Unresolved Scales Research Articles

Reconstructing the Ocean State Using Argo Data and a Data-Driven Method

Abstract Over the past two decades, Argo profiles have provided unprecedented insight into the global patterns of space and time variability of ocean temperature and salinity, significantly reducing associated uncertainties. However, analyzing such assessments during the pre-Argo period remains challenging due to the scarcity of observations in many regions. From the Argo period, a set of dominant three-dimensional patterns can be estimated using Empirical Orthogonal Function (EOF) analysis, which helps to fill in observational gaps. From the associated principal components, temporal fluctuations can be observed, aiming to build a catalog of possible ocean state trajectories. To map pre-Argo observations, EOFs are used in a data assimilation framework that uses this catalog to feed an analog prediction and provide reanalysis. In this study, a new data-driven interpolation method called RedAnDA (Reduced-space Analog Data Assimilation) was tested in the tropical Pacific Ocean. RedAnDA was first validated through an Observing System Simulation Experiment (OSSE) approach, using synthetic observations extracted from a model simulation. It was subsequently applied to a real historical dataset and compared with other available reanalysis products. Overall, the reconstructed temperature field showed variability consistent with the OSSE true field and other reanalysis products in the real data application. Further improvements are needed to optimally estimate uncertainty, but RedAnDA already combines valuable information about state predictability, observational sampling, and unresolved scale issues.

Particle image velocimetry in the impeller of a centrifugal pump: Relationship between turbulent flow and energy loss

Turbulent flows play a dominant role in the operation of centrifugal pumps, which find widespread use in industrial settings and various aspects of human life. The dissipation rate of turbulent kinetic energy emerges as a key parameter within these devices, with its local values exerting a significant influence on centrifugal pump performance. Recent advances in particle image velocimetry (PIV) techniques have expanded the ability to analyze complex turbulent flows across a broad spectrum of scales. In this context, this paper aims to deepen our understanding of the turbulent flow field and its correlation with energy loss in centrifugal pump impellers. To achieve this, experiments were conducted using PIV on a transparent pump operating under different conditions. Statistics of the turbulent flow were then obtained from phase-ensemble averages of velocities, vorticity, turbulence production, and local dissipation of turbulent kinetic energy. To overcome the limited spatial resolution constraint of PIV, the large-eddy PIV (LES-PIV) method was employed to estimate the local dissipation rate. In this method, it is assumed that the motion of larger scales is measured by the PIV technique, while the smaller scales (unresolved scales) are modeled by a sub-grid scale model, calculated from the strain rate tensors obtained from the measured fields. Energy losses in the impeller were studied using two methodologies: (i) a conventional method based on power measurements, and (ii) an alternative approach based on the budget of turbulent kinetic energy. Our results reveal that turbulent loss caused by turbulence production is the main source of energy loss in the pump impeller, and it is particularly pronounced in low-flow operating conditions characterized by large-scale structures. On the other hand, in situations where flow rates exceed the best efficiency point (BEP) condition, the predominant flow structures are marked by small-scale features, mainly attributed to local dissipation of turbulence. Our findings clarify the characteristics of energy losses in centrifugal pump impellers and their relationship with the turbulent flow field, and, in addition, providing a methodology for calculating the local turbulent dissipation rate and its limitations when derived from PIV measurements.

Atlantic Meridional Overturning Circulation Decline: Tipping Small Scales under Global Warming.

The Atlantic circulation is a key component of the global ocean conveyor that transports heat and nutrients worldwide. Its likely weakening due to global warming has implications for climate and ecology. However, the expected changes remain largely uncertain as low-resolution climate models currently in use do not resolve small scales. Although the large-scale circulation tends to weaken uniformly in both the low-resolution and our high-resolution climate model version, we find that the small-scale circulation in the North Atlantic changes abruptly under global warming and exhibits pronounced spatial heterogeneity. Furthermore, the future Atlantic Ocean circulation in the high-resolution model version expands in conjunction with a sea ice retreat and strengthening toward the Arctic. Finally, the cutting-edge climate model indicates sensitive shifts in the eddies and circulation on regional scales for future warming and thus provides a benchmark for next-generation climate models that can get rid of parametrizations of unresolved scales.

Numerical analysis of a parallel triple-jet of liquid-sodium in a turbulent forced convection regime

In the present study, we have applied a combined wall-resolving dynamic Large-Eddy Simulation (LES) (for the velocity field) and Direct Numerical Simulation (DNS) (for the temperature field) approach for mixing of parallel triple-jets with different temperatures of liquid sodium in a turbulent forced convection regime. Because of the high thermal conductivity of sodium (a low-Prandtl fluid), we adopted the dynamic Smagorinsky subgrid closure for the unresolved velocity scales, while the thermal scales are fully resolved. Furthermore, the Time-dependent Reynolds-Averaged Navier-Stokes (T-RANS) approach with the high-Reynolds number variant (i.e. with the wall functions as boundary conditions along solid boundaries) of the four-equation eddy viscosity model (k−ε−kθ−εθ) was applied. The fine-mesh LES/DNS provided a close agreement with the experimental data for both velocity and temperature fields (for both first- and second-moments). In contrast, the coarse-mesh LES/DNS overestimated the turbulent kinetic energy profiles at different distances from the inlet plane. The T-RANS results confirmed a good agreement with the mean streamwise velocity and turbulent kinetic energy, as well as the mean temperature profiles. Finally, the analysis of power spectral density distributions of the temperature signal revealed that all simulation techniques captured a dominant flow frequency originating from the induced Kelvin-Helmholtz instabilities between the side and central jets. The presented combined dynamic LES/DNS approach is recommended for future simulations of the turbulent forced convection flows of low Prandtl fluids, especially if thermal fatigue effects need to be predicted correctly.

Learning about structural errors in models of complex dynamical systems

Complex dynamical systems are notoriously difficult to model because some degrees of freedom (e.g., small scales) may be computationally unresolvable or are incompletely understood, yet they are dynamically important. For example, the small scales of cloud dynamics and droplet formation are crucial for controlling climate, yet are unresolvable in global climate models. Semi-empirical closure models for the effects of unresolved degrees of freedom often exist and encode important domain-specific knowledge. Building on such closure models and correcting them through learning the structural errors can be an effective way of fusing data with domain knowledge. Here we describe a general approach, principles, and algorithms for learning about structural errors. Key to our approach is to include structural error models inside the models of complex systems, for example, in closure models for unresolved scales. The structural errors then map, usually nonlinearly, to observable data. As a result, however, mismatches between model output and data are only indirectly informative about structural errors, due to a lack of labeled pairs of inputs and outputs of structural error models. Additionally, derivatives of the model may not exist or be readily available. We discuss how structural error models can be learned from indirect data with derivative-free Kalman inversion algorithms and variants, how sparsity constraints enforce a “do no harm” principle, and various ways of modeling structural errors. We also discuss the merits of using non-local and/or stochastic error models. In addition, we demonstrate how data assimilation techniques can assist the learning about structural errors in non-ergodic systems. The concepts and algorithms are illustrated in two numerical examples based on the Lorenz-96 system and a human glucose-insulin model.

Modeling the Subgrid-Scale Scalar Variance: A Priori Tests and Application to Supersaturation in Cloud Turbulence

Abstract The subgrid-scale (SGS) scalar variance represents the “unmixedness” of the unresolved small scales in large-eddy simulations (LES) of turbulent flows. Supersaturation variance can play an important role in the activation, growth, and evaporation of cloud droplets in a turbulent environment, and therefore efforts are being made to include SGS supersaturation fluctuations in microphysics models. We present results from a priori tests of SGS scalar variance models using data collected in turbulent Rayleigh–Bénard convection in the Michigan Tech Pi chamber for Rayleigh numbers Ra ∼ 108–109. Data from an array of 10 thermistors were spatially filtered and used to calculate the true SGS scalar variance, a scale-similarity model, and a gradient model for dimensionless filter widths of h/Δ = 25, 14.3, and 10 (where h is the height of the chamber and Δ is the spatial filter width). The gradient model was found to have fairly low correlations (ρ ∼ 0.2), with the most probable values departing significantly from the one-to-one line in joint probability density functions (JPDFs). However, the scale-similarity model was found to have good behavior in JPDFs and was highly correlated (ρ ∼ 0.8) with the true SGS variance. Results of the a priori tests were robust across the parameter space considered, with little dependence on Ra and h/Δ. The similarity model, which only requires an additional test filtering operation, is therefore a promising approach for modeling the SGS scalar variance in LES of cloud turbulence and other related flows.

Modulation of flow, heat, and mass transfer in turbulent double-diffusive convection

The present study addresses numerical simulations of the double-diffusive convection in a turbulent regime. The characteristic concentration and temperature Prandtl numbers (PrC = 700 and PrT = 7) correspond to the typical seawater properties. Here, we applied the Large-Eddy Simulation (LES) approach, with relatively simple subgrid turbulence closures of momentum, concentration, and temperature for the unresolved scales. The wall-resolved LES approach proved to work well on significantly coarser numerical mesh than used in recent Direct Numerical Simulation (DNS) studies of Yang et al. (2016) in the intermediate range of working parameters, 107 ≤ RaC ≤ 109, 0 ≤ RaT ≤ 106, which covers the quasi-Rayleigh-Bénard, fingering, and damping flow regimes. A good agreement between concentration and temperature Nusselt numbers is obtained. The instantaneous Nusselt numbers distribution revealed a significant impact of the imposed strong thermal stratification (RaT = 106) in comparison to the neutral case (RaT = 0).

The impact of operating temperatures on the fluctuating flow field and precessing vortex core in cyclone separator using large-eddy simulations

The present study evaluates numerically the impact of the operating temperature of gas on the cyclone performance viz. the pressure drop, collection efficiency, and flow field details at an inlet velocity, Uin=15 m/s. The gas temperature in a range of 273–1073 K is considered to significantly vary the fluid density and viscosity. For an in-depth analysis, we use advanced closure large-eddy simulation (LES) with the standard Smagorinsky model for treating the unresolved scales. LES can accurately provide additional details on the precessing vortex core phenomena that give rise to enhanced fluctuations in the core region of the cyclone. Apart from the traditional fast Fourier transformation analysis to evaluate the periodicity in the signal, we also perform continuous wavelet transformation and empirical mode decomposition operation on the temporal velocity signals for a better understanding of the flow instabilities—the signals reveal variations of frequency components with time, indicating a non-stationary behavior. It has been observed that an increase in the gas temperature causes lateral contraction of the inner vortex followed by the reduction in its precessional frequency about the cyclone axis with a significantly increased level of noise in the spectra. Furthermore, both pressure losses and collection efficiency largely reduce due to the weakening of swirling strength and enhancement in the fluctuating velocity components with an increase in the gas temperature.

Multiscale model reduction for incompressible flows

We introduce a projection-based model reduction method that systematically accounts for nonlinear interactions between the resolved and unresolved scales of the flow in a low-dimensional dynamical systems model. The proposed method uses a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart–Landau equation. The leading-order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modelling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows.

A deep reinforcement learning framework for dynamic optimization of numerical schemes for compressible flow simulations

Marginal or under-resolved simulations of compressible flow configurations that often occur in practical applications classically are enabled by administering sufficient numerical dissipation to keep the simulation stable. Such measures, however, often are physically inconsistent due to non-selectively altering of dynamics across scales. Sustaining physically consistent large scale dynamics requires the numerical solution to effectively model non-resolved small scale dynamics. In this work, we propose a general deep-reinforcement-learning framework for devising an agent to interact with high-resolution scheme in order to balance dissipation and dispersion such that physically consistent modeling of non-resolved scales is achieved. A densely distributed reward function without involving labeled data is defined. The agent is trained on low-resolution uniform grids that capture the dominant flow structures. We demonstrate that it can be applied directly to high-resolution simulations without the need for retraining or fine-tuning, thereby, demonstrating significantly improved modeling performance compared to empirically designed high-resolution schemes. The proposed methodology opens a new path for self-adaptive numerical solutions whose truncation errors act as physically consistent model for unresolved scales of widely differing flow configurations.

Lagrangian large eddy simulations via physics-informed machine learning

High-Reynolds number homogeneous isotropic turbulence (HIT) is fully described within the Navier-Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as large eddy simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of subgrid scales on the resolved scales. Here, we take an alternative approach and design LES heuristics stated in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus L-LES, is described by equations generalizing the weakly compressible smoothed particle hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from direct numerical simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The subgrid-scale contributions are modeled separately with physical constraints to account for the effects from unresolved scales. We build the resulting model under the differentiable programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our L-LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.

Stabilized bi-cubic Hermite Bézier finite element method with application to gas-plasma interactions occurring during massive material injection in tokamaks

Development of a numerical tool based upon the high-order, high-resolution Galerkin finite element method (FEM) often encounters two challenges: First, the Galerkin FEMs give central approximations to the differential operators and their use in the simulation of the convection-dominated flows may lead to the dispersion errors yielding entirely wrong numerical solutions. Secondly, high-order, high-resolution numerical methods are known to produce high wave-number oscillations in the vicinity of shocks/discontinuities in the numerical solution adversely affecting the stability of the method. We present the stabilized finite element method for plasma fluid models to address the two challenges. The numerical stabilization is based on two strategies: Variational Multiscale (VMS) and the shock-capturing approach. The former strategy takes into account (the approximation of) the effect of the unresolved scales onto resolved scales to introduce upwinding in the Galerkin FEM. The latter adaptively adds the artificial viscosity only in the vicinity of shocks. These numerical stabilization strategies are applied to stabilize the bi-cubic Hermite Bézier FEM in the computational framework of the nonlinear magnetohydrodynamics (MHD) code JOREK. The application of the stabilized FEM to the challenging simulation of Shattered Pellet Injection (SPI) in JET-like plasma is presented. It is shown that the developed numerical stabilization model improves the stability of the underlying numerical algorithm and the computational cost required to reveal the complex physics is reduced. The physical and numerical models presented can be used to perform expensive simulations of the plasma applications in large computational domains such as JET, and ITER.

Predicting unresolved scales interactions with 3D neural networks in homogeneous isotropic turbulence

Interactions between the different scales of motion featured by any turbulent flow are currently mathematically and numerically intractable. Instead, numerical reduced models, such as Large Eddy Simulations (LES), have been proposed: large-scale motions are resolved using the large eddy equations whereas small-scale influence is modeled through the subgrid stress tensor and injected into the large-scale dynamics. In this paper, we develop the learning of this tensor from the raw 3D filtered velocity field diced into sub-cubes whose length is turbulence-induced. We used the U-net convolutional neural network architecture. The performance is assessed using component-wise correlations, PDF and contours comparisons. We extended our a priori analyses to monitor the impacts of such predictions on momentum and kinetic energy evolution. Our model is shown to perform well considering velocity fields extracted from 150% more turbulent simulations.

Variational multiscale super‐resolution: A data‐driven approach for reconstruction and predictive modeling of unresolved physics

AbstractThe variational multiscale (VMS) formulation formally segregates the evolution of the coarse‐scales from the fine‐scales. VMS modeling requires the approximation of the impact of the fine‐scales in terms of the coarse‐scales. In linear problems, our formulation reduces the problem of learning the subscales to learning the projected element Green's function basis coefficients. For this approximation, a special neural‐network structure—the variational super‐resolution N‐N (VSRNN)—is proposed. The VSRNN constructs a super‐resolution model of the unresolved scales as a sum‐of‐the‐products of individual functions of coarse‐scales and physics‐informed parameters. Combined with a set of locally nondimensional features obtained by normalizing the input coarse‐scale and output subscale basis coefficients, the VSRNN provides a general framework for the discovery of closures for different Galerkin discretizations. By training it on a sequence of projected data and using the subscale to compute the continuous Galerkin subgrid terms, and the super‐resolved state to compute the discontinuous Galerkin fluxes, we improve the optimality and the accuracy of these methods for the convection‐diffusion and linear advection problems. Finally, we demonstrate that the VSRNN allows generalization to out‐of‐sample initial conditions and nondimensional numbers.

A multifidelity deep operator network approach to closure for multiscale systems

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed “in-the-loop” training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier–Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

Effect of the N3LO three-nucleon contact interaction on p−d scattering observables

Chiral effective field theory allows one to express nuclear physics observables in terms of low-energy constants (LECs), representing physics at unresolved scales. In this work the authors focus on the two-nucleon contact interaction at fourth order (N3LO) and use a unitary transformation to reduce the number of involved LECs. This procedure induces a three-nucleon ($3N$) interaction depending on five unconstrained LECs. Those can be used, in association with a two-nucleon ($2N$) interaction, to fit very accurate data on polarization observables at low energies, in particular the $p$-$d$ ${A}_{y}$ polarization asymmetry. Thereby, adjusting the induced $3N$ N3LO LECs can solve the long-standing ${A}_{y}$ puzzle.

Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling

We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs while significantly increasing the ROM accuracy.

Exploratory analysis of citizen observations of hourly precipitation over Scandinavia

Abstract. We present a comparison between Netatmo hourly precipitation amounts and observations of the same quantity from weather stations managed by national meteorological services, the latter used as reference values. The empirical distributions of the crowdsourced observations in the surroundings of reference stations are used to assess accuracy and precision of crowdsourced data. We found that reference values are typically within the distribution of the crowdsourced data. However, as the amount of precipitation increases, the spread of the crowdsourced distribution increases and the reference values are more and more frequently found towards the right tail of the distribution. These results indicate that accuracy and precision of crowdsourced data change as precipitation increases. We have studied the sensitivity of our results to the size of the neighbourhood chosen around the reference stations and we show that by aggregating the values over those neighbourhoods, crowdsourced data can be trusted in determining precipitation occurrence. We have assessed the variability of precipitation within small neighbourhoods (of radius 1, 3 and 5 km) and we provide estimates on the basis of the precipitation amounts. Our study quantifies the variability of hourly precipitation over small regions, of the size of the so-called “unresolved spatial scales” in limited area models, based on three years of data collected at several places in Scandinavia.

Predictability and forecast skill of solar irradiance over the contiguous United States

Current solar forecast verification processes place much attention on performance comparison of a group of competing methods. However, forecast verification ought to further answer how the best method within the group performs relative to the best-possible performance which one can attain under that forecasting situation, which makes the quantification of predictability and forecast skill immediately relevant. Unfortunately, the literature on the quantification of relative performance of solar irradiance has hitherto been lacking, and very few studies have focused on the spatial distributions of predictability and forecast skill of solar irradiance. The predictability and forecast skill of an atmospheric process depend on two concepts: (1) the growth of initial error in unresolved scale of motion, and (2) the forecast performance of the standard of reference. Based upon this formalism, predictability and forecast skill of solar irradiance in the United States are quantified and mapped. Through this study, a couple of common misconceptions in regard to irradiance predictability are refuted, and the original formulation of skill score revived.

Ultra-high-resolution observations of persistent null-point reconnection in the solar corona

Magnetic reconnection is a key mechanism involved in solar eruptions and is also a prime possibility to heat the low corona to millions of degrees. Here, we present ultra-high-resolution extreme ultraviolet observations of persistent null-point reconnection in the corona at a scale of about 390 km over one hour observations of the Extreme-Ultraviolet Imager on board Solar Orbiter spacecraft. The observations show formation of a null-point configuration above a minor positive polarity embedded within a region of dominant negative polarity near a sunspot. The gentle phase of the persistent null-point reconnection is evidenced by sustained point-like high-temperature plasma (about 10 MK) near the null-point and constant outflow blobs not only along the outer spine but also along the fan surface. The blobs appear at a higher frequency than previously observed with an average velocity of about 80 km s−1 and life-times of about 40 s. The null-point reconnection also occurs explosively but only for 4 minutes, its coupling with a mini-filament eruption generates a spiral jet. These results suggest that magnetic reconnection, at previously unresolved scales, proceeds continually in a gentle and/or explosive way to persistently transfer mass and energy to the overlying corona.