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NASA NTRS2010

Variations of Cloud and Radiative Properties of Boundary-layer and Deep Convective Systems with Sea Surface Temperature Anomalies

Gridded monthly-mean satellite data contain compositing information from different cloud system types and clear-sky environments. To isolate the variations of cloud physical properties of an individual cloud system type with its environment, orbital data are needed. In this study, we will analyze the variations of cloud and radiative properties of boundary-layer clouds and deep convective cloud systems with sea surface temperature (SST) anomalies. We use Terra-CERES (Clouds and the Earth s Radiant Energy System) Level 2 data to classify distinct cloud objects defined by cloud-system types (deep convection, boundary-layer cumulus, stratocumulus and overcast clouds), sizes, geographic locations, and matched large-scale environments. This analysis method identifies a cloud object as a contiguous region of the Earth with a single dominant cloud-system type. It determines the shape and size of the cloud object from the satellite data and the cloud-system selection criteria. The statistical properties of the identified cloud objects are analyzed in terms of probability density functions (PDFs) of a single property or joint PDFs between two properties. The SST anomalies are defined as the differences from five-year annual-cycle means. Individual cloud objects are sorted into one of five equal size subsets, with the matched SST anomalies ranging from the most negative to the most positive values, for a given size category of deep convective cloud objects, boundary-layer cumulus, stratocumulus and overcast cloud objects. The PDFs of cloud and radiative properties for deep convective cloud objects (between 30 S and 30 N) are found to largely similar among the five SST anomaly subsets except for the lowest SST anomaly subset. The different characteristics from this SST anomaly subset may be related to some cloud objects resulting from equatorward movement of extratropical cloud systems. This result holds true for all three different size categories (measured by equivalent diameters of 100-150, 150-300 and > 300 km). The PDFs of cloud and radiative properties for boundary-layer cloud types are more variable for macrophysical properties than for microphysical properties, due probably to different large-scale environments where these cloud objects are formed. Implications of these results for the cloud feedback will also be discussed.

Xu, Kuan-Man↗

DOE OSTI2025

Statistical evaluation of microscale stress conditions leading to void nucleation in the weak shock regime

Here, we investigate the heterogeneity of the stress state driven by anisotropic deformation response at the single crystal level through five statistical volume element (SVE) calculations of polycrystalline BCC tantalum. This work focuses on grain boundaries as a prominent material defect type prone to void nucleation based upon experimental observations of predominantly intergranular void nucleation in this material. The SVEs are constructed to be statistically representative of larger volumes of material and are meshed such that mean and standard deviation of grain size and orientation information is reconstructed. The computational meshes feature hexahedral (brick) elements and smooth conformal grain boundaries where significant stress concentration is known to occur, a tail effect of interest in the extreme events process of dynamic ductile damage. An existing micromechanical crystallographic plasticity model shown to capture the single crystal behavior of BCC tantalum well is used to perform the polycrystal calculations. The model includes representation of the non-Schmid effect of non-planar screw dislocation kinetics in tantalum. A three-dimensional stress state time profile predicted by damage modeling of a flyer plate impact experiment is applied as boundary conditions to each SVE. Resulting grain boundary stress state statistics are strongly non-Gaussian. Significant structural evolution is observed within the compressive hold before unloading into tension in the stress profile. Strong angular dependence of grain boundary traction magnitude with shock direction is observed. Non-Schmid effects continue to suggest their influence on propensity of microstructural defect types to nucleate voids. A general void nucleation criterion is proposed using probability theory. The general framework is specified to polycrystalline BCC tantalum in the weak shock regime to include the SVE calculations and literature molecular dynamics calculations of grain boundary void nucleation strength. Probability density functions (PDFs) are used to describe the interaction between the local stress state heterogeneity and the distributed grain boundary void nucleation strength state. A causation entropy maximization procedure removes the requirement for ad hoc selection of a PDF functional form and provides a rigorous procedure for data-based PDF determination. The resulting physically informed PDF describes the spatial appearance frequency of nucleated voids as a function of applied macroscale pressure. Lower length scale physics are thus packaged in a precise and computationally efficient way to provide computational plasticity insight to macroscale dynamic ductile damage models.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Conditional Pseudo-Reversible Normalizing Flow for Surrogate Modeling in Quantifying Uncertainty Propagation

We introduce a conditional pseudo-reversible normalizing flow (PR-NF) that directly learns conditional probability distributions from noisy physical models to efficiently quantify both forward and inverse uncertainty propagation. Traditional surrogate modeling approaches approximate only the deterministic component of physical models, requiring separate noise characterization and computationally expensive sampling methods for inverse problems. Here, in this work, we develop the conditional PR-NF model to directly learn and efficiently generate samples from the conditional probability density functions (PDFs). The training process utilizes dataset consisting of input-output pairs without requiring prior knowledge about the noise and the function. Once trained, our model efficiently generates samples from conditional PDFs for any input within the training domain. Moreover, the pseudo-reversibility feature allows for the use of fully connected neural network architectures, which simplifies the implementation and enables theoretical analysis. We provide a rigorous convergence analysis of the conditional PR-NF model, showing its ability to converge to the target conditional PDF using the Kullback−Leibler divergence. To demonstrate the effectiveness of our method, we apply it to several benchmark tests and a real-world geologic carbon storage problem.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Region and cloud regime dependence of parametric sensitivity in E3SM atmosphere model

Abstract The Department of Energy (DOE)’s Energy Exascale Earth System Model (E3SM), including its atmosphere model (EAM), has many relatively new features. In a previous study we conducted a systematic parametric sensitivity analysis for EAM based on short, perturbed parameter ensemble (PPE) simulations, mainly focusing on global mean climate features and metrics. While parameter values in global climate models are generally invariant in space and time, model response to parameters perturbation may vary by regions and climate regimes, which motivates the need to better understand the EAM model behaviors and physics at regional scale and process level. In this study, using the same set of PPE simulations and a similar sensitivity analysis framework, we identify parameters that cause largest sensitivities over different regions and compare model responses in fast atmospheric processes to the parameters across different cloud regimes for several important cloud-related fidelity metrics. We find that cloud forcing has opposite response to some parameters over mid-latitude vs. tropical land. We also analyze how the parametric sensitivity varies as stratocumulus transitions to shallow convection and to deep convection over ocean. Low cloud forcing and shortwave cloud forcing in the subtropical eastern Pacific are most sensitive to the parameters controlling the width of the probability density function (PDF) of the subgrid vertical velocity ( w’ ) ( gamma ) and the damping of the w’ skewness ( c8 ) near the coast but become more sensitive to the parameter affecting the damping of the w’ variance ( c1 ) further offshore. Detailed interpretation of the spatial dependence of parametric sensitivity is provided. We also investigate how the parametric sensitivity evolves with prediction duration. This study improves our process-level understanding of cloud physics and parameterization and provides insights for developing more advanced regime-aware parameterization schemes in global climate model.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

Abstract In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

Mathematics↗

DOE OSTI2022

Using ultrasonic attenuation in cortical bone to infer distributions on pore size

Here, in this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models’ forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the “true” microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

PYSIMFRAC: A Python library for synthetic fracture generation and analysis

In this paper, we introduce PYSIMFRAC, an open-source python library for generating 3-D synthetic fracture realizations, integrating with fluid simulators, and performing analysis. PYSIMFRAC allows the user to specify one of three fracture generation techniques (Box, Gaussian, or Spectral) and perform statistical analysis including the autocorrelation, moments, and probability density functions of the fracture surfaces and aperture. This analysis and accessibility of a python library allows the user to create realistic fracture realizations and vary properties of interest. In addition, PYSIMFRAC includes integration examples to two different pore-scale simulators and the discrete fracture network simulator, dfnWorks. The capabilities developed in this work provides opportunity for quick and smooth adoption and implementation by the wider scientific community for accurate characterization of fluid transport in geologic media. We present PYSIMFRAC along with integration examples and discuss the ability to extend PYSIMFRAC from a single complex fracture to complex fracture networks.

58 GEOSCIENCES↗

DOE OSTI2021

Machine learning assisted modeling of mixing timescale for LES/PDF of high-Karlovitz turbulent premixed combustion

Accurate modeling of mixing in the transported probability density function (PDF) method remains a great challenge, especially for turbulent premixed combustion under extreme conditions such as high Karlovitz number Ka. Recently, a power-law based mixing timescale model was developed for the large-eddy simulations (LES)/PDF modeling of high-Ka number turbulent premixed flames. It is found in this work that the power-law mixing timescale model is highly sensitive to the model parameters. It is thus critically needed to develop accurate calibration of these model parameters. The empirical specification of the model parameters developed in Zhang et. al. is found to be inadequate for accurate modeling of the mixing timescale. Here, machine learning is introduced as an attractive alternative in this work for the specification of the model parameters. A high-Ka number DNS jet flame is used as the training and validation of the machine learning models. The choices of the input parameters are discussed and compared for the machine learning models. The effect of differential molecular diffusion on mixing is examined by including the effect of the Lewis number in the training of the machine learning models. The performance of different machine learning algorithms is compared for the specification of the mixing model parameters. Overall, excellent performance of the machine learning models is observed for assisting the mixing modeling. The feasibility, interpretability, applicability, generality, and portability of using machine learning are discussed in general to provide a perspective on applying data-driven machine learning for turbulent combustion modeling studies.

42 ENGINEERING↗

DOE OSTI2022

An indirect approach to optimize the reaction rates of thermal NO formation for diesel engines

With stringent emission regulations, it has become more important for modern diesel engine manufacturers to accurately predict engine-out nitrogen oxide (NO x ) emissions across a wide range of operating conditions. Thermal NO is the major source of engine-out NO x in modern diesel engines. For thermal NO formation, several earlier studies have recommended the forward and reverse reaction rate coefficients of the rate-limiting reaction (O + N 2 ⇌ NO + N). However, due to deficiencies in sub-models and inadequacies of reduced chemical mechanisms to represent diesel combustion, these recommended values more often than not need to be adjusted in reduced order combustion models to accurately predict engine-out NO x . Hence, in this work a systematic and computationally efficient approach has been proposed to streamline the process of determining the optimum reaction rate coefficients. Here, to develop the optimization approach, four different production diesel engines with different operating conditions in terms of speed, load, and exhaust-gas recirculation have been considered. Numerical simulations have been performed using a detailed zero-dimensional velocity-composition-frequency transported probability density function (0D-VCF-tPDF) model that uses hundreds of notional particles to capture in-cylinder stratification. Four different combinations of hydrocarbon and NO x chemical mechanisms were used to represent chemistry. It was found that for the rate-limiting reaction, the pre-exponent factors (A f1 , A r1 ) and activation energies (E A,f1 , E A,r1 ) of the forward and reverse reaction rates follow a linear band in A f1 - E A,f1 and A r1 - E A,r1 space where predicted engine-out NO x match the measured values closely. By encompassing such bands from different engines and considering constraints on activation energies, a reduced search domain of pre-exponent factors and activation energies was constructed that is expected to be applicable to any diesel engine. Eventually, computationally efficient three-line and one-line search approaches were proposed to determine the optimum values of the pre-exponent factors and activation energies that led to a minimum error between measured and predicted engine-out NO x . Finally, these three-line and one-line NO x optimization approaches were applied to a fifth production diesel engine for which the 0D-VCF-tPDF model showed a very good predictive performance in terms of predicting peak pressure, 50% burn rate, and engine-out NO x when compared to measured and 3D-CFD values.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2024

A Eulerian three-fluid flow framework for the study of fuel dispersal behavior under loss-of-coolant accident conditions

As the nuclear industry explores new fuel designs to accommodate increased burnup, studying fuel behavior during loss-of-coolant accidents is essential for ensuring the safe operation of light-water reactors. Here, this study presents a computational model to simulate the intricate three-phase flow dynamics of fuel dispersal following cladding breach, where solid fuel particles and carrier fission gases interact with the surrounding liquid or gas medium. The model utilizes a Eulerian three-fluid framework to depict the phase interaction within interpenetrating continua, treating fragmented fuel as granular material. Closure relationships for bulk-flow properties of the solid phase and detailed interfacial exchange terms in high-temperature, high-speed gas-solid-liquid flow scenarios are proposed. The model was validated using adiabatic experimental results in the literature, focusing on characterizing fuel dispersal behavior during loss-of-coolant accidents conditions. Settlement characteristics of the solids, quantified by the probability distribution of equivalent particles, closely matched probability density functions reported in experimental studies. Key highlights of this study include the theoretical description of a coupled three-phase flow with phase change, interpretation of the high-pressure boundary conditions and insights into the transient behavior of gas-solid-liquid phase dynamics during a simulated high-speed dispersal event.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

Adversarial sampling of unknown and high-dimensional conditional distributions

Many engineering problems require the prediction of realization-to-realization variability or a refined description of modeled quantities. In that case, it is necessary to sample elements from unknown high-dimensional spaces with possibly millions of degrees of freedom. While there exist methods able to sample elements from probability density functions (PDF) with known shapes, several approximations need to be made when the distribution is unknown. In this paper the sampling method, as well as the inference of the underlying distribution, are both handled with a data-driven method known as generative adversarial networks (GAN), which trains two competing neural networks to produce a network that can effectively generate samples from the training set distribution. In practice, it is often necessary to draw samples from conditional distributions. When the conditional variables are continuous, only one (if any) data point corresponding to a particular value of a conditioning variable may be available, which is not sufficient to estimate the conditional distribution. This work handles this problem using an a priori estimation of the conditional moments of a PDF. Herein, two approaches, stochastic estimation, and an external neural network are compared for computing these moments; however, any preferred method can be used. The algorithm is demonstrated in the case of the deconvolution of a filtered turbulent flow field. It is shown that all the versions of the proposed algorithm effectively sample the target conditional distribution with minimal impact on the quality of the samples compared to state-of-the-art methods. Additionally, the procedure can be used as a metric for the diversity of samples generated by a conditional GAN (cGAN) conditioned with continuous variables.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Kernel learning backward SDE filter for data assimilation

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. Further, to characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Load-Displacement Relation and Gap Distribution Between Rough Surfaces: Partial Differential Equations Approach

We develop a theoretical model to predict the load-displacement relation and probability density function for gaps between contacting rough surfaces. We derive partial differential equations from the previous model (Joe et al., 2018), and extend them to the non-adhesive contact problem. The predictions of the present theory are compared with numerical results using the Green's function molecular dynamics algorithm, and a good agreement is obtained.

adhesion↗

DOE OSTI2024

Determining reference standard strength for neutron-irradiated reduced activation ferritic/martensitic steel F82H by Bayesian method

The deterministic approach widely adopted in the design of structural components relies on systematically defined design limits using empirically determined safety factors. However, this approach is not always appropriate because structures are subjected to a variety of loads in the practical environment, which may result in excessively conservative design limits. In recent years, a more rigorous probabilistic approach that incorporates material strength distributions has become an important solution. In the probabilistic approach, the probability density functions of material strength properties underpin the design criteria. Here, the objective of this study is to identify the density distribution functions that best describe tensile properties of irradiated F82H to define a reference strength for DEMO design. Due to the limited number of existing data, this study specifically employs a Bayesian prediction method based on Monte Carlo simulations to determine a material reference value with statistical reliability and to investigate its effectiveness. For example, the dependence of tensile properties of 300 °C irradiated materials on irradiation damage and the range predicted by 95% Bayesian estimation was evaluated. As a statistical model for the dose dependence of statistical parameters, the normal distribution exhibited a better fit for 0.2% proof strength and tensile strength, whereas the distribution of total elongation data gave comparable reference values for both the normal and Weibull distribution models. Both models gave comparable criteria for the distribution of total elongation data. The Weibull model also gave better results for uniform elongation. The function best describing the model was a logarithmic law for both 0.2% proof strength and tensile strength, while a power law for both total and uniform elongation, which allowed for more comprehensive data prediction of irradiation data with statistical accuracy for DEMO reactor design.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Pseudorapidity dependence of anisotropic flow and its decorrelations using long-range multiparticle correlations in Pb–Pb and Xe–Xe collisions

The pseudorapidity dependence of elliptic (v 2 ), triangular (v 3 ), and quadrangular (v 4 ) flow coefficients of charged particles measured in Pb–Pb collisions at a centre-of-mass energy per nucleon pair of $\sqrt{S_{NN}}$ = 5.02 TeV and in Xe–Xe collisions at $\sqrt{S_{NN}}$ = 5.44 TeV with ALICE at the LHC are presented. The measurements are performed in the pseudorapidity range − 3.5 < η < 5 for various centrality intervals using two- and multi-particle cumulants with the subevent method. The flow probability density function (p.d.f.) is studied with the ratio of flow coefficient v 2 calculated with four- and two-particle cumulant, and suggests that the variance of flow p.d.f. is independent of pseudorapidity. The decorrelation of the flow vector in the longitudinal direction is probed using two-particle correlations. The results measured with respect to different reference regions in pseudorapidity exhibit differences, argued to be a result of saturating decorrelation effect above a certain pseudorapidity separation, in contrast to previous publications which assign this observation to non-flow effects. The results are compared to 3 + 1 dimensional hydrodynamic and the AMPT transport model calculations. Neither of the models is able to simultaneously describe the pseudorapidity dependence of measurements of anisotropic flow and its fluctuations. The results presented in this work highlight shortcomings in our current understanding of initial conditions and subsequent system expansion in the longitudinal direction. Therefore, they provide input for its improvement.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Collective effect of thermal plumes on temperature fluctuations in a closed Rayleigh–Benard convection cell

Here, we report a systematic study of the collective effect of thermal plumes on the probability density function (PDF) P(δT) of temperature fluctuations δT(t) in turbulent Rayleigh-Benard convection. By decomposing δT(t) into four basic fluctuation modes associated with single and multiple warm and cold plumes and a turbulent background, we derive an analytic form of P(δT) based on the convolutions of the five independent modes. To test the derived form of P(δT) in the multiple-plume regions, where the thermal plumes are heavily populated, we conduct time series measurements of temperature fluctuations in two convection cells; one is a vertical thin disk and the other is an upright cylinder of aspect ratio unity. For a given normalized position in most regions of the convection cell, all of the measured PDFs P(δT) for different Rayleigh numbers fall onto a single master curve, once δT is normalized by its rms value σ T . It is found that the measured P(δT / σT ) at different locations along the symmetric horizontal and vertical axes of the convection cells can all be well described by the derived form of P(δT / σT ). The fitted values of the parameters associated with the number of plumes in multiple plume clusters and their relative strengths and degrees of intermittency are closely linked to the spatial distribution of thermal plumes and local dynamics of the large-scale circulation in a closed convection cell. Our work thus provides a unified theoretical approach for understanding scalar PDFs in a turbulent field, which is very useful not only for the present study but also for the study of many turbulent mixing problems of practical interest.

42 ENGINEERING↗

DOE OSTI2021

Boundary zonal flows in rapidly rotating turbulent thermal convection

Recently, in Zhang et al. ( Phys. Rev. Lett. , vol. 124, 2020, 084505), it was found that, in rapidly rotating turbulent Rayleigh–Bénard convection in slender cylindrical containers (with diameter-to-height aspect ratio $\varGamma =1/2$ ) filled with a small-Prandtl-number fluid ( ${Pr}\approx 0.8$ ), the large-scale circulation is suppressed and a boundary zonal flow (BZF) develops near the sidewall, characterized by a bimodal probability density function of the temperature, cyclonic fluid motion and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount ( ${>}60\,\%$ ) of the total heat transport for ${Pr} < 1$ , but decreases rather abruptly for larger ${Pr}$ to approximately $35\,\%$ . In this work, we show that the BZF is robust and appears in rapidly rotating turbulent Rayleigh–Bénard convection in containers of different $\varGamma$ and over a broad range of ${Pr}$ and ${Ra}$ . Furthermore, tirect numerical simulations for Prandtl number $0.1 \leq {\textit {Pr}} \leq 12.3$ , Rayleigh number $10^7 \leq {Ra} \leq 5\times 10^{9}$ , inverse Ekman number $10^{5} \leq 1/{\textit {Ek}} \leq 10^{7}$ and $\varGamma = 1/3$ , 1/2, 3/4, 1 and 2 show that the BZF width $\delta _0$ scales with the Rayleigh number ${Ra}$ and Ekman number ${\textit {Ek}}$ as $\delta _0/H \sim \varGamma ^{0} Pr^{\{-1/4, 0\}} {Ra}^{1/4} {\textit {Ek}}^{2/3}$ ( $\{{\textit {Pr}}<1, {\textit {Pr}}>1\}$ ) and with the drift frequency scales as $\omega /\varOmega \sim \varGamma ^{0} Pr^{-4/3} {Ra}\,{\textit {Ek}}^{5/3}$ , where $H$ is the cell height and $\varOmega$ the angular rotation rate. The mode number of the BZF is 1 for $\varGamma \lesssim 1$ and $2 \varGamma$ for $\varGamma = \{1,2\}$ independent of ${Ra}$ and ${Pr}$ . The BZF is quite reminiscent of wall mode states in rotating convection.

42 ENGINEERING↗

DOE OSTI2022

Analysis of scale-dependent kinetic and potential energy in sheared, stably stratified turbulence

Budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\ell$ in sheared, stably stratified turbulence are analysed using a filtering approach. Competing effects in the flow are considered, along with the physical mechanisms governing the energy fluxes between scales, and the budgets are used to analyse data from direct numerical simulation at buoyancy Reynolds number $Re_b=O(100)$ . The mean TKE exceeds the TPE by an order of magnitude at the large scales, with the difference reducing as $\ell$ is decreased. At larger scales, buoyancy is never observed to be positive, with buoyancy always converting TKE to TPE. As $\ell$ is decreased, the probability of locally convecting regions increases, though it remains small at scales down to the Ozmidov scale. The TKE and TPE fluxes between scales are both downscale on average, and their instantaneous values are correlated positively, but not strongly so, and this occurs due to the different physical mechanisms that govern these fluxes. Moreover, the contributions to these fluxes arising from the sub-grid fields are shown to be significant, in addition to the filtered scale contributions associated with the processes of strain self-amplification, vortex stretching and density gradient amplification. Probability density functions (PDFs) of the $Q,R$ invariants of the filtered velocity gradient are considered and show that as $\ell$ increases, the sheared-drop shape of the PDF becomes less pronounced and the PDF becomes more symmetric about $R=0$ .

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗