Gregory L. Eyink

A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence

A public database system archiving a direct numerical simulation (DNS) data set of isotropic, forced turbulence is described in this paper. The data set consists of the DNS output on 10243 spatial points and 1024 time samples spanning about one large-scale turnover time. This complete 10244 spacetime history of turbulence is accessible to users remotely through an interface that is based on the Web-services model. Users may write and execute analysis programs on their host computers, while the programs make subroutine-like calls that request desired parts of the data over the network. The users are thus able to perform numerical experiments by accessing the 27 terabytes (TB) of DNS data using regular platforms such as laptops. The architecture of the database is explained, as are some of the locally defined functions, such as differentiation and interpolation. Test calculations are performed to illustrate the usage of the system and to verify the accuracy of the methods. The database is then used to analyze a dynamical model for small-scale intermittency in turbulence. Specifically, the dynamical effects of pressure and viscous terms on the Lagrangian evolution of velocity increments are evaluated using conditional averages calculated from the DNS data in the database. It is shown that these effects differ considerably among themselves and thus require different modeling strategies in Lagrangian models of velocity increments and intermittency.

Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer

Abstract We outline a proof and give a discussion at a physical level of an assertion of Onsagers: namely, that a solution of incompressible Euler equations with Holder continuous velocity of order h > 1 3 conserves kinetic energy, but not necessarily if h ≤ 1 3 . We prove the result under a “∗-Holder condition” which is somewhat stronger than usual Holder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for a ∗-Holder solution with exponenth in the range 0

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a “thermostat” constructed according to Gauss’ principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Youngs expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohms transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.

The joint cascade of energy and helicity in three-dimensional turbulence

Three-dimensional (3D) turbulence has both energy and helicity as inviscid constants of motion. In contrast to two-dimensional (2D) turbulence, where a second inviscid invariant—the enstrophy—blocks the energy cascade to small scales, in 3D there is a joint cascade of both energy and helicity simultaneously to small scales. It has long been recognized that the crucial difference between 2D and 3D is that enstrophy is a nonnegative quantity whereas the helicity can have either sign. The basic cancellation mechanism which permits a joint cascade of energy and helicity is illuminated by means of the helical decomposition of the velocity into positively and negatively polarized waves. This decomposition is employed in the present study both theoretically and also in a numerical simulation of homogeneous and isotropic 3D turbulence. It is shown that the transfer of energy to small scales produces a tremendous growth of helicity separately in the + and − helical modes at high wave numbers, diverging in the limi...

Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence

We study Onsagers theory of large, coherent vortices in turbulent flows in the approximation of the point-vortex model for two-dimensional Euler hydrodynamics. In the limit of a large number of point vortices with the energy perpair of vortices held fixed, we prove that the entropy defined from the microcanonical distribution as a function of the (pair-specific) energy has its maximum at a finite value and thereafter decreases, yielding the negative-temperature states predicted by Onsager. We furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy, and, under regularity assumptions, obeys the Joyce-Montgomery mean-field equation. We also prove that, under appropriate conditions, the vorticity distribution is the same as that for the canonical distribution, a form of equivalence of ensembles. We establish a large-fluctuation theory for the microcanonical distributions, which is based on a level-3 large-deviations theory for exchangeable distributions. We discuss some implications of that property for the ergodicity requirements to justify Onsagers theory, and also the theoretical foundations of a recent extension to continuous vorticity fields by R. Robert and J. Miller. Although the theory of two-dimensional vortices is of primary interest, our proofs actually apply to a very general class of mean-field models with long-range interactions in arbitrary dimensions.

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.

The renormalization group method in statistical hydrodynamics

This paper gives a first principles formulation of a renormalization group (RG) method appropriate to study of turbulence in incompressible fluids governed by Navier–Stokes equations. The present method is a momentum‐shell RG of Kadanoff–Wilson type based upon the Martin–Siggia–Rose (MSR) field‐theory formulation of stochastic dynamics. A simple set of diagrammatic rules are developed which are exact within perturbation theory (unlike the well‐known Ma–Mazenko prescriptions). It is also shown that the claim of Yakhot and Orszag (1986) is false that higher‐order terms are irrelevant in the e expansion RG for randomly forced Navier–Stokes (RFNS) with power‐law force spectrum F(k)=D0k−d+(4−e). In fact, as a consequence of Galilei covariance, there are an infinite number of higher‐order nonlinear terms marginal by power counting in the RG analysis of the power‐law RFNS, even when e≪4. The difficulty does not occur in the Forster–Nelson–Stephen (FNS) RG analysis of thermal fluctuations in an equilibrium NS fl...

Exact results on stationary turbulence in 2D: consequences of vorticity conservation

We establish a series of exact results for a model of stationary turbulence in two dimensions: the incompressible Navier-Stokes equation in a periodic domain with stochastic force white-noise in time. Essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics. Our main results are as follows: (1) we show the blow-up of mean energy as ∼l02eν for ν→0 when there is no IR-dissipation at the large length-scale l0; (2) with an additional IR-dissipation, we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for ν→0, assuming finite mean energy in the limit; (3) we rigorously establish the balance equations for the energy and vorticity invariants in the 2D steady-state and the forward cascade of the higher-order vorticity invariants assuming finite mean values; (4) we derive exact inequalities for scaling exponents in the 2D enstrophy range, as follows: if 〈|Δlω|p〉 ∼ lζp, then ζ2 ≤ 23 and ζp ≤ 0 for p ≥ 3. If the minimum Holder exponent of the vorticity hmin < 0, then we establish a 2D analogue of the refined similarity hypothesis which improves these bounds. The most novel and interesting conclusion of this work is the connection established between “intermittency” in 2D and “negative Holder singularities” of the vorticity: we show that the latter are necessary for deviations from the 1967 Kraichnan scaling to occur.

Resonant interactions in rotating homogeneous three-dimensional turbulence

Direct numerical simulations of three-dimensional homogeneous turbulence under rapid rigid rotation are conducted for a fixed large Reynolds number and a sequence of decreasing Rossby numbers to examine the predictions of resonant wave theory. The theory states that ‘slow modes’ of the velocity, with zero wavenumber parallel to the rotation axis (kz = 0), will decouple at first order from the remaining ‘fast modes’ and solve an autonomous system of two-dimensional Navier–Stokes equations for the horizontal velocity components, normal to the rotation axis, and a two-dimensional passive scalar equation for the vertical velocity component, parallel to the rotation axis. The Navier–Stokes equation for three-dimensional rotating turbulence is solved in a 128 3 mesh after being diagonalized via ‘helical decomposition’ into normal modes of the Coriolis term. A force supplies constant energy input at intermediate scales. To verify the theory, we set up a corresponding simulation for the two-dimensional Navier–Stokes equation and two-dimensional passive scalar equation to compare them with the slow-mode dynamics of the three-dimensional rotating turbulence. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by two-dimensional Navier–Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically averaged horizontal velocity field from three-dimensional Navier–Stokes converges to the velocity field from two-dimensional Navier–Stokes, as measured by the energy in their difference field. Likewise, the vertically averaged vertical velocity from three-dimensional Navier–Stokes converges to a solution of the two-dimensional passive scalar equation. The slow-mode energy spectrum approaches k −5/3 h , where kh is the horizontal wavenumber, and, as in two dimensions, energy flux becomes closer to constant the greater the rotation rate. Furthermore, the energy flux directly into small wavenumbers in the kz = 0 plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. The simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.

Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Ficks law.