[17] The forces acting on a unit volume with density *ρ* can be written in detail as below [*Cargill*, 2004; *Chen and Krall*, 2003; *Wu et al.*, 2004]: (1) The Lorentz force, ; the first, , can be thought of as due to a gradient of magnetic pressure ; and the second part, , is magnetic tension; (2) The pressure gradient, *F*_{p} = −∇*p*; (3) the aerodynamic drag force, *F*_{D} = −*ρ*_{e}*AC*_{D}(*V*_{i} − *V*_{e})|*V*_{i} − *V*_{e}|/*τ*; (4) the gravitational force, *F*_{g} = −∇*ϕ*, where *τ* and *A*are the volume and the cross-sectional area of the CME,*C*_{D} is the drag coefficient and subscript *i*, *e* refers to quantities inside and external, to the CME, respectively. In our simulation, *C*_{D} is set as 1.0 [*Cargill*, 2004; *Cargill et al.*, 1994; *Schmidt and Cargill*, 2000; *Temmer et al.*, 2012]. *τ* is the volume which is defined as being a region where the relative density is not less than 0.5. *ϕ* is the gravitational potential. Here, we only studied the forces on radial direction attribution to the acceleration and deceleration of the CMEs on radial direction. And the *r*-component of the pressure gradient at point (*i*, *j*, *k*) is determined by: . The total force *f* acting on a unit volume is given as: *f* = *F*_{L} + *Fp* + *F*_{D} + *F*_{g}. The forces are calculated at every 45 min.

#### 3.2. Case 2: Interaction of Two CMEs

[22] In this section, we present simulation results for the interaction of two CMEs.

[23] Figure 4shows the curves that portray (a) the distance-time profiles (b) the radial speed-time profile and (c) the speed-distance profile of the two CMEs interaction effects from*t* = 0 to 60 h. The second eruption is initiated 10 h after the launch of the first one. The first in contact is ∼30 h (see Figure 5e and 6e in *Shen et al.* [2011b]). At ∼40 h, CME2 catch up with CME1. Because the CMEs' initial speeds are much higher than the ambient solar wind speed, at the initial time, both of the two CMEs transfer momentum to solar wind, and they begin to decelerate rapidly. At ∼37 h, the two CMEs get closer, while the speed of CME1 is less than that of CME2. CME1 gains momentum from CME2, then, CME1 begins to accelerate from 37 h to 45 h and after 45 h, both of them decelerate together. This momentum transfer is same as what happened due to the momentum conservation (snow-plow model) proposed by*Tappin* [2006]. As the faster, denser CME2 propagates near CME1, it slows down due to the interaction with CME1 by the momentum transfer, and then the slower CME1 ahead of it is accelerated.

[24] The solid lines with different colors in Figure 5 and Figure 6 show the detailed breakdown of the six forces in the radial direction which contributes to the acceleration and deceleration of CME1 and CME2, respectively, at different time intervals from the initiation. The black curves in Figure 5 and Figure 6 show the sum of all the forces which represent the total force acting on CME1 and CME2, respectively. In the earlier stage of the evolution, the total force is dominated by the positive pressure gradient and the negative aerodynamic drag force, especially the drag force, because the CME speed is much faster than the solar wind speed, thus the total force is negative toward Sun in the initial time, it causes both CMEs to decelerate.

[25] From Figure 5, we find that as CME1 propagates into the heliosphere, all the magnitude of the forces acting on CME1 become much smaller in comparison to the total force during the initiation phase. Because of the interaction between the two CMEs, when the two CMEs get closer and closer around 37 h, the pressure, the velocity and the magnetic field of CME2 are all higher than that of CME1, the pressure, velocity and the magnetic field at the CME1 front will increase because of the momentum and energy transfer from CME2 to CME1. So near 37 h, an obvious bump appears in the pressure and the magnetic pressure; and a dip appears in the drag force and the magnetic tension, based on the definition of the forces. The total force begins to turn from negative to positive, which is consistent with the speed curve in Figure 4.

[26] It should be noted that the dashed lines in Figure 5a–5f show the forces acting on the single CME1 at different time intervals, without the effects of CME2. In the early stage (i.e., Figures 5a–5c), the solid lines and the dashed lines are identical, which means that the two CMEs interaction has not engaged yet as shown in Figure 4. From the difference between the solid lines and the dashed lines of the two cases, we find that the total force of the single CME case remains negative from *t* = 0 to 60 h, and the CME speed keeps declining. While in the interaction case, CME1 begins to accelerate and the total force changes from negative to positive near 37 h. This can be explained by the momentum transfer between the two CMEs. In the interaction case, when the two CMEs get closer and closer around 37 h, while the CME1 speed is less than the CME2 speed, the CME1 gains momentum from CME2, so CME1 begins to accelerate after 37 h.

[27] In Figure 6, also shows that all the forces acting on CME2 declined rapidly as CME2 propagates into the heliosphere, and the total force begins to turn from negative to positive near 15 h, which happens much earlier than that for CME1, it can be noted from Figure 6c. It should be mentioned that there exist differences between the two CMEs; when CME1 propagates into the background solar wind, it removes some of the background's mass. Therefore, CME2 does not propagate into the original background solar wind, but into a disturbed medium, less dense, faster and more magnetized [*Lugaz et al.*, 2005; *Xiong et al.*, 2009]. As shown in Figure 6cthe positive total force lasted only for about 5 h, and then turned into negative again, which is consistent with the small dip around 16 h in the speed-time curve ofFigure 4. The total force remained negative until near 40 h. After 40 h, the total force becomes positive, lasting for no more than 10 h, and then turns negative. From the corresponding speed-time curve inFigure 4, we find that the speed keeps on decreasing and the slope of the curve is also decreased. From Figure 6e, we notice that between 34 and 37 h, there is a significant negative contribution of the Lorentz force to the total force acting on CME2, and the magnetic pressure keep on decrease while the magnetic tension has an obvious dip between 32 and 38 h which is probably because that the shape of the flux rope behind the shock of CME2 changed obviously after the interaction of CME1 and CME2, as shown in Figure 7, which describe the 3D flux rope of CME2 at 32 h (a) and 38 h (b), respectively. This kind of negative magnetic force contributing to the deceleration of CME during the CME-CME interaction event was also reported by*Temmer et al.* [2012].

[28] The dashed lines in Figures 6a–6e show the forces acting on the single CME2 during different time intervals, without interaction with CME1. It should note in Figure 6a, the solid lines and dashed lines are identical at the first hour, because these two CMEs have not engaged yet, which also can be indicated by the shock front distance versus time curve (Figure 10a). From the difference between the two cases, we find that the total force of a single CME case remains negative, and the CME speed keeps on declining. In the interaction case, the total force of CME2 remains negative most of the time, and it turns to positive mainly around 18 and 40 h, which corresponds with the small bump of the speed and the decrease of the slope in the speed-time curve.

[29] In order to understand further the acceleration and deceleration of these two CMEs, we have plotted the temporal evolution of the ratio of the Lorentz force and the thermal pressure force to the aerodynamic drag force and the gravitational force of (a) CME1 and (b) CME2, as shown in Figure 8. Frequently, the sum of the forces which contributed to the deceleration is more pronounced than the forces which contributed to the acceleration for CME1 and CME2, as demonstrated in this simulation shown in Figure 8.

[30] To study the interaction between the two CMEs, we made a comparison of the simulation results for three different cases: double CMEs, CME1 only and CME2 only, with all other conditions remaining the same.

[31] Figure 9 and 10give the temporal evolution of (a) the CME time-distance curve (b) the radial velocity as a function of time, (c) the acceleration/deceleration as a function of time, and (d) the acceleration/deceleration-speed profile, for CME1 and CME2, respectively, with and without interaction. InFigure 9, the front of CME1 in the interaction case moves faster than that in the noninteraction case, and the influence of CME2 to the moving speed of the CME1 front primarily happens after time *t* = 40 h, when the shock of CME2 overtakes the shock of CME1 and merges into a combined shock. Figure 9c and 9d, clearly indicates that the effects of interaction causes the change from deceleration to acceleration. After the period of interaction, CME1 appears to decelerate because the speed of CME1 is still faster than the background solar wind speed. It is worth to note that the relationship of radial velocity and acceleration/deceleration shows a complex feature in Figure 9d, this feature could be attributed to interaction of two CMEs as also indicated in Figures 9b and 9c.

[32] In Figure 10 the difference between the two curves happens much earlier than in Figure 9. After 10 h, the heliocentric distance of the CME2 front for the interaction case increases more quickly than that of a single CME2. This is probably because when CME1 propagates into the background solar wind, the property of the solar wind has been disturbed, namely, some of the mass has been removed and the magnetic field also changes. As a consequence, CME2 does not propagate into the original background solar wind but into a disturbed solar wind medium.

[33] By comparison with observational deduced total force given by *Gopalswamy et al.* [2000], the dashed lines in Figure 11 and Figure 12 show the forces acting on a unit volume of CME1 and CME2 with density *ρ* which could be defined by *F*_{gopal} = *ρa.* Here *a* is the acceleration of the CMEs, then, we obtain:

where *a* is expressed in m/s^{2} and *u* in km/s. The solid lines in Figure 11 and 12 denote the total forces acting on a unit volume for CME1 and CME2 in our simulation. The forces deduced from Gopalswamy's expression (equation (5)) is simply by multiplying the density at a specific time and space with the acceleration (*F*_{gopal} = *ρa*). In their expression, they did not specifically indicate whether it was a single CME or multiple CMEs, but, these effects should have been included in the observed velocity. As pointed out by *Gopalswamy et al.* [2000], if *a* = 0 in equation (5), a critical speed *u*_{c} = 405 km/s can be obtained, which is remarkably close to the asymptotic solar wind speed in the equatorial plane. The CMEs speed in our simulation is much faster than *u*_{c}, as shown as Figure 6. Therefore, *F*_{gopal} always remains negative, which can be seen from the dashed lines in Figures 11 and 12. Some qualitative resemblance can be found from these two figures. As shown in Figure 11, the forces acting on CME1 from the two methods almost increase from initiation to 36 h, both have a little bump near 46 h and begin to increase at 48 h. Figure 12 shows that the forces acting on CME2 from the two methods almost increase from initiation to 15 h, from 20 to 30 h, with a little bump around 44 h, and begin to increase at 48 h.