Evolution of the Earth's Magnetosheath Turbulence: A statistical study based on MMS observations

Composed of shocked solar wind, the Earth's magnetosheath serves as a natural laboratory to study the transition of turbulence from low Alfv{\'e}n Mach number, $M_\mathrm{A}$, to high $M_\mathrm{A}$. The simultaneous observations of magnetic field and plasma moments with unprecedented high temporal resolution provided by NASA's \textit{Magnetospheric Multiscale} Mission enable us to study the magnetosheath turbulence at both magnetohydrodynamics (MHD) and sub-ion scales. Based on 1841 burst-mode segments of MMS-1 from 2015/09 to 2019/06, comprehensive patterns of the spatial evolution of magnetosheath turbulences are obtained: (1) from the sub-solar region to the flanks, $M_\mathrm{A}$ increases from $<$ 1 to $>$ 5. At MHD scales, the spectral indices of the magnetic-field and velocity spectra present a positive and negative correlation with $M_\mathrm{A}$. However, no obvious correlations between the spectral indices and $M_\mathrm{A}$ are found at sub-ion scales. (2) from the bow shock to the magnetopause, the turbulent sonic Mach number, $M_{\mathrm{turb}}$, generally decreases from $>$ 0.4 to $<$ 0.1. All spectra steepen at MHD scales and flatten at sub-ion scales, representing a positive/negative correlations with $M_\mathrm{turb}$. The break frequency increases by 0.1 Hz when approaching the magnetopause for the magnetic-field and velocity spectra, while it remains at 0.3 Hz for the density spectra. (3) In spite of some differences, similar results are found for the quasi-parallel and quasi-perpendicular magnetosheath. In addition, the spatial evolution of magnetosheath turbulence is found to be independent of the upstream solar wind conditions, e.g., the Z-component of the interplanetary magnetic field and the solar wind speed.