The paper continues the study of the evolution of interplanetary (IP) shocks in the solar wind and their propagation through the magnetosheath. We compare profiles of the magnetic field and plasma parameters observed experimentally by several spacecraft in the solar wind and magnetosheath with profiles of the same parameters generated by two MHD numerical models (the global 3D MHD BATS-R-US model and a local magnetosheath model). The experimental magnetosheath data are well reproduced by the model profiles even though the profiles exhibit complicated structure caused by the interaction of the IP shock with Earth's bow shock and magnetopause. The good agreement of experimental data with the model results suggests that kinetic effects play a minor role in these interactions.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 The speed of a shock and its geometry are key parameters used as input by many numerical models that predict the arrival of interplanetary (IP) shocks at Earth. These IP shocks are typically associated with either co–rotating interaction regions or coronal mass ejections, which propagate supersonically in the interplanetary medium. IP shocks are often precursors of large–scale heliospheric structures which can produce large geomagnetic disturbances.
 The shock normal direction, propagation speed, and arrival time are the quantities that control the efficiency of an interaction between the shock and Earth' magnetosphere. These shock characteristics can be determined from different methods, either from observations at a single spacecraft or from simultaneous multiple spacecraft measurements. Several single–spacecraft techniques have been developed to derive these characteristics from a magnetohydrodynamic (MHD) approximation (e.g., magnetic coplanarity, velocity coplanarity, different mixed data methods, and least–square Rankin–Hugoniot (R–H) technique). Szabo  compared the four–spacecraft timing method with the least–square R–H technique and found that the four–spacecraft timing method is not generally superior in accuracy. On the other hand, Koval et al.  used four spacecraft to determine shock characteristics and argued that the most reliable and least noisy parameters that can be derived from spacecraft observations are usually the time of the shock arrival at the spacecraft location and its coordinates. They also demonstrated that the shock front in the magnetosheath is inclined and that this inclination results in a delay of the shock arrival at the magnetopause compared to the assumption of planar propagation that follows from the Stahara and Spreiter  model supported by Szabo . However, we note that their model uses a gasdynamic approximation and that the magnetopause location is fixed and thus the validity of these predictions is limited.
 In this paper, we present one experimental example of a non–planar IP shock propagating through the magnetosheath and compare the measured density, velocity, and magnetic field profiles with two numerical MHD models. We discuss the forms of these profiles and the influence of the magnetopause reaction on model results.
 In late May 17 and early May 18, 1999, SOHO, ACE, Wind, and Interball–1 were located in the solar wind and registered the passage of an IP shock; Geotail, in the magnetosheath, observed a corresponding shock–like discontinuity. The times of the shock transitions and the spacecraft locations are summarized in Table 1. Figure 1 shows the spacecraft geometry and model positions of Earth's bow shock [Jerab et al., 2005] and magnetopause [Petrinec and Russell, 1996].
Table 1. Locations of the Spacecraft and the Times of the IP Shock Passagea
GSE Location of Spacecraft
Time of Arrival, UT
t is the temporal resolution of data used for the IP shock identification.
Figure 2 shows the ACE, Wind, and Interball observations of the IP shock transition. The simultaneous jumps of the magnetic field magnitude, proton number density, and bulk velocity indicate that the observed IP shock is a fast forward shock. The three bottom panels in Figure 2 present the magnetic field magnitude, proton density, and bulk speed measured by Geotail in the magnetosheath. The jumps of the magnetosheath parameters are not as sharp as in the solar wind.
 To study the propagation of the IP shock toward the magnetosheath, the shock normal orientation and speed in the solar wind must be determined. We used magnetic field and plasma data from the ACE and Wind spacecraft and applied the R–H relations to determine the local shock normal and speed. With four spacecraft, we can also estimate the shock normal and speed as global shock parameters from the times of shock arrival at each spacecraft. Table 2 shows the derived shock parameters and Figure 1 shows the shock orientations: local shock normals computed from the R–H relations are denoted by arrows, while the global shock plane computed from the shock arrival times at each spacecraft is represented by a dashed line.
Table 2. Parameters of the IP Shock (Shock Normals, Speeds in km/s (v1 in the Earth's Frame and v2 in the Plasma Frame), Alfvénic, and Fast Magnetosonic Mach Numbers in the Solar Wind and Magnetosheath
(−0.990, 0.005, 0.139)
(−0.996, −0.087, −0.003)
4 S/C timing
(−0.985, −0.063, −0.159)
(−0.961, 0.276, 0.000)
 The R–H relations can also be applied to the magnetosheath measurements of Geotail. The derived parameters are presented in Table 2 and the computed IP shock orientations and locations are summarized in Figure 3. The straight lines show the shock planes estimated from ACE, Wind, and from multi–spacecraft measurements at the time of the Geotail shock observation. The arrow at the Geotail location points along the shock normal computed from the Geotail data. Figure 3 also shows that if the shock planarity was the same in the magnetosheath as in the solar wind, the shock would arrive at Geotail 147, 95, or 35 s (with a maximum value of standard deviation of 74 s for the middle value) earlier than actually observed, based on the shock normals and speeds determined from Wind, four spacecraft timing, and ACE data, respectively. The heavy curved line is an approximation of the IP shock front in the magnetosheath under assumption that the four–spacecraft method provides accurate IP shock parameters. The shock orientation in the magnetosheath differs significantly from that in the solar wind and the IP shock propagates much slower through the magnetosheath. However, the shock speeds in the plasma frame are almost equal in the solar wind and magnetosheath (Table 2). Therefore, the decrease of the shock speed in the magnetosheath can be attributed to a smaller bulk flow velocity as noted by Koval et al. .
3. MHD Modeling
 In order to gain a deeper insight into the nature of the IP shock propagation and its interaction with the bow shock and the magnetosheath in its pre– and post–shock states, we used a global BATS-R-US model [Gombosi et al., 2002] and a local magnetosheath model [Samsonov, 2006].
 The BATS-R-US (a Block Adaptive–Tree Solar–wind Roe–type Upwind Scheme) model solves fully conservative MHD equations and employs a split B = B0 + B1, with B0 being the analytically defined Earth dipole field and B1 the first–order (time–varying) part of the magnetic field [e.g., Gombosi et al., 2002]. BATS-R-US uses a high–resolution finite-volume approximate Riemann solver scheme for calculation of ideal MHD equations. As initial conditions, BATS-R-US fills the simulation box with the solar wind plasma density, flow velocity, and interplanetary magnetic field (IMF) conditions at a start time and adds the tilted dipole field. In a set–up phase, a stationary state is computed from the initial state forming the near–Earth's magnetosphere, magnetosheath, and magnetotail structures. After this phase, the time–dependent simulation is started and realistic magnetospheric and magnetotail conditions are achieved 10–15 min after starting the simulation. In regions where large spatial gradients exist, the resolution is enhanced, with a minimum grid spacing of 0.25 RE.
 The numerical magnetosheath model of Samsonov [Samsonov, 2006] is based on the interaction of the supersonic solar wind with a parabolic obstacle and uses non–stationary 3–D MHD equations. Two coordinate systems are used, parabolic coordinates in the main part of the numerical box and spherical coordinates near the subsolar region, in order to avoid the singularity of the parabolic coordinates at the Sun–Earth line. Both coordinate systems intersect and the values from inner points of one system are applied to determine the boundary conditions for the other system.
 This model uses the TVD Lax-Friedrichs II-order scheme [e.g., Toth and Odstrcil, 1996] for the calculations. The normal magnetic field is to be imposed at the inner boundary of our simulation box using condition div B = 0 and the field is corrected after a few time steps by the projection scheme to hold the constraint divB = 0. The original code of Samsonov  uses a fixed, impenetrable magnetopause as the inner boundary of the simulation box and, consequently, the normal velocity and IMF components are set to zero at this boundary. This simplified approach corresponds to a situation when the magnetopause does not react (or reacts very slowly) to a pressure pulse. In order to account for a magnetopause reaction, the code was modified to allow plasma outflow through the inner boundary. Thus the normal velocity at the boundary is proportional to the increase of the total plasma pressure. This method is discussed by A. A. Samsonov et al. (Numerical MHD modeling of propagation of interplanetary shock through the magnetosheath, submitted to Journal of Geophysical Research, 2005) and we point out here that it corresponds to an immediate magnetopause reaction to the pressure pulse. The real reaction would be probably somewhere in between, but a comparison of these two cases should elucidate the influence of the magnetopause reaction on the IP shock propagation in the magnetosheath.
 We use the MHD simulations to predict magnetic field and plasma parameter profiles at Geotail and compare these predictions with observations in Figure 4. The black solid lines show the Geotail data averaged to 15 and 50 s for the magnetic field and ion flow, respectively. BATS-R-US computations are shown by dashed lines; red and blue lines show the two results from the local magnetosheath model – with solid (red) and movable (blue) inner boundaries. The comparison with the data reveals a good coincidence between the predicted and observed times of the shock passage, indicating that the models predict the observed deceleration of the shock in the magnetosheath. Differences in timing among models are small and they are attributed to the differing grid resolutions. The main difference between the models is the behavior of the IMF BZ component (last panel) that is nearly zero in the BATS-R-US computation. Profiles provided by the local magnetosheath model are more similar to the observations. The fluctuations observed in the post–shock interval are probably caused by the magnetopause reaction and thus their forms differ in the two versions of the local model.
 All the modeled pre–shock magnetosheath values in Figure 4 are very close to those observed. The local magnetosheath model (both versions) gives a better match, probably because this model uses an approximation of the magnetopause surface obtained from the Shue et al.  empirical model, whereas the BATS-R-US magnetopause is built self–consistently under simplified assumptions. The same is true for the post–shock interval.
 The most interesting results were obtained for the interval immediately following the IP shock ramp (0100 – 0112 UT). All models predict an overshoot in the plasma velocity profile that is consistent with the observations. The observed density overshoot, however, is much larger than the model predictions. The most distinct overshoot is predicted by the local magnetosheath model with a solid obstacle (red line in Figure 4), whereas the overshoot is missing in the same model with the movable obstacle (blue line). This difference suggests that the presence of the overshoot is connected with the magnetopause reaction to the pressure pulse. The height of the overshoot is underestimated in all models and thus it may be connected with non–MHD effects.
 The BATS-R-US profiles of the magnetic field magnitude, BX and BY are consistent with observations, whereas both versions of the local magnetosheath model predict a smoother rise than observed at the shock. The behavior of IMF BZ predicted by the BATS-R-US model differs from that observed throughout the whole interval, as noted above.
 The differences in the timing of the magnetosheath features between the local and global model results may be because the local model uses ACE as a solar wind monitor, whereas Wind was used as the input for the global model. We propagate the solar wind data using spacecraft positions and actual IP shock speed and then, the timing was slightly adjusted using INTERBALL–1 as a second monitor. Other solar wind features were propagated using the separation of a particular spacecraft from the simulation box along the Sun–Earth's line and the actual value of solar wind velocity. Since Wind and ACE are separated by ∼35 RE in the perpendicular direction, the solar wind inputs slightly differ for the global and local simulations.
 We analyzed propagation of an IP shock through the solar wind and magnetosheath. Multi–spacecraft observations of the shock passage in the solar wind indicate that the shock parameters are similar at different locations. However, the shock front significantly deflects and decelerates in the magnetosheath. The deceleration of the shock is also confirmed by the 35–147 s difference between the predicted and observed times of the shock arrival. The decrease of the shock speed is manifested by the smaller bulk flow velocity in the magnetosheath compared to that in the solar wind. Such IP shock deceleration was suggested by the statistical study of Koval et al. . Since the data and model results provide a self–consistent picture, we prefer this interpretation over an alternative explanation based on a curved shock front in the solar wind. The models also reproduce the observed pre– and post–shock conditions as well as the steepness of the shock front in the magnetosheath. We did not find any significant differences in the plasma parameters predicted by the BATS-R-US model and two versions of local magnetosheath model. A comparison of two versions of the local model showed that the overshoot of the plasma density is caused by a delay in the magnetopause reaction to the increase of the upstream pressure.
 A surprising result is that, although the magnetic field magnitude is described very well by BATS-R-US, this model strongly underestimated the IMF BZ component.
 All the basic features of the IP shock propagation in the magnetosheath were modeled successfully with the grid resolution used (∼0.25 RE for BATS-R-US and ∼1000 km for the local model in the investigation region) and a further decrease of the grid spacing did not bring any notable improvement.
 Finally, the good overall agreement between the measured and modeled data suggests that MHD effects dominate over kinetic effects even in the vicinity of the magnetopause. A further improvement of MHD results probably requires a better description of the solar wind input, possibly through the use of data from all available solar wind monitors.
 The authors thank the CDA Web service for data from Wind and ACE and acknowledge the CCMC and the originators of the BATS-R-US model. The present work was supported by the Czech Grant Agency under contracts 205/05/0170 and 202/03/H162, by the INTAS grant for young scientists 03-55-1034, by RFBR grant 03-05-64865, and by the USA NSF grants ATM-0207775 and ATM-0203723.