Collision of an Interplanetary Shock Wave with the ... - Springer Link

ISSN 0015-4628, Fluid Dynamics, 2014, Vol. 49, No. 2, pp. 270–287. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A.S. Korolev, E.A. Pushkar, 2014, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2014, Vol. 49, No. 2, pp. 148–168.

Collision of an Interplanetary Shock Wave with the Earth’s Bow Shock. Hydrodynamic Parameters and Magnetic Field A. S. Korolev and E. A. Pushkar Moscow State Industrial University (MSIU), ul. Avtozavodskaya 16, Moscow, 115280 Russia e-mail: [email protected] Received January 28, 2013

Abstract—Hydrodynamic parameters and magnetic field generated in each of the waves in neighborhood of the Earth’s bow shock when an interplanetary shock wave impinges on it and propagates along its surface are found in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model. The interaction pattern is constructed in the quasi-steady-state formulation as a mosaic of exact solutions, obtained by means of a computer, to the Riemann problem of breakdown of a discontinuity between the states downstream of the impinging wave and the bow shock on the traveling line of intersection of their fronts. The calculations are carried out for typical parameters of the quiescent solar wind and the interplanetary magnetic field in the Earth’s orbit when the plane front of a shock wave moves along the Sun-Earth radius with various given velocities. The solutions obtained can be used to interpret measurements carried out by spacecraft in the solar wind and in neighborhood of the Earth’s magnetosphere. Keywords: solar wind, interplanetary shock wave, Earth’s magnetosphere, bow shock, magnetosheath, three-dimensional interaction between shock waves. DOI: 10.1134/S001546281402015X

At present, the Wind, SOHO, and ACE spacecraft are continuously measuring the state of the solar wind and interplanetary magnetic field in the neighborhood of the Lagrange point L1 , where the Sun’s and Earth’s attraction forces are balanced, at distances of approximately 250 Earth’a radii RE from the Earth. A particular attention is focused on sudden sharp perturbations identified with shock waves, rotational and tangential discontinuities in the solar wind (Fig. 1) and their relation with subsequent manifestations of these perturbations in the outer magnetosphere, magnetosheath, and in the neighborhood of the Earth’s bow shock (approximately in a hour) recorded hear by groups of THEMIS, Cluster, and Double Star spacecraft [1–6]. These investigations are necessary to forecast cosmic weather which affects the state of the Earth’s magnetosphere (for example, such phenomena as sudden storm commencements, magnetic substorms, and sudden geomagnetic impulses [4–5]). In certain cases, the results of numerical MHD simulations obtained by means of various methods [4–8] are used to analyze flow generated in the magnetosheath and in the neighborhood of the magnetopause when an interplanetary shock wave or another strong solar wind discontinuity arrives here after interaction with the bow shock. However, as a rule, these methods have insufficient spatial resolution so that several MHD waves merge together and cannot always be identified, for example, slow or Alfvén waves are combined with the contact discontinuity [7]. In order to interpret the measurements more exactly and in detail, it is necessary to use the solution of the problem of interaction between a discontinuity propagating in the solar wind stream and the Earth’s bow shock Sb [9–16]. In [9, 10] a quasi-steady-state method for constructing such solutions within the framework of magnetohydrodynamics of an ideally conducting medium was first proposed on the base of the solution 270

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Fig. 1. Interplanetary shock wave S f developed in the solar wind as a result of a chromospheric flare or a coronal mass ejection on the Sun S, and force lines of the interplanetary magnetic field Bsw . Spacecraft are schematically shown in the neighborhood of the Lagrange point L1 and the Earth’s bow shock Sb and in the magnetosheath between Sb and the magnetopause m which is the boundary of the magnetosphere M (shown in section with an image of the Earth’s magnetic field); broken curve corresponds to the Earth’s (E) orbit.

of the Riemann problem of breakdown of a discontinuity between the states downstream of the interacting waves on the traveling line of intersection of their fronts. The wave flow pattern and the dependences of the physical parameters of the medium and magnetic field were obtained as functions of the angle of inclination of Sb to the solar wind velocity Vsw in the two-dimensional plane-polarized formulation corresponding to the line of intersection of the plane containing Vsw and the interplanetary magnetic field Bsw with the surface of Sb [9–11]. For its 3D surface a series of features and the wave structure of the three-dimensional interaction pattern were found in the non-plane-polarized formulation in [12–16]. The present study, which is a continuation of [14, 16], contains pictorial illustrations of the gas dynamic parameters and magnetic field, convenient for the further use and presented as functions of the coordinates of points of contact of the impinging shock wave with the bow shock in its neighborhood. The density and magnetic field strength distributions are constructed as functions of the latitude and the longitude of points on the surface of Sb in all the waves appearing in the neighborhood of its surface. They are analyzed with reference to “catastrophic” flow restructurings [12, 14] and changes in the wave flow pattern and intensities of the comprising waves [16]. These solutions can be used to obtain the boundary conditions for calculating flow in the magnetosheath and successive impact on the Earth’s magnetosphere. 1. FORMULATION OF THE PROBLEM We will consider the problem of impingement of the plane front of a fast MHD shock wave S f propagating from the Sun with a velocity VS f with respect to the solar wind on the terrestrial bow shock Sb within the framework of the ideal magnetohydrodynamic model [17]. We will treat Sb as a fast MHD shock wave of variable intensity standing off from a blunt obstacle in the supersonic solar wind stream. This obstacle is the Earth’s magnetosphere M with the boundary in the form of the magnetopause m (Figs. 1 and 2). The shape of Sb is determined by the state of solar wind and the interplanetary magnetic field Bsw , and the shape and conditions of flow past the magnetopause. The shape of Sb is assumed to be specified as a convex surface FLUID DYNAMICS

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Fig. 2. Locations of the front of an interplanetary shock wave S f i (i = 1, 2) and curve LS f Sb of intersection of S f with the surface of the bow shock Sb when the S f front travels. The magnetosheath is the region between Sb the magnetopause m. Negative and positive values of y correspond to the dawn and dusk flanks, respectively. The tangential discontinuity catastrophe KT takes place on the curve KT (a). Wave pattern (regular solution) developed in the neighborhood of a point of intersection of S f and Sb (b).

whose example may be a paraboloid or one of the sheets of a two-sheeted hyperboloid [18, 19]. We will assume that the vector Bsw is inclined to Vsw at an angle ψsw and, to be specific, lies in the plane of the ecliptic (Fig. 1, the xy plane in Fig. 2a), while the normal nS f to the front S f is aligned with the Sun-Earth radius being collinear to the vectors Vsw and VS f (Vsw ∥VS f ) [14]. In the Cartesian coordinate system xyz moving with the Earth (the x axis is directed toward the Sun, the y axis lies in the plane of the ecliptic, and the z axis is perpendicular to the plane of the ecliptic) the bow shock is presumed to be steady-state. The location of each of its elements, which has a normal nSb and can be approximated by the tangential plane to the surface of Sb , is specified by two angular coordinates, namely, the latitude α (angle of inclination of the element of Sb to Vsw ) and the longitude τ (angle of inclination of nSb to the plane of the ecliptic) [12, 14], so that nSb = (cos α , sin α cos τ , sin α sin τ ) (Fig. 2a). The shape of the surface of Sb can be specified by means of the distance of a surface element from the x axis: r = R(α , τ ). In the present study we does not concretize R(α , τ ). We will neglect asymmetry of the shape of Sb (which is resulted from the fact that ψsw ∕= 0), i.e., presume that r = R(α ). The function R(α ) is assumed to be a convex monotonically increasing function of the angle 90∘ –α when α decreases from α = 90∘ (subsolar point on Sb ) to the characteristic values α ∗ = α (τ ) on flanks of Sb . The latter correspond to the angles of inclination of the fast magnetosonic characteristics [17] to Vsw in the steady-state solar wind stream. The lines of constant longitude τ = const represent the meridians on the surface of Sb passing through its vertex (α = 90∘ ). In Fig. 2a the “dawn” and “dusk” flanks are located to “east” and “west” (view from the Sun) from the midday meridian τ = ±90∘ . The meridian τ = 0∘ corresponds to intersection of the plane of the ecliptic xy with the dusk flank (y > 0), while the meridian τ = 180∘ to intersection with the dawn flank. The longitudes 0 < τ < 180∘ correspond to the north side of the surface of Sb , while 180∘ < τ < 360∘ to the south side. FLUID DYNAMICS

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In the undisturbed solar wind stream ahead of Sb (on its windward side) the density ρsw , the pressure psw , Vsw , and Bsw are known; therefore, all the flow parameters and the magnetic field can be determined on the outer boundary of the magnetosheath (on the leeward side of Sb ) as functions of α and τ from the relations on the oblique steady-state non-plane-polarized MHD shock wave [13, 20, 21]. For this purpose it is necessary to go over into an instantaneous local coordinate system moving with a point (α , τ ) on the curve LS f Sb of intersection of the fronts of S f and Sb [13–15]. After contact of the fronts of S f and Sb at the subsolar point (α = 90∘ , the angle τ cannot be determined), a discontinuity between the states downstream of Sb and S f appears at each point on the curve LS f Sb . Flow downstream of LS f Sb developed in breakdown of this discontinuity [17, 20] is assumed to consist of two combinations of waves traveling in opposite directions, namely, those penetrating into the magnetosheath and propagating toward the magnetosphere (denoted by subscript f (forward)) and those reflected from Sb to the solar wind disturbed by the shock wave S f (denoted by subscript “b” (backward)) (Fig. 2b). Each of the wave trains contains a fast wave (shock wave S+ or rarefaction wave R+ [22]), a non-plane-polarized Alfvén (rotational) discontinuity A, and a slow wave (S− or R− ) [13–15, 22]. The states downstream of the slow waves are separated by a contact discontinuity C on which six independent boundary conditions, namely, the no-flow condition (vn = 0) and the velocity, pressure, and magnetic field continuity conditions, must be satisfied [17, 20]. The shock wave S f travels along the surface of Sb with a supersonic velocity; therefore, the process of collision of S f and Sb can be considered locally. The establishment of flow in the interaction between S f and Sb is assumed to occur in a fairly short time which is negligible as compared with the characteristic traveling time. Since the normal nS f is assumed to be aligned with Vsw , the entire problem is determined by five dimensionless parameters. These are the gasdynamic Mach number of the solar wind Msw = ∣Vsw ∣/a0 , where √ gaskinetic and magnetic pressures in the solar wind β = 8π psw /B2sw (or a0 = γ psw /ρsw , the ratio of the √ √ the Alfvén number N = aA /a0 ≡ 2/γβ , where aA = ∣Bsw ∣/ 4πρsw is the Alfvénic velocity), the angle ψsw between the vectors Vsw and Bsw , the Mach number of the interplanetary shock wave MS f = ∣VS f ∣/a0 , and the specific heat ratio γ = 5/3. The state downstream of Sb at the point of interaction between S f and Sb (local problem) is given by the angles α and τ . In our investigation, we will specify parameters corresponding to mean values in the quiescent solar wind in the Earth’s orbit, namely, Vsw = 390 km/s, ∣Bsw ∣ = 6.2 nT, the proton temperature 1.2 × 105 K, the particle concentration 11 cm−3 , and aAsw ≈ 55 km/s [23, 24] from which we can find the dimensionless quantities Msw = 8, N = 1.1 (β ≈ 1), and ψsw = 45∘ . The solar wind shock wave S f can appear as a result of different events on the Sun, namely, flares, chromospheric coronal mass ejections, etc., the observed values of MS f lying on the interval 1.3 ≤ MS f ≤ 10 [23]. In order to investigate the process of interaction between Sb and S f we take the gasdynamic Mach numbers MS f = 2, 3, 5, and 8 [14, 16] which are typical for weak, medium, and strong interplanetary shock waves. In Table 1 we have presented the values of the density ρS f , the pressure PS f , and the magnetic field strength ∣B∣S f in the state behind S f , divided by their values in the solar wind, as functions of MS f . From Table 1 we can see how the velocity of S f affects the gasdynamic parameters and the magnetic field downstream of S f . The nature of the interaction between S f and Sb depends on the angle θ = 90∘ –α between their fronts, the intensities of S f and Sb , and the orientation of the magnetic field. The interaction is almost one-dimensional in the latitudes α ∈ [75∘ , 90∘ ], where Sb is strong. Two-dimensional effects begin to manifest themselves when α ≤ 70∘ and are significantly strengthened with decrease in the latitude α and the corresponding increase in θ . In this case S f has a fixed intensity determined by its direction of propagation with respect to the interplanetary magnetic field Bsw , its velocity (Mach number), and dependent also on the solar wind parameters and ∣Bsw ∣, while Sb has a variable intensity since, firstly, its local Mach number depends on α : MSb = Msw sin α and, secondly, the inclination of the magnetic field to the normal nSb , which affects significantly the current intensity of Sb , is a function of the latitude α and the longitude τ on the curve LS f Sb (Fig. 2a). FLUID DYNAMICS

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Fig. 3. Polar coordinate system α , τ for diagrams in Figs. 4–10; the longitudes τ are plotted near the corresponding meridians (y and z axis); r is the distance from the surface Sb to the x axis of the xyz coordinate system in Fig. 2a

Table 1 MS f

ρS f

PS f

∣B∣S f

2

1.62

2.38

1.5

3

2.4

6.66

2.

5

3.24

24.4

2.5

8

3.67

71.9

2.8

In Fig. 3 we have reproduced the polar coordinate system used in Figs. 4–10. The origin corresponds to the projection of the vertex of Sb coinciding with the subsolar point on Sb , at which α = 90∘ , to the yz plane. The lines of equal latitudes α = const (“parallels”) are concentric circles of radii r with centers at the origin. These radii are proportional to the distance R of a point on the surface Sb from the x axis (Fig. 2a): r = kR(α ), where k is a scale factor. The curves LS f Sb of intersection of the S f and Sb fronts (Fig. 2a) are projected to these circles. The meridians τ = const are the rays issuing from the origin. In Figs. 4–10 we have reproduced distributions of quantities over the surface Sb in projection to the yz plane (Fig. 2a) in the coordinate system α , τ (Fig. 3). No regular solution was found in white spots. Near the outside left and lower pictures in Figs. 4–10 we have plotted the values of the latitudes (in degrees) corresponding to the parallels α = const in Fig. 3. The level lines of the depicted parameters are constructed and its own scale of correspondence of the grey colors (from black for a minimum to white for a maximum) to values of this parameter is given near each of the pictures. 2. VARIATIONS OF PARAMETERS OF THE MEDIUM AND THE MAGNETIC FIELD ON Sb The steady states of the medium and the magnetic field are significantly inhomogeneous and anisotropic on the leeward side of the bow shock Sb (Fig. 4). Approximate equality of the gaskinetic and magnetic 2 as compared pressures psw and ∣Bsw ∣2 /8π (β ≈ 1) and predomination of the dynamic pressure pdyn = ρswVsw 2 with the gaskinetic (and magnetic) pressure pdyn /psw = γ Msw ≈ 106 is characteristic of the solar wind in the Earth’s orbit. Nevertheless, even so small, at first sight, contribution of the magnetic field (magnetic pressure is less than 1% of the dynamic one) leads to an appreciable transformation of the distributions FLUID DYNAMICS

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Fig. 4. Dimensionless density ρSb = ρ /ρsw , pressure PSb = p/psw , magnetic field strength ∣B∣Sb = ∣B∣/∣B∣sw (a–c), magnetic field components −Bx , By , Bz divided by ∣B∣sw (d–f), velocity of the medium −∣V∣Sb and its components Vy , Vz divided by a0sw (g–i) downstream of Sb in the undisturbed state; the solar wind parameters: the particle concentration 11 cm−3 , the proton temperature 1.2 × 105 K, ∣Bsw ∣ = 6.2 nT, Vsw = 390 km/s, a0sw = 49 km/s, and aA ≈ 55 km/s.

of parameters of the medium behind the Sb front which are not axisymmetric (Fig. 4). This is due to the magnetohydrodynamic nature of flow. For example, the density distribution (Fig. 4a), which is symmetric about the plane of the ecliptic (meridians τ = 0 180∘ ), has approximate central symmetry about a point on Sb at which the jump in density is maximum ρSb /ρsw ≈ 3.65 (latitude α ≈ 70∘ ) and is displaced toward the dawn flank from the subsolar point at ∼ 15∘ . The asymmetry of the density variations, which reaches 15–20% on the dawn-dusk flanks [25], was confirmed in spacecraft measurements [26]. The pressure distribution (Fig. 4b) is also almost cylindrically symmetric about the maximum point (α ≈ 80∘ , τ = 180∘ ). The density distribution (Fig. 4a) is plateau-shaped as compared with the pressure distribution (Fig. 4b) since Sb is a strong shock wave in a wide vicinity of the subsolar point (α > 40∘ ) and in this zone the jump in density is close to the possible maximum (∼ (γ + 1)/(γ − 1)ρsw at M = ∞), whereas the pressure has no limitations for increase with increase in Vsw . The maximum velocity deceleration to ∼ 2a0sw takes place almost at the subsolar point (Fig. 4g). The effects of dawn-dusk asymmetry are due to significantly different influence of the magnetic field on Sb since the bow shock is quasi-parallel and quasi-perpendicular, respectively, on these flanks (with respect to the angle between the normal and the magnetic field) [27]. The angle between the vectors nSb and Bsw is crucial for variations in the solar wind parameters on Sb . The “center” of the ∣BSb ∣ distribution is the point α = 45∘ , τ = 180∘ on the surface Sb (Fig. 4c) at which Bsw is perpendicular to the Sb front, the magnetic field strength does not vary (BSb = Bsw = (∣Bsw ∣ cos ψsw , ∣Bsw ∣ sin ψsw , 0)), and the electric current flowing along the Sb front is equal to zero. The component Bz (Fig. 4f) varies in a radically different way as compared with Bx and By (Figs. 4d and 4e). Its variations are antisymmetric about the plane of the ecliptic, in which Bz ≡ 0 and Sb is planepolarized, and have clearly expressed maximum and minimum on the north and south sides of Sb on the dawn flank when α ≈ 45∘ (τ ≈ 135 and 225∘ , respectively (Fig. 4f)). The “pass” relief is characteristic of Bz (Fig. 4f). The curve Bz = 0 is crescent-shaped and passes through the points α = 13∘ , τ = ±75∘ and FLUID DYNAMICS

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Table 2 MS f

Sf

S′f

S(or R)−f

C

Sb′

S(or R)− b

2

0.62

0.6–1.3

−0.35–0.1

−1.4–0.2

0–4

−0.2–0.25

3

1.4

1.4–3

−0.5–0.2

−2.5–0.5

0–6

−0.3–0.7

5

2.24

2.8–5

−0.3–0.8

−2.5–1.5

0.5–6

−0.6–0.5

8

2.67

4.8–6.8

−0.4–1.0

0–3.5

1.5–5.5

−0.8–0.2

α = 45∘ , τ = 0∘ (pass point). Here, the interplanetary magnetic field is almost parallel to the surface Sb . Significant effect on the behavior of ∣BSb ∣ and Bx , By , and Bz components (Figs. 4c–f) exerts the fact that their variations tend to zero as the angles of inclination of Sb approach the angles of inclination of fast magnetosonic characteristics on the outer boundary of the domain plotted in Fig. 4 (for MS f = 2 this boundary is close to the boundary of existence of the regular solution [10, 14, 16], see Fig. 2b) since the intensity of Sb tends to zero here. Variations in the velocity components Vy and Vz on the leeward side of Sb (Figs. 4h and 4i) are related with the geometry (shape) of Sb associated with supersonic flow past a blunt obstacle (Earth’s magnetosphere). This leads to deviations of the velocity in the directions of the y and z axes which are opposite in sign on the dusk flank and the north side of Sb as compared with the dawn flank and the south side of Sb . In absolute value their behavior on the dawn and dusk flanks (Figs. 4h and 4i) differs only slightly due to the predominant effect of the dynamic pressure as compared with the effect of the magnetic field. The inhomogeneities of the density, pressure, velocity, and, in particular, magnetic field distributions on the leeward side of Sb demonstrated by Fig. 4 play a significant role in the interaction between S f and Sb . It is precisely these inhomogeneities determine the wave pattern and the intensities of waves generated at the point of interaction [16]. 3. VARIATIONS IN THE DENSITY IN WAVES GENERATED IN THE NEIGHBORHOOD OF THE POINT OF INTERACTION BETWEEN S f AND Sb The density varies in different ways when a shock wave S f of different intensity impinges on Sb (Fig. 5 and Table 2). In Table 2 we have given the boundaries of variations in the dimensionless density Δρ /ρsw reached in the waves S f , S′f , S(R)−f , Sb′ , and S(R)− b and on the contact discontinuity C for the values of MS f considered. Different variations in the density result both from the quantitative differences between changes in the parameters in the initial shock Sb on the dawn and dusk flanks and from the qualitative differences between the nature of interaction of S f and Sb on these flanks due to different orientations of Bsw with respect to their fronts [14, 16] leading to strengthening of asymmetry in the density distribution. As compared with the initial S f (Table 2), the density variations in the shock wave S′f (Fig. 2b), which arises during transformation of S f in the nonlinear process of its interaction with Sb , and, being in the lead among the waves penetrating into the magnetosheath, will be called, for the sake of brevity, by “refracted” shock, are strengthened, the jump in density in S′f increasing significantly at the very edge of the dawn flank near the boundary of existence of the regular solution [9, 10, 14, 16] in a fairly wide neighborhood of the plane of the ecliptic 135∘ < τ < 225∘ (Figs. 5a–5d). The greater MS f , the greater variations in the density in this zone: as compared with S f , increase by more than two times when MS f = 2 and 3 and almost by three times when MS f = 8. When MS f = 2 and 3, the density varies only slightly in S′f at the edge of the dawn flank, as compared with S f (Table 2 and Figs. 5a–5d), since a regular interaction on this flank exists up to the boundary of existence of Sb on which Sb is transformed into a characteristic and variations in all the quantities tend to zero on it. When MS f = 5 and 8, the regular interaction domain becomes narrower and the lower limit of the density variations in S′f becomes significantly greater than its jump in S f , since in this case S f interacts always with a fairly strong Sb which have a significant jump in density. As MS f increases, FLUID DYNAMICS

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′ Fig. 5. Variations in the dimensionless density Δρ /ρsw in S′f , S(or R)−f , on C, in S(or R)− b , and Sb (from top to bottom in each column) when MS f = 2, 3, 5, and 8 (from left to right). Notation of waves see in Fig. 2b

. a nonmonotonicity of the density variations in S′f develops during its propagation from the subsolar point (Figs. 5a–5d). Qualitatively different variations in the density on the dawn and dusk flanks are caused by the “quasiperpendicular” and “quasi-parallel” nature of the interaction between S f and Sb (with respect to the angle between the reference-frame velocity of the local coordinate system and Bsw) [28]. In the case of quasi-perpendicular interaction the medium is compressed by the magnetic field whose strength varies only slightly, whereas in the case of quasi-parallel interaction the magnetic field strength increases considerably but no additional compression of the medium is observed. A rotational discontinuity A f moves in the wake of S′f toward the magnetopause. On A f the density does not vary; however, variations in the magnetic field and the velocity create the state through which a slow wave propagates. This wave can be a slow shock S−f or a rarefaction wave R−f [16]. The wave flow pattern developed depends significantly on MS f and was described in detail in [16]. In the case of weak shock waves S f (MS f = 2 and 3, Figs. 5e and 5f) the density increases only slightly in S−f and decreases considerably in R−f (Table 2). The rarefaction waves R−f develop in the most part of interaction. Stronger shock waves S f (MS f = 5 and 8, Figs. 5g and 5h) lead to weakening R−f , contraction FLUID DYNAMICS

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of their existence domain, and strengthening S−f (Table 2). Thus, increase in the intensity of S f leads to an additional considerable increase in the density in S−f . When MS f = 2, the tangential discontinuity catastrophe KT [12, 14, 16], which takes place on the dusk flank at the edge of the regular interaction domain, has almost no effect on the density variations: ∣Δρ ∣ < 0.025ρsw on the curve KT and the width of the zone downstream of the curve KT is less than 0.6∘ in latitude [14]. When MS f = 3, changes in the density on KT become appreciable, flow S′f A f S−f CSb− Ab Sb′ is replaced − ′ by flow S′f A f R−f CR− b Ab Sb [14, 16] as Sb is crossing the curve KT (Fig. 2a) [14, 16], S f being transformed − suddenly in R f of finite intensity, the density decreases by ∣Δρ ∣ ≈ 0.23ρsw in the neighborhood of the plane ′ ∘ of the ecliptic (Fig. 5f). The width of the zone with flow S′f A f R−f CR− b Ab Sb is ∼ 3 in the latitude and occupies a sector of ∣τ ∣ < 38∘ in the longitude. The catastrophes KT manifest themselves ever more clearly when MS f = 5 and 8 (Figs. 5g and 5h). It is precisely on the curve KT (Fig. 2a) the maximum and minimum values of ΔρS− and ΔρR− given in Table 2 are reached. f

f

Changes in the density taking place in S f (Table 1), Sb , S′f , and S−f (or R−f ) lead to values of the density in the states downstream of the corresponding waves whose limits are given in Table 3. The density distributions in the states downstream of S′f (Figs. 6a–6d) and slow waves (Figs. 6e–6h) show: 1) successive strengthening of asymmetry of the density distributions over the dawn and dusk flanks as MS f increases from values ∼ 15–20% for MS f = 2 (Figs. 6a, 6e) to 30% for MS f = 8 (Figs. 6d, 6h); 2) displacement to the dawn flank and simultaneous increase in the maximum densities from the value ∼ 5ρsw reached in the latitude α ≈ 70∘ (τ = 180∘ ) when MS f = 2 (Figs. 6a, 6e) to the value ∼ 10ρsw in the latitude α ≈ 40∘ (τ = 180∘ ) when MS f = 8 (Figs. 6d, 6h); 3) formation of a region in the magnetosheath with a local minimum density ∼ 8.5ρsw in the neighborhood of α ≈ 70∘ (τ = 0∘ ) (Fig. 6h); 4) formation of a zone with a sharp drop in the density and the pressure as a result of the tangential discontinuity catastrophe KT occurring on a certain curve on which the self-electric field reverses sign. With increase in MS f the curve KT is displaced toward the subsolar point but remains fairly far from it (Fig. 2a). We note that this zone, which is narrow in the angular coordinates (α , τ ) (Figs. 5 and 6), may be quite extended over the surface Sb on the dusk flank (Fig. 2a). The formation of a zone in the magnetosheath on the dawn flank with considerable (> 25% when MS f = 5 and 8) increase in the density as compared with the dusk flank (Figs. 6c, 6d, 6g, 6h) is only slightly related with the dynamic impact of the shock wave S′f on the magnetosheath, which has no so strongly expressed asymmetry [16]. This effect is mainly determined by the presence of the magnetic field of radically different orientation on these flanks, different action of the ponderomotive forces exerted on the medium, and redistribution of the electric currents among the waves. Variations in the density in the slow waves, which have the magnetohydrodynamic nature, are significantly more tightly coupled with their dynamic impact on the medium but the dynamic impact is considerably weaker [16]; therefore, scales of the density variations are smaller almost by an order of the magnitude (Table 3). Change in the action of the ponderomotive forces manifests itself most clearly in the formation of a zone on the dusk flank which is narrow in the latitude (angle of inclination of the Sb front to Vsw ) but extended in the longitude and spatial coordinate along the normal to the plane of ecliptic and related with vanishing the electric field and reversing its sign. On the boundaries of this zone the density drops jumpwise in slow waves of finite intensity (jump ∼ 1.5ρsw ) when the type of the slow waves changes from shock to rarefaction wave. The formation of the jump in density Δρ ∣C = ρS(R)− − ρS(R)− on the contact discontinuity separating f b the waves traveling toward the magnetopause and those traveling toward the Sun (Table 3, Figs. 5i–5l) represents an important aspect of the impact of S f on Sb and the magnetosheath. We note the transition from Δρ ∣C < 0 (the higher density in the state 4′ (Fig. 2b) on the side of C with the normal directed to the Sun) in a wide neighborhood of the subsolar point when MS f = 2, 3, and 5 (Figs. 5i–5k) to Δρ ∣C > 0 over the entire surface of Sb when MS f = 8 (Fig. 5l). In this connection we can propose a hypothesis that in the case of a fairly strong shock wave (MS f = 8), once S′f and S−f (or R−f ) have crossed the magnetosheath and FLUID DYNAMICS

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Table 3 MS f

Sb

Sf

S′f

S(or R)−f

Δρ ∣C

S(or R)− b

Sb′

Sb′ /S′f

2

1–3.67

1.62

1.5–5

1.5–5

−1.4–0.2

1.5–6

1.5–6

0.92–3.7

3

1–3.67

2.4

2.5–6

2.5–6

−2.5–0.5

2–9

2–8

0.83–3.33

5

1–3.67

3.24

4–8

4–8

−2.5–1.5

3–10

4–9

1.23–2.77

8

2–3.67

3.67

7–10

6.5–9.5

0–3.5

4–9

5.5–9

1.49–2.45

Table 4 MS f

Sf

S′f

S(or R)−f

Sb′

S(or R)− b

2

0.5

0–0.9

−0.1–0.8

0–3.5

−1.2–0.4

3

1.0

0–1.8

−0.2–1.4

0–4.5

−2–0.5

5

1.5

−0.5–3

−1.5–1.5

−0.5–4.5

−2–1.5

8

1.8

0–4.5

−4–3

0–4

−1–3.5

reached the magnetopause, the inverse jump in density −Δρ ∣C can affect the magnetopause and lead to its expansion, whereas, when MS f = 2, 3, and 5, the magnetopause can be compressed over the head-on part of Sb , where Δρ ∣C < 0, and expanded on the flanks, where Δρ ∣C > 0 (Figs. 5i–5k). From Table 3 and Figs. 6m–6p it follows that the bow shock Sb′ is weakened only slightly under the impact of S f when MS f = 2 and the relative scale of the density variations in Sb′ is conserved with allowance for increase in the density by a factor of 1.62 in S f . When MS f = 3 and 5 the relative change in the density ρSb′ /ρS f corresponds to decrease in the solar wind Mach number to Msw′ ≈ 5.1 and 4, whereas MS f = 8 leads to considerable weakening of Sb′ in which the density variations correspond to Msw′ ≈ 3. With increase in MS f the density variations in Sb′ become more and more symmetric as compared with Sb . This is caused by decrease in the Alfvén number N in S f : in the fast shock wave the magnetic field strength and the magnetic pressure increase in a significantly smaller extent as compared with increase in the gaskinetic pressure [20]. 4. VARIATIONS IN THE MAGNETIC FIELD STRENGTH S′f

propagates through the magnetosheath, variations of ∣B∣ (Table 4, Figs. 7a–7d) partially replicate As Δ∣B∣ in Sb (it is necessary to subtract unity from the values of ∣B∣Sb given in Fig. 4c to find Δ∣B∣Sb ). However, they have considerable differences since, firstly, Δ∣B∣S f ′ ↛ 0 at the “limb” of the regular interaction domain and, secondly, which is of importance, the values Δ∣B∣S f ′ increase by 1.8–2.5 times as S f becomes stronger (MS f increases) (Table 4). The maximum increase in ∣B∣ in S′f (Figs. 7a–7d) is observed in the same zones, where ∣B∣Sb reaches a maximum (Fig. 4c). Thus, heavy electric currents induced in Sb are redistributed during the interaction in these zones so that they now flow inside S′f and are carried away by it toward the magnetopause. In this case ∣B∣S f ′ can become smaller than ∣B∣S f (Table 4). This also indicates that the electric current flowing in S f is redistributing during the interaction so that it flows after interaction in other waves and the fact that Δ∣B∣S f ′ ≈ 0 on the dawn flank in the neighborhood of the plane of the ecliptic (Figs. 7a–7d) means that the front of S′f propagates here almost along the magnetic field. An interesting effect consists in the opposite of variations in ρS f ′ and ∣B∣S f ′ , namely, the density increases in the maximum extent in the neighborhood of those zones in the magnetosheath on the dawn flank, where the magnetic field varies only slightly (Figs. 6a–6d and 7a–7d). The opposite is also valid: in those zones in which the density varies only slightly in the undisturbed magnetosheath, for example, on the dusk flank, Δ∣B∣S f ′ varies in the maximum extent. FLUID DYNAMICS

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′ Fig. 6. Dimensionless density ρ /ρsw in the states downstream of S′f , S(or R)−f , S(or R)− b , and Sb (from top to bottom in each column) when MS f = 2, 3, 5, and 8 (from left to right).

The magnetic field strength is the vector quantity. As compared with the density and the pressure, it varies in a more complex way (Figs. 8–10). For example, nonmonotonic variations of Bx in S′f are possible (Figs. 8a, 8b). We note that the component By , generated in S′f (Figs. 9a–9d) on the dusk flank, is greater by several times than Bx (Figs. 8a–8d). The boundaries of variations in By almost coincides with the boundaries of variations in ∣B∣S f ′ , i.e. By makes a main contribution to ∣B∣S f ′ . The distributions of Bz generated in S′f are qualitatively similar (Figs. 10a–10d) and their shape almost coincides with the distribution of Bz downstream of Sb (Fig. 4f). Only the scales of the variations are different. They are equal to ±0.5∣B∣sw , ±1.5∣B∣sw , ±2.5∣B∣sw , and ±4∣B∣sw when MS f = 2, 3, 5, and 8, respectively. An Alfvén discontinuity A f (Figs. 8e–8h, 9e–9h, and 10e–10h) and a slow wave (Table 4, Figs. 7e–7h, 8i–8l, 9i–9l, and 10i–10l) propagate through the magnetosheath disturbed by S′f . The influence of A f and S−f (or R−f ) on variations in the magnetic field is commensurable with the influence of S′f and the scales of variations in the magnetic field components in S′f (Figs. 8–10a–d), A f (Figs. 8–10e–h), and slow waves (Figs. 7e–h and Figs. 8–10i–l) are of the same order. The crucial factors for A f and S−f (or R−f ) is the state of the medium and, in particular, the magnetic field distribution arising as a result of propagation of S′f through the inhomogeneous anisotropic state downstream of Sb (Fig. 4). It is turned out that, as distinct from the distribution ρ (α , τ ) downstream of S′f which becomes more and more asymmetric as MS f increases and has a sharp increase in the density at the very edge of the dawn flank (Figs. 5a–d), the distributions B(α , τ ) downstream of S′f practically conserve the shape of B(α , τ ) behind Sb (Figs. 4d–4f) for all the components Bx , By , and Bz ; however, each of the components has its own “strengthening factor” which increases with MS f from a value close to unity when MS f = 2 to ≈ 1.4 for Bx and ≈ 3 for By and Bz when MS f = 8 (Table 5). FLUID DYNAMICS

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′ Fig. 7. Variations in the magnetic field strength ∣B∣/∣B∣sw in S′f , S(or R)−f , S(or R)− b , and Sb . The same notation as in Fig. 6.

Table 5 MS f

Sb

Sf

S′f

S(R)−f = C = S(R)− b

Sb′

2

1–3.5

1.5

1–4.5

1–5

1–5

3

1–3.5

2.0

1–5

1.5–6.5

2–6.5

5

1–3.5

2.5

1–6.5

2–7.5

2.5–7.5

8

1–3.5

2.8

1–8

2.5–7.5

2.5–6.5

Generation of B(α , τ ) and variations of the Bx , By , and Bz components in A f are caused by the planepolarized interaction between S f and Sb on the curve of intersection of the surface Sb with the plane of the ecliptic and the absence of the plane-polarized A f up to a certain latitude α ∗ (≈ 35–32∘ depending on MS f ) and its continuous splitting-off from R−f of maximum intensity in this latitude α ∗ as a result of a local catastrophe KL0 [14, 16] (Figs. 8e–8h and 8i–8l, 9e–9h and 9i–9l, and 10e–10h and 10i–10l) which takes place when the velocities of A f and R−f coincide. Then ∣A f ∣ increases along the meridian τ = 180∘ , as S f propagates along Sb (latitude decreases: α < α ∗ ), ∣A f ∣ reaches a maximum and then decreases so that A f disappears due to merging with R−f at the edge of the domain of existence of the regular interaction when MS f = 2, 3, and 5 as a result of the second local catastrophe KL0 [14, 16] (Figs. 8e–8g, 8i–8k, 9e–9g, 9i–9k, and 10e–10g, 10i–10k). It is precisely the presence of the local catastrophes KL0 on the meridian τ = 180∘ at certain latitudes on the dawn flank determines the global character of variations in the magnetic field in A f and slow waves R−f on this flank (Figs. 8e–8h and 8i–8l, 9e–9h and 9i–9l, and 10e–10h and 10i–10l). FLUID DYNAMICS

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Fig. 8. Variations in the component Bx /∣B∣sw of the magnetic field strength in the waves S′f , A f , S(or R)−f , S(or R)− b , Ab , and Sb′ (from top to bottom in each column) when MS f = 2, 3, 5, and 8 (from left to right).

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Fig. 9. Variations in the component By /∣B∣sw of the magnetic field strength in the waves. The same notation as in Fig. 8.

Variations of the magnetic field in A f and slow waves on the dusk flank has another character and they are caused by the quasi-parallelism of the magnetic field and the velocity. When MS f = 2 and 3 the distribution B(α , τ ) varies only slightly in A f , whereas when MS f = 5 and 8 the distribution B(α , τ ) changes significantly in A f , but only in the neighborhood of the boundary of the regular interaction (Figs. 8e–8h, 9e–9h, and 10e–10h). In this zone a catastrophic flow restructuring (catastrophe KT [14]) takes place. It is accompanied by sudden change in the magnetic field in A f and slow waves (Figs. 8e–8h and 8i–8l, 9e–9h and 9i–9l, and 10e–10h and 10i–10l) [14]. As MS f increases, the slow rarefaction wave R−f is transformed into a slow shock wave S−f . This leads to the following change in the character of the behavior of B: nonmonotonic variations of Bx and By in R−f (initial increase and successive decrease) when MS f = 2 and 3 are replaced by decrease in Bx and By FLUID DYNAMICS

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Fig. 10. Variations in the component Bz /∣B∣sw of the magnetic field strength in the waves. The same notation as in Fig. 8.

in S−f when MS f = 5 and 8 up to the curve KT , on which the flow is restructured and R−f suddenly appear downstream of the curve KT and all the components of the magnetic field change jumpwise (Figs. 8e–8h and 8i–8l, 9e–9h and 9i–9l, and 10e–10h and 10i–10l).

5. VARIATIONS IN THE MAGNETIC FIELD STRENGTH AND THE DENSITY IN REFLECTED WAVES The waves succeeded the transformed bow shock Sb′ can be measured by spacecraft [1, 4]. In S f the angle ψsw between Vsw and Bsw increases (since By increases); therefore, the region on Sb′ in which ∣B∣sw does not vary is displaced from the latitude α = 45∘ (Fig. 4c) toward lower latitudes at the very edge of the FLUID DYNAMICS

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dawn flank and the region with small variations in ∣B∣sw expands in the latitude (Figs. 7m–7p). The region on Sb′ in which ∣B∣ reaches the maximum values is displaced from the dusk flank toward the subsolar point (Figs. 7m–7p). Variations in ∣B∣ in slow waves proceed nonmonotonically with increase in MS f (Figs. 7i–7p). Variations in the density in slow waves succeed variations in the magnetic field and also depend nonmonotonically on MS f (Figs. 5i–5p). Appreciable decrease in ρ in R− b on the dusk flank occurring when MS f = 2 and 3 (Figs. 5m, 5n) with a crescent-shaped minimum is replaced by a less appreciable annular-shaped decrease in the density when MS f = 5 and 8 (Figs. 5o, 5p). For these values of MS f the minimum densities are reached at the very edge of the dusk flank downstream of the curve KT (Figs. 5o, 5p), where ∣B∣ increases sharply so that the freezing-in integral ∣B∣/ρ = const [17] is not valid due to the three-dimensional non-plane-polarized nature of the interaction, namely, a zone of rarefied plasma with strong magnetic field is formed on the dusk flank downstream of Sb′ . A comparison of variations in the components of B in Ab (Figs. 8–10q–t) and A f (Figs. 8–10e–h) and − “reflected” S(R)− b and “refracted” S(R) f slow waves (Figs. 8–10i–10l, 10m–10p) makes it possible to assert that the global pattern of variations in the magnetic field on the dawn and dusk flanks is mainly dictated by the presence of local catastrophes KL− on the dawn flank (when MS f = 2 and 3 at latitudes α ≈ 34 and 26∘ , respectively) related with splitting the switch-off slow shock wave Sb−∗ on its combination with a plane-polarized Ab [12, 14, 16] or the tangential discontinuity catastrophe KT on the dusk flank. In this case the maximum and minimum values of the Bx and By components are reached to west and east (at the higher and lower latitude) from the corresponding point of the catastrophe KL− on the meridian τ = 180∘ , while the Bz component changes jumpwise to the opposite value in the perpendicular direction with respect to the meridian τ = 180∘ (with respect to the plane of the ecliptic) (Figs. 8–10). A similar effect takes place on the curve KT . Summary. The constructed diagrams of variations in the density and the magnetic field strength give a clear representation of global variations in the state of the medium and the magnetic field in all the waves generated in the process of interaction between the Earth’s bow shock Sb and the front of an interplanetary shock wave S f propagating with various given velocities along the Sun-Earth radius. The dawn-dusk asymmetry of the action of S f on the flanks of Sb is determined by different orientation of the magnetic field with respect to the interacting waves. The velocity of S f and, consequently, its intensity are of importance both for the qualitative flow restructuring and for the quantitative variation in the parameters of the medium and magnetic field. The maximum increase in the density in the refracted shock wave S′f is reached on the dawn flank, where the magnetic field strength is almost aligned with the normal to Sb . The dawn-dusk asymmetry of variations in the density reaches 30% when MS f = 8. The increase in the density in S′f is partially compensated by its variations in slow waves S(R)−f so that in the zones, in which the density increases to the highest extent in S′f , subsequently the density decreases in a slow rarefaction wave R−f , whereas in the zones, in which the density in S′f increases initially to a lesser extent, the medium is additionally compressed by a slow shock wave S−f . Rotational discontinuities and slow waves affect significantly the magnetic field. On the dawn flank, their behavior and, respectively, variations in the magnetic field are related with the possibility of coincidence of the velocities of Alfvén discontinuities and slow magnetosonic waves at certain points on Sb on the curve of its intersection with the plane determined by the vectors of the undisturbed magnetic field and the solar wind velocity. On the dusk flank this is related to merging of five waves (Alfvénic, slow magnetosonic, and entropy characteristics) and vanishing the self-electric field on a certain curve orthogonal to the abovementioned plane. This leads to sudden change in these waves and results in a sharp change in the physical parameters of the medium (increase in the density and the pressure and decrease in the magnetic field strength) in the neighborhood of the contact discontinuity separating the groups of waves propagating in the wake of S′f and those traveling downstream of Sb′ , the contact discontinuity being transformed in a tangential discontinuity. FLUID DYNAMICS

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The solutions constructed can be used to interpret the measurements, carried out on spacecraft located between Sb and the magnetopause, of perturbations of the medium developed as a result of the impact of an interplanetary shock wave and can manifest themselves in the form of a sudden storm commencement or geomagnetic impulses in the Earth’s magnetosphere. The authors wish to thank A.G. Kulikovskii, G.A. Lyubimov, and A.B. Vatazhin for their interest in the study, valuable comments, and fruitful discussions and D.O. Levchenko for his assistance in the graphic representation of the pictures. The work was carried out with support from the Russian Foundation for Basic Research (project No. 1101-00235a) and a Russian Federation President’s Grant for the Support of Leading Science Schools (project No. NSh-1303.2012.1). REFERENCES ˇ anková, “Response of Magnetospheric Boundaries to the Interplanetary Shock: 1. L. Pˇrech, Z. Nˇemeˇcek, and J. Safr´ Themis Contribution,” Geophys. Res. Lett. 35, No. 17, L17S02, DOI: 10.1029/2008GL033593 (2008). 2. N.C. Maynard, C.J. Farrugia, D.M. Ober, W.J. Burke, M. Dunlop, F.S. Mozer, H. Rème, P. Décréau, and K.D. Siebert, “Cluster Observations of Fast Shocks in the Magnetosheath Launched as a Tangential Discontinuity with a Pressure Increase Crossed the Bow Shock,” J. Geophys. Res. 113, A10212, DOI: 10.1029/2008JA013121, 21 p. (2008). 3. K. Keika, R. Nakamura, W. Baumjohann, V. Angelopoulos, K. Kabin, K.H. Glassmeier, D.G. Sibeck, W. Magnes, H.U. Auster, K.H. Fornaçon, J.P. McFadden, C.W. Carlson, E.A. Lucek, C.M. Carr, I. Dandouras, and R. Rankin, “Deformation and Evolution of Solar Wind Discontinuities through their Interactions with the Earth’s Bow Shock,” J. Geophys. Res. 114, A00C26, DOI: 10.1029/2008JA013481 (2009). 4. K. Keika, R. Nakamura, W. Baumjohann, V. Angelopoulos, P.J. Chi, K.H. Glassmeier, M. Fillingim, W. Magnes, H.U. Auster, K.H. Fornaçon, G.D. Reeves, K. Yumoto, E.A. Lucek, C.M. Carr, and I. Dandouras, “Substorm Expansion Triggered by a Sudden Impulse Front Propagating from the Dayside Magnetopause,” J. Geophys. Res. 114, A00C24, DOI: 10.1029/2008JA013445 (2009). 5. A.A. Samsonov, D.G. Sibeck, N.V. Zolotova, H.K. Biernat, S.-H. Chen, L. Rastaetter, H.J. Singer, and W. Baumjohann, “Propagation of a Sudden Impulse Through the Magnetosphere Initiating Magnetospheric Pc5 Pulsations,” J. Geophys. Res. 116, A10216, No. 10, DOI: 10.1029/2011JA016706 (2011). 6. G. Pallocchia, A.A. Samsonov, M.B. Bavassano Cattaneo, M.F. Marcucci, H. Rème, C.M. Carr, and J.B. Cao, “Interplanetary Shock Transmitted into the Earth’s Magnetosheath: Cluster and Double Star Observations,” Ann. Geophys. 28, 1141–1156 (2010). ˇ anková, “Numerical MHD Modeling of Propagation of Interplanetary 7. A.A. Samsonov, Z. Nˇemeˇcek, and J. Safr´ Shock through the Magnetosheath,” J. Geophys. Res. 111, A08210, DOI: 10.1029/2005JA011537 (2006). 8. A.A. Samsonov, “Propagation of Inclined Interplanetary Shock through the Magnetosheath,” J. Atmos. Sol. Terr. Phys. 73, No. 1, 30–39, DOI: 10.1016/j.jastp.2009.10.014 (2011). 9. E.A. Pushkar, A.A. Barmin, and S.A. Grib, “Investigation in the MHD Approximation of Impingement of a Solar Wind Shock Wave on the Earth’s Bow Shock,” Geomagnetizm i Aeronomiya 31, No. 3, 522–525 (1991). 10. A.A. Barmin and E.A. Pushkar, “Magnetohydrodynamic Description of the Process of Collision of a Solar Wind Shock Perturbation with the Bow Shock,” Fluid Dynamics 27 (4), 560–572 (1992). 11. A.A. Barmin and E.A. Pushkar, “Two-dimensional MHD Model of the Interaction between Strong Discontinuities and Its Cosmophysical Applications,” Tr. Mat. Inst. RAN im. V.A. Steklova, 223, 87–101 (1998). 12. E.A. Pushkar, “Three-Dimensional MHD Model of Collision of a Solar Wind Shock Wave with the Earth’s Bow Shock,” in: Problems of Modern Mechanics: to the 85th Anniversary of Academician G.G. Chernyi [Collected Articles] (Moscow University Press, Izd-vo “Omega-L”, Moscow, 2008) [in Russian], pp. 461–475. 13. E.A. Pushkar, “Collision of a Solar Wind Shock Wave and the Earth’s Bow Shock in a Strong Interplanetary Magnetic Field: Three-Dimensional Magnetohydrodynamic Model,” Izv. MGIU, Informatsionnye Tekhnologii i Modelirovanie, No. 1 (10), 46–74 (2008). 14. E.A. Pushkar, “Three-Dimensional Magnetohydrodynamic Description of the Process of Collision of a Solar Wind Shock Wave and the Earth’s Bow Shock,” Fluid Dynamics 44, No. 6, 917–930 (2009). FLUID DYNAMICS

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15. E.A. Pushkar, “Three-Dimensional Magnetohydrodynamic Description of the Process of Impingement of a Solar Wind Rotational Discontinuity on the Earth’s Bow Shock,” Fluid Dynamics 46 (2), 308–326 (2011). 16. E.A. Pushkar and A.S. Korolev, “Collision of a Solar Wind Shock Wave with the Earth’s Bow Shock. Wave Flow Pattern,” Tr. Mat. Inst. RAN im. V.A. Steklova, 281, 199–214 (2013). 17. A.G. Kulikovskii and G.A. Lyubimov, Magneto-Hydrodynamics, (Addison-Wesley, Reading, Mass., 1965). 18. J. Merka, A. Szabo, J. A. Slavin, and M. Peredo, “Three-Dimensional Position and Shape of the Bow Shock and their Variation with Upstream Mach Numbers and Interplanetary Magnetic Field Orientation,” J. Geophys. Res. 110, No. 4, A04202, DOI: 10.1029/2004JA010944 (2005). 19. M.I. Verigin, “Location and Shape of Planetary Shocks: Gasdynamic and MHD Aspects,” in: Solar-Terrestrial Relations and Electromagnetic Foreshocks. Proceedings of the IIIrd Intern. Conf., Paratunka, Kamchatka Reg., 16–21 Aug. 2004 (IKIR DVO RAN, Petropavlovsk-Kamch., 2004) [in Russian], Vol. 2, pp. 49–68. 20. E.A. Pushkar, “Oblique Non-Plane-Polarized MHD Shock Waves and their Interaction,” Fluid Dynamics 34 (4), 567–579 (1999). 21. E.A. Pushkar, “Oblique Non-Plane-Polarized MHD Shock Waves,” in: Appendix 5 of A.G. Kulikovskii and G.A. Lyubimov, Magnetohydrodynamics, 2nd Ed. (Logos, Moscow, 2005) [in Russian], pp. 291–303. 22. E.A. Pushkar, “Simple Steady-State Waves in an Arbitrary Magnetic Field,” in: Appendix 4 in A.G. Kulikovskii and G.A. Lyubimov, Magnetohydrodynamics, 2nd Ed. (Logos, Moscow, 2005) [in Russian], pp. 285–290. 23. A.J. Hundhausen, Coronal Expansion and Solar Wind (Springer, New York, etc., 1972). 24. V.B. Baranov and K.V. Krasnobaev, Hydrodynamic Theory of Cosmic Plasma (Nauka, Moscow, 1977) [in Russian]. 25. S.A. Grib and E.A. Pushkar, “Asymmetry of Nonlinear Interactions of Solar MHD Discontinuities with the Bow Shock,” Geomagnetizm i Aeronomiya, 46, No. 4, 442–448 (2006). 26. K.I. Paularena, J.D. Richardson, M.A. Kolpak, C.R. Jackson, G.L. Siscoe, “A Dawn-Dusk Density Asymmetry in Earth’s Magnetosheath,” J. Geophys. Res. 106, No. A11. 25377–25394 (2001). 27. K.A. Anderson, “A Review of Upstream and Bow Shock Energetic-Particle Measurements,” Nuovo Cimento 2, No. 6, P. 747–771 (1979). 28. A.A. Barmin and E.A. Pushkar, “Shock Wave Intersection in Magnetohydrodynamics,” Fluid Dynamics 26 (3), 428–437 (1991).

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