A new perspective on turbulent Galactic magnetic fields through
  comparison of linear polarisation decomposition techniques
  We compare two rotationally invariant decomposition techniques on linear
polarisation data: the spin-2 spherical harmonic decomposition in two opposite
parities, the $E$- and $B$-mode, and the multiscale analysis of the gradient of
linear polarisation, $|\nabla \mathbf{P}|$. We demonstrate that both
decompositions have similar properties in the image domain and the spatial
frequency domain. They can be used as complementary tools for turbulence
analysis of interstellar magnetic fields in order to develop a better
understanding of the origin of energy sources for the turbulence, the origin of
peculiar magnetic field structures and their underlying physics. We also
introduce a new quantity $|\nabla EB|$ based on the $E$- and $B$-modes and we
show that in the intermediate and small scales limit $|\nabla EB| \simeq
|\nabla \mathbf{P}|$. Analysis of the 2.3 GHz S-band Polarization All Sky
Survey (S -PASS) shows many extended coherent filament-like features appearing
as 'double-jumps' in the $|\nabla \mathbf{P}|$ map that are correlated with
negative and positive filaments of $B$-type polarisation. These local
asymmetries between the two polarisation types, $E$ and $B$, of the non-thermal
Galactic synchrotron emission have an influence on the $E$- and $B$-mode power
spectra analyses. The wavelet-based formalism of the polarisation gradient
analysis allows us to locate the position of $E$- or $B$-mode features
responsible for the local asymmetries between the two polarisation types. In
analysed subregions, the perturbations of the magnetic field are trigged by
star clusters associated with HII regions, the Orion-Eridanus superbubble and
the North Polar Spur at low Galactic latitude.
