Anomalous Dissipation Research Articles

On Anomalous Dissipation in Plasma of Dusty Mercury’s Exosphere

The anomalous dissipation related to the effect of charging of dust particles that gives rise to new physical phenomena, effects, and mechanisms represents one of the main specific features of dusty plasma that makes it different from conventional plasma containing no charged dust particles. We analyze the process of anomalous dissipation in the context of description of the dynamics of dust particles in dusty plasma of the Mercury’s exosphere. An analytical description of oscillations of a dust particle above the surface of Mercury is presented. The frequency of charging of dust particles that characterizes the anomalous dissipation determines the damping of such oscillations. It is demonstrated that the anomalous dissipation is important for substantiation of the model of levitating dust particles that is used for description of dusty plasma above Mercury. The results of numerical simulations that justify the use of the discussed model are presented.

Three dimensional branching pipe flows for optimal scalar transport between walls

Abstract We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximises the transport of a passive temperature between two differentially heated walls for a given power supply budget. This problem in the community is also known as ‘wall-to-wall optimal transport’ and can be reduced to optimizing the choice of the divergence-free velocity field in the advection-diffusion equation subject to an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established that the transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco and Doering (2017 Phys. Rev. Lett. 118 264502) and Doering and Tobasco (2019 Commun. Pure Appl. Math. 72 2385–448) constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction. This correction appears to arise due to a topological obstruction inherent to two-dimensional steady flows. In this paper, we present a novel design of three-dimensional ‘branching pipe flows’ that bypasses this obstruction and, consequently, eliminates this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Our flows resemble the ones obtained in previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara and Shimizu (2018 J. Fluid Mech. 851 R4). We also discuss the implications of this result to the heat transfer problem in Rayleigh–Bénard convection and the problem of anomalous dissipation in a passive scalar.

A Josephson–Anderson relation for drag in classical channel flows with streamwise periodicity: Effects of wall roughness

The detailed Josephson–Anderson relation equates the instantaneous work by pressure drop over any streamwise segment of a general channel and the wall-normal flux of spanwise vorticity spatially integrated over that section. This relation was first derived by Huggins for quantum superfluids, but it holds also for internal flows of classical fluids and for external flows around solid bodies, corresponding there to relations of Burgers, Lighthill, Kambe, Howe, and others. All of these prior results employ a background potential Euler flow with the same inflow/outflow as the physical flow, just as in Kelvin's minimum energy theorem, so that the reference potential incorporates information about flow geometry. We here generalize the detailed Josephson–Anderson relation to streamwise periodic channels appropriate for numerical simulation of classical fluid turbulence. We show that the original Neumann b.c. used by Huggins for the background potential creates an unphysical vortex sheet in a periodic channel, so that we substitute instead Dirichlet b.c. We show that the minimum energy theorem still holds and our new Josephson–Anderson relation again equates work by pressure drop instantaneously to integrated flux of spanwise vorticity. The result holds for both Newtonian and non-Newtonian fluids and for general curvilinear walls. We illustrate our new formula with numerical results in a periodic channel flow with a single smooth bump, which reveals how vortex separation from the roughness element creates drag at each time instant. Drag and dissipation are thus related to vorticity structure and dynamics locally in space and time, with important applications to drag-reduction and to explanation of anomalous dissipation at high Reynolds numbers.

On the Support of Anomalous Dissipation Measures

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For LtqLxr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q_tL^r_x$$\\end{document} suitable Leray–Hopf solutions of the d-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d-$$\\end{document}dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure Ps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}^{s}$$\\end{document}, which gives s=d-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s=d-2$$\\end{document} as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity.

Regularity for a Fractional Liquid Crystal Model with Anomalous Dissipation and Thermal Effects

Regularity for a Fractional Liquid Crystal Model with Anomalous Dissipation and Thermal Effects

Scaling laws and exact results in turbulence *

In this note, we address the validity of certain exact results from turbulence theory in the deterministic setting. The main tools, inspired by the work of Duchon and Robert (2000 Nonlinearity 13 249–55) and Eyink (2003 Nonlinearity 16 137), are a number of energy balance identities for weak solutions of the incompressible Euler and Navier–Stokes equations. As a consequence, we show that certain weak solutions of the Euler and Navier–Stokes equations satisfy deterministic versions of Kolmogorov’s 45 , 43 , 415 laws. We apply these computations to improve a recent result of Hofmanova et al (2023 arXiv:2304.14470), which shows that a construction of solutions of forced Navier–Stokes due to Bruè et al (2023 Commun. Pure Appl. Anal.) and exhibiting a form of anomalous dissipation satisfies asymptotic versions of Kolmogorov’s laws. In addition, we show that the globally dissipative 3D Euler flows recently constructed by Giri et al (2023 arXiv:2305.18509) satisfy the local versions of Kolmogorov’s laws.

Spheres and fibers in turbulent flows at various Reynolds numbers

We perform fully coupled numerical simulations using immersed boundary methods of finite-size spheres and fibers suspended in a turbulent flow for a range of Taylor-Reynolds numbers 12.8<Reλ<442 and solid mass fractions 0≤M≤1. Both spheres and fibers reduce the turbulence intensity with respect to the single-phase flow at all Reynolds numbers, with fibers causing a more significant reduction than the spheres. The particles' effect on the anomalous dissipation tends to vanish as Re→∞. A scale-by-scale analysis shows that both particle shapes provide a “spectral shortcut” to the flow, but the shortcut extends further into the dissipative range in the case of fibers. Multifractal spectra of the near-particle dissipation show that spheres enhance dissipation in two-dimensional sheets, and fibers enhance the dissipation in structures with a dimension greater than one and less than two. In addition, we show that spheres suppress vortical flow structures, whereas fibers produce structures which completely overcome the turbulent vortex stretching behavior in their vicinity. Published by the American Physical Society 2024

Onsager's ‘ideal turbulence’ theory

In 1945–1949, Lars Onsager made an exact analysis of the high-Reynolds-number limit for individual turbulent flow realisations modelled by incompressible Navier–Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea that such flows are well described by distributional or ‘weak’ solutions of ideal Euler equations. 1/3 Hölder singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov $4/5$ th law. The approach is closely related to renormalisation group methods in physics and envisages ‘conservation-law anomalies’, as discovered later in quantum field theory. There are also deep connections with large-eddy simulation modelling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his $1/3$ Hölder singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or ‘convex integration’ techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting ‘spontaneously stochastic’ behaviour of high-Reynolds-number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space–time local phenomena. This theory thus makes testable predictions and offers new perspectives on large-eddy simulation in the presence of solid walls.

Gyre turbulence: Anomalous dissipation in a two-dimensional ocean model

Gyre turbulence: Anomalous dissipation in a two-dimensional ocean model

Turbulent Flows Are Not Uniformly Multifractal.

Understanding turbulence rests delicately on the conflict between Kolmogorov's 1941 theory of nonintermittent, space-filling energy dissipation characterized by a unique scaling exponent and the overwhelming evidence to the contrary of intermittency, multiscaling, and multifractality. Strangely, multifractality is not typically envisioned as a local flow property, variations in which might be clues exposing inroads into the fundamental unsolved issues of anomalous dissipation and finite time blowup. We present a simple construction of local multifractality and find that much of the dissipation field remains surprisingly monofractal àla Kolmogorov. Multifractality appears as small islands in this calm sea, its strength growing logarithmically with the local fluctuations in energy dissipation-a seemingly universal feature. These results suggest new ways to understand how singularities could arise and provide a fresh perspective on anomalous dissipation and intermittency. The simplicity and adaptability of our approach also holds great promise in applications ranging from climate sciences to medical data analysis.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

Layer separation of the 3D incompressible Navier–Stokes equation in a bounded domain

We provide an unconditional L 2 upper bound for the boundary layer separation of Leray–Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution u ν and a fixed (laminar) regular Euler solution u ¯ with similar initial conditions and body force. We show an asymptotic upper bound C | | u ¯ | | L ∞ 3 T on the layer separation, anomalous dissipation, and the work done by friction. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit.

Absence of anomalous dissipation of enstrophy for 3D incompressible Navier-Stokes equations

A valid method of studying the absence of anomalous dissipation of enstrophy for long-time averaged solutions of the forced and damped fluid equations in two dimensions was introduced in Constantin and Ramos (2007) [8]. Under the blueprint of their work, we will prove the absence of anomalous dissipation of enstrophy for the axisymmetric flow of the 3D damped and incompressible Navier-Stokes equations in this paper.

Tracking dynamo mechanisms from local energy transfers: Application to the von Kármán sodium dynamo

Motivated by the observation that dynamo is a conversion mechanism between kinetic and magnetic energy, we develop a new approach to unravel dynamo mechanism based on local (in space, scale, and time) energy budget describing dissipation and scale-by-scale energy transfers. Our approach is based upon a new filtering approach that can be used effectively for any type of meshes, including unstructured ones. The corresponding formalism is very general and applies to any geometry or boundary conditions. We further discuss the interpretation of these energy transfers in the context of fast dynamo and anomalous dissipation. We apply it to the results from direct numerical simulations of the von Kármán Sodium setup (referred to as VKS) using a finite element code, showing dynamo action for two types of impellers (steel or soft iron) in the magnetic field growth and saturation phases. Although the two types of dynamo hardly differ from the mean-field theory point of view (the velocity fields are the same in both cases), the locality of our formalism allows us to trace the origin of the differences between these two types of dynamo: for steel impellers, the dynamo is due to the transfer of velocity energy both in the bulk and in the vicinity of the impellers, whereas for soft iron impellers, the dynamo effect mainly comes from the rotation of the blades. We finally discuss possible signatures of precursors to anomalous dissipation and fast dynamo, which could become relevant in the inviscid limit.

Microscopic Expression of Anomalous Dissipation in Passive Scalar Transport

Microscopic Expression of Anomalous Dissipation in Passive Scalar Transport

Anomalous Dissipation and Lack of Selection in the Obukhov–Corrsin Theory of Scalar Turbulence

The Obukhov–Corrsin theory of scalar turbulence [21, 54] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov’s K41 theory of fully developed turbulence [47]. The scaling analysis of Obukhov and Corrsin from 1949 to 1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation. Our approach relies on a fine quantitative analysis of the interaction between the spatial scale of the solution and the scale of the Brownian motion which represents diffusion in a stochastic Lagrangian setting. This provides a direct Lagrangian approach to anomalous dissipation which is fundamental in order to get detailed insight on the behavior of the solution. Exploiting further this approach, we also show that for a velocity field in Cα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\alpha $$\\end{document} of space and time (for an arbitrary 0≤α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le \\alpha < 1$$\\end{document}) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation. This is motivated by the fundamental open problem of the selection of solutions of the Euler equations as vanishing-viscosity limit of solutions of the Navier-Stokes equations and provides a complete negative answer in the case of passive advection.

To the Theory of Decaying Turbulence

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension d&gt;2. This solution family is equivalent to a fractal curve in complex space Cd with random steps parametrized by N Ising variables σi=±1, in addition to a rational number pq and an integer winding number r, related by ∑σi=qr. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in Rd space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size N→∞ or its chemical potential μ→0. The system with fixed N has different asymptotics at odd and even N→∞, but the limit μ→0 is well defined. The energy dissipation rate is analytically calculated as a function of μ using methods of number theory. It grows as ν/μ2 in the continuum limit μ→0, leading to anomalous dissipation at μ∝ν→0. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as t−λ, with a continuous spectrum of indexes λ in the local limit μ→0. The spectrum is determined by a resolvent, which is represented as an infinite product of 3⊗3 matrices depending of the element of the Euler ensemble.

Vortex preconditioning of the 2021 sudden stratospheric warming: barotropic–baroclinic instability associated with the double westerly jets

Abstract. This study explores the abrupt split of the polar vortex in the upper stratosphere prior to a recent sudden stratospheric warming event on 5 January 2021 (SSW21) and the mechanisms of vortex preconditioning by using the Modern-Era Retrospective Analysis for Research and Applications version 2 (MERRA2) global reanalysis data. SSW21 is preceded by the highly distorted polar vortex that was initially displaced from the pole but eventually split at the onset date. Vortex splitting is most significant in the upper stratosphere (1 hPa altitude) accompanied by the anomalous growth of westward-propagating planetary waves (PWs) of zonal wavenumber (ZWN) 2 (WPW2). While previous studies have suggested the East Asian trough as a potential source for the abnormal WPW2 growth, the prominent westward-propagating nature cannot be explained satisfactorily by the upward propagation of the quasi-stationary ZWN2 fluxes in the troposphere. More importantly, WPW2 exhibits an obvious in situ excitation signature within the barotropically and baroclinically destabilized stratosphere, dominated by the easterlies descending from the stratopause containing the WPW2 critical levels. This suggests that the vortex split is attributed to the WPW2 generated in situ within the stratosphere via instability. Vortex destabilization is achieved as the double-jet structure consisting of a subtropical mesospheric core and a polar stratospheric core develops into SSW21 by encouraging the anomalous dissipation of the upward-propagating tropospheric ZWN1 PWs. This double-jet configuration is likely a favorable precursor for SSW onset, not only for the SSW21 but generally for most SSWs, through promoting the anomalous growth of unstable PWs as well as the enhancement of the tropospheric PW dissipation.

Fractional-order mathematical model of single-mass rotor dynamics and stability

Ensuring the vibration reliability of rotary machines is based on the reliable mathematical model of rotor movement. Considering the critical factors leading to a stability loss is an important problem in up-to-date machinery. However, the presence of fractional-order hydrodynamic features of gap seals and the effect of anomalous energy dissipation lead to a need to clarify existing rotordynamic models by considering the fractional-order terms. Therefore, this article deals with the comprehensive modeling approach to study forced oscillations of a single-mass rotor and check its dynamic stability considering the fractional-order origin of the damping force. Moreover, the impact of fractional order on the damping factor and dynamic stability of whirl movement considering the circulation force in gap seals, is still not fully explored. This problem adds particular importance to the presented research. As a result, a single-mass flexible rotor model was extended by considering the fractional-order damping force. Moreover, based on the discovered asymptotic property of the Mittag-Leffler function, the impact of fractional order on the amplitude-frequency response was also determined. Finally, boundaries for dynamic stability loss were identified considering the fractional-order damping force. Overall, the study allowed the development and substantiation of a more generalized scientific and methodological approach to ensuring the vibration reliability of rotary machines.

Topological Vortexes, Asymptotic Freedom, and Multifractals

In this paper, we study the Kelvinons, which are monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, which is in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS.