Simulating the Solar Wind‐Magnetosphere Interaction During the Matuyama‐Brunhes Paleomagnetic Reversal

During a paleomagnetic reversal, Earth's geomagnetic field is about 10% of today's magnitude and its topology is far more complex than dipolar. Revealing the solar wind energy transmission during the reversal is of importance for understanding the historic space environment. Using a global magnetohydrodynamic model implemented with a data‐reconstructed time‐dependent paleomagnetic field, this study simulates the solar wind‐magnetosphere interactions during the last Matuyama‐Brunhes reversal that occurred about 780,000 years ago. As the magnetosphere shrinks in the midst of the reversal, the stand‐off distance of the subsolar magnetopause approaches as close as 3 Re. Moreover, multiple magnetic reconnection sites emerge due to the irregular geomagnetic configuration, opening up many channels on the magnetopause surface for the solar wind to directly access. The power transmission efficiency through the magnetopause increases to 18%.

. It has been suggested that the dipole moment is reduced and the high-order multipoles are significant during the last M-B reversal (Leonhardt & Fabian, 2007;Singer et al., 2019). In order to understand how the geomagnetic field changes in the reversal period influence the magnetospheric configuration and impact on the geospace environment, previous studies have used simplified geomagnetic field configuration for the transitional reversal time. For example, by appropriately scaling today's geomagnetic field models, Vogt and Glassmeier (2001) provided scaling relations for magnetospheric current systems. Such a method of using a scaled dipole field is further used in other studies to examine the magnetosphere-ionosphere coupling and the dependences on the interplanetary magnetic field (IMF) (e.g., , Zieger, Vogt, Ridley, & Glassmeier, 2006. H. Wang et al. (2017) found that reduced geomagnetic fields generally lead to an increase of the thermospheric mass density but a decrease of electron density at auroral latitudes. Some studies also investigated the effects of equatorial dipole magnetosphere  or quadrupole configurations (Vogt et al., , 2007 and suggested that high-energy particles in the MeV range are more likely to penetrate to about 40 • altitude atmosphere during the reversal. In recent years, a superposition of dipole and quadrupole fields has been proposed to be a possible representation during a geomagnetic reversal (K. H. Tsareva, 2019). Stadelmann et al. (2010) considered several possible combinations of dipole and quadrupole paleomagnetic fields (including axisymmetric paleomagnetic fields) and illustrated the particle (in the GeV range) impact regions and cutoff latitudes against kinetic energy.
However, neither the scaled/reduced dipole fields nor multipole fields can realistically represent the paleomagnetic field during the polarity reversal. Therefore, in order to reveal the solar wind-magnetosphere interaction during the reversal stage, we use a global magnetohydrodynamic (MHD) model implemented with a more realistic intrinsic geomagnetic field, which is characterized by the data-reconstructed model "IMMAB4 (final inverse model of M-B)" (Leonhardt & Fabian, 2007). The Space Weather Modeling Framework (SWMF) model is used to simulate the last M-B reversal and explore how the energy transfer from the solar wind into the magnetosphere is dependent on the structure and strength of the intrinsic field.

Model Description
To simulate the solar wind-magnetosphere interaction during the paleomagnetic reversal, we use the SWMF (Tóth et al., 2005(Tóth et al., , 2012) that integrates several first-principle physics-based models, including a global MHD model Block-Adaptive Tree Solar-wind Roe Upwind Scheme (BATS-R-US; Powell et al., 1999) and an ionospheric potential solver . The integrated model is suitable for solving the geospace circulation dynamics and has been extensively validated (e.g., H. Wang et al., 2008;Ridley et al., 2016;Yu et al., 2010Yu et al., , 2015. The BATS-R-US code solves the global magnetospheric using ideal MHD equations from the upstream solar wind to the magnetosphere with the inner boundary at r = 2.0 Re and the upstream outer boundary at 32 Re. For r < 4 Re, the spatial resolution is 1/8 Re, and the outer region has a coarser resolution (e.g., 1/4 Re for r < 8 Re). The electric potential is obtained based on constant conductance (∑ P = 4.0, ∑ H = 0.25) and field alignment currents (FACs) calculated at r = 2.5 Re are mapped to the ionosphere. The electric potential is used to specify the inner boundary velocity of the MHD code.

Setup of Paleomagnetic Field and Solar Wind Conditions
The last geomagnetic reversal, that is, the M-B reversal, occurred approximately 780,000 years ago. Based on paleomagnetic data, Leonhardt and Fabian (2007) developed a Bayesian inversion technique to reconstruct the spherical harmonic expansion of the transitional field from 795 ka to 764 ka. Their data-reconstructed model "IMMAB4" can be used to specify the geomagnetic field at any location in space for the M-B reversal. However, one caveat of the model that should be noted is that it may not represent the complete complexity of the M-B reversal because it ultimately relies on sparse global data coverage, and the M-B reversal may not fully represent the behavior of reversals overall.
We implement the corresponding spherical harmonic coefficients of "IMMAB4" in BATS-R-US (see these coefficients in Table S1 and comparisons of different components in Figure S1 in Supporting Information S1). Due to the limitation of high-resolution data, only the interval between 782 ka and 767 ka is relatively reliable with sufficient data coverage. Singer et al., (2019) proposed in a recent study that the M-B reversal was complicated with various magnetic field changes from 795 to 773 ka. Therefore, in order to attain a more reliable field from the IMMAB4 model, we mainly focus on three different phases of the reversal period within 782 ka and 773 ka (Figure 1).
At 794 ka, the geomagnetic field is approximately a dipole with its magnetic north pole in the northern hemisphere. At 765 ka, the magnetic north pole is in the southern hemisphere, similar to today's configuration. At these two phases, the dipolar components of the Earth's magnetic field are ∼8.78 × 10 22 A m 2 and ∼9.15 × 10 22 A m 2 , respectively, both slightly larger than in AD 2020 at 7.70 × 10 22 A m 2 (Alken et al., 2021). However, in the midst of the reversal (i.e., 774.5 ka), the geomagnetic field no longer exhibits a dipole field, but rather a more complicated and irregular configuration. Local magnetic field loops occur within the same hemisphere. At this epoch, the dipolar field component is ∼4.78 × 10 21 A m 2 , only about 5.5% of that prior to the reversal.
The M-B reversal event lasted for around 22 kyr, much longer than the 11-year solar cycle (Friis-Christensen & Lassen, 1991). We, therefore, drive the BATS-R-US model with constant solar wind and IMF conditions during the paleomagnetic reversal, assuming that the fluctuations in interplanetary conditions are negligible within such a long era. We set the solar wind conditions with V xsw = 400 km/s, N sw = 5 cm −3 , T = 10 5 K, and IMF B z = −10 nT, while V ysw = V zsw = 0 km/s, and B x = B y = 0 nT. With such driving conditions, we conduct 20 simulations from 794 ka to 765 ka, each embedded with a specific intrinsic geomagnetic field determined from the spherical harmonic model "IMMAB4." Each simulation is run for 100,000 steps to arrive at a steady state of the magnetosphere.

Calculation of Energy Transfer Through the Magnetopause
To understand the solar wind-magnetosphere interaction during the reversal, we apply a streamlined method similar to Palmroth et al. (2003) and C. Wang et al. (2014) to identify the magnetopause surface and then estimate the power transfer of the solar wind across the surface.
The total power gain through the magnetopause (SWP m.p. ) is defined as follows (Palmroth et al., 2003): where W m is the mechanical power, W p is the electromagnetic power, dS is the area of the surface element, ̂ is the normal unit vector, K p is the electromagnetic energy flux (i.e., Poynting vector), and K m is the mechanical energy flux. These energy fluxes are defined as The global geomagnetic field distribution during the Matuyama-Brunhes reversal period using the data-reconstructed model "IMMAB4" (Leonhardt & Fabian, 2007). The X-axis and Y-axis represent geographic longitude and latitude, respectively.
where P is the thermal pressure, B is the magnetic field just outside of the magnetopause, V is the velocity, E is the electric field, and U is the total energy density, defined as The above three terms on the right are the thermal energy density, kinetic energy density, and magnetic energy density, respectively, with γ = 5/3 the polytropic index, and ρ the mass density. The total power gain through the magnetopause (SWP ) is calculated by integrating the energy flux (i.e., K p and K m ) above the entire magnetopause surface from the subsolar magnetopause to X GSM = −40 Re in the tail (GSM: geocentric solar magnetospheric coordinates). The positive value indicates inward energy transfer through the magnetopause, and vice versa. In addition, the total kinetic energy of the solar wind impinging on the magnetopause per unit time, or the total effective solar wind power (SWP e ), can be calculated as : where A is the projected cross-section area of the magnetopause surface in the GSM Y-Z plane, and are the upstream (non-disturbed) solar wind mass density and velocity, respectively.

Result
Figure 2 (left) shows a 3D view of the magnetopause surface at different phases during the M-B reversal. The magnetosphere shrinks while undergoing the geomagnetic field reversal, and expands again after the reversal. Figure 2 (right) shows the 3D magnetic field configuration. Before 776 ka, magnetic reconnection occurs at high latitudes where the diverted southward IMF is antiparallel to the Earth's magnetic field. As the reversal progresses, the magnetopause gradually approaches the Earth while the high-latitude magnetic reconnection remains because the geomagnetic field is still manifested as a dipolar field, despite the weakened magnetic field strength. As the size of the magnetosphere decreases, the geomagnetic field exhibits an irregular configuration and many geomagnetic field lines loop locally within the same hemisphere, for example, at 774.5 ka. Multiple reconnection sites emerge where the local "dipolar loops" encounter the antiparallel IMF. When the geomagnetic field slowly turns back into a normal dipole, the reconnection appears on the dayside magnetopause as well as in the magnetotail (i.e., at 773.5 ka), allowing for the familiar Dungey configuration (Dungey, 1961). Under such a state, the magnetosphere continues its expansion as the magnitude of the dipole gradually strengthens.
As illustrated in Figure 3a, the magnetosphere experiences three distinct stages during this polarity reversal event: pre-reversal phase before 779 ka, reversal phase (779 ka-769 ka), and post-reversal phase after 769 ka. In the pre-reversal phase, the stand-off distance of the subsolar point of the magnetosphere smoothly and monotonically decreases from 10 Re to 7 Re, while in the post-reversal phase, the subsolar point gradually migrates outward from 6 Re. In between the two phases, the stand-off distance undergoes drastic changes: rapidly dropping from 7 Re to 3 Re first, and then increasing to 4.5 Re at 774 ka, followed by a quasi-steady "plateau" period for about 2.5 kyr before increasing to 6 Re again. Such a complicated reaction of the magnetosphere indicates that the shape of the magnetosphere depends strongly on the intrinsic geomagnetic field. Figure 3b shows that, in the pre-reversal phase, the total solar wind power gain through the magnetopause surface (SWP m.p. , the results from Equation 1) remains nearly constantly at a level of 10 11 W, comparable to today's power transfer under northward IMF conditions (e.g., Lu et al., 2013). It only enhances near the end of this phase at 779 ka. In contrast, as the geomagnetic dipole moment decreases and the magnetosphere shrinks, the effective solar wind power (SWP e , the results from Equation 5) shown in Figure 3c, representing the upstream solar wind kinetic energy impinging on the maximum cross-section of the magnetopause in the YZ plane, monotonically decreases with the same trend as a dipolar moment but at a time-varying rate. This means that the dipole component has a more significant impact on the Earth's magnetosphere than higher-order components. This pre-reversal SWP e at a level of 10 13 W is consistent with the theory developed in Blackman and Tarduno (2018). The ratio between SWP m.p. and SWP e , that is, the power transmission efficiency, gradually increases to 1% and then briefly fluctuates around 3% near the end of this pre-reversal phase (Figure 3d).
During the reversal phase, as the geomagnetic dipole moment is significantly reduced, the SWP e in Figure 3b dramatically drops to the minimum at 775 ka, and then returns to its previous values, followed by a transitional decrease at 771 ka. After that, it returns to its initial level again. Such variation follows a similar trend of the subsolar stand-off distance. On the other hand, the SWP m.p. in Figure 3c reaches its minimum earlier than the time when the subsolar point shifts to the closest, and then rapidly increases by more than a factor of two until the magnetospheric configuration nearly rests at the "plateau" around 774 ka. Later on, it briefly drops and gradually increases again until 769 ka.
The ratio between the two powers, that is, the power transmission efficiency in Figure 3d, dramatically soars from 3% to 18% at 775 ka. The largest percentage occurs when the SWP m.p. temporarily maximizes. Such high efficiency lasts for about 2.5 kyr before leveling off. This intense power transmission rate appears right in the midst of the reversal when the intrinsic geomagnetic field is substantially irregular and when multiple channels, due to more reconnection sites, are cut open on the magnetopause surface. Despite the small size of the magnetosphere, more open channels provide the solar wind with more access into the geospace system, and hence a relatively larger efficiency. However, although the ratio of the dipole to quadrupole components at the two moments (marked by two vertical green lines in the subplot in Figure 3b) is similar, the power transmission efficiency is quite different. Note that at these two moments, the dipole moment reverses its polarity. Therefore, it is the direction of the dipole component with respective to the IMF Bz that impacts significantly on the power transmission efficiency. Near the end of the reversal, as the geomagnetic dipole moment grows, the magnetosphere configuration restores to be more dipolar and continues to inflate. Those multiple energy channels on the magnetopause surface gradually disappear, suppressing the pathways for energy transmission, resulting in a decreased power transmission efficiency.
Previous studies reported that for a typical intense geomagnetic storm (−300 < SYM-H ≤ −100 nT; SYM-H: 1-min high-resolution storm index), the largest power transmission efficiency is 11.8%-13.2% Li et al., 2012). These large efficiencies are highly dependent on the solar wind electric field and dayside reconnection rate (Li et al., 2010). In this study, the peak efficiency is 18%, much larger than that during an intense storm. This suggests that with a paleomagnetic configuration of the polarity reversal, the power transmission efficiency during a magnetic storm is significantly improved.
In the post-reversal phase after 771 ka, the increase of the geomagnetic dipole moment allows the magnetosphere to gradually recover to the pre-reversal size, and both SWP e and SWP m.p. smoothly increase when the magnetopause inflates and encounters more solar wind energy. The SWP m.p. reaches 10 12 W, comparable to the level in a normal magnetosphere under southward IMF conditions (e.g., C. Wang et al., 2014). However, the power transmission efficiency decreases to 10% and is lower than that during the reversal.
After the solar wind energy transmits through the magnetopause, the amount of energy flowing down to the Earth is of particular importance for the upper atmosphere heating and mass escape (Palmroth et al., 2006). Therefore, we calculate the perturbation Poynting flux S p at r = 2.5 Re during the reversal event using the following formula (Vanhamäki et al., 2012): where E is the electric field, and = − is the magnetic field perturbation, with the intrinsic geomagnetic field at R = 2.5 Re. Figure 4 shows both the downward Poynting flux and the field-aligned currents (FACs) J || at r = 2.5 Re in the northern hemisphere. Before the polarity reversal at 776 ka, as the magnetic reconnection occurs at high latitudes, a pair of strong NBZ-sense FACs and a pair of weak Region-1-sense FACs emerge near the north pole, similar to today's FACs pattern under northward IMF conditions (Ganushkina et al., 2018). The total downward Poynting flux is relatively small, consistent with northward IMF. During the reversal time at 774.5 ka and 774.2 ka, both FACs and Poynting flux are more intense. The global distribution of FACs and Poynting flux extend across all latitudes, even near the equator. This happens when multiple reconnection sites occur on the magnetopause and the magnetic field configuration is substantially non-dipolar, suggesting that the entire near-Earth environment would have intense energy deposition. After 773.5 ka, a pair of Region-1-sense FACs appears and enhances at high latitudes. Equatorward of that, weak Region-2 FACs are also visible. The Poynting flux flows down towards the Earth dominantly in the polar cap region as well as mid-latitudes at both dawn and dusk. From 768 ka on, both FACs and Poynting flux are consistent with today's nominal Dungey configuration. This marks the restoration of the dipolar configuration and the end of reversal.
The Poynting flux and FACs are more intense after the reversal. As the total energy gain through the magnetopause keeps growing following the expansion of the magnetosphere during the post-reversal period, it is not surprising to see stronger Poynting flux deposited downward to the Earth's atmosphere since the configuration is similar to southward IMF today. However, during the reversal, the worldwide and irregular energy transfer channels cause the upper atmosphere to have more globally distributed energy sources, meaning that the upper atmosphere probably would experience relatively random heating across the globe. Knowing whether such a condition is favorable for energy dissipation over the entire upper atmosphere is significant for understanding atmospheric expansion and loss. This will be investigated in our future studies.

Summary and Discussion
It has been proposed that the geomagnetic field can positively help with the preservation of the atmosphere (and ultimately the retention of water) through the shielding effect (Tarduno et al., 2014;Blackman & Tarduno, 2018). During geomagnetic field reversals, the field is likely to be irregular and weakened (Valet et al., 2012;Bono et al., 2019). Previous numerical studies simply assumed a reduced dipole magnetic moment or multipolar magnetic field to represent the geomagnetic configuration in the polarity reversal. This study incorporates a more realistic paleomagnetic model into a global magnetosphere MHD model BATS-R-US to unveil the solar wind-magnetosphere interactions and associated solar wind energy transfer during the entire M-B reversal event. We also note that the realistic M-B reversal may be more complex than what the model can represent because the pelomagnetic field model highly relied on historic data sets that unfortunately lacked global availability (see more details in Leonhardt and Fabian (2007)).
With the modeling results, some major findings are summarized as follows. Prior to the reversal process, the paleomagnetic field is dipolar with its north pole in the northern hemisphere as its magnetic moment becomes smaller. It then undergoes a period without any clear dipolar structure, with multiple local loop configurations appearing randomly. Finally, the magnetic north pole emerges in the southern hemisphere as the magnetic moment continuously increases.
With such unique changes in the geomagnetic field, the magnetosphere undergoes drastic changes during the reversal. The stand-off distance of the subsolar magnetopause shifts toward the Earth and approaches 3 Re, as the field weakens. In such a condition, if the solar wind dynamic pressure were to change significantly to further compress the magnetosphere (e.g., during a Coronal Mass Ejections event), the Earth's atmosphere may be directly exposed to the solar wind. It has been discussed whether a smaller or larger magnetosphere can protect the planet's atmosphere (Moore & Horwitz, 2007;Tarduno et al., 2014). Blackman and Tarduno (2018) believed that the competition between inflow velocity (from stellar wind) and collection area (from the magnetosphere) determines whether the magnetosphere can protect the planetary atmosphere. In our work, the polarity reversal of the Earth's intrinsic magnetic field not only changes the size of the magnetosphere but also affects the solar wind power transmission efficiency. According to previous studies, this can undoubtedly affect the protective role of the magnetosphere on the Earth's atmosphere.
In the midst of the reversal, multiple reconnection sites appear where local dipole-like magnetic fields emerge and encounter anti-parallel IMF. The ratio between SWP m.p. and SWP e ramps up to 18% and keeps at this high level for about 2.5 kyr. This high efficiency is approximately 10 times larger than the pre-reversal level and twice as much as the post-reversal level, implying that the geomagnetic reversal enables elevated power transmission efficiency. As shown in the in-plane subplot in Figure 3b, we find that the rising of the power transmission efficiency occurs when the quadrupole component of the geomagnetic field is dominant over the dipole component and when the dipole component starts to increase after its declination in the pre-reversal phase. During this time period, the magnetosphere starts to expand gradually with an increasing solar wind energy impinging on the magnetopause, and the dipole component starts to recover as the dominant component with an opposite polarity. These reasons may help increase the magnetic reconnection rate on the dayside and contribute to the enhancement of power transmission efficiency.
Multiple open channels over the magnetopause surface at this stage provide more chances for solar wind energy to enter the magnetosphere and deposit towards the Earth. The enhanced Poynting flux down to the upper atmosphere appears globally, indicating that a large amount of solar wind energy is directly transmitted into the lower latitudes and even equatorial regions, which may suggest global heating of the atmosphere of the Earth.
The stand-off distance of the magnetopause compressed to 3 Re during the M-B reversal, which is within the normal plasmapause boundary in today's terrestrial magnetosphere (Laakso & Jarva, 2001). Therefore, the plasmasphere, filled with cold plasma from the ionosphere, would be greatly affected or may disappear. Moreover, the near-Earth magnetospheric configuration is totally disturbed when the geomagnetic fields are highly non-dipolar. Hence different circumstances would be expected in the ring currents, radiation belts, and other regions that heavily rely on the magnetic field configuration. With regional magnetic loop structures, the drifting charged particles may not be trapped around the Earth. The global-scale ring current may not even exist during the reversal phase as suggested by Vogt and Glassmeier (2000). In the near future, we plan to apply a particle tracing technique to investigate the near-Earth particle motions under the reversal configuration.