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Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicitydriven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.

We formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales . In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the 'bare' velocity and magnetic fields are 'renormalized' by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. Particular attention is given to the effects of these subscale terms on the balances of the quadratic invariants of ideal incompressible MHD-energy, cross-helicity and magnetic helicity. At large coarse-graining length-scales, the direct dissipation of the invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we prove a very strong version of the Woltjer-Taylor conjecture on conservation of magnetic helicity. Our bounds show that forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.

Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouville's theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the direction of the cascades, e.g., generalizations of the well-known ideal MHD inverse cascade of magnetic helicity. After a similar analysis is conducted for XMHD by inspecting second order structure functions and absolute equilibrium states, a couple of interesting results emerge. When cross helicity is taken to be ignorable, the inverse cascade of injected magnetic helicity also occurs in the Hall MHD range-this is shown to be consistent with previous results in the literature. In contrast, in the inertial MHD range, viz at scales smaller than the electron skin depth, all spectral quantities are expected to undergo direct cascading. The consequences and relevance of our results in space and astrophysical plasmas are also briefly discussed.

Lewis Fry Richardson proposed his famous picture of turbulent flows in 1922, where the kinetic energy is transferred from large-scale to small-scale structures until the viscosity converts it into heat. This cascade idea, also known as the forward energy cascade, is now widely accepted and is treated as the cornerstone of not only turbulent modeling, but also global circulation models of the ocean and atmosphere. In this work, the Filter-Space-Technique is applied to the oceanic flow field provided by the CMEMS reanalysis model to quantify the scale-to-scale energy flux. A rich dynamical pattern associated with different scales is observed. More precisely, either positive or negative fluxes are observed, indicating the direction of the energy cascade, where the energy is transferred from large-scale structures to small-scale ones or vice versa. High-intensity energy exchange is found mainly in the Western Boundary Current Systems and Equatorial Counter Currents. For the latter case, a wavelike pattern is observed on the westward travel. Moreover, strong seasonal variation is evident for some scales and regions. These results confirm the existence of forward and inverse cascades and rich regional dynamics.

This study presents discoveries on the spectral trends of energy cascading in turbulent buoyant plumes. A high-resolution large eddy simulation was used to simulate thermal and buoyant gas plumes. Turbulent plumes have positive helicity throughout the spatial development process. The helicity and turbulent kinetic energy spectra reveal consistent trends and demonstrate a deviation from the classical Kolmogorov’s inertial spectra at high wave numbers. Additional insight into turbulence physics has been confirmed in this study: the forward cascading of energy exists only at higher wavenumbers, whereas the flux of energy and helicity flows from smaller to large-scale structures at the lower wavenumbers.

Urban heat islands (UHIs) constitute one of the most conspicuous anthropogenic impacts on local climates, characterized by elevated land surface temperatures in urban areas compared to surrounding rural regions. This study represents a novel and comprehensive effort to characterize the spectral signature of SUHI through the lens of the two-dimensional (2D) turbulence theory, with a particular focus on identifying energy cascade regimes and their climatic modulation. The theory of two-dimensional (2D) turbulence, first described by Kraichnan and Batchelor, predicts two distinct energy cascade regimes: an inverse energy cascade at larger scales (low wavenumbers) and a direct enstrophy cascade at smaller scales (high wavenumbers). These cascades can be detected and characterized through spatial power spectra analysis, offering a scale-dependent understanding of the SUHI phenomenon. Despite the theoretical appeal, empirical validation of the 2D turbulence hypothesis in urban thermal landscapes remains scarce. This study aims to fill this gap by analyzing the spatial power spectra of land surface temperatures across 14 cities representing diverse climatic zones, capturing varied urban morphologies, structures, and materials. We analyzed multi-decadal LST datasets to compute spatial power spectra across summer and winter seasons, identifying spectral breakpoints that separate large-scale energy retention from small-scale dissipative processes. The findings reveal systematic deviations from classical turbulence scaling laws, with spectral slopes before the breakpoint ranging from ~K−1.6 to ~K−2.7 in winter and ~K−1.5 to ~K−2.4 in summer, while post-breakpoint slopes steepened significantly to ~K−3.5 to ~K−4.6 in winter and ~K−3.3 to ~K−4.3 in summer. These deviations suggest that urban heat turbulence is modulated by anisotropic surface heterogeneities, mesoscale instabilities, and seasonally dependent energy dissipation mechanisms. Notably, desert and Mediterranean climates exhibited the most pronounced small-scale dissipation, whereas oceanic and humid subtropical cities showed more gradual spectral transitions, likely due to differences in moisture availability and convective mixing. These results underscore the necessity of incorporating turbulence theory into urban climate models to better capture the scale-dependent nature of urban heat exchange. The observed spectral breakpoints offer a diagnostic tool for identifying critical scales at which urban heat mitigation strategies—such as green infrastructure, optimized urban ventilation, and reflective materials—can be most effective. Furthermore, our findings highlight the importance of regional climatic context in shaping urban spectral energy distributions, necessitating climate-specific urban design interventions. By advancing our understanding of urban thermal turbulence, this research contributes to the broader discourse on sustainable urban development and resilience in a warming world.

After more than a decade of experiments generating and studying the physics of quantized vortices in atomic gas Bose—Einstein condensates, research is beginning to focus on the roles of vortices in quantum turbulence, as well as other measures of quantum turbulence in atomic condensates. Such research directions have the potential to uncover new insights into quantum turbulence, vortices, and superfluidity and also explore the similarities and differences between quantum and classical turbulence in entirely new settings. Here we present a critical assessment of theoretical and experimental studies in this emerging field of quantum turbulence in atomic condensates.

This paper presents a summary of the optimization design process for a multi-objective, two-level engineering problem, utilizing the generalized inverse cascade method under uncertainty. The primary objective is to enhance the vibration isolation performance of a mounting system, considering the influence of uncertain factors on its stiffness. The focus is on determining the value range of the design variables at the bottom layer, ensuring that the design goal is met with a specified confidence level. To illustrate the application of this methodology, the optimization design of a powertrain mount is used as a case study. A data-driven approach is adopted, establishing a quantitative mapping relationship between mount stiffness, force transmission rate, modal decoupling rate, and other design indicators. This is achieved through the development of a CRBM-DBN approximate model, which combines Conditional Restricted Boltzmann Machines (CRBMs) and a Deep Belief Network (DBN). Additionally, an intelligent optimization algorithm and interval search technology are employed to determine the optimal design interval for the mount stiffness. Simulation and experimental verification are conducted using selected parameter combinations. The results demonstrate notable improvements in the vibration isolation performance, modal decoupling rate, and vehicle NVH performance when compared to the original state. These findings provide valuable insights for the interval optimization design of similar multi-objective, as well as two-level engineering problems, serving as useful references for future research and applications.

An overview is presented of several diverse branches of work in the area of effectively 2D fluid equilibria which have in common that they are constrained by an infinite number of conservation laws. Broad concepts, and the enormous variety of physical phenomena that can be explored, are highlighted. These span, roughly in order of increasing complexity, Euler flow, nonlinear Rossby waves, 3D axisymmetric flow, shallow water dynamics, and 2D magnetohydrodynamics. The classical field theories describing these systems bear some resemblance to perhaps more familiar fluctuating membrane and continuous spin models, but the fluid physics drives these models into unconventional regimes exhibiting large scale jet and eddy structures. From a dynamical point of view these structures are the end result of various conserved variable forward and inverse cascades. The resulting balance between large scale structure and small scale fluctuations is controlled by the competition between energy and entropy in the system free energy, in turn highly tunable through setting the values of the conserved integrals. Although the statistical mechanical description of such systems is fully self-consistent, with remarkable mathematical structure and diversity of solutions, great care must be taken because the underlying assumptions, especially ergodicity, can be violated or at minimum lead to exceedingly long equilibration times. Generalization of the theory to include weak driving and dissipation (e.g., non-equilibrium statistical mechanics and associated linear response formalism) could provide additional insights, but has yet to be properly explored.

The knowledge of the observed inverse energy cascade in typhoon boundary layers holds significant implications for understanding the formation, enhancement, and deformation of typhoons. This study reveals that the observed inverse energy cascade originates from the rapid rotation of typhoons. The transition from the direct energy cascade regime to the inverse energy cascade regime can be characterized by the Zeman length scale. The turbulence structures behave as two dimensional above the Zeman length scale. Through symmetry analysis, it is discovered that the ratio of the inverse energy cascade flux to the direct energy cascade flux is proportional to the turbulence Rossby number with a power of −2. These findings offer a framework for incorporating the inverse energy cascade into the turbulence parameterizations and improving typhoon modeling. Plain Language Summary The process of the energy flowing inversely to large scales in typhoon boundary layers is important for the evolution of typhoons. This study reveals that this process originates from the rotation of typhoons. This study also establishes a theoretical relation between the ratio of the inverse energy cascade flux to the direct energy cascade flux and the turbulence Rossby number. These findings have practical implications for improving the turbulence parameterizations in numerical models. Key Points The observed inverse energy cascade in typhoon boundary layers originates from the rapid rotation of typhoons The Zeman length scale can characterize the transition from the direct energy cascade regime to the inverse energy cascade regime The ratio of the inverse energy cascade flux to the direct energy cascade flux is controlled by the turbulence Rossby number