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The large-scale structure of many turbulent flows encountered in practical situations such as aeronautics, industry, meteorology is nowadays successfully computed using the Kolmogorov–Kármán–Howarth energy cascade picture. This theory appears increasingly inaccurate when going down the energy cascade that terminates through intermittent spots of energy dissipation, at variance with the assumed homogeneity. This is problematic for the modelling of all processes that depend on small scales of turbulence, such as combustion instabilities or droplet atomization in industrial burners or cloud formation. This paper explores a paradigm shift where the homogeneity hypothesis is replaced by the assumption that turbulence contains singularities, as suggested by Onsager. This paradigm leads to a weak formulation of the Kolmogorov–Kármán–Howarth–Monin equation (WKHE) that allows taking into account explicitly the presence of singularities and their impact on the energy transfer and dissipation. It provides a local in scale, space and time description of energy transfers and dissipation, valid for any inhomogeneous, anisotropic flow, under any type of boundary conditions. The goal of this article is to discuss WKHE as a tool to get a new description of energy cascades and dissipation that goes beyond Kolmogorov and allows the description of small-scale intermittency. It puts the problem of intermittency and dissipation in turbulence into a modern framework, compatible with recent mathematical advances on the proof of Onsager’s conjecture.

In three-dimensional turbulence there is on average a cascade of kinetic energy from the largest to the smallest scales of the flow. While the dominant idea is that the cascade occurs through the process of vortex stretching, evidence for this is debated. Here we show theoretically and numerically that vortex stretching is in fact not the main contributor to the average cascade. The main contributor is the self-amplification of the strain-rate field, and we provide several arguments for why its role must not be conflated with that of vortex stretching. Numerical results, however, indicate that vortex stretching plays a more important role during fluctuations of the cascade about its average behaviour. We also resolve a paradox regarding the differing role of vortex stretching on the energy cascade and energy dissipation rate dynamics.

The dynamics of spherical laser-induced cavitation bubbles in water is investigated by plasma photography, time-resolved shadowgraphs and sensitive single-shot probe beam scattering that portrays the transition from initial nonlinear to late linear oscillations. The frequency of late oscillations yields the bubble's gas content. Numerical simulations with an extended Gilmore model using plasma size as input and oscillation times as fit parameter provide insights into experimentally not accessible bubble parameters and shock wave emission. Model extensions include a term covering the initial shock-driven acceleration of the bubble wall, an automated method determining shock front position and pressure decay and a complete energy balance for the partitioning of absorbed laser energy into vaporization, bubble and shock wave energy and dissipation through viscosity and condensation. These tools are used for analysing a scattering signal covering 102 oscillation cycles from a bubble with 36 μm maximum radius produced by a plasma with 1550 K average temperature. Predicted bubble wall velocities during expansion agree well with experimental data. Upon first collapse, most energy was stored in the compressed liquid around the bubble and radiated away acoustically. The collapsed bubble contained more vapour than gas and had a pressure of 13.5 GPa. The decay of the rebound shock wave pressure with radius r was initially $\\mathrm{\\ \\propto }{r^{ - 1.8}}$, and energy dissipation at the shock front heated the liquid near the bubble wall to temperatures above the superheat limit. The shock-induced temperature rise reduces damping during late bubble oscillations. Damping in first collapse increases significantly for small bubbles with less than 10 μm radius.

The recent shift in Arctic ice conditions from prevailing multi-year ice to first-year ice will presumably intensify fall-winter sea ice freezing and the associated salt flux to the underlying water column. Here, we conduct a dual modeling study whose results suggest that the predicted catastrophic consequences for the global thermohaline circulation (THC), as a result of the disappearance of Arctic sea ice, may not necessarily occur. In a warmer climate, the substantial fraction of dense water feeding the Greenland-Scotland overflow may form on Arctic shelves and cascade to the deep basin, thus replenishing dense water, which currently forms through open ocean convection in the sub-Arctic seas. We have used a simplified model for estimating how increased ice production influences shelf-basin exchange associated with dense water cascading. We have carried out case studies in two regions of the Arctic Ocean where cascading was observed in the past. The baseline range of buoyancy-forcing derived from the columnar ice formation was calculated as part of a 30-year experiment of the pan-Arctic coupled ice-ocean general circulation model (GCM). The GCM results indicate that mechanical sea ice divergence associated with lateral advection accounts for a significant part of the interannual variations in sea ice thermal production in the coastal polynya regions. This forcing was then rectified by taking into account sub-grid processes and used in a regional model with analytically prescribed bottom topography and vertical stratification in order to examine specific cascading conditions in the Pacific and Atlantic sectors of the Arctic Ocean. Our results demonstrate that the consequences of enhanced ice formation depend on geographical location and shelf-basin bathymetry. In the Pacific sector, strong density stratification in slope waters impedes noticeable deepening of shelf-origin water, even for the strongest forcing applied. In the Atlantic sector, a 1.5× increase of salt flux leads to a threefold increase of shelf-slope volume flux below the warm core of Atlantic water. This threefold increase would be a sufficient substitute for a similar amount of dense water that currently forms in the Greenland, Iceland, and Norwegian (GIN) seas but is expected to decrease in a warming climate.

Triaxial loading and unloading tests on marble specimens under different stress paths were conducted to investigate the characteristics of energy evolution in rock deformation process. Results show that tensile failure occurred in rock specimens under uniaxial compression, while shear failure dominated under triaxial loading and unloading. The energy storage limit of rock specimens under triaxial loading was higher than that under uniaxial compression or triaxial unloading. A nonlinear energy evolution model of rock was established based on the interaction mechanism of energy accumulation and energy dissipation. Results from the theoretical model are in good agreement with the test results. Since the evolution of energy was characterised by bifurcation and chaos, a strain value corresponding to the energy iterative growth factor at the first bifurcation of the energy equation was chosen as the initiation criterion for rock failure. The critical strain accounted for 77%, 72–76%, and 72–81% of the peak strain under uniaxial compression, triaxial loading, and unloading, respectively.

Significance Efficient calculation of the properties of atoms, molecules, and solids on the computer requires a semilocal approximation to the density functional for the exchange-correlation energy, which becomes thereby a single integral over three-dimensional space. A recent, strongly tightened lower bound on the exchange energy has been built into the approximation “meta-generalized gradient approximations made very simple,” or MGGA-MVS, with accurate results for heats of formation, energy barriers, and weak interactions of molecules, and for lattice constants of solids. This would not have been possible without the use of a third ingredient (the local kinetic energy density) in addition to the standard two (the local electron density and its gradient). This third ingredient permits accurate energies even with the drastically tightened bound. Because of its useful accuracy and efficiency, density functional theory (DFT) is one of the most widely used electronic structure theories in physics, materials science, and chemistry. Only the exchange-correlation energy is unknown, and needs to be approximated in practice. Exact constraints provide useful information about this functional. The local spin-density approximation (LSDA) was the first constraint-based density functional. The Lieb–Oxford lower bound on the exchange-correlation energy for any density is another constraint that plays an important role in the development of generalized gradient approximations (GGAs) and meta-GGAs. Recently, a strongly and optimally tightened lower bound on the exchange energy was proved for one- and two-electron densities, and conjectured for all densities. In this article, we present a realistic “meta-GGA made very simple” (MGGA-MVS) for exchange that respects this optimal bound, which no previous beyond-LSDA approximation satisfies. This constraint might have been expected to worsen predicted thermochemical properties, but in fact they are improved over those of the Perdew–Burke–Ernzerhof GGA, which has nearly the same correlation part. MVS exchange is however radically different from that of other GGAs and meta-GGAs. Its exchange enhancement factor has a very strong dependence upon the orbital kinetic energy density, which permits accurate energies even with the drastically tightened bound. When this nonempirical MVS meta-GGA is hybridized with 25% of exact exchange, the resulting global hybrid gives excellent predictions for atomization energies, reaction barriers, and weak interactions of molecules.

To investigate the quantification of the extent of damage by considering the energy during rock failure, the pattern of energy dissipation and energy conversion, and the stress–energy mechanism for induced rock failure were analysed under cyclic loading/unloading. Based on damage mechanics, rock mechanics, and energy conservation theory, the test data were analysed. The results showed that the characteristics of hard rock compression are small deformation, high energy, and sudden failure; an elastic–plastic damage constitutive model and a stress–energy–rigidity–damage multi-criteria model for rock failure were established for hard rock. We compared the numerical curves and the experimental curves and found that they coincide. Rock failure is a combination of the results of elastic strain accumulation and dissipation by stress propagation. The key to inducing the energy storage capacity of rock before failure is closely related to the rock damage evolution. The pattern of energy release and dissipation through stress during rock failure was revealed from the perspective of energy using the constitutive model and multi-criteria model established for rock failure; these theoretical studies are very helpful in elucidating the mechanism of rock failure.

The deformation and failure of rocks result from the dissipation and release of their internal energy. The energy evolution throughout the processes of deformation and failure in rock is a critical research topic. The triaxial cyclic loading and unloading tests under five confining pressures were carried out on high-temperature rock samples to investigate the influences of the confining pressure (σ3) and temperature (T) on their energy evolution and distribution characteristics. The energy densities of rock samples under various confining pressures were calculated by determining the area between the loading and unloading curves, including axial energy densities (u10, u1e, u1d) and circumferential strain energy density (u30). The energy accumulation and dissipation and the effect of σ3 and T on the energy distribution laws of loaded rock samples were analysed. The characteristic energy density (u1t) was used to analyse the accumulation, dissipation and release of energy of the loaded rock sample. u1t increased with the increase in σ3 and decreased with the increase in T. Furthermore, u30 increased with the increase in σ3, which effectively limited the energy dissipation and release due to fracture or failure of the rock sample. With the increase in T, the circumferential strain of the rock sample increased, which led to an increase in u30. At the pre-peak stage, energy accumulation characterised the energy behaviour of the loaded rock sample, and the proportion of the elastic energy density (k1e) was large. At the post-peak stage, energy release and dissipation characterised the energy behaviour of the loaded rock sample, the dissipated energy density proportion (k1d) increased gradually, and the change law for k1e and k1d was considerably affected by the confining pressure and temperature effect. The dissipated energy density of the loaded rock sample was used to establish the energy damage variable and analyse the evolution law of the dissipated energy damage variable of the high-temperature rock sample with σ3 and T. The results of this study can provide guidance for the research on high-temperature rock damage mechanisms and prevention of dynamic disasters of rock underground engineering.

The Town Energy Balance (TEB) urban climate model has recently been improved to more realistically address the radiative effects of trees within the urban canopy. These processes necessarily have an impact on the energy balance that needs to be taken into account. This is why a new method for calculating the turbulent fluxes for sensible and latent heat has been implemented. This method remains consistent with the “bigleaf” approach of the Interaction Soil–Biosphere–Atmosphere (ISBA) model, which deals with energy exchanges between vegetation and atmosphere within TEB. Nonetheless, the turbulent fluxes can now be dissociated between ground-based natural covers and the tree stratum above (knowing the vertical leaf density profile), which can modify the vertical profile in air temperature and humidity in the urban canopy. In addition, the aeraulic effect of trees is added, parameterized as a drag term and an energy dissipation term in the evolution equations of momentum and turbulent kinetic energy, respectively. This set of modifications relating to the explicit representation of the tree stratum in TEB is evaluated on an experimental case study. The model results are compared to micrometeorological and surface temperature measurements collected in a semi-open courtyard with trees and bordered by buildings. The new parameterizations improve the modeling of surface temperatures of walls and pavements, thanks to taking into account radiation absorption by trees, and of air temperature. The effect of wind speed being strongly slowed down by trees is also much more realistic. The universal thermal climate index diagnosed in TEB from inside-canyon environmental variables is highly dependent and sensitive to these variations in wind speed and radiation. This demonstrates the importance of properly modeling interactions between buildings and trees in urban environments, especially for climate-sensitive design issues.

The effects of internal waves (IWs), externally forced by high-frequency wind, on energy pathways are studied in submesoscale-resolving numerical simulations of an idealized wind-driven channel flow. Two processes are examined: the direct extraction of mesoscale energy by externally forced IWs followed by an IW forward energy cascade to dissipation and stimulated imbalance, a mechanism through which externally forced IWs trigger a forward mesoscale to submesoscale energy cascade to dissipation. This study finds that the frequency and wavenumber spectral slopes are shallower in solutions with high-frequency forcing compared to solutions without and that the volume-averaged interior kinetic energy dissipation rate increases tenfold. The ratio between the enhanced dissipation rate and the added high-frequency wind work is 1.3, demonstrating the significance of the IW-mediated forward cascades. Temporal-scale analysis of energy exchanges among low- (mesoscale), intermediate- (submesoscale), and high-frequency (IW) bands shows a corresponding increase in kinetic energy E k and available potential energy APE transfers from mesoscales to submesoscales (stimulated imbalance) and mesoscales to IWs (direct extraction). Two direct extraction routes are identified: a mesoscale to IW E k transfer and a mesoscale to IW APE transfer followed by an IW APE to IW E k conversion. Spatial-scale analysis of eddy–IW interaction in solutions with high-frequency forcing shows an equivalent increase in forward E k and APE transfers inside both anticyclones and cyclones.