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Shape of the equatorial magnetopause affected by the radial interplanetary magnetic field
Planetary and Space Science 148 (2017) 28–34
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Shape of the equatorial magnetopause affected by the radial interplanetary magnetic field ankova, Z. Nemecek *, G. Pi, L. Prech, J. Urb K. Grygorov, J. Safr ar Charles University, Faculty of Mathematics and Physics, V Holesovickach 2, 180 00 Prague, Czech Republic
A R T I C L E I N F O
A B S T R A C T
Keywords: Magnetopause location Radial interplanetary magnetic field Magnetopause deformation
The ability of a prediction of the magnetopause location under various upstream conditions can be considered as a test of our understanding of the solar wind-magnetosphere interaction. The present magnetopause models are parametrized with the solar wind dynamic pressure and usually with the north-south interplanetary magnetic field (IMF) component. However, several studies pointed out an importance of the radial IMF component, but results of these studies are controversial up to now. The present study compares magnetopause observations by five THEMIS spacecraft during long lasting intervals of the radial IMF with two empirical magnetopause models. A comparison reveals that the magnetopause location is highly variable and that the average difference between the observed and predicted positions is þ0:7 RE under this condition. The difference does not depend on the local times and other parameters, like the upstream pressure, IMF north-south component, or tilt angle of the Earth dipole. We conclude that our results strongly support the suggestion on a global expansion of the equatorial magnetopause during intervals of the radial IMF.
1. Introduction The magnetopause is an obstacle varying in a size and shape in a response to changes of the solar wind and interplanetary magnetic field (IMF). Its location is determined by a pressure balance between the total pressure in the magnetosheath and the pressure in the magnetosphere. The magnetospheric pressure is usually dominated by the magnetic pressure but a situation is more complicated in the magnetosheath. On the other hand, the magnetosheath pressure is closely related to the solar wind dynamic pressure and this relation is a base of all empirical magnetopause models developed in the course of past decades. The first model of Fairfield (1971) considered only a balance between the upstream dynamic pressure and dipole magnetic field of the Earth but its successors (e.g., Sibeck, 1990; Petrinec and Russell, 1996; Shue et al., 1997, 1998) introduced an influence of IMF via a dependence of the magnetopause location on the magnitude and polarity of the IMF BZ component. These models use different surfaces to describe the magnetopause shape, and parametrize this surface with the upstream pressure and IMF BZ . This approach is acceptable for the equatorial magnetopause ankov but Safr a et al. (2005) have shown significant magnetopause indentations caused by the magnetic field depression in the cusp regions. This fact was later reflected in the Lin et al. (2010) model.
The overall performance of a majority of magnetopause models is ankova et al., 2002), the statistical uncertainty of a comparable (Safr prediction of the magnetopause location is ±1 RE at the subsolar region and it increases toward flanks. The prediction uncertainties are usually attributed to errors in a determination of upstream conditions or to magnetopause surface deformation caused by surface waves (e.g., Grygorov et al., 2016, and references therein), and by a transient phenomena ankova et al., like IMF discontinuities (e.g., Tkachenko et al., 2011; Safr 2012) or interplanetary shocks (Nemecek et al., 2011). On the other hand, when IMF is nearly aligned with the solar wind velocity, a whole dayside magnetopause is behind the quasiparallel bow shock and foreshock pressure fluctuations can penetrate through the subsolar bow shock into the magnetosheath and influence the magnetopause location as reported by Fairfield et al. (1990). The authors also noted a transient magnetopause motion and its significant inflation. Merka et al. (2003) reported large amplitudes of magnetopause oscillations during intervals of the radial IMF based on a limited statistics and MHD modeling. The authors suggested that the radial IMF orientation would lead to a change of the shape of the whole magnetopause surface. A subsolar part of the magnetopause would move outward, whereas the flanks would be compressed and the magnetopause would attain a bullet-like shape.
* Corresponding author. E-mail address: [email protected] (Z. Nemecek). https://doi.org/10.1016/j.pss.2017.09.011 Received 12 June 2017; Received in revised form 18 September 2017; Accepted 26 September 2017 Available online 28 September 2017 0032-0633/© 2017 Elsevier Ltd. All rights reserved.
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The case studies of Suvorova et al. (2010) and Jelínek et al. (2010) based on THEMIS observations brought a new evidence that the magnetopause lies significantly farther from the Earth than it is expected from models. Suvorova et al. (2010) suggested a global inflation of the magnetopause at dayside and magnetotail locations. These findings are consistent with a large statistical study by Dusík et al. (2010) that demonstrated a systematic expansion of the magnetopause location by 1:7 RE for cone angles (the angle between the IMF and solar wind velocity vectors) close to 0o and 180o . Samsonov et al. (2012) analyzed distributions of different pressure components (dynamic, thermal, and magnetic pressures) in the magnetosheath along the Sun-Earth line for northward and radial IMF orientations. The authors argued that the temperature anisotropy in the magnetosheath controls the pressure distribution near the magnetopause and that the total pressure exerting on the subsolar magnetopause depends on the IMF orientation. For the radial IMF, they observed the total pressure significantly lower than the solar wind dynamic pressure and it would lead to a magnetopause expansion. However, MHD models do not properly describe kinetic effects that are connected with a presence of the foreshock in front of the dayside bow shock during radial IMF intervals. To improve a prediction of the magnetopause position by global MHD models, Samsonov et al. (2017) suggested the method of a modification of upstream parameters that reflects a decrease of the total pressure in the magnetosheath for the IMF orientation obtained in MHD simulations and THEMIS observations. A location of the mid-latitude magnetopause under a radial IMF was studied by Huang et al. (2015) using the Cluster data and both Shue et al. (1997) and Boardsen et al. (2000) models. The authors found a dawn-dusk asymmetry with the dawn magnetopause compressed but the dusk magnetopause expanded if the observations are compared with the Shue et al. (1997) model. However, a majority of magnetopause crossings used in this study was observed at magnetic latitudes exceeding 35o where the influence of the cusp is important. For this reason, a comparison with the Boardsen et al. (2000) model is more relevant and this comparison strongly depressed the observed magnetopause asymmetry. Above named studies do not confirm suggestions of Merka et al. (2003) on a bullet-like magnetopause shape and support a hypothesis on the global magnetopause expansion at all local times. In order to clarify this contradiction, Park et al. (2016) analyzed a correlation of changes of the magnetic field at the geostationary orbit under different IMF orientations. Their study brings a further evidence on a global expansion of the equatorial magnetopause during intervals of a radial IMF but the geostationary orbit is a rather far from the nominal magnetopause and the changes of magnetopause locations are only one of many possible sources of geostationary magnetic field variations, thus such proof is only indirect. Although radial IMF is usually treated as an exception, the analysis of Pi et al. (2014) revealed that the long intervals of continuous radial IMF (longer than 4 h) occupy 10–15% of the year with a slight preference for the solar minimum. Consequently, the understanding of the magnetopause behavior under this IMF orientation can significantly improve our ability of predictions of its location, thus the real magnetopause shape under the radial IMF is a key point. The mentioned statistical studies of the magnetopause shape under radial IMF suffer with the determination of the IMF cone angle. This angle is usually computed from observations of a distant solar wind monitor and propagated toward the bow shock. To demonstrate the errors of this method, Fig. 1 (top) shows a comparison of the cone angles computed from the propagated Wind IMF observations with the cone angles determined from THEMIS B and C measurements. The figure shows only data from intervals when THEMIS was farther than 5 RE from the model bow shock and on the IMF line that was either not connected to the bow shock or the angle between the IMF and bow shock normal was larger than 50o . This selection ensures that the THEMIS magnetic field was not affected by foreshock fluctuations. The cone angles computed from 5-minute IMF and velocity averages are shown by small dots; the large symbols stand for medians in 3o bins. The figure shows a large
Fig. 1. A comparison of cone angles computed from the propagated Wind IMF observations and the cone angles registered by THEMIS B and C. (top) For a whole set of THEMIS observations from 2007 to 2016; (bottom) the same plot for long intervals of a radial IMF orientation.
uncertainty of a determination of the IMF direction affecting the magnetopause from IMF observations at the L1 point. The uncertainty can be significantly reduced if only long-lasting intervals of a radial IMF are used (Fig. 1, bottom). This figure has the same format as Fig. 1 (top); the points were taken from intervals when the Wind IMF cone angle was lower than 30o for more than 4 h continuously (i.e., long radial IMF events in terms of Pi et al. (2014)). Since this IMF orientation leads to a strong foreshock in front of the whole dayside bow shock, a majority of THEMIS cone angles were computed from the measurements affected by the foreshock fluctuations. In spite of this fact, an overall quality of the cone angle prediction is better than that demonstrated in Fig. 1 (top). For this reason, the present study is based on the analysis of magnetopause crossings observed by five THEMIS spacecraft during prolonged intervals of the radial IMF. The second and even more important reason for usage of prolonged intervals of a nearly stable radial IMF orientation is the time that the magnetosphere needs for the reaction to changes of upstream conditions. This reaction can be as long as 15 minutes (Nemecek et al., 2011). As an example, we show in detail one case of simultaneously observed magnetopause crossings by three spacecraft largely separated in local times. Furthermore, using a larger set of similar observations (28 events with multiple magnetopause crossings), we provide a short statistical 29
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whereas S97 is parametrized with the upstream dynamic pressure and IMF BZ component only, the L10 model uses also the dipole tilt angle as an input parameter. Although THEMIS are orbiting close to the equatorial plane, the flank crossings could be influenced by the magnetopause indentation in the cusp region for extreme values of the dipole tilt angle and this indentation is missing in the S97 model. The S97 and L10 models differ also in computations of the upstream pressure. Whereas S97 uses the proton dynamic pressure, L10 also accounts for the alpha particle content and IMF pressure. The main tool of our analysis is a comparison of the distance of the observed crossing from the Earth center, ROBS with that obtained from the model, RMOD . RMOD was computed at the intersection of the model magnetopause with a line defined by the Earth center and a particular crossing. Note that we do not use a separation along the normal to the magnetopause surface but the difference between both methods is negligible in the subsolar region and only slightly larger at flanks. A distribution of basic parameters of identified magnetopause crossings is shown in Fig. 2. Fig. 2a presents that a number of crossings with large negative dipole tilt angles (southern cusp tilted toward the Sun) are rather small and thus an influence of the magnetopause indentation would be depressed. Since the dayside parts of the THEMIS orbits are in the southern hemisphere, the positive dipole tilt angle moves the cusp indentation away from the observed crossings. Fig. 2b demonstrates a typical feature of long intervals of the radial IMF—a low upstream pressure (Pi et al., 2014). The dashed vertical line in this panel shows the median value of upstream pressures for all magnetopause crossings identified in the THEMIS data in course of 2007–2016 (PSW ¼ 1:7 nPa), whereas the dotted line shows the median pressure (PSW ¼ 1:2 nPa) for our set. A small IMF BZ component (Fig. 2c) is a consequence of the conditions for the event selection but Fig. 2d reveals that although we selected the intervals with minimum excursions of the cone angle to the interval 30o 150o , a large portion of crossings was observed for these cone angles. We will return to this point later. The full set of crossings used in the study is introduced in Fig. 3. A part of orbits of the THEMIS spacecraft corresponding to intervals of the radial IMF are shown by color lines and the crossings are indicated by the crosses. The figure reveals that only THEMIS B and C (blue and red, respectively) are able to observe really distant crossings because the apogees of other three spacecraft are too low. Nevertheless, a great majority of crossings is observed out of the depicted magnetopause that was computed for the pressure of 1.3 nPa, IMF BZ ¼ 0 and zero tilt angle of the Earth dipole using the S97 (dotted line) and L10 (dashed line) models.
study in which we estimate the magnetopause shape for radial IMF and calculate average differences between the observed and predicted magnetopause positions. Our analysis is in a general agreement with previous studies and shows that the subsolar as well as flank magnetopause expands in the outward direction. 2. Data selection A statistical study of Dusík et al. (2010) used 5-minute averages of IMF components propagated from Wind for a computation of the cone angle (the angle between IMF vector and XGSM axis). Such approach suffers with aforementioned large uncertainties because the IMF direction is often modified by Alfven waves that propagate in the solar wind. To depress this effect, we have limited our study to long ( > 2 hours) intervals of the radial IMF orientation. However, the intervals of a pure radial IMF occur only 10% of the time, thus we allowed short (up to 15 min) excursions of the cone angle to larger values. We have chosen 15 min as a compromise between our intention to analyze only long radial IMF intervals and the number of such intervals with magnetopause crossings. A decrease of this value to 10 min leads to the decrease of the total duration of available intervals to about one half but, as we checked, it does not change the results of our analysis. Since our goal is to investigate a quasi-steady magnetopause shape, we have compared our preliminary list of intervals with the list of KelvinHelmholtz waves collected by Kavosi and Raeder (2015) and excluded the overlaps. Finally, we have identified 26 intervals with the cone angle lower than 30o (or greater than 150o ) that contain more than 350 magnetopause crossings observed by THEMIS in a course of 2008–2013. The aggregate duration of these intervals is about 83 h. The list of crossings was complemented with the solar wind dynamic pressure and IMF components. We used THEMIS B and C observations whenever they were located in the solar wind out of the foreshock and Wind for the rest of cases. All upstream data were lagged on a propagation time estimated by a two-step procedure based on the actual solar ankov wind velocity (Safr a et al., 2002) and we used 5 min averages centered around the estimated time. The coordinates of crossings were aberrated to account for the Earth's orbital motion. Magnetopause crossings were identified by a visual inspection of plots of the THEMIS magnetic field (Auster et al., 2008) and plasma parameters (McFadden et al., 2008) and a model prediction of the magnetopause location was computed for each crossing. Suvorova and Dmitriev (2015) concluded that the Lin et al. (2010) model is capable to predict the magnetopause position under a radial IMF, thus we applied two models: Shue et al. (1997) and Lin et al. (2010) (S97 and L10, hereafter). Both models are based on fits to the observed crossings but
Fig. 2. Histograms of basic parameters of magnetopause crossings included in this study: (a) the dipole tilt angle; (b) the solar wind dynamic pressure, PSW ; (c) IMF BZ ; and (d) the cone angle. In the panel (b), the dashed vertical line shows the median value of upstream pressures for all THEMIS magnetopause crossings through 2007–2016 and the dotted line stands for the median pressure in our set of the radial events. 30
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Fig. 3. Projections of observed magnetopause crossings by all THEMIS probes onto the XY (left) and XZ (right) planes in the GSM coordinate system. S97 (dotted) and L10 (dashed) models for BZ ¼ 0, PSW ¼ 1:2 nPa, and zero dipole tilt angle are given for reference. A part of THEMIS orbits corresponding to intervals of the radial IMF are shown by color lines and magnetopause crossings are indicated by the crosses in a corresponding color.
3. Case study
orientation simplifies an identification of the magnetopause crossings. The first two panels present distances of the registered crossings from L10 and S97, respectively. The crossings were observed in course of a whole depicted interval by THEMIS A, whereas B and C probes crossed the magnetopause several times but only within limited time periods. The reason can be seen in the left part of Fig. 5. THEMIS A with apogee of about 12 RE skimmed the dusk magnetopause but THEMIS B and C with apogees of 55 and 26 RE , respectively, passed the magnetopause region very quickly. Nevertheless, both these spacecraft crossed the dawn magnetopause at similar MLT and their observations are complementary. First two panels of Fig. 4 reveal several features typical for the magnetopause behavior under a radial IMF:
This section presents magnetopause crossings observed by three THEMIS spacecraft during a long interval (more than 6 h) of the radial IMF on August 31, 2009. The main reason for a selection of this event is that two spacecraft observed the magnetopause in very different local times simultaneously and it gives us a possibility to distinguish between the global expansion and deformation of the magnetopause surface. The basic observations are shown in Fig. 4. The colors denote the spacecraft (THEMIS A in orange, THEMIS B in red, THEMIS C in blue, and Wind in black); note that the Wind data were lagged on the estimated propagation time. The IMF BZ (penultimate panel) was very weak and mostly negative and it resulted in the negative BZ observed during magnetosheath intervals registered by THEMIS (third panel). This magnetic field
The L10 and S97 models provide very similar predictions.
Fig. 4. An example of a series of magnetopause crossings observed by THEMIS A (yellow), THEMIS B (red), and THEMIS C (blue) on August 31, 2009. From top to bottom: differences in observed and predicted of magnetopause crossing locations (in RE ) for the L10 model; for the S97 model; the BZ component registered by THEMIS A (orange), B (red), and C (blue); IMF cone angle; IMF BZ component; and solar wind dynamic pressure, PSW (last three panels show Wind data).
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Fig. 5. Projections of THEMIS probes (August 31, 2009) onto the XY (top) and XZ (bottom) coordinates at the same time interval as in Fig. 4. The dotted and dashed lines show the magnetopause shape according to S97 and Lin10, respectively, under BZ ¼ 0, PSW ¼ 1:2 nPa, and the actual dipole tilt angle of 18o . The colors mark a particular THEMIS probe: THA–orange; THB–red; and THC–blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
All crossings were observed at larger distances from the Earth than the models predict; the difference may reach 2.2 RE . A large dispersion of the points in first two panels of Fig. 4 demonstrates a large variability of the magnetopause location under this IMF orientation. No crossing was observed between 1520 and 1550 UT; Wind observations suggest the Parker spiral-like orientation from 1530 to 1615 UT. At about 1105 UT, THEMIS C observed the dawn magnetopause at about nominal position but THEMIS A crossed the dusk magnetopause about 1.7 RE farther from the Earth than both models predict. The opposite observations can be seen around 1510 UT when THEMIS B identified the dawn magnetopause at a larger distance from the Earth than THEMIS A at dusk did. These observations indicate that the magnetopause was expanded in all local times covered by THEMIS measurements under the radial IMF orientation. Moreover, a number of crossings as well as a comparison of THEMIS A with THEMIS B and C shows that the magnetopause location is highly variable and these variations are not caused by changes of the upstream pressure (last panel in Fig. 4). We cannot exclude that this variability is caused by foreshock structures (spontaneous hot flow anomalies, foreshock cavities, foreshock bubbles, etc.) but these structures are localized and there was no spacecraft monitoring the magnetosheath or foreshock upstream of THEMIS A. Fig. 6. ROBS RMOD projected onto the normalized magnetopause surface. The length of segments shows the difference between the observation and prediction of a particular magnetopause crossing and the segment color represents an actual value of PSW . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4. Discussion A case study presented in the previous section suggests a global magnetopause expansion due to the radial IMF orientation. The same conclusion follows from Fig. 6 where the magnetopause crossings observed during our all long radial IMF intervals are shown. In Fig. 6, the curved line stands for the S97 equatorial magnetopause surface. The lengths of line segments beginning at this surface correspond to a deviation of each particular crossing from the model and their colors mark the values of upstream pressures at the time of a particular crossing. The figure shows only several crossings that were observed inbound the nominal magnetopause; a great majority of crossings (317 from 350 for S97, and 241 from 350 for L10) indicates up to 2 RE magnetopause expansion that does not depend on the local time and upstream pressure. The IMF BZ component is the second parameter that controls the magnetopause shape and position. Since our set contains several crossings that were observed under non-negligible BZ , we present an analysis of a possible BZ influence for S97 in Fig. 7 (left). A value of the IMF BZ component is represented by the color scale; diamonds, stars and triangles stand for dawn, subsolar (9–15 MLT) and dusk crossings,
respectively. The dimensions of symbols provide a rough estimate of the upstream dynamic pressure. This plot is rather complex but it can be clearly seen that even the crossings observed under a relatively large negative IMF BZ (from 1:5 to 2:5 nT, dark blue) are located far upstream of the model magnetopause and the same is true for the crossings observed under a positive IMF BZ (red points). The crossings located significantly closer to the Earth than the model predicts were observed predominantly at flanks (triangles and diamonds) but the same is valid for the most distant crossings. It suggests that a location of the flank magnetopause is more variable, however, this fact is not surprising. The right panel in Fig. 7 shows the comparison of observations and L10 model predictions. The distribution of crossings for small (0o 30o ) cone angles is similar as that in the left panel but the both positive and negative extreme values of ROBS RMOD are large. The distribution of ROBS RMOD for large (150o 180o ) cone angles is again wider for L10 than for the S97 model but the distribution is centered around þ0:3 RE , 32
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Fig. 7. ROBS RMOD as a function of the IMF cone angle. The colors present values of IMF BZ ; different symbols (diamonds, stars, and triangles stand for dawn, subsolar (9–15 MLT), and dusk crossings, respectively) indicate the actual location of a particular spacecraft; and their size determines approximate values of PSW . (a) For the Shue et al. (1997) and (b) for Lin et al. (2010) models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
whereas that for S97 at about þ0:7 RE . Fig. 7 suggests that the magnetopause location is more variable for the low ( < 30o ) cone angles, i.e., for the IMF BX component pointing toward the Earth. Although the statistics is limited, we think that this result is reliable and that it is a consequence of a transformation of the radial IMF downstream of the bow shock. A structure of the magnetosheath magnetic field and its temporal evolution under a radial IMF was investigated by Pi et al. (2017). According to this study, negative IMF BX converts to negative BZ in the southern magnetosheath (see Fig. 6 in the mentioned paper). This magnetic field orientation leads to erosion of the magnetopause and THEMIS that is orbiting in the southern hemisphere would observe the magnetopause closer to the Earth than the model predicts. This effect would be enhanced if the original IMF would have a (small) negative BZ component and it is generally consistent with Fig. 7. However, a long-lasting combination of subsolar reconnection in the southern hemisphere and lobe reconnection in the northern hemisphere would change a topology of the magnetic field within the magnetosheath boundary layer at THEMIS latitudes (Pi et al., 2017) and this effect would displace the magnetopause farther from the Earth. Based on these considerations, we think that a larger variability of the magnetopause for negative IMF BX (cone angles 0o 30o ) seen in Fig. 7 is only apparent because a spacecraft crossing the magnetopause above the equator would observe a large spread of the crossings during positive IMF BX intervals (cone angles 150o 180o ). Our study is based on long radial IMF periods when we allow only a short excursion of the cone angle to the interval from 30 to 150o . However, a number of crossings that were observed for these cone angles (72 crossings between vertical dashed lines in Fig. 7) is rather large; much larger than a proportion of non-radial time periods in relation to a whole analyzed time. We think that this effect is connected with the low THEMIS apogee. A majority of crossings was observed by THEMIS A, D and E with the apogee bellow 12 RE . This altitude is lower than a typical magnetopause stand-off distance for radial IMF conditions because such IMF orientation is usually complemented with the low upstream pressure (Pi et al., 2014, and Fig. 2b in the present paper). The magnetopause crossings are thus observed mainly during changes of the IMF toward a
less radial orientation. The presented summary plots (Fig. 7) use both S97 and Lin10 models for a determination of the nominal magnetopause location. Their results are very similar, only a core of the distribution shifts to lower values of differences. However, the spread of ROBS RMOD is larger for the L10 model, probably due to uncertainties connected with the high-altitude cusp location. This location is affected by other factors like IMF BY (e.g., Merka et al., 2003) that are not included in models. 5. Conclusion The results of this short study of a location of the magnetopause during long lasting intervals of radial IMF can be summarized as it follows: The radial IMF causes an expansion of the dayside as well as flank magnetopause in the investigated range of XGSE coordinate (from 5 to þ15 RE ). The magnitude of this expansion can reach þ2 RE regardless of the local times. Median values of the difference between the location of the observed and predicted magnetopause are þ0:7 RE , and þ0:3 RE for the S97 and L10 models, respectively. The magnetopause location is highly variable under a radial IMF, partly due to the pressure and IMF fluctuations originated in the foreshock that can penetrate into the magnetosheath (e.g., BlancoCano et al., 2009; Gutynska et al., 2015) and result in a transient magnetopause motion (Sibeck et al., 2000; Shue et al., 2009; Korotova et al., 2011), and partly due to effects connected with a transformation of the radial magnetic field orientation in the magnetosheath (Pi et al., 2017). Finally, we can conclude that the results on the magnetopause expansion are in a general agreement with Dusík et al. (2010); Suvorova and Dmitriev (2015) and Park et al. (2016). The conclusion of Merka et al. (2003) on a bullet-like shape of the magnetopause during the radial
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IMF was based on a very limited set of dayside observations and thus it could be affected by the mentioned large variability of the magnetopause location. Acknowledgements The authors acknowledge the NASA contract NAS5-02099 and V. Angelopoulos for use of data from the THEMIS mission. Specifically, C. W. Carlson and J. P. McFadden for use of ESA data and K. H. Glassmeier, U. Auster and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR). The present work was supported by the Czech Science Foundation under Contract 17-06065S. References Auster, H.U., et al., 2008. The THEMIS fluxgate magnetometer. Space Sci. Rev. 141, 235–264. https://doi.org/10.1007/s11214-008-9365-9. Blanco-Cano, X., Omidi, N., Russell, C.T., 2009. Global hybrid simulations: foreshock waves and cavitons under radial interplanetary magnetic field geometry. J. Geophys. Res. 114, A01216. Boardsen, S.A., Eastman, T.E., Sotirelis, T., Green, J.L., 2000. An empirical model of the high-latitude magnetopause. J. Geophys. Res. 105, 23193. Granko, G., Safr ankova, J., Nemecek, Z., Jelínek, K., 2010. IMF cone angle Dusík, S., control of the magnetopause location: statistical study. Geophys. Res. Lett. 37, L19103. Fairfield, D.H., 1971. Average and unusual locations of the Earth's magnetopause and bow shock. J. Geophys. Res. 76, 6700–6716. Fairfield, D.H., Baumjohann, W., Paschmann, G., Luehr, H., Sibeck, D.G., 1990. Upstream pressure variations associated with the bow shock and their effects on the magnetosphere. J. Geophys. Res. 78, 3731–3744. ankova, J., Prech, L., Pi, G., Shue, J.-H., 2016. KelvinGrygorov, K., Nemecek, Z., Safr Helmholtz wave at the subsolar magnetopause boundary layer under radial IMF. J. Geophys. Res. Space Phys. 121 (10), 9863–9879. Gutynska, O., Sibeck, D.G., Omidi, N., 2015. Magnetosheath plasma structures and their relation to foreshock processes. J. Geophys. Res. 120, 7687–7697. Huang, T., Wang, H., Shue, J.-H., Cai, L., Pi, G., 2015. The dayside magnetopause location during radial interplanetary magnetic field periods: Cluster observation and model comparison. Ann. Geophys. 33, 437–448. ankova, J., Shue, J.-H., Suvorova, A.V., Sibeck, D.G., 2010. Jelínek, K., Nemecek, Z., Safr Thin magnetosheath as a consequence of the magnetopause deformation: THEMIS observations. J. Geophys. Res. 115, A10203. Kavosi, S., Raeder, J., 2015. Ubiquity of Kelvin–Helmholtz waves at earths magnetopause. Nat. Commun. 6, 7019. Korotova, G.I., Sibeck, D.G., Weatherwax, A., Angelopoulos, V., Styazhkin, V., 2011. THEMIS observations of a transient event at the magnetopause. J. Geophys. Res. 116, A07224. Lin, R.L., Zhang, X.X., Liu, S.Q., Wang, Y.L., Gong, J.C., 2010. A three-dimensional asymmetric magnetopause model. J. Geophys. Res. 115, A04207. https://doi.org/ 10.1029/2009JA014235. McFadden, J.P., Carlson, C.W., Larson, D., Ludlam, M., Abiad, R., Elliott, B., Turin, P., Marckwordt, M., Angelopoulos, V., 2008. The THEMIS ESA plasma instrument and in-flight calibration. Space Sci. Rev. 141, 277. https://doi.org/10.1007/s11214-0089440-2.
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Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Shape of the equatorial magnetopause affected by the radial interplanetary magnetic field ankova, Z. Nemecek *, G. Pi, L. Prech, J. Urb K. Grygorov, J. Safr ar Charles University, Faculty of Mathematics and Physics, V Holesovickach 2, 180 00 Prague, Czech Republic
A R T I C L E I N F O
A B S T R A C T
Keywords: Magnetopause location Radial interplanetary magnetic field Magnetopause deformation
The ability of a prediction of the magnetopause location under various upstream conditions can be considered as a test of our understanding of the solar wind-magnetosphere interaction. The present magnetopause models are parametrized with the solar wind dynamic pressure and usually with the north-south interplanetary magnetic field (IMF) component. However, several studies pointed out an importance of the radial IMF component, but results of these studies are controversial up to now. The present study compares magnetopause observations by five THEMIS spacecraft during long lasting intervals of the radial IMF with two empirical magnetopause models. A comparison reveals that the magnetopause location is highly variable and that the average difference between the observed and predicted positions is þ0:7 RE under this condition. The difference does not depend on the local times and other parameters, like the upstream pressure, IMF north-south component, or tilt angle of the Earth dipole. We conclude that our results strongly support the suggestion on a global expansion of the equatorial magnetopause during intervals of the radial IMF.
1. Introduction The magnetopause is an obstacle varying in a size and shape in a response to changes of the solar wind and interplanetary magnetic field (IMF). Its location is determined by a pressure balance between the total pressure in the magnetosheath and the pressure in the magnetosphere. The magnetospheric pressure is usually dominated by the magnetic pressure but a situation is more complicated in the magnetosheath. On the other hand, the magnetosheath pressure is closely related to the solar wind dynamic pressure and this relation is a base of all empirical magnetopause models developed in the course of past decades. The first model of Fairfield (1971) considered only a balance between the upstream dynamic pressure and dipole magnetic field of the Earth but its successors (e.g., Sibeck, 1990; Petrinec and Russell, 1996; Shue et al., 1997, 1998) introduced an influence of IMF via a dependence of the magnetopause location on the magnitude and polarity of the IMF BZ component. These models use different surfaces to describe the magnetopause shape, and parametrize this surface with the upstream pressure and IMF BZ . This approach is acceptable for the equatorial magnetopause ankov but Safr a et al. (2005) have shown significant magnetopause indentations caused by the magnetic field depression in the cusp regions. This fact was later reflected in the Lin et al. (2010) model.
The overall performance of a majority of magnetopause models is ankova et al., 2002), the statistical uncertainty of a comparable (Safr prediction of the magnetopause location is ±1 RE at the subsolar region and it increases toward flanks. The prediction uncertainties are usually attributed to errors in a determination of upstream conditions or to magnetopause surface deformation caused by surface waves (e.g., Grygorov et al., 2016, and references therein), and by a transient phenomena ankova et al., like IMF discontinuities (e.g., Tkachenko et al., 2011; Safr 2012) or interplanetary shocks (Nemecek et al., 2011). On the other hand, when IMF is nearly aligned with the solar wind velocity, a whole dayside magnetopause is behind the quasiparallel bow shock and foreshock pressure fluctuations can penetrate through the subsolar bow shock into the magnetosheath and influence the magnetopause location as reported by Fairfield et al. (1990). The authors also noted a transient magnetopause motion and its significant inflation. Merka et al. (2003) reported large amplitudes of magnetopause oscillations during intervals of the radial IMF based on a limited statistics and MHD modeling. The authors suggested that the radial IMF orientation would lead to a change of the shape of the whole magnetopause surface. A subsolar part of the magnetopause would move outward, whereas the flanks would be compressed and the magnetopause would attain a bullet-like shape.
* Corresponding author. E-mail address: [email protected] (Z. Nemecek). https://doi.org/10.1016/j.pss.2017.09.011 Received 12 June 2017; Received in revised form 18 September 2017; Accepted 26 September 2017 Available online 28 September 2017 0032-0633/© 2017 Elsevier Ltd. All rights reserved.
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The case studies of Suvorova et al. (2010) and Jelínek et al. (2010) based on THEMIS observations brought a new evidence that the magnetopause lies significantly farther from the Earth than it is expected from models. Suvorova et al. (2010) suggested a global inflation of the magnetopause at dayside and magnetotail locations. These findings are consistent with a large statistical study by Dusík et al. (2010) that demonstrated a systematic expansion of the magnetopause location by 1:7 RE for cone angles (the angle between the IMF and solar wind velocity vectors) close to 0o and 180o . Samsonov et al. (2012) analyzed distributions of different pressure components (dynamic, thermal, and magnetic pressures) in the magnetosheath along the Sun-Earth line for northward and radial IMF orientations. The authors argued that the temperature anisotropy in the magnetosheath controls the pressure distribution near the magnetopause and that the total pressure exerting on the subsolar magnetopause depends on the IMF orientation. For the radial IMF, they observed the total pressure significantly lower than the solar wind dynamic pressure and it would lead to a magnetopause expansion. However, MHD models do not properly describe kinetic effects that are connected with a presence of the foreshock in front of the dayside bow shock during radial IMF intervals. To improve a prediction of the magnetopause position by global MHD models, Samsonov et al. (2017) suggested the method of a modification of upstream parameters that reflects a decrease of the total pressure in the magnetosheath for the IMF orientation obtained in MHD simulations and THEMIS observations. A location of the mid-latitude magnetopause under a radial IMF was studied by Huang et al. (2015) using the Cluster data and both Shue et al. (1997) and Boardsen et al. (2000) models. The authors found a dawn-dusk asymmetry with the dawn magnetopause compressed but the dusk magnetopause expanded if the observations are compared with the Shue et al. (1997) model. However, a majority of magnetopause crossings used in this study was observed at magnetic latitudes exceeding 35o where the influence of the cusp is important. For this reason, a comparison with the Boardsen et al. (2000) model is more relevant and this comparison strongly depressed the observed magnetopause asymmetry. Above named studies do not confirm suggestions of Merka et al. (2003) on a bullet-like magnetopause shape and support a hypothesis on the global magnetopause expansion at all local times. In order to clarify this contradiction, Park et al. (2016) analyzed a correlation of changes of the magnetic field at the geostationary orbit under different IMF orientations. Their study brings a further evidence on a global expansion of the equatorial magnetopause during intervals of a radial IMF but the geostationary orbit is a rather far from the nominal magnetopause and the changes of magnetopause locations are only one of many possible sources of geostationary magnetic field variations, thus such proof is only indirect. Although radial IMF is usually treated as an exception, the analysis of Pi et al. (2014) revealed that the long intervals of continuous radial IMF (longer than 4 h) occupy 10–15% of the year with a slight preference for the solar minimum. Consequently, the understanding of the magnetopause behavior under this IMF orientation can significantly improve our ability of predictions of its location, thus the real magnetopause shape under the radial IMF is a key point. The mentioned statistical studies of the magnetopause shape under radial IMF suffer with the determination of the IMF cone angle. This angle is usually computed from observations of a distant solar wind monitor and propagated toward the bow shock. To demonstrate the errors of this method, Fig. 1 (top) shows a comparison of the cone angles computed from the propagated Wind IMF observations with the cone angles determined from THEMIS B and C measurements. The figure shows only data from intervals when THEMIS was farther than 5 RE from the model bow shock and on the IMF line that was either not connected to the bow shock or the angle between the IMF and bow shock normal was larger than 50o . This selection ensures that the THEMIS magnetic field was not affected by foreshock fluctuations. The cone angles computed from 5-minute IMF and velocity averages are shown by small dots; the large symbols stand for medians in 3o bins. The figure shows a large
Fig. 1. A comparison of cone angles computed from the propagated Wind IMF observations and the cone angles registered by THEMIS B and C. (top) For a whole set of THEMIS observations from 2007 to 2016; (bottom) the same plot for long intervals of a radial IMF orientation.
uncertainty of a determination of the IMF direction affecting the magnetopause from IMF observations at the L1 point. The uncertainty can be significantly reduced if only long-lasting intervals of a radial IMF are used (Fig. 1, bottom). This figure has the same format as Fig. 1 (top); the points were taken from intervals when the Wind IMF cone angle was lower than 30o for more than 4 h continuously (i.e., long radial IMF events in terms of Pi et al. (2014)). Since this IMF orientation leads to a strong foreshock in front of the whole dayside bow shock, a majority of THEMIS cone angles were computed from the measurements affected by the foreshock fluctuations. In spite of this fact, an overall quality of the cone angle prediction is better than that demonstrated in Fig. 1 (top). For this reason, the present study is based on the analysis of magnetopause crossings observed by five THEMIS spacecraft during prolonged intervals of the radial IMF. The second and even more important reason for usage of prolonged intervals of a nearly stable radial IMF orientation is the time that the magnetosphere needs for the reaction to changes of upstream conditions. This reaction can be as long as 15 minutes (Nemecek et al., 2011). As an example, we show in detail one case of simultaneously observed magnetopause crossings by three spacecraft largely separated in local times. Furthermore, using a larger set of similar observations (28 events with multiple magnetopause crossings), we provide a short statistical 29
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whereas S97 is parametrized with the upstream dynamic pressure and IMF BZ component only, the L10 model uses also the dipole tilt angle as an input parameter. Although THEMIS are orbiting close to the equatorial plane, the flank crossings could be influenced by the magnetopause indentation in the cusp region for extreme values of the dipole tilt angle and this indentation is missing in the S97 model. The S97 and L10 models differ also in computations of the upstream pressure. Whereas S97 uses the proton dynamic pressure, L10 also accounts for the alpha particle content and IMF pressure. The main tool of our analysis is a comparison of the distance of the observed crossing from the Earth center, ROBS with that obtained from the model, RMOD . RMOD was computed at the intersection of the model magnetopause with a line defined by the Earth center and a particular crossing. Note that we do not use a separation along the normal to the magnetopause surface but the difference between both methods is negligible in the subsolar region and only slightly larger at flanks. A distribution of basic parameters of identified magnetopause crossings is shown in Fig. 2. Fig. 2a presents that a number of crossings with large negative dipole tilt angles (southern cusp tilted toward the Sun) are rather small and thus an influence of the magnetopause indentation would be depressed. Since the dayside parts of the THEMIS orbits are in the southern hemisphere, the positive dipole tilt angle moves the cusp indentation away from the observed crossings. Fig. 2b demonstrates a typical feature of long intervals of the radial IMF—a low upstream pressure (Pi et al., 2014). The dashed vertical line in this panel shows the median value of upstream pressures for all magnetopause crossings identified in the THEMIS data in course of 2007–2016 (PSW ¼ 1:7 nPa), whereas the dotted line shows the median pressure (PSW ¼ 1:2 nPa) for our set. A small IMF BZ component (Fig. 2c) is a consequence of the conditions for the event selection but Fig. 2d reveals that although we selected the intervals with minimum excursions of the cone angle to the interval 30o 150o , a large portion of crossings was observed for these cone angles. We will return to this point later. The full set of crossings used in the study is introduced in Fig. 3. A part of orbits of the THEMIS spacecraft corresponding to intervals of the radial IMF are shown by color lines and the crossings are indicated by the crosses. The figure reveals that only THEMIS B and C (blue and red, respectively) are able to observe really distant crossings because the apogees of other three spacecraft are too low. Nevertheless, a great majority of crossings is observed out of the depicted magnetopause that was computed for the pressure of 1.3 nPa, IMF BZ ¼ 0 and zero tilt angle of the Earth dipole using the S97 (dotted line) and L10 (dashed line) models.
study in which we estimate the magnetopause shape for radial IMF and calculate average differences between the observed and predicted magnetopause positions. Our analysis is in a general agreement with previous studies and shows that the subsolar as well as flank magnetopause expands in the outward direction. 2. Data selection A statistical study of Dusík et al. (2010) used 5-minute averages of IMF components propagated from Wind for a computation of the cone angle (the angle between IMF vector and XGSM axis). Such approach suffers with aforementioned large uncertainties because the IMF direction is often modified by Alfven waves that propagate in the solar wind. To depress this effect, we have limited our study to long ( > 2 hours) intervals of the radial IMF orientation. However, the intervals of a pure radial IMF occur only 10% of the time, thus we allowed short (up to 15 min) excursions of the cone angle to larger values. We have chosen 15 min as a compromise between our intention to analyze only long radial IMF intervals and the number of such intervals with magnetopause crossings. A decrease of this value to 10 min leads to the decrease of the total duration of available intervals to about one half but, as we checked, it does not change the results of our analysis. Since our goal is to investigate a quasi-steady magnetopause shape, we have compared our preliminary list of intervals with the list of KelvinHelmholtz waves collected by Kavosi and Raeder (2015) and excluded the overlaps. Finally, we have identified 26 intervals with the cone angle lower than 30o (or greater than 150o ) that contain more than 350 magnetopause crossings observed by THEMIS in a course of 2008–2013. The aggregate duration of these intervals is about 83 h. The list of crossings was complemented with the solar wind dynamic pressure and IMF components. We used THEMIS B and C observations whenever they were located in the solar wind out of the foreshock and Wind for the rest of cases. All upstream data were lagged on a propagation time estimated by a two-step procedure based on the actual solar ankov wind velocity (Safr a et al., 2002) and we used 5 min averages centered around the estimated time. The coordinates of crossings were aberrated to account for the Earth's orbital motion. Magnetopause crossings were identified by a visual inspection of plots of the THEMIS magnetic field (Auster et al., 2008) and plasma parameters (McFadden et al., 2008) and a model prediction of the magnetopause location was computed for each crossing. Suvorova and Dmitriev (2015) concluded that the Lin et al. (2010) model is capable to predict the magnetopause position under a radial IMF, thus we applied two models: Shue et al. (1997) and Lin et al. (2010) (S97 and L10, hereafter). Both models are based on fits to the observed crossings but
Fig. 2. Histograms of basic parameters of magnetopause crossings included in this study: (a) the dipole tilt angle; (b) the solar wind dynamic pressure, PSW ; (c) IMF BZ ; and (d) the cone angle. In the panel (b), the dashed vertical line shows the median value of upstream pressures for all THEMIS magnetopause crossings through 2007–2016 and the dotted line stands for the median pressure in our set of the radial events. 30
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Fig. 3. Projections of observed magnetopause crossings by all THEMIS probes onto the XY (left) and XZ (right) planes in the GSM coordinate system. S97 (dotted) and L10 (dashed) models for BZ ¼ 0, PSW ¼ 1:2 nPa, and zero dipole tilt angle are given for reference. A part of THEMIS orbits corresponding to intervals of the radial IMF are shown by color lines and magnetopause crossings are indicated by the crosses in a corresponding color.
3. Case study
orientation simplifies an identification of the magnetopause crossings. The first two panels present distances of the registered crossings from L10 and S97, respectively. The crossings were observed in course of a whole depicted interval by THEMIS A, whereas B and C probes crossed the magnetopause several times but only within limited time periods. The reason can be seen in the left part of Fig. 5. THEMIS A with apogee of about 12 RE skimmed the dusk magnetopause but THEMIS B and C with apogees of 55 and 26 RE , respectively, passed the magnetopause region very quickly. Nevertheless, both these spacecraft crossed the dawn magnetopause at similar MLT and their observations are complementary. First two panels of Fig. 4 reveal several features typical for the magnetopause behavior under a radial IMF:
This section presents magnetopause crossings observed by three THEMIS spacecraft during a long interval (more than 6 h) of the radial IMF on August 31, 2009. The main reason for a selection of this event is that two spacecraft observed the magnetopause in very different local times simultaneously and it gives us a possibility to distinguish between the global expansion and deformation of the magnetopause surface. The basic observations are shown in Fig. 4. The colors denote the spacecraft (THEMIS A in orange, THEMIS B in red, THEMIS C in blue, and Wind in black); note that the Wind data were lagged on the estimated propagation time. The IMF BZ (penultimate panel) was very weak and mostly negative and it resulted in the negative BZ observed during magnetosheath intervals registered by THEMIS (third panel). This magnetic field
The L10 and S97 models provide very similar predictions.
Fig. 4. An example of a series of magnetopause crossings observed by THEMIS A (yellow), THEMIS B (red), and THEMIS C (blue) on August 31, 2009. From top to bottom: differences in observed and predicted of magnetopause crossing locations (in RE ) for the L10 model; for the S97 model; the BZ component registered by THEMIS A (orange), B (red), and C (blue); IMF cone angle; IMF BZ component; and solar wind dynamic pressure, PSW (last three panels show Wind data).
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Fig. 5. Projections of THEMIS probes (August 31, 2009) onto the XY (top) and XZ (bottom) coordinates at the same time interval as in Fig. 4. The dotted and dashed lines show the magnetopause shape according to S97 and Lin10, respectively, under BZ ¼ 0, PSW ¼ 1:2 nPa, and the actual dipole tilt angle of 18o . The colors mark a particular THEMIS probe: THA–orange; THB–red; and THC–blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
All crossings were observed at larger distances from the Earth than the models predict; the difference may reach 2.2 RE . A large dispersion of the points in first two panels of Fig. 4 demonstrates a large variability of the magnetopause location under this IMF orientation. No crossing was observed between 1520 and 1550 UT; Wind observations suggest the Parker spiral-like orientation from 1530 to 1615 UT. At about 1105 UT, THEMIS C observed the dawn magnetopause at about nominal position but THEMIS A crossed the dusk magnetopause about 1.7 RE farther from the Earth than both models predict. The opposite observations can be seen around 1510 UT when THEMIS B identified the dawn magnetopause at a larger distance from the Earth than THEMIS A at dusk did. These observations indicate that the magnetopause was expanded in all local times covered by THEMIS measurements under the radial IMF orientation. Moreover, a number of crossings as well as a comparison of THEMIS A with THEMIS B and C shows that the magnetopause location is highly variable and these variations are not caused by changes of the upstream pressure (last panel in Fig. 4). We cannot exclude that this variability is caused by foreshock structures (spontaneous hot flow anomalies, foreshock cavities, foreshock bubbles, etc.) but these structures are localized and there was no spacecraft monitoring the magnetosheath or foreshock upstream of THEMIS A. Fig. 6. ROBS RMOD projected onto the normalized magnetopause surface. The length of segments shows the difference between the observation and prediction of a particular magnetopause crossing and the segment color represents an actual value of PSW . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4. Discussion A case study presented in the previous section suggests a global magnetopause expansion due to the radial IMF orientation. The same conclusion follows from Fig. 6 where the magnetopause crossings observed during our all long radial IMF intervals are shown. In Fig. 6, the curved line stands for the S97 equatorial magnetopause surface. The lengths of line segments beginning at this surface correspond to a deviation of each particular crossing from the model and their colors mark the values of upstream pressures at the time of a particular crossing. The figure shows only several crossings that were observed inbound the nominal magnetopause; a great majority of crossings (317 from 350 for S97, and 241 from 350 for L10) indicates up to 2 RE magnetopause expansion that does not depend on the local time and upstream pressure. The IMF BZ component is the second parameter that controls the magnetopause shape and position. Since our set contains several crossings that were observed under non-negligible BZ , we present an analysis of a possible BZ influence for S97 in Fig. 7 (left). A value of the IMF BZ component is represented by the color scale; diamonds, stars and triangles stand for dawn, subsolar (9–15 MLT) and dusk crossings,
respectively. The dimensions of symbols provide a rough estimate of the upstream dynamic pressure. This plot is rather complex but it can be clearly seen that even the crossings observed under a relatively large negative IMF BZ (from 1:5 to 2:5 nT, dark blue) are located far upstream of the model magnetopause and the same is true for the crossings observed under a positive IMF BZ (red points). The crossings located significantly closer to the Earth than the model predicts were observed predominantly at flanks (triangles and diamonds) but the same is valid for the most distant crossings. It suggests that a location of the flank magnetopause is more variable, however, this fact is not surprising. The right panel in Fig. 7 shows the comparison of observations and L10 model predictions. The distribution of crossings for small (0o 30o ) cone angles is similar as that in the left panel but the both positive and negative extreme values of ROBS RMOD are large. The distribution of ROBS RMOD for large (150o 180o ) cone angles is again wider for L10 than for the S97 model but the distribution is centered around þ0:3 RE , 32
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Fig. 7. ROBS RMOD as a function of the IMF cone angle. The colors present values of IMF BZ ; different symbols (diamonds, stars, and triangles stand for dawn, subsolar (9–15 MLT), and dusk crossings, respectively) indicate the actual location of a particular spacecraft; and their size determines approximate values of PSW . (a) For the Shue et al. (1997) and (b) for Lin et al. (2010) models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
whereas that for S97 at about þ0:7 RE . Fig. 7 suggests that the magnetopause location is more variable for the low ( < 30o ) cone angles, i.e., for the IMF BX component pointing toward the Earth. Although the statistics is limited, we think that this result is reliable and that it is a consequence of a transformation of the radial IMF downstream of the bow shock. A structure of the magnetosheath magnetic field and its temporal evolution under a radial IMF was investigated by Pi et al. (2017). According to this study, negative IMF BX converts to negative BZ in the southern magnetosheath (see Fig. 6 in the mentioned paper). This magnetic field orientation leads to erosion of the magnetopause and THEMIS that is orbiting in the southern hemisphere would observe the magnetopause closer to the Earth than the model predicts. This effect would be enhanced if the original IMF would have a (small) negative BZ component and it is generally consistent with Fig. 7. However, a long-lasting combination of subsolar reconnection in the southern hemisphere and lobe reconnection in the northern hemisphere would change a topology of the magnetic field within the magnetosheath boundary layer at THEMIS latitudes (Pi et al., 2017) and this effect would displace the magnetopause farther from the Earth. Based on these considerations, we think that a larger variability of the magnetopause for negative IMF BX (cone angles 0o 30o ) seen in Fig. 7 is only apparent because a spacecraft crossing the magnetopause above the equator would observe a large spread of the crossings during positive IMF BX intervals (cone angles 150o 180o ). Our study is based on long radial IMF periods when we allow only a short excursion of the cone angle to the interval from 30 to 150o . However, a number of crossings that were observed for these cone angles (72 crossings between vertical dashed lines in Fig. 7) is rather large; much larger than a proportion of non-radial time periods in relation to a whole analyzed time. We think that this effect is connected with the low THEMIS apogee. A majority of crossings was observed by THEMIS A, D and E with the apogee bellow 12 RE . This altitude is lower than a typical magnetopause stand-off distance for radial IMF conditions because such IMF orientation is usually complemented with the low upstream pressure (Pi et al., 2014, and Fig. 2b in the present paper). The magnetopause crossings are thus observed mainly during changes of the IMF toward a
less radial orientation. The presented summary plots (Fig. 7) use both S97 and Lin10 models for a determination of the nominal magnetopause location. Their results are very similar, only a core of the distribution shifts to lower values of differences. However, the spread of ROBS RMOD is larger for the L10 model, probably due to uncertainties connected with the high-altitude cusp location. This location is affected by other factors like IMF BY (e.g., Merka et al., 2003) that are not included in models. 5. Conclusion The results of this short study of a location of the magnetopause during long lasting intervals of radial IMF can be summarized as it follows: The radial IMF causes an expansion of the dayside as well as flank magnetopause in the investigated range of XGSE coordinate (from 5 to þ15 RE ). The magnitude of this expansion can reach þ2 RE regardless of the local times. Median values of the difference between the location of the observed and predicted magnetopause are þ0:7 RE , and þ0:3 RE for the S97 and L10 models, respectively. The magnetopause location is highly variable under a radial IMF, partly due to the pressure and IMF fluctuations originated in the foreshock that can penetrate into the magnetosheath (e.g., BlancoCano et al., 2009; Gutynska et al., 2015) and result in a transient magnetopause motion (Sibeck et al., 2000; Shue et al., 2009; Korotova et al., 2011), and partly due to effects connected with a transformation of the radial magnetic field orientation in the magnetosheath (Pi et al., 2017). Finally, we can conclude that the results on the magnetopause expansion are in a general agreement with Dusík et al. (2010); Suvorova and Dmitriev (2015) and Park et al. (2016). The conclusion of Merka et al. (2003) on a bullet-like shape of the magnetopause during the radial
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