Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence

We examine the probability distribution function (PDF) of the energy injection rate (power) in numerical simulations of stationary two-dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating 2D turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured PDFs with the theoretical calculations and briefly discuss how the power PDF might change with other forcing mechanisms

A Uniform Mathematical Representation of Logic and Computation.

The current models of computation share varying levels of correspondence with actual implementation schemes. They can be arranged in a hierarchical structure depending upon their level of abstraction. In classical computing, the circuit model shares closest correspondence with physical implementation, followed by finite automata techniques. The highest level in the abstraction hierarchy is that of the theory of computation.Likewise, there exist computing paradigms based upon a different set of defining principles. The classical paradigm involves computing as has been applied traditionally, and is characterized by Boolean circuits that are irreversible in nature. The reversible paradigm requires invertible primitives in order to perform computation. The paradigm of quantum computing applies the theory of quantum mechanics to perform computational tasks.Our analysis concludes that descriptions at lowest level in the abstraction hierarchy should be uniform across the three paradigms, but the same is not true in case of current descriptions. We propose a mathematical representation of logic and computation that successfully explains computing models in all three paradigms, while making a seamless transition to higher levels of the abstraction hierarchy. This representation is based upon the theory of linear spaces and, hence, is referred to as the linear representation. The representation is first developed in the classical context, followed by an extension to the reversible paradigm by exploiting the well-developed theory on invertible mappings. The quantum paradigm is reconciled with this representation through correspondence that unitary operators share with the proposed linear representation. In this manner, the representation is shown to account for all three paradigms. The correspondence shared with finite automata models is also shown to hold implicitly during the development of this representation. Most importantly, the linear representation accounts for the Hamiltonians that define the dynamics of a computational process, thereby resolving the correspondence shared with underlying physical principles.The consistency of the linear representation is checked against a current existing application in VLSI CAD that exploits the linearity of logic functions for symbolic representation of circuits. Some possible applications and applicability of the linear representation to some open problems are also discussed

An Experimental Investigation of Nonequilibrium Physics and Dynamical Systems in Turbulent Fluids.

Experiment 1 studies finite system size effects on temporal energy flux fluctuations in three-dimensional (3D) incompressible turbulence. The measured instantaneous energy flux shows that the turbulent energy transfer proceeds towards small spatial scales on average but frequently reverses direction (backscatter) to travel towards larger scales. The frequency of backscatter events is studied experimentally and through simulations. In Experiment 2 the third-order Eulerian structure function is measured for compressible turbulence on a free surface for the first time, and is found to scale linearly in space and agrees well with Kolmogorov's theory of 1941 (K41). K41 predicts the second-order Lagrangian structure function should scale linearly in time. However the experimental measurements show it instead scales as a power-law with exponent 1/2. Experiment 3 concerns measurement of entropy production rate in steady-state compressible turbulence. The analysis relies on the recent theory of Falkovich and Fouxon. The entropy rate is expected to equal the time integral of the lagrangian velocity divergence correlation function with a negative prefactor. The experimental results are found to disagree with this prediction. In addition, if the system is highly chaotic (follows SRB statistics), the system's entropy rate equals the sum of its Lyapunov exponents. The measured entropy rate agrees well with the sum of Lyapunov exponents obtained from simulations by Boffetta et. al. under flow conditions similar to the experiment. Experiment 4 presents a test of the Steady-State Fluctuation Theorem of Gallavotti and Cohen for entropy rate statistics collected from the individual lagrangian trajectories of experiment 3. The entropy rate statistics show excellent agreement with the Fluctuation Theorem within a limited interval of the probability distributions and limited window of averaging times

Universal Scaling Laws for Shear Induced Dilation in Frictional Granular Media

Compressed frictional granular matter cannot flow without dilation. Upon forced shearing to generate flow, the amount of dilation may depend on the initial preparation and a host of material variables. On the basis of both experiments and numerical simulations we show that as a result of training by repeated compression-decompression cycles the amount of dilation induced by shearing the system depends only on the shear rate and on the (pre-shearing) packing fraction. Relating the rheological response to structural properties allows us to derive a scaling law for the amount of dilation after $n$ cycles of compression-decompression. The resulting scaling law has a universal exponent that for trained systems is independent of the inter-granules force laws, friction parameters and strain rate. The amplitude of the scaling law is analytically computable, and it depends only on the shear rate and the asymptotic packing fraction.Comment: 8 pages, 10 figures, Published Versio

Curvature-induced stiffening of a fish fin

How fish modulate their fin stiffness during locomotive manoeuvres remains unknown. We show that changing the fin's curvature modulates its stiffness. Modelling the fin as bendable bony rays held together by a membrane, we deduce that fin curvature is manifested as a misalignment of the principal bending axes between neighbouring rays. An external force causes neighbouring rays to bend and splay apart, and thus stretches the membrane. This coupling between bending the rays and stretching the membrane underlies the increase in stiffness. Using analysis of a 3D reconstruction of a Mackerel (Scomber japonicus) pectoral fin, we calculate the range of stiffnesses this fin is expected to span by changing curvature. The 3D reconstruction shows that, even in its geometrically flat state, a functional curvature is embedded within the fin microstructure owing to the morphology of individual rays. Since the ability of a propulsive surface to transmit force to the surrounding fluid is limited by its stiffness, the fin curvature controls the coupling between the fish and its surrounding fluid. Thereby, our results provide mechanical underpinnings and morphological predictions for the hypothesis that the spanned range of fin stiffnesses correlates with the behaviour and the ecological niche of the fish

Geographic smoothing of solar photovoltaic electric power production in the Western USA

We examined the geographic smoothing of solar photovoltaic generation from 15 utility-scale plants in California, Nevada, and Arizona and from 19 commercial building installations in California. This is the first comparison of geographic smoothing from utility-scale and building-mounted PV and the first examination of solar PV smoothing in this region. Our research questions were (1) how does geographic smoothing of commercial building rooftop PV compare to that of utility scale PV?, (2) is the geographic smoothing found for utility-scale plants the same for the western US as in India?, and (3) how does the geographic smoothing for PV compare to that of wind? By examining the power output of these generators in the frequency domain, we quantified the smoothing obtained by combining the output of geographically separated plants. We found that utility-scale and commercial rooftop plants exhibited similar geographic smoothing, with 10 combined plants reducing the amplitude of fluctuations at 1 h to 18%–28% of those seen for a single plant. We find that combining a few PV sites together reduces fluctuations, but that the point of quickly diminishing returns is reached after ∼5 sites, and that for all the locations and plant sizes considered, PV does not exhibit as much geographic smoothing as is seen for combining wind plants. We present preliminary theoretical arguments for why geographic smoothing of PV plants is less effective than that for wind plants. The slope of the high-frequency part of the PV power spectrum can at best be geographically smoothed (steepen) to an asymptotic spectrum of f−2. This limit for PV has considerably less smoothing than that for wind\u27s geographic smoothing, shown theoretically and from observed data to be f−2.33

Stiffness of the human foot and evolution of the transverse arch

The stiff human foot enables an efficient push-off when walking or running, and was critical for the evolution of bipedalism(1-6). The uniquely arched morphology of the human midfoot is thought to stiffen it(5-9), whereas other primates have flat feet that bend severely in the midfoot(7,10,11). However, the relationship between midfoot geometry and stiffness remains debated in foot biomechanics(12,13), podiatry(14,15) and palaeontology(4-6). These debates centre on the medial longitudinal arch(5,6) and have not considered whether stiffness is affected by the second, transverse tarsal arch of the human foot(16). Here we show that the transverse tarsal arch, acting through the inter-metatarsal tissues, is responsible for more than 40% of the longitudinal stiffness of the foot. The underlying principle resembles a floppy currency note that stiffens considerably when it curls transversally. We derive a dimensionless curvature parameter that governs the stiffness contribution of the transverse tarsal arch, demonstrate its predictive power using mechanical models of the foot and find its skeletal correlate in hominin feet. In the foot, the material properties of the inter-metatarsal tissues and the mobility of the metatarsals may additionally influence the longitudinal stiffness of the foot and thus the curvature-stiffness relationship of the transverse tarsal arch. By analysing fossils, we track the evolution of the curvature parameter among extinct hominins and show that a human-like transverse arch was a key step in the evolution of human bipedalism that predates the genus Homo by at least 1.5 million years. This renewed understanding of the foot may improve the clinical treatment of flatfoot disorders, the design of robotic feet and the study of foot function in locomotion

Variability of the Wind Turbine Power Curve

Wind turbine power curves are calibrated by turbine manufacturers under requirements stipulated by the International Electrotechnical Commission to provide a functional mapping between the mean wind speed v ¯ and the mean turbine power output P ¯ . Wind plant operators employ these power curves to estimate or forecast wind power generation under given wind conditions. However, it is general knowledge that wide variability exists in these mean calibration values. We first analyse how the standard deviation in wind speed σ v affects the mean P ¯ and the standard deviation σ P of wind power. We find that the magnitude of wind power fluctuations scales as the square of the mean wind speed. Using data from three planetary locations, we find that the wind speed standard deviation σ v systematically varies with mean wind speed v ¯ , and in some instances, follows a scaling of the form σ v = C × v ¯ α ; C being a constant and α a fractional power. We show that, when applicable, this scaling form provides a minimal parameter description of the power curve in terms of v ¯ alone. Wind data from different locations establishes that (in instances when this scaling exists) the exponent α varies with location, owing to the influence of local environmental conditions on wind speed variability. Since manufacturer-calibrated power curves cannot account for variability influenced by local conditions, this variability translates to forecast uncertainty in power generation. We close with a proposal for operators to perform post-installation recalibration of their turbine power curves to account for the influence of local environmental factors on wind speed variability in order to reduce the uncertainty of wind power forecasts. Understanding the relationship between wind’s speed and its variability is likely to lead to lower costs for the integration of wind power into the electric grid