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Solar Wind—Magnetosphere Coupling During Radial Interplanetary Magnetic Field Conditions: Simultaneous Multi‐Point Observations

In‐situ spacecraft missions are powerful assets to study processes that occur in space plasmas. One of their main limitations, however, is extrapolating such local measurements to the global scales of the system. To overcome this problem at least partially, multi‐point measurements can be used. There are several multi‐spacecraft missions currently operating in the Earth's magnetosphere, and the simultaneous use of the data collected by them provides new insights into the large‐scale properties and evolution of magnetospheric plasma processes. In this work, we focus on studying the Earth's magnetopause (MP) using a conjunction between the Magnetospheric Multiscale and Cluster fleets, when both missions skimmed the MP for several hours at distant locations during radial interplanetary magnetic field (IMF) conditions. The observed MP positions as a function of the evolving solar wind conditions are compared to model predictions of the MP. We observe an inflation of the magnetosphere (∼0.7 RE), consistent with magnetosheath pressure decrease during radial IMF conditions, which is less pronounced on the flank ( < 0.2 RE). There is observational evidence of magnetic reconnection in the subsolar region for the whole encounter, and in the dusk flank for the last portion of the encounter, suggesting that reconnection was extending more than 15 RE. However, reconnection jets were not always observed, suggesting that reconnection was patchy, intermittent or both. Shear flows reduce the reconnection rate up to ∼30% in the dusk flank according to predictions, and the plasma β enhancement in the magnetosheath during radial IMF favors reconnection suppression by the diamagnetic drift.

where r is the radial distance to the Earth's center, and θ is the solar zenith angle. α and r 0 are found empirically as a function of IMF B z and solar wind dynamic pressure. The predictions of this model are similar to the predictions by Petrinec and Russell (1996) (PR96), another widely used axisymmetric model. Case and Wild (2013) estimated, using more than 2,700 crossings of the Cluster spacecraft (polar orbit), spanning more than 8 years, that on average these two models tend to overestimate the radial distance between the MP and the Earth center by ∼1 R E (9%).
Since the S98 and PR96 models are axisymmetric, they cannot account for cusp indentations, and are expected to produce deviations at high latitudes. The model reported by Lin et al. (2010) (L2010) is another empirical model, where the asymmetry of the MP and the effect of the dipole tilt are considered. As additional inputs, it uses the IMF magnetic pressure (P m ) and the dipole tilt (Φ). They employed 2,708 MP crossings from multiple spacecraft to build their model, which uses 21 free parameters. Case and Wild (2013) estimated, using the same database mentioned above, that the radial MP distance was underestimated, on average, by ∼0.25 R E (2.3%). Other non-axisymmetric models present in the literature are, for instance, Boardsen et al. (2000) and Wang et al. (2013). Samsonov et al. (2016) performed an exhaustive model comparison, including eight empirical models and seven MHD models. They concluded that empirical models yield differences in radial distance of the order of 1 R E between themselves. Depending on the solar wind upstream conditions, different models may provide better predictions than others, whose accuracy also depends on the MP latitude. For instance, the L10 model provides the best predictions for the case B z = 0, and these predictions are very close to MHD models. They also noted that none of the models is designed to account for radial IMF conditions, when the MP location drifts toward the Sun (D. Fairfield et al., 1990;Merka et al., 2003).
Radial IMF conditions (IMF cone angles <25 • or >155 • ), represent ∼15% of observations at 1 AU (Pi et al., 2014;Suvorova et al., 2010), although they have received much less attention than northward and southward IMF conditions. For radial IMF, a quasi-parallel bow shock in the subsolar region is formed, resulting in lower magnetic pressure exerted on the magnetosphere. In addition, the dynamic pressure of the solar wind is usually small for radial IMF (P d < 1.5 nPa) (e.g., Park et al., 2016), plus the magnetosheath dynamic pressure becomes even smaller than in the solar wind, partly due the increase of reflected ions in the quasi-parallel bow shock. Therefore, the total pressure that the magnetosphere experiences is much smaller than for IMF cone angles close to 90°, and as a result the MP expands toward the Sun. Merka et al. (2003), based on a two-point MP observation event, proposed a bullet-shaped expansion of the magnetosphere for radial IMF, featuring an expansion toward the Sun in the subsolar region and thinning in the flanks. By contrast, Dusik et al. (2010) proposed a global expansion of the magnetosphere during radial IMF, featuring an inflation both in the subsolar region and in the flanks, based on statistical observations (∼6,500 MP crossings from THEMIS) during radial IMF. Dusik et al. (2010) also reported that the PR96 empirical model tends to underestimate the radial position of the MP, in particular when the IMF has a large radial component, from ∼0.3 R E for cone angle of 90° to ∼1.7 R E for cone angle close to 0° or 180°. They attributed it to a decrease in the effective dynamic pressure exerted at the boundary. Samsonov et al. (2012) studied the effective total pressure reduction over the MP using MHD simulations and THEMIS observations. They concluded that the total pressure exerted near the subsolar MP is reduced by ∼24% when the IMF cone angle is close to 0° or 180°. Suvorova and Dmitriev (2015) compared various MP models and concluded that for low solar wind dynamic pressure conditions (P d < 0.3 nPa), typical of radial IMF conditions, L2010 model performed better than S98 and PR96 models, although none of these models could account for the magnetosheath P d reduction with respect to P d in the solar wind for radial IMF.

10.1029/2021JA029506
3 of 17 The coupling between the Earth's magnetosphere and the solar wind is largely controlled by magnetic reconnection, which is most efficient during southward IMF conditions, i.e., the magnetic flux density reconnected per unit time maximizes. The amount of energy transferred to the Earth's magnetosphere system depends on the efficiency of this coupling, which is governed by both the reconnection rate and the extent of the X line. Cassak and Shay (2007) found scaling relations of the reconnection rate for asymmetric reconnection, which have been tested both using numerical simulations and statistical observations. The denser magnetosheath dominates the hybrid Alfvén velocity and controls, to a large extent, the reconnection rate (e.g., Borovsky, 2008;Borovsky et al., 2013;Lavraud & Borovsky, 2008;Fuselier et al., 2017). In the presence of cold ions of ionospheric origin, the outer dayside magnetosphere sometimes can have densities similar to magnetosheath densities, which also impact the reconnection rate (Borovsky & Denton, 2006;Dargent et al., 2020;Fuselier, Mukherjee, et al., 2019;Fuselier et al., 2021;Walsh et al., 2013).
The extent of the X line at the MP has been constrained using spacecraft conjunctions by a number of studies, most of them during southward IMF conditions. There have been various studies that made use of simultaneous multi-point observations during southward IMF, and have reported extended X line lengths at the MP, with measured minimum lengths ranging from two to nine Earth radii (R E ), and potentially extending longer distances (Berchem et al., 2008;Dunlop et al., 2011;Fear et al., 2009;Kitamura et al., 2016;Marchaudon et al., 2005;Phan et al., 2000). Similarly, Phan et al. (2006) reported an X line extending at least 8 R E during B y IMF. On the other hand, Walsh et al. (2017) used simultaneous (less than 1 min) observations of the MP on two THEMIS spacecraft separated by 3.9 Earth radii in the Y GSM direction. They found signatures of reconnection (jets) only in one of the spacecraft, challenging the model of an extended X line as predicted by MHD global simulations. The situation they found is consistent with either spatially patchy reconnection or a spatially limited X line. Reconnection switching on and off in time is not consistent with their observations owing to the simultaneity of the measurements. The IMF was southward but the cone angle for this event was ∼50°.
Although what controls the extent of the X line at the MP is not fully understood, there are two mechanisms that are expected to suppress magnetic reconnection locally: shear flows and diamagnetic drifts along the reconnection jet direction. Cowley and Owen (1989) indicated that magnetic reconnection should be suppressed if the flow shear velocity parallel to the jet direction exceeds twice the Alfvén speed of the magnetosheath. La Belle-Hamer et al. (1995) suggested that twice the largest Alfvén speed (magnetosphere or magnetosheath) would be the critical speed for determining reconnecting suppression by shear flows. For symmetric reconnection, Cassak and Otto (2011) found that if the shear flow exceeds the Alfvén speed, reconnection is suppressed. Their simulations provided a scaling law for the reconnection rate where E and E 0 correspond to the reconnecting electric field with and without correction for the shear flow reduction, v s is the shear flow speed in the outflow direction, and v A is the Alfvén speed.
More recently, Doss et al. (2015) extended the formulation in Equation 2 to the case of asymmetric magnetic reconnection. They showed, using two-fluid simulations, that asymmetric reconnection may be more difficult to suppress by shear flows when the asymmetry is large, as it is the case at the MP: where E asym and E 0,asym correspond to the resulting reconnecting electric field with and without correction for the shear flow in asymmetric reconnection, v A,asym is the hybrid Alfvén speed (Cassak & Shay, 2007), ρ is the mass density, B is the magnetic field strength, and subscripts 1 and 2 stand for each region adjacent to the reconnecting current sheet. This prediction has been shown to hold in particle-in-cell simulations (CDoss et al., 2016). Equation 3 may have implications for our current understanding on how planetary magnetospheres interact with the solar wind. For instance in Saturn, shear flow suppression has been considered a major suppression mechanism by e.g., Desroche et al. (2013). However, Sawyer et al. (2019) did not find evidence of reconnection suppression by shear flows at Saturn.
Another mechanism that is known to be able to suppress magnetic reconnection is the diamagnetic drift of the reconnection X line (along the outflow direction) due to pressure gradients across the current sheet. The condition for reconnection suppression is that the diamagnetic drift speed exceeds the Alfvén velocity (Swisdak et al., 2003(Swisdak et al., , 2010. This suppression condition is often expressed as where Δβ is the change in plasma β across the current sheet, L is the current sheet width, d i is the ion skin depth and θ is the magnetic field shear angle across the current sheet at the reconnection site. Vernisse et al. (2020) noted that, strictly speaking, Δβ should be calculated using only the normal to the current sheet component of the pressure tensor (P nn in LMN coordinates) and the guide field component of the magnetic field (B M in LMN coordinates), although typically the total plasma β is considered. Studying this suppression mechanism is important because it has implications for how long the reconnection X line can be at the MP, for instance. It also indicates which magnetosheath conditions are less favorable for reconnection to take place. Tests of reconnection suppression by the diamagnetic drift at the MP of Earth (Phan et al., 2013) and Saturn (S. Fuselier et al., 2020) have been largely successful. Equation 4 indicates that this suppression mechanism is at work mainly for large guide field configurations or large β asymmetries in the plasma inflow.
In this work, we make use of quasi simultaneous observations of the MP near the subsolar region and the dusk flank. We first compare the actual location of the MP to various models and study its inflation and shape due to the reduced effective pressure exerted by the magnetosheath during radial IMF conditions. Using simultaneous crossings at distant regions allows us to observe the relative inflation in the two distant regions without the need of assumptions on the magnetosheath time evolution. Then, we study the occurrence of reconnection at each region and discuss our findings. This manuscript is organized as follows. In Section 2, we describe the magnetospheric multiscale (MMS) and Cluster orbits during the MP conjunction, its configuration and the main plasma properties during the event. In Section 3, we compare our observations to two model predictions of the MP location simultaneously in the flank and the subsolar region. In Section 4, we assess the occurrence of magnetic reconnection based on observations and compare it to the predictions of the reconnection suppression mechanisms. Finally, in Section 5, we discuss and summarize the main findings of this study.

Description of the MMS-Cluster MP Conjunctions on November 28, 2016
The Cluster mission (Escoubet et al., 2001) was launched in 2001 into an elliptical polar orbit with the aim of surveying multiple magnetospheric regions. It is composed of four identical spacecraft that have been flying in multiple configurations, e.g., tetrahedron or string of pearls, at different length-scales, from few km (electron scale) to several thousand km (MHD scale). In this work, we use measurements from the Flux-Gate magnetometer (Balogh et al., 1997), and Cluster Ion Spectrometry-Composition Distribution Function (CIS-CODIF) (Reme et al., 2001).
The MMS mission (Burch et al., 2015) was launched in 2015 with the aim of studying magnetic reconnection at the Earth's MP and magnetotail, with a focus on the associated kinetic-scale processes. It is a suite of four identical spacecraft flying in tetrahedron formation, to distinguish time from spatial variations. Each spacecraft has several instruments to measure plasma parameters. In this work, we use the flux gate magnetometers (Russell et al., 2014) and Fast Plasma Instrument (FPI) to measure electrons and ions (Pollock et al., 2016).
On November 28, 2016, both the Cluster and MMS fleets were skimming the MP simultaneously for several hours. Cluster was in the dusk flank near the terminator and MMS was near the subsolar region, at roughly (0, 15, 0) and (8, 5, 1) Earth radii (R E ) in GSE coordinates, respectively. The Cluster and MMS position in the interval 09:00-18:00 UT is shown in Figures 1a-1c, in the GSE XZ, XY, and YZ planes, respectively. C1 and C2 were at 0.5 R E of separation and C3 and C4 at 0.4 R E of separation, and the distance between the two groups was of ∼1.1 R E . On the other hand, all four MMS spacecraft were in close (∼10 km) tetrahedron formation. For the rest of this work, all MMS measurements are taken from MMS1 and are representative of the other MMS spacecraft observations. During the MMS-Cluster conjunction studied here, the solar wind speed was roughly 400 km/s (not shown), and the IMF was dominated by GSE X component (B IMF ≃ B x , Figures 1d and 1e). The solar wind conditions remained roughly stable between 09:00 -14:00 UT. After that time, there is a B field rotation in Y and the dynamic pressure started increasing, from ∼1.5 nPa at 14:00 UT to more than 3 nPa at 18:00UT (Figure 1f), and the IMF cone angle (θ CA ) started fluctuating.
The next two panels show an overview of the observations made by MMS. Figure 1g shows MMS measured magnetic field in GSE coordinates. When MMS is in the magnetosphere, near the subsolar region, B is dominated by B z ≃ 40 nT. Figure 1h shows the FPI ion omnidirectional spectrogram observed by MMS. The magnetosphere regions show high-energy ions at several keV, corresponding to the dayside plasma sheet population. A cold ion component of ionospheric origin is also detected by FPI most of the time in the magnetosphere, at few tens of eV (visible between 14:00 -14:30 UT in Figure 1h). In the magnetosheath, the ion energies are of the order of several hundred eV to few keV. Figure 1i shows B field measurements in the dusk flank from C4 during the same time interval. B z is positive at times when Cluster is in the magnetosphere, and B m ≃ 30 nT, where subscript m stands for magnetosphere. Figure 1j shows the CODIF H + omnidirectional spectrogram measured by C4. It corresponds to the unique ion measurement available on the cluster fleet during the conjunction. The magnetospheric plasma sheet ion population, with energies above 10 keV, shows similar density and temperature in the flank (Cluster) and in the subsolar region (MMS). The magnetosheath ion population, on the other hand, shows lower density in the flank (not shown). Vertical black lines correspond to the times when a conjunction between any of the Cluster and MMS spacecraft was identified. We define the conjunctions when both the MMS fleet and at least one of the Cluster spacecraft cross the MP current sheet within an interval of less than 5 min. Using this criterion, we identify 15 conjunctions that are summarized in Table 1, corresponding to red numbers and vertical black lines in Figure 1e. Some of the conjunctions correspond to full crossings and some to partial crossings. Some of them are clean, single crossings, but others may correspond to multiple crossings within a short (less than 5 min) time interval.

Location and Shape of the MP
The observations of the MP reported in Table 1 allow us to test current models of the MP simultaneously at distant locations. We focus on two empirical models: S98 (Shue et al., 1998) and L10 (Lin et al., 2010). These models do not depend on IMF cone angle, and to account for the effect of the extended radial IMF observed during the conjunction, we use the effective magnetosheath pressure reduction reported by Samsonov et al. (2012), scaled linearly as a function of the IMF cone angle (θ CA ): where θ CA varies between 0 -π/2. This reduction in pressure was estimated using observations near the subsolar MP, with most data points fulfilling the condition √ 2 + 2 < 10 . In the following, we compare the two MP models with and without applying this correction (subscript c and no subscript, respectively), to test these results simultaneously both in the subsolar region and in the flank. Table 2 shows the upstream solar wind conditions from the OMNI database, i.e., propagated to the bow shock (P d , B z , B x /B) and the value of the dipole tilt (Φ) for the 15 crossings reported in Table 1. The magnetic pressure P m is negligible (i.e., lower than 0.1 nPa) for all the events. OMNI makes use of the Advanced Composition Explorer (ACE) magnetometer (Smith et al., 1998) and ion detector (McComas et al., 1998) for inferring the solar wind parameters mentioned above. Based on the noise figures provided for the magnetometer, the uncertainty in the magnetic field-derived quantities (P m , B z , B x /B) is less than 1%. For the dynamic pressure (P d ), we consider it is comparable to the energy resolution of the ion instrument, that  Table 1.    is 5%. These uncertainties in the solar wind parameters correspond to roughly ±0.1 R E uncertainty in the model predictions of the MP position, both in the subsolar and flank regions (not shown). If we consider 10% uncertainty in the P d measurement, the associated uncertainty of the models becomes roughly ±0.2 R E . Using these input values, we computed the MP location for S98 and L10 models, with and without the correction defined in Equation 5 (subscript c for corrected pressure) suggested by Samsonov et al. (2012). Table 2 also shows the distance of MMS constellation and C4 to the MP models. A negative sign corresponds to r model < r sc . The distance between the observed location of the MP and the location predicted by each model are summarized in Figure 2. The mean distance over the 15 simultaneous crossings is plotted using circles, and the error bars correspond to ±0.2 R E , i.e., 10% uncertainty in P d . At the flank MP, both S98 and L10 match the measured MP position within the error bars. In the subsolar region, the models S98 and L10 underestimate the MP position by ∼0.8 R E and ∼0.6 R E on average, respectively, the corrected model S98 c underestimates the MP position by ∼0.4 R E and and L10 c provides a correct estimate within the error bars. Therefore, the corrections for radial IMF yield better results in the subsolar region, with the model L10 c as the most accurate one. On the other hand, the corrections for radial IMF in the flank overestimate the measured MP position by ∼0.4 R E on average, while the models without correction provide accurate MP positions within the error bars. The corrections in the pressure exerted during radial IMF were obtained using spacecraft data near the subsolar region, and we confirm the validity of the calculations by reported by Samsonov et al. (2012) in that region. In addition, we note that these corrections cannot be extrapolated to the flanks. Figure 3a shows the MMS (red) and C4 (blue) orbits during the 9-hr interval. Red and blue dots correspond to each of the 15 MP crossings of Tables 1 and 2 for MMS and C4, respectively. The black and green curves correspond to the S98 and L10 MP models corresponding to the solar wind conditions at the beginning of the time interval in Figure 1. Figures 3b-3g show details of crossings 2, 13 and 15 and the MP models for the solar wind conditions at the time of each event, for MMS (red) and C4 (blue).

Magnetic Reconnection at the Subsolar and Dusk Flank MP
Next, we take the events of Table 1 that have full MP crossings for both MMS and C4 (i.e., events 3, 5, 6, 8, 9, 10, 11, and 15) and apply minimum variance analysis (MVA) to the magnetic field. The N direction obtained in the subsolar region and in the flank is roughly consistent with the MP model predictions, except    for event 9, when the obtained N-direction at the subsolar region (MMS) is roughly in the Z GSE direction. This can be either to a poor performance of MVA (intermediate to minimum eigenvalue ratio is ∼3 for this case) or reconnection 3D effects. We find it difficult to interpret this event and it is discarded. For each of the non-discarded events, we search for observational evidence of ongoing reconnection based on two criteria: presence of reconnection jets in the L direction and the existence of electron only Low Latitude Boundary layer (eLLBL) earthward of the MP (Gosling et al., 1990). We also estimate and compare the conditions on both sides of the MP (magnetosphere, sp, and magnetosheath, sh) simultaneously in the subsolar region (MMS) and at the dusk flank (C4), which allow us to test the theoretical conditions for reconnection suppression discussed in the introduction (Equations 3 and 4).  (Gosling et al., 1990;Vines et al., 2017). The eLLBL in Figure 4j cannot be distinguished at the time scale of the plot, please refer to Figure S7 in Supporting Information S1 of the supplemental material for a detailed observation of the eLLBL, which is not obvious for this event. The low energy electrons observed in Figure 4j after 17:49 UT are associated to cold ions of ionospheric origin ( Figure 4j) and do not correspond to the eLLBL (e.g., Toledo-Redondo et al., 2021). We also search for jets in ion velocity (black lines in Figures 4c and 4h) of the order of the Alfvén velocity (listed in Table 3), which would indicate ongoing reconnection. For event 15, the data is not conclusive. Two possible narrow reconnection jets are observed at ∼17:48:45UT (Figure 4c, cluster) and ∼17:46:36 UT (Figure 4h, MMS), although their peak velocity in the L direction is less than 50% of the predicted Alfvén velocity. The blue-shaded regions correspond to the reference time interval (15 s) for inferring magnetosheath quantities, and the red-shaded regions correspond to the reference time interval (15 s) for inferring magnetospheric quantities. Ion velocities estimated by CIS-CODIF on C4 are not reliable in the magnetosphere due to the low counts, so they have been masked in panel c. We assume that velocity in the flank magnetosphere is negligible compared to flank magnetosheath velocity.
The same analysis explained in Figure 4 for event 15 has been applied to events 3, 5, 6, 8, 10 and 11, and their corresponding Figures are provided in Figures S1-S6 in Supporting Information S1. The reference magnetosheath and magnetosphere intervals adjacent to the MP crossings allow us to test the theoretical predictions of reconnection suppression by shear flows and the diamagnetic drift. The main parameters (Land N-direction, magnetic field and density, hybrid Alfvén velocity, shear flow velocity, Δβ and B clock angle) are provided in Table S1 in Supporting Information S1. Table 3 summarizes the results of the expected reduction in reconnection rate due to shear flows, ( ∕ 0) , whether reconnection is expected to be suppressed by the diamagnetic drift of the X line, and the observed reconnection signatures (jets and eLLBL).

Observational Evidence of Reconnection
The eLLBL is observed in all MMS crossings, although for event 15 the identification is ambiguous (see Figure S7 in Supporting Information S1), indicating that reconnection was taking place in the dayside region during the encounter. The eLLBL is also observed by C4 in the flank toward the end of the encounter, for events 11 and 15. This suggests that reconnection may be at work in the flank during the late hours of the encounter.
In addition, events 6, 10, and 11 show clear reconnection jets (v jet > 5v A ) in the subsolar region (see Figures  S3, S5, and S6 in Supporting Information S1), and possibly events 5, 8, and 15 (v jet < 0.5v A ) (See Figure S2 and S4 in Supporting Information S1 and Figure 4). The direction of five of the jets (southward) is consistent with the expected location of the X line according to the maximum magnetic shear model (Trattner  , 2007). Overall, the combination of eLLBL and jet identification suggests that reconnection was at work near the subsolar region during the whole encounter, while in the flank reconnection was at work after ∼13 UT. Clear jet signatures are not identified for all subsolar crossings, but this may be due to various reasons, including intermittent occurrence of reconnection, or the X line being close to the spacecraft position, as for the electron diffusion region event observed by MMS the same day at ∼07 UT (Genestreti et al., 2018).

Suppression of Magnetic Reconnection by Shear Flows
In the subsolar region (MMS observations), the L direction corresponds roughly to GSE Z for all the crossings, while the N direction is a combination of GSE X and GSE Y. On the other hand, the L direction is not stable in the dusk flank (C4 observations), with L changing between GSE -X and GSE Z. The N direction in the dusk flank is roughly in GSE Y and GSE X. Table 3 indicates that in the subsolar region, the observed shear flows in the L direction are smaller than the hybrid Alfvén velocity, resulting in negligible (less than 2%) expected reconnection rate reduction (E/E 0 ) asym , according to Equation 3 (Doss et al., 2015). On the other hand, the shear flow velocity in the L direction is of the same order or larger than the hybrid Alfvén velocity in the dusk flank for all the events in Table 3, resulting in variable expected reconnection reductions, 0.71 (E/E 0 ) asym < 0.98.  Table 1). The yellow-shaded regions mark the time interval used to apply minimum variance analysis to the current sheet crossing and obtain the LMN coordinate system for each spacecraft. Blue-shaded and redshaded regions mark the intervals used as reference for the asymptotic conditions of the magnetosheath and the magnetosphere, respectively. Note. See Table S1 in Supporting Infromation S1 for additional information of the computed values. a Hybrid Alfvén velocity (Cassak & Shay, 2007). b Shear flow speed parallel to the outflow (L) direction. c Expected reduction in reconnection rate due to shear flows, see Equation 3. d Diamagnetic drift of the X line, see Equation 4.

Suppression of Magnetic Reconnection by Diamagnetic Drift
We test the Swisdak condition (Equation 4) at each MP crossing from Table 3 Table S1 and Figures S1-S6 in Supporting Information S1). Their associated error bars correspond to the instrument uncertainties in measuring the plasma pressure, approximated by the channel energy widths of CODIF and FPI (16% and 15%, respectively). The uncertainty associated to the magnetic pressure measurement is less than 1%. The B rotation angle is taken in the plane perpendicular to the MP normal, i.e., the plane that contains L and M directions, computed using MVA on magnetic field data. The associated error in estimating the B rotation angle due to the magnetometers uncertainty is less than 1°. Uncertainty associated to the LMN coordinate system estimation can result in larger errors for the B rotation angle, but these are difficult to quantify.
We find that reconnection is expected to be suppressed for 6 of the crossings: 5, 6, 8 and 11 in the flank, and 3 and 5 in the subsolar region. In the flank, we did not identify reconnection signatures for crossings 5, 6, 8, therefore the theory is consistent with our observations for these crossings. For event 11 (blue square in Figure 5), however, we did observe the eLLBL plus a possible jet (v < 0.5v A ). In the subsolar region, event 3 (red circle in Figure 5) has eLLBL associated to it, and event 5 (red inverted triangle in Figure 5) has both eLLBL and a possible jet (v < 0.5v A ). We attribute this discrepancy with the theory to the fact that the Swisdak test is applied using plasma conditions at the spacecraft crossing, not at the X line location, which is unknown. These results suggest that the reconnection site was not close to Cluster for during event 11 and not close to MMS during events 3 and 5. The plasma β in the subsolar magnetosheath are most of the time well above 1 (red points in Figure 5). For reconnection to take place with such large magnetosheath plasma β, moderate to large B rotation angles are required (roughly larger than 90° for Δβ = 2, see Figure 5). The clock angles and the Δβ are in general smaller in the flank (Cluster observations, blue) than in the subsolar magnetosphere (MMS observations, red).  Park et al. (2016) analyzed 19 years of magnetospheric magnetic field data at geosynchronous orbit and cross-correlated it with magnetic field data of the solar wind at 1 AU. They found that for radial IMF conditions, the magnetospheric magnetic field was systematically smaller than for northward IMF conditions, over all magnetic local times and regardless of season or magnetic latitude. This result is consistent with the model of global expansion of the magnetosphere during radial IMF (Dusik et al., 2010). Our results in Figure 2 are consistent with an expansion of the order of 0.6-0.8 R E near the subsolar region. At the flanks, we do not observe a deviation of the MP position predicted by the models S98 and L10, assuming an uncertainty of ±0.2 R E .

Discussion and Conclusion
The persistent observation of the eLLBL in MMS data indicates that reconnection was at work in the subsolar region. This result is supported by the identification of reconnection jets in events 6, 10 and 11, and possibly in events 5, 8, and 15. By contrast, no jet signatures are present for event 3. The variability of jet observations has two possible explanations: MMS was close to the X line during some of the events, as for the event reported by Genestreti et al. (2018) few hours before, or reconnection was intermittent in time.
Evidence for reconnection in the dusk flank is also present for events 11 and 15. This is consistent both with an X line extending from the MMS to the C4 location, i.e., more than 15 R E , or with patchy reconnection involving multiple X lines. On the other hand, reconnection seems not to be at work in the flank MP near the C4 location for events 3, 5, 6, and 8.
While the L direction in the subsolar region is roughly in the GSE Z direction, in the flank is often oriented in the GSE X direction, i. e.., the direction of the magnetosheath flow. The predicted reconnection rate reductions due to shear flows in the flank are E/E 0 = 2% -29% depending on the event, while the reconnection rate reduction is negligible (i.e., E/E 0 < 2%) in the subsolar region. We note, however, that these calculations consider magnetosphere and magnetosheath references at the spacecraft location, while the conditions at the X line may be different, in particular the L direction.
During radial IMF conditions, the magnetosheath dynamic pressure becomes low, and the magnetic pressure that the magnetosheath exerts on the MP becomes even lower, resulting in an enhanced magnetosheath plasma β (e.g., Le & Russell, 1994;Suvorova et al., 2010;Suvorova & Dmitriev, 2016). The dynamic pressure in the magnetosheath is lower than in the solar wind during radial IMF owing to the quasi-parallel bow shock that is formed in the subsolar region and to the shorter size of the magnetosheath. The resulting magnetosheath β enhancement favors suppression of magnetic reconnection by the diamagnetic drift, as illustrated in Figure 5. However, these results are evaluated at the spacecraft location, not at the X line. In addition, accurate evaluation of Equation 4 requires reliable LMN coordinates. While the L direction determination is robust for our events, the N direction was less robust. The eigenvalue ratio of the intermediate to minimum direction (l 2 /l 3 ) resulting from MVA was small (∼3) for some of the events, which can result in errors in determining the effective B rotation angle across the MP. The magnetosheath magnetic field orientation and strength is variable during the encounter, as expected behind a quasi-parallel bow shock. Overall, radial IMF conditions may favor time-varying conditions at the MP, which may result in intermittent and spatial and time varying magnetic reconnection. More analysis of radial IMF events is needed to confirm these results.
To summarize, we analyzed an equatorial MP conjunction between MMS (subsolar region) and Cluster (dusk flank) during radial IMF conditions, enabling us to study the meso-scale of the MP using simultaneous in-situ measurements. Our results indicate that the magnetosphere inflates under radial IMF in the subsolar region (∼0.7 R E ), while changes in the flank are 0.2 R E , suggesting a MP deformation in addition to the inflation. Magnetic reconnection was at work in the subsolar region for the whole encounter based on the observed eLLBL, although reconnection jets were not always clearly identified. In the flank, reconnection was at work for the last hours of the encounter, suggesting that the extent of the X line could be larger than 15 R E . However, the magnetosheath B is variable during radial IMF, and this may lead to patchy and non-steady magnetic reconnection at the MP.

Data Availability Statement
The omni database is publicly available at https://omniweb.gsfc.nasa.gov/. The MMS database is publicly available at https://lasp.colorado.edu/mms/sdc/public/. The Cluster database is publicly available at https://csa.esac.esa.int/.