The Magnetic Field Geometry of Small Solar Wind Flux Ropes Inferred from Their Twist Distribution

Abstract

This work extends recent efforts on the force-free modeling of large flux rope-type structures (magnetic clouds, MCs) to much smaller spatial scales. We first select small flux ropes (SFRs) by eye whose duration is unambiguous and which were observed by the Solar Terrestrial Relations Observatory (STEREO) or Wind spacecraft during solar maximum years. We inquire into which analytical technique is physically most appropriate, augmenting the numerical modeling with considerations of magnetic twist. The observational fact that these SFRs typically do not expand significantly into the solar wind makes static models appropriate for this study. SFRs can be modeled with force-free methods during the maximum phase of solar activity. We consider three models: (i) a linear force-free field ($\nabla\times \mathbf{B} = \alpha \mathbf{B}$) with a specific, prescribed constant $\alpha$ (Lundquist solution), and (ii) with $\alpha$ as a free constant parameter (“Lundquist-alpha” solution), and (iii) a uniform twist field (Gold–Hoyle solution). We retain only those cases where the impact parameter is less than one-half the flux rope (FR) radius, $R$, so the results should be robust (29 cases). The SFR radii lie in the range $[{\approx}\,0.003, 0.059]~\mbox{AU}$. Comparing results, we find that the Lundquist-alpha and uniform twist solutions yielded comparable and small normalized $\chi^{2}$ values in most cases. This means that analytical modeling alone cannot distinguish which of these two is better in reproducing their magnetic field geometry. We then use Grad–Shafranov (GS) reconstruction to analyze these events further in a model-independent way. The orientations derived from GS are close to those obtained from the uniform twist field model. We then considered the twist per unit length, $\tau$, both its profile through the FR and its absolute value, applying a graphic approach to obtain $\tau$ from the GS solution. The results are in better agreement with the uniform twist model. We find $\tau$ to lie in the range $[5.6, 34]~\mbox{turns}/\mbox{AU}$, i.e. much higher than typical values for MCs. The GH model-derived $\tau$ values are comparable to those obtained from GS reconstruction. We find that the twist per unit length, $L$, is inversely proportional to $R$, as $\tau\approx0.17/R$. We combine MC and SFR results on $\tau(R)$ and give a relation that is approximately valid for both sets. Using this, we find that the axial and azimuthal fluxes, $F_{z}$ and $F_{\phi}$, vary as ${\approx}\,2.1 B_{0} R^{2}$ (${\times}10^{21}~\mbox{Mx}$) and $F_{\phi}/L \approx0.36 B_{0} R$ (${\times}10^{21}~\mbox{Mx}/\mbox{AU}$). The relative helicity per unit length is $H/L \approx0.75 B_{0}^{2} R^{3}$ (${\times}10^{42}~\mbox{Mx}^{2}/\mbox{AU}$).