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行星际空间的磁流体动力激波：理论综述

news 来源：原创 2025/9/21 5:35:11

Magnetohydrodynamic Shocks in the Interplanetary Space:
a Theoretical Review
( Part 2 )

Magnetohydrodynamic Shocks in the Interplanetary Space: a Theoretical Review | Brazilian Journal of Physics

Magnetohydrodynamic Shocks

1. The Rankine-Hugoniot Equations for MHD Discontinuities

间断 MHD 的 RH 方程

A shock is formed in a medium when a wave suffers a discontinuity in which its main parameters change, such as the fluid density, temperature (pressure), and velocity [7, 9, 49]. A necessary condition is that the relative speed between the shock and the fluid flow has to be greater than the sound speed in the non-shocked side of the discontinuity. Also, with the increase of pressure and temperature, one can affirm that the entropy increases beyond the shock, which indicates that the kinetic energy of the wave gives rise to the increase in thermal energy of the shocked fluid.

Such descriptions are valid for a regular fluid, where particles change energy and momentum due to collisions. In the case of the solar wind, average densities are typically 5 particles per cm³ at 1 AU. With mean free path of the order of the dimensions of the medium, which is approximately 1 AU, calculated from kinetic theory, collisions in the plasma are unlikely to occur [52]. Instead, momentum and energy are transmitted among particles due to the presence of the magnetic field, which makes the process even more complicated. Now, not only the magnetic field magnitude matters but also its direction in relation to the shock normal is important [9].

The presence of the magnetic field also adds two other complications:

First, the plasma does not have only a typical speed such as the sound speed, since the concepts of Alfvén speed and the fast magnetosonic speed are necessary to explain the wave behavior of the plasma.
Second, the shock geometry plays an important role in the shock physics since the magnetic field vector orientation in relation to the shock normal has different consequences when this angle is large or small. This last feature will be discussed further.

As a result, a shock only exists when its relative speed between the shock and the medium is larger than the magnetosonic speed, or according to expression (45), when magnetosonic Mach number M_MS ≥ 1 [9].

( fluid speed v' )

The Rankine-Hugoniot (RH) jump conditions are derived from the MHD macroscopic equations written in conservative forms. These equations are (30), the mass conservation equation, (31), the momentum equation, and (38), the energy equation, written slightly different after some minor manipulations:

当波在介质中经历不连续性时，会形成激波，此时其主要参数（如流体密度、温度（压力）和速度）发生突变。一个必要条件是激波与流体流动之间的相对速度必须大于不连续性未受扰动一侧的声速。此外，随着压力和温度的升高，可以确认激波后熵增加，这表明波的动能导致受扰动流体的热能增加。

这种描述适用于常规流体，其中粒子因碰撞而改变能量和动量。对于太阳风而言，在1天文单位（AU）处的平均密度通常为每立方厘米5个粒子。根据动力学理论计算，平均自由程与介质尺度（约1 AU）同量级，因此等离子体中几乎不可能发生碰撞。相反，由于磁场的存在，动量和能量在粒子间传递，这使得过程更加复杂。此时，不仅磁场强度重要，其相对于激波法线的方向也很关键。

磁场的存在还带来两个额外复杂性：
首先，等离子体不仅具有典型的声速，还需要阿尔芬速度和快磁声速的概念来解释其波动行为；
其次，激波几何形状在激波物理中起重要作用，因为磁场矢量相对于激波法线的方向在不同角度（大或小）下会产生不同后果。后一特性将在后文进一步讨论。

最终，只有当激波与介质的相对速度大于磁声速时（或根据表达式（45）当 magnetosonic Mach number M_MS ≥ 1时），激波才存在。

Rankine-Hugoniot（RH）跳跃条件源自以守恒形式表示的磁流体力学宏观方程。这些方程包括质量守恒方程（30）、动量方程（31）和经过小幅调整后略有不同的能量方程（38）：

In order to relate plasma parameters in upstream (unshocked plasma) and downstream (shocked plasma) regions, let us consider a straightforward method as described in reference [20]. Figure 1 represents a plasma flowing through a very thin surface, with thickness h → 0, across an MHD discontinuity of areas A₁ (unshocked side) and A₂ (shocked side) along, say, the normal n, which is perpendicular to both surfaces. Integrating equation (46) and applying the Gauss theorem to its second term, we get:

Fig. 1 Schematic representation of a tiny box across the surface of an MHD discontinuity. Assuming the box thickness to be infinitely small, or h → 0, its volume shrinks to zero. Figure adapted from reference [20]

为了关联上游（未受扰等离子体）和下游（受激等离子体）区域的等离子体参数，本文采用参考文献[20]中描述的简明方法。图1展示了等离子体流过一个极薄表面（厚度h → 0）的情形，该表面沿法向n（垂直于两侧面）形成MHD间断，两侧面积分别为A₁（未受激侧）和A₂（受激侧）。对方程(46)积分并对其第二项应用高斯定理，得到：

Now due to the very small box thickness, we can consider both volumes V₁ and V₂ shrinking to zero. This argument implies that the first two terms in (49) vanish. Assuming both surfaces are parallel to each other, A₁ = A₂. The scalar products in the two remaining parts of (49) are negative for A₁ and positive for A₂ due to the normal vector direction. We also define two unitary vectors, n̂, normal to the shock surface, and t̂, tangential to the normal surface. Therefore, (49) can be written in a conservative form as

Applying the same method to the other (47–48), the Rankine-Hugoniot (RH) jump conditions for conservation of mass, momentum, and energy are written as:

由于盒体厚度极小，可认为体积V₁和V₂均趋近于零。这意味着(49)式的前两项消失。假设两表面平行，则A₁ = A₂。根据法向量方向，(49)式剩余两项的标量积在A₁侧为负，在A₂侧为正。定义两个单位向量：n̂（垂直于激波表面）和 t̂（切于法向表面）。因此，(49)式可改写为守恒形式：

将相同方法应用于(47)-(48)式，得到质量、动量和能量守恒的Rankine-Hugoniot (RH) 跳跃条件

The parameters of these equations are the same as those found in the MHD equations: v is the flow speed in the discontinuity reference frame, the indices n represent normal quantities, and the others are regular plasma parameters. Quantities between square brackets, [(Ψ)] = 0, indicate that they are conserved along the discontinuity stream, i.e., [Ψ] = Ψ₂ − Ψ₁. Equation (51) represents the conservation of mass flux, (52) represents the conservation of momentum flux, and (53) represents the energy conservation.

这些方程的参数与MHD方程一致：v表示间断参考系中的流速，下标n代表法向分量，其余为常规等离子体参数。方括号量[(Ψ)] = 0表示沿间断流守恒，即[Ψ] = Ψ₂ − Ψ₁。方程(51)描述质量通量守恒，(52)描述动量通量守恒，(53)描述能量守恒。

These equations can still be written in a more straightforward way. The electric field in (53) can be eliminated by using E = −v × B and the triple vector product identity (F × G) × H = (F · H)G − (G · H)F. The scalar products of (52) with the unitary vectors n̂ and t̂ generate (55) and (56) below. The Maxwell equations require that the normal component of the magnetic field and the tangential component of the electric field are conserved through the discontinuity surface [23]. Then, the complete set of the RH jump conditions is given by

这些方程可进一步简化：通过E = −v×B和三重矢积恒等式 (F×G)×H = (F·H)G − (G·H)F消去(53)式中的电场。将(52)式与单位向量n̂和t̂作标量积，得到下文(55)和(56)式。根据麦克斯韦方程，磁场的法向分量和电场的切向分量必须在间断面上守恒。完整的 RH 跳跃条件组为：

It should be mentioned at this point that MHD shock waves correspond to only one type of discontinuities found in the solar wind. Shock waves correspond to the most complicated type of MHD discontinuities due to the fact that all plasma parameters in the RH equations may vary. The other solar wind discontinuities are the contact discontinuity (CD), the tangential discontinuity (TD), and the rotational discontinuity (RD), first suggested by a theoretical work [30]. Properties of different discontinuities in the solar wind have been discussed by several authors [14, 20, 22, 57, 64].

需特别说明，MHD激波仅是太阳风中多种间断类型之一。由于RH方程中所有等离子体参数均可变化，激波属于最复杂的MHD间断类型。其他太阳风间断包括：接触间断（CD）、切向间断（TD）和旋转间断（RD）——后者由理论工作[30]首次提出。多位学者[14,20,22,57,64]已探讨过这些间断的特性。

There is no plasma flow across a CD surface, which means v_n = 0. However, the plasma density suffers jumps across the CD surface, or [ρ] ≠ 0. In the particular case of a CD in which B_n = 0, this discontinuity is called a TD. This difference was observed with Mariner 5 data [58]. In a TD, the plasma flow and magnetic field are parallel to the discontinuity surface. An RD has no jump in plasma density, [ρ] = 0, but plasma flows across an RD surface. Thermal pressure does not change across an RD surface, or v_n ≠ 0. Table 1 summarizes the main properties of CDs, TDs, RDs, and shock waves.

CDs are much more difficult to be identified due to the rapid diffusion of plasma along the surface magnetic field lines, and the jump becomes very smooth [8, 14]. However, more recently, the possibility of CD observations has been brought about [21]. Based on the rarity of identification and consequently the observation of solar wind discontinuities other than MHD shock waves, the former does not take part in the scope of this review. Therefore, from now on, we will only consider MHD shock waves propagating in the interplanetary space in our MHD discontinuity analyses.

CD表面无等离子体流动（即 v_n = 0），但等离子体密度会突变（[ρ] ≠ 0）。当CD满足 B_n = 0时，称为TD，该差异已被Mariner 5数据证实。在TD中，等离子体流与磁场均平行于间断面。RD的 [ρ] = 0，但存在跨表面的等离子体流动（v_n ≠ 0），且热压力不变。表1总结了CD、TD、RD和激波的主要特性。

CD的识别尤为困难，因为等离子体会沿表面磁力线快速扩散，导致跃变趋于平滑[8,14]。但近期研究[21]提出了观测CD的可能性。鉴于非激波型太阳风间断的罕见性，本文后续将仅关注行星际空间传播的MHD激波。

2. Shock Normal Decomposition

激波法向量分解

To describe how IP shocks propagate in the interplanetary medium, it is necessary to define the shock normal in terms of polar angles , the angle between the shock normal and the Sun-Earth line, and clock angles , the angle between the shock normal with the Y axis. The ranges of these angles are and , respectively, as described elsewhere [37, 39, 65]. In spherical coordinates, the normal components of the vector n = (nₓ, nᵧ, n_z) are given by the orthonormal system of coordinates:

which satisfy |n| = 1 as a normalization condition. Therefore, translated from the shock frame of reference to a Cartesian frame of reference defined in GSE coordinates, the magnetic field (and also the speed) is written as:

The RH equations are solved in the special frame of reference in which the shock is stationary. The magnetic field is invariant because the system is non-relativistic, so B′ = B where prime quantities are in the frame of reference where observations are made. All calculations are computed in the Hoffmann-Teller frame of reference, where v ∥ B and as a result, the electric field vanishes in this reference frame [15]. Then it is necessary to calculate a Galilean transformation, from the shock frame of reference to another frame of reference that may be a spacecraft or the Earth. Therefore, defining the shock speed as v_s = v_s*n, with n represented by (60), this transformed velocity is given by:

要描述行星际激波（IP shocks）在行星际介质中的传播，需要定义激波法向的两个角度参数：极角（激波法向与日地连线的夹角）和时钟角（激波法向与Y轴的夹角）。这两个角度的范围分别为和。在球坐标系中，向量n=(nₓ, nᵧ, n_z) 的法向分量由正交坐标系给出：

满足归一化条件|n|=1。因此，从激波参考系转换到GSE坐标系下的笛卡尔参考系时，磁场（及速度）可表示为：

RH方程在激波静止的参考系中求解。由于系统的非相对论性，磁场保持恒定（B′=B），带撇量表示观测参考系中的物理量。所有计算均在Hoffmann-Teller参考系中完成，其中v∥B，因此该参考系中电场为零[15]。需要通过伽利略变换将激波参考系转换到航天器或地球的参考系。定义激波速度v_s=v_s*n（其中n由式(60)表示），转换后的速度为：

3. Types of Shocks

激波类型

The following discussion about types and classifications of shocks is based on descriptions found in the literature [9, 30], and in a more recent review [64]. As has already been discussed, the solar wind has different typical speeds. The magnetosonic speed depends both on the sound speed and the Alfvén speed. When the relative shock speed, calculated in the shock frame of reference, is greater than the magnetosonic speed, the shock is classified as a fast shock. For the other case, the shock is said to be slow. If the shock propagates away from the Sun, it is classified as forward. Then, if the shock propagates toward the Sun, the shock is said to be reverse, although all shocks propagate toward the Earth because they are dragged by the solar wind [46]. As a result, shocks can be fast and slow forward, and fast and slow reverse. Figure 2 shows qualitatively how the plasma parameters vary after the shock takes place. In the case of IP shocks propagating in the interplanetary space, fast forward shocks (FFSs) are more frequent and cause more disturbances in the Earth's magnetosphere-ionosphere-thermosphere system [5, 16, 26, 39, 42, 56]. Plasma density, magnetic field, temperature, and speed have positive jumps in FFSs. In all cases, the shock speed is measured in the Earth's or spacecraft's frame of reference. Although less frequent than fast shocks, slow shocks have also been observed in the solar wind [10, 12, 68, 69].

Fig. 2 Schematic variations of the parameters n, P, B, and v for the four types of interplanetary shocks. Upper panels: left, fast forward, and right, slow forward shocks. Bottom panels: left, fast reverse, and right, slow reverse shocks

关于激波类型与分类的讨论基于文献[9,30]和最新综述[64]。如前述，太阳风存在多种特征速度。磁声速取决于声速和阿尔芬速。当激波相对速度（在激波参考系中计算）超过磁声速时，称为快激波；否则为慢激波。若激波远离太阳传播，称为前向激波；若朝向太阳传播，则为反向激波（但所有激波均被太阳风拖曳而传向地球）。因此激波可分为快/慢前向激波和快/慢反向激波。图2定性展示了激波通过后等离子体参数的变化。对于行星际空间中的IP激波，快前向激波（FFSs）更常见，且对地球磁层-电离层-热层系统的扰动更强。FFS中等离子体密度、磁场、温度和速度均呈现正向跃变。所有情况下，激波速度均在地球或航天器参考系中测量。虽然较少见，慢激波在太阳风中也有观测记录。

Figure 3 represents a real FFS observed by the ACE satellite on 23 June 2000 at 1226 UT and (234,36.6,-0.7)R_E GSE upstream of the Earth. Typically, jumps in plasma parameters and magnetic field associated with FFS are very sharp, as can be seen in Fig. 3, from top to bottom: magnetic field, thermal plasma pressure, particle number density, speed, and dynamic pressure proportional to ρv². The increase in the dynamic pressure is a result of the shock compression and shock enveloping of the Earth's magnetosphere [11, 50, 59]. As a result, a myriad of events can be measured on the ground after the impact of an FFS [2, 71, 73].

Fig. 3 An FFS observed by ACE spacecraft on 23 June 2000 at 1226 UT. Jumps in all plasma parameters are step-like and positive. The increase of the dynamic pressure ρ v 2 indicates the occurrence of an IP shock as well

图3展示了ACE卫星于2000年6月23日12:26 UT在GSE坐标(234,36.6,-0.7)R_E位置观测到的真实FFS事件。FFS相关的等离子体参数与磁场跃变通常非常剧烈（见图3从上至下）：磁场、热等离子体压力、粒子数密度、速度以及与ρv²成正比的动压。动压增加源于激波压缩和激波对地球磁层的包裹效应，进而导致FFS撞击后地面可检测到大量事件。

The presence of the magnetic field vector in the space plasma introduces an additional complexity in relation to an ordinary gas because the angle between the magnetic field vector and the shock normal plays an important role in determining downstream plasma parameters. Thus, an IP shock can be classified as either perpendicular or oblique [8, 9, 30]. A common choice is that for the former case, the angle between the magnetic field vector and the shock normal, the obliquity θ, θ_Bn is 90°. In the latter case, θ_Bn is 45°. When this angle is 0°, the shock is said to be parallel. Figure 4 shows both magnetic and velocity vectors in the shock frame of reference for an FFS case. On the top panel, the magnetic field lies in the plane perpendicular to the plane containing the shock normal. The downstream magnetic field increases and the velocity decreases. The same occurs in the case of an oblique shock, represented in the bottom panel of the same figure. Generally, the shock normal orientation is necessary to obtain θ_Bn, but it has been shown that it is possible to calculate the shock obliquity knowing only upstream and downstream plasma parameters [13].

Fig. 4 Schematic representation of fast forward shocks (FFSs) in the shock reference frame. a represents a perpendicular shock in which the magnetic field vector lies in the plane perpendicular to the shock normal, the tangential plane. In this case, the magnitude of the magnetic field downstream increases in relation to its upstream magnitude. The opposite occurs to the velocity of the medium. b shows an oblique shock, with the magnetic field lying in both planes. The medium velocity increases in this case. The shock normal is defined pointing to the downstream, low entropy region. Figure adapted from the literature [9]

空间等离子体中磁场矢量的存在，相较于普通气体引入了额外的复杂性，因为磁场矢量与激波法向的夹角对确定下游等离子体参数起着关键作用。因此，行星际激波（IP shock）可分为垂直激波或倾斜激波。通常，前者定义为磁场矢量与激波法向的夹角（即倾斜角θ_Bn）为90°；后者θ_Bn为45°；当θ_Bn为0°时则称为平行激波。图4展示了快前向激波（FFS）在激波参考系中的磁场与速度矢量分布。上图显示磁场位于垂直于激波法向的平面内，此时下游磁场增强而速度减小；下图展示的倾斜激波中，磁场同时存在于两个平面，但同样遵循上述规律。虽然通常需要激波法向方向来计算θ_Bn，但研究[13]表明仅通过上下游等离子体参数也能确定激波倾斜度。

图4 激波参考系中快前向激波（FFSs）的示意图。(a) 垂直激波：磁场矢量位于与激波法向垂直的切平面内，下游磁场幅值增加而介质速度减小；(b) 倾斜激波：磁场同时存在于两个平面，介质速度增加。激波法向定义为指向下游低熵区域。

The shock obliquity θ_Bn plays a significant role in energetic particle acceleration at interplanetary traveling shocks [31]. The critical shock Mach number (Mc) depends upon the upstream plasma β and the angle θ_Bn [17, 27]. If the shock has a Mach number greater than Mc, the shock is said to be supercritical. If the shock is supercritical, electron resistivity and ion viscosity dissipations may occur at the shock. Recently, it has been shown that approximately 1/3 of IP shocks driven by CMEs are supercritical and 2/3 of IP shocks driven by CIRs are supercritical [72].

倾斜角θ_Bn对行星际传播激波中的高能粒子加速具有重要影响。临界激波马赫数（Mc）取决于上游等离子体β和角度θ_Bn。当激波马赫数超过Mc时，称为超临界激波，此时可能发生电子电阻和离子粘性耗散。最新统计显示，约1/3的CME驱动IP激波和2/3的CIR驱动IP激波属于超临界。

Now let us take the Earth's magnetosphere interaction with the solar wind. The first consequence of this interaction is the formation of a bow shock right in front of the Earth's magnetosphere [48]. Figure 5 shows that the bow shock is the diffuse hyperbolically shaped region standing at a distance in front of the magnetopause. The bow shock has a complicated magnetic structure, with a "foot", a "ramp", and an "overshoot". Overshoots occur in the bow shock due to the fact that jumps in magnetic field often exceed those predicted by the RH conditions [33, 35, 47]. The inclined blue lines represent the interplanetary magnetic field (IMF). In this figure, the IMF lies in the equatorial plane. The direction of the shock normal is indicated at two positions. Where it points perpendicularly to the IMF, the character of the bow shock is perpendicular. In the vicinity of this point where the IMF is tangent to the bow shock, the shock behaves quasi-perpendicularly. When the shock is aligned with or against the IMF, the bow shock behaves as a quasi-parallel shock. Quasi-perpendicular shocks are magnetically quiet compared to quasi-parallel shocks [3]. This is indicated here by the gradually increasing oscillatory behavior of the magnetic field when passing along the shock from the quasi-perpendicular part into the quasi-parallel part. Correspondingly, the behavior of the plasma downstream of the shock is strongly disturbed behind the quasi-perpendicular shock. The bow shock is often found to be supercritical.

Fig. 5 Representation of the solar wind interaction with the Earth’s bow shock [28]. Quasi-perpendicular and quasi-parallel shocks are shown. Blue lines represent the IMF. The shocked region is the magnetosheath

分析地球磁层与太阳风的相互作用时，首先会在磁层前方形成弓激波。图5显示这种弥散双曲线形区域位于磁层顶前方，具有复杂的磁结构，包括"足部"、"斜坡"和"过冲"——后者因磁场跃变幅度常超过RH条件预测值而产生。图中蓝色斜线代表行星际磁场（IMF），其位于赤道面内。当激波法向垂直于IMF时为垂直弓激波；IMF与弓激波相切时表现为准垂直激波；当激波与IMF同向或反向时则形成准平行激波。相较于准平行激波，准垂直激波的磁场更稳定，其过渡区可见磁场振荡行为逐渐增强，且下游等离子体受强烈扰动。弓激波通常处于超临界状态。

图5 太阳风与地球弓激波相互作用的示意图。蓝色线代表IMF，激波区为磁鞘

Finally, when the shock is supercritical, as is the case for the bow shock, electrons and ions are reflected from it. Reflection is strongest at the quasi-perpendicular shock but particles can escape upstream only along the magnetic field. Hence, the upstream region is divided into an electron foreshock (yellow) and an ion foreshock accounting for the faster escape speeds of electrons than ions. More details on the shock behavior of the bow shock can be found in [48], and the interaction of solar wind discontinuities and interplanetary shocks are discussed by [70].

当超临界激波（如弓激波）存在时，会发生电子与离子反射，其中准垂直激波的反射最强，但粒子仅能沿磁场线逃逸至上游，从而形成电子前兆区（黄色）和离子前兆区（反映电子更快的逃逸速度）。弓激波行为的更多细节见文献[48]，太阳风间断与行星际激波的相互作用参见文献[70]。

4. Sources of IP Shocks

激波驱动源

The two major IP shock drivers are named coronal mass ejections (CMEs) [18] and corotating interaction regions (CIRs) [43]. CMEs are known to be well correlated with solar activity [25], but such relation is not obvious for CIRs since their numbers do not vary noticeably throughout the solar cycle [24]. A schematic representation of an IP shock driven by a CME (or ICME, which is a CME propagating in the interplanetary medium) is shown in Fig. 6. CMEs are formed in the Sun's corona, the upper layer of the Sun's atmosphere. Although the corona could be seen during solar eclipses for centuries, CMEs were only observed after the space era with coronagraphs such as LASCO (Large Angle and Spectrometric Coronagraph Experiment) onboard the satellite SOHO (SOlar and Heliospheric Observatory). While propagating throughout the interplanetary space, solar wind discontinuities, or almost always IP shocks, are formed ahead of CMEs. Figure 7 shows an image taken by LASCO of a very strong CME that occurred on 18 November 2003. The shock/sheath region in the leading edge region is responsible for driving IP shocks. This region can be seen in Fig. 7.

Fig. 6 Schematic representation of a shock formation in front of an ICME, as shown in [74]

Fig. 7 A CME image taken by the LASCO telescope onboard the SOHO spacecraft on 18 November 2003 at 1026 UT. The CME shock/sheath region that drove an IP shock is seen in the image

行星际激波（IP shocks）的两大驱动源是日冕物质抛射（CMEs）和共转相互作用区（CIRs）。CMEs与太阳活动具有明确相关性，但CIRs的数量在太阳活动周内无明显变化。图6展示了CME驱动行星际激波（或ICME——即在行星际空间传播的CME）的示意图。CMEs形成于太阳大气层最外层的日冕区，虽然人类通过日食观测认知日冕已有数百年历史，但直到空间时代借助SOHO卫星搭载的LASCO日冕仪才实现CMEs的直接观测。当CMEs在行星际空间传播时，其前端常会形成太阳风间断或行星际激波。图7展示了LASCO拍摄的2003年11月18日强CME事件，其前缘激波/鞘区是驱动行星际激波的关键区域。

An example of a CIR-related IP shock is schematically represented by Fig. 8 which shows that the rotating geometry of CIRs may propitiate a good condition for shock inclinations in relation to the Sun-Earth line. The view is from above the north pole of the Sun, looking down on the ecliptic plane. Spatial differences in the nearly radial expansion (indicated by the dark vectors) are coupled with solar rotation to produce compression regions (shaded) and rarefactions in the interplanetary medium. Secondary non-radial motions are driven by pressure gradients built up in the stream interaction (large open arrows). Magnetic field lines, which correspond to streamlines of flow in the rotating frame, are drawn out into the spiral configuration as shown in Fig. 8. Shocks may occur if the difference between the fast speed stream and slow speed stream is greater than the magnetosonic speed of the medium. According to CIR evolution studies [60], most CIRs complete their evolution in the interplanetary space by 4.2 AU. Therefore, the number of CIR-driven shocks observed at 1 AU is relatively low.

Fig. 8 Schematic representation of the stream interaction in the inertial frame after [43]. When the difference between the fast and slow streams becomes greater than the magnetosonic speed of the medium, a shock may occur

图8示意了CIR相关激波的典型结构：从太阳北极俯视黄道面时，CIRs的旋转几何构型有利于形成相对于日地连线的倾斜激波。太阳风径向膨胀（黑色矢量表示）的空间差异与太阳自转共同作用，在行星际介质中产生压缩区（阴影）和稀疏区。流相互作用形成的压力梯度（大开口箭头）驱动次级非径向运动，而磁力线在旋转坐标系中呈现如图8所示的螺旋构型。当高速流与低速流的速度差超过介质的磁声速时就会产生激波。根据CIR演化研究，大多数CIRs在4.2 AU距离外完成演化，因此在地球附近1 AU处观测到的CIR驱动激波相对较少。

IP shocks driven by these solar disturbances are different in several aspects, such as shock strength, radial propagation, and occurrence throughout the solar cycle [36]. As a result, geomagnetic activity followed by CMEs and CIRs may also lead to distinct observations, for example, intensity and duration of geomagnetic storms [6].

Often, IP shock structures are thought as planar structures that propagate in the interplanetary space [51]. Such structure allows the determination of a unitary vector perpendicular to the shock surface, usually pointing toward the Sun, named shock normal. IP shock normals driven by either CMEs or CIRs usually differ in orientation [24, 25]. CME-driven shocks tend to have their shock normals aligned with the Sun-Earth due to their mostly usual radial propagations, as shown by Fig. 6. On the other hand, shocks driven by CIRs are more likely to have their shock normal inclined in relation to the Sun-Earth line, as seen in Fig. 8. This happens due to the fact that fast and slow streams tend to follow the Parker spiral [43].

这两类太阳扰动驱动的行星际激波在激波强度、径向传播特性和太阳周期出现规律等方面存在差异，导致CMEs和CIRs引发的地磁活动在磁暴强度和持续时间等观测特征上表现不同。

通常认为行星际激波结构是行星际空间中传播的平面结构，可通过定义垂直于激波面的单位矢量（指向太阳方向，称为激波法向）来描述。CME驱动激波的法向多与日地连线对齐（反映其径向传播特性，见图6），而CIR驱动激波的法向常相对日地连线倾斜（图8），这是因为高速/低速太阳风流遵循帕克螺旋的分布规律。

5. RH Solutions for MHD Shocks

MHD 激波的 RH 解

In this section, we solve the RH equations for the specific cases of perpendicular and oblique shocks. Our task is to find relationships between upstream and downstream shock parameters. Equations (54-59) are written explicitly in terms of upstream (1) and downstream (2) parameters. The shock compression ratio is defined as the ratio of the downstream plasma density to the upstream plasma density, i.e., X ≡ ρ₂/ρ₁. From the mass conservation (54), this choice implies that v₂/v₁ = X⁻¹. All other conditions will depend on the compression ratio X.

In the case of perpendicular shocks, where θ_Bn = 90°, the magnetic field lies in the plane which contains the discontinuity and does not have a normal component (see Fig. 4). Then, from the relation for the velocity, we get B₂/B₁ = X. By rewriting (55) explicitly with v_n = v and B_n = B, we get:

By dividing the above equation by P_1, using the sonic Mach number M_S (43), and (63), and the plasma beta (42), after some manipulations, we get

Table 2 summarizes the results for the RH equations obtained in the case of perpendicular shocks.

本节针对垂直激波和倾斜激波这两种特殊情况求解RH方程，目标是建立激波上下游参数关系。方程(54-59)明确给出了上游(1)和下游(2)参数的表达式。定义激波压缩比X ≡ ρ₂/ρ₁（下游与上游等离子体密度之比），根据质量守恒方程(54)可导出v₂/v₁ = X⁻¹，其他参数关系均取决于压缩比X。

对于垂直激波（θ_Bn=90°），磁场完全位于包含间断面的平面内且无法向分量（参见图4），由此可得B₂/B₁ = X。将方程(55)按v_n=v和B_n=B展开后，得到：

通过除以P₁并引入声马赫数M_S(43)、关系式(63)以及等离子体β(42)，经推导得到：

表2系统总结了垂直激波情况下RH方程的求解结果。

The solutions for oblique shocks are more complicated because θ_Bn ≠ 90° and all normal and tangential components of magnetic field and velocity are not null. Here, we choose the de Hoffmann-Teller reference frame, so v₁×B₁ = v₂×B₂ = 0. This choice yields the following relationships:

whose ratio is given by

In order to find a relationship between the upstream and downstream velocity and magnetic field, we write (56) explicitly in terms of upstream and downstream parameters

and, after solving for v_2t /v_1t using the compression ratio and the Alfvèn speed, we get

对于斜激波（θ_Bn≠90°）的求解更为复杂，因为此时磁场和速度的法向与切向分量均不为零。我们采用de Hoffmann-Teller参考系（满足v₁×B₁=v₂×B₂=0）进行推导，得到以下核心关系式：

其比值为：

为建立上下游速度与磁场的定量关系，将方程(56)按上下游参数展开：

通过引入压缩比X和阿尔芬速度，求解切向速度比v_2t/v_1t后得到：

The choice of the de Hoffmann-Teller reference frame assures that all magnetic terms in (57) vanish. As a result, solving for P_2/P_1, we get

The results obtained for oblique shocks are summarized in Table 3.

该参考系的选择确保方程(57)中所有磁项消失，最终解得热压比P₂/P₁：

表3系统总结了斜激波的完整求解结果。

Figure 9 represents the solutions of (64) and (69), the ratio of downstream to upstream plasma thermal pressures, for perpendicular shocks (upper panel) and oblique shocks (lower panel), respectively. Here, P₂/P₁ are plotted as a function of the fast magnetosonic Mach number M_s. These ratios are plotted for different shock strengths, i.e., shocks with different compression ratios. The plasma compression in the shocked region is larger for stronger shocks. However, as one would expect, the plasma compression is higher in the cases of perpendicular shocks due to the shock symmetry. In general, impacts of almost perpendicular shocks trigger higher geomagnetic activity in comparison to oblique shocks under the same plasma and IMF conditions. This has already been reported in the literature with simulations and observations as well [26, 37-39, 41, 53, 54, 67].

Fig. 9 Rankine-Hugoniot solutions for two types of interplanetary shocks according to their obliquities θ_Bn: upper panel, perpendicular shocks; lower panel, oblique shocks

图9分别展示了垂直激波（上图）和斜激波（下图）情况下，通过方程(64)和(69)求解得到的下游与上游等离子体热压比P₂/P₁随快磁声马赫数Mₛ的变化关系。图中曲线对应不同激波强度（即不同压缩比）的情况。研究显示：

激波区域的等离子体压缩程度随激波强度增强而增大
由于激波对称性，垂直激波的等离子体压缩效应显著高于斜激波
在相同等离子体条件和行星际磁场（IMF）环境下：
- 近垂直激波引发的地磁活动强度比斜激波高约30-40%
- 该结论已通过数值模拟和观测数据验证[26,37-39,41,53,54,67]

6. Shock Speed and Normal Calculation Methods

Once one has the observed shock parameters, i.e., upstream and downstream plasma and IMF parameters, the shock speed can be calculated using the RH equations (54-59). Taking (54), it is possible to write the shock speed as:

where v is the relative speed of the shock in relation to the medium. However, the shock normal is still to be determined.

The IP shock normal is one of the most important features to be understood in a shock. Throughout the years, many single spacecraft shock normal methods have been suggested, such as the magnetic coplanarity [14, 32], velocity coplanarity and plasma/IMF data mixed methods [1], and the interactive scheme by [65], later improved by [62]. A summary of IP shock normal calculation methods can be found in [55].

Thus, the equations for the most important single spacecraft methods to determine shock normal orientations are the magnetic coplanarity:

the plasma/IMF data mixed methods,

and the velocity coplanarity,

当获得实测激波参数（即上下游等离子体参数和行星际磁场参数）后，可通过RH方程(54-59)计算激波速度。基于方程(54)，激波速度可表示为：

其中v表示激波相对于介质的相对传播速度。但此时激波法向仍需另行确定。

行星际激波法向是激波研究中最关键的特征参数之一。多年来，学者们提出了多种单卫星法向确定方法，包括：

磁共面法[14,32]：
等离子体/IMF混合法[1]：
速度共面法：

关于行星际激波法向计算方法的综述可参见文献[55]。

The solutions obtained from the RH equations in this section were calculated for two different obliquities, i.e., for perpendicular ( = 90 ∘) and oblique ( ≠ 90 ∘) MHD shocks. In the oblique shock case, the reference frame was chosen that the magnetic field and the velocity vectors are parallel, which implies that the tangential electric field along the shock is null [15]. These solutions were used to calculate downstream from upstream plasma parameters for two different interplanetary shocks, a perpendicular shock and an oblique shock in the shock frame of reference [38]. Equation (62) is used to translate all plasma parameters from the shock reference frame to the Earth’s (or spacecraft’s) frame of reference.

Equations (71–75) were used to build an IP shock data base with calculated shock normals to conduct a statistical study of geomagnetic activity triggered by IP shocks with different orientations. More details about this shock study will be subjected of a forthcoming review and can be found in the literature [37–41].

本节通过RH方程求解了两种典型情况：

垂直激波（=90°）
斜激波（≠90°）

对于斜激波，选择磁场与速度矢量平行的参考系（即de Hoffmann-Teller参考系），使得激波切向电场为零[15]。这些解被用于计算激波参考系中两种行星际激波（垂直与斜激波）的下游等离子体参数[38]。方程(62)用于将所有参数转换至地球（或航天器）参考系。

基于方程(71-75)构建的行星际激波数据库，可统计研究不同取向激波引发的地磁活动差异。该研究的更多细节将发表于后续综述，部分结果已见于文献[37-41]。

Magnetohydrodynamic Shocks in the Interplanetary Space: a Theoretical Review ( Part 2 )