Optimal Fitting of the Freidberg Solution to In Situ Spacecraft Measurements of Magnetic Clouds

Abstract

We report, in detail, an optimization approach for fitting a three-dimensional (3D) magnetic cloud (MC) model to in situ spacecraft measurements. The model, dubbed the Freidberg solution, encompasses 3D spatial variations in a generally cylindrical geometry, as derived from a linear force-free formulation. The approach involves a least-squares minimization implementation with uncertainty estimates from magnetic field measurements. We present one case study of the MC event on 22 May 2007 to illustrate the method and demonstrate the satisfying result of the minimum reduced $\chi^{2}\lesssim1$, obtained from the Solar and TErrestrial RElations Observatory (STEREO) Behind spacecraft measurements. In addition, since the Advanced Composition Explorer (ACE) spacecraft at Earth crossed the STEREO Behind solution domain with an appropriate separation distance, the result from the optimally fitted Freidberg solution along the ACE spacecraft path is compared with the actual measurements of magnetic field components. A correlation coefficient of 0.89 is obtained between the two sets of data.

Appendix: The DeHoffmann–Teller (HT) Analysis

The DeHoffmann–Teller (HT) analysis is used to determine an HT frame from the time-series data, following the approach given by Khrabrov and Sonnerup (1998). One advantage of an HT frame is that by definition, in the HT frame, the electric field vanishes, then it follows from the Faraday law, the time-dependence of the magnetic induction $\mathbf{B}$ should also vanish. Therefore the magnetic field can be regarded to be stationary in time, consistent with the MC model assumptions.

Namely, for a time-series interval containing magnetic field $\mathbf{B}^{(m)}$ and plasma bulk flow velocity $\mathbf{V}^{(m)}$ measurements ($m=1,2, \ldots, M$) in the spacecraft frame, a constant HT frame velocity $\mathbf{V}_{HT}$ is obtained by minimizing (Khrabrov and Sonnerup, 1998):

$$ D_{HT}=\frac{1}{M}\sum_{m=1}^{M}|(\mathbf{V}^{(m)}-\mathbf{V}_{HT}) \times\mathbf{B}^{(m)}|^{2}. $$

(7)

The quality of an HT frame is demonstrated by the component-wise plots of $\mathbf{E}_{c}=\mathbf{V}\times\mathbf{B}$ versus $\mathbf{E}_{HT}=\mathbf{V}_{HT}\times\mathbf{B}$, and $\mathbf{v'}=\mathbf{V}-\mathbf{V}_{HT}$ versus the Alfvén velocity $\mathbf{V}_{A}$. The latter (so-called Walén plot) also indicates the relative importance of the inertia force (i.e. $\rho\mathbf{v'}\cdot\nabla\mathbf{v'}$) compared to the Lorentz force in the HT frame. Two metrics, the correlation coefficient for the former, and the slope of a linear regression line for the latter, are calculated, respectively.

For the STB MC interval given in Figure 2, Figure 9 shows the HT analysis results with the HT frame velocity $\mathbf{V}_{HT}=[ 440.16, -36.54, -0.08099]$ km/s in RTN coordinates. The corresponding correlation coefficient and regression line slope are 0.9990 and −0.084, respectively. For the STA MC interval marked in Figure 6 (right), the corresponding correlation coefficient and regression line slope are 0.9978 and 0.033, respectively, with $\mathbf{V}_{HT}= [482.66, 38,47, -16.28]$ km/s in RTN coordinates. Therefore for both cases, the assumption of time-stationary quasi-static equilibrium is satisfied when the analysis was carried out in the corresponding HT frame (with the correlation coefficient $\approx1.0$ and the magnitude of Walén slope $\ll1.0$).

Figure 9

HT analysis results for the STB MC interval. Left panel: Component-wise plot of $\mathbf{E}_{HT}$ versus $\mathbf{E}_{c}$. Right panel: Component-wise plot of $\mathbf{v'}$ versus the Alfvén velocity in km/s. A linear regression line is drawn with the slope denoted in the top left corner.