Magnetic field strength from turbulence theory (I): Using differential
  measure approach (DMA)
  The mean plane-of-sky magnetic field strength is traditionally obtained from
the combination of polarization and spectroscopic data using the
Davis-Chandrasekhar-Fermi (DCF) technique. However, we identify the major
problem of the DCF to be its disregard of the anisotropic character of MHD
turbulence. On the basis of the modern MHD turbulence theory we introduce a new
way of obtaining magnetic field strength from observations. Unlike the DCF, the
new technique uses not the dispersion of the polarization angle and line of
sight velocities, but increments of these quantities given by the structure
functions. To address the variety of the astrophysical conditions for which our
technique can be applied, we consider the turbulence in both media with
magnetic pressure larger than the gas pressure corresponding e.g. to molecular
and the gas pressure larger than the magnetic pressure corresponding to the
warm neutral medium. We provide general expressions for arbitrary admixture of
Alfv\'en, slow and fast modes in these media and consider in detail the
particular cases relevant to diffuse media and molecular clouds. We
successfully test our results using synthetic observations obtained from MHD
turbulence simulations. We demonstrate that our Differential Measure Approach
(DMA), unlike the DCF, can be used to measure the distribution of magnetic
field strengths, can provide magnetic field measurements with limited data and
is much more stable in the presence of large scale variations induces of
non-turbulent nature. In parallel, our study uncover the deficiencies of the
earlier DCF research.
