The first complete estimation of the compressible energy cascade rate |εC| at magnetohydrodynamic (MHD) and sub-ion scales is obtained in the Earth's magnetosheath using Magnetospheric MultiScale (MMS) spacecraft data and an exact law derived recently for compressible Hall MHD turbulence. A multi-spacecraft technique is used to compute the velocity and magnetic gradients, and then all the correlation functions involved in the exact relation. It is shown that when the density fluctuations are relatively small, |εC| identifies well with its incompressible analogue |εI| at MHD scales but becomes much larger than |εI| at sub-ion scales. For larger density fluctuations, |εC| is larger than |εI| at every scale with a value significantly higher than for smaller density fluctuations. Our study reveals also that for both small and large density fluctuations, the non-flux terms remain always negligible with respect to the flux terms and that the major contribution to |εC| at sub-ion scales comes from the compressible Hall flux.Introduction. Turbulence is a universal phenomenon observed from quantum to astrophysical scales [see e.g., 1-3]. It is mainly characterized by a nonlinear transfer (or cascade) of energy from a source to a sink. In astrophysical plasmas, fully developed turbulence plays a major role in several physical processes such as accretion flows around massive objects, star formation, solar wind heating or energy transport in planetary magnetospheres [e.g., 4-7]. In particular, the Earth's magnetosheath (MS) -the region of the solar wind downstream of the bow shock [8] -provides a unique laboratory to investigate compressible plasma turbulence. Indeed, a key feature of the MS plasma is the high level of density fluctuations in it, which can reach up to 100% of the background density [9][10][11], in contrast to the solar wind where it is ∼ 20% [12,13]. Thanks to the high time resolution data provided by the Magnetospheric Multi-Scale (MMS) mission in the Earth's MS [14], we are now able for the first time to measure the compressible energy cascade rate ε C from the magnetohydrodynamic (MHD) inertial range down to the sub-ion scales.
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