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At least 631 records · Page 35

SAM-ML: Integrating data-driven closure with nuclear system code SAM for improved modeling capability

Advanced reactors often involve complicated thermal-fluid (T-F) phenomena. Modeling such phenomena with the traditional one-dimensional (1-D) system code is a challenging task. The System Analysis Module (SAM), a modern nuclear system code, has developed a coarse mesh multi-dimensional (multi-D) flow model to capture the spatial effect of T-F phenomena in advanced reactors. As a coarse mesh solver, constitutive relations are required for SAM's multi-D model for unresolved fine-scale physics, such as turbulence. Here this work presents a novel approach that integrates neural networks as data-driven closure for SAM's multi-D flow model. The data-driven closure is trained with fine-resolution data to ensure its accuracy while maintaining a coarse mesh setup to ensure its efficiency and consistency with SAM. We demonstrate the applicability of this SAM-ML capability in an open volume thermal stratification problem, where a neural network model serves as the eddy viscosity closure. A customized interface between the neural network and SAM is developed to ensure flexible and efficient data exchange. The SAM-ML results demonstrate superior performance compared to SAM's built-in zero-equation eddy viscosity closure. The case study shows that although the generalization capability of the data-driven closure still needs to be improved for different transient case or different geometric setup, SAM -ML demonstrates good potential for challenging simulation problems with improved accuracy and computational efficiency.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

The growth of short cracks in 4340 steel and aluminum-lithium 2090

The objectives were to investigate crack initiation characteristics and short crack growth behavior for Al-Li 2090 and for 4340 steel and to evaluate the ability of a closure-based crack-growth model to predict fatigue crack growth rates and total fatigue lives for the steel. Single-edge-notched tension specimens of each alloy were used to obtain the short crack growth rate information via an acetate replica technique. In addition to constant amplitude loading, tests on the steel were conducted using the Felix/28 variable amplitude spectrum (a shortened form of a standard loading sequence for fixed or semi-rigid helicopter rotors). The short crack growth rates were compared to those for long cracks grown under similar loading conditions. Metallurgical features associated with crack initiation are discussed. For Al-Li 2090 under R = -1 loading, the short cracks grew well below the long crack threshold and grew at acute angles to the loading axis. For 4340 steel under constant amplitude loading at R = 0.5 and 0 and for the Felix/28 spectrum loading, short-crack growth rates agreed well with long-crack growth rates, even near the long-crack threshold. A slight short-crack effect, growth below the long-crack threshold, was observed at R = -1. Fatigue lives were found to depend on the size and type of initiation site, especially for the Felix/28 loading sequence. A semi-empirical crack-growth model incorporating crack-closure effects was used to predict crack growth rates and total fatigue lives of notched 4340 steel specimens. An initial defect size and shape typical of those identified in this steel was assumed for the life predictions. For all loading conditions, reasonable agreement was found between measured and predicted values for both crack growth rates and fatigue lives.

Swain, M. H.↗

Mesh Convergence Requirements for Composite Damage Models

The ability of the finite element method to accurately represent the response of objects with intricate geometry and loading renders the finite element method as an extremely versatile analysis technique for structural analysis. Finite element analysis is routinely used in industry to calculate deflections, stress concentrations, natural frequencies, buckling loads, and much more. The method works by discretizing complex problems into smaller, simpler approximations that are valid over small uniform domains. For common analyses, the maximum size of the elements that can be used is often be determined by experience. However, to verify the quality of a solution, analyses with several levels of mesh refinement should be performed to ensure that the solution has converged. In recent years, the finite element method has been used to calculate the resistance of structures, and in particular that of composite structures. A number of techniques such as cohesive zone modeling, the virtual crack closure technique, and continuum damage modeling have emerged that can be used to predict cracking, delaminations, fiber failure, and other composite damage modes that lead to structural collapse. However, damage models present mesh refinement requirements that are not well understood. In this presentation, we examine different mesh refinement issues related to the representation of damage in composite materials. Damage process zone sizes and their corresponding mesh requirements will be discussed. The difficulties of modeling discontinuities and the associated need for regularization techniques will be illustrated, and some unexpected element size constraints will be presented. Finally, some of the difficulties in constructing models of composite structures capable of predicting transverse matrix cracking will be discussed. It will be shown that to predict the initiation and propagation of transverse matrix cracks, their density, and their saturation may require models that are significantly more refined than those that have been contemplated in the past.

Davila, Carlos G.↗

Summary of Analytical Results from Samples Supporting Tank Closure Cesium Removal (TCCR) Batch 3 and Modeling Results for Cs Loading on CST

Savannah River Remediation (SRR) is currently operating the Tank Closure Cesium Removal (TCCR) process to remove 137 Cs from tank waste supernate using an ion exchange process. As part of that process, Savannah River National Laboratory (SRNL) receives and analyzes samples in support of the qualification of each batch to be processed. SRNL recently received supernate samples retrieved from Tank 10H as well as in-tank batch contact samples for characterization in support of qualifying Batch 3 for processing through the TCCR unit. Some results from analysis of those samples have been previously reported. This report documents the remaining analyses of the in-tank batch contact samples as well as the results of ZAM (Zheng, Anthony, Miller) isotherm modeling performed for comparison to the measured results. Results of the additional analyses include analysis of the loading of other radionuclides besides 137 Cs on the crystalline silicotitanate (CST) contained within the in-tank batch contact test samples. Results from those analyses revealed the next highest contributor to the activity on the CST was 90 Sr with an average loading of 2.76E+08 dpm/g CST compared to 3.56E+10 dpm/g CST for the 137Cs. Isotopes of plutonium were also detected on the samples. ZAM modeling was performed using the measured composition of the Tank 10H Batch 3 qualification samples. The modeling predicted a maximum Cs loading approximately 2.2x higher than the measured result. This is a slightly lower ratio (expected/measured) compared to what was observed for the prior TCCR in-tank batch contact testing performed for Batches 1A and 2 where the ZAM results were 2.7-2.8x higher than the measured values.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Simulated Precipitation Diurnal Variation With a Deep Convective Closure Subject to Shallow Convection in Community Atmosphere Model Version 5 Coupled With CLUBB

In order to improve the physical consistency between shallow and deep convection, we modify the deep convective closure in the Community Atmosphere Model version 5 (CAM5) coupled with a third-order turbulence closure parameterization (i.e., Cloud Layers Unified by Binormals; CLUBB). The revised closure reserves a portion of the total convective available potential energy for shallow convection via utilizing the heating and moistening profiles from CLUBB to distribute moisture and energy between shallow and deep convection. Simulations at two resolutions (i.e. 2° and 0.5°, respectively) are conducted to investigate the impacts of convective closure on the simulated precipitation diurnal variations. Results from low-resolution simulations show that the revised closure suppresses deep convection until the lower troposphere is sufficiently moistened by shallow convection, which improves the precipitation diurnal variation simulations compared to the default closure, with the precipitation diurnal peak over tropical lands delayed from 12LST to 19LST. The revised closure better simulates the diurnal variations for precipitation over the Asian monsoon region, such as the delayed precipitation onset, but still fails to well capture the nighttime peak for precipitation there. This deficiency is alleviated to some extent when applying the revised closure in highresolution simulations, but nighttime precipitation is still underestimated probably because key processes responsible for nighttime convection are missing. Altogether, our results indicate that establishing the consistency between shallow and deep convection is critical for the precipitation diurnal cycle simulations.

54 ENVIRONMENTAL SCIENCES↗

Mineral Dust Optical Properties for Remote Sensing and Global Modeling: A Review

Dust plays a key role in many Earth system processes and is ubiquitous in the Martian atmosphere. Various intensive field campaigns, laboratory analyses, space-based remote sensing missions, and global modeling efforts aim to characterize dust optical properties. This is a bountiful time for dust scientists, and yet the interpretation of retrievals and comparison to models remains complicated by various conflicting assumptions that are part of each algorithm. For example, the conversion of satellite radiance measurements into products like aerosol optical depth for model evaluation depends upon aerosol properties like particle size and shape that are often prescribed and not part of the retrieval. Conversely, the model calculation of aerosol optical depth often uses different assumptions. The goal of this review is to first document algorithmic assumptions by various satellite retrieval products and models, and identify where there is consistency and where there are differences. In general, the differences documented in this paper reflect uncertainties resulting from incomplete observational characterization of dust aerosols and limitations in our understanding. Second, we note what observations might reduce uncertainties in our knowledge and bring greater consistency to retrievals and models, allowing for a more rigorous and harmonious comparison. The lack of comprehensive and realistic shape models for dust is an outstanding issue, such that closure between forward modeling from particle refractive index, shape, and size and observed optical properties cannot be achieved. Limitations in the computational methods that must be applied to model scattering from complex shapes also makes accurate optical modeling for dust challenging. Field observations indicate the persistence of coarse and giant dust particles at higher altitudes and farther downwind from their source than previously expected. Remote sensing retrieval algorithms based on observations at visible wavelengths have limited sensitivity to these particles and generally do not consider them, although a recent product based on longwave radiances is encouraging. Current measurements of the refractive index of bulk dust and fundamental dust minerology components such as hematite vary widely, inhibiting attempts to represent the variability in dust optical properties and forcing, as expected from different major dust source regions on Earth that have varying mineralogical composition. Some remote sensing retrieval algorithms allow for limited refractive index variability in their inversion solutions through mixing with other fine mode aerosol models, or optimizing the single scattering albedo, but Earth system models surveyed for this paper assume a globally uniform, size-invariant refractive index. Although no Martian dust samples have yet been returned to Earth, remote sensing observations indicate that Martian dust is globally homogenous in composition, and a single spectral refractive index assumption has been widely adopted to represent Martian dust. The lack of comprehensive, statistically representative measurements of dust particle microphysical properties (size distribution, morphology, complex index of refraction spectra, internal structure heterogeneity), and the resulting optical properties, limits our ability to verify the fidelity of these assumptions. A chain of measurements is needed, ranging from characterizing individual dust mineralogy components (e.g., pure hematite and goethite) to in situ sampling of complex atmospheric aerosol mixtures. Such results could be applied to both remote sensing retrievals that characterize the optical properties of the total aerosol burden in the atmosphere from total radiance measurements, and to global models that represent the total aerosol burden in the atmosphere by building it up from the balance of individual aerosol sources and sinks.

Aeolian dust↗

Internal Nozzle Flow Simulations of Gasoline-Like Fuels under Diesel Operating Conditions

Spray formation in internal combustion engines with direct injection is strictly correlated with internal nozzle flow characteristics, which are in turn influenced by fuel physical properties and injector needle motion. This paper pre-sents a series of 3D simulations that model the in-nozzle flow in a 5-hole mini-sac diesel injector. Two gasoline-like naphtha fuels, namely full-range and light naphtha, were tested under operating conditions typical of diesel applica-tions and were compared with n-dodecane, selected from a palette used as diesel surrogates. Validated methodolo-gies from our previous work were employed to account for realistic needle motion. The multi-phase nature of the problem was described by the mixture model assumption with the Volume of Fluid method. Cavitation effects were included by means of the Homogeneous Relaxation Model and turbulence closure was achieved with the Standard k-ε model in an Unsteady Reynolds-Averaged Navier-Stokes formulation. The results revealed that injector perfor-mance and propensity to cavitation are influenced by the fuel properties. Analyses of several physical quantities were carried out to highlight the fuel-to-fuel differences in terms of mass flow rate, discharge coefficients, and fuel vapor volume fraction inside the orifices. A series of parametric investigations was also performed to assess the fuel response to varied fuel injection temperature, injection pressure, and cross-sectional orifice area. For all cases, the strict correlation between cavitation magnitude and saturation pressure was confirmed. Owing to their higher volatil-ity, the two gasoline-like fuels were characterized by higher cavitation across all the simulated conditions. Occur-rence of cavitation was mostly found at the needle seat and at the orifice inlets during the injection event’s transient, when very small gaps exist between the needle and its seat. This behavior tended to disappear at maximum needle lift, where cavitation was absent for all fuels. Differences in mass flow rate between the naphtha fuels and n-dodecane were measured and ascribed to the different densities of the three fuels. Nevertheless, they were found to be smaller than expected, owing to the lower viscosity of the gasoline-like fuels. This beneficial influence of the lower viscosity was shown to be less effective at higher temperature, where the relative viscosity differences de-creased.

Torelli, R.↗

Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow

Turbulent channel flow and homogeneous shear flow have served as basic building block flows for the testing and calibration of Reynolds stress models. A direct theoretical connection is made between homogeneous shear flow in equilibrium and the log-layer of fully-developed turbulent channel flow. It is shown that if a second-order closure model is calibrated to yield good equilibrium values for homogeneous shear flow it will also yield good results for the log-layer of channel flow provided that the Rotta coefficient is not too far removed from one. Most of the commonly used second-order closure models introduce an ad hoc wall reflection term in order to mask deficient predictions for the log-layer of channel flow that arise either from an inaccurate calibration of homogeneous shear flow or from the use of a Rotta coefficient that is too large. Illustrative model calculations are presented to demonstrate this point which has important implications for turbulence modeling.

Abid, R.↗

Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow

Turbulent channel flow and homogeneous shear flow have served as basic building block flows for the testing and calibration of Reynolds stress models. A direct theoretical connection is made between homogeneous shear flow in equilibrium and the log-layer of fully-developed turbulent channel flow. It is shown that if a second-order closure model is calibrated to yield good equilibrium values for homogeneous shear flow it will also yield good results for the log-layer of channel flow provided that the Rotta coefficient is not too far removed from one. Most of the commonly used second-order closure models introduce an ad hoc wall reflection term in order to mask deficient predictions for the log-layer of channel flow that arise either from an inaccurate calibration of homogeneous shear flow or from the use of a Rotta coefficient that is too large. Illustrative model calculations are presented to demonstrate this point which has important implications for turbulence modeling.

Abid, R.↗

Troughs on Martian Ice Sheets: Analysis of Their Closure and Mass Balance

At the Copenhagen workshop on Martian polar processes, Ralf Greve commented that the flow regime surrounding scarps and troughs of the Martian polar ice sheets cannot be modeled using traditional "plan view" ice-sheet models. Such models are inadequate because they typically use reduced equations that embody certain simplifications applicable only to terrestrial ice sheets where the upper ice sheet surface is smooth. In response to this suggestion, we have constructed a 2-dimensional, time dependent "side view" (two spatial dimensions: one horizontal, one vertical) model of scarp closure that is designed to overcome the difficulties described by Greve. The purpose of the model is to evaluate the scales of stress variation and styles of flow closure so as to estimate errors that may be encountered by "plan view" models. We show that there may be avenues whereby the complications associated with scarp closure can be overcome in "plan view" models through appropriate parameterizations of 3-dimensional effects. Following this, we apply the flow model to simulate the evolution of a typical scarp on the North Polar Cap of Mars. Our simulations investigate: (a) the role of "radiation trapping" (see our companion abstract) in creating and maintaining "spiral-like" scarps on the ice sheet, (b) the consequences of different flowlaws and ice compositions on scarp evolution and, in particular, scarp age, and (c) the role of dust and debris in scarp evolution.

Fountain, A.↗

Interface learning in fluid dynamics: Statistical inference of closures within micro–macro-coupling models

Many complex multiphysics systems in fluid dynamics involve using solvers with varied levels of approximations in different regions of the computational domain to resolve multiple spatiotemporal scales present in the flow. The accuracy of the solution is governed by how the information is exchanged between these solvers at the interface and several methods have been devised for such coupling problems. In this article, we construct a data-driven model by spatially coupling a microscale lattice Boltzmann method (LBM) solver and macroscale finite difference method (FDM) solver for reaction-diffusion systems. The coupling between the micro-macro solvers has one to many mapping at the interface leading to the interface closure problem, and we propose a statistical inference method based on neural networks to learn this closure relation. The performance of the proposed framework in a bifidelity setting partitioned between the FDM and LBM domain shows its promise for complex systems where analytical relations between micro-macro solvers are not available.

42 ENGINEERING↗

RELAP-7 Closure Correlations

The RELAP-7 code is the next generation nuclear reactor system safety analysis code being developed at the Idaho National Laboratory (INL). The code is based on the INL’s modern scientific software development framework, MOOSE (Multi-Physics Object Oriented Simulation Environment). The overall design goal of RELAP-7 is to take advantage of the previous thirty years of advancements in computer architecture, software design, numerical integration methods, and physical models. The end result will be a reactor systems analysis capability that retains and improves upon RELAP5’s and TRACE’s capabilities and extends their analysis capabilities for all reactor system simulation scenarios. The RELAP-7 code utilizes the well-posed 7-equation two-phase flow model for compressible two-phase flow. Closure models used in the TRACE code has been reviewed and selected to reflect the progress made during the past decades and provide a basis for the colure correlations implemented in the RELAP-7 code. This document provides a summary on the closure correlations that are currently implemented in the RELAP-7 code. The closure correlations include sub-grid models that describe interactions between the fluids and the flow channel, and interactions between the two phases.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

PORFLOW modeling of Vadose zone flow and transport for the E-area intermediate level vault

In support of the E-Area Performance Assessment, a two-dimensional model of water flow and radionuclide transport through the E-Area Intermediate Level Vault (ILV) and local vadose zone has been developed using the PORFLOWTM software. The purpose of the model is to calculate flux to the water table for radionuclides eluted from the ILV during its operational life, the period of institutional control, and times following site closure. Results of model calculations will be used by a three-dimensional PORFLOW model of transport through the aquifer to determine radionuclide concentrations at a hypothetical 100 meter well and at the site boundary where contaminated groundwater could be assessed by members of the public following site closure. While newly developed, the model structure closely follows that used in the 2008 PA while incorporating a refined computational mesh, updated material properties for the vadose zone soil and vault concrete, revised infiltration rates, and new solid-liquid distribution coefficients. The model also addresses degradation of concrete hydraulic properties by blending soil and concrete water retention curves over a 500 year period. That is, the hydraulic properties of ILV concrete start out as fresh concrete at the time of site closure and degrade to soil properties over a 500 year time period. This approach has not been used previously in E-Area PA’s. The model has been used to calculate water flow through the ILV vadose zone and, on a trial basis, transport of a limited number of radionuclides from the waste region inside the ILV to the water table. Figure ES-1 provides a view of the model and an example of water flow after the concrete has substantially degraded. Figure dimensions are in centimeters. The red area in the left hand side figure is the waste disposal region within the ILV. Colored bands in the figures indicate different soil and concrete regions. Figure ES-2 shows a calculation of flux to the water table (mol/yr) for U-234 and its one-year half-life decay chain (U-234 → Th-230 → Ra-226 → Pb-210). Pb-210 flux is too small to show on the same scale used for Th-230 and Ra-226 in Figure ES-2. The model has also been used to make a preliminary evaluation of the release of tritium from TPBAR disposal containers that will be place in the ILV. A separately calculated tritium release rate as a function of time was used as a source in the PORFLOW model. The TPBAR disposal schedule and tritium release rate differ from the values used in the 2008 PA. Results from these initial trial runs are compared to calculations made for the 2008 PA ILV analysis. In brief, a new PORFLOW model of flow and transport through the ILV vadose zone has been developed and tested. The model is ready to be used for PA calculations.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Mechanisms of fatigue crack retardation following single tensile overloads in powder metallurgy aluminum alloys

In ingot metallurgy (IM) alloys, the number of delay cycles following a single tensile overload typically increases from a minimum at an intermediate baseline stress intensity range, Delta-K(B), with decreasing Delta-K(B) approaching threshold and increasing Delta-K(B) approaching unstable fracture to produce a characteristic 'U' shaped curve. Two models have been proposed to explain this behavior. One model is based on the interaction between roughness and plasticity-induced closure, while the other model only utilizes plasticity-induced closure. This article examines these models, using experimental results from constant amplitude and single overload fatigue tests performed on two powder metallurgy (PM) aluminum alloys, AL-905XL and AA 8009. The results indicate that the 'U'-shaped curve is primarily due to plasticity-induced closure, and that the plasticity-induced retardation effect is through-thickness in nature, occurring in both the surface and interior regions. However, the retardation effect is greater at the surface, because the increase in plastic strain at the crack tip and overload plastic zone size are larger in the plane-stress surface regions than in the plane-strain interior regions. These results are not entirely consistent with either of the proposed models.

Bray, G. H.↗

High Fidelity BWR Fuel Simulations

This report describes the Consortium for Advanced Simulation of Light Water Reactors (CASL) work conducted for completion of the Thermal Hydraulics Methods (THM) Level 3 milestone THM.CFD.P13.03: High Fidelity BWR Fuel Simulation. High fidelity computational fluid dynamics (CFD) simulation for Boiling Water Reactor (BWR) was conducted to investigate the applicability and robustness performance of BWR closures. As a preliminary study, a CFD model with simplified Ferrule spacer grid geometry of NUPEC BWR Full-size Fine-mesh Bundle Test (BFBT) benchmark has been implemented. Performance of multiphase segregated solver with baseline boiling closures has been evaluated. Although the mean values of void fraction and exit quality of CFD result for BFBT case 4101-61 agreed with experimental data, the local void distribution was not predicted accurately. The mesh quality was one of the critical factors to obtain converged result. The stability and robustness of the simulation was mainly affected by the mesh quality, combination of BWR closure models. In addition, the CFD modeling of fully-detailed spacer grid geometry with mixing vane is necessary for improving the accuracy of CFD simulation.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

ZAM Modeling Study to Support the Tank Closure Cesium Removal (TCCR) 1A Unit

Currently at the Savannah River Site (SRS), the Tank Closure Cesium Removal (TCCR) is an “at-tank” process designed to remove cesium from aqueous tank waste. Cesium will be removed by ion exchange using the engineered IONSIV® R9120 form of Crystalline Silicotitanate (CST) media. The current TCCR design has two columns online in a lead-lag configuration to optimize media usage and achieve the target decontamination. Once the lead column is saturated with cesium, it will be removed from service, the lag column will rotate into the lead position, and a new column with fresh ion-exchange media will be placed into the lag position. The TCCR process for cesium removal from Tank 10H is detailed in X-SOW-H- 00002. Demonstration of the system began in early calendar year 2019 with two batches of salt solution generated by dissolving saltcake in Tank 10H, followed by processing of these batches through the TCCR system. A third TCCR Tank 10H dissolved saltcake batch is scheduled for processing soon. Upon completion of the demonstration with Tank 10H dissolved saltcake, Tank 9H salt solution will be transferred to Tank 10H and subsequently processed through the TCCR unit with new CST media (referred to as R9120-B 30x60) added to new IX columns. The TCCR processing campaign of Tank 9H salt solution is referred to as TCCR-1A .

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Implications of a weakening N = 126 shell closure away from stability for r -process astrophysical conditions

The formation of the third r-process abundance peak near A ∼ 195 is highly sensitive to both nuclear structure far from stability and the astrophysical conditions that produce the heaviest elements. In particular, the N = 126 shell closure plays a crucial role in shaping this peak. Experimental data hints that the shell weakens as proton number departs from Z = 82, a trend largely missed by global mass models. To investigate its impact on r-process nucleosynthesis, we employ both standard global models with strong closures and modified Duflo-Zuker (DZ) models that reproduce the weakening, combined with three sets of β − -decay rates. Strong shell closures generate sharply peaked abundances, whereas weakened closures consistent with the experimental trend produce broader, flatter patterns. Accurately reproducing the solar third peak under weakened shell strength requires sufficiently neutron-rich conditions that significant fission occurs, and slower decay rates. These results demonstrate that a weakening N = 126 shell closure away from stability imposes significant constraints on the astrophysical environments of the r-process and underscores the need for precise mass measurements and improved characterization of β − -decay properties in this region.

N = 126 closed shells↗

Second Moment Closure Near the Two-component Limit

The purpose of this paper is to explore some wider implications of the two-component limit for both single point turbulence models and spectral closure theories. Although the two-component limit arises most naturally in inhomogeneous problems like wall-bounded turbulence, the analysis will be restricted to homogeneous turbulence. But since homogeneous turbulence is the crucial case for realizability, the conclusions will nevertheless be applicable to modeling. Th essential point of our argument is that whereas the evolution of the stochastic velocity field is Markovian because it is governed by the Navier-Stokes equations, the exact stress evolution equation is not Markovian because it is unclosed. This property of moment evolution has been stressed by Kraichnan (1959). We will show that modeling stress evolution at the two-component limit with a closure that is Markovian in the stresses alone leads to basic inconsistencies in single-point modeling and, perhaps surprisingly, in spectral modes as well.

Rubinstein, Robert↗