The nature of Els\"{a}sser variables in compressible MHD

The Els\"{a}sser variables are often used in studies of plasma turbulence, in helping differentiate between MHD waves propagating parallel or anti-parallel to the main magnetic field. While for pure Alfv\'en waves in a homogeneous plasma the method is strictly valid, we show that compressible, magnetoacoustic waves are in general described by both Els\"{a}sser variables. Furthermore, in a compressible and inhomogeneous plasma, the pure MHD waves (Alfv\'en, fast and slow) are no longer normal modes, but waves become linearly coupled or display mixed properties of Alfv\'en and magnetoacoustic nature. These waves are necessarily described by both Els\"{a}sser variables and therefore the Els\"{a}sser formalism cannot be used to strictly separate parallel and anti-parallel propagating waves. Nevertheless, even in an inhomogeneous plasma, for a highly Alfv\'enic wave the Els\"{a}sser variable corresponding to the propagation direction appears still dominating. We suggest that for Alfv\'enic waves, the relative amplitude of Els\"{a}sser variables depends on the local degree of inhomogeneity and other plasma and wave properties. This finding has implications for turbulence studies in inhomogeneous and compressible plasmas, such as the solar corona and solar wind.