Nicholas Wynn Watkins

Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and deer

The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface. This well known finding was followed by similar inferences about the search strategies of deer and bumblebees. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer, microzooplankton, grey seals, spider monkeys and fishing boats. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood and Akaike weights. We apply this to the four original deer and bumblebee data sets, finding that none exhibits evidence of Lévy flights, and that the original graphical approach is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool. Our results question the strength of the empirical evidence for biological Lévy flights.

A simple avalanche model as an analogue for magnetospheric activity

The power law dependence of the power spectrum of auroral indices, and in-situ magnetic field observations in the earths geotail, may be evidence that the coupled solar wind-magnetospheric system exhibits scale free self organised criticality and can to some extent be described by avalanche models. In contrast, the intensity of, and time interval between, substorms both have well defined probability distributions with characteristic scales. We present results from a simple cellular automaton that models avalanches in a one dimensional “sandpile”; here we examine the simplest case of constant inflow. This model generates a probability distribution of energy discharges due to internal reorganization that is a power law implying SOC, whereas systemwide discharges (flow of “sand” out of the system) form a distinct group which do not exhibit SOC. The energy dissipated in a systemwide discharge follows a probability distribution with a well defined mean, as does the time interval between one systemwide discharge and the next. Internal and external avalanches can therefore in principle be identified with distinct processes in the dynamic geotail. If so, the avalanche model places restrictions on the class of physical process that may be invoked to explain the observed geomagnetic dynamics.

Evidence for a solar wind origin of the power law burst lifetime distribution of the AE indices

In this paper we examine the claim that the power law distribution of burst lifetimes in the AE index is evidence that the magnetosphere is a Self-Organized Critical (SOC) system. To do this we compare the burst lifetime distributions of the AU and |AL| indices with those of the υBs and e solar wind input functions. We show for the first time that both the υBs and e burst lifetime distributions are of power law form with an exponential cut-off, consistent with the solar wind being an SOC system. Furthermore, the power law of the e burst lifetime distribution is not significantly different to that of the AU and |AL| indices, indicating that this scale-free property of the AE indices could arise from the solar wind input and may not be an intrinsic property of the magnetospheric system. We discuss the implications of this result for SOC theories of the magnetosphere.

Finite size scaling in the solar wind magnetic field energy density as seen by WIND

Statistical properties of the interplanetary magnetic field fluctuations can provide an important insight into the solar wind turbulent cascade. Recently, analysis of the Probability Density Functions (PDF) of the velocity and magnetic field fluctuations has shown that these exhibit non-Gaussian properties on small time scales while large scale features appear to be uncorrelated. Here we apply the finite size scaling technique to explore the scaling of the magnetic field energy density fluctuations as seen by WIND. We find a single scaling sufficient to collapse the curves over the entire investigated range. The rescaled PDF follow a non Gaussian distribution with asymptotic behavior well described by the Gamma distribution arising from a finite range Levy walk. Such mono scaling suggests that a Fokker-Planck approach can be applied to study the PDF dynamics. These results strongly suggest the existence of a common, nonlinear process on the time scale up to 26 hours.

Power law distributions of burst duration and interburst interval in the solar wind: Turbulence or dissipative self-organized criticality?

We calculate for the first time the probability density functions (PDFs) P of burst energy e, duration T and inter-burst interval τ for a known turbulent system in nature. Bursts in the earth-sun component of the Poynting flux at 1 AU in the solar wind were measured using the MFI and SWE experiments on the NASA WIND spacecraft. We find P (e) and P (T) to be power laws, consistent with self-organised criticality (SOC). We find also a power law form for P (τ) that distinguishes this turbulent cascade from the exponential P (τ) of ideal SOC, but not from some other SOC-like sandpile models. We discuss the implications for the relation between SOC and turbulence.

Scaling in long term data sets of geomagnetic indices and solar wind ϵ as seen by WIND spacecraft

[4] In this paper we use a larger 10-year data set for the AE indices to obtain a more accurate statistical determination of the functional form of the PDF of fluctuations over a more extensive dynamic range, including characterization of extremal events up to 10 standard deviations for the first time. We apply structure functions to characterize and compare both the low and higher order moments for all quantities. A 4-year subset of the index data, corresponding to the same period in the solar cycle as that used to produce � , is used to facilitate this comparison. We then verify these results by direct examination of the fluctuation PDF using the full 10-year AE indices dataset.

Robustness of collective behaviour in strongly driven avalanche models: Magnetospheric implications

The hypothesis of self-organised criticality (SOC) predicts that certain open dissipative systems evolve to a critical state where all energy release statistics display power law distributions for event occurrence, size and duration. This has motivated “sandpile” simulations of magnetospheric energy confinement and release events (“avalanches”), previous examples of which have taken the limit where energy inflow (“fuelling”) is slow relative to dissipation, and either uniform or random. However the magnetospheric system has both slow and fast periods mixed together in observations, and naturally modulated fuelling. We have developed an avalanche model with variable, modulated fuelling rate. The power law form for the distribution of energy release events is the least ambiguous current indicator of SOC; we show that this is preserved for the large avalanches in such a system under both constant and varying loading and so such systems are remarkably efficient at eliminating small scale information about their fuelling.

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 Levy flight.

Robustness of Estimators of Long-Range Dependence and Self-Similarity under non-Gaussianity

Long-range dependence (LRD) and non-Gaussianity are ubiquitous in many natural systems such as ecosystems, biological systems and climate. However, it is not always appreciated that the two phenomena may occur together in natural systems and that self-similarity in a system can be a superposition of both phenomena. These features, which are common in complex systems, impact the attribution of trends and the occurrence and clustering of extremes. The risk assessment of systems with these properties will lead to different outcomes (e.g. return periods) than the more common assumption of independence of extremes. Two paradigmatic models are discussed that can simultaneously account for LRD and non-Gaussianity: autoregressive fractional integrated moving average (ARFIMA) and linear fractional stable motion (LFSM). Statistical properties of estimators for LRD and self-similarity are critically assessed. It is found that the most popular estimators can be biased in the presence of important features of many natural systems like trends and multiplicative noise. Also the LRD and non-Gaussianity of two typical natural time series are discussed.

Testing the SOC hypothesis for the magnetosphere

As noted by Chang, the hypothesis of self-organised criticality provides a theoretical framework in which the low dimensionality seen in magnetospheric indices can be combined with the scaling seen in their power spectra and with observed plasma bursty bulk flows. As such, it has considerable appeal, describing the aspects of the magnetospheric fuelling:storage:release cycle which are generic to slowly-driven, interaction-dominated, thresholded systems rather than unique to the magnetosphere. In consequence, several recent numerical “sandpile” algorithms have been used with a view to comparison with magnetospheric observables. However, demonstration of SOC in the magnetosphere will require further work in the definition of a set of observable properties which are the unique “fingerprint” of SOC. This is because, for example, a scale-free power spectrum admits several possible explanations other than SOC. A more subtle problem is important for both simulations and data analysis when dealing with multiscale and hence broad-band phenomena such as SOC. This is that finite length systems such as the magnetosphere or magnetotail will by definition give information over a small range of orders of magnitude, and so scaling will tend to be narrow band. Here we develop a simple framework in which previous descriptions of magnetospheric dynamics can be described and contrasted. We then review existing observations which are indicative of SOC, and ask if they are sufficient to demonstrate it unambiguously, and if not, what new observations need to be made?