Turbulence-Driven Corrugation of Collisionless Fast-Magnetosonic Shocks

TL;DR

The paper develops a moving‑interface linear MHD formulation to quantify how upstream turbulence corrugates collisionless fast‑magnetosonic shocks, showing that the surface response is governed by an impedance $\mathcal{Z}(\omega,\boldsymbol{k}_{\perp})$ and is strongly enhanced near grazing when the transmitted fast mode has vanishing normal group speed $v_{g,n2}$. Upstream fluctuations act through a scalar drive $\mathcal{S}$, and the corrugation spectrum is shaped by a Lorentzian resonance $\mathcal{Z}\approx C\,v_{g,n2}+i\Gamma$, yielding a resonance cone in $(k_{\perp},k_{n2})$-space and a surface response that mirrors the upstream Alfvénic and compressive content with obliquity through $\cos^2\theta_{Bn}$ and $\sin^2\theta_{Bn}$. The authors quantify how corrugations scale with compression, $\beta$, and obliquity, derive the integrated power and coherence length $L_{\parallel}$, and connect surface dynamics to particle injection via a linear reaction–diffusion closure, predicting along‑front hot‑spot spacing $\lambda_{\parallel}\propto v_{corug}$ and recurrence time $\Delta t_{patch}\propto \kappa/U_{n1}^2$. The results provide a physically motivated baseline for interpreting heliospheric and SNR shock observations, including type II radio fine structure and elongated X‑ray stripes, while highlighting the need for nonlinear and kinetic extensions to capture feedback on the base state and transport properties.

Abstract

Collisionless fast-magnetosonic shocks are often treated as smooth, planar boundaries, yet observations point to organized corrugation of the shock surface. A plausible driver is upstream turbulence. Broadband fluctuations arriving at the front can continually wrinkle it, changing the local shock geometry and, in turn, conditions for particle injection and radiation. We develop a linear-MHD formulation that treats the shock as a moving interface rather than a fixed boundary. In this approach the shock response can be summarized by an effective impedance determined by the Rankine-Hugoniot base state and the shock geometry, while the upstream turbulence enters only through its statistics. This provides a practical mapping from an assumed incident spectrum to the corrugation amplitude, its drift along the surface, and a coherence scale set by weak damping or leakage. The response is largest when the transmitted downstream fast mode propagates nearly parallel to the shock in the shock frame, which produces a Lorentzian-type enhancement controlled by the downstream normal group speed. We examine how compression, plasma $β$, and obliquity affect these corrugation properties and discuss implications for fine structure in heliospheric and supernova-remnant shock emission.

Figures (6)

• Figure 1: Schematic of the interface formulation in the shock rest frame. A turbulent upstream flow convects broadband fluctuations toward an initially planar shock with normal $\hat{\boldsymbol n}$, which is displaced into a corrugated surface described by $\zeta(\boldsymbol{x}_\perp,t)$. The local shock normal is then $\hat{\boldsymbol n}-\nabla_{\!\perp}\zeta$.
• Figure 2: Schematic resonance geometry for transmitted downstream fast–magnetosonic modes. For fixed frequency and tangential wavevector $\boldsymbol{k}_\perp$, the fast branch has a locus in $(k_{n2},\boldsymbol{k}_\perp)$ where the normal group speed in the shock frame, $v_{g,n2}$, vanishes. This "resonance cone" (green mesh) marks directions where wave packets dwell near the interface and the interfacial impedance $\mathcal{Z}$ is minimized; the closer an orange ray lies to this cone, the stronger the Lorentzian enhancement of the surface response $|\mathcal{T}|^2 = 1/|\mathcal{Z}|^2$.
• Figure 3: Injected broadband driver used in all runs. One–dimensional perpendicular spectrum $P(k_\perp)$ showing the total (solid), Alfvénic (dashed), and compressive (dash–dotted) components. A dotted line indicates a $k_\perp^{-5/3}$ guide. Spectra are accumulated from $N=6\times10^4$ upstream samples with a fixed compressive power fraction $\chi_C$. $k_\perp$ in $L^{-1}$ and power in arbitrary units.
• Figure 4: Wave–vector diagnostics that relate the upstream driver to the transmitted fast branch and to the surface response. Left column mapping of total wavenumber magnitude $(|k_1|\!\rightarrow\!|k_2|)$ for $\theta_{Bn}=30^\circ$ (top row) and $60^\circ$ (bottom row). Middle column mapping of the normal components $(|k_{n1}|,|k_{n2}|)$ for the same angles and rows. Right column surface weight $W$ in the $(k_{\perp},|k_{n2}|)$ plane (susceptibility map). Color in the left and middle columns shows counts per hexbin, normalized by the median count in each panel. Color in the right column shows $\sum W$ per hexbin, normalized by the panel median. Results use the same $N=6\times10^4$ upstream samples as Figure \ref{['fig:driver']} and the base state $\beta=0.1$, $M_f\simeq1.5$.
• Figure 5: Real–space slices of the scalar field in planes that include the shock normal. Panels show normalized scalar–field amplitude in the $(n,t_1)$ left column and $(n,t_2)$ right column for $\theta_{Bn}=30^\circ$ top row and $60^\circ$ bottom row. The solid black curve is the corresponding slice of the synthesized surface $\zeta$ and the vertical dashed line marks $n=0$. Upstream lies to the left of the curve and downstream to the right. Colors are normalized by the absolute maximum within each panel; contour labels indicate the plotted levels.