2.1 Coarse-grained energy equations

Our starting point is incompressible three-dimensional (3-D) homogeneous MHD turbulence. The primary dynamical variables are then the fluctuation velocity $\boldsymbol {u} (\boldsymbol {x}, t)$ and the fluctuation magnetic field $\boldsymbol {b} (\boldsymbol {x},t)$, where we measure the latter in Alfvén speed units: $\boldsymbol {b} / \sqrt {4{\rm \pi} \rho } \to \boldsymbol {b}$, with $\rho$ the uniform mass density. We consider situations with no mean magnetic field since these are more likely to exhibit global isotropy. The governing equations, with allowance for hyper-dissipation, are

(2.1)\begin{gather} \partial_t u_i+ \partial_j \left( u_i u_j \right) ={-} \partial_i \left( p + \frac{b^2}{2} \right) + \partial_j \left( b_i b_j \right) + \nu_\alpha ({-}1)^{\alpha +1} {\nabla}^{2\alpha} u_i + F_i , \end{gather}

(2.2)\begin{gather}\partial_t b_i + \partial_j \left (b_i u_j \right) = \partial_j \left( u_i b_j \right)+ \mu_\alpha ({-}1)^{\alpha +1} {\nabla}^{2\alpha} b_i, \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0 , \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{b} = 0 . \end{gather}

Here, $p$ is the pressure, $\boldsymbol {F}$ is a (large-scale) velocity forcing, $\nu _\alpha$ and $\mu _\alpha$ are the hyper-viscosity and hyper-resistivity, and $\alpha$ denotes the power of the Laplacian operator employed in the hyper-dissipation. Standard Laplacian dissipation corresponds to the case $\alpha =1$.

The MHD variables, and equations, may be spatially coarse-grained using a suitable filtering field, $G^\ell (\boldsymbol {r} )$ (Germano Reference Germano1992; Aluie Reference Aluie2017). The role of $G^\ell (\boldsymbol {r} )$ is to strongly suppresses structure at scales less than the filtering scale $\ell$. For example, the filtered velocity field is

(2.4)\begin{equation} \bar{\boldsymbol{u}}^\ell(\boldsymbol{x}) = \int \,\mathrm{d} \boldsymbol{r} \, G^\ell (\boldsymbol{r} ) \, \boldsymbol{u} ( \boldsymbol{x} + \boldsymbol{r}) . \end{equation}

This can be interpreted as a weighted average of $\boldsymbol {u}$ centred on the position $\boldsymbol {x}$. The weighting function $G^\ell$ decays very rapidly to zero at distances greater than a few $\ell$ from $\boldsymbol {x}$ and satisfies some other weak restrictions, such as smoothness and having a volume integral of unity. Filtering is a linear operation and commutes with differentiation, properties of which we will make considerable use below. From § 2.2 onwards, we will specialise to a Gaussian filter, but in this section, a specific choice of filter is not needed.

Coarse-graining of (2.1)–(2.2) introduces four SGS stress tensors, $\tau ^\ell (\cdot, \cdot )$, associated with the advective-type nonlinear terms (those containing a $\partial _j$). These each have the form

(2.5)\begin{equation} \tau^\ell (f_i, g_j) = \overline{f_i g_j}^\ell - \overline{f_i}^\ell \, \overline{g_j}^\ell , \end{equation}

where $\boldsymbol {f}$ and $\boldsymbol {g}$ are the solenoidal vectors appearing in a $g_j \partial _j \, f_i$ advective-type term. We remark that with this notation, the advecting field is the second argument in a $\tau ^\ell (\cdot,\cdot )$.

To obtain the equations governing the (pointwise) evolution of the coarse-grained kinetic energy $E^\ell _u (\boldsymbol {x},t) = \tfrac {1}{2} \bar {\boldsymbol {u}}^\ell \boldsymbol {\cdot } \bar {\boldsymbol {u}}^\ell$ and magnetic energy $E^\ell _b (\boldsymbol {x},t) = \tfrac {1}{2} \bar {\boldsymbol {b}}^\ell \boldsymbol {\cdot } \bar {\boldsymbol {b}}^\ell$, one filters (2.1)–(2.2) and then multiplies by $\bar {\boldsymbol {u}}^\ell$ and $\bar {\boldsymbol {b}}^\ell$, respectively (e.g. Zhou & Vahala Reference Zhou and Vahala1991; Kessar et al. Reference Kessar, Balarac and Plunian2016; Aluie Reference Aluie2017; Offermans et al. Reference Offermans, Biferale, Buzzicotti and Linkmann2018; Alexakis & Chibbaro Reference Alexakis and Chibbaro2022). The result can be written as

(2.6)\begin{gather} \partial_t {E}^\ell_u + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{{\mathcal{J}}}_u ={-} \varPi^{I,\ell} - \varPi^{M,\ell} - {\mathcal{W}}^\ell - {\mathcal{D}}_{u} + \bar{\boldsymbol{u}}^\ell \boldsymbol{\cdot} \bar{\boldsymbol{F}}^\ell, \end{gather}

(2.7)\begin{gather}\partial_t {E}^\ell_b+ \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{{\mathcal{J}}}_b ={-} \varPi^{A,\ell} - \varPi^{D,\ell} + {\mathcal{W}}^\ell - {\mathcal{D}}_{b} , \end{gather}

where the $\boldsymbol {{\mathcal {J}}}$ terms account for the spatial transport of energy and the $\varPi^\ell$ terms embody energy fluxes (i.e. transfer across scale $\ell$). Our sign convention for the definitions of the $\varPi^\ell$ (see below) means that $\varPi^\ell > 0$ corresponds to forward transfer of energy, i.e. to scales smaller than $\ell$. The ${\mathcal {D}}$ terms represent (hyper-)dissipative effects. Also present is the resolved scale conversion (RSC) term, ${\mathcal {W}}^\ell = \bar {b}^\ell_i \bar {b}^\ell_j \partial _j \bar {u}^\ell_i$, here expressed as in Aluie (Reference Aluie2017). This appears with opposite sign in each equation, and represents an exchange between large-scale kinetic and magnetic energies. An important point is that it is not an energy flux term since it does not involve energy transfer across scale $\ell$. It supports several interpretations including as: (i) the rate of work done on the large-scale flow by the large-scale Lorentz force and (ii) the energy gained by $\bar {\boldsymbol {b}}^\ell$ as it is distorted by $\bar {\boldsymbol {u}}^\ell$ or vice versa. Detailed forms for the spatial transport currents, $\boldsymbol {{\mathcal {J}}}_u, \boldsymbol {{\mathcal {J}}}_b$, depend on the form employed for ${\mathcal {W}}^\ell$ and are available elsewhere (e.g. Kessar et al. Reference Kessar, Balarac and Plunian2016; Aluie Reference Aluie2017; Offermans et al. Reference Offermans, Biferale, Buzzicotti and Linkmann2018; Alexakis & Chibbaro Reference Alexakis and Chibbaro2022), while the problem of Galilean invariance has been addressed by Offermans et al. (Reference Offermans, Biferale, Buzzicotti and Linkmann2018).

Our primary interest herein centres on the pointwise energy flux (at scale $\ell$) terms, denoted by $\varPi ^\ell (\boldsymbol {x}, t)$, together with their volume averages, $\left \langle \varPi ^\ell \right \rangle$. For any choice of filter, these can be expressed in terms of contractions of filtered gradient tensors and SGS stress tensors:

(2.8)\begin{gather} \varPi^{I,\ell} ={-} \frac{\partial{\bar{u}_i^\ell}}{\partial{x_j}} \, \tau^\ell(u_i,u_j) , \end{gather}

(2.9)\begin{gather}\varPi^{M,\ell} = \frac{\partial{\bar{u}_i^\ell}}{\partial{x_j}} \, \tau^\ell(b_i,b_j) , \end{gather}

(2.10)\begin{gather}\varPi^{A,\ell} ={-} \frac{\partial{\bar{b}_i^\ell}}{\partial{x_j}} \, \tau^\ell(b_i,u_j) , \end{gather}

(2.11)\begin{gather}\varPi^{D,\ell} = \frac{\partial{\bar{b}_i^\ell}}{\partial{x_j}} \, \tau^\ell(u_i,b_j). \end{gather}

Like the $\tau ^\ell ( \cdot, \cdot )$, these arise in connection with the four advection type nonlinearities in (2.1)–(2.2) that we refer to as the Inertial, Maxwell (meaning from the Lorentz force), Advection and Dynamo terms. Note the capitalisation. Taken together with (2.6) and (2.7), these definitions of the $\varPi^\ell$ terms mean that the interpretation of the direction of an energy flux does not depend on which flux it is. This is why (2.9) and (2.11) lack a leading minus sign. Specifically, a positive value for any one of these fluxes corresponds to transfer of energy from scales greater than $\ell$ to scales smaller than $\ell$. Clearly, $\varPi ^{I,\ell } + \varPi ^{M,\ell }$ is the net flux of $E^\ell _u$, and $\varPi ^{A,\ell } + \varPi ^{D,\ell }$ that for $E^\ell _b$. As is well known, $\varPi ^{A,\ell }$ and $\varPi ^{D,\ell }$ have a common origin and may be readily combined to obtain the magnetic energy flux associated with the curl of the induced electric field.

We remark that energy transfer in MHD turbulence has traditionally been discussed in a Fourier-space approach (e.g. Dar, Verma & Eswaran Reference Dar, Verma and Eswaran2001; Verma Reference Verma2004; Alexakis, Mininni & Pouquet Reference Alexakis, Mininni and Pouquet2005; Mininni, Montgomery & Pouquet Reference Mininni, Montgomery and Pouquet2005; Teaca et al. Reference Teaca, Verma, Knaepen and Carati2009; Linkmann et al. Reference Linkmann, Sahoo, McKay, Berera and Biferale2017; Verma Reference Verma2019). The filtering approach used here results in equivalent expressions for the mean inertial flux if the filter is chosen to be a Galerkin projector. However, in MHD, the two approaches differ in important details. In the Fourier-space approach, the transfer terms originating from the Lorentz force (momentum equation) and the field-line stretching term (induction equation) retain contributions that only involve a coupling among resolved scales. That is, the RSC term ${\mathcal {W}}^\ell$ is not separated out from the fluxes. This is the reason these terms do not vanish in the limit of $k\to \infty$ in the Fourier-based approach, in contrast to the terms defined in (2.8)–(2.11) using the SGS stresses, which therefore are genuine flux terms. As a consequence, the Fourier-based definitions can result in misinterpretations of the multi-scale nature of kinetic-to-magnetic energy conversion and of the role of the Lorentz force in inducing kinetic energy transfer across scales. For further details, see the works of Aluie (Reference Aluie2017) and Offermans et al. (Reference Offermans, Biferale, Buzzicotti and Linkmann2018). An in-depth discussion of the SGS-based definition of energy flux and a comparison with the Fourier-based approach is supplied by Aluie & Eyink (Reference Aluie and Eyink2009).

2.2 Gaussian filter

Rather remarkably, the choice of a Gaussian filter enables an analytic determination of the SGS stresses and fluxes in terms of field gradients with contributions from the resolved scales and the subfilter scales. In the remainder of the paper, we therefore employ an isotropic Gaussian filter

(2.12)\begin{equation} G^\ell (\boldsymbol{r} ) = \frac{1}{(2{\rm \pi}\ell^2)^{3/2}} \exp \left( -\frac{|\boldsymbol{r}|^2}{2 \ell^2} \right) . \end{equation}

2.3 Exact expressions for the $\tau ^\ell$ and $\varPi ^\ell$

Employing the Fourier transform of the Gaussian filter (2.12), Johnson (Reference Johnson2020, Reference Johnson2021) showed that the filtered version of an arbitrary field $\boldsymbol {u}(\boldsymbol {x},t)$ satisfies a diffusion equation,

(2.13)\begin{equation} \frac{\partial{ \bar{u}^\ell_j}}{\partial{(\ell^2)}} =\frac{1}{2} \nabla^2 \bar{u}^\ell_j, \quad \left.\bar{u}^\ell_j \right\vert_{\ell=0} = u_j( \boldsymbol{x}, t ), \end{equation}

where $\ell ^2$ is the time-like variable. It was further shown that the associated SGS stress tensor, $\tau ^\ell (u_i, u_j )$, obeys a forced version of this diffusion equation. The forcing term is $\bar {A}^\ell _{ik} \bar {A}^\ell _{jk}$, where $\bar {A}^\ell _{ik} = \partial _k \bar {u}^\ell _i$ is the gradient tensor for the filtered field. An exact solution for $\tau ^\ell ( u_i, u_j)$ was obtained that depends on $\bar {A}^\phi _{ik}$ for all scales $\phi \le \ell$. Substituting this into the equation for the SGS Inertial flux, (2.8), produces an exact solution for this flux corresponding to the exact summation of the perturbation series proposed by Eyink (Reference Eyink2006).

Happily, this approach is readily extended to MHD and may be used to calculate the elements contained in (2.8)–(2.11). Below, we outline how to achieve this for the particular case of the magnetic energy subflux (2.10) that originates with the advection term in the induction equation (i.e. $\boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {\nabla } \boldsymbol {b}$). More details, plus the general case of three distinct solenoidal fields, are available in Appendix A; see also Appendix C and Capocci et al. (Reference Capocci, Johnson, Oughton, Biferale and Linkmann2023).

We seek an exact solution for $\tau ^\ell ( b_i, u_j) = \overline {b_i u_j}^\ell - \overline {b_i}^\ell \overline {u_j}^\ell$. Clearly, $\overline { b_i u_j }^\ell$ will also satisfy (2.13) since that equation holds for any (Gaussian) filtered field. Together with the product rule expansion of $\partial ( \bar {b}^\ell _i \bar {u}^\ell _j ) / \partial {\ell ^2}$, this yields

(2.14)\begin{equation} \left( \frac{\partial{}}{\partial{\ell^2}} - \frac{1}{2} \nabla^2 \right) \tau^\ell (b_i, u_j) = \bar{B}^\ell_{ik} \bar{A}^\ell_{jk} , \quad \tau^{\ell=0} (b_i, u_j) = 0 , \end{equation}

where $\bar {B}^\ell _{ik} = \partial _k \bar {b}^\ell _i$ is the gradient tensor for $\bar {b}^\ell _i$. The solution can be written in the form

(2.15)\begin{equation} \tau^\ell (b_i, u_j) =\ell^2 \, \bar{B}^\ell_{ik} \bar{A}^\ell_{jk} + \int_0^{\ell^2} \,\mathrm{d}\theta\left( \overline{\overline{B}_{ik}^{\sqrt{\theta}} \, \overline{A}_{jk}^{\sqrt{\theta}} }^\phi -\overline{\overline{B}_{ik}^{\sqrt{\theta}}}^\phi \, \overline{\overline{A}_{jk}^{\sqrt{\theta}}}^\phi\right), \end{equation}

where $\phi = \sqrt {\ell ^2 - \theta }$. The first term on the right-hand side is a ‘single-scale’ piece as it contains only resolved scale terms (cf. Clark, Ferziger & Reynolds Reference Clark, Ferziger and Reynolds1979). In terms of other work, it corresponds to the nonlinear model employed by Leonard (Reference Leonard1975), Borue & Orszag (Reference Borue and Orszag1998) and Meneveau & Katz (Reference Meneveau and Katz2000), and is the first-order term in the expansion of Eyink (Reference Eyink2006). It is also the leading-order term of the power law expansion in the filter limit going to zero relative of any filter kernel with finite moments (cf. § 13.4.4 of Pope Reference Pope2000).

The second term, involving an integral over all scales smaller than $\ell$, is manifestly a multi-scale contribution that includes subfilter-scale field gradients. Note that the integrand in (2.15) can itself be written as an SGS stress, but one based on the field gradients rather than the fields themselves: $\tau ^\phi ( \bar {B}_{ik}^{\sqrt {\theta }}, \bar {A}_{jk}^{\sqrt {\theta }} )$.

Contracting with $\bar {B}_{ij}^\ell$ provides an exact expression for (2.10):

(2.16)\begin{align} \varPi^{A,\ell} & ={-} \ell^2 \, \bar{B}_{ij}^\ell \bar{B}_{ik}^\ell \bar{A}_{jk}^\ell - \bar{B}_{ij}^\ell \int_0^{\ell^2} \,\mathrm{d}\theta \left( \overline{\overline{B}_{ik}^{\sqrt{\theta}} \,\overline{A}_{jk}^{\sqrt{\theta}} }^\phi - \overline{\overline{B}_{ik}^{\sqrt{\theta}}}^\phi \, \overline{\overline{A}_{jk}^{\sqrt{\theta}}}^\phi \right) \end{align}

(2.17)\begin{align} & = \varPi^{A,\ell}_s + \varPi^{A,\ell}_m , \end{align}

where the subscripts $s$ and $m$ denote the single- and multi-scale contributions, respectively. It is evident that all the SGS energy fluxes, (2.8)–(2.11), and also other SGS fluxes (e.g. for helicities), can be written strictly in terms of (multi-scale) gradients of the velocity and magnetic vector fields. Appendix A contains further details. When discussing the individual SGS energy fluxes in § 4, we will make regular reference to (A3), the generalised form of (2.17).

The tensor contractions present in (2.17) can be expressed as the trace of the matrix products involved, after appropriate use of the transpose operation (superscript $t$). For example, $\bar {B}^\ell _{ij} \bar {B}^\ell _{ik} \bar {A}^\ell _{jk} = \mathrm {Tr}\left \{ { (\bar {\boldsymbol {B}}^\ell )^t \,\bar {\boldsymbol {B}}^\ell (\bar {\boldsymbol {A}}^\ell )^t} \right \}$.

Further insight into the physics of the scale-space flux $\varPi ^{A,\ell }$ may be extracted by expressing each gradient tensor as the sum of its index-symmetric and index-antisymmetric components. Let us write $S_{ij} = ( A_{ij} + A_{ji} ) /2$ and ${\varOmega }_{ij} = ( A_{ij} - A_{ji} ) / 2$, respectively the (velocity) rate-of-strain tensor and the rotation rate tensor, with ${\varOmega }_{ij} = -\epsilon _{ijk} {\omega }_k /2$ in terms of the vorticity ${\omega }_k = \epsilon _{ijk} \partial _i {u}_j$. Similarly, $B_{ij} = \varSigma _{ij} + {J}_{ij}$, where the non-zero elements of $J_{ij} = ( B_{ij} - B_{ji} ) / 2 = -\epsilon _{ijk} j_k/2$ are essentially the components of the electric current density $\boldsymbol {j} = \boldsymbol {\nabla } \times \boldsymbol {b}$ and the magnetic strain-rate tensor $\varSigma _{ij} = (B_{ij} + B_{ji})/2$. We will of course require the filtered versions of all of these quantities.

Ostensibly, this decomposition gives eight single-scale and eight multi-scale sub-fluxes; see (A7). However, properties of the trace of particular products of symmetric or antisymmetric matrices mean some of these may vanish, cancel or be equivalent. In the present case, one obtains, in connection with the single-scale contributions,

(2.18)\begin{align} \mathrm{Tr}\left\{ { \left(\bar{\boldsymbol{\mathsf{B}}}^\ell\right)^t \, \bar{\boldsymbol{\mathsf{B}}}^\ell \, \left(\bar{\boldsymbol{\mathsf{A}}}^\ell\right)^t} \right\} & = \mathrm{Tr}\left\{ { \left( \bar{\boldsymbol{\varSigma}}^\ell - \bar{\boldsymbol{\mathsf{J}}}^\ell \right) \left( \bar{\boldsymbol{\varSigma}}^\ell + \bar{\boldsymbol{\mathsf{J}}}^\ell \right) \left( \bar{\boldsymbol{\mathsf{S}}}^\ell - \bar{\boldsymbol{\varOmega}}^\ell \right) } \right\} \end{align}

(2.19)\begin{align} & = \mathrm{Tr}\left\{ { \bar{\boldsymbol{\varSigma}}^\ell \bar{\boldsymbol{\varSigma}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell - \bar{\boldsymbol{\mathsf{J}}}^\ell \bar{\boldsymbol{\mathsf{J}}}^\ell \, \bar{\boldsymbol{\mathsf{S}}}^\ell + 2 \bar{\boldsymbol{\varSigma}}^\ell \bar{\boldsymbol{\mathsf{J}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell } \right\}, \end{align}

so that there are only three distinct subfluxes; see Appendix A for details. We write the single-scale flux for the Advection term as

(2.20)\begin{equation} \varPi^{A,\ell}_s =\varPi^{A, \ell}_{s, \varSigma \varSigma S}+ \varPi^{A, \ell}_{s, J J S} + 2\varPi^{A, \ell}_{s, \varSigma J S}, \end{equation}

where $\varPi ^{A,\ell }_{s, PQR} = -\ell ^2 \mathrm {Tr}\left \{ {(\bar {\boldsymbol {\mathsf{P}}}^\ell )^t \bar { \boldsymbol {\mathsf{Q}}}^\ell (\bar { \boldsymbol {\mathsf{R}}}^\ell )^t} \right \}$ and each of $\boldsymbol {\mathsf{P}}, \boldsymbol {\mathsf{Q}},\boldsymbol {\mathsf{R}}$ are either symmetric or antisymmetric tensors.

The multi-scale flux contributions which explicitly contain subfilter-scale fluctuations can also be so decomposed, with the details given in Appendix A. The particular forms for $\varPi ^{A,\ell }_m$ are discussed in § 4.3.

4.1 Inertial flux and comparison with hydrodynamics

The exact decomposition of the Inertial term $\Pi^{I,\ell}$, (2.8), is

(4.1)\begin{equation} \varPi^{I,\ell} = \varPi^{I,\ell}_{s, SSS} + \varPi^{I,\ell}_{m,SSS} + \varPi^{I,\ell}_{s, S \varOmega \varOmega} + \varPi^{I,\ell}_{m,S \varOmega \varOmega} + \varPi^{I,\ell}_{m, S \varOmega S} , \end{equation}

where $\Pi^{I,\ell}_{s,S \Omega S}$ is not included as it is identically zero. This is the special case of (A3) where all the fields are the velocity and naturally it coincides with the original decomposition provided by Johnson (Reference Johnson2020) for Navier–Stokes turbulence. It is thus of interest to investigate whether there are differences between the HD and MHD instances of (4.1), and how these might arise.

Figure 3 compares these two cases. For the hydrodynamic case, panel (a), there is a relatively clear and extended plateau for the total flux (and some of the subfluxes) corresponding to an inertial range. In contrast, the MHD case shown in panel (b) lacks such a plateau and the individual subfluxes are mostly much smaller than their hydrodynamic counterparts. Indeed, only $\langle \Pi^{I,\ell}_{m,SS\Omega} \rangle$ has values comparable to its hydrodynamic counterpart, albeit with a different functional form, being negative for almost all $k$. Intriguingly, this term is the only one that does not vanish point-wise in two-dimensional (2-D) turbulence, as discussed by Johnson (Reference Johnson2021).Footnote ² A depletion of the inertial flux in MHD turbulence has been observed by Alexakis (Reference Alexakis2013) and Yang et al. (Reference Yang, Linkmann, Biferale and Wan2021) for configurations with large-scale electromagnetic forcing and by Offermans et al. (Reference Offermans, Biferale, Buzzicotti and Linkmann2018) for a saturated dynamo at lower Reynolds number. We have verified that the relation of Betchov (Reference Betchov1956), $\langle \Pi^{I,\ell}_{s,SSS} \rangle = 3 \langle \Pi^{I,\ell}_{s,S\Omega \Omega} \rangle$, holds for both datasets, as it must.

Figure 3.

Contributions to the Inertial energy flux $\langle \Pi^{I,\ell} \rangle$ for (a) the HD dataset H2 and (b) the MHD dataset A1, as a function of the (non-dimensionalised) reciprocal scale $\ell$, i.e. $\pi \eta_\alpha / \ell$. All fluxes are normalised by the mean total energy dissipation rate . for the dataset. Filled symbols correspond to single-scale contributions while hollow markers indicate multi-scale contributions. Error bars are for one standard error. Both panels indicate that $\langle \Pi^{I,\ell}_{m,SSS} \rangle \approx \langle \Pi^{I,\ell}_{m,S \Omega \Omega} \rangle$. In panel (b), the purple arrow locates $k \eta_\alpha = 5.4 \times 10^{-2}$ (equivalent to $kL_u \approx 14$), the value used for the probability density functions (p.d.f.s) shown in figures 4, 7 and 10.

Figure 4.

Standardised p.d.f.s of the MHD Inertial subfluxes $\Pi^{I,\ell}_X$ at $k\eta_\alpha = 5.4 \times 10^{-2}$ (equivalent to $kL_u =14$) from dataset A1, where $X$ identifies the specific subflux. (a) Single-scale fluxes; (b) multi-scale fluxes.

Also of interest is that there appears to be an approximate multi-scale analogue of the Betchov relation, with $\langle \Pi^{I,\ell}_{m,SSS} \approx \langle \Pi^{I,\ell}_{m,S\Omega \Omega} \rangle$. For hydrodynamics, this was already noted by Johnson (Reference Johnson2021) and justified by Yang et al. (Reference Yang, Zhou, Xu and He2023). Evidently, this approximate degeneracy also holds in this MHD situation, although the smallness of the terms makes this difficult to appreciate from figure 3(b).

Having discussed mean fluxes, we now examine some statistical properties of their pointwise contributions. Figure 4 presents standardised probability density functions (p.d.f.s) of the MHD Inertial subfluxes at $k\eta_\alpha = 5.4 \times 10^{-2}$. It is evident that the distributions are strongly non-Gaussian and exhibit very wide tails, with fluctuations at tens of standard deviations. (As we shall see, this is a common characteristic for all the MHD energy fluxes and subfluxes.) Most importantly, we observe strong backscatter in all subfluxes. That is, the depletion of the mean Inertial flux results from cancellations between forward and inverse transfer events made possible by an increase in backscatter events. Similar observations for the total Inertial term have been reported by Offermans et al. (Reference Offermans, Biferale, Buzzicotti and Linkmann2018), where a comparison between the kinematic, nonlinear and saturated stages of the dynamo had been carried out. The variance, skewness and kurtosis for each p.d.f. are reported in table 2. Although the p.d.f. of $\Pi^{I,\ell}_{s,S\Omega \Omega}$, which corresponds to the scale-local vortex stretching, has more asymmetrical tails than those of $\Pi^{I,\ell}_{s,SSS}$, the skewness for the latter is nonetheless larger. The term $\Pi^{I,\ell}_{s,S \Omega S}$ does not show up in figure 4 because it is identically zero due to the symmetries of the tensors involved in the corresponding trace of (A3). In contrast, panel (b) shows that . is the most negatively skewed p.d.f. among the Inertial ones.

Table 2.

Values of variance $\left(\sigma_X^Y\right)^2$, skewness $S_X^Y= \left\langle \left( \Pi^{Y,\ell}_X - \langle \Pi^{Y,\ell}_X\rangle \right)^3 \right\rangle/\left(\sigma_X^Y\right)^3$ and kurtosis $K_X^Y= \left\langle \left( \Pi^{Y,\ell}_X - \langle \Pi^{Y,\ell}_X\rangle \right)^4 \right\rangle/\left(\sigma_X^Y\right)^4$ for the subflux p.d.f.s shown in figures 4–10, where $X$ indicates the subflux identifier and $Y$ denotes the term identifier.

Figure 5.

Two-dimensional sketch of current-sheet thinning and strain rate amplification by the latter. A current sheet, $\boldsymbol{\mathsf{J}}$, is stretched by (a) large-scale strain $\boldsymbol{\mathsf{S}}$, into a magnetic shear layer (b, red arrows). This induces a stretching of the magnetic flux tubes in the sheet. By conservation of magnetic flux, the magnetic field strength at the thereby generated smaller scales increases. That is, magnetic energy is transferred from large to small scales. The resulting magnetic shear layer has an associated magnetic strain rate field, $\textsf{$\mathit{\Sigma}$}$, whose principal axes (solid blue arrows) are at $.$ to those of the large-scale strain rate tensor (straight black arrows). As the magnetic shear will align with the extensional direction of the (velocity) strain rate tensor, this causes the fluid to be accelerated along these extensional directions and slowed down in the compressional directions, thereby generating a stronger rate-of-strain field across smaller scales, $\boldsymbol{\mathsf{S}'}$, indicated by the dashed arrows. The principal axes of the large-scale strain rate tensor are denoted by $\boldsymbol{\mathsf{e}}_1$ in one extensional direction and $\boldsymbol{\mathsf{e}}_3$ in one compressional direction and analogously for the small-scale strain rate.

In connection with the approximate degeneracy between $\Pi^{I,\ell}_{m,SSS}$ and $\Pi^{I,\ell}_{m,S\Omega \Omega}$ discussed above, figure 4(b) reveals that their p.d.f.s coincide up to events with a standardised probability density of $10^{-7}$, potentially indicating that the approximate identity is true not just on average but also for higher-order moments, although further analysis is required.

4.2 Maxwell flux

The decomposition of the energy flux associated with the Lorentz force, which we refer to as the Maxwell term, $\Pi^{M,\ell}$ of (2.9), contains an extra (single-scale) term with respect to the Inertial flux. Specifically, from (A3), we obtain

(4.2)\begin{equation} \varPi^{M,\ell} = \varPi^{M,\ell}_{s,S \varSigma \varSigma} + \varPi^{M,\ell}_{m,S \varSigma \varSigma} + \varPi^{M,\ell}_{s, S J J} + \varPi^{M,\ell}_{m, S J J} + \varPi^{M,\ell}_{s, S J \varSigma} + \varPi^{M,\ell}_{m, S J \varSigma} . \end{equation}

Terms of type $S \Sigma \Sigma$ can be associated with strain rate amplification by magnetic shear, while terms of type $SJJ$ correspond to current-filament stretching that is analogous to vortex stretching in HD. The last two terms are of type $SJ\Sigma$ and describe the back-reaction of the magnetic field on the flow or, more specifically, how the velocity strain rate is modified (typically amplified) in connection with a current-sheet thinning process. As we shall see, this is by far the dominant process. It proceeds as follows (figure 5). First, a current sheet is stretched by large-scale straining motions into a magnetic shear layer, in a process similar to vortex thinning in HD (Kraichnan Reference Kraichnan1976; Chen et al. Reference Chen, Ecke, Eyink, Rivera, Wan and Xiao2006; Johnson Reference Johnson2021). This results in a stretching of the magnetic flux tubes in the sheet. By conservation of magnetic flux, the magnetic field strength at the thereby generated smaller scales must increase. That is, magnetic energy is transferred from large to small scales. (We will revisit this process in § 4.3 in the context of the inter-scale transfer of magnetic energy.) The magnetic rate-of-strain field associated with the resulting magnetic shear layer now accelerates fluid along its extensional directions and slows it down in the compressional directions, thereby generating a stronger rate-of-strain field across smaller scales.

It is instructive to consider the process in two dimensions, in analogy to the vortex thinning of 2-D HD. In the reference frame of the rate-of-strain tensor at scale $\ell$, the associated terms are

(4.3)\begin{gather} \varPi^{M,\ell}_{s,SJ\varSigma}(\boldsymbol{x}) = 2 \lambda_S^\ell(\boldsymbol{x}) \, \lambda^{\ell}_\varSigma (\boldsymbol{x}) \, \bar{j}^\ell (\boldsymbol{x}) \sin{2\psi(\boldsymbol{x})}, \end{gather}

(4.4)\begin{gather}\varPi^{M,\ell}_{m,SJ\varSigma}(\boldsymbol{x}) = 2 \int_0^{\ell^2} \,\mathrm{d}\theta \int_{\mathbb{R}^3} G^\phi(\boldsymbol{r}) \, \lambda_S^\ell(\boldsymbol{x}) \, \lambda^{\sqrt{\theta}}_\varSigma (\boldsymbol{x} + \boldsymbol{r}) \, \bar{j}^{\sqrt{\theta}} (\boldsymbol{x} + \boldsymbol{r}) \sin{2\psi(\boldsymbol{x} + \boldsymbol{r})} \,\mathrm{d} \boldsymbol{r} , \end{gather}

where $\pm \lambda_S$ and $\pm \lambda_\Sigma$ are the eigenvalues of the velocity and magnetic rate-of-strain tensors, respectively, $\psi$ the angle between the respective eigenvectors, and $j$ the out-of-plane component of the current density. As can be seen from these formulae, a maximum energy transfer occurs when the principal axes of the magnetic rate-of strain tensor have a $\pm 45^{\circ}$ angle to those of the velocity rate-of-strain tensor. Depending on the sign of the out-of-plane current density and $\sin(2\psi)$, (4.3) and (4.4) result in a direct or an inverse energy transfer. Figure 5 presents a schematic depiction of the process. A direct energy transfer occurs if the angle between the principal axes of velocity and magnetic strain-rate tensors is in the same rotational direction as the out-of-plane current. A similar, albeit less straightforward, assessment is possible in three dimensions, where

(4.5)\begin{equation} \varPi^{M,\ell}_{s,SJ\varSigma}(\boldsymbol{x}) = 2 \ell^2 \bar{S}_{ij}^\ell \bar{J}^\ell_{ik}\bar{\varSigma}^\ell_{kj} = 2 \ell^2 \sum_{i=1}^3 \sum_{j=1}^3 \lambda^\ell_i \mu_j^\ell\cos^2{\psi_{ij}} , \end{equation}

and similarly for the multi-scale term. Here, $\mu_j^\ell$ is the $j-$th eigenvalue of the symmetric part of the product matrix $\overline{{J}}^\ell_{ik}\overline{\Sigma}^\ell_{kj}$ (a contribution to the Maxwell SGS stress tensor) and $\psi_{ij}$ the angle between the $i-$th eigenvector of $\bar{\boldsymbol{\mathsf{S}}}^\ell$ and the $j-$th eigenvector of the symmetric part of $\overline{J}^\ell_{ik}\overline{\Sigma}^\ell_{kj}$. For a forward cascade of kinetic energy, $\langle \Pi^{M,\ell} \rangle > 0$, which implies that $\overline{S}^\ell_{ij} \, \overline{J}^\ell_{ik} \overline{\Sigma}^\ell_{kj}$ should be preferentially positive. Due to the presence of the cosine squared factor, this implies that the principal axes of the rate-of-strain tensor and those of the subscale stress must preferentially align, resulting in a stretching of the magnetic flux tubes along the extensional directions of the strain-rate tensor, as discussed above.

The subfluxes on the right-hand side of (4.2) and the total Maxwell flux are shown in figure 6, as a function of (non-dimensionalised) reciprocal $\ell$. We see immediately that the net energy transfer proceeds from large scales to small scales with the total Maxwell flux $\langle \Pi^{M,\ell} \rangle$ being the dominant energy subflux for MHD, carrying approximately $80\%$ of the total energy dissipation rate at its peak. At large scales, the major contribution is from the multi-scale term $\langle \Pi^{M,\ell}_{m,SJ\Sigma}\rangle$, switching to its single-scale partner, $\langle \Pi^{M,\ell}_{s,SJ\Sigma}\rangle$, as the dissipation scale is approached. All remaining terms in (4.2) are negligible. Summarising the mean Maxwell flux behaviour, we may say that the net kinetic energy transfer in MHD proceeds by the back-reaction of the magnetic field on the flow during the aforementioned current-sheet thinning process, while the contribution from current filament stretching and strain-amplification by magnetic shear are negligible.

Figure 6.

Contributions to the Maxwell energy flux .. Data are from dataset A1 and normalised by the mean total energy dissipation rate .. Error bars indicate one standard error.

The p.d.f.s for the Maxwell energy fluxes are shown in figure 7. The predominance of $\langle \Pi^{M,\ell}_{m,SJ\Sigma} \rangle$ in the direct cascade can be also appreciated by examining its p.d.f., which is the most positively skewed among the multi-scale terms of figure 7; see also table 2 for p.d.f. moments. As can be seen from the data shown in figure 7, while all terms of type $S\Sigma \Sigma$ and $SJJ$ nearly vanish on average, fluctuations of approximately 60 standard deviations are not uncommon, and their multi-scale contributions show considerable backscatter. That is, the mean effects of current-filament stretching and strain-amplification by magnetic shear on the flow are negligible because of cancellations between pointwise forward and inverse transfers, and both these processes result in extreme backscatter events.

Figure 7.

Standardised p.d.f.s for Maxwell subfluxes $\Pi_X^{M,\ell}$, at $k\eta_\alpha = 5.4 \times 10^{-2}$, from dataset A1, where $X$ represents the subflux identifier. (a) Single-scale fluxes; (b) multi-scale fluxes. Note that the p.d.f.s for $\Pi^{M,\ell}_{m,S\Sigma \Sigma}$ and $\Pi^{M,\ell}_{m,SJJ}$ are approximately coincident.

In Appendix B, we show that the averages of the subfluxes $\Pi_{m,S\Sigma \Sigma}^{M,\ell}$ and $\Pi_{m,S J J}^{M,\ell}$ can be connected via an exact Betchov-like relation that holds for all homogeneous flows,

(4.6)\begin{equation} \langle \varPi^{M,\ell}_{m,S\varSigma \varSigma} \rangle = \langle \varPi^{M,\ell}_{m,S J J }\rangle + 2\langle\varPi_{m,\varOmega \varSigma J}^{M,\ell} \rangle . \end{equation}

Note that the final term, $\Pi_{m,\Omega \Sigma J}^{M,\ell}$, does not appear in the Maxwell flux, (4.2), and our numerical results indicate $\Pi_{m,S\Sigma \Sigma}^{M,\ell} \approx \Pi_{m,SJJ}^{M,\ell}$, see figure 6. Physically, we may interpret this approximate identity as indicating that the net ‘strain-production’ by magnetic shear is almost equal to the net strain production by current-filament stretching. However, we stress again that these contributions to the interscale kinetic energy transfer are negligible. Further discussion on terms associated with strain production and current-filament stretching is provided in Appendix B.

4.3 Advection and dynamo fluxes

In this section, we focus on the decomposition of both the Advection term, $\Pi^{A,\ell}$, and the Dynamo term, $\Pi^{D,\ell}$, as defined in (2.10)–(2.11). As is well known, these two SGS fluxes share the same physical origin, namely the induced (fluctuation) electric field, and this is associated with certain symmetries and equivalences between the Advection and Dynamo subfluxes.

From the application of (A3), we find

(4.7)\begin{align} \varPi^{A,\ell} & = \varPi^{A,\ell}_{s, \varSigma \varSigma S} + \varPi^{A,\ell}_{m, \varSigma \varSigma S} + \varPi^{A,\ell}_{m, \varSigma \varSigma \varOmega} + \varPi^{A,\ell}_{s, \varSigma J S} + \varPi^{A,\ell}_{m, \varSigma J S} + \varPi^{A,\ell}_{s, \varSigma J \varOmega}\nonumber\\ & \quad + \varPi^{A,\ell}_{m, \varSigma J \varOmega} + \varPi^{A,\ell}_{s, J \varSigma S} + \varPi^{A,\ell}_{m, J \varSigma S} + \varPi^{A,\ell}_{s, J \varSigma \varOmega} + \varPi^{A,\ell}_{m, J \varSigma \varOmega} + \varPi^{A,\ell}_{s, J J S} + \varPi^{A,\ell}_{m, J J S} + \varPi^{A,\ell}_{m, J J\varOmega} , \end{align}

(4.8)\begin{align} \varPi^{D,\ell} & = \varPi^{D,\ell}_{s,\varSigma S \varSigma} + \varPi^{D,\ell}_{m,\varSigma S \varSigma} + \varPi^{D,\ell}_{s,\varSigma S J} + \varPi^{D,\ell}_{m,\varSigma S J} + \varPi^{D,\ell}_{m,\varSigma \varOmega \varSigma} + \varPi^{D,\ell}_{s,\varSigma \varOmega J} + \varPi^{D,\ell}_{m,\varSigma \varOmega J}\nonumber\\& \quad + \varPi^{D,\ell}_{s,J S \varSigma} + \varPi^{D,\ell}_{m,J S \varSigma} + \varPi^{D,\ell}_{s, J S J} + \varPi^{D,\ell}_{m, J S J} + \varPi^{D,\ell}_{s, J \varOmega \varSigma} + \varPi^{D,\ell}_{m, J \varOmega \varSigma} + \varPi^{D,\ell}_{m,J \varOmega J} , \end{align}

where we do not list terms that vanish identically, see Appendices A and F.

Figure 8 displays most of the Advection and Dynamo (sub)fluxes in separate panels. For clarity, only the subfluxes relevant to the discussion below and to the net energy flux are shown. The cyclic property of the trace can be used to show that some terms vanish identically and that each Advection subflux (both single-scale and multi-scale) is equal to (plus or minus) a partner Dynamo subflux; see the subfluxes expressions in Appendix F. For this reason, in figure 8, the following subflux pairs are not displayed since they are opposite in sign, $\Pi_{s,\Sigma \Sigma S}^{A,\ell}$ and $\Pi_{s,\Sigma S \Sigma}^{A,\ell}$ as well as $\Pi_{s,\Sigma J \Omega}^{A,\ell}$ and $\Pi_{s,\Sigma \Omega J}^{A,\ell}$ together with their multi-scale counterparts. Hence, these cancel pairwise and make no contribution to the net magnetic energy flux $\Pi^{A,\ell} + \Pi^{D,\ell}$; see Appendix F. In contrast, $\langle \Pi^{A,\ell}_{s,JJS} \rangle$ and $\Pi^{D,\ell}_{s,JSJ}$ and the related multi-scale terms, are equal and thus do contribute to the net flux.

Figure 8.

Decomposed fluxes for the (a) Advection term $\langle \Pi^{A,\ell}\rangle$ and (b) Dynamo term $\langle \Pi^{D,\ell}\rangle$. Fluxes are normalised by the mean total energy dissipation rate $\varepsilon$ for dataset A1. In panel (a), $\langle \Pi^{A,\ell}_{s,\Sigma J S} \rangle$ and $\langle \Pi^{A,\ell}_{s,J \Sigma S} \rangle$ perfectly coincide. The error bars indicate one standard error.

Note, too, that the two Betchov relations (B6)–(B10) can provide another source of symmetry, or approximate symmetry.

The physical interpretation of the respective terms is very similar to what has been discussed for the Maxwell flux, except that now the effect of the flow on the magnetic field must be considered. Recall from § 4.2 that the Maxwell flux terms of type $SJ\Sigma$ correspond to the stretching and, by incompressibility, thinning of current sheets into magnetic shear layers, and that the back-reaction on the flow induced by this process is responsible for the bulk of the kinetic energy transfer to smaller scales. As we shall see, and as expected from the discussion of current-sheet thinning in § 4.2, this process also transfers most magnetic energy from large to small scales. However, in contrast to the Maxwell flux situation (dominated by a multi-scale term), here it is two of the Advection single-scale terms, $\Sigma J S$ and $J \Sigma S$, that carry most of the magnetic energy flux.

Focusing on the Advection term, from figure 8(a), we observe that the net flux is everywhere positive and peaked at smaller scales, roughly at the end of the inertial range. Recall that the Maxwell flux is peaked at larger scales (figure 6). Due to the cyclic property of the trace, the terms $\Pi^{A,\ell}_{s,\Sigma J S}$ and $\Pi^{A,\ell}_{s,J\Sigma S}$ are equal while, because of the symmetry of the tensors involved, the subfluxes $\Pi^{A,\ell}_{s,JJ\Omega}$ and $\Pi^{A,\ell}_{s,\Sigma \Sigma \Omega}$ together with $\Pi^{D,\ell}_{s,J\Omega J}$ and $\Pi^{D,\ell}_{s, \Sigma \Omega \Sigma}$ are identically zero. It is also apparent from figure 8(a) that the remaining Advection terms make negligible contributions to the net flux. We note that $\Sigma \Sigma S$ type terms correspond to magnetic shear amplification due to straining motions and those of type $JJS$ to current filament stretching – the analogue to vortex stretching in HD – as depicted in figure 9. Terms of type $\Sigma J \Omega$ encode bending of magnetic field lines into current filaments by rotational flow, i.e. a change in the magnetic-field geometry induced by vortical motion. Thus, the net Advection term is primarily due to single-scale contributions, being approximately equal to $2\, \Pi^{A,\ell}_{s,\Sigma J S}$, and it carries approximately $40\%$ of the total energy flux at its peak.

Figure 9.

Sketch of current-filament stretching. A current filament $\boldsymbol{\mathsf{J}}$, with associated magnetic field $\mathbf{b}$, is (a) stretched by large-scale strain, $\boldsymbol{\mathsf{S}}$ with two contractional and one extensional direction into (b) a longer and thinner filament along the extensional direction. This induces a stretching of the magnetic flux tubes in the filament. By conservation of magnetic flux, the magnetic field strength at the thereby generated smaller scales increases (red arrows). That is, magnetic energy is transferred from large to small scales. This process is analogous to vortex stretching in hydrodynamic turbulence. The principal axes of the large-scale strain rate tensor are denoted by ${\boldsymbol{\mathsf{e}}}_1$ in the extensional direction and ${\boldsymbol{\mathsf{e}}}_2$ and ${\boldsymbol{\mathsf{e}}}_3$ in the contractional directions.

For the Dynamo term, $\Pi^{D,\ell}$, we observe that the net flux is almost flat in the inertial range, substantially positive definite for these scales (figure 8b) and responsible for approximately $15\%$ of the total energy flux. All subfluxes, except $\Pi^{D,\ell}_{s,\Sigma SJ}$ shown in yellow and $\Pi^{D,\ell}_{s,JS\Sigma}$ indicated by the filled purple symbols, are negligible. However, using the cyclic property once again, one can show that $\Pi^{D,\ell}_{s,\Sigma SJ} = -\Pi^{D,\ell}_{s,J S \Sigma}$; hence, these two terms cancel out and do not contribute to the net flux. In contrast to the Advection term, the major contributions to the net Dynamo term are from subleading single and multi-scale subfluxes of various types, with each contributing only 1–2 % to the total energy flux adding up to a total of approximately $15\%$ of the total energy flux. In summary, single-scale current-sheet thinning is the dominant process transferring magnetic energy across scales. It solely originates from the advective term in the induction equation.

Consider now the p.d.f.s. As a consequence of the symmetries between Dynamo and Advection subfluxes, only the Advection p.d.f.s are shown in figures 10(a) and 10(c), respectively for the single-scale terms and the multi-scale terms. It is striking that the p.d.f. of $\Pi^{A,\ell}_{s,\Sigma J \Omega}$, which describes the bending of magnetic field lines into current filaments by rotational flow, is by far the most strongly fluctuating with huge fluctuations of more than 100 standard deviations, also showing strong magnetic backscatter. In contrast, the aforementioned current-sheet thinning process, that is mostly responsible for the forward cascade of magnetic energy, shows weak magnetic backscatter. The other Advection subflux p.d.f.s, both single-scale and multi-scale, span a range comparable with those associated with the p.d.f.s for the Inertial and Maxwell fluxes. Moreover, all the multi-scale fluxes (Advection and Dynamo) have quite similar p.d.f.s, and thus so do the net multi-scale flux p.d.f.s (figure 10d). In the case of the net single-scale p.d.f.s, the Advection–Dynamo agreement is still good in the cores of the distributions, but there is a significant difference at larger negative fluctuations (figure 10b).

Figure 10.

Standardised p.d.f.s for dataset A1 at $k \eta_\alpha = 5.4 \times 10^{-2}$. (a) Single-scale Advection subfluxes $\Pi^{A,\ell}_X$, where $X$ represents the subflux identifier, and (b) net single-scale fluxes for the Dynamo ($\Pi_s^{D,\ell}$) and Advection terms ($\Pi^{A,\ell}_s$). In the $x$-axis titles for panels (b) and (d), $Y=A$ or $D$, as appropriate. Note that $\Pi^{A,\ell}_{s,\Sigma J S} = \Pi^{A,\ell}_{s,J S \Sigma}$ (see figure 8). (c,d) As for panels (a,b) except for the multi-scale subfluxes of $\Pi^{A,\ell}$ and $\Pi^{D,\ell}$. Note that panels (c,d) have the same $x$-axis range as figures 4 and 7.

4.4 Total energy flux, .

In the preceding subsections, we analysed the four contributions to the incompressible MHD energy flux, finding that each of them may be reasonably well approximated using just some of the subfluxes. These approximations may now be assembled to give an approximate form for the mean total MHD energy flux. Specifically, we suggest that

(4.9)\begin{align} \left\langle \varPi^{\ell} \right\rangle & = \left\langle \varPi^{I, \ell} \right\rangle + \left\langle \varPi^{M, \ell} \right\rangle + \left\langle \varPi^{A, \ell} \right\rangle + \left\langle \varPi^{D, \ell} \right\rangle \approx{-} 2 \langle \varPi^{M,\ell}_{s,SJ\varSigma} \rangle - \langle \varPi^{M,\ell}_{m,SJ\varSigma} \rangle \end{align}

(4.10)\begin{align} & = 4 \ell^2 \left\langle \mathrm{Tr}\left\{ { (\bar{\boldsymbol{S}}^\ell)^t \bar{\boldsymbol{J}}^\ell (\bar{\boldsymbol{\varSigma}}^\ell)^t} \right\} \right\rangle \end{align}

(4.11)\begin{align} & \quad + 2\left\langle \int_0^{\ell^2} \,\mathrm{d}\theta \, \mathrm{Tr}\left\{ {\left( \bar{\boldsymbol{S}}^\ell \right)^t \left( \overline{ \bar{\boldsymbol{J}}^{\sqrt{\theta}} \left(\bar{\boldsymbol{\varSigma}}^{\sqrt{\theta}} \right)^{t} }^\phi - \overline{\bar{\boldsymbol{J}}^{\sqrt{\theta}}}^\phi\, \overline{ \left( \bar{\boldsymbol{\varSigma}}^{\sqrt{\theta}} \right)^{t} }^\phi \right)} \right\} \right\rangle \end{align}

is a suitable expression. Note that, after using cyclic properties of the trace, this is expressed only in terms of two of the Maxwell subfluxes, one single-scale and one multi-scale. These are discussed in § 4.2 and we remind the reader that all subflux definitions can be found in Appendix F. Interestingly, the terms on the right-hand side of (4.11) are part of a group of terms that remain non-zero in 2-D MHD; see Appendix C. We intend to explore this intriguing feature in future work.

Equation (4.11) demonstrates that the mean total energy flux in MHD turbulence is largely given by the stretching and thinning of current-sheets into magnetic shear layers by large-scale strain, resulting in a transfer of magnetic energy from large to small scales. In addition, there is a back-reaction of this process on the flow, whereby the ensuing magnetic strain-rate field accelerates fluid along its extensional directions and slows it down in the compressional directions, thereby generating a stronger strain-rate field across smaller scales, as shown schematically in figure 5.

4.5 Current-sheet thinning and magnetic reconnection

As a means of inter-scale energy transfer, the current-sheet thinning process can only proceed as described in an inertial range, i.e. at scales where Joule and viscous dissipation are negligible. As the dissipative scales are approached, Alfvén's theorem ceases to hold and magnetic reconnection can occur, with associated changes to the topology of the magnetic field. Interestingly, models of MHD reconnection are geometrically very similar to the inertial-range energy transfer process described here. For example in the Sweet–Parker model (Parker Reference Parker1957; Sweet Reference Sweet1958), a current sheet is thinned by a flow that pushes magnetic field lines closer and closer together. Eventually, when the distance between the field lines approaches scales where Joule dissipation becomes important, the topological conservation of the magnetic field is broken and magnetic field-lines reconnect. That is, the continuation of the current-sheet thinning process described herein, and shown conceptually in figure 5, to smaller and smaller scales can lead naturally to magnetic reconnection.