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 length scales. 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 interscale 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 nonlocal triads, extending the work of Eyink [Physica D 207, 91 (2005)] 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 [Phys. Fluids 9, 1728 (1966); J. Fluid Mech. 47, 525 (1971)]. The faster decay predicted by Kraichnan is argued to be the consequence of additional cancellations in the signed contributions to transfer from nonlocal triads after averaging over space. This picture is supported by data from a 5123 pseudospectral simulation of Navier–Stokes turbulence with phase-shift dealiasing.

We prove that interscale transfer of kinetic energy in compressible turbulence is dominated by local interactions. In particular, our results preclude direct transfer of kinetic energy from large-scales 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 cospectrum, 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 subgrid scale 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.

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\char22{}or, equivalently, fluxes of Els\"asser energies\char22{}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 10243 simulation of forced MHD turbulence.

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 because of 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 quasi-spectral, 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 Countercurrent have a marked downscale transfer. Nevertheless, on average an upscale transfer is a reasonable mean description of the extratropical energy scale transfer over regions of O(10³) km in size.