Download Ion Distribution in the Foreshock

Ion Distribution in the Foreshock
Thesis submitted in partial fulfillment of the requirement for the degree of master of
science in the Faculty of Natural Sciences
Submitted by: Michael Liverts
Advisor: Prof. M. Gedalin
Department of Physics
Faculty of Natural Sciences
Ben-Gurion University of the Negev
October, 2008
Abstract
The foreshock region is the first signature of the interaction of solar wind with a
planet’s plasma environment when approaching its bow shock. Part of its structure and
dynamic is determined by instabilities, which are created by the interaction of the solar
wind with backstreaming ions. Prominent examples are the field aligned and gyrophase
bunched ion beams which are observed in the foreshock of the Earth bow shock. These
ions are apparently produced at the shock front by the same non-adiabatic mechanism
which is responsible for the downstream ion gyration and ion reflection. The process
and the resulting distributions are sensitive to the angle between the shock normal and
the upstream magnetic field, the upstream ion temperature, the cross-shock potential and
Mach number. If the shock is one-dimensional and stationary, the number of escaping ions
rapidly drops with the increase of the angle. Specifically, upon encountering the shock a
fraction of the ions is specularly reflected, which due to their large velocity perpendicular
to the magnetic field, gyrate back into the shock. The details of the reflection mechanism
should determine what part of the initial distribution is reflected and what is the spatial
and phase-space distribution of the escaping ions.
One of the mechanisms proposed for their production is non-specular reflection at
the shock front. We study the distributions which are formed at the stationary quasiperpendicular shock front within the same process which is responsible for the generation
of reflected ions and transmitted gyrating ions. The test particle motion analysis in a
model shock allows one to identify the parameters which control the efficiency of the
process and the features of the escaping ion distribution. These parameters are: the
angle between the shock normal and the upstream magnetic field, the ratio of the ion
thermal velocity to the flow velocity upstream, and the cross-shock potential. A typical
distribution of escaping ions exhibits a bimodal pitch-angle distribution (in the plasma
rest frame).
2
Contents
1 Introduction
5
1.1
Collisionless Shocks in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The Earth’s Bow Shock and its Foreshock Region . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Shock structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.1
Shock geometry and strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.2
Shock types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.3
Rankine-Hugoniot jump conditions for MHD . . . . . . . . . . . . . . . . . . . . .
11
1.4
Scientific Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.5
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2 Analytical approach
16
2.1
Ion motion in the shock front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Reflection mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.3
Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
Upstream equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.2
Downstream equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4
Reflection conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.5
Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3 Numerical Analysis
30
3.1
Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.1
Ion Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.2
Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.2.3
Energy and Pitch angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Escaping Efficiency vs. Plasma Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.3
4 Summary
46
3
4.1
Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Solution of motion equations
46
52
4
Chapter 1
Introduction
1.1
Collisionless Shocks in General
The everyday notions about shock waves originate in our knowledge and experience related
to supersonic airplanes and blasts of explosion. In an ordinary gas the collisions between
the gas particles transfer the momentum and energy and allow the sound wave to exist.
The sound wave propagation through a medium is an adiabatic process. After the sound
wave has passed, the medium (the gas) regains its original state since the process is
reversible. The velocity of the sound wave is determined by the parameters of the medium
(i.e., density and pressure). On the other hand, when a disturbance travels through the
medium with a velocity larger than the speed of the sound, a shock wave is generated.
A shock wave differs significantly from the sound wave because it affects the medium
irreversibly. Every shock wave rises the temperature and density of the medium, while the
supersonic flow is decelerated to subsonic flow regarded from the frame of the shock wave.
The study of shock waves began at the end of the nineteenth century with gas dynamics.
In the 1940’s the understanding of shock waves improved substantially when the aircraft
jet engine was developed. Interest in fusion plasmas and (thermo)nuclear explosions in
the upper atmosphere during the 1950’s gave new impulse to shock wave research. Later,
when spacecraft were developed, the study of the space surrounding our planet became
possible by means of measurements. It was discovered that the interplanetary space is
dominated by a magnetized, tenuous, high-velocity plasma flow: the solar wind. The solar
5
Figure 1.1: A schematical picture of Earth’s bow shock and magnetosphere. As the solar
wind plasma encounters the terrestrial magnetosphere a shock wave is generated. At
the shock the solar wind is decelerated to subsonic velocity. The plasma is heated and
compressed, while the magnetic field magnitude increases.
wind is a neutral mixture of dissociated electrons and nucleii (mostly protons). Because of
its very low density and high velocities, the collisions between the particles are extremely
rare. This kind of plasma, as the solar wind, is called collisionless plasma. The discovery
of the Earth’s bow shock demonstrated that shock waves can exist in collisionless plasmas.
When the supersonic solar wind reaches the Earth’s surrounding and interacts with its
magnetic field a shock wave, the bow shock is formed. The bow shock slows down the
solar wind to subsonic speed, while the plasma is heated and its density and the magnetic
field magnitude increases. Since the solar wind flow is continuous, the Earth’s bow shock
is a ”standing” shock wave regarded from our planet. Other planets in the solar system
were also reached by spacecraft and the existence of bow shocks in front of these planets
was demonstrated. The shocks, however, are not limited to the solar system, since the
Universe is dominated by plasma flows. Wherever there are plasma flows, there are also
6
shock waves. Supernovae explosions also produce shocks. Collisionless shocks have their
scientific importance in their own right, but also because they are involved in a very wide
range of phenomena.
1.2
The Earth’s Bow Shock and its Foreshock Region
The Earth’s bow shock is a natural laboratory where the physics of collisionless plasma can
be investigated under ideal conditions. The conditions are ideal because the bow shock is
always present and its distance from the Earth’s center at the subsolar point is ∼ 15Re
(where Re is the Earth radius; 1Re = 6370km). This distance can be relatively easily
reached by spacecraft orbiting around the Earth. Therefore we can accumulate enough
data about the plasma and the electromagnetic fields at the bow shock to investigate it
in detail. Since the laws of physics are valid all over the Universe, the understanding
of processes at the Earth’s bow shock can help us to understand processes in regions
unavailable for direct measurements. Because of its scientific importance the Earth’s bow
shock is the most intensively studied nonlinear wave. Despite the fact that it has been
under investigation for more than three decades and a substantial amount of data and
knowledge has been gathered, there are still unanswered fundamental questions. One of
these questions is related to the ability of the bow shock to reflect particles. The exact
mechanism of how the ions are reflected, and which ions are involved in the reflection
process is not fully understood. The primary goal of this thesis is to contribute to the
understanding of ion reflection processes at Earth’s bow shock.
Even at an early stage of bow shock investigation it became clear that the shock
encounters fall into two groups. In one group the shock encounters were identified as
clean, localized, well defined transitions between the upstream and downstream regions
(the term upstream refers to the supersonic plasma, while the term downstream refers
to the slowed, heated and shocked turbulent plasma state). Sometimes shock transitions
presented a turbulent and noisy appearance, characterized by the presence of large amplitude magnetic fluctuations which made it difficult to identify a well defined transition
between downstream and upstream regions. A correlation was found between the two
7
Figure 1.2: The most common geometrical structure of the region in front of the Earth’s
bow shock in the GSE (Geocentric Solar Ecliptic coordinate system) x − y plane. The x
axis points to the Sun, while the y axis is in the ecliptic plane. Both types of shocks are
present simultaneously.
appearances and the θBn angle (i.e. the angle between the shock surface normal direction
and the magnetic field direction). Results showed that a quasi-parallel shock (i.e. when
θBn ≤ 45◦ ) presents an extended, turbulent transition, while a quasi-perpendicular shock
(i.e. when θBn ≥ 45◦ ) shows a well-defined, localized jump of plasma parameters between
the upstream and downstream regions. In present work we shall focus on the latter type of
shocks. Figure 1.2 presents the geometrical configuration of the region in front of the bow
shock in the GSE (Geocentric Solar Ecliptic) coordinate system. The GSE coordinate
system has its x axis pointing from Earth towards the Sun, its y axis is chosen to be in
the ecliptic plane pointing towards dusk (thus opposing planetary motion). Its z axis is
parallel to the ecliptic pole. Relative to an inertial system, this has a yearly rotation. Because of the bow shock curvature, the quasi-parallel and the quasi-perpendicular shocks
are simultaneously present, independent of the interplanetary magnetic field direction.
The region of space upstream of the bow shock, magnetically connected to the shock and
filled with particles backstreaming from the shock is known as the foreshock.
Before discussing in detail the foreshock region, we need to introduce a few concepts
related to ion motion in a magnetic field. In electromagnetic field the ion motion is
8
determined by the Lorentz force (in the following we use SI units):
~ + ~v × B)
~
F~L = q(E
~ and B
~ are the
where F~L is the Lorentz force, q is the ion charge, ~v is the ion velocity, E
electric and magnetic fields, respectively. In the frame of reference of the moving plasma
~ = 0. We represent the ion velocity in the following, one
the electric field vanishes, i.e. E
component is parallel, the other is perpendicular to the magnetic field:
~v = v~k + v~⊥
(1.1)
where v~|| is the parallel and v~⊥ is the perpendicular component. Taking into consideration
~ = 0 and Eq. 1.1, the Lorentz force can be written as:
E
~
F~L = q(v~|| + v~⊥ ) × B
where
~ =0
v~|| × B
Solving this, we find that the ion gyrates around the magnetic field line with a radius
rL =
mv~⊥
qB
where rL is the Larmor radius and B is the magnetic field magnitude. It should be noted,
that the latter derivation is valid only in case of constant magnetic field, otherwise the
particle is subjected to drifts, due to the dependence of trajectory curvature on particle
coordinate. In addition, the ion can move parallel to the magnetic field with v~|| . The
combination of the two movements results in a helicoidal trajectory around the magnetic
field line while the absolute velocity of the ion remains constant.
Another convenient parameter of the ion movement is the pitch angle, the angle
between the magnetic field direction and the ion velocity vector:
tan ψ =
v⊥
vk
where ψ is the pitch angle. The pitch angle shows the ratio of the velocity components
perpendicular and parallel to the magnetic field.
9
1.3
Shock structure
Shocks and discontinuities are transition layers where the plasma properties change from
one equilibrium state to another. The relation between the plasma properties on both
sides of a shock or a discontinuity can be obtained from the conservative form of the
magnetohydrodynamic (MHD) equations, taking into account the conservation of mass,
momentum, energy and of ∇ · B. Planetary bow shock have upstream regions called
foreshocks, created by energetic particles that travel upstream from the bow shock.
1.3.1
Shock geometry and strength
An important factor influencing the shock behavior is the shock geometry, i.e., direction of
the upstream magnetic field. It is measured using an angle θBn between the field and the
shock normal. Accordingly, θBn = 0◦ gives parallel shock, and θBn = 90◦ perpendicular.
Oblique shock is something in between the latter cases and can be divided into quasiparallel θBn < 45◦ and quasi-perpendicular θBn > 45◦ . Shock strength is related to the
amount of energy processed by the shock and is measured with the Mach number M .
Bow shock is an example of the high-Mach shocks (Alfven Mach number, MA ∼ 1.5 − 10),
while interplanetary shocks are of the low-Mach type.
1.3.2
Shock types
Shocks are transition layers across which the particles are transported. There are three
types of shocks in MHD:
• Slow shocks - Plasma pressure increases, magnetic field strength decreases, magnetic
field bends towards normal
• Intermediate shocks - Magnetic field rotation of 180◦ in plane of shock, density jump
only in anisotropic plasma
• Fast shocks - Plasma pressure and field strength increase at shock, magnetic field
bends away from normal
10
Fast shocks are the most typical in solar system plasmas.
The type of shocks depends on relative magnitude of the upstream velocity in the
frame moving with the shock with respect to some characteristic speed (Mach number).
Those characteristic speeds, slow and fast magnetosonic speeds, are related to the Alfvén
speed, vA and the sonic speed, cS as follows:
·
¸
q
1
2
2
2
2
2 2
2 2
(cS + vA ) − (cS + vA ) − 4cS vAn
vslow =
2
¸
·
q
1
2
2
2
2 2
2 2
2
vfast =
(cS + vA ) + (cS + vA ) − 4cS vAn
2
where vAn is the normal component of the Alfvén speed. The normal component of the
slow shock propagates with velocity vslow in the frame moving with the upstream plasma,
that of the intermediate shock with velocity vAn and that of the fast shock with velocity
vfast .
1.3.3
Rankine-Hugoniot jump conditions for MHD
The jump conditions across an MHD shock or a discontinuity are referred to as the
Rankine-Hugoniot equations for MHD. In the frame moving with the shock/discontinuity,
those jump conditions can be written as
ρvn = const
Bn = const
ρvn2 + p +
Bt2
= const
2µ0
~ t Bn
B
= const
µ0
µ
¶
~ t · ~vt )
γ p v2
vn Bt2 Bn (B
+
−
= const
ρvn +
γ −1ρ
2
µ0
µ0
ρvn~vt −
~ t = const
(~v × B)
~ are the plasma density, velocity, (thermal) pressure and magnetic field
where ρ, ~v , p, B
respectively. The subscripts n and t refer to the normal and tangential components of a
vector (with respect to the shock/discontinuity front).
11
1.4
Scientific Background and Motivation
A very prominent feature at the Earth’s bow shock is the presence of backstreaming
accelerated ions. Different ion distributions are seen depending on the angle θBn between
the interplanetary magnetic field and the shock normal. In the quasi-parallel region where
θBn < 45◦ , a more or less isotropic (diffuse) ion distribution is found, and in the quasiperpendicular region for θBn > 45◦ , a collimated ion beam emerges, and reflected gyrating
ions are seen. Although significant progress has been made in understanding of the ions
behavior in the foreshock region (through numerical simulations and observations), the
underlying production mechanisms are still being debated. Pickup ions constitute an
upstream ion population with high effective temperature, so that some ions may have very
low velocities as they enter the shock front. These ions are trapped by cross-shock electric
field and magnetic field in the shock transition layer (ramp) and surf along the shock
front, thus acquiring substantial energies. Some of these ions have favorable velocities
to escape into upstream, after becoming de-trapped. Although observational evidence of
pickup He has been recently found [Oka et al., 2002], the fluxes of pickup ions at the
Earth bow shock are too low to be responsible for the observed escaping ion densities.
Moreover, the observed escaping field-aligned beams are exclusively made of protons.
Typical upstream β = 8πnT /B 2 for solar wind protons is not especially high, so that the
above mechanism is unlikely. Yet, observations reveal presence of escaping ions in the
foreshock of the terrestrial bow shock [Gosling et al., 1978; Paschmann et al., 1980, 1981;
Ipavich et al., 1981; Thomsen et al., 1983; Gurgiolo et al., 1981, 1983; Meziane et al.,
1997, 2001, 2004; Möbius et al., 2001]. Several distinct distributions of escaping ions are
attributed to different shock geometries. The diffuse distributions are observed upstream
of a quasi-parallel shock [Gosling et al., 1978; Paschmann et al., 1980; Ipavich et al.,
1981]. Numerical simulations of quasi-parallel shocks [Winske and Leroy, 1984; Lee, 1984]
result in similar distributions. The field-aligned beams are thought to be produced by
quasi-perpendicular part of the shock which is also usually thought to be the main source
of gyrating distributions.
Observations show that field-aligned beams and gyrating ions appear in the fore12
shock. The mechanism of the generation of the escaping ion distribution is not quite
understood yet. The early proposed models of adiabatic ion reflection [Sonnerup, 1969;
Paschmann et al., 1980] are not supported by observations [Meziane et al., 2005, 2007].
The leakage from magnetosheath does not appear to be the main source for reflected
ions upstream the shock (Edmiston et al. 1982, Tanaka et al. 1983, Kucharek et al.
2004). Non-specular reflection, where the reflected ions gyrate at the shock front and escape upstream, due to the dc fields in the transition layer, has been proposed as a viable
mechanism for backstreaming ion generation. Leroy and Winske [1983] performed hybrid
simulations of oblique shocks with Alfvenic Mach number M = 6 and β = 0.5 and found
no backstreaming ions for the angle between the shock normal and upstream magnetic
field θ > 45◦ , which made them to conclude that non-specular reflection is not efficient
for quasi-perpendicular shocks. Burgess and Schwartz [1984] analyzed ion trajectories
in a planar discontinuity model of a quasi-perpendicular shock and showed that some
ions escape after having a multiple encounter with the shock front. Hybrid simulations
by Burgess [1987] have shown that backstreaming ions are generated for θ > 45◦ also
provided β is sufficiently high. Despite the physical transparency of the proposed mechanism, other processes have been in the center of attention during the last decades. At
the same time, observations revealed escaping ions at shocks with M ≈ 3 and β ≈ 0.1.
Recently, Kucharek et al. [2004] presented convincing arguments in favor of the escaping
ion generation at the shock ramp, within the same process which is responsible for ion
reflection and downstream gyration. Oka et al. [2005] have succeeded in modelling an
observed shock and partial reproduction of the upstream ion population, on the basis of
tracing ion motion in a stationary shock profile. Yet, the details of the process are not
completely clear, as well as the relation of the distribution to the shock structure and
possible non-stationarity. Möbius et al. [2001] interpreted a part of these distributions as
specularly reflected ions which had been pitch-angle scattered after reflection. Meziane
et al. [2004] found that the field-aligned and gyrating beams are spatially separated: the
guiding centers of the gyrating ions are closer to the shock. Fuselier et al. [1986] have
shown that gyrating beams are associated with ultra-low frequency magnetic oscillations
[see also Meziane et al., 1997, 2001, 2005, for recent detailed observations]. Production of
13
large pitch-angle (up to 60◦ ) gyrophase bunched distributions are explained by nonlinear
wave-particle interaction [Mazelle et al., 2000, 2003; Mazelle et al., 2007]. Recent hybrid
simulations [Blanco-Cano et al., 2006] have been able to produce both populations of gyrophase bunched ions (one at the bow shock and the other type farther in the foreshock).
Other recent Cluster observations [Möbius et al., 2001; Kucharek et al., 2004] brought
to the conclusion that the escaping ions are produced at the ramp due to effective pitch
angle scattering. The latter is usually attributed to the wave-particle interaction. However, substantial production of the foreshock ion beams is observed at low Mach number
(M ∼ 3) shocks which may be almost laminar without noticeable waves in the ramp vicinity [Gedalin et al., 2000]. Moreover, ions spend too little time in the quasi-perpendicular
ramp vicinity to be critically affected by waves there (it is different from quasi-parallel
shocks with their extended transition). It is, therefore, natural to investigate further the
possibility of the ”pitch angle scattering” by the stationary fields [Burgess and Schwartz,
1984; Burgess, 1987].
1.5
Thesis Outline
In this work we study ion distributions which are produced by non-specular reflection
in a stationary one-dimensional shock front. Our approach is based on the test particle
analysis in a model shock front. At the first sight, this approach is less efficient than
selfconsistent numerical simulations [Burgess, 1987] or even using measured magnetic
field for the shock model [Oka et al., 2005]. However, the advantage of our approach is
that a) we separate the effects related to stationary fields from those due to any nonstationarity, and b) varying shock parameters we can identify those parameters which
control the efficiency of the reflection and the shape of the escaping ion distributions.
The objectives of the present thesis are not to provide numerical of visual data which
could be immediately compared with observations but to study physics of the formation
of the escaping ion distributions, due to the ion motion in the stationary fields of a shock
front, to understand limitations of the proposed mechanism, to thoroughly investigate the
interaction of upcoming solar wind ions with bow shock, to understand the mechanisms
14
of reflection, to identify the fraction of escaping ions from the initial velocity distribution,
and to reveal the plasma parameters responsible for generation of escaping ions. This
study is carried out in following steps:
Chapter 2 covers the analytical approach to the problem. The magnetic field profile
is approximated by a discontinuity between upstream and downstream regions. The
expected electric field along the shock normal is considered by electric potential. The
motion of particles governed by Newton-Lorentz equations is considered separately on
both sides and matched at the shock front concerning the potential barrier. The conditions
for generation of escaping ions are deduced.
Chapter 3 shows the numerical simulation of the problem. The results are compared
with ones estimated analytically. The magnetic field model profiles are used. Two qualitative examples of low-Mach (M = 3) and high-Mach (M = 7) shocks are considered.
Several mechanisms responsible for the escaping process are shown. The dependence of
ions escaping efficiency on plasma parameters (the ratio of the ion thermal velocity to the
flow velocity upstream, the cross-shock potential and the angle between the shock normal
and the upstream magnetic field) is discussed.
Finally, Chapter 4 presents the conclusions produced from the analysis and suggestions for future work on the subject.
15
Chapter 2
Analytical approach
2.1
Ion motion in the shock front
In this section we provide a qualitative description of the ion motion in the shock front.
Assuming the shock normal to be along x-axis, we have the upstream magnetic field
~ u = Bu (cos θBn ; 0; sin θBn ) (hereinafter refer to θ = θBn ). Inside the shock transition
B
layer Bx = Bu cos θ = const, while By and Bz depend on x. Generally, the non-coplanar
component By is typically small and nonzero only inside the transition itself and vanishes
when Bz achieves its downstream asymptotic value, Bd = Bu (cos θ, 0, R sin θ), where R
is downstream to upstream magnetic field ratio. For a simplistic approach, in present
chapter, the By component is neglected. A potential component of the electric field
Ex = −dφE /dx appears inside the transition, while Ey = const and Ez = 0. The overall
potential drop φE is a sizable fraction of the incident ion energy mi Vu2 /2. The ion motion
in the quasi-perpendicular shock front has been thoroughly analyzed by Gedalin [1996] for
the low-temperature core, and by Lee et al. [1996], Zilbersher and Gedalin [1997] for the
hot tail of the incident ion distribution. For the qualitative description of the ion motion
it is essential that the transition layer (ramp in what follows) is substantially thinner than
the convective ion gyroradius and that the cross-shock electric field is high, while ahead
of the ramp and behind it the typical scale of inhomogeneity is of the order or exceeds
the gyroradius and the cross-shock electric field is negligible. In these conditions the ion
motion inside the ramp is governed by the cross-shock electric field while outside the ramp
16
magnetic gyration plays the main role.
We assume the cross-shock potential (the potential drop at the transition layer) to
be φE = s(mi Vu2 /2)/e, where s < 1. An ion which enters the shock with the velocity
~v = (vx , vy , vz ) is only weakly deflected by the magnetic field but strongly decelerated by
Ex . In a good approximation, if mi vx2 /2 > eφE , the ion will cross the ramp and leave
p
with the velocity w
~ d = ( vx2 − 2eφE /mi , vy , vz ) into the downstream. Unless this velocity
coincides with the downstream drift velocity of the plasma flow, the ion starts to gyrate.
In case of strong gyration an ion can return back to the ramp and cross it in reverse
direction. The chances for such reflection should increase with the decrease of the crossshock potential and increase of the magnetic compression R. It might be that a reflected
ion appears again in the upstream region with the velocity which allows it to escape the
shock front. An ion which has a low velocity at the entry to the front, mi vx2 /2 < eφE ,
would be nearly specularly reflected and cannot cross the ramp. The specularly reflected
ion leaves the ramp into upstream with the velocity w
~ u = (−vx , vy , vz ). The motion of
such ion is a drift due to the convective electric field, motion along the magnetic field, and
gyration. A gyrating-reflected ion or a specularly reflected ion can return to the shock
front and follow once again a trajectory of the kind described above. Eventually, some
ions may, e.g., be reflected specularly, cross the ramp, gyrate behind the ramp, cross
it again in the reverse direction, and escape into the foreshock provided that the final
velocity is favorable for the last step.
2.2
Reflection mechanisms
Following the above description, the ion escaping process can consist of infinite number
of complicated mechanisms of specular reflections and transitions across the shock front.
However, only three fundamental mechanisms can be highlighted:
1. The ion transports through the shock from upstream to downstream
2. The ion gyrates in downstream and transits through the shock in reverse direction
into upstream
17
3. The ion is specularly reflected off the shock in upstream, due to mi vx2 /2 < eφE
Any of the expected ion motions near the shock front can be easily composed using the
above elementary mechanisms in various combinations. Since the mechanisms of particle/shock front interaction are one of our foremost objectives and will be widely discussed
throughout the present paper, we shall introduce here some additional abbreviations in
order to simplify the issue of naming the mechanisms as follows: G - the ion transports
through the shock and gyrates back into the upstream, R - the ion is reflected off the
cross-shock potential, T - the ion is transmitted into the downstream and E - the ion
escapes. As example, the following mechanism - ”ion approaches the shock front, reflects
back to the upstream, gyrates in upstream, transports into the downstream, gyrates in
downstream, transports back into the upstream and finally escapes” we shall call RGEmechanism.
2.3
Equations of motion
Due to a complicated form of the magnetic field profile across the shock, the analytical
study of these processes seems to be hardly possible, therefore we consider a more simple
case of the magnetic field, the discontinuity model of the shock, i.e the step-like jump of
the magnetic field Bz component at x = 0. Due to the singularity at x = 0, in return for
electric field x-component, which following Gedalin [1996] is Ex ∝ dBz /dx, we consider
a potential drop φE at the shock front transition layer. First, we shall solve the ion
motion equations for both upstream and downstream regions separately and then derive
analytically the reflection conditions which depend on the cross-shock potential φE , the
magnetic compression ratio R and the angle between the upstream magnetic field and
the shock normal θ. As well known, the ion motion is described by the Newton-Lorentz
equations in the following form (here Ez and By are zero):
e
mi v̇x = vy Bz
c
(2.1)
e
mi v̇y = eEy + (vz Bx − vx Bz )
c
(2.2)
18
e
mi v̇z = − vy Bx
c
(2.3)
By normalizing the variables as follows:
~
B
= ~b,
Bu
~v
= ~u,
Vu
Ωu t = τ
where B~u = Bu (cos θ, 0, sin θ) is the upstream (x < 0) magnetic field, V~u = Vu (1, 0, 0)
is the upstream plasma velocity (solar wind) and Ωu = eBu /mi c is the upstream ion
cyclotron frequency. One should note that Ex component is omitted since Ex → ∞ at
x = 0, however it will be considered when the equations of upstream and downstream are
matched, by using the potential drop φE instead. Hence, the equations (2.1)-(2.3) can be
rewritten in dimensionless form as follows:
u̇x = uy bz
(2.4)
u̇y = ey + (uz bx − ux bz )
(2.5)
u̇z = −uy bx
(2.6)
where ey = cEy /Vu Bu = sin θ and the derivation is with respect to τ .
The detailed steps of solving the equations (2.4)-(2.6) are demonstrated in Appendix,
wherefrom the solutions are:
1
[ey bz + ux0 b2x + uz0 bx bz −
2
+ bz
p
− (bz ey − ux0 b2z + uz0 bx bz ) cos( b2x + b2z τ ) +
p
p
+ uy0 bz b2x + b2z sin( b2x + b2z τ )]
(2.7)
p
1
2
2
)
cos(
+
b
[u
(b
b2x + b2z τ ) +
y0
z
x
b2x + b2z
p
p
+
b2x + b2z (ey + uz0 bx − ux0 bz ) sin( b2x + b2z τ )]
(2.8)
1
[−ey bx + ux0 bx bz + uz0 b2z +
2
+ bz
p
+ (bx ey − ux0 bx bz + uz0 b2x ) cos( b2x + b2z τ ) −
p
p
− uy0 bx b2x + b2z sin( b2x + b2z τ )]
(2.9)
ux =
b2x
uy =
uz =
b2x
19
Integrating ux gives
1
[uy0 bz + (ey bz + ux0 b2x + uz0 bx bz )τ −
2
+ bz
p
1
− p
(bz ey − ux0 b2z + uz0 bx bz ) sin( b2x + b2z τ ) −
b2x + b2z
p
− uy0 bz cos( b2x + b2z τ )]
x = x0 +
b2x
(2.10)
The obtained expression for x(t) coordinate is responsible for location of the ion along the
shock normal. Locating the shock front at x = 0 makes the upstream and the downstream
regions to be at x < 0 and x > 0 respectively. Equations (2.7)-(2.10) are the common set
of solutions for Eqs.(2.4)-(2.6). Since the magnetic fields vary across the discontinuity of
the shock front, equations (2.7)-(2.10) can be rewritten for each of the regions separately
as shown below.
2.3.1
Upstream equations
In the upstream region the magnetic field is ~bu = (cos θ, 0, sin θ) and the electric field is
ey = sin θ, therefore equations (2.7)-(2.10) take the form:
ux = (ux0 cos2 θ + uz0 sin θ cos θ + sin2 θ) + (uy0 sin θ) sin τ +
+ (ux0 sin2 θ − uz0 sin θ cos θ − sin2 θ) cos τ (2.11)
uy = (−ux0 sin θ + uz0 cos θ + sin θ) sin τ + (uy0 ) cos τ
(2.12)
uz = (ux0 sin θ cos θ + uz0 sin2 θ − sin θ cos θ) − (uy0 cos θ) sin τ
+ (−ux0 sin θ cos θ + uz0 cos2 θ + sin θ cos θ) cos τ
(2.13)
x = x0 + uy0 sin θ + (ux0 cos2 θ + uz0 sin θ cos θ + sin2 θ)τ + (ux0 sin2 θ −
− uz0 sin θ cos θ − sin2 θ) sin τ − (uy0 sin θ) cos τ
which can be rewritten as
ux = Vux + Su V⊥u cos (τ − αu )
20
(2.14)
V⊥u
uy = −Su
sin (τ − αu )
sin θ
uz = Vuz − Su V⊥u cot θ cos (τ − αu )
x = xu + Vux τ + Su V⊥u sin (τ − αu )
where
Vux = ux0 cos2 θ + uz0 sin θ cos θ + sin2 θ
Vuz = ux0 sin θ cos θ + uz0 sin2 θ − sin θ cos θ
q
V⊥u =
sin2 θ(u2y0 + (uz0 cos θ + sin θ − ux0 sin θ)2 )
xu = x0 + uy0 sin θ
αu = arctan (
uy0
)
ux0 sin θ − uz0 cos θ − sin θ

 −1, if sign(u sin θ) = −sign(sin α ),
y0
u
Su =
 1,
if sign(uy0 sin θ) = sign(sin αu ).
2.3.2
Downstream equations
In the downstream region the magnetic field is ~bd = (cos θ, 0, R sin θ) and the electric field
is ey = sin θ, therefore equations (2.7)-(2.10) take the following form:
ux1 cos2 θ + Ruz1 sin θ cos θ + R sin2 θ
+
cos2 θ + R2 sin2 θ
p
Ruy1 sin θ
√
+
sin ( cos2 θ + R2 sin2 θτ ) +
2
2
2
cos θ + R sin θ
2
p
R ux1 sin2 θ − Ruz1 sin θ cos θ − R sin2 θ
cos
(
cos2 θ + R2 sin2 θτ )
+
cos2 θ + R2 sin2 θ
ux =
uy =
p
−Rux1 sin θ + uz1 cos θ + sin θ
√
sin
(
cos2 θ + R2 sin2 θτ ) +
2
2
2
cos θ + R sin θ
p
+uy1 cos (
21
cos2 θ + R2 sin2 θτ )
(2.15)
(2.16)
Rux1 sin θ cos θ + R2 uz1 sin2 θ − sin θ cos θ
−
cos2 θ + R2 sin2 θ
p
uy1 cos θ
− √
sin
(
cos2 θ + R2 sin2 θτ ) +
2
2
2
cos θ + R sin θ
p
−Rux1 sin θ cos θ + uz1 cos2 θ + sin θ cos θ
√
+
cos2 θ + R2 sin2 θτ )
cos
(
2
2
2
cos θ + R sin θ
uz =
(2.17)
Ruy1 sin θ
ux1 cos2 θ + Ruz1 sin θ cos θ + R sin2 θ
x = x0 +
+
τ+
cos2 θ + R2 sin2 θ
cos2 θ + R2 sin2 θ
p
R2 ux1 sin2 θ − Ruz1 sin θ cos θ − R sin2 θ
cos2 θ + R2 sin2 θτ ) −
+
sin
(
2
2
2
3/2
(cos θ + R sin θ)
p
Ruy1 sin θ
cos2 θ + R2 sin2 θτ )
(2.18)
−
cos
(
2
2
2
cos θ + R sin θ
Here we introduce some additional notations and rewrite equations (2.15)-(2.18) as
ux = Vdx + Sd V⊥d cos (ωτ − αd )
r
cot2 θ d
V sin (ωτ − αd )
uy = −Sd 1 +
R2 ⊥
uz = Vdz − Sd
V⊥d cot θ
cos (ωτ − αd )
R
x = xd + Vdx τ + Sd
V⊥d
sin (ωτ − αd )
ω
where
Vdx
ux0 cos2 θ + Ruz0 sin θ cos θ + R sin2 θ
=
cos2 θ + R2 sin2 θ
Rux1 sin θ cos θ + R2 uz0 sin2 θ − sin θ cos θ
cos2 θ + R2 sin2 θ
s
R2 sin2 θ((uz0 cos θ + sin θ − Rux0 sin θ)2 + u2y0 (cos2 θ + R2 sin2 θ))
d
V⊥ =
(cos2 θ + R2 sin2 θ)2
p
ω = cos2 θ + R2 sin2 θ
Vdz =
Ruy1 sin θ
θ + R2 sin2 θ
√
uy1 cos2 θ + R2 sin2 θ
)
αd = arctan (
Rux0 sin θ − uz0 cos θ − sin θ

 −1, if sign(u sin θ) = −sign(sin α ),
y0
d
Sd =
 1,
if sign(u sin θ) = sign(sin α ).
xd = x0 +
cos2
y0
d
22
Figure 2.1: Ion upstream position function. The presence of downstream conditions is
temporarily omitted.
2.4
Reflection conditions
Following section 2.2, the propagation of ion in the transition layer can be considered as
a consequent combination of fundamental mechanisms. The number of such steps can be
very large therefore we shall focus on the most elementary ones. Nevertheless, the more
complicated cases would be easily combined from ones we analyze here. We shall define
τ = 0 at each of the following steps with corresponding initial velocities deduced from the
previous. The ions with initial velocity ~u0 are launched from the position x = −1(Vu /Ωu ).
Assuming that the arriving ion (~u1 ) has not enough energy to cross the potential barrier it
is reflected off the shock front with ~u2 = (−ux1 , uy1 , uz1 ). Then in the next step (upstream
gyration) the position expression can be written as
xu (τ ) = uy2 sin θ + Vux τ + Su V⊥u sin (τ − αu )
Analyzing this expression one already finds a mandatory requirement for escaping, i.e. a
condition for ion to stay in the upstream region. The position expression consists of three
terms: constant, linear and waveform (sine). The obligatory condition that ion is moving
in negative-x direction yields that its trajectory is of the form (or similar) as shown in
figure 2.1. Obviously, for ion to stay in the upstream one should require Vux < 0 and the
23
Figure 2.2: Ion position function
first sine maximum at (τmax = αu + 3π/2) to be negative, or
uy2 sin(θ) + Vux (αu + 3π/2) − Su V⊥u < 0
One should notice that this is the only common requirement for ions to escape when
entering the upstream region. However, if the above conditions are not fulfilled the ion is
obliged to return to the shock front, where it is again subjected to the cross-shock potential
barrier. In case of reiterated energy lack, the ion is again reflected into the upstream and
the above described steps are repeated. However, the ion can arrive to the shock with
enough energy to overcome the potential drop, i.e. u2x4 > s. Then the ion is transmitted
p
into the downstream, where its initial velocities become ~u3 = ( u2x4 − s, uy4 , uz4 ) or
p
~u3 = ( u2x1 − s, uy1 , uz1 ), if the ion had enough energy at the first arrival. In addition,
the condition for maximum velocity when entering the shock can be deduced from the
position function xu (τ ), which requires the ion to arrive to the shock in appropriate time.
As shown on figure 2.2, the steepest slope will be obtained at time τ = αu + π, therefore
if the ion during this time is positioned in the vicinity of the shock front, i.e. xu (τ ) → 0
its x velocity component is the largest and it has more chances to overcome the potential
barrier. The corresponding condition can be written as
uy2 sin θ + Vux (αu + π) → 0
In downstream, the position function xd (τ ) is modified by taking into account the magnetic
field compression ratio R (see Eq. 2.18). Again the position function xd (τ ) is useful in
24
analyzing the motion in the downstream. The first and decent condition for ion to arrive
back to the shock front is Vdx < 0, however this requires uz3 < −(tan2 θ + ux3 /R),
which demands large and nearly impossible negative velocities in z-direction for quasiperpendicular shocks. Therefore one should consider the Vdx > 0 case, but require the
first minimum of the sine term (at τmin = (αd + 3π/2)/ω) to be negative. Then, the
condition can be written as
Ruy3 sin θ
αd + 3π/2
V⊥d
+
V
−
S
<0
dx
d
ω2
ω
ω
When ion gyrates back to the shock with velocity ~u4† , it crosses the potential barrier in reverse direction, i.e. from downstream to upstream, and accelerates through the transition
q
layer into the upstream, where the velocity becomes ~u5 = (− u2x4† + s, uy4† , uz4† ).
To summarize table 2.1 illustrates the sufficient conditions for solar wind upcoming
ions to be reflected from or transmitted through the shock transition layer. Combining a
Name
Conditions
Upstream Reflection (R)
u2x1 < s
Upstream Escape (E)
Vux < 0 and uy1 sin(θ) + Vux (αu + 3π/2) − Su V⊥u < 0
Downstream Gyration (G) Ruy0 sin θ/ω 2 + Vdx (αd + 3π/2)/ω − Sd V⊥d /ω < 0
Table 2.1: Transition/Reflection conditions.
necessary number of these steps in any possible sequence one obtains the trajectory of ion
across the shock front and conditions for it to escape into the upstream. The consequent
chain of such conditions is shown on diagram 2.3. Based on this diagram/algorithm a
corresponding code was constructed, where the above conditions defined whether the ion
escapes into the upstream or transmits through the shock into the downstream. Initially,
NT OT = 105 ions were positioned at x = 0 with uniformly distributed velocities ux1 =
[0, 2], uy1 = [−1, 1] and uz1 = [−1, 1]. Then, each ion was traced along its trajectory
governed by equations (2.14) and (2.18) in following steps:
25
Figure 2.3: Transition/Reflection conditions diagram.
1. The ions were checked for ability to cross the potential barrier, i.e. the ”Upstream
Reflection” condition. Here, the velocity distribution was split into 2 fractions, i.e.
~u2 - reflected ions and ~u3 transmitted ions.
2. Then, the reflected fraction was checked for ”Upstream Escape” condition. If it
was fulfilled the ions were considered as escaped ones. If not, they moved in the
upstream, corresponding to (2.14), until their arrival back to the shock front with a
new set of velocities ~u4 . These velocities were again checked for ”Upstream Escape”
condition.
3. As well, the fraction of initially transmitted ions, ~u3 , was checked for ”Downstream
Gyration” condition. If one was fulfilled the ions moved in downstream, corresponding to (2.18), until their arrival back to the shock front with a new set of velocities
~u5 . These velocities were again checked for ”Upstream Escape” condition.
At the beginning of each step the initial velocity distributions are defined from the step
before, as shown schematically in Fig. 2.4. Indeed, after such finite combination of steps, a
fraction of ions still remains in the vicinity of the shock, and yet not identified as escaping
or transmitting ones. However, as will be shown, the major part of the ions can be well
defined during these steps and the above presented conditions become a useful instrument
in determination of ions population.
26
Figure 2.4: Schematics of ion trajectory in steps and corresponding velocities.
2.5
Analytical results
The described in the above section steps were carried out varying shock parameters, R,
s and θ. As a qualitative example, we choose a set of parameters s = 0.7, R = 3 and
θ = 50◦ . Since low-Mach shocks are almost laminar without noticeable waves in the ramp
vicinity, this working set of parameters reminds a shock wave of M ∼ 3.
(a)
(b)
(c)
Figure 2.5: Initial uniform velocity distribution of escaping ions (s = 0.7, R = 3 and
θ = 50◦ ) initially located at x = −1, completing the: RE-mechanism (blue), RGEmechanism (red) and RGGE-mechanism (black). (a) x-y components, (b) x-z components
and (c) y-z components. Circles illustrate thermal velocities of Maxwellian distribution.
The results show that under these conditions, a number of mechanisms through
which ions escape is well finite. In particular, tracing RE, RGE and RGGE mechanisms
was enough to catch nearly all the ions that have escaped into the upstream.
27
Since in further numerical analysis we shall use the more complicated magnetic field
profile, consisting of foot, ramp and overshoot with corresponding finite lengths along x
at the shock transition layer, the ions will be initially located away from the front into the
upstream, i.e. x = −1. Therefore, Fig. 2.5 illustrates the mentioned mechanisms by ions
initial uniform velocity distributions, at x = −1. As one can see the mechanisms are well
separated in the velocity space. Indeed, the ability of ion to escape entirely depends on its
initial velocity. Ions that have larger thermal velocities are capable to escape through REmechanism, which is a specular reflection. However, ions with smaller vT rather escape
through RGE and RGGE. Therefore, in order to estimate the real fraction of escaping
ions one has to trace the particles with Maxwellian initial distribution, i.e.
µ
¶
N
(~u − ~vu )2
f (~v |v~u , vT ) =
exp −
(2πvT2 )3/2
2vT2
where
~vu = (1, 0, 0),
r
vT =
β
2M 2
2µ0 nT
B2
and β =
Since the realistic solar wind is initialized Maxwellian, only a part of these ions will be in14
12
10
vT=0.25
8
6
vT=0.14
4
2
vT=0.1
0
0
2
4
6
8
10
M
Figure 2.6: Relation of plasma β and Mach number M for different thermal velocities vT
volved in the escaping process. The circles correspond to thermal velocities (or Maxwellian
distribution deviation) and show what mechanisms do take place in the escaping. As one
can see on Fig. 2.5 larger vT involves larger fractions of ions. Consequently, Fig. 2.5 and
therein data can be used as a rough estimate of the quantity of escaping ions and the
corresponding mechanisms of reflection, for a given solar wind thermal velocity or plasma
28
parameters β and M , since vT =
p
β/2M 2 . Fig. 2.6 shows the coupling of β and M for
a corresponding vT . As a result, using the mentioned set of parameters, and introducing
β ∼ 0.35 and M = 3, we have estimated that 0.1NT OT ions escaped upstream.
In addition, Fig.2.7 shows the energy spectrum of ions that were specularly reflected
by the cross-shock potential on their primary encounter. The energy of ions that had not
Figure 2.7: Energy spectrum of ions gyrating in upstream. Initial (red) and after second
arrival to the shock (black)
enough energy to overcome the potential barrier were reflected and forced to gyrate in
the upstream. Their initial energy upon the encounter is shown by red. The same ions
after the gyration on their second arrival to the shock is shown by black. As one can
see, the newly arrived ions (after their gyration in the upstream) consist of ones that can
now successively transmit through the shock, i.e. u2x > s, while the others stay in the
upstream.
29
Chapter 3
Numerical Analysis
3.1
Simulation Setup
In what follows we study the ion motion in a model shock profile, following the principles
of Gedalin [1996]. The magnetic fields are chosen in the form:
(3.1)
bx = cos θ
by = k1
dbz
dx
(3.2)
·
bz
µ
µ
¶¶
x + Df − 3Dr
= sin θ 1 + 0.5(Rf − 1) 1 + tanh 3
+
Df
µ
µ ¶¶
µ
¶
3x
2(x − Do )2
+ 0.5(Rr − Rf ) 1 + tanh
+ (Ro − Rr ) exp −
+
Dr
Do2
¶¶¸
µ
µ
3(x − Do − Dd )
+ 0.5(Rd − Rr ) 1 + tanh
Dd
(3.3)
where the coefficient k1 = 0.001 is chosen so that the non-coplanar component of the
magnetic field by remains small throughout the shock and bz ≈ 3by at x = 0. Much of
the earlier observational and theoretical studies of collisionless shocks was focused on the
physics of quasi-perpendicular shocks. Based on these studies, it was found that at high
Mach numbers the magnetic structure of the shock consists of a foot (region of slight
enhancement in the upstream field), ramp (region of sharpest field gradient), and a series
of overshoots and undershoots. Whereas, at low Mach numbers the shock structure is a
30
sharp discontinuity between the up- and down-stream regions. We shall focus in present
section on two representatives of low and high Mach shocks, in particular M = 3 and
M = 7 respectively.
As an example of the corresponding magnetic fields we used data from the fluxgate
magnetometer (FGM) on board of the CLUSTER II spacecraft. A detailed description of
the instruments may be found in Réme et al. (1997) and Möbius et al. (1998). During the
operational phase of the CLUSTER mission, the spacecraft encountered a huge number
of bow shock crossings under a variety of different plasma conditions. Figure 3.1 shows a
shock crossings at 24 January and 31 March 2001 correspondingly, which is also discussed
in a paper by Möbius et al. (2001) and Kucharek et al. (2004) among the early results of
the CLUSTER mission. The top panel refers to high-Mach M = 7 and the bottom panel
to low-Mach M = 3 shock.
Figure 3.1: Cluster spacecraft measurements of magnetic fields. Top panel - M = 7,
bottom - M = 3 shock.
Based on these data and equations (3.1-3.3), the examples of resulting magnetic
field profiles used in following simulation are demonstrated on Fig. 3.2 for (a) M = 3 low Mach and (b) M = 7 - high Mach number shock.
The electric fields are shown on Fig. 3.3 and chosen in the form:
ex = −k2
dbz
dx
31
(a)
(b)
Figure 3.2: Magnetic field profiles for (a) low Mach number shock (M = 3) and (b) high
Mach number shock (M = 7).
ey = sin θ
ez = 0
where k2 is chosen from the requirement
Z ∞
−
ex dx = 0.5s
⇒
k2 =
−∞
s
2(Ro − 1) sin θ
The parameters used in simulation were [Rf = 1, Rr = 3, Ro = 3, Rd = 3, Df = 0.01, Dr =
0.01, Do = 0.01, Dd = 0.15], for Mach number M = 3 and [Rf = 1.5, Rr = 5, Ro = 6, Rd =
3, Df = 0.3, Dr = 0.01, Do = 0.1, Dd = 0.15], for Mach number M = 7.
Ions starting from x = −1 are traced during their motion throughout the shock
front. The ions trajectory is obtained by solving the ordinary differential equation with
initial conditions (velocities) using the Runge-Kutta explicit 4th order iterative method.
Every particle of the total 105 was traced individually.
3.2
3.2.1
Simulation Results
Ion Trajectories
Figures 3.4-3.8 show the trajectories of only escaping ions in x-z plane. The escaping
ions are not reflected specularly (due to the potential only), nor adiabatically (magnetic
32
5
5
0
0
−5
−5
−10
−10
ex
ex
−15
−20
−15
−25
−20
−30
−25
−35
−40
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
−30
−0.4
0.4
−0.3
−0.2
−0.1
x
0
0.1
0.2
0.3
0.4
x
(a)
(b)
Figure 3.3: Electric field ex for (a) low Mach number shock (M = 3) and (b) high Mach
number shock (M = 7).
bottle), but due to the non-adiabatic ion motion in the shock front and multiple crossings.
It is seen from figures 3.4 and 3.5 that in case of a low-Mach shock (M = 3) the ions
are first reflected by the cross-shock potential, then gyrate once in the transition layer.
Afterwards some of them (71%) escape while others (29%) with high gyration velocities
return to the ramp and finally escape when reflected by the cross-shock potential again.
To summarize, in case of M = 3 shock, ions (0.1NT OT ) escape through 2 mechanisms:
RGE - reflect from the cross-shock potential → gyrate → escape and RGRE - reflect from
the cross-shock potential → gyrate → reflect again → escape.
In case of a high-Mach shock (M = 7) all the ions are first reflected by the cross-shock
potential while still inside the overshoot region, without completely crossing it. These ions
further gyrate in the transition layer. Hereinafter, there are 3 possible scenarios of ions
further motion. Some (86%) make one more similar gyration in downstream and escape
(see Fig. 3.6b), others (12%) make a similar gyration but having high gyration velocities,
return to the front, reflect from the cross-shock potential again (see Fig. 3.7c) and escape,
the residual fraction (2%) is twice reflected from the cross-shock potential and escapes
into the upstream as well (see Fig. 3.8c). To summarize, in case of M = 7 shock, ions
(0.04NT OT ) escape through 3 mechanisms: RGGE - reflect from the cross-shock potential
→ gyrate → gyrate → escape, RGGRE - reflect from the cross-shock potential → gyrate
33
Figure 3.4: Trajectories of 50 escaping ions in x-z plane propagating through low-Mach
shock (M = 3). (a) Trajectories of ions escaped through RGE-mechanism. (b) Zoomed
trajectories near the transition layer. (c) Zoomed trajectories of the cross-shock potential
reflection process.
Figure 3.5: Trajectories of 50 escaping ions in x-z plane propagating through low-Mach
shock (M = 3). (a) Trajectories of ions escaped through RGRE-mechanism. (b) Zoomed
trajectories near the transition layer. The second reflection is observed. (c) Zoomed
trajectories of the first cross-shock potential reflection process.
→ gyrate → reflect again → escape and RGRRE - reflect from the cross-shock potential
→ gyrate → double reflect → escape.
Only those ions eventually escape which are first reflected by the cross-shock potential while still inside the ramp or/and overshoot, without completely crossing it. Some
of them escape while having large gyration velocities, thus constituting a gyrating beam.
Others have substantially lower gyration velocities.
34
Figure 3.6: Trajectories of 50 escaping ions in x-z plane propagating through high-Mach
shock (M = 7). (a) Trajectories of ions escaped through RGGE-mechanism. (b) Zoomed
trajectories near the transition layer. (c) Zoomed trajectories of the first cross-shock
potential reflection process.
Figure 3.7: Trajectories of 50 escaping ions in x-z plane propagating through high-Mach
shock (M = 7). (a) Zoomed trajectories of the first cross-shock potential reflection
process. (b) Trajectories of ions escaped through RGGRE-mechanism. (c) Zoomed trajectories of the second cross-shock potential reflection process.
The mechanisms of the transmitted ions are less complicated. They either gyrate
once in the transition layer due to reflection from the cross-shock potential and escape
into the downstream (see Figs. 3.9b and 3.10b for M = 3 and M = 7 respectively) or
directly transmit having enough energy to overcome the potential barrier (see Figs. 3.9c
and 3.10c for M = 3 and M = 7 respectively). To summarize, ions transmit into the
downstream through 2 mechanisms: TE - transport trough the transition layer → escape
35
Figure 3.8: Trajectories of 50 escaping ions in x-z plane propagating through high-Mach
shock (M = 7). (a) Zoomed trajectories of the first cross-shock potential reflection
process. (b) Trajectories of ions escaped through RGRRE-mechanism. (c) Zoomed trajectories of the second double cross-shock potential reflection process.
downstream and RTE - reflect from the cross-shock potential → transport → escape
downstream. Total fractions of transmitted ions are 0.9NT OT for M = 3 (TE - 99% and
RTE - 1%) and 0.96NT OT for M = 7 (TE - 82% and RTE - 18%).
Figure 3.9: Trajectories of 50 transmitted ions in x-z plane propagating through low-Mach
shock (M = 3). (a) Both mechanisms of transmitted ions. (b) Zoomed trajectories of ions
transmitted through RTE-mechanism near the transition layer. (c) Zoomed trajectories
of ions transmitted through TE-mechanism near the transition layer.
As one can see the beam of RTE ions is clearly separated from the TE ions. These
ions, after being pulled back into the upstream, move in the downstream region close
to the shock front while the directly transmitted are widely scattered and propagate
36
Figure 3.10: Trajectories of 50 transmitted ions in x-z plane propagating through highMach shock (M = 7). (a) Both mechanisms of transmitted ions. (b) Zoomed trajectories
of ions transmitted through RTE-mechanism near the transition layer. (c) Zoomed trajectories of ions transmitted through TE-mechanism near the transition layer.
downstream in various directions.
3.2.2
Velocity Distribution
The distributions are built by catching ions in five layers, each of the width ∆ = 0.1(Vu /Ωu ),
separated by the distance 2.5(Vu /Ωu ), so that evolution of gyrating distributions should
be clearly seen. The closest layer is well away at the distance 10(Vu /Ωu ) from the shock
front for both upstream and downstream. The velocities are transformed into the plasma
rest frame. The velocities v1 and v2 below are the two projections onto the plane perpendicular to the upstream and downstream magnetic fields, so that v2 is the component
perpendicular to the shock normal too. The exact transformation looks as follows:
v1 = −(ux − 1) sin θ + uz cos θ
v 2 = uy
(3.4)
q
vk = (ux − 1) cos θ + uz sin θ
v12 + v22
v⊥ =
(3.5)
Figures 3.11 - 3.14 show the five distributions of ions in v2 -v1 and vk -v⊥ planes correspondingly.
Most of the ions have relatively high perpendicular velocities v⊥ ≈ 1.2 and concentrate near a certain gyrophase, without any noticeable changes with the distance from
37
Figure 3.11: Upstream distribution for M = 3 in v2 -v1 plane at five layers (x = −10,
−12.5, −15, −17.5 and −20) respectively.
Figure 3.12: Upstream distribution for M = 3 in vk -v⊥ plane at five layers (x = −10,
−12.5, −15, −17.5 and −20) respectively.
Figure 3.13: Upstream distribution for M = 7 in v2 -v1 plane at five layers (x = −10,
−12.5, −15, −17.5 and −20) respectively.
the shock front. A separate inner ring population with v⊥ ≈ 0.5, vk ≈ −3 and much
lower density is present on all plots. As well, figures 3.12 and 3.14 show that most of the
ions have their parallel velocities around vk ≈ −2.3 for M = 3 shock and vk ≈ −2.5 for
M = 7 shock. One should however note, that ions with higher parallel velocities would
encounter the v1 -v2 plane more rapidly, therefore a spacecraft located at any distance upstream would possibly measure larger fluxes of rather backstreaming particles with larger
38
Figure 3.14: Upstream distribution for M = 7 in vk -v⊥ plane at five layers (x = −10,
−12.5, −15, −17.5 and −20) respectively.
parallel velocities (FABs).
Figure 3.15: Upstream distribution at x = −10(Vu /Ωu ) for different mechanisms of escaping (RGE, RGRE, RGGE, RGGRE, RGRRE). (a) M = 3, v2 -v1 plane, (b) M = 3,
vk -v⊥ plane, (c) M = 7, v2 -v1 plane and (d) M = 7, vk -v⊥ plane.
Additional information is provided in Figure 3.15. Here, the distribution at x =
−10(Vu /Ωu ) is shown for the different mechanisms of escaping. As can be seen the outer
ring population is formed by a mechanism of a single encounter with the cross-shock
potential, i.e. RGE for M = 3 and RGGE for M = 7. The inner population with
lower perpendicular velocities is formed by mechanisms RGRE for M = 3, RGGRE and
RGRRE for M = 7. These results demonstrate that the multiple encounter with the
cross-shock potential leads to formation of rather field-aligned beams (FABs), while the
RGE or RGGE mechanisms are responsible for gyrating beams of escaping ions.
The downstream distribution of transmitted ions is shown on Fig. 3.16. In similar,
the transformation into v1 -v2 and vk -v⊥ planes was made according to (3.4) and (3.5),
thus for a corresponding angle of θ = 74.4◦ . As well, there was no significant dependence
of distribution on the distance from the shock front, therefore the distributions are shown
39
Figure 3.16: Downstream distribution at x = 10(Vu /Ωu ) for different mechanisms of
transmission (TE, RTE). (a) M = 3, v2 -v1 plane, (b) M = 3, vk -v⊥ plane, (c) M = 7,
v2 -v1 plane and (d) M = 7, vk -v⊥ plane.
for a single layer at x = 10(Vu /Ωu ). The two major mechanisms of transmission (TE and
RTE) are clearly separated in both v2 -v1 and vk -v⊥ planes. As can be seen the directly
transmitted ions (TE) have very low parallel velocities, but are widely distributed in the
perpendicular direction. This yields that most ions after being transmitted through the
shock rather gyrate near the front than propagate into the downstream. The other, much
smaller fraction of ions (RTE), that was initially reflected from the cross-shock potential,
has negative parallel velocities vk ≈ −1.3 for M = 3 and vk ≈ −0.7 for M = 7 and as
well gyrates with v⊥ ≈ 0.7.
Figure 3.17: Initial velocity distribution (grayscale) for M = 3 shock and fractions of
escaping ions via different mechanisms (RGE - blue and RGRE - red) at the launching
point x = −1(Vu /Ωu ). (a) ux0 -uy0 plane, (b) ux0 -uz0 plane and (c) uy0 -uz0 plane.
Figures 3.17 and 3.18 show the initial velocity distribution of ions at the launching
40
point x = −1(Vu /Ωu ). Escaping ions appear from the same regions in initial velocity
space. In all plots the velocity regions of different mechanisms overlap with each other,
which shows that the origin of escaping process entirely depends on ions initial velocities.
As observed, ions preferably escape with negative y and z initial velocities, however this
condition depends on initial distance from the shock front. Finally, comparing Fig. 3.17
with Fig. 2.5 one can find good agreement between the analytical and numerical treatments for a case of low-Mach (M = 3) shock. The graphs demonstrate identically both
the types of mechanisms and the velocity distributions.
Figure 3.18: Initial velocity distribution (grayscale) for M = 7 shock and fractions of
escaping ions via different mechanisms (RGGE - blue, RGGRE - red and RGRRE green) at the launching point x = −1(Vu /Ωu ). (a) ux0 -uy0 plane, (b) ux0 -uz0 plane and
(c) uy0 -uz0 plane.
3.2.3
Energy and Pitch angle
We shall also use the ”energy” E (ratio of the ion energy to the incident ion energy), and
the pitch-angle defined as follows:
2
E = vk2 + v⊥
and
ψ = tan−1
v⊥
|vk |
(3.6)
Figure 3.19 shows energies and pitch angles of ions escaped through the above mentioned
mechanisms at the distance x = −10(Vu /Ωu ). In case of M = 3 shock, as can be seen from
Figs. 3.19a and 3.19b, there are two populations: high energy ions with E ≈ 9.5 and pitch
41
angles ψ < 15◦ and lower energy population with E ≈ 7 and pitch angles 15◦ < ψ < 30◦ .
In case of M = 7 shock, Figs. 3.19c and 3.19d demonstrate that there are two similar
populations: high energy ions with E ≈ 9.5 and pitch angles 5◦ < ψ < 20◦ and lower
energy population with E ≈ 7.5 and pitch angles 15◦ < ψ < 30◦ . As shown in all plots
each population is formed by a corresponding mechanism of escaping. Ions escaping via a
single encounter with cross-shock potential (i.e RGE and RGGE) are with lower energies
and larger pitch-angles, while the multiple encounter with the potential produces higher
energies and lower pitch angles population.
Figure 3.19: Upstream distributions of energy and pitch angle at x = −10(Vu /Ωu ) for
different mechanisms of escaping. M = 3 shock: (a) E and (b) ψ, M = 7 shock: (c) E
and (d) ψ.
Far away from the shock front in upstream or downstream regions ions mean energy
remains constant. Eventually, particles gain energy while interacting with the transition
layer. The energy jump of escaping ions can be explained by following. We shall look on
the y-direction of ion motion, since neither x or z component of the electric field do not
produce any work in average on the escaping ion. In case of Ex the ion brakes, but is
similarly accelerated on its way back to the upstream; the Ez component is simply zero.
Just before the encounter with the shock front, the ion’s energy slowly varies depending
on its y-direction of propagation (whether it is along the Ey component or not). The
energy behaves in oscillatory manner and remains constant in average. However, when the
particle is transmitted into the downstream it meets larger Bz component and therefore
gyrates via smaller gyroradius in x-y plane. The amplitude of oscillations in y-direction
is reduced and therefore there is a difference between the work done on the particle in
upstream and downstream regions. As a result, in case of appropriate velocity phase,
42
the ion is energized while gyrating in the transition layer and the larger the number of
gyrations the more energy it can gain.
Figure 3.20: Downstream distributions of energy and pitch angle at x = 10(Vu /Ωu ) for
different mechanisms of transmission. M = 3 shock: (a) E and (b) ψ, M = 7 shock: (c)
E and (d) ψ.
Figure 3.20 demonstrates energies and pitch angles of ions that have transmitted
through the shock. As can be seen, the population of ions that was initially reflected
by the cross-shock potential is well separated from the fraction of directly transmitted.
Despite the small amount of such ions, they have larger energies and lower pitch angles for
both M = 3 and M = 7 shocks. Ions transmitted through TE-mechanism have energies
E < 1 and pitch angles ψ > 50◦ in the plasma rest frame.
3.3
Escaping Efficiency vs. Plasma Parameters
In this section we provide the results of numerical simulations with varying shock/plasma
parameters. The dependence of escaping ions fraction on such parameters as plasma β,
cross-shock potential s and angle θ are presented on Figs.(3.21-3.22) for M = 3 and M = 7
shocks respectively.
Simulations were performed for various magnitudes of parameters, thus one of them
was fixed in similar to the discussed simulation above, i.e. β = 0.35, s = 0.7 and θ = 50◦
(shown by green dot on all the plots). Apparently, the number of escaping ions grows
with larger β, larger s and smaller θ. The escaping process appears to be very sensitive to
the angle between the shock normal and the magnetic field. There were no ions observed
in upstream for angle θ > 60◦ . Within these angles 45◦ < θ < 60◦ , the number of escaping
43
Figure 3.21: Dependence of total fraction of escaping ions on plasma/shock parameters:
(a) cross-shock potential - s for different β, (b) shock normal/magnetic field angle θ for
different s, (c) plasma β for different θ. M = 3.
Figure 3.22: Dependence of total fraction of escaping ions on plasma/shock parameters:
(a) cross-shock potential - s for different β, (b) shock normal/magnetic field angle θ for
different s, (c) plasma β for different θ. M = 7.
ions intensively grows with the cross-shock potential ratio s. Plasma β, as well, controls
the number of escaping ions, larger β, more ions in upstream for both M = 3 and M = 7
shocks. One can notice some irregular dependence in Fig. 3.22a. Here, the number of
escaping ions increases with s, however at s > 0.85 it has inverse dependence on β. A
similar point can be found in Fig. 3.21a for M = 3 shock at s ' 0.99. This feature can be
explained as follows: smaller s enforce less ions to be specularly reflected from the crossshock potential since larger fraction is capable to transmit through the barrier. Therefore
larger thermal velocities, subsequently β, would force more ions to escape. However,
after some ”critical” value of s the number of escaping ions decreases with increasing β.
Fig. 3.23 illustrates the fractions of ions that are initially reflected by the cross-shock
potential barrier (shown by green). As one can see at large enough s the ions with more
44
narrow distributions (or small β) are totally reflected by the potential barrier while those
of larger β are cut and ions still manage to transmit (red) into the downstream on their
first encounter with the shock front. Following the discussed above conditions, the first
specular reflection is a necessary mechanism for ions to escape, therefore at smaller β the
number of escaping ions would increase. In addition, at larger Mach number shocks a part
of ions is specularly reflected not in the ramp but in the overshoot region, therefore the
effective s (enough to initially reflect the ion in the ramp/overshoot) should be smaller.
This is the reason for the smaller value of ”critical” scr , i.e., at M = 7, scr = 0.85 while
at M = 3, scr = 0.99.
Figure 3.23: Fractions of initially reflected from the cross-shock potential (green) and
transmitted (red) ions during the first encounter with a shock front
45
Chapter 4
Summary
4.1
Discussions and Conclusions
We have analyzed both analytically and numerically ion trajectories in low and high Mach
number quasi-perpendicular shock fronts, using model profiles for the corresponding magnetic and cross-shock electric fields. We have considered some fundamental conditions and
constructed an appropriate algorithm to be used in order to predict the number of escaping
ions. These data, obtained analytically, was briefly checked with numerical simulations
and demonstrated a good correlation. However, such analytical analysis can be used
for only low-Mach shocks, since its structure is similar to the necessary approximation
(discontinuity).
We have traced a large number of particles for two types of shocks: low-Mach
(M = 3) and high-Mach (M = 7) number. It was found that ions either escape or
transmit through the shock via a finite number of mechanisms which are well-defined
and can be easily distinguished. Since the downstream field amplitude is similar for
both types of the mentioned shocks, the only difference between them is the transition
layer structure. Despite the fact that the ramp width is only a small fraction of the
convective ion gyroradius, the ion motion appears to be sensitive to the details of the
field distribution in the ramp. In case of M = 7 shock the ions escape through more
complicated mechanisms and encounter the shock front more frequently, thus acquire
larger spectrum of energies compared with M = 3 shock.
46
We have found that all escaping ions originate from the ion population which is
initially reflected by the cross-shock potential while still inside the ramp, without crossing
the ramp completely. If ion distribution in the solar wind is shifted Maxwellian, the
number of ions which are stopped in the ramp, ns , is given by
nu
ns = √
2πvT
Z
where σ = vu /vT =
(2eφ/mi )1 /2
2
−(vx −vu )2 /2vT
e
0
nu
dvx = √
2π
Z
√
( s−1)σ
e−v
2 /2
dv
−σ
p
2M 2 /β. For a given s, the larger σ the lower is ns , and the decrease
is rather steep. We conclude that higher Mach number and/or lower β shocks should be
less efficient in producing escaping ions. This is supported by direct numerical analysis
above. It should be noted that the above conclusion assumes the shock stationarity. This
assumption may be too strong for high Mach number shocks, and the reflection process
would be modified by temporal variations of the shock front structure. This issue is
beyond the scope of the present work. Ion escape should be suppressed in shocks with
large angles between the shock normal and the upstream magnetic field. Indeed, during
the first reflection inside the ramp a reflected ion acquires velocity along the magnetic
field. This parallel velocity is proportional to cos θ. If this component is too small,
the ion is efficiently convected across the ramp and further downstream. Reduction of
cos θ, therefore, reduces the chances of an ion to acquire velocity favorable for escape into
upstream. This conclusion is also supported by direct numerical analysis.
To compare the presented results with observational data we have considered the
work by Kucharek et al. (2004) where the authors have investigated shock crossings of
a high and low Mach number shocks with multi-spacecraft measurements. In particular,
they have investigated the spatial and temporal evolution of the proton distribution during
these bow shock crossings.
In Fig. 4.1 the distributions observed upstream and downstream of the high Mach
number shock and the corresponding distributions obtained during the crossing of a low
Mach number shock are demonstrated. The left half of this figure shows the total magnetic
field (top panel) and the two distributions for the high Mach number shock (24 January
2001, 05:40 UT), as seen by SC1. In the right half of this figure the distributions and the
magnetic field magnitude for the low Mach number crossing on 31 March at 18:02 UT
47
Figure 4.1: Color-coded velocity distribution in the vk -v⊥ plane in the shock ramp and
downstream of the Earths bow shock for 31 March (right-M = 3) and for 24 January
(left-M = 11) 2001. The observation intervals are indicated by the shaded bar in the
corresponding panel of the magnetic field.
is shown. At the shock ramp one finds the solar wind and the gyrating ion distribution.
While Fig. 4.1 shows velocities of the ions at the ramp only, which yet cannot be classified
as escaping ions, Fig. 4.2 shows velocity distribution at the ramp and far upstream of the
shock. As one can see when ions leave the transition layer the fraction of the escaping
ions is clearly identified.
Figure 4.2: Multi-spacecraft observations during crossing of the Earths bow shock on 2
January (M = 5) 2002. The top panel shows the magnetic field and lower panel shows
the color-coded velocity distribution in the vk -v⊥ plane during a bow shock crossing.
48
Both gyrating and field-aligned ions populations are observed and the range of measured velocities is of the order of the obtained results. While the results can be qualitatively compared, a more direct comparison is difficult. For instance, the electric potential
which has presented itself as an important parameter for ion reflection is not considered
in the referenced work. It should be also noted that the found distributions would be
observed if the shock fronts were planar. Earth’s bow shock is curved, and measured escaping ion distributions at different positions are produced by reflection at different places
at the shock front. Since the reflection process is very sensitive to the angle between the
shock normal and upstream magnetic field (while the shock structure is not expected to be
so sensitive), comparison of the derived results with observations would require a rather
complicated convolution of the reflection features, shock shape, and detector position.
This task is beyond the scope of the present work. However, some indications of what
Figure 4.3: Distributions of vk , v⊥ , E and ψ vs. z at x = −5(Vu /Ωu ).
could be expected are given by Figure 4.3 which shows the parallel velocity vk , perpendicular velocity v⊥ , energy E and pitch-angle ψ of the ions as a function of their displacement
z along the shock front, as measured at some distance, x0 = −5(Vu /Ωu ), from the ramp
of the shock (M = 3). Let us imagine, that only the stripe near z = 0 at the shock
front produces escaping ions. In this case a spacecraft at x = x0 , z = −26 would observe
only ions with lower perpendicular velocities (FABs), lower energy and pitch-angles. A
spacecraft at x = x0 , z = −19 would observe ions with high perpendicular/parallel velocities, high energies and higher pitch angles, and the two distributions would be not
observed simultaneously. It is important to mention that the origin of two populations is
the mechanisms through which ions escape. The ions that had a multiple encounter with
49
the shock are well separated from those that had a single one. Figure 4.3 shows that these
distributions should not overlap, in agreement with observations [Meziane et al., 2004,
Kucharek et al., 2004].
In our numerical analysis there were no escaping ions for θ > 60◦ . Oka et al. [2005]
made an attempt to adjust the shock parameters (in particular, the cross-shock potential
to s = 0.55) to the flux to reflected ions, but have not adjusted them to the downstream ion
distribution. Variation of the parameters would affect the reflection efficiency. However,
the dependence on the angle is particularly strong, and too drastic increase of β or s would
be necessary to substantially shift the limiting reflection angle. Our analysis shows that
stationary one-dimensional shocks cannot produce escaping ions when θ > 60◦ , unless our
parameter space is grossly wrong. The only observations so far which relate generation of
field-aligned beams to highly oblique shocks are reported by Kucharek et al. [2004]. It is
possible that shock rippling [Moullard et al., 2006], which changes the local shock normal,
or shock non-stationarity are responsible for the reflection in this case. To summarize,
we have shown that generation of escaping ions is naturally explained as a result of the
ion motion in the stationary and one-dimensional structure of a low Mach number quasiperpendicular shock front. Using test-particle numerical analysis in a model shock front
(which is not a self-consistent simulation) we found the following features of the process:
• The non-specular reflection is very sensitive to the a) angle between the shock normal
and upstream magnetic field, b) ion β (or, more precisely, the ratio of the thermal
ion velocity to the upstream flow velocity), and c) cross-shock potential. The latter,
used here as a free parameter, should be intimately related to the shock structure.
• In general, escaping ions bunch in the phase space in two regions: a) v|| ≈ −3 and
ψ . 15◦ , and b) v|| ≈ −2.5 and ψ ≈ 25◦ . Relative densities in these regions depend
on the shock parameters.
• The ions escape through a well-defined finite number of mechanisms, that are individually responsible for generation of each of the observed populations.
Indeed, more research is necessary to have distributions which could be compared
with observations. All above, however, neither disqualifies the non-specular reflection
50
as the basic mechanism of escaping ion generation, nor reduces the importance of the
knowledge of the features of the distributions produced by this mechanism.
Finally, a manuscript considering partial results reported in present thesis was prepared and recently published in the Journal of Geophysical Research - Space Physics (for
reference see [11]).
51
Appendix A
Solution of motion equations
One can rewrite Eqs.(2.4)-(2.6) in the following vector form:
~u˙ = B~u + E
(A.1)
where B and E are the following matrices:



0
bz 0
0






B =  −bz 0 bx  , E =  ey



0 −bx 0
0





We define ~u = Tw,
~ where T is a transformation matrix which is chosen so that the
Eq.(A.1) can be rewritten as:
w
~˙ = Dw
~ + T−1 E
where D is a diagonal matrix

0
0
0

p

D = T−1 BT =  0 − −b2x − b2z
0

p
0
0
−b2x − b2z





Hence, we rewrite the equations (2.4)-(2.6) as:
ẇx = 0
ẇy = −
(A.2)
p
ey bx
−b2x − b2z wy + p
2 −b2x − b2z
(A.3)
52
p
ey bx
−b2x − b2z wz − p
2 −b2x − b2z
ẇz =
(A.4)
Solving the equations (A.2)-(A.4) one obtains
wx = C1
√2 2
wy = C2 e−ı bx +bz τ −
wz = C3 eı
√
b2x +b2z τ
−
e y bx
2(b2x + b2z )
ey bx
2(b2x + b2z )
Since ~u = Tw
~ we obtain
√2 2
√2 2
ey bz
bx
bz
C1 − (C2 e−ı bx +bz τ + C3 eı bx +bz τ ) + 2
bz
bx
bx + b2z
ux =
p
√2 2
√2 2
b2x + b2z
uy =
(C2 e−ı bx +bz τ − C3 eı bx +bz τ )
bx
ı
uz = C1 + C2 e−ı
√
b2x +b2z τ
(A.5)
(A.6)
√2 2
ey bx
+ C3 eı bx +bz τ − 2
bx + b2z
For τ = 0 equations (A.5)-(A.7) correspond to initial velocities, where
ux0 =
uy0
bx
bz
ey bz
C1 − (C2 + C3 ) + 2
bz
bx
bx + b2z
(A.7)
p
ı b2x + b2z
=
(C2 − C3 )
bx
uz0 = C1 + C2 + C3 −
(A.8)
e y bx
+ b2z
(A.9)
b2x
Solving the equation set (A.7)-(A.9) we obtain the constants of integration
C1 = bz
ux0 bx + uz0 bz
b2x + b2z
C2 = −bx
C3 = bx
p
(ey + uz0 bx − ux0 bz ) −b2x − b2z + uy0 (b2x + b2z )
3
2(−b2x − b2z ) 2
p
−(ey + uz0 bx − ux0 bz ) −b2x − b2z + uy0 (b2x + b2z )
3
2(−b2x − b2z ) 2
Introducing these constants back into the equations (A.5)-(A.7) one obtains the solutions
presented and used in section 2.3.
53
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