Model of super-Alfv\'enic MHD turbulence and structure functions of polarization

Abstract

MHD turbulence driven at velocities higher than the Alfv\'en velocity, i.e., super-Alfv\'enic 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 $D^\phi(R)= \left\langle\sin^2(\phi_1-\phi_2) \right\rangle$, where $\phi$ 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-Alfv\'enic turbulence, the spectral slope of $D^\phi(\mathbf{R})$ 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 $D^\phi(R)$ spectral slope in super-Alfv\'enic turbulence and validate our predictions with numerical simulations. We demonstrate that for super-Alfv\'enic turbulence, the structure function $D^\phi (R)$ gets shallower with the increase of $M_A$. Our study makes $D^\phi (R)$ a valuable diagnostic of super-Alfv\'enic 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 $D^\phi$. We discuss the implications of our findings for turbulence and magnetic field studies in the intracluster and interstellar media.