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.

This work presents a rigorous framework based on coarse-graining to analyze highly compressible turbulence. We show how the requirement that viscous effects on the dynamics of large-scale momentum and kinetic energy be negligible -an inviscid criterion-naturally supports a density weighted coarse-graining of the velocity field. Such a coarse-graining method is already known in the literature as Favre filtering; however its use has been primarily motivated by appealing modeling properties rather than underlying physical considerations. We also prove that kinetic energy injection can be localized to the largest scales by proper stirring, and argue that stirring with an external acceleration field rather than a body force would yield a longer inertial range in simulations. We then discuss the special case of buoyancy-driven flows subject to a spatially-uniform gravitational field. We conclude that a range of scales can exist over which the mean kinetic energy budget is dominated by inertial processes and is immune from contributions due to molecular viscosity and external stirring.

We prove that inter-scale transfer of kinetic energy in compressible turbulence is dominated by local interactions. In particular, our results preclude direct transfer of kinetic energy from largescales directly to dissipation scales, such as into shocks, in high Reynolds number turbulence as is commonly believed. Our assumptions on the scaling of structure functions are weak and enjoy compelling empirical support. Under a stronger assumption on pressure dilatation co-spectrum, we show that mean kinetic and internal energy budgets statistically decouple beyond a transitional "conversion" range. Our analysis establishes the existence of an ensuing inertial range over which mean SGS kinetic energy flux becomes constant, independent of scale. Over this inertial range, mean kinetic energy cascades locally and in a conservative fashion, despite not being an invariant.

A coarse-graining framework is implemented to analyze nonlinear processes, measure energy transfer rates and map out the energy pathways from simulated global ocean data. Traditional tools to measure the energy cascade from turbulence theory, such as spectral flux or spectral transfer rely on the assumption of statistical homogeneity, or at least a large separation between the scales of motion and the scales of statistical inhomogeneity. The coarse-graining framework allows for probing the fully nonlinear dynamics simultaneously in scale and in space, and is not restricted by those assumptions. This paper describes how the framework can be applied to ocean flows.Energy transfer between scales is not unique due to a gauge freedom. Here, it is argued that a Galilean invariant subfilter scale (SFS) flux is a suitable quantity to properly measure energy scale-transfer in the Ocean. It is shown that the SFS definition can yield answers that are qualitatively different from traditional measures that conflate spatial transport with the scale-transfer of energy. The paper presents geographic maps of the energy scale-transfer that are both local in space and allow quasispectral, or scale-by-scale, dynamics to be diagnosed. Utilizing a strongly eddying simulation of flow in the North Atlantic Ocean, it is found that an upscale energy transfer does not hold everywhere. Indeed certain regions, near the Gulf Stream and in the Equatorial Counter Current have a marked downscale transfer. Nevertheless, on average an upscale transfer is a reasonable mean description of the extra-tropical energy scale-transfer over regions of O(10 3 ) kilometers in size.

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.

We investigate the scale-locality of subgrid-scale ͑SGS͒ energy flux and interband energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments, and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial range. Interband energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of "local transfer by nonlocal triads," with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all individual wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, nonlocal triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the Ϫ4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan's abridged Lagrangian-history direct-interaction approximation and test-field model closures. We support our results with numerical data from a 512 3 pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We also discuss a rigorous counterexample of Eyink ͓Physica D 78, 222 ͑1994͔͒, which showed that nonlocal wavenumber triads may dominate in the sharp spectral flux ͑but not in the SGS energy flux for graded filters͒. We show that this mathematical counterexample fails to satisfy reasonable physical requirements for a turbulent velocity field, which are employed in our proof of scale locality. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation modeling of turbulent flows.

We introduce a novel approach to scale-decomposition of the fluid kinetic energy (or other quadratic integrals) into band-pass contributions from a series of lengthscales. Our decomposition is based on a multiscale generalization of the "Germano identity" for smooth, graded filter kernels. We employ this method to derive a budget equation that describes the transfers of turbulent kinetic energy both in space and in scale. It is shown that the inter-scale energy transfer is dominated by local triadic interactions, assuming only the scaling properties expected in a turbulent inertial-range. We derive rigorous upper bounds on the contributions of non-local triads, extending the work of Eyink (2005) 1 for low-pass filtering. We also propose a physical explanation of the differing exponents for our rigorous upper bounds and for the scaling predictions of Kraichnan (1966Kraichnan ( ,1971) 2,3 . The faster decay predicted by Kraichnan is argued to be the consequence of additional cancellations in the signed contributions to transfer from non-local triads, after averaging over space. This picture is supported by data from a 512 3 pseudospectral simulation of Navier-Stokes turbulence with phase-shift dealiasing.Recent numerical studies 4,5,6,7,8,9 have largely supported these conclusions about energy flux. All of these groups verify the individual dominance of nonlocal triadic interactions but conclude that the net energy flux from summation over triads is dominated by local interactions. In particular, the studies 7,8,9 calculated the locality function W (s) of Kraichnan 2,3 which measures the contribution to energy flux from wavenumber triads (k, q, p) with a fixed scale-ratio s = min{k, q, p}/med{k, q, p}. They verify to reasonable accuracy the prediction of Kraichnan that W (s) ∝ s 4/3 for s 1, in agreement with some earlier DNS results 14,15 . Note that this decay of nonlocal contributions in s is even faster than that proved in Eyink (2005) 1 , which gives W (s) = O(s 2/3 ) as a rigourous upper bound. The studies 7,8,9 find also no difference between graded and sharp filters, at least with bands defined for a geometric sequences of scales.Nevertheless, debate continues on this important issue. In addition to using low-pass filtering as in Eyink (2005) 1 , the numerical works 4,5,6,7,8,9 have studied the problem also by using band-pass filtering. These studies employed both sharp-spectral and graded kernels to define modal bands. The relevant nonlinear transfer functions for band-pass fields are no longer energy flux but interband energy transfers. In this setting, one group 4,5,6 has claimed to verify the importance of nonlocal triadic interactions. Their DNS results at resolutions up to 2048 3 support earlier observations of "local transfer by nonlocal triads" 10,11,12,13 . This really should be called ultralocal transfer because it is

The physical nature of compressible turbulence is of fundamental importance in a variety of astrophysical settings. We investigate the question: "At what scales does the mechanism of pressure-dilatation operate?" and present the first direct evidence that mean kinetic energy cascades conservatively beyond a transitional "conversion" scale range despite not being an invariant of the dynamics. We use high-resolution 1024 3 subsonic and transonic simulations. The key quantity we measure is the pressure-dilatation cospectrum, E PD (k), where we show that it decays at a rate faster than k −1 in wavenumber in at least the subsonic and transonic regimes. This is sufficient to imply that mean pressure-dilatation acts primarily at large scales and that kinetic and internal energy budgets statistically decouple beyond a transitional scale range. However, we observe that small-scale dynamics remains highly compressible locally in space and that the statistical decoupling in the energy budgets is unrelated to the existence of a subsonic scale range. Our results suggest that an extension of Kolmogorov's inertial-range theory to compressible turbulence is possible.