Local approach to the study of energy transfers in incompressible magnetohydrodynamic turbulence

We present a local approach to the study of scale-to-scale energy transfers in magnetohydrodynamic (MHD) turbulence. This approach is based on performing local averages of the physical fields, which amounts to filtering scales smaller than some parameter $\ell$. A key step in this work is the derivation of a local Kármán-Howarth-Monin relation which provides a local form of Politano and Pouquet's 4/3 law, without any assumption of homogeneity or isotropy. Our approach is exact and nonrandom, and we show its connection to the usual statistical results of turbulence. Its implementation on data obtained via a three-dimensional direct numerical simulation of the forced incompressible MHD equations from the John Hopkins turbulence database constitutes the main part of our study. First, we show that the local Kármán-Howarth-Monin relation holds well. The space statistics of local cross-scale transfers are studied next, their means and standard deviations being maximum at inertial scales and their probability density functions (PDFs) displaying very wide tails. Events constituting the tails of the PDFs are shown to form structures of strong transfers, either positive or negative, which can be observed over the whole available range of scales. As $\ell$ is decreased, these structures become more and more localized in space while contributing to an increasing fraction of the mean energy cascade rate. Finally, we highlight their quasi-one-dimensional (filamentlike) or quasi-two-dimensional (sheetlike or ribbonlike) nature and show that they appear in areas of strong vorticity or electric current density.

Figure 1

Typical slices of the physical fields in the $(x,y)$ plane of forcing for the snapshot considered in our analysis. For vector fields, arrows represent the in-plane component and the colormap represents the out-of-plane component. The vector fields have been normalized by their norm.

Figure 2

Kármán-Howarth-Monin relation as a function of scale $\ell$. (a) Balance between the left- and right-hand sides of Eq. (25). (b) Contribution of each term constituting (25) to the balance displayed in (a). Note that the solid lines, which represent terms with a plus superscript, are almost superimposed with the symbols representing the corresponding terms with the minus superscript. (c) and (e) Local balance between the left- and right-hand sides of Eq. (10) at the same point. (d) and (f) Contribution of each term constituting (10) to the balance displayed in (c) and (e), respectively.

Figure 3

Typical slices of ${\mathrm{\Pi }}_{\ell }^T$ in the same plane as in Fig. 1 at three different scales: (a) close to the injection scale, (b) in the inertial range, and (c) in the dissipative range. The mean and variance of ${\mathrm{\Pi }}_{\ell }^T$ have been set to zero and unity, respectively, at all three scales for better comparison. We observe that local transfers are organized into structures existing over a wide range of scales.

Figure 4

(a) Global scale-to-scale transfers as a function of scale $\ell$. (b) Standard deviation ${\sigma }_{\ell }^T$ of ${\mathrm{\Pi }}_{\ell }^T$ as a function of scale $\ell$. (c) Probability distribution functions of the local scale-to-scale transfers ${\mathrm{\Pi }}_{\ell }^T$. The scales represented are, from top to bottom, $\ell /L=0.0078$, 0.0352, 0.125, 0.352, and 0.996. All curves have been arbitrarily shifted vertically for the sake of clarity, and their mean and variance have been respectively set to zero and unity.

Figure 5

(a) Estimate of the fraction of space occupied by events which deviate more than $n\times {\sigma }_{\ell }^T$ from their mean as a function of $n$. (b) Estimate of the contribution to global scale-to-scale transfers of events which deviate more than $n\times {\sigma }_{\ell }^T$ from their mean as a function of $n$. (c) Fraction of space in which a given fraction of the global scale-to-scale transfers is contained. The scales represented are $\ell /L=0.0078$ [blue (top) solid line], 0.0352 [red (top) dashed line], 0.125 [yellow (middle) solid line], 0.352 [purple (bottom) dashed line], and 0.996 [green (bottom) solid line].

Figure 6

Three-dimensional organization of cross-scale transfer structures at $\ell /L=0.125$ (inertial scale) for $n=4$. We observe that the structures are (b) sheetlike or ribbonlike or (c) filamentlike.

Figure 7

Slices of (a) the vorticity component ${\omega }_{y,\ell }$ and (b) the electric current density $j_{z,\ell }$ in the same plane as in Figs. 1 and 3, at $\ell /L=0.125$ (inertial scale). Comparing with Fig. 3, we observe that structures of scale-to-scale transfers are located in regions of either high vorticity or vertical currents.