Angular dependence of third-order law in anisotropic MHD turbulence

TL;DR

The paper addresses how anisotropy in MHD turbulence, driven by a mean magnetic field, biases energy-dissipation estimates obtained from the third-order law when observations have limited directional sampling. It develops and tests three angle-averaging schemes in DNS of 3D incompressible MHD for varying $B_0$, analyzes the Yaglom flux $Y_\ell^{\pm}$ and the divergence of the energy-flux vector, and compares with axisymmetric models (Podesta, Galtier) while validating with virtual-spacecraft data. The key finding is that the azimuthally averaged third-order statistic at $\theta \approx 60^\circ$ closely reproduces the full direction-averaged dissipation rate across $0 \le B_0/b_{rms} \le 5$, with modest errors that grow with $B_0$; this angle corresponds to a near-zero polar contribution to the flux-divergence, consistent with a Mean Value Theorem of Integrals argument. The results provide a practical guideline for solar-wind measurements and other turbulent plasmas, suggesting that targeting observations near $60^\circ$ relative to the mean field yields robust energy-transfer estimates even with limited angular sampling, and that axisymmetric models can capture essential polar dependencies for $Y_\theta$ and $Y_\rho$.

Abstract

In solar wind turbulence, the energy transfer/dissipation rate is typically estimated using MHD third-order structure functions calculated using spacecraft observations. However, the inherent anisotropy of solar wind turbulence leads to significant variations in structure functions along different observational directions, thereby affecting the accuracy of energy-dissipation rate estimation. An unresolved issue is how to optimise the selection of observation angles under limited directional sampling to improve estimation precision. We conduct a series of MHD turbulence simulations with different mean magnetic field strengths, $ B_0 $. Our analysis of the third-order structure functions reveals that the global energy dissipation rate estimated around a polar angle of $ θ= 60^\circ$ agrees reasonably with the exact one for $ 0 \le B_0/b_{rms} \le 5 $, where $b_{rms}$ denotes the root-mean-square magnetic field fluctuation. The speciality of $60^\circ$ polar angle can be understood by the Mean Value Theorem of Integrals, since the spherical integral of the polar-angle component ($\widetilde{T_θ}$) of the divergence of Yaglom flux is zero, and $\widetilde{T_θ}$ changes sign around 60$^\circ$. Existing theory on the energy flux vector as a function of the polar angle is assessed, and supports the speciality of $60^\circ$ polar angle. The angular dependence of the third-order structure functions is further assessed with virtual spacecraft data analysis. The present results can be applied to measure the turbulent dissipation rates of energy in the solar wind, which are of potential importance to other areas in which turbulence takes place, such as laboratory plasmas and astrophysics.

Figures (9)

• Figure 1: Current intensity ($J=|\nabla\times \textbf{b}|$) in domain cross-sections that are (a,b,c) parallel--perpendicular ($z$--$x$) and (d,e,f) strictly perpendicular ($x$--$y$) with respect to the mean magnetic field, for $B_0=$ 0 (panels a,d), 2 (b,e), and 5 (c,f). Note the increasing alignment of structures with the mean field direction seen in the top row.
• Figure 2: Normalized longitudinal third-order structure functions, $Y_\ell=(Y^+_\ell + Y^-_\ell)/2$, for various types of angle averaging. (a) Azimuthal average using method II, $\widetilde{ Y_\ell}$ in Eq. \ref{['eq:3rd_order_sf_2D_lag']}, for the $B_0=5$ simulation at indicated values of $\theta$. The solid line with triangles represents the average of these and is equivalent to the direction-averaged profile obtained via method III, Eq. \ref{['eq:3rd_order_sf_1D_lag']}: $-3(\overline{ Y_\ell^+}+\overline{ Y_\ell^-})/({8\varepsilon \ell})$. (b) Direction-averaged (Eq. \ref{['eq:3rd_order_sf_1D_lag']}, lines with triangle symbols) and azimuthally averaged at fixed $\theta = 60^\circ$ (Eq. \ref{['eq:3rd_order_sf_2D_lag']}, lines without triangle symbols) for the $B_0=2$ and 5 simulations. Recall $\theta$ is the angle between the lag vector and the external mean magnetic field, $B_0 \boldsymbol{e}_z$. Time-averaging has been employed in all cases; see Table \ref{['table:setup']}. In panel (b), the results for $B_0=2$ are shifted by -0.1 to avoid visual clutter with the $B_0=5$ results.
• Figure 3: Top row: image plots of normalized divergence of (azimuthally averaged) energy flux $\widetilde{\boldsymbol{Y}}$. Bottom row: selected cuts at constant $\theta$ from the top row. Columns are for (i): $B_0 =0$; (ii): $B_0 =2$; (iii): $B_0 = 5$. Recall $\theta$ is the angle between the lag $\boldsymbol{\ell}$ and $\boldsymbol{B}_0$. Black dashed arcs indicate inertial range boundaries, i.e., normalized lag lengths $\ell^*_\parallel, \ell^*_\perp \in [6,55]$, as identified using the direction-averaged form of the third-order law, i.e., method III, Eq. \ref{['eq:3rd_order_sf_1D_lag']}, with $-3(\overline{ Y_l^+}+\overline{ Y_l^-})/({8\varepsilon l})$ above a threshold, here 0.9.
• Figure 4: Image plots of the (top row) lag contribution $\widetilde{T_\ell}$ and (bottom row) polar contribution $\widetilde{T_\theta}$ to the normalized divergence of the (azimuthally averaged) energy flux for $B_0=0$ (panels a,d), 2 (b,e), and 5 (c,f). Black dashed arcs indicate inertial range boundaries, i.e., normalized lag lengths $\ell^*_\parallel,\ell^*_\perp \in [6,55]$, as identified using the direction-averaged form of the third-order law, i.e., method III, Eq. \ref{['eq:3rd_order_sf_3D_lag']}, with $-3(\overline{ Y_\ell^+}+\overline{ Y_\ell^-})/({8\varepsilon \ell})$ above a threshold, here 0.9. Plotted quantities have been averaged over time and $\phi$.
• Figure 5: Verification of the Podesta (blue, stars) and Galtier (red, circles) models for $\boldsymbol{Y}$. The dashed, dash-dotted, and solid curves are results for the $Y_\rho, Y_\ell, Y_\theta$ components, respectively. Curves without symbols are obtained directly from DNS data. (a) $B_0= 0$, (b) $B_0 = 2$, and (c) $B_0 = 5$.