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A compact exact law for compressible isothermal Hall magnetohydrodynamic turbulence

Published online by Cambridge University Press:  19 April 2021

Renaud FerrandOpen the ORCID record for Renaud Ferrand[Opens in a new window],
Sébastien GaltierOpen the ORCID record for Sébastien Galtier[Opens in a new window] and
Fouad Sahraoui
Renaud Ferrand*
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128 Palaiseau Cedex, France
Sébastien Galtier
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128 Palaiseau Cedex, FranceInstitut Universitaire de France, 75005 Paris, France
Fouad Sahraoui
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128 Palaiseau Cedex, France
*
†Email address for correspondence: renaud.ferrand@lpp.polytechnique.fr
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Abstract

Using mixed second-order structure functions, a compact exact law is derived for isothermal compressible Hall magnetohydrodynamic turbulence with the assumptions of statistical homogeneity, time stationarity and infinite kinetic/magnetic Reynolds numbers. The resulting law is written as the sum of a Yaglom-like flux term, with an overall expression strongly reminiscent of the incompressible law, and a pure compressible source. Being mainly a function of the increments, the compact law is Galilean invariant but is dependent on the background magnetic field if one is present. Only the magnetohydrodynamic source term requires multi-spacecraft data to be estimated whereas the other components, which include those introduced by the Hall term, can be fully computed with single-spacecraft data using the Taylor hypothesis. These properties make this compact law more appropriate for analysing both numerical simulations and in situ data gathered in space plasmas, in particular when only single-spacecraft data are available.

Keywords

plasma nonlinear phenomenaspace plasma physics

Information

Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021, 905870220
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

1. Introduction

Although having already been studied extensively in several areas of physics, turbulent flows are not yet fully understood because the nonlinear nature of the problem makes it difficult to derive fundamental laws. Most of these laws take the form of exact relations, mathematical expressions that relate the energy cascade rate to a third-order correlation function of the fields under the assumptions of statistical homogeneity, stationarity and high Reynolds numbers. A well-known result is the four-fifths law derived using a tensorial method by Kolmogorov (Reference Kolmogorov1941) for incompressible hydrodynamic turbulence, and the four-thirds law derived in a modern way through the use of a two-point correlation function only (Monin Reference Monin1959; Antonia et al. Reference Antonia, Ould-Rouis, Anselmet and Zhu1997). Following this simpler derivation, other fundamental laws were derived for plasma models such as incompressible magnetohydrodynamics (IMHD) (Politano & Pouquet Reference Politano and Pouquet1998), which allowed for the study of astrophysical turbulent plasmas such as the solar wind (Sorriso-Valvo et al. Reference Sorriso-Valvo, Marino, Carbone, Noullez, Lepreti, Veltri, Bruno, Bavassano and Pietropaolo2007; MacBride, Smith & Forman Reference MacBride, Smith and Forman2008; Marino et al. Reference Marino, Sorriso-Valvo, Carbone, Noullez, Bruno and Bavassano2008; Stawarz et al. Reference Stawarz, Smith, Vasquez, Forman and MacBride2009; Osman et al. Reference Osman, Wan, Matthaeus, Weygand and Dasso2011; Galtier Reference Galtier2012). This plasma model was then refined by adding the Hall effect (IHMHD) (Galtier Reference Galtier2008; Banerjee & Galtier Reference Banerjee and Galtier2017; Hellinger et al. Reference Hellinger, Verdini, Landi, Franci and Matteini2018; Ferrand et al. Reference Ferrand, Galtier, Sahraoui, Meyrand, Andrés and Banerjee2019) as a way of describing near and sub-ion scales, which are not described by ideal MHD, and allow for multi-scale studies of space plasmas (Bandyopadhyay et al. Reference Bandyopadhyay, Sorriso-Valvo, Chasapis, Hellinger, Matthaeus, Verdini, Landi, Franci, Matteini and Giles2020).

In addition to the incompressible models, exact laws have also been derived that take compressibility into account both for non-magnetized fluids (Galtier & Banerjee Reference Galtier and Banerjee2011; Banerjee & Galtier Reference Banerjee and Galtier2014; Banerjee & Kritsuk Reference Banerjee and Kritsuk2017; Ferrand et al. Reference Ferrand, Galtier, Sahraoui and Federrath2020) and magnetized plasmas (Banerjee & Galtier Reference Banerjee and Galtier2013; Andrés & Sahraoui Reference Andrés and Sahraoui2017; Andrés, Galtier & Sahraoui Reference Andrés, Galtier and Sahraoui2018; Banerjee & Kritsuk Reference Banerjee and Kritsuk2018). The use of a compressible description of astrophysical flows is relevant for the solar wind (Banerjee et al. Reference Banerjee, Hadid, Sahraoui and Galtier2016; Hadid, Sahraoui & Galtier Reference Hadid, Sahraoui and Galtier2017) but also for highly compressible media, such as the interstellar medium, where supersonic turbulence is expected to play a key role in star-forming structures (Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Arzoumanian et al. Reference Arzoumanian, André, Didelon, Könyves, Schneider, Men´shchikov, Sousbie, Zavagno, Bontemps and Di Francesco2011; Kritsuk, Wagner & Norman Reference Kritsuk, Wagner and Norman2013; Federrath Reference Federrath2016). It is also relevant to study the sub-ion scales of a subsonic turbulent plasma. Indeed, while the influence of compressibility at MHD scales has already been demonstrated in turbulent flows of the magnetosheath of the Earth (Hadid et al. Reference Hadid, Sahraoui, Galtier and Huang2018), it was recently shown that small density fluctuations can significantly enhance the compressible cascade rate, with respect to its incompressible estimate at sub-ion scales, even when the two cascade rates coincide at the large (MHD) scales (Andrés et al. Reference Andrés, Sahraoui, Galtier, Hadid, Ferrand and Huang2019).

For a given model, the expression of the law is not unique. This is the case with incompressible hydrodynamics, where both the four-fifths and the four-thirds laws exist and were derived by different methods. Likewise, there is a tensorial derivation in IMHD that yields a four-fifths-like law (Yoshimatsu Reference Yoshimatsu2012) that is different from that derived by Politano & Pouquet (Reference Politano and Pouquet1998), and four different laws have been derived in IHMHD (Ferrand et al. Reference Ferrand, Galtier, Sahraoui, Meyrand, Andrés and Banerjee2019). In particular, it was shown in the latter study that using the appropriate structure function allows one to derive a simplified expression of the exact law. Owing to the size and complexity of the law already available for the compressible isothermal Hall MHD (CHMHD) model (Andrés et al. Reference Andrés, Galtier and Sahraoui2018), which will hereinafter be called the A18 law, we aim to apply a similar method to that used by Ferrand et al. (Reference Ferrand, Galtier, Sahraoui, Meyrand, Andrés and Banerjee2019) (hereafter F19) to derive a CHMHD law with an expression that is significantly simpler. Deriving such a compact expression could ease the subsequent in situ data analysis by simplifying the computation and obtaining a more precise estimate of the heating rate.

In § 2, we present our derivation based on the three-dimensional isothermal CHMHD equations. We then discuss the new exact law in detail in § 3 and compare it with the previous law and its incompressible counterpart. We finally provide the conclusion of the study in § 4.

2. Derivation of the compact CHMHD law

2.1. Compressible Hall MHD equations

Hall MHD is a model often used in astrophysics in the context of cold and dense regions of protoplanetary discs (Kunz & Lesur Reference Kunz and Lesur2013), crusts of neutron stars (Gourgouliatos & Cumming Reference Gourgouliatos and Cumming2014), fast magnetic reconnection (Bhattacharjee Reference Bhattacharjee2004), high-frequency waves and turbulence in magnetospheric physics (Belmont & Rezeau Reference Belmont and Rezeau2001; Sahraoui et al. Reference Sahraoui, Pincon, Belmont, Rezeau, Cornilleau-Wehrlin, Robert, Mellul, Bosqued, Balogh and Canu2003; Sahraoui, Galtier & Belmont Reference Sahraoui, Galtier and Belmont2007) and in the solar wind (Galtier Reference Galtier2006), and laboratory experiments such as the Madison Plasma Dynamo (Cooper et al. Reference Cooper, Wallace, Brookhart, Clark, Collins, Ding, Flanagan, Khalzov, Li and Milhone2014) or the Wisconsin Plasma Astrophysics Laboratory (Forest et al. Reference Forest, Flanagan, Brookhart, Clark, Cooper, Désangles, Egedal, Endrizzi, Khalzov and Li2015) experiments. In the context of weakly collisional plasmas, the limits of the validity of Hall MHD have been discussed in the literature (e.g. Howes Reference Howes2009).

In this paper, we shall consider three-dimensional compressible isothermal Hall MHD, for which the equations are (Galtier Reference Galtier2016)

(2.1)\begin{gather} \partial_t \rho + \boldsymbol\nabla \boldsymbol{\cdot} (\rho \boldsymbol{u}) = 0, \end{gather}
(2.2)\begin{gather}\rho ( \partial_t \boldsymbol{u} + \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol\nabla\boldsymbol{u} ) ={-} \boldsymbol\nabla P+ \boldsymbol{J} \times\boldsymbol{B} + {\boldsymbol d_{\nu}} + {\boldsymbol f} , \end{gather}
(2.3)\begin{gather}\partial_t \boldsymbol{B} = \boldsymbol\nabla \times (\boldsymbol{u} \times \boldsymbol{B}) - \lambda \boldsymbol\nabla \times (\boldsymbol{J}_c \times \boldsymbol{B}) + {\boldsymbol d_{\eta}}, \end{gather}
(2.4)\begin{gather}\boldsymbol\nabla \boldsymbol{\cdot} \boldsymbol{B} = 0, \end{gather}

where $\rho$ is the mass density, $\boldsymbol {u}$ is the velocity, $P$ is the pressure, $\boldsymbol {B}$ is the magnetic field, $\boldsymbol {J}=(\boldsymbol \nabla \times \boldsymbol {B})/\mu _{0}$ is the current density, ${\boldsymbol f}$ is a stationary homogeneous external force acting at large scales, $\boldsymbol {J}_c=\boldsymbol {J}/\rho$ is the normalized current, and $\lambda =m_i/q_e$ with $m_{i}$ is the ion mass and $q_e$ is the magnitude of the electron charge. The dissipation terms are

(2.5)\begin{gather} {\boldsymbol d_{\nu}} = \nu \varDelta \boldsymbol{u} + \frac{\nu}{3} \boldsymbol\nabla \theta, \end{gather}
(2.6)\begin{gather}{\boldsymbol d_{\eta}} = \eta \varDelta \boldsymbol{B}, \end{gather}

with $\theta = \boldsymbol \nabla \boldsymbol {\cdot } \boldsymbol {u}$ is the dilatation, $\nu$ is the dynamic viscosity and $\eta$ is the magnetic diffusivity. We use the isothermal closure $P= c_s^2 \rho$, where $c_{s}$ is the speed of sound constant.

We introduce the spatial increment $\boldsymbol \ell$ connecting two points $\boldsymbol {x}$ and $\boldsymbol {x}'$ in the physical space as $\boldsymbol {x}'=\boldsymbol {x}+\boldsymbol \ell$ and we define, for any given field $\xi$, $\xi = \xi (\boldsymbol {x})$ and $\xi ' = \xi (\boldsymbol {x}')$. We also define the field increment $\delta \xi = \xi '-\xi$ and the mean value $\bar {\delta } \xi = (\xi + \xi ')/2$. For this system, the equation of energy conservation reads as

(2.7)\begin{equation} \partial_{t} \langle E^{\textrm{tot}} \rangle = \langle \boldsymbol{u} \cdot {\boldsymbol d_{\nu}} \rangle + \frac{1}{\mu_0} \langle \boldsymbol{B} \cdot {\boldsymbol d_{\eta}} \rangle + \langle \boldsymbol{u} \cdot {\boldsymbol f} \rangle, \end{equation}

where $\langle \rangle$ is an ensemble average, $E^{\textrm {tot}} = \rho u^{2}/2 + B^{2}/(2\mu _{0}) + \rho e$ is the total energy and $e= c_{s}^{2} \ln (\rho /\rho _{0})$ is the work component of internal energy ($\rho _{0}=\langle \rho \rangle$ is the average density). Note that this definition limits the variations of the internal energy to the work done by the pressure force, the heat component owing to entropy variation being neglected (like in the case of incompressible turbulence). Thus, the total energy is conserved. We define the mean rate of total energy injection as $\varepsilon = \langle \boldsymbol {u} \cdot {\boldsymbol f} \rangle$. By using this quantity, (2.7) reduces in the stationary regime to

(2.8)\begin{equation} \left\langle \boldsymbol{u} \cdot {\boldsymbol d_{\nu}} + \frac{1}{\mu_0} \boldsymbol{B} \cdot {\boldsymbol d_{\eta}} \right\rangle={-}\varepsilon . \end{equation}

Note that we also have the relation

(2.9)\begin{equation} \langle \boldsymbol{u} \cdot {\boldsymbol d_{\nu}} \rangle + \frac{1}{\mu_0} \langle \boldsymbol{B} \cdot {\boldsymbol d_{\eta}} \rangle ={-} \nu \left\langle {\boldsymbol w}^2 + \frac{4}{3} \theta^2 \right\rangle - \mu_{0} \eta \left\langle \boldsymbol{J}^2 \right\rangle, \end{equation}

where ${\boldsymbol w} = \boldsymbol {\nabla } \times \boldsymbol {u}$ is the vorticity, which gives the expression of the mean rate of total energy dissipation.

2.2. Two-point correlators

Next, we shall derive analytically a dynamical equation for the following mixed structure functions

(2.10)\begin{equation} \langle S \rangle = \left\langle \frac{1}{2} \bar{\delta} \rho |\delta \boldsymbol{u}|^2+ \frac{1}{2\mu_0} |\delta \boldsymbol{B}|^2 + \frac{1}{2} \delta \rho \delta e \right\rangle. \end{equation}

First, we note that for homogeneous turbulence, we have the relations

(2.11)\begin{gather} \langle \bar \delta \rho \vert \delta \boldsymbol{u} \vert^2 \rangle = \langle \rho u^2 \rangle - 2 \langle \bar \delta \rho \boldsymbol{u} \cdot \boldsymbol{u}' \rangle + \tfrac{1}{2} \langle \rho u'^2 + \rho' u^2 \rangle, \end{gather}
(2.12)\begin{gather}\langle \delta \rho \delta e \rangle = 2 \langle \rho e \rangle - \langle \rho e' + \rho' e \rangle, \end{gather}
(2.13)\begin{gather}\langle \vert \delta \boldsymbol{B} \vert^2 \rangle = 2 \langle B^{2} \rangle - 2 \langle \boldsymbol{B} \cdot \boldsymbol{B}' \rangle. \end{gather}

We recall that, under our formalism, we have for any entity A the relation $\partial _x A' = \partial _{x'} A = 0$. We also have the relation $\langle \boldsymbol \nabla '\boldsymbol {\cdot }\rangle =-\langle \boldsymbol \nabla \boldsymbol {\cdot }\rangle =\boldsymbol \nabla _{\boldsymbol \ell } \boldsymbol {\cdot }\langle \rangle$, where $\boldsymbol \nabla _{\boldsymbol \ell }$ denotes the derivative operator along the increment $\boldsymbol \ell$. Using these relations, we obtain from (2.1)–(2.4) the following expressions

(2.14)\begin{align} \partial_t \langle \rho \boldsymbol{u} \cdot \boldsymbol{u}' \rangle &= \left\langle \rho \boldsymbol{u} \cdot \left( - \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' \boldsymbol{u}' - \frac{1}{\rho'} \boldsymbol\nabla' P' + \boldsymbol{J}_c' \times \boldsymbol{B}' \right) \right\rangle \nonumber\\ &\quad + \langle \boldsymbol{u}' \cdot (- \boldsymbol\nabla \boldsymbol{\cdot} ( \rho \boldsymbol{u} \boldsymbol{u} ) - \boldsymbol\nabla P + \boldsymbol{J} \times \boldsymbol{B}) \rangle + \left\langle \boldsymbol{u}' \cdot {\boldsymbol d_\nu} + \boldsymbol{u}' \cdot {\boldsymbol f} + \frac{\rho}{\rho'} \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &= \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - \rho (\boldsymbol{u} \cdot \boldsymbol{u}') \delta \boldsymbol{u} + P \boldsymbol{u}' - \rho e' \boldsymbol{u} \rangle + \langle \rho \theta' (\boldsymbol{u} \cdot \boldsymbol{u}') \rangle \nonumber\\ &\quad + \langle \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') + \boldsymbol{u}' \cdot (\boldsymbol{J} \times \boldsymbol{B}) \rangle \nonumber\\ &\quad + \left\langle \boldsymbol{u}' \cdot {\boldsymbol d_\nu} + \boldsymbol{u}' \cdot {\boldsymbol f} + \frac{\rho}{\rho'} \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle, \end{align}
(2.15)\begin{align} \partial_t \langle \rho u'^2 \rangle &= \langle 2 \rho \boldsymbol{u}' \cdot \partial_t \boldsymbol{u}' + u'^2 \partial_t \rho \rangle \nonumber\\ &= \left\langle - \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' (\rho u'^2) - 2 \frac{\rho}{\rho'} \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' P' + 2 \frac{\rho}{\rho'} \boldsymbol{u}' \cdot (\boldsymbol{J}' \times \boldsymbol{B}')\right\rangle \nonumber\\ &\quad + \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle \rho u'^2 \boldsymbol{u} \rangle + \left\langle 2 \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &= \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle -\rho u'^2 \boldsymbol{u}' + \rho u'^2 \boldsymbol{u} \rangle + \left\langle \rho u'^2 \theta' -2\frac{\rho}{\rho'} \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' P' + 2 \frac{\rho}{\rho'} \boldsymbol{u}' \cdot (\boldsymbol{J}' \times \boldsymbol{B}') \right\rangle \nonumber\\ &\quad + \left\langle 2 \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle, \end{align}
(2.16)\begin{align} \partial_t \langle \rho e' \rangle &= \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - \rho e' \delta \boldsymbol{u} - P \boldsymbol{u}' \rangle + \langle \rho e' \theta' \rangle, \end{align}
(2.17)\begin{align} \partial_t \langle \boldsymbol{B} \cdot \boldsymbol{B}' \rangle &= \langle \boldsymbol{B} \cdot (\boldsymbol{B}' \boldsymbol{\cdot} \boldsymbol\nabla' \boldsymbol{u}' - \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' \boldsymbol{B}' - \boldsymbol{B}' \theta') \rangle+ \langle \boldsymbol{B}' \cdot (\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol\nabla \boldsymbol{u} - \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol\nabla \boldsymbol{B} - \boldsymbol{B} \theta) \rangle \nonumber\\ &\quad - \lambda \langle \boldsymbol{B}' \boldsymbol{\cdot} ( \boldsymbol\nabla \times (\boldsymbol{J}_c \times \boldsymbol{B})) + \boldsymbol{B} \boldsymbol{\cdot} ( \boldsymbol\nabla' \times (\boldsymbol{J}_c' \times \boldsymbol{B}')) \rangle + \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle \nonumber\\ &= \langle \boldsymbol\nabla' \boldsymbol{\cdot} ((\boldsymbol{B} \cdot \boldsymbol{u}') \boldsymbol{B}') - \boldsymbol\nabla' \boldsymbol{\cdot} ((\boldsymbol{B} \cdot \boldsymbol{B}') \boldsymbol{u}') + \boldsymbol\nabla \boldsymbol{\cdot} ((\boldsymbol{B}' \cdot \boldsymbol{u}) \boldsymbol{B}) - \boldsymbol\nabla \boldsymbol{\cdot} ((\boldsymbol{B} \cdot \boldsymbol{B}') \boldsymbol{u}) \rangle \nonumber\\ &\quad - \lambda \langle \boldsymbol\nabla \boldsymbol{\cdot} ((\boldsymbol{J}_c \cdot \boldsymbol{B}') \boldsymbol{B}) - (\boldsymbol{J}_c \boldsymbol{\cdot} \boldsymbol\nabla) (\boldsymbol{B} \cdot \boldsymbol{B}') - (\boldsymbol{B} \cdot \boldsymbol{B}') \boldsymbol\nabla \boldsymbol{\cdot} \boldsymbol{J}_c \rangle \nonumber\\ &\quad - \lambda \langle \boldsymbol\nabla' \boldsymbol{\cdot} ((\boldsymbol{J}_c' \cdot \boldsymbol{B}) \boldsymbol{B}') - (\boldsymbol{J}_c' \boldsymbol{\cdot} \boldsymbol\nabla') (\boldsymbol{B} \cdot \boldsymbol{B}') - (\boldsymbol{B} \cdot \boldsymbol{B}') \boldsymbol\nabla' \boldsymbol{\cdot} \boldsymbol{J}_c' \rangle \nonumber\\ &\quad + \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle \nonumber\\ &= \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{u} + (\boldsymbol{B} \cdot \boldsymbol{u}') \boldsymbol{B}' - (\boldsymbol{B}' \cdot \boldsymbol{u}) \boldsymbol{B} \rangle \nonumber\\ &\quad - \lambda \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{J}_c + (\boldsymbol{J}_c' \cdot \boldsymbol{B}) \boldsymbol{B}' - (\boldsymbol{J}_c \cdot \boldsymbol{B}') \boldsymbol{B} \rangle \nonumber\\ &\quad + \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle. \end{align}

From these relations, we find

(2.18)\begin{align} \partial_t \langle S \rangle &= \partial_t \langle E^{\textrm{tot}} \rangle + \partial_t \langle E^{b} \rangle \nonumber\\ &\quad - \frac{1}{2} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle - 2\bar{\delta} \rho (\boldsymbol{u} \cdot \boldsymbol{u}') \delta \boldsymbol{u} - \rho e' \boldsymbol{u}' + \rho' e \boldsymbol{u}+ \frac{1}{2} \rho u'^2 \boldsymbol{u}' \right. \nonumber\\ &\quad \left.- \frac{1}{2}\rho u'^2 \boldsymbol{u} - \frac{1}{2} \rho' u^2 \boldsymbol{u} + \frac{1}{2} \rho' u^2 \boldsymbol{u}' \right \rangle \nonumber\\ &\quad -\frac{1}{2} \langle \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') + \rho' \boldsymbol{u}' \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) + \boldsymbol{u}' \cdot (\boldsymbol{J} \times \boldsymbol{B}) + \boldsymbol{u} \cdot (\boldsymbol{J}' \times \boldsymbol{B}') \rangle \nonumber\\ &\quad +\frac{1}{4} \left\langle \rho u'^2 \theta' +\rho' u^2 \theta - 2\frac{\rho}{\rho'} \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' P' - 2\frac{\rho'}{\rho} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol\nabla P \right. \nonumber\\ &\quad + \left. 2 \frac{\rho}{\rho'} \boldsymbol{u}' \cdot (\boldsymbol{J}' \times \boldsymbol{B}') + 2 \frac{\rho'}{\rho} \boldsymbol{u} \cdot (\boldsymbol{J} \times \boldsymbol{B}) \right\rangle \nonumber\\ &\quad -\frac{1}{2} \langle (\rho \theta' +\rho' \theta) (\boldsymbol{u} \cdot \boldsymbol{u}') \rangle - \frac{1}{2} \langle \rho e' \theta' + \rho' e \theta \rangle \nonumber\\ &\quad -\frac{1}{\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{u} + (\boldsymbol{B} \cdot \boldsymbol{u}') \boldsymbol{B}' - (\boldsymbol{B}' \cdot \boldsymbol{u}) \boldsymbol{B} \rangle \nonumber\\ &\quad -\frac{\lambda}{\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{J}_c - \lambda (\boldsymbol{J}_c' \cdot \boldsymbol{B}) \boldsymbol{B}' + \lambda (\boldsymbol{J}_c \cdot \boldsymbol{B}') \boldsymbol{B} \rangle \nonumber\\ &\quad -\frac{1}{2} \left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &\quad +\frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) + \frac{\rho'}{\rho} \boldsymbol{u} \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) \right\rangle - \frac{1}{\mu_0} \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle. \end{align}

We can simplify the previous expression by stating that

(2.19)\begin{gather} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle -2\bar \delta \rho \boldsymbol{u} \cdot \boldsymbol{u}' \delta \boldsymbol{u} + \tfrac{1}{2} \rho u'^2 \boldsymbol{u}' - \tfrac{1}{2} \rho u'^2 \boldsymbol{u} + \tfrac{1}{2} \rho' u^2 \boldsymbol{u}' - \tfrac{1}{2} \rho' u^2 \boldsymbol{u} \right\rangle \nonumber\\ \quad + \left\langle \tfrac{1}{2} \rho u^2 \theta' + \tfrac{1}{2} \rho' u'^2 \theta \right\rangle, \end{gather}
(2.20)\begin{gather} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' P' \right\rangle + \left\langle \frac{\rho'}{\rho} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} P \right\rangle = \langle \rho \boldsymbol{u}' \boldsymbol{\cdot} \boldsymbol\nabla' e' + \rho' \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol\nabla e \rangle \nonumber\\ = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle \rho e' \boldsymbol{u}' - \rho' e \boldsymbol{u} \rangle - \langle \rho e' \theta' + \rho' e \theta \rangle, \end{gather}
(2.21)\begin{gather} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - (\boldsymbol{u}' \cdot \boldsymbol{B}) \boldsymbol{B}' + (\boldsymbol{u} \cdot \boldsymbol{B}') \boldsymbol{B} - (\boldsymbol{u} \cdot \boldsymbol{B}') \boldsymbol{B}'+ (\boldsymbol{u}' \cdot \boldsymbol{B}) \boldsymbol{B} \rangle, \end{gather}
(2.22)\begin{gather}\boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle -2 (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{u} + B^2 \boldsymbol{u}' - B'^2 \boldsymbol{u} \rangle, \end{gather}
(2.23)\begin{gather}\langle (\rho \theta' + \rho' \theta) \vert\delta \boldsymbol{u} \vert^2 \rangle = \langle -2 (\rho \theta' + \rho' \theta) (\boldsymbol{u} \cdot \boldsymbol{u}') + \rho \theta' u'^2 + \rho \theta' u^2 + \rho' \theta u'^2 + \rho' \theta u^2 \rangle, \end{gather}
(2.24)\begin{gather}\langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c \rangle = \langle B'^2 \delta \boldsymbol{J}_c + B^2 \delta \boldsymbol{J}_c - 2 (\boldsymbol{B} \cdot \boldsymbol{B}') \delta \boldsymbol{J}_c \rangle, \end{gather}
(2.25)\begin{gather}\boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle (\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - (\boldsymbol{B}' \cdot \boldsymbol{J}_c) \boldsymbol{B}' + (\boldsymbol{B}' \cdot \boldsymbol{J}_c)\boldsymbol{B} - (\boldsymbol{B} \cdot \boldsymbol{J}_c') \boldsymbol{B}' + (\boldsymbol{B} \cdot \boldsymbol{J}_c') \boldsymbol{B} \rangle. \end{gather}

The introduction of these expressions into (2.18) gives

(2.26)\begin{align} \partial_t \langle S \rangle &= \partial_t \langle E^{\textrm{tot}} \rangle + \partial_t \langle E^{b} \rangle - \frac{1}{2} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} \right \rangle + \frac{1}{4} \langle (\rho \theta' + \rho' \theta) \vert\delta \boldsymbol{u} \vert^2 \rangle \nonumber\\ &\quad -\frac{1}{2} \langle \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') + \rho' \boldsymbol{u}' \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) - \rho \boldsymbol{u}' \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') - \rho' \boldsymbol{u} \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) \rangle \nonumber\\ &\quad -\frac{1}{2} \langle \boldsymbol{u}' \cdot (\boldsymbol{J} \times \boldsymbol{B}) + \boldsymbol{u} \cdot (\boldsymbol{J}' \times \boldsymbol{B}') \rangle \nonumber\\ &\quad -\frac{1}{2\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} - 2 (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} - B^2 \boldsymbol{u}' + B'^2 \boldsymbol{u} - 2(\boldsymbol{u} \cdot \boldsymbol{B}') \boldsymbol{B}' + 2(\boldsymbol{u}' \cdot \boldsymbol{B}) \boldsymbol{B} \right\rangle \nonumber\\ &\quad +\frac{\lambda}{2\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c - B'^2 \delta \boldsymbol{J}_c - B^2 \delta \boldsymbol{J}_c- 2 (\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} \nonumber\\ &\quad - 2(\boldsymbol{B}' \cdot \boldsymbol{J}_c) \boldsymbol{B}' +2(\boldsymbol{B} \cdot \boldsymbol{J}_c') \boldsymbol{B} \rangle \nonumber\\ &\quad -\frac{1}{2} \left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &\quad +\frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) + \frac{\rho'}{\rho} \boldsymbol{u} \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) \right\rangle - \frac{1}{\mu_0} \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle. \end{align}

As we have, for any vector field $\boldsymbol {X}$, the relation $\boldsymbol \nabla (\boldsymbol {X}\cdot \boldsymbol {X}) = 2\boldsymbol {X}\times (\boldsymbol \nabla \times \boldsymbol {X}) + 2(\boldsymbol {X}\boldsymbol {\cdot }\boldsymbol \nabla )\boldsymbol {X}$, we find the following expressions

(2.27)\begin{gather} \langle \boldsymbol{u}' \cdot (\boldsymbol{J} \times \boldsymbol{B}) + \boldsymbol{u} \cdot (\boldsymbol{J}' \times \boldsymbol{B}') \rangle = \frac{1}{\mu_0}\langle \boldsymbol{u}' \boldsymbol{\cdot} ((\boldsymbol\nabla \times \boldsymbol{B}) \times \boldsymbol{B}) + \boldsymbol{u} \boldsymbol{\cdot} ((\boldsymbol\nabla' \times \boldsymbol{B}') \times \boldsymbol{B}') \rangle \nonumber\\ = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \frac{B^2}{2} \boldsymbol{u}' - (\boldsymbol{B} \cdot \boldsymbol{u}') \boldsymbol{B} - \frac{B'^2}{2} \boldsymbol{u} + (\boldsymbol{B}' \cdot \boldsymbol{u}) \boldsymbol{B}' \right\rangle, \end{gather}
(2.28)\begin{gather} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle (\boldsymbol{B} \cdot \boldsymbol{J}_c') \boldsymbol{B} \rangle ={-} \langle \boldsymbol{J}_c' \cdot ((\boldsymbol{B} \boldsymbol{\cdot} \boldsymbol\nabla) \boldsymbol{B}) \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \frac{B^2}{2} \boldsymbol{J}_c' \right\rangle + \mu_0 \langle \boldsymbol{J}_c' \cdot (\boldsymbol{B} \times \boldsymbol{J}) \rangle, \end{gather}
(2.29)\begin{gather}\boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle (\boldsymbol{B}' \cdot \boldsymbol{J}_c) \boldsymbol{B}' \rangle = \langle \boldsymbol{J}_c \cdot ((\boldsymbol{B}' \boldsymbol{\cdot} \boldsymbol\nabla') \boldsymbol{B}') \rangle = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \frac{B'^2}{2} \boldsymbol{J}_c \right\rangle - \mu_0 \langle \boldsymbol{J}_c \cdot (\boldsymbol{B}' \times \boldsymbol{J}') \rangle. \end{gather}

By stating that

(2.30)\begin{equation} \partial_t \langle E^{b} \rangle ={-} \left\langle \boldsymbol{J} \cdot \left( \eta \mu_0 \boldsymbol{J} - \boldsymbol{u} \times \boldsymbol{B} + \frac{1}{ne} \boldsymbol{J} \times \boldsymbol{B}\right) \right\rangle = \frac{1}{\mu_0} \langle \boldsymbol{B} \cdot {\boldsymbol d_\eta} \rangle - \langle \boldsymbol{u} \cdot (\boldsymbol{J} \times \boldsymbol{B}) \rangle, \end{equation}

we find

(2.31)\begin{align} \partial_t \langle S \rangle &= \partial_t \langle E^{\textrm{tot}} \rangle - \frac{1}{2} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} \right \rangle + \frac{1}{4} \langle (\rho \theta' + \rho' \theta) \vert\delta \boldsymbol{u} \vert^2 \rangle \nonumber\\ &\quad - \frac{1}{2} \langle \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') \nonumber\\ &\quad + \rho' \boldsymbol{u}' \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) - \rho \boldsymbol{u}' \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') - \rho' \boldsymbol{u} \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) + 2 \boldsymbol{u} \cdot (\boldsymbol{J} \times \boldsymbol{B}) \rangle \nonumber\\ &\quad - \frac{1}{2\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} - 2 (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} - \frac{B^2}{2} \boldsymbol{u}' + \frac{B'^2}{2} \boldsymbol{u} - (\boldsymbol{u} \cdot \boldsymbol{B}') \boldsymbol{B}' + (\boldsymbol{u}' \cdot \boldsymbol{B}) \boldsymbol{B} \right\rangle \nonumber\\ &\quad - \frac{\lambda}{2\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \langle - \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c + 2 (\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} \rangle + {\lambda} \langle \boldsymbol{J}_c \cdot (\boldsymbol{B}' \times \boldsymbol{J}') + \boldsymbol{J}_c' \cdot (\boldsymbol{B} \times \boldsymbol{J}) \rangle \nonumber\\ &\quad - \frac{1}{2} \left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &\quad + \frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) + \frac{\rho'}{\rho} \boldsymbol{u} \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) \right\rangle - \frac{1}{\mu_0} \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} - \boldsymbol{B} \cdot {\boldsymbol d_\eta} \rangle. \end{align}

Finally, we introduce the relations

(2.32)\begin{align} \langle 2 \delta \rho \delta \boldsymbol{u} \cdot \bar{\delta} (\boldsymbol{J}_c \times \boldsymbol{B}) \rangle &= \langle \rho' \boldsymbol{u}' \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) + \rho' \boldsymbol{u}' \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') - \rho' \boldsymbol{u} \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) \nonumber\\ &\quad - \rho' \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') - \rho \boldsymbol{u}' \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) \nonumber\\ &\quad - \rho \boldsymbol{u}' \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') + \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c \times \boldsymbol{B}) + \rho \boldsymbol{u} \cdot (\boldsymbol{J}_c' \times \boldsymbol{B}') \rangle, \end{align}
(2.33)\begin{align} \langle \delta (\boldsymbol{J} \times \boldsymbol{B}) \cdot \delta \boldsymbol{J}_c \rangle & = \langle \boldsymbol{J}_c \cdot (\boldsymbol{B}' \times \boldsymbol{J}') + \boldsymbol{J}_c' \cdot (\boldsymbol{B} \times \boldsymbol{J}) \rangle, \end{align}

to obtain

(2.34)\begin{align} \partial_t \langle S \rangle &= \partial_t \langle E^{\textrm{tot}} \rangle - \frac{1}{2} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} \right \rangle + \frac{1}{4} \langle (\rho \theta' + \rho' \theta) \vert\delta \boldsymbol{u} \vert^2 \rangle - \langle \delta \rho \delta \boldsymbol{u} \cdot \bar{\delta} (\boldsymbol{J}_c \times \boldsymbol{B}) \rangle \nonumber\\ &\quad - \frac{1}{2\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} - 2 (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} - \lambda \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c + 2 \lambda(\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} \right \rangle \nonumber\\ &\quad + {\lambda} \langle \delta (\boldsymbol{J} \times \boldsymbol{B}) \cdot \delta \boldsymbol{J}_c \rangle \nonumber\\ &\quad - \frac{1}{2} \left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) \right\rangle \nonumber\\ &\quad + \frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot ({\boldsymbol d_\nu'} + {\boldsymbol f'}) + \frac{\rho'}{\rho} \boldsymbol{u} \cdot ({\boldsymbol d_\nu} + {\boldsymbol f}) \right\rangle - \frac{1}{\mu_0} \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} - \boldsymbol{B} \cdot {\boldsymbol d_\eta} \rangle. \end{align}

2.3. Locality of interaction

In the inertial range and in a stationary state (the zeroth law of turbulence is used), we can simplify the expressions for the forcing and dissipative terms. The forcing is assumed to act on large scales only, thus its spatial variations should remain small and cross-terms such as $\boldsymbol {u} \cdot {\boldsymbol f'}$ are expected to behave like $\boldsymbol {u} \cdot {\boldsymbol f}=\varepsilon$. However, dissipation acts at small scales only, thus cross-terms like $\boldsymbol {u} \cdot {\boldsymbol d'_\nu }$ or $\boldsymbol {B} \cdot {\boldsymbol d'_\eta }$ will be uncorrelated and of null statistical mean. Then, using (2.8), the following simplifications can be made

(2.35)\begin{gather} \langle \boldsymbol{B}' \cdot {\boldsymbol d_{\eta}} + \boldsymbol{B} \cdot {\boldsymbol d'_{\eta}} \rangle \simeq 0, \end{gather}
(2.36)\begin{gather}-\frac{1}{2}\left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot {\boldsymbol d_\nu} + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot {\boldsymbol d_\nu'} \right\rangle \simeq 0, \end{gather}
(2.37)\begin{gather}-\frac{1}{2} \left\langle \left(1+ \frac{\rho'}{\rho} \right) \boldsymbol{u}' \cdot {\boldsymbol f} + \left(1+\frac{\rho}{\rho'}\right) \boldsymbol{u} \cdot {\boldsymbol f'} \right\rangle \simeq{-}2 \varepsilon, \end{gather}
(2.38)\begin{gather}\frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot {\boldsymbol f'} + \frac{\rho'}{\rho} \boldsymbol{u} \cdot {\boldsymbol f} \right\rangle \simeq \varepsilon, \end{gather}
(2.39)\begin{gather}\frac{1}{2} \left\langle \frac{\rho}{\rho'} \boldsymbol{u}' \cdot {\boldsymbol d_\nu'} + \frac{\rho'}{\rho} \boldsymbol{u} \cdot {\boldsymbol d_\nu} \right\rangle + \frac{1}{\mu_0} \langle \boldsymbol{B} \cdot {\boldsymbol d_\eta} \rangle \simeq{-} \varepsilon. \end{gather}

Finally, the exact law for CHMHD turbulence reads as

(2.40)\begin{align} - 4 \varepsilon &= \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} + \frac{1}{\mu_0} \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} - \frac{2}{\mu_0} (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} \right \rangle \nonumber\\ &\quad + \frac{\lambda}{\mu_0} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left\langle 2 (\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} - \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c \right \rangle \nonumber\\ &\quad - \frac{1}{2} \langle (\rho \theta' + \rho' \theta) \vert \delta \boldsymbol{u} \vert^2 \rangle + 2 \langle \delta \rho \delta \boldsymbol{u} \cdot \bar{\delta} (\boldsymbol{J}_c \times \boldsymbol{B}) \rangle - {2 \lambda} \langle \delta (\boldsymbol{J} \times \boldsymbol{B}) \cdot \delta \boldsymbol{J}_c \rangle. \end{align}

One can remark that the stationarity assumption used here implies neglect of the time derivatives of $S$ and $E^{\textrm {tot}}$ in (2.34), which are related, but not identical, to their counterparts in A18. Consequently, it might be useful to test, through direct numerical simulations, the convergence between the two laws and how this might be impacted by the stationarity assumption.

3. Discussion

Relation (2.40) is the main result of the paper. A comparison with the previous A18 law shows a drastic reduction in the number of terms (from 35 to 9). The present compact law can be written in the form

(3.1)\begin{equation} {-}4 \varepsilon = \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} ({\boldsymbol F}_\textrm{MHD} + {\lambda}{\boldsymbol F}_\textrm{Hall})+ S_\textrm{MHD} + {\lambda} S_\textrm{Hall}, \end{equation}

where, by definition,

(3.2)\begin{gather} {\boldsymbol F}_\textrm{MHD} = \left \langle \bar{\delta} \rho \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} + \frac{1}{\mu_0} \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{u} - \frac{2}{\mu_0} (\delta \boldsymbol{u} \cdot \delta \boldsymbol{B}) \delta \boldsymbol{B} \right \rangle, \end{gather}
(3.3)\begin{gather}{\boldsymbol F}_\textrm{Hall} = \frac{1}{\mu_0} \left\langle 2 (\delta \boldsymbol{B} \cdot \delta \boldsymbol{J}_c) \delta \boldsymbol{B} - \vert \delta \boldsymbol{B} \vert^2 \delta \boldsymbol{J}_c \right \rangle, \end{gather}
(3.4)\begin{gather}S_\textrm{MHD} ={-} \frac{1}{2} \langle (\rho \theta' + \rho' \theta) \vert \delta \boldsymbol{u} \vert^2 \rangle + 2 \langle \delta \rho \delta \boldsymbol{u} \cdot \bar{\delta} (\boldsymbol{J}_c \times \boldsymbol{B}) \rangle, \end{gather}
(3.5)\begin{gather}S_\textrm{Hall} ={-} 2 \langle \delta (\boldsymbol{J} \times \boldsymbol{B}) \cdot \delta \boldsymbol{J}_c \rangle. \end{gather}

The CHMHD law (2.40) contains two Yaglom-like flux terms that cover both the MHD and Hall scales (${\boldsymbol F_{\textrm {MHD}}}$ and ${\boldsymbol F_{\textrm {Hall}}}$, respectively) with an overall shape strongly reminiscent of the incompressible law, a situation not observed with the A18 law (Andrés et al. Reference Andrés, Sahraoui, Galtier, Hadid, Dmitruk and Mininni2018). It is possible to recover the incompressible law in the associated limit after a renormalization by the mass density (and by taking $\rho \to \rho _0$). To do so, we need to use the mean rate of the total energy dissipation per unit of mass ($\bar \varepsilon = \varepsilon / \rho _0$) and the Alfvén units for the magnetic field (${\boldsymbol b} = \boldsymbol {B} /\sqrt {\mu _0 \rho _0}$). We also need to replace $\lambda$ with the ion skin depth $d_i$, which has the relation $d_i=\lambda /\sqrt {\mu _0 \rho _{0}}$. Similar to the flux, the two source terms cover both the MHD and Hall scales ($S_\textrm {MHD}$ and $S_\textrm {Hall}$, respectively). In the incompressible limit, it is straightforward to show that $S_\textrm {MHD}$ tends to zero, as this term is compressible by nature, so that only ${\boldsymbol F_{\textrm {MHD}}}$ remains at the MHD scale. At the Hall scale, a few lines of calculation are required, where we find that the sum of the flux and source is

(3.6)\begin{equation} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} {\boldsymbol F}_\textrm{Hall} + S_\textrm{Hall} = \tfrac{1}{2} \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} {\boldsymbol F}_\textrm{Hall} = d_i \boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle (\delta \boldsymbol{b} \cdot \delta \boldsymbol{J}) \delta \boldsymbol{b} - \tfrac12 \vert \delta \boldsymbol{b} \vert^2 \delta \boldsymbol{J} \right \rangle, \end{equation}

as was demonstrated by Ferrand et al. (Reference Ferrand, Galtier, Sahraoui, Meyrand, Andrés and Banerjee2019), which ultimately leads to the incompressible Hall MHD exact relation

(3.7)\begin{equation} {-}4 \bar \varepsilon =\boldsymbol\nabla_{\boldsymbol\ell} \boldsymbol{\cdot} \left \langle \vert\delta \boldsymbol{u}\vert^2 \delta \boldsymbol{u} + \vert \delta \boldsymbol{b} \vert^2 \delta \boldsymbol{u} - 2 (\delta \boldsymbol{u} \cdot \delta \boldsymbol{b}) \delta \boldsymbol{b} + d_i \left[ (\delta \boldsymbol{b} \cdot \delta \boldsymbol{J}) \delta \boldsymbol{b} - \tfrac12 \vert \delta \boldsymbol{b} \vert^2 \delta \boldsymbol{J} \right] \right \rangle . \end{equation}

Unlike the A18 law, the new expression (2.40) does not depend explicitly on the (work part of the) internal energy. We also note that only the source $S_\textrm {MHD}$ exhibits the divergence of a field (through the dilatation), whereas most of the source terms in the A18 law have such a dependence. As the precise evaluation of the divergence of a field is difficult when dealing with space plasma data because it requires a three-dimensional mapping of the system to be computed, which in turn calls for multi-spacecraft data, we conclude that the new law represents an important advance for the analysis of space plasma turbulence. This comment is particularly true considering that the entire Hall part of the law, which is the most important part when studying compressible plasma turbulence at small scales, can here be fully calculated. As the MHD source has been shown to have little influence on the energy cascade (Andrés et al. Reference Andrés, Sahraoui, Galtier, Hadid, Dmitruk and Mininni2018), not being able to compute $S_\textrm {MHD}$ for single-spacecraft missions should not significantly alter the estimation of the energy cascade rate. One should note however that this statement assumes that a direct measurement of the current is available (through ion and electron moments). If not, then multiple spacecraft are still required to compute the current as the curl of the magnetic field.

As expected, the compact law is Galilean invariant; however, and unlike the incompressible HMHD law (3.7), the compressible expression shows an explicit dependence on a mean background magnetic field ${\boldsymbol B_0}$ if one is initially present. In this case, a new term appears on the right-hand side of (3.1), which reads

(3.8)\begin{align} S_\textrm{B0} &= 2 {\boldsymbol B_0} \cdot \langle \delta \rho (\delta \boldsymbol{u} \times \bar \delta \boldsymbol{J}_c) \rangle - 2 \lambda {\boldsymbol B_0} \cdot \langle \delta \rho (\delta \boldsymbol{J}_c \times \bar \delta \boldsymbol{J}_c) \rangle \nonumber\\ &\simeq{-}2 {\boldsymbol B_0} \cdot \langle \delta \rho (\delta \boldsymbol{u}_e \times \bar \delta \boldsymbol{u}_e) \rangle, \end{align}

where the second line corresponds to the small-scale limit (where the assumption $\boldsymbol {J}_c \propto \boldsymbol {u}_e$ is used, where $\boldsymbol {u}_e$ is the electron velocity). We see that this term goes trivially to zero in the incompressible limit. Note that, even in the absence of an explicit dependence in ${\boldsymbol B_0}$, this term is still expected to have an impact on the energy cascade by the way it shapes the nonlinear dynamics (Wan et al. Reference Wan, Oughton, Servidio and Matthaeus2012; Meyrand & Galtier Reference Meyrand and Galtier2013; Oughton et al. Reference Oughton, Wan, Servidio and Matthaeus2013).

4. Conclusions

In this paper, we have detailed the derivation of a new exact law for three-dimensional compressible isothermal Hall MHD turbulence. This law stems from the analysis of mixed second-order structure functions for energy increments under the assumptions of statistical homogeneity, time stationarity and infinite kinetic/magnetic Reynolds numbers. The law is a function of the plasma velocity, the magnetic field and the plasma density. It can depend explicitly on a uniform magnetic field but it is Galilean invariant. The strength of this new law is its size, as it has only 9 terms instead of the 35 terms in the previous A18 law (Andrés et al. Reference Andrés, Sahraoui, Galtier, Hadid, Dmitruk and Mininni2018), which makes it much easier to manipulate. Furthermore, this law has the advantage of being almost fully applicable to single-spacecraft data (assuming the Taylor hypothesis), except for the MHD source term $S_\textrm {MHD}$ that requires a three-dimensional mapping of the velocity field. This property will certainly allow for a more complete analysis of the new data from the recent NASA Parker Solar Probe (Bandyopadhyay et al. Reference Bandyopadhyay, Goldstein, Maruca, Matthaeus, Parashar, Ruffolo, Chhiber, Usmanov, Chasapis and Qudsi2020), as well as the ESA Solar Orbiter mission, which should lead to a better understanding of the ion-scale turbulence in the solar wind turbulence.

Editor Steve Tobias thanks the referees for their advice in evaluating this article.

Funding

This research received no specific grant from any funding agency, commercial or not-for-profit sectors.

Declaration of interests

The authors report no conflict of interest.

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