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Shock Wave Interactions with Viscosity Observed after the Coronal Mass Ejection Activities Occurred on December 18, 1999 and April 4, 2001

The release of magnetic field and plasma from the solar atmosphere (i.e. coronal mass ejectionsCMEs and solar wind) resulting from solar magnetic activity can produce shock waves and geomagnetic storms. Shock waves are known to occur while the solar ejected particles alter from the supersonic to the subsonic regime. Especially, in the supersonic case for the flow of compressible gas interaction of shock waves with viscosity plays a key role for space weather broadcasts. Therefore, the major objective of this paper was to search the outcome of viscosity in the shocks subsequently detected after the CMEs occurred on December 18, 1999 and April 4, 2001 by using the previous modelling study of [1].


INTRODUCTION
The solar atmosphere consisting of the photosphere, chromosphere, corona and the solar wind acceleration layers extend from the solar surface in the order of 10 0 , 10 1 , 10 1 -10 2 and 10 3 Mm's respectively. The corona as the outermost part of the solar atmosphere is placed above the solar chromosphere layer. The temperature in the corona changes suddenly from a few thousand to a few million Kelvins [2] in the form of plumes, loops and streamers.
The corona is especially interesting with its complex magnetically 'closed' and 'open' structures. According to Priest [3] interaction between the magnetic field and the plasma characterises which kind of phenomena will occur. Closed magnetic loops located below the coronal streamers, may sometimes expand and coronal mass ejections (CME) in the form of an enormous plasma cloud can be ejected to the interplanetary space [4] and [5]. On the other hand, coronal holes result from the open plasma structures. In that case, a fast stream of plasma occurring in the fast solar wind spreads to the interplanetary space from the holes in the solar corona [6]. At such high coronal temperatures [3], the plasma will no more be constrained to the Sun gravitationally and will grow through the interplanetary medium at supersonic speeds as a solar wind. In his pioneering coronal expansion model, [7] predicted that high-speed solar wind can be observed near our Earth. Later, measurements of the solar wind parameters and improvements achieved in the theory led to the acceptance of the original idea. In the recent times, plumes appearing outside the coronal holes were suggested as possible reasons of the solar wind [8]. Interactions of these supersonic solar winds with the local interplanetary medium result in a shock wave. Amongst several ways to produce shocks in the solar wind are blast waves emitted from the Sun, CMEs and the interactions between the fast and slow streams [9]. Indeed, they may cause some changes in the physical conditions related to compression, heat, and changes in the magnetic field. Cavus and Kazkapan [10] studied the Kelvin-Helmholtz instability in the solar atmosphere and estimated the values of radial speed vary between 380 km/s and 780 km/s, for slow and fast solar winds respectively.
Another feature of the Sun is that it generates a continuous outflow of particles in the form of a high-speed solar wind which creates a shock wave in the sunward side after the collision with the planet's atmosphere. Shock waves result when particles in the solar wind are emitted at velocities 350-700 km/s [11], much higher than 100 km/s, the speed of sound in the interstellar medium [12] and [13]. Shocks arising from some of CMEs and solar winds were detected through the project of Solar and Heliospheric Observatory/The Large Angle and Spectrometric Coronagraph (SOHO/LASCO) and published by Stepanova and Kosovichev [14]. The expanding ejects travelling faster than ahead or behind the ambient gas will be a reason of a shock ahead and present a decreasing distribution of speed within ejects [15]. These shock properties were associated with the density compression features by Kilpua et al. [16].
Eselevich and Eselevich [17] showed that the CME's ahead frontal structure forms a disturbed region due to the CME's interaction with the undisturbed solar wind. The size of this region gradually increases as the CME travels away from the Sun and a narrow discontinuity region is observed to form at the disturbed zone of the front. Characteristics of this disturbed zone are similar to a piston shock which are collisional at the radial distances r < 6R Sun and collisionless for r > 6R Sun (where, R Sun denotes the solar radius, and r is the distance from the centre of the Sun).
Some case studies of shock waves treated the complex entropy behaviour across the shock wave. For example, [18] studied entropy profile through the shock without viscosity and heat conduction effects. They showed that entropy increases in the shock front up to its maximum at the centre and then diminishes in the other half of the shock front. Even though this seems to violate the 2 nd law of thermodynamics, it is still valid for the whole system, since entropy increases in the downstream region of the shock wave. Later, [19] studied entropy behaviour across the shock waves in a typical dusty gas precedent of the Navier-Stokes equations. He has found that the entropy distribution has its greatest value within the shock front and is increasing over the shock wave with respect to the upstream Mach number and the particle density. As a case study, [20] studied entropy in the shock wave that occurred after the CME of 12/12/2006 by the model described in [1].
The NASA-Advanced Composition Explorer (ACE) spacecraft routinely observe the events mentioned above. The ACE observatory, a spinning spacecraft (5 rpm), will orbit around the Sun-Earth L1 libration point (i.e. 240 times of the Earth radius). In order to get rid of the effects of the Earth's magnetic field, the ACE spacecraft has travelled almost a1.5 million km from the Earth. As another example [15] and [16] studied the shock waves that occurred after the CMEs of 18 February 1999 and 28 April 2001, which were accompanied by a flare and coronal waves. In the present paper various models of [1], [20][21][22][23] are applied to the shock wave that happened after these CMEs. Necessary values for the physical parameters are taken from the ACE mission and used as an upstream condition.
Initially, the model of [24] intended to study and predict the arrival of the shock waves on Earth. Unlike their study, the major aim of this article is to search the effects of viscous flows for the shock wave that happened after these two CMEs. In order to describe such shock processes, the Navier-Stokes equations are solved mathematically by means of the hydrodynamic model explained in [1]. In this modelling approach, viscous behaviour of a gas is considered as a function of the Reynolds number [1], [25] and [26]. In section 3, the downstream characteristics of the shock waves that occurred after the CMEs of February 18 1999 (hereafter CME18/02/1999) and April 28 2001 (hereafter CME28/04/2001) will be given. Results will be compared with other works in Section 4, and the conclusion will be driven after the discussions.

Physical Parameters
Structure of the solar atmosphere is characterised with respect to the dominant roles played by the complex plasma and magnetic pressures, described by the plasma β parameter which is the ratio of the plasma pressure to the magnetic pressure. In other words, plasma pressure dominates over magnetic pressure, if β is greater than 1, and for the contrary case magnetic pressure dominates over that of the plasma. This ratio varies as a function of the magnetic field, reaches β>>1 values in the solar wind acceleration region, considered as infinity by Gary [27], and in the work of Matthaeus et al. [28] changes from 44 to infinity. In this context, importance of gas pressure in the dynamical modelling of the solar wind is well revised by Gonzales-Esparza et al. [29].
The CME interval is identified by the recombination of the enhanced density, temperature and velocity profiles. In the present work, upstream parametric values for different shocks that happened after the CME18/02/1990 and CME28/04/2001 are taken from the ACE spacecraft, listed in Table 1 [15] and [16].
They are employed in the model of [1] in order to examine effects of viscosity in these shocks.
In many cases, behaviour of the density data gives a guide to the occurrence and arrival of the shocks [24]. In Table 1, velocities are estimated as 390 km/s and 445 km/s for the cases of CME18/02/1999and CME28/04/2001, respectively. Since the local sound speed in the interplanetary medium is about 100 km/s [13], the shock wave should come into existence in this region, with temperatures around 1x10 5 Kelvin and 5x10 4 Kelvin respectively, at the start (Table 1).

Basic Formulae
The plasma β parameter defined as, gas mag p p   (1) have values higher than one in the solar wind [27][28][29][30], since, the gas pressure of the plasma is dominant over the magnetic pressure, at such high coronal temperatures. This reduces the formulation of the wind problem to the hydrodynamic case explained in the pioneering work of [31] and later [32].
Thus, for a more generalised viscous shock in steady flow [1] obtained: In equation (2) [33], for a collisionless shock γ ranges as 5/3 < γ < 3. Then using the Rankine-Hugoniot jump formulas relations, [34] found the downstream physical parameter values. In which, they describe the relationship between the states on both sides of a shock wave in fluids. Similar to Cavus and Kurt [20], changes in entropy (S 2 -S 1 ) can be evaluated in terms of the pressure ratios to the compression rate as follows:

Downstream Reynolds Numbers in the Solar Wind
Consequently, the Reynolds number with (Re>>1) values in the solar wind, plays a key role on the dynamics of this region. It is indicated to jump to 10 12 and 10 14 in the solar wind speeding up region by [8] and [35].  Fig. 1, is decreasing as the upstream Mach number increases and equals to unity (i.e. Re 1 =Re 2 ) for M 1 2. The latter can be considered as a transition region from weak to strong shocks, since for weak shocks M 1 <2, and for strong shocks M 1 >2 [1] and [36].
In the right panel of Fig. 1, variation of Re 2 according to Re 1 for various M 1 values, is given for the case of a monatomic gas of =5/3 [21]. In the same manner described by Borovsky and Funsten [8] and Veselovsky [35] Reynolds number Re 2 can be deduced and seen to increase with increasing Re 1 ending with greater values for smaller M 1 .

MODELLING RESULTS FOR THE SHOCKS HAPPENED AFTER THE CME18/02/1999 AND CME28/04/2001
Some particular solutions of the equations (2-3) and the Rankine-Hugoniot jump formulas [34] were adapted to a symbolic and numeric computing environment Maple 9.5, in order to derive the downstream parameter values for the shock waves driven by the CME18/02/1999 and CME28/04/2001. The downstream physical parameters thus found are represented either in Table 2 or in Figures (Figs. 2-8). The upstream Reynolds number used in our solar wind calculations is taken as 10 13 [8,37] and [38].  Table 2 as a function of M 1 . As before these are expressed as a function of ratios for the Reynolds numbers (Re 2 /Re 1 ),  (i.e. density ratios, n 2 /n 1 ), velocity ratio (u 2 /u 1 ), ratios of temperature (T 2 /T 1 ), and the Mach numbers ratio (M 2 /M 1 ) together with the entropy difference (S 2 -S 1 ), all obtained from the solutions of equation (2) and the application of the jump conditions. For the case of M 1 =2.045, equating the Reynolds numbers ratio to unity, the critical Mach number for the turning point is found (see Fig. 1 Kelvin, respectively. In Fig. 4 variation of the downstream velocity u 2 is depicted as a function of M 1 (left panel) and Re 2 /Re 1 (right panel). The upstream values directly taken from Table 1, are both observed to have decreasing tendencies. Unlike T 2 , changes in u 2 are large for the weak shock case (i.e. M 1 <2), compared to the changes in the strong shocks (M 1 >2).  5 shows changes in some parameters with respect to the entropy differences, where the cross symbols representing temperature ratios of the downstream to the upstream values (T 2 /T 1 ), are observed to increase with the entropy difference. The empty squares depict the compression ratio (n 2 /n 1 ) exhibit a similar behaviour with the sound speed, increases with S 2 -S 1 , whereas the Re 2 /Re 1 ratio, (empty triangles) have a decreasing tendency. On the other hand, the downstream to upstream velocity ratios (u 2 /u 1 ) represented by plus signs is also decreasing with increasing entropy differences. All of these ratios are unity for S 2 -S 1 =0 (i.e. the isentropic case). Finally, all of these ratios become unity for the isentropic case (S 2 -S 1 =0), which means that no shock occurs for the case with no compression (=1).
Downstream density variation is presented in Fig.  6 as a function of the entropy difference where S 2 -S 1 <2.44 corresponds to the weak shock (M 1 <2) region in Table 2. As expected the downstream density variations (n 2 ) increase with entropy differences S 2 -S 1 but these variations are small for further increase of the S 2 -S 1 . On the other hand, from the S 2 -S 1 dependence of T 2 shown in Fig. 7 tends to increase with increasing entropy differences. The T 2 changes are small for the weak shock region (S 2 -S 1 <2.44) compared to the changes in the strong shocks (S 2 -S 1 >2.44).
From Fig. 8 we observe a decreasing tendency of the downstream velocity (u 2 ) for greater entropy differences (S 2 -S 1 ) as expected, whereas u 2 variations are small for larger S 2 -S 1 differences. Again the very weak shocks are closely isentropic when S 2 is nearly equal to its upstream value given in Table 2 and this variation is high when M 1 >>2 happens for strong shocks.

DISCUSSION AND CONCLUSION
Investigation of CME driven shocks waves from the Sun to the interplanetary space is important for space weather forecasting since sufficient energy is released very rapidly capable to produce a ''fast'' CME which can drive an interplanetary shock [15]. On the other hand, understanding behaviour and change of the physical parameters, describing these phenomena still stays a very sophisticated task relevant to the present observational facts.
As far as we know when a CME explodes in the corona, besides the interactions with the ambient interplanetary gas, more complex magnetic and thermal energy processes occur. Even though the magnetic pressure dictates near the Sun, gas pressure becomes more dominant beyond the Sun. Then, the hydrodynamical modelling approach is adequate for the study and evaluation of the CME driven shock in the solar wind [27][28][29] and [39].
With this context in mind, two different shock waves that occurred after the CME18/02/1999 and CME28/04/2001 are studied here. Evolution of the shock propagation in the surrounding space is analysed by means of a 1-D hydrodynamical model, parameterised with respect to the Reynolds number effects. From our results we can draw the following conclusions presented as items: -Comparing our result, with that of [15] we deduced that the downstream plasma density 10.3 cm -3 fits well to M 1 4.4 in the present model presented by Figure 2.  Fig. 4, we observe a increasing tendency of the downstream velocity (u 2 ) for greater values of  1 / 2 as expected.
-The compression rates () of these shock waves are greater than 3.5 for the two cases.
-When we employ the upstream Mach numbers 4.4 and 4.9, and the upstream velocities given in Table 1, the sound speeds in the interplanetary medium are found as 89 km/s and 91 km/s for the two cases which are comparable to the works of [12] and [13], who estimated this value within the range 90-100 km/s. -The entropy difference, S 2 -S 1 , is found to increase with increasing upstream Mach number M 1 . Similar to [20] and [40] very weak shocks (i.e. M 1 <1.2) turn into become nearly isentropic for the increasing values Reynolds number ratios (Re 2 /Re 1 ).

-
Entropy difference S 2 -S 1 has a tendency to increase with increasing compression rate (), which means that the downstream density itself is increasing with the entropy difference. -Downstream temperature also shows an increasing trend with greater entropy differences.
-On the other hand, unlike the  and temperature ratio variations, the entropy differences S 2 -S 1 decrease with increasing fluid velocity ratios.