Energy Cascade Rate in Compressible Hall-MHD: Theory vs. MMS Observations

Energy cascade rate derived from Incompressible MHD theory (IMHD) has been widely used to study inertial range solar wind (SW) turbulence in view of better understanding the longstanding problem of SW heating. In recent years the IMHD theory was extended to compressible flows described within the isothermal approximation [1,2] and applied to investigate the role of density fluctuations in inertial range SW and magnetosheath turbulence [3]. Here we discuss the extension of those theoretical and observational models to describe sub-ion scale cascade using new incompressible and compressible Hall-MHD models (resp. IHMHD and CHMHD) [4,5]. We will discuss the application of these new models to the MMS in-situ observations in the magnetosheath and the SW. Comparing the results obtained from IHMHD and CHMHD highlights the importance of density fluctuations at the sub-ion scale, even when the inertial range is nearly incompressible, in agreement with theoretical expectation. We will discuss other features such as the link between the &fluid & sub-ion cascade and &kinetic & features like plasma instabilities and kinetic damping processes.

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