The Most Intense Electron-Scale Current Sheets in the Solar Wind

Abstract

Previous analysis of magnetohydrodynamic-scale currents in high-speed solar wind near 1 AU suggests that the most intense current-carrying structures occur at electron scales and are characterized by average current densities on the order of $1~\mbox{pA}/\mbox{cm}^{2}$. Here, this prediction is verified by examining the effects of the measurement bandwidth and/or measurement resolution on the analysis of synthetic solar wind signals. Assuming Taylor’s hypothesis holds for the energetically dominant fluctuations at kinetic scales, the results show that when $\nu_{c}\gg \nu_{b}$, where $\nu_{c}$ is the measurement bandwidth and $\nu_{b} \approx 1/3~\mbox{Hz}$ is the break frequency, the average scale of the most intense fluctuations in the current density proxy is approximately $1/\nu_{c}$, and the average peak current density is a weakly increasing function that scales approximately like $\nu_{c}^{0.1}$.

Appendix A: ACF of the Derivative of a Stationary Process

The mean of a stationary process $x(t)$ is constant. The ACF of a zero-mean stationary process $x(t)$ is $R_{x}(\tau)=\langle x(t+\tau)x(t)\rangle$. Hence,

$$ \frac{\mathrm{d}R_{x}}{\mathrm{d}\tau}= \bigl\langle x'(t+\tau)x(t)\bigr\rangle = \bigl\langle x'(t)x(t-\tau)\bigr\rangle , $$

(A.1)

where the last equality follows from stationarity. Likewise,

$$ \frac{\mathrm{d}^{2}R_{x}}{\mathrm{d}\tau^{2}}= -\bigl\langle x'(t)x'(t-\tau)\bigr\rangle = -\bigl\langle x'(t+\tau)x'(t)\bigr\rangle . $$

(A.2)

The quantity on the far right-hand side of Equation (A.2) is $-R_{x'}(\tau)$, that is, minus one times the ACF of $x'(t)=\mathrm{d}x/{\mathrm{d}}t$. Thus

$$ R_{x'}(\tau)=-\frac{\mathrm{d}^{2}R_{x}}{\mathrm{d}\tau^{2}}. $$

(A.3)

The Fourier transform of this equation yields $S_{x'}(\omega)=\omega^{2}S_{x}(\omega)$. For an alternate derivation see Equation (9-109) in Papoulis (1984).