We investigate how observations of synchrotron intensity fluctuations can be used to probe the sonic and Alfvénic Mach numbers of interstellar turbulence, based on mock observations performed on simulations of magnetohydrodynamic turbulence. We find that the structure function slope and a diagnostic of anisotropy that we call the integrated quadrupole ratio modulus both depend on the Alfvénic Mach number. However, these statistics also depend on the orientation of the mean magnetic field in the synchrotron emitting region relative to our line of sight, and this creates a degeneracy that cannot be broken by observations of synchrotron intensity alone. We conclude that the polarization of synchrotron emission could be analyzed to break this degeneracy, and suggest that this will be possible with the Square Kilometre Array.

Linearly polarized emission is described, in general, in terms of the Stokes parameters Q and U , from which the polarization intensity and polarization angle can be determined. Although the polarization intensity and polarization angle provide an intuitive description of the polarization, they are affected by the limitations of interferometric data, such as missing single-dish data in the u-v plane, from which radio frequency interferometric data is visualized. To negate the effects of these artefacts, it is desirable for polarization diagnostics to be rotationally and translationally invariant in the Q-U plane. One rotationally and translationally invariant quantity, the polarization gradient, has been shown to provide a unique view of spatial variations in the turbulent interstellar medium when applied to diffuse radio frequency synchrotron emission. In this paper we develop a formalism to derive additional rotationally and translationally invariant quantities. We present new diagnostics that can be applied to diffuse or point-like polarized emission in any waveband, including a generalization of the polarization gradient, the polarization directional curvature, polarization wavelength derivative, and polarization wavelength curvature. In Paper II we will apply these diagnostics to observed and simulated images of diffuse radio frequency synchrotron emission.

We have investigated the magneto-ionic turbulence in the interstellar medium through spatial gradients of the complex radio polarization vector in the Canadian Galactic Plane Survey (CGPS). The CGPS data cover 1300 square-degrees, over the range 53 • ≤ ≤ 192 • , −3 • ≤ b ≤ 5 • with an extension to b = 17.5 • in the range 101 • ≤ ≤ 116 • , and arcminute resolution at 1420 MHz. Previous studies found a correlation between the skewness and kurtosis of the polarization gradient and the Mach number of the turbulence, or assumed this correlation to deduce the Mach number of an observed turbulent region. We present polarization gradient images of the entire CGPS dataset, and analyze the dependence of these images on angular resolution. The polarization gradients are filamentary, and the length of these filaments is largest towards the Galactic anti-center, and smallest towards the inner Galaxy. This may imply that small-scale turbulence is stronger in the inner Galaxy, or that we observe more distant features at low Galactic longitudes. For every resolution studied, the skewness of the polarization gradient is influenced by the edges of bright polarization gradient regions, which are not related to the turbulence revealed by the polarization gradients. We also find that the skewness of the polarization gradient is sensitive to the size of the box used to calculate the skewness, but insensitive to Galactic longitude, implying that the skewness only probes the number and magnitude of the inhomogeneities within the box. We conclude that the skewness and kurtosis of the polarization gradient are not ideal statistics for probing natural magneto-ionic turbulence.

Diagnostics of polarized emission provide us with valuable information on the Galactic magnetic field and the state of turbulence in the interstellar medium, which cannot be obtained from synchrotron intensity alone. In Paper I (Herron et al. 2017b), we derived polarization diagnostics that are rotationally and translationally invariant in the Q-U plane, similar to the polarization gradient. In this paper, we apply these diagnostics to simulations of ideal magnetohydrodynamic turbulence that have a range of sonic and Alfvénic Mach numbers. We generate synthetic images of Stokes Q and U for these simulations, for the cases where the turbulence is illuminated from behind by uniform polarized emission, and where the polarized emission originates from within the turbulent volume. From these simulated images we calculate the polarization diagnostics derived in Paper I, for different lines of sight relative to the mean magnetic field, and for a range of frequencies. For all of our simulations, we find that the polarization gradient is very similar to the generalized polarization gradient, and that both trace spatial variations in the magnetoionic medium for the case where emission originates within the turbulent volume, provided that the medium is not supersonic. We propose a method for distinguishing the cases of emission coming from behind or within a turbulent, Faraday rotating medium, and a method to partly map the rotation measure of the observed region. We also speculate on statistics of these diagnostics that may allow us to constrain the physical properties of an observed turbulent region.

Previous studies have shown that star formation depends on the driving of molecular cloud turbulence, and differences in the driving can produce an order of magnitude difference in the star formation rate. The turbulent driving is characterised by the parameter ζ, with ζ = 0 for compressive, curl-free driving (e.g. accretion or supernova explosions), and ζ = 1 for solenoidal, divergence-free driving (e.g. Galactic shear). Here we develop a new method to measure ζ from observations of synchrotron emission from molecular clouds. We calculate statistics of mock synchrotron intensity images produced from magnetohydrodynamic simulations of molecular clouds, in which the driving was controlled to produce different values of ζ. We find that the mean and standard deviation of the log-normalised synchrotron intensity are sensitive to ζ, for values of ζ between 0 (curl-free driving) and 0.5 (naturally-mixed driving). We quantify the dependence of zeta on the direction of the magnetic field relative to the line of sight. We provide best-fit formulae for ζ in terms of the log-normalised mean and standard deviation of synchrotron intensity, with which ζ can be determined for molecular clouds that have similar Alfvénic Mach number to our simulations. These formulae are independent of the sonic Mach number. Signal-to-noise ratios larger than 5, and angular resolutions smaller than 5% of the cloud diameter, are required to apply these formulae. Although there are no firm detections of synchrotron emission from molecular clouds, by combining Green Bank Telescope and Very Large Array observations it should be possible to detect synchrotron emission from molecular clouds, thereby constraining the value of ζ.