Do we have critical balance in compressible MHD turbulence?

The concept of critical balance in MHD turbulence theory suggests that the ratio between the nonlinear and Alfvén wave propagation times, denoted as χ, is approximately equal to unity across scales, resulting in a scale-dependent anisotropy described by the Goldreich-Sridhar relation (k_ll~k_⊥^2/3). The validity of the critical balance postulate is crucial for explaining the underlying physical mechanisms behind various astrophysical and space phenomena. However, recent simulations and space observations reveal that the distribution of χ exhibits a broad wing and is not peaked at unity for all three MHD modes, which challenges the foundations of the Goldreich-Sridhar theory of MHD turbulence. Motivated by the prevalence of non-zero low-frequency fluctuations observed in compressible MHD turbulence, we analytically derive and numerically verify a new form of "generalized critical balance" that is applicable to all three MHD modes. In particular, the compressible modes exhibit distinct scale-independent anisotropic amplitudes, particularly in low β regime. We also discuss the implications of our findings for recent space and astrophysical observations.