行星際激波在日球層中的傳播：Propagation of Interplanetary Shocks in the Heliosphere （第一部分）

行星際激波在日球層中的傳播：Propagation of Interplanetary Shocks in the Heliosphere （第二部分）- Chapter 3: Solar and heliospheric physics

行星際激波在日球層中的傳播：Propagation of Interplanetary Shocks in the Heliosphere （第三部分）- Chapter 4: Analysis methods

行星際激波在日球層中的傳播：Propagation of Interplanetary Shocks in the Heliosphere （第四部分）- Chapter 5: Discussion & Chapter 6: Summary and Conclusions

Propagation of Interplanetary Shocks in the Heliosphere?
[ Chapter 1 & Chapter 2 ]

[PDF: arXiv]

本文保留原文及參考文獻，參考文獻詳見

行星際激波在日球層中的傳播：Propagation of Interplanetary Shocks in the Heliosphere （參考文獻部分）-CSDN博客

? ?Abstract? ?

Interplanetary shocks are one of the crucial dynamic processes in the Heliosphere. They accelerate particles into a high energy, generate plasma waves, and could potentially trigger geomagnetic storms in the terrestrial magnetosphere disturbing significantly our technological infrastructures. In this study, two IP shock events are selected to study the temporal variations of the shock parameters using magnetometer and ion plasma measurements of the STEREO?A and B, the Wind, Cluster fleet, and the ACE spacecraft. The shock normal vectors are determined using the minimum variance analysis (MVA) and the magnetic coplanarity methods (CP).

During the May 07 event, the shock parameters and the shock normal direction are consistent. The shock surface appears to be tilted almost the same degree as the Parker spiral, and the driver could be a CIR. During the April 23 event, the shock parameters do not change significantly except for the shock θBn angle, however, the shape of the IP shock appears to be twisted along the transverse direction to the Sun-Earth line as well. The driver of this rippled shock is SIRs/CIRs as well. Being a fast-reverse shock caused this irregularity in shape.

行星際激波是日球層內關鍵的動力學過程之一。它們能將粒子加速至高能狀態，產生等離子體波，并可能觸發地球磁層中的磁暴，對地球技術基礎設施造成顯著干擾。本研究選取了兩次IP激波事件，基于STEREO-A/B衛星、Wind衛星、Cluster衛星編隊及ACE衛星的磁強計與離子等離子體測量數據，分析激波參數的時序變化特征。通過最小方差分析法（MVA）和磁共面法（CP）確定了激波法線方向。

在5月7日事件中，激波參數與激波法線方向呈現一致性。激波面傾斜角度與帕克螺旋結構近似，其驅動源可能為共轉相互作用區（CIR）。4月23日事件中，除激波θBn角外其他參數變化較小，但該IP激波在垂直于日地連線的橫向上呈現出扭曲形態。該波紋狀激波的驅動源同樣為流相互作用區/共轉相互作用區（SIRs/CIRs），其作為快速逆向激波的特性導致了這種不規則形態。

? ?Chapter 1? ?Introduction? ?

The solar corona is hotter than the photosphere, the chromosphere, and the transient layers beneath it. As a result, the high temperatures ionize atoms, creating a plasma of free-moving electrons and ions, known as the solar wind.?Historically, (Parker, 1958) ?predicted the existence of the solar wind?and coined the term "solar wind". He deducted it based on German astronomer Ludwig Bierman’s observation of how the comet tail always points away from the Sun (Biermann, 2013). ?The existence of the solar wind was confirmed by the Mariner 2 spacecraft?(Snyder and Neugebauer, 1965). ?The solar wind is a collisionless plasma, and it flows at both supersonic and super-Alfvénic speed, meaning they exceed the Alfvén speed, which is the speed of magnetohydrodynamic waves in a plasma. ?A shock wave is where a fluid changes from supersonic to subsonic speed. Therefore, the fast-moving solar wind tends to create a shock on its journey.?Hence, ?interplanetary (IP) shocks are common through the heliosphere, which is a bubble-like region of space surrounding the Sun and extending far beyond the orbits of the planets and is filled with the solar wind. There are a few varieties of shocks such as planetary bow shocks, shocks that are risen due to the stream interaction regions (SIR), which is called co-rotation interaction region (CIR) when extending beyond 1 AU, and coronal mass ejection (CME) driven shocks. ?IP shocks are one of the main and efficient accelerators of energetic particles?(Tsurutani and Lin, 1985; Keith and Heikkila, 2021). ?These accelerated particles can cause disturbances to the geomagnetic field and are hazardous to astronauts and satellites.?(IP) shocks driven by (CMEs) precede large geomagnetic storms (Gonzalez et al., 1994). ?Large geomagnetic storms can damage oil and gas pipelines and interfere with electrical power infrastructures.?GPS navigation and high-frequency radio communications are also affected by ionosphere changes brought on by geomagnetic storms (Cid et al., 2014) and can cause internet disruptions around the world for many months (Jyothi, 2021). ?Therefore, IP shocks are important in determining and understanding space weather.

【背景介紹】

太陽日冕的溫度高于光球層、色球層及其下方的過渡層。?高溫使原子電離，形成由自由移動的電子和離子組成的等離子體，即太陽風。歷史上，Parker（1958）?預測了太陽風的存在并創造了"太陽風"這一術語，其理論源于德國天文學家Ludwig Bierman對彗尾總是背向太陽這一現象的觀測。?太陽風的存在后被Mariner 2號探測器證實?。?太陽風是一種無碰撞等離子體，以超音速和超阿爾芬速度流動?（即超過 Alfvén速度，即等離子體中磁流體動力學波的傳播速度）。激波是流體從超音速變為亞音速的過渡區域，因此高速運動的太陽風在傳播過程中會產生激波。?由此，日球層（充滿太陽風、延伸至行星軌道之外的泡狀空間區域）中普遍存在行星際（IP）激波，包括行星弓激波、流相互作用區（SIR，延伸超過1天文單位時稱為共轉相互作用區CIR）激波和日冕物質拋射（CME）驅動激波。?IP激波是高效的高能粒子加速器?，?這些粒子會擾動地磁場，威脅宇航員和衛星安全。CME驅動的IP激波往往引發強烈地磁暴。研究IP激波對空間天氣預報至關重要。

The main goal of this thesis is to study and determine parameters such as IP shock normals, upstream and downstream plasma parameters (magnetic field, density, temperature, velocity), and how they vary in their temporal evolution.?There are several methods for determining the shock normal vector (Paschmann and Daly, 1998). ?In this thesis, the minimum variance analysis (MVA) and the magnetic coplanarity method (CP) are used.?These two methods are primarily utilized because they require solely magnetic field data. ?The data are from NASA’s twin Solar Terrestrial Relations Observatory Ahead (STEREO?A) and Behind (STEREO?B) (Kaiser et al., 2008), the Wind (Ogilvie and Desch, 1997), and Advanced Composition Explorer (ACE) spacecraft Stone et al. (1998c), and ESA’s four identical Cluster constellation satellites–Cluster-1 (C1), Cluster-2 (C1), Cluster-3 (C3), and Cluster-4 (C4) (Escoubet et al., 1997).?The temporal resolution of the magnetometers of the heliospheric (and the Cluster) spacecraft is significantly higher than the plasma instruments because the variations are quite slow in the heliosphere. ?So, any agreement between the two methods indicates relatively accurate shock normal vectors?(Facskó et al., 2008, 2009, 2010). ?Here, two events are studied, one is May 07, 2007, and the other is April 23.?The data selection year, 2007, is special because location-wise it was the year when twin STEREO?A and B spacecraft were closer to the Sun-Earth line until their gradual separation from each other in the following years. Later, the spatial separation is so high that it is hard to distinguish the spatial and temporal changes. ?Hence, shocks during this period are proper to study shock propagation and their temporal developments in the case of using these spacecraft.

【本文工作】

本論文的核心目標是研究IP激波法向量、上下游等離子體參數（磁場、密度、溫度、速度）演化特征。?現有多種激波法向量計算方法，?本文采用最小方差分析（MVA）和磁共面法（CP）?——這兩種僅需磁場數據的方法。?數據源自NASA的STEREO-A/B雙星、Wind衛星、ACE探測器及ESA的Cluster四星編隊。由于日球層變化緩慢，磁強計時間分辨率顯著高于等離子體儀器，?故兩種方法結果的一致性可確保法向量精度?。?選取2007年5月7日與4月23日兩次事件——該年STEREO雙星尚未大幅偏離日地連線，空間分離度適宜研究激波時空演化。

?Chapter 2? introduces the basics of plasma physics and magnetohydrodynamics, which are the governing equations of these heliosphere phenomena.??
?Chapter 3? discusses a brief description of the Sun, the solar wind, and the interplanetary magnetic field.?
? Chapter 4? is about instrumentation, database, and methods.??
?Chapter 5? presents the results and discusses them.??
?Chapter 6? is the summary and conclusions.

? ?Chapter 2? ?Basics of Magnetohydrodynamics? ?

? ▍2.1 Plasma? ▍

The term plasma for this state of matter was coined by Irvin Langmuir after its similarity with the blood plasma carrying white and red cells (Tonks, 1967). A plasma is a set of charged particles made up of an equal number of free carriers for positive and negative charges. Having nearly the same number of opposite charges ensures that the plasma appears quasi-neutral from the outside. Free particle carriers mean the particles inside a plasma must have enough kinetic (thermal) energy to overcome the potential energies of their nearest neighbors, which means a plasma is a hot and ionized gas. There are the three basic criteria for a plasma (Baumjohann and Treumann, 2012; Chapters 1-4).

等離子體?（plasma）這一物質狀態術語由歐文·朗繆爾提出，靈感源于其與攜帶血細胞的血漿的相似性。等離子體是由數量相等的正負自由電荷載體組成的帶電粒子集合，這種電荷平衡使其在宏觀上呈現準電中性。自由粒子意味著其動能（熱能）必須足以克服鄰近粒子的勢能，因此等離子體本質上是高溫電離氣體。

等離子體的判定需滿足三個基本準則：

first criterion

The first criterion is a test-charged particle is clouded by its opposite-charged particles, canceling the electric field of the test particle. This is known as the Debye shielding and its so?called Debye?lenght, λ_D, is defined as follows:

where ?_0 is the free space permittivity, k_B the Boltzmann constant, T_e = T_i the electron and ion temperature, ne the electron plasma density, and e electric charge.

To ensure the quasineutrality of a plasma, the system length L must be greater than the Debye length

1 德拜屏蔽效應：測試電荷會被異性電荷云包圍，抵消其電場。德拜長度λ_D定義為：

其中ε?為真空介電常數，k_B為玻爾茲曼常數，T_e = T_i為電子/離子溫度，n_e為電子密度，e為元電荷。

?系統尺度L必須大于λD以維持準電中性?（L ? λ_D）。

second criterion

The second criterion is since the Debye shielding results from the collective behavior of particles inside the Debye sphere with the radius λ_D, the Debye sphere must contain enough particles

which indicates the mean potential energy of particles between their nearest neighbors must be less than their mean individual energies.

2 德拜球粒子數條件：德拜屏蔽是由半徑為λ_D的德拜球內粒子的集體行為引起的，德拜球（半徑λ_D）內需包含足夠粒子：

該條件要求粒子間平均勢能低于單個粒子平均動能。

third criterion

The third criterion is the collision time scale τ of the system is greater than the inverse of the plasma ω?1_p , electron ??1_e , and ion ??1_i cyclotron gyrofrequencies.

By solving each particle, plasma dynamics can be described, but this approach is too difficult and inefficient. Therefore, there are certain approximations, depending on the corresponding problems. Magnetohydrodynamics is one such approximation that instead of taking account of individual particles, the plasma is assumed as a magnetized fluid.

3 ?時間尺度條件：系統碰撞時間 τ 需大于等離子體振蕩頻率倒數（ω?1_p）、電子回旋頻率倒數（Ω?1_e）和離子回旋頻率倒數（Ω?1_i）：

? ▍2.2 The single fluid MHD??▍

In this section, the magnetohydrodynamics (MHD) equations are briefly introduced without too much detail and derivations.?The following formulizations are based on (Baumjohann and Treumann, 2012)[page 138-158], (Murphy, 2014), (Freidberg, 2014) and (Antonsen Jr., 2019). ?Magnetohydrodynamics (MHD) was developed by Hannes Alfvén (Davidson, 2002).??MHD equations are the result of coupling the Navier-Stokes equations (the fluid equations) to the Maxwell equations.??In the MHD, the plasma is treated as a single fluid with macroscopic parameters such as average density, temperature, and velocity.?Since plasma consists of generally two species of particles, namely electrons, and ions, the different species should be handled together. ?Hence, the single fluid approximation is utilized.

單流體磁流體力學（MHD）理論體系

本節簡要介紹磁流體力學（MHD）方程，不涉及過多細節與推導。以下公式化表述基于以下文獻：(Baumjohann and Treumann, 2012)[第138-158頁]、(Murphy, 2014)、(Freidberg, 2014) 和 (Antonsen Jr., 2019)。

?磁流體力學（MHD）由 Hannes Alfvén（Davidson, 2002）發展而來。MHD方程是納維-斯托克斯方程（流體方程）與麥克斯韋方程組耦合的結果。在MHD中，等離子體被視為具有宏觀參數（如平均密度、溫度、速度）的單一流體。?

由于等離子體通常由電子和離子兩種粒子組成，需將不同組分統一處理，?因此采用單流體近似。?

-- Single fluid variables

Mass density

Considering the mass m_e of an electron is significantly lower than the mass m_i of an ion, m_e << m_i :

Momentum density

where v, u_e, u_i are plasma, electron, ion velocities respectively, and ρ is the plasma density, n_e, n_i are the electron and ion number densities.

Current density

where q_e and q_i are electron and ion charges.

Total pressure

-- 單流體變量定義?

質量密度

考慮到電子質量m_e遠小于離子質量m_i，即m_e << m_i：

動量密度

其中v、u_e、u_i分別表示等離子體、電子和離子速度，ρ為等離子體密度，n_e、n_i為電子和離子數密度。

電流密度

其中q_e和q_i分別為電子和離子電荷。

總壓力

-- Single fluid MHD equations

By using the single fluid variables, the single fluid MHD equations can be defined as follows:

The continuity equation for the mass density

The momentum equation

where ρ_e denotes the charge density, and E, B electric and magnetic fields.

The generalized Ohm’s law

where η is magnetic diffusivity , and ηJ is the resistive term.

Amp`ere’s law

where μ_0 is the vacuum magnetic permeability, ?_0 is vacuum permittivity.

Faraday’s law

magnetic field divergence constraint

--?單流體MHD方程組

使用單流體變量，可定義單流體MHD方程組如下：?

質量密度的連續性方程：?

動量方程：?

其中ρ_e表示電荷密度，E、B分別表示電場和磁場。

廣義歐姆定律：?

為磁擴散率，ηJ為電阻項。

安培定律：?

其中μ_0為真空磁導率，ε_0為真空介電常數。

法拉第定律：

磁場無散約束：

The MHD approximations are valid when the characteristic speed of the system is much slower than the speed of light,

the characteristic frequency, ω, must be smaller than the ion cyclotron frequency ω_i

the characteristic scale length L of the system must be longer than the mean free path r_gi of ion gyroradius

the characteristic scale times must be larger than the collision times as stated in equation 2.4.

當滿足以下條件時，MHD近似成立：?

系統特征速度遠小于光速，

特征頻率ω必須小于離子回旋頻率 ω_i，

系統特征尺度 L 必須大于離子回旋半徑 r_gi 的平均自由程，

特征時間尺度必須大于如方程2.4所述的碰撞時間。

? ▍2.3 MHD wave modes? ▍

Plasma is considered a ??highly conductive fluid??, which consists of ??charged particles??. Therefore, ??MHD waves?? in plasma fundamentally arise from two distinct wave speeds: the ??sound speed?? of a fluid and the ??Alfvén speed??, which is due to the ??magnetic field pressure and tension??. Their combination gives rise to ??magnetosonic waves??. Thus, in MHD there are three wave modes–namely ??slow magnetosonic??, the ??shear-Alfvénic?? and the ??fast magnetosonic waves?? (Fitzpatrick, 2014) in addition to the sound wave, as seen in ??Figure 2.1??.

Figure 2.1: Phase velocities of the three MHD waves. (Figure is from Fitzpatrick, 2014; Figure 7.1)

等離子體被視為一種高導電性流體，由帶電粒子組成。因此，等離子體中的磁流體力學（MHD）波主要源于兩種不同的波速：流體的聲速和由磁場壓力與張力產生的阿爾芬速度。這兩種速度的結合產生了磁聲波。因此，在MHD中，除了聲波外，還存在三種波動模式——慢磁聲波、剪切阿爾芬波和快磁聲波，如圖2.1所示。

The derivation of the waves can be obtained from the linearized MHD dispersion relation.

sound wave

The sound wave is due to a compressible fluid and the wave is longitudinal. The sound speed is defined as:

where γ is the polytropic index and in space plasma, it is in the range 1 < γ < 5/3 (Livadiotis and Nicolaou, 2021), k_B is the Boltzmann constant, T is temperature, mi is mass of a particle species.

Mach number

The Mach number, a ratio of flow to the (thermal) sound speed

Alfvén speed

The shear Alfvén wave is incompressible and transverse. The Alfvén speed is defined as follows:

here B2/μ??is the magnetic pressure and ρ is density.

magnetic Mach number

Similarly, the magnetic Mach number, which is defined as a ratio between the flow speed V_f low and the Alfvénic speed V_A, is defined as follows:

magnetosonic waves

The magnetosonic waves are as follows:

where b term is as follows:

聲速

波動方程的推導可以從線性化的MHD色散關系中獲得。聲波源于可壓縮流體，是一種縱波。聲速定義為：

其中：

????????γ 是多變指數，在空間等離子體中范圍為 1 < γ < 5/3；
????????k_B 是玻爾茲曼常數；
????????T 是溫度；
????????m_i 是粒子質量。

馬赫數?（流動速度與聲速之比）

Alfvén 速度

剪切阿爾芬波是不可壓縮的橫波。阿爾芬速度定義為：

其中：

????????B2/μ? 表示磁壓

????????ρ 是密度

磁馬赫數?（流動速度與阿爾芬速度之比）

磁聲波

其中b項為：

The equation 2.22 has two terms. The term containing (+) is the fast magnetosonic wave while the one containing (-) is the slow magnetosonic wave. When V_A >> V_s or V_s >> V_A in 2.23, as well as the wave propagation direction, is nearly perpendicular to the magnetic field (cos2θ << 1),

slow magnetosonic wave speed (simplified)

fast magnetosonic wave (simplified)

fast magnetosonic Mach number

方程2.22包含兩項：含有(+)的項代表快磁聲波，含有(-)的項代表慢磁聲波。當滿足V_A >> V_s或V_s >> V_A（如式2.23所示），且波傳播方向與磁場近乎垂直（cos2θ << 1）時：

簡化的慢磁聲波速度

簡化的快磁聲波速度

快磁聲波馬赫數

? ▍2.4 MHD discontinuities??▍

When plasmas of different properties collide, they reach equilibria, resulting in the boundaries separating neighboring plasma regions (Baumjohann and Treumann, 2012; Chapter 8). These boundary regions are called discontinuities, and in astrophysics, heliopause and magnetopause are examples of these discontinuities. Through discontinuities, the field and plasma parameters change abruptly, but these sudden changes satisfy a few conditions, namely the Rankine-Hugoniot jump conditions. To derive the jump conditions, it is suitable to integrate the conservation laws across the discontinuity. Therefore, it is better to write the single fluid MHD equations 2.2 in conservative form, assuming that the two sides of the discontinuity satisfy an ideal single-fluid MHD. Following some notations and derivations of (Baumjohann and Treumann, 2012; Chapter 8), the ideal single-fluid MHD equations in conservative forms are defined below:

where P the plasma pressure, B the magnetic field magnitude,B the magnetic field vector, μ0 the vacuum permeability, and I the identity tensor.

當不同性質的等離子體碰撞時，會形成平衡態邊界。?這些邊界區域稱為間斷面，在天體物理中，?太陽風層頂（heliopause）和磁層頂（magnetopause）就是典型的間斷面實例。雖然場和等離子體參數在間斷面上發生突變，但這些突變必須滿足Rankine-Hugoniot跳躍條件。

為推導跳躍條件，需要：

?將單流體MHD方程（式2.2）改寫為守恒形式
假設間斷面兩側滿足理想單流體MHD條件

根據Baumjohann和Treumann的推導，理想MHD的守恒形式方程組包含以下物理量：

其中

?P：等離子體壓力
?B：磁場強度（標量）
?B：磁場矢量
?μ?：真空磁導率
?I：單位張量

The induction equation

The divergence of the magnetic field

The energy conservation equation

where for slowly variable fields μ0j = ? × B and the ideal Ohm’s law E = ?? × B are implemented as well as neglecting charges ρE = 0, and w = c_v*P/ρ*k_B is the internal enthalpy. For completion, the equation of state is set:

where p is the scalar pressure.

Choosing a co-moving reference frame with discontinuity, a steady-state situation is assumed that all the time-dependent terms are canceled, leaving the flux terms in the conservation laws.

A discontinuity causes the plasma parallel to the discontinuity to stay the same while the plasma perpendicular to the discontinuity changes. For these reasons, a two-dimensional function S(x)=0 can be used to characterize the discontinuity surface, and the normal vector to the discontinuity, n, is defined as follows

To the direction of the n the vector derivative has only one component ? = n( ?/?n). After these considerations, integrating over the discontinuity for the flux terms has to be done.

磁感應方程：?

磁場散度方程：?

能量守恒方程：?

其中對于緩變場，采用μ?j = ? × B和理想歐姆定律?E = -? × B，并忽略電荷項 ρE = 0，w = c_v*P/ρ*k_B表示內焓。為完整起見，狀態方程設為：

其中p為標量壓強。

選擇與間斷面共動的參考系，假設穩態情況下所有時間相關項被消除，僅保留守恒定律中的通量項。

間斷面會導致平行于間斷面的等離子體保持不變，而垂直于間斷面的等離子體發生變化。因此，可用二維函數S(x)=0來表征間斷面，其法向量n定義為：

在n方向上，矢量導數僅有一個分量 ? = n(?/?n)。基于這些考慮，必須對通量項在間斷面上進行積分。

Remembering only ??two perpendicular sides of the discontinuity?? contribute twice to the integration, a quantity ??X crossing S?? becomes

where parenthesis ??[X]?? indicates the ??jump of a quantity X??. Now replacing the ??conservation laws of the ideal one-fluid MHD?? with the ??jump conditions??, the ??Rankine-Hugoniot conditions?? are defined as follows:

From the equations above, the ??normal component of the magnetic field?? is continuous across all discontinuities, which leads to its ??jump condition vanishing??.

Also ??normal direction mass flow?? is continuous:

Splitting the fields between the ??normal and tangential components??, the remaining ??jump conditions?? are derived:

where subscript ??t?? and ??n?? denote ??normal and tangential components?? respectively. The ??equations 2.39 and 2.40?? demonstrate that the ??normal components of the magnetic field??, as well as the ??mass flow??, are constant across the discontinuity. The ??Rankine-Hugoniot conditions?? provide the ??four types of MHD discontinuities?? (Baumjohann and Treumann, 2012; Chapter-8), namely ??contact discontinuity??, ??tangential discontinuity??, ??rotational discontinuity??, and ??shocks?? as the values are defined in ??Table 2.1??.

Table 2.1: Some properties of MHD discontinuities. (The values are from Oliveira, 2015)

考慮到間斷面的兩個垂直邊對積分有雙重貢獻??，物理量??X穿過S??可表示為

其中括號??[X]??表示物理量??X的躍變??。現將??理想單流體磁流體力學守恒定律??替換為??躍變條件??，??Rankine-Hugoniot條件??定義如下：

由上述方程可知，??磁場的法向分量??在所有間斷面保持連續，因此其??躍變條件為零??。

同時??法向質量流??也是連續的：

將場量分解為??法向和切向分量??后，可導出剩余??躍變條件??：

其中下標??t??和??n??分別表示??切向和法向分量??。??方程2.39和2.40??表明??磁場的法向分量??以及??質量流??在間斷面保持恒定。??Rankine-Hugoniot條件??給出了??四種磁流體力學間斷面類型??，即??接觸間斷面??、??切向間斷面??、??旋轉間斷面??和??激波??，其具體參數定義見??表2.1??。

Contact discontinuity

Contact discontinuity is determined by the condition when there is no normal mass flow across discontinuities, which means from the Rankine-Hugoniot conditions v_n = 0. When B_n ?= 0 or [B_n] = 0, the only quantity that experiences a change across the discontinuity is the density, [ρ] ?= 0. Due to these constraints, the plasma on the two sides of the discontinuity are attached and connected by the normal component of the magnetic field, As a result, they flow together at the same tangential speed. This is known as the contact discontinuity.

接觸間斷（Contact discontinuity）?

接觸間斷的判定條件是在間斷面上不存在法向質量流動，即根據Rankine-Hugoniot條件有v_n=0。當B_n≠0或[B_n]=0時，?唯一發生變化的物理量是密度[ρ]≠0。由于這些約束條件，?間斷兩側的等離子體通過磁場的法向分量相互連接，從而以相同的切向速度共同運動。這種現象稱為接觸間斷。

Tangential discontinuity

Tangential discontinuity is when B_n = 0 in addition to v_n = 0, only non-trivially satisfied jump condition is the total pressure balance in the equation 2.41, which indicates that a discontinuity is created between two plasma with total pressure balance on both sides and no mass and magnetic fluxes are crossing across the discontinuity while all other quantities are changing. This is the tangential discontinuity.

切向間斷（Tangential discontinuity）?

切向間斷的條件是B_n=0且v_n=0，此時唯一非平凡的跳躍條件是方程2.41中的總壓力平衡。?這表明在兩個等離子體之間形成了保持總壓力平衡的間斷面，且沒有質量和磁通量穿過間斷面，而其他所有物理量都發生變化。這就是切向間斷。

Rotational discontinuity

Rotational discontinuity is formed when there is a continuous normal flow velocity [v_n] = 0 with a non-vanishing B_n ?= 0 such that from the equations 2.42 and 2.43 the tangential velocity and the magnetic field can only change together, especially the equation 2.43 indicates they rotate together when crossing the discontinuity.

旋轉間斷（Rotational discontinuity）?

旋轉間斷的形成需要連續的法向流速[v_n]=0和非零的B_n≠0，根據方程2.42和2.43，?切向速度和磁場必須同步變化，特別是方程2.43表明它們在穿過間斷面時會共同旋轉。

Shock

Shocks Unlike the previous three discontinuities, shocks are irreversible in that the entropy increases (Goedbloed et al., 2019). And one of the main distinctive characteristics of shocks is normal fluxes of the Rankine-Hugionot conditions take non-zero values, ρv_n ?= 0, and the density ρ is discontinuous.

激波（Shocks）?

激波與前三種間斷不同，?激波是不可逆的，其熵會增加?。?激波的主要特征之一是Rankine-Hugoniot條件中的法向通量取非零值（ρv_n≠0）?，且密度ρ不連續。

Consequently of the ??three wave modes??, there are corresponding ??three shocks?? – ??fast shock (FS)??, ??intermediate shock (IS)??, and ??slow shock (SS)?? (Tsurutani et al., 2011). When the ??fast wave speed?? is greater than the ??upstream magnetosonic speed 2.25??, the ??fast shocks (FSs)?? are formed. Similarly, when the ??shear-Alfvén wave speed?? is greater than the ??upstream Alfvén velocity 2.20??, the ??intermediate shocks (ISs)?? are formed, and when the ??slow wave speed?? is greater than the ??upstream (thermal) sound speed 2.18??, the ??slow shocks (SSs)?? are formed with their respective ??Mach numbers greater than at least 1??.

The ??three MHD waves 2.3?? are ??anisotropic??, which means their ??speeds depend on the angle between wave propagation direction and the upstream (unshocked) magnetic field??. An illustration of this is the ??Parker spiral propagation?? of the solar wind such that the ??interplanetary magnetic field (IMF)?? hits, for example, the Earth from the morning side, it creates ??parallel and perpendicular shocks?? concerning the ??shock normal?? and the ??upstream magnetic field B?? as shown in ??Figure 2.2??.

Figure 2.2: The solar wind interaction with the Earth’s bow shock.(Figure is from Oliveira, 2015; Figure 2-5)

由此產生的三種波模式對應著三類激波：快激波（FS）、中間激波（IS）和慢激波（SS）。?當快波速度超過上游磁聲速（式2.25）時形成快激波（FSs）?；當剪切阿爾芬波速度超過上游阿爾芬速度（式2.20）時形成中間激波（ISs）?；當慢波速度超過上游聲速（式2.18）時形成慢激波（SSs）?，且對應的馬赫數至少大于1。

這三種MHD波（式2.3）具有各向異性，即其傳播速度取決于波傳播方向與上游（未受擾動）磁場方向的夾角。?這種現象的典型表現是太陽風的帕克螺旋傳播——例如當行星際磁場（IMF）從晨側撞擊地球時，會形成與激波法線和上游磁場B平行或垂直的激波，如圖2.2所示。

Hence, depending on the shock normal angle θ_Bn , the shocks can be geometrically classified as parallel, θ_Bn = 0? , perpendicular, θ_Bn = 90? , oblique, 0? < θ_Bn < 90? , quasi-parallel, 0? < θ_Bn < 45? , and quasi-perpendicular, 45? < θBn < 90? (Chao and Hsieh, 1984; Johlander et al., 2022).

因此，根據激波法向角θ_Bn，激波在幾何上可分為：?