The first complete estimation of the compressible energy cascade rate $|{ϵ}_{\mathrm{C}}|$ at magnetohydrodynamic (MHD) and subion scales is obtained in Earth’s magnetosheath using Magnetospheric MultiScale spacecraft data and an exact law derived recently for compressible Hall MHD turbulence. A multispacecraft 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, $|{ϵ}_{\mathrm{C}}|$ identifies well with its incompressible analog $|{ϵ}_{\mathrm{I}}|$ at MHD scales but becomes much larger than $|{ϵ}_{\mathrm{I}}|$ at subion scales. For larger density fluctuations, $|{ϵ}_{\mathrm{C}}|$ is larger than $|{ϵ}_{\mathrm{I}}|$ 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 nonflux terms remain always negligible with respect to the flux terms and that the major contribution to $|{ϵ}_{\mathrm{C}}|$ at subion scales comes from the compressible Hall flux.

A comparison is made between several existing exact laws in incompressible Hall magnetohydrodynamic turbulence in order to show their equivalence, despite stemming from different mathematical derivations. Using statistical homogeneity, we revisit the law proposed by Hellinger et al. and show that it can be written, after being corrected by a multiplicative factor, in a more compact form implying only flux terms expressed as increments of the turbulent fields. The Hall contribution of this law is tested and compared to other exact laws derived by Galtier and Banerjee & Galtier using direct numerical simulations of three-dimensional electron MHD turbulence with a moderate mean magnetic field. We show that the studied laws are equivalent in the inertial range, thereby offering several choices on the formulation to use depending on the needs. The expressions that depend explicitly on a mean (guide) field may lead to residual errors in estimating the energy cascade rate; however, we demonstrate that this guide field can be removed from these laws after mathematical manipulation. Therefore, it is recommended to use an expression independent of the mean guide field to analyze numerical or in situ spacecraft data.

Three-dimensional direct numerical simulations are used to study the energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. Our analysis is guided by a two-point exact law derived recently for this problem in which flux, source, hybrid and mixed terms are present. The relative importance of each term is studied for different initial subsonic Mach numbers $M_{S}$ and different magnetic guide fields $\boldsymbol{B}_{0}$ . The dominant contribution to the energy cascade rate comes from the compressible flux, which depends weakly on the magnetic guide field $\boldsymbol{B}_{0}$ , unlike the other terms whose moduli increase significantly with $M_{S}$ and $\boldsymbol{B}_{0}$ . In particular, for strong $\boldsymbol{B}_{0}$ the source and hybrid terms are dominant at small scales with almost the same amplitude but with a different sign. A statistical analysis undertaken with an isotropic decomposition based on the SO(3) rotation group is shown to generate spurious results in the presence of $\boldsymbol{B}_{0}$ , when compared with an axisymmetric decomposition better suited to the geometry of the problem. Our numerical results are compared with previous analyses made within situmeasurements in the solar wind and the terrestrial magnetosheath.

The exact law for fully developed homogeneous compressible magnetohydrodynamics (CMHD) turbulence is derived. For an isothermal plasma, without the assumption of isotropy, the exact law is expressed as a function of the plasma velocity field, the compressible Alfvén velocity, and the scalar density, instead of the Elsasser variables used in previous works. The theoretical results show four different types of terms that are involved in the nonlinear cascade of the total energy in the inertial range. Each category is examined in detail, in particular, those that can be written either as source or flux terms. Finally, the role of the background magnetic field ${\mathbf{B}}_0$ is highlighted and a comparison with the incompressible MHD (IMHD) model is discussed. This point is particularly important when testing this exact law on numerical simulations and in situ observations in space plasmas.

The scaling of the turbulent spectra provides a key measurement that allows us to discriminate between different theoretical predictions of turbulence. In the solar wind, this has driven a large number of studies dedicated to this issue using in situ data from various orbiting spacecraft. While a semblance of consensus exists regarding the scaling in the magnetohydrodynamic (MHD) and dispersive ranges, the precise scaling in the transition range and the actual physical mechanisms that control it remain open questions. Using the high-resolution data in the inner heliosphere from the Parker Solar Probe mission, we find that the sub-ion scales (i.e., at the frequency f ∼ [2, 9] Hz) follow a power-law spectrum f ^α with a spectral index α varying between −3 and −5.7. Our results also show that there is a trend toward an anticorrelation between the spectral slopes and the power amplitudes at the MHD scales, in agreement with previous studies: the higher the power amplitude the steeper the spectrum at sub-ion scales. A similar trend toward an anticorrelation between steep spectra and increasing normalized cross helicity is found, in agreement with previous theoretical predictions about the imbalanced solar wind. We discuss the ubiquitous nature of the ion transition range in solar wind turbulence in the inner heliosphere.

The first computations of the compressible energy transfer rate from ∼0.2 up to ∼1.7 au is obtained using Parker Solar Probe (PSP), Time History of Events and Macroscale Interactions during Substorms (THEMIS), and Mars Atmosphere and Volatile EvolutioN (MAVEN) observations. Using a recently derived exact relation for isothermal magnetohydrodynamics turbulence, the compressible energy cascade rate, ε _C , is computed for hundred of events at different heliocentric distances, for time intervals when the spacecrafts were in the pristine solar wind. The observational results show moderate increases of ε _C with respect to the incompressible cascade rate, ε _I . Depending on the level of compressibility in the plasma, which reaches up to 25% at PSP’s perihelion, the different terms in the exact compressible relation are shown to have different impacts on the total cascade rate, ε _C . Finally, the observational results are connected with the local ion temperature and the solar wind heating problem.