Download Nonlinear waves and shocks in relativistic two-fluid hydrodynamics

Nonlinear waves and shocks in relativistic
two-fluid hydrodynamics
Thesis submitted in partial fulfillment of the requirement for the degree of master of
science in the Faculty of Natural Sciences
Submitted by: Lev Haim
Advisor: Prof. M. Gedalin
Department of Physics
Faculty of Natural Sciences
Ben-Gurion University of the Negev
August, 2009
Abstract
Studies of nonlinear waves in relativistic plasmas are important in connection with
collisionless shocks which are expected to form in a number of astrophysical objects dominated by relativistic plasmas. We study stationary one-dimensional nonlinear waves propagating in the direction perpendicular to the external magnetic field, within the two-fluid
approach. The research covers the cold and hot pair plasma cases as well as electron-ion
plasma. In the symmetric pair plasma the absence of the electric field along the propagation direction allows to derive a pseudopotential-type equation for the magnetic field or
density in the cold as hot cases as well. We reproduce the well-known results regarding
compressive solitons and show that the pair plasma allows also rarefactive solitary wave
solutions. In the case of electron-ion plasma we invoke the assumption of the quasineutrality in the species rest frame and solve the equations in the approximation of the
weak longitudinal electric field. This approach allows to derive a single pseudopotential
equation for the density. The equation is solved numerically in a wide range of plasma
parameters. It is found that the compressive solitons are similar to those found for the
pair plasma and do not exist for Mach numbers exceeding some critical Mach number.
There are no rarefactive solitons in the electron-ion plasma in this approximation.
2
Contents
1 Introduction
4
1.1
Scientific Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Multi-Fluid Hydrodynamics Model
9
2.1
4-Representation and Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2.1
Energy, Momentum, State & Maxwell’s Equations . . . . . . . . . . . . . . . . . .
10
One-dimensional Stationary Oblique Equations . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
3 Two Fluid MHD: Perpendicular Case
14
3.1
Two Fluid Perpendicular Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2
Normalization and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.3
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4 Electron-Positron Plasma
19
4.1
Cold Plasma Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.2
Cold Plasma Non Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.3
Hot Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3.1
Singularity & Maximum Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.3.2
Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Magnetic Profile & Pair Plasma Summarize . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.4
5 Electron-Ion Plasma
39
5.1
Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Magnetic Profile
& Electron-Ion Plasma Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Discussions & Conclusions
44
48
52
3
Chapter 1
Introduction
1.1
Scientific Background and Motivation
Observations have shown that active galactic nuclei (AGNs), quasars (QSOs), micro
quasars (µQSOs) and gamma-ray bursts (GRBs) eject relativistic jets [Pearson & Zensus
1987; Biretta et al. 1999; Junor et al. 1999; Mirabel & Rodriguez 1998; Kulkarni 1999].
In spite of the drastic difference of their characteristic scales and powers, it is believed
that the activities of these objects, such as the relativistic jet ejection, are supported
commonly by the drastic phenomena of accretion disks around black holes [Mirabel &
Rodriguez 1998]. However, the distinct mechanism of the activity is not confirmed yet.
Models with interaction between the plasma and magnetic field in the very strong gravity
of the black hole are thought to be most promising. A black hole magnetosphere is the
region where the plasma interact with the magnetic field of the black hole, which consist
of an accretion disk and corona around a black hole and it is thought to be composed
of various kinds of plasma. In the case of an AGNs, it is suggested that the disk is
made of electron-ion plasma [Ford et al. 1994], the core of the relativistic jet is mainly
of electron-positron (pair) plasma [Wardle et al. 1998], and the corona is both of them,
while the actual components of such plasmas have not been conformed observationally
yet. Propagating relativistic shock waves are the leading candidates for the origin of the
emission from GRBs and their afterglows [Piran 1999, 2000; Mészáros 2000 ] as well as
of the spectral flares and apparent superluminal motions exhibited by the blazar class
4
of AGNs [Ulrich, Maraschi, & Urry 1997]. The energy for the accelerated, synchrotron
emitting particles is though to come from the kinetic energy and Poynting flux of relativistic jets emanating from energizing compact objects. Shocks waves in these relativistic
flows are most plausible candidate for conversion of flow energy into random energy of the
synchrotron emitting particles. In the case of blazars, the measurement of high optical
linear polarization in some objects directly supports the synchrotron interpretation [ Angel & Stockman 1980]. In the case of GRBs there are two confirmed detections of optical
polarization [Covino et al. 1999; Wijers et al. 1999; Rol et al. 2000], which, despite the
relatively low measured values , appear to be consistent with a synchrotron origin. In
both blazars and microquasars the case for synchrotron radiation is supported also by
measurements of radio polarization. These shock waves are expected to be collisionless
[Granot & Königl 2001].
Nonlinear waves were extensively studied in the framework of magnetohydrodynamics (MHD) and two-fluid hydrodynamics. A typical analysis sequence is as follows: onedimensionality assumption, model equation construction, and stationary solution analysis
with particular attention to solitons. Although deviations from one-dimensionality are significant in some cases, as e.g. in the case of whistler self-focusing [Karpman et al. 1992] ,
it is widely believed that in most cases one-dimensional waves are a good approximation.
In most papers on nonlinear waves in MHD and two-fluid hydrodynamics cold plasmas are
considered [Kennel & Pellat 1976; Aslop & Arons 1988]. Both [Kennel & Pellat 1976] and
[Aslop & Arons 1988] have concluded that fast waves propagating perpendicular to the
magnetic field are restricted to low amplitudes in relativistic limit and have shown that
compressive solitons can exist in electron-positron plasma as well as ion-electron plasma,
respectively, in the regime where the flow of cold plasma is perpendicular to the magnetic
field. Moreove, [Kennel & Pellat 1976] have assumed that for electron ion plasmas particle
inertia is determined by wave amplitude, not by the rest masses in the relativistic limit.
p
Hence, have found a critical Mach number, that is MC = 2(1 + γo ), and largest possible
∗
magnetic field amplitude when MA = MC , that is Bmax
= 1 + 2/γo . The importance of
the parallel/perpendicular distinction is clear when we consider the motion of a particle
that is initially headed away from the shock. In the de Hoffman-Teller frame the motion
5
has two parts: an unimpeded motion along the direction of the magnetic field, and a
gyration around (i.e., transverse to) it. In the case of a parallel shock, the field lines pass
through the shock, and a particle’s motion along the field will carry the particle through
(and away from) the shock relatively easily. On the other hand, at a perpendicular shock,
the field lines are parallel to the shock surface, and so particle motion along the magnetic
field does not let the particle pass away from the shock. Indeed, the particle gyration at a
perpendicular shock brings the particle back to the shock. There is a further simple conclusion to be drawn from the parallel/perpendicular distinction. Because particle motion
in the normal direction is ”easier” at the parallel shock, compared with at the perpendicular shock, then we can expect that the magnetic field does not play an important role
in the former case. However, in the context of a collisionless plasma, the only way for
dissipation to occur is via field-particle processes, and it is certain that here the fields will
play a crucial role [Kivelson & Russell 1996 ]. Moreover, assuming that magnetic field
is isotropic in the wave frame, which moves relatively to laboratory frame with velocity
~v . Thus, according to Lorentz field transformations, in the laboratory frame magnetic
field parallel to the flow is not change where as transverse magnetic field is increased by
Lorentz factor. Hence, in this paper we restrict our self to the perpendicular case.
Investigations of various aspects of relativistic hot electron-positron/ion plasma have
bean carried out. But there is little progress in this trend due to the mathematical
complexity and difficulty in the understanding of the phenomena. In addition, it should
be noted that consideration of the problem is very complicated at relativistic temperatures
and also when the particles have relativistic hydrodynamical velocities. Nevertheless, a
number of papers dealt with a hot plasma. Using the water-bag model for the distribution
function and neglected the electron inertia, electron internal energy and pressure in the
face of relativistic proton energy scale [Chiueh & Lai 1991], those assumption generally
are wrong and can not be used in hot relativistic case. Latter conformed that small
amplitude soliton of sech2 type do exist. Moreover, the peak amplitude for cold ionp
electron plasma of the charge density is npeak = [2 γo2 σ(1 + σ)/(γo2 − 1) − 1 − σ]−1 , while
for the hot case npeak is of order γo , where γo is the Lorentz factor corresponding to
the mean fluid velocity far upstream (sub index o) and σ = (uA )/c2 , with uA being the
6
proper Alfvenic speed. Considerable deviation from the above steps scenario was done
[Medvigy & Loeb 2001]. Showing that no stationary, continuous shock solution exist for
hot relativistic pair plasmas. Moreover, they have found that soliton solutions exist only
for Bo2 /8πme no c2 γo2 1, where me is electron mass and Bo /γo plasma’s self magnetic field
defined at the proper reference frame with charge density no . Above references confirmed
the existence of the compressive (bright) solitons. Existence of a different nature of
solitary wave, rarefactive (dark) solitons were noticed [Lakhina & Verheest 1997], as well
as compressive (bright) solitons in relativistic hot pair plasma, where an expansion for all
variables were used to deduce a KdV equation, which implied that in ultrarelativistic pair
plasmas perpendicularly propagating solitons of sech2 type have a compressive/rarefactive
nature. However, typical length scales or peak amplitudes of those solitons were not found.
It should be noted that compressive and rarefactive terms are not really appropriate for
the changes in magnetic field but more suitable for changes in a charge density. Dark
solitons are known in optical systems [Farina & Bulanov and references in them], and
they have been observed and discussed also in the Bose-Einstein condensate [Farina &
Bulanov and references in them].
The objective of the present research is to study the properties of the nonlinear
stationary one-dimensional waves (NS1DW) in relativistic magnetized plasmas. The main
emphasis is on the case where the plasma flow is perpendicular to the magnetic field.
1.2
Thesis Outline
In Chapter 2, we derive general equations and relations for NS1DW from the multifluid relativistic hydrodynamics with finite nonzero temperature and arbitrary number of
species, emphasizing the differences between the oblique and perpendicular propagation
of the wave with respect to the magnetic field. In Chapter 3, we reduce our set of
oblique multi-fluid equations to the two-fluid perpendicular equations, meaning θ = π/2
and s = 1, 2, where 1 stands for positive electric charge e.g. positrons or ions depends
on the type of plasma and 2 stands for negative electric charge e.g. electrons. The new
normalized set of equations reveal seven constants of the motion. The first is continuity of
7
charge density. This relates the plasma density to the x component of the velocity. The xcomponent of the current is zero with the consequence that both species speeds along the
x axis are equal by virtue of quasineutrality. The energy and x, y momentum equations,
where z component was left out for the sake of convenience, are used to construct four
conservations laws, which correspond to conservation of energy flux, transverse momentum
flux, longitudinal momentum flux and electric potential flux. Faraday’s law requires that
the component of the electric field, transverse to the direction of propagation, is constant.
In perpendicular case Gauss’s law requires that the component of the magnetic field in
the direction of propagation be zero. The conditions for existence of soliton structure
are derived in chapter 4, which covers the electron positron plasma whereas chapter 5
covers the electron ion plasma. Hot and cold regimes for each plasma are investigated in
chapters 4 and 5 using Sagdeev’s pseudopotential method. Compressive and rarefactive
soliton solutions are found. Chapter 6 presents main conclusions of the analysis and
suggestions for future work on the subject.
8
Chapter 2
Multi-Fluid Hydrodynamics Model
In this section we derive fully nonlinear equations from the multi-fluid hydrodynamics. We
derive a complete set of equations with finite nonzero temperature and arbitrary number
of species.
2.1
4-Representation and Continuity Equation
We choose Minkowski metrics with positive signature, gαβ = [1, −1, −1, −1]. From now
on all four dimensional tensor indexes will be denoted by Greek letters α, β, ... and take
the values t, x, y, z. Here, Usα is the four velocity for each species defined as
Usα = (γs , γs~vs /c)
where ~vs is the plasma velocity that measured in the lab frame and U α Uα = 1. The index
s denotes different species. Continuity equation [Lichnerowicz 1967] reads
∂α (ns Usα ) = 0
where ∂α =
1
~
∂ ,∇
c t
(2.1)
and ns is the proper particle density while Ñs = γs ns is particle
density at the laboratory frame. Using those notations we get
~ s γs~vs ) = 0
∂t (ns γs ) + ∇(n
(2.2)
9
2.2
Energy-Momentum Tensor
The Energy-Momentum Tensor [Landau & Lifshitz 1986] for an ideal fluid is
Tβα = ( + p)U α Uβ − pδβα
(2.3)
where , p are the energy density and the pressure, respectively, in the rest frame of
the fluid. We adopt the ideal fluid description for each species and do not specify the
species index unless it is necessary. In this section and its subsections the species index
is suppressed, because each species fulfill all equations independently. In what follows we
shell use also the notation w = ( + p)/n. The motion (Euler) equation for the charged
fluid is
∂β T αβ = qUβ F αβ
(2.4)
where Fαβ is the electromagnetic field tensor [Landau & Lifshitz 1986], defined as




0
Ex
Ey
Ez
0 −Ex −Ey −Ez









 −Ex
E
0
−B
B
0
−Bz By 
x
z
y 
αβ




Fαβ = 

, F = 
 Ey Bz
 −Ey Bz
0
−Bx 
0
−Bx 




Ez −By Bx
0
−Ez −By Bx
0
Together with continuity equation we have
nU α ∂α (wUβ ) − δβα ∂α p + qnU α Fαβ = 0
(2.5)
It should be noted that the usual Einstein summation convention is been used.
2.2.1
Energy, Momentum, State & Maxwell’s Equations
Multiplying the electromagnetic field tensor, Fαβ , by four velocity, U α , we obtain:
γ
~
U α Fαt = − ~v · E
c
~ + ~v × B/c]
~
U α Fαi = γ[E
(2.6)
(2.7)
where i is a spatial index, that is i = x, y, z.
Energy and Momentum equations are obtained using equations (2.5)-(2.7):
10
Setting β = t, we obtain Energy Equation, that is
h
i
~ (wγ) − ∂t p − qnγ~v · E
~ =0
nγ ∂t (wγ) + ~v · ∇
(2.8)
Setting β = i, we obtain Momentum Equations, that is
i
nγ h
~
~ + qnγ[E
~ + ~v × B/c]
~
− 2 ∂t (wγ~v ) + ~v · ∇ (wγ~v ) − ∇p
=0
c
(2.9)
State Equation
Multiplying equation (2.5) by four velocity, U α , and using U β ∂α Uβ = 0 with continuity
equation, Eq.(2.1) we have
1
α
nU ∂α w − ∂α p = 0
n
Defining the convective derivative D = U α ∂α one has
Dp = nDw
(2.10)
If we assume that pressure has a polytropic behavior p = CnΓ then
= mc2 n +
1
p
Γ−1
and
w=
Γ p
+p
= mc2 +
n
Γ−1n
(2.11)
Above equations are valid provided that the flow does not come to a halt, that is v > 0.
Maxwell’s Equations
The derived equations should be completed with Maxwell’s Equations:
X
~
1 ∂E
~ ×B
~ = 4π
qs Ñs~vs +
∇
c s
c ∂t
(2.12)
~
~ ×E
~ = − 1 ∂B
∇
c ∂t
~
~
∇·B = 0
X
~ ·E
~ = 4π
∇
qs Ñs
(2.13)
(2.14)
(2.15)
s
11
In above equations magnetic field, electric field, plasma velocity and charge density are
measured in an arbitrary reference frame which will henceforth designate as the laboratory
frame.
2.3
One-dimensional Stationary Oblique Equations
From now on and to the end of the paper we are restricting our equations to stationary
one-dimensional waves, meaning we let everything depend on x, that is (∂/∂t) = (∂/∂y) =
~o = Bo (cos θ, 0, sin θ) be the magnetic field in some reference point.
(∂/∂z) = 0. Let us B
We will normalize all variables with the fluid parameters in this reference point, where we
define the rest-frame density n0 and 4-velocity u = (γo , uo , 0, 0) for each species. We are
especially interested in soliton solutions, where the plasma state is asymptotically uniform
P
at x → ±∞. In this case the reference point is at x → −∞, where s qs nos γos = 0 and
P
qs nos uαos = 0. In what follows subscript ⊥ denotes components perpendicular to the
~ ⊥ = (0, By , Bz ), and subscript o denotes constant components far
x direction, e.g. B
upstream. Using above notations the complete set of the equations takes the following
form
∂x (ns usx ) = 0
→
ns usx = const ≡ Js
dps = ndws
h
i
~⊥
Js ∂x (ws γs ) = qs ns usx Ex + ~u⊥s · E
Js
1
∂x (ws usx ) = qs ns γs Ex + (~u⊥s × B⊥ ) · x̂ − ∂x ps
c2
c
Js
usx
Bo cosθ
∂x (ws~u⊥s ) = qs ns γs E⊥ +
(x̂ × B⊥ ) +
(~u⊥s × x̂)
c2
c
c
X
~ ⊥ = 4π
(x̂∂x ) × B
qs ns~u⊥s
c s
~ ⊥ = const
→
E
X
= 4π
qs ns γs
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
~⊥ = 0
(x̂∂x ) × E
(2.22)
∂x Ex
(2.23)
s
X
qs ns usx = 0
(2.24)
s
12
Those are multi-fluid one dimensional stationary oblique equations, which should be accompanied with state equation, we shell assume that the pressure have a polytropic behavior p = CnΓ . Now it is easy to emphasize the difference in our equations between the
oblique case and the perpendicular case. The main difference is an additional term at the
transverse momentum equation, Eq.(2.20). θ = 0 and θ = π/2 corresponds to parallel and
perpendicular cases, respectively, where it is easy to see that in the latter case parallel
magnetic field is equal to zero.
Combining these equations one has (we omit indices for simplicity)
d X
c
~⊥ × B
~ ⊥ )] = 0
[
nwγux + ~xˆ · (E
dx
4π
(2.25)
or
X
nwγux +
c ˆ ~
~ ⊥ ) ≡ Q = const
~x · (E⊥ × B
4π
(2.26)
Further one has
X
X
X
d X
~⊥ + (
~⊥
(
nwux~u⊥ ) = (
qnγ)E
qn~u⊥ ) × ~xˆBx + (
qnux )~xˆ × B
dx
~⊥ d
d X
E
Bx d ~
(
nwux~u⊥ ) =
Ex +
B⊥
dx
4π dx
4π dx
X
~ ⊥ Bx B
~⊥
Ex E
~ = const
nwux~u⊥ −
−
≡S
4π
4π
(2.27)
(2.28)
(2.29)
Similarly one also has
X
(nwu2x + p) −
2
Ex2 − B⊥
≡ P = const
8π
(2.30)
Now multiply the equation
d
~ ⊥)
(nwu2x + p) = qnγEx + qn~xˆ · (~u⊥ × B
dx
(2.31)
by wux /q and sum up to get
X wux d
X
X
~ ⊥ × ~xˆ)
nwγux )Ex + (
nwγ~u⊥ ) · (B
(nwu2x + p) = (
q dx
(2.32)
Using (2.26) and (2.29) one gets
X wux d
~
~
~ ×B
~ ⊥ ) − Ex~xˆ · E⊥ × B⊥ = QEx
(nwu2x + p) − ~xˆ · (S
q dx
4π
13
(2.33)
Chapter 3
Two Fluid MHD: Perpendicular
Case
In this section we reduce our set of oblique multi-fluid equations to the two-fluid perpendicular equations, meaning θ = π/2 and s = 1, 2, where 1 stands for positive electric
charge e.g. positrons or ions depends on the type of plasma and 2 stands for negative
electric charge e.g. electrons, q1 = −q2 = q. The new set equations will be normalized
and used to construct four conservations laws, which correspond to conservation of energy,
transverse momentum, longitudinal momentum and electric potential.
3.1
Two Fluid Perpendicular Equations
We are restricted to a 1 + 2 dimension analysis (one spatial coordinate x and two velocity
components ux̂ and uys ŷ). Although a finite constant uzs , z component of the velocity, in
principle exists, it can be trivially incorporated and we shell leave it out in this analysis for
the sake of convenience. Latter and transverse momentum equation (2.20) implies that the
z component of the electric field is equal to zero as well as the y component of the magnetic
field, that is Ez = By = 0. Finally, we consider only situations in which the magnetic field
~ = (0, 0, Bz ), and no variations in the z and y direction is
is always in the z direction, B
allowed. But the electric field begins to develop the x component with in the soliton where
as according Faraday’s law (2.22) the y component of the electric field remain constant.
14
It follows from longitudinal momentum equation (2.19) that Ey =
uo
B
cγo o
= βo Bo , given
the boundary conditions at infinity. Charge conservation requires
X
qs ns usx = 0 → n1 ux1 = n2 ux2
s
We add the assumption of quasineutrality (verified aposteriori), n1 = n2 = n. This is
obviously valid exactly for a symmetric pair plasma but is an approximation for electronion plasmas. As a result, ux1 = ux2 = u.
Equations (2.16)-(2.23) take the following form:
nu = no uo
(3.1)
Γ s ps
Γs − 1 n
qs n [uEx + uys Ey ]
h
uys i
qs n γs Ex +
Bz − ∂x ps
c i
h
u
qs n γs Ey − Bz
c
4πqn
−
(uy1 − uy2 )
c
4πqn (γ1 − γ2 )
ws = ms c2 +
no uo ∂x (ws γs ) =
no uo
∂x (ws u) =
c2
no u o
∂x (ws uys ) =
c2
∂x Bz =
∂x Ex =
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
Those are two-fluid one dimensional stationary perpendicular equations. It should be
noticed that our analysis is carried out in the rest frame of the soliton.
Lets us summarize the boundary condition far upstream, that is x → −∞:
x
n
p s , ws
~u1 = (u, uy1 , 0)





















~u2 = (u, uy2 , 0) 




~ = (0, 0, Bz ) 

B




~
E = (Ex , βo Bo , 0) 
x → −∞
no
pos , wos
→
~uo1 = (uo , 0, 0)
~uo2 = (uo , 0, 0)
~ o = (0, 0, Bo )
B
~ o = (0, βo Bo , 0)
E
15
3.2
Normalization and Definitions
The phase (and group) speed of the wave in the rest frame of the plasma is cβA , defined
[Aslop & Arons 1988; Gedalin 1993 ] by
βA2 =
Bo2 /4πno wo γo2
σ
=
2
2
1 + Bo /4πno wo γo
1+σ
where
σ=
2
/4π)
Poynting flux
(cβo )(Brz
Bo2
=
=
plasma energy flux
(cβo )(no wo )
4πno wo γo2
Here, σ is Lorenz invariant quantity, since Brz = Bo /γo is the strength of the upstream
magnetic field (fields transformations from [Landau & Lifshitz 1986]) which is measured
in an Proper/Rest reference frame and wo = wo1 + wo2 is total enthalpy.
Constructing the momentum of the particle uA , when the particle’s speed equal the wave
speed, where
√
cβA
=
c
uA = γA vA = γA cβA = p
σ
1 − βA2
Here, uo is the upstream momentum of the particle, that is
uo = γo vo = γo cβo
Shocks and solitons are expected to form when uo ≥ uA . One might expect, therefore,
the structure of solitary waves to expressed in terms of the magnetosonic Mach number
MA , defined by
2
uo
u2
γ 2β 2
2
= 2o = o o
MA =
uA
cσ
σ
(3.8)
We proceed by expressing the equations in terms of normalized variables,that is
Ws =
ws
,
wo
W = W1 + W2 ,
N=
B=
Ps =
ps
= N Γs ,
po1
Bz
,
Bo
E=
Ex
,
Bo
P = P 1 + P2 ,
n
,
no
U=
wo = wo1 + wo2
u
,
uo
x = XL,
βs =
16
8πpos
,
Bo2
uys
Vs =
c
r
wo
L=
4πno q 2
β = β1 + β2
where L is the typical length scale. Equations (3.1)-(3.7) take the following form:
NU = 1
(3.9)
Ws = Wos +
∂X (Ws γs ) =
βo2 γo2 ∂X (Ws U ) =
∂X (Ws Vs ) =
∂X B =
∂X E =
Γs
Γs − 1
βs σγo2
2
N Γs −1 − 1
(3.10)
√ qs
σ [γo E + N Vs ]
q
√ qs
βs σγo2
γo σ N [γs E + Vs B] −
∂ X Ps
q
2
√ qs
σ N [γs − γo U B]
q
N
− √ (V1 − V2 )
γo σ
N
√ (γ1 − γ2 )
γo σ
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
and the normalized boundary condition far upstream, that is X → −∞, are:



X → −∞
X






N
1




U , Vs 
1, 0
→

B, E 
1, 0





Ps , P 
1, 2



2

2
ms c
W , W 
+ Γs βs σγo , 1
s
3.3
wo
Γs −1
2
Conservation Laws
The new normalized set equations (3.9)-(3.15) can be used to construct four conservation
laws.
W1 γ1 + W2 γ2 = γo [1 + σ (1 − B)]
B 2 − E 2 = 1 + β1 (1 − P1 ) + β2 (1 − P2 ) +
W1 V1 + W2 V2 = σγo E
√
2γo2 σ(1 + σ)Φ = − W12 − W22 1 + βo2 γo2 U 2 ,
(3.16)
2βo2
σ
(1 − W U )
(3.17)
(3.18)
E = −dΦ/dX
(3.19)
Those are conservation laws of energy, transverse momentum, longitudinal momentum
and electric potential, respectively. First three equations are easily obtained by summing
17
up the equations, (3.11), (3.12) and (3.13) respectively, for all species and making use
of normalized Ampere’s law Eq(3.14) together with normalized Gauss’s law Eq(3.15) the
resultant equation must be integrated once. Last equation (3.19) is more complex, so we
specify how it was obtained in more details. First step is to replace the pressure term
in longitudinal momentum equation (3.12) by
βs σγo2
∂X Ps
2
= N ∂X Ws . Second step is to
multiply the resultant equation by qs U Ws and summing up it for all species. In the last
step we use conservation of the energy flux (3.16) and the transverse momentum flux
(3.18) so that the resultant equation can be integrated once. It should be noticed that
we set the constant electric potential far upstream Φo so it eliminates the constant value
after integration.
18
Chapter 4
Electron-Positron Plasma
In this chapter we examine relativistic pair plasmas in the perpendicular regime using the
pseudopotential method. We first study a cold plasma and generalize further onto the hot
plasma case. We reproduce classical results for cold plasmas using our model as well as
some new results (dark solitons). Dark solitons have been covered by the literature but
not in an magnetized plasma [Pillay & Bharuthram 1992; Lee 2008]. Before proceeding
to our analysis we briefly describe the pseudopotential method.
Figure 4.1: The typical shape of the pseudopotential well that permits solitary wave
solutions. φinitial is the value far upstream, x → −∞. φeq is the root of Ψ = 0 and φcn is
the root of ∂Ψ/∂φ = 0.
In the pseudopotential theory, it is well known that for the solitary wave to exist the
19
pseudopotential should have the form shown in Fig. 4.1 [Lee 2008], the shape of which
can be characterized by the following conditions:
I. Ψ(φinitial ) = 0
II. ∂Ψ(φinitial )/∂φ = 0
III. ∂ 2 Ψ(φinitial )/∂φ2 < 0
IV. there exist φ = φeq such that Ψ(φeq ) = 0 and no singularities Ψ (φinitial < φ < φeq )
It should be noticed that compressive (bright) solitons are corresponds to φinitial < φeq
while rarefactive (dark) solitons are corresponds to φinitial > φeq .
4.1
Cold Plasma Classical Solutions
In a symmetric pair plasma ms = m, Ws = W , γs = γ, uy1 = −uy2 = V , Wx = 0. The
derived equations take the form
1
N
N
√ 2V
∂X B =
γo σ
γ = γo [1 + σ (1 − B)]
2β 2
B 2 = 1 + o (1 − U )
σ
U =
(4.1)
(4.2)
(4.3)
(4.4)
where
γ=
p
1 + βo2 γo2 U 2 + V 2
(4.5)
and W = 1/2 for the cold case. It is easy to derive the equation for the magnetic field in
the form
(∂X B)2 = −Ψ̃(B)
(4.6)
where the pseudopotential U (B) is a function of the magnetic field only:
h
i2 σ
2
2
2 2
2
γo [1 + σ(1 − B)] − 1 + βo γo 1 + 2β 2 (1 − B )
o
Ψ̃(B) ≡ −
h
i
2
γo2
σ 1 + 2βσ2 (1 − B 2 )
4
o
20
(4.7)
Then equation (4.6) becomes formally analogous to the equation of motion of a particle
in the a force field of potential Ψ̃(B), where the role of time and positions are played
by X and B, respectively. For solitary wave to exist conditions I-IV should be satisfied.
Condition I, Ψ̃(1) = 0, is automatically satisfied by the initial conditions as well as
condition II, ∂B Ψ̃(1) = 0. Condition III, ∂BB Ψ̃(1) < 0, which is often called the soliton
condition by itself, yields a well known result, Mach number should be higher than one,
that is MA2 > 1. Condition IV, Ψ(B) = 0, yields three possible roots, which are marked
with subscript one two and max:
B1 = B2 = 1,
Bmax =
2
γo
q
MA2 − 1 + γo2 − 1
Those are well behaved as long as the singularity in the denominator of equation (4.6)
is avoided. The denominator is simply a squared plasma velocity, U 2 , so the singularity
corresponds to U → 0 and N → ∞. Choosing B = Bmax we can find such Mach number
which sets the plasma velocity to be a zero, U = 0, using equation (4.4):
MC =
p
2(1 + γo )
(4.8)
this is the maximum Mach number permissible for solitary solutions. Using this critical
Mach number MC we can find the largest possible magnetic field amplitude.
∗
Bmax
= 1 + 2/γo
(4.9)
∗
= 3 when γo = 1.
In the non relativistic limit (4.8) and (4.9) reduces to MC = 2 and Bmax
While the relativistic limits permits an apparently much larger range of permissible Mach
numbers
1 < MA <
p
2(1 + γo )
than the non relativistic limit, this hole range corresponds to solitary waves of very small
maximum amplitudes.
1 < B ∗ < 1 + 2/γo
Thus, fast waves propagating perpendicular to magnetic field are restricted to low amplitudes in the relativistic limit.
21
Figure 4.2: Pseudopotential as function of magnetic field, where γo = 2 for all curves.
Dashed upper and dashed lower curves are corresponding to cases when there are no
solitons, MA < 1 and MA > MC , respectively. The other two corresponds to 1 < MA <
MC but different mach numbers. In all cases .
Figure (4.2) emphasize the classical soliton (compressive) solution for γo = 2 so
√
√
MC = 6 then in the range of 1 < MA < 6 solitons solutions exists. Moreover, in this
range it is easy to see that the amplitude rises from B = 1 at x → +∞ to maximum
magnetic filed Bmax and then returns to B = 1 at x → −∞, not reaching the singularity
when the denominator of the pseudopotential equal to zero or the plasma velocity. The
compressive solitons from the right side of figure (4.2) is well known and fully described
in the literature [e.g. Kennel & Pellat 1976]. But what happens when the magnetic field
reaches zero, the figure (4.2) tell us that the magnetic field continue to decrease, however
we know this is wrong, because the magnetic field cannot be negative, so there should be
a cutoff when magnetic field reaches zero.
22
4.2
Cold Plasma Non Classical Solutions
The problem when the magnetic field reaches zero can be avoid if we express every parameters as function of charge density, because charge density have positive values when
B = 0. Equation (4.6) will take the following form:
(∂X N )2 = −Ψ(N )
(4.10)
where
Ψ(N ) ≡ −
γo2 [1
+
γo2 −1
2 (1
MA
− B)] − 1 +
γo2 γo2 −1 1
2 N2
4 MA
2
γo2 −1
N2
(∂N B)2
(4.11)
The potential is pure function of the density (the prefixes pseudo and charge were neglected for simplicity), because the magnetic field is pure function of the density as well
as the derivative of the magnetic field with respect to density.
2MA2
1
2
B = 1+ 2
1−
γo
N
2
MA 1
∂N B =
γo2 BN 2
(4.12)
(4.13)
For solitary wave to exist conditions I-IV should be satisfied. Condition I, Ψ(1) = 0,
is automatically satisfied by the initial conditions as well as condition II, ∂N Ψ(1) = 0.
Condition III, ∂N N Ψ(1) < 0, yields non surprising result, that is MA2 > 1. Condition IV,
Ψ(N ) = 0, yields four possible roots, which are marked with subscript 1, 2, max and min:
N1 = 0
N2 = 1
Nmax =
Nmin =
MA2
p
= N (Bmax )
2 − MA2 − 2γo2 + 2 γo2 (−1 + MA2 + γo2 )
1
1+
γo2
2
2MA
N2 is the usual one which we could expect where as N1 is non physical. Nmax correspond
to Bmax which is not surprising. Nmin obtained by setting the magnetic field to be zero,
because Ψ ∝ (∂N B)−1 ∝ B. Those are well behaved as long as the singularity in the
denominator of equation (4.10) is avoided. The denominator is proportional to plasma
23
velocity, U , so the singularity corresponds to U → 0 and N → ∞ as in previous section,
p
so the critical Mach number is the same, MC = 2(1 + γo ), this is the maximum Mach
number permissible for solitary solutions.
Using this critical Mach number MC we can find the largest possible magnetic field
amplitude, density and the smallest possible density.
∗
∗
∗
=
= ∞, Nmin
= 1 + 2/γo , Nmax
Bmax
4(1 + γo )
(2 + γo )2
(4.14)
∗
∗
= 8/9
= 3 and Nmin
In the non relativistic limit (4.8) and (4.14) reduces to MC = 2, Bmax
when γo = 1. While the relativistic limits permits an apparently much larger range of
√
permissible Mach numbers and density, that is 1 < MA < 2γo and N > 4γo−1 , than
the non relativistic limit, this whole range corresponds to solitary waves of very small
√
maximum amplitudes , 1 < B ∗ < 1 + 2/γo . Setting γo = 2 and 1 < MA < 6 we obtain
two solitons, compressive for N > 1 and rarefactive for 0 < N < 1 figure (4.3).
(a)
(b)
Figure 4.3: Two curves for different MA and γo = 2. Each curve represents two solitons
compressive from the right and rarefactive from the left. (a) Zoom in on 0 < N < 1.
Rarefactive solitons and (b) Zoom in on 1 < N < 6. Compressive solitons.
In figure (4.3) the intersection points with x axis are corresponding to Nmin from
the left of N = 1 and to Nmax from the right of N = 1, which are increased when
Mach number is growing up. It should be noted that the only condition for existence of
compressive solution is that the Mach number should be higher than one and less than
the critical Mach number, 1 < MA < MC , as demonstrated in figure(4.4a) by upper solid
24
curve. While for rarefactive solutions the Mach number should be higher than one only,
MA > 1, and can exceed the critical mach number as it shown in figure(4.4b) by lower
solid curve. However, for Mach numbers that less then one there are no solitons of any
kind, (4.4) dashed curve.
(a)
(b)
Figure 4.4: Same Plot for different N where γo = 3 and MA = 0.9, 2, 4. Each curve
represent different regime. Dashed curve for MA < 1 no solitons of any kind. Solid upper
curve for 1 < MA < MC both rarefactive and compressive solitons are shown. Solid lower
curve for MA > MC only rarefactive solitons is shown. (a) Zoom in on 0 < N < 1 and
(b) Zoom in on 1 < N < 6.
To understand why the rarefactive solution is less restricted one has to remember
that the restriction of MA < MC is coming from the singularity of the pseudopotential
which corresponds to N → ∞ where as the rarefactive solution corresponds to Nmin <
N < 1. Finally, we describe in details the magnetic profile, figure(4.5). Compressive
solitary waves exist when 1 < MA < MC , in this regime magnetic field amplitude rises
from B = 1 at x → +∞ to maximum magnetic filed Bmax and then returns to B = 1
at x → −∞ where the first derivative of the magnetic field with respect to normalized
distance at B = Bmax is zero, that is ∂X B(Bmax ) = 0, meaning magnetic profile looks like
a parabola in the vicinity of Bmax . Increasing MA leads to the increase in the amplitude
of the maximum magnetic field until MA → MC . Moreover, the profile has cusp behavior
∗
∗
at B = Bmax
, because [∂X B(Bmax
)]2 > 0.
25
Rarefactive solitary waves exist when MA > 1, in this regime magnetic field amplitude decreases from B = 1 at x → +∞ to zero magnetic filed and then returns to B = 1
at x → −∞. The profile has cusp behavior at B = 0, because [∂X B(0)]2 > 0 according
to Eq(4.6).
Figure 4.5: Magnetic profile sketch. The upper curve is compressive soliton while the
lower is rarefactive soliton.
4.3
Hot Plasma
In the hot pair plasma both species have identical ratio of pressure far upstream to the
magnetic pressure far upstream, that is β1 = β2 = β/2, because we have assumed that
both species have the same pressure far upstream, po1 = po2 = po , and same polytropic
indexes, Γ1 = Γ2 = Γ . Hence, both species have same enthalpy, W1 = W2 = W/2 and
far upstream Wo1 = Wo2 = Wo /2. The equations read
N
√ 2V
γo σ
W γ = γo [1 + σ (1 − B)]
∂X B =
(4.15)
(4.16)
2βo2
W
2
Γ
B = 1 + β(1 − N ) +
1−
σ
N
2
Γ βσγo
W = 1+
N Γ−1 − 1
p Γ−1 2
γ =
1 + βo2 γo2 /N 2 + V 2
26
(4.17)
(4.18)
(4.19)
Using ∂X B = (∂X N )(∂N B) it is easy to derive:
(∂X N )2 = −Ψ(N )
(4.20)
where Ψ(N ) is the pseudopotential
Ψ(N ) ≡
γo2 [1 +
γo2 −1
2 (1
MA
− B)]2 − W 2 1 +
2 2 −1
W2
o
− γ4o γM
2 N2
A
2
(∂N B)
γo2 −1
N2
≡
f1 (N )
f2 (N )
(4.21)
where
Γ βγo2 (γo2 − 1)
Γ−1
N
−
1
Γ−1
2MA2
Γβγo2 (γo2 − 1) Γ−2
∂N W =
N
2MA2
2MA2
W
2
Γ
B = 1 + β(1 − N ) + 2
1−
γo
N
2
2MA N ∂N W − W
1
Γ−1
−ΓβN
− 2
∂N B =
2B
γo
N2
W = 1+
(4.22)
(4.23)
(4.24)
(4.25)
For solitary wave to exist conditions I-IV should be satisfied. Condition I, Ψ(1) = 0,
is automatically satisfied by the initial conditions. For condition II we need the first
derivative of the pseudopotential, that is
f10 (N )f2 (N ) − f1 (N )f20 (N )
f22 (N )
γo2 − 1
γo2 − 1
0
2
f1 (N ) = 2γo 1 +
(1 − B) (−)
∂N B
MA2
MA2
2
γo2 − 1
2 (γo − 1)
∂
W
+
2W
−2W 1 +
N
N2
N3
2 2
γ γ − 1 W2
f20 (N ) = −∂N o o 2
(∂N B)2
4 MA N 2
Ψ0 (N ) =
(4.26)
(4.27)
(4.28)
Prime is derivative respect to the density. It easy to see that f20 (1) = 0, because Ψ(1) = 0.
Imposing initial conditions together with Eq(4.23) and Eq(4.25) we obtain that f10 (1) = 0.
Hence, condition II is fulfil, ∂N Ψ(1) = 0. For condition III we need the second derivative
27
of the pseudopotential, that is
0
f1 (N )f20 (N )
f100 (N ) f10 (N )f20 (N )
+
(4.29)
∂N N Ψ(N ) = −
+
f2 (N )
f22 (N )
f22 (N )
2
2γo2 (γo2 − 1)
γo2 − 1
γo − 1
2
00
f1 (N ) = −
1+
(1 − B) ∂N N B −
(∂N B)
MA2
MA2
MA2
γo2 − 1 γo2 − 1
2
−2 1 +
(∂
W
)
+
W
∂
W
+
4W
∂N W
N
N
N
N2
N3
2W N 3 ∂N W − 3N 2 W 2
2
(4.30)
+2(γo − 1)
N6
where the second derivative of the enthalpy and the magnetic field are produced from
Eq(4.23) and Eq(4.25), respectively, that is
∂N N W =
Γ(Γ − 2)βγo2 (γo2 − 1) Γ−3
N
2MA2
(4.31)
2B∂N N B = −2 (∂N B)2 − Γ(Γ − 1)βN Γ−2
N 2 ∂N W − 2N W
2MA2 N ∂N N W − ∂N W
−
− 2
γo
N2
N4
(4.32)
f 00 (1)
Remembering that f1 (N ) = f10 (N ) = 0 at N = 1, so ∂N N Ψ(1) = − f12 (1) . Imposing
boundary condition we obtain
γo2 − 1
Γβγo4
Γβγo4
00
2
2
f1 (1) = 2 2 2 MA −
MA − 1 −
γo MA
2
2
2
Γβγo4
γo2 − 1
2
M
−
f2 (1) =
A
2γo2 MA2
2
(4.33)
(4.34)
Condition III, ∂N N Ψ(1) < 0, is fulfilled at two regimes, that is
a. MA2 −
Γβγo4
2
> 0 and MA2 − 1 −
Γβγo4
2
>0
b. MA2 −
Γβγo4
2
< 0 and MA2 − 1 −
Γβγo4
2
<0
The β parameter is related to the magnetic field. In order to characterize the plasma
temperature we introduce T =
β=
po1 +po2
,
2mc2 no
so that
2MA2
T
Γ
2
2
γo (γo − 1) 1 + Γ−1
T
28
The two regimes ( a and b ) take the following form:
a. 1 − T /T ∗ > 0 and MA2 (1 − T /T ∗ ) − 1 −
ΓT
Γ−1
>0
b. 1 − T /T ∗ < 0 and MA2 (1 − T /T ∗ ) − 1 −
ΓT
Γ−1
<0
where
1
T∗
=Γ
h
γo2
γo2 −1
−
1
Γ−1
i
. We do not deal specially with the second regime, (b), because
if 1 − T /T ∗ < 0 then condition III is fulfilled regardless the Mach number. According
to the first regime, (a), the minimum Mach number can be found for solitary waves to
exists, that is
s
MA > MA,min ≡
Γ
1 + Γ−1
T
(1 − T /T ∗ )
(4.35)
if T ∗ < 0 or T ∗ > T > 0, where former is true for Γ < 2 and γo2 >
true when Γ ≥ 2 or Γ < 2 and γo2 <
1
.
2−Γ
1
2−Γ
while latter is
Equation (4.35) can be reduced to the cold case
by setting T = 0, that is MA > 1. Let us summarize when condition III, ∂N N Ψ(1) < 0, is
fulfilled :
III.a) T /T ∗ > 1 no restriction on MA
III.b) Γ < 2, γo2 >
1
,
2−Γ
MA > MA,min
III.c) Γ < 2, γo2 <
1
,
2−Γ
MA > MA,min , T ∗ > T
III.d) Γ ≥ 2, MA > MA,min , T ∗ > T
However, condition IV, Ψ(N ) = 0, can not be solved analytically. Nevertheless, we
can plot the pseudopotential using Mathematica for different set of parameters, that is
Γ, T, γo and MA , which be able to predict if there are solitary waves. For this purpose
we present four examples, where Γ, T, γo remain constants while MA vary. Each example
is exhibit one of the four cases of condition III.
29
4.3.1
Singularity & Maximum Amplitude
We can determine the maximum density, that is Nmax , at the center of a soliton, X = 0,
by setting the pseudopotential to zero, that is f1 (Nmax ) = 0 from Eq(4.21). However,
the exact expression of Nmax can not be found as function of T, Γ, MA and γo , since
f1 (Nmax ) = 0 is transcendental equation, in contrast to the cold case where Nmax =
h
i−1
p
MA2 2 − MA2 − 2γo2 + 2 γo2 (−1 + MA2 + γo2 ) . Numerically solving it, one should remember that according to condition IV (defined in chapter 4) Nmax ≤ Nsp , here Nsp is a
singular point where pseudopotential diverge. Moreover, as MA increases, Nmax increases
figures (4.6c, 4.7c, 4.8c) and (4.9a) until Nmax = Nsp the corresponding MA is MC .
We can determine the singular point, that is Nsp , by letting denominator of pseudopotential be equal to zero, f2 (N ) ∝ [(∂N B)W/N ]2 = 0 from Eq(4.21). Magnetic field is
restricted by the pressure term in Eq(4.24), thus maximum density must be finite, in contrast to the cold case where it diverge. Hence, W/N 6= 0 and the only term which can be
zero is the derivative of magnetic filed with respect to the density, that is ∂N B(Nsp ) = 0,
so Nsp is a solution to
N Γ (γo2 − 1)(Γ − 2) + N Γ+2 (Γ − 1) =
(γo2 − 1)(Γ − 1)
N
ΓT
(4.36)
This transcendental equation cannot be solved by algebraic methods. Nevertheless, for
Γ ≥ 2 the left and right functions of Eq(4.36) are monotonically increasing functions,
thus the intersection point corresponds to Nsp . It is easy to see that MA has no influence
on Nsp , while increasing γo or decreasing T enlarges the value of Nsp . Moreover, for
1 < Γ < 2, as N increases the left side of Eq(4.36) decreases monotonically, reaches a
q 2
o −1)(2−Γ)
minimum at N = (γ(Γ−1)(Γ+2)
, and then increases monotonically eventually intersecting
with the right side of Eq(4.36). Hence, for ultrarelativistic plasma, γo 1, Nsp is of order
γo as well as Nmax when MA = MC .
30
4.3.2
Numerical Analysis
Example 1 (IIIb)
(a)
(b)
(c)
Figure 4.6: Pseudopotential as function of density which fulfill IIIb, where Γ = 1.8, T =
0.7, MA,min ≈ 1.49, MC ≈ 2.143, γo = 3 and MA = 1.4, 1.85, 2, 2.5 corresponding to
curves 1,2,3,4 respectively. (a) Zoom in on 0 < N < 1. Rarefactive solitons exist when
MA > MA,min curves 2,3,4, (b) Zoom in on 1 < N < 2. Compressive soliton exists when
MA,min < MA < MC , moreover, as MA increases Nmax increases curves 2,3. (c) Zoom in
on 1 < N < 12. Nsp ≈ 2.29 for all curves .
31
Example 2 (IIIc)
(b)
(a)
(c)
Figure 4.7: Pseudopotential as function of density which fulfill IIIc, where Γ = 1.9, T =
0.2, MA,min ≈ 1.19, MC ≈ 1.875, T ∗ ≈ 37.89, γo = 3 and MA = 1.1, 1.7, 1.8, 2.5
corresponding to curves 1,2,3,4 respectively. (a) Zoom in on 0 < N < 1.1. Rarefactive
solitons exist when MA > MA,min curves 2,3,4, (b) Zoom in on 1 < N < 2.5. Compressive
soliton exists when MA,min < MA < MC , moreover, as MA increases Nmax increases curves
2,3. (c) Zoom in on 1 < N < 20. Nsp ≈ 3 for all curves .
32
Example 3 (IIId)
(b)
(a)
(c)
Figure 4.8: Pseudopotential as function of density which fulfill IIId, where Γ = 2, T =
0.9, MA,min ≈ 1.19, MC ≈ 2.491, T ∗ ≈ 7.5, γo = 4 and MA = 1.6, 2, 2.1, 2.5 corresponding to curves 1,2,3,4 respectively. (a) Zoom in on 0 < N < 1.1. Rarefactive solitons
exist when MA > MA,min curves 2,3,4, (b) Zoom in on 1 < N < 1.5. Compressive soliton
exists when MA,min < MA < MC , moreover, as MA increases Nmax increases curves 2,3.
(c) Zoom in on 1 < N < 15. Nsp ≈ 2.03 for all curves .
33
Example 4 (IIIa)
(a)
(b)
Figure 4.9: Pseudopotential as function of density which fulfill IIIa, where Γ = 2.1, T =
0.9, T ∗ ≈ 0.88, γo = 1.7 and MA = 0.8, 1.5, 2 corresponding to curves 1,2,3 respectively.
(a) Zoom in on 0 < N < 1.1. Rarefactive solitons do not exists, since condition IV is not
fulfilled for all curves and Nsp ≈ 0.944. (b) Zoom in on 1 < N < 4. Compressive soliton
exists regardless the value of MA , as MA increases Nmax increases for all curves.
4.4
Magnetic Profile & Pair Plasma Summarize
Cold Pair Plasma
• Compressive solitons emerge when 1 < MA < MC =
p
2(1 + γo ), figure (4.4a),
rarefactive solitons emerge when MA > 1, figure (4.4b).
• As MA increases, the magnetic field amplitude of the compressive soliton increases,
figure (4.10a), until R = MA /Mc = 1, where cusp appears (see figure (4.11a)), since
∗
[∂x B(Bmax
)]2 > 0. The minimum magnetic field for rarefactive solitons is zero and
the magnetic field profile has a cusp for any value of MA , (figure (4.10b)), since
[∂x B(0)]2 > 0.
• The maximum magnetic field and density for compressive solitons when R 6= 1 are
q
MA2
2
2
2
p
Bmax =
MA − 1 + γo − 1 , Nmax =
γo
2 − MA2 − 2γo2 + 2 γo2 (−1 + MA2 + γo2 )
34
• The largest possible magnetic field amplitude is reached when R = 1
∗
Bmax
= 1 + 2/γo
∗
which corresponds to U → 0 and Nmax
→ ∞. The non relativistic limit permits
a much larger range of magnetic field amplitudes, B ≤ 3. Fast waves propagating
perpendicular to magnetic field are restricted to low amplitudes in the relativistic
limit.
(a)
(b)
Figure 4.10: Magnetic profiles for γo = 10 with different Mach numbers, R = MA /MC =
1, 0.8, 0.6, 0.4 . (a) Compressive solitons and (b) Rarefactive solitons.
(a)
(b)
Figure 4.11: Zooming on Magnetic profiles for γo = 10 at B = Bmax . (a) For R = 1 there
is a cusp behavior (b) For R 6= 1 (e.g. R = 0.8) there is no cusp, since ∂X B(Bmax ) = 0.
35
Hot Pair Plasma
• For solitary waves to exists conditions I-IV should be satisfied. Condition III is
fulfilled in two different regimes (see examples 1-4 from Chapter 4), that is
a) T /T ∗ > 1 - no rarefactive solitons (figure (4.9a)), compressive solitons emerge
regardless the value of MA , figure (4.9b). We do not deal in detail with this
regime, since it valid for Γ ≥ 2 or when Γ < 2 and γo2 <
1
.
2−Γ
b) T /T ∗ < 1 and MA > MA,min - rarefactive solitons emerge when MA > MA,min
while compressive solitons exist when MA,min < MA < MC , (see examples 1-3).
r
h
i
where
1
T∗
=Γ
γo2
γo2 −1
−
1
Γ−1
and MA,min ≡
Γ
T
1+ Γ−1
.
(1−T /T ∗ )
Latter can be reduced to the
cold case by setting T = 0, that is MA,min = 1. There is a critical Mach number MC ,
figures (4.6)-(4.8), however it can not be found analytically because of complexity
of our equations.
∗
has a finite value at R = 1, in contrast to the cold case where
• Maximum density Nmax
it diverges. This maximum density increases with the increase of γo or decrease of
∗
T . Moreover, for ultrarelativistic plasma γo 1, Nmax
is of order γo .
• As MA increases while Γ, γo and T remain constant, the magnetic field amplitude
increases (figure (4.12a)), until R = 1 for compressive solitons. At R = 1 the
magnetic profile has a cusp which is related to the singularity of the pseudopotential
there. An increase of MA leads to the decrease of the width of the soliton at half
maximum of the magnetic field for compressive solitons and rarefactive solitons as
well (figure (4.12b)).
• As γo increases while Γ, R and T remain constant, the magnetic field amplitude
decreases, figure (4.13a,b). Moreover, same increase of γo leads to increase of the
width of rarefactive solitons figure (4.13c), while the width of the compressive solitons decreases.
• Decreasing T and keeping Γ, R and γo as constants leads to the increase in the
magnetic field amplitude (figure (4.14a)). Moreover, same decrease leads to the
36
increase of the width for compressive solitons while the width of the rarefactive
solitons does not change, figure (4.14b).
• For the reasons above we can confirm that fast waves propagating perpendicular to
magnetic field are restricted to low amplitudes in the hot relativistic limit as well
as cold.
(a)
(b)
Figure 4.12: Magnetic profiles for γo = 10, Γ = 4/3, T = 1 and different Mach numbers
where MA,min = 1.169 and MC = 3.579. (a) Compressive solitons for MA = 1.8, 2, 3 (b)
Rarefactive solitons for MA > 1.16.
It should be noted that for comparison of the the typical length scales (in physical
units) of the hot solitons with the cold ones, one must multiply the width by the factor
q
Γ
1 + Γ−1
T , because of the normalization of X.
37
(a)
(b)
(c)
Figure 4.13: Magnetic profiles for R = 0.5, Γ = 4/3, T = 1 and different γo . (a) Compressive solitons, (b) Zoom in on the center of a compressive solitons, no cusp behavior.
and (c) Rarefactive solitons.
(a)
(b)
Figure 4.14: Magnetic profiles for γo = 10, Γ = 4/3, R = 0.5 and different T . (a)
Compressive solitons (b) Rarefactive solitons .
38
Chapter 5
Electron-Ion Plasma
In this chapter we examine relativistic electron-ion plasmas in the perpendicular regime
using the pseudopotential method. All equations, (3.9)-(3.19), and boundary conditions
that were deduced in chapter 3 are applicable here, as well as conditions for existence of
solitons, I-IV, from chapter 4.
W1 γ1 + W2 γ2 = γo [1 + σ (1 − B)]
(5.1)
B 2 − E 2 = 1 + β1 (1 − N Γ1 ) + β2 (1 − N Γ2 ) +
W1 V1 + W2 V2 = σγo E
√
2γo2 σ(1 + σ)Φ = − W12 − W22 1 + βo2 γo2 /N 2
Γs βs σγo2
N Γs −1 − 1
Ws = Wos +
Γs − 1 2
N
∂X B = − √ (V1 − V2 )
γo σ
N
√ (γ1 − γ2 )
∂X E =
γo σ
and



X





N, U , Vs 


B, E




Ps , P 




Ws , W 
X → −∞
1 ,1 , 0
→
1, 0
1, 2
c2
ms
wo
+
Γs βs σγo2
Γs −1 2
39
, 1
2βo2
σ
(1 − W/N )
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
The lower indexes 1 and 2 stand for ion and electron, respectively. First, let us express
the normalized electric field, E, as function of density and it’s derivatives with respect
to normalized distance, using the expression for the electric potential, Eq(5.4), and E =
−∂X Φ.
E = (∂X N )∂N
(W12 − W22 ) (1 + γo2 βo2 /N 2 )
√
≡ (∂X N )(∂N Φ)
2γo2 σ (1 + σ)
(5.8)
Using Ampere’s law, Eq(5.6), and conservation of longitudinal momentum, Eq(5.3), we
can express V1 and V2 as function of N, E, and ∂X B.
γo σ
W2
V1 =
E−√
∂X B
W
σN
γo σ
W1
V2 =
∂X B
E+√
W
σN
(5.9)
(5.10)
Besides the quasineutrality assumption we shell assume that magnetic pressure is much
bigger than electric pressure, that is B 2 E 2 , so the conservation of transverse momentum reads, Eq(5.2):
B 2 = 1 + β1 (1 − N Γ1 ) + β2 (1 − N Γ2 ) +
2βo2
(1 − W/N )
σ
(5.11)
and
2βo2 N ∂N W − W
1
Γ1 −1
Γ2 −1
∂N B =
−Γ1 β1 N
− Γ2 β2 N
−
2B
σ
N2
Γs βs σγo2 Γs −2
N
∂N Ws =
2
(5.12)
(5.13)
Moreover, we can express V1 and V2 as function of N and ∂X N , that is
Vs = gs ∂X N
(5.14)
where
g1
g2
γo σ
≡
∂N Φ −
W
γo σ
≡
∂N Φ +
W
W2
√
∂N B
σN
W1
√
∂N B
σN
(5.15)
(5.16)
Using the definition of Lorentz factor, γs2 = 1 + βo2 γo2 /N 2 + Vs2 , in conservation of energy,
Eq(5.1) together with equation (5.14) we obtain
q
q
W1 1 + βo2 γo2 /N 2 + g12 (∂X N )2 + W2 1 + βo2 γo2 /N 2 + g22 (∂X N )2 = γo [1 + σ (1 − B)]
40
Squaring above equation twice to eliminant the roots we have
Λ(∂X N )4 − 2Π(∂X N )2 + Ξ = 0
(5.17)
where
Λ(N ) = (W12 g12 − W22 g22 )2
βo2 γo2
2
2
2
2
(W1 + W2 ) (W12 g12 + W22 g22 )
Π(N ) = γo [1 + σ(1 − B)] − 1 +
2
N
2 2
β γ
+2W12 W22 1 + o 2o (g12 + g22 )
N
2
βo2 γo2
2
2
2
2
Ξ(N ) = γo [1 + σ(1 − B)] − 1 +
(W1 + W2 )
N2
2
βo2 γo2
2
2
−4W1 W2 1 +
N2
(5.18)
(5.19)
(5.20)
Equation (5.17) is usual quadratic equation with respect to (∂X N )2 having the following
solutions
2
(∂X N ) =
Π±
√
Π2 − ΛΞ
Λ
We can neglect the plus solution of equation (5.17) because it does not satisfy the boundary conditions, that is Ξ = 0 and Π > 0 when N = 1. Finely
(∂X N )2 = −Ψ(N )
where Ψ(N ) is the pseudopotential which is a pure function of density, defined as
√
Π − Π2 − ΛΞ
Ψ(N ) ≡ −
(5.21)
Λ
For solitary wave to exist conditions I-IV should be satisfied. Condition I, Ψ(1) = 0,
is automatically satisfied by the initial conditions as well as condition II, Ψ0 (1) = 0,
because Ξ = 0, Π > 0 and Ξ0 = 0 when N = 1. The latter is somewhat tedious but
straightforward (see Appendix A). Prime symbolize derivative with respect to the density.
For condition III we need to differentiate twice the pseudopotential, Eq(5.21), and impose
initial conditions, thus
Ψ00 (1) = −
Ξ00 (1)
2Π(1)
(5.22)
41
According to Eq(5.19) Π(1) is positive, that is
2
2
>0
+ Wo2 go2
Π(1) = 2γo2 Wo1 Wo2 Wo1 go1
and
2
2
Ξ00 (1) = 4γo4 (Wo1
+ Wo2
)δ(1)
where (for details see Appendix A)
βo2
γo4
γo4
2
2
δ(1) =
MA − (β1 Γ1 + β2 Γ2 ) MA − 1 − (β1 Γ1 + β2 Γ2 )
MA2 γo2
2
2
The only requirement that condition III, Ψ00 (1) < 0, will be fulfilled is δ(1) > 0. We
express βs in terms of Ts =
β1 =
pos
,
ms c2 no
2T1 /γo2 σ
1
1 + Γ1Γ−1
T1 + µ1 1 +
so that
Γ2
T
Γ2 −1 2
,
Above relations are easily obtained using
δ(1) =
β2 =
4π
Bo2
=
2T2 /γo2 σ
1
µ 1 + Γ1Γ−1
T1 + 1 +
1
no wo γo2 σ
Γ2
T
Γ2 −1 2
where µ = m1 /m2 . Hence,
βo2 2 2
G
M
−
1/G
A
γo2
where
G≡1−
γo2
γo2 −


µT1 Γ1 + T2 Γ2
 1 µ 1 + Γ1 T + 1 +
Γ1 −1 1
Γ2
T
Γ2 −1 2

(5.23)
Condition III, is fulfilled in two cases:
a. G < 0
b. G > 0 and MA2 > 1/G
In the cold limit, Ts = 0, where G = 1 only the second case of condition III is valid,
hence MA > 1. Condition IV, Ψ(N ) = 0, can not be solved analytically in the hot
limit. Nevertheless, we can plot the pseudopotential using Mathematica for different set
of constant values, that is Γ1 , Γ2 , T1 , T2 , γo and MA , which be able to predict if there are
solitary waves. However, in the cold limit when Ts = 0 condition IV, Ψ(N ) = 0, can be
42
solved analytically. According to Eq(5.21) there is such N which set the pseudopotential
to zero, when Ξ = 0, where
βo2 γo2
2
2
·
Ξ(N ) = γo [1 + σ(1 − B)] − 1 +
N2
βo2 γo2
2
2
2
γo [1 + σ(1 − B)] − 1 +
(Wo1 − Wo2 )
N2
2β 2
B 2 = 1 + o (1 − 1/N )
σ
It is sufficient to find roots of the first brackets of Eq(5.24), since Wos =
(5.24)
(5.25)
ms
m1 +m2
and
m1 m2 , by squaring them once together with Eq(5.25), obtaining
#
2
2 "
1
σ
1
1−
+ 1 + 2σ − 4(1 + σ) 2 = 0
N
N
βo
using the definition of σ =
βo2 γo2
2 ,
MA
solutions to above are
N = 1
Nmax =
MA2 −
(5.26)
MA2
2s2 +
(5.27)
2γo s
those are the roots of Ξ = 0, where we have the neglected the negative root, since N > 0,
where
s2 = MA2 + γo2 − 1
and the magnetic field at N = Nmax , according to Eq(5.25) is
Bmax =
2s
−1
γo
Critical Mach number can be found by setting the denominator of the pseudopotential
to zero, Π = 0. In the cold limit Π ∝ N −3 f (γo , Wo1 , Wo2 , MA ). Hence, the singularity
corresponds to N → ∞ and U → 0 or f = 0. The latter is true for negative MA , so it
may be neglected. While the former leads to the same MC which were obtained in pair
plasma, that is
MC =
p
2(1 + γo )
which in turn leads to same results that where obtain in pear plasma at least for compressive solitons.
43
5.1
Numerical Analysis
Example 1 (Cold Plasma)
The setup is as follows: Ts = 0, γo = 2, m1 /m2 = 2000 and MA = 0.9, 2, 2.5 so MC ≈ 2.45.
According to condition III there should be no solitons for MA = 0.9 < 1 see figure (5.1a),
continuous curve. Compressive solitons do exist for MA = 2 < MC , since conditions I-IV
are fulfilled, as demonstrated in figure (5.1) by dashed curve, where the intersection of the
x axis is corresponds to Nmax = 6.86. For MA = 2.5 > MC there are no solitary waves see
figure (5.1b), dotted-dashed curve, since Λ = 0. In our model for cold electron ion plasma
we can not confirm or disconfirm existence of rarefactive solitons, since the assumption
of B 2 E 2 is no longer valid see figure (5.2a), where E 2 /B 2 is plotted as function of N .
(a)
(b)
Figure 5.1: Same Plot for different N where γo = 2, Ts = 0. Dashed, dotted-dashed
and continuous curves corresponds to MA = 2, 2.5, 0.9, respectively. (a) Zoom in on
0 < N < 1.5 and (b) Zoom in on 1 < N < 7.
As a result to our assumption there is a minimum density, Nmin =
1
1+
2
γo
2M 2
A
, which cor-
responds to zero magnetic field. Otherwise, according to Eq(5.25) magnetic field becomes
imaginary. Hence, the curves’ cutoff in figure (5.1a) which occurs when N = Nmin .
44
The maximum deviation of E 2 /B 2 , which will be defined as η, in the regions where
there are compressive solitons, 1 < MA < MC , is a function of two parameters MA and
γo , that is η(MA , γo ). When γo = 2 the deviation η ∼ 1% for MA = 1 and η ∼ 10% for
MA = MC see figure (5.2c), dashed curves. Thus, enlargement of MA causes enlargement
of η, breaking down our assumption.
(a)
(b)
(c)
Figure 5.2: Same Plot for different N where T = 0, γo = 2. Lower dashed, continuous
and upper dashed curves corresponds to MA = 1, 2, MC , respectively. (a) Zoom in on
0 < N < 1, (b) Zoom in on 1 < N < 7 and (c) Zoom in on 1 < N < 70.
However, small enlargement of γo causes a significant decreasing of η. When γo = 5
the deviation η ∼ 0.006% for MA = 1 and η ∼ 0.73% for MA = MC see figure (5.3),
dashed curves.
(a)
(b)
(c)
Figure 5.3: Same Plot for different N where T = 0, γo = 5. Lower dashed, continuous
and upper dashed curves corresponds to MA = 1, 2, MC , respectively. (a) Zoom in on
0 < N < 1, (b) Zoom in on 1 < N < 7 and (c) Zoom in on 1 < N < 70.
45
Example 2 (Hot Plasma, G < 0)
As we mentioned before the roots of Ψ(N ) = 0 in hot plasma can not be found because
of complexity of our equations. However, it can be done graphically. In accordance with
condition III there are two possible regimes, G < 0 and G > 0, MA2 > 1/G. The former is
treated in this section while the letter in the next one. In electron ion plasma µ → 2000
so G, Eq(5.23), highly depend on ions’ temperature and polytropic index. In our model
for cold electron ion plasma we can not confirm or disconfirm existence of compressive
solitons, since the assumption of B 2 E 2 is no longer valid see Fig.(5.4), when G < 0
which corresponds to high Γ1 > 2, Fig.(5.4a), or low γo → 1, Fig.(5.4d), there is no
dependents on Mach number. However, there is no rarefactive solitons.
(a)
(b)
(c)
(d)
Figure 5.4: Plot for G < 0 where , Γ2 = 1.8, T2 = 0.8, µ = 2000. Dashed and continuous
curves corresponds to E 2 /B 2 and Ψ(N ), respectively. (a) Const γo = MA = 2, T1 = 0.9
and Γ1 = 2.7, 2.9 (b) Const MA = 2, T1 = 0.9, Γ1 = 3 and γo = 1.5, 3 (c) Const
γo = 2, T1 = 0.9, Γ1 = 3 and MA = 2, 3 (d) Const MA = 2, T1 = 0.8, Γ1 = 1.8 and
γo = 1.2, 1.4 .
46
Example 3 (Hot Plasma, G > 0)
When G > 0 there is a minimum Mach number for condition III to be satisfied, that
is MA2 > 1/G, demonstrated in figure (5.5) where T1 = T2 = 1, Γ1 = Γ2 = 1.6 and
γo = 4 so G = 0.53. According to figure (5.5c) when MA = 2.5 and η ∼ 0.1%, there is no
compressive soliton because the singularity of the pseudopotential, Λ = 0, is encountered
before condition IV is fulfilled. Hence, implies on existence of critical Mach number.
Compressive solitons do exist for MA = 2 < MC and η ∼ 0.04%, since conditions I-IV
are fulfilled, as demonstrated in figure (5.5b). In our model for cold electron ion plasma
we can not confirm or disconfirm existence of rarefactive solitons, since the assumption
of B 2 E 2 is no longer valid see figure (5.5d).
(a)
(b)
(c)
(d)
Figure 5.5: Plot for G > 0 where T1 = T2 = 1, Γ1 = Γ2 = 1.6, γo = 4 so G = 0.53 and
MA = 1.3, 2, 2.5. (a) No rarefactive solitons in 0 < N < 1, (b) Zoom in on 1 < N < 2.5
where there is compressive soliton for MA = 2 (c) Zoom in on 1 < N < 7 singularity of
Ψ, (d) deviation from E 2 /B 2 1 .
47
5.2
Magnetic Profile
& Electron-Ion Plasma Summarize
Cold Electron-Ion Plasma
• In our model for cold electron ion plasma we can not confirm or disconfirm existence
of rarefactive solitons, since the assumption of B 2 E 2 is no longer valid, figure
(5.1a).
• Compressive solitons emerge when 1 < MA < MC =
p
2(1 + γo ), figure (5.1b).
• Enlargement of MA causes to enlargement of η, meaning for braking down of our assumption, B 2 E 2 . However, small enlargement of γo causes a significant decrease
of η, figure (5.2) and figure (5.3).
• Increasing MA leads to increase in the magnetic field amplitude and decrease of
the soliton width, figure (5.6), until R = 1 thus causing to cusp behavior, since
∗
)]2 > 0.
[∂x B(Bmax
• Increasing γo and keeping R as constant leads to decrease in the magnetic field
amplitude and the soliton width, figure (5.7).
• The largest possible magnetic field amplitude
∗
Bmax
= 1 + 2/γo
is attain when R = 1, which corresponds to U → 0 and N → ∞ is identical to the
Pair plasma case. It is reasonable that the particle inertia be determined by the
wave amplitude, not by the rest masses in the relativistic limit. Thus confirming
that fast waves propagating perpendicular to magnetic field are restricted to low
amplitudes in the relativistic limit.
48
Figure 5.6: Magnetic profiles for γo = 10 with different Mach numbers, R = MA /MC =
0.99, 0.8, 0.6, 0.4 , where for R = 0.99 there is cusp. While l = 0.7, 0.8, 1.1, 1.7 from the
highest to the lowest R, respectively.
(a)
(b)
(c)
Figure 5.7: Magnetic profiles for R = 0.5 with different γo , γo = 101 , 102 , 103 , 104 . And
l = 0.69, 0.7, 0.75, 1.3 from the highest to the lowest γo , respectively. (a) Compressive
solitons for γo = 10, 102 , (b) Compressive solitons for γo = 103 , 104 and (c) There is no
cusp for any γo .
49
Hot Electron-Ion Plasma
• For solitary waves to exists conditions I-IV should be satisfied. Condition III is
fulfilled in two different regimes (see examples 2,3 from Chapter 5), that is
a) G < 0 - No rarefactive solitons figure (5.4), because the singularity of the
pseudopotential is encountered before condition IV is fulfilled. No compressive
solitons since the assumption is not valid, figure (5.4).
b) G > 0 and MA2 > 1/G - No rarefactive solitons since the assumption, B 2 E 2
is not valid, figure (5.5d). Compressive solitons emerge when MA,min < MA <
MC , figure (5.5b,c).
γo2
where G ≡ 1 − γ 2 −1 o
µ
µT1 Γ1 +T
2 Γ2
Γ1
T1
1+ Γ −1
1
Γ2
T2
+1+ Γ −1
2
and MA,min ≡
p
1/G. Latter can be
reduced to the cold case by setting T = 0, that is MA,min = 1. It is easy to see that
G is controlled by Ions’ parameters till T2 ∝ µT1 . There is a critical Mach number
MC , figure (5.5), however it can not be found analytically because of complexity of
our equations.
• Magnetic profile for compressive soliton has a cusp when MA = MC , this fact is not
accompanied by any of our figures, but can easily be explained. The pseudopotential
has a singularity when B = Bmax and MA = MC , thus (∂X B)2 6= 0.
• According to Table 5.1, Increasing Te , Ti or γo lowers the magnetic field amplitude,
thus we confirm that fast waves propagating perpendicular to magnetic field are
restricted to low amplitudes in the hot relativistic limit as well as cold.
• According to Table 5.1, the critical Mach number is highly influenced by γo .
• According to Table 5.2, the width of the soliton at half maximum of the magnetic
field amplitude, l, strongly depends on Ti . But, when Te is bigger than µTi , the
electron temperature becoming more significant.
50
γo = 102 , η < 10−4 %
γo = 101 , η < 0.02%
Ti
10−1
Te
10−1
102
10−1
104
1.07563 1.07091 1.05624
3.571
102
3.541
3.632
1.05556 1.05556 1.05555
3.650
104
3.650
3.650
1.05552 1.05552 1.05552
3.651
3.651
3.651
102
104
γo = 104 , η < 10−9 %
10−1
102
1.01255 1.01295 1.01668
1.0003
11.572
11.804
13.741
174.007 190.786 276.701
1.017
1.017
1.017
1.00079 1.00079 1.00079
13.898
13.898
13.898
282.234 282.235 282.249
1.01701 1.01701 1.01701
1.00079 1.00079 1.00079
13.906
282.54
13.906
13.906
1.00036 1.00076
282.54
Table 5.1: The First Raw divided to three which correspond to different γo and having
maximum deviation η. The second raw and the first column is Te and Ti , respectively.
Each sell have two values where the upper is the maximum permitted magnetic field
ampletude and the lower value is the critical Mach number.
γo = 101 , η < 10−2 %
γo = 102 , η < 10−5 %
γo = 104 , η < 10−10 %
10−1
102
104
10−1
102
104
10−1
102
104
10−1
2.5
2.3
11.7
0.99
0.86
3.96
0.8
0.7
3
102
57.4
57.4
57.8
18.8
18.7
17.9
104
574.4 574.4
574.1
188.2
188
187.8
Ti
Te
104
14.09 14.1
141
140
13.4
140
Table 5.2: The First Raw divided to three which correspond to different γo and having
maximum deviation η. The second raw and the first column is Te and Ti , respectively.
Each cell represents a ratio between the width of the hot soliton and the width of the cold
soliton.
51
282.54
Chapter 6
Discussions & Conclusions
First, let us remind to the reader that we have analyzed stationary one-dimensional nonlinear waves propagating in the direction perpendicular to the external magnetic field,
within the two-fluid approach and have covered the cold and hot pair plasma cases as well
as electron-ion plasma. The quasineutrality assumption in the species rest frame allowed
us to derive a pseudopotential-type equation for the magnetic field or density in the cold
as hot cases as well. We reproduce the well-known results regarding compressive solitons,
that are summarized in detail at section 4.4, and show that the pair plasma allows also
rarefactive solitary wave solutions, which are absent in the literature for the relativistic
magnetized pair plasma. In the case of electron-ion plasma we derive a single pseudopotential equation for the density by solving the equations in the approximation of the weak
longitudinal electric field. The equation is solved numerically in a wide range of plasma
parameters, and summarized in section 5.2. It is found that the compressive solitons are
similar to those found for the pair plasma and do not exist for Mach numbers exceeding
some critical Mach number or inferior than some minimum Mach number. There are
no rarefactive solitons in the electron-ion plasma in this approximation and shock like
solution as well.
Stability of the solitons is an important issue that deserve attention, but has not
been addressed in this study . Moreover, examining some features e.g correction of perpendicular propagation equations to the oblique case, generalization into the anisotropic
plasma theory onto generally relativistic cases, consideration to non stationarity together
52
with non one-dimensionality and deviation from quasineutrality are adequate and have a
fundamental importance for understanding better the structure of relativistic nonlinear
waves and shocks in magnetized plasma.
53
Appendix A
A.1
This section is devoted to calculation of Ξ(1), Ξ0 (1) and Ξ00 (1).
2
2
βo2 γo2
βo2 γo2
2
2
2
2
2
2
(W1 + W2 ) − 4W1 W2 1 +
Ξ(N ) = γo [1 + σ(1 − B)] − 1 +
N2
N2
= (c1 − c2 c3 )(c1 + c2 c4 ) = c5 c6
where
c1 = γo2 [1 + σ(1 − B)]2 ,
βo2 γo2
N2
c1 (1) = γo2
,
c2 (1) = γo2
c3 = W 2
,
c3 (1) = 1
c4 = (W1 − W2 )2
,
c4 (1) = (Wo1 − Wo2 )2
c5 = (c1 − c2 c3 )
,
c5 (1) = 0
c6 = (c1 + c2 c4 )
,
2
2
)
+ Wo2
c6 (1) = 2γo2 (1 − 2Wo1 Wo2 ) = 2γo2 (Wo1
c2 = 1 +
First derivative:
Ξ0 (N ) = c05 c6 + c5 c06
c01 = −2γo2 [1 + σ(1 − B)]σB 0 ,
2 2
where
c01 (1) = −2γo2 σB 0
c02 = −2 βNo γ3o
,
c02 (1) = −2βo2 γo2
c03 = 2W W 0
,
c03 (1) = 2W 0 (1)
c05 = c01 − c02 c3 − c2 c03
,
c05 (1) = 0
Second derivative Ξ00 (N ) = c005 c6 + c05 c06 + c05 c06 + c5 c006
c001 = 2γo2 [σ 2 B 02 − [1 + σ(1 − B)]σB 00 ] ,
2 2
where
c001 (1) = 2σγo2 σB 0 2 − B 00
c002 = 6 βNo γ4o
,
c002 (1) = 6βo2 γo2
c003 = 2W 02 + 2W W 00
,
c003 (1) = 2W 0 2 (1) + 2W 00 (1)
c005 = c001 − c002 c3 − 2c02 c03 − c2 c003
,
c005 (1) = 2γo2 δ(1)
where δ(1) = σ 2 B 0 2 − σB 00 − 3βo2 + 4βo2 W 0 − W 0 2 − W 00 . So
Ξ(1) = c5 (1)c6 (1) = 0
Ξ0 (1) = c05 (1)c6 (1) + c5 (1)c06 (1) = 0
2
2
Ξ00 (1) = c005 (1)c6 (1) + c05 (1)c06 (1) + c05 (1)c06 (1) + c5 c006 (1) = 4γo4 (Wo1
+ Wo2
)δ(1)
54
A.2
This section is devoted to calculation of Ws0 (1), Ws00 (1), W 0 (1) and W 00 (1).
2
s βs σγo
Ws = Wos + ΓsΓ−1
N Γs −1 − 1 , Ws (1) = Wos
2
2
Ws0 = Γs βs σγ2o N Γs −2
2
Ws00 = Γs βs (Γs − 2) σγ2o N Γs −3
2
,
Ws0 (1) = Γs βs σγ2o
,
Ws00 (1) = Γs (Γs − 2)βs σγ2o
2
So
W (1) = 1
σγo2
2
σγo2
(κ2 − 2κ1 )
=
2
W 0 (1) = κ1
W (1)00
where κ1 = β1 Γ1 + β2 Γ2 and κ2 = β1 Γ21 + β2 Γ22
A.3
This section is devoted to calculation of B 0 (1), B 00 (1) and δ(1).
B 2 = 1 + β1 (1 − N Γ1 ) + β2 (1 − N Γ2 ) +
2
B
B
0
00
2βo2
(1 − W/N )
σ
0
N −W
−β1 Γ1 N Γ1 −1 − β2 Γ2 N Γ2 −1 − 2βσo W N
2
=
2B
02
−2B − β1 Γ1 (Γ1 − 1)N Γ1 −2 − β2 Γ2 (Γ2 − 1)N Γ2 −2 −
=
2B
So
B(1) = 1
β 2 γ 2 κ1
B 0 (1) = o − o
σ
2
2
2
2
κ2 γo2 2βo2
κ1 γo
βo
γo2 κ1
00
2
B (1) =
(1 + 3βo ) −
−
−
−
2
2
σ
σ
2
Using all above we obtain:
2
2
δ(1) = σ 2 B 0 − σB 00 − 3βo2 + 4βo2 W 0 − W 0 − W 00 =
σ σ 4 2 βo4
4
2 2
= βo + κ1 + γo κ1 +
− βo γo κ1 − βo2
2
4
σ
βo2
κ1 γo4
κ1 γo4
2
2
=
MA −
MA − 1 −
MA2 γo2
2
2
55
2βo2
σ
W 00
N
−
2W 0
N2
−
2W
N3
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