3.1. CME Initiation
 Detailed descriptions of the 28 March 2001 event have been reported by Wang et al. [2003, 2004]. For completeness we will summarize some of the highlights for this event.
 The first halo CME was visible in LASCO/C2 at 01:27 UT on 28 March 2001. The projected speed according to the LASCO CME catalog is 427 km/s. This CME was considered to be associated with a C5.6 X-ray flare which erupted from AR9401 (N20E22) at 01:29 UT. The second halo CME was visible in C2 at 12:50 UT on the same day. The projected speed is 519 km/s according to the LASCO CME catalog. An M4.3 X-ray flare beginning at 11:21 UT was detected accompanying this event, located in the AR9393 (N18E02). The interval of the two CMEs' initiation was 11.38 h. If their projected speeds were representative of the speeds along the Sun-Earth direction, the second CME was moving faster than the first one. Therefore, it could overtake the earlier one and form a MultiMC as was observed by ACE [Wang et al., 2003]. On 31 March 2001, a very intense forward shock arrived at the L1 point (1.5 × 109 m from Earth sunward) at 00:20 UT on the basis the ACE spacecraft observation. Then, the first magnetic cloud was observed from 05:05 to 10:15 UT and the second one was observed during 12:35–21:40 UT. This MultiMC event caused the largest geomagnetic storm with Dst value of −387 nT during the 23th solar maximum (2000–2001) [Wang et al., 2003].
 To simulate this two-CME event, two high-density, high-velocity and high-temperature magnetized plasma blobs are superimposed successively on a background steady state solar wind medium and a disturbed solar wind medium. The magnetized plasma blob model given by Chané et al. [2005, 2006, 2008] is a kind of very simple non force free flux rope model for CME initiation, which has relative simple type and can reproduce some features about the magnetic cloud; moreover, the best fit parameters of the CME initial state can be determined to get a relative close comparison with the ACE data at the L1 point. In our previous paper [Shen et al., 2011], we used the magnetized plasma blob model to simulate the 4 April 2000 event, and get a relative close comparison with the ACE data at the L1 point. In the plasma blob model, the CME can be launched at a certain velocity, in a given direction and are further characterized by a given density, radial velocity, temperature, magnetic field strength and magnetic polarity. The initial CME magnetic field and the background wind magnetic field can have the same or the opposite polarity. It can be called as an inverse and a normal CME, as described by Chané et al. [2005, 2006].
 The density, radial velocity and temperature profiles of the initial perturbation are defined as follows:
where, acme is the radius of the initial plasma blob, a(r, θ, ϕ) denotes the distance from the center of the initial plasma blob, which can be written as , and (rcme, θcme, ϕcme) is the position of the initial blob center. Here ρmax, vmax and Tmax are the maximum density, radial velocity and temperature in the plasma bubble added on top of the background solar wind, respectively.
 The initial magnetic field of the perturbation in r and θ direction can be defined as [Shen et al., 2011]
is the magnetic flux function.
 This initial perturbation will be given by the following relation:
where ρ0, vr0, T0, Br0 and Bθ0 are the background values of the density, radial velocity, temperature, magnetic field in radial direction and in meridional direction calculated in section 2.
 In our simulation, the radius of the two plasma blobs acme is set as 0.5 Rs and the center of the initial plasma blobs is situated at 3.5 Rs. The second plasma blob is initiated 10 h after the launch of the first one. The other parameters are given as following:
 1. For the first CME (CME1), the initial plasma blob is launched in direction of θcme = 20° and ϕcme = 158° (N20E22 event); the maximum density (ρmax), radial velocity (vmax) and temperature (Tmax) are set as 1.2 × 109 cm−3, 1200 km/s and 4 × 106K, respectively; ψ0 is set as 2.0 to obtain the initial maximum magnetic field strength as ∼6 × 105 nT;
 2. For the second CME (CME2), the initial plasma blob is launched in direction of θcme = 18° and ϕcme = 178° (N18E02 event); the maximum density (ρmax), radial velocity (vmax) and temperature (Tmax) are set as 1.5 × 109 cm−3, 1500 km/s and 5 × 106K, respectively; ψ0 is set as −2.4 to obtain the initial maximum magnetic field strength as ∼8 × 105 nT, with an inverse magnetic polarity compared with CME1.
 The parameters of the magnetized plasma blob are chosen to agree with the observed values of initial latitude, longitude and approximate speed of the two CMEs (assuming the vave of the plasma blob approximately equal to 1/3 vmax) from SOHO/LASCO, the maximum of the shock speed, numerical density and the changing mode of Bz (N-S-N-S) at the L1 point observed by ACE as the best fit as possible. This initiation model will yield the driving force to launch a CME. The introduction of the additional heating by raising the temperature of the plasma blob can generate the driving force to launch a CME, since the pressure force is calculated from p = ρRT, which was also mentioned in our previous paper [Shen et al., 2011].
 Figure 3 displays the 3-D intuitive views of the isosurfaces with three values of the radial velocity vr and the initial magnetic field lines of CME1 (Figure 3a) and CME2 (Figure 3b) initialization on the background solar wind, including zooming in on the plasma blob. It should be pointed out that to emphasize the initial plasma blob, we only shows three kinds of color corresponding to three levels of the isosurface of the radial velocity vr, which are vr = 1000, 800, 600 km/s in Figure 3a and vr = 1400, 1000, 600 km/s in Figure 3b, without showing the other levels of color contours. Figures 3a and 3b show that the maximum value of the radial velocity appeared at the center of the initial plasma blob in the two CMEs initialization, and the magnetic field of CME2 has initially the inverse polarity compared with CME1.
Figure 3. Three-dimensional views of the isosurface of the radial velocity vr and the initial magnetic field lines of the CME initialization for (a) the first CME and (b) the second CME, including zooming in on the plasma blob.
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3.2. Dynamical Evolution, Overtaking, and Interaction of the Two CMEs
 In this section, the simulation results are presented. To identify the shock front, we use the value of the relative density ((ρ − ρo)/ρo) being 2.0 as the criterion to identify the position of the shock front. Then we define the distance of the shock front by using the maximum value of the heliocentric distance in the shock front plane. Figure 4 shows a time-height plot of the shock front distance of the two CMEs from t = 0 to 57.3 h. The second eruption is initiated 10 h after the launch of the first one, at which time the distance of the CME1's shock front is at 45.3 Rs.
Figure 4. Comparison of the shock front heliocentric distance between the first CME (dashed line) and the second CME (solid line).
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 Figures 5 and 6 show the 3-D propagation of the simulated CMEs at 40 min (Figures 5a and 6a), 5 h (Figures 5b and 6b), 10 h 40 min (40 min after CME2 launched) (Figures 5c and 6c), 20 h (Figures 5d and 6d), 32 h (Figures 5e and 6e), 40 h (Figures 5f and 6f), 50 h (Figures 5g and 6g) and 60 h (Figures 5h and 6h) after the launch of CME1. Figure 5 represents the relative density ((ρ − ρo)/ρo) and magnetic field lines, and Figure 6 shows the radial velocity and magnetic field lines. The magnetic field topology is represented by the rod-shaped magenta lines in both Figures 5 and 6. Because the initial heliolatitudes of the two CMEs are similar (θ = 20° and 18°), a 2-D distribution of the relative density at a constant latitude angle of 18° is given in Figure 7 to provide another view of the two CMEs at t = 30 h (Figure 7a), 32 h (Figure 7b), 36 h (Figure 7c) and 40 h (Figure 7d).
Figure 5. Three-dimensional representation of the CMEs (140° < ϕ < 180°) shown (a) 40 min, (b) 5 h, (c) 10 h 40 min (40 min after the second CME launched), (d) 20 h, (e) 32 h, (f) 40 h, (g) 50 h, and (h) 60 h after the first CME initiation. The solid rod-shaped lines are magnetic field lines, and the color codes represent the relative density ((ρ −ρ0)/ ρ0). (Axis units given in Rs.)
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Figure 6. Three-dimensional representation of the CMEs (140° < ϕ < 180°) shown (a) 40 min, (b) 5 h, (c) 10 h 40 min (40 min after the second CME launched), (d) 20 h, (e) 32 h, (f) 40 h, (g) 50 h, and (h) 60 h after the first CME initiation. The solid rod-shaped lines are magnetic field lines, and the color codes represent the radial velocity magnitude. (Axis units given in Rs.)
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Figure 7. Evolution of the two-dimensional relative density ((ρ −ρ0)/ ρ0) contours of the constant latitude angle of θ = 18° at four consecutive times. (Axis units given in Rs.)
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 The shock fronts with a high velocity and relatively high density are clearly visible in Figures 5, 6, and 7. At time t = 40 min, the flux rope of CME1 still remain an almost circular shape, but is quickly stretched to a pancake shape after t = 5 h. The same thing also happens to CME2. At t = 20 h, the distance between the shock fronts of the two CMEs is 25.7 Rs and the two flux ropes get closer. At t = 32 h, the shock front of CME2 reaches the trailing edge of CME1. Then from 32–40 h the shock of CME2 penetrates through the body of CME1. At time t = 39 h, the two shock fronts are only 2.37 Rs apart, and from time t = 40 h, the two shocks begin to merge to a stronger combined shock, seen from Figures 4 and 7 and from Figures 5e, 5f, 6e, and 6f. In Figures 5e–5h and 6e–6h, after time t = 32 h, the flux rope of CME2 with relative low density and high velocity overtakes the flux rope of CME1, and the oblique collision occurs between the two flux ropes. It induces obvious deformation and compression of the two flux ropes, and the second flux rope develops its shape from circular-like structure to a flat structure. However, there are not coherent spiral-shaped magnetic field lines in Figures 5 and 6, which can be explained by that the absence of an initial axial component Bϕ may destabilize the imposed magnetic field.
 To study the interaction between the two CMEs, we make a comparison of the simulation results among three different cases: double CMEs, CME1 only and CME2 only, with other conditions being the same. Figures 8a and 8b show the comparison results of the temporal evolution of the heliocentric distances of the shock fronts from near the Sun to 213 Rs for CME1 and CME2, respectively. In Figures 8a and 8b the shock fronts of CME1 and CME2 in the interaction case moves faster than those in noninteraction case. In Figure 8a the influence of CME2 to the moving speed of the shock front of CME1 primarily happens after time t = 40 h, when the shock of CME2 overtakes the shock of CME1 and merging into one combined shock. While in Figure 8b the difference of the two curves happens much earlier than in Figure 8a. After time about t = 10 h, the heliocentric distance of the shock front of CME2 in the interaction case increases more quickly than that of only CME2 case. This is probably because, when CME1 propagates into the background solar wind, it removes some of the background's mass. Associated with the propagation of the CME, there is as well a disruption of the solar wind, especially of the background magnetic field. As a consequence, CME2 does not propagate into the background solar wind, but into a disturbed medium, less dense, faster and more magnetized. This kind of analysis was also made by Lugaz et al. .
Figure 8. (a) Temporal evolution of shock front distance of the first CME with (black line) and without (red line) interaction with the second CME and (b) temporal evolution of shock front distance of the second CME with (black line) and without (red line) interaction with the first CME. For the black line in Figure 8b the time reference has been shifted by 10 h so that time t = 0 h corresponds to the initiation of the second CME.
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 Figure 9 plots the velocity (v; Figure 9a) and z component magnetic field (Bz; Figure 9b) at the L1 point from time t = 40 h to 100 h of the double-CME case (black line), of only CME1 (red line), of only CME2 (initial time t = 0 h, to be compared with only CME1 case) (green line) and of only CME2 shifted by 10 h (initial time t = 10 h, to be compared with double-CME case) (blue line). From comparing the red line and green line of Figure 9a, the arrival time of the shock at the L1 point is 70.4 h and 61.9 h of the only CME1 case and only CME2 case after the CME initiation, respectively, which demonstrates that the initial velocity of the CMEs can obviously influence the arrival time of the shock at the L1 point. Meanwhile, we can compare the arrival time at the L1 point of the shock using two CMEs initiation (black line) with using only one CME initiation (red line and blue line), since the exact same parameters are used in both cases to generate the CME. We notice that the shock arriving at the L1 point ∼67.4 h after the launch of CME1, whereas in the case without interaction, the shock arrives at the L1 point 70.4 h for only CME1 case and 71.9 h for only CME2 case. So the arrival time of the shock with two CMEs interaction is the shortest one among the three cases, which is also consistent with the result of Figure 8. At the shock, the solar wind speed jumps from about 457 km/s to over 650 km/s for two CMEs interaction case, to about 550 km/s for only CME1 case and to about 610 km/s for only CME2 case, respectively, which indicates that the shock of the two CMEs merges into one stronger combined shock.
Figure 9. Temporal evolution of (a) velocity (v) and (b) z component magnetic field (Bz) at the L1 point of the double-CME case (black line), only the first CME (red line), only the second CME (initial time t = 0 h, to be compared with only the first CME case) (green line), and only the second CME shifted by 10 h (initial time t = 10 h, to be compared with double-CME case) (blue line).
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 In Figure 9b, for the z component of the magnetic field, the profiles for the three cases at the L1 point are different: (1) becoming northward first, then changing to southward, later changing to northward again, and finally tuning southward for a long time for two CMEs interaction case; (2) becoming northward first, then changing to southward, and finally turning to near 0 for only CME1 case; and (3) becoming southward first, then changing to northward, later changing to southward slightly, and finally turning to near 0 for only CME2 case. The maximum of the south component of magnetic field, Bs, is about 34 nT for two CMEs interaction case, while only about 5 nT for only CME1 case and about 13 nT for only CME2 case. The fact that the large duration of a strong Bs appearing in the double-CME interaction cases, does not appear in the only CME1 case or the only CME2 case suggests the importance of the interaction between two CMEs in causing major geomagnetic storms.
3.3. Properties of the Simulated CMEs at L1 Point and Comparison With ACE Data
 Figure 10 shows the comparison of the computed plasma and field parameters at the L1 point in Figure 10b with the observed MultiMC of 31 March to 1 April 2001 shown in Figure 10a. Figure 10 depicts the plots of total, z component magnetic field in GSE coordinate system, number density, temperature, velocity and plasma β at the L1 point. At the L1 point, the two flux ropes have evolved to show some of the characteristics commonly associated with MultiMC, namely two high magnetic field strength regions separated by a region of increased β, smooth variation of the magnetic field in each cloud, a shorter duration of the first cloud compare to the second one, low proton density and temperature in both clouds and a high-velocity profile [Wang et al., 2003; Lugaz et al., 2005].
Figure 10. A comparison of the MHD simulation of the magnetic field and plasma parameters with the measured (ACE spacecraft) magnetic field and solar wind parameters at the L1 point in 2001. (a) Measured parameters by ACE, top to bottom: magnetic field strength ∣B∣ (nT), Bz (nT) at GSE coordinate system, proton density (cm−3), proton temperature (K), velocity (km/s), and plasma β. (b) Simulation parameters, top to bottom: magnetic field strength ∣B∣ (nT), Bz (nT) at GSE coordinate system, number density (cm−3), plasma temperature (K), velocity (km/s), and plasma β. The blue vertical solid lines indicate the arrival time at L1 of the shock, the red vertical dashed lines denote the beginning and ending of the first MC, and the green vertical dashed line denotes the beginning of the second MC.
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 As seen in Figure 10b, the simulated shock reaches L1 at DOY of 89.86, marked with the blue vertical solid line, and the two magnetic clouds are preceded by a very strong shock with high density, temperature and velocity. The first compressed magnetic cloud (MC1) arrives at L1 during DOY of 90.22–90.40, marked with red vertical dashed line, and after a short time, the second overexpanded magnetic cloud (MC2) reaches L1 from DOY of 90.49, marked with green vertical dashed line. As the shock passes L1, the density increases from 16 cm−3 to 67 cm−3, the temperature increases by a factor of over two and the velocity jumps from about 457 km/s to over 650 km/s. The velocity first decreases slightly behind the shock, but then increases again to a higher value of ∼700 km/s in the MC2 than in the MC1. This obvious increase in velocity in the rear part of plasma is specific to this model of CME interaction, and was not found in only CME1 or only CME2 case, as seen from Figure 9a. At the sheath and MC1, the density drops from over 60 cm−3 to ∼5 cm−3 and the temperature decreases from ∼4 × 105 K to ∼1.6 × 105 K. In MC2, both the density and temperature remain the relative low values of 4–6 cm−3 and 1.5 × 105–2 × 105 K, respectively, and the velocity remains relative high value of 680–740 km/s.
 The maximum magnetic field strength of 46 nT is obtained in the MC1 at DOY of 90.35, 12 h after the shock's arrival. In the connection region between MC1 and MC2, the magnetic field reaches a minimum value of 6.5 nT. The MC2 has a maximum magnetic field strength of 26.3 nT. As seen from Figures 9b and 10b, when the first flux rope arrives at the L1 point, Bz turns northward for about 8 h reaching a maximum value of 11.5 nT. Then, Bz smoothly rotates southward to reach a minimum value of −34 nT in MC1, and the magnetic field remains southward for 3.7 h. As the second flux rope passes L1, the magnetic field turns to northward again for 3.4 h, reaching a maximum value of 16.8 nT, and then rotates southward for more than 16 h, reaching a minimum value of −12.5 nT. In the connection region between MC1 and MC2, the value of plasma β increases by a factor of ∼17, corresponding to the low magnetic field. Such high-β interaction regions were also described by Wang et al.  for three different MultiMC events in March–April 2001 and by Lugaz et al.  for simulating this 28 March 2001 MultiMC event.
 Comparing our simulation result with ACE data, we find that in spite of the simple CME model used, our simulation is in good qualitative agreement with the data. The two MCs are preceded by a very strong shock with high density, temperature and velocity. The maximum values of the velocity and density almost display realistic values of over 700 km/s and 67 cm−3, respectively. The transit time of MC1 from DOY of 90.22 to 90.40, and the start time of MC2 of DOY of 90.49 are approximately reproduced. The density, temperature and plasma β in the two MCs remain relative low value of almost below 20 cm−3, 2 × 105 K and around 0.1, respectively, and the velocity in the two MCs has relative high value of 600–700 km/s, which are consistent with the values of density, temperature and velocity in MC1 and MC2 of observational data. The two MCs are separated by a region with a relative large β (βmax > 1, while around 0.1 in the clouds) and low magnetic field strength (∣B∣min < 10 nT compared with values larger than 20 nT in the clouds). For the z component of the magnetic field, the simulated and measured profiles at the L1 point are similar, becoming northward first, then changing to southward, later changing to northward again, and finally turning southward. And the southward component of the magnetic field reaches a maximum value of 34 nT in MC1 and remains southward for a long time, which lead to a peak value of Dst of −387 nT [Wang et al., 2003].
 Some quantitative disagreement between our simulation and reality is to be expected. The shock center characterized by the maximum value of the velocity arrives 3.6 h earlier in the simulation. The maximum value of the magnetic field strength and temperature in the simulation in only ∼64% and ∼50% of the ACE data, respectively. An obvious increase in magnetic field strength at the shock is absent and the magnetic field is weak in the simulated sheath, compared with the observational data. The simulated velocity of MC2 is higher than the measured data. The relative high temperature and density in the connection region between the two MCs aren't attained in the simulated profile, but can be seen in the measured profile. So the simulated maximum value of the plasma β in the connection region is much lower than the observational data.
 One limit causing the inconsistency is that our present model, like many others already mentioned, is only a single-fluid (proton) model, which cannot account for the high electron temperature in the CMEs and the anticorrelation between the electron temperature and density. The magnetic field by our model is always week compared to observations, which is also a result of the initial synoptic magnetic field being too weak at the poles. Meanwhile, the absence of an initial axial component Bϕ may destabilize the flux rope and make it more prone to being deformed and incoherent, and can also influence other parameters, such as the mismatch of the higher velocity in MC2, the lower-density and lower-plasma β between the clouds, compared with the observation data. Moreover, there exist two other extremely important and still unsolved limits as pointed out by Dryer [1998, 2007] and now realized by many other modelers [Fry et al., 2001; Odstrcil et al., 2004; Feng et al., 2011]. One is the uncertainty of the initial realistic solar wind and interplanetary magnetic field (IMF) background conditions, and the other is the uncertainty of the appropriate solar observations used to “mimic” solar flare/filament and CME initiation. To some extent, our establishment of using more observational data such as photospheric magnetic fields by SOHO/MDI and the brightness recorded in SOHO/LASCO to constrain the model is an attempt to reduce the uncertainty of the initial realistic solar wind. But, how to “mimic” solar flare/filament and CME initiation based on the solar observations is a challenge.
 We believe that more solar and interplanetary observations will be able to minimize these uncertainties. For example, the recently launched Solar Dynamic Observatory (SDO) will help us understand the Sun's magnetic changes. SDO will tell us more about how the magnetic field is generated and structured, and how the stored magnetic energy is released into the heliosphere and geospace. STEREO observations can provide new insights into the 3-D structure of CMEs and their evolution in the heliosphere which can directly be compared with existing models and simulations. Comprehensive data and analysis with multiple spacecraft (such as SDO, STEREO, SOHO, ACE, WIND, or other future missions) will probably help us develop the ability of including physically realistic coronal heating modules into 3-D MHD codes, improve the determination of the structure of the ambient solar wind, and further numerically characterize the 3-D propagation of CMEs through the heliosphere, as mentioned by Feng et al. [2010, 2011].