Magnetic reconnection is an important process in driving the dynamics of the Earth’s magnetosphere (e.g. Eastwood et al., 2017 and references therein). When the interplanetary magnetic field (IMF) is southward (IMF B z < 0 ), it allows for enhanced plasma entry into the magnetosphere by opening up magnetic field lines on the dayside and closing on the nightside (Dungey, 1961). In reality, reconnection is inherently three-dimensional and non-steady (Fu et al., 2015; Wang et al., 2020), causing the formation of Flux Transfer Events (FTEs). They are thought to transfer magnetic flux and hence energy into the magnetosphere when they are “open”: connected to either of the planet’s poles and to the IMF. Though the origin and evolution of FTEs are not fully understood, they have been observed to have a significant effect on magnetospheric dynamics and space weather.

FTEs are identified in spacecraft observations by their characteristic bipolar signature in magnetic field data, typically in the component normal to the magnetopause surface (Farrugia et al., 2016; Russell & Elphic, 1978). This signature may be indicative of rotational magnetic field structures called flux ropes. While flux ropes exist in a variety of locations, such as on the solar surface or within coronal mass ejections, FTEs are specifically flux ropes generated on at the Earth’s magnetopause, and range from 0.6 R E to 4.4 R E in size (Fear et al., 2007) They consist of twisted magnetic field structures which can persist due to their force-free nature. Furthermore, they typically contain plasma features due to reconnection (hot and high velocity plasma populations), variation in the magnetic field strength (Fear et al., 2008; Paschmann et al., 1982), and force-free magnetic fields (Farrugia et al., 2016).

With the majority of FTEs observed on the dayside, little is known about their transition around the magnetosphere. The prevailing theory is that once reconnection forms an FTE, it is accelerated along the magnetopause surface towards the cusps by the J → × B → force (Fedder et al., 2002; Russell & Elphic, 1978). However, FTEs have been observed far into the magnetotail ( x ≃ 67   R E ), where their axis has been rotated from aligned in the azimuthal direction (parallel to the x G S E − y G S E plane) to aligned with the z G S E axis (Eastwood et al., 2012).

One of the main difficulties in understanding FTEs and their impact is due to the relative sparseness of spacecraft observations. Multi-spacecraft missions such as the Magnetospheric Multiscale Mission (MMS), Time History of Events and Macro-scale Interactions during Substorms (THEMIS), and Cluster have improved our understanding of FTE size and motion (Farrugia et al., 2016; Fear et al., 2005, Fear et al., 2008). In an FTE observed by MMS, it was found that there were two distinct electron populations within the FTE: one characteristic of originating from the magnetosphere, and the other from the magnetosheath, suggesting a single FTE can have a complex magnetic topology (Kacem et al., 2018). Whilst auroral signatures of FTEs have been observed, they are rarely accompanied with a spacecraft in the correct position to observe the FTE structure. Hence, global simulations of the magnetosphere, which capture the magnetosphere as a whole, can be very useful in understanding a number of FTE features, giving insights into their formation, evolution, topology and impact on the magnetosphere.

A number of papers have used global simulations to study FTEs and flux ropes. Though it is known that MHD cannot capture the full physics required for reconnection, it has been shown that global simulations do capture the location of reconnection at the dayside magnetopause relatively well (Komar et al., 2015). Raeder (2006) used OpenGGCM to investigate the effect of dipole tilt on the generation of FTEs. Under a strongly southward IMF, the FTEs were formed by multiple x-line reconnection, and was modulated by the dipole tilt. Cardoso et al. (2013)’s simulation of a steady, strong South- and duskward IMF, found elbow-shaped flux tubes. Other flux tubes were found to be interlinked with field lines of different topology. Perez et al. (2018) further reported a study showing a number of flux ropes spontaneously forming under constant solar wind parameters.

The majority of simulations studying this problem have used idealized solar wind events, with inflow parameters usually kept constant. In this article, the Gorgon code (Ciardi et al., 2007; Mejnertsen et al., 2016, 2018) is used to simulate flux ropes that are formed in response to a real interval of strongly southward IMF solar wind as observed by ACE and Cluster on March 31, 2001 (Maksimovic et al., 2003). By using a global simulation approach, the full three-dimensional structure of different flux ropes are found as a function of time. This allows their properties to be established, and their generation, transport and fate to be studied. Furthermore, their magnetic topology can be calculated, from which the amount of flux transferred into the magnetosphere can be inferred. The manuscript is organized as follows. In Methodology, we discuss the methodology, including the Gorgon MHD code and the properties of the solar wind on the day of interest. Overview of the Event Simulation provides an overview of the Simulation, and Identification of Flux Ropes Through Field Line Tracing and Topology Mapping describes in detail the methodology used to identify and study flux ropes at the magnetopause. Results presents the results of the flux rope analysis, and conclusions are summarized in Conclusion.

The work in this article also uses the same solar wind parameters as in Mejnertsen et al. (2018), from March 31, 2001 17:14:00 – 19:34:00 UT. This interval was the subject of a case study by Maksimovic et al. (2003), who used four spacecraft Cluster observations to characterize the motion of the bow shock in response to variation in the solar wind inflow. Whereas Mejnertsen et al. (2018) focused on the motion of the outer boundaries over the whole 2 hour period, but reduced to the ecliptic and noon-midnight plane, in this chapter the focus is on the formation of FTEs over a shorter 20 min timeframe. For this, the full three-dimensional state of the simulation is sampled at 10 s intervals.

The full solar wind input to the simulation is shown in Figure 1. However, now the focus is on the analysis interval from t = 190   min to 210   min as is indicated by the grey shaded region. The start of the simulated period (t = 0) is 15:14:00 UTC, and so this corresponds to 18:24:00 – 18:44:00 UTC. During this time, the solar wind number density drops from 30   / c m 3 to 10   / c m 3 , as seen in panel (a). The solar wind speed stays constant at approximately 600   km / s (panel b). However, there is a significant v y component, averaging approximately 100   km / s . The strength of the IMF (panel 3) was larger than typical IMF strength, ranging between 20   nT and 25   nT . The dominating IMF component is the B z component, with B z starting at approximately − 30   nT and reducing in magnitude slightly near the end. During this interval, the B y component drops to approximately zero. With a negligibly small constant B x , the IMF is predominantly southward, with no dawn-dusk component. The final panel 4) shows the sub-solar magnetopause and bow shock positions, as calculated by the methods in Mejnertsen et al. (2018). As expected, the reduction of solar wind number density causes the magnetopause distance to increase, from approximately 6.5   R E to 7.5   R E . This causes the bow shock position to also increase in distance. The magnetopause position is much more variable than the bow shock position (Mejnertsen et al., 2018).

We first discuss the overall dynamics of the magnetopause during the 20 min interval of interest. The plasma number density in the ecliptic plane is shown in Figure 2 at 1.5 min intervals from t = 190   min to 206.5   min. The solar wind streaming in from the left of each subplot first passes through the bow shock, shown by the increase in mass density. It flows around the magnetopause, which corresponds to the sharp cut-off in mass density. The magnetopause and bow shock are approximately conical, but oriented at an angle to the Sun-Earth line: this is due to the significant v y component in the solar wind. At t = 190   min (panel 1), the magnetopause appears relatively smooth: there are only a few small structures, indicated by the bulges in the sharp density cut-off corresponding to the magnetopause. At this time, the solar wind number density is also at its highest, leading to the bow shock and magnetopause being most compressed, with a relatively thin magnetosheath. As time increases, the solar wind number density decreases, as can be seen by the decreasing magnitude of the color bar scales. This increases the magnetopause and bow shock standoff distances, and increases the magnetosheath width, which is most easily seen by comparing panels 1, 4, 7 and 10. As the number density decreases, more coherent structure forms on the magnetopause. On the dawn side, y G S E < 0 , there is a rippling of the magnetopause surface (indicated by the white dotted ellipse on Figure 2) which persists throughout each panel in Figure 2 with varying amplitude. On the dusk side, y G S E > 0 , distinct bulges in the plasma can be seen propagating down the flanks of the magnetopause; an example of one is shown in Figure 2 by the white dotted ellipse. These bulges are much more sporadic and intermittent than on the dawn side. An example of these bulges can be seen at t = 193   min, at position x = 5   R E , y = 10   R E , which propagates along the magnetopause surface before disappearing in the tail.

This same variability in the structure of the magnetosphere can also be seen in the current density magnitude, as shown in Figure 3. The magnetopause current density magnitude is largest in the sub-solar region, and decreases down the flanks. Many of the structures seen in the number density (Figure 2) are also visible here. On the dawn side, the same oscillatory feature can be seen (indicated by the white dotted circle on Figure 3), as well as the bulges moving down the dusk-side flank. More structure can also be seen on the sub-solar magnetopause. For example, at t = 194.5   min (panel 4), the magnetopause current sheet exhibits local variations in current density strength.

Figure 2 and Figure 3 show that there are dynamic processes occurring on the magnetopause surface, which propagate down the flanks of the magnetosphere. We now discuss these features in more detail, finding that they are in fact caused by flux ropes associated with magnetic reconnection on the dayside magnetopause. We now describe the methodology used to identify flux ropes in the simulation before examining their properties in more detail.

In order to visualize magnetic field lines and understand the features of the magnetopause, in particular to identify flux ropes, streamlines are calculated from the magnetic field vector field. This is performed for two reasons. The first is to establish the existence of flux ropes based on identifying twisted field line structures, and the second is to examine flux rope topology. Flux ropes may be open (connected at one end to the ionosphere with the other end in the solar wind), closed (connected at both ends to either the ionosphere), or entirely contained within the solar wind.

The quality of stream-tracing is heavily dependent on its starting (seed) point. By under-sampling the magnetic field, visualizations can miss vital structure and dynamics. Ideally, there would be multiple seed points for every grid cell to ensure that every flux rope of interest is captured. In typical simulation grid sizes, this is computationally prohibitive. To render stream-tracing feasible, it is only performed in regions of interest, such as on the dayside or the nightside flank, as is shown in Figure 4.

Since global simulations provide the full three-dimensional path of field lines, their topology can be found by categorizing the field line according to their end point location. In cases where both ends of the field line reaches the outer boundary, the field line is said to be part of the IMF, and is colored red. When both ends of the field line touch the inner boundary, it is said to be closed, and is colored blue. Open field lines occur when one end connects to the inner boundary, and the other to the outer boundary. Open field lines are further classified by which pole they reach: North (magnetic South) is purple, and South (magnetic North) is green. The final type of field line identified by the algorithm are so-called incomplete field lines, where one or both ends do not reach a simulation boundary. Computationally, the stream-tracer assign a finite number of steps to the streamline, so when the field line has not reached a boundary, it has likely run out of steps.

In order to filter out flux ropes from background magnetic field lines, the amount a field line twists or rotates is also computed. This measure of twist provides a way to isolate the twisted flux rope fields. The total rotation, Λ , is given by the angle between subsequent magnetic field directions along a field line, Λ = ∑ i cos − 1 ( − B ^ ( s i ) ⋅ B ^ ( s i + Δ s ) ) ,     where s is the path along the field line and Δ s is the stream-tracer step size. A perfectly straight field line will thus have Λ = 0 . A perfectly circular field line will have Λ   = 2 π = 360 ° after completing one revolution.

We note that the computation time can be decreased even further by removing the need to store every point in the field line and saving only the large-scale properties of each field line. This is used in topology mapping, where, for every seed point of interest, the field line is calculated to determine where it is connected to. An example of this is shown by the slice shown in Figure 4: a streamline is calculated from every position on that slice, and its topology is saved. This is then visualized on the slice.

By combining the topology mapping algorithm with calculations of the total rotation, Λ , regions where flux ropes have formed can be identified. Visualizing only field lines which exhibit flux rope characteristics gives a clearer picture of flux rope dynamics, whilst also finding all the flux ropes which meet the rotation criterion. The general method in plotting these flux ropes is as follows. For every timestep, the total rotation is calculated at cells in the simulation grid in the region of interest (e.g. the dayside). For each seed point where the Λ is large enough, usually Λ > 8 π , their field line is drawn and its topology calculated. The threshold value of 8 π was chosen through trial-and-error, as it allowed for effective filtering of non flux rope like field lines (e.g. draped field lines in the magnetosheath), whilst still including flux rope structures.

An illustration of the method is shown in Figure 5. In panel a, the field lines are drawn using seeds spread on a uniform spherical grid. Whilst it does show some structure - for example, the magnetic separator can be inferred by the region separating the South (green) from the North (purple) field lines, it is difficult to see flux ropes, and the IMF has been filtered out as otherwise it would obscure the view of the magnetopause. In contrast, panel b shows the field lines filtered by the rotation method. Flux ropes can more easily be seen, and since the IMF field is not filtered out, examples of all four field topologies is visible.

In this work, we have studied the evolution of flux ropes generated by a real solar wind event using the Gorgon global MHD simulation. The event, observed by Cluster on 2001-03-31, creates several flux ropes on the dayside, which propagate along magnetopause flanks. These flux ropes were identified using a novel magnetic field line filtering method based on computing their total rotation. While the method allowed for effective identification of tightly wound flux ropes, it may miss smaller flux ropes.

The flux ropes are formed on the dayside magnetopause surface, where the strongly southward IMF reconnects with the magnetospheric field. Overlapping the highly warped magnetopause current layer, there exists a complex interlaced set of field lines whose topologies vary from closed to open to completely IMF. This is due to time-dependent reconnection along multiple x-lines on the magnetopause. Out of this complex interwoven dayside field region emerge stand-alone flux ropes which propagate along the magnetopause surface. However, the latitude of their path is dependent on their topology. Flux ropes which are completely open - they are solely connected to a single pole (North/purple or South/green) - tend to move along the magnetopause at high latitudes. These flux ropes would be the equivalent to the classic flux rope evolution picture where they move towards the cusps. No flux ropes were observed to go directly over the cusps.

Flux ropes containing both closed and open field lines connected to both the North and South poles were also generated on the dayside and remain along the ecliptic plane at low latitudes throughout their evolution. In the classic cusp flux rope picture, the J → × B → force acts to move the flux rope over the poles. Since this flux rope is connected to both poles, the two opposite curvature forces cancel out, carrying it along ecliptic plane in the direction of the magnetosheath plasma flow. The flux ropes were found to be force free, with the thermal pressure balancing the J → × B → force. These flux ropes move along the duskward flanks toward the tail, with their axis remaining parallel to the z direction as they move along the flank. However, once they move past the terminator, field lines present in the flux rope begin to reconnect non-locally in the nightside magnetotail current sheet. This alters the topology of the flux rope, detaching it from the magnetosphere. The flux rope-like structure then persists until unravelling ∼1 min later.

This behavior is interesting because it shows that flux rope topology can change during its lifetime as a consequence of reconnection at locations far from the flux rope itself. It may also suggest that flux ropes on the flanks of the magnetopause may be more likely to be topologically disconnected from the ionosphere, (although this is tensioned against a possibly shorter lifetime). FTEs have previously been shown to create ionospheric signatures (e.g. Sandholt et al. (1986); Milan et al. (1999); Wild et al. (2001); Fear et al. (2009)), and so these ionospheric signatures may only partially represent the magnetopause dynamics, particularly away from the dayside. This behavior has further implications for the amount of flux that flux ropes transfer into the magnetosphere. Further work is required to establish how common these changes in topology are for different solar wind conditions, so as to better constrain the extent to which flux rope topologies observed on the dayside correspond to those subsequently found on the nightside.

In the simulated event, no flux ropes were found to transit along the dawnward flank. One potential reason for this is that the flux ropes on the dawnside magnetopause exhibit a total rotation of less than the threshold 8π, and were filtered out. The other reason could be due to the specific solar wind conditions, whose combination of solar wind and IMF parameters could predispose the flux ropes to transit along the duskward flanks. To fully determine the impact of solar wind and IMF parameters on the direction of transit, a thorough study of different solar wind conditions should be performed.Resistivity is an important aspect in simulating reconnection events and flux rope dynamics. In the Gorgon code, a low resistivity is inherent in the simulation due to numerical resistivity caused by the grid resolution. Previous studies of reconnection in Global MHD simulations have found that enhanced resistivity, either through coarse grid resolutions or an artificial resistivity, prevents the formation of FTEs and causes them to diffuse away rapidly (Komar et al., 2013; Raeder, 2006). With the introduction of an artificial resistivity in the Gorgon code, we would expect similar results with fewer flux ropes generated. Those that do generate are likely to dissipate due to resistive diffusion: this is likely unphysical due to the collisionless nature of magnetospheric plasmas.

One potentially fruitful avenue for future work may be to focus on comparing multi-spacecraft observations of flux ropes (e.g. MMS, THEMIS and Cluster) with global simulations using the same solar wind input, particularly where the local topology can be measured. This would allow the evolution of flux rope topology to be explored experimentally. These results may also be relevant for the SMILE mission, where imaging of the dayside magnetopause is expected to shed new light on the role of flux ropes and FTEs in global magnetospheric dynamics.