Variability of Draped Interplanetary Magnetic Field in the Subsolar Martian Ionosphere

In the absence of an intrinsic magnetic field, the solar wind interacts directly with the Martian atmosphere, and the interplanetary magnetic field (IMF) drapes around the ionosphere. With the aid of multi-instrument measurements from Mars Atmosphere and Volatile Evolution (MAVEN), we investigate the altitudinal profiles of the draped magnetic field in the subsolar Martian ionosphere. Our analysis of 141 profiles found that they have a median profile with a nearly constant field strength of about 30 nT above 160 km. However, these profiles also show large variability, for example, with a standard deviation of 16 nT at 250 km. Timescale analysis found that magnetic field transport with downward ion plasma motion dominates the draping of IMF at high altitudes. At low altitudes, magnetic diffusion takes over due to increasing collisions. At even lower altitudes, approaching the ionospheric peak, the draped field is quickly damped owing to high conductivity. We used a 1D model simulation to confirm the above processes involved and further indicate that undulating magnetic fields with oscillation periods longer than tens of minutes can penetrate deeply into the ionosphere. This causes the large variability of draped field observed down to below 200 km. The diverse profiles of draped field observed by MAVEN are thus likely due to varying IMF conditions.


Introduction
When a planet lacks a global dynamo magnetic field, interaction with the solar wind (SW) leads to the formation of an induced magnetosphere, and the interplanetary magnetic field (IMF) drapes around the ionosphere (Nagy et al. 2004;Brain 2006;Kallio et al. 2011).For Mars, the SW plasma interacts directly with the ionosphere and creates several regions with distinct plasma and field properties, as encountered at Venus and Titan (e.g., Ma et al. 2004;Modolo et al. 2016).However, this picture is further complicated by the presence of strong magnetic fields in crustal anomalies (e.g., Acuna et al. 1999), which may protect the atmosphere from direct SW interaction in localized regions.As a result, minimagnetospheres can be formed associated with strong crustal remanent magnetism on Mars, as well as at the terrestrial Moon (e.g., Ma et al. 2002;Fang et al. 2010Fang et al. , 2015)).
Since the SW dynamic pressure generally exceeds the thermal plasma pressure in the upper Martian ionosphere (Sánchez-Cano et al. 2020), the IMF penetrates into the ionosphere.Above the Martian exobase, which is located at around 170-200 km altitude (Jakosky et al. 2017), the draped magnetic field may provide ionospheric particles access to the solar wind.This is particularly important in the context of ionospheric escape at Mars.Studies have found that the variability of draped magnetic field is driven in large part by upstream solar wind conditions.When coronal mass ejections (CMEs) arrived at Mars, draped magnetic field lines were observed at much deeper altitudes in the ionosphere than in quiet conditions (Xu et al. 2018(Xu et al. , 2019)).In a statistical study, Fowler et al. (2019) further found that the draped magnetic field penetrates deeper when the normal SW dynamic pressure is stronger.They accordingly suggested that draped magnetic field should be present down to 200 km altitude or below during extreme events.However, fewer studies have been devoted to the deep penetration of IMF in the Martian ionosphere, hence we are still very much in the dark as to how deep into the ionosphere the magnetic field can penetrate and by what means the variability of draped field is influenced by the upstream IMF.Moreover, it is still unclear whether small-scale magnetic structures in the IMF might be transported into the ionosphere and form the fossil structures supposed by Hamil et al. (2022).
Models based on ionospheric physics and chemistry have been established to investigate the SW interaction with Mars down to ionospheric peak altitudes (e.g., Dong et al. 2014Dong et al. , 2015;;Ma et al. 2014b;Modolo et al. 2016).However, the simulation below the ionopause is not well constrained due to the lack of simultaneous measurements of the magnetic field and ionospheric plasma at low altitudes.The situation has changed since the arrival of the Mars Atmosphere and Volatile Evolution (MAVEN) spacecraft at the red planet on 2014 September 21 (Jakosky et al. 2015).MAVEN provides measurements of the properties of both the field and plasma from distant regions in the unperturbed SW down to ionospheric regions approaching 120 km in altitude during isolated Deep Dip campaigns.
In this study, we use a multi-instrument MAVEN data set to investigate the draped magnetic field and its variability below 400 km in the Martian ionosphere.The data set contains magnetic field measurements by the Magnetometer (MAG; Connerney et al. 2015b), ion density measurements by the Neutral Gas and Ion Mass Spectrometer (NGIMS; Mahaffy et al. 2015), ion temperature measurements by the Suprathermal and Thermal Ion Composition (STATIC; McFadden et al. 2015), as well as electron temperature measurements by the Langmuir Probe and Waves (LPW; Andersson et al. 2015).In addition, we established a 1D model to explore the penetration of variable IMF, which may contribute to the variability of draped field in the ionosphere.Special concern is devoted to the subsolar regions of Mars away from strong crustal magnetic anomalies to avoid complexity due to reconnection between draped and crustal magnetic fields.The study is also in view of a direct comparison to the SW interaction with unmagnetized Venus as inferred from the data accumulated by a similar set of instruments on board the Pioneer Venus Orbiter (PVO; e.g., Russell et al. 1979;Brace et al. 1980;Taylor et al. 1980).

MAVEN Observations of Draped Magnetic Field
The crustal magnetic model of Langlais et al. (2019 was used to exclude regions of strong crustal magnetic fields.We selected MAVEN orbits with low crustal magnetic fields by choosing those with maximum vertical and horizontal crustal magnetic components less than 30 nT below 500 km.We then analyzed MAVEN data from these orbits to investigate the induced/draped magnetic field. Figure 1(a) shows an example of an inbound orbit (#5922).The crustal components from the L2019 model are further subtracted from the MAVEN observed magnetic field to calculate the draped magnetic field.As shown in Figure 1(b), the draped magnetic field remains constant above 200 km and decreases rapidly below 160 km.This profile may represent typical draped magnetic morphology in a planet's atmosphere without a global dynamo magnetic field.The IMF permeates the atmosphere and is finally dissipated in the conductive ionosphere.Model studies on Venus's atmosphere had produced similar magnetic profiles (Shinagawa et al. 1987;Shinagawa & Cravens 1989).
The draped magnetic field on the dayside of Mars exhibits high variability when we examine more orbits.Figure 2 shows six additional examples, all of which are mainly horizontal magnetic fields that agree well with the characteristics of the draped magnetic field.However, the profiles show very different altitudinal variations.For example, a peak of magnetic field appears at 200 km for orbit #2134, but at 150 km for orbit #2282 (in panels (a) and (b)).The magnetic field increases with altitude in orbit #2710, but decreases with altitude in orbit #6057 (in panels (c) and (f)).The profile of orbit #3126 is similar to Venus's magnetic field measured by Pioneer Venus Orbiter (Shinagawa & Cravens 1988) (in panel (d)).The profile of orbit #5928 exhibits a local maximum at 170 km, which shares similarities with the model result of Ma et al. (2014a) (in panel (e)).
We then collected all profiles from MAVEN orbits with low crustal magnetic fields until the end of 2022.The location of these profiles is denoted by black dots in Figure 3.Of these, 141 profiles with solar zenith angle (SZA) <30°are selected for statistical analysis.The limitation of SZA <30°is based on the fact that the penetration of draped magnetic field depends on the normal SW dynamic pressure, which is proportional to cos 2 (SZA) (Zhang & Luhmann 1992;Fowler et al. 2019), and the draped field is mainly significant at low SZA (Dubinin et al. 2021).As denoted by red dots in Figure 3, the qualified profiles are clustered at mid-latitude in the northern hemisphere and at some longitudes around the equator.
For each of the 141 profiles, we also removed the crustal component to obtain draped magnetic field profiles, as mentioned above.The data points comprising the draped field profiles were divided into six altitude groups: 120 km < h < 140 km, 140 km < h < 160 km, 160 km < h < 180 km, 180 km < h < 200 km, 200 km < h < 220 km, and 220 km < h < 240 km. Figure 4 illustrates the distribution of draped field strength for different groups, which shows peaks at 20-40 nT for h > 160 km and at 10-20 nT for h < 140 km.A long tail is present, which extends to ∼100 nT for h > 140 km whereas it shrinks at lower altitudes.Figure 5 shows the median profile and standard deviation of the 141 profiles.The median field remains constant for h > 160 km and decreases rapidly at lower altitudes, while the turning point of the standard deviation is at about 140 km.It is worth noting that the median profile is similar to the example profile in Figure 1, meaning that the median profile may represent ordinary profiles of the draped field.Profiles for larger SZA have a similar shape but with smaller magnetic field strength, in agreement with the weak draped field expected outside the subsolar region.We will not show these profiles or go into details due to limited space.

Timescales of Magnetic Field Diffusion and Transport
Penetration of draped magnetic field into the ionosphere can be understood in terms of timescales of magnetic field diffusion, magnetic field transport, ion transport, and ion chemistry.Considering the 1D magnetic diffusion equation is the diffusive coefficient of the magnetic field and ( ( )) is the ion velocity.m e and m i are the electron and + O 2 mass, e is the electron charge, n e = n i assumes that the electron and ion number densities are equal, μ 0 is the permeability of vacuum, T i and T e are the ion and electron temperatures, and k B is the Boltzmann constant.ν en and ν ei are the electron-neutral and electron-ion collision frequencies, and ν in is the ion-neutral collision frequency.We can calculate magnetic diffusion timescale as and the magnetic transport timescale as . The ion transport timescale, τ ion−trans , should be equal to the magnetic transport timescale, because of the "frozen term" ( ( )) . Leaving alone photoionization that is fast enough, we consider recombination of + O 2 for the chemical timescale, which is 2 .The recombination coefficient α and collision frequencies mentioned above are adopted from Schunk & Nagy (2009).
In the calculation of the timescales, magnetic field strength B is obtained from the median profile presented in Figure 5. Figure 6 shows an example for the MAVEN DD2 period with orbits near the subsolar point (SZA ∼15°), with the density and temperature data from MAVEN observations and empirical models, as denoted in the figure caption.The timescales as a function of altitude are depicted in Figure 6(d).τ B−diff decreases rapidly as altitude declines, from 10 5 s at 250 km to less than 1 minute at 120 km, reflecting increased magnetic diffusion efficiency at low altitude, where denser neutral atmosphere causes larger ν en and thus a larger diffusion coefficient D B .This is also related to the altitudinal dependence of the parallel electrical conductivity, since D B is in inverse proportion to the conductivity.On the other hand, τ B−trans varies in the opposite direction, from 10 1 s at 250 km to 10 6 s at 120 km.The analysis indicates that transport dominates the draping of magnetic field at high altitudes.The magnetic field moves downward and experiences little attenuation owing to fewer collisions at high altitudes.However, deep in the ionosphere (below 200 km), the diffusion process takes control, and frequent ion-neutral collisions cause the magnetic field to dissipate quickly.This analysis qualitatively explains the mean profile observed in Figure 5, with magnetic field strength almost constant at high altitudes but decreasing quickly deep in the ionosphere.
For h < 220 km, which is the so-called photochemical equilibrium region, the recombination timescale of + O 2 is shorter than both the transport and diffusion timescales.Photochemical reactions hence dominate in this region, where neither transport nor diffusion could change the ion profile significantly.In contrast, above 220 km, the ion profile is controlled by the transport process.

Simulation of Magnetic Field Penetrating into the Ionosphere
We constructed a 1D model to explore the draped magnetic field observed in the subsolar Martian ionosphere below 250 km.This model is analogous to the early models implemented to describe the SW interaction with the Venusian ionosphere (e.g., Cravens et al. 1984;Luhmann et al. 1984;Phillips et al. 1984;Shinagawa et al. 1987).We are especially interested in reproducing magnetic field variability observed around 160-200 km.This indicates that the assumption of a singlespecies ionosphere could be favorably used because the Martian ionosphere is exclusively dominated by + O 2 at these altitudes (Benna et al. 2015).Without loss of generality, we further assume that the magnetic field is strictly horizontal, and its strength varies with altitude only.For simplicity, horizontal variations of the ionospheric plasma are ignored as well.Therefore, the model is one-dimensional in nature.It is established according to Equation (1), where the ion velocity is calculated by collision balance: ´-e z 250 km 500 km 2 ; hence B declines exponentially with altitude as depicted by the line for t = 0 s in Figure 7(b).The profiles of ion density and electron and ion temperature used in the calculation are obtained from Figure 6.
Figure 7 shows the temporal evolution of the ion velocity and magnetic field profiles from t = 0 s to t = 6000 s.Due to magnetic field transport and diffusion, the magnetic field penetrates from the upper boundary to lower altitudes.After 2000 s, the simulation nearly reaches a steady state.The B profiles in Figure 7(b) show a rapid attenuation below 180 km, which agrees well with the median profile in Figure 5.Meanwhile, the ion velocity depicted in Figure 7(a) is of the order of 20 m s −1 , which is much slower than the speed of sound.This means that the collision balance condition holds for Equation (2), thus it is appropriate to omit the ∂W i /∂t and W i ∂W i /∂z terms on the left side.
The variance of magnetic field profiles is further investigated by imposing an undulating magnetic field at the upper  5. We then run the model with the oscillation periods T = 10 4 , 5 × 10 3 , 2 × 10 3 , 10 3 , 5 × 10 2 , 2 × 10 2 , 100, 10, and 1 s.With the initial B profile set to the steady-state profile in Figure 7, the simulated profiles with time t are depicted in Figure 8.It is clearly shown that the penetration depth of the undulating field depends on the oscillation period.The rapidly varying upper boundary B, for example, with T < 10 s, mainly affects the altitudes above 220 km.In contrast, for T = 10 4 s, the magnetic fields below 160 km are largely diverse, due to the influence of undulating magnetic field at the upper boundary.
For each panel in Figure 8, deviation from the initial profile (t = 0) is collected to calculate a standard deviation profile, which is displayed in Figure 9.The standard deviation at the upper boundary is fixed to ( ) . For long-period magnetic field undulation, the standard deviation is large above 180 km, but decreases rapidly at lower altitudes.This is generally consistent with the observed standard deviation profile in Figure 5.

Discussion and Conclusion
Statistics of MAVEN observations in the subpolar Martian ionosphere show that the draped magnetic field has nearly constant strength above 200 km altitude and decreases rapidly below 160 km, as presented in Figure 5.The invariable magnetic field strength with altitude, however, seemingly contradicts the result of Hamil et al. (2019).Figure 4 of Hamil et al. (2019) shows decreasing magnetic pressure in the northern hemisphere as altitude increases.This discrepancy could be attributed to the existence of a crustal field.In our statistics, if the crustal field is not subtracted from the MAVEN magnetic measurements, / dB dz will also be obviously negative.Moreover, the crustal field is a gradient field, and its magnetic pressure is completely balanced by magnetic tension, thus it will not drive the plasma on its own.When calculating ion motion driven by magnetic pressure, the crustal field should be removed to obtain a realistic ion velocity.This also stresses the necessity of removing the crustal field to get the draped magnetic field.
A careful inspection of the median profile in Figure 5 may find a slightly negative / ¶ ¶ B z above 380 km.The negative gradient is possibly caused by deceleration of the solar wind as it passes through the ionopause.The ionopause on Mars is located at 300-400 km, where the dynamic pressure of the solar wind is balanced by the total pressure of the ionosphere (Girazian et al. 2019;Sánchez-Cano et al. 2020).According to the frozen solution = , deceleration of the solar wind leads to increasing magnetic field strength.It is thus plausible to attribute the negative gradient of the draped magnetic field to decelerated downward plasma flow.
The timescale analysis based on Equation (1) indicates that the penetration of magnetic field is controlled by the transport term ( ) above 170 km, and by the diffusion term z below that altitude.At high altitudes, the atmosphere is nonconducting, so diffusion is negligible and the magnetic field passes through the atmosphere without dissipation.Then, considering a steady state with , where B 0 and W i,0 are the magnetic field strength and plasma velocity at the upper boundary.The solution implies that constant or slowly varying B corresponds to a stable downward plasma velocity W i .The invariable magnetic field strength with altitude observed above 170 km hence can be attributed to the downward magnetic/plasma transport.The stable downward plasma velocity is depicted in Figure 7(a), which also shows nearly zero plasma velocity at low altitudes, owing to the dense atmosphere and thus frequent ion-neutral collisions.It is important to note that the plasma velocity, in this case, includes contributions from both magnetic and thermal pressure gradients, though the former dominates.Meanwhile, down to the ionospheric peak, collisions increase and diffusion becomes important.Inspecting the steady-state solution again for this region, from ( ) . As altitude decreases, collisions and D B increase, and the magnetic field dissipates and decreases to zero rapidly.Magnetic energy is converted into ion thermal energy, and the heating rate can be estimated as The simulated draped field of Mars has a profile similar to previous results for Venus (e.g., Cravens et al. 1984;Shinagawa et al. 1987), which is expected since both planets are weakly magnetized.However, there are still differences.On Mars, the magnetic field strength is invariable with altitude above the "turning point" at around 200 km, and below that point it is largely attenuated.The profile thus has a shape like a   reverse "L."However, PVO observations and models show that the magnetic field strength on Venus exhibits a local minimum above the "turning point" (e.g., Luhmann et al. 1980;Shinagawa et al. 1987).The profile shape is like an "S."The differences might be attributed to stronger IMF around Venus (B ∼ 100 nT).A stronger magnetic field will produce excessive magnetic pressure, causing the plasma to accelerate significantly during the downward transport.Due to the frozen solution , the magnetic field is "diluted" as the plasma velocity increases, resulting in a local minimum.The IMF around Mars, on the other hand, is much weaker (B ∼ 30 nT). Figure 7 depicts a gentle flow of plasma caused by smaller magnetic pressure, and ∂ z W i = 0 holds almost everywhere.Consequently, the magnetic field remains invariable with altitude above the dissipation region on Mars.
The draped magnetic field exhibits significant variability, as demonstrated by the standard deviation profile in Figure 5.The median magnetic field strength is approximately 30 nT above an altitude of 200 km, with a standard deviation of around 16 nT.The observed variability is replicated in the simulation by introducing a sinusoidal component to the upper boundary magnetic field.Additionally, the penetration depth of the magnetic field is dependent on the oscillation period, suggesting that slowly varying IMF, for example, with a period of tens of minutes, could penetrate deeply into the ionosphere to an altitude of 180 km.The MAVEN spacecraft requires several minutes to fly through the Martian ionosphere.This duration is longer than τ B−diff or τ B−trans , whichever is smaller, for most altitude regions; it is also comparable to the period of IMF disturbances that can penetrate downward to 180 km.Therefore, MAVEN could be observing a patchwork of varying draped magnetic fields at different times, unless the upstream IMF remains stable for an extended period.This may help to explain the diverse profiles observed as shown in Figure 5.
The undulating magnetic field at the upper boundary causes a varying pressure gradient, which drives ions back and forth in the vertical direction.The ion velocity W i in our simulation for different magnetic oscillation periods is depicted in Figure 10.When the magnetic field in upstream SW decreases, ions will be thrown outside by the larger fossil field that has previously been deposited, like a compressed spring being released.This might serve as an ion escape channel, which we will not go deeply into in this article.It is necessary to mention that a major limitation of our model is the assumption of collision balance.According to , the speed of sound c s in the Martian atmosphere is around 1 km s −1 .Figure 10 shows that the ion velocity gets larger for shorter periods of oscillating magnetic field, and the velocity can approach and exceed the speed of sound when the period is shorter than 200 s.This is caused by rapidly varying B that creates a larger pressure gradient in a narrower region, which obviously violates the assumption of collision balance.However, this assumption will overestimate the ion velocity, so the actual velocity and the penetration depth of the magnetic field should be even smaller for short oscillation periods.For the same reason, our model does not account for the very smallscale fluctuations that are ubiquitous in the observed profiles (see Figures 1 and 2).These fluctuations may be proton cyclotron waves or magnetosonic waves, which have been observed widely in the Martian dayside atmosphere (Connerney et al. 2015a;Fowler et al. 2018).In our model, these waves are filtered out since we apply the collision balance condition in Equation (2).However, simulating a complete momentum equation including ∂/∂t terms would be timeconsuming, and the small-scale fluctuations are beyond the scope of this paper.Future studies may consider these waves to gain a more comprehensive understanding of the draped magnetic field.
In conclusion, MAVEN observations have revealed the profile shape of the draped magnetic field in the subsolar Martian ionosphere.The processes involved can be summarized as follows.
1.The strength of the draped magnetic field is nearly constant above 160 km due to magnetic field transport from above with downward ion plasma motion.2. Magnetic diffusion takes over at low altitudes with increasing ion-neutral collisions, until the draped field quickly dissipates in the more conductive ionosphere below 160 km. 3. Undulating IMF with a period longer than tens of minutes can penetrate deeply into the ionosphere, probably explaining the observed large variability of the draped field down to below 200 km altitude.

Figure 1 .
Figure 1.Example profile from MAVEN inbound orbit #5922 (2017-10-18, SZA ∼29°).(a) MAVEN observations of the radial (B r , in red) and horizontal (B h , in blue) components (solid lines), and the crustal magnetic components from the L2019 model (dashed lines).(b) Draped magnetic field calculated by subtracting the crustal components from MAVEN observed magnetic field.The black line denotes the total strength of the draped magnetic field.

Figure 3 .
Figure3.Geographic location of draped magnetic field profiles observed by MAVEN.The black dots denote the location of profiles free of crustal magnetic anomalies (crustal field <30 nT below 500 km altitude), among which 141 profiles with SZA <30°are marked in red.
and (2) are solved with an upper boundary at 250 km, where B(t, 250 km) = 30 nT according to Figure5.The lower boundary is at 120 km with B(t, 120 km) = 0.The initial magnetic field profile is set to B(t = 0, z) = 30 nT ( )/

Figure 4 .
Figure 4. Distribution of draped magnetic field at different altitudes for the 141 selected profiles.

Figure 5 .
Figure 5. Median profile (black line) and standard deviation (red line) of the 141 draped magnetic field profiles.The 141 individual profiles are plotted as thinner gray lines.

Figure 6 .
Figure 6.Observational and modeled density and temperature profiles during the MAVEN DD2 period.(a) Median profiles of + O 2 , O + , and CO 2 + density from MAVEN observations.The dashed line is a fitted curve for total ion density using the function from Sánchez-Cano et al. (2013).(b) Profiles of neutral species (CO 2 , O, N 2 , CO) from Wu et al. (2021).(c) Electron, ion, and neutral temperature from Wu et al. (2021).(d) Calculated timescales of magnetic field diffusion, transport, and + O 2 recombination.

Figure 7 .
Figure 7. Temporal variations of model simulated profiles.(a) Profiles of ion velocity.The profiles for t = 0 s, 50 s, and 100 s are divided by 50, 3, and 3, respectively, to compare with others.(b) Profiles of magnetic field strength.

Figure 9 .
Figure 9. Altitudinal profiles of the standard deviation of the simulated magnetic field.The profiles in each panel of Figure 8 are used to calculate the standard deviation for different oscillation periods (T) at the upper boundary.