The influence of Parker spiral on the reflection-driven turbulence

TL;DR

The paper addresses how Parker spiral geometry modifies reflection-driven turbulence (RDT) and heating in the solar wind by extending Dmitruk et al.'s RDT phenomenology to a Parker-spiral mean field and testing it with 3D expanding-box MHD simulations. It introduces an expanding-box framework, uses Elsässer variables, and analyzes how the Parker spiral reshapes perpendicular outer scales ${\ell_{\perp,\mathrm{T}}}$ and ${\ell_{\perp,\mathrm{N}}}$, thereby keeping the nonlinearity-to-expansion ratio ${\chi_{\rm exp}}$ above unity longer and sustaining energy dissipation. The results show that while the fundamental RDT dynamics persist in PS geometry, the azimuthal field causes 3D eddy deformation that curtails indefinite pancake formation, leading to ongoing heating and strongly imbalanced turbulence out to larger heliocentric distances; spectra, switchbacks, and compressibility diagnostics provide concrete observational predictions. These findings offer a path to reconcile solar-wind heating with in-situ observations and furnish testable signatures for upcoming spacecraft data.

Abstract

The solar wind is observed to undergo substantial heating as it expands through the heliosphere, with measured temperature profiles exceeding those expected from adiabatic cooling. A plausible source of this heating is reflection-driven turbulence (RDT), in which gradients in the background Alfvén speed partially reflect outward-propagating Alfvén waves, seeding counter-propagating fluctuations that interact and dissipate via turbulence. Previous RDT models assume a radial background magnetic field, but at larger radii the interplanetary field is known to be twisted into the Parker Spiral (PS). Here, we generalize RDT phenomenology to include a PS, using three-dimensional expanding-box magnetohydrodynamic (MHD) simulations to test the ideas and compare the resulting turbulence to the radial-background-field case. We argue that the underlying RDT dynamics remain broadly similar with a PS, but the controlling scales change: as the azimuthal field grows it "cuts across" perpendicularly stretched, pancake-like eddies, producing outer scales perpendicular to the magnetic field that are much smaller than in the radial-background case. Consequently, the outer-scale nonlinear turnover time increases more slowly with heliocentric distance in PS geometry, weakening the tendency (seen in radial-background models) for the cascade to 'freeze' into quasi-static, magnetically dominated structures. This allows the system to dissipate a larger fraction of the fluctuation energy as heat, also implying that the turbulence remains strongly imbalanced (with high normalized cross-helicity) out to larger heliocentric distances. We complement our heating results with a detailed characterization of the turbulence (e.g., spectra, switchbacks, and compressive fractions) providing a set of concrete predictions for comparison with spacecraft observations.

Figures (17)

• Figure 1: Schematic of an expanding plasma parcel in the solar wind. (Top) Geometry of the expanding box with axes aligned to radial ($x$), azimuthal ($y$), and normal ($z$) directions. The solar wind initially flows radially outward, while the box expands transversely. The mean magnetic field $\bar{\bm B}$ (green) is radial near the Sun but rotates into a Parker spiral with angle $\Phi$ as $a(t)$ increases, so that $B_y/B_x \sim a(t)$. Wavevectors $\bm{k}$ are shown at angle $\vartheta$ to $\bar{\bm B}$, with azimuthal ($k^{(Y)}$) and normal ($k^{(Z)}$) components indicated. (Bottom) Comparison of eddy evolution under (a) purely radial and (b) Parker-spiral expansion. In the radial case, the perpendicular scale $\ell_\perp$ grows uniformly with expansion. In the PS case, rotation of the mean field changes $\ell_\perp$ creating 3-D anisotropy of the eddies.
• Figure 2: Evolution of key parameters for waves under solar-wind expansion, plotted versus expansion factor $a$. Thick black: radial field ($\Phi_0=0^\circ$). Colored: Parker spiral cases ($\Phi_0=2^\circ\text{--}20^\circ$, darker colors correspond to larger $\Phi_0$). (a) Wave amplitude $z^+/v_{\rm A}$ stays roughly constant for the purely radial case, and in the Parker spiral (PS) case remains constant initially but then decays as $\propto a^{-1}$ once the azimuthal component becomes significant and $v_{\rm A}\propto a^0$. (b) Obliquity $\sin\vartheta(a)$: the angle between $\bm k$ and $\bar{\bm B}$ initially decreases and then increases again as the mean field rotates azimuthally, producing a clear inflection with a PS. (c) Expansion–cascade parameter $\chi_{\rm exp} = (k_\perp\,z^+_\perp)/(\dot a/a)$ for the out-of-plane case ($\varphi = \pi/2$): turbulence is sustained while $\chi_{\rm exp} \gtrsim 1$; $\chi_{\rm exp}$ initially decays as $\propto a^{-1}$ for both radial and Parker spiral cases, but for PS it starts increasing and eventually flattens as $a^{0}$ once the azimuthal component becomes dominant. (d) $\chi_{\rm exp}$ for in-plane wave vectors: shown for $\varphi=\pi$ (solid) and $\varphi=0$ (dashed); in both cases, the wave vector lies in the $x$--$y$ plane. For $\varphi=0$, the wave passes through purely parallel propagation.
• Figure 3: Snapshots of the Elsässer fields $|\bm z_\perp^\pm|$ perpendicular to magnetic field in the $y$-$z$ plane. The top two rows (a) show different stages of expansion are shown for the A05-$\Phi_0\!=\!0^\circ$ with a radial $\overline{\bm B}$ simulation at $a \approx 6$ (left), $a \approx 22.35$ (middle), and $a \approx 50.35$ (right). These snapshots illustrate the turbulent evolution from an initially imbalanced regime to a magnetically dominated and balanced phase. The bottom two rows (b) show the snapshots of the Elsässer fields in the PS case for A05-$\Phi_0\!=\!1.5^\circ$. Elsässer fields $|\bm z_\perp^\pm|$ are shown at three stages of expansion: $a \approx 11$ (left), $a \approx 19.7$ (middle), and $a \approx 67.7$ (right) with the spiral angles of $\Phi \approx 16.1^\circ, \; 27.3^\circ, \; 60.57^\circ$ respectively. The system remains turbulent, with distinctive anisotropic structural features due to the component of the mean magnetic field along $y$. Note that in each panel, fluctuations are normalized by their rms value of each time.
• Figure 4: 3D visualizations of the Elsässer fields $|\bm z_\perp^\pm|$ perpendicular to the magnetic field at $a \approx3.5, \; 11.5,\; 21$ for HR-$\Phi_0\!=\!0^\circ$ and HR-$\Phi_0\!=\!4^\circ$ simulations. White arrow shows the direction of mean magnetic field in PS run.
• Figure 5: Evolution of normalized Elsässer wave action energies, $\tilde{E}^\pm(a)/\tilde{E}^+_0$, as a function of $a$ . Solid curves denote outward wave-action energy ($\tilde{E}^+$) and dashed curves denote inward wave-action energy ($\tilde{E}^-$); line colors correspond to different initial PS angles. Panel (a) shows the high–resolution runs (HR-$\Phi_0\!=\!0^\circ$ and HR-$\Phi_0\!=\!4^\circ$). In the radial case ($\Phi_0\!=\!0^\circ$), the outward energy decays approximately as $a^{-0.6}$ up to $a\sim15$, after which $\tilde{E}^+$ flattens and rises slightly, indicating that turbulent heating effectively shuts off. In the PS runs, the outward energy continues to decay to larger $a$, maintaining an imbalanced cascade and sustained dissipation for longer. The inset displays the normalized cross helicity $\sigma_c$ for these runs. Panel (b) shows A05-$\Phi_0\!=\!0^\circ,2^\circ,5^\circ$ runs with similar initial $\chi_{\rm exp,0}$ as panel (a) but reduced amplitude $\rm A=0.5$. The lower amplitude, near-RMHD case decays slightly more slowly than the high amplitude case, roughly as $\tilde{E}^+\propto a^{-0.5}$, possibly because the higher amplitude spherical-polarized fluctuations help to make reflection more efficient. (c) MR-$\Phi_0\!=\!0^\circ,2^\circ,5^\circ$ runs with higher initial $\chi_{\rm exp,0}$, which leads to steeper decay following approximately $\tilde{E}^+\propto a^{-0.8}$. The runs with larger spiral angles (e.g., $\Phi_0\!=\!5^\circ$) were terminated at smaller distances due to numerical instabilities discussed in \ref{['Numerical issues']}. Note that the first snapshot $(a=1)$ is omitted to improve visualization, since the inward-propagating mode $\tilde{z}^-$ is negligible at that stage.