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. We consider three models: (i) linear force-free field ($\bigtriangledown\times$ B = $\alpha (r) $ B) with a specific, prescribed constant $\alpha$ (Lundquist solution), and (ii) with $\alpha$ as a free constant parameter (Lundquist-alpha solution), (iii) uniform twist field (Gold-Hoyle solution). We retain only those cases where the impact parameter is less than one-half the FR radius, $R$, so the results should be robust (29 cases). The SFR radii lie in the range [$\sim$ 0.003, 0.059] AU. Comparing results, we find that the Lundquist-alpha and uniform twist solutions yielded comparable and small normalized $\chi^2$ values in most cases. We then use Grad-Shafranov (GS) reconstruction to analyze these events further. We then considered the twist per unit length, $\tau$, both its profile through the FR and its absolute value. We find $\tau$ to lie in the range [5.6, 34] turns/AU. The GH model-derived $\tau$ values are comparable to those obtained from GS reconstruction. We find that twist unit length ($L$) is inversely proportional to $R$, as $\tau \sim 0.17/R$. We combine MC and SFR results on $\tau (R)$ and give a relation which is approximately valid for both sets. The axial and azimuthal fluxes, $F_z$ and $F_\phi$, vary as $\approx 2.1 B_0 R^2 \times10^{21}$ Mx and $F_{\phi}/L \approx 0.36 B_0 R \times10^{21}$Mx/AU. The relative helicity per unit length, $H/L \approx 0.75 B_0^2 R^3$$\times 10^{42}$ Mx$^2$/AU.