Wave-Number Spectra and Intermittency in the Terrestrial Foreshock Region

Wave-number spectra of magnetic field fluctuations are directly determined in the terrestrial foreshock region (upstream of a quasiparallel collisionless shock wave) using four-point Cluster spacecraft measurements. The spectral curve is characterized by three ranges reminiscent of turbulence: energy injection, inertial, and dissipation range. The spectral index for the inertial range spectrum is close to Kolmogorov’s slope, $-5/3$. On the other hand, the fluctuations are highly anisotropic and intermittent perpendicular to the mean magnetic field direction. These results suggest that the foreshock is in a weakly turbulent and intermittent state in which parallel propagating Alfvén waves interact with one another, resulting in the phase coherence or the intermittency.

Figure 1

Magnetic field fluctuations exhibiting burstlike small scale fluctuation recorded by Cluster 1 on 18 February 2002. Time unit is given with hours and minutes (HH:MM) in universal time (UT).

Figure 2

Total power as a function of wave-number magnitude $k$. The wave numbers corresponding to ion gyroradius ($k_{\mathrm{gy}}$) and ion inertial length ($k_{\mathrm{in}}$) are also displayed. The vertical bar on the left denotes the confidence interval of the wave power for 95% reliability with 16 degrees of freedom, and the horizontal bar is the grid size in the wave-number space (equidistant in logarithmic scale).

Figure 3

Spectral energy density of noncompressive fluctuations ($P_{\perp }$) as function of wave number parallel ($k_{\parallel }$) with respect to the ambient magnetic field (top panel) and compensated spectrum with $\alpha =5/3$ (bottom panel).

Figure 4

Spectral energy density of compressive fluctuations as function of perpendicular wave number (top panel) and compensated spectrum with $\alpha =5/3$ (bottom panel).

Figure 5

Probability distribution functions (PDFs) of the magnetic fields for the compressive ($\delta B_{\parallel }$) and the noncompressive ($\delta B_{\perp }$) parts. Solid curves in gray and black represent PDFs superposed for four $s/c$ data and PDFs averaged over four $s/c$ data, respectively. Dashed curves are the corresponding Gaussian-fit around $\delta B=0$.