Earth s bow shock and magnetopause in the case of a field-aligned upstream flow: Observation and model comparison

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A7, 1269, doi: /2002ja009697, 2003 Earth s bow shock and magnetopause in the case of a field-aligned upstream flow: Observation and model comparison J. Merka 1 and A. Szabo Laboratory for Extraterrestrial Physics, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA J. Šafránková and Z. Němeček Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic Received 24 September 2002; revised 31 October 2002; accepted 16 January 2003; published 3 July [1] On 5 May 1996 the Interball-1 and IMP 8 spacecraft crossed the bow shock boundary. The upstream conditions were special in two factors: (1) the interplanetary magnetic field was anti-parallel to the solar wind flow within 15 and (2) the conditions were stable for a prolonged period (9 hours). At the nose of the magnetosphere, the Interball-1 data revealed that the magnetopause was farther outward by 2 R E than model predictions and the subsolar magnetosheath was unusually thin, at most 10% of the magnetopause standoff distance. Both results stand in contrast to predictions of existing magnetopause/bow shock models. Assuming a hyperboloidal (paraboloidal) shock wave, the calculated shock s standoff distance was 13.7 (13.6) R E, and the focus was located on the x axis at 4.5 (4.2) R E. On the basis of the IMP 8 observation, the bow shock flares significantly less than MHD simulations predict for a field-aligned bow shock at the magnetospheric flanks. This study discusses differences between the observations and existing MHD bow shock simulations for field-aligned upstream flow. Furthermore, it is suggested that the flowaligned IMF orientation causes a significant change of the magnetopause shape into a bullet-like obstacle. INDEX TERMS: 2154 Interplanetary Physics: Planetary bow shocks; 2728 Magnetospheric Physics: Magnetosheath; 2724 Magnetospheric Physics: Magnetopause, cusp, and boundary layers; 2784 Magnetospheric Physics: Solar wind/magnetosphere interactions; KEYWORDS: bow shock, magnetopause, magnetosheath thickness, field-aligned flow, IMP 8, Interball-1 Citation: Merka, J., A. Szabo, J. Šafránková, and Z. Němeček, Earth s bow shock and magnetopause in the case of a field-aligned upstream flow: Observation and model comparison, J. Geophys. Res., 108(A7), 1269, doi: /2002ja009697, Introduction [2] The positions, shape, and motion of Earth s bow shock have been extensively studied for the last four decades. Although many bow shock models have been developed [e.g., Spreiter et al., 1966; Fairfield, 1971; Formisano, 1979; Němeček and Šafránková, 1991; Farris and Russell, 1994; Cairns and Lyon, 1995; Peredo et al., 1995], they still do not sufficiently describe the observed bow shock, especially for unusual conditions in the solar wind [Šafránková et al., 1999; Merka et al., 2003]. [3] The existing theories apply to a magnetosphere immersed in a uniform, steady state solar wind plasma/ field environment, which, of course, is an idealized and simplified picture of the highly variable reality. Therefore when comparing the observations to current theories or models one expects a scatter of the bow shock locations 1 On leave from Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic. Copyright 2003 by the American Geophysical Union /03/2002JA from the model predictions. In general, in order to compare model predictions with observations, large numbers of observed bow shock crossings are required. However, during periods of prolonged steady solar wind conditions, individual crossings can be readily compared to the various model predictions. [4] In the gasdynamic simulations of Spreiter et al. [1966], the sheath thickness msh (and equivalently the bow shock standoff distance a s ) is a function of the freestream sonic Mach number M S and magnetopause standoff distance a mp that closely resembles results for an aerodynamic flow around a blunt obstacle. Rizzi [1971] presented MHD simulations for the special case when the magnetic field is parallel to the flow (q =0 ) and the magnetopause obstacle is prescribed with hard, infinitely conducting boundary conditions. Here and below q is the angle between the incident solar wind velocity v sw and magnetic field B sw. (MHD symmetries permit q to be constrained to the interval [0, 90 ], that is, both parallel and antiparallel orientations of v sw and B sw correspond to q = 0.) The simulations showed that the bow shock location is very similar to the gasdynamic solution when the Alfvén Mach number M A is high. However, when M A diminishes the shock wave SMP 2-1

2 SMP 2-2 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW Figure 1. Locations of the IMP 8, Interball-1, and Wind spacecraft on 5 May 1996 between 19 and 23 UT (IMP 8 and Interball-1) or between 12 and 24 UT (Wind). The arrows point in the direction of the spacecraft movement. The solid and dashed lines depict positions of the model bow shock [Formisano, 1979] and magnetopause [Shue et al., 1998], respectively. Note different spatial scales on the axes. location moves closer to the magnetopause in the nose region than does the gasdynamic one but moves substantially farther away from the magnetospheric tail downstream from Earth. Rizzi [1971] demonstrated a good agreement of his simulations with the Mariner-5 data from the Venus flyby [Bridge et al., 1967], even though the observed field alignment was more than 15. A brief summary of Rizzi s [1971] results can be found in Spreiter and Rizzi [1974]. [5] Slavin et al. [1996] studied near-simultaneous bow shock crossings by Wind and IMP 8 when the magnetic field made an angle <20 to the Sun-Earth line. They found that the radial distance of the shock at both spacecraft was only 80 85% of that predicted by models. Motivated by this event, Slavin et al. [1996] fitted a shock shape to a sample of 19 crossings with q 20 and on this basis suggested that magnetosheath thickness may decrease by 10% as the IMF becomes increasingly flow aligned. [6] Cairns and Grabbe [1994] developed an MHD theory for the bow shock standoff distance a s and the thickness msh of the magnetosheath predicting that msh /a mp should depend strongly on q, M A, and M S for M A ] 6. The prediction is based on the assumption that msh /a mp is proportional to the density ratio r sw /r msh at the subsolar shock. [7] The MHD simulations of Cairns and Lyon [1995] based on the theory of Cairns and Grabbe [1994] showed only a weak dependence of the bow shock position on the IMF orientation for q =45 and 90. Using the same code for q =0 and 20, Cairns and Lyon [1996] found that both ms /a mp and a s /a mp depend very strongly on q for low M A ] 6, with remnant effects possibly still existing at M A 10. For a given M A the shock is more distant for higher q, while ms /a mp and a s /a mp increase with decreasing M A for q ^ 20 but decrease with decreasing M A for q 0. The q = 0 results confirm and extend the previous work of Spreiter and Rizzi [1974]. This behavior is opposite to the conventional expectation and is not yet understood. The work of Cairns and Lyon [1996] demonstrated that models with ms =a mp / a s =a mp ¼ mx þ k have robust applicability but that the quantities m and k vary with q. In equation (1), X is the upstream to downstream density ratio r sw /r msh at the shock. The MHD simulations of ð1þ Cairns and Lyon [1995] obtained values m = 3.4, k = 0.4. The Cairns and Lyon [1995] model remains useful and viable for q > 45 but becomes inaccurate for q < Furthermore, the more recent work of Grabbe [1997] proposed that the linear relation (1) is valid only when M A 3 and X < [8] This paper presents a study of multiple bow shock crossings encountered by Interball-1 and IMP 8 spacecraft on 5 May 1996, when the IMF was nearly aligned with the incident solar wind flow (q 10 ) and the upstream conditions were very stable for 9 h. These observations will demonstrate serious deficiencies in existing magnetopause and bow shock models/theories since estimates of the magnetosheath thickness, based on Interball-1 measurements, are <10% of the magnetopause distance and both spacecraft observed the shock much closer to Earth than predicted. 2. Data Set 2.1. Upstream Measurements [9] The Wind spacecraft serves as the common solar wind monitor in our study (see Figure 1). Figure 2 displays the Figure 2. Magnetic field and plasma parameters observed by Wind on 5 May Note the relatively steady conditions between 1500 and 2400 UT. The highfrequency features in the IMF are due to direct bow shock connection. Note very steady conditions for 9 h (between the vertical lines).

3 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW SMP 2-3 Figure 3. Magnetic field and plasma parameters observed by IMP 8 on 5 May IMF strength and its components, proton density n p, thermal velocity v th, solar wind bulk velocity v, and both horizontal f (east-west) and vertical w (north-south) deflections from the radial flow (f =tan 1 v y /v x, and w =tan 1 v z /v x )for UT on 5 May The data were obtained using the plasma (SWE) and magnetic field (MFI) instruments on board the Wind spacecraft [Ogilvie et al., 1995; Lepping et al., 1995]. Figure 2 shows the MFI data with 3-s time resolution, whereas the SWE data are with 95-s resolution. Note that the 3-s MFI data have been rescaled to the time step of the SWE data before using them for calculations of model predictions. In GSE coordinates, the satellite was located at (63.9, 21.7, 4.6) R E and (59.3, 23.6, 4.6) R E at times 1200 and 2400 UT, respectively, that is, close enough to both the Earth and the Sun-Earth line to expect a high-degree correlation between Wind and near-earth solar wind observations (see Figure 1). [10] During the second half of 5 May 1996 the upstream measurements exhibit very steady conditions for 9 h in both plasma and magnetic field data as delimited by two vertical lines in Figure 2. During UT the IMF B Y and B Z components exhibitfast variations, probably because of wave activity associated with the bow shock. Note that the high-frequency variations around the running mean value are less than ±1 nt and thus not important in our study. For the study of the bow shock and the magnetopause we are most concerned of solar wind variations on the manyminute scale length. The selected interval exhibits very slow changes in the magnetofluid parameters on this scale length. Specifically, the largest peak-to-peak variation in the IMF is in the B Y component (3 nt), significantly smaller than usual or even than during the preceding hours. The plasma parameters show a similar steady behavior. [11] Figure 3 shows data from IMP 8 during the same period as Figure 2. A comparison of Figures 2 and 3 reveals the same features and parameter values observed by both the Wind and IMP 8 spacecraft. Although not shown in the presented figures, even the ion flux calculated from Wind measurements matches closely the INTERBALL-1 observations (see Figure 4). Taking advantage of this fact, we shall use the Wind upstream measurements in the model calculations to obtain directly comparable results at both the Interball-1 and the IMP 8 locations Interball-1 [12] Between 1853 and 2059 UT on 5 May 1996 the Interball-1 spacecraft moved from the solar wind into the magnetosphere. Its position in GSE coordinates was (13.1, 5.3, 4.4) and (10.5, 4.5, 4.9) R E at the beginning and end of the specified time interval, respectively (see Figure 1). The estimate of boundaries is based on the analysis of the 1- s averaged ion flux data measured by the Faraday cup instrument (VDP) [Šafránková etal., 1997]. In addition, the 2-min magnetic field key parameter data from CDAWeb were used [Nozdrachev et al., 1998]. [13] An event was identified as a shock crossing if a simultaneous sharp drop in the anti-sunward and increase of perpendicular fluxes were found in the data (see Figure 4). We found seven unambiguous bow shock crossings in time interval UT. [14] At 2019 UT, both anti-sunward and perpendicular fluxes diminished and magnetic field intensity rose, increased to 30 nt as a signature of a magnetopause crossing. Interball-1 crossed this boundary seven times from 2019 to 2059 UT IMP 8 [15] As shown in Figure 3, IMP 8 crossed bow shock three times between 2135 and 2245 UT, at ( 9.1, 27.5, Figure 4. Magnetic field and ion flux (arbitrary units) observed by Interball-1 on 5 May The displayed flux directions were anti-sunward and perpendicular to the Sun- Earth line.

4 SMP 2-4 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW Table 1. Bow Shock Properties at the IMP 8 Location a Time, UT r/r 1 B/B 1 T/T 1 v/v b >2.75 b 7.93 b 0.89 b a Various shock properties as observed by IMP 8 on 5 May 1996, and from numerical simulations by Rizzi [1971]. Values were obtained from graphical results of a simulation run with parameters M A1 = 10, M S1 = 10, and g = 5/3. b Values are from Rizzi [1971]. 16.4) GSE to ( 9.8, 27.0, 16.6) GSE R E (see Figure 1). Both 15-s magnetic field and 60-s plasma data clearly display multiple transitions from the solar wind to magnetosheath and vice versa. For a detailed description of the magnetic field and plasma experiments on board the IMP 8 satellite see Mish and Lepping [1976] and Bellomo and Mavretic [1978], respectively. [16] Table 1 summarizes the downstream/upstream ratios of selected parameters (plasma density, magnetic field strength, temperature and velocity) at the bow shock crossings. The 1 index denotes upstream parameters, whereas no index means downstream parameters (e.g., r/r 1 = r msh /r sw ). The upstream (downstream) parameters were taken as averages on 5-min intervals 1 min upstream (downstream) of the shock crossings. For comparison with the IMP 8 observations, the table displays results of numerical simulation for a field-aligned flow with upstream parameters M A = 10, M S = 10, and g = 5/3. The values were deduced from plots presented by Rizzi [1971] at the approximate location of the IMP 8 observed bow shock crossings. During the IMP 8 observation of the three consecutive bow shock crossings, the upstream Mach numbers M A and M S were 8 9 (see Figure 5). 3. Method of Analysis [17] For comparison with the Interball-1 and IMP 8 bow shock observations, we have selected widely used models in this study. Our primary focus will be on the CL95 model [Cairns and Lyon, 1995] because it explicitly describes the bow shock dependence on the IMF orientation, that is, on the q angle. In contrast, the other models of Formisano [1979] and Němeček and Šafránková [1991] (referred as F79 and NS91, respectively) do not take into account the interplanetary magnetic field at all or simply scale the bow shock with the field magnitude. [18] Implementations of the F79, NS91, and CL95 models have been described by Merka et al. [2003]. However, here we introduce a few variations of the original CL95 model. Cairns and Lyon [1996] performed three-dimensional, global MHD simulations of solar wind flow onto a prescribed magnetopause obstacle and found that the bow shock s nose location a s is a strong function of the IMF cone angle q and the Alfvén Mach number M A. Their results suggest that the coefficients m and k in equation (1) strongly depend on angle q when q <45 [see Cairns and Lyon, 1996, Figure 3]. However, for the specific value of the density ratio X = 0.25 at the shock (i.e., the maximum of r 1 /r), they calculated that the relative magnetosheath thickness msh /a mp was 0.25 regardless of the IMF orientation. Thus one can extend the CL95 model even for q <45 by introducing a linear dependence of m on q: a s ¼ ð6:75q 1:9ÞX þ 0:25ð6:9 6:75qÞ; ð2þ a mp where q is expressed in radians. The new coefficients m(q) and k(q) were derived under the assumption that the slope of curves changes linearly with q as suggested by Figure 3 of Cairns and Lyon [1996]. If q 2 [45, 90 ], the original formula [Cairns and Lyon, 1995] is used: a s a mp ¼ 3:4 X þ 0:4: [19] The other modification addresses the bow shock shape. Cairns et al. [1995] surmised a self-similar expansion of the bow shock to obtain a relation for the flaring parameter b s, which described b s as a function of single Figure 5. Bow shock crossings observed by IMP 8, along with various upstream and downstream parameters. For detailed description see section 4. ð3þ

5 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW SMP 2-5 parameter p sw, the solar wind dynamic pressure. However, the assumption of self-similar expansion leads to b s being a function of the bow shock standoff distance a s and the distance x between the parabola focus and the origin of the used coordinate system (see Appendix A). [20] Furthermore, the bow shock asymptotically approaches the magnetosonic Mach cone [e.g., Landau and Lifshitz, 1959; Spreiter et al., 1966; Slavin et al., 1984]. Therefore the Mach cone angle a controls the shape s eccentricity (equation (A8)), and this effect should be included in the calculations of the bow shock position as, for example, in the study by Bennett et al. [1997]. To calculate the asymptotic Mach cone angle a the implicit analytical solution of Verigin et al. [2003] is employed in the present paper. Although < 1.02 during the time interval under study, the hyperboloidal shock shape is assumed in the variants of CL95 model because of the shape s general applicability. [21] The CL95 model predicts only the bow shock standoff distance a s and therefore we still need to find the position of parabola/hyperbola s focus x to be able to use equations (A3)/(A7). The predicted shock standoff distances a s were computed for more than 3600 IMP 8 observed bow shock crossings and the corresponding x values were derived. The database description can be found in Merka et al. [2003]. The resulting positions of the paraboloid and hyperboloid foci are 1.3 ± 5.5 R E (median is 1.8 R E ) and 1.6 ± 5.4 R E (median is 2.1 R E ), respectively. In spite of significant variations, the results are consistent with previously published values [e.g., Slavin and Holzer, 1981; Cairns et al., 1996]. The primary reason for this spread is the steady state nature of the model (and, in fact, of all current bow shock models) in contrary to unsteady conditions in the solar wind, which lead to permanent motions of the shock boundary. However, with reasonable confidence we can still use the median values of x. [22] In the following paragraphs the variants of CL95 model are referred to as CL95h and CL95ht, where both variants incorporate a hyperboloidal shock shape with x = 2.1, and in addition, CL95ht is based on equations (2) and (3), expressing the q dependence. The CL95 model and its variants require knowledge of the obstacle (magnetopause) standoff distance a mp. The magnetopause model by Shue et al. [1998] (referred as S98) provided us with values of a mp. A 4% Helium abundance in the solar wind and g = 5/3 are assumed in our calculations. 4. Results [23] Profiles of various plasma and field parameters, and radial distance of the magnetopause and bow shock at the Interball-1 location are displayed in Figure 6. From bottom to top, the panels present (1) solar wind ram pressure p sw and angle q (Figure 6a); (2) CL95, CL95h, CL95th predicted radial distance of the bow shock in the direction of Interball-1 compared with the spacecraft position (note that the CL95 and CL95h curves overlap) (Figure 6b); (3) same as Figure 6b using F79 and NS91 models (Figure 6c); (4) S98 predicted radial (in the direction of Interball-1) and standoff distances (r mp and a mp ) of the magnetopause (Figure 6d); (5) anti-sunward and perpendicular ion flux observed by Interball-1 (Figure 6e); (6) magnetic field Figure 6. Bow shock and magnetopause crossings observed by Interball-1, along with various upstream and downstream parameters. For detailed description see section 4. magnitude measured by Interball-1 (Figure 6f); (7) upstream Mach numbers (Figure 6g); (8) magnetospheric magnetic field measured by GOES-8 and GOES-9 at geosynchronous orbit and compared with model magnetic field values (dotted lines) provided by a Tsyganenko 2002 (referred as T02) magnetospheric magnetic field model [Tsyganenko, 2002a, 2002b] (Figure 6h). Upstream parameters displayed in Figures 6a 6d and Figure 6g are based on Wind observations shifted by 12.5 min, which is the average propagation time of the solar wind between the Wind and Interball-1 positions during this time period. This time lag was computed by a multistep approximation based on Wind plasma velocity measurements. [24] The orange and grey areas in Figure 6 mark time intervals when Interball-1 was in the magnetosheath and magnetosphere, respectively. Seven bow shock crossings were encountered before 2000 UT and seven magnetopause

6 SMP 2-6 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW Table 2. Magnetopause Radial Distances Observed by Interball-1 and Computed Using the S98 Model Time, UT Interball-1, R E Shue et al. [1998], R E Ratio crossings after this time. The whole time interval is characterized by very stable upstream conditions with p sw 1.6 npa, nearly anti-parallel orientation of IMF (q ] 10 ), and magnetosonic Mach number M MS 6. Furthermore, the overall stability of the upstream conditions is reflected in the small difference (<5 nt) between the T02 model predictions and GOES observations. The individual GOES observational variations are consistent with the motion of the spacecraft in the magnetosphere. This implies a very stationary magnetopause location. [25] The multiple bow shock crossings were observed within 27 min and can be attributed to small solar wind pressure oscillations even though a closer look at Figure 6 might suggest that the bow shock moves in the opposite direction than the p sw changes should cause (see crossing at 19 UT). However, the pressure changes only by ]0.4 npa, which means that we are not able to attribute a pressure jump to a particular bow shock crossing (i.e, magnetosphere compression/expansion). Furthermore, the time shift of Wind data to Interball-1 may be a few minutes different than the computed lag (12.5 min) because of a combination of various effects as, for example, measurement errors of v sw and/or because the two spacecraft were on different streamlines. On the other hand, the 0.4 npa changes in the dynamic pressure result in the model bow shock shift by 0.5 R E (see Figure 6), which could explain the observation of multiple bow shock crossings by Interball-1. Note that all bow shock models provide virtually the same shock movement suggesting again that p sw is the dominant parameter in this case. [26] Comparing directly the spacecraft radial distance to bow shock model predictions, we find the best agreement for F79, CL95, and CL95h models. NS91, which is based on the empirical F79 model by adding a dependency on the IMF magnitude, overestimates the shock position by 1 R E, suggesting that the magnetic field magnitude is not an important parameter in this case. [27] Supposedly the best model, CL95ht, apparently underestimates the shock radial distance by 1 R E. However, we note that CL95, CL95h, and CL95ht depend on the magnetopause standoff distance a mp, see equations (2) and (3). Furthermore, the exceptionally steady upstream (and magnetosphere) conditions justify the assumption that the magnetopause was located approximately at the same position at the times of bow shock and magnetopause observations by Interball-1. Figure 6 shows a significant difference between the S98 predictions r mp and actual observations of the magnetopause radial distance. The first three magnetopause crossings were located farther by 2 R E (20%) than the S98 model predicts (Table 2). Because Interball-1 crossed the dayside magnetopause not far from the subsolar point, we can scale up the predicted standoff distance a mp by 20% to approximate the actual subsolar magnetopause location. Using this obtained magnetopause standoff distance, predictions provided by CL95, CL95h, and CL95ht place the bow shock farther by the same factor 1.2 (see equations (2) and (3)), which leads to bow shock radial distances greater than 16.5 R E. This makes all three variants of the CL95 model incorrect and brings the CL95ht s predictions closest to the observed shock positions. [28] Two hours after Interball-1 observed the last bow shock crossing, the IMP 8 satellite entered the magnetosheath on the magnetospheric flank. Figure 5 shows this and two later bow shock crossings. From bottom to top, the panels display (1) solar wind dynamic pressure p sw (Figure 5a); (2) relative IMF orientation as angle q (Figure 5b); (3) predictions of the CL95, CL95h, and CL95ht bow shock models using IMP 8 upstream measurements (Figure 5c); (4) same as Figure 5c with the Wind spacecraft acting as solar wind monitor (Figure 5d); (5) predictions of the shock position by the F79 and NS91 models using IMP 8 or Wind upstream data (Figure 5e); (6) magnetic field magnitude and east-west orientation of the solar wind flow observed by IMP 8 (Figure 5f); (7) upstream Mach numbers measured by IMP 8 (Figure 5g); (8) same as Figure 5g using Wind data (Figure 5h). Wind data were shifted by a 7.5 min time lag that was estimated by comparing solar wind features observed by both Wind and IMP 8. Computing the average propagation time of the solar wind between Wind and IMP 8 positions on the basis of the Wind measured solar wind speed alone yields a time lag of 18 min and indicates solar wind front orientations other than purely perpendicular to the Sun-Earth line. Although we have chosen the former time lag in this study, using the latter would not change the results due to steady upstream conditions. [29] Detailed comparison of Figures 5 and 6 reveals that the upstream conditions were practically identical in both cases. We can again classify the upstream conditions as average, or with slightly lower ram pressure, except for the field-aligned orientation of the solar wind flow. In the case of IMP 8, all bow shock models place the shock by more 4 R E farther than observed. Note that the MHD-based models provide better predictions than the empirical models F79 and NS91. [30] Because of the steady upstream conditions the last Interball-1 and first IMP 8 crossings can be considered as simultaneous and therefore the two unknown parameters a s, x of a hyperboloidal (paraboloidal) shock shape can be computed using equations (A1) (A8). For the hyperboloid (paraboloid) we obtained the bow shock standoff distance a s = 13.7(13.6) R E and focus location x = 4.4(4.2) R E.An average Mach cone angle a = 10 (M MS = 5.8) was estimated that lead to a hyperbola eccentricity = [31] Figure 7 displays locations of the Interball-1 and IMP 8 observed bow shock crossings in the solar wind flow-aligned frame. In this frame, the solar wind flows in the X direction. For a comparison, CL95ht s predictions of the bow shock shape and position are shown. Two black lines depict the bow shock at times when the last Interball-1 and first IMP 8 bow shock crossings occurred. On the other hand, the orange lines (or band) present variations of the model bow shock during the time interval UT on 5

7 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW SMP 2-7 Figure 7. Downstream mapping of the IMF magnetic field lines from Wind in the solar wind flowaligned frame between 12 and 24 UT on 5 May 1996 compared to CL95ht bow shock predictions. The red stars mark the actual shock crossings by Interball-1 and IMP 8. May The red (green) curve is a bow shock with hyperboloidal (paraboloidal) shape as derived from Interball-1 and IMP 8 observations (see previous paragraph). In spite of Wind s distant upstream location (X GSE 60 R E ), the spacecraft measured foreshock electrons for almost the whole interval UT on 5 May 1996 (Figure 8). The electron pitch angle distributions plotted in Figure 8 were provided by the 3DP plasma experiment on Wind [Lin et al., 1995]. Note that the solar wind, and hence the electron halo component, is streaming anti-parallel to the IMF (180 ). Bursts below 90 pitch angle, which start at 1250 UT, represent electrons reflected from the bow shock. Assuming that all solarwind features traveled with the instantaneous bulk speed v sw, electron trajectories were mapped downstream along magnetic field lines using the propagation method described by Filbert and Kellogg [1979], Etcheto and Faucheux [1984], and Cairns et al. [1997]. The result is displayed in Figure 7, where field lines with (without) foreshock electrons are yellow (blue). Although the yellow and blue lines partially overlap in Figure 7, the transition between those two types of field lines asymptotically approaches the bow shock computed according to the formulas given in the present paper rather than the CL95htmodel s predictions. This can be considered as an indirect evidence of the correctness of our suggested bow shock shape. Note that the yellow and blue lines are curved because of the projection effect in the cylindrical coordinates. 5. Discussion [32] For MHD formulations we know that the angle q between the IMF and solar wind velocity plays an important role in the determination of the bow shock position and shape. In this paper, we describe a case where all solar wind Figure 8. Electron pitch angle distributions measured by the Wind spacecraft at three energies. Bursts below the 90 pitch angle represent electrons reflected from the bow shock.

8 SMP 2-8 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW Figure 9. Magnetopause deformation into a bullet-like shape due to the pressure decrease at the subsolar point in the case of field-aligned upstream flow. and IMF input parameters are within an average or nominal range except for the parameter q corresponding to a nearly flow-aligned field, a situation where MHD models predict the largest q effect. In the described case we have indeed found significant deviations between the magnetopause and bow shock observations and current models that do not take into consideration the q parameter. [33] Specifically, Interball-1 observations found the magnetopause farther outward by 20% than predicted by S98 model. This is a significant difference that cannot be simply explained by the error of prediction of the particular model. Šafránková et al. [2002] compared Interball and Geotail observations with seven magnetopause models and found that the difference between investigated models was smaller than the error of prediction caused by the factors not included in models. Thus one of the factors influencing the magnetopause location could be the angle q between the incident solar wind velocity and IMF direction. [34] Figure 9 presents the suggested shock/magnetopause configuration for the field-aligned upstream flow in comparison with a magnetopause typically used in MHD simulations, which is prescribed with hard, infinitely conducting boundary conditions [i.e., Rizzi, 1971; Cairns and Lyon, 1995]. The magnetopause nose location was determined from Interball-1observations. However, only indirect clues suggest a smaller magnetosphere cross-section: the magnetospheric magnetic field model [Tsyganenko, 2002a, 2002b] excellently agrees with GOES-8 and 9 field measurements at the geosynchronous orbit and does not show an expansion, or contraction for that matter, of the dayside magnetosphere. Therefore nonsimilar shape change and constant volume of the dayside magnetosphere can be surmised, resulting in a bullet-like shape in Figure 9. At this point, we can only speculate about a physical cause for the suggested magnetopause deformation. Possibly, the solar wind is deflected around the obstacle causing rarefaction at the subsolar point and resulting in a decreased magnetic field strength, thus reducing the solar wind pressure on the subsolar boundary. This idea is hardly anything more than a speculation and its purpose is to initiate a discussion because no theoretical work has been done in describing the magnetosphere configuration for the case of a field-aligned upstream flow. Existing magnetosphere/magnetopause models focus on the influence of the IMF B Z component rather than on the IMF orientation itself. [35] On the other hand, MHD simulations of the bow shock shape and position have been presented for the special case when the magnetic field is parallel to the flow [Rizzi, 1971; Cairns and Lyon, 1996] that can be compared with our two-point bow shock observation. Figure 10 presents the gasdynamic bow shock of Spreiter et al. [1966], MHD simulations for M S = 10, M A =2.5byRizzi [1971] and the observed shock wave shape normalized to the magnetopause standoff distance. Furthermore, the anticipated magnetopause is displayed together with the hard, infinitely conducting boundary from MHD simulations [Rizzi, 1971]. In order to compare the various bow shock shapes, the magnetopause surfaces were scaled to the same standoff distance in Figure 10. Note that with increasing upstream Alfvénic Mach number Rizzi s bow shock moves toward the gasdynamic solution. On the basis of MHD theory [Rizzi,1971; Cairns and Lyon, 1996], the minimum magnetosheath thickness is 15% of a magnetopause standoff distance a mp. In contrast, Interball-1 and IMP 8 observations suggest a magnetosheath thickness <10% of a mp. In addition, the upstream Mach numbers were not extremely low (2.5) as required by MHD simulations to significantly reduce the sheath thickness at the subsolar point. In contrary to the MHD simulations, the observed shock wave for field-aligned flow does not flare out at the magnetospheric flanks and its shape resembles the gasdynamic solution (see Figure 10), which could be due to an overall thinner magnetosheath. The differences between MHD theories and observations presented above require a change in our understanding of the magnetospheric response to fieldaligned solar wind flows. 6. Summary [36] On 5 May 1996 the Interball-1 and IMP 8 spacecraft crossed the bow shock boundary nearly at the same time. The upstream conditions were special in two ways: (1) the interplanetary magnetic field was anti-parallel to the solar wind flow within 15 and (2) the conditions were stable for a prolonged period (9 h). [37] At the magnetosphere nose the Interball-1 data revealed that the magnetopause was more distant by 2 R E than predicted by the S98 model and the subsolar magnetosheath thickness was at most 10% of the magnetopause standoff distance, significantly thinner than expected. [38] Assuming a hyperboloidal (paraboloidal) shock wave, the calculated shock standoff distances were 13.7 (13.6) R E and the foci were located on the x axis at 4.5 (4.2) R E ; these values are consistent with previously published results [see Slavin and Holzer, 1981; Cairns et al., 1996, and references therein]. These shocks flare significantly less

9 MERKA ET AL.: MAGNETOPAUSE AND BOW SHOCK FOR FIELD-ALIGNED FLOW SMP 2-9 antiparallel to the x axis. Equation (A1) can be expressed in form where the flaring parameter b s is x ¼ a s b s y 2 þ z 2 ; ða2þ b s ¼ 1 4ða s xþ ; ða3þ in contrast with b s used by Cairns et al. [1995] that was not a function of the bow shock standoff distance. [41] However, the bow shock rather should be described as a hyperboloid that asymptotically approaches the magnetosonic Mach cone as indicated by both theory and observations [e.g., Landau and Lifshitz, 1959; Spreiter et al., 1966; Slavin et al., 1984]. Assuming the same symmetry as for the parabola the hyperboloidal shock shape is ðx ða þ a s ÞÞ 2 a 2 r2 b 2 ¼ 1: ða4þ The ratio b/a follows from the shock surface asymptoting toward the magnetosonic Mach cone: b ¼ tan a; a ða5þ Figure 10. A qualitative comparison of the gasdynamic [Spreiter et al., 1966], MHD field-aligned [for M S = 10 and M A = 2.5; Rizzi, 1971], and observed field-aligned bow shocks. The MHD magnetopause is by Rizzi [1971], and the field-aligned one is being guessed from Interball-1 observations. then MHD simulations predict for field-aligned bow shocks at the magnetospheric flanks (see Figure 10). [39] This peculiar magnetopause/bow shock behavior is not understood currently and will require the development of new theories describing how the Earth s magnetosphere reacts to different IMF orientations. Simultaneous, multispacecraft observations of the bow shock and magnetopause are needed to improve our view of the shape and position of both boundaries for field-aligned upstream flows. Appendix A: Parabola and Hyperbola as a Bow Shock Shape [40] The bow shock can be described as an open secondorder surface in three dimensions. Assuming cylindrical symmetry about the incident solar wind flow the paraboloidal shock shape is r 2 ¼ 2pðx a s Þ; ða1þ where r 2 = y 2 + z 2, the semifocal chord p =2(a s x), a s is the bow shock standoff distance, and x is the position of the focus on the x axis. A coordinate system is used where the Earth is in the origin and the solar wind flow is where a is the asymptotic Mach cone angle. Usually, the asymptotic Mach cone angle a is expressed as a ¼ sin 1 ð1=m MS Þ: ða6þ However, Verigin et al. [2003] recently derived an implicit analytical solution, which enables the calculation of the asymptotic MHD Mach cone angle at any clock angle for an arbitrary M S, M A, and q set. [42] Geometrical properties of hyperbola then lead to a ¼ a s x 1 ; ða7þ where hyperbola s eccentricity e depends on the Mach cone angle a as 2 ¼ 1 þ tan 2 a: If we use equation (A6), then equation (A8) leads to M 2 MS 2 ¼ MMS 2 1 : ða8þ ða9þ From the last equation it readily follows that, for high magnetosonic Mach numbers, the bow shock shape is well described by a paraboloid. For example, shapes with eccentricities > 1.02 require M MS ] 5. [43] Acknowledgments. This work was performed while the author J. Merka held a National Research Council Research Associateship Award at NASA/GSFC. We are grateful to D. G. Sibeck and N. A. Tsyganenko for their fruitful suggestions.

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Szabo, Laboratory for Extraterrestrial Physics, NASA Goddard Space Flight Center, Code 696, Greenbelt, MD 20771, USA. ( jan.merka@gsfc.nasa.gov; adam.szabo@gsfc.nasa.gov) Z. Němeček and J. Šafránková, Faculty of Mathematics and Physics, Charles University, V Holesovickach 2, , Praha 8, Czech Republic. (zdenek.nemecek@mff.cuni.cz; jana.safrankova@mff.cuni.cz)