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

Abstract

We briefly review helicity dynamics, inverse and bidirectional cascades in fluid and magnetohydrodynamic (MHD) turbulence, with an emphasis on the latter. The energy of a turbulent system, an invariant in the nondissipative case, is transferred to small scales through nonlinear mode coupling. Fifty years ago, it was realized that, for a two‐dimensional fluid, energy cascades instead to larger scales and so does magnetic excitation in MHD. However, evidence obtained recently indicates that, in fact, for a range of governing parameters, there are systems for which their ideal invariants can be transferred, with constant fluxes, to both the large scales and the small scales, as for MHD or rotating stratified flows, in the latter case including quasi‐geostrophic forcing. Such bidirectional, split, cascades directly affect the rate at which mixing and dissipation occur in these flows in which nonlinear eddies interact with fast waves with anisotropic dispersion laws, due, for example, to imposed rotation, stratification, or uniform magnetic fields. The directions of cascades can be obtained in some cases through the use of phenomenological arguments, one of which we derive here following classical lines in the case of the inverse magnetic helicity cascade in electron MHD. With more highly resolved data sets stemming from large laboratory experiments, high‐performance computing, and in situ satellite observations, machine learning tools are bringing novel perspectives to turbulence research. Such algorithms help devise new explicit subgrid‐scale parameterizations, which in turn may lead to enhanced physical insight, including in the future in the case of these new bidirectional cascades.

Coupling Large Eddies and Waves in Turbulence: Case Study of Magnetic Helicity at the Ion Inertial Scale

“…Such two-signed constant fluxes have been found as well in oceanic data [126] and in numerical models of the atmosphere [127]. Similarly, bi-directional cascades were analyzed in the case of MHD turbulence both in two dimensions and in three dimensions (see the reviews in [30,32] and references therein). Further study of the role of H C and of the Reynolds number in the dynamics of Hall MHD is reserved for future work.…”

“…In particular, it can make them less efficient in the presence of a strong Hall current, as found here for P M < 0. One could also look at these questions from the slightly less-demanding problem, from a numerical stand-point, of electron MHD (or EMHD [32,107,130,132]), in which one only deals with the evolution of the magnetic induction. EMHD is the limit of Hall MHD that obtains for small velocities and large ion inertial scales, and is known to have an inverse cascade of magnetic helicity [130,133].…”

“…This illustrates and confirms the important role of the interplay between large and small scales in the dynamics of turbulent flows. arXiv:2001.11625v1 [physics.flu-dyn] 31 Jan 2020 and large scales play an essential role in estimating the efficiency of mixing in such flows [28][29][30], and it is found to vary linearly with the control parameter, namely the Froude number [31,32].…”

A Century of Nonlinearity in the Geosciences

“…Still, it does give rise to both inverse and dual cascades, which are important in GFD as well as in MHD. Helicity dynamics and bidirectional cascades are discussed in this issue by Pouquet et al (2019). A particularly important application is to astrophysics in general and to the solar wind in particular (e.g., Pouquet et al, 2017).…”

Kinetic-magnetic energy exchanges in rotating magnetohydrodynamic turbulence

“…Previous studies have shown that MHD turbulence decay is slower than the decay of non-magnetic isotropic turbulence [36][37][38][39]. For instance, in non-helical isotropic MHD turbulence the total energy decays as E(t) ∼ t −1 [40,41].…”