Spatial Scaling Exponent Research Articles

Scaling law of structure function of Richtmyer–Meshkov turbulence

The scaling law of the structure function of Richtmyer–Meshkov (RM) turbulence is investigated both numerically and theoretically. High-fidelity simulations with a minimum-dispersion, adaptive-dissipation scheme are first performed. Results show that the mixing width experiences an exponential growth and the turbulent kinetic energy has a visible $-3/2$ spectrum. The scalar field exhibits a greater degree of intermittency than the velocity field, and also the small-scale statistics suffer a larger influence of large scales. Visible differences in the scaling law of the structure function among the RM turbulence and other types of turbulence are observed, which reveal the unique characteristic of RM turbulence. A phenomenological theory, which gives the spatial and temporal scaling laws of the structure functions of velocity and scalar of RM turbulence, is developed for the first time by introducing an external agent. The spatial scaling exponents of structure functions from simulation deviate from the Kolmogorov exponents, but are quite close to the RM-modified anomalous exponents. This demonstrates the validity of the present phenomenological theory. The temporal scaling exponents of structure functions first meet the RM-modified anomalous exponents, and then approach the Kolmogorov–Obukhov–Corrsin non-intermittent ones.

Annual Cycles of Sea Ice Motion and Deformation Derived From Buoy Measurements in the Western Arctic Ocean Over Two Ice Seasons

AbstractData collected by two buoy arrays that operated during the ice seasons of 2014/2015 and 2016/2017 were used to characterize annual cycles of ice motion and deformation in the western Arctic Ocean. An anomalously strong and weak Beaufort Gyre in 2014/2015 and 2016/2017 induced generally anticyclonic and cyclonic sea ice drift during 2014/2015 and 2016/2017, respectively. Cyclonic ice motion resulted in higher contributions of ice divergence to total ice deformation in 2016/2017 than in 2014/2015. In 2014, the autumn ice concentration and multiyear ice coverage were higher than in 2016; consequently, the response of ice motion to wind forcing was weak, and less ice deformation was observed in autumn 2014. During the autumn‐winter transition, the ice‐wind speed ratio, ice deformation rate and its spatial and temporal scaling exponents, and localization of ice deformation decreased markedly in both 2014/2015 and 2016/2017 as a result of freeze‐up and consolidation of ice floes. Such dynamic behavior was maintained through to spring with the further thickening of ice cover. Ice deformation increased due to weakened ice strength as summer approached. The amplitude of the annual cycle of ice deformation rate in the western Arctic Ocean in 2014/2015 and especially in 2016/2017 was larger than that observed during the Surface Heat Budget of the Arctic Ocean (SHEBA) program in 1997/1998. We attribute this phenomenon to ice loss during the recent summers, especially of thick multiyear ice.

Corrigendum: Spatial and temporal characterisation of sea-ice deformation

Correcting a sign error results in no changes to the key conclusions of Hutchings and others (2011). However, there is an improved agreement with previous work. Mean total sea-ice deformation scales log linearly with distance and the scaling exponent was found to be dependent on time. We find a linear relationship between the temporal scale and spatial scaling exponent, for timescales of an hour to a day. Extrapolating to the timescales of deformation resolved by RADARSAT, we find total deformation and distance scale with an exponent of between −0.16 and −0.19.

Multiple regression and inverse moments improve the characterization of the spatial scaling behavior of daily streamflows in the Southeast United States

AbstractUnderstanding the spatial structure of daily streamflow is essential for managing freshwater resources, especially in poorly gaged regions. Spatial scaling assumptions are common in flood frequency prediction (e.g., index‐flood method) and the prediction of continuous streamflow at ungaged sites (e.g. drainage‐area ratio), with simple scaling by drainage area being the most common assumption. In this study, scaling analyses of daily streamflow from 173 streamgages in the southeastern United States resulted in three important findings. First, the use of only positive integer moment orders, as has been done in most previous studies, captures only the probabilistic and spatial scaling behavior of flows above an exceedance probability near the median; negative moment orders (inverse moments) are needed for lower streamflows. Second, assessing scaling by using drainage area alone is shown to result in a high degree of omitted‐variable bias, masking the true spatial scaling behavior. Multiple regression is shown to mitigate this bias, controlling for regional heterogeneity of basin attributes, especially those correlated with drainage area. Previous univariate scaling analyses have neglected the scaling of low‐flow events and may have produced biased estimates of the spatial scaling exponent. Third, the multiple regression results show that mean flows scale with an exponent of one, low flows scale with spatial scaling exponents greater than one, and high flows scale with exponents less than one. The relationship between scaling exponents and exceedance probabilities may be a fundamental signature of regional streamflow. This signature may improve our understanding of the physical processes generating streamflow at different exceedance probabilities.

Renormalization group approach to 1D cellular automata with large updating neighborhoods

We study self‐similarity in one‐dimensional probabilistic cellular automata (PCA) by applying a real‐space renormalization technique to PCA with increasingly large updating neighborhoods. By studying the flow about the critical point of the renormalization, we may produce estimates of the spatial scaling properties of critical PCA. We find that agreement between our estimates and experimental values are improved by resolving correlations between larger blocks of spins, although this is not sufficient to converge to experimental values. However, applying the technique to PCA with larger neighborhoods, and, therefore, more renormalization parameters, results in further improvement. Our most refined estimate produces a spatial scaling exponent, found at the critical point of the five‐neighbor PCA, of ν = 1.056 which should be compared to the experimental value of ν = 1.097. © 2014 Wiley Periodicals, Inc. Complexity 21: 206–213, 2015

Early androgen treatment influences the pattern and amount of locomotion activity differently and sexually differentially in an animal model of ADHD

Higher testosterone level in males is one of the most obvious possibilities for the development of a clear gender difference in ADHD. The present study focused on the influence of excessive androgen exposure in the developmental stage on the hyperactivity feature of ADHD. The study used the spontaneously hypertensive rat (SHR) as an animal model. The amount of locomotion activity previously used as an over-activity measure in the SHR has resulted in a complicated picture. While the general activity level of SHR was significantly higher than its progenitor—the Wistar Kyoto rat (WKY), comparative differences with the Wistar rat could be observed only under certain experimental conditions. The present study applied the scaling approach to assess open field behaviors from a qualitative aspect. Although SHR and Wistar rats showed similar locomotion amounts, movement patterns differed significantly, as indicated by the spatial scaling exponent. Androgen treatment during the early postnatal developmental stage significantly increased total path lengths only in the male SHR. Effects of the hormone manipulation were not expressed in the scaling measurement. The scaling approach conclusively provides a different aspect of open field behaviors and also reacts differently as the total path length to excessive early testosterone exposure.

Effect of Temporal Sampling on Inferred Rainfall Spatial Statistics

Abstract On the basis of temporally sampled data obtained from satellites, spatial statistics of rainfall can be estimated. In this paper, the authors compare the estimated spatial statistics with their “true” or ensemble values calculated using 5 yr of 15-min radar-based rainfall data at a spatial domain of 512 km × 512 km in the central United States. The authors conducted a Monte Carlo sampling experiment to simulate different sampling scenarios for variable sampling intervals and rainfall averaging periods. The spatial statistics used are the moments of spatial distribution of rainfall, the spatial scaling exponents, and the spatial cross correlations between the sample and ensemble rainfall fields. The results demonstrated that the expected value of the relative error in the mean rain-rate estimate is zero for rainfall averaged over 5 days or longer, better temporal sampling produces average fields that are “less noisy” spatially, an increase in the sampling interval causes the sampled rainfall to be increasingly less correlated with the true rainfall map, and the spatial scaling exponent estimators could give a bias of 40% or less. The results of this study provide a basis for understanding the impact of temporal statistics on inferred spatial statistics.

Earthquakes Descaled: On Waiting Time Distributions and Scaling Laws

Recently, several authors have used waiting time distributions for large earthquake data sets to draw conclusions regarding the physics of earthquake processes. We show, theoretically and by simulation, that a characteristic kink in observed waiting time distributions does not have the physical significance of separating correlated and uncorrelated earthquakes. It also follows from our discussion that the Omori law is not trivially related to a proposed scaling law and that caution must be taken before the spatial scaling exponent of the law is interpreted as a fractal dimension of seismicity.

Behavioral organization is independent of locomotor activity in 129 and C57 mouse strains

Assessing locomotor behavior is a standard methodology to characterize the behavioral phenotype of a genetic manipulation. Typically, levels of locomotor activity are measured using various methods that are based on the frequency of photobeam breaks or distance traveled as assessed by video-tracking systems. Locomotor behavior, however, is multi-dimensional and reflects the combined influences of multiple processes. Here, we examine the number of independent dimensions of locomotor behavior in mice based on measures derived from a video-tracking system. In addition, we test the hypothesis that locomotor behavior varies substantially across mouse strains. 84 mice were tested for 30 min in a 41 × 41 cm enclosure. Based on previous investigations in rats, we also assessed the spatial and dynamical aspects of locomotor behavior using the spatial scaling exponent, d, and the dynamical entropy, h. A principal component analysis and a one-way repeated measure ANOVA were conducted. C57 mouse strains differ substantially from 129 mouse strains on almost all measures of locomotor behavior. The principal component analysis revealed that two independent factors influence this set of measures. The first factor reflects the amount or level of locomotor activity, the second factor quantifies the degree of spatial and dynamical organization of behavior. These strain differences and the existence of at least two independent dimensions when measuring locomotor behavior may help to parse the effects of gene manipulations relative to strain differences in mutant mice.

Differential Movement Patterns But Not Amount of Activity in Unconditioned Motor Behavior of Fischer, Lewis, and Sprague–Dawley Rats

Histocompatible Fischer 344 and Lewis rats have been shown to differ on a wide variety of behavioral, chemical, and molecular measures. This investigation aimed to clarify strain differences in unconditioned motor behavior with respect to the amount and patterns of movements. Twenty female Fischer 344, Lewis, and Sprague–Dawley were tested in the Behavioral Pattern Monitor for 30 min. The locomotor activity and movement patterns, quantified by counts of photobeam breaks and the spatial scaling exponent, d, were assessed. The level of locomotor activity did not differ significantly between Fischer, Lewis, and Sprague–Dawley rats. In contrast, movement patterns differed significantly between the strains. Specifically, Sprague–Dawley rats exhibited significantly more straight movements than both Fischer and Lewis rats. Moreover, Lewis rats showed significantly more straight movements compared to Fischer rats during the first 10 min in the enclosures. Differences in movement patterns across strains may provide an important behavioral variable to further explore the genetic and developmental aspects of behavior.

Isolation rearing affects sequential organization of motor behavior in post-pubertal but not pre-pubertal Lister and Sprague–Dawley rats

Objective: Dopaminergic and serotonergic modulation potently influences the sequential organization of rat movements in a simple unconditioned motor paradigm. Rats reared in social isolation post-weaning differ profoundly from their socially reared litter mates on behavioral, neurochemical, and neuroanatomical measures. This investigation examined (1) whether social isolation significantly affects the sequential organization of rat movements, (2) whether these changes occur at different ages, and (3) whether these changes differ across strains. Method: male Lister and Sprague–Dawley rats reared in isolation post-weaning and socially reared controls were tested at 2 and 8 weeks post-weaning, in the Behavioral Pattern Monitor for 30-min sessions. The amount of activity and the spatial patterns of movements as measured by both the spatial scaling exponent and the fluctuation spectrum of local spatial scaling exponents were assessed in 10-min intervals. Results: Habituation of locomotor activity was significantly attenuated in isolation reared rats during the 30-min sessions irrespective of strain. Spatial patterns of movements were significantly affected by isolation rearing in movements in post-pubertal but not pre-pubertal Lister and Sprague–Dawley rats. The spatial scaling exponent and the fluctuation spectrum analysis revealed a shift towards straight, distance-covering, and repetitive movements rather than a complex re-organization of the behavioral repertoire. Conclusions: Isolation rearing profoundly affects the sequential organization of movements in post-pubertal rats, suggesting that emerging behavioral dysfunctions parallel developmentally those found in patients with schizophrenia.

Long-range interactions in sequences of human behavior

Measures derived from the thermodynamic formalism for dynamical systems are applied to human behavior to determine the degree of interaction within sequences of choices. Sequences of 500 binary choices generated by 22 human subjects are analyzed using autocorrelation and mutual information functions, as well as the fluctuation spectrum of local dynamical entropies or local spatial scaling exponents. The main results are as follows: (1) choices generated by the subjects are interdependent on short, midrange, and large scales relative to the 500 choices generated in the paradigm; (2) human behavior is characterized by large fluctuations of dynamical and geometrical choice characteristics; and (3) these geometric and dynamical characteristics can be related to a few competing principles.

Perturbation theory for the δ-correlated model of passive scalar advection near the Batchelor limit

The third-order correlation function of the scalar field advected by a Gaussian random velocity, with a spatial scaling exponent 2-\ensuremath{\epsilon}, and in the presence of a mean gradient, is calculated perturbatively in \ensuremath{\epsilon}\ensuremath{\ll}1. This expansion corresponds to the regime close to Batchelor's advection by linear diffeomorphisms. The scaling exponent is found to be equal to 1 in dimensions 2 and 3, up to corrections smaller than O(\ensuremath{\epsilon}), implying an anomalous scaling of the third-order correlation function and the persistence of small scale anisotropy.

Development of an affordable hi-resolution activity monitor system for laboratory animals

We describe a hardware and software system for recording and analyzing the spatial and temporal pattern of locomotor activity of laboratory animals. The system offers maximal spatial resolution 500-fold greater than existing light beam monitors. An infrared motion analysis systems (MacReflex, Qualysis) simultaneously tracks the location of up to 20 subjects (identified by reflective markers) to within 0.04 mm at a rate of up to 50 Hz. Macintosh software provides measures of distance traveled, amount of area traversed, number of position changes (microevents), average time between movements, number of left and right turns, number of forward movements and reversals, as well as temporal and spatial scaling exponents. This system was validated by comparing these parameters to direct observer scoring of video tapes and other commercially available activity monitors. Our findings show that applying reflective markers to the subjects does not significantly alter activity levels. The effect of pharmacological manipulation with d-amphetamine is provided to show the value of the different activity parameters. The main advantages of this system are very high spatial resolution, capacity to monitoring up to 20 animals simultaneously at reasonable cost, and lack of sensitivity of the system to ambient lighting. The main limitation is the need to apply reflective markers.

Development of an animal model of fluoxetine akathisia

1. 1. Akathisia describes the pattern of intense inner restlessness often associated with neuroleptic and antidepressant treatment. 2. 2. The authors postulated that drug-induced akathisia would be characterized by more position changes and less time spent immobile, in the absence of significant increase in ambulation. In contrast, a psychomotor stimulant would produce both activation and ambulation. 3. 3. Procedures and instruments were developed to test this hypothesis. Adult rats were habituated for 72 hours to the testing environment, and their precise pattern of movements was tracked and recorded (10 reading per second; resolution 0.04 mm) by an infrared motion analysis system. Activity was recorded for a 90 min period after a single injection of sub-stereotypic doses of d-amphetamine (0, 0.3, 1.0 mg/kg) or racemic fluoxetine (0, 3.0, 10.0, 20.0, or 30.0 mg/kg, s.c.). 4. 4. Amphetamine produced both activation and ambulation. Activation was indicated by a decrease in time spent immobile, and an increase in the temporal scaling exponent, which reflects the degree the animal is “acting” in its environment, and the number of position changes. Enhanced locomotion was inferred from marked increases in both the total distance traversed and the ratio of forward movements-to-reversals and a decrease in the spatial scaling exponent, indicative of a less complex and more linear movement pattern. 5. 5. Fluoxetine caused animals to spend more time active, but exerted little effect on locomotion. Activation was indicated by a decrease in time spent immobile and an increase in the temporal scaling exponent and number of position changes. Fluoxetine failed to significantly effect either the ratio of forward movements-to-reversals or the spatial scaling exponent. 6. 6. These findings provide an operational definition and methodology that can be used to differentiate between psychostimulant effects and akathisic effects. This approach may have utility for screening drugs for akathisic potential, for exploring underlying mechanisms, and for developing novel treatments.

Quantitative assessment of the microstructure of rat behavior: I, f(d), the extension of the scaling hypothesis.

Previous studies demonstrated that drug effects on the movement sequences of rats in unconditioned motor activity paradigms can be quantified by scaling measures that describe the average relationship between a variable of interest and an experimental parameter. However, rats engage in a wide variety of geometrically distinct movements that can be influenced differentially by drugs. In this investigation, the extended scaling approach is presented to capture quantitatively the relative contributions of geometrically distinct movement sequences to the overall path structure. The calculation of the spectrum of local spatial scaling exponents, f(d), is based on ensemble methods used in statistical physics. Results of the f(d) analysis confirm that the amount of motor activity is not correlated with the geometrical structure of movement sequences. Changes in the average spatial scaling exponent, d, correspond to shifting the entire f(d) function, and indicate overall changes in path structure. With the extended scaling approach, straight movement sequences are assessed independently from highly circumscribed movements. Thus, the f(d) function identifies drug effects on particular ranges of movement sequences as defined by the geometrical structure of movements. More generally, the f(d) function quantifies the relationship between microscopically recorded variables, in this paradigm consecutive (x,y) locations, and the macroscopic behavioral patterns that constitute the animal's response topography.

Three independent factors characterize spontaneous rat motor activity

A scaling approach was introduced recently to assess sequential and geometrical aspects of animal behavior. This study describes the relationship between different measures of unconditioned motor activity of rats. Specifically, the amount of motor activity was assessed using both a traditional photobeam break measure, counts, and the temporal scaling exponent, alpha, which describes the ratio of fast to slow behavioral micro-events. The sequential characteristics of the behavior were assessed by the dynamical entropy, h, describing the degree of unpredictability of future movements. The geometrical characteristics of rat motor activity were quantified by the spatial scaling exponent, d. Exploratory activity was measured by counting rearings and holepoking responses. A factor analysis of these measures was conducted based on results from 137 drug-naive animals that were tested for 1 h in the Behavioral Pattern Monitor. Three independent factors account for 77% of the variance. These factors can be described as the 'amount of activity', 'sequential response organization', and 'exploratory activity'. The factor loadings support the initial hypothesis that the geometrical structure of rat motor activity, i.e. the spatial scaling exponent d, varies independently from the amount of activity, i.e. counts. In addition, the distribution of these measures did not deviate significantly from normality suggesting that the z-scores of these variables, which have been used previously in the d-alpha plane description, are able to indicate significant changes of behavior. These results suggest that unconditioned motor activity is influenced by at least three independent factors. The independent assessment of these factors may contribute significantly to the understanding of the neural substrates involved in the organization of unconditioned behavior.

Lagrangian turbulence and the Brownian motion paradox

The unique properties of three-dimensional hydrodynamic turbulence depend on the nature of the long-range time correlations as well as the spatial correlations. Although Kolmogorov’s second similarity hypothesis predicts a power-law spatial scaling exponent for the Eulerian velocity fluctuations in agreement with experiments, it also leads, via the Lagrangian velocity time structure function relationship, to particle dispersion predictions that are inconsistent with enhanced diffusion. Recently, a new computational technique has been developed which can generate random power-law correlated fields in any number of dimensions with unlimited scale range. This new method is used to explore the consequences of a proposed set of assumptions about the nature of the time correlations and their relationship to the spatial correlations. In particular, the Brownian motion paradox is examined and it is shown that it can be resolved if the time domain constraint part of Kolmogorov’s second hypothesis is relaxed and replaced with an assumption of space-time isotropy. The proposed modification preserves the observed one-dimensional k−5/3 spatial energy spectrum, allows for enhanced diffusion consistent with Richardson’s law, is consistent with Taylor’s frozen turbulence assumption under the appropriate conditions, and yields an ω−5/3 frequency spectrum for the velocity fluctuations in a frame at rest with respect to the turbulence.