Model of super-Alfvénic MHD turbulence and structure functions of polarization

TL;DR

This work develops an analytical framework and numerical validation for the polarization-angle structure function $D^\phi(R)$ as a diagnostic of super-Alfvénic turbulence. It shows that LOS integration across $l_A$-sized domains and the presence of a mean field render the PA slope shallower as $M_A$ grows, with explicit expressions for $D_0^\phi(R)$ and its LOS-summed form, and introduces the Spectrum of Directions $\mathcal{F}_\phi$ as a complementary diagnostic. The authors validate predictions with high-$\beta$ 3D MHD simulations and discuss practical implications for inferring $M_A$ (and thus magnetic-field strength via MM2) from polarization observations in galaxy clusters and molecular clouds, as well as the use of $D^p$ and polarization-gradients as synergistic probes. The results provide a robust polarization-based toolkit to study turbulence magnetization, offering a path to quantify magnetic-field strength and structure in environments where the magnetic field is dynamically subdominant at injection but impacts small-scale fluctuations.

Abstract

MHD turbulence driven at velocities higher than the Alfvén velocity, i.e., super-Alfvénic turbulence, is widely spread in astrophysical environments, including galaxy clusters and molecular clouds. For statistical studies of such turbulence, we explore the utility of the polarization angle structure functions $D^φ(R)= \left\langle\sin^2(φ_1-φ_2) \right\rangle$, where $φ$ denotes the polarization angle measured at points separated by a projected distance $\mathbf{R}$ on the plane of the sky. Lazarian, Yuen and Pogosyan, 2022, showed that in the case of super-Alfvénic turbulence, the spectral slope of $D^φ(\mathbf{R})$ differs from that of the underlying magnetic fluctuations, limiting its applicability for field strength estimation with known techniques. In this work, we provide an analytical framework that explains the modification of the $D^φ(R)$ spectral slope in super-Alfvénic turbulence and validate our predictions with numerical simulations. We demonstrate that for super-Alfvénic turbulence, the structure function $D^φ(R)$ gets shallower with the increase of $M_A$. Our study makes $D^φ(R)$ a valuable diagnostic of super-Alfvénic turbulence and opens a way to obtain $M_A$ from observations. We also explore numerically the structure function of the polarization degree and the spectrum of the polarization directions, the latter being the Fourier transform of $D^φ$. We discuss the implications of our findings for turbulence and magnetic field studies in the intracluster and interstellar media.

Figures (10)

• Figure 1: Spectra of velocity (left) and magnetic field under different $M_A$ conditions. The dashed line represents the expected scaling for Kolmogorov turbulence $E_k \sim k^{-5/3}$ over the inertial range. The magnetic field spectrum in sub-Alfvénic regime is similar to that of velocity, but in super-Alfvénic regime, the spectrum becomes shallower, indicating that magnetic energy is subdominant for large scales. The dashed red and blue line represents the transition wavenumber $k_A$ for $M_A\approx1.5$ and $M_A\approx3.0$, respectively.
• Figure 2: $D^\phi$ for super-Alfvénic turbulence (orange line) compared to the $D^\phi$ for an individual domain (blue line).
• Figure 3: Polarization properties when measured perpendicular to the mean magnetic field. Left and middle column: 2D maps of the Stokes parameters $Q$ and $U$ in sub-Alfvénic $M_A=0.8$ (top row), trans-Alfvénic $M_A=1.5$ (middle row), and super-Alfvénic $M_A=3.0$ (middle row) conditions. Right column: the structure function of polarization angle $D^\phi$. $l_{\rm diss}$ is the numerical dissipatio scale and $l_A$ is the transition scale to strong turbulence regime. The dashed line represents the expected scaling for Kolmogorov-type turbulence, and the dotted dashed line is the expectation of Eq. \ref{['eq:DfSuperA']}. The subpanels illustrate the structure functions as the separation $R \to 0$.
• Figure 4: A comparison of the polarization angle's structure function $D^\phi$ in $M_A=1.5$ (left) and $M_A=3.0$ (right) conditions. The solid line means the Stokes parameters $Q$ and $U$ are constructed in the dust polarization manner, while the dashed line represents the synchrotron polarization.
• Figure 5: Projected magnetic field and polarization directions for turbulence with $M_A=3$.