Pseudo-nonstationarity in the scaling exponents of finite-interval time series

The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series

Signatures of dual scaling regimes in a simple avalanche model for magnetospheric activity

Recently, the paradigm that the dynamic magnetosphere displays sandpile-type phenomenology has been advanced, in which energy dissipation is by means of avalanches which do not have an intrinsic scale. This may in turn imply that the system is in a self-organised critical (SOC) state. Indicators of internal processes are consistent with this, examples are the power-law dependence of the power spectrum of auroral indices, and in situ magnetic field observations in the earth's geotail. However substorm statistics exhibit probability distributions with characteristic scales. In this paper we discuss a simple sandpile model which yields for energy discharges due to internal reorganisation a probability distribution that is a power-law, whereas systemwide discharges (flow of “sand” out of the system) form a distinct group whose probability distribution has a well defined mean. When the model is analysed over its full dynamic range, two regimes having different inverse power-law statistics emerge. These correspond to reconfigurations on two distinct length scales: short length scales sensitive to the discrete nature of the sandpile model, and long length scales up to the system size which correspond to the continuous limit of the model. The latter are anticipated to correspond to large-scale systems such as the magnetosphere. Since the energy inflow may be highly variable, the response of the sandpile model is examined under strong or variable loading and it is established that the power-law signature of the large-scale internal events persists. The interval distribution of these events is also discussed

Rhythm and Randomness in Human Contact

There is substantial interest in the effect of human mobility patterns on opportunistic communications. Inspired by recent work revisiting some of the early evidence for a L\'evy flight foraging strategy in animals, we analyse datasets on human contact from real world traces. By analysing the distribution of inter-contact times on different time scales and using different graphical forms, we find not only the highly skewed distributions of waiting times highlighted in previous studies but also clear circadian rhythm. The relative visibility of these two components depends strongly on which graphical form is adopted and the range of time scales. We use a simple model to reconstruct the observed behaviour and discuss the implications of this for forwarding efficiency

Extremum statistics: a framework for data analysis

Recent work has suggested that in highly correlated systems, such as sandpiles, turbulent fluids, ignited trees in forest fires and magnetization in a ferromagnet close to a critical point, the probability distribution of a global quantity (i.e. total energy dissipation, magnetization and so forth) that has been normalized to the first two moments follows a specific non-Gaussian curve. This curve follows a form suggested by extremum statistics, which is specified by a single parameter a (a = 1 corresponds to the Fisher-Tippett Type I (“Gumbel”) distribution). Here we present a framework for testing for extremal statistics in a global observable. In any given system, we wish to obtain a, in order to distinguish between the different Fisher-Tippett asymptotes, and to compare with the above work. The normalizations of the extremal curves are obtained as a function of a. We find that for realistic ranges of data, the various extremal distributions, when normalized to the first two moments, are difficult to distinguish. In addition, the convergence to the limiting extremal distributions for finite data sets is both slow and varies with the asymptote. However, when the third moment is expressed as a function of a, this is found to be a more sensitive method

Scaling collapse and structure functions: identifying self-affinity in finite length time series

Empirical determination of the scaling properties and exponents of time series presents a formidable challenge in testing, and developing, a theoretical understanding of turbulence and other out-of-equilibrium phenomena. We discuss the special case of self affine time series in the context of a stochastic process. We highlight two complementary approaches to the differenced variable of the data: i) attempting a scaling collapse of the Probability Density Functions which should then be well described by the solution of the corresponding Fokker-Planck equation and ii) using structure functions to determine the scaling properties of the higher order moments. We consider a method of conditioning that recovers the underlying self affine scaling in a finite length time series, and illustrate it using a Lévy flight

Tapinoma nigerrimum as safeguard for Italian myrmecofauna against Argentine ant

The Hurst effect plays an important role in many areas such as physics, climate and finance. It describes the anomalous growth of range and constrains the behavior and predictability of these systems. The Hurst effect is frequently taken to be synonymous with Long-Range Dependence (LRD) and is typically assumed to be produced by a stationary stochastic process which has infinite memory. However, infinite memory appears to be at odds with the Markovian nature of most physical laws while the stationarity assumption lacks robustness. Here we use Lorenz's paradigmatic chaotic model to show that regime behavior can also cause the Hurst effect. By giving an alternative, parsimonious, explanation using nonstationary Markovian dynamics, our results question the common belief that the Hurst effect necessarily implies a stationary infinite memory process. We also demonstrate that our results can explain atmospheric variability without the infinite memory previously thought necessary and are consistent with climate model simulations

Statistical characteristics of total electron content intensifications on global ionospheric maps

Global ionospheric total electron content (TEC) maps exhibit TEC intensifications and depletions of various sizes and shapes. Characterizing key features on TEC maps and understanding their dynamic coupling with external drivers can significantly benefit space weather forecasting. However, comprehensive analysis of ionospheric structuring over decades of TEC maps is currently lacking due to large data volume. We develop feature extraction software based on image processing techniques to extract TEC intensification regions, that is, contiguous regions with sufficiently elevated TEC values than surrounding areas, from global TEC maps. Applying the software to the Jet Propulsion Laboratory Global Ionospheric Map data, we generate a TEC intensification data set for years 2003–2022 and carry out a statistical study on the number and strength of TEC intensifications. We find that the majority of the TEC maps (about 86%) are characterized with one or two intensification(s), while the rest of the TEC maps have three or more intensifications. Both the number and strength of TEC intensifications exhibit semi-annual variation that peaks near equinoxes and dips near solstices, as well as an annual asymmetry with larger values around December solstice compared to June solstice. The number and strength of intensifications increase with enhanced solar extreme-violet irradiance. The strength of intensifications also increases with elevated geomagnetic activity, but the number of intensifications does not. In addition, the number of intensifications is not correlated with the strength of intensifications

Topology of turbulence within collisionless plasma reconnection

In near-collisionless plasmas, which are ubiquitous in astrophysics, entropy production relies on fully-nonlinear processes such as turbulence and reconnection, which lead to particle acceleration. Mechanisms for turbulent reconnection include multiple magnetic flux ropes interacting to generate thin current sheets which undergo reconnection, leading to mixing and magnetic merging and growth of coherent structures, unstable reconnection current layers that fragment and turbulent reconnection outflows. All of these processes act across, and encompass, multiple reconnection sites. We use Magnetospheric Multi Scale four-point satellite observations to characterize the magnetic field line topology within a single reconnection current layer. We examine magnetopause reconnection where the spacecraft encounter the Electron Diffusion Region (EDR). We find fluctuating magnetic field with topology identical to that found for dynamically evolving vortices in hydrodynamic turbulence. The turbulence is supported by an electron-magnetohydrodynamic (EMHD) flow in which the magnetic field is effectively frozen into the electron fluid. Accelerated electrons are found in the EDR edge where we identify a departure from this turbulent topology, towards two-dimensional sheet-like structures. This is consistent with a scenario in which sub-ion scale turbulence can suppress electron acceleration within the EDR which would otherwise be possible in the electric field at the X-line

Intrinsic ELMing in ASDEX Upgrade and global control system-plasma self-entrainment

It is well established that edge localized modes can be entrained to the frequency of applied global magnetic perturbations. These perturbations are delivered to the plasma using the vertical control system field coil currents. These field coils are part of an active control system that is required to maintain the plasma in a steady state. We perform time domain timeseries analysis of natural ELMing when there are no applied perturbations in the ASDEX Upgrade tokamak. We find that the plasma can transition into a state in which the control system field coil currents continually oscillate and are synchronized with oscillations in characteristic plasma parameters such as plasma edge position and total MHD energy. These synchronous oscillations have a one-to-one correlation with the naturally occurring ELMs; the ELMs all occur when the control system coil current is around a specific temporal phase. Large and small ELMs may be distinguished by the amplitude of inward movement of the edge following an ELM. Large ELMs are then found to occur preferentially around a specific temporal phase of the vertical position control coil current. Small ELMs are most likely in antiphase to this. The large and small natural ELMs occur at the opposite extrema of the oscillations in the control system vertical position control coil current. The control system coil current phase may thus provide a useful parameter to order the observed ELM dynamics. We have identified a class of natural ELMing which is a self-entrained state, in which there is a continual non-linear feedback between the global plasma dynamics and the active control system that is intrinsic to the cyclic dynamics of naturally occurring ELMs. Control system-plasma feedback thus becomes an essential component for integration into future models of natural ELM dynamics

Response to “Limitations in the Hilbert Transform Approach to Locating Solar Cycle Terminators” by R. Booth

Booth (Solar Phys.296, 108, 2021; hereafter B21) is essentially a critique of the Hilbert transform techniques used in our paper (Leamon et al., Solar Phys.295, 36, 2020; hereafter L20) to predict the termination of solar cycles. Here we respond to his arguments; our methodology and parameter choices do extract a mathematically robust signature of terminators from the historical sunspot record. We agree that the attempt in L20 to extrapolate beyond the sunspot record gives a failed prediction for the next terminator of May 2020, and we identify both a possible cause and remedy here. However, we disagree with the B21 assessment that the likely termination of Solar Cycle 24 is two years after the date predicted in L20, and we show why