Dual constant-flux energy cascades to both large scales and small scales

Abstract

In this paper, we present an overview of concepts and data concerning inverse cascades of excitation towards scales larger than the forcing scale in a variety of contexts, from two-dimensional fluids and wave turbulence, to geophysical flows in the presence of rotation and stratification. We briefly discuss the role of anisotropy in the occurrence and properties of such cascades. We then show that the cascade of some invariant, for example the total energy, may be transferred through nonlinear interactions to both the small scales and the large scales, with in each case a constant flux. This is in contrast to the classical picture, and we illustrate such a dual cascade in the context of atmospheric and oceanic observations, direct numerical simulations and modeling.We also show that this dual cascade of total energy can in fact be decomposed in some cases into separate cascades of the kinetic and potential energies, provided the Froude and Rossby numbers are small enough. In all cases, the potential energy flux remains small, of the order of 10% or less relative to the kinetic energy flux. Finally, we demonstrate that, in the small-scale inertial range, approximate equipartition between potential and kinetic modes is obtained, leading to an energy ratio close to one, with strong departure at large scales due to the dominant kinetic energy inverse cascade and piling-up at the lowest spatial frequency, and at small scales due to unbalanced dissipation processes, even though the Prandtl number is equal to one.

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

“…The presence of waves affects directly turbulent flows in lessening the rate at which energy is dissipated. It has been observed in several instances (Alexakis, 2013;Campagne et al, 2016;Feraco et al, 2018;Maffioli et al, 2016;Pouquet et al, 2017), within the case of rotating stratified flows, a clear linear dependence of the measured dissipation on the control parameter, the Froude number in an intermediate regime of eddy-wave interactions Pouquet et al, 2018). On the other hand, the Rossby number controls the amount of energy going to the large scales in a simultaneous inverse cascade, whereas for MHD the control parameter is the magnitude of the large-scale imposed magnetic field.…”

“…RST is strongly anisotropic (see, e.g., Cambon & Jacquin, ; Cambon & Scott, ; Favier et al, ). In such flows, similar to the MHD case, there is also a bidirectional constant‐flux system of energy cascades to both the small scales and the large scales (Marino et al, ; Pouquet & Marino, ; Pouquet et al, ). One can summarize these results, by analogy to MHD, in the following manner.…”

“…Buoyancy Reynolds numbers vary between roughly 31 and 313 (black curve), with Reynolds numbers high enough that the small-scale fluxes are rather well developed and approximately constant. As analyzed in Pouquet and Marino (2013), Marino et al (2015), and Pouquet et al (2017), the ratio of the forward to inverse flux varies as [RoFr] −1 as long as the Rossby number is below unity and for buoyancy Reynolds numbers  B above a threshold of order 10. Note that the analysis of the forced case was also shown to be compatible with the energy flux to the small scales being proportional to Fr, when the (total) energy flux to the large scale was inversely proportional to the Rossby number Ro: at fixed Froude number, the stronger the rotation, the more the energy flows to the large scales.…”

A Century of Nonlinearity in the Geosciences

“…Helicity dynamics and bidirectional cascades are discussed in this issue by Pouquet et al (). A particularly important application is to astrophysics in general and to the solar wind in particular (e.g., Pouquet et al, ).…”

Introduction to Focus Issue: Two-Dimensional Turbulence

“…Another interesting insight into transitions between 2D and 3D turbulence is presented in the review by Biferale, Buzzicotti, and Linkmann, 4 where a synthetic two-dimensional-threecomponent turbulence is considered. The subject of a split energy cascade presented by Musacchio and Boffetta 3 is to be compared with the dual constant-flux energy cascade in the review by Pouquet et al 5 The latter is a broad review where dual cascades are presented in the context of atmospheric and oceanic observations, direct numerical simulations, and modeling. New data are presented for rotated stratified turbulence, and the subject is further explored in the review of Oks et al 6 Here the focus is on the non-monotonic dependence of the strength of the inverse cascade on the relative role of rotation versus stratification and suppression of resonant wave triads as a mechanism for that.…”