Impact of Solar Wind Depression on the Dayside Magnetosphere under Northward Interplanetary Magnetic Field

We present a follow up study of the sensitivity of the Earth's magnetosphere to solar wind activity using a particles-in-cell model [Baraka and Ben Jaffel, 2007], but here during northward IMF. The formation of the magnetospheric cavity and its elongation is obtained with the classical structure of a magnetosphere with parallel lobes. An impulsive disturbance is then applied to the system by changing the bulk velocity of the solar wind to simulate a decrease in the solar wind dynamic pressure followed by its recovery. In response to the imposed disturbance, a gap [abrupt depression] in the incoming solar wind plasma appears moving toward the Earth. The gap's size is a ~15 RE and is comparable to the sizes previously obtained for both Bz<0 and Bz =0. During the initial phase of the disturbance, the dayside magnetopause (MP) expands slower than the previous cases of IMF orientations as a result of the depression. The size of the MP expands nonlinearly due to strengthening of its outer boundary by the northward IMF. Also, during the initial 100 {\Delta}t, the MP shrank down from 13.3 RE to ~9.2 RE before it started expanding; a phenomenon that was also observed for southern IMF conditions but not during the no IMF case. As soon as they felt the solar wind depression, cusps widened at high altitude while dragged in an upright position. For the field's topology, the reconnection between magnetospheric and magnetosheath fields is clearly observed in both northward and southward cusps areas. Also, the tail region in the northward IMF condition is more confined, in contrast to the fishtail-shape obtained in the southward IMF case. An X-point is formed in the tail at ~110 RE compared to ~103 RE and ~80 RE for Bz =0 and Bz<0 respectively. Our findings are consistent with existing reports from many space observatories for which predictions are proposed to test furthermore our simulation technique.


Introduction 2
Chapman and Ferraro [1931 and 1932] were the first to discuss the existence of 3 a boundary to the Earth's magnetic field. In the 1950's the concept of the solar wind was 4 developed. In the early 1960's, Explorer 10 and 12 provided the first measurements of 5 this boundary [Cahill and Amazeen, 1963], called the magnetopause. Since then, we 6 learned that the interaction of the solar wind with the Earth's magnetic field leads to a  Gosling et al, 1991]. In the resulting magnetospheric configuration, the magnetopause 13 appears as the boundary adjacent to the magnetosphere that is directly exposed to the 14 variable external solar wind. 15 is the response of the dayside MP of the Earth's magnetosphere for which available in 1 situ observations and theoretical simulations are quite abundant [Paschmann et al., 2 1979;Sonnerrup et al.,1990;Gosling et al.,1991; Baraka and Ben-Jaffel, 2007;Zhang et 3 al., 2009]. This region was also selected because fields and plasmas in the dayside 4 magnetosphere tend to be more ordered, and thereby make field line mapping more 5 tractable. In addition, dayside dynamics tend to be directly driven by solar wind forcing, 6 a property that should assist in studying its sensitivity to the solar wind variations 7 [Murr, 2004]. 8 In our previous work, we used a PIC code to study the interaction of the solar 9 wind with the Earth's magnetosphere during a depression in the solar wind dynamic 10 pressure that was artificially created within the flow of the plasma traveling along the 11 Sun-Earth-direction [Baraka and Ben-Jaffel, 2007]. This effect was applied for two 12 conditions of IMF, Bz =0 and Bz <0 respectively. In those studies, during the initial phase 13 of the interaction with the traveling gap, the outer boundary of the magnetopause 14 expanded sunward, which led to the break-up of its structure in the absence of IMF, 15 while it sustained its bullet shape in the case of southward IMF. Also, in response to the 16 generated gap, low density ensembles of plasma reversed direction and moved 17 sunward against the stream in Bz =0 case. By contrast, this reversal also took place 18 when Bz <0 with much lower density than the case Bz =0. We interpreted this reversal as 19 a result of a gradient in the dynamic pressure from the MP toward the gap region that 20 induced a sunward oriented mechanical force, which stopped these tiny clouds and 21 then reversed their direction of motion [Mishin, 1993;Baraka and Ben-Jaffel, 2007]. We 22 also proposed some similarities between our gap's effect and hot flow anomalies (HFA), 23 particularly plasma flow out of the Sun-Earth line [Sibeck et al 1994[Sibeck et al , 1999. 24 Furthermore, the orientation of the cusps was found to be highly affected by the It is important to remark that all quantities mentioned above correspond to the initial 1 unperturbed plasma before the application of any magnetic field. These values are also 2 consistent with the fluid description of the solar wind plasma initially injected. However, as 3 we shall see, with the magnetospheric cavity formation, most regions of the magnetosphere 4 will host hotter and/or smaller density plasma. This means the Debye length in those regions 5 will be much larger than assumed for the initial plasma (see section 2.2 for more details). 6 Besides kinetic effects, it is important to remark that for magnetospheric conditions, 7 the plasma is assumed collisionless. However, when using PIC simulation codes, collision 8 occurs all the time because of the spatial grid used to handle fields and charge and current 9 deposit. In a PIC code, a charge diffuses occupying a volume with a charge distribution 10 defined by a shape function (Langdon, 1971). The numerical collisions lead to the so-called 11 aliasing or grid-heating problem. A lot of work has been done in the past to clarify the onset 12 of the grid heating and its dependency on the plasma parameters (Langdon, 1971;Hockney & 13 Eastwood, 1981 and references therein). From these studies, it appears that the two main 14 parameters are the Debye length ( ) and the plasma frequency (ω p ). In the case of 15 electrostatic simulations (no magnetic field applied) of Maxwellian plasma, Hockney & 16 Eastwood derived a set of and ω p values for which grid heating is acceptable. In addition, 17 Langdon (1971) showed that the grid heating occurs by the assumed electrostatic plasma 18 until an asymptotic value of λ D ~0.3 is reached. As shown by Hockney & Eastwood, 1981 19 (their fig. 9-6 on page 320), our initial and time evolution of (λ D , ω p ) ~(0.11, 0.89) are 20 compatible with their optimum path. In addition, we remind that a 3Δx3Δx3Δ shape function 21 of particles is applied on all quantities projected on the grid, a very efficient way to damp 22 instabilities. Obviously, the smoothing is done at the cost of lower spatial resolution, an 23 effect acceptable when dealing with large scales, as it is the case here. We will discuss more 24 in detail this problem in the next section to compare the grid heating derived in this study 1 when a magnetic field is applied. Before the application of any solar wind disturbance, we need to set-up a steady 5 state magnetosphere under northward IMF condition that has the structure expected 6 from past observations. Results are presented in the same way as in our previous study, 7 in order to comparatively conclude on similarities and differences between the current 8 study (Bz >0) and the previous ones (Bz =0 and Bz <0 respectively) [ Baraka and Ben-9 Jaffel, 2007]. 10 The simulation box has the same dimensions of (155,105,105) Δ as in our 11 previous study; where Δ = 1 RE. It is loaded with 2x10 6 electron-ion pairs, where Earth 12 is located at (60,52,53) RE. In all graphs, this location is centered at (0, 0, 0) RE, which 13 makes follow up of interpretations more convenient. Starting from a box filled by pairs 14 of electron-ion macro-particles, the Earth's dipole field is switched on, letting the 15 system evolves with time up to 900 Δt with an impinging northward IMF component Bz 16 =0.2. The IMF strength is weak but strong enough to provide the expected 17 macrostructure of the magnetosphere after the prescribed elapsed time. Initially, the 18 Debye length is λDe,i=(0.11, 0.11), respectively for electrons and ions. These values are 19 consistent with the multi-fluid description of the frozen-in magnetized solar wind flow 20 initially assumed. After the planetary and solar magnetic field have been applied, the 21 plasma configuration changed so that it is not clear from the only density distribution 22 shown in Fig. 1 if the PIC code provides a kinetic or fluid description of the particles. To 23 answer this question, we derived a 2D distribution of the Debye length that corresponds 24 to the plasma structure obtained for the steady state configuration displayed in Fig. 1A.
As shown in Fig. 1B, the Debye length contours have a nice distribution that 1 progressively evolve from the multi-fluid description of the impinging solar wind flow 2 into a kinetic description of the magnetospheric plasma. For example, in the 3 magnetosheath nose region, we estimated that the Debye length should be ~0.45. In the 4 current sheet area, the Debye length is found as large as ~1.37. In the cusp regions, we 5 estimate that the Debye length is ~1.17. Finally, in the far tail, we estimate that the 6 Debye length should be around ~0.92. All these values tend to support that our 7 simulation is a kind of hybrid description of the plasma from multi-fluid to kinetic's 8 handling of the particles that is naturally accounted for from the dayside to the 9 nightside of the simulation box. As shown in Fig. 1B, in most of the magnetospheric 10 regions, particles are described kinetically by our PIC code. 11 Besides kinetic properties, numerical stability of a PIC code should also be 12 addressed. As remarked in section 2.1, grid heating is one of the most discussed issues 13 in PIC simulations. In the case of electrostatic simulations (no magnetic field applied) of 14 Maxwellian plasma, Hockney & Eastwood derived a set of λD and ωp values for which 15 grid heating is acceptable, if not negligible. We may summarize their results remarking 16 that for plasma with λ D < 0.3, grid aliasing should induce a numerical heating that 17 depends on the shape function assumed for the particles on the grid but that should 18 bring the plasma to an asymptotic state with λD = 0.3-0.5 (Langdon, 1971). According to 19 Hockney, 1971, this artificial heating may corrupt the output of PIC simulation of 20 electrostatic and stable Maxwellian plasma if not handled with care through an 21 adequate selection of the particles shape function. In the present case, we started with function for particles as used here. Using the plasma properties in the region upstream 1 of the magnetosphere, we estimated that after 900 time steps, the Debye length is λD,i 2 ~0.14 for ions and λD,e ~0.25 for electrons in the unperturbed solar wind plasma. In 3 addition, numerical heating seems uniformly distributed over the simulation box and 4 smoothly increasing with time until a saturation value λD,e ~0.25 of the Debey length is 5 obtained for electrons. Our results tend to support that the grid aliasing problem is 6 negligible for the heavier ions but follows the same trend for electrons as derived by 7 past studies (Langdon, 1971, Hockney & Eastwood, 1981. For the large structure of the 8 magnetosphere studied here, the numerical noise should not therefore substantially 9 affect our results. 10 Besides the discussion on the numerical stability of our code, it is also important 11 to compare the magnetosphere so far obtained as a steady state configuration to past 12 observations. In the unperturbed solar wind plasma upstream of the magnetosphere 13 structure, we estimated that the Alfven velocity is VA = 0.08c, the Alfvénic mach number 14 is MA ~ 6.1, and the corresponding magneto-acoustic Mach number is Mm = 5.2. The 15 plasma parameter is β ~ 0.35 with a critical Mach number ~ 2.3 (Edmiston & Kennel, 16 1984). For this set of plasma parameters, the Earth's bow shock positions along the OX 17 and OY axis are expected from observations at ~(14.8 RE, 29 RE) respectively (Peredo et 18 al., 1995). The magnetopause is expected around ~10.6 RE along the nose direction 19 (Peredo, 1995;Merka, 2005). It follows that the bow shock to magnetopause standoff 20 distances ratio should be ~1.4 as derived from past observations and models (Fairfield 21 et al., 2001). For comparison, our bow shock positions are ~(15, 29) RE for sunward and 22 duskward, and the magnetopause nose standoff distance is ~10.8 RE, quite consistently 23 with past models and observations. In addition, the magnetosphere morphology 24 obtained here is also compatible with past observations and models in terms of plasmas and fields distributions, particularly the lobes that are slightly flare out and the fishtail 1 configuration of the magnetotail (Petrinec and Russell, 1996). All these results tend to 2 support that our PIC code when using the appropriate scaled plasma parameters 3 describe quite accurately the larges scales of the magnetosphere, an achievement that 4 provides us with the right tool to investigate in the following the impact of a solar wind 5 disturbance on the steady state configuration obtained at step time 900 Δt. The total pressure curve shows a real solar wind depression. Such input solar wind 20 conditions could be described as a depression consequently followed by compression in 21 the solar wind. It may also be fulfilled by two successive shocks and/or discontinuities 22 that are separated by a lapse of time and that may hit the magnetosphere in a way 23 similar to the gap's effect described here. Such disturbances are common events in the 1 2 3

3.
Results: Response of the Magnetosphere to a Depression in the Solar 4

Wind Flow 5
Results are presented in the same way as in our previous study [Baraka and Ben 6 -Jaffel, 2007], in order to comparatively conclude on similarities and differences 7 between the current study (Bz >0) and the previous ones (Bz =0 and Bz <0 respectively) 8

Plasma Density Distribution under Northward IMF condition 15
A snapshot of the system taken at 1001 Δt shows the formed gap that appears as 16 a planar structure between x=-47 and x=-32 RE (see Fig. 3  In Figure 3(B), taken at 1100 Δt, the downstream edge of the generated gap is 3 now approaching the dayside magnetopause, where it forms a concave layers over the 4 magnetosheath as a cover. This cover has an apparent thickness along the x-direction of 5 ~5.5 RE. Its concavity can be explained by the incoming plasma, which strongly feels the 6 dipole magnetic pressure on the Sun-Earth line, whilst both ends (toward poles) 7 proceed almost steadily along the x-axis. Inside the gap, clouds of plasma are seen all 8 over and are more uniformly distributed than the previous step time. Apparently, at this 9 particular time the magnetopause standoff position is almost ~-11.5 RE in the x 10 direction, compared with~13.5 RE for Bz =0 and ~12.3 RE for Bz <0. In the night-side, a 11 plasma structure of a relative thickness of ~ 8 RE is seen starting at x~10 RE. This 12 structure has a ring shape and corresponds to the so-called trapping regions that 13 connect cusps to the equatorial plasma sheet. Both cusps are clearly observed dayside 14 oriented and slightly wider in the poleward direction. At both cusps' peaks, located at 15 (x=-4, z=9) and (x=3,z=7) RE respectively for the northern and southern hemispheres, 16 high-density plasma is observed. In addition, other tiny clouds of plasma are observed 17 inside the inner magnetosphere. However, the noise level and the rather moderate 18 number of particles assumed in the simulated plasma makes it difficult to conclude on 19 the origin of these clouds in these particular regions. 20 The density distribution of the system, taken at step time 1175 Δt (75 Δt later), 21 shows that the gap is apparently filled by plasma with its downstream edge now almost 22 located along the planet position (0,0,0) (see Fig. 3(C)). By coincidence of the selected 23 time, the new standoff position of the magnetopause at the subsolar point reads ~- 18 24 RE, almost the position of the upstream edge of the gap (see the upright dashed-dotted line in Figure 3(C)). On the contrary, at the same step time, the magnetopause was 1 broken up at ~-15.5 RE for Bz =0 with a huge cut that opened sunward, while it 2 expanded keeping its classical shape up to ~-17 RE for Bz <0. 3 The trapping region is seen as a configuration of clouds of plasma that resembles a ring 4 shape (~7 RE thick) observed around ~ 15 RE nightside from Earth (along x-axis). The 5 northward and the southern cusps are dragged in an upright position, but showing a 6 much-extended poleward region compared to a much thinner equatorward region. The 7 thinning of the cusps shape due to the low dynamic pressure of the gap region and its 8 widening by the northward IMF poleward as obtained here confirm and extend 9 previous studies [Burch, 1973;Yamauchi et al., 1996]. Farther away, the cavity 10 structure of the magnetotail can be seen along x out to ~45 RE (45 RE from planet). 11 Beyond that distance, in contrast to the cases for Bz =0 and Bz <0, the magnetotail 12 becomes filled up with plasma.

Magnetic Field Lines Distribution under Northward IMF condition 4
In the following, we describe the magnetic field's topology corresponding to the 5 plasma density distributions shown in  Figure 4A, shows that the 2 reconnection between the magnetosheath and magnetosphere fields appears clearly, 3 extending over a few Earth radii and confirming its large-scale structure (see Figure 5). 4 It follows that our simulation PIC code offers an unprecedented opportunity to study in 5 3D the start-up and time evolution of such reconnection events under northward IMF. 6 Another signature of a large scale reconnection is also visible in the magnetotail region 7 beyond 44 RE along the x-axis. These results confirm the general configuration that is 8 expected and observed for northward IMF [Burch, 1973;Hasegawa et al., 2008]. 9 In Figure 4 Aside from the large scales changes noticed during the travel of the gap from the 10 day to night side of the magnetosphere, in the following we discuss the expansion/ 11 recovery phases of the magnetopause during the depression/compression of the solar 12 wind dynamic pressure, but this time for northward IMF. With the two cases of 13 vanishing and southward IMF, the following analysis should complete our sensitivity 14 study of the magnetospheric response for the whole range of IMF conditions. 15 In Figure 6, panels A, B, and C represent the expansion/recovery phase for B z >0 as 16 measured through the size of the magnetopause along the three main axis. This size is 17 estimated respectively from Earth's position along the x-axis, dawn to dusk for the y-axis, 18 and south to north for the z-axis. We remind that no tilt was assumed for the planet and that 19 x, y, and z represents the Sun-Earth, dawn-dusk, and south-north lines respectively. To we estimate the position of that edge from the Earth's location. 23 In panel A of Figure 6, the magnetopause starts expanding from 11 R E up to 13 R E (~2 1 R E jump) between 1099 and 1107 Δt, then it shrank from 13Re down to 12 R E between 2 1107Δt and 1115Δt. Subsequently, the MP linearly expands from the former position up to 3 ~20 R E , a size that it reaches at about 1160 Δt. Then, apparently, the system relaxes for 6 Δt, 4 before the MP starts contracting, thusly entering the recovery phase. The later phase starts as 5 soon as the upstream edge of the abrupt depression hits the expanded nose of the 6 magnetopause. The MP then recovers linearly to its original length at 1212 Δt. It is important 7 to notice that the time variation of the magnetopause's position appears as two distinct 8 phases, one for the expansion and the second for the recovery. This property confirms our 9 claim that the gap disturbance could be sketched as two distinct velocity edges affecting the 10 magnetosphere simultaneously but separately. 1223.2 Δt. Afterward the system recovers very fast as the abrupt depression is over at 1235.7 20

Δt. 21
In panel C, the expansion/recovery of the magnetopause size in shown in the z-22 direction. In contrast to aforementioned directions, the MP size remains relatively stable at ~ 23 still relatively far to be felt by the MP boundary in z-direction. Then, the system expands 1 linearly up to 28.555 R E at 1190.7 Δt, before it relaxes over ~37 Δt until 1228 Δt The 2 recovery phase is linearly attained when the MP reached its initial position (19 R E ) at about 3 1240 Δt. 4

4.
What can we learn about the Magnetosphere under Northward IMF 5 using a PIC code? 6 For the purpose of consistency with our previous study, in the following, we try 7 to compare our results with existing observations and models, with a focus on the 8 novelty a PIC code brings in understanding the solar wind interaction with the Earth 9 magnetosphere under northward IMF. Here, we stress that the idea of carrying out 10 these case studies is to separate effects (.i.e solar wind parameters, ionosphere, 11 currents systems…etc.) before going into detailed global cases. For example, some 12 principal parameters are not taken into account in our current work, such as 13 ionosphere-magnetosphere coupling, planet tilt, rotational currents, IMF fluctuation 14 values and so on. In our endeavor to carry out these tasks, the dayside magnetosphere 15 (magnetopause) was targeted. In the past, the magnetopause motion was reported 16 often oscillating more or less uniformly in and out a reference position [Formisano et 17 al., 1979]  From this it follows that the primary source of the magnetopause motion is the 23 change in the dynamic pressure of the solar wind that generate waves and instabilities (Kelvin-Helmholtz) that propagate all over the magnetosphere, particularly on the 1 flanks where deformation could be large [Fairfield et al., 2003]. Magnetic reconnection 2 north and south of cusp regions may induce disturbances that could also distort the MP 3 shape. Other processes may trigger magnetopause motion Earthward even when the 4 dynamic pressure is constant, like erosion [Fairfield, 1971, Sibeck, 1991, 19945 Tsyganenko and Sibeck, 1994] or expansion of the MP due to HFA [Sibeck, 1999]. 6 Here, we consider a new process driven by an abrupt depression (gap) in the 7 solar wind ram pressure that we describe and follow in time using a PIC code. As shown 8 in section 2.3, this disturbance could also be realized as two distinct events of the two 9 velocity edges separated by a lapse of time. By contrast with previous cases, the 10 propagation of such feature through the Earth's magnetosphere has never been 11 considered in the past with northward IMF [Baraka and Ben-Jaffel, 2007]. Moreover, the 12 scales of changes in the magnetosphere expected from the depression are far beyond 13 the linear regime of oscillations around a reference structure considered in the past. 14 15

Code Picture 17
All results so far presented confirm that our updated version of the 3D macro-particle 18 code (PIC) is able to generate the macrostructure of the Earth's magnetosphere and was able 19 to simulate a strong depression of the solar wind under northward IMF conditions, 20 confirming and extending our initial report on vanishing or southward IMF cases. Similarly 21 to our previous study, the applied depression in the SW flow results in the formation of a gap 22 in the plasma flow that has almost the same size of ~15 R E as obtained for the other IMF 23 conditions. The gap has a planar shape with sharp boundaries perpendicular to the x-axis moving Earthward. In the real solar wind, such depressions with low plasma density on large 1 scales are not unusual events as low-density sub-Alfvénic flows are observed often [Chisham 2 et al., 2000;Usmanov et al., 2005]. In addition, many events in the ACE solar wind data 3 showing two simultaneous velocity peaks separated by a lapse of time may correspond to the 4 disturbance simulated here. 5 First, we notice a remarkable accumulation of plasma clouds of density above the 6 noise level distributed all along the cavity of the magnetosphere, probably particles diffusing 7 into the magnetosphere, and along the upstream edge of the generated gap as shown in Figure  8 3. This picture is far from the MHD fluid simulation where the magnetospheric structure 9 appears as smooth boundaries as far as neither diffusion nor reconnection are allowed, unless 10 artificially added. Despite its scaling limitations and the corresponding noise level, our PIC 11 code renders better the kinetic nature of this interaction and may help better understand the 12 formation and evolution of large scale structures of the magnetosphere. 13 Another large-scale feature of interest in this study is the cusp region with its behavior 14 in response to the solar wind disturbance. Before and after the arrival of the event, the 15 orientation of the cusps was classical and fits rather well with observations (eg Fig. 3) 16 [Burch, 1973;Yamauchi et al., 1995]. However, when the disturbance was right over the 17 planet position covering the whole cusps region, it was remarkable to see how fast the cusps 18 responded to the depression inside the gap and moved anti-sunward, becoming almost in an 19 upright position. Furthermore, they appear very thin in the bottom end at low altitude but 20 much more extended at high altitude, confirming the key role of the solar wind dynamic 21 pressure in controlling the cusps shape and motion [Yamauchi et al., 1995[Yamauchi et al., , 1996Pitout et al., 22 2006]. In the future, a sensitivity study of the cusps properties in response to the solar wind 23 dynamic pressure for different IMF conditions, particularly when the IMF switches from one direction to another, should provide a reasonable background to understand the main 1 processes that drive their formation and time evolution [Pitout et al., 2009]. 2 One important feature that our PIC simulation obtains naturally is magnetic 3 reconnection, particularly at high latitudes on northward and southern hemispheres. In 4 a recent study, [Lin and Wang, 2006] showed that under a purely northward 5 interplanetary magnetic field (IMF), magnetic reconnection in both northward and 6 southern hemispheres leads to a continued formation of newly closed field lines on the 7 dayside, a feature that we observe in our simulation (see Figure 4(A) and 5). Also, 8 recent observations by Cluster and Themis confirm the existence of the double 9 reconnection predicted at high latitudes in north and south cusps areas (indicated by 10 small circles in Figures 4(A)), with persisting X points that our PIC simulations recover 11 nicely as shown in Figure 5  northward IMF seems to agree with these findings, where the magnetotail ceased to flare out after ~14.5 RE. In addition, the magnetotail is seen shifted toward south and its 1 length is shortened. This result is consistent with the occurrence of the magnetic 2 reconnection at the high latitude mantle , Zheng et al., 2005. Our 3 PIC code thus offers a great opportunity to study these large-scale features, which in 4 return could help define better the code scaling to the real world. The other useful side of the PIC code is its capability to follow in time the evolving 9 system. However, such a time scale is not clear as far as other key parameters, like the 10 particle masses or number densities, do not correspond to the real magnetosphere. We 11 claimed in section 2 that using the Courant condition may help define a time scale, yet this 12 has to be proven to compare with observations. Here, we stress that our time scale works only 13 for the purpose the code was build for: the formation and evolution of large scales (i.e. 14 macrostructure of the Earth magnetosphere) [Buneman 1993. Therefore, as In the following, we thus provide more quantitative results that we propose as 5 predictions for the future and when possible to compare to existing observations. It has been 6 noted that the magnetopause expanded non-linearly in all IMF cases along noon, dusk and 7 north direction (x,y, and z). On the other hand, soon after the abrupt depression effect is over, 8 they all recovered almost linearly in 55 Δt for the x-axis, 35 Δt for the y-axis and 48 Δt for the 9 z-axis, respectively. Quantitatively, the magnetopause expansion rate along the x-direction 10 took place with a speed equal to ~ 0.18 in the code units corresponding to an equivalent 11 velocity of 360 km.s -1 , and recovers at a speed rate equal ~0.16 (~320 km. s -1 ) afterward. 12 Moreover the expansion rate of the MP along the y-direction is 0.177 (~354 km.s -1 ) and the 13 corresponding recovery speed is ~0.31 (~ 620 km.s -1 ). Additionally, the expansion rate as 14 seen in the z-direction showed speed equivalent to 0.12 (~ 240 km. s -1 ), whilst the 15 corresponding recovery speed reaches the value ~0.45 (~900 km.s -1 ). In conclusion, the 16 expansion phase in x-direction is the fastest compared to the expansion in y and z-direction, 17 probably because the subsolar point of the magnetopause is closer to the generated 18 disturbance than the other two boundaries in y and z-directions. On contrary, the recovery 19 phase of the magnetopause in y and z-direction is very much faster than in x-direction. 20 Because the disturbance is effectively generated as a planar cut along x-direction, the elapsed 21 time for the gap to over pass the magnetopause along x-is longer than that elapsed along y 22 and z. This remark is important because it stresses the need to account for a "filling factor" of 23 the disturbance in any analysis of the impact of a strong disturbance (compression or depression) on the whole magnetospheric system. This "filling factor" must be defined 1 through a careful analysis using tools like a PIC code.  Table 1 should be compared with observational data in the near future.

Summary 1
In this paper, we present a follow up study on the sensitivity of the Earth's 2 magnetosphere to the solar wind activity as initiated by [Baraka and Ben-Jaffel,  o One of the findings of this study is that, regardless of the orientation of the 18 IMF, the initial general shape of the solar wind disturbance remains almost 19 unchanged for all IMF values when interacting with the magnetosphere (B z 20 =0, B z <0 and B z >0) . Between the two edges of the disturbance, we noticed 21 that the density of plasma is much larger for B z >0. Also, more dense plasma 22 populates the cavity at the magnetotail, which results in a southward tail shift 23 ( Figure 3(C)); which is in turn a sign of reconnection nearby that region. The 1 nightside magnetosphere is more confined than the other two cases for zero 2 and south IMF. 3 o The recovery phase always takes place separately but faster that the expansion 4 phase. It was also noted that the whole expansion/recovery phases take less 5 than a few minutes when using our time scale, consistently with most 6 observations on the motion and oscillation of the magnetopause that report the 7 same time scale of a few minutes. shown in Figure 1A. From magnetosheath to magnetotail, the Debye length in the 6 magnetosphere is ~> 0.5, ensuring a kinetic description of the particles. 7 8 Figure 2 shows the normalize plasma parameters drawn within the gap. This figure is 9 plotted in 100-step time, soon after the disturbance is applied to the system. In five 10 panels the solar parameters are shown in the following order, speed, number density, 11 temperature, magnetic field strength and total pressure. The x axis is reversed time 12