Self-organization of local streamline structures and energy transfer rate in compressible plasma turbulence

TL;DR

This work addresses how local streamline topology governs energy transfer in compressible and incompressible plasma turbulence. It combines a 3D fully compressible Hall-MHD simulation with coarse-graining and Favre filtering to analyze subgrid-scale energy transfer $\pi_{sgs}$ in terms of gradient-tensor geometric invariants (GTGIs) $P$, $Q$, and $R$, and their discriminant $\Delta$. The key finding is a clear topology-driven energy cascade in nearly-incompressible regions, where direct transfer occurs for strain-dominant and stable-vortical structures and inverse transfer for unstable-vortical structures; in compressible regions, the transfer direction is largely governed by the sign of $P$ (volumetric compression or expansion), with reduced topological selectivity. This work advances our ability to diagnose and interpret turbulence in space plasmas and informs the design of multi-scale in-situ missions that can leverage GTGI diagnostics to understand energy transfer across scales.

Abstract

We examine how local streamline topology and energy cascade rate self-organize in plasma turbulence for both compressible and incompressible regimes. Using a fully-compressible Hall-magnetohydrodynamic simulation, we quantify the subgrid-scale energy transfer and analyze its relationship to streamline structures by means of grandient tensor geometric invariants of the velocity field. Our results highlight how streamline topology is crucial for diagnosing turbulence: for nearly-incompressible fluctuations the energy is primarily transferred to smaller scales through strain-dominated and stable-vortical structures, while is back-transferred towards larger scales through unstable-vortical structures. Compressible fluctuations, on the contrary, do not show a clear topological selection of the energy transfer since the overall direction of the local cascade rate is found to be determined by the sign of $-\nabla\cdot u$ (plasma volumetric compression or expansion).

Figures (5)

• Figure 1: Sketch of the local streamline topologies in the $RQ$-plane for the three cases $P=0$ (left), $P>0$ (middle), and $P<0$ (right). Thick line is the discriminant surface $\Delta$. Red line marks the separation between stretching and compressing regions, and the dashed line indicate the separation between stable and unstable topologies. Shaded areas indicate topologies associated with local volumetric contraction (yellow) and expansion (green) emerging in compressible cases only.
• Figure 2: PDF of the first GTGI $P=-\nabla\cdot\bm{u}$. Shaded areas indicate $P\ll0$, $P\gg0$ (blue), and $P\sim0$ (red) regimes. Vertical lines indicate the median values $\bar{P}_\ll=-0.06$, $\bar{P}_0=-1.2\times10^{-3}$, and $\bar{P}_\gg=0.07$ inside each subset.
• Figure 3: Joint PDFs of $Q$ and $R$ GTGIs (a) and averaged energy transfer rate in the $RQ$-plane (b) for $P\simeq 0$. GTGI values are normalized to powers of the enstrophy $\langle\omega^2\rangle$. Thick black line indicate the discriminant for $\bar{P}_0=-1.2\times10^{-3}$ and the dotted line indicate the $\bar{P}_0 Q-R=0$ curve.
• Figure 4: Joint PDFs of $Q$ and $R$ GTGIs (a) and averaged energy transfer rate in the $RQ$-plane (b) for $P\gg 0$. Thick black lines is the discriminant line obtained for $\bar{P}_\gg=0.07$ and the dotted line indicate the $\bar{P}_\gg Q-R=0$ curve. Joint PDFs of $Q$ and $R$ GTGIs (c) and averaged energy transfer rate in the $RQ$-plane (d) for $P\ll 0$. Thick black lines is the discriminant line obtained for $\bar{P}_\ll=-0.06$ and the dotted line indicate the $\bar{P}_\ll Q-R=0$ curve. GTGI values are normalized to powers of the enstrophy $\langle\omega^2\rangle$.
• Figure 5: Reduced 1D perpendicular and parallel spectra of magnetic field fluctuations w.r.t the mean magnetic field. Dashed and dot-dashed lines represent typical slopes expected in inertial and Hall ranges that serve as reference.