A Kinetic Route to Helicity-Constrained Decay

Abstract

Through 2D3V PIC simulations of freely decaying sub-ion turbulence, intermittent localized regions with $\mathbf{E} \cdot \mathbf{B} \neq 0$ are found to be statistically associated with reductions in the magnitude of magnetic helicity while evolving in the early electron-scale interaction phase. Motivated by this behavior, we propose a source-compensated, history-dependent helicity density that satisfies an exact local balance identity by construction, enabling Saffman-type two-point correlation integrals which, under standard flux-decorrelation assumptions, can exhibit intermediate-scale plateaus that are roughly time-independent. In our simulations we demonstrate such plateaus to remain approximately invariant even as the usual Saffman helicity integral plateau value $I_H$ evolves during the early kinetic stage. Under approximate single-scale self-similarity, the plateau behavior of the magnetic integral is consistent with the 2D decay constraint $BL \sim \text{const}$. For initially net-helical configurations, we observe rapid development of mixed-signed magnetic helicity patches and a decrease of the global fractional helicity, such that the decay over the kinetic interval is again most consistent with the cancellation-dominated scaling constraint.

Figures (8)

• Figure 1: Numerical evidence of Eqs. \ref{['eq:EJsign_scaling']} and \ref{['eq:sign']} for all simulations presented in this work (see Table \ref{['tab:table1']} in Appendix \ref{['appA']}). (a) plots $\langle \mathbf{E} \cdot \mathbf{B}/(\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha) \rangle_{V_{\text{sys}}}$ at times spaced by $\Delta t \approx [6.84 \times 10^{-3}, 7.98 \times 10^{-4}] \; t_A$ for $m_i/m_e = 25$ and $\approx 1836$, respectively. Here, $\langle \cdot \rangle_{V_{\text{sys}}}$ denotes an average over the entire spatial domain $V_{\text{sys}}$, and included as the black dashed line is when $\langle \mathbf{E} \cdot \mathbf{B}/(\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha) \rangle_{V_{\text{sys}}} = 0$. (b) shows $\mathbf{E} \cdot \mathbf{B}/(\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha)$ over the entire domain for the strongest outlier shown in (a) ($m_i/m_e = 25$, $\sigma_0 = 0$, at time $t = 0.1094 \; t_A$). (c) zooms in on (b) over an $\approx [100 \times 50] \; d_{e0}$ subdomain (with $d_{e0} = d_e (n_0) = (m_e c^2/4\pi n_0 e^2)^{1/2}$ and $n_0$ the initial, uniform ion and electron number density) that consisted of the largest negative $\mathbf{E} \cdot \mathbf{B}/(\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha)$ average value. Included in (c) are contours where $\mathbf{E} \cdot \mathbf{B}/(\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha) = 0$ (in magenta). We find that even in this situation in which Eq. \ref{['eq:sign']} appears to be contradicted most significantly, the majority of the subdomain consists of patches in which Eq. \ref{['eq:sign']} is valid. For example, $\mathbf{E} \cdot \mathbf{B}$ and $\mathbf{E}^{\prime} \cdot \mathbf{J}/\alpha$ share the same sign over nearly $70 \%$ of the box, whereas they differ in sign over approximately $30 \%$ of the subdomain.
• Figure 2: Evolution of $E_{\text{rms}}^2/B_{\text{rms}}^2$ for all 2D3V simulations presented in this work. Included as the black dashed line is when $E_{\text{rms}}^2/B_{\text{rms}}^2 = 1$.
• Figure 3: Evolution of $f_{S^+}$, the fraction of structures bounded by $A_z$ contours for which $S \equiv \text{sgn}(H_{V_s}) \text{sgn}(H_{C_s}) = + 1$, for all 2D3V PIC simulations presented in this work. Indicated by the solid and dash-dotted black lines for initially net-helical and nonhelical configurations, respectively, are the times plotted in Fig. \ref{['fig:contour_align']}, with annotations indicating the corresponding subplot letter. The black dashed line indicates $f_{S^+} = 0.5$.
• Figure 4: Representation of the structure-by-structure helicity sign alignment in our 2D3V PIC simulations. Each panel shows the out-of-plane flux function $A_z (x,y)$ (red-blue colormap) together with closed $A_z = \text{const}$ contours (flux surfaces) that define individual coherent magnetic structures. For each structure $s$, we compute the structure-integrated helicity proxies $H_{V_s}=\int_{V_s} dV \; \mathbf{A} \cdot \mathbf{B}$ (Eq. \ref{['eq:HVs']}) and $H_{C_s}=\int_{V_s} dV \; \mathbf{J} \cdot \mathbf{B}$ (Eq. \ref{['eq:HCs']}), and color the structure boundary by the sign alignment $S \equiv \text{sgn}(H_{V_s}) \text{sgn}(H_{C_s}) \in \{ \pm 1 \}$ (Eq. \ref{['eq:contour_sign']}; magenta denotes aligned signs, $S = +1$, and green denotes anti-aligned signs, $S = -1$). Shown are three representative times (early (left column)/intermediate (middle column)/late (right column), indicated by the solid and dash-dotted black lines in Fig. \ref{['fig:contour_align_plot']}) for each of the four simulations ($m_i/m_e = 25$ (top rows) and $\approx 1836$ (bottom rows) and $\sigma_0 = 1$ (first and third rows) and $0$ (second and fourth rows)). At early times—when sub-ion nonideality and reconnection-driven interactions are strongest—structure boundaries are predominantly magenta, indicating that a large majority of coherent structures satisfy $\text{sgn}(H_{V_s}) \simeq \text{sgn}(H_{C_s})$. At later times the population becomes mixed, showing comparable magenta/green, and in some cases majority green. This transition coincides with the regime in which $I_H$ (see, e.g., Eq. \ref{['eq:IHdef']}) becomes approximately stationary (see, e.g., Fig. \ref{['fig:Saffman_evol_theory']}), whereas the early-time decline of $I_H$ occurs during the high-alignment phase.
• Figure 5: Time evolution of $I_{\mathscr{L} e}$ (solid red), $I_{\mathscr{L}, 1}$ (dash-dotted red), $I_H$ (solid blue), and $B_{\text{rms}}^4$ (solid purple; multiplied by a proportionality constant equivalent to $2\pi L_h^4 (t = 0)$, indicated by the lower left annotation in the left panel) in globally nonhelical simulations with $m_i/m_e = 25$ (left) and $\approx 1836$ (right). Included (dotted black) is the calculated value of $I_{\mathscr{L}, 1} (t = 0) = I_H (t = 0)$ ($= I_{\mathscr{L} e} (t = 0)$ for initially zero flow), derived in Appendix \ref{['appB']}. Also included are the power-law scalings: $\propto t^{-0.87}$ for $m_i/m_e = 25$ and $\propto t^{-0.89}$ for $m_i/m_e \approx 1836$ (dash-dotted black, fitted using the $B_{\text{rms}}^4$ data) and $\propto t^0$ (dashed black). The insets zoom in on the vertical axis range $\in [1/2, 2] \times 16/9$.