Interplay between Alfv\'en and magnetosonic waves in compressible magnetohydrodynamics turbulence

Using spatio-temporal spectra we show direct evidence of excitation of magnetosonic and Alfv'en waves in three-dimensional compressible magnetohydrodynamic turbulence at small Mach numbers. For the plasma pressure dominated regime, or high $\beta$ regime (with $\beta$ the ratio between fluid and magnetic pressure), and for the magnetic pressure dominated regime, or low $\beta$ regime, we study magnetic field fluctuations parallel and perpendicular to a guide magnetic field ${\mathbf{\text{B}}}_0$. In the low $\beta$ case we find excitation of compressible and incompressible fluctuations, with a transfer of energy towards Alfv'enic modes and to a lesser extent towards magnetosonic modes. In particular, we find signatures of the presence of fast magnetosonic waves in a scenario compatible with that of weak turbulence. In the high $\beta$ case, fast and slow magnetosonic waves are present, with no clear trace of Alfv'en waves, and a significant part of the energy is carried by two-dimensional turbulent eddies.