An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence

Abstract

We propose an alternative formulation for the exact relations in three-dimensional homogeneous turbulence using two-point statistics. Our finding is illustrated with incompressible hydrodynamic, standard and Hall magnetohydrodynamic turbulence. In this formulation, the cascade rate of an inviscid invariant of turbulence can be expressed simply in terms of mixed second-order structure functions. Besides the usual variables like the velocity u, vorticity ω, magnetic field b and the current j, the vectors u × ω, u × b and j × b are also found to play a key role in the turbulent cascades. The current methodology offers a simple algebraic form which is specially interesting to study anisotropic space plasmas like the solar wind, with in principle a faster statistical convergence than the classical laws written in terms of third-order correlators.

Magnetohydrodynamic and kinetic scale turbulence in the near-Earth space plasmas: a (short) biased review

“…This is because these laws allow one to better understand the turbulence dynamics, to evidence turbulence inertial range in numerical and experimental data and to estimate the energy cascade rate in turbulent flows. The original idea developed by Kolmogorov (1941) for Navier-Stokes equations was generalized to plasmas described within various theoretical frameworks: incompressible MHD (IMHD) (Politano and Pouquet 1998) (hereafter PP98), incompressible Hall-MHD (IHMHD) (Galtier 2006b;Banerjee and Galtier 2017;Hellinger et al 2018;Ferrand et al 2019), compressible MHD (CMHD) (Banerjee and Galtier 2013) (hereafter BG13) (see also the new derivation of the same law using the classical variables , and instead of the Elsässer variables in Andrés and Sahraoui (2017)), compressible Hall-MHD (Andrés et al 2018a) and incompressible twofluid model (Andrés et al 2016). The PP98 model has been widely applied to in situ measurements in the SW to estimate the amount of the energy that is cascaded from large-to-small scale where it is expected to be dissipated into plasma heating (Smith et al 2006;Sorriso-Valvo et al 2007;MacBride et al 2008;Marino et al 2008;Carbone et al 2009;Stawarz et al 2009;Coburn et al 2014;Banerjee et al 2016).…”

Statistics of incompressible hydrodynamic turbulence: An alternative approach

“…Usually, the mean flux term F ≡ |δu| 2 δu along is identified as the flux of kinetic energy through scales. It is worth mentioning that in the new alternative derivation to compute the energy cascade rate from Banerjee and Galtier [19] (see Sec. II C), there is no projection along the increment direction and the expression only depends in the two-point mixed structure functions.…”

“…Following Banerjee and Galtier [19], here we give an schematic description of the derivation of the alternative exact relation for fully developed homogeneous and incompressible turbulence. The alternative Navier-Stokes Eq.…”

Determination of energy flux rate in homogeneous ferrohydrodynamic turbulence using two-point statistics

“…Hence energy flux rate of which can be used in correlation function. The two-point symmetric correlation function of energy [17] can be written as:…”

“…Following [17,18], in this paper, we derive an exact relation corresponding to the total energy conservation in the so-called inertial zone of incompressible ferrofluid turbulence.…”