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We present an analysis of the Navier-Stokes equations based on a spatial filtering technique to elucidate the multi-scale nature of fully developed turbulence. In particular, the advection of a band-pass-filtered small-scale contribution by larger scales is considered, and rigorous upper bounds are established for the various dynamically active scales. The analytical predictions are confirmed with direct numerical simulation data. The results are discussed with respect to the establishment of effective large-scale equations valid for turbulent flows., 14 pages, 6 figures

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows non-uniqueness of solutions to the transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$ drift, which is smooth except at one point in time. We also provide three sufficient conditions for anomalous dissipation provided solutions to the inviscid equation become singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications., Comment: It was pointed out to us by E. Bru\`{e} and Q-H. Nguyen that Conjecture 1.7, as stated, was false. The new version contains a modification of this conjecture which emerged after discussions with them

We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are H\"{o}lder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. Our proof makes use of a probabilistic interpretation of the dissipation of solutions to the advection diffusion equation., Comment: 17 pages, 3 figures

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution $\omega_0$, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of $\omega_0$ can occur. Since kinetic energy $\mathsf{E}$ is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy $\mathsf{E}_0$ corresponding to $\omega_0$. The set of said vorticity fields, denoted by $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (e.g. straight channel or annulus), we exploit all the conserved quantities and the characterizations of $\overline{\mathcal{O}_{\omega_0}}^*\cap \{ {\mathsf E}= {\mathsf E}_0\}$ to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in $L^1$ of vorticity but do not weakly converge to them in the long time limit., Comment: 36 pages

We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincar\'{e} and stochastic Navier-Stokes-Poincar\'{e} equations respectively. The stochastic Euler-Poincar\'{e} equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems., Comment: 26 pages

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [Chapter IX, ��96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $C^{1, {\frac{1}{2}} }$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $C^{1, {\frac{1}{2}} }$ one-sided cusps while the pressure and normal velocity remain $C^2$; as such, we term this surface a weak contact discontinuity., 150 pages, 15 figures, typos corrected

Material elements – which are lines, surfaces, or volumes behaving as passive, non-diffusive markers – provide an inherently geometric window into the intricate dynamics of chaotic flows. Their stretching and folding dynamics has immediate implications for mixing in the oceans or the atmosphere, as well as the emergence of self-sustained dynamos in astrophysical settings. Here, we uncover robust statistical properties of an ensemble of material loops in a turbulent environment. Our approach combines high-resolution direct numerical simulations of Navier-Stokes turbulence, stochastic models, and dynamical systems techniques to reveal predictable, universal features of these complex objects. We show that the loop curvature statistics become stationary through a dynamical formation process of high-curvature folds, leading to distributions with power-law tails whose exponents are determined by the large-deviations statistics of finite-time Lyapunov exponents of the flow. This prediction applies to advected material lines in a broad range of chaotic flows. To complement this dynamical picture, we confirm our theory in the analytically tractable Kraichnan model with an exact Fokker-Planck approach.

We study stationary free boundary configurations of an ideal incompressible magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply connected domains, we prove that if the magnetic field and velocity field are never commensurate, the only possible domain for any such equilibria is a disk, and the velocity and magnetic field are circular. We give examples of non-symmetric equilibria occupying a domain of any shape by imposing an external magnetic field generated by a singular current sheet charge distribution (external coils). Some results carry over to 3D axisymmetric solutions. These results highlight the importance of external magnetic fields for the existence of asymmetric equilibria., Comment: revised version. 18 pages, 3 figures

Flexibility and rigidity properties of steady (time-independent) solutions of the Euler, Boussinesq and Magnetohydrostatic equations are investigated. Specifically, certain Liouville-type theorems are established which show that suitable steady solutions with no stagnation points occupying a two-dimensional periodic channel, or axisymmetric solutions in (hollowed out) cylinder, must have certain structural symmetries. It is additionally shown that such solutions can be deformed to occupy domains which are themselves small perturbations of the base domain. As application of the general scheme, Arnol'd stable solutions are shown to be structurally stable., 35 pages, 3 figures

We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $\Omega\subset \mathbb{R}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity $u\in L^3(0,T;B_{3}^{1/3, c_0})$. On an arbitrary thin neighborhood of the boundary, we assume boundedness of velocity and pressure and, at the boundary, we assume continuity of wall-normal velocity. We also prove two theorems which establish that the global viscous dissipation vanishes in the inviscid limit for Leray--Hopf solutions $u^\nu$ of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width $O(\nu^{\min\{1,\frac{1}{2(1-\sigma)}\}})$ when $u\in L^3(0, T; B_3^{\sigma, c_0})$ in the interior for any $\sigma\in [1/3,1]$. The first theorem assumes continuity of the velocity in the boundary layer whereas the second assumes a condition on the vanishing of energy dissipation within the layer. In both cases, strong $L^3_tL^3_{x,loc}$ convergence holds to a weak solution of the Euler equations. Finally, if a strong Euler solution exists in the background, we show that equicontinuity at the boundary within a $O(\nu)$ strip alone suffices to conclude the absence of anomalous dissipation., Comment: 23 pages, 1 figure. Theorem 3 added in version 2. Sharpened the interior regularity assumption on the velocity in Theorems 1,2 and 3 in version 3

An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This ‘Lagrangian fluctuation–dissipation relation’ equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated with initial scalar values and internal scalar sources, as these are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of ‘Lagrangian spontaneous stochasticity’ or persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. Previous work on the Kraichnan (Phys. Fluids, 1968, vol. 11, pp. 945–953) model of turbulent scalar advection has shown that anomalous scalar dissipation is associated in that model with Lagrangian spontaneous stochasticity. There has been controversy, however, regarding the validity of this mechanism for scalars advected by an actual turbulent flow. We here completely resolve this controversy by exploiting the fluctuation–dissipation relation. For either a passive or an active scalar advected by any divergence-free velocity field, including solutions of the incompressible Navier–Stokes equation, and away from walls, we prove that anomalous scalar dissipation requires Lagrangian spontaneous stochasticity. For passive scalars, we prove furthermore that spontaneous stochasticity yields anomalous dissipation for suitable initial scalar fields, so that the two phenomena are there completely equivalent. These points are illustrated by numerical results from a database of homogeneous isotropic turbulence, which provide both additional support to the results and physical insight into the representation of diffusive effects by stochastic Lagrangian particle trajectories.

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also `nearly' quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$, to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$--symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad-Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing., Comment: 24 pages, 2 figures, accepted version

We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and non-local diffusion and isentropic pressure. The local viscosity is allowed to degenerate as the density approaches vacuum. The method provides a tool to propagate initial regularity of classical solutions provided no vacuum has formed and serves as an alternative to the classical energy method. We obtain a series of global well-posedness results for state laws in previously uncovered cases including $p(\rho) = c_p \rho$. As an application we prove global well-posedness of collective behavior models with pressure arising from agent-based Cucker-Smale system., Comment: 16 pages

We investigate dissipative anomalies in a turbulent fluid governed by the compressible Navier-Stokes equation. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. In the limit of high Reynolds and P\'eclet numbers, the flow realizations are found to be described as distributional or "coarse-grained" solutions of the compressible Euler equations, with standard conservation laws broken by turbulent anomalies. The anomalous dissipation of kinetic energy is shown to be due not only to local cascade, but also to a distinct mechanism called pressure-work defect. Irreversible heating in stationary, planar shocks with an ideal-gas equation of state exemplifies the second mechanism. Entropy conservation anomalies are also found to occur by two mechanisms: an anomalous input of negative entropy (negentropy) by pressure-work and a cascade of negentropy to small scales. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required to sustain the cascades. We compare our approach with alternative theories and empirical evidence. It is argued that the "Big Power-Law in the Sky" observed in electron density scintillations in the interstellar medium is a manifestation of a forward negentropy cascade, or an inverse cascade of usual thermodynamic entropy.

We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $L^2$ weak limit of Leray--Hopf weak solutions of the Navier-Stokes equations on any bounded domain $\Omega\subset \mathbb{R}^d$, $d= 2,3$ is a weak solution of the Euler equations. This holds for both no-slip and Navier-friction conditions with viscosity-dependent slip length. The result allows for the emergence of non-unique, possibly dissipative, limiting weak solutions of the Euler equations., Comment: Remark 3 added and minor changes incorporated after revision. Accepted to J. Nonlinear Science

We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott-Mann-Gaw��dzki relation, sometimes described as a "Lagrangian analogue of the 4/5ths law". In particular, we prove that for any space-time $L^3$ weak solution of the Euler equations, the Lagrangian forward/backward dispersion measure matches on to the energy defect in the sense of distributions. For strong limits of $d\geq3$ dimensional Navier-Stokes solutions the defect distribution coincides with the viscous dissipation anomaly. The Lagrangian formula shows that particles released into a $3d$ turbulent flow will initially disperse faster backward-in-time than forward, in agreement with recent theoretical predictions of Jucha et. al (2014). In two dimensions, we consider strong limits of solutions of the forced Euler equations with increasingly high-wavenumber forcing as a model of an ideal inverse cascade regime. We show that the same Lagrangian dispersion measure matches onto the anomalous input from the infinite-frequency force. As forcing typically acts as an energy source, this leads to the prediction that particles in $2d$ typically disperse faster forward in time than backward, which is opposite to what occurs in $3d$. Time-asymmetry of the Lagrangian dispersion is thereby closely tied to the direction of the turbulent cascade, downscale in $d\geq 3$ and upscale in $d=2$. These conclusions lend support to the conjecture of Eyink & Drivas (2015) that a similar connection holds for time-asymmetry of Richardson two-particle dispersion and cascade direction, albeit at longer times., 16 pages. Some claims in the proof of Theorem 1 are rigorously justified. Accepted to J. Nonlinear Science

We develop a first-principles theory of relativistic fluid turbulence at high Reynolds and Péclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a nonperturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for nonrelativistic turbulence, with hydrodynamic fields in the inertial range described as distributional or “coarse-grained” solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation laws of smooth Euler solutions but are afflicted with dissipative anomalies in the balance equations of internal energy and entropy. The anomalies are shown to be possible by exactly two mechanisms, local cascade and pressure-work defect. We derive “4/5th-law” type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required for their not vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find to be broken by our coarse-graining regularization but which is restored in the limit where the regularization is removed, similar to relativistic lattice quantum field theory. In the formal limit as speed of light goes to infinity, we recover the results of previous nonrelativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of nonrelativistic kinetic energy.

We derive here Lagrangian fluctuation–dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting time at points where scalar values are imposed. These concepts are illustrated both by analytical results for the problem of pure heat conduction and by numerical results from a database of channel-flow turbulence, which also demonstrate the scalar mixing properties of near-wall turbulence. As an application of the fluctuation–dissipation relation, we examine for wall-bounded flows the relation between anomalous scalar dissipation and Lagrangian spontaneous stochasticity, i.e. the persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. In Part I of this series, we showed that spontaneous stochasticity is the only possible mechanism for anomalous dissipation of passive or active scalars, away from walls. Here it is shown that this remains true when there are no scalar fluxes through walls. Simple examples show, on the other hand, that a distinct mechanism of non-vanishing scalar dissipation can be thin scalar boundary layers near the walls. Nevertheless, we prove for general wall-bounded flows that spontaneous stochasticity is another possible mechanism of anomalous scalar dissipation.

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus ${\mathbb T}^d$, assuming that the solutions have norms for Besov space $B^{\sigma,\infty}_3({\mathbb T}^d),$ $\sigma\in (0,1],$ that are bounded in the $L^3$-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form $O(\nu^{(3\sigma-1)/(\sigma+1)}),$ vanishing as $\nu\to0$ if $\sigma>1/3.$ A consequence is that Onsager-type "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit $\nu\to 0$, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions $u$ which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For $\sigma\in (1/3,1)$ the anomalous dissipation vanishes and the weak Euler solutions may be spatially "rough" but conserve energy., Comment: 14 pgs; v2: reorganized main results and added additional technical details to proofs, v3 accepted in Nonlinearity

In situ spacecraft data on the solar wind show events identified as magnetic reconnection with wide outflows and extended "X lines," 10(3)-10(4) times ion scales. To understand the role of turbulence at these scales, we make a case study of an inertial-range reconnection event in a magnetohydrodynamic simulation. We observe stochastic wandering of field lines in space, breakdown of standard magnetic flux freezing due to Richardson dispersion, and a broadened reconnection zone containing many current sheets. The coarse-grain magnetic geometry is like large-scale reconnection in the solar wind, however, with a hyperbolic flux tube or apparent X line extending over integral length scales.

We develop a Lagrangian approach to conservation-law anomalies in weak solutions of inviscid Burgers equation, motivated by previous work on the Kraichnan model of turbulent scalar advection. We show that the entropy solutions of Burgers possess Markov stochastic processes of (generalized) Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale. This property is shown to guarantee dissipativity of conservation-law anomalies for general convex functions of the velocity. The backward stochastic Burgers flows with these properties are not unique, however. We construct infinitely many such stochastic flows, both by a geometric construction and by the zero-noise limit of the Constantin-Iyer stochastic representation of viscous Burgers solutions. The latter proof yields the spontaneous stochasticity of Lagrangian trajectories backward in time for Burgers, at unit Prandtl number. It is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for Burgers. We also study linear transport of passive densities and scalars by inviscid Burgers flows. We show that shock solutions of Burgers exhibit spontaneous stochasticity backward in time for all Prandtl numbers, implying conservation-law anomalies for linear transport. We discuss the relation of our results for Burgers with incompressible Navier-Stokes turbulence, especially Lagrangian admissibility conditions for Euler solutions and the relation between turbulent cascade directions and time-asymmetry of Lagrangian stochasticity., minor revisions, paper will appear in JSP

We derive an exact equation governing two-particle backwards mean-squared dispersion for both deterministic and stochastic tracer particles in turbulent flows. For the deterministic trajectories, we probe the consequences of our formula for short times and arrive at approximate expressions for the mean-squared dispersion which involve second order structure functions of the velocity and acceleration fields. For the stochastic trajectories, we analytically compute an exact $t^3$ contribution to the squared separation of stochastic paths. We argue that this contribution appears also for deterministic paths at long times and present direct numerical simulation results for incompressible Navier-Stokes flows to support this claim. We also numerically compute the probability distribution of particle separations for the deterministic paths and the stochastic paths and show their strong self-similar nature., Comment: 5 pages, 4 figures