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The first estimation of the energy cascade rate ${|\epsilon_C|}$ ofmagnetosheath turbulence is obtained using the CLUSTER and THEMIS spacecraftdata and an exact law of compressible isothermal magnetohydrodynamicsturbulence. ${|\epsilon_C|}$ is found to be of the order of ${10{^{-13}}J.m^{3}.s^{-1}}$, at least two orders of magnitude larger than its value in thesolar wind (order of ${10{^{-16}} J.m^{3}.s^{-1}}$ in the fast wind). Two typesof turbulence are evidenced and shown to be dominated either by incompressibleAlfv\'enic or magnetosonic-like fluctuations. Density fluctuations are shown toamplify the cascade rate and its spatial anisotropy in comparison withincompressible Alfv\'enic turbulence. Furthermore, for compressiblemagnetosonic fluctuations, large cascade rates are found to lie mostly near thelinear kinetic instability of the mirror mode. New empirical power-laws areevidenced and relate ${|\epsilon_C|}$ to the turbulent Mach number and theinternal energy. These new finding have potential applications in distantastrophysical plasmas that are not accessible to in situ measurements.

Compressible magnetohydrodynamic turbulence in the Earth's magnetosheath: estimation of the energy cascade rate using in situ spacecraft data