Asymmetrical Looping Magnetic Fields and Marsward Flows on the Nightside of Mars

As the interplanetary magnetic field (IMF) carried by the solar wind encounters the martian atmosphere, it tends to pile up and drape around the planet, forming looping magnetic fields and inducing marsward ion flows on the nightside. Previous statistical observations revealed asymmetrical distribution features within this morphology; however, the underlying physical mechanism remains unclear. In this study, utilizing a three‐dimensional multi‐fluid magnetohydrodynamic simulation model, we successfully reproduce the asymmetrical distributions of the looping magnetic fields and corresponding marsward flows on the martian nightside. Analyzing the magnetic forces resulting from the bending of the IMF over the polar area, we find that the asymmetry is guided by the orientation of the solar wind motional electric field (ESW). A higher solar wind velocity leads to enhanced magnetic forces, resulting in more tightly wrapped magnetic fields with an increased efficiency in accelerating flows as they approach closer to Mars.


Introduction
A stream of charged particles, accompanied by the interplanetary magnetic field (IMF), is emitted radially from the sun and travels continuously throughout our solar system, forming what is termed the solar wind.The effects exerted by the solar wind stream on planets and other celestial bodies in its path depend upon the properties of the obstacles encountered, as well as prevailing solar wind conditions (e.g., Schunk & Nagy, 2009).For magnetized planets such as Earth, the strong intrinsic dipole magnetic field effectively diverts the approaching solar wind plasma, resulting in the formation of a cavity known as the magnetosphere (Chapman & Ferraro, 1930).Unmagnetized celestial bodies lacking a substantial atmosphere, such as Earth's moon, are able to directly absorb the impinging solar wind ions into their dayside surface (Ness et al., 1968).For other planets that possess extended atmosphere but lack an intrinsic global magnetic field, such as Venus, the primary obstacles that divert the incident solar wind plasma are magnetospheres induced by the interactions between the frozen-in IMF and the conducting ionospheres (Bertucci et al., 2011).Although Mars exhibits regions of strong localized crustal magnetic fields, primarily in the southern hemisphere (Acuña et al., 1999;Connerney et al., 2005), it belongs to this final category of solar system objects.
Due to the absence of a global intrinsic magnetic field at Mars, the IMF carried by the solar wind tends to directly interact with the martian upper atmosphere, inducing currents in the electrically-conducting ionosphere.Consequently, these currents give rise to additional magnetic fields, leading to the formation of an induced magnetosphere that effectively deflects the solar wind flow around Mars.It is well established that magnetic fields pile-up on the induced magnetosphere, followed by draping and stretching toward the nightside, forming an induced magnetotail characterized by two lobes separated by a current sheet (e.g., Luhmann et al., 2004;McComas et al., 1986).Therefore, the configuration of the induced magnetosphere is determined by the orientation of a highly time-variable IMF.In order to investigate induced features related to the direction of the IMF, spacecraft observation data are usually displayed in the Mars Solar Electric (MSE) coordinate system with X MSE antiparallel to the upstream solar wind velocity (V SW ), Y MSE along the cross-flow magnetic field component of the IMF (B IMF ), and Z MSE pointing in the direction of the solar wind motional electric field (E SW = V SW × B IMF ) (Dubinin et al., 1996;Moore et al., 1990;Russell et al., 1995).In general, the MSE coordinate system is able to effectively depict the induced magnetosphere controlled by the orientation of the E SW , eliminating asymmetry features arising from the martian crustal magnetic field (Dubinin et al., 2021(Dubinin et al., , 2023)).
Employing the MSE coordinate system, the inner martian induced magnetosphere is comprehensively characterized in terms of numerous significant phenomenological features related to ionospheric ion dynamics and magnetic field topologies.Ions extracted from the ionosphere (Dubinin et al., 2021) tend to accelerate by the E SW forming ion plume in the dayside +E SW hemisphere (Dong et al., 2015;Fang et al., 2010;Luhmann, 1990), while ion trail filled by dense slow-moving plasmas tends to shift to the nightside E SW hemisphere (Dubinin et al., 2019;Inui et al., 2019).This shift of the nightside ionosphere is associated with the frozen magnetic field lines bending, wrapping (Du et al., 2013;Rong et al., 2014;Zhang, C. et al., 2022a;Zhang, T.L. et al., 2010) and even closing in the near-martian induced magnetotail, termed as "looping" magnetic field structures (Chai et al., 2016(Chai et al., , 2019)).The formation of closed looping magnetic fields on the martian nightside is facilitated by magnetic reconnection, accompanied by marsward flows.Notably, asymmetrical characteristics of these structures between the ±E SW hemispheres in the MSE coordinate system has been observed (Brain et al., 2015;Harada et al., 2015aHarada et al., , 2015bHarada et al., , 2017)).However, the majority of these studies primarily concentrate on the statistical observations obtained from a single orbiting satellite, thereby confining their ability to provide comprehensive theoretical explanations for these physical phenomena.
In this study, we take advantage of our well-developed multi-fluid Hall magnetohydrodynamic (MHD) model (Li et al., 2022a(Li et al., , 2022b;;Li et al., 2023) to investigate the underlying physical mechanisms responsible for the asymmetrical distributions of looping magnetic field structures and their corresponding marsward flows.Furthermore, we conduct an investigation into the impact of the solar wind velocity magnitude on these asymmetries.The subsequent sections of this paper are structured as follows: Section 2 presents the model description, Section 3 provides simulation results, and Section 4 concludes our study.

Model Description
In order to accurately simulate the interaction between incident solar wind and the martian upper atmosphere, a comprehensive model must incorporate the self-consistent evolution of plasma dynamics for each species, as governed by the Vlasov equation, along with consideration of the electromagnetic field environment controlled by Maxwell's equations.Unfortunately, computational power limitations often render it impractical to employ a three-dimensional fully kinetic approach for solving the model.However, based on multi-fluid assumption which treats each species as a distinct fluid coupled to the magnetic induction equation including the Hall term, the Hall-MHD model has been developed and proved to be a computationally efficient yet reliable approach for simulating the large-scale electromagnetic processes that govern solar wind-Mars interactions, including the ion fluid dynamics of each species (Li et al., 2020(Li et al., , 2022a(Li et al., , 2022b;;Najib et al., 2011;Song et al., 2023).
Our well-developed three-dimensional four-species multi-fluid Hall-MHD model is a numerical framework that accurately captures the temporal evolution of individual ion species (i.e., H + , O + 2 , O + , CO + 2 ), described by a set of discrete Navier-Stokes (NS) equations equivalent to a Vlasov system.Subsequently, an uncoupled magnetic induction equation is computed using a second-order MacCormack scheme to derive the evolution of electromagnetic environments (see details in Li et al., 2023).Furthermore, the NS equations incorporate physical collisions and chemical reactions among all species as source terms, ensuring a self-consistent development of the martian ionosphere.This is achieved by introducing a 1D neutral density profile (Bougher et al., 2001) and considering the solar extreme ultraviolet flux (Huestis, 2001).The neutral atoms undergo ionization due to solar radiation and subsequently interact with energetic electrons or charge exchanges with solar wind and planetary ions, resulting in the formation of ionospheric ions.The dynamics of an individual ion fluid in the electromagnetic field is described by the Hall-MHD momentum equation: where ρ s , u s , p s , n s and q s are the individual mass density, velocity, pressure, number density and charge of the ions respectively.I is the identity matrix.p e = ∑ i=ions p i , n e = ∑ i=ions n i are the pressure and number density of electrons.E and B are the electric and magnetic fields.The electric current density can be obtained from The source term δM s δt on the right side of Equation ( 1a) represents the variation of the plasma flow momentum due to the physical collisions and chemical reactions among all the species (Li et al., 2020).
In this study, our simulations are conducted within the classical Mars-centered Solar Orbital (MSO) coordinate system.The X-axis points from Mars toward the Sun, the Z-axis is perpendicular to the X-axis and positive toward the north celestial pole, while the Y-axis completes the right-handed coordinate system.A curvilinear mesh consisting of 960,000 cells encompasses 24R M ≤ X ≤ 8R M , 16R M ≤ Y and Z ≤ 16R M , where R M represents the martian radius (R M ≈ 3396 km).The smallest cells are located at the inner boundary of the computation domain, with a grid size of 10 km.From the inner to the outer boundary, there is a non-linear increase in grid size up to approximately 1R M located at the outer boundary.The inner boundary is defined as 100 km above the martian surface, where the ion velocities and densities are assumed to be zero and the photochemical equilibrium values respectively, while the H + is considered to be 0.3 times that of the solar wind densities (adopted from Dong et al., 2014).
Given that a reversal sign of the B y component in the martian upstream and downstream regions corresponds to a wrapping of the magnetic fields (Dubinin et al., 2019), it is justifiable to employ the three upstream solar wind conditions listed in Table 1, which represent two distinct classes of E SW orientation (Case 1 and Case 2) as well as solar wind velocity magnitude (Case 1 and Case 3).In order to emphasize the looping features in the martian induced magnetosphere, based on these simulations conducted without considering crustal field sources, this study investigates the impacts and mechanisms of the orientation of E SW and the magnitude of solar wind velocity on the asymmetrical distributions of looping magnetic field structures and corresponding marsward flows on the martian nightside.

Simulation Results
The interaction between the solar wind and martian atmosphere leads to the formation of various magnetic field topologies (Li, G et al., 2023;Xu et al., 2017): draped field lines with both foot points connected to the solar wind, open field lines with one end intersecting the martian collisional ionosphere and the other end connected to the solar wind, and closed field lines with both foot points in the ionosphere.To investigate the asymmetrical distribution of closed looping magnetic fields on the martian nightside under the three distinct solar wind conditions, we traced several field lines with both ends embedded within the inner boundary of the nightside after reaching a quasi-steady state in our simulation results.These field lines are depicted as orange lines in Figure 1.Since the IMF lies in the XY plane and B y is specified as positive in simulation Case 1, it can be inferred that the MSO and MSE coordinate systems align perfectly for this particular scenario.It is evident from the left column panels of Figure 1 that closed loops of the magnetic field lines form in the northern hemisphere and subsequently shift toward the southern hemisphere (refer to panels (1b) and (1d)), which is consistent with previous observations (Chai et al., 2019;Dubinin et al., 2019;Zhang, Rong, et al., 2022).However, by assuming the same velocities and temperatures of different ion species, the previously-reported single fluid MHD approach fails to reproduce such a comparable shift (Li, G et al., 2023), suggesting that ion  Li et al., 2023) Solar wind velocity (km/s) Interplanetary magnetic field (nT) Case 1 ( 400, 0, 0) (0, 3, 0) Case 2 ( 400, 0, 0) (0, 3, 0) Case 3 ( 800, 0, 0) (0, 3, 0) dynamics may potentially exert a significant influence on this process (to be demonstrated subsequently).The symmetrical nature of the looping magnetic fields with respect to Y MSO = 0 is evident (panel (1c)), indicating a potential shift in conjunction with the martian magnetotail current sheet when considering the IMF B x component.Furthermore, the looping magnetic field lines exhibit a tendency to wrap the planet more tightly in the southern hemisphere.In simulation Case 2 where B y < 0, the orientations of Z MSO and Z MSE are diametrically opposed.The field lines are shifted to the opposite hemisphere compared to Case 1, as depicted in the middle column panels of Figure 1, implying that the configuration of the closed looping magnetic field lines is guided by the orientation of the E SW .When the solar wind velocity is doubled while IMF B y remain unchanged compared to Case 1, the trend in the shift direction of looping magnetic field lines is observed as depicted in the right column panels of Figure 1 for Case 3 and is consistent with the shift in Case 1.However, under conditions where the solar wind velocity is doubled, the flare angle of looping magnetic field lines in Case 3 exhibits a decrease (refer to panels (3c) and ( 3d)), particularly in the southern hemisphere compared to Case 1, suggesting that under conditions of high solar wind velocity, the field lines tend to be shifted farther away from +E SW hemisphere and tightly wrap around the martian nightside E SW hemisphere.
To illustrate the influence of upstream E SW orientation and solar wind velocity on the asymmetrical distribution of looping magnetic fields more specifically, we present simulation results depicting B y with an opposite direction to the upstream IMF and a magnitude exceeding 1 nT on the martian nightside, as shown in Figure 2. Results are presented in the XY plane at Z MSO = 0.75R M ( 0.75R M ) to represent northern (southern) hemisphere, and in the XZ and YZ plane at Y MSO = 0R M and X MSO = 1.2RM respectively.For Case 1, when upstream IMF B y is positive, from panels (1a), ( 1b) and (1c), a significantly higher magnitude and distribution range of negative B y on the nightside southern hemisphere compared to the northern hemisphere are observed, suggesting that the classical draping configuration dominates in the martian nightside northern hemisphere, while magnetic fields tend to tightly wrap around Mars in the nightside southern hemisphere.This asymmetry, resulting from the displacement of magnetic fields, has also been observed by MAVEN in a study conducted by Zhang, Rong, et al. (2022), who constructed an average three-dimensional configuration of magnetic field lines using more than 6 years' of in situ magnetic field data.Another intriguing characteristic is the well-observed clockwise rotation (when viewed from the sun) of the magnetic fields (Figure 2(1d)), which exhibit positive B y in the northern solar wind and magnetosheath, while displaying negative B y in the magnetotail.These simulation results provide support for the proposed looping structure of the magnetic field by Chai et al. (2019).For the case of negative upstream IMF B y in Case 2, the looping magnetic fields are shifted toward the northern hemisphere, resulting in an opposite asymmetry compared to Case 1 (refer to panels (2a), ( 2b) and ( 2c)).The clockwise rotation of the magnetic fields is also observed for Case 2 (Figure 2(2d)), where B y is negative in the southern solar wind and magnetosheath yet positive in the magnetotail.The comparison between Case 3 and Case 1 elucidates the impact of solar wind velocity on the looping magnetic fields observed on the martian nightside.When the solar wind velocity is doubled while maintaining the IMF B y orientation, it can be observed that the shifting direction of looping magnetic fields remains unchanged (refer to right column panels in Figure 2).However, a notable increase in the magnitude of B y on the nightside, particularly in the southern hemisphere, indicates a tighter wrap of magnetic field lines around the planet under higher solar wind velocity conditions.Therefore, it can be inferred that the displacement direction of looping magnetic fields depends on the E SW orientation, while the flare angle of wrapping field lines tends to be governed by the velocity of the solar wind.
When magnetized solar wind plasma encounters the martian atmosphere, the IMF piles up and drapes around the planet.Given the bending of magnetic field lines during this process, magnetic forces (J × B), which can be decomposed into magnetic tension and magnetic pressure gradient force, tend to emerge over the polar region, exerting an influence on the plasma and subsequently impacting the configuration of the ionosphere as well as the frozen-in magnetic fields.To gain a deeper understanding of the impact of the J × B forces, the top panels of Figure 3 provide distributions of magnetic forces derived from the three simulation cases, specifically highlighting the direction of forces as being opposite to the E SW .Additionally, corresponding vectors are depicted on each panel.It should be pointed that the electric current density in our model is obtained from In the case of representing E SW along the +Z MSO direction (Case 1), it is evident from panel 1a that J × B in Z MSO direction forms over the pole of the northern hemisphere.However, the J × B in the direction opposite to the E SW emerges over the pole of the opposite hemisphere when there is a change in sign of the cross-flow component of upstream IMF for Case 2 (as depicted in panel 2a).Under a condition of doubled solar wind velocity compared to Case 1, the distribution of J × B with E SW direction in Case 3 remains nearly unchanged (panel 3a), however, the corresponding magnitude exhibits an amplified value.Therefore, it is reasonable to deduce that the dominant distribution and magnitude of J × B in the opposite direction to the E SW are governed by both the upstream IMF B y orientation and the solar wind speed value individually.The J × B has the capability to induce shifts in the plasma and frozen-in magnetic field configurations toward E SW magnetotail.This scenario supports that magnetic reconnection is more prone to occur in the martian E SW magnetotail (Harada et al., 2017).It should be noted that the formation of looping magnetic fields does not necessarily rely on magnetic reconnection.Additional mechanisms contributing to the formation of looping magnetic fields are beyond the scope of our study and deserve further investigations.
To illustrate the impact of J × B on propelling ionospheric plasma in the martian nightside, the middle and bottom panels of Figure 3 present the averaged slow ion velocities below 40 km/s and O + dense ion densities exceeding 1 cm 3 obtained from the three simulation cases.It is widely acknowledged that the acceleration of ions with high velocities can be attributed to be accelerated by the E SW , resulting in the formation of plume structures within the +E SW hemisphere (Dong et al., 2015;Inui et al., 2019).It is obvious that the trajectory of the ion trail, consisting of dense and slowly moving plasma, is guided by the J × B force, which opposes the direction of the E SW .This confirms the underlying physical mechanism assumed by Dubinin et al. (2019) and Chai et al. (2019).However, as a result of the more self-consistent evolution of the martian ionosphere in the MHD model, the derived ion trail demonstrates improved agreement with MAVEN observations compared to the hybrid simulations (Dubinin et al., 2019).Furthermore, it should be noted that the day-to-night flow speed experiences an increase when subjected to higher solar wind velocity condition (refer to panel 3b).Consequently, this leads to a more elongated ion trail compared to Case 1 (as depicted in panel 3c).
Upon comparing Figures 2(1c) and 3(1b) for simulation Case 1, it becomes evident that the B y component undergoes a sign change on the martian nightside.This signifies the wrapping of magnetic field lines around Mars, facilitating their subsequent reconnection with the emergence of marsward flows, which are also observable in the remaining two cases.In order to investigate the relationship between distributions of marsward ion flows and looping magnetic fields, we extracted O + ion velocity at various X-plane slices ranging from 1.1R M to 1.6R M with a spacing interval of 0.1R M , then eliminated any V X values smaller than zero (indicating martian tailward flow) and revealed the presence of marsward flows on the martian nightside, as depicted in the top panels of Figure 4.In the case where the E SW is aligned with +Z MSO (panel (1a)), the predominant occurrence of marsward ion flows is observed in the southern hemisphere, with a gradual shift of the northernmost point toward the southern hemisphere.However, the marsward flows become manifest and shifts toward the opposite hemisphere in response to the reversal of the orientation of the E SW (panel (2a)).Such an asymmetrical distribution of the marsward flow pattern on the martian nightside was also reported by observations in the MSE coordinate system (Dubinin et al., 2019;Hara et al., 2017).Considering that the orientation of the E SW also governs the asymmetrical distribution of looping magnetic fields, it is reasonable to deduce that closed looping magnetic fields tend to manifest in the E SW hemisphere accompanied by the generation of marsward flows, while classical draping magnetic field lines prevail in the +E SW hemisphere.Additionally, under doubled solar wind velocity conditions as shown in panel (3a), it appears that the marsward flows compared to the Case 1 exhibit a reduction in magnitude as they further from the martian midnight.To illustrate the influence of the solar wind velocity on the marsward ion flows, panels (1b) and (3b) show the contour plots with vector of marsward flow (V X ≥ 0) at Z MSO = 0.75R M extracted from simulations Case 1 and Case 3 separately.Comparing the two panels, it is obvious that under doubled solar wind condition, the marsward flows exhibit a reduction in both magnitude and extent, with the exception of the region closer to Mars.Since the sling-shot effect on the marsward flow included by reconnection is attribute to highly curved magnetic field lines (Lavraud et al., 2007), in order to provide a more specific illustration of the discrepancy, panel (c) displays the variation in marsward velocities and Hall electric fields (E H ) along Y MSO = 0R M (purple solid line in panel (3b)) at the plane of Z MSO = 0.75R M derived from both simulation cases.Under conditions of lower solar wind velocity (Case 1), the marsward plasma vortex tends to form at a greater distance from the planet and is subsequently accelerated by a relatively larger E H , indicating that reconnections are more likely to occur further away from Mars in this scenario.However, as return flows approach Mars under higher solar wind velocity condition (Case 3), the velocities exceed those obtained from Case 1, which can be attributed to a relatively larger E H resulting from tightly wrapped magnetic field lines.Furthermore, it is evident that a negative marsward E H can be observed in Case 1, which arises from the dominance of magnetic pressure gradient forces over magnetic tension forces within that particular region.Therefore, it can be deduced that the marsward plasma flows prevail in the E SW hemisphere, which are believed to originate from magnetic reconnection occurring at a relatively greater distance from Mars and subsequently experience efficient acceleration due to tightly wrapping magnetic field lines closer to Mars.The distance at which marsward flows induced by reconnection occur, as well as the efficiency of acceleration exerted by wrapping magnetic field lines, are determined by the magnitude of solar wind velocity.

Conclusions
In this study, we have utilized a multi-fluid Hall-MHD model to investigate the morphology and potential mechanisms underlying asymmetrical distributions of looping magnetic fields and corresponding marsward flows on the martian nightside.Additionally, we have explored the impact of solar wind velocity on these structures.
Comparisons with previous statistical observations demonstrate that our model well reproduced the asymmetrical characteristics, including the ion trail formed on the nightside of the planet.
The simulation results suggest that the asymmetry distributions are governed by the E SW orientation.Due to the bending of magnetic fields caused by the IMF drapes around Mars, magnetic forces in the E SW direction tend to emerge over the pole of the +E SW hemisphere, resulting in the formation of a trail composed of dense slowmoving ions and leading to a displacement of frozen-in looping magnetic field lines accompanied by marsward flows toward the E SW hemisphere.
When the velocity of the solar wind is doubled, there is a substantial enhancement in magnetic forces in the direction opposite to E SW .For the ionospheric ions, this augmentation of forces results in an elevation of the dayto-night flow velocity, propelling heavy ions deeper into the nightside and prolonging the presence of the ion trail.
For the looping magnetic fields, the stronger magnetic forces lead to a shift of the frozen-in magnetic fields farther away from +E SW hemisphere and a more tightly wrapped configuration of magnetic field lines around martian nightside E SW hemisphere.For the marsward ion flow, the tightly wrapping magnetic field lines result in an enhanced velocity of the flow as it approaches Mars and a decreased acceleration of such flow as it induced by reconnection.
In summary, the asymmetrical distributions of looping magnetic fields and corresponding marsward ion flows are attributed to their deflection by magnetic forces toward E SW direction.The magnitude of these asymmetries is modulated by variations in solar wind velocity.

Figure 1 .
Figure 1.Looping magnetic field lines on the martian nightside for the three simulation cases, depicted in (a) 3D representation, (b) XZ plane, (c) XY plane and (d) YZ plane.

Figure 2 .
Figure 2. Color plots of B y with an opposite direction to the upstream interplanetary magnetic field and a magnitude exceeding 1 nT for the three simulation cases, overlapping with magnetic field vectors, in (a) XY plane (Z MSO = 0.75R M ) (b) XY plane (Z MSO = 0.75R M ) (c) XZ plane (Y MSO = 0R M ) and (d) YZ plane (X MSO = 1.2RM ).The black dashed and dotted lines in panels (d) represent the shadow of Mars.The color of each case is in accordance with the color bar presented in the top panel.

Figure 3 .
Figure 3. Color plots of (a) magnetic forces in the direction opposite to the E SW , (b) average ion velocities smaller than 40 km/s and (c) O + ion densities above 1 cm 3 in the XZ plane for the three simulation cases, overlapping with vectors of corresponding magnetic forces and average ion velocities in panels (a) and (b) respectively.

Figure 4 .
Figure 4. (a) Color plots of marsward O + flow (V X ≥ 0) obtained from the three simulation cases in YZ plane from 1.1R M to 1.6R M with a spacing interval of 0.1R M .(b) Color with vector plots of marsward O + flow (V X ≥ 0) obtained from Case 1 and Case 3 in Z MSO = 0.75R M XY plane.(c) Profiles of the X components of O + velocity and Hall electric field along martian midnight (purple solid line in panel (3b)) for Case 1 (red) and Case 3 (green).

Table 1
Upstream Solar Wind Condition Settings of Simulation Cases (The Remaining Parameters Are Kept Consistent With