and the magnetosphere

M A S T E R ' S T H E S I S

Magnetic reconnection structure including flow in the magnetosheath

and the magnetosphere

Henrik Wiberg

Luleå University of Technology MSc Programmes in Engineering

Space Engineering

Department of Applied Physics and Mechanical Engineering Division of Fluid Mechanics

2010:165 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--10/165--SE

This thesis aims to investigate the reconnection structure and the plasma flow at the mag- netopause, both on the magnetosheath and the magnetosphere side. Ongoing magnetic reconnection is assumed and not the onset of reconnection. A reconnection structure incorporating an Alfv´en discontinuity, a contact discontinuity and a slow mode shock is used. Shock jump conditions are then used to derive expressions for the magnetic field and the plasma flow on both sides of the magnetopause, as well as the angles of the different discontinuities. MATLAB is used to produce plots of the structure by varying the velocity gradient and the lowest order flow velocity.

iii

I wish to take this opportunity to thank a few persons that have been of great help while I wrote this thesis. My supervisor, Prof. Hans O. ˚Akerstedt has helped me a lot every step of the way. To all my Research Trainee colleagues I would like to say thanks for the great time we had during our year together. Last I would like to give a special thanks to Sara, for standing by me all this time.

Henrik Wiberg

v

Chapter 1 – Introduction 1

1.1 Space Plasmas and the Solar Wind . . . . 1

1.2 The Terrestrial Magnetosphere . . . . 3

1.3 Magnetic Reconnection . . . . 4

1.4 Scope of the Thesis . . . . 7

Chapter 2 – Model and Calculations 9 2.1 Setup of the model . . . . 9

2.2 Perturbation Analysis . . . . 16

2.3 Angles . . . . 21

Chapter 3 – Results and Discussion 25 3.1 Reconnection structures . . . . 25

Appendix A – MATLAB code 35 A.1 Reconnection.m . . . . 35

A.2 North reconnection.m . . . . 38

A.3 South reconnection.m . . . . 51

Introduction

1.1 Space Plasmas and the Solar Wind

Matter comes in four different states; solid, liquid, gas and plasma. The plasma state differs from the gas state by the fact that it contains ions and electrons, instead of neutral atoms. Plasmas are quasi-neutral, meaning that ion and electron densities are roughly the same. A plasma can be produced by heating a gas to such temperatures that the atoms become ionized. Plasma can also be produced by subjecting a gas to high energy radiation, such as UV-light or X-rays. There are numerous environments where plasmas occur; from the interior of stars to the ionosphere of planets; from interplanetary space to laboratories. The plasmas in these environments display many differences, but they all obey the same basic equations. Thus plasma physics is a very broad and diverse research area, with many applications. From understanding the local space environment to the achievement of nuclear fusion, the value of plasma research is great.

1.1.1 The Solar Wind

The solar atmosphere, unlike Earth’s atmosphere, is not in a radiative equilibrium.

Plasma is blown away from the sun in the solar wind. The existence of the solar wind was postulated in the early 20th century by Kirkeland, who thought of it as a connection between sunspots and auroral activity. The first substantial evidence came with the ob- servation of comet tails. The tail of a comet was thought to extend radially away from the sun, due to the pressure from light acting on the dust in the tail. However, the tail was shown to consist of two tails, one dust tail and one ion tail. The ion tail deviates from the radial direction by several degrees, while the dust tail extends radially. The orientation of the ion tail can be explained by a plasma flow from the sun.[1]

The solar wind has a velocity ranging from a few hundred km s⁻¹ to about 1000 km s⁻¹. The density of the plasma flow is typically about 5 particles per cm³ at 1 AU from the sun.[2] The solar wind consists of two types of flows; the fast and slow solar winds.

1

Figure 1.1: Schematic view of the spiral shaped magnetic field due to the Sun’s rotation.

Image courtesy of Kivelson and Russell (1995)

The fast wind originates from coronal holes at high solar latitudes, where plasma can exit the solar atmosphere along open field lines. The slow wind comes from lower latitudes and is more turbulent and variable than the fast wind.

As the fast solar wind travels outwards, it catches up with the slow wind and this creates velocity shears and shock structures in the flow.

The low density of the solar wind should suggest that it cannot be treated as a fluid, since the collisional mean-free path is about 1 AU for ions, [2]. But magnetic and electric fields act on the plasma and assumes the role of collisions.

1.1.2 The Interplanetary Magnetic Field

The magnetic field that is generated inside the sun is very complex near the sun, but further away it becomes more or less radial. The solar wind plasma has a near-infinite conductivity, and this causes the magnetic field to be “frozen-in” into the plasma. The solar wind thus drags the magnetic field along with it. The sun rotates around its axis, causing the magnetic field to be wound up into a spiral shape, see Figure 1.1. Far out in the heliosphere, the magnetic field assumes an almost toroidal shape. [1]

1.2 The Terrestrial Magnetosphere

The magnetic field of the Earth is generated inside the planet in a dynamo process. The field near the Earth can be adequately approximated by a dipole. At higher altitudes however, the interaction with currents and the solar wind distorts its shape.

The magnetosphere is shaped by the interaction between the terrestrial magnetic field and the magnetized plasma of the solar wind. See Figure 1.2 for a detailed view of Earth’s magnetosphere.

1.2.1 The Magnetopause

The solar wind exerts a pressure on the Earth’s magnetic field. Where the magnetic pressure from the magnetosphere balances the dynamic pressure of the solar wind, lies a boundary called the magnetopause. The magnetopause is not a thin discontinuity, although it is sometimes regarded as one. Rather, it is an extended region several hundred kilometers thick. When the magnetic field in the solar wind has a component anti-parallel to the magnetic field of the Earth, magnetic reconnection can occur and allow for plasma to enter the magnetosphere. The open field lines that result from this are swept past the Earth. For much more on the magnetopause, see for example [3].

1.2.2 The Magnetic Tail

There are two regions in the dayside magnetosphere where the magnetic field vanishes and allows particles to move into the magnetosphere. These regions are the polar cusps and they separate the closed magnetic field lines of the dayside magnetosphere from the open field lines on night side. The field lines that are swept to the night side form the boundary of the magnetotail, which extends for more than 100 Earth radii.

In the magneto tail, a current sheet develops because the oppositely directed field lines are pressed together. This current sheet allows for magnetic reconnection to occur and accelerate plasma towards the Earth. Thus, the field lines that were ’opened’ at the magnetopause now closes in the magnetotail, preserving the dipole field.[1, Kallenrode (2004)]

1.2.3 The Bow Shock and the Magnetosheath

In front of the magnetopause is a shock structure called the bow shock. It forms when the supersonic plasma of the solar wind is decelerated to subsonic speeds. The solar wind does not flow through the magnetopause, so enough space must exist to allow the plasma to flow around it. This region is called the magnetosheath. Since the plasma flow is slowed down, kinetic energy is converted to heat. The plasma in the magnetosheath is thus hotter and denser than the plasma ahead of the bow shock. This jump in the density is consistent with the Rankine-Hugoniot shock jump conditions. As the plasma

Figure 1.2: Schematic view of the Earth’s magnetosphere. From[5, Fujimoto (2006)].

flow reaches the flanks of the magnetopause, it is again accelerated to supersonic speeds.

[4]. [6]

1.3 Magnetic Reconnection

The conductivity of the solar wind plasma is very high, allowing the magnetic field of the sun to move with the plasma flow. The induction equation

∂B

∂t = ∇×(u×B) + η∇²B, (1.1)

can be simplified to

∂B

∂t = ∇×(u×B) (1.2)

in the solar wind, since the resistivity is essentially zero. When the solar wind encounters the magnetosphere, a current sheet develops when the magnetic field in the solar wind has a component opposite to the magnetospheric field. In this current sheet, the conductivity is no longer high, so the equation (1.1) becomes

∂B

∂t = η∇²B, (1.3)

which is a diffusion equation. The magnetic fields on each side of the sheet thus diffuses into the current sheet and reconnects with each other. Magnetic reconnection provides one mechanism for exchanging plasma between the magnetosphere and the solar wind.

Figure 1.3: The Sweet-Parker reconnection model.

There are several other mechanisms for this, but reconnection is regarded as the most important one. Theories describing reconnection can be divided into slow or fast models.

Early theories, e.g. the Sweet-Parker model is an example of a slow model. Later theories have mostly been concerned about describing fast reconnection. Magnetic reconnection occurs in a multitude of different environments; it is the driving force behind solar flares, it governs much of the dynamics of the magnetosphere and it is also present in fusion plasmas where it is one of many obstacles for achieving stable fusion reactions.

1.3.1 The Sweet-Parker Model

The Sweet-Parker model of reconnection involves an extended current sheet, into which the magnetic field diffuses. The plasma is accelerated sideways to the Alfv´en speed. This configuration gives a reconnection rate that is much too low to be the mechanism behind solar flares. [7, Priest & Forbes (2000)]. This discrepancy drove researcher to find a faster model for reconnection.

1.3.2 Petschek’s Model

A faster model for reconnection was proposed by Petschek in 1964, at a symposium on solar flares. [8, Biskamp (1997)] His model includes a much smaller diffusion region and two pairs of shocks, which extends in different directions and stand in the plasma flow once a steady state has been achieved, see Figure 1.4. The shocks provide an additional way for converting magnetic energy into kinetic energy and heat, besides the diffusion region. In fact, in Petschek’s model, most of the energy is converted in the shocks and most of the plasma flow does not enter the diffusion region at all.

Petschek’s model was seen as the complete solution to fast reconnection for about two decades, until new models emerged and advanced computer simulations could be performed. It’s status now is more like a special case of more general reconnection theories. [7, Priest & Forbes (2000)]

Figure 1.4: The geometry of Petschek’s model. The central shaded region is the diffusion region and the other two are the outflow regions where hot plasma is expelled. From [7, Priest & Forbes (2000)]

Figure 1.5: Overview of magnetic reconnection, form [3, Song]

1.4 Scope of the Thesis

The scope of this thesis is to find a description of the solar wind plasma and magnetic field across the terrestrial magnetopause and in the magnetosphere during active magnetic reconnection. Previous work has focused on the flow and magnetic field structure ahead of the magnetopause. This thesis focuses on the crossing of the magnetopause and the magnetosphere flow. To determine the plasma flow and magnetic field structure in the magnetosphere, the total pressure on both sides of the magnetopause is assumed to be equal. From this assumption, along with the ideal Ohm’s law and momentum equation, the magnetosphere parameters are calculated.

Model and Calculations

2.1 Setup of the model

2.1.1 Coordinate system and shock structure

The coordinate system used in this model is a curvilinear one with the x-axis normal and the z-axis tangential to the magnetopause. The analysis is done in a region north of the subsolar point, with a northward flow velocity component. A southward IMF is also assumed. The shock structure at the reconnection site consists of an Alfv´en wave, a contact discontinuity and a slow-mode shock, see Figure 2.1. The region where the reconnection is assumed to occur has a anomalous resistivity.

2.1.2 Jump conditions

The quantities in the different regions are denoted as

X, in region 1, (2.1)

X⁰, in region 2, (2.2)

X˜⁰, in region 3, (2.3)

X, in region 4.˜ (2.4)

A jump across a discontinuity is written as

[X]^b_a = X_b− X_a. (2.5)

The following jump conditions must be satisfied, according to [9].

[ρu_n] = 0, (2.6)

[B_n] = 0, (2.7)

[u × B] = 0, (2.8)

9

Figure 2.1: The coordinate system used in the model. The x-axis points normal to the magnetopause and the z-axis is tangent to the magnetopause.

Figure 2.2: Overview of the structure of the model. Three shock waves extend from the central diffusion region. A denotes an Alfv´en wave, C a contact discontinuity and S a slow-mode shock. The region denoted by 1 is the magnetosheath region and number 4 is the magnetosphere region.

where u is the plasma velocity, B the magnetic field and ρ the plasma density. The pressure on both sides of the magnetopause must balance, hence



p + B² 2µ₀



, (2.9)

where the first term is the plasma pressure and the second is the magnetic pressure. For the three discontinuities, the following additional conditions apply.

Alfv ´en discontinuity

[ρ] = 0, (2.10a)

[u_n] = 0, (2.10b)

B_t² = 0. (2.10c)

The tangential component of B rotates in the plane of the discontinuity, which is there- fore also called a rotational discontinuity. When the magnetosphere is open, it may be described as a rotational discontinuity.

Contact discontinuity

There is no jump in plasma density so

[ρ] = 0. (2.11a)

The tangential component of the flow velocity is continuous, giving

[ut] = 0. (2.11b)

The magnetic field is also continuous, thus

[B] = 0. (2.11c)

Finally, the normal component of the plasma flow is zero, so

u_n= 0. (2.11d)

This is why it is called a contact discontinuity; there is no flow across it.

Slow-mode shock

[ρ] 6= 0, (2.12a)

u_n6= 0. (2.12b)

2.1.3 The jumps

Alfv ´en discontinuity

For the first discontinuity we have, according to eqs (2.10a) - (2.10c). From these we get un = ± B_n

√µ₀ρ, (2.13)

where the positive case is used for the jump across the Alfv´en discontinuity. and u⁰_n= u_n= B_n

√µ₀ρ (2.14)

For the northern case, ˆn = ˆx − α^Nˆz and thus

u^N_n = u·ˆn = u_x− αu_z. (2.15) With u_z = U₀, (2.14) becomes

u^N_x − α^NU₀ = u⁰_x^N− α^NU₀⁰ (2.16) For the southern part, we have ˆn = ˆx + α^Sˆz, so

u^S_x− α^SU₀ = u⁰_x^S− α^SU₀⁰. (2.17) Since [B_n] = 0 we have, for the northern and southern case respectively

B_x^N− α^NB₀ = B_x^0N− α^NB₀⁰. (2.18) B_x^S+ α^SB₀ = B_x^0S+ α^SB₀⁰. (2.19) By choosing

u^∗_t = u_t− u_HT, (2.20)

where u_HT is the de Hoffmann-Teller velocity, satisfying E + u_HTB_n = 0, and

u^∗_tB_n− u_nB_t = 0, (2.21)

we get

u_HT = u_t− u_nB_t

B_n (2.22)

(2.13) in (2.22) gives

u_HT = u_t− B_t

√µ₀ρ. (2.23)

With u_t = U₀, B_t = B₀ and ρ = µ₀ = 1, we get

u^N_HT = U₀− B₀, (2.24)

for the northern part. The southern case is

u^S_HT = U₀+ B₀. (2.25)

The equation E + u_HTB_n= 0 gives

B_x^N = − E

U₀− B₀ + α^NB₀, (2.26)

and

B_x^S = − E

U₀+ B₀ + α^SB₀, (2.27)

for the northern and southern part, respectively. For region 2, we have

u⁰_HT = U₀⁰ − B⁰₀, (2.28)

But u⁰_HT = u_HT, so

U0− B0 = U₀⁰ − B⁰₀, (2.29) giving

U₀^0N= U0− B₀+ B₀⁰, (2.30) for the northern case and

U₀^0S = U₀+ B₀− B₀⁰, (2.31) for the southern case. Given that u_n6= 0 and [u_n] = 0, we have

u⁰_x^N− α^NU₀^0N= u^N_x − α^NU₀, (2.32) and

u⁰_x^S+ α^SU₀^0S= u^S_x+ α^S. (2.33) Thus

u⁰_x^N= u_x+ α^N(B₀⁰ − B₀), (2.34) u⁰_x^S = u_x− α^S(B⁰₀− B₀), (2.35) for the northern and southern case, respectively. For the magnetic field, we have [B_n] = 0, or

B_x^0N− α^NB₀⁰ = B_x^N− α^NB₀, (2.36) B_x^0S− α^SB₀⁰ = B_x^S− α^SB0, (2.37) which becomes, for the northern and southern case,

B_x^0N= B_x^N+ α^N(B₀⁰ − B₀), (2.38) B_x^0S= B_x^S− α^S(B₀⁰ − B₀). (2.39)

Contact discontinuity

For the contact discontinuity, the normal vector is ˆ

n = ˆx + θ^Nˆz, (2.40)

for the northern part and

ˆ

n = ˆx − θ^Sˆx, (2.41)

for the southern part. The condition [u_n] = 0 and u_n= 0 gives

˜

u⁰_x^N+ θ^NU˜₀⁰ = 0 = u⁰_x^N+ θ^NU₀⁰, (2.42)

˜

u⁰_x^S− θ^SU˜₀⁰ = 0 = u⁰_x^S− θ^SU₀⁰. (2.43) The condition that B is continuous gives

B˜⁰_x B˜₀⁰



= B_x⁰ B₀⁰



⇒ (2.44)

B˜_x⁰ = B_x⁰, (2.45)

B˜₀⁰= B₀⁰. (2.46)

Ohm’s law gives

E + un×Bt+ ut×Bn= 0. (2.47)

Since E is constant, we have

−u⁰_nB_t⁰+ u⁰_tB_n⁰ = −˜u⁰_nB˜_t⁰ + ˜u⁰_tB˜_n⁰. (2.48) The conditions [B_n] = 0 and u⁰_n= ˜u⁰_n= 0 then gives

u⁰_tB_n⁰ = ˜u⁰_tB_n⁰. (2.49) Hence u⁰_t= ˜u⁰_t and thus

U₀⁰= ˜U₀⁰, (2.50)

u⁰_x= ˜u⁰_x. (2.51)

Slow mode shock

In regions 3 and 4, ρ = ˜ρ 6= 1. The condition [ρu_n] = 0 gives

˜

u⁰_x^N+ ˜α ˜U₀^0N= ˜u_x+ ˜α ˜U₀, (2.52)

for the northern part and

˜

u⁰_x^S− ˜α ˜U₀^0S= ˜u_x− ˜α ˜U₀ (2.53) for the south. The momentum equation in the ˆt direction is

˜ ρ



˜ u_n∂ ˜u_t

∂n



= −∂ ˜p

∂t

| {z }

=0

+ ˜B_n∂ ˜B_t

∂n, (2.54)

⇒ ˜ρ˜u_n∂ ˜u_t

∂n = B˜_n∂ ˜B_t

∂n, (2.55)

which may be written as

˜

ρ˜u_n[˜u_t] = ˜B_nh ˜B_ti

. (2.56)

Since (˜u − uHT)× ˜B = 0,

−˜utB˜n+ u_HTB˜n+ ˜unB˜t = 0, (2.57) and hence

˜

u_t= u_HT+ ˜u_n B˜_t

B˜_n. (2.58)

Equation (2.58) in the momentum equation gives

˜

ρ˜u_n ∂ ˜u_n

∂n B˜_t

B˜_n + ˜u_n ∂

∂n ˜B_t

B˜_n

!!

= ˜B_n∂ ˜B_t

∂n, (2.59)

which reduces to

˜ ρ˜u²_n

B˜_n

∂ ˜B_t

∂n = ˜B_n∂ ˜B_t

∂n. (2.60)

We can then solve for ˜u_n to get

˜

u_n = −B˜_n

√ρ˜ (2.61)

The momentum equation then reduces to

−p

˜

ρ [˜u_t] =h ˜B_ti

, (2.62)

or simply

−p

˜

ρ( ˜U₀− ˜U₀⁰) = ˜B₀ − ˜B₀⁰, (2.63) which solves for ˜U₀⁰ as

U˜₀^0N = ˜U₀⁰ⁿ+

B˜0− ˜B₀^N

√ρ˜ = U₀^0N, (2.64)

U˜₀^0S = ˜U₀^S+

B˜₀− ˜B^S₀

√ρ˜ = U₀^0S, (2.65)

As stated earlier

U₀^0N= U₀− B₀+ B₀^0N, (2.66) so we get

U₀− B₀+ B₀^0N= ˜U₀+

B˜₀− ˜B₀^0N

√ρ˜ , (2.67)

and

U₀+ B₀− B⁰₀^S= ˜U₀+

B˜₀− ˜B₀^0S

√ρ˜ . (2.68)

Solving for B₀⁰ then gives

B₀^0N =

√ρ˜

√ρ + 1˜

U˜0+ B0− U0+ B˜₀

√ρ˜

!

, (2.69)

and

B₀^0S = U₀ + B₀− ˜U₀ − B˜₀

√ρ˜

! √

˜

√ ρ

˜

ρ + 1. (2.70)

Now, to find ˜U₀, we start with

(˜u_t− u_HT) ˜B_n− ˜u_nB˜_t= 0, (2.71) which gives

˜

u_t= u_HT+u˜nB˜t

B˜_n . (2.72)

Using the expression in (2.61), we get

˜

ut= u_HT− B˜_t

√ρ˜. (2.73)

For ˜u_t = ˜U₀ and ˜B_t= ˜B₀ and for u_HT = U₀− B₀ U˜₀ = U₀− B₀− B˜₀

√ρ˜. (2.74)

When performing these calculations the lowest order term for the magnetic field in the magnetosphere, ˜B0, has been considered to be known already as ˜B0 = ˜B0ˆz.

2.2 Perturbation Analysis

Now that the zeroth order terms for the magnetic field and plasma flow is known for the four regions, we can construct the higher order terms as well. In this section, the

procedure used in [10] will be presented. The magnetic field and flow velocity is perturbed around a uniform field and velocity as

u = u₀+ u₁+ u₂+ O(E^3/2), (2.75) B = B₀ + B₁+ B₂+ O(E^3/2), (2.76) where the quantities are scaled as E^1/2. The lowest orders are just

u₀ = U₀ˆz, (2.77)

B0 = B0ˆz. (2.78)

The equations used to find expressions for the higher orders are the MHD equation of motion

(u·∇) u = ∇p + (B·∇) B, (2.79)

and Ohm’s law

E + u×B = 0. (2.80)

(2.79) becomes, to lowest order

∇p₀ = 0, (2.81)

⇒ = p₀ = C = 0. (2.82)

The constant C may be chosen as zero. For the order O(E^1/2) we have

(U0·∇) u1 = −∇p1+ (B0·∇) B1. (2.83) In the z-direction, we have

U0

∂u_1z

∂z = −∂p₁

∂z + B0

∂B_1z

∂z , (2.84)

for the equation of motion, and

U₀B_1x − u_1xB₀ = 0, (2.85)

B_1x = B₀

U₀u_1x, (2.86)

for Ohm’s law. ∇·B = 0 and ∇·u = 0 gives

∂B_1z

∂z = −B₀ U₀

∂u_1x

∂x , (2.87)

∂u_1x

∂x = −∂u_1z

∂z , (2.88)

giving

∂B_1z

∂z = B₀ U₀

∂u_1z

∂z , (2.89)

B_1z = B₀

U₀u_1z. (2.90)

The flow velocities for order O(E^1/2) are described as

u1x = −Qx, (2.91)

u_1z = Qz, (2.92)

where Q is the velocity gradient. With this information at hand, we can find an expression for the pressure. The equation of motion gives

U₀Q = ∂p1

∂z +B₀²

U₀Q, (2.93)

which then gives the pressure as

p₁ = B²₀ U₀² − 1



U₀Qz. (2.94)

For the order O(E¹), (2.79) becomes

(u₁·∇) u₁+ (U₀·∇) u₂ = −∇p₂+ (B₁·∇) B₁+ (B₀·∇) B₂. (2.95) The x-component is

u_1x∂u_1x

∂x + u_1z∂u_1x

∂z

| {z }

= 0

+U₀∂u_2x

∂z (2.96)

= −∂p₂

∂x + B_1x∂B_1x

∂x + B_1z∂B_1x

∂z

| {z }

= 0

+B₀∂B_2x

∂z .

Inserting the O(E^1/2) expressions gives Q²x + U₀∂u_2x

∂z = −∂p₂

∂x +B₀²

U₀²Q²x + B₀∂B_2x

∂z . (2.97)

The z-component gives in the same manner Q²z + U₀∂u_2z

∂z = −∂p₂

∂z +B₀²

U₀²Q²z + B₀∂B_2z

∂z . (2.98)

Ohm’s law

E + u×B = 0, (2.99)

becomes

E + U₀×B₂+ u₁×B₁+ u₂×B₀ = 0. (2.100) Since E = Eˆy, there is only a y-component:

E + U₀B_2x− Q²B0

U₀xz + Q²B0

U₀xz − u_2xB₀, (2.101) E + U₀B_2x− u_2zB₀ = 0.

Solving for the velocity gives

u_2x= E B₀ + U₀

B₀B_2x. (2.102)

Using ∇·u2 = 0 results in

∂u_2x

∂x = −∂u_2z

∂z , (2.103)

and ∇·B = 0 gives

∂B_2x

∂x = −∂B_2z

∂z . (2.104)

From these we get an expression for u_2z: u_2z = U₀

B₀B_2z. (2.105)

Looking again at (2.97) and (2.98), we have

U0

∂u2x

∂z = −∂p2

∂x − Q²x + B₀²

U₀²Q²x + B0

∂B2x

∂z , (2.106)

U₀∂u_2z

∂z = −∂p₂

∂z − Q²z +B₀²

U₀²Q²z + B₀∂B_2z

∂z . (2.107)

By defining

P₂ = p₂+ Q²

2 x²+ z²



1 −B₀² U₀²



, (2.108)

we get

U₀∂u_2x

∂z = −∂P₂

∂x + B₀∂B_2x

∂z , (2.109)

U₀∂u_2z

∂z = −∂P₂

∂z + B₀∂B_2z

∂z . (2.110)

Using (2.102) and (2.105) in the above equations gives U₀²

B0

∂B_2x

∂z = −∂P₂

∂x + B₀∂B_2x

∂z , (2.111)

U₀² B₀

∂B2z

∂z = −∂P2

∂z + B₀∂B2z

∂z , (2.112)

which may be solved for P₂ as

P₂ =



B₀− U₀² B₀



B_2z. (2.113)

The O(E¹) expression for the magnetic field is found by using the Poisson integral [10]

as follows.

B_2x= 1 π

Z −

−L

B_2x^S (0, z) x

x²+ (z − ζ)²dζ + 1 π

Z L



B_2x^N(0, z) x

x²+ (z − ζ)²dζ (2.114) and

B2z = 1 π

Z −

−L

B_2x^S (0, z) z − ζ

x²+ (z − ζ)²dζ + 1 π

Z L



B_2x^N(0, z) z − ζ

x² + (z − ζ)²dζ, (2.115) where B_2x^N(0, z) and B_2x^S (0, z) are the expressions in (2.26) and (2.27). Evaluating these integrals gives

B2x(x, z) = −1 πB_2x^S



arctanz + 

x − arctanz + L x



− 1

πB_2x^N



arctanz − L

x − arctanz −  x



, (2.116)

B_2z(x, z) = −1

πB_2x^S ln x²+ (z + )² x²+ (z + L)²



− 1

πB_2x^N ln x²+ (z − L)² x²+ (z − )²



. (2.117)

For the other regions, the higher order terms are constructed in the same manner as described above, while keeping in mind to replace the zeroth order terms with their corresponding terms for the specific region. For the magnetosphere region, the O(E¹)

terms for the magnetic field are B˜_2x(x, z) = −1

π B˜_2x^S



arctanz + 

x − arctanz + L x



− 1

π B˜_2x^N



arctanz − L

x − arctanz −  x



, (2.118)

B˜_2z(x, z) = −1 π

B˜_2x^S ln x²+ (z + )² x²+ (z + L)²



− 1

π

B˜_2x^N ln x²+ (z − L)² x²+ (z − )²



, (2.119)

where ˜B_2x^S = _U^−E

0−B0 + ˜α ˜B₀ and ˜B_2x^N = _U^−E

0+B0 − ˜α ˜B₀.

2.3 Angles

The procedure to find the angles of the different discontinuities is presented here. First, the pressure on both sides of the magnetopause must be equal, so we have

P₂ = p₂+ B₀B_2z = ˜p₂+ ˜B₀B˜_2z = ˜P_s, (2.120)



2 − U₀² B₀²



B₀ −E u^S

HT

− E u^N

HT

+ α^S+ α^N
B₀



= 2 − U˜₀² B˜₀²

!

B˜₀ −E

˜

u^S_HT − E

˜

u^N_HT + 2 ˜α ˜B₀



. (2.121)

Using

˜

u^S_HT = ˜U₀+ B˜₀

√ρ˜, (2.122)

˜

u^N_HT = ˜U₀− B˜0

√ρ˜, (2.123)

we get



2 − U₀² B₀²

 B₀

 −E

U₀ + B₀ − E

U₀− B₀ + α^S+ α^N
B₀



= 2 − U˜₀² B˜₀²

!

B˜₀ −E U˜0 +^√B˜0_ρ˜

− E

U˜0− ^√B˜0_ρ˜ + 2 ˜α ˜B₀

!

. (2.124)