Energy Cascade Rate Measured in a Collisionless Space Plasma with MMS Data and Compressible Hall Magnetohydrodynamic Turbulence Theory

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.

Figure 1

Histogram for (a) the number density, velocity, and magnetic fluctuations and (b) the ion ${\beta }_i$ parameter, respectively.

Figure 2

(a) Magnetic field amplitude (black) and number density (blue), as a function of time. (b) Magnetic field spectra compensated by the power laws $f$ (blue) and $f^{5/3}$ (red), as a function of the frequency $f$ (solid bold line is the average spectrum obtained using a sliding window). (c) Energy cascade rates estimated from incompressible MHD (light blue), incompressible HMHD (orange), compressible MHD (blue), and compressible HMHD (red), as a function of the time lag $\tau$. Vertical dash-dotted and dotted gray lines in (b) and (c) correspond to the Taylor shifted ion skin depth and Larmor radius.

Figure 3

Same plots as in Fig. 2 but for another time interval.

Figure 4

Mean ratio of the compressible to incompressible cascade rates $|{ϵ}_{\text{CHall}}|/|{ϵ}_{\text{IHall}}|$ as a function of the mean ratio $|{ϵ}_{\mathrm{CMHD}}|/|{ϵ}_{\mathrm{IMHD}}|$. The color bar indicates the mean relative density fluctuations per event (72 events are analyzed) in the MS plasma. The error bars are the corresponding standard deviation in the MHD and subion ranges. The stars correspond to cases in Figs. 2 and 3.

Figure 5

Mean ratio of the nonflux terms to the compressible cascade rates in the subion scales $|{ϵ}_{\text{CHall}}^{\text{nonflux}}|/|{ϵ}_{\text{CHall}}|$ as a function of ratio in the MHD scales $|{ϵ}_{\mathrm{CMHD}}^{\text{nonflux}}|/|{ϵ}_{\mathrm{IMHD}}|$. The color bar indicates the mean level of density fluctuations per event (72 events are analyzed) in the MS plasma. The error bars are the corresponding standard deviation in the MHD and subion ranges. The stars correspond to cases in Figs. 2 and 3.