Measurement of the Electric Fluctuation Spectrum of Magnetohydrodynamic Turbulence

Magnetohydrodynamic (MHD) turbulence in the solar wind is observed to show the spectral behavior of classical Kolmogorov fluid turbulence over an inertial subrange and departures from this at short wavelengths, where energy should be dissipated. Here we present the first measurements of the electric field fluctuation spectrum over the inertial and dissipative wave number ranges in a $\beta \gtrsim 1$ plasma. The $k^{-5/3}$ inertial subrange is observed and agrees strikingly with the magnetic fluctuation spectrum; the wave phase speed in this regime is shown to be consistent with the Alfvén speed. At smaller wavelengths $k{\rho }_i\ge 1$ the electric spectrum is enhanced and is consistent with the expected dispersion relation of short-wavelength kinetic Alfvén waves. Kinetic Alfvén waves damp on the solar wind ions and electrons and may act to isotropize them. This effect may explain the fluidlike nature of the solar wind.

Figure 1

Wavelet and time series data of solar wind turbulence. From the top down, the five panels show (a) the wavelet spectrogram of $E_y$, as a function of $k{\rho }_i$, (b) a similar wavelet spectrogram of $B_z$, (c) the $X$ and $Y$ components of the measured electric field, (d) the vector magnetic field, (e) plasma ion density, (f) plasma ion $\beta$, and (g) the Alfvén Mach number. This entire interval was used for the spectral analysis of $E_y$ and $B_z$. The spectral breakpoints are called out.

Figure 2

Power spectral density of electric $\delta E_y$ and magnetic fluctuations $\delta B_z$ as a function of frequency, computed from FFT (black) and Morlet wavelet (red) algorithms. The FFT spectrum of the electric field (upper panel) shows the effect of notch filters and residual spin-tone data.

Figure 3

The wavelet (upper) and FFT (lower) power spectra of $E_y$ (green) and $B_z$ (black) binned as a function of wave number $k{\rho }_i$ (and offset for clarity) in panel (a). The electric spectra are multiplied by factors to lie atop the magnetic spectra. The spectrum is Kolmogorov $k^{-5/3}$ over the interval $k{\rho }_i\in (0.015,0.45)$; a spectral breakpoint occurs for both $E_y$ and $B_z$ at $k{\rho }_i\approx 0.45$. A second breakpoint occurs for the electric spectrum at $k{\rho }_i\approx 2.5$ above which the electric spectrum is more exponential. Panel (b) shows the ratio of the electric to magnetic spectra in the plasma frame; the average Alfvén speed (${\overline{v}}_A\approx 40\mathrm{km}/\mathrm{s}$) is shown as a horizontal line. The red line is a fitted dispersion curve, discussed in the text. Panel (c) shows both the cross coherence of $\delta E_y$ with $\delta B_z$ (as blue dots with error bars) and the correlation between the electric and magnetic power (as black dots).