fractional kinetics. This possibility is demonstrated by using a typical example from plasma physics, a three-dimensional resistive pressure-gradient-driven turbulence model. In a toroidal geometry and with a low-pressure gradient, this model shows the emergence of quasicoherent structures. In this situation, the isosurfaces of the velocity stream function have a web structure with filamentary surfaces emerging from the outer region of the torus and covering the inner region. The filamentary surfaces can result in stochastic jets of particles that cause a "topological instability." In such a situation, particle transport along the surfaces is of the anomalous superdiffusion type.

Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach.

The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail.

Quiet-time statistics of electrostatic turbulent fluxes from the JET tokamak and the W7-AS and TJ-II stellarators.

The statistics of the quiet times between successive turbulent flux bursts measured at the edge of the JET tokamak and the W7-AS and TJ-II stellarators are analyzed in search for evidence of self-organized critical behavior. The results obtained are consistent with what would be expected in the situation where the underlying plasma is indeed in a near critical state.

Quiet-time statistics: a tool to probe the dynamics of self-organized-criticality systems from within the strong overlapping regime.

A method is presented that allows one to obtain information about the underlying dynamics of a self-organized-criticality system even when the strong-overlapping or hydrodynamic regime (in which individual avalanches are no longer distinguishable) is the only one amenable of probing. The method is based on the analysis of the statistics of the lapses of time between activity bursts or quiet times. The case of a randomly driven running sandpile is used to illustrate the use and capabilities of this technique.