The Grad–Shafranov Reconstruction of Toroidal Magnetic Flux Ropes: Method Development and Benchmark Studies

Abstract

We develop an approach of the Grad–Shafranov (GS) reconstruction for toroidal structures in space plasmas, based on in situ spacecraft measurements. The underlying theory is the GS equation that describes two-dimensional magnetohydrostatic equilibrium, as widely applied in fusion plasmas. The geometry is such that the arbitrary cross-section of the torus has rotational symmetry about the rotation axis, $Z$, with a major radius, $r_{0}$. The magnetic field configuration is thus determined by a scalar flux function, $\Psi$, and a functional $F$ that is a single-variable function of $\Psi$. The algorithm is implemented through a two-step approach: i) a trial-and-error process by minimizing the residue of the functional $F(\Psi)$ to determine an optimal $Z$-axis orientation, and ii) for the chosen $Z$, a $\chi^{2}$ minimization process resulting in a range of $r_{0}$. Benchmark studies of known analytic solutions to the toroidal GS equation with noise additions are presented to illustrate the two-step procedure and to demonstrate the performance of the numerical GS solver, separately. For the cases presented, the errors in $Z$ and $r_{0}$ are $9^{\circ}$ and 22%, respectively, and the relative percent error in the numerical GS solutions is smaller than 10%. We also make public the computer codes for these implementations and benchmark studies.

Appendices

Appendix A: Calculation of $R$ for a Given $Z$ at $O'$

We present one approach here of calculating the array $R$ for each point denoted by a vector $\mathbf{r}_{sc}$ along the spacecraft path across the torus, for a given $Z$-axis of components $(Z_{r},Z_{t},Z_{n})$ at location $O'$, as illustrated in Figure 2. This is the distance between the origin $O$, given by the vector $\mathbf{O}$ and $\mathbf{r}_{sc}$ (note all vectors are given in the $({r}_{sc},t,n)$ coordinates):

$$ R=|\mathbf{r}_{sc} - \mathbf{O}|. $$

(12)

Then, the key step is to derive $\mathbf{O}$ for each $\mathbf{r}_{sc}$, realizing that it is changing along $Z$, except when $Z$ is perpendicular to $\mathbf{r}_{sc}$. It is trivial for the special case when all $O$s coincide with one point along $Z$ (becoming $O'$ when $Z$ is perpendicular to the $(r_{sc},t)$ plane). So the following is for a general case and for $Z_{t}\ne0$.

From the known fact that both $O$ and $O'$, denoted by vector components $(r_{o},t_{o},n_{o})$ and $(r',t',n')$, respectively, are along $Z$, it follows

$$ \frac{r'-r_{o}}{Z_{r}}=\frac{t'-t_{o}}{Z_{t}}=\frac{n'-n_{o}}{Z_{n}}. $$

For $Z_{t}\ne0$, we obtain

$$ r_{o}=r' - \frac{Z_{r}}{Z_{t}}\bigl(t'-t_{o} \bigr) $$

(13)

and

$$ n_{o}=n' - \frac{Z_{n}}{Z_{t}}\bigl(t'-t_{o} \bigr). $$

(14)

By substituting them into $(\mathbf{r}_{sc} -\mathbf{O})\cdot \hat{Z}=0$ and rearranging the terms, we obtain

$$ t_{o}=\frac{(\mathbf{r}_{sc} -\mathbf{r}_{op})\cdot{\hat{Z}}}{|Z|^{2}/Z_{t}} + t', $$

(15)

where quantities on the right-hand side are all known with $\mathbf{r}_{op}=(r',t',n')$. Then the vector $\mathbf{O}$ is fully determined from Equations 13 and 14 above. So is the array of $R$ from Equation 12 along the spacecraft path.

Similar set of formulas can be obtained for the cases of $Z_{r}\ne 0$ or $Z_{n}\ne0$.

Appendix B: The Numerical GS Solver

The numerical GS solver for the toroidal GS reconstruction is in direct analogy to the straight-cylinder case (see, e.g. Hau and Sonnerup, 1999), i.e. the approach by the Taylor expansion, using the GS Equation 3 for evaluating the second-order derivative in $\theta$.

To lay out the implementation of the numerical scheme in the code, we denote $u_{i}^{j}=\Psi$ and $v_{i}^{j}=B_{r}$, where the indices $i$ and $j$ represent uniform grids along directions $r$ and $\theta$, with grid sizes $h$ and $\Delta\theta$, respectively. We set $\Delta\theta=0.01h$, and $\theta^{j}=(j-j_{0})\Delta\theta+\theta_{0}$ ($j=1$ to $n_{y}$), where the index of the grid at $\theta=\theta_{0}$, i.e. along the projected spacecraft path, is denoted $j_{0}$. Changing $j_{0}$ will allow the spacecraft path where the initial data are derived to shift away from the center line of the computational domain. Then, the solutions to the GS equation can be obtained through usual Taylor expansions in $\theta$ (truncated at the second-order term with respect to $\Psi$), both upward and downward from the initial line ($\theta=\theta_{0}$). For example, for the upper half annular region $j\ge j_{0}$, noting the relations $\frac{\partial \Psi}{\partial\theta}=rRB_{r}$, $\frac{\partial\Psi}{\partial r}=RB_{\theta}$, and $R=R_{0}+r\cos\theta$, we obtain (further denoting $rhs=-FF'$, as a known function of $u$ via the functional fitting $F(\Psi)$, see, e.g., Figure 9)

$$\begin{aligned} u_{i}^{j+1} =&u_{i}^{j}+ \bigl(-v_{i}^{j} r_{i} R_{i}\bigr) \Delta\theta+ \frac{1}{2}a_{i}^{j}\Delta \theta^{2} r_{i}^{2}, \end{aligned}$$

(16)

$$\begin{aligned} v_{i}^{j+1} =&v_{i}^{j}+\Delta\theta \biggl(-a_{i}^{j}\frac{r_{i}}{R_{i}}+\frac{r_{i}\sin \theta^{j} v_{i}^{j}}{R_{i}} \biggr), \end{aligned}$$

(17)

where the term $a_{i}^{j}$ involves the second-order derivative in $\theta$ and is evaluated via the GS equation,

$$ a_{i}^{j}=rhs_{i}^{j}- \biggl( \frac{\partial^{2} u}{\partial r^{2}} \biggr)_{i}^{j}+\sin\theta^{j} v_{i}^{j} - \biggl(\frac{1}{r_{i}}-\frac{\cos\theta^{j}}{R_{i}} \biggr) \biggl(\frac{\partial u}{\partial r} \biggr)_{i}^{j}. $$

As usual, the partial derivatives in $r$ are evaluated by second-order centered finite difference for inner grid points and one-sided finite difference for boundary points.

Also similar to the usual straight-cylinder case, smoothing of the solution at each step is necessary to suppress the growth of the numerical error. The same scheme is applied as follows to inner grid points only (Hu, 2001; Hu and Sonnerup, 2002) and for the upper-half domain ($j\ge j_{0}$),

$$ \tilde{u}_{i}^{j}=\frac{1}{3}\bigl[k_{1} u_{i+1}^{j}+k_{2} u_{i}^{j}+k_{3} u_{i-1}^{j}\bigr], $$

where the coefficients are $k_{1}=k_{3}=f_{y}$, and $k_{2}=3-2f_{y}$, with

$$ f_{y}=\min \biggl\{ 0.7, \frac{\theta^{j}-\theta_{0}}{\theta^{n_{y}}-\theta_{0}} \biggr\} . $$

The same applies to $v$ and similarly to the lower-half domain.

Appendix C: The Hodograms for the Cases of Submerged Spacecraft Paths

These are the cases that cannot be dealt with by the toroidal GS reconstruction technique developed here. These have traditionally been analyzed by a fitting method to fit the spacecraft measurements along its embedded path to a theoretical toroidal flux rope model (see, e.g. Marubashi et al., 2015). As we discussed earlier and demonstrate further below, the “projected” spacecraft path takes a peculiar shape and the measured magnetic field components possess certain features, as indicated by the associated hodogram pairs obtained from the usual minimum-variance analysis (Sonnerup and Scheible, 1998).

We again demonstrate these cases by using the analytic solutions presented in Section 4. However, here the spacecraft path is especially taken, not to exit into the “hole” of the torus, but to be along the green line in Figure 1. Two such cases are presented in Figure 15: (a) the spacecraft path is perpendicular to $Z$ so that the “projected” path is double-folded onto itself, resulting in a situation where the spacecraft is entering and exiting the cross section along the same path, but is only half-way through, and (b) the spacecraft path is traversing along a slanted path, resulting in a warped non-overlapping path across about half of the cross section. For both cases, the magnetic field components change in time and show clear features of symmetry or antisymmetry, and possess significant radial components, persistently ${\sim}\,10~\mbox{nT}$ throughout the intervals. This is because the spacecraft nearly encounters the same set of field lines during its inbound and outbound passages and because of the up-down symmetry in these cases. These features are clearly demonstrated by the corresponding hodogram pairs shown in Figure 16. Especially in Case (a), the $B_{1}$ versus $B_{2}$ hodogram exhibits a nearly closed loop, while the other one is double-folded because of a completely folded path. Case (b) also displays significant rotation in $B_{1}$, about 180 degrees. It is worth noting that this type of pattern in Case (a) is rarely reported in in situ magnetic field measurements, except for the case of Romashets and Vandas (2003), where a nearly 360 degree rotation in the magnetic field was seen in the MC interval. In other words, we caution that for this type of configuration of a glancing pass by a spacecraft through a torus, the magnetic field signatures as demonstrated here need to be considered for a proper modeling of these configurations.

Figure 15

Cases of submerged spacecraft paths. (a) Path perpendicular to $Z$ and (b) a slanted path. In each subplot, the upper panel shows the analytic solution and the projected spacecraft path in red in the same format as in Figure 4, while the lower panel shows the magnetic field components along such a path (see legend of Figure 8).

Figure 16

Hodogram pairs for Cases (a) and (b), respectively. The magnetic field components are projected onto the maximum, intermediate, and minimum variance directions, $B_{1}$, $B_{2}$, and $B_{3}$, respectively, with corresponding eigenvalues, $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$. The diamond and cross mark the beginning and end of the data interval.

The current implementation of the numerical GS solver cannot solve for a solution over a significant portion of the cross section because the “projected” spacecraft path is no longer along a single constant coordinate dimension, i.e. that of $\theta\approx\theta_{0}=\mathrm{const}.$, across the whole cross-sectional domain. A word of caution is that when interpreting the measured time series in the $(r_{sc},t,n)$ coordinates, they have to be taken along the actual spacecraft path $\mathbf{r}_{sc}$ shown in Figure 1, not the “projected” paths on the $(R,Z)$ plane shown in Figure 15. Another important observation from these preliminary analyses is that the field rotation is more significant, as indicated by the hodogram pairs in these cases of a “glancing” passage of the spacecraft, contrary to general perceptions one may have. Although this provides proof-of-merits of flux rope model fitting to in situ spacecraft data under the toroidal geometry, we urge that such fitting better be done in the way of Equation 6 with the mathematical rigor of proper uncertainty estimates for quantitative and more objective assessment of the goodness-of-fit.