Exact relations in fully developed turbulence: energy cascade rate from the MHD to the ion-scales

Nahuel Andres
Institute of Astronomy and Space Physics, UBA-CONICET, Buenos Aires, Argentina
Exact laws derived for incompressible magnetohydrodynamics (IMHD) turbulence have been widely used to gain insight into the problem of solar wind (SW) heating through the estimation of the turbulent energy cascade rate. While the incompressibilty assumption can, to some extent, be justified to address large scale SW turbulence where alfv\'enic fluctuations dominate, it is likely to fail to accurately describe sub-ion scale physics, as well as other more compressible plasmas such as planetary magnetospheres or the interstellar medium. Here, we will review a set of recent analytical and numerical results obtained for compressible flows within the isothermal closure. First, we will discuss the new exact law derived for compressible MHD (CMHD) and emphasize the major differences with IMHD, in particular the role of the mean (background) magnetic field and plasma density. In the next step, we will discuss the extension of the laws to compressible Hall-MHD (CHMHD) and discuss the physics brought up by the new terms due to the Hall current. The incompressiblity limit is further studied using a more compact form that include only increments of the turbulent fields and compared to previous derivations. The validation of the various exact laws are done using 3D direct numerical simulations (GHOST code for the compressible flows and TURBO for the incompressible models). Potential applications of the models to estimate the energy cascade rate of turbulence over a broad range of scales that span both the inertial and sub-ion (dispersive) ranges in spacecraft data will be discussed.