On the utility of flux rope models for CME magnetic structure below 30 $R_{\odot}$

The Type 1 simulated CME magnetic field profiles have a local flux rope geometry that corresponds to the classic, bipolar flux rope/magnetic cloud orientation (also defined as a ‘low-inclination flux rope’ in Palmerio et al., 2018). A bipolar flux rope axis orientation is essentially parallel to the ecliptic plane (or to the $R$–$T$ plane in RTN coordinates) and points in more-or-less the $\pm\bm{\hat{t}}$ direction. Thus, the normal field component either starts positive and smoothly transitions to negative (i.e. North-to-South) over the course of the flux rope transient or vice-versa (South-to-North). The tangential field component is approximately zero at the leading edge flux rope boundary but smoothly increases to its maximum magnitude at the center of the flux rope with positive (negative) values pointing West (East) and then smoothly decreases back to approximately zero at the trailing edge boundary. These “horizontal” flux rope orientations result in the large-scale, coherent magnetic field rotations that can be classified as NWS, NES, SWN, or SEN using the Bothmer & Schwenn (1998) and Mulligan et al. (1998) convention.

Figure 4(a) shows two of the four Type 1 synthetic spacecraft CME encounters, S1 and P5, that represent these classic bipolar flux rope profiles. Each synthetic spacecraft time series is shown on its own row. From left to right, first we show a 3D visualization of the spacecraft position (yellow sphere currently sampling the yellow field line), a set of adjacent field lines (dark gray) traced at $0.25\,R_{\odot}$ intervals in the radial direction either side of the spacecraft point, and a set of light gray field lines indicating the approximate MHD flux rope boundary. The next two columns are the time series of the vector magnetic field profiles as $(\,|\bm{B}|,\,\delta,\,\lambda\,)$ in the center and $(\,B_{R},\,B_{T},\,B_{N}\,)$ on the right. In each of the time series panels, we have indicated the flux rope ejecta boundaries as vertical lines and plotted each of the in-situ flux rope model fits: LFF–blue, GH–green, and CCS–orange. The parameter values for each models’ fit to S1 and P5 are given in Table A.1.

The Type 1 flux rope profiles ought to be the simplest and most straightforward to fit with the in-situ flux rope models. In general, each of the flux rope models captures some or even most of the overall trend in the coherent field rotation, but there are aspects of the MHD profiles that certain flux rope models cannot reproduce. Specifically, the asymmetry in the $|\bm{B}|$ profile—where the peak/maximum value is more toward the front of the ejecta than the center. The LFF, GH, and CCS models are all symmetric, by construction. This aspect is also seen in the RTN components. All of the models underestimate the amplitude of the initial negative $B_{N}$ components because their $B_{N}$ profiles are symmetric with respect to the center of the spacecraft crossing. This is a well known shortcoming of models with cylindrical symmetry.

The other feature worth mentioning is that the MHD profiles, even in the simplest bipolar flux rope orientations, have a non-zero $B_{R}$ component. While this does not significantly affect the overall structure of the large-scale field rotation, it does indicate that the idealized, in-situ flux rope models will not result in a perfect fit. In A, the formulas for each model’s field components are given and every model has a radial component of zero (in their local flux-rope frame). Thus, to generate a non-zero $B_{R}$ profile, it must originate from a tilt or angle of the flux rope symmetry axis with respect to the RTN coordinates and/or from a trajectory that passes either above or below the flux rope cylinder symmetry axis. In other words, an orientation of $\phi_{0}=90^{\circ}$, $\theta_{0}=0^{\circ}$, which points the $z_{\rm FR}$ axis parallel to the RTN $\pm\bm{\hat{t}}$ direction will always give $B_{R}=0$ for an impact parameter of $p_{0}=0$. This is also true for a completely vertical flux rope orientation of $\theta_{0}=\pm 90^{\circ}$, which we discuss below.

The Type 2 magnetic field profiles correspond to a local unipolar flux rope/magnetic cloud orientation (also defined as a ‘high-inclination flux rope’ in Palmerio et al., 2018). In a unipolar flux rope event, the axis is essentially perpendicular to the $R$–$T$ plane, aligned with the $\pm\bm{\hat{n}}$ direction—or at least it makes a significant angle so the axis is highly inclined. In these cases the normal field component is predominately North or South, while the bipolar rotation signature of positive-to-negative (negative-to-positive) is in the West-to-East (East-to-West) direction. These “vertical” flux rope orientations result in the large-scale, coherent magnetic field rotations that can be classified as WNE, WSE, ENW, or ESW in the Bothmer & Schwenn (1998) and Mulligan et al. (1998) convention. We note that, in general, unipolar South magnetic clouds tend to drive the most intense geomagnetic storms (e.g. Zhang et al., 2004), however our particular MHD coordinate transform results in unipolar North configurations.

Figure 4(b) shows two examples of the Type 2 synthetic spacecraft CME encounters, S6 and P7, that represent classic unipolar flux rope orientations in the same format as Figure 4(a). The parameter values for each models’ fit to S6 and P7 are also given in Table A.1. Visually, the quality of the in-situ flux rope model fits to the unipolar MHD time series seem comparable to those for the classic bipolar events in Figure 4(a). In general, the large-scale magnetic structure of the coherent field rotations are reasonably well captured in the flux rope models, with some components matching better than others (e.g. the $B_{T}$ profiles of S6 look better than those of P7 but the $B_{N}$ profiles look similar).

The fit parameters for the P7 encounter are essentially identical across models (given the typical parameter uncertainties) indicating a fairly robust reconstruction. For example, the axial tilt $\theta_{0}$ values are all within $70^{\circ}$–$80^{\circ}$, the impact parameters are all ${\sim}20$%, and each model determines the flux rope size to be $R_{c}\approx 0.02$ AU. The S6 encounter fits are also pretty consistent in certain parameters (e.g. $\theta_{0}$, $R_{c}$), however the LFF and CCS models give impact parameters $p_{0}/R_{c}$ of only a few % whereas GH gives 23%. The P7 fits are apparently a bit better than the S6 fits, in that there is less variation between models in the orientation parameters. This is likely due to the S6 time series having more $|\bm{B}|$ asymmetry than P7. However, as we will see in Section 4.2.2, taking the variation between different models as an assessment of fit quality may not encompass the whole picture.

The next category of flux rope encounter we are investigating in the MHD simulation and the flux rope model fits are profiles that we have described as having a “problematic orientation.” What this means is that the flux rope axis makes a significant angle within the spacecraft’s $R$–$T$ plane but is still has a relatively low inclination, i.e. the flux rope parameter angle representing tilt out of the $R$–$T$ plane, $\theta_{0}$, is small, but the azimuthal angle, $\phi_{0}$, has a large departure from 90^∘ or 270^∘. This results in a spacecraft trajectory that includes a significant component along or parallel to the magnetic flux rope axis. This type of flux rope orientation has been discussed by Marubashi (1997), who sketched the geometry (see their Figure 3) and presented two examples of magnetic cloud observations consistent with this interpretation. Since then, there has been a number of well-observed flux rope ICME events that have similar morphology and interpretation (Owens et al., 2012). This scenario has also been recently considered in the context of PSP CME encounters (Möstl et al., 2020).

Figure 5(a) shows the Type 3 problematic orientation events seen by observers S3 and P6. In these examples, the significant angle the (local) flux rope axis makes with respect to the $\pm\bm{\hat{t}}$ direction can be seen in the constant-latitude plane rendering of the MHD simulation $B_{\theta}$ values (in the red-to-white-to-blue color scheme). Here the red (blue) colors represent positive (negative) values, illustrating the bipolar or twist component of the flux rope. The local flux rope axis, therefore, can be considered as the $B_{\theta}=0$ point of the sign change, i.e. essentially the midpoint of the central white stripe between the red-to-blue transition. The intersection of the meridional plane ($R$–$N$) and the latitudinal plane ($R$–$T$) shows the $\bm{\hat{r}}$ direction. Since the angle between the flux rope axis (white stripe) and $\bm{\hat{r}}$ is not 90^∘, there will be some contribution of both the axial and azimuthal fields of the flux rope to the spacecraft’s $B_{R}$ time series.

The flux rope parameter values for S3 and P6 are given in Table A.2. Every in-situ model fit for S3 gives $\phi_{0}$ values with anywhere from 20^∘–70^∘ departure from the classic bipolar orientation of $\phi_{0}=90^{\circ}$, while the model fits to P6 give $\phi_{0}$ values with a 5^∘–60^∘ departure. The model profiles shown in the central and right columns of Figure 5(a) do not give the impression of especially good or especially bad flux rope fits, even when compared to the classic Types 1 and 2 of Figure 4. However, there is much more variation in the fit parameters (and derived quantities) between models for these cases than in those previous events. For example, in the S3 (P6) fits, the flux rope radius, $R_{c}$, ranges from 0.013–0.045 AU (0.018–0.037 AU), largely due to the variation in the impact parameter. In the LFF and CCS models, the impact parameter $p_{0}/R_{c}$ is found to be on the order of 50% for both S3 and P6, but the visual representation of the spacecraft in Figure 5(a) shows this is clearly not the case. Thus, despite the more-or-less adequate fits to the MHD time series components, these in-situ flux rope fits should be considered more uncertain than those of the classic types.

Our final category of synthetic observer flux rope encounters is the problematic impact parameter type. These events correspond to spacecraft trajectories that pass a substantial distance from the central symmetry axis of the ICME flux rope. In terms of in-situ flux rope model parameters, the Type 4 events are usually characterized by normalized impact parameters of the order $p_{0}/R_{c}\gtrsim 0.50$. The effects of the large impact parameter on the flux rope magnetic field signatures is typically to decrease the magnitude of every component as well as to reduce the overall rotation that the $\bm{B}$ vector makes during the ICME interval. It is well known that the quality of the in-situ flux rope model fits also suffers when the impact parameter becomes large. For example, in one of the first papers to compare different flux rope model fits to MHD simulation data, Riley et al. (2004) showed that when the impact parameter was small, essentially every model gave similar and consistent fitting results, whereas for a large impact parameter there was much more variation between model fits and none of the models did particularly well at reproducing the actual distorted, elliptical shape of the MHD ejecta cross-section.

Figure 5(b) shows the Type 4 events, S8 and P4. The S8 encounter is rotated in the same manner as the classic unipolar cases. Here we can see the $\delta$ profile has a single, central peak at $\sim$70^∘ corresponding to the unipolar $B_{N}$ component profile. In both of the MHD visualizations, the large impact parameter puts the S8 and P4 spacecraft trajectories through their flux ropes much closer to the outer layers. Hence, the dark gray field lines are seen to wrap around the flux rope axis much more than in Figure 4 or 5(a) where they were essentially tracing field lines through the central axis.

The in-situ flux rope model fit parameters are fairly consistent for S8, i.e. each give an axis inclination angle of $\sim$67^∘, impact parameters between 50–60% of $R_{c}$, and $R_{c}$ values of 0.036 AU. The flux rope model fit parameters for the P4 synthetic trajectory yield consistent values for some aspects of the eject (e.g. $\theta_{0}$, $R_{c}$) but considerably more variation in others (e.g. $\phi_{0}$, $p_{0}$). Qualitatively, the S8 time series has a larger field strength and more of a coherent rotation in the angular profiles and/or component profiles. The P4 time series shows almost no rotation in the $(\delta,\lambda)$ angles.

An important feature of both the Type 3 and Type 4 problematic encounter profiles is that the set of $\chi^{2}$ error norms—defined in Equation 23 and used in the optimization of the model fit parameters through its minimization—are not quantitatively worse than those of Types 1 and 2. It is also not immediately obvious that a given set of model fits to a Type 3 or 4 time series are, by visual inspection, qualitatively worse than the Type 1 or 2 fits. In a certain sense, each set of model fits for the Type 3 and Type 4 encounters are worse than the corresponding set of model fits for each Type 1 and Type 2 encounter, but we would have to use a “quality of fit” metric that is includes the variation of the models’ best-fit parameters within each set.

One of the primary uses of the in-situ flux rope models is to estimate the large-scale ICME flux rope geometric properties and orientation. The parameters describing the flux rope model cylinder axis direction are the two orientation angles: the azimuthal angle $\phi_{0}$ in the $R$–$T$ plane and the elevation angle $\theta_{0}$. The third parameter required to fully specify the 3D orientation with respect to the observing spacecraft is the impact parameter $p_{0}$, which describes how close the spaceraft trajectory passes to the flux rope cylinder axis. The radial size of the cylindrical cross-section, $R_{c}$, can then be calculated as a function of the three model orientation parameters and the observed radial velocity time series—specifically, the ejecta duration and an average $V_{r}$ during the interval (e.g. see equation A4 in Lynch et al. 2005).

In order to compare the in-situ flux rope model fit parameters with the MHD simulation “ground truth,” we need to estimate equivalent cylinder geometry parameters from the MHD data cubes. Our procedure for estimating the MHD version of $\{\,\phi_{0},\,\theta_{0},\,p_{0}/R_{c},\,R_{c}\,\}$ is as follows. The flux rope size, $R_{c}$, is the most straightforward; we choose left ($L$) and right ($R$) boundary features in the $r$ and $\theta$ directions based on the mass density and current density structure of the flux rope to obtain

The normalized impact parameter can then be obtained by estimating the center of the MHD ejecta cross-section as the mid-point of the $L$, $R$ boundary features, ($r_{m}$, $\theta_{m}$), where $r_{m}=(r_{L}+r_{R})/2$, $\theta_{m}=(\theta_{L}+\theta_{R})/2$, and using the coordinates of the synthetic observing spacecraft. This yields

where $\theta_{\rm obs}(t^{*})$ is the observer latitude at the times shown in each of the MHD visualization panels of Figures 4, 5, B.1, and B.2.

The cylinder axis orientation parameters, ($\phi_{0}^{\rm MHD}$, $\theta_{0}^{\rm MHD}$), are estimated from two planar cuts centered on the estimated midpoint. First, the azimuthal angle, $\phi_{0}^{\rm MHD}$, is determined from the spatial orientation of the $B_{\theta}=0$ contour in the $r$–$\phi$ plane at the latitude midpoint, $\theta_{m}$. The $\phi_{0}^{\rm MHD}$ angle is defined with respect to the $+\bm{\hat{r}}$ direction and the $\pm$180^∘ ambiguity is resolved by choosing the direction of the positive axial field. Similarly, the elevation angle, $\theta_{0}^{\rm MHD}$, is determined from the spatial orientation of the $B_{r}=0$ contour on a spherical wedge at the radial midpoint $r_{m}$ and defined as elevation above or below the $r$–$\phi$ plane at $\theta_{m}$ (again, in the direction of positive axial field). For both angles, we fit a straight line to the spatial points of the $\{\,B_{\theta},\,B_{r}\,\}=0$ contours on their respective 2D planes, using the position $(r_{m},\,\theta_{m},\,\phi_{\rm obs})$ as the origin of a local RTN Cartesian coordinate system and calculate ($\phi_{0}^{\rm MHD}$, $\theta_{0}^{\rm MHD}$) from these linear fits. For the rotated cases (S2, S6, S8, P3, P7), we have applied the same procedure as in the non-rotated MHD data and then converted these angles into the corresponding cylinder axis orientation values for the local (rotated) coordinate system.

Figure 6 shows the differences between the best-fit in-situ flux rope model parameters $\{\,\phi_{0},\,\theta_{0},\,p_{0}/R_{c},\,R_{c}\,\}$ and the corresponding estimates of their MHD equivalent. The three in-situ models and the MHD values are plotted as different symbols: MHD–black full squares; LFF–blue diamonds; GH–green triangles; CCS–orange $\times$’s. The error bars are representative 1-$\sigma$ parameter uncertainties derived from the LFF model (e.g., see Lepping et al., 2003; Lynch et al., 2005). We have applied these to every model and the MHD values. The parameter uncertainties are taken to be $\sigma_{\phi}=30^{\circ}$, $\sigma_{\theta}=20^{\circ}$, $\sigma_{(p0/Rc)}=0.30$, and $\sigma_{Rc}=0.10\,R_{c}$. The MHD version of the flux rope cylinder parameters for each synthetic spacecraft’s encounter with the simulation ejecta are also listed in Tables A.1 and A.2.

The cylinder axis orientation angles ($\phi_{0}$, $\theta_{0}$) are well approximated by every in-situ flux rope model for both the Type 1 classic bipolar profiles and the Type 2 unipolar profiles. While it looks like there is large disagreement in the $\phi_{0}$ values for the Type 2 events, recall that for highly-inclined flux ropes ($\theta_{0}=\pm 90^{\circ}$) the azimuthal angle becomes essentially degenerate. The Type 3 problematic orientation profiles show more agreement between the in-situ flux rope models and the MHD estimates for $\theta_{0}$ than $\phi_{0}$. While the overall scatter is the greatest for the Type 4 problematic impact parameter profiles, the flux rope elevation angles for the S8 and P4 profiles are the closest to the MHD values.

The (normalized) impact parameters ($p_{0}/R_{c}$) are reasonably consistent between the in-situ flux rope models and the MHD estimates for both the Type 1 and 2 events, although we note that the flux rope model’s impact parameter uncertainty of 30% is the largest of the parameters examined here. For the problematic Type 3 and 4 profiles, the impact parameters are statistically well matched for some events (S5, P8, S8) and not for others (S3, P6, P2, P4). The flux rope cross-section radius, $R_{c}$, shows the greatest disagreement between the MHD estimates and the in-situ flux rope estimates. Five of the eight flux rope model fits for the classic Type 1 and Type 2 profiles systematically underestimate the MHD flux rope size (S1, S7, P5, S2, P3), while the other three agree reasonably well (P1, S6, P7). Similar to the impact parameter performance, the flux rope model $R_{c}$ estimates for the problematic Type 3 and 4 profiles show more variation and certain fits return obviously incorrect answers (e.g. GH for S3, P6, S4; CCS for S4; every model for P2, etc).

Another method of determining how much coherent rotation the magnetic field vector experiences during a particular event interval is to examine the hodograms of each of the different components, i.e. $B_{R}$–$B_{T}$, $B_{R}$–$B_{N}$, and $B_{T}$–$B_{N}$. Hodograms have been used since the first generation of interplanetary magnetic field measurements (Klein & Burlaga, 1982; Berchem & Russell, 1982), but they are now being used fairly regularly in the analysis of ICME flux rope ejecta (e.g. Bothmer & Schwenn, 1998; Nieves-Chinchilla et al., 2018). The general form of the hodograms for magnetic flux ropes are that two of the three component-pair plots show little-to-no coherent structure while the third shows a large, relatively smooth rotation. In the local flux rope frame coordinate, it is obvious that the axial and azimuthal fields will be the pair of components with the coherent rotation while neither will vary with the radial component (defined as zero in the mathematical formulations of A). This can also be understood as relating to the directions of minimum, intermediate, and maximum variance in the magnetic field over the ejecta interval (specificially, the eigenvectors of the variance space matrix one obtains with Minimum Variance Analysis, e.g. see Rosa Oliveira et al., 2020, 2021, and references therein). Herein, we will consider only the $B_{T}$–$B_{N}$ hodograms as these are expected to be the directions of maximum and intermediate variance, respectively (or vice versa for the rotated, unipolar cases).

Figure 7 shows the $B_{T}$–$B_{N}$ hodogram plots for each of the synthetic spacecraft samples of the 3D MHD flux rope ejecta shown in Figures 4 and 5, organized by profile type: 7(a) Type 1 bipolar profiles (S1, P5), 7(b) Type 2 unipolar profiles (S6, P7), 7(c) Type 3 problematic orientation profiles (S3, P6), and 7(d) Type 4 problematic impact parameter profiles (S8, P4). In each panel the MHD simulation data are shown with black squares and each of the in-situ flux rope models are shown in their respective color-scheme: LFF (blue), GH (green), and CCS (orange). The remaining eight observers’ hodogram plots are shown in Figure B.3. There are a number of interesting features in Figure 7: first, the hodogram representation of the field rotations in the MHD simulation data on their own; second, the hodogram representation of each of the in-situ flux rope models; and third, in the comparison of the two.

Considering the MHD simulation data, we see that the Type 1 and Type 2 curves make a more continuous circular arc than the Type 3 and Type 4 curves. In each panel the $B_{T}$:$B_{N}$ aspect ratio is 1:1 so that the relative shapes between panels can be more easily compared. The Type 3 and 4 hodogram curves tend to be less circular and have considerably more small-scale bumps, wiggles, and/or sharp discontinuities, i.e. the curves are less smooth. However, the Type 1 and 2 hodograms are not completely smooth either, as there is still some substructure along the large-scale arcs.

The hodograms for the in-situ flux rope models are generally smoother than the simulation data, as one might expect from analytic expressions. In every case, the LFF and GH models give continuous circular arcs, often with a significant amount of overlap, although the GH starting and ending points tend to not extend as far as the LFF ones. The CCS hodograms tend to make more of a V-shape, which makes sense given the linear relationship of both the azimuthal and axial field components with radius from the cylinder center ($B_{\varphi}\,,B_{z}\propto\rho$; see Equation 20) and these components map to $B_{T}$ and $B_{N}$ in the syntheric observer’s RTN coordinates.

The $B_{T}$–$B_{N}$ hodogram plots are a complementary way of evaluating the quality of the in-situ flux rope model fits, as well as how close the observational (simulation) data themselves are to the form of an idealized flux rope structure. While there was some indication in the Figure 4, 5 time series that encounter Types 1 and 2 were “better fit” than Types 3 and 4, the hodogram vizualizations of the coherent field rotations show more difference in the sizes, shapes, and positions of the in-situ model fits with respect to the simulation data and more structural differences between the classic and problematic events. The same trends are also seen in the hodograms of Figure B.3.

Given the somewhat unexpected disparity between the MHD and in-situ flux rope models’ estimates for the flux rope sizes, $R_{c}$, we have examined the time evolution of each model’s $R_{c}$ values. It may be the case that, despite a consistent offset between the MHD and flux rope model sizes, the estimates of the flux rope expansion rate(s) are similar. Our previous analyses have treated each synthetic spacecraft trajectory as an independent CME event encounter, but here we now treat each event encounter as a a measurement of the same CME, just at different times.

Figure 8 plots the $R_{c}$ values from each of our trajectories (from Figure 6(d)) with respect to their temporal midpoints, $t_{m}=(t_{s}+t_{e})/2$, to yield a series of $R_{c}(t_{m})$ samples. The points for each flux rope model and the MHD values follow our usual convention (MHD–black full squares; LFF–blue diamonds; GH–green triangles; CCS–orange $\times$’s). We use the same uncertainty estimate of $\sigma_{Rc}/R_{c}=10\%$. The linear fit to each flux rope model and the MHD values are shown as the solid lines in the associated color. We define the expansion velocity as $V_{\rm exp}=\partial R_{c}/\partial t$ and convert the slope of each linear fit from $R_{\odot}$/hr to km/s. These values, along with their parameter uncertainties, are listed in the Figure 8 legend.

There are two points to highlight in these results. First, the MHD, LFF, and CCS models all yield remarkably similar $V_{\rm exp}$ values on the order of 60–90 km/s. The exception is the GH model $R_{c}(t)$ values, which give a linear fit with a negative slope—largely biased by the two problematic orientation points (S3, P6) at $(t-145)\gtrsim 11$ hr. The second point is that the mean $V_{\rm exp}$ consensus for the MHD, LFF, and CCS models is consistent with the observed expansion speeds of flux rope CMEs in the STEREO/COR2 coronagraph observations during solar minimum. The multi-viewpoint STEREO/COR2 CME catalog (Vourlidas et al., 2017) shows the average CME expansion velocity is $V_{\rm exp}\sim 100$ km/s for the three years of solar minimum (2007–2009). This period is when the majority of streamer blowout CMEs have an easily identifiable flux rope morphology (Vourlidas & Webb, 2018). Given that our MHD simulation is essentially one giant streamer-blowout eruption, the agreement with the observed $V_{\rm exp}$ is encouraging. Additionally, our $V_{\rm exp}$ values are also entirely consistent with the in-situ velocity profiles within flux rope ICMEs measured throughout the inner heliosphere (e.g. Lugaz et al., 2020, and references therein). This also implies that the LFF and CCS flux rope models are “good enough” to be able to capture a significant portion of the MHD ejecta’s expansion, despite the fact these are both static models and there is apparently some systematic disagreement in the estimated CME flux rope sizes.

An important implication of flux rope expansion in the extended solar corona is that the magnetic structure must not be in equilibrium, i.e. it is not force-free. The force-free in-situ flux rope models (LFF, GH) have, by definition, $\bm{j}\parallel\bm{B}$ so the Lorentz force, $\bm{j}\times\bm{B}$, is identically zero. It is instructive, therefore, to investigate how appropriate this assumption is in our MHD simulation’s time series, and evaluate just how non-force free the CCS model fits end up being.

A number of authors have examined this issue via different methodologies. One way of quantifying the departure from a force-free state is to examine the ratio of perpendicular-to-parallel currents $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\parallel}\,|$ (e.g. Möstl et al., 2009) or perpendicular-to-total current $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\rm total}\,|$ (e.g. Nieves-Chinchilla et al., 2016). This current density decomposition, $\bm{j}_{\rm total}=\bm{j}_{\perp}+\bm{j}_{\parallel}$, is defined with respect to the magnetic field direction and is therefore obtained via

Given that the force-free component of the current density ($\bm{j}_{\parallel}$) is parallel to $\bm{B}$, these current ratios are simply the equivalent of examining the misalignment angle between the $\bm{j}$ and $\bm{B}$ vectors, defined as $\sin{\vartheta}=|\,\bm{j}\times\bm{B}\,|/(\,|\,\bm{j}\,|\,|\,\bm{B}\,|\,)$.

Herein, we use the Nieves-Chinchilla et al. (2016) ratio of the perpendicular-to-total current densities because $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\rm total}\,|=\sin\vartheta$ whereas the Möstl et al. (2009) ratio gives $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\parallel}\,|=\tan\vartheta$. In practice, the force-free threshold chosen by Möstl et al. (2009) of $\tan\vartheta<0.30$ can be applied to the $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\rm total}\,|$ ratio with almost no discernible difference, i.e., a maximum of $0.75^{\circ}$ in the resulting values of $\sin^{-1}(0.30)\approx\tan^{-1}(0.30)\approx 17^{\circ}$.

Figure 9(a) shows a representative example of the $|\,\bm{j}_{\perp}\,|/|\,\bm{j}_{\rm total}\,|$ ratio (and its equivalent misalignment angle $\vartheta$) derived from the MHD simulation data as a function of time seen by the P5 synthetic observer. The orange W-shaped profile shows the misalignment angle of the non-force free CCS flux rope model, $\sin\vartheta(t)^{\,\rm CCS}$, determined by the best-fit model parameters (see A and Equation 22). The red dotted horizontal line denotes the misalignment angle threshold $\sin(0.30)=17.45^{\circ}$ and the blue dashed vertical lines indicate the flux rope boundaries at start time $t_{s}$ and end time $t_{e}$.

The largest non-force free regions within the CME flux rope are clearly at the boundaries, i.e. at the CME–solar wind interface, but the misalignment angle is not uniformly distributed throughout the magnetic ejecta. Using the Möstl et al. (2009) and Nieves-Chinchilla et al. (2016) thresholding, we find that 51% (19/37 data points) of the P5 ejecta interval identified in Figure 9(a) can be considered approximately force-free, whereas the CCS flux rope model fit over the same interval yields 32% (12/37 data points).

We have also calculated the mean and standard deviation of the misalignment angle from the MHD $\bm{j}(t)\times\bm{B}(t)$ time series directly over each of the flux rope ejecta intervals. Figure 9(b) plots $\langle\,\vartheta\,\rangle\pm\sigma_{\vartheta}$ for each of the CME flux rope intervals in our set of synthetic time series. The MHD values are shown as the black squares and the analogous values derived from the CCS model profiles are shown as the orange $\times$’s. The large $\sigma_{\vartheta}$ values are a consequence of the misalignment angle structure at the flux rope boundaries (as seen in Figure 9(a)).

A somewhat complementary approach has been employed by Subramanian et al. (2014) in their study of the self-similar expansion of CMEs observed in STEREO/SECCHI coronagraph data and how that relates to the overall Lorentz “self-force” as a driver of the CME eruption and/or propagation. Subramanian et al. (2014) used an expression for the $\bm{j}\times\bm{B}$ misalignment angle

to evaluate the range of values for $\vartheta$ inferred by the observed CME expansion. Here, $x_{01}$ is the first zero of the Bessel function $J_{0}(x)$ and $\kappa$ is the observed self-similar expansion parameter, defined as the ratio of the flux rope’s minor radius to its major radius. This expression for the misalignment angle was derived by Kumar & Rust (1996) under the assumption that the flux rope’s magnetic structure has only a small departure from the LFF Bessel function solution.

We have performed the same calculation, defining the self-similar parameter for each of our MHD CME samples as $\kappa=R_{c}/r_{m}$, using the $R_{c}$ values from Figure 8 and the midpoint distances $r_{m}$ (from Section 4.2.1). We obtain an average self-similar parameter of $\langle\kappa\rangle=0.357\pm 0.042$, which is in excellent agreement with the Subramanian et al. (2014) observational values (e.g. see their Table 1). This self-similar parameter yields an average misalignment angle of $\langle\,\vartheta\,\rangle^{\rm KR96}=8.5^{\circ}\pm 1.0^{\circ}$ over our set of 16 encounters and each of the individual $\vartheta^{\rm KR96}$ values are also shown in Figure 9(b) as the purple + symbols.

For most of the synthetic encounters, the MHD and CCS values are consistent within the statistical spread, however it is worth highlighting that both of these are always greater than the corresponding Kumar & Rust (1996) values. Similar to our previous findings, the CCS flux rope model results are more consistent—with each other and with the MHD values—over the subset of Type 1 and 2 events. The other important takeaway from Figure 9(b) is that the percentage of each MHD flux rope interval that can be considered force-free is greater than the corresponding percentage derived from the CCS model fits. In other words, for 12 of our 16 profiles, the MHD intervals have more points below the “force-free threshold” than the CCS fits to the same intervals.

There is some ambiguity in exactly what range of misalignment angles can be considered “nearly force-free.” Subramanian et al. (2014) obtained a similar range of $\langle\,\vartheta\,\rangle^{\rm KR96}$ angles (5^∘–10^∘) and suggested these values were sufficiently large to demonstrate the “nearly force-free” assumption of Kumar & Rust (1996) was inconsistent with coronagraph observations—despite being comfortably below the Möstl et al. (2009) and Nieves-Chinchilla et al. (2016) threshold (${\sim}17^{\circ}$) used in the in-situ flux rope analyses. Overall, we conclude that the MHD flux rope profiles obtained by our synthetic observers, while obviously not force-free, are in some sense, more force free than the in-situ CCS flux rope model fits to the same MHD profiles and less force free than the Kumar & Rust (1996) assumption of a small deviation from the LFF model structure.

There is a direct quantitative relationship between the reconnection flux in the corona and the magnetic flux swept by the flare ribbons (Forbes & Priest, 1983; Qiu et al., 2007; Kazachenko et al., 2017). Magnetic reconnection rapidly adds flux to the erupting magnetic structure(s) during the eruption (Welsch, 2018) resulting in the formation of coherent, twisted flux rope structures in agreement with remote-sensing and in-situ CME observations (e.g. Bothmer & Schwenn, 1998; Vourlidas et al., 2013; Hu et al., 2014). In addition to the relationship between eruptive flare reconnection and CME acceleration (Jing et al., 2005; Qiu et al., 2010), properties of the magnetic field—handedness, orientation, and magnetic flux content—are important quantities for making the CME–ICME connection (e.g. Qiu et al., 2007; Démoulin, 2008; Hu et al., 2014; Palmerio et al., 2017; Gopalswamy et al., 2017, 2018).

The toroidal/axial flux $\Phi_{t}$ and poloidal/twist flux $\Phi_{p}$ of the in-situ flux rope models are given by

in cylindrical flux rope coordinates $(\rho,\,\varphi,\,z)$. In these models with circular cross-sections, $R_{c}(\varphi)=R_{c}$ and the area is simply $\pi R_{c}^{2}$. The poloidal/twist flux calculation involves integrating $dz$ along the flux rope axis over the entire length of the ICME. Since the axial length $L$ of interplanetary flux ropes are generally unknown, it is common to either estimate their value as 2–2.5 times the radial distance of the observation (e.g. Leamon et al., 2004; Pal et al., 2021) or just calculate the poloidal/twist flux per unit length, $\Phi_{p}/L$.

Figure 10 plots $\Phi_{t}$ and $\Phi_{p}/L$ calculated for each in-situ flux rope model. In terms of the model parameters (or quantities derived from the model fit parameters), the toroidal/axial fluxes are calculated as:

We refer the reader to A for the precise definitions of each of the flux rope model parameters but note here the general form of $\Phi_{t}\sim$ constant $\times\,(B\,\pi R^{2})$. The poloidal/twist fluxes (per unit length) are calculated as:

Here we note the general form of $\Phi_{p}/L\sim$ constant $\times\,B\,R$. The uncertainties for the in-situ flux rope model magnetic flux quantities are obtained via the usual fractional error addition: $\sigma_{t}/\Phi_{t}\approx 22\%$ and $\sigma_{p}/(\Phi_{p}/L)\approx 14\%$.

To evaluate the in-situ flux rope model fluxes, we calculate a $\Phi_{t}$ and $\Phi_{p}/L$ from the MHD simulation data cube at each time indicated in Figures 4, 5, B.1, and B.2. The toroidal/axial flux is determined by integrating $B_{\phi}\,dA$ for $B_{\phi}\geq 0$ over a spherical wedge in the $r$–$\theta$ plane encompassing the flux rope cross-section. The poloidal/twist flux (per unit length) is obtained by integrating $|B_{\theta}|\,dr$ in our spherical wedge for the whole range of $\theta$ values and averaging the positive ($B_{\theta}>0$) and negative ($B_{\theta}<0$) values. The MHD flux estimates are shown as black squares in Figure 10. The mean $\Phi_{t}$ and $\Phi_{p}/L$ values for the Type 1 and Type 2 events are $\langle\Phi_{t}^{\rm MHD}\rangle=1.95\pm 0.16\times 10^{22}$ Mx and $\langle(\Phi_{p}/L)^{\rm MHD}\rangle=1.13\pm 0.22\times 10^{10}$ Mx cm^-1. These averages are shown in the corresponding left Figure 10 panels as the solid horizontal lines with $\pm 1\sigma$ shown as the dashed lines. The yellow horizontal line shows the Lynch et al. (2019) reconnection flux estimate $\Phi_{\mathrm{rxn}}/L=1.64\times 10^{10}$ Mx cm^-1 from the flare ribbon area at $t=152$ hr and approximating the flux rope axis length as $L=2\pi R_{\mathrm{CME}}$ with $R_{\mathrm{CME}}=25R_{\odot}$.

We note that the Type 3 and 4 MHD flux values that deviate the most from the Type 1 and 2 average values in one or both quantities, i.e. S3, P6, S4, and P2, have their estimates at the largest simulation times (155.67, 154.50, 155.67, and 153.0 hr, respectively). As seen in Figures 5 and B.2, for each of these synthetic observers, the leading edge or a nearby section of the magnetic ejecta has passed through the $r=30R_{\odot}$ outer boundary, so it seems reasonable to expect both the toroidal/axial and poloidal/twist fluxes to be a bit lower in these cases.

The right panel of Figure 10 shows the same data (with the same colors and plot symbols) as the left panels, but now as a two-dimensional distribution of $\left(\Phi_{t},\Phi_{p}/L\right)$. The $\langle\Phi_{t}^{\rm MHD}\rangle$ and $\langle(\Phi_{p}/L)^{\rm MHD}\rangle$ values are the vertical and horizontal black lines. In order to assess the overlap of the in-situ flux rope model flux values with the corresponding MHD flux estimates, we fit a 2D Gaussian to the smoothed, 2D histogram of the in-situ flux rope model points using the standard IDL function gauss2Dfit.pro. The red contours show the resulting Gaussian distribution at 0.5-$\sigma$ intervals between 0.5 and 3.0 standard deviations. The 1-$\sigma$ width of the distribution function in the $\Phi_{t}$ direction is $\sigma_{t}=0.525\times 10^{21}$ Mx whereas in the $\Phi_{p}/L$ direction it is $\sigma_{p}=0.325\times 10^{10}$ Mx cm^-1. The tilt of the contour ellipses show that the variables $\left(\Phi_{t},\Phi_{p}/L\right)$ are not independent (as expected given their dependence on the same model parameters). We have also manually constructed a similar 2D Gaussian from the classic-profile MHD flux averages with their respective standard deviations given above. The MHD-average Gaussian distribution is shown as the black elliptical contours (at 0.5-$\sigma$ intervals) centered on $\left(\langle\Phi_{t}^{\rm MHD}\rangle,\langle(\Phi_{p}/L)^{\rm MHD}\rangle\right)$.

The overlap between the in-situ model fit and MHD-average distributions presented in Figure 10 provides another way to visualize the flux rope model performance. The MHD distribution contours (black) fall almost entirely within the 1$\sigma$ and 2$\sigma$ contours (red) of the in-situ flux rope model distribution. The peak of the in-situ flux rope model distribution lies below the peak of the MHD distribution in both toroidal and poloidal flux coordinates (also seen in the clustering of the individual model fit points in the lower-left quadrant rather than a more uniform distribution across each quadrant). This suggests that the in-situ flux rope models are, at least statistically, underestimating both components of the CME flux content: the toroidal/axial flux by $\sim$60%, i.e., $\langle\Phi_{t}^{\rm MHD}\rangle\approx 1.63\,\langle\Phi_{t}^{\rm FR}\rangle$, and the poloidal/twist flux by $\sim$25%, i.e., $\langle(\Phi_{p}/L)^{\rm MHD}\rangle\approx 1.26\,\langle(\Phi_{p}/L)^{\rm FR}\rangle$. However, the MHD flux estimates for the problematic Type 3 and Type 4 events are also lower than the classic-profile averages, in roughly the same direction and proportion as the in-situ flux rope model points.