Energy Bounds from Relative Magnetic Helicity in Spherical Shells

TL;DR

This work proves that, in a spherical-shell domain with a potential reference field, nonzero relative helicity $H_{ m R}$ imposes a lower bound on magnetic energy, analogous to the closed-domain Arnol'd inequality, via an intrinsic expression $H_{ m R}=orall_V oldsymbol{A}^*oldsymbol{ullet}oldsymbol{B}\,dV$. By exploiting the poloidal–toroidal decomposition and Green’s-function representations, the authors express $H_{ m R}$ purely in terms of $oldsymbol{B}$ and derive a global bound $|H_{ m R}|\, leq rac{\,\, ext{}} ext{ }{ ext{}} rac{\, ext{}}{2}\pi r_1 \,igl\\|oldsymbol{B}\bigr\ Vert^2$, yielding $\|oldsymbol{B}\bigr riangle^2 \ ightarrow W_1$. They further decompose helicity into current-carrying and volume-threading parts to obtain a bound on the free energy $\|oldsymbol{B}_{ m J}\|^2 \, leq W_2$ and a refined bound $W_3$ when the potential component $\boldsymbol{B}_{ m p}$ is nonzero. Extending to local scales, the paper introduces a magnetic partition and the unsigned helicity $\bar{H}_{\Pi}$, giving $\|oldsymbol{B}\|^2 \,\geq W_4 = 2\bar{H}_{\Pi}/(\pi r_1)$, which tightens energy constraints in mixed-helicity coronae. The strongest bound arises in the field-line partition limit, $\overline{H}$, with $W_4=2\overline{H}/(\pi r_1)$, and is robust even when $H_{ m R}$ vanishes. Numerical demonstrations on both analytical linear force-free fields and data-driven coronal models show that $H_{ m R}$ and especially $\bar{H}$ constrain energy and dynamics in the solar corona, highlighting the practical relevance for predicting eruptions and understanding magnetic topology.

Abstract

Relative magnetic helicity is commonly used in solar physics to avoid the well known gauge ambiguity of standard magnetic helicity in magnetically open domains. But its physical interpretation is difficult owing to the invocation of a reference field. For the specific case of spherical shell domains (with potential reference field), relative helicity may be written intrinsically in terms of the magnetic field alone, without the need to calculate the reference field or its vector potential. We use this intrinsic expression to prove that non-zero relative helicity implies lower bounds for both magnetic energy and free magnetic energy, generalizing the important Arnol'd inequality known for closed-field magnetic helicity. Further, we derive a stronger energy bound by spatially decomposing the relative helicity over a magnetic partition of the domain to obtain a new ideal invariant which we call unsigned helicity. The bounds are illustrated with analytical linear force-free fields (that maximize relative helicity for given boundary conditions) as well as a non-potential data-driven model of the solar corona. These bounds confirm that both relative helicity and the unsigned helicity can influence the dynamics in the solar corona.

Figures (7)

• Figure 1: Geometry for Sections \ref{['sec:intrinsicA']} and \ref{['sec:intrinsicHR']} -- in particular, the intrinsic expression \ref{['eq:hwinding']} for $H_R$. Note that $\boldsymbol{x}=r\boldsymbol{e}_r$ and $\boldsymbol{x}'=r\boldsymbol{e}_r'$.
• Figure 2: Linear force-free fields matching $B_r$ from \ref{['eq:pfss']} on both $S_0$ and $S_1$. Panel (a) shows the ratio $W_1/\|\boldsymbol{B}\|^2$ in energy bound \ref{['eq:bound1']} for these fields, as a function of (normalized) $\alpha$. Curves represent domains with different $r_0/r_1$, with colored dots showing the maximizing solution in each case. Panel (b) shows the flux surfaces for these maximizing solutions. Dashed vertical lines in (a) show the lowest resonant values of $\alpha$ (Appendix \ref{['app:lfff']}).
• Figure 3: Normalized (a) free energy $\|\boldsymbol{B}_{\rm J}\|^2$ and (b) current-carrying helicity $H_{\rm J}$, for the linear force-free solutions in Figure \ref{['fig:lfff-bounds']}. Panels (c) and (d) show the ratios $W_2/\|\boldsymbol{B}_{\rm J}\|^2$ and $W_3/\|\boldsymbol{B}_{\rm J}\|^2$ in energy bounds \ref{['eq:bound2']} and \ref{['eq:bound3b']}, respectively. Curves represent the same domains ($r_0/r_1$) as Figure \ref{['fig:lfff-bounds']}, and colored dots all show the fields maximizing the original bound $W_1/\|\boldsymbol{B}\|^2$.
• Figure 4: An example partition $\Pi$ of the spherical shell $V$ into magnetic subvolumes $V_1, \ldots, V_{12}$, in this case for an axisymmetric magnetic field. Each subvolume $V_i$ has an associated sub-helicity, $h_i$, with $\sum_{i=1}^{12}h_i = H_{\rm R}$.
• Figure 5: Field line helicity for the same day (2014 December 15) in a PFSS extrapolation (a, b), and two magneto-frictional simulations: T0 (c, d) and TU0.1 (e, f) from yeates2024cycle24. The right column shows $\mathcal{A}$ as a function of footpoint location on $r=r_0$, with the photospheric neutral line shown dashed. The dotted vertical line shows the longitude of the viewpoint in the left column, where (selected) field lines are shown colored by $\mathcal{A}$. The color scale is capped at the same value in all plots; actual maxima are $0.4\times 10^{22}\,\mathrm{Mx}$, $1.3\times 10^{22}\,\mathrm{Mx}$ and $3.3\times 10^{22}\,\mathrm{Mx}$.