Compressible Magnetohydrodynamic Turbulence in the Earth’s Magnetosheath: Estimation of the Energy Cascade Rate Using in situ Spacecraft Data

The first estimation of the energy cascade rate $|{\epsilon }_C|$ of magnetosheath turbulence is obtained using the Cluster and THEMIS spacecraft data and an exact law of compressible isothermal magnetohydrodynamics turbulence. The mean value of $|{\epsilon }_C|$ is found to be close to ${10}^{-13}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, at least 2 orders of magnitude larger than its value in the solar wind ($\sim {10}^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ in the fast wind). Two types of turbulence are evidenced and shown to be dominated either by incompressible Alfvénic or compressible magnetosoniclike fluctuations. Density fluctuations are shown to amplify the cascade rate and its spatial anisotropy in comparison with incompressible Alfvénic turbulence. Furthermore, for compressible magnetosonic fluctuations, large cascade rates are found to lie mostly near the linear kinetic instability of the mirror mode. New empirical power-laws relating $|{\epsilon }_C|$ to the turbulent Mach number and to the internal energy are evidenced. These new findings have potential applications in distant astrophysical plasmas that are not accessible to in situ measurements.

Figure 1

(a), (e) The magnetic field modulus (black) and fluctuations (blue), (b), (f) ion number density (black) and density fluctuations (blue), and (c),(g) total plasma $\beta$. (d) and (h) the corresponding magnetic compressibility. The inset is the PSD of $\delta B$ and the corresponding power-law fit in the inertial range (blue).

Figure 2

The energy cascade rates computed using BG13 (red) and PP98 (black) for the same (a) Alfvénic and (b) magnetosoniclike events of Fig. 1.

Figure 3

Histograms of $⟨|{\epsilon }_I|⟩$ (red) and $⟨|{\epsilon }_C|⟩$ (blue) computed using the PP98 and BG13 exact laws for the (a) Alfvénic and (b) magnetosoniclike events.

Figure 4

Compressible energy cascade rate as a function of the turbulent Mach number for the Alfvénic (a) and magnetosoniclike (b) events. The black line represents a least square fit of the data, $\alpha$ is the slope of the power-law fit. The error bars represent the standard deviation of $|{\epsilon }_C|$.

Figure 5

Normalized mean compressible magnetic (blue) and kinetic (red) binned energies to the internal energy as a function of the binned compressible energy cascade rate for the Alfvénic (a) and magnetosonic (b) events.

Figure 6

Compressible and incompressible energy cascade rates $⟨|{\epsilon }_C|⟩$ and $⟨|{\epsilon }_I|⟩$ as a function of the mean angle ${\mathrm{\Theta }}_{\mathbf{V}\mathbf{B}}$ and the total energy (colored bar) for the Alfvénic (a) and magnetosonic (b) events. The blue line is the ratio $\mathcal{R}=⟨|{\epsilon }_C|⟩/⟨|{\epsilon }_I|⟩$.

Figure 7

Compressible energy cascade rate $⟨|{\epsilon }_C|⟩$ averaged into bins of proton temperature anisotropy ($T_{\perp }/T_{\parallel }$) vs ${\beta }_{\parallel }$ for (a) the Alfvénic and (b) magnetosoniclike events. The dashed lines correspond to the mirror, the Alfvénic ion cyclotron (AIC), and fire hose linear instabilities thresholds.