Spontaneous Stochasticity Amplifies Even Thermal Noise to the Largest Scales of Turbulence in a Few Eddy Turnover Times.

How predictable are turbulent flows? Here, we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the nonuniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. These universal statistics are predictable, even though individual flow realizations are not. Any small-scale noise vanishing slowly enough with increasing Reynolds number can trigger spontaneous stochasticity, and here we show that thermal noise alone, in the absence of any larger disturbances, would suffice. If confirmed for Navier-Stokes turbulence, our findings would imply that intrinsic stochasticity of turbulent fluid motions at all scales can be triggered even by unavoidable molecular noise, with implications for modeling in engineering, climate, astrophysics, and cosmology.

Dissipation-range fluid turbulence and thermal noise.

We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation-range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an "effective field theory" valid below some high-wave-number cutoff Λ, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million, comparable to that in Earth's atmospheric boundary layer, the strongest turbulent excitations observed in our simulation penetrate down to a length scale of about eight microns, still two orders of magnitude greater than the mean free path of air. However, for longer observation times or for higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.

The Onsager theory of wall-bounded turbulence and Taylor's momentum anomaly.

We discuss the Onsager theory of wall-bounded turbulence, analysing the momentum dissipation anomaly hypothesized by Taylor. Turbulent drag laws observed with both smooth and rough walls imply ultraviolet divergences of velocity gradients. These are eliminated by a coarse-graining operation, filtering out small-scale eddies and windowing out near-wall eddies, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of the Navier-Stokes equation and contain, in addition to the usual turbulent stress, also an inertial drag force modelling momentum exchange with unresolved near-wall eddies. Using an Onsager-type argument based on the principle of renormalization group invariance, we derive an upper bound on wall friction by a function of Reynolds number determined by the modulus of continuity of the velocity at the wall. Our main result is a deterministic version of Prandtl's relation between the Blasius [Formula: see text] drag law and the 1/7 power-law profile of the mean streamwise velocity. At higher Reynolds, the von Kármán-Prandtl drag law requires instead a slow logarithmic approach of velocity to zero at the wall. We discuss briefly also the large-eddy simulation of wall-bounded flows and use of iterative renormalization group methods to establish universal statistics in the inertial sublayer. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind.

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.

Diffusion approximation in turbulent two-particle dispersion.

We solve an inverse problem for fluid particle pair statistics: we show that a time sequence of probability density functions (PDFs) of separations can be exactly reproduced by solving the diffusion equation with a suitable time-dependent diffusivity. The diffusivity tensor is given by a time integral of a conditional Lagrangian velocity structure function, weighted by a ratio of PDFs. Physical hypotheses for hydrodynamic turbulence (sweeping, short memory, mean-field) yield simpler integral formulas, including one of Kraichnan and Lundgren (K-L). We evaluate the latter using a space-time database from a numerical Navier-Stokes solution for driven turbulence. The K-L formula reproduces PDFs well at root-mean-square separations, but growth rate of mean-square dispersion is overpredicted due to neglect of memory effects. More general applications of our approach are sketched.

Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence.

The idea of 'frozen-in' magnetic field lines for ideal plasmas is useful to explain diverse astrophysical phenomena, for example the shedding of excess angular momentum from protostars by twisting of field lines frozen into the interstellar medium. Frozen-in field lines, however, preclude the rapid changes in magnetic topology observed at high conductivities, as in solar flares. Microphysical plasma processes are a proposed explanation of the observed high rates, but it is an open question whether such processes can rapidly reconnect astrophysical flux structures much greater in extent than several thousand ion gyroradii. An alternative explanation is that turbulent Richardson advection brings field lines implosively together from distances far apart to separations of the order of gyroradii. Here we report an analysis of a simulation of magnetohydrodynamic turbulence at high conductivity that exhibits Richardson dispersion. This effect of advection in rough velocity fields, which appear non-differentiable in space, leads to line motions that are completely indeterministic or 'spontaneously stochastic', as predicted in analytical studies. The turbulent breakdown of standard flux freezing at scales greater than the ion gyroradius can explain fast reconnection of very large-scale flux structures, both observed (solar flares and coronal mass ejections) and predicted (the inner heliosheath, accretion disks, γ-ray bursts and so on). For laminar plasma flows with smooth velocity fields or for low turbulence intensity, stochastic flux freezing reduces to the usual frozen-in condition.

Synchronization of chaos in fully developed turbulence.

We investigate chaos synchronization of small-scale motions in the three-dimensional turbulent energy cascade, via pseudospectral simulations of the incompressible Navier-Stokes equations. The modes of the turbulent velocity field below about 20 Kolmogorov dissipation lengths are found to be slaved to the chaotic dynamics of larger-scale modes. The dynamics of all dissipation-range modes can be recovered to full numerical precision by solving small-scale dynamical equations with the given large-scale solution as an input, regardless of initial condition. The synchronization rate exponent scales with the Kolmogorov dissipation time scale, with possible weak corrections due to intermittency. Our results suggest that all sub-Kolmogorov length modes should be fully recoverable from numerical simulations with standard, Kolmogorov-length grid resolutions.

Suppression of particle dispersion by sweeping effects in synthetic turbulence.

Synthetic models of Eulerian turbulence like so-called kinematic simulations (KS) are often used as computational shortcuts for studying Lagrangian properties of turbulence. These models have been criticized by Thomson and Devenish (2005), who argued on physical grounds that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields, in particular for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution obtained from the Gaussian integration-by-parts identity. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a time-dependent pair diffusivity tensor of the form K(ij)(r,t)=S(ij)(r)τ(r,t), where S(ij)(r) is the structure-function tensor and τ(r,t) is an effective correlation time of velocity increments. Crucially, this is found to be the minimum value of three times: the intrinsic turnover time τ(eddy)(r) at separation r, the overall evolution time t, and the sweeping time r/v(0) with v(0) the rms velocity. We study the diffusion model numerically by a Monte Carlo method. With inertial ranges like the largest achieved in most current KS (about 6 decades long), our model is found to reproduce the t(9/2) power law for pair dispersion predicted by Thomson and Devenish and observed in the KS. However, for much longer ranges, our model exhibits three distinct pair-dispersion laws in the inertial range: a Batchelor t(2) regime, followed by a Kraichnan-model-like t(1) diffusive regime, and then a t(6) regime. Finally, outside the inertial range, there is another t(1) regime with particles undergoing independent Taylor diffusion. These scalings are exactly the same as those predicted by Thomson and Devenish for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion of Thomson and Devenish (2005) that sweeping effects make Lagrangian properties of KS fundamentally differ from those of hydrodynamic turbulence for very extended inertial ranges.

Stochastic flux freezing and magnetic dynamo.

Magnetic flux conservation in turbulent plasmas at high magnetic Reynolds numbers is argued neither to hold in the conventional sense nor to be entirely broken, but instead to be valid in a statistical sense associated to the "spontaneous stochasticity" of Lagrangian particle trajectories. The latter phenomenon is due to the explosive separation of particles undergoing turbulent Richardson diffusion, which leads to a breakdown of Laplacian determinism for classical dynamics. Empirical evidence is presented for spontaneous stochasticity, including numerical results. A Lagrangian path-integral approach is then exploited to establish stochastic flux freezing for resistive hydromagnetic equations and to argue, based on the properties of Richardson diffusion, that flux conservation must remain stochastic at infinite magnetic Reynolds number. An important application of these results is the kinematic, fluctuation dynamo in nonhelical, incompressible turbulence at magnetic Prandtl number (Pr(m)) equal to unity. Numerical results on the Lagrangian dynamo mechanisms by a stochastic particle method demonstrate a strong similarity between the Pr(m)=1 and 0 dynamos. Stochasticity of field-line motion is an essential ingredient of both. Finally, some consequences for nonlinear magnetohydrodynamic turbulence, dynamo, and reconnection are briefly considered.

Scale locality of magnetohydrodynamic turbulence.

We investigate the scale locality of cascades of conserved invariants at high kinetic and magnetic Reynold's numbers in the "inertial-inductive range" of magnetohydrodynamic (MHD) turbulence, where velocity and magnetic field increments exhibit suitable power-law scaling. We prove that fluxes of total energy and cross helicity-or, equivalently, fluxes of Elsässer energies-are dominated by the contributions of local triads. Flux of magnetic helicity may be dominated by nonlocal triads. The magnetic stretching term may also be dominated by nonlocal triads, but we prove that it can convert energy only between velocity and magnetic modes at comparable scales. We explain the disagreement with numerical studies that have claimed conversion nonlocally between disparate scales. We present supporting data from a 1024{3} simulation of forced MHD turbulence.

Fluctuation dynamo and turbulent induction at small Prandtl number.

We study the Lagrangian mechanism of the fluctuation dynamo at zero Prandtl number and infinite magnetic Reynolds number, in the Kazantsev-Kraichnan model of white-noise advection. With a rough velocity field corresponding to a turbulent inertial range, flux freezing holds only in a stochastic sense. We show that field lines arriving to the same point which were initially separated by many resistive lengths are important to the dynamo. Magnetic vectors of the seed field that point parallel to the initial separation vector arrive anticorrelated and produce an "antidynamo" effect. We also study the problem of "magnetic induction" of a spatially uniform seed field. We find no essential distinction between this process and fluctuation dynamo, both producing the same growth rates and small-scale magnetic correlations. In the regime of very rough velocity fields where fluctuation dynamo fails, we obtain the induced magnetic energy spectra. We use these results to evaluate theories proposed for magnetic spectra in laboratory experiments of turbulent induction.

Matrix exponential-based closures for the turbulent subgrid-scale stress tensor.

Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the recent fluid deformation approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other nonlinear scrambling terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy.

Equation-free implementation of statistical moment closures.

We present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex, large-scale, nonlinear systems. Building on recently developed equation-free methods, this approach numerically integrates the closure dynamics, the equations of which may not even be available in closed form. Although closure dynamics introduce statistical assumptions of unknown validity, they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model. The numerical closure approach can in principle be applied to a wide class of nonlinear problems, including strongly coupled systems (either deterministic or stochastic) for which there may be no scale separation. We demonstrate the equation-free approach for implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic partial differential equation.

Is the Kelvin theorem valid for high Reynolds number turbulence?

The Kelvin-Helmholtz theorem on conservation of circulation is supposed to hold for ideal inviscid fluids and is believed to be play a crucial role in turbulent phenomena. However, this expectation does not take into account singularities in turbulent velocity fields at infinite Reynolds number. We present evidence from numerical simulations for the breakdown of the classical Kelvin theorem in the three-dimensional turbulent energy cascade. Although violated in individual realizations, we find that circulation is still conserved in some average sense. For comparison, we show that Kelvin's theorem holds for individual realizations in the two-dimensional enstrophy cascade, in agreement with theory. The turbulent "cascade of circulations" is shown to be a classical analogue of phase slip due to quantized vortices in superfluids, and various applications in geophysics and astrophysics are outlined.

Subgrid-scale modeling of helicity and energy dissipation in helical turbulence.

The subgrid-scale (SGS) modeling of helical, isotropic turbulence in large eddy simulation is investigated by quantifying rates of helicity and energy cascade. Assuming Kolmogorov spectra, the Smagorinsky model with its traditional coefficient is shown to underestimate the helicity dissipation rate by about 40%. Several two-term helical models are proposed with the model coefficients calculated from simultaneous energy and helicity dissipation balance. The helical models are also extended to include dynamic determination of their coefficients. The models are tested a priori in isotropic steady helical turbulence. Together with the dynamic Smagorinsky model and the dynamic mixed model, they are also tested a posteriori in both decaying and steady isotropic helical turbulence by comparing results to direct numerical simulations (DNS). The a priori tests confirm that the Smagorinsky model underestimates SGS helicity dissipation, although quantitative differences with the predictions are observed due to the finite Reynolds number of the DNS. Also, in a posteriori tests improvement can be achieved for the helicity decay rate with the proposed models, compared with the Smagorinsky model. Overall, however, the effect of the new helical terms added to obtain the correct rate of global helicity dissipation is found to be quite small. Within the small differences, the various versions of the dynamic model provide the results closest to the DNS. The dynamic model's good performance in capturing mean kinetic energy dissipation at the finite Reynolds number of the simulations appears to be the most important aspect in accounting also for accurate prediction of the helicity dissipation.

Physical mechanism of the two-dimensional inverse energy cascade.

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45 degrees. This "skew-Newtonian" stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.

Cascade of circulations in fluid turbulence.

Kelvin's theorem on conservation of circulations is an essential ingredient of Taylor's theory of turbulent energy dissipation by the process of vortex-line stretching. In previous work, we have proposed a nonlinear mechanism for the breakdown of Kelvin's theorem in ideal turbulence at infinite Reynolds number. We develop here a detailed physical theory of this cascade of circulations. Our analysis is based upon an effective equation for large-scale coarse-grained velocity, which contains a turbulent-induced vortex force that can violate Kelvin's theorem. We show that singularities of sufficient strength, which are observed to exist in turbulent flow, can lead to nonvanishing dissipation of circulation for an arbitrarily small coarse-graining length in the effective equations. This result is an analog for circulation of Onsager's theorem on energy dissipation for singular Euler solutions. The physical mechanism of the breakdown of Kelvin's theorem is diffusion of lines of large-scale vorticity out of the advected loop. This phenomenon can be viewed as a classical analog of the Josephson-Anderson phase-slip phenomenon in superfluids due to quantized vortex lines. We show that the circulation cascade is local in scale and use this locality to develop concrete expressions for the turbulent vortex force by a multiscale gradient expansion. We discuss implications for Taylor's theory of turbulent dissipation and we point out some related cascade phenomena, in particular for magnetic flux in magnetohydrodynamic turbulence.

Physical mechanism of the two-dimensional enstrophy cascade.

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.

Recovering isotropic statistics in turbulence simulations: the Kolmogorov 4/5th law.

One of the main benchmarks in direct numerical simulations of three-dimensional turbulence is the Kolmogorov prediction for third-order structure functions with homogeneous and isotropic statistics in the infinite Reynolds number limit. Previous direct numerical simulations (DNS) techniques to obtain isotropic statistics have relied on time-averaging structure functions in a few directions over many eddy-turnover times, using forcing schemes carefully constructed to generate isotropic data. Motivated by recent theoretical work, which removes isotropy requirements by spherically averaging the structure functions over all directions, we will present results which supplement long-time averaging by angle-averaging over up to 73 directions from a single flow snapshot. The directions are among those natural to a square computational grid, and are weighted to approximate the spherical average. We use this angle-averaging procedure to compare the statistically steady flows generated by two different forcing schemes in a periodic box. Our results show that despite the apparent differences in the two flows, their isotropic components, as measured by the Kolmogorov laws, are essentially identical. This procedure may be used to investigate the isotropic part of the small-scale statistics of any quantity of interest. The averaging process is inexpensive, and for the Kolmogorov 4/5 law, reasonable results can be obtained from a single snapshot of data. This implies consistency with the recently derived local versions of the Kolmogorov laws, which do not require long-time averages.

Kolmogorov's third hypothesis and turbulent sign statistics.

The breakdown of turbulent eddies can be characterized by sets of "multipliers," defined as ratios of velocity increments at successively smaller scales. These quantities were introduced by Kolmogorov, who hypothesized their self-similar statistics and independence at distant scales. Here we report experimental and numerical results on the statistics of these multipliers, for both their magnitude and sign. We show that the multipliers at adjacent scales are not independent but that their correlations decay rapidly in scale separation. New scaling laws are thereby predicted and verified for both roughness and sign of turbulent velocity increments. The sign oscillations per cascade step are found to decrease at points of increasing roughness or singularity of the velocity.