1. Introduction

Space and astrophysical plasmas are often characterised by multiscale fluctuations, in most cases, well described in the framework of turbulence (Bruno & Carbone Reference Bruno and Carbone2013). Two standard ways to disentangle the multiscale properties of the turbulent fluctuations in space are scale-filtering representations and lag-dependent structure functions.

Structure functions comprising the increments of velocity at different locations at the same time are a primary focus in hydrodynamic turbulence. In magnetohydrodynamic (MHD) turbulence, the magnetic field $\boldsymbol{b}$ must also be taken into account and the mixed Elsässer fields $\boldsymbol{z}^\pm = \boldsymbol{u} \pm \boldsymbol{b}$ are often used to describe Alfvénic fluctuations (Elsasser Reference Elsasser1950), where $\boldsymbol{b} = \boldsymbol{B}/\sqrt {\mu _0 \rho }$ is the magnetic field transformed in velocity units, $\mu _0$ is the permeability of vacuum and $\rho$ is the plasma mass density. The increment of $\boldsymbol{z}^\pm$ is $\delta \boldsymbol{z}^\pm (\boldsymbol{x},\boldsymbol{r}) = \boldsymbol{z}^\pm (\boldsymbol{x}+\boldsymbol{r}) - \boldsymbol{z}^\pm (\boldsymbol{x})$ , where $\boldsymbol{x}$ is the spatial location and $\boldsymbol{r}$ is a spatial lag with length $r$ . The structure functions in the form $\langle (\delta \boldsymbol{z}^\pm )^n(\delta \boldsymbol{z}^\mp )^m \rangle$ , where $\langle \boldsymbol{\cdot }\rangle$ represents the ensemble average (which is equivalent to spatial average in homogeneous turbulence), are the statistical quantities we use to construct the dynamic differential equations. Among them, the MHD von Kármán-Horwath equation (Politano & Pouquet, Reference Politano and Pouquet1998a , Reference Politano and Pouquetb ), derived from the MHD equations, is the foundation of our statistical analysis of MHD turbulence:

(1.1) \begin{align} \frac {1}{4}\frac {\partial }{\partial t}\langle (\delta \boldsymbol{z}^\pm )^2 \rangle = -\frac {1}{4}\boldsymbol{\nabla} _r\boldsymbol{\cdot }\langle \delta \boldsymbol{z}^\mp |\delta \boldsymbol{z}^\pm |^2 \rangle + \frac {1}{2}\nu {\nabla} ^2_r\langle (\delta \boldsymbol{z}^\pm )^2 \rangle - \varepsilon ^\pm , \end{align}

where $\nu$ is the kinematic viscosity (here, we use equal viscosity $\nu$ and resistivity $\mu$ ), $\varepsilon ^\pm$ is the energy dissipation rate corresponding to $\boldsymbol{z}^\pm$ and $\boldsymbol{\nabla} _r = ({\partial }/{\partial r_i})\hat {\boldsymbol{e_i}}$ is the gradient operator with respect to the separation vector $\boldsymbol{r}$ . From left to right, the terms represent the time rate of change of the global energy of $\delta \boldsymbol{z}^\pm$ , the nonlinear energy transfer and the energy dissipation term, and $\varepsilon ^\pm$ is the global mean dissipation rate. In the ideal case, e.g. with infinite large Reynolds numbers, the time rate of change of large-scale energy (the left-most term) dominates at the energy-containing scales, the nonlinear energy transfer (the first term on the right-hand side; rhs) mainly impacts at intermediate scales in the so-called inertial range, and the energy dissipation term (the second term on the rhs) primarily dominates at the dissipation scale.

Based on the energy conservation in the MHD von Kármán–Horwath equation, the plateau of the nonlinear energy transfer term approaches values close to $\varepsilon = (1/2)(\varepsilon ^+ + \varepsilon ^-)$ upon the establishment of a well-separated MHD inertial range. In this case, the effects of the time rate of change term and the dissipative term are negligible, which gives rise to the widely used third-order law,

(1.2) \begin{align} \boldsymbol{\nabla} _r\boldsymbol{\cdot }\boldsymbol{Y}^\pm = -4\varepsilon ^\pm , \end{align}

where $\boldsymbol{Y}^\pm = \langle \delta \boldsymbol{z}^\mp |\delta \boldsymbol{z}^\pm |^2 \rangle$ denotes the Yaglom flux. This can be further simplified when the isotropic assumption is used (Politano & Pouquet Reference Politano and Pouquet1998a , Reference Politano and Pouquetb ):

(1.3) \begin{align} Y^\pm _\parallel (r) = \langle \delta z^\mp _\parallel \delta z^\pm _i \delta z^\pm _i \rangle = -\frac {4}{d}\varepsilon ^\pm r, \end{align}

where $\delta z_\parallel = \delta \boldsymbol{z}\boldsymbol{\cdot }\boldsymbol{r}/r$ and $d$ is the space dimension. Note that $\varepsilon ^\pm$ is the globally averaged dissipation rate, which is constant and scale ( $r$ ) independent.

To study the local properties of energy transfer, Sorriso-Valvo et al. (Reference Sorriso-Valvo, Marino, Lijoi, Perri and Carbone2015, Reference Sorriso-Valvo, Carbone, Perri, Greco, Marino and Bruno2018a , Reference Sorriso-Valvo, Perrone, Pezzi, Valentini, Servidio, Zouganelis and Veltrib ) proposed that the unaveraged $\delta z^\mp _\parallel \delta z^\pm _i \delta z^\pm _i$ could be indicative of the local energy transfer. That is,

(1.4) \begin{align} \langle \delta z^\mp _\parallel \delta z^\pm _i \delta z^\pm _i\rangle \rightarrow \delta z^\mp _\parallel (\boldsymbol{x},r) \delta z^\pm _i(\boldsymbol{x},r) \delta z^\pm _i(\boldsymbol{x},r) \end{align}

and

(1.5) \begin{align} -\frac {4}{d}\varepsilon ^\pm r \rightarrow -\frac {4}{d}\varepsilon ^\pm (\boldsymbol{x}) r. \end{align}

Based on these, Sorriso-Valvo et al. (Reference Sorriso-Valvo, Marino, Lijoi, Perri and Carbone2015, Reference Sorriso-Valvo, Carbone, Perri, Greco, Marino and Bruno2018a , Reference Sorriso-Valvo, Perrone, Pezzi, Valentini, Servidio, Zouganelis and Veltrib ) define the local energy transfer rate (LET) as

(1.6) \begin{align} \xi ^\pm _r = -\frac {d}{4}\frac {\delta z^\mp _\parallel \delta z^\pm _i \delta z^\pm _i}{r}. \end{align}

So the local total (magnetic + kinetic) energy transfer rate across scale $r$ is

(1.7) \begin{align} \xi _r = \frac {1}{2}(\xi ^+_r + \xi ^-_r). \end{align}

The third-order law and the related LET have been widely used because they are easy to implement even from single-spacecraft observations (Coburn et al. Reference Coburn, Forman, Smith, Vasquez and Stawarz2015; Hadid et al. Reference Hadid, Sahraoui and Galtier2017, Reference Hadid, Sahraoui, Galtier and Huang2018; Sorriso-Valvo et al. Reference Sorriso-Valvo, Carbone, Perri, Greco, Marino and Bruno2018a , Reference Sorriso-Valvo, Perrone, Pezzi, Valentini, Servidio, Zouganelis and Veltrib ; Bandyopadhyay et al. Reference Bandyopadhyay2020; Andrés et al. Reference Andrés, Sahraoui, Hadid, Huang, Romanelli, Galtier, DiBraccio and Halekas2021, Reference Andrés, Sahraoui, Huang, Hadid and Galtier2022; Simon & Sahraoui Reference Simon and Sahraoui2021; Romanelli et al. Reference Romanelli, Andrés and DiBraccio2022, Reference Romanelli, Andrés, DiBraccio, Verniero, Gruesbeck, Szabo, Espley and Halekas2024; Brodiano, Dmitruk & Andrés Reference Brodiano, Dmitruk and Andrés2023; Marino & Sorriso-Valvo Reference Marino and Sorriso-Valvo2023). They have also been used to estimate the energy dissipation rate in collisionless plasmas (Yang et al. Reference Yang, Shi, Wan, Matthaeus and Chen2016, Reference Yang, Wan, Matthaeus, Sorriso-Valvo, Parashar, Lu, Shi and Chen2018, Reference Yang, Matthaeus, Roy, Roytershteyn, Parashar, Bandyopadhyay and Wan2022; Manzini Reference Manzini2021; Vásconez et al. Reference Vásconez, Foldes, Marino, Manobanda, Lévêque, Telloni and Sorriso-Valvo2026), wherein an exact expression of dissipation in terms of viscosity and resistivity is absent. For example, LET is used in particle-in-cell (PIC) simulations (Yang et al. Reference Yang, Wan, Matthaeus, Sorriso-Valvo, Parashar, Lu, Shi and Chen2018; Matthaeus et al. Reference Matthaeus, Yang, Wan, Parashar, Bandyopadhyay, Chasapis, Pezzi and Valentini2020). Although LET is not rigorously derived from the MHD equations, its association with energy transfer has been shown by Aluie & Eyink (Reference Aluie and Eyink2010), where the coarse-grained energy flux of the Elsässer field can be expressed in terms of the increments: $-\varPi ^\pm _\ell \simeq \delta z^\mp (\ell ) [\delta z^\pm (\ell )]^2$ , where $\delta z^\pm (\ell )$ is an abbreviation of $\delta z^\pm (\boldsymbol{x},\ell ) = \text{sup}_{|\boldsymbol{r}|\lt \ell }|\delta z^\pm (\boldsymbol{x},\boldsymbol{r})|$ and $\delta z^\pm (\boldsymbol{x},\boldsymbol{r})$ is the increment of the Elsässer field over a separation $\boldsymbol{r}$ . Note that $\delta z^\pm (\ell )$ of Aluie & Eyink (Reference Aluie and Eyink2010) is different from the $\delta z^\pm (\ell )$ used in the LET, though related. In addition, LET is physically and practically appealing. For example, LET was used to identify regions of interest for studying kinetic dissipative processes (Sorriso-Valvo et al. Reference Sorriso-Valvo2019). This heuristic premise motivates us to investigate the LET in more detail. Therefore, in this work, we compare LET values $\xi _r$ across scales and select the most intense ones, i.e. those at which LET has the strongest impact, to compare with the pointwise dissipation rate $\varepsilon (\boldsymbol{x})$ .

In addition to the lag-dependent structure function representation, the separation of large-scale structures from turbulence can be achieved using scale-filtering techniques (Leonard Reference Leonard, ed. and Munn1975; Germano Reference Germano1992). We employ the scale-filter technique on quantity $f(\boldsymbol{x},t)$ , $\bar {f}_r(\boldsymbol{x},t) = \int f(\boldsymbol{x}^\prime ,t)G_r(\boldsymbol{x} - \boldsymbol{x}^\prime )\text{d}\boldsymbol{x}^\prime ,$ where $\bar {f}_r(\boldsymbol{x},t)$ is the filtered field at scale $r$ and $G_r(\boldsymbol{x}) = G(\boldsymbol{x}/r)/r^3$ is the filtering kernel. To ensure the magnitude of the field remains unchanged, it is required that $\int G(\boldsymbol{x})\,\text{d}\boldsymbol{x} = 1$ . Here, we select the normalised boxcar window function as $G(\boldsymbol{x})$ . By applying the scale filtering techniques to the velocity and magnetic fields in MHD turbulence, we can derive the equations for resolved large-scale kinetic and magnetic energy equations, see for example, Aluie (Reference Aluie2017) and Yang (Reference Yang2019), for both the incompressible and compressible MHD equations. The term arising from the filter operation is the scale-filtered energy flux $\varPi _r$ , which accounts for the energy across scales due to nonlinearities.

For the incompressible MHD turbulence in which we are interested here, the equations of the low-pass filtered kinetic energy and magnetic energy are

(1.8) \begin{align} \frac {\partial \bar {E}^{u}_{r}}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{J}^u_r = -\varPi ^u_r - T_r - D^u_r, \end{align}

(1.9) \begin{align} \frac {\partial \bar {E}^{b}_{r}}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{J}^b_r = -\varPi ^b_r + T_r - D^b_r, \end{align}

where $\bar {E}^u_r = ({1}/{2})\bar {u}^2$ is the large-scale kinetic energy, $\bar {E}^b_r = (1/2)\bar {b}^2$ is the large-scale magnetic energy, $\boldsymbol{J}^u_r = \bar {E}^u_r\bar {\boldsymbol{u}} - \bar {\boldsymbol{b}}\bar {\boldsymbol{b}}\boldsymbol{\cdot }\bar {\boldsymbol{u}} + (\boldsymbol{\tau }^u_r+\boldsymbol{\tau }^b_r)\boldsymbol{\cdot }\bar {\boldsymbol{u}} + \bar {P}\bar {\boldsymbol{u}} - (1/2)\nu \boldsymbol{\nabla }\bar {u}^2$ is the spatial transport of large-scale kinetic energy, $\boldsymbol{J}^b_r = \bar {E}^b_r\bar {\boldsymbol{u}} -\boldsymbol{\mathcal{E}}_r\times \bar {\boldsymbol{b}} - (1/2)\mu \boldsymbol{\nabla }\bar {b}^2$ is the spatial transport of large-scale magnetic energy, $\boldsymbol{\tau }^u_r = \overline {(\boldsymbol{uu})} - \bar {\boldsymbol{u}}\bar {\boldsymbol{u}}$ and $\boldsymbol{\tau }^b_r = -[\overline {(\boldsymbol{bb})} - \bar {\boldsymbol{b}}\bar {\boldsymbol{b}}]$ are the sub-grid-scale stresses, $\boldsymbol{\mathcal{E}}_r = \overline {(\boldsymbol{u}\times \boldsymbol{b})} - \bar {\boldsymbol{u}}\times \bar {\boldsymbol{b}}$ is a sub-grid-scale electric field, $\bar {P}$ is the scale-filtered total pressure, $\varPi ^u_r = -(\boldsymbol{\tau }^u_r + \boldsymbol{\tau }^b_r):\boldsymbol{\nabla }\bar {\boldsymbol{u}}$ is the kinetic energy flux across scale $r$ due to the sub-scale stresses, $\varPi ^b_r = -\boldsymbol{\mathcal{E}}_r\boldsymbol{\cdot }(\boldsymbol{\nabla }\times \bar {\boldsymbol{b}})$ is the magnetic energy flux across scale $r$ due to the sub-scale electromotive force, $T_r = \boldsymbol{\nabla }\bar {\boldsymbol{u}}:\bar {\boldsymbol{b}}\bar {\boldsymbol{b}}$ is the conversion between coarse-grained kinetic energy and magnetic energy due to large-scale magnetic line stretching, $D^u_r = \nu \boldsymbol{\nabla }\bar {\boldsymbol{u}}:(\boldsymbol{\nabla }\bar {\boldsymbol{u}})^{\text{T}}$ is the viscous dissipation term of large-scale kinetic energy, and $D^b_r = \mu \boldsymbol{\nabla }\bar {\boldsymbol{b}}:(\boldsymbol{\nabla }\bar {\boldsymbol{b}})^{\text{T}}$ is the Ohmic dissipation term. Therefore, the total scale-filtered energy flux is

(1.10) \begin{align} \varPi _r = \varPi ^u_r + \varPi ^b_r. \end{align}

The scale-filtering representation is a framework for investigating the mechanisms underlying the energy transfer in turbulence. Scale filter is based on real space coarse-graining at a specified length scale, which is widely used in large eddy simulation computational methods (Meneveau & Katz Reference Meneveau and Katz2000; Pope Reference Pope2004), in analytical theory (Eyink Reference Eyink2002; Aluie Reference Aluie2013), and recently in promising applications to turbulence in plasma models (Yang et al. Reference Yang, Matthaeus, Parashar, Haggerty, Roytershteyn, Daughton, Wan, Shi and Chen2017; Camporeale et al. Reference Camporeale, Sorriso-Valvo, Califano and Retinò2018; Eyink Reference Eyink2018; Yang et al. Reference Yang, Wan, Matthaeus, Sorriso-Valvo, Parashar, Lu, Shi and Chen2018; Cerri & Camporeale Reference Cerri and Camporeale2020; Teaca et al. Reference Teaca, Gorbunov, Told, Bañón Navarro and Jenko2021; Manzini et al. Reference Manzini, Sahraoui, Califano and Ferrand2022; Foldes et al. Reference Foldes, Cerri, Marino and Camporeale2024) and in anisotropic fluid turbulence (Foldes et al. Reference Foldes, Marino, Cerri and Camporeale2025a ). Recent works (Manzini, Sahraoui & Califano Reference Manzini, Sahraoui and Califano2024; Roy et al. Reference Roy, Bandyopadhyay, Adhikari, Yang and Matthaeus2025) using MMS data in the Earth’s magnetosheath show the scale-filtered energy flux across scales and find evidence of a balance over two decades in scales between the energy cascade and dissipation rates.

Note that $\varPi _r$ is rigorously derived from the low-pass filtered MHD equations. Its impacts also vary across scales like the LET, since other terms may also influence the energy transfer at different scales. Therefore, we also compare $\varPi _r$ at different scales, select the one with the most intense value, and make a comparison among $\varPi _r$ , LET ( $\xi _r$ ) and the point-wise dissipation rate $\varepsilon (\boldsymbol{x})$ .

The viscous ( $\varepsilon _\nu =2\nu S^u_{ij}S^u_{ij}$ ) and ohmic ( $\varepsilon _\mu =2\mu S^b_{ij}S^b_{ij}$ ) dissipation rates are the most accurate measure of energy dissipation at each point, where $S_{ij}^u = (1/2)(({\partial u_j}/{\partial x_i}) + ({\partial u_j}/{\partial x_i}))$ and $S_{ij}^b = (1/2)(({\partial b_j}/{\partial x_i}) + ({\partial b_j}/{\partial x_i}))$ are velocity and magnetic strain rate tensors. The dissipation rate of the Elsässer energy can be defined in a similar way, $\varepsilon ^\pm (\boldsymbol{x}) = 2\nu ^+S^\pm _{ij}S^\pm _{ij} + 2\nu ^-S^\pm _{ij}S^\mp _{ij}$ , where $\nu ^\pm =(1/2)(\nu \pm \mu )$ and $S^\pm _{ij}=(1/2)(({\partial z^\pm _j}/{\partial x_i})+({\partial z^\pm _i}/{\partial x_j}))$ . In our case, $\nu = \mu$ , so $\nu ^+ = \nu$ and $\nu ^- = 0$ , so

(1.11) \begin{align} \varepsilon ^\pm (\boldsymbol{x})= 2\nu S^\pm _{ij}S^\pm _{ij}. \end{align}

The total (viscous + ohmic) dissipation should then be

(1.12) \begin{align} \varepsilon ({\boldsymbol{x}})= \frac {1}{2}(\varepsilon ^+ (\boldsymbol{x})+ \varepsilon ^-(\boldsymbol{x})). \end{align}

The local dissipation rate in hydrodynamic turbulence was found to be intermittent (Meneveau & Sreenivasan Reference Meneveau and Sreenivasan1991; Stolovitzky, Kailasnath & Sreenivasan Reference Stolovitzky, Kailasnath and Sreenivasan1992), which has also been observed in the solar wind (Matthaeus et al. Reference Matthaeus, Wan, Servidio, Greco, Osman, Oughton and Dmitruk2015; Wan et al. Reference Wan, Matthaeus, Roytershteyn, Parashar, Wu and Karimabadi2016). To take into account the effect of intermittency, Kolmogorov revised his earlier theory and proposed the refined similarity hypothesis (RSH) (Kolmogorov Reference Kolmogorov1962), following the suggestions of Landau and Oboukhov (Oboukhov Reference Oboukhov1962; Landau & Lifshitz Reference Landau and Lifshitz1987), wherein a locally averaged energy dissipation rate, $\varepsilon _r$ was used:

(1.13) \begin{align} \varepsilon _r(\boldsymbol{x}) = \frac {3}{4\pi r^3}\int _{|\boldsymbol{x}^\prime | \leqslant r} \varepsilon (\boldsymbol{x}+\boldsymbol{x}^\prime )\,\text{d}\boldsymbol{x}^\prime , \end{align}

where the integration is taken over a sphere of radius $r$ centred on the given point $\boldsymbol{x}$ . In our work, a cubical average is used, that is, the integration in (1.13), is taken over a cubic region centred on the given point $\boldsymbol{x}$ ,

(1.14) \begin{align} \varepsilon _r(\boldsymbol{x}) = \frac {1}{(r+1)^3} \sum _{i=-r/2\Delta x}^{r/2\Delta x}\sum _{j=-r/2\Delta y}^{r/2\Delta y}\sum _{k=-r/2\Delta z}^{r/2\Delta z} \varepsilon (\boldsymbol{x} + (i\Delta x, j\Delta y, k\Delta z)), \end{align}

where $\boldsymbol{x}$ is the centre point, $r$ is the size of the cube in grid units and $\Delta x, \Delta y, \Delta z$ are grid spacings.

Despite the apparent parallels among several diagnostics of the energy transfer and dissipation processes in plasma turbulence, the understanding of the extent to which these diagnostics differ from each other is limited. To appreciate the physical content of the following comparisons, it is necessary to understand both the differences and the similarities of the structure function formulation of LET in (1.6) and the scale-filtering formulation in (1.10), and their associations with the dissipation rate in (1.12). The scale-decomposed formulations (1.6) and (1.10) share common elements in the physics they represent. Both are expressions of energy transfer across scale $\ell$ . These elements are in direct association with energy dissipation based on the energy transfer process. For example, when a well-separated inertial range is present, the energy flux terms should reach a plateau at the energy dissipation rate. In summary, one would expect that the energy transfer rate obtained from third-order structure functions (1.6) and the energy transfer flux in terms of scale filters (1.10) share global similarities, which has also been validated by Yang et al. (Reference Yang, Matthaeus, Roy, Roytershteyn, Parashar, Bandyopadhyay and Wan2022).

The present study aims to address the local properties of energy transfer, i.e. a systematic comparison among point-wise dissipation rate $\varepsilon (\boldsymbol{x})$ , LET $\xi _r$ in (1.6) and the scale-filtered energy flux $\varPi _r$ in (1.10). Detailed treatments based on the three aforementioned formulations are useful to describe the extent to which these dissipation/transfer flux proxies are correlated and their domain of validity. Data from MHD simulations will be primarily used, which provide complete velocity and magnetic field, and enable us to calculate the point-wise dissipation rate $\varepsilon$ . Since $\varepsilon$ is directly computed from viscosity, resistivity, and velocity and magnetic gradients, it is the most accurate quantity to reflect the energy dissipation at every point. Previous studies have noted that the spatial distribution of $\varepsilon$ is intermittent (Biskamp Reference Biskamp2003). Therefore, we will also calculate the locally averaged dissipation rate to smooth out some extreme values and to evaluate the effect of dissipation rate at scales larger than the dissipation scale. MHD simulation data will also enable us to accurately calculate energy-transfer terms across scales, such as LET and the scale-filtered energy flux $\varPi _r$ , and to analyse their scale dependence. The characteristics of dissipation rate $\varepsilon$ , LET $\xi _r$ and $\varPi _r$ have already been discussed separately in previous studies; our work here is to juxtapose them together and compare their local and scale-dependent properties.