Scaling Of Structure Functions Research Articles

The Cloud Factory II: gravoturbulent kinematics of resolved molecular clouds in a galactic potential

ABSTRACTWe present a statistical analysis of the gravoturbulent velocity fluctuations in molecular cloud complexes extracted from our ‘Cloud Factory’ Galactic-scale interstellar medium (ISM) simulation suite. For this purpose, we produce non-local thermodynamic equilibrium 12CO J = 1 − 0 synthetic observations and apply the principal component analysis (PCA) reduction technique on a representative sample of cloud complexes. The velocity fluctuations are self-consistently generated by different physical mechanisms at play in our simulations, which include Galactic-scale forces, gas self-gravity, and supernova feedback. The statistical analysis suggests that, even though purely gravitational effects are necessary to reproduce standard observational laws, they are not sufficient in most cases. We show that the extra injection of energy from supernova explosions plays a key role in establishing the global turbulent field and the local dynamics and morphology of molecular clouds. Additionally, we characterize structure function scaling parameters as a result of cloud environmental conditions: some of the complexes are immersed in diffuse (interarm) or dense (spiral-arm) environments, and others are influenced by embedded or external supernovae. In quiescent regions, we obtain time-evolving trajectories of scaling parameters driven by gravitational collapse and supersonic turbulent flows. Our findings suggest that a PCA-based statistical study is a robust method to diagnose the physical mechanisms that drive the gravoturbulent properties of molecular clouds. Also, we present a new open source module, the pcafactory, which smartly performs PCA to extract velocity structure functions from simulated or real data of the ISM in a user-friendly way.

Scaling and Anisotropy of Solar Wind Turbulence at Kinetic Scales during the MMS Turbulence Campaign

Abstract We employ data from the 2019 Magnetospheric Multiscale (MMS) Solar Wind Turbulence Campaign to evaluate scaling and anisotropy of second-order magnetic field structure functions and scale-dependent kurtosis from a single data interval. Prior results have been based either on the Taylor approximation, which conflates time and space dependence, or on averaging many data sets, which introduces sample-to-sample variation of plasma parameters. The present results overcome these systematic effects. We find that for length scales between 57 and 201 km the fluctuations are anisotropic with a preferred orientation of 60°–70° to the magnetic field direction. The peak kurtosis also lies in this angular range. This confirms anisotropy of the spectrum and suggests that coherent structures such as current sheets are oriented nearly perpendicular to the magnetic field direction. Both the anisotropy and the kurtosis decrease approaching the ion inertial length, here ≈91 km.

Scalar Power Spectra and Scalar Structure Function Evolution in the Richtmyer–Meshkov Instability Upon Reshock

Abstract The Richtmyer–Meshkov instability of a twice-shocked gas interface is studied using high-speed planar laser-induced fluorescence in the Wisconsin Shock Tube Laboratory's vertical shock tube. The initial condition is a shear layer with broadband diffuse perturbations at the interface between a helium–acetone mixture and argon. This initial condition is accelerated by a shock of nominal strength M = 1.9, and then accelerated again by the transmitted shock that reflects off the end wall of the tube. Three individual experiments are analyzed, the energy spectrum and the structure functions of the light gas mole fraction field are calculated and compared.

Hadron and lepton tensors in semileptonic decays including new physics

We extend our general framework for semileptonic decay, originally introduced in N. Penalva et al. [Phys. Rev. D100, 113007 (2019)], with the addition of new physics (NP) tensor terms. In this way, all the NP effective Hamiltonians that are considered in lepton flavour universality violation (LFUV) studies have now been included. Besides, we now also give general expressions that allow for complex Wilson coefficients. The scheme developed is totally general and it can be applied to any charged current semileptonic decay, involving any quark flavors or initial and final hadron states. We show that all the hadronic input, including NP effects, can be parametrized in terms of 16 Lorentz scalar structure functions. In the second part of this work, we use this formalism to obtain the complete NP effects in the $\Lambda_b\to \Lambda_c \tau\bar\nu_\tau$ semileptonic decay, where LFUV, if finally confirmed, is also expected to be seen. We stress the relevance of the center of mass (CM) $d^2\Gamma/(d\omega d\cos\theta_\ell)$ and laboratory (LAB) $d^2\Gamma/(d\omega dE_\ell)$ differential decay widths. While models with very different strengths in the NP terms give the same differential $d\Gamma/d\omega$ and total decay widths for this decay, they predict very different numerical results for some of the $\cos\theta_\ell$ and $E_\ell$ coefficient-functions that determine the above two distributions. Thus, the combined analysis of the CM $d^2\Gamma/(d\omega d\cos\theta_\ell)$ and LAB $d^2\Gamma/ (d\omega dE_\ell)$ differential decay widths will help clarifying what kind of NP is a better candidate in order to explain LFUV.

Scaling of Second-Order Structure Functions in Turbulent Premixed Flames in the Flamelet Combustion Regime

The second-order velocity structure function statistics have been analysed using a DNS database of statistically planar turbulent premixed flames subjected to unburned gas forcing. The flames considered here represent combustion for moderate values of Karlovitz number from the wrinkled flamelets to the thin reaction zones regimes of turbulent premixed combustion. It has been found that the second-order structure functions exhibit the theoretical asymptotic scalings in the dissipative and (relatively short) inertial ranges. However, the constant of proportionality for the theoretical asymptotic variation for the inertial range changes from one case to another, and this value also changes with structure function orientation. The variation of the structure functions for small length scale separation remains proportional to the square of the separation distance. However, the constant of proportionality for the limiting behaviour according to the separation distance square remains significantly different from the theoretical value obtained in isotropic turbulence. The disagreement increases with increasing turbulence intensity. It has been found that turbulent velocity fluctuations within the flame brush remain anisotropic for all cases considered here and this tendency strengthens towards the trailing edge of the flame brush. It indicates that the turbulence models derived based on the assumptions of homogeneous isotropic turbulence may not be fully valid for turbulent premixed flames.

Isotropic Scaling Features Measured Locally in the Solar Wind Turbulence with Stationary Background Field

Abstract The scaling anisotropy is crucial to interpret the nonlinear interactions in solar wind turbulence. Previous observations provide diverse results and the structure function analyses are also reported to be an approach to investigate the scaling anisotropy based on a local magnetic field. However, the determination of the sampling angle with respect to the local background magnetic field requires that the observed time series for the average are time stationary. Whether or not this required time stationarity is compatible with the measurements has not been investigated. Here we utilize the second-order structure function method to study the scaling anisotropy with a time-stationary background field. We analyze 88 fast solar wind intervals each with time durations ≥ 2 days measured by WIND spacecraft in the period 2005–2018. We calculate the local magnetic field as the average of the time series whose time stationarity is fulfilled by our criterion ϕ < 10° (ϕ is the angle between the two averaged magnetic fields after cutting into two halves). We find for the first time the isotropic scaling feature of the magnetic-trace structure functions with scaling indices and , respectively, with the local magnetic field parallel and perpendicular to the solar wind velocity directions. The scaling for the velocity-trace structure functions is also isotropic and the indices are and . We also find that with increasing ϕ threshold to 90°, the scaling index of the magnetic-trace structure function in the parallel direction decreases to −0.81, while the rms of the instantaneous angle between magnetic field and solar wind velocity increases up to 45° at the timescale 150 s, indicating a mix of perpendicular measurements into parallel ones at large scales.

Similarities between the structure functions of thermal convection and hydrodynamic turbulence

In this paper, we analyze the scaling of velocity structure functions of turbulent thermal convection. Using high-resolution numerical simulations, we show that the structure functions scale similar to those of hydrodynamic turbulence, with the scaling exponents in agreement with the predictions of She and Leveque [“Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. 72, 336–339 (1994)]. The probability distribution functions of velocity increments are non-Gaussian with wide tails in the dissipative scales and become close to Gaussian in the inertial range. The tails of the probability distribution follow a stretched exponential. We also show that in thermal convection, the energy flux in the inertial range is less than the viscous dissipation rate. This is unlike in hydrodynamic turbulence where the energy flux and the dissipation rate are equal.

A note on acoustic turbulence

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$ and the $p$th-order structure function scales as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, $\unicode[STIX]{x1D716}$ being the mean energy dissipation per unit mass, $d$ the mean distance between the shocks and $r$ the separation distance. However, for the acoustic field, $\unicode[STIX]{x1D716}$ should be replaced by $\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as $\langle \unicode[STIX]{x1D6FF}u_{r}^{3}\rangle =-C(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712})r$, where $C$ takes the value $12/5(\unicode[STIX]{x1D6FE}+1)$ in the weak shock limit, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ being the ratio between the specific heats at constant pressure and constant volume.

Turbulent lithosphere deformation in the Tibetan Plateau.

In this work, we show that the Tibetan Plateau deformation demonstrates turbulence-like statistics, e.g., spatial invariance across continuous scales. A dual-power-law behavior is evident to show the existence of two possible conservation laws for the enstrophy-like cascade in the range 500≲r≲2000km and kinetic-energy-like cascade in the range 50≲r≲500km. The measured second-order structure-function scaling exponents ζ(2) are similar to their counterparts in the Fourier scaling exponents observed in the atmosphere, where in the latter case the earth's rotation is relevant. The turbulent statistics observed here for nearly zero-Reynolds-number flow can be interpreted by the geostrophic turbulence theory. Moreover, the intermittency correction is recognized with an intensity close to that of the hydrodynamic turbulence of high-Reynolds-number turbulent flows, implying a universal scaling feature of very different turbulent flows. Our results not only shed new light on the debate regarding the mechanism of the Tibetan Plateau deformation but also lead to new challenges for the geodynamic modeling using Newton or non-Newtonian models because the observed turbulence-like features have to be taken into account.

Cascade of turbulent energy and scalar variance in DC electrokinetic turbulence

Abstract Turbulent flow can be generated by DC electrokinetic (EK) mechanism resulted from the electric conductivity and permittivity variation in fluids, as has been demonstrated by Varshney et al (2016), where a −1.4 slope of velocity power spectrum is observed. Here, we theoretically investigate the scaling exponents of both velocity and scalar structure functions in DC EK turbulence. The scaling exponents of the second order structure functions of velocity and scalar in the so-called electric body force (EBF) dominant subrange are 2/5 (equivalent to the −7/5 slope of velocity power spectrum) and 4/5 respectively. The theory directly explains the experimental results of Varshney et al. (2016). Based on Karman–Howarth equation with forcing terms, the third order structure function of velocity which is the representative of energy transport process is investigated in the DC EK turbulence. Compared with the conventional hydrodynamic turbulence, the energy transport in DC EK turbulence has an additional scaling subrange and is governed by additional small length scales.

Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence

We develop first-principles theory of kinetic plasma turbulence governed by the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates. Following an exact renormalization-group approach pioneered by Onsager, we demonstrate the existence of a "collisionless range" of scales (lengths and velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations are satisfied in a "coarse-grained sense". Entropy conservation may nevertheless be violated in that range by a "dissipative anomaly" due to nonlinear entropy cascade. We derive "4/5th-law" type expressions for the entropy flux, which allow us to characterize the singularities (structure-function scaling exponents) required for its non-vanishing. Conservation laws of mass, momentum and energy are not afflicted with anomalous transfers in the collisionless limit. In a subsequent limit of small gyroradii, however, anomalous contributions to inertial-range energy balance may appear due both to cascade of bulk energy and to turbulent redistribution of internal energy in phase space. In that same limit the "generalized Ohm's law" derived from the particle momentum balances reduces to an "ideal Ohm's law", but only in a coarse-grained sense that does not imply magnetic flux-freezing and that permits magnetic reconnection at all inertial-range scales. We compare our results with prior theory based on the gyrokinetic (high gyro-frequency) limit, with numerical simulations, and with spacecraft measurements of the solar wind and terrestrial magnetosphere.

Scaling and collapse of conditional velocity structure functions in turbulent premixed flames

A new scale-sensitive physical-space conditional analysis is outlined and used to examine the scaling and collapse of velocity structure functions in turbulent premixed flames. The conditioning is based on local instantaneous temperatures in the premixed flame, and structure function scaling and collapse are examined using Kolmogorov-type dimensional arguments and scaling relations. Both longitudinal and lateral structure functions are computed using the local flame normal and tangent as reference directions. The analysis is based on data from direct numerical simulations of unconfined statistically-planar flames at three different intensities of turbulence in the premixed reactants. The analysis shows that as the turbulence intensity increases and for locations near the unburnt reactants, conditional structure functions approach the rN/3 inertial range scaling predicted by Kolmogorov, where N is the structure function order. Using conditionally-calculated scaling variables, it is further shown that structure functions throughout the flame increasingly collapse as the turbulence intensity increases, with a more complete collapse observed for longitudinal structure functions and for small r within the analytic (or dissipative) range. These results suggest that, at sufficiently high intensities, Kolmogorov-type scaling laws and dimensional arguments may retain some validity in premixed flames, provided that scaling variables are computed on a conditional basis for different temperatures.

Transition to ultimate Rayleigh–Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions

In turbulent Rayleigh–Bénard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra\approx 10^{13}{-}10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB convection. In this paper, we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB convection, employing extended self-similarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached-eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug et al., J. Fluid Mech., vol. 830, 2017, pp. 797–819). We find no ESS scaling at $Ra$ below the transition and in the near-wall region. However, beyond the transition and for large enough wall distance $z^{+}>100$, we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra=10^{11}$ and well developed at $Ra=10^{14}$, thus providing strong evidence that the observed transition in the global Nusselt number at $Ra\approx 10^{13}$ indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts, extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in three-dimensional turbulent channel flow.

Disappearance of Anisotropic Intermittency in Large-amplitude MHD Turbulence and Its Comparison with Small-amplitude MHD Turbulence

Abstract Multi-order structure functions in the solar wind are reported to display a monofractal scaling when sampled parallel to the local magnetic field and a multifractal scaling when measured perpendicularly. Whether and to what extent will the scaling anisotropy be weakened by the enhancement of turbulence amplitude relative to the background magnetic strength? In this study, based on two runs of the magnetohydrodynamic (MHD) turbulence simulation with different relative levels of turbulence amplitude, we investigate and compare the scaling of multi-order magnetic structure functions and magnetic probability distribution functions (PDFs) as well as their dependence on the direction of the local field. The numerical results show that for the case of large-amplitude MHD turbulence, the multi-order structure functions display a multifractal scaling at all angles to the local magnetic field, with PDFs deviating significantly from the Gaussian distribution and a flatness larger than 3 at all angles. In contrast, for the case of small-amplitude MHD turbulence, the multi-order structure functions and PDFs have different features in the quasi-parallel and quasi-perpendicular directions: a monofractal scaling and Gaussian-like distribution in the former, and a conversion of a monofractal scaling and Gaussian-like distribution into a multifractal scaling and non-Gaussian tail distribution in the latter. These results hint that when intermittencies are abundant and intense, the multifractal scaling in the structure functions can appear even if it is in the quasi-parallel direction; otherwise, the monofractal scaling in the structure functions remains even if it is in the quasi-perpendicular direction.

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

We develop first-principles theory of relativistic fluid turbulence at high Reynolds and P\'eclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for non-relativistic turbulence, with hydrodynamic fields in the inertial-range described as distributional or "coarse-grained" solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation-laws of smooth Euler solutions but are afflicted with dissipative anomalies in the balance equations of internal energy and entropy. The anomalies are shown to be possible by exactly two mechanisms, local cascade and pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required for their non-vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find is broken by our coarse-graining regularization but which is restored in the limit that the regularization is removed, similar to relativistic lattice quantum field theory. In the formal limit as speed of light goes to infinity, we recover the results of previous non-relativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of non-relativistic kinetic energy.

Cascades and Dissipative Anomalies in Compressible Fluid Turbulence

We investigate dissipative anomalies in a turbulent fluid governed by the compressible Navier-Stokes equation. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. In the limit of high Reynolds and P\'eclet numbers, the flow realizations are found to be described as distributional or "coarse-grained" solutions of the compressible Euler equations, with standard conservation laws broken by turbulent anomalies. The anomalous dissipation of kinetic energy is shown to be due not only to local cascade, but also to a distinct mechanism called pressure-work defect. Irreversible heating in stationary, planar shocks with an ideal-gas equation of state exemplifies the second mechanism. Entropy conservation anomalies are also found to occur by two mechanisms: an anomalous input of negative entropy (negentropy) by pressure-work and a cascade of negentropy to small scales. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required to sustain the cascades. We compare our approach with alternative theories and empirical evidence. It is argued that the "Big Power-Law in the Sky" observed in electron density scintillations in the interstellar medium is a manifestation of a forward negentropy cascade, or an inverse cascade of usual thermodynamic entropy.

Spontaneous symmetry breaking, conformal anomaly and incompressible fluid turbulence

We propose an effective conformal field theory (CFT) description of steady state incompressible fluid turbulence at the inertial range of scales in any number of spatial dimensions. We derive a KPZ-type equation for the anomalous scaling of the longitudinal velocity structure functions and relate the intermittency parameter to the boundary Euler (A-type) conformal anomaly coefficient. The proposed theory consists of a mean field CFT that exhibits Kolmogorov linear scaling (K41 theory) coupled to a dilaton. The dilaton is a Nambu-Goldstone gapless mode that arises from a spontaneous breaking due to the energy flux of the separate scale and time symmetries of the inviscid Navier-Stokes equations to a K41 scaling with a dynamical exponent z=frac{2}{3} . The dilaton acts as a random measure that dresses the K41 theory and introduces intermittency. We discuss the two, three and large number of space dimensions cases and how entanglement entropy can be used to characterize the intermittency strength.

Scaling Features of High‐Latitude Geomagnetic Field Fluctuations at Swarm Altitude: Impact of IMF Orientation

AbstractThis paper attempts to explore the statistical scaling features of high‐latitude geomagnetic field fluctuations at Swarm altitude. Data for this study are low‐resolution (1 Hz) magnetic data recorded by the vector field magnetometer on board Swarm A satellite over 1 year (from 15 April 2014 to 15 April 2015). The first‐ and second‐order structure function scaling exponents and the degree of intermittency of the fluctuations of the intensity of the horizontal component of the magnetic field at high northern latitudes have been evaluated for different interplanetary magnetic field orientations in the GSM Y‐Z plane and seasons. In the case of the first‐order structure function scaling exponent, a comparison between the average spatial distributions of the obtained values and the statistical convection patterns obtained using a Super Dual Auroral Radar Network dynamic model (CS10 model) has been also considered. The obtained results support the idea that the knowledge of the scaling features of the geomagnetic field fluctuations can help in the characterization of the different ionospheric turbulence regimes of the medium crossed by Swarm A satellite. This study shows that different turbulent regimes of the geomagnetic field fluctuations exist in the regions characterized by a double‐cell convection pattern and in those regions near the border of the convective structures.

Structure Function Analysis of Water Vapor Simulated with a Convection-Permitting Model and Comparison to Airborne Lidar Observations

Abstract The spatial scale dependence of midlatitude water vapor variability in the high-resolution limited-area model COSMO is evaluated using diagnostics of scaling behavior. Past analysis of airborne lidar measurements showed that structure function scaling exponents depend on the corresponding airmass characteristics, and that a classification of the troposphere into convective and nonconvective layers led to significantly different power-law behaviors for each of these two regimes. In particular, scaling properties in the convective air mass were characterized by rough and highly intermittent data series, whereas the nonconvective regime was dominated by smoother structures with weaker small-scale variability. This study finds similar results in a model simulation with an even more pronounced distinction between the two air masses. Quantitative scaling diagnostics agree well with measurements in the nonconvective air mass, whereas in the convective air mass the simulation shows a much higher intermittency. Sensitivity analyses were performed using the model data to assess the impact of limitations of the observational dataset, which indicate that analyses of lidar data most likely underestimated the intermittency in convective air masses due to the small samples from single flight tracks, which led to a bias when data with poor fits were rejected. Though the quantitative estimation of intermittency remains uncertain for convective air masses, the ability of the model to capture the dominant weather regime dependence of water vapor scaling properties is encouraging.

Scaling of velocity and scalar structure functions in ac electrokinetic turbulence.

Electrokinetic (EK) turbulence or electrohydrodynamic (EHD) turbulence has been recently achieved in different fluids under both ac [G. Wang et al., Lab Chip 14, 1452 (2014)10.1039/C3LC51403J; Phys. Rev. E 93, 013106 (2016)10.1103/PhysRevE.93.013106] and dc electric fields [A. Varshney et al., Soft Matter 12, 1759 (2016)10.1039/C5SM02316E]. Here, through dimensional analysis, scaling laws of both velocity and electric conductivity structure functions in the forced cascade region of ac EK turbulence can be predicated (similar to Bolgiano-Obukhov scaling law in turbulent Rayleigh-Bénard convection), in either macroscale or microscale flows. In the forced cascade region, EK force, which relies on the direct cascade of conductivity structures, injects energy directly into a wide spectral region to sustain the flow disturbance. The scaling exponents of the second-order velocity and conductivity structures are 2/5 and 4/5, respectively. In addition to the scaling regions, two characteristic small length scales are derived for both weak and strong electric body forces, respectively. This theoretical investigation can significantly enhance our understanding of EK or EHD turbulence while forced by an ac electric field. It can further broaden our understanding of the forced cascade region of forced turbulence and make the manipulation of the turbulent cascade process more flexible and controllable.