# Three-dimensional Multispecies Simulation of the Solar Wind Interaction with Mars Under Different Interplanetary Magnetic Field Orientations

Yun Li, Haoyu Lu, Jinbin Cao, Shibang Li, Christian Mazelle, Guokan Li

► To cite this version:

Yun Li, Haoyu Lu, Jinbin Cao, Shibang Li, Christian Mazelle, et al. Three-dimensional Multispecies Simulation of the Solar Wind Interaction with Mars Under Different Interplanetary Magnetic Field Orientations. The Astrophysical Journal, 2021, 921 (2), pp.139. 10.3847/1538-4357/ac1ce5. hal-04939692

HAL Id: hal-04939692

https://hal.science/hal-04939692v1

Submitted on 3 Aug 2025

**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Three-dimensional Multispecies Simulation of the Solar Wind Interaction with Mars Under Different Interplanetary Magnetic Field Orientations

Yun Li¹,², Haoyu Lu¹,², Jinbin Cao¹,², Shibang Li¹, Christian Mazelle³, and Guokan Li¹
¹School of Space and Environment, Beijing, People's Republic of China; Lvhy@buaa.edu.cn

²Key Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology, Beijing, People's Republic of China

³Institute de Recherche en Astrophysique et Planétologie, CNRS—University of Toulouse—CNES, Toulouse, France
Received 2021 May 12; revised 2021 July 31; accepted 2021 August 10; published 2021 November 9

## Abstract

Without the intrinsic magnetic field, the solar wind interaction with Mars can be significantly different from the interaction with Earth and other magnetized planets. In this paper, we investigate how a global configuration of the magnetic structures, consisting of the bow shock, the induced magnetosphere, and the magnetotail, is modulated by the interplanetary magnetic field (IMF) orientation. A 3D multispecies numerical model is established to simulate the interaction of solar wind with Mars under different IMF directions. The results show that the shock size including the subsolar distance and the terminator radius increases with Parker spiral angle, as is the same case with the magnetotail radius. The location and shape of the polarity reversal layer and inverse polarity reversal layer in the induced magnetotail are displaced to the $y < 0$ sector for a nonzero flow-aligned IMF component, consistent with previous analytical solutions and observations. The responses of the Martian global magnetic configuration to the different IMF directions suggest that the external magnetic field plays an important role in the solar wind interaction with unmagnetized planets.

*Unified Astronomy Thesaurus concepts: Mars (1007); Solar-planetary interactions (1472); Magnetohydrodynamics (1964)*

## 1. Introduction

The solar wind interacts with non-absorbing obstacles to form the collisionless bow shock (BS). The nature of the planetary BS depends on the type of interaction (whether there is a magnetosphere) and the solar wind conditions near the planetary obstacle (Russell 1985). For the magnetized planets, such as Earth, the size of the shock is related to the intrinsic magnetic moment of the planet. For the unmagnetized planets such as Venus and Mars, the solar wind interacts with the planets' atmosphere to generate BSs smaller in size than magnetized planets. The size of the shock determines the scale of some physical processes occurring near the obstacle, and the spatial distance demanded for the deceleration and deflection of the solar wind flow (Bertucci et al. 2005a). It is of great importance to study the shape and size of the BS.

Venus, Mars, and Titan are the typical examples of unmagnetized bodies with an atmosphere. Unlike Venus and Mars, the upstream condition of Titan is the Saturn's magnetospheric flow instead of the solar wind most of the time (Thomsen 2013). Titan was observed in the supersonic and super-Alfvénic solar wind for the first time by the Cassini spacecraft after 9 yr in the Saturn system (Bertucci et al. 2015). The observations revealed that the solar wind interaction with Titan resembles the case with Venus, Mars, and comets in many ways. A distinct transitional region is located between the shocked solar wind and the bodies' atmosphere/ionosphere. In situ observations and numerical simulations have identified several plasma boundaries in the region, such as the induced magnetosphere boundary (IMB) for Mars and Venus (e.g., Nilsson et al. 2006; Shan et al. 2015), the magnetic pileup boundary (MPB) on Venus, Titan, and Comet Halley (e.g., Bertucci et al. 2003; Snowden et al. 2007; Delva et al. 2014), and the Martian or Venusian magnetopause (MP) (e.g., Zhang et al. 1991a; Verigin et al. 1996; Zhang et al. 2008a; Echim et al. 2011).

These boundaries with different academic terms refer to the same physical structure (e.g., Vignes et al. 2000; Dubinin et al. 2008; Ramstad et al. 2017). This boundary separates the magnetosheath from the induced magnetosphere (IM) (e.g., Kallio et al. 2008a, 2008b; Zhang et al. 2008a, 2008b) or the so-called magnetic barrier (Zhang et al. 1991a) or the magnetic pileup region (MPR) (e.g., Vennerstrom et al. 2003; Espley et al. 2004).

The lack of core convection to drive a global dynamo may account for the absence of the intrinsic magnetic field on these unmagnetized obstacles (Nimmo 2002; Brain et al. 2016). Acuña et al. (1998) reported the local magnetic field of crust sources on Mars for the first time. The strongest crustal magnetic field is located in the southern hemisphere with an intensity of more than 1500 nT at the height of 100 km from the surface of Mars (Acuña et al. 1999). The remanence strength decreases rapidly at altitudes of 1300–1400 km, above which the magnetic field is almost completely determined by the solar wind (Brain et al. 2003). The magnetic field in the region close to Mars has three forms: the crust magnetic field forming a closed loop, the open magnetic field connecting the crust magnetic field with the interplanetary magnetic field (IMF), and the draped magnetic field (Brain et al. 2007). In areas with weak remanence, the solar wind interaction with Mars is similar to the case with Venus and comets.

Early observations of the magnetic field and plasma revealed the existence of BS on Mars (Slavin & Holzer 1982). This BS decelerates the solar wind flow from supersonic velocity to subsonic velocity (Yi et al. 1999). Downstream from the shock is the magnetosheath, a region with low magnitude and high variable magnetic field. It was suggested that there was not enough space to thermalize the solar wind for the narrow extent of the Martian magnetosheath comparable to the gyroradius of solar wind protons (Moses et al. 1988). A well-defined IMB was detected by a series of spacecraft, e.g., the pioneering Phobos 2
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The total strength $|B|$ and velocity $|V|$ for the Parker spiral angle of 56° in the equatorial plane (the upper panel) and the meridianal plane (the lower panel). The white lines with arrows indicate the directions of the induced magnetic field and velocity. The outer black dashed line represents the mean location of bow shock and the inner black solid line is for the mean MPB by Vignes et al. (2000).</figcaption></figure>

(Sauer et al. 1992), the following Mars Global Surveyor (MGS) (Vignes et al. 2000), and the latest Mars Atmosphere and Volatile Evolution (MAVEN) mission (Matsunaga et al. 2017). The IMB is distinguished by an increase of the magnetic field intensity by a factor of two to four (Acuña et al. 1998; Bertucci et al. 2005b).

Vignes et al. (2002) used linear regression and least squares methods to fit the positions of BS for 553 shock-crossing events. Their statistical results showed that the crustal field has no evidential influence on the position of BS. Mazelle et al. (2004) found a slight asymmetry in the BS position between the northern and southern hemispheres. The cross section of the shock located in the dawn and dusk planes is asymmetric and is affected by the direction of the planet's magnetic field (Zhang et al. 1991b, 1991c). Numerical simulation results indicated that, when IMF direction changes from satisfying the classical Parker spiral angle to the Y direction, the BS wave moves 2% outward (Ma et al. 2004).

The magnetotail is a main channel for the escaping plasma, as a result of which, the configuration of the magnetic field topology may lead to significant changes to the ion loss from the unmagnetized planets. Although some studies referred to the draping of IMF around unmagnetized planets, only a few studies were performed to clarify this problem in detail. The IMF carried by the solar wind is suspended near the ionosphere on the day side of Mars and stretched into a double-lobe

induced magnetotail with the opposite magnetic polarity on the night side (Brain et al. 2007). Bertucci et al. (2011) reviewed the induced magnetospheric boundaries and the magnetotails of the three distinct unmagnetized objects: Mars, Venus, and Titan. The results showed a common feature that the dual-lobe magnetotail with a central cross-tail current sheet is modulated by the direction of the upstream magnetic field.

The exposure of Titan in the varied ambient magnetic field provided by Saturn's magnetosphere leads to a different configuration of Titan's IM (Simon et al. 2013). Observational results showed that the polarity reversal layer (PRL) between the magnetic lobes is deviated into the northern/southern hemisphere in the case of the positive/negative finite flow-aligned magnetic field component. Romanelli et al. (2014) obtained the magnetic field topology by analytical solution of ideal MHD equations with the magnetized plasma flowing around an unmagnetized spherical obstacle. Another layer named by the inverse polarity reversal layer (IPRL) was identified by a change of the sign in the flow-aligned magnetic field component opposite to that of the PRL. In addition, they calculated the deviation distance of the IPRL as a function of the orientation of the background magnetic field. Further observations (Romanelli et al. 2015) made it possible to confirm the displacement of the Martian magnetic lobes with a nonzero flow-aligned IMF component by MGS magnetometer data.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** The sonic Mach number $M_s$ (the left panel) and magnetosonic Mach number $M_{MS}$ (the right panel) for the Parker spiral angle of 56° in the equatorial plane (the upper panel) and the meridional plane (the lower panel). The outer black dashed line represents the mean location of bow shock and the inner black solid line is for the mean MPB by Vignes et al. (2000).</figcaption></figure>

This study adopts a 3D multispecies MHD model to investigate the global configuration of the magnetic field associated with the solar wind plasma flowing around Mars, including the BS, the IM, and the magnetotail. Four cases with Parker spiral angles of 10°, 30°, 56°, and 90° are chosen to see the joint responses of these magnetic structures to the change of the IMF orientation. The locations and shapes of the PRL and IPRL in the induced magnetic lobes are presented from our simulation results. Besides, we show the draping magnetic field line in the 3D Cartesian system for the different IMF directions.

## 2. Model Equations

Computational models of the global Martian plasma environment mainly include a hybrid model and a series of models based on the extended MHD formulations. Both types of models have their strengths and their limitations. The hybrid model combines fully kinetic ions and a massless-electron fluid description (Winske & Omidi 1995; Holmstrom & Wang 2015), but it can be considered as a conducting fluid without specifying its various individual species. The traditional MHD model treats the space plasma as a kind of quasi-neutral magnetized fluid of single species. It is useful in the description of some features

observed in the solar wind interaction with planetary magnetosphere/ionosphere (Lu et al. 2013; Li & Lu 2019). However, it is hard to estimate the number density of each single species of the plasma including two or more components. The multi-fluid MHD model regards the plasma as a combination of multiple fluid species with the continuity, momentum, and energy equations of each species considered (Najib et al. 2011; Li et al. 2020). Since the multi-fluid MHD model decouples the Navier-Stokes equations for fluid and the induced equation for magnetic field, it may be difficult to describe the typical types of large-scale MHD wave or mode, including the incompressive Alfvén wave, compressive fast and slow magnetoacoustic waves, etc. The multispecies model calculates the densities of the solar wind protons and all the major ion species in the Martian ionosphere, as well as the plasma bulk velocities and energies, by assuming that the plasma is collisional and the velocities and temperatures of the ion species are approximately the same (Ma et al. 2004).

In this study, a three-dimensional multispecies model is used to simulate the solar wind interaction with Mars. The four ion species adopted in our model include $H^+$, $O_2^+$, $O^+$, and $CO_2^+$. We use the decoupling method to solve the mass density of
different components, that is, to solve the continuous equation for each component separately, and to solve the single momentum equation and energy equation for the mass population composed of four components. The three main neutral components of Mars atmosphere are CO₂, H, and O. The four main ions are formed through a series of ionospheric chemical reactions. These chemical reactions are mainly categorized into three types: photoionization (impact ionization by superthermal electrons is not included for the day side ionosphere), charge exchange, and recombination reactions. The detailed chemical reactions are adopted from Ma et al. (2004).

In our model, the decoupled continuous equation of the four ion species is as follows:

$$ \frac{\partial \mathbf{W}_s}{\partial t} + \nabla \cdot \mathbf{F}_s = \mathbf{Q}_s \qquad (1) $$

where Ws is the vector formed by the mass density of each component, and Fs is the tensor flux formed by the momentum vector of each component, which is expressed as follows:

$$ \mathbf{W}_s = [\rho_1 \rho_2 \rho_3 \rho_4]^T \qquad (2) $$

$$ \mathbf{F}_s = [\rho_1 \mathbf{u} \rho_2 \mathbf{u} \rho_3 \mathbf{u} \rho_4 \mathbf{u}]^T \qquad (3) $$

where ρ₁, ρ₂, ρ₃, and ρ₄ represent the mass densities of the four ion components H⁺, O₂⁺, O⁺, and CO₂⁺, respectively. Qs is the vector formed by the source terms of each component, and the expression is as follows:

$$ \mathbf{Q}_s = [S_1 - L_1 S_2 - L_2 S_3 - L_3 S_4 - L_4]^T \qquad (4) $$

where Si and Li respectively represent the terms of production and loss of the species i.

The governing equation is as follows:

$$ \frac{\partial \mathbf{W}}{\partial t} + \nabla \cdot \mathbf{F} = \mathbf{Q}. \qquad (5) $$

The conservative vector W composed of eight conserved variables is as follows:

$$ \mathbf{W} = [\rho \rho \mathbf{u} \mathbf{B} \varepsilon]^T \qquad (6) $$

where ρ is the total mass density of the four ion components:

$$ \rho = \sum_{i=1}^{4} \rho_{i}. \qquad (7) $$

The expressions for the conserved flux F and the source term Q of the main MHD equation are as follows:

$$ \mathbf{F} = \begin{bmatrix} \rho \mathbf{u} \\ \rho \mathbf{u} \mathbf{u} + \left(p + \frac{1}{2} B^2\right) \mathbf{I} - \mathbf{B} \mathbf{B} \\ \mathbf{u} \mathbf{B} - \mathbf{B} \mathbf{u} \\ \left(\varepsilon + p + \frac{1}{2} B^2\right) \mathbf{u} - \left(\mathbf{B} \cdot \mathbf{u}\right) \mathbf{B} \end{bmatrix} \qquad (8) $$

$$ \mathbf{Q} = \begin{bmatrix} S_1 + S_2 + S_3 + S_4 - L_1 - L_2 - L_3 - L_4 \\ \rho \mathbf{G} - \rho \nu \mathbf{u} - \left(\sum_{i=1}^{4} L_i\right) \mathbf{u} \\ \mathbf{0} \\ Q_E \end{bmatrix} \qquad (9) $$

where the total energy density ε is:

$$ \varepsilon = \frac{1}{2} \sum_{i=1}^{4} \rho_i u^2 + \frac{p}{\gamma - 1} + \frac{1}{2} B^2. \qquad (10) $$

The expression of the energy source term is as follows (Ma et al. 2004):

$$ \begin{aligned} Q_E = \rho \mathbf{u} \cdot \mathbf{G} & - \frac{1}{2} \left( \sum_{i=1}^{4} L_i \right) u^2 - \rho \nu u^2 \\ & + \frac{k T_o}{\gamma - 1} \left( \sum_{i=1}^{4} \frac{S_i}{m_i} \right) - \frac{p}{\gamma - 1} \left( \frac{\sum_{i=1}^{4} \frac{L_i}{m_i}}{\sum_{i=1}^{4} \frac{\rho_i}{m_i}} \right) \\ & - 4 \times 10^{-11} \sum_{i=1}^{3} n_i \sum_{i=1}^{4} \frac{\rho_i}{m_i} \exp \left( \frac{10(T_p - 6000)}{T_p} \right) \end{aligned} \qquad (11) $$

where ν is the ion-neutral collision frequency, T_o is the temperature of newborn ions assigned the same value as that of the local neutral atmosphere, and T_p is the plasma temperature taken as T_p = p/(2 * Σi=1^4 ρi/mi) for the assumption of equal temperature between ions and electrons. The last term in Equation (11) is added to allow the ionospheric temperature to maintain a reasonable state.

### 3. Initial and Boundary Conditions

The model adopts the Mars-Solar-Orbital (abbreviated MSO) coordinate system with the x-axis pointing toward the Sun, the z-axis parallel to the normal direction of Mars orbit plane and the y-axis completing the right-hand coordinate system. The calculation domain is defined by -24 R_M < X < 8 R_M, -16 R_M < Y, Z < 16 R_M, where R_M represents the Mars radius.

The inner boundary of the grid is set at a distance of 100 km from the surface of Mars for the self-consistent solution of ionosphere. The densities of O₂⁺, O⁺, and CO₂⁺ at the inner boundary are set by the equilibrium values of photochemical reactions, while the H⁺ density is 30% of the proton density of solar wind (Taylor et al. 1980). Assuming that the fluid is inviscid, the normal velocity value at the boundary is set to zero. The number densities of the three neutral atmospheric components H, O, and CO₂ follow the exponential distribution with height H, i.e., n_s' = n_s'0 exp(-H/H_s'), where n_s'0 is the number density of each neutral component s' and H_s' is the corresponding scale height on the Mars surface.

The number density of 4 cm⁻³ and the speed of 400 km s⁻¹ along the x-axis are adopted to represent the average initial conditions of the upstream solar wind. Based on the statistical results from DiBraccio et al. (2018), the IMF is located on the ecliptic plane for most of the cases with an average magnitude of 3.3 nT. Moreover, the Parker spiral angle is observed to be a broad bimodal distribution in the range from -180° to +180° with two peak values, consistent with the latest statistical results by Liu et al. (2021).

To study the influence of the IMF orientation on the solar wind interaction with Mars, the magnitude of the IMF is set at a fixed value of 3 nT. The Parker spiral angles that the magnetic field direction satisfies in the x-y plane are assumed to be 10°, 30°, 56°, and 90°, respectively. Among the four IMF directions, the Parker spiral angle of 90° indicates the IMF toward the +y direction. The Parker spiral angle of 56° is mostly used to represent the upstream
<figure><img src="image_5.png" /><figcaption>**Figure 3.** The induced magnetic field strength in the x-y plane for the four IMF directions: (a) 10°, (b) 30°, (c) 56°, and (d) 90°.</figcaption></figure>

solar wind IMF condition. The Parker spiral angle of 10° and 30° are for the cases where the IMF is quasi-parallel to the solar wind flow with the positive $B_y$ component.

4. Simulation Results

The simulated distributions of the induced magnetic field and velocity for the Parker spiral angle of 56° in the x-y plane and x-z plane are presented in Figure 1. The induced magnetic field produced by the motion of the electrically conducting plasma is obtained by solving the magnetic induction equation. The white lines with arrows indicate the directions of the induced magnetic field and velocity. The outer black dashed line represents the mean location of BS and the inner black solid line is for the mean MPB by Vignes et al. (2000). The BS is the shared evident feature in both the magnetic field and velocity results for a jump of the intensity by three to four times. In addition, the simulated BS positions fit quite well with the MGS observations by Vignes et al. (2000). The magnetic pileup region is clearly visible in the magnetic field contour plots for the field strength two to three times higher than that in the magnetosheath. The white lines with arrows show the magnetic field draping around the magnetic field pileup region in the x-y plane. As for the velocity, the plasma flows downstream and toward the terminator on the day side in both the x-y and x-z plane. These features of the plasma environment around Mars from our simulation results agree

quite well with the previous investigations (Ma et al. 2004; Nagy et al. 2004).

The sonic Mach number $M_S$ and magnetosonic Mach number $M_{MS}$ for the Parker spiral angle of 56° are shown in Figure 2. The dashed line and the solid line represent the mean location of BS and MPB by Vignes et al. (2000), respectively. The distribution of $M_{MS}$ and $M_S$ are quite similar except for on the tail region in the x-y plane, where the Alfvén speed $V_A$ is important for the high strength of the induced magnetic field. Bow shock is also an evident feature to separate the solar wind from the magnetosheath in the $M_S$ and $M_{MS}$ distributions. Heavy ions play a role within the MPB region due to their large mass density.

Figure 3 shows the induced magnetic field strength in the x-y plane for the four IMF directions: (a) 10°, (b) 30°, (c) 56°, and (d) 90°. The BS is formed on the day side as the solar wind flows around the planet obstacle. In the inner region, an IM is clearly visible where the magnetic field piles up as high as ~55 nT in Figures 3(c) and (d). A slight asymmetry exists along the dawn and dusk sides of the BS. For the Parker spiral angle of 90°, however, no asymmetry is observed on the dawn and dusk sides of the BS. As shown on Figures 3(a) and (b) for the cases of smaller Parker spiral angles, the BS and the IM shrink closer to the planet as well as the magnetotail. When the IMF is almost parallel to the solar wind flow, the IM is weaker with the peak strength of about 35 nT.
<figure><img src="image_6.png" /><figcaption>Figure 4. The induced magnetic field strength in the x–z plane for the four IMF directions: (a) 10°, (b) 30°, (c) 56°, and (d) 90°.</figcaption></figure>

The induced magnetic field strength in the x–z plane for the four IMF directions is shown in Figure 4. Comparing the results between Figures 3 and 4, it can be seen that the region with strong magnetic field extends to larger solar zenith angle on the equatorial plane than that on the meridian plane. No obvious asymmetry is observed between the north and south hemispheres. The inner boundary of the draping magnetic is much further from the planet on the north and south pole regions. However, the draping magnetic field is pressed around the planet surface closely to form the induced magnetotail in the x–y plane.

Figure 5 presents the results of the induced magnetic field strength in the y–z plane for the four IMF directions. The cross section of the BS on the terminator plane is almost a circular with equal distances between the dawn-dusk sides and the north-south poles for the small Parker spiral angles (Figures 5(a) and (b)). For the large Parker spiral angles (Figures 5(c) and (d)), the cross section is an ellipse with the major axis parallel to the z-axis and the short axis parallel to the y-axis. The dawn-dusk asymmetry is more clearly observed on the terminator plane compared with the results on the equatorial plane. The shock position on the dawn side is closer to the planet than that on the duskside, which is consistent with previous results (Ma et al. 2004). Figures 4(a) and (b) present the magnetosheath region with the weaker strength of the induced magnetic field on the dawn side. Conclusively, the cross-section area of the BS increases with the Parker spiral angles.

The total strength $|B|$ and the three components of the induced magnetic field along the subsolar line for the Parker spiral angle of 56° are shown in Figure 6. The magnitude of the induced magnetic field is three times higher than that of the IMF at around $1.65 R_M$ in the subsolar region. The magnetic field continues to pile up and reaches to a peak value higher than 50 nT at around $1.2 R_M$. As the dominant component of the draping magnetic, $B_y$ almost keeps the same magnitude as the total $|B|$, while $B_x$ and $B_z$ are very small.

Figure 7 shows the pressure profiles along the subsolar line for the four IMF Parker spiral angles, including the magnetic pressure $P_B$ (the red line), the dynamic pressure $P_{Dyn}$ (the green line) of the solar wind, and the thermal pressure $P_T$ (the blue line). Previous studies assumed that the total pressure keeps conserved in the solar wind interaction with Mars or Venus, with the dominant component alternated by the solar wind dynamic pressure upstream the BS, the thermal pressure in the magnetosheath, and the magnetic pressure closer to the planet, respectively (Zhang et al. 1991a; Crider et al. 2003; Brain et al. 2010; Holmberg et al. 2019).

The simulated location of BS is based on the balance between the solar wind dynamic pressure and the thermal pressure on the subsolar region, consistent with the average BS position from the MGS observations (Modolo et al. 2016). The thermal pressure in the magnetosheath balances the magnetic pressure in the magnetic pileup region at the MPB (Ma et al. 2014). As presented in Figure 7, the dynamic pressure is the
<figure><img src="image_7.png" /><figcaption>**Figure 5.** The induced magnetic field strength in the y-z plane for the four IMF directions: (a) 10°, (b) 30°, (c) 56°, and (d) 90°.</figcaption></figure>

dominant component far from the planet with a magnitude of 0.5 nPa corresponding to the incident solar wind conditions. There is a fast transition from dynamic to thermal pressure at BS, below which there is a gradual transition from thermal to magnetic pressure. As the IMF orientation changes from quasi-parallel to perpendicular to the solar wind flow, the change of the magnitude of $P_T$ is slight. However, the peak magnetic pressure drops significantly for the case with a Parker spiral angle of 10°, only one-third of that for the case of a Parker spiral angle of 90°. Therefore, it can be concluded that the IMF quasi-parallel to the solar wind flow impedes the evolution of the magnetic pressure at the lower altitude around the ionosphere, leading to a much weaker IM. For the four IMF directions, the locations of BS and MPB are listed in Table 1. In the subsolar region, the shock position moves from 1.50 to 1.70 $R_M$ and the MPB from 1.22 to 1.42 $R_M$ as the IMF direction changes from quasi-parallel to the solar wind flow to along the positive y-axis.

As shown in Figure 8, the magnetotail configuration in the x-y plane is strongly affected by the four IMF directions: (a) 10°, (b) 30°, (c) 56°, and (d) 90°. The magnetotail is a region consisting of two magnetic lobes with the two opposite directions of the $B_x$ component, respectively. As the IMF direction changes from quasi-parallel to the solar wind flow to along the positive y-axis, the tail radius and the magnitude of the induced magnetic field increase significantly. A clear boundary of about 2.5 $R_M$ from the center x-axis of the tail

exists in Figure 8(c), separating the strong $B_x$ component with a magnitude of ~15 nT from the weaker background magnetic field at $2 R_M$ downstream. This result is consistent with that by DiBraccio et al. (2018) for the same IMF direction.

Locations with $B_x$ component 0 are drawn by the two white lines in Figures 8(a), (b), and (c), referred to as PRL and IPRL, respectively. These two layers present the location where the $B_x$ component reverses its polarity. For the case with the IMF along the y-axis, the magnetic field topology of the induced magnetotail is mirror-symmetric with the x-axis and separated by (PRL) in the center of the two magnetic lobes (Figure 8(d)). However, for the cases with nonzero flow-aligned IMF component, the mirror-symmetry is damaged by the generation of the IPRL. The PRL at the tail region far away from the Mars surface is shifted to the y<0 sector with the $B_x$ component parallel to the flow. The slight displacements from the y=0 axis show no much difference for the three IMF directions. The IPRL is distinctly shifted to further distance from the center of the magnetotail for lager Parker spiral angles. For the case with the Parker spiral angle of 10°, the two white lines indicating the PRL and IPRL respectively intersect at around -16 $R_M$. It is easy to infer that in further tail beyond this distance, the presence of Mars in the magnetized solar wind flow causes little influence on the IMF deformation. In Figures 8(b) and (c), the two lines with zero $B_x$ component do not cross in our simulation range, suggesting the extension of the induced dual-lobe magnetotail to further distance downstream.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The total strength $|B|$ and the three components of the induced magnetic field along the subsolar line for the Parker spiral angle of 56°.</figcaption></figure>

The draping magnetic field lines for the different IMF directions with the Parker spiral angles of (a) 10°, (b) 30°, (c) 56°, (d) 90° is presented by the same orientation view in the 3D Cartesian system in Figure 9. The red ball surrounded by the magnetic field line in the four panels represents the Mars surface. In the 3D plot, it can be seen clearly that the magnetic field piles up and drape around the day side of Mars. In the downstream region, as the Parker spiral angle of the IMF increases from 10° to 56°, the magnetic field is stretched more evidently for the longer distance between the two points where the $B_x$ component changes its direction. For the case with the IMF long the +y-axis, the magnetic field forms a symmetric dual-lobe with the $B_x$ component pointing toward/outward Mars.

**5. Conclusion and Discussion**

We present the results of the solar wind interaction with Mars by simulating our 3D multispecies model for the four cases with different IMF directions. Both the BS and MPB are further from the planet as the IMF direction changes from

quasi-parallel to the solar wind flow to along the positive y-axis. The cross section of the shock on the terminator plane is enlarged with the Parker spiral angle increasing from 10° to 90°, as well as the radius of the induced magnetotail. The consistent behavior of the BS and the magnetotail indicates the unity of the global magnetic structures on the day side and the tail region downstream. For the case with the IMF along the y-axis, the flow-aligned component changes its sign at the $y=0$ plane. For the cases with nonzero flow-aligned IMF component, the PRL and IPRL are displaced to $y < 0$ sector, forming an asymmetric magnetic field topology in the tail region. Our simulation results are consistent with previous analytical solutions and observations (Simon et al. 2013; Romanelli et al. 2014, 2015). In conclusion, the IMF orientation plays an important role in the whole picture of these united magnetic structures as a result of the solar wind interaction with unmagnetized planets.

It is commonly believed that the Martian magnetotail consisting of two lobes with oppositely oriented magnetic fields is separated by a cross-tail current sheet. Based on MAVEN observations in
<figure><img src="image_9.png" /><figcaption>**Figure 7.** The pressure profiles along the subsolar line for the four IMF Parker spiral angles, including the magnetic pressure $P_B$ (the red line), the dynamic pressure $P_{Dyn}$ (the green line) of the solar wind and the thermal pressure $P_T$ (the blue line). The small dots stand for the location of the radial grid points along the subsolar line. The locations of BS and MPB are marked by the dashed line with arrow.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** The $B_x$ component of the induced magnetic field for the different IMF directions with the Parker spiral angles of (a) 10°, (b) 30°, (c) 56°, and (d) 90°, respectively, in the x-y plane.</figcaption></figure>

the tail region, Grigorenko et al. (2017) found a thin current layer inserted into a thicker one for the cases with small local normal magnetic field component $B_N$. For cases with larger $B_N$, it was

observed that the thickness of the current sheet increases while the inserting current layer vanishes. Their explanation for this phenomenon is the quasi-adiabatic ion dynamics for the formation
<figure><img src="image_11.png" /><figcaption>**Figure 9.** The draping magnetic field lines for the different IMF directions with the Parker spiral angles of (a) 10°, (b) 30°, (c) 56°, and (d) 90° shown by the same orientation view in the 3D Cartesian system.</figcaption></figure>

<table><caption>**Table 1**
The Magnetic Field Characteristics Under Different IMF Directions</caption><thead><tr><th>IMF direction</th><th>10°</th><th>30°</th><th>56°</th><th>90°</th></tr></thead><tbody><tr><td>BS (R<sub>M</sub>)</td><td>1.50</td><td>1.53</td><td>1.65</td><td>1.70</td></tr><tr><td>MPB (R<sub>M</sub>)</td><td>1.22</td><td>1.30</td><td>1.36</td><td>1.42</td></tr></tbody></table>

of a current layer consisting of oxygen ions in the region peripheral to the current sheet. Our simulation results might provide another interesting explanation for this phenomenon with the PRL and IPRL denoting the location of currents as well. As

the cross-flow IMF component increases, i.e., the Parker spiral angle changes from 10° to 90°, the IPRL gradually weakens and finally disappears, leaving the single PRL alone separating the two magnetic lobes.

Dubinin et al. (2011) reported that the plasma sheet consists of energetic ions characterized by a few hundred electronvolts. These ions are energized by the $J \times B$ force in the center of the tail for the strong magnetic shear stress contributed by the draping IMF. The displacement of the induced magnetic lobes caused by the nonzero flow-aligned IMF component might has an influence on the particle acceleration in the tail region. This effect should be taken into account for the estimation of the escape rate of planetary ions. It is noteworthy that not only the
IMF direction evidently displaces the induced tail, but a weak global dipole magnetic field or strong crustal magnetic fields on the south hemisphere also lead to a twist of the Marian induced tail (Kallio et al. 2008b; DiBraccio et al. 2018).

This work was supported by the B-type Strategic Priority Program of the Chinese Academy of Sciences (grant No. XDB41000000) and the pre-research projects on Civil Aerospace Technologies No. D020103 and D020105 funded by China's National Space Administration (CNSA), and the National Natural Science Foundation of China (NSFC) under grant No. 42074214.

## ORCID iDs

Shibang Li https://orcid.org/0000-0001-9175-0007

Christian Mazelle https://orcid.org/0000-0001-5332-9561

## References

Acuña, M. H., Connerney, J. E. P., Ness, N. F., et al. 1999, *Sci*, 284, 790

Acuña, M. H., Connerney, J. E. P., Wasilewski, P., et al. 1998, *Sci*, 279, 1676

Bertucci, C., Duru, F., Edberg, N., et al. 2011, *SSR*, 162, 113

Bertucci, C., Hamilton, D. C., Kurth, W. S., et al. 2015, *GRL*, 42, 193

Bertucci, C., Mazelle, C., & Acuña, M. H. 2005a, *ASR*, 36, 2066

Bertucci, C., Mazelle, C., Acuña, M. H., et al. 2005b, *JGRSP*, 110, A01209

Bertucci, C., Mazelle, C., Slavin, J. A., et al. 2003, *GRL*, 30, 1876

Brain, D., Barabash, S., Boesswetter, A., et al. 2010, *Icar*, 206, 139

Brain, D. A., Bagenal, F., Acuña, M. H., et al. 2003, *JGRA*, 108, 1424

Brain, D. A., Bagenal, F., Ma, Y. J., et al. 2016, *JGRP*, 121, 2364

Brain, D. A., Lillis, R. J., Mitchell, D. L., et al. 2007, *JGRSP*, 112, A09201

Crider, D. H., Vignes, D., Krymskii, A. M., et al. 2003, *JGR*, 108, 1461

Delva, M., Bertucci, C., Schwingenschuh, K., et al. 2014, *P&SS*, 96, 125

DiBraccio, G. A., Luhmann, J. G., Curry, S. M., et al. 2018, *GRL*, 45, 4559

Dubinin, E. M., Fraenz, M., Fedorov, A., et al. 2011, *SSR*, 162, 173

Dubinin, E. M., Fraenz, M., Woch, J., et al. 2008, *AnGeo*, 26, 3511

Echim, M., Maggiolo, R., Keyser, J. D., et al. 2011, *P&SS*, 59, 1028

Espley, J. R., Cloutier, P. A., Brain, D. A., et al. 2004, *JGRA*, 109, A07213

Grigorenko, E. E., Shuvalov, S. D., Malova, H. V., et al. 2017, *JGRSP*, 122, 10176

Holmberg, M. K. G., André, N., Garnier, P., et al. 2019, *JGRSP*, 124, 8564

Holmstrom, M., & Wang, X. D. 2015, *P&SS*, 119, 43

Kallio, E., Barabash, S., Janhunen, P., et al. 2008b, *P&SS*, 56, 823

Kallio, E., Zhang, T. L., Barabash, S., et al. 2008a, *P&SS*, 56, 796

Li, S. B., Lu, H. Y., Cui, J., et al. 2020, *EPP*, 4, 23

Li, Y., & Lu, H. Y. 2019, *APJ*, 875, 47

Liu, D., Rong, Z. J., Gao, J. W., et al. 2021, *ApJ*, 911, 113

Lu, H. Y., Cao, J. B., Zhou, M., et al. 2013, *JGRSP*, 118, 6019

Ma, Y. J., Fang, X., Nagy, A. F., et al. 2014, *JGRSP*, 119, 1272

Ma, Y. J., Nagy, A. F., Sokolov, I. V., et al. 2004, *JGRSP*, 109, A07211

Matsunaga, K., Seki, K., Brain, D. A., et al. 2017, *JGRSP*, 122, 9723

Mazelle, C., Winterhalter, D., Sauer, K., et al. 2004, *SSRv*, 111, 115

Modolo, R., Hess, S., Mancini, M., et al. 2016, *JGRSP*, 121, 6378

Moses, S. L., Coroniti, F. V., & Scarf, F. L. 1988, *GRL*, 15, 429

Nagy, A. F., Winterhalter, D., Sauer, K., et al. 2004, *SSRv*, 111, 33

Najib, D., Nagy, A. F., Tóth, G., et al. 2011, *JGRSP*, 116, A05204

Nilsson, H., Carlsson, E., Gunell, H., et al. 2006, *SSRv*, 126, 355

Nimmo, F. 2002, *Geo*, 30, 987

Ramstad, R., Barabash, S., Futaana, Y., et al. 2017, *JGRSP*, 122, 7279

Romanelli, N., Bertucci, C., Gómez, D., et al. 2015, *JGRSP*, 120, 7737

Romanelli, N., Gomez, D., Bertucci, C., et al. 2014, *APJ*, 789, 43

Russell, C. T. 1985, *GMS*, 35, 109

Sauer, K., Roatsch, T., Motschmann, U., et al. 1992, *JGRSP*, 97, 6227

Shan, L., Lu, Q., Mazelle, C., et al. 2015, *P&SS*, 109, 37

Simon, S., Treeck, S. C., Wennmacher, A., et al. 2013, *JGRSP*, 118, 1679

Slavin, J. A., & Holzer, R. E. 1982, *JGR*, 87, 10285

Snowden, D., Winglee, R., Bertucci, C., et al. 2007, *JGRSP*, 112, A12221

Taylor, H. A., Brinton, H. C., Bauer, R. E., et al. 1980, *JGR*, 85, 7765

Thomsen, M. F. 2013, *GRL*, 40, 5337

Vennerstrom, S., Olsen, N., & Purucker, M. 2003, *GRL*, 30, 1369

Verigin, M., Apathi, I., Kotova, G., et al. 1996, *CosRe*, 34, 551

Vignes, D., Acuña, M. H., Connerney, J. E. P., et al. 2002, *GRL*, 29, 1328

Vignes, D., Mazelle, C., Rme, H., et al. 2000, *GRL*, 27, 49

Winske, D., & Omidi, N. 1995, *JGRA*, 100, 11923

Yi, Y., Kim, E., Kim, Y. H., et al. 1999, *JASS*, 16, 139

Zhang, T. L., Delva, M., Baumjohann, W., et al. 2008a, *JGRE*, 113, E00B20

Zhang, T. L., Delva, M., Baumjohann, W., et al. 2008b, *P&SS*, 56, 790

Zhang, T. L., Luhmann, J. G., & Russell, C. T. 1991a, *JGR*, 96, 11145

Zhang, T. L., Schwingenschuh, K., Lichtenegger, H., et al. 1991b, *JGRA*, 96, 11265

Zhang, T. L., Schwingenschuh, K., Russell, C. T., et al. 1991c, *GRL*, 18, 127