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N. W. Watkins

Publications and source records attributed to N. W. Watkins.

At least 19 recordsLinked to original sources
arXiv2021

The Sun's magnetic (Hale) cycle and 27 day recurrences in the $aa$ geomagnetic index

We construct a new solar cycle phase clock which maps each of the last 18 solar cycles onto a single normalized epoch for the approximately 22 year Hale (magnetic polarity) cycle, using the Hilbert transform of daily sunspot numbers (SSN) since 1818. The occurrences of solar maxima show almost no Hale cycle dependence, confirming that the clock is synchronized to polarity reversals. The odd cycle minima lead the even cycle minima by ~ 1.1 normalized years, whereas the odd cycle terminators (McIntosh et al. (2019)) lag the even cycle terminators by ~ 2.3 normalized years. The mimimum-terminator interval is thus relatively extended for odd cycles and shortened for even ones. We re-engineer the Sargent(1985,2021) R27 index and combine it with our epoch analysis to obtain a high time resolution parameter for 27 day recurrence in aa, . This reveals that the transition to recurrence, that is, to an ordered solar wind dominated by high speed streams, is fast, occurring within 2-3 solar rotations or less. It resolves an extended late declining phase which is approximately twice as long on even Schwabe cycles as odd. Galactic Cosmic Ray flux rises in step with but then stays high. Our analysis also identifies a slow timescale trend in SSN that simply tracks the Gleissberg cycle. We find that this trend is in phase with the slow timescale trend in the modulus of sunspot latitudes, and in antiphase with that of the R27 index.

astro-ph.SR↗
arXiv2019

First-passage and first-hitting times of Levy flights and Levy walks

For both Levy flight and Levy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Levy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ("leapovers"), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Levy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.

cond-mat.stat-mech↗
arXiv2018

A parallel workload has extreme variability in a production environment

Writing data in parallel is a common operation in some computing environments and a good proxy for a number of other parallel processing patterns. The duration of time taken to write data in large-scale compute environments can vary considerably. This variation comes from a number of sources, both systematic and transient. The result is a highly complex behavior that is difficult to characterize. This paper further develops the model for parallel task variability proposed in the paper "A parallel workload has extreme variability" (Henwood et. al 2016). This model is the Generalized Extreme Value (GEV) distribution. This paper further develops the systematic analysis that leads to the GEV model with the addition of a traffic congestion term. Observations of a parallel workload are presented from a High Performance Computing environment under typical production conditions, which include traffic congestion. An analysis of the workload is performed and shows the variability tends towards GEV as the order of parallelism is increased. The results are presented in the context of Amdahl's law and the predictive properties of a GEV models are discussed. A optimization for certain machine designs is also suggested.

cs.DC↗
arXiv2017

Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection

We investigate the utility of the convex hull of many Lagrangian tracers to analyze transport properties of turbulent flows with different anisotropy. In direct numerical simulations of statistically homogeneous and stationary Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD Boussinesq convection a comparison with Lagrangian pair dispersion shows that convex hull statistics capture the asymptotic dispersive behavior of a large group of passive tracer particles. Moreover, convex hull analysis provides additional information on the sub-ensemble of tracers that on average disperse most efficiently in the form of extreme value statistics and flow anisotropy via the geometric properties of the convex hulls. We use the convex hull surface geometry to examine the anisotropy that occurs in turbulent convection. Applying extreme value theory, we show that the maximal square extensions of convex hull vertices are well described by a classic extreme value distribution, the Gumbel distribution. During turbulent convection, intermittent convective plumes grow and accelerate the dispersion of Lagrangian tracers. Convex hull analysis yields information that supplements standard Lagrangian analysis of coherent turbulent structures and their influence on the global statistics of the flow.

physics.flu-dyn↗
arXiv2016

A parallel workload has extreme variability

In both high-performance computing (HPC) environments and the public cloud, the duration of time to retrieve or save your results is simultaneously unpredictable and important to your over all resource budget. It is generally accepted ("Google: Taming the Long Latency Tail - When More Machines Equals Worse Results", Todd Hoff, highscalability.com 2012), but without a robust explanation, that identical parallel tasks do take different durations to complete -- a phenomena known as variability. This paper advances understanding of this topic. We carefully choose a model from which system-level complexity emerges that can be studied directly. We find that a generalized extreme value (GEV) model for variability naturally emerges. Using the public cloud, we find real-world observations have excellent agreement with our model. Since the GEV distribution is a limit distribution this suggests a universal property of parallel systems gated by the slowest communication element of some sort. Hence, this model is applicable to a variety of processing and IO tasks in parallel environments. These findings have important implications, ranging from characterizing ideal performance for parallel codes to detecting degraded behaviour at extreme scales.

cs.DC↗
arXiv2015

The global build-up to intrinsic ELM bursts seen in divertor full flux loops in Jet

A global signature of the build-up to an intrinsic ELM is found in the phase of signals measured in full flux azimuthal loops in the divertor region of JET. Full flux loop signals provide a global measurement proportional to the voltage induced by changes in poloidal magnetic flux; they are electromagnetically induced by the dynamics of spatially integrated current density. We perform direct time-domain analysis of the high time-resolution full flux loop signals VLD2 and VLD3. We analyze plasmas where a steady H-mode is sustained over several seconds, during which all the observed ELMs are intrinsic; there is no deliberate intent to pace the ELMing process by external means. ELM occurrence times are determined from the Be II emission at the divertor. We previously found that the occurrence times of intrinsic ELMs correlate with specific phases of the VLD2 and VLD3 signals. Here, we investigate how the VLD2 and VLD3 phases vary with time in advance of the ELM occurrence time. We identify a build-up to the ELM in which the VLD2 and VLD3 signals progressively align to the phase at which ELMs preferentially occur, on a ~ 2 -5ms timescale. At the same time, the VLD2 and VLD3 signals become phase synchronized with each other, consistent with the emergence of coherent global dynamics in the integrated current density. In a plasma that remains close to a global magnetic equilibrium, this can reflect bulk displacement or motion of the plasma. This build-up signature to an intrinsic ELM can be extracted from a time interval of data that does not extend beyond the ELM occurrence time, so that these full flux loop signals could assist in ELM prediction or mitigation.

physics.plasm-ph↗
arXiv2014

Anomalous dispersion of Lagrangian particles in local regions of turbulent flows revealed by convex hull analysis

Local regions of anomalous particle dispersion, and intermittent events that occur in turbulent flows can greatly influence the global statistical description of the flow. These local behaviors can be identified and analyzed by comparing the growth of neighboring convex hulls of Lagrangian tracer particles. Although in our simulations of homogeneous turbulence the convex hulls generally grow in size, after the Lagrangian particles that define the convex hulls begin to disperse, our analysis reveals short periods when the convex hulls of the Lagrangian particles shrink, evidence that particles are not dispersing simply. Shrinkage can be associated with anisotropic flows, since it occurs most frequently in the presence of a mean magnetic field or thermal convection. We compare dispersion between a wide range of statistically homogeneous and stationary turbulent flows ranging from homogeneous isotropic Navier-Stokes turbulence over different configurations of magnetohydrodynamic turbulence and Boussinesq convection.

physics.flu-dyn↗
arXiv2013

Similarity Analysis of Macroecology

We perform a full similarity analysis of an idealized ecosystem using Buckingham's $Π$ theorem to obtain dimensionless similarity parameters given that some (non- unique) method exists that can differentiate different functional groups of individuals within an ecosystem. We then obtain the relationship between the similarity parameters under the assumptions of (i) that the ecosystem is in a dynamically balanced steady state and (ii) that these functional groups are connected to each other by the flow of resource. The expression that we obtain relates the level of complexity that the ecosystem can support to intrinsic macroscopic variables such as density, diversity and characteristic length scales for foraging or dispersal, and extrinsic macroscopic variables such as habitat size and the rate of supply of resource. This expression relates these macroscopic variables to each other, generating commonly observed macroecological patterns; these broad trends simply reflect the similarity property of ecosystems. We thus find that details of the ecosystem function are not required to obtain these broad macroecological patterns this may explain why they are ubiquitous. Departures from our relationship may indicate that the ecosystem is in a state of rapid change, i.e., abundance or diversity explosion or collapse. Our result provides normalised variables that can be used to isolate the trend in one ecosystem variable from another, providing a new method for isolating macroecological patterns in data. A dimensionless control parameter for ecosystem complexity emerges from our analysis and this will be a control parameter in dynamical models for ecosystems based on energy flow and conservation and will order the emergent behaviour of these models.

physics.bio-ph↗
arXiv2009

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 non-stationarity. 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 goes to infinity for certain statistical estimators; however, the convergence to this behaviour 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 find that in particular for heavy tailed processes, for realizable N, one is far from this 1/N limiting behaviour. We propose a semi-empirical 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.

physics.data-an↗
arXiv2009

Macroscopic control parameter for avalanche models for bursty transport

Similarity analysis is used to identify the control parameter $R_A$ for the subset of avalanching systems that can exhibit Self- Organized Criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when the number of excited degrees of freedom is maximal, is found to occur when $R_A \to 0$. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R_A$, with the same $R_A \to 0$ exponent, if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical ($R_A$ finite) systems.

math-ph↗
arXiv2009

On the kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts

Levy flights and fractional Brownian motion (fBm) have become exemplars of the heavy tailed jumps and long-ranged memory widely seen in physics. Natural time series frequently combine both effects, and linear fractional stable motion (lfsm) is a model process of this type, combining alpha-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete have for many years been modelled using the fully fractional (FF) kinetic equation for the continuous time random walk (CTRW), with power laws in the pdfs of both jump size and waiting time. We derive the analogous kinetic equation for lfsm and show that it has a diffusion coefficient with a power law in time rather than having a fractional time derivative like the CTRW. We discuss some preliminary results on the scaling of burst "sizes" and "durations" in lfsm time series, with applications to modelling existing observations in space physics and elsewhere.

math-ph↗
arXiv2008

Control parameters in turbulence, Self Organized Criticality and ecosystems

From the starting point of the well known Reynolds number of fluid turbulence we propose a control parameter $R$ for a wider class of systems including avalanche models that show Self Organized Criticality (SOC) and ecosystems. $R$ is related to the driving and dissipation rates and from similarity analysis we obtain a relationship $R\sim N^{β_N}$ where $N$ is the number of degrees of freedom. The value of the exponent $β_N$ is determined by detailed phenomenology but its sign follows from our similarity analysis. For SOC, $R=h/ε$ and we show that $β_N<0$ hence we show independent of the details that the transition to SOC is when $R \to 0$, in contrast to fluid turbulence, formalizing the relationship between turbulence (since $β_N >0$, $R \to \infty$) and SOC ($R=h/ε\to 0$). A corollary is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R$ with unchanged exponent if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical systems. We propose a conceptual model ecosystem where $R$ is an observable parameter which depends on the rate of throughput of biomass or energy; we show this has $β_N>0$, so that increasing $R$ increases the abundance of species, pointing to a critical value for species 'explosion'.

physics.bio-ph↗
arXiv2007

Synthesis of scaling exponents: a fractional Levy model

Mandelbrot introduced the concept of fractals to describe the non-Euclidean shape of many aspects of the natural world. In the time series context he proposed the use of fractional Brownian motion (fBm) to model non-negligible temporal persistence, the "Joseph Effect"; and Levy flights to quantify large discontinuities, the "Noah Effect". In space physics, both effects are manifested in the intermittency and long-range correlation which are by now well-established features of geomagnetic indices and their solar wind drivers. In order to capture and quantify the Noah and Joseph effects in one compact model we propose the application of the "bridging" fractional Levy motion (fLm) to space physics. We perform an initial evaluation of some previous scaling results in this paradigm, and show how fLm can model the previously observed exponents. We suggest some new directions for the future.

physics.space-ph↗
arXiv2005

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. Using the example of a Lévy flight, we demonstrate a method for conditioning the time series that recovers the underlying self affine scaling in a finite length time series.

physics.flu-dyn↗
arXiv2005

Scaling and Commonality in Anomalous Fluctuation Statistics in Models for Turbulence and Ferromagnetism

Recently, Portelli et al (2003) have semi-numerically obtained a functional form of the probability distribution of fluctuations in the total energy flow in a model for fluid turbulence. This follows earlier work suggesting that fluctuations in the total magnetization in the 2D X-Y model for a ferromagnet also follow this distribution. Here, starting from the scaling ansatz that is the basis of the turbulence model we analytically derive the functional form of this distribution and find its single control parameter that depends upon the scaling exponents and system size of the model. Our analysis allows us to identify this explicitly with that of the X-Y model, and suggest a possible generalization.

cond-mat.stat-mech↗
arXiv2003

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

We study scaling in fluctuations of the geomagnetic indices (AE, AU, and AL) that provide a measure of magnetospheric activity and of the epsilon parameter which is a measure of the solar wind driver. Generalized structure function (GSF) analysis shows that fluctuations exhibit self-similar scaling up to about 1 hour for the AU index and about 2 hours for AL, AE and epsilon when the most extreme fluctuations over 10 standard deviations are excluded. The scaling exponents of the GSF are found to be similar for the three AE indices, and to differ significantly from that of epsilon. This is corroborated by direct comparison of their rescaled probability density functions.

physics.space-ph↗
arXiv2002

Scaling of solar wind epsilon and the AU, AL and AE indices

We apply the finite size scaling technique to quantify the statistical properties of fluctuations in AU, AL and AE indices and in the epsilon parameter that represents energy input from the solar wind into the magnetosphere. We find that the exponents needed to rescale the probability density functions (PDF) of the fluctuations are the same to within experimental error for all four quantities. This self-similarity persists for time scales up to ~4 hours for AU, AL and epsilon and up to ~2 hours for AE. Fluctuations on shorter time scales than these are found to have similar long-tailed (leptokurtic) PDF, consistent with an underlying turbulent process. These quantitative and model-independent results place important constraints on models for the coupled solar wind-magnetosphere system.

physics.space-ph↗