Spatiotemporal Properties of Compressible Magnetohydrodynamic Turbulence from Space Plasma

Abstract

Previous studies have established that a weak-to-strong transition occurs in Alfvenic magnetohydrodynamic (MHD) turbulence as energy cascades from large to small scales. However, the spatiotemporal (frequency-wavenumber) properties of compressible MHD turbulence involving all eigenmodes, which encode the strength of nonlinear interactions, remain difficult to characterize observationally. Consequently, whether a similar weak-to-strong transition occurs in compressible turbulence remains elusive. Using a novel multi-spacecraft, polarization-based mode-decomposition technique with measurements from the Cluster spacecraft in Earth's magnetosheath, we obtain spatiotemporal power spectra of all MHD eigenmodes and present the first quantitative assessment of nonlinear frequency broadening. Our results show that slow modes exhibit a weak-to-strong transition, evolving from wave-like peaks to frequency-broadened spectra as nonlinearity increases, whereas fast modes remain weakly turbulent with narrow peaks near their eigenfrequencies. Both Alfvenic and compressible fluctuations contribute significantly to low-frequency, large-scale quasi-two-dimensional structures. These findings provide a comprehensive observational characterization of compressible turbulence across mode composition, spatiotemporal scales, and weak-strong turbulence regimes, with implications for energetic particle transport, turbulent dynamos, plasma heating, and solar wind-magnetosphere coupling.

Figures (9)

• Figure 1: Fluctuations in the $\hat{k}\hat{b}_0$ coordinates. Alfvénic magnetic and velocity fluctuations ($\delta B_A$ and $\delta V_{\rm A}$) are polarized perpendicular to the $\hat{k}\hat{b}_0$ plane, whereas compressible fluctuations ($\delta B_C$ and $\delta V_C$) are polarized within the $\hat{k}\hat{b}_0$ plane. $\theta$ is the angle between $\hat{\mathbf{k}}$ and $\hat{\mathbf{b}}_0$, and $\eta$ is the angle between timing-derived ${\mathbf{k}}_M$ and SVD-derived $\hat{\mathbf{k}}$.
• Figure 2: Overview of fluctuations in Earth's magnetosheath. (a) Magnetic field. (b) Proton bulk velocity. (c) Proton density from CIS-HIA and electron density from PEACE and WHISPER. (d) Temperature. (e) Plasma $\beta$ (the ratio of thermal to magnetic pressures). (f) Spectral slopes ($\alpha$) of trace velocity and magnetic power spectra. (g) Spectral slopes ($\alpha$) of density power spectra. (f,g) The horizontal dashed lines indicate $\alpha=-5/3$ and $-3/2$. (h) Turbulent Alfvén Mach number ($\delta V_{\rm rms}/V_{\rm A}$) and half of the relative amplitudes of the magnetic field ($\delta B_{\rm rms}/(2B_0)$). Unless otherwise noted, all observations are from Cluster-1. The analysis is restricted to the interval between the two vertical dashed lines.
• Figure 3: Spatiotemporal power spectra of magnetic fluctuations in the plasma flow frame. (a,b,d) $f_{\mathrm{rest}}-k_\perp$ distributions of Alfvénic power ($\hat{P}_{\rm BA}$), total compressible power ($\hat{P}_{\rm BC}$), and non-fast compressible power ($\hat{P}_{\rm BC,non\text{-}fast}$). White dotted curves denote the scaling $f_{\rm rest} \propto k_\perp^{2/3}$. Black dotted (dashed) lines denote the fast-mode dispersion relations at $\theta=90^\circ$ ($0^\circ$). (c) $f_{\mathrm{rest}}-k$ distribution of fast-mode power ($\hat{P}_{\rm BC,fast}$). (e) $f_{\mathrm{rest}}-k_\parallel$ distribution of $\hat{P}_{BA}$. Pink dotted line denotes the Alfvén-mode dispersion relation at $\theta=0^\circ$. White solid and dashed curves indicate the frequency of the mean power spectrum ($f_{\rm P_m}$) and the corresponding $\pm\sigma$ confidence intervals ($f_{\rm P_{m\pm\sigma}}$), where $\sigma$ is the standard deviation of the power. (f) Phase correlation between fluctuating magnetic field strength ($\delta |\mathbf{\tilde{B}}|$) and PEACE electron density ($\delta \tilde{N}_e$), measured by Cluster-2. In-phase (anti-phase) regions correspond to phase differences within $\pm80^\circ$ around $0^\circ$ ($180^\circ$), leaving a narrow transition region. (g,h) $k_\parallel-k_\perp$ distributions of $\hat{P}_{\rm BC,fast}$ and $\hat{P}_{\rm BC,non\text{-}fast}$. Blue dotted curve denotes the scaling $k_{\rm \perp} \propto k_\perp^{2/3}$. Black dashed curves denote isotropic contours at $k=2\times10^{-4}$, $5\times10^{-4}$, and $8\times10^{-4}$$\rm km^{-1}$. Panels with the same format are normalized by the same factor, and values smaller than $10^{-6}$ are set to NaN. The data are binned into $N_f\times N_{k_\parallel}\times N_{k_\perp}=50\times50\times50$ with a threshold angle $\eta\leq10^\circ$.
• Figure 4: Alfvénic magnetic power at selected ($k_\parallel,k_\perp$) in the plasma flow frame. (a) $k_\parallel-k_\perp$ distribution of $\hat{P}_{\rm BA}$, where $\hat{P}_{\rm BA}=P_{\rm BA}/P_{\rm BA,max}$ is normalized by the maximum over all ($k_\parallel,k_\perp$) bins. Blue contour marks isotropy at $k=10^{-4}$$\rm km^{-1}$. Black dotted lines denote the scaling $k_{\rm \parallel} \propto k_\perp^{2/3}$. (b) $k_\parallel-k_\perp$ distribution of the Alfvénic nonlinearity parameter $\chi_{\rm BA}$. (c,d) $P_{\rm BA}(f_{\rm rest})$ at fixed $k_\parallel$, with color indicating $k_\perp$. The propagation angle $\theta= \rm arctan(k_\perp/k_\parallel)$. Dotted lines indicate Alfvén frequencies $f_A$ calculated with $k_\parallel=8.2\times10^{-5}$ and $1.8\times10^{-4}$$\rm km^{-1}$. Pink-shaded regions indicate the quasi-zero-frequency power $P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_{\mathrm{min}})=\frac{\int_{f_{\mathrm{min}}}^{5\times10^{-4} \rm Hz}P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}})df_{\mathrm{rest}}}{5\times10^{-4} \rm Hz-f_{\mathrm{min}}}$ and Alfvén-wave-like power $P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_A)= \frac{\int_{0.9f_{\mathrm{A}}}^{1.1f_{\mathrm{A}}}P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}})df_{\mathrm{rest}}}{0.2f_{\mathrm{A}}}$. (e,f) $P_{\rm BA}(f_{\rm rest})$ at fixed $k_\perp$, with color indicating $k_\parallel$. Colored dotted lines show the corresponding $f_A$, and black dotted lines indicate $f_{\rm A0}$ from $k_{\parallel,0}\sim7\times10^{-5}$$\rm km^{-1}$ (Fig. 3(b) of Zhao et al. Zhao2024a). The data are binned into $N_f\times N_{k_\parallel}\times N_{k_\perp}=200\times12\times12$ with $\eta\leq30^\circ$ to improve statistical robustness.
• Figure 5: Dependence of $\frac{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_{\mathrm{min}})}{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_A)}$ on the Alfvénic nonlinearity parameter $\chi_{BA}$. The quasi-zero-frequency power $P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_{\mathrm{min}})$ and Alfvén-wave-like power $P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_A)$ are indicated by the pink-shaded regions in Figs. \ref{['fig:4']}(c,d). The minimum frequency is $f_{\rm min}=4/t_{\rm win}$. Points with $k_\parallel \leq k_{\parallel,0}$ ($k_\parallel > k_{\parallel,0}$) are marked by asterisks (pluses), respectively. Pink horizontal line indicates $\frac{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_{\mathrm{min}})}{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_A)}=1$, and the black dashed line indicates $\frac{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_{\mathrm{min}})}{P_{\rm BA}(\mathbf{k},f_{\mathrm{rest}}\sim f_A)}\propto \chi_{BA}$. Only data with $\chi_{\rm BA}>10^{-2}$ are included to reduce uncertainties.