Download Introduction to Plasma Physics

Introduction to Plasma Physics
Emilia Kilpua and Hannu Koskinen
HK, 11.12.2015
2
Contents
1 Introduction
3
1.1
General definition and occurrence . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Brief history of plasma physics . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Levels of description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Basic Definitions and Parameters
9
2.1
Formation of the plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Quasi-neutrality in plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4
Plasma response to electromagnetic fields . . . . . . . . . . . . . . . . . . 16
2.5
Collective behavior and collisions . . . . . . . . . . . . . . . . . . . . . . . 18
2.6
Plasma conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7
Plasma definition: A summary . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8
Exercises: Basic Definitions and Parameters . . . . . . . . . . . . . . . . . 26
3 Single Particle Motion
9
29
3.1
Motion in a static, uniform magnetic field . . . . . . . . . . . . . . . . . . 29
3.2
Motion in constant perpendicular electric and magnetic fields . . . . . . . 33
3.3
General drift velocity due to a force perpendicular to magnetic field . . . 35
3.4
Particle motion in non-uniform electric fields . . . . . . . . . . . . . . . . 36
3.5
Particle motion in non-uniform magnetic fields . . . . . . . . . . . . . . . 38
3.6
Examples of particle motion in simple geometries . . . . . . . . . . . . . . 48
3.7
Exercise: Single Particle Motion . . . . . . . . . . . . . . . . . . . . . . . 53
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CONTENTS
4 Kinetic Plasma Description
57
4.1
Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2
Time evolution of distribution functions . . . . . . . . . . . . . . . . . . . 61
4.3
Solving the Vlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4
Examples of distribution functions . . . . . . . . . . . . . . . . . . . . . . 67
4.5
Exercises: Kinetic Plasma Description . . . . . . . . . . . . . . . . . . . . 71
5 Macroscopic Plasma Equations
73
5.1
Macroscopic transport equations . . . . . . . . . . . . . . . . . . . . . . . 73
5.2
Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3
Magnetohydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . 77
5.4
Exercises: Macroscopic Plasma Equations . . . . . . . . . . . . . . . . . . 82
6 Magnetohydrodynamics
85
6.1
MHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2
Magnetic field evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3
Frozen-in condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4
MHD waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5
Magnetic reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.6
Magnetohydrostatic equilibrium and stability . . . . . . . . . . . . . . . . 104
6.7
Force-free magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.8
Exercises: Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 111
7 Cold plasma waves
113
7.1
General form of the dispersion equation . . . . . . . . . . . . . . . . . . . 113
7.2
Wave propagation in non-magnetized plasma . . . . . . . . . . . . . . . . 115
7.3
Wave propagation in magnetized plasma . . . . . . . . . . . . . . . . . . . 118
7.4
Exercises: Cold Plasma Waves . . . . . . . . . . . . . . . . . . . . . . . . 128
8 Warm plasma
131
8.1
Warm plasma dispersion equation . . . . . . . . . . . . . . . . . . . . . . . 131
8.2
Langmuir wave and the ion sound wave . . . . . . . . . . . . . . . . . . . 132
8.3
On plasma stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.4
Exercises: Warm Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
CONTENTS
9 Appendix
iii
141
9.1
Useful vector identities and theorems . . . . . . . . . . . . . . . . . . . . . 141
9.2
Maxwell equations and useful concepts of electrodynamics . . . . . . . . . 142
9.3
Basic concepts of wave propagation . . . . . . . . . . . . . . . . . . . . . . 143
9.4
The Maxwellian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 147
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CONTENTS
Preface
This course is an introduction to basic concepts and methods of plasma physics. It
provides basics for further studies of laboratory, fusion, space and astrophysical plasma
phenomena. The most important goals of these lectures are to:
• familiarize the reader with the main concepts and phenomena of plasma physics
• give an overview of the importance and applicability of plasma physics
• teach the basic mathematical tools and approaches used in plasma physics
After a brief introduction to fundamental plasma properties, following topics are
discussed: motion of charged particles in the electromagnetic field, kinetic plasma description, macroscopic plasma quantities and equations, magnetohydrodynamics, Alfvén
waves, cold plasma waves, warm plasmas.
Most of these lectures deal with plasma in so high-temperatures that the plasma
is practically fully ionized with only a small effect arising from neutral particles. We
will also limit the discussion to non-relativistic plasmas with temperatures ranging from
about a few eV to a few hundred keV. Quantum mechanical effects are also neglected
since the interactions distances are usually much longer than the de Broglie wavelength.
Plasma physics is based on the main fields of classical physics: electrodynamics, mechanics and statistical physics. These lectures require good understanding of bachelorlevel basic physics and solid skills in undergraduate-level mathematical methods of
physics (e.g., vector calculus and analysis, Fourier analysis).
1
2
CONTENTS
Chapter 1
Introduction
We start with a brief introduction to the idea of plasma as a state of matter before we
go to the more technical treatment of plasma physics.
1.1
General definition and occurrence
“What is plasma?” This is a natural question to ask at the beginning of plasma physics
lectures. However, as we will soon find out, this question is not trivial and it is difficult
to give an exact definition of the plasma state. We refer at this point to a following
practical description of plasma:
Plasma is quasi-neutral gas with so many free charges that collective electromagnetic
phenomena are important to its physical behavior.
Two key aspects of plasma can be found from this definition: 1) Due to the presence
of free charges plasma responds strongly to electromagnetic fields, and 2) in plasma
collective long-range interactions dominate. These characteristics of plasma lead to a
wide variety of interesting phenomena distinct from neutral gases, including collective
shielding of individual charges, a large variety of new wave modes, and transfer of energy
from waves to particles (damping of oscillations) and vice versa (plasma instabilities).
Plasma is ubiquitous in the universe. It is speculated that more than 99% of baryonic
matter in the universe is in the plasma state. Thus, plasma physics is a necessary tool in
space physics and in many astrophysical problems. What we mean by “space physics”
needs a little explanation. Space physics investigates physical phenomena in space of
which it is possible, at least in principle, to get detailed in-situ observational information.
Thus its domain is mainly the solar system including the studies of the Sun, the solar
wind, and the magnetospheres, ionospheres, and upper atmospheres of the Earth and
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CHAPTER 1. INTRODUCTION
other planets. Astrophysical plasma physics includes, in turn, the studies of plasmas and
plasma processes farther in the universe, e.g., stars from the Sun-like objects to neutron
stars, black hole accretion discs, and the interstellar medium.
Figure 1.1: Examples of plasmas near and far. Top row: Sun captured by ultraviolet
light emitted by ionized helium atoms (Courtesy: SOHO/NASA), artist impression of
an active galactic nucleus (Courtesy: Alfred Kamajan), lightning (Courtesy: NOAA).
Bottom row: plasma welding (Courtesy: Pro-Fusion), Joint European Torus (JET)
fusion experiment (Courtesy: AFP/Getty Images), plasma thruster (Courtesy: NASA).
Plasmas in our immediate environment are much less common, but they exist. In
particular, technological applications of plasma physics are numerous, including thermonuclear fusion research, neon-lights, plasma displays, sterilizing of certain medical
products, and plasma processing of semiconductors and materials (e.g., etching and
welding). Visible examples of natural plasmas in the near-Earth environment are auroras and lightning.
1.2
Brief history of plasma physics
The word “plasma” originates from Greek where it means something molded. British
scientist Sir William Crookes was the first to appreciate plasma as the fourth state of
matter. He investigated the conduction of electricity in low pressure gases in electrical
discharge tubes (where air is ionized by applying a high voltage). In 1879 Crookes
published an article “On Radiant Matter” in The Popular Science Monthly where he
stated: “So distinct are these phenomena from anything which occurs in air or gas at
1.2. BRIEF HISTORY OF PLASMA PHYSICS
5
ordinary tension, that we are led to assume that we are here brought face to face with
matter in a fourth state or condition, a condition as far removed from the state of gas
as a gas is from a liquid.”.
The term “plasma” was coined a few decades later by American physicists Lewi
Tonks and Irving Langmuir. They conducted one of the first plasma experiments with
electric discharge tubes. These experiments already led to many important discoveries
concerning the basic properties of plasma, such as the shielding of charge and plasma
oscillations.
A significant part of early plasma physics dealt with space and astronomical phenomena. A particular interest for space plasma physics in the early 1900s was radio
broadcasting. Radio waves are reflected from the ionosphere, the partially ionized layer
of the upper atmosphere, which enables transfer of waves over long distances. The
efforts to understand radio communication led to the development of the theory how
electromagnetic waves propagate through non-uniform magnetized plasmas.
Figure 1.2: A few great minds of plasma physics. Top: William Crookes (Courtesy: Library of congress), Hannes Alfvén (Courtesy: Royal Institute of Technology, Stockholm),
Irving Langmuir (Courtesy: IEEE).
Another main branch in early space plasma physics dealt with examining the connection between solar activity and the disturbances in the Earth’s magnetic field. A
most remarkable theory in the early 1900s in the field of solar–terrestrial studies was
Sidney Chapman’s and Vincenzo Ferraro’s suggestion that magnetospheric storms are
caused when magnetized plasma clouds ejected from the Sun envelop the Earth’s magnetosphere.
In the 1940s the Swedish scientist Hannes Alfvén developed the formalism of magnetohydrodynamic (MHD) theory. MHD treats plasma as a conductive fluid that can
support magnetic fields. Over the years MHD has developed to one of the main tools of
plasma physics. Hannes Alfvén has probably been the most influential individual in the
history of plasma physics. His contributions to plasma physics are numerous including
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CHAPTER 1. INTRODUCTION
theories describing the behavior of aurora, the radiation belts, now known as Van Allen
belts, the effect of magnetic storms on the Earth’s magnetic field and cosmic electrodynamics. Perhaps one of his best-known ideas is the theory of low-frequency magnetohydrodynamic waves, now known as Alfvén waves, in magnetized plasma. The basic mode
of the Alfvén wave propagates along magnetic field and it is the fundamental mode to
transfer magnetic disturbances in plasma. Hannes Alfvén was granted the Nobel prize
in 1970 for his work on “fundamental work and discoveries in magnetohydro-dynamics
with fruitful applications in different parts of plasma physics”.
Radio astronomy started to develop in the 1930s when Karl Jansky observed radio
waves coming from the direction of the Milky Way. The Second World War brought
rapid developments in radio- and microwave technologies, which opened a new window
to the Universe at those radio frequencies that penetrate through the atmosphere. Part
of the radiation is bremsstrahlung in hot astrophysical plasmas but it soon turned out
that all radio emissions could not be explained in this way. In the mid-1950s Vitaly
Ginzburg argued that radio emissions from, e.g., the Crab nebula, i.e., the remainder
of the supernova observed in 1054, must be synchrotron radiation by electrons gyrating
in the strong magnetic field of the neutron star. This was an important milestone as it
indicated the central role of the magnetic fields in cosmic plasma physics.
Modern plasma physics can be said to have originated after the Second World War
and has expanded to several directions. A few main branches are briefly discussed below.
Space Plasma Physics. The space age began with the launch of Sputnik in 1957. The
US satellite Explorer 1 was launched a year later and its sole scientific instrument, a
Geiger counter by James Van Allen, discovered radiation belts around the Earth. These
belts were later named Van Allen belts. Space exploration quickly expanded from the
vicinity of the Earth further out in the heliosphere. The spacecraft have passed-by all
planets of our solar system. Some of them have become artificial satellites of Mercury,
Venus, Mars, Jupiter and Saturn and carried landers with them. Mankind has now even
reached beyond the solar system when Voyager 1 entered the interstellar space in 2013.
The most detailed data is, however, obtained from the near-Earth space and from the
Sun. Numerical simulations have become an integral part in modern space research besides observations. The observational network in space physics is relatively sparse and
simulations are needed to fill this gap. In addition, simulations provide new physical insight to many space physics problems that can be tested by observations. Understanding
of space plasma physics is also necessary for space technology, ranging from designing,
manufacturing and testing scientific instruments to developing new propulsion systems
for faster and more cost-effective space travel and exploration.
Plasma Astrophysics. As said earlier, almost all baryonic matter in the Universe
is in plasma state. While the Sun is often considered to belong to the topics of space
plasma physics, it is also a very typical star. The Sun is entirely in the plasma state and
detailed understanding of its plasma physics, e.g., the dynamo process creating its cyclically varying magnetic field, can readily be transferred to studies of other magnetically
active cool stars. Current topics of plasma astrophysics include neutron star magneto-
1.3. LEVELS OF DESCRIPTION
7
spheres, formation of astrophysical jets, accretion discs of black holes, acceleration of
cosmic rays in astrophysical shock waves, emission of electromagnetic radiation from
radio frequencies to X- and gamma rays. It is clear that also in astrophysical context
simulations using the most powerful computers today have become an essential tool.
Controlled Fusion Research. The early fusion experiments were conducted already
in 1930s but it was only after the Second World War when the interest in fusion research
really sparked. The development of nuclear fission weapons raised interest also in fusion
weapon technologies. It was proposed that fusion reaction could be controlled to make
an effective reactor. Since then fusion research has quickly expanded as an important
international enterprise with several large experimental facilities being constructed with
the goal to develop a relatively clean and abundant energy source. The most prominent
current effort is the International Thermonuclear Experimental Reactor (ITER), which
is being constructed near Cadarache in the Southern France. When finished, ITER
will be the world’s largest tokamak nuclear fusion reactor. Much of the fusion research
nowadays is involved in studying how extremely hot plasma can be stabilized for long
enough to attain sustained effective fusion and the tokamak geometry is the currently
favored approach in the large-scale devices. Another approach to controlled fusion is to
create the required hot and dense plasma state using intense lasers.
1.3
Levels of description
Plasma processes are often extremely complicated and their spatial and temporal scales
vary by many orders of magnitude. Plasmas exhibit diverse characteristics, their temperatures, densities and ionization degree can differ greatly as well as the importance
of collisions and electromagnetic forces to the behavior of plasma. Thus, different levels
of description are used to tackle different types of problems (see Figure 1.3). Different approaches can also provide alternative insights to understanding a given plasma
phenomenon.
Although collective behavior is a fundamental property of plasma, single particle
description (or an exact microphysical description) is the first step in understanding of
the processes occurring in plasma. It is often a necessary approach, for example, when
studying cosmic rays or energetic particles in the Van Allen radiation belts. In this
approach the task is to solve the equation of motion (F = ma) for a charged particle.
Only in a few special cases the motion can be solved analytically and typically (e.g., in
time varying and curved magnetic fields) approximations or direct numerical calculations
are needed.
The next step is the kinetic theory. It is a statistical approach to average out individual particle orbits and treat the motion of a large number of particles in form of
a distribution function. However, the detailed knowledge the particle distribution as
a function of location and velocity is needed and in this sense kinetic theory is still
8
CHAPTER 1. INTRODUCTION
Figure 1.3: Levels of plasma descriptions.
microscopic. The core of the kinetic treatment is to determine the velocity distribution functions and their evolution for each plasma species. From velocity distribution
functions one can calculate macroscopic plasma variables, such as the bulk speed, temperature and density. The kinetic approach can deal with non-Maxwellian distributions
and it is often the required approach when studying plasma waves and instabilities.
In many cases it is not necessary to know the exact evolution of distribution functions,
but it is sufficient to determine how macroscopic plasma variables behave in time and
space. The evolution of these parameters are determined by means of macroscopic fluid
approach, the equations of which are analogous to the equations of hydrodynamics.
However, the effects of electromagnetic fields on the charge particles and often different
behavior of electrons and ions in a plasma make plasma fluid equations more complex
than hydrodynamic equations. In fluid description the velocity distributions of each
species are often implicitly assumed to be Maxwellian.
The simplest description of plasma is the one-fluid or magnetohydrodynamic (MHD)
theory. Although a very crude approximation, MHD is a widely applicable theory and
can be used to describe many plasma physical phenomena. Due to simplicity and computational effectiveness it is one of the main tools for global numerical simulations.
Sometimes, a combination of different approaches are used. For examples, in hybrid
simulations electrons can be described as a fluid and ions either as individual particles
or in terms of distribution functions.
Chapter 2
Basic Definitions and Parameters
2.1
Formation of the plasma
Plasma is generally considered as the fourth state of matter because it arises as the
next natural step from solid to liquid to gas, when the temperature is increased (Figure
2.1). For example, when ice is heated its crystalline bonds are broken and it changes to
water (liquid state). If more heat is added the molecular bindings break first, followed
by independent H2 O molecules separating into hydrogen and oxygen atoms (gas state).
In order to achieve plasma, even more heat has to be added to dissociate the atoms into
electrons and positive ions. At some point the fraction of the atoms that are ionized
becomes large enough that the collective electromagnetic forces take over the behavior
of the system (plasma state).
Figure 2.1: Plasma is considered as the fourth state of matter.
Adding even more heat would finally break nuclear bonds (energies > MeV) and
quark–quark bonds (energies > 175 MeV) resulting in quark-gluon plasma. Such an
exotic plasma state dominated the universe just after the Big Bang and may exist in the
core of neutron stars. Experiments on CERN’s Large Hadron Collider are studying the
properties of quark–gluon plasma. However, this is beyond the scope of normal plasma
physics courses and will not be treated in these lectures.
According to our practical plasma definition there has to be “enough free charges”.
But how much is enough? There is no unique phase transition point when a gas turns to
a plasma, but a rough guideline is that already 0.1% degree of ionization typically gives
clear plasma properties and 1% ionization means almost perfect conductivity. Thus,
9
10
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
plasma state is achieved after a remarkably small fraction of ionization. Partially ionized
plasmas can be found, for example, in ionospheres, neon-lights, and gas-discharge tubes.
Examples of fully ionized plasmas include fusion plasmas and most of space plasmas, for
example, the solar wind, solar corona and magnetosphere. At the end of this Chapter
we will provide a more quantitative definition of a plasma state.
The degree of ionization for a gas in thermal equilibrium can be calculated from
Saha’s equation:
ni
T 3/2
= 3 × 1027
exp(−U/T ) ,
(2.1)
nn
ni
where ni is the ion number density ([ni ] = m−3 ), nn the neutral number density, T
temperature ([T ] = eV) and U the ionization energy ([U ] = eV, 1 eV ≈ 11604 K) , i.e. the
energy that is required to remove the outermost electron from the atom. From Saha’s
equation it is clear that the ionization degree increases rapidly with the temperature
(Exercise 2.1). Note that to maintain the plasma state there has to be a balance of
ionization and recombination. This means that either the ionization source must be
continuous and strong enough, or the recombination rate must be low.
Contemplate: Why is the degree of ionization in Eq. 2.1 inversely proportional to
ion density? Using the literature find out the principle behind the derivation of Saha’s
equation and determne the units of the factor 3 × 1027 (it is not dimensionless!). You
can also try to calculate its value.
Apart from heating, ionization can be achieved by applying large local electric fields
or by exposing the matter to ionizing radiation such as strong laser light, ultraviolet
light, or X-rays. In fact, it is possible to produce plasma even from solid state. An
example of low-temperature plasma sustained by solar EUV light and energetic particle
precipitation is the Earth’s ionosphere. Also the solar photosphere, that is the layer
from which most of the solar irradiation emerges, is at a temperature of less than 6000 K
that is well below the ionization energy of the photospheric gas. In that case the source
of ionization is the heat coming from below the solar surface.
2.2
Quasi-neutrality in plasma
Plasma consist of a mixture of positively and negatively charged particles, but overall
plasma is quasi-neutral. This means that the positive and negative charges must have
approximately equal charge densities:
X
s
ρqs =
X
n s qs = 0 ,
(2.2)
s
where ρqs is the charge density ([ρqs ] = C m−3 ), ns the number density ([ns ] = m−3 ) and
qs the electric charge of the species s. For plasma consisting of electrons and one singly
charged ion species:
ne (−e) + ni (+e) = e(ni − ne ) ,
(2.3)
2.2. QUASI-NEUTRALITY IN PLASMA
11
where ne is the electron number density, ni the ion number density and e the elementary
charge (e = 1.6022 × 10−19 C). We see from Eq. 2.3 that if we in this case require
quasi-neutrality, electron and ion number densities must be equal, i.e., ne = ni .
A significant fraction of “free” electrons makes plasma electrically conductive. In
fact, plasma is typically an exceptionally good conductor. When temperatures are high
and densities low, collisions are rare, and thus, the resistivity is very small. If an electric
field is introduced in plasma, electrons quickly rearrange themselves and the electric
field is neutralized. As a consequence, no significant large-scale electric field can exist in
the rest frame of the unmagnetized plasma. The ability of plasma to shield out applied
electric fields is one of its fundamental characteristics.
Contemplate: While the resistivity of plasma can be negligible, plasma is not a superconductor. Why?
Although plasma is neutral in large scales, deviations from charge neutrality can
develop in shorter scales. Let’ us now look more quantitatively a distance over which
quasi-neutrality is true. Suppose that a positive point charge qT is introduced into an
otherwise quasi-neutral plasma. The “bare” Coulomb potential of the test charge is
qT /4π0 r, where r is the distance from qT . Negative electrons are attracted to qT and
they form a neutralizing cloud around it (Figure 2.2) modifying its Coulomb potential.
Figure 2.2: Debye shielding of a test charge qT .
Let us compute the approximate form of the modified Coulomb potential and the
thickness of the neutralizing electron cloud. The electrostatic potential φ can be derived
from the Poisson equation:
∇ · E = −∇2 φ = −ρtot (x)/0 ,
(2.4)
where 0 is the permittivity of free space (≈ 8.8542 × 10−12 A s V−1 m−1 ) and the charge
density ρtot is composed from the contribution of qT (ρT ) and the polarization of the
quasi-neutral plasma as a response to qT (ρpol ):
ρtot (x) = ρT δ(x − xT ) + ρpol (x) .
(2.5)
12
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
Here δ is Dirac’s delta function (recall its properties!). It ensures that the charge density
ρT vanishes outside x = xT .
We assume that the plasma is sufficiently close to the thermal equilibrium so that
its density can be given by the Boltzmann distirbution:
ns = n0s exp (−
qs φ
),
kB Ts
(2.6)
where n0s is the equilibrium number density in the absence of qT .
For a gas to be in the plasma state the constituent electrons and ions must be
unbound. This means that we must require that the random thermal energy must be
much greater than the average electrostatic energy. Thus, we can assume that qs φ kB Ts and expand Eq. 2.6 as:
ns ' n0s (1 −
qs φ
1 qs2 φ2
+
2 T 2 + ...) .
kB Ts 2 kB
s
(2.7)
The polarization charge density now becomes:
ρpol =
X
s
where
P
s
n s qs ≈
X
ns0 qs0 −
X n0s q 2
s
s
s
kB Ts
φ=−
X n0s q 2
s
s
kB Ts
φ,
(2.8)
ns0 qs0 = 0 due to quasi-neutrality, see Eq. 2.2. Inserting Eq. 2.8 into the
Poisson equation (Eq. 2.4) the potential turns out to be (Exercise 2.2):
φ=
r
qT
exp (−
).
4π0 r
λD
(2.9)
The factor λD in Eq. 2.9 is called the Debye length:
λ2D = 0
X kB Ts
s
n0s qs2
.
(2.10)
When ions are much colder than electrons, the ion term can be dropped from the definition of the Debye length.
Figure 2.3 shows how the shielded and “bare” Coulomb potentials of qT differ from
each other. When the distance from qT is much smaller than λD the Coulomb potential
is recovered. For distances much larger than λD the potential shows exponential decay,
i.e. it decays much faster than the bare Coulomb potential. Thus, the Debye length
is the distance over which significant charge separations (and electric fields) can occur
in plasma. Intuitively, Debye length is the limit beyond which the thermal speed of
particles is high enough to escape from the Coulomb potential of qT (see Exercise 2.3).
Electric field due to qT is restricted within a sphere having the radius given by λD .
2.2. QUASI-NEUTRALITY IN PLASMA
13
Figure 2.3: The “bare” Coulomb potential
and the shielded potential for two different
Debye lengths compared (λD1 > λD2 ).
Contemplate: How does the Debye-length change with density and temperature? Try
to give a physical explanation for this behavior
Table 2.2 shows some typical values for Debye length. Note that while in many
applications/domains (tokamak, Earth’s ionosphere, solar corona) the charged regions
do not exceed one millimeter, in some space plasmas (e.g., solar wind) Debye length can
have macroscopic values. See Exercise 2.4 for comparing typical sizes of the spacecraft
and the Debye length of the medium they are measuring. Exercise 2.5 investigates the
form of the potential in the vicinity of a spherical conductor immersed in a plasma (e.g.,
a spherical electric probe measuring the properties of the solar wind).
Table 2.1: Typical values of Debye length in different plasma environments.
Plasma
Solar core
Gas discharge tube
Tokamak
Ionosphere
Solar wind
Interstellar medium
Intergalactic medium
Debye length [m]
10−11
10−4
10−4
10−3
10
10
105
Using the Debye length we can formulate a more quantitative criterion for the ionized
gas to be in a plasma state. First, to guarantee the quasi-neutrality the plasma system
has to have a size L of several Debye lengths:
λD L .
(2.11)
14
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
Otherwise, significant charge separations can arise and the plasma becomes dominated
by the boundary effects.
Second, in order for the Debye length to be a statistically valid concept there have
to be enough particles within the Debye sphere. The number of particles in the Debye
sphere is given by
4
ND = n × πλ3D
(2.12)
3
and thus for the ionized gas to be plasma it is required that
ND 1 .
(2.13)
This criterion also guarantees that collective long-range interactions between charged
particles dominate over binary interactions.
In many cases a parameter omitting the factor 4π/3 is used:
Λ = n0 λ3D .
(2.14)
It is called the plasma parameter.
In Exercise 2.6 the condition Λ 1 it used to prove that in a plasma the kinetic
energy is larger than the Coulomb potential energy. This was an essential assumption
in the derivation of the Debye length.
2.3
Plasma frequency
Let’s now investigate the dynamic response of plasma to a small perturbation. Imagine
that a fraction of electrons are slightly displaced with respect to ions (Figure 2.4). The
charge separation gives rise to an electric field that tries to restore the plasma quasineutrality. As a consequence, electrons are accelerated by the electric field back towards
their original positions. Due to their inertia the electrons will overshoot and start to
oscillate around the equilibrium position with a specific frequency. Electron oscillations
convert continuously electrostatic energy to kinetic energy and back again keeping the
total energy conserved. This kind of electron plasma oscillations were first observed by
Irving Langmuir and Levy Tonks in a low pressure discharge tube filled with mercury vapor. Their original article can be found from http://www.columbia.edu/ mem4/ap6101/
Next, we derive the frequency of this electron oscillation as a response to a small
electric field E1 . We make the cold plasma approximation, i.e. assume there is no
thermal motion. Ions are so heavy that they are practically unaffected, and we can
consider them as a fixed background. E1 is caused by a small perturbation n1 in the
electron density:
2.3. PLASMA FREQUENCY
15
Figure 2.4: Electric field introduced to a plasma by a slight electron–ion displacement.
ni = n0
(2.15)
ne = n0 + n1 (r, t) .
(2.16)
The continuity equation of the electron density is:
∂ne
+ ∇ · (ne u) = 0 ,
(2.17)
∂t
where ne = n0 + n1 and the velocity caused by perturbation is u = u1 . Obviously,
derivatives of the equilibrium quantities (here n0 ) vanish and if we omit all second order
terms (i.e. multiples of two small perturbation quantities, here n1 u1 ) we get
∂n1
+ n 0 ∇ · u1 = 0 .
∂t
(2.18)
The electric field causes a force F = qE1 , and thus the equation of motion for
electrons is:
∂u1
me
= −eE1 ,
(2.19)
∂t
and E1 is determined from the Gauss law:
en1
∇ · E1 = −
.
(2.20)
0
Taking ∂/∂t of Eq 2.18 and using Eq 2.20 we obtain
∂ 2 n1
n0 e2
+
(
)n1 = 0 .
∂t2
0 me
(2.21)
Eq. 2.21 is the equation for a standing wave with the angular frequency called the
electron plasma frequency:
n0 e 2
2
ωpe
=
(2.22)
0 me
16
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
Plasma frequency gives the most fundamental time scale in plasma. Usually the term
refers to the electron plasma frequency. Ions are affected by the same electric field as
electrons, but due to their much larger mass their oscillation is much slower than the
electron oscillation (justifies our assumption of the fixed ion background). If the cold
plasma approximation is relaxed (i.e. non-zero temperatures), the oscillation propagates
as a wave in a plasma as we will see later.
2.4
Plasma response to electromagnetic fields
Plasmas respond strongly to electromagnetic fields, and therefore, the effect of electric
and magnetic forces is critical to understanding the behavior of plasma. A charged
particle in a plasma moves under the influence of the Lorentz force and thus its equation
of motion is:
dv
m
= F = q(E + v × B),
(2.23)
dt
where E is the electric field, B the magnetic field and v the velocity of the charged
particle. Note that the electric and magnetic fields that are used to calculate the Lorentz
force arise from all particles in the plasma and include also external (applied) electric
and magnetic fields. Thus, it is clear that calculation of the motion of a large number
of plasma particles is an immense problem.
Figure 2.5: Left: The electric field of the Lorentz force accelerates positive and negative
charges to opposite directions. Right: The magnetic part of the Lorentz force changes
the path of the particle. The direction of the bending can be inferred applying the
right-hand rule to the vector product.
The electric field in the Lorentz force accelerates positive and negative charges in
opposite directions (left-hand part of Figure 2.5). The magnetic part of the Lorentz
force is always perpendicular to the particle’s velocity (right-hand part of Figure 2.5).
Thus, magnetic field can only change the path of the particle, but it cannot do work on
the charge (you can see this easily by calculating the power v · F). It is often stated that
time-varying magnetic field is used to accelerate particles, but in fact, it is the induced
electric field that is responsible for the acceleration.
2.4. PLASMA RESPONSE TO ELECTROMAGNETIC FIELDS
17
Figure 2.5 illustrates that in a static and homogeneous magnetic field the charged
particles perform a circular motion about the magnetic field lines. The angular frequency
of this motion for species α is
qα B
.
(2.24)
ωcα =
mα
The corresponding frequency is fcα = ωcα /2π. This “Larmor motion” is investigated in
detail in Chapter 3. It is seen from above that due to their lower mass electrons spin
much faster around the magnetic field than ions. Numerically for electrons and protons
≈ 28 × B (nT)
≈ 1.5 × 10−2 · B (nT)
fce (Hz)
fcp (Hz)
.
It is important for plasma physics that charged particles can move relatively freely
along the magnetic field lines, but their motion perpendicular to the magnetic field is
much more restricted. As a consequence, the magnetic field “binds” plasma particles
together.
The Lorentz force per unit volume acting on charge density ρq and electric current
density J = ρq v is given by
f = ρq E + J × B .
(2.25)
Now v · f = J · E represents the power per unit volume acting on the moving charges.
Depending on the sign of J·E power is either extracted from the fields and used as acceleration or heating of the particles, or vice versa. Which way the energy is transformed
can be found by considering the conservation law of the electromagnetic energy known
as the Poynting theorem.
Recall from your electrodynamics course that the energy densities of electric and
magnetic fields are given by
wE =
wM
=
1
E·D
2
1
H·B,
2
(2.26)
(2.27)
where D = E is the electric displacement field and H = B/µ the magnetic field intensity. is the permittivity and µ the permeability of the medium in consideration
(in vacuum 0 and µ0 ). Define the Poynting vector as S = E × H. From Maxwell’s
equations it is an easy exercise to to derive
∇ · S = −E · J − E ·
∂D
∂B
−H·
.
∂t
∂t
(2.28)
The Poynting theorem is the integral of this expression over volume V
−
Z
V
J · E d3 r =
Z
V
∇ · S d3 r +
Z
V
∂
(wE + wM ) d3 r .
∂t
(2.29)
18
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
The left-hand side is the work performed by the electromagnetic
field per unit time (i.e.,
H
power) in volume V. The first term on the right-hand side is ∂V S · da, i.e., the energy
flux per unit time through the surface ∂V. Thus the Poynting vector gives the flux of
electromagnetic energy density. The last term on the right-hand side expresses the rate
of change of the electromagnetic energy in volume V.
Assuming that there is no energy flux through the surface, the Poynting theorem
states that if J · E > 0 the energy of the electromagnetic field in the volume V decreases
in time, i.e., the energy is transferred to the particles. In the opposite case (J · E < 0)
the particles lose energy to the electromagnetic field. A rule-of-thumb is to interpret the
inequality sign materialistically; if it is “larger than”, the matter gains.
2.5
Collective behavior and collisions
One of the most distinct features separating plasmas from neutral gases is the way
particles interact with each other. In a neutral gas particles interact primarily through
direct binary collisions, where individual collisions lead to large deflections and can be
considered “strong”. In plasma charged particles interact predominantly through the
Coulomb force. These “Coulomb collisions” are long-range and each charged particle
interacts simultaneously with a large number of charged particles. The Coulomb collisions
are “weak” in the sense that vast majority of the collisions cause only minor deflections.
This is the key to the collective behavior.
Figure 2.6: Plasmas are divided to collisionless and collisional plasmas. Note that the
very high temperature of the fully ionized plasmas makes them in many cases effectively
collisionless.
Collisional plasmas can be divided into partially and fully ionized plasmas. Important
regions of fully ionized plasmas where collisions are frequent are the stellar interiors.
Examples of partially ionized plasmas are stellar photospheres and the ionized layers
of the upper atmospheres, the ionospheres, of planets. In weakly ionized plasma the
dominant type of collisions are those between charged particles and neutral atoms and
molecules. When the ionization degree increases the Coulomb collisions between charged
particles become dominant.
Often high temperature and tenuous plasmas are practically collisionless. Physically
this means that the time between collisions, or the mean free path, becomes longer than
2.5. COLLECTIVE BEHAVIOR AND COLLISIONS
19
the temporal or spatial scales of problems under study. For example, in the solar wind
the effective mean free path between collisions is of the order the distance from the Sun
to the Earth. It is important to understand that plasma being collisionless does not mean
that electromagnetic interactions between plasma particles would become negligible. On
the contrary, they dominate the plasma behavior. At the collisionless limit it is, however,
sufficient to consider the effect of average electromagnetic fields on the particles instead
of individual collisions.
2.5.1
Collisions with neutral particles
Charged particles interact with neutrals through direct collisions. A key quantity to
determine how often collisions occur is the effective cross-section σc that expresses the
likelihood of the interaction between two particles. For binary collisions σc is simply
given as πd20 , i.e. it is the cross-section of a neutral atom or molecule, whose radius is
d0 . If nn is the number density of neutral particles and hvi is the average speed of the
particles, we can calculate the collision frequency:
νn = nn πd20 hvi
(2.30)
and the average mean free path:
lmf p =
hvi
1
=
.
νn
nn πd20
(2.31)
For binary collisions νn gives the frequency between individual collisions and the lmf p
the average distance particles travel between two collisions.
Note that binary collisions may be of very variable nature. They may be elastic,
where two particles bounce off each other retaining their identities and energy states, or
inelastic in which case the kinetic energy of a colliding particle is transferred to internal
energy of the neutral particle or molecular ion of the plasma. Inelastic collisions can
thus lead to recombination, excitation, ionization, and charge exchange, which all are
important processes in space plasmas. Auroras are examples of the excitation process of
neutral molecules in the Earth’s atmosphere due to precipitating high-energy electrons
(Figure 2.7). Important charge-exchange processes are collisions where a high-energy
proton collides with a slow atom. As a result of the charge-exchange a high-energy
neutral atom (ENA) and a low-energy ion are formed:
+
p+
f ast + Hslow → HEN A + p
+
p+
f ast + Oslow → HEN A + O .
(2.32)
Examples of the first of these are the interactions of interstrellar hydrogen with solar
wind protons and inner magnetospheric protons with the hydrogen geocorona around
the Earth. The latter process is important when the solar wind interacts with the
atmospheres of unmagnetized planets Venus and Mars. ENA imaging provides a useful
20
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
Figure 2.7: Left) Precipitating electrons can excite the molecules into a higher energy
state. Auroras form when exited atoms emit photons. Different colors of auroras depend
on the energy levels of molecules present in the atmospheric layer the electrons can penetrate. Image Courtesy: Jouni Jussila. Right) Large ribbon of ENA emission detected
by the IBEX satellite. Image Courtesy: NASA/IBEX/Heerikhuisen et al.
tool to image the electric currents carried by ions around the Earth, Jupiter and Saturn.
The right-hand panel of Figure 2.7 shows a huge ribbon imaged by the NASA Interstellar
Boundary Explorer (IBEX) spacecraft. The ribbon was found in 2009 and its origin has
kept scientists puzzled. Currently, it is though that the ribbon is a reflection of particles
bouncing off a galactic magnetic field.
In plasmas collisions between neutrals and charged particles cannot happen too frequently. Otherwise, the behavior of the substance would be controlled by ordinary
hydrodynamic forces than by electromagnetic forces. If ω is the plasma frequency and
τc the mean time between collisions with neutrals, for plasma the condition
ωτc > 1
(2.33)
must be fulfilled.
2.5.2
Coulomb collisions between charged particles
For Coulomb collisions the determination of the collision cross-section is a difficult task.
Coulomb collisions are weak and they rarely result in large deflections. This is because
each charged plasma particle interacts with many far-away charges simultaneously, while
closer encounters where the path would deflect significantly are much less common (see
Exercise 2.7 to demonstrate this for a fully ionized plasma). However, the cumulative
effect of many small Coulomb collisions can deviate the path significantly. For Coulomb
collisions the collision frequency (νc ) is the measure of the frequency with which the
particle trajectory is deviated by 90◦ due to many successive Coulomb interactions and
lmf p is the distance traveled until such a deflection has accumulated. Here we give only
a very rough quantitative inspection of Coulomb collisions in plasma.
2.5. COLLECTIVE BEHAVIOR AND COLLISIONS
21
electron
Figure 2.8: Coulomb collisions are
long-range interactions between
charged particles and in plasma
they result typically only to a
small deflection in the particle
path.
ion (at rest)
dc
gc
Figure 2.8 shows an electron that is interacting with a positive ion. Since ions are
much more massive than electrons we can assume that the ion is at rest. The path of
the electron is a hyperbola, which far from the ion can be approximated by two straight
lines at an angle γc .
The distance dc in Figure 2.8 is called the impact parameter and it describes the
closest approach distance between the electron and the ion. To estimate dc we investigate
the Coulomb force on the electron
Fc = −
e2
.
4π0 d2c
(2.34)
The electron feels this force over the time τ = dc /ve , during which its momentum changes
by the amount τ |Fc |:
e2
|4(me ve )| ≈
.
(2.35)
4π0 ve dc
For large angle collisions (γc ' 90◦ ) this will be of the same order of magnitude as the
total momentum of the electron me ve . Thus, we get:
dc ≈
e2
,
4π0 me ve2
(2.36)
and the approximation for the cross-section is:
σc = πd2c ≈
e4
,
16π20 m2e hve i4
(2.37)
where ve has been replaced with the average speed hve i. The collision frequency becomes:
νei = ne σc hve i ≈
ne e4
.
16π20 m2e hve i3
(2.38)
The average speed of the electrons can be replaced by their average thermal energy
kB Te = me hve i2 /2. Rewriting this using the plasma frequency we get:
22
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
√
4
2 ωpe
νei ≈
64π ne
kB Te
me
−3/2
.
(2.39)
4 ∝ n2 , the collision frequency is proportional to the density and inversely
Because ωpe
e
proportional to T 2/3 .
This equation is only a rough approximation since most of the collisions are small
angle collisions. A correction for small angle deflections (not derived here) using the
plasma parameter Λ = ne λ3D gives:
νei ≈
ωpe ln Λ
.
32π Λ
(2.40)
and the electron mean free path is:
lmf p =
Λ
2ωpe λD
hve i
=
≈ 64πλD
.
νe
νe
ln Λ
(2.41)
Contemplate: Why does the collision frequency increase with the decreasing temperature?
Electrons also collide with each other in plasma. For electron-electron collisions the
Coulomb force is repulsive, and thus the colliding electron is deflected away from the
target. Electron–electron collisions are more complex to deal with than electron–ion
collisions since we cannot assume anymore that the scattering electron is at rest. Since
the Coulomb force is of the same order of magnitude, the deflection will be about the
same amount as for electron–ion collisions. In addition, ions collide with electrons and
other ions. Due to much larger mass of an ion their momentum gain or loss in not
significant when they interact with an electron. We can approximate:
νee ≈ νei
(2.42)
νie ≈ (me /mi )νee
νii ≈
q
me /mi νee
Contemplate: Explain why the electron collision frequency is much larger than the
ion–electron collision frequency.
2.6. PLASMA CONDUCTIVITY
2.6
23
Plasma conductivity
Because collisions change the momentum of the particles, they introduce a term corresponding to friction in the equation of motion (Eq. 2.23):
m
dv
= q(E + v × B) − mνc (v − u) ,
dt
(2.43)
where νc is the collision frequency, irrespective whether the electron collisions occur
between neutrals or charged particles. u is the velocity of the collision targets.
2.6.1
Conductivity in non-magnetized plasma
Let us first investigate non-magnetized plasma and choose the coordinate system where
all collision targets are at rest We can also assume that all electrons have the same
velocity ve , i.e. the plasma is cold (remember that the temperature arises from the
velocity spread). Assume furthermore that the system has reached a static state, i.e.,
electrons have already been accelerated to the velocity where the Coulomb force and the
collisions balance each other, i.e., dv/dt = 0. Now the solution to Eq. 2.43 is:
E=−
me νc
ve .
e
(2.44)
Electrons carry the current density:
J = −ene ve ,
(2.45)
Inserting Eq. 2.45 into Eq. 2.44 we obtain the relationship between the electric
current density and the electric field, i.e., Ohm’s law :
J=
ne e2
E
me νc
σ=
ne e2
.
me νc
(2.46)
and the conductivity is
(2.47)
The inverse of conductivity σ is called resistivity (η).
Contemplate: We have learned that that plasma is hot but here we assume it to be
cold. What does this mean? We will return to this question when we discuss waves in
the cold plasma approxiamtion in Chapter 7.
For fully ionized plasma νc is the electron-ion collision frequency given by Eq. 2.40.
24
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
This yields the Spitzer resistivity:
η=
me ωpe ln Λ
.
ne e2 32π Λ
(2.48)
Using the definitions of ωpe and λD one can write this as:
1/2
η=
1
e2 me
ln Λ .
32π20 (kB T )3/2
(2.49)
This shows that the resistivity has only a weak dependence on electron density through Λ.
This means that if an electric field is applied to plasma the electric current is independent
of the number of current carriers (electrons) as long as there are enough of them, which
is not always the case in low-density plasmas as we discuss in the next section in the
context of magnetized plasma. A simple way to heat plasma is to pass an electric current
through it. However, according to Spitzer’s formula resistivity is inversely proportional
to the temperature. This means that when temperature increases, resistivity drops fast
and plasma becomes such a good conductor that the Ohmic heating is not effective
anymore.
Contemplate: Why does Spitzer’s resistivity not significantly depend on the electron
density? And why is it inversely proportional to the temperature?
2.6.2
Conductivity in magnetized plasma
If a magnetic field is present in the plasma, the conductivity is generally a tensor quantity because charged particles move in different ways perpendicular and parallel to the
magnetic field. The equation of motion in a steady state situation can now be written
as:
0 = −ne e(E0 + ve × B0 ) − νc me ne ve .
(2.50)
By computing the current as in Eq. 2.45 we can rewrite this equation as:
J = σ0 E −
σ0
J × B.
ne e
(2.51)
To calculate J choose B = Bez . Using the electron gyrofrequency ωce (Eq. 2.24) we
can write:
ωc
Jx = σ0 Ex − Jy
νc
ωc
Jy = σ0 Ey + Jx
(2.52)
νc
Jz = σ0 Ez .
2.6. PLASMA CONDUCTIVITY
25
And thus:
ωc νc
νc2
σ0 Ex − 2
σ0 Ey
2
2
2
νc + ωce
νc + ωce
νc2
ωc νc
=
σ0 Ey + 2
σ0 Ex
2
2
2
νc + ωce
νc + ωce
= σ0 Ez .
Jx =
Jy
Jz
(2.53)
This set of equations describes a matrix equation between J and E
−
J=→
σ · E,
(2.54)
where the conductivity tensor is (Exercise 2.8):

σP

→
−
σ =  σH
0
−σH
σP
0

0

0 .
σk
(2.55)
The elements of the conductivity tensor, assuming for simplicity only one ion population,
are given by:
σP
σH
σk
νc2
σ0
2
νc2 + ωce
ωce νc
=
σ0
2
2
νc + ωce
ne e2
= σ0 =
.
me νc
=
(2.56)
The elements of the conductivity tensor depend both on the collision and gyro frequencies. σP is known as the Pedersen conductivity. It gives the conductivity in the
direction of the electric field E⊥ perpendicular to the magnetic field. The Hall conductivity σH is the conductivity perpendicular to both the ambient magnetic and electric
fields. The magnetic field-aligned conductivity σk is the same as the classical collisional
conductivity in the absence of magnetic field. In collisionless plasmas it is typically several orders of magnitude larger than the perpendicular conductivities, meaning that the
electrons can quickly rearrange to cancel any electric field parallel to B and the electric
field in a plasma is typically perpendicular to the magnetic field. However, if there are
not enough current carriers available, finite Ek can arise to accelerate the electrons to
a large enough current. Such a structure is often described as an electric double layer.
Parallel electric fields have been identified at a few thousand kilometers above the auroras with potential drops of several kilovolts, which corresponds to the energy of the
electrons causing main auroral light.
26
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
2.7
Plasma definition: A summary
As a summary we will gather together three conditions that a gas must satisfy to be in
a plasma state:
• Collective interactions dominate over binary interactions: There has to be enough
charged particles within a Debye sphere, ne λD 1
• Plasma is quasi-neutral: The size of the plasma system L has to be larger than
the Debye’s length, λD L.
• Neutral collision frequency must be smaller than the collective inertial response
frequency in plasma (i.e., the plasma frequency): ωτc > 1 (electromagnetic forces
dominate).
In Exercise 2.9 basic plasma parameters (plasma frequency, electron gyro frequency,
Debye length and plasma parameter) are calculated and compared for different regimes,
while in Exercise 2.10 the plasma state is investigated using the three conditions given
above. These exercises demonstrates a wide range of conditions plasma may exist.
2.8
Exercises: Basic Definitions and Parameters
1. The degree of ionization is described by the Saha equation
ni
= 3 × 1027 T 3/2 n−1
i exp(−U/T ) ,
nn
where ni is the number density of ions and nn of neutral atoms, T temperature
in eV, and U the ionization energy. Assume that the dominating species in the
ionosphere is O+ and their density 1011 m−3 and temperature 0.3 eV. The ionization energy of oxygen is 13.62 eV. What is the ionization degree of this plasma?
Calculate the ionization degree also for temperatures of 0.1 eV, 0.2 eV, and 0.5
eV. You will notice that the ionization degree increases rapidly as a function of
temperature!
2. Derive the formula
qT
r
ϕ=
exp −
4π0 r
λD
for the screened potential qT of a test charge in a plasma with Boltzmann’s density
distribution: nα (r) = nα0 exp(−qα ϕ(r)/kB Tα ).
Some hints: 1) Use e−x ' 1 − x when substituting the densities into Coloumb’s
law and make use of quasi-neutrality. 2) Make also use of spherical symmetry to
write
1 d
dϕ
r2
.
∇2 ϕ = 2
r dr
dr
2.8. EXERCISES: BASIC DEFINITIONS AND PARAMETERS
27
After solving the differential equation require that the solution approaches the
Coulomb potential of qT when r → 0 and remains finite at all distances.
Debye screening is considered as the most fundamental property of plasma.
3. An alternative derivation of Debye’s length and further insight to its meaning:
Consider two infinite parallel plates at x = ±d, set at potential φ = 0. The space
between them is uniformly filled by a gas of density n particles of charge q.
(a) Using Poisson’s equation, show that the potential distribution between the
plates is
nq 2
(d − x2 ) .
φ=
20
(b) Show that for d > λD , the energy needed to transport a particle from a plate
to the midplane is greater than the average kinetic energy of particles
4. What is the size of a typical spacecraft used to measure plasmas in the solar wind
and magnetosphere? (You can look this up on the Web.) Do spacecraft disturb
the medium they are trying to measure? What would you expect should happen
to a spacecraft passing through plasma?
5. A spherical conductor of radius a is immersed in a plasma and charged to a potential φ0 . The electrons remain Maxwellian and move to form a Debye shield,
but the ions are stationary during the time frame of the experiment. Assuming
φ0 e kB Te , derive an expression for the potential as a function of r in terms of
a, φ0 and λD . (Hint: Assume a solution of the form (exp(br))/r).)
6. Prove that g 2/3 (g is the inverse of the plasma parameter Λ) is proportional to
the ratio of the average Coulomb potential energy between two electrons and the
average kinetic energy of electrons.
Note: Since the plasma condition requires that Λ 1 this means that in plasma the
average kinetic energy between electrons is much larger than the average potential
energy. The result is intuitive since electrons are“free” in a plasma. The kinetic
energy being much larger than the potential energy is also a central assumption
when deriving the Debye length, see Exercise 2.2.
7. Show that in a fully ionized plasma the frequency of small-angle collisions is much
larger than the frequency of large-angle collisions. What plasma parameter gives
the order of magnitude of the relation between the small- and large-angle collisions?
This result allows us to describe the Coulomb collisions simply by the Lorentz force
and we do not usually need to calculate the (very) complicated collision integral.
8. Derive the elements of the conductivity tensor

σP

σ =  σH
0
−σH
σP
0

0

0 
σk
28
CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS
starting from the equation of motion
E + ve × B = −
me νc
ve .
e
Sketch the electric field components in the plane perpendicular to the magnetic
field.
9. Calculate the electron plasma frequency, electron gyro frequency, Debye length,
and plasma parameter for the following plasmas (note the units!)
(a) Fusion device:
Te ≈ 100 keV, ne ≈ 1016 cm−3 , B ≈ 1 T
(b) Ionosphere at 300 km altitude:
Te ≈ 0.1 eV, ne ≈ 106 cm−3 , B ≈ 50000 nT
(c) Solar wind at 1 AU:
Te ≈ 10 eV, ne ≈ 10 cm−3 , B ≈ 5 nT
(d) Core of the Sun:
Te ≈ 1 keV, ne ≈ 1026 cm−3 , no magnetic field
(e) Neutron star environment:
Te ≈ 100 keV, ne ≈ 1012 cm−3 , B ≈ 108 T
Contemplate different ranges these plasma parameters can have in different environments.
10. Consider a spherical container of 1.5 m radius filled with completely ionized hydrogen gas. Ions are are assumed cold.
(a) The electron density in the container is set to 4 × 1023 m−3 , i.e., couple of
percent of atmospheric number density on ground level. What should the
temperature be (upper and lower limits) so that the gas would behave like a
plasma?
(b) If the temperature is set to 27◦ C, what should be the electron density so that
the plasma conditions are fulfilled?
(c) Let us mix some neutral particles to the gas, when the temperature is set to
27◦ C. The collision frequency nun between charged particles and neutrals is
< νn >= nn σn < v >,
where nn is neutral atom number density, collision cross-section σn = 10−19
m2 and < v > is the average thermal velocity of ionized particles. How much
should neutral gas be added so that the gas dynamics would be controlled by
collisions instead of collective electromagnetic interactions?
Explain also why in a) and b) the gas ceases to be plasma if the temperature and
density are lower/higher than the obtained limits. It is helpful to revise how the
Debye length and plasma parameter vary as a function of temperature and density!
Chapter 3
Single Particle Motion
Plasma is composed of a large number of charged particles that move under the influence of electromagnetic fields. The electric and magnetic fields can be either external
(applied fields) or generated by the charged particles themselves. A large part of collective plasma phenomena can be understood (even quantitatively) in terms of single
particle motion. After all, plasma behaviour is based, ultimately, on the motions of
its constituent particles. Single particle description is a very useful approach in studies
of high energy particles in low density plasma where collisions are infrequent and the
external magnetic and electric fields are much stronger than the fields generated by the
motion of charged the particles themselves.
Our task in this chapter is to solve the equation of motion for a charged particle:
m
dv
= q(E + v × B) + Fnon−EM ,
dt
(3.1)
where Fnon−EM govern non-electromagnetic forces.
We will start by investigating how a charged particle moves in the simplest magnetic
field configuration, the static and uniform magnetic field in the absence of an electric
field. Then we will proceed to examine more complicated magnetic and electric fields, in
particular, to determine how the motion of charged particles is affected by spatial and
temporal field gradients.
3.1
Motion in a static, uniform magnetic field
Assume first that the electric field E = 0, there are no non-electromagnetic forces, and
the magnetic field B is constant. Now the equation of motion (Eq. 3.1) of a charged
particle is:
m
dv
= q(v × B) .
dt
29
(3.2)
30
CHAPTER 3. SINGLE PARTICLE MOTION
Taking the scalar product of this with the velocity v and noting that (v × B) · v = 0,
we obtain:
dv
d mv 2
m
·v = (
) = 0,
(3.3)
dt
dt 2
which shows that the kinetic energy and the speed are both constant for a particle in a
static magnetic field.
Next we determine the trajectory of the particle. Let us choose the coordinate system
so that B = Bêz . The components of Eq. 3.2 are:
mv̇x = qBvy
mv̇y = −qBvx
(3.4)
mv̇z = 0 .
The velocity component parallel to the magnetic field (vz ) is constant, i.e., the particle
moves at a constant speed along the magnetic field. This is because the v × B-force
has no component parallel to the magnetic field. We concluded above (Eq. 3.3) that
the total speed of the particle is also a constant, and hence, the absolute value of the
velocity perpendicular to the magnetic field (v⊥ ) must also be constant.
Taking the second time derivatives of the perpendicular velocity components gives:
v̈x = −ωc2 vx
v̈y = −ωc2 vy .
(3.5)
Eq. 3.5 describes a simple harmonic oscillator at the Larmor (or cyclotron, or gyro)
frequency:
qB
ωc =
.
(3.6)
m
As the Larmor frequency is inversely proportional to the mass of the particle, electrons gyrate much faster than ions. For an electron ωc = 1.76×1011 B, while for a proton
ωc = 9.58 × 107 B, where the unit of the gyro frequency is rad s−1 and the magnetic field
is given in Teslas.
Solving the spatial coordinates of the particle (Exercise 3.1) we see that it performs
a circular motion in the xy-plane:
The radius of this gyro motion is Larmor (or cyclotron, or gyro radius):
rL =
mv⊥
,
|q|B
(3.7)
3.1. MOTION IN A STATIC, UNIFORM MAGNETIC FIELD
31
q
where v⊥ = vx2 + vy2 is the particle velocity perpendicular to the magnetic field. The
Larmor radius is proportional to the mass of the particle and inversely proportional to
the magnetic field magnitude. Hence, electrons have much smaller Larmor radii than
ions (see Exercise 3.2 for comparing Larmor frequencies and radii for particles in different
plasma regions).
Therefore, the particle motion is divided into two components:
1. Linear motion along the magnetic field at a constant speed (v|| = constant)
2. Circular motion in the plane perpendicular to the magnetic field. The center of
this circular motion is called the guiding center (GC).
Figure 3.1: The trajectory of a charged particle in space in a homogeneous magnetic
field is a helix.
Combining these motions we will see that the particle is gliding along the magnetic
field while making a circle in the plane perpendicular to the magnetic field (i.e., its GC
follows the field line). The trajectory of a charged particle is thus a helix (Figure 3.1).
The pitch angle of this helix is defined as:
tan α = v⊥ /vk ,
(3.8)
α = arcsin(v⊥ /v) = arccos(vk /v) .
(3.9)
and thus
In many applications it is convenient to omit the relatively fast circular motion of
the particle around the magnetic field as the main interest is to investigate the motion
of the GC. Hannes Alfvén was the first to introduce the idea of this guiding center
approximation.
The frame of reference where v|| = 0 is called the guiding center system (GCS).
The GC approximation is valid if the applied magnetic field varies slowly in space
and in time when compared to the Larmor motion:
B
−1
rL /L 1
(3.10)
∂B/∂t ωc ,
(3.11)
32
CHAPTER 3. SINGLE PARTICLE MOTION
where L is the length scale of the inhomogeneity in the magnetic field.
The ions gyrate in the left-handed sense and the electrons in the right-handed sense
around the magnetic field. We see from Eq. 3.7 that the Larmor radius is inversely
proportional to magnetic field magnitude (rL ∝ B −1 ), i.e., the stronger the magnetic
field, the more tightly particles are “bound” to the magnetic field. We also see that
since rL ∝ m the Larmor radius is much smaller for electrons than for ions (Figure 3.2).
Thus, electrons are bound more tightly to magnetic fields than ions. This means that
ions lose more easily the guidance of the magnetic field than electrons.
Figure 3.2: Larmor orbits of positive ions and negative
electrons in a magnetic field.
In the GCS gyrating charges form small current loops that are associated with the
magnetic moment:
2
µ = πrL
I=
2B
2
1 q 2 rL
1 mv⊥
W⊥
=
=
.
2 m
2 B
B
(3.12)
The directions of the gyro motion of positive and negative charges is always such that
the magnetic moment is opposite to the externally imposed magnetic field (Exercise 3.3).
This means that plasma particles tend to reduce the applied magnetic field, and therefore,
plasma is diamagnetic (Figure 3.3).
µ
A = p rL2
Figure 3.3: Charged particles which gyrate in a
magnetized plasma form small current loops. The
associated magnetic moment µ is always opposite
the magnetic field B.
I +
rL
B
3.2. MOTION IN CONSTANT PERPENDICULAR ELECTRIC AND MAGNETIC FIELDS33
In vector form this is:
q
µ = r L × v⊥ .
2
(3.13)
When a large number of particles is present, the magnetization M is defined as the
magnetic moment per unit volume:
M=−
X
ns hµs ib̂ ,
(3.14)
s
where < µs > is the average magnetic moment of the particle species s,
Magnetization contributes to the current density in plasma. Any circulation in the
magnetization field gives rise to a magnetization current density that can be calculated
as:
JM = ∇ × M .
(3.15)
Note that the magnetization current density is often distinguished from the “true current
density” due to the motion of “free charges” in the medium.
3.2
Motion in constant perpendicular electric and magnetic fields
Next we consider the particle motion in spatially uniform, static electric and magnetic
fields. The equation of motion 3.1 is now:
m
dv
= q(E + v × B) .
dt
(3.16)
The component of the equation along the magnetic field is:
mv̇k = qEk ,
(3.17)
where vk and Ek are the velocity and electric field components parallel to the magnetic
field. This equation describes acceleration along the magnetic field at a constant rate.
Due to high conductivity in the plasma free charges react quickly to qEk . As a
consequence, the electric field component parallel to the magnetic field is typically close
to zero in plasma. However, as discussed in Section 2.6.2 “double layers” (i.e., sustained
parallel electric fields) can arise in some situations, e.g., if there are not enough charge
carriers to maintain the continuity of the electric current.
Let us then investigate the perpendicular component of Eq. 3.16.
mv̇⊥ = q(E⊥ + v⊥ × B) .
(3.18)
By choosing the frame-of-reference appropriately we can eliminate the electric field
from Eq. 3.18 (remember that when conductor (e.g., plasma) moves in a magnetic
34
CHAPTER 3. SINGLE PARTICLE MOTION
field, the observed electric field depends on the frame-of-reference). We can use the
non-relativisic Lorentz transformation (i.e., setting γ = 1):
E
0
0
B
= E + vE × B
= B.
(3.19)
0
Here E is the electric field in the non-moving frame (e.g., the spacecraft frame), E is
the electric field in the moving frame, which we take to be the plasma rest frame, and
vE is the velocity of the frame-of-reference moving perpendicular to E and B (i.e., the
drift velocity of the GC). Since the electric field must vanish in the plasma rest-frame
0
E = E + vE × B = 0, which can be inverted as (see Exercise 3.4):
vE =
E×B
.
B2
(3.20)
This is called the E × B-drift velocity.
In the moving GC frame-of-reference the equation of motion is thus:
0
dv
0
m ⊥ = qv⊥ × B ,
(3.21)
dt
which simply corresponds to a particle moving in a static uniform magnetic field. Hence,
we see that the motion of a particle in the original (non-moving) frame consist of Larmor
motion around the magnetic field and the drift of the GC at the velocity vE perpendicular
to the magnetic and electric fields. The E × B-drift of electrons and ions is shown in
Figure 3.4 in the special case when parallel velocity vk = 0. If the particle has a velocity
component parallel to the magnetic field the GC glides across the magnetic field and the
actual path of the particle in 3-D space is a slanted helix. For a detailed derivation of
the trajectory of a charged particle in constant and perpendicular electric and magnetic
fields see Exercise 3.5.
ion
electron
Figure 3.4: E × B-drift for electrons and ions in constant electric and magnetic fields
for the parallel velocity vk = 0.
It is important that the E × B-drift speed does not depend on the mass, charge or
the velocity of the particle. This means that when many particles are present the whole
3.3. GENERAL DRIFT VELOCITY DUE TO A FORCE PERPENDICULAR TO MAGNETIC FIELD35
plasma drifts at the same speed. Hence, no electric current arises from the E × B-drift
because there is no relative drift of electrons and positive ions. E × B-drift is also very
slow when compared to the particle gyro motion around the magnetic field. In Exercise
3.6 the E × B-drift speed is calculated for an electron in the auroral ionosphere and
compared to its Larmor motion.
Contemplate: For a more complete physical picture consider the energy gain and loss
of a particle during its gyro orbit around B under the influence of constant perpendicular
E. This also explains why the drift is at the constant speed, although the effect of the
force F = qE is either accelerating or decelerating. See also Exercise 3.5.
3.3
General drift velocity due to a force perpendicular to
magnetic field
The particle motion in constant electric and magnetic fields can be generalized to drifts
due to a general constant external force perpendicular to the magnetic field:
dv⊥
q
F⊥
= (v⊥ × B) +
.
dt
m
m
(3.22)
Assuming that the GC drift vD is caused by the force F⊥ we can make the transformation
0 +v :
v⊥ = v⊥
D
0
dv⊥
q 0
q
F⊥
= (v⊥
× B) + (vD × B) +
.
(3.23)
dt
m
m
m
In the GCS two last terms have to cancel each other (see the previous section) and:
vD =
F⊥ × B
.
qB 2
(3.24)
Note that this treatment assumes that F/qB c (otherwise the GC approximation
is no longer valid).
A common example is the gravitational force F = mg, which causes the gravitational
drift:
vg =
mg×B
.
q B2
(3.25)
Note that now vg depends on the sign of the particle’s charge. Hence the ions and
electrons will drift in the opposite directions and a net current density is produced. The
physical reason for the gravitational drift is in the change in the Larmor radius as ions
and electrons gain and lose energy as they move in the gravitational field (i.e., due to
36
CHAPTER 3. SINGLE PARTICLE MOTION
changes in v⊥ ). In most cases of interest the magnitude of the gravitational drift is
negligible to other GC drifts, but it has importance in some regions e.g., in the Earth’s
ionosphere and in the solar atmosphere.
It is also interesting to note that the gravitational drift is, in the same way as the
E×B-drift, not in the direction of the gravitational force, but perpendicular to it. Figure
3.5 illustrates a consequence of these two drifts by demonstrating what happens when
a wave-like ripple develops onto a horizontal surface between plasma (up) and vacuum
(below). The gravitational drift is in the horizontal direction and it separates electrons
and ions creating a small electric field. Figure 3.5 shows that the direction of the electric
field is such that the associated E × B-drift is upwards where the layer has already
moved upward due to a ripple, and downward where the layer is already downward. As
a consequence, the ripple grows and result in an instability known as the Rayleigh-Taylor
instability.
plasma
vg (ions)
B
E‰B
E‰B
Fg=mg
-
-
-
-
+
E
+
+
E
-
+
-
-
-
+
E
+
+
+
vacuum
E‰B
Figure 3.5: Illustration of how gravitational and electric drifts lead together to the
Rayleigh-Taylor instability.
We have now investigated particle motion in uniform magnetic and electric fields
and found expressions for the GC drifts. In the following sections we will investigate the
GC drifts in inhomogeneous fields, i.e., we allow either electric field or magnetic field
to vary in space or in time. Now the equation of motion becomes too difficult to solve
analytically. We will generally assume that the changes in the fields are small when
compared to the Larmor motion.
3.4
Particle motion in non-uniform electric fields
Understanding the behavior of a charged particle in non-uniform electric fields is important, as the response of plasma determines the properties of electromagnetic wave
propagation.
3.4. PARTICLE MOTION IN NON-UNIFORM ELECTRIC FIELDS
3.4.1
37
Spatially varying electric field
We now assume that the magnetic field is uniform and the electric field in the x-direction
varying sinusoidally in the y-direction:
E = E0 cos(ky)êx .
(3.26)
Such an electric field can arise for example due to a wave motion. The equation of
motion for the charged particle is now:
m
dv
= q[E + v × B]
dt
(3.27)
If electric field is weak we can use the undisturbed orbit to estimate Ex and average over
a gyro cycle. We also consider the case of a small Larmor radius, krL 1. The result
is a small correction to the E × B-drift (Exercise 3.7):
1
E×B
2
1 − k 2 rL
vE =
2
B
4
.
(3.28)
The physical reason for this is that a charged particle with its GC near the maximum
electric field spends a significant time in a region of weaker electric field, and thus,
experiences weaker E × B-drift. The correction term depends on the second derivative
of E, and we can generalize:
1 2 2 E×B
vE = 1 − rL
∇
.
4
B2
(3.29)
Note that since electrons and ions have different Larmor radii, this drift is charge
dependent.
3.4.2
Time varying electric fields
Let us next assume that the electric field is uniform in space, but it varies sinusoidally
in time:
E = E0 exp (iωt)êx .
(3.30)
If changes are slow when compared to the particles gyro period (∂/∂t ωc ) we find
the polarization drift (Exercise 3.8):
vP = −
m dvE
1 dE⊥
×B=
.
2
qB dt
ωc B dt
(3.31)
38
CHAPTER 3. SINGLE PARTICLE MOTION
The polarization drift separates particles with different charges and masses and gives
rise to a polarization current (JP = nqvP ). Due to the large mass ratio between electrons
and ions this current is carried mostly by the ions.
It is instructive to contemplate the physics behind the polarization drift. Assume
that an ion is at rest in a magnetic field and that an electric field E is suddenly applied.
The ion will start to move in the direction of E. While gaining speed the particle starts
to feel the Lorentz force qv × B, and consequently starts to E × B-drift. If E will
vary sinusoidally in time as assumed above, E will be reversed after some time and the
particle starts to E × B-drift in the opposite direction.
In the advanced space plasma physics course we will investigate also the cases when
the rate of change in the electric field is at the same order as the gyro frequency of the
particle (ω ∼ ωc ) and high-frequency electric fields (ω ωc ). In the former case it is
found that particles are in the resonance with the wave (see Chapter 7).
3.5
Particle motion in non-uniform magnetic fields
Real magnetic fields can be homogeneous only locally. Both the strength and direction
of the magnetic field varies and this gives rise to important drift motions.
3.5.1
Drift due to a magnetic field gradient
Assume first that the magnetic field lines are straight, but allow the magnitude of the
magnetic field vary in space. The basic motion is again the gyro motion, but now
the particle experiences small field variations as it gyrates. As is seen from Figure 3.6
the GC drift arises because the Larmor radius varies due to changes in the magnetic
field magnitude in different regions. It is also obvious from the figure that the drift is
perpendicular to both B and ∇B.
We assume that the magnetic field is only weakly inhomogeneous and can thus use
three-dimensional Taylor expansion near the GC (indicated by the subscript 0):
B(r) ' B0 + r · (∇B)0 + ...
(3.32)
This expansion requires that rL /L 1, where L is the lenght scale of the field gradient.
The Lorentz force at the GC (i.e., we consider the GC as if it were the drifting
particle) can be calculated as an average over one gyro radius, by using the orbit in the
homogeneous magnetic field from Section 3.1 and the Taylor expansion given above.
After some calculations (Exercise 3.9) one obtains:
3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS
39
Figure 3.6: The Larmor radius of a particle gyrating in a magnetic field whose magnitude
varies in space changes. This results in a GC drift perpendicular both to B and ∇B.
Figure shows the trajectory for a postively charged particle.
F = −µ∇B .
(3.33)
Parallel to the magnetic field this yields the acceleration:
dvk
µ
= − ∇k B .
dt
m
(3.34)
Perpendicular to the magnetic field one obtains from Eq. 3.24 the gradient drift:
vG =
µ
W⊥
B × (∇B) =
B × (∇B) .
qB 2
qB 3
(3.35)
Thus vG depends both on the perpendicular energy and the charge of the particle.
Thus, the gradient drift contributes to the net plasma current.
3.5.2
Drift due to a curved magnetic field
Next, we assume that the density of magnetic field lines is constant but they are curved
with a constant radius of curvature RC (positive inward). Now a drift arises from the
centrifugal force FC felt by the particle as it moves in the magnetic field:
40
CHAPTER 3. SINGLE PARTICLE MOTION
FC = −
mwk2
RC
n̂ ,
(3.36)
where wk is the particle’s speed along the magnetic field. In practice a sufficient accuracy
is wk ' vk . n̂ is the unit vector in the direction of RC . FC is again perpendicular to the
magnetic field and can be inserted to Eq. 3.24. We obtain:
vC = −
mvk2 n̂ × B
q RC B 2
.
(3.37)
Figure 3.7: A charged particle in a curved magnetic field.
The radius of curvature is RC .
Now we need to express n̂ in terms of B. If ds is a small displacement along the
magnetic field, from Figure 3.7 we find that ds = RC dφ and db̂ = n̂dφ. Dividing these
gives:
db̂
n̂
=
.
(3.38)
ds
RC
Since d/ds denotes the derivative along the magnetic field, it can be replaced with (b̂·∇).
Thus:
dB
= (b̂ · ∇)B .
(3.39)
ds
We now obtain the expression for the curvature drift:
vC =
mvk2
qB 4
B × (B · ∇)B .
(3.40)
Similar to vG (Eq. 3.35), we see that vC depends on the charge of the particle,
but while vG depends on the perpendicular energy, the curvature drift depends on the
parallel energy (explain why).
Assuming that there are no local currents (∇ × B = 0), we can write Eq. 3.40 in a
similar form as the equation for the gradient drift:
3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS
mvk2
vC =
qB 3
B × ∇B .
41
(3.41)
Now it is possible to combine vG ja vC :
vGC =
W⊥ + 2Wk
W
(1 + cos2 α)n̂ × t̂ ,
B × ∇B =
3
qB
qBRC
(3.42)
where t̂ k B ja n̂ k RC are unit vectors, and α is the pitch angle.
Contemplate: Investigate the differences between the E × B-drift and the combined
gradient and curvature drift (Eq. 3.42). For example, contemplate whether these drifts
affect primarily low or high-energy particles, do they give rise to an electric current and
can they change the energy of the particle.
3.5.3
Drift due to a time varying magnetic field
Next we allow the magnetic field vary in time. We discussed in Section 3.1 that the
magnetic field cannot do work on a charged particles, but the electric field induced by
a time variable magnetic field (∇ × E = −∂B/∂t) can accelerate/decelerate particles
(Figure 3.8). The particle’s velocity perpendicular to the magnetic field can be written
as v⊥ = dl/dt, where l is the length element of the path along the particle’s trajectory.
Let us take the scalar product of the equation of motion (Eq. 3.1 when Fnon−EM = 0)
with v⊥ :
E
B
Ñ´E = -
¶B
¶t
Figure 3.8: A charged particle is accelerated by an electric field that
is induced by a time-varying magnetic field. If the time variations are
slow when compared with the particle’s gyro motion, the magnetic
moment of the particle stays constant.
dW⊥
= q(E · v⊥ ) .
dt
During one gyration the particle gains energy
4W⊥ = q
Z 2π/ωc
0
(3.43)
E · v⊥ dt .
(3.44)
Assuming slow temporal changes we can replace the time integral by a line integral over
a closed loop and use the Stokes law:
4W⊥ = q
I
C
E · dl = q
Z
S
(∇ × E) · dS = −q
Z
S
∂B
· dS ,
∂t
(3.45)
42
CHAPTER 3. SINGLE PARTICLE MOTION
where dS = n dS, n is the normal vector of the surface with the direction defined by the
positive circulation of the loop C. For small variations of the field ∂B/∂t → ωc 4B/2π,
where 4B is the amount by which the magnetic field changes during one Larmor orbit.
Note that here the magnetic field changes are assumed to be so slow that the Larmor
radius of the particle is not changed significantly during one Larmor orbit. Thus, we
obtain:
1
2
4W⊥ = |q|ωc rL
4B = µ4B .
2
(3.46)
4W⊥ = µ4B + B4µ
(3.47)
On the other hand
and thus 4µ = 0. Hence, in a slowly time varying magnetic field the magnetic moment
µ is conserved although the inductive electric field accelerates the particle throughout
its Larmor orbit.
3.5.4
Adiabatic invariants
We found in the previous section that if the magnetic field is varying slowly in time,
the magnetic moment µ of the charged particle stays constant. In statistical mechanics
the quantity related to a (nearly) periodic motion that stays constant when the system
changes slowly, temporally or spatially, is called an adibatic invariant.
In Hamiltonian mechanics it is shown that if q and p are the canonical coordinate
and momentum of the system and the motion is nearly periodic, the closed integral of p
over one period in q
I
I=
p dq
(3.48)
is an adiabatic invariant. This statement requires a proof that we will not discuss here
(see, e.g., classical mechanics textbooks by Goldstein or Landau and Lifshitz).
The momentum of a particle in the electromagnetic field is p = mv + qA, where A is
the magnetic vector potential and the canonical variables related to the motion perpendicular to the magnetic field are p⊥ ja rL . Using the Stokes theorem and assuming that
the magnetic field and the particle’s perpendicular velocity do not change significantly
during one Larmor gyration we obtain:
I
I =
p⊥ · drL =
I
Z 2πrL
=
0
mv⊥ · drL + q
Z
mv⊥ dl + q
Z
B · dS
S
2
= 2πmv⊥ rL − |q|BπrL
=
2πm
µ,
|q|
i.e., the magnetic moment µ is an adiabatic invariant.
(∇ × A) · dS
S
(3.49)
3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS
43
When conserved µ is called the first adiabatic invariant in plasma physics.
Also the magnetic flux that is enclosed by the particle with its Larmor motion,
2
Φ = BπrL
=
2πm
µ,
q2
(3.50)
is constant. When the magnetic field increases, the Larmor radius decreases and consequently the enclosed flux stays constant.
reference
point, B0
mirror
point, Bm
B
weak B
strong B
Figure 3.9: A charged particle moving towards stronger magnetic field.
Now let us investigate a charged particle that is moving towards a stronger and
stronger magnetic field (Figure 3.9). As the magnetic field increases the Larmor radius
of the particle gets smaller and smaller and its gyro frequency increases. We assume that
the magnetic field changes so slowly that µ stays constant. Since µ = W⊥ /B, it is clear
that to keep µ constant the perpendicular energy (W⊥ ), and hence the perpendicular
velocity (v⊥ ), of the particle increase with the increasing magnetic field. Because in the
GC approximation the total kinetic energy is conserved, the parallel energy Wk and the
parallel velocity vk must decrease. See Exercise 3.10 for a demonstration how the parallel
velocity of a particle varies in a simple magnetic bottle configuration. W⊥ can increase
until Wk → 0. In Section 3.5.1 we discussed that the magnetic field whose magnitude
varies in space causes the force F = −µ∇k B. In this context this force is called the
mirror force and it slows down the GC motion and finally when all parallel energy has
vanished, it turns the particle around, in other words, the particle gets “mirrored”. The
physical origin of the mirror force arises from the convergence of the magnetic field lines.
The Lorentz force has a component opposite to the direction of convergence.
The converging magnetic field regions have many important applications. A magnetic
bottle is composed of two magnetic mirrors (not necessary of equal strengths) placed
facing each other. A charged particle can be trapped within the bottle. Magnetic bottles
have been used in various laboratory experiments to confine plasma (Figure 3.10), and
they are also found in natural plasmas. For example, the Earth’s dipole field forms a
44
CHAPTER 3. SINGLE PARTICLE MOTION
huge magnetic bottle where charged particles bounce between the mirror points in the
northern and southern hemisphere (Figure 3.11, see also Exercise 3.11). The high energy
particles trapped in this bottle form the Van Allen radiation belts. Another example of
a natural magnetic bottle is a solar coronal loop.
Figure 3.10: Two magnetic mirrors facing each other form a magnetic bottle. In practical
plasma confinement experiments much more complicated coil geometries are used to
improve the confinement conditions.
trapped particle
mirror point
North
ions
electrons
magnetic field line
South
Figure 3.11: The Earth’s dipole field forms a large magnetic bottle. The electrons
gradient and curvature drift eastward and ions westward carrying a net westward current
around the Earth.
For many practical purposes it is interesting to know which particles are mirrored
and remain trapped in the magnetic bottle and which can escape from it (Exercise
3.12). If a particle has too much parallel energy with respect to the maximum magnetic
field magnitude in the converging field, it will escape from the bottle. For example, if a
particle in the Van Allen radiation belt hits the Earth’s atmosphere before it is reflected,
it will be lost. Write the perpendicular velocity in terms of the pitch angle: v⊥ = v sin α.
3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS
45
Now the magnetic moment can be expressed as:
µ=
mv 2 sin2 α
.
2B
(3.51)
µ is assumed to be constant and on the other hand v 2 ∝ W is also a constant. Thus,
we find a relation between the pitch angle and the magnetic field magnitude at two
locations:
B1
sin2 α1
=
.
(3.52)
2
B2
sin α2
From the definition of the pitch angle we see that at the mirror point α → 90◦ , as
Wk → 0. If B2 in Eq. 3.52 is the mirror field Bm (see Figure 3.9) then sin α2 = 1.
Therefore, the strength of the mirror field Bm depends on the particle’s pitch angle at
the reference point (subscript 0):
sin2 α0 = B0 /Bm .
(3.53)
If B0 is the weakest magnetic field in the bottle and Bm is the weaker of the mirror
fields, the particle will be trapped in the bottle if:
s
arcsin
B0
≤ α0 ≤ 180 − arcsin
Bm
s
B0
,
Bm
(3.54)
Otherwise the particle is lost from the bottle. It is said to be in the loss-cone.
If the magnetic field does not change much during the time the charged particle
bounces back and forth between the magnetic mirror, the bounce motion is nearly periodic. The bounce period τb is obtained from the formula:
τb = 2
Z s0
m
sm
2
ds
=
vk (s)
v
Z s0
m
ds
sm
(1 − B(s)/Bm )1/2
,
(3.55)
0
where s is the arc length along the GC orbit and sm and sm are the coordinates of the
mirror points. Note that the bounce period is defined over the whole bounce motion
back and forth. The GC approximation is valid if τb ωc−1 . Thus, the condition to
consider the bounce motion as nearly periodic is more restrictive than in the case of
Larmor motion:
dB/dt
τb
1.
(3.56)
B
If this condition is fulfilled, there is an associated adiabatic invariant, which in plasma
physics is called the second adiabatic invariant
I
J=
pk ds ,
where pk = mvk for a non-relativistic particle.
(3.57)
46
CHAPTER 3. SINGLE PARTICLE MOTION
To directly prove the invariance of J in a general case is a formidable task. The
complete proof is given by Northrop [1963]. The textbook by Goldston and Rutherford
[1995] presents the proof for time-independent fields, which is long enough.
Also the drift across the magnetic field may be nearly-periodic if the field is sufficiently symmetric, as e.g., in the quasi-dipolar planetary magnetic fields.
The corresponding third adiabatic invariant is the magnetic flux through the closed
contour defined by the GC drift:
I
Φ=
A · ds ,
(3.58)
where A is the vector potential of the magnetic field and ds is the arc element along the
drift path of the GC. The drift period τd has to fulfill τd τb τL . The invariant is
weaker than µ and J because much slower changes in the field can break the invariance
of Φ.
In the Earth’s magnetosphere µ is often a good invariant. J is invariant for particles
that spend at least some time in the magnetic bottle defined by the nearly dipolar field of
the Earth. Φ is constant for energetic particles in the trapped radiation belts. However,
any or all of the invariances can be broken by perturbations to the system.
Figure 3.12: Three adiabatic invariants and corresponding motions in the Earth’s magnetic field.
The table below and Figure 3.12 present all three adiabatic invariants and associated
motions.
3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS
Invariant
magnetic
moment µ
longitudinalinvariant J
flux invariant
Φ
Speed
gyro motion
v⊥
parallel velocity
of GC wk
perpendicular velocity
of GC w⊥
Time-Scale
gyro period
τL = 2π/ωc
bounce period
τb
drift period
τd
47
Validity
τ τL
τ τb τL
and µ constant
τ τd τb τL
and µ and J constant
Every invariant has its characteristics energy conversion mechanism. First, let’s
consider a particle drifting across the magnetic field from the field B1 towards B2 with
B2 > B1 so that its magnetic moment µ (i.e. the first adiabatic invariant) is conserved.
Such a drift can be caused e.g. by the E × B drift.
The conservation of µ leads to adiabatic heating:
W⊥2
B2
=
,
W⊥1
B1
(3.59)
An example of adiabatic acceleration is given in Exercise 3.13.
Next, consider a particle bouncing between two magnetic mirrors, conserving J (i.e.
the
second adiabatic invariant). Moving the mirror points closer to each other causes
H
ds to decrease. To compensate this, vk , and thus, Wk must increase. This is called
Fermi acceleration.
Enrico Fermi introduced this mechanism to explain the acceleration of cosmic rays
to very high energies (107 − 1010 eV) in the magnetic fields of the universe. A typical
galactic cosmic ray has wandered around in the galaxy for millions of years. The radius
of the Milky Way is of the order of 100 000 light years, and thus the particle has had
a lot of time to “collide” with magnetic field structures in the galaxy that have a wide
range of velocities. Note that in a given reference frame (e.g., ours) the particle either
gains or loses energy when it gets deflected by a magnetic structure (e.g., a mirror). As a
result, the velocity distribution of the seed population widens and finally some particles
end up at very high energies.
The modern version of Fermi acceleration, believed to be responsible for the acceleration of galactic cosmic rays, no longer relies on the conservation of the second adiabatic
invariant in a distribution of moving magnetic mirrors. Instead, particles are assumed to
be accelerated in shock waves generated in supernova explosions by a mechanism called
diffusive shock acceleration. In this model, particles gain energy by repeatedly crossing
a single shock front from one side to the other (details not discussed in these lectures).
Finally, if the magnetic flux through the closed contour particle’s drift encloses stays
constant (i.e., third adiabatic invariant), the total energy can change if the drift shells are
48
CHAPTER 3. SINGLE PARTICLE MOTION
compressed or expanded. As a summary, the characteristic energy conversion mechanism
for each invariant are
µ: W⊥ changes when Larmor radius (that is, |B|) changes
J: Wk changes when streching or contracting the magnetic bottle
Φ: W changes when compressing or expanding drift shells
3.6
Examples of particle motion in simple geometries
Real magnetic field configurations in laboratories and in space are usually so complicated
that numerical integration of the equation of motion is required. In this section we briefly
discuss the motion in two simple but in practice very important geometries: the dipole
field and the field of a current sheet.
3.6.1
Motion in a dipole field
Calculation of charged particle motion in the dipole field is an important application
of the orbit theory. Within the distances 2–7 RE from the Earth’s center the dipole
is a reasonably good approximation of the geomagnetic field and all particles except
high-energy cosmic rays behave adiabatically as long as their orbits are not disturbed
by collisions or time-varying electromagnetic fields.
In the following we use “geomagnetically” defined spherical coordinates. The dipole
moment ME is in the origin and points toward the south. Latitude (λ) is zero at
the equator and increases toward the north. Longitude (φ) increases toward the east
from a given reference longitude. The SI unit of ME is A m2 . ME is often replaced
by k0 = µ0 ME /4π, which is also referred to as the dipole moment. The strength and
orientation of the terrestrial dipole moment varies slowly and must be taken into account
in time scales of space climate. For our purposes sufficiently accurate approximations
are
ME = 8 × 1022 A m2
k0
= 8 × 1015 Wb m (SI : Wb = T m2 )
= 8 × 1025 G cm3 (Gaussian units, G = 10−4 T)
3
= 0.3 G RE
(RE ' 6370 km)
The last (non-SI) expression is useful in practice because the dipole field on the surface
of the Earth varies in the range 0.3–0.6 G.
The dipole field is an idealization where the source current is assumed to be shrunk
into a point at the origin. The source of a planetary or stellar magnetic field is actually
a finite, even large, region within the body giving rise to a whole sequence of higher
multipoles. When moving away from the source the non-dipolar (quadrupole, octupole,
etc.) contributions vanish faster than the dipole. Outside the source the field is a
potential field (B = −∇Ψ). The potential for the dipole is
1
sin λ
Ψ = −k0 · ∇ = −k0 2 .
r
r
(3.60)
3.6. EXAMPLES OF PARTICLE MOTION IN SIMPLE GEOMETRIES
49
It is a standard exercise in elementary electromagnetism to show that
B=
1
[3(k0 · er )er − k0 ] ,
r3
(3.61)
from which
Br = −
2k0
sin λ
r3
k0
cos λ
r3
= 0.
Bλ =
Bφ
(3.62)
The magnitude of the magnetic field is
B=
k0
(1 + 3 sin2 λ)1/2
r3
(3.63)
r = r0 cos2 λ ,
(3.64)
and the equation for the field line is
where r0 is the distance from the dipole to the point where the field line crosses the
dipole equator. In dipole calculations we also need the length of the line element
ds = (dr2 + r2 dλ2 )1/2 = r0 cos λ(1 + 3 sin2 λ)1/2 dλ .
(3.65)
The geometric factor (1 + 3 sin2 λ)1/2 = (4 − 3 cos2 λ)1/2 appears frequently in dipole
expressions.
Every dipole field line is uniquely determined by its (constant) longitude φ0 and
the distance r0 . A useful quantity is the L parameter L = r0 /RE . For a given L the
corresponding field line reaches the surface of the Earth at the latitude
1
λe = arccos √ .
L
(3.66)
The field magnitude along a given field line as a function of latitude is
B(λ) = [Br (λ)2 + Bλ (λ)2 ]1/2 =
k0 (1 + 3 sin2 λ)1/2
.
cos6 λ
r03
(3.67)
For the Earth, we find
k0
0.3
3 × 10−5
=
G
=
T.
L3
L3
r03
(3.68)
At the equator on the surface of the Earth the dipole field is 0.3 G, and at the poles 0.6 G
(i.e., 30 and 60 µT), respectively. The observable geomagnetic field has considerable
deviations from this because the dipole is not quite in the center of the Earth, the
source is not a point, and the conductivity of the Earth is not uniform.
50
CHAPTER 3. SINGLE PARTICLE MOTION
The guiding center approximation can be applied if the particle’s Larmor radius is
much smaller than the curvature radius of the field defined by RC = |d2 r/ds2 |−1 , which
for a static dipole field is
RC (λ) =
r0
(1 + 3 sin2 λ)3/2
cos λ
.
3
2 − cos2 λ
(3.69)
In terms of the particle’s rigidity mv⊥ /|q|, we write
∇⊥ B = mv⊥ ∝ mv⊥ ,
rL B |q|RC B
|q|r0 B
(3.70)
and thus, the GC approximation is valid if
mv⊥
r0 B .
|q|
(3.71)
Contemplate: Rigidity is a widely used concept in cosmic ray studies. Consider
two otherwise identical cosmic ray particles, but with different momenta. Which one is
affected more when travelling through a magnetic field, the one with more momentum,
or one with less? And why? Write the rigidity in terms of the Larmor radius.
The dipole field is a magnetic bottle and the energetic particles trapped in the bottle
around the Earth or magnetized planets are said to form trapped radiation. Let λm be
the mirror latitude of a trapped particle and let the subscript 0 refer to the equatorial
plane. Then the equatorial pitch angle of the particle is
sin2 α0 =
B0
cos6 λm
=
.
B(λm )
(1 + 3 sin2 λ)1/2
(3.72)
This shows that the mirror latitude does not depend on L, but the mirror altitude does.
If λe is the latitude where the field line intersects the surface of the Earth and if
λe < λm , the particle hits the Earth before mirroring and is lost from the bottle. In
reality the loss takes place in the upper atmosphere at an altitude that depends on the
particle’s energy, i.e., on how far it can penetrate before it is lost by collisions. The
critical pitch angle in the equatorial plane is (Exercise 3.14)
sin2 α0l = L−3 (4 − 3/L)−1/2 = (4L6 − 3L5 )−1/2 .
(3.73)
The particle is in the loss-cone, if α0 < α0l . For the derivation of the loss cone size as a
function of latitude see Exercise 3.15.
The conservation of the second adiabatic invariant requires that the bounce period
is much shorter than the variations in the magnetic field. For example, in the inner
magnetosphere the bounce times of 1-keV electrons are a few seconds and of 1-keV
protons a few minutes. During magnetospheric disturbances typical time scales of the
field changes are minutes. Thus under such conditions J is a good invariant for electrons
but not for protons or heavier ions.
3.6. EXAMPLES OF PARTICLE MOTION IN SIMPLE GEOMETRIES
51
Both the gradient and curvature of the dipole field are directed toward the planet.
In the dipole field of the Earth positively charged ions drift to the west and electrons to
the east.
Because ∇ × B = 0, we find for vGC
vGC
=
W
(1 + cos2 α)
qBRC
=
(1 + 3 sin2 λ)1/2
3mv 2 r02 cos5 λ(1 + sin2 λ)
2
2
−
sin
α
0
2qk0
cos6 λ
(1 + 3 sin2 λ)2
(3.74)
"
#
.
Particles with 90◦ -pitch angle have zero parallel velocities, and hence, stay at the equator.
The gradient drift velocity in this special case is derived in Exercise 3.16.
3.6.2
Particle motion in a current sheet geometry
The single particle approach is also useful when describing charged particle motion near
a current sheet. When two regions of oppositely directed magnetic fields are brought
together, a sheet of current must arise according to Ampère’s law (∇ × B = µ0 J) to
account for the change in the magnetic field. An example of a current sheet in space
plasmas is the tail current sheet in the Earth’s extended magnetotail, where a current
arises to separate the oppositely directed magnetic fields in the southern and northern tail
lobes. An even larger-scale current sheet is the “heliospheric current sheet” that extends
to the whole heliosphere and separates the opposite magnetic fields in the southern and
northern solar hemispheres. Current sheets have also a key role in many solar phenomena
(e.g., solar flares and coronal mass ejections), in the interaction between the solar wind
and the Earth’s magnetosphere, and in fusion experiments.
Previously in this section we have generally assumed that the Larmor radius of a
charged particle that gyrates in the magnetic field is small compared to the length
scale of the field gradients. This assumption allowed us to use the GC approximation
and Taylor expansion around the GC to estimate the particle’s orbit. However, near
a thin current sheet there can be large field gradients over short spatial distances, and
hence, neither the GC approximation nor the invariance of µ are no longer valid. Many
physically interesting phenomena (instabilities, magnetic reconnection) arise near strong
and thin current sheets where particle motion becomes chaotic and non-adiabatic.
The simplest model to describe the magnetotail current sheet is the Harris model for
one- and two-dimensional configurations (Figure 3.13). In the two-dimensional Harris
current sheet the magnetic field is of the form
z
ex + Bn ez ,
L
B = B0 tanh
(3.75)
where B0 and Bn are constant, Bn B0 and L is the characteristic thickness of the
current sheet. If Bn = 0, the field is one-dimensional. The magnetic field magnitude
52
CHAPTER 3. SINGLE PARTICLE MOTION
changes from a constant value far away from the current sheet (B0 ) as a hyberbolic
tangent accross the sheet. The electric current points toward the positive y-axis and is
Jy =
z
B0
sech2
µ0 L
L
.
(3.76)
Figure 3.13: left) One-, and right) two-dimensional Harris current sheet.
Examples of orbits near a current sheet are given in Figure 3.14. Outside the current
sheet the motion is normal Larmor motion, but in the vicinity of the current sheet the
motion is more complicated. The monotonic motion in the ±y-direction is called Speiser
motion. Particles in the Speiser motion carry most of the current in the current sheet.
They do not conserve the magnetic moment.
Figure 3.14: Trajectories of positively charged particles near the 1-dimensional Harris
current sheet.
Contemplate: Explain qualitatively the particle trajectories shown in Figure 3.14.
Consider the effect of gradient drift and note how density and magnetic field changes
from the current sheet outwards. (In the Advanced Space Plasma Physics course we will
also look the particle motion more quantitatively).
3.7. EXERCISE: SINGLE PARTICLE MOTION
3.7
53
Exercise: Single Particle Motion
1. Consider the case with a uniform magnetic field with no background electric field.
The Lorentz force on a charged particle is F = qv × B. Starting from this, derive
solution for the spatial coordinates of the particle in detail. Study how positively
and negatively charged particles rotate in the magnetic field according to this
solution.
2. Derive numerical scaling formulas for the gyro frequency ωg and gyro radius rg
using electron mass, electron charge and km/s as scaling units (i.e., express particle
mass, charge and velocity in these units) and calculate them (assuming vk = 0) for
(a) a 10-keV electron in the Earths magnetic field with B = 500 nT and plasma
electron density n0 = 100 cm−3
(b) a solar wind proton with bulk velocity 400 km −1 , B = 5 nT and n0 = 5 cm−3
(c) a 1-keV He+ ion in a sunspot, where B = 0.05 T and n0 = 109 cm−3 .
Compare the gyro frequencies with the plasma frequencies in the corresponding
plasmas.
3. A charged particle rotating in a magnetic field generates an electric current. Calculate the generated magnetic moment and show that the magnetic moment vector antiparallel to the magnetic field vector B for both positively and negatively
charged particles. Show also that the magnetic moment can be written as µ =
W⊥ /B.
4. Show that the electric drift velocity
vE =
E×B
B2
can be obtained from the Lorentz-transformed electric field E = E + vE × B = 0.
5. Study a particle (mass m and charge q) in static and homogeneous electric E =
E0 ey and magnetic B = B0 ez fields. If the particle is at rest at t = 0, show that
it follows a cycloid orbit:
E0
x(t) =
B0
y(t) =
!
1
t−
sin(ωg t)
ωg
E0 q
(1 − cos(ωg t))
ωg B0 |q|
Draw the orbits for a positive and a negative particle. Prove that the average
kinetic energy of the drift motion equals the average potential energy drop of the
particle in the electric field during half a cycloid orbit.
54
CHAPTER 3. SINGLE PARTICLE MOTION
6. Calculate Larmor radius, Larmor period and E × B-drift speed for an electron
(energy 0.1 eV) in auroral ionosphere where magnetic field is 50 000 nT. Assume
that the electric field is perpendicular to the magnetic field and has a magnitude
of 20 mV m−1 . How far does the electron drift during one Larmor period?
7. Consider a charged particle in a homogeneous magnetic field B = B0 ez and inhomogeneous electric field E = E0 cos(ky) ex , here B0 and E0 are constant and
krL << 1. Show that the drift speed is
vE =
E×B
2
(1 − k 2 rL
/4).
B2
Give a physical interpretation for krL << 1 and compare its validity and drift
speeds for electrons and protons.
8. Starting from the equation of motion derive the polarization drift
vP =
1 dE⊥
.
ωc B dt
9. Derive the force
F = −µ∇B .
on the guiding center of a charge (with magnetic moment µ) moving in an inhomogeneous but straight magnetic field B. (Feel free to use text books in plasma
physics.)
10. Let us study a magnetic bottle with B(z) = B0 (1 + (z/a0 )2 ) (sketch this field
configuration!). Using conservation of energy and first adiabatic invariance, show
that a particle (mass m), which is mirroring between points −zm and zm , has a
longitudinal velocity
s
vk =
2µB0
m
s
zm
a0
2
−
z
a0
2
,
where µ is the magnetic moment. What is the particle velocity a the centre of
the bottle (z = 0, where B = B0 ) and at the mirror points (z = zm and B =
B0 (1 + (zm /a0 )2 ) )?
11. Draw a picture that shows how electron and ion orbits look like when the particles
bounce between the two mirror points in the Earth’s dipole field. Pay attention to
the gyro motion around the mirror points. How does the gyro radius change along
the orbit?
12. A group of charged particles with an isotropic velocity distribution is placed in a
magnetic bottle with a mirror ratio of Rm = Bm /B0 = 4. There are no collisions,
so the particles in the loss cone simply escape and the rest remain trapped. What
fraction of the particles is trapped? (Hint: Try to figure out what fraction of space
the loss cone fills.)
3.7. EXERCISE: SINGLE PARTICLE MOTION
55
13. A proton with 1-keV kinetic energy and vk = 0 in a uniform magnetic field B =
0.1 T is accelerated adiabatically as B is slowly increased to 1 T. The proton then
makes an elastic collision with a heavy particle and changes direction so that
v⊥ = vk . The magnetic field is then adiabatically decreased back to 0.1 T. What
is the proton’s kinetic energy now?
14. Starting from the expressions for the components of the magnetic dipole field
Br , Bλ and Bφ make a detailed derivation of the expression for the particle pitch
angle at the equatorial plane as a function of its mirror latitude B(λm )
sin2 α0 =
B0
cos6 λm
=
B(λm )
(1 + 3 sin2 λm )1/2
and show that the loss-cone width in terms of the L-parameter is given by
sin2 α0 = L−3 (4 − 3/L)−1/2 = (4L6 − 3L5 )−1/2 .
15. Derive the width of the loss cone as a function of latitude along a magnetic field
line (from the equator towards Earth). Draw the loss cone size as a function of
latitude for L = 6. Study how the loss cone size varies between the equator and
higher latitudes and as the function of L.
16. Calculate the gradient-drift velocity of a 90◦ -pitch angle particle in a dipolar magnetic field. Start with equations
vD =
F⊥ × B
qB 2
and F = −µ∇B. Before you start writing down equations, think very carefully
what the pitch-angle assumption means.
56
CHAPTER 3. SINGLE PARTICLE MOTION
Chapter 4
Kinetic Plasma Description
As the first step to understand plasma we studied how individual particles behave in
electric and magnetic fields. However, the definition of plasma requires that there has
to be a large number of particles within a Debye sphere and that the plasma system has
the size of several Debye lengths (Eq. 2.2). In plasma charged particles that move in
the applied electric and magnetic fields generate their own fields. The computation of
the motion of all plasma particles from Maxwell’s equations and the Lorentz force would
be an immense task. The kinetic plasma approach has its roots in statistical physics,
representing the behaviour of a large collection of particles using distribution functions
in configuration and velocity space. In a case of plasma one needs to include Maxwell’s
equations in the formulation of the theory.
Kinetic theory is one of the most challenging areas of plasma physics. Here we introduce the most central concepts only. Fortunately, the fluid description (see Section 1.3
and Chapter 6) is sufficiently comprehensive to describe a large part of observed plasma
phenomena. However, the fluid approach loses the detailed information on distribution
of the plasma constituents in the velocity space. For instance, the kinetic description is
relevant in situations where significant deviations from the local thermodynamic equilibrium arise and plasma particle species are non-Maxwellian. In particular in hot and
tenuous plasmas there are not enough collisions to drive plasma towards Maxwellian
distribution. Kinetic theory must also be used when one considers phenomena occurring
at short spatial (smaller than Debye length or Larmor radius) or temporal (faster than
gyro or plasma frequency scales) scales. In addition, description of kinetic processes,
such as instabilities and wave-particle interactions, require the knowledge of the velocity
space effects.
4.1
Distribution function
The dynamical state of a particle in a plasma at time t can be described by its position:
r = xêx + yêy + zêz
57
(4.1)
58
CHAPTER 4. KINETIC PLASMA DESCRIPTION
and velocity:
v = vx êx + vy êy + vz êz .
(4.2)
Combining information of the particle’s position and velocity gives its location (r, v) in
a 6-dimensional phase space (Figure 4.1). The infinidesimal volume element of phase
space is d3 r d3 v.
v
(r,v)
d3v
Figure 4.1: 6-dimensional phase space
d3r
r
In statistical physics the single particle distribution function f (r, v, t) expresses the
number density in a 6-dimensional phase space element at time t. Hence, f is a function
of seven independent variables. The units of number density in the configuration space
is m−3 and in the velocity space is (m s−1 )−3 , thus the units of the distribution function
are [f ] =m−6 s3 .
An example of a domain where kinetic processes prevail is the solar wind. The solar
wind has low density (on average 5 cm−3 ), it is collisionless and is composed of different
particle species. For examples, wave-particle interactions play an important role in the
solar wind and temperature anisotropies drive kinetic instabilities. Although large-scale
solar wind variations can be understood in terms of single-fluid approach, micro-scale
processes may affect the local solar wind properties. Figure 4.2 illustrates typical velocity
distribution functions (VDF) in the solar wind.
The distribution function needs to be normalized. The most intuitive normalization
is to require that the integration of the distribution function over the 6-dimensional
phase space volume V gives the total particle number N .
Z
f (r, v, t) d3 rd3 v = N .
(4.3)
V
The average density in spatial volume V is hni = N/V . However, the density can
usually vary with space and time, and thus, the particle number density is defined as
the zero order velocity moment of the distribution function
Z
n(r, t) =
f (r, v, t) d3 v .
(4.4)
Note that in statistical and mathematical physics the distribution function is often
normalized to 1. This is also a logical normalization, because then f (r, v, t) gives the
probability to find the particle at location r with velocity v at time t in the 6-dimensional
phase space (or if integrated over the whole phase space it states that the probability to
find the particle somewhere in the phase space is 1).
4.1. DISTRIBUTION FUNCTION
59
Figure 4.2: Velocity distribution function (VDF) measured in the solar wind [Stevark
et al., JGR, 2009]. VDF in the solar wind exhibits three components, 1) thermal core,
2) non-Maxwellian halo with isotropic pitch angle distribution, and 3) non-Maxwellian
strahl that features an electron beam propagation along the magnetic field.
As an example let us investigate the Maxwellian velocity distribution function (see
Appendix 9.4):
!
3/2
m
mv 2
f (v) = n
exp −
,
(4.5)
2πkB T
2kB T
where m is the mass of the particle and density n = hni is assumed to be constant.
Using the result:
Z ∞
√
exp(−x2 ) dx = π
(4.6)
−∞
it is easy to show (Exercise 4.1) that the integral of the Maxwellian distribution over the
3-dimensional velocity space gives n ([n] = m−3 ). The average and root-mean-square
velocities for a Maxwellian velocity distribution are calculated in Exercise 4.2.
The definition of the particle density as an integral of the distribution function illustrates how macroscopical parameters can be expressed as velocity moments of the
distribution function:
R
f d3 v ;
R
vf d3 v ;
R
vvf d3 v .
Note that vv is a cartesian tensor, whose components are vi vj . Velocity moments
depend on time and space. Because in plasma different particle species have often
different distribution functions we distinguish them using Greek subscripts.
60
CHAPTER 4. KINETIC PLASMA DESCRIPTION
The first-order moment yields the particle flux
Z
Γα (r, t) =
vfα (r, v, t) d3 v .
(4.7)
Its SI units are (m−3 )(m s−1 ) = m−2 s−1 , which shows that the particle flux is the
number of particles that traverse through a unit surface in a unit time. Dividing this by
particle density we get the average, or bulk, velocity at a given location:
R
vfα (r, v, t) d3 v
Vα (r, t) = R
,
3
fα (r, v, t) d v
(4.8)
from which we can further determine the current density:
Jα (r, t) = qα Γα (r, t) = qα nα Vα (r, t) .
(4.9)
The second order moment gives parameters that are related to the square of the
velocity such as pressure and kinetic energy. In plasma physics pressure is a tensor
quantity (particles are likely to have different velocities parallel and perpendicular to
the magnetic field) and it is defined to depend on how much the particle velocities
deviate from the average velocity Vα :
Pα (r, t) = mα
Z
(v − Vα )(v − Vα )fα (r, v, t) d3 v ,
(4.10)
which in a case if spherical symmetry reduces to Pα = pα I , where I is the unit tensor
and pα the scalar pressure:
mα
(v − Vα )2 fα (r, v, t) d3 v = nα kB Tα (r, t) .
(4.11)
pα (r, t) =
3
Here we have introduced the concept of temperature Tα . In the frame moving with the
velocity Vα , i.e., where Vα = 0, the temperature is given by:
Z
3
mα v 2 fα (r, v, t)d3 v
R
kB Tα (r, t) =
,
2
2
fα (r, v, t) d3 v
R
(4.12)
which for a Maxwellian distribution is the temperature of classical thermodynamics. In
collisionless plasmas equilibrium distributions may be far from Maxwellian and, consequently, temperature is a non-trivial concept. Temperature can be understood in terms
of the width of the distribution function, but only for the Maxwellian distribution function there is a unique level where to determine the width to correspond to the classical
definition of temperature.
The chain of moments continues to higher orders. The third order introduces the
heat flux, i.e., temperature multiplied by velocity. Higher moments can be calculated,
but do not have a simple physical interpretation. In plasma physics higher moments
than the heat flux are seldom needed.
Contemplate: Write the distribution function of electrons that are all moving at the
same velocity V0 and the ions are all at rest. Write also equations for the electric current
and the pressure tensor
4.2. TIME EVOLUTION OF DISTRIBUTION FUNCTIONS
4.2
61
Time evolution of distribution functions
To determine how particle distribution functions evolve in space and time we need the
appropriate equations of motion. We start by assuming that the number of particles
in the 6-dimensional phase space remains constant. We will investigate a small plasma
element and follow its motion. Each point in the plasma element moves according to
equations:
dr
dv
F
=v ;
= ,
(4.13)
dt
dt
m
where F describes the forces that influence the system. The number of particles within
a volume V of a 6-dimensional phase space is:
Z
f (r, v, t) d3 r d3 v .
N=
(4.14)
V
Long-range forces affect in a similar way to all particles in the plasma element, but
short range forces, typically resulting from collisions, can scatter particles in and out
from the phase space element. Here we consider only long-range forces. Figure 4.3
illustrates the evolution of a phase space plasma element under the influence of longrange forces. All particles will be accelerated by the same force, and the phase space
density at time t2 will be the same as at time t1 . Only if there are collisions the density
can change.
plasma
element
v
t1
v
t2
r
r
Figure 4.3: A plasma element retains its density in 6-dimensional phase space as it moves
under the influence of long-range forces.
The conservation of particle number in volume V that moves with the particles is
given by the continuity equation:
0=
∂N
+ ∇u · (N U) =
∂t
Z V
∂f
+ ∇u · (f U)
∂t
d3 rd3 v ,
(4.15)
where U = (ẋ, v̇) = (v, F/m), and ∇u is the 6-dimensional gradient operator, whose
components are the the components of the gradients in the configuration and velocity
spaces (∂/∂r, ∂/∂v) The first term on the left-hand side depends on the change in density
at each phase space point and the latter depends on how V changes with the motion so
62
CHAPTER 4. KINETIC PLASMA DESCRIPTION
that the change of the volume does not change the total number of particles within it.
As the conservation of particles has to apply for all phase space elements, we obtain:
∂f
+ ∇u · (f U) = 0 .
∂t
(4.16)
If the force F does not depend on the velocity (remember that U = (ẋ, v̇)) we can write
the above equation as:
∂f
∂f
F ∂f
+v·
+
·
= 0.
(4.17)
∂t
∂r
m ∂v
The Coulomb and gravitational forces do not depend on velocity, but the magnetic
part of the Lorentz force does. However, fortunately:
∂
· (v × B) = 0 ,
∂v
(4.18)
granting that Equation (4.17) applies also to the Lorentz force.
Thus, we have arrived at an equation that describes the evolution of the distribution
function under the influence of long-range forces. This is called the Vlasov equation:
∂f
q
∂f
∂f
+v·
+ (E + v × B) ·
= 0.
∂t
∂r
m
∂v
(4.19)
It was formulated by the Soviet theoretical physicist Anatoly Alexandrovich Vlasov in
the late 1930s.
In classical statistical physics the particle collisions are important and the equation
corresponding to the Vlasov equation is the Boltzmann equation:
∂f
∂f
F ∂f
+v·
+
·
=
∂t
∂r
m ∂v
∂f
∂t
,
(4.20)
c
where the term on the right hand side describes the change of the distribution function
due to individual collisions.
The Vlasov equation is sometimes called as the “collisionless Boltzmann equation”.
Ludwig Boltzmann derived the collision term (∂f /∂t)c for strong short-range interactions. In plasma physics Coulomb interactions are mainly long-range and weak. Therefore in plasma physics the average interactions between the particles are included in the
Boltzmann equation through the external Lorentz force:
∂f
∂f
q
∂f
+v·
+ (E + v × B) ·
=
∂t
∂r
m
∂v
∂f
∂t
.
c
(4.21)
4.3. SOLVING THE VLASOV EQUATION
63
The collision term includes only large-angle collisions between charged particles and
the possible collisions with neutrals, including charge-exchange processes. The general
calculation of the Boltzmann collision term (∂f /∂t)c is a tedious task. Fortunately, hot
and low density plasmas can often be considered collisionsless and the Vlasov equation
is the appropriate approach.
The simplest situation taking account the collisions arises when the collisions occur
predominantly with neutrals. In this case the collision term can be approximated by the
Krook model :
∂f
= −νc (f − f0 ) ,
(4.22)
∂t c
where f0 is the equilibrium distribution and νc is the constant average collision frequency.
In Exercise 4.3 the conductivity is determined for unmagnetized, homogeneous and timeindependent plasma where collisions are taken into account using the Krook model.
Taking into account the effect of long-range Coulomb interactions results in so-called
Fokker-Planck equations. Their derivation is also a rather difficult task and beyond the
scope of this book.
4.3
Solving the Vlasov equation
The Vlasov equation is not easy to solve. It must, of course, be done under the constraint to fulfill Maxwell’s equations because the source terms of Maxwell’s equations
(ρ, J) are determined by the distribution function, which, in turn, evolves according to
the Vlasov equation. Furthermore, the force term in the Vlasov equation is nonlinear.
Thus the Vlasov equation can be solved analytically only for small perturbations when
linearization is possible.
We investigate here only the simplest case, where there are no background electric or
magnetic fields. Let us also assume 2-dimensional phase space (x, v), and that the plasma
distribution function is homogeneous and depends on speed only f0 (v). We consider how
the plasma responds to a small perturbation. This corresponds to the setting in Section
2.3 when we derived the formula for the plasma frequency. We consider again the electron
motion only and assume ions as a fixed background. Since plasma starts to oscillate, we
assume that the perturbation will cause an electric field of the form of a plane wave:
E(x, t) = Ê exp[i(kx − ωt)] .
(4.23)
If we denote the small perturbation to the distribution functioon by f1 , the distribution function that enters to the Vlasov equation is f = f0 + f1 and
∂f
∂f
e ∂f
+v
− E
= 0.
∂t
∂x m ∂v
(4.24)
In the case of small perturbations the linearization is possible, which means that
only the first order terms of small perturbations will be considered. Since f0 is the
64
CHAPTER 4. KINETIC PLASMA DESCRIPTION
equilibrium solution of the Vlasov equation the sum of the zero order terms is trivially
zero. The first-order Vlasov equation thus becomes:
∂f1
∂f1
e ∂f0
+v
− E
= 0.
∂t
∂x
m ∂v
(4.25)
From Maxwell’s equations we only need:
0 ∇ · E = ρ = −e
Z
f1 d3 v ,
(4.26)
which in the 1-dimensional case simplifies to the form:
0
∂E
= −e
∂x
Z ∞
−∞
f1 dv .
(4.27)
Vlasov tried to solve these equations in the end of the 1930s by assuming that the
perturbation of the distribution function also has the form of a plane wave:
f1 (x, v, t) = fˆ1 (v) exp[i(kx − ωt)] ,
(4.28)
which is practically the same as using Fourier transformations in space and time. With
this assumption the linearized Vlasov equation is reduced to:
e ∂f0
−i(ω − kv)fˆ1 − Ê
= 0,
m ∂v
(4.29)
ieÊ ∂f0 /∂v
.
fˆ1 =
m ω − kv
(4.30)
with the solution:
By inserting this to the Coulumb law (Eq. 4.27) we obtain:
Z ∞
2
ie Ê
fˆ1 dv = −
ik0 Ê = −e
m
−∞
Z ∞
∂f0 /∂v
−∞
ω − kv
dv ,
(4.31)
from which we can cancel Ê.
If we know the equilibrium distribution f0 we can calculate the relation between the
wave number and frequency related to the perturbation caused by the electric field,
i.e., we have found the dispersion equation:
e2 1
D(k, ω) ≡ 1 +
m0 k
Z ∞
∂f0 /∂v
−∞
ω − kv
dv = 0 .
(4.32)
4.3. SOLVING THE VLASOV EQUATION
65
Charge density ρ can be considered as the internal property of plasma and thus the
Maxwell equation can be written as: ∇ · D = 0, where D = 0 D(k, ω)E.
We can find the electron plasma wave we encountered in Section 2.3 by considering
the dispersion relation at the long-wavelegth limit (ω kv). Now we can expand the
denominator in the integral:
1
1
kv k 2 v 2 k 3 v 3
= + 2 + 3 + 4 + ...
ω − kv
ω ω
ω
ω
(4.33)
For instance, by using the 1-dimensional Maxwellian distribution (for a complete derivation see Exercise 4.4) :
m
f0 (v) = n
2πkB T
1/2
mv 2
exp −
2kB T
!
(4.34)
and taking into account only the leading terms, the dispersion equation reads:
ωp2
1− 2
ω
where vth =
2
3k 2 vth
1+
2ω 2
!
= 0,
(4.35)
p
2kB T /m is the electron thermal speed .
Assuming infinite wavelength
(k → 0) the solution is the standing plasma oscillation
p
at the frequency ωp = ne2 /(m0 ) we found in Section 2.3. For finite wavelengths and
finite temperatures there is a small correction in the dispersion relation:
3
2
.
ω 2 ≈ ωp2 + k 2 vth
2
(4.36)
The wave is now dispersive (see Appendix 9.3), i.e., propagates with different (finite)
speeds at different frequencies. It is called the Langmuir wave. In Chapter 8 we will
derive the same dispersion relation starting from the warm plasma theory.
If the denominator in Eq. 4.32 cannot be expanded, the integral is not straightforward to evaluate. If frequency ω is real, there is a singularity at v = ω/k along the
path of integration. In most situations frequencies are not real since waves in plasma
are typically either damped by collisions or amplified by some instability mechanism.
Inserting ω = ωr + iωi in Eq. 4.30 we see that
f1 ∝ exp (−iωr t) exp (ωi t).
If ω is complex, the singularity is not along the real axis.
(4.37)
66
CHAPTER 4. KINETIC PLASMA DESCRIPTION
Vlasov did not find the correct way of dealing with the singularity. Lev Landau
realized in 1946 that because the perturbation must begin at some instant, the problem can be treated as an initial value problem and, instead of a Fourier transform, a
Laplace transform in the time domain can be applied. Once the initial transients of the
perturbation have faded away, the asymptotic solution gives the intrinsic properties of
the plasma, i.e., the dispersion equation. The exact Landau solution is complicated and
technically beyond the scope of these lectures (for the full treatment see, e.g., Koskinen,
2011). The final result includes an imaginary part γ ∝ (∂f /∂v)v=ω/k . The value of γ
thus depends on the form of the distribution function and it determines whether the
perturbation
E = E0 exp[−i(ω + iγ)t] ∝ exp(γt)
(4.38)
results in a growing or damped wave solution:
γ > 0:
γ < 0:
energy from wave to particles
energy from wave to particles
→ growing wave (instability)
→ damped wave.
For instance, for the Maxwellian distribution γ is:
γ=−
1/2
π
ωpe
8
1
3
exp − 2 2 −
3
3
2
k λD
2k λD
!
(4.39)
For the Maxwellian distribution γ is negative and the perturbation will damp (Figure
4.4). This phenomenon is known as the Landau damping. The damping is a genuine collective effect characteristic for plasmas and important in describing how energy transfers
from plasma particles to wave modes and vice versa. The particles that have a velocity
close to the phase velocity vph of the wave interact strongly with the wave (“resonance”)
and can exchange energy.
f(v)
slower particles
faster particles
0
vph
Figure 4.4: In a Maxwellian distribution, there are always more low than high energy
particles, and thus, wave loses more energy than gains back.
Landau’s original solution was not immediately accepted. The wave damping without
energy dissipation by collisions has been one of the most astounding results of plasma
4.4. EXAMPLES OF DISTRIBUTION FUNCTIONS
67
physics. This unexpected result was discovered through a purely mathematical analysis
but it was not experimentally verified in laboratories until the 1960s.
4.4
Examples of distribution functions
We have previously considered primarily the one-dimensional Maxwellian distribution:
m
f0 (v) = n
2πkB T
1/2
mv 2
exp −
2kB T
!
.
(4.40)
It is a distribution towards which the gas thermalizes due to collisions. Note that the
power in the normalization factor depends on the degrees of freedom in the velocity
space. Each degree of freedom contributes a factor of 1/2. For a 3-dimensional isotropic
distribution the power is simply 3/2. Figure 4.5 illustrates an isotropic Maxwellian
velocity distribution function in the velocity space.
Figure 4.5: Isotropic Maxwellian velocity distribution function. The right-hand picture
shows contours of constant f in the velocity phase. The horizontal (vertical) axis shows
the velocity component perpendicular (parallel) to the magnetic field.
Many hot and tenuous plasmas are collisionless and cannot be described by a Maxwellian
distribution. However, in many cases the Maxwellian is a reasonable starting point. For
example, the whole distribution may be moving with respect to the observer. If we
denote this velocity with V0 the 3-dimensional distribution function is
m
f (v) = n
2πkB T
3/2
m(v − V0 )2
exp −
2kB T
!
.
(4.41)
This is called the drifting Maxwellian distribution (Figure 4.6).
The magnetic field has a significant effect on how charged particles move in the
plasma. In particular, the magnetic field drives plasma towards anisotropy with different
properties perpendicular and parallel to the magnetic field. For instance, assume that
particles are trapped within a magnetic bottle and in the center of the bottle their
68
CHAPTER 4. KINETIC PLASMA DESCRIPTION
Figure 4.6: Velocity distribution function (left) and countours of constant f for a
Maxwellian distribution that drifts perpendicular to the magnetic field at the velocity
V0 (drifting Maxwellian).
distribution is Maxwellian. If the bottle is contracted, the mirror points move closer
together resulting in the increase of the pitch angles. This will flatten the isotropic
distribution in the direction parallel to the magnetic field and stretch it in the direction
perpendicular to the magnetic field (parallel velocity of the particles will decrease, while
perpendicular velocities increase). The resulting anisotropic distribution is often called
a pancake distribution. In turn, if the bottle will be stretched the mirror points move
further away and the distribution will be stretched parallel to the magnetic field to a
cigar-shaped distribution.
Anisotropic plasma can have a Maxwellian distribution both parallel and perpendicular to the magnetic field but with different temperatures Tk ja T⊥ . Now the distribution
function will be
f (v⊥ , vk ) =
n
1/2
T⊥ Tk
m
2πkB
3/2
2
mvk2
mv⊥
exp −
−
2kB T⊥ 2kB Tk
!
.
(4.42)
As the perpendicular velocity space is 2-dimensional, the normalization factor has the
power 2 × 1/2 = 1 for T⊥ (the width of the distribution is assumed to be the same in
all perpendicular directions, the distribution is said to be gyrotropic), whereas there is
only one degree of freedom in the parallel direction.
Anisotropic plasma may also move accross the magnetic field, for example due to the
E × B-drift or the gradient drift (Figure 4.7). Now the Maxwellian distribution reads:
f (v⊥ , vk ) =
n
1/2
T⊥ Tk
m
2πkB
3/2
mvk2
m(v⊥ − v0⊥ )2
exp −
−
2kB T⊥
2kB Tk
!
.
(4.43)
As discussed in Section 3.5.4 there are always some particles that can escape from the
magnetic bottle. In the absence of a mechanism that would replenish the lost particles
the distribution becomes a loss-cone distribution.
4.4. EXAMPLES OF DISTRIBUTION FUNCTIONS
69
Figure 4.7: Drifting pancake
distribution
Contemplate: Sketch a loss-cone distribution.
Another important special case is a field-aligned beam whose 3-dimensional distribution function is
f (v⊥ , vk ) =
n
1/2
T⊥ Tk
m
2πkB
3/2
2
m(vk − v0k )2
mv⊥
exp −
−
2kB T⊥
2kB Tk
!
.
(4.44)
It is often convenient to present the distribution function as a function of energy
instead of velocity. If all energy is kinetic, the energy is simply obtained from W =
mv 2 /2. In the case the particles are in the external electric potential field U = −qϕ the
total energy of particles is W = mv 2 /2 + U and the Maxwellian distribution function is
m
f (v) = n
2πkB T
3/2
W
exp −
kB T
.
(4.45)
This can be written as the energy distribution:
2(W − U )
g(W ) = 4π
m3
1/2
f (v) .
(4.46)
For derivation of the above form in a case U = 0 see Exercise 4.5. The normalization
factor is determined by requiring that the integration of the energy distribution over all
energies gives the density.
A very important distribution function in space plasmas is the so-called kappa distribution. Distribution functions are often nearly Maxwellian at low energies, but they
decrease more slowly at high energies. At higher energies the distribution is described
better by a power law than by an exponential decay of the Maxwell distribution. Such
a behaviour is not surprising if we remember that the Coulomb collisional frequency
decreases with increasing temperature as ∝ T −2/3 (see Section 2.5). Hence, it takes
longer time for fast particles to reach Maxwellian distribution than for slow particles.
The kappa-distribution has the form (Figure 4.8):
m
fκ (W ) = n
2πκW0
3/2
Γ(κ + 1)
W
1+
Γ(κ − 1/2)
κW0
−(κ+1)
.
(4.47)
70
CHAPTER 4. KINETIC PLASMA DESCRIPTION
Figure 4.8: Maxwell and Kappa distributions as a function of energy.
Here is W0 is the energy at the peak of the distribution and Γ is the gamma function of
mathematics. When κ 1 the kappa distribution is close to the Maxwellian distribution
(Exercise 4.6). When κ is smaller but > 1 the distribution has a high-energy tail.
Velocity and energy distribution functions cannot be measured directly. Instead, the
observed quantity is the particle flux to the detector (an example given in Exercise 4.7).
Particle flux is defined as the number density of particles multiplied by the velocity
component normal to the surface. We define the differential flux of particles traversing
a unit area per unit time, unit solid angle (in spherical coordinates the differential solid
angle is dΩ = sin θdθdφ) and unit energy as J(W, Ω, α, r, t). The units of J are normally
given as (m2 sr s eV)−1 . Note that in literature cm is often used instead of m and,
depending on the actual energy range considered, electron volts are often replaced by
keV, MeV, or GeV. Thus it is important to pay attention to the correct factors of 10 in
data displays!
Let us conclude this discussion by finding out how differential flux and distribution
function are related to each other. We can write the number density in a differential
velocity element (in spherical coordinates d3 v = v 2 dv dΩ) as dn = f (α, r, t) v 2 dv dΩ.
By multiplying this with v we obtain another expression for the differential flux
f (α, r, t) v 3 dv dΩ. Comparing with our earlier definition of the differential flux we obtain:
J(W, Ω, r, t) dW dΩ = f v 3 dv dΩ .
(4.48)
Since dW = mv dv we can write the relationship between the differential flux and
the distribution function as:
v2
J(W, Ω, r, t) = f .
(4.49)
m
4.5. EXERCISES: KINETIC PLASMA DESCRIPTION
4.5
71
Exercises: Kinetic Plasma Description
1. Integrate the Maxwellian distribution function over three-dimensional velocity
space.
2. (a) Using the Maxwellian velocity distribution function
3/2
m
f (v) = n
2πkB T
m
exp −
2πkB T
calculate the average velocity hvi
p
(b) Calculate also the root-mean-square velocity vrms
= hv 2 i. What is the
2
corresponding average kinetic energy hEi = m v /2 ?
Hint: It is useful to remember that
Z ∞
= exp −x2 dx =
√
π.
−∞
3. Consider electrons in an unmagnetized (B = 0) homogeneous (∂/∂r = 0), timeindependent (∂/∂t = 0) plasma in a weak constant electric field. Assume that the
equilibrium distribution of the electrons is Maxwellian and take the collision into
account using the relaxation time approximation known also as the Krook model
∂f
∂t
= −νc (f − f0 ) .
c
ne2
.
mνc
Show that the conductivity of this plasma is given by σ =
4. Insert the one-dimensional Maxwellian into
D(k, ω) ≡ 1 +
e2 1
m0 k
Z ∞
∂f0 /∂v
−∞
ω − kv
dv = 0
and derive the dispersion equation for the Langmuir wave
5. Starting again from the simple Maxwellian velocity distribution show that the
Maxwellian energy distribution (or Boltzmann distribution) becomes
2n
f (W ) = √
π
s
W
W
exp
3
(kB T )
kB T
.
6. Show that for large κ the kappa distribution approaches the Maxwellian distribution.
72
CHAPTER 4. KINETIC PLASMA DESCRIPTION
7. Let us consider energetic particle measurements by satellites. The following differential fluxes (particles/s keV) were obtained: 569000, 3850, 137, 4.52 corresponding
to energies 30 keV, 80 keV, 240 keV and 800 keV.
(a) Plot the measurements as log(flux) vs. log(energy). What can you say about
the spectrum?
(b) Use the least squares method to fit a power-law spectrum of type f (E) =
f0 E −γ to the data. What is the spectral index γ and f0 ?
(c) How many particles did the satellite measure altogether at the energy range
50–200 keV?
Chapter 5
Macroscopic Plasma Equations
In the two previous chapters we have covered the microscopic and kinetic plasma descriptions. The Boltzmann and Vlasov equations derived in Chapter 4 can be considered
as the basic equations of plasma physics. In many cases it is not necessary to know the
exact evolution of the distribution function, rather we are interested in the macroscopic
(and measurable) properties of the plasma (density, flow velocity, temperature, pressure,
etc.) and their evolution in space and time. This can be achieved by taking velocity
moments of the Boltzmann and Vlasov equations. The resulting macroscopic variables,
such as density, velocity and pressure, are functions of the position and time only. This
is the fluid (or macroscopic, see Figure 1.3) plasma description. In the fluid theory, the
time evolution of macroscopic parameters is determined by means of fluid equations that
are analogous to, but generally more complicated than, the equations of hydrodynamics
There are different levels of fluid theories. Multifluid theories consider the plasma
particle species independently. For example, the two-fluid model has separate equations
for electron and ion fluids. The simplest and most important macroscopic model is called
magnetohydrodynamics (MHD). MHD combines one-fluid (hydrodynamic and Lorentz
force) effects and the Maxwell equations.
Historically, the development did not proceed from microphysical to fluid theories.
The development of plasma physics in the 1930s and 1940s started from the physics
of neutral gases and fluids, and magnetic terms were added to the equations of hydrodynamics. This led to the equations of MHD. Only later the equations of MHD were
derived from the microscopic theory.
5.1
Macroscopic transport equations
For the needs of many applications we could start from the Vlasov equation, but retaining
the collision term gives us a more complete macroscopic theory. When not needed, the
73
74
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
collision effects can be dropped at the macroscopic level. We start by taking the velocity
moments of the Boltzmann equation for plasma particle species α:
∂fα
∂fα
∂fα
qα
(E + v × B) ·
+v·
+
=
∂t
∂r
mα
∂v
5.1.1
∂fα
∂t
.
(5.1)
c
Continuity equation (the zeroth moment)
We first integrate Eq. 5.1 over the velocity space. For physical distributions f → 0,
as |v| → ∞, and the force term vanishes in the integration. If there are no ionizing,
recombining, or charge-exchange collisions, the zero-order moment of the collision term
is also zero. The integral of the first term of Eq. 5.1 yields the time derivative of density.
The second term is of the first order in velocity. The integration gives:
Z
v·
∂fα 3
d v =∇·
∂r
Z
vfα d3 v = ∇ · (nα Vα ) ,
(5.2)
and we have found the equation of continuity
∂nα
+ ∇ · (nα Vα ) = 0 .
∂t
(5.3)
Continuity equations for charge or mass densities are obtained by multiplying Eq.
5.3 by qα or mα , respectively:
∂ρmα
+ ∇ · (ρmα Vα ) = 0
∂t
∂ρqα
+ ∇ · Jα = 0 .
∂t
(5.4)
(5.5)
The equation of continuity is an example of the general form of a conservation law
∂F
+ ∇ · G = 0.
∂t
(5.6)
where F is the density of a physical quantity and G the associated flux.
5.1.2
Equation of motion (the first moment)
Multiply Eq. 5.1 by mα v and integrate over v (Exercise 5.1). This yields the momentum
transport equation, which is the macroscopic equation of motion.
5.1. MACROSCOPIC TRANSPORT EQUATIONS
nα mα
∂Vα
+ nα mα Vα · ∇Vα − nα qα (E + Vα × B) + ∇ · Pα
∂t
Z ∂fα
d3 v .
= mα v
∂t c
75
(5.7)
Equation of motion couples plasma velocity to number density. The term Vα · ∇Vα
and the pressure tensor Pα arise from the term with vv and moving the nabla-operator
(∇ = ∂/∂r) outside the integral. In the pressure tensor the diagonal elements represent what we normally understand as pressure, while the off-diagonal elements represent
shearing or tension in the medium. The divergence of Pα contains information of inhomogeneity and viscosity of the plasma. The Lorentz force term does not integrate to
zero. The average electric and magnetic fields in the Boltzmann equation are determined
by both internal and external sources (ρext , Jext ) and fulfill the Maxwell equations for
the average plasma properties.
X nα qα
ρext
∇·E =
+
0
0
α
X
1 ∂E
+ µ0
∇×B = 2
nα qα Vα + µ0 Jext .
(5.8)
c ∂t
α
Because collisions transport momentum between different plasma populations, the
collision integral does not vanish, except for collisions between the same type of particles.
The collision term is a complicated function of velocity. A useful approximation related
to the Krook model is
Z
mα
∂fα
v
∂t
d3 v = −
c
X
mα nα (Vα − Vβ ) hναβ i ,
(5.9)
β
where hναβ i is the average collision frequency of particle species α with particles of
species β.
The momentum equation relates the fluid velocity to the density gradient and electromagnetic forces acting on the fluid element, but not on the single particles anymore.
Note that the momentum equation has a close relationship to the Navier-Stokes equation
of hydrodynamics. In neutral hydrodynamics the only forces that are acting to fluid are
the pressure and viscous forces.
5.1.3
Energy equation (the second moment)
Next, let us calculate the moments over vv. The second velocity moment yields the
energy or heat transport equation (conservation law of energy). Integration is now quite
a tedious process. We write here the equation in the form:
76
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
3
nα k B
2
∂Tα
+ Vα · ∇Tα + pα ∇ · Vα
∂t
!
2
∂
n
m
V
α
α
α
= −∇ · Hα − (Pα0 · ∇) · Vα +
,
∂t
2
c
(5.10)
where the isotropic part of the pressure pα I is written on the left-hand side and the
non-isotropic part Pα0 on the right-hand side. The relation between the scalar pressure
pα and temperature Tα is assumed to be that of an ideal gas pα = nα kB Tα , The thirdorder term is Hα and it describes the heat flux. The energy equation states that the
temperature of plasma can increase due to compressive flow (∇ · Vα < 0), dissipation
due to flow gradient induced stress ((Pα0 · ∇) · Vα ), collisional energy exhange, and due
to divergence of the heat flux (sources or sinks).
Now we have macroscopic equations separately for each plasma species. In a real
plasma several species co-exist. The simplest description of the real plasma consists
of electrons and protons (two-fluid model). The separate fluid components interact
through collisions and electromagnetic interaction. Continuity equation and momentum
transport equations are valid separately for both fluids. In addition to electrons and
protons, there may be a variety of heavier ions, as well as neutral particles, which may
contribute to plasma dynamics through collisions, including charge-exchange processes.
Sometimes it is also necessary to consider different species of the same type of particles;
e.g., in the same spatial volume there may be two electron populations of widely different
temperatures or average velocities. Such situations often give rise to plasma instabilities
to be discussed in Chapter 8.
5.2
Equations of state
The equation for the heat flux is found by taking the third velocity moment of the Boltzmann equation. This would lead to an equation with the fourth-order contribution, and
so on. This is because the Boltzman equation includes both the zeroth and first order
velocity. The chain of equations with increasing order of velocity (and with increasing
complexity) must be truncated at some point to form a closed system of transport equations. In many practical problems this is made in the second order, either by neglecting
the heat flux, or by substituting the energy equation by an equation of state. Here physical insight is essential. Krall and Trivelpiece [1973] state this: “The fluid theory, though
of great practical use, relies heavily on the cunning of its user”.
The simplest of closed system is the cold plasma model. It contains the conservation
equations for mass and momentum and the related macroscopic variables are the density
and bulk velocity. As the temperature is taken to be zero, the pressure tensor is zero.
5.3. MAGNETOHYDRODYNAMIC EQUATIONS
77
The particle distribution function becomes the delta function that is centred at the bulk
velocity: f (r, v, t) = a δ(v − V(r, t)), where a is the dimensional normalization factor.
In the case of warm plasma the thermal effects can be taken into account by considering, e.g., isothermal or adiabatic approach. We assume non-viscous plasma, i.e.,
the non-diagonal elements of the pressure tensor are zero. Let us further assume that
plasma is isotropic, i.e., the diagonal elements of Pα are equal, and thus, the pressure
tensor can be replaced by a scalar pressure. Thus, in the momentum transport equation
the term ∇ · P degenerates to ∇p. The macroscopic variables appearing in this case are
the number density n, the bulk velocity V, and the scalar pressure p.
In isothermal plasma T = T0 = constant and the equation of state is:
p = nkB T0 .
(5.11)
In isothermal plasma changes of plasma parameters are so slow that the system has
time to thermalize during the time-scale of the change. In collisionless plasmas this is
typically not a very good assumption.
The opposite limit is that the changes occur so fast that there is no heat exchange
between the considered plasma element and its surroundings. Hence, we can set the the
heat flux to zero (∇ · H = 0). The resulting adiabatic equation of state can be derived
rather easily from the heat flux equation by assuming scalar pressure, using the density
continuation equation and writing d/dt = ∂/∂t + V · ∇. The result is
3 dT
dn
n
=T
,
2 dt
dt
(5.12)
which gives the relations:
T = T0
n
n0
γ−1
; p = p0
n
n0
γ
,
(5.13)
where γ = cp /cv is the polytropic index , which in the adiabatic case is known as the
adiabatic constant. In statistical mechanics it is shown that for monoatomic adiabatic
gas γ = (f + 2)/f , where f is the number of degrees of freedom. Thus, for 3-dimensional
ideal gas γ = 5/3. In non-adiabatic cases γ 6= 5/3, e.g., for isothermal process γ = 1, for
the isobaric process γ = 0, and for the isometric (constant density) γ → ∞. Plasmas
are not always isotropic. For instance, if a strong magnetic field is present or if there
are not enough collisions to maintain the isotropic velocity distribution the pressure is
anisotropic. In addition, pressure tensor does not even need to be diagonal.
5.3
Magnetohydrodynamic equations
We have now derived macroscopic fluid equations for each plasma species. Next we will
combine these equations to a one-fluid theory called magnetohydrodynamics (MHD).
78
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
MHD is probably the most widely known plasma theory. In MHD the plasma is considered as a single fluid in the centre-of-mass (CM) frame. This is a well-motivated
approach in collision-dominated plasmas, where the collisions constrain the plasma particles to follow each other closely and thermalize the distribution toward a Maxwellian,
which makes the interpretation of velocity moments straightforward. MHD works also
remarkably well in collisionless tenuous space plasmas, e.g., when studying the largescale interaction of the solar wind, the magnetosphere and the ionosphere of the Earth
(Figure 5.1). However, great care should be exercised both with interpretation and approximations. To some extent the electromagnetic forces take the role of collisions, e.g.,
constraining the motion across the magnetic field. This picture is, however, not complete
because the motion along the magnetic field is unconstrained in a homogeneous plasma.
Figure 5.1: The Earth’s magnetosphere simulated by the global magnetosphereionosphere simulation GUMICS-4. The colours indicate the plasma density. The small
inset shows the ionospheric conductivity from the electrostatic ionospheric module coupled to the MHD-based GUMICS-4. Courtesy: GUMICS team at FMI.
5.3.1
MHD transport equations
The single-fluid variables are defined as:
mass density
ρm (r, t) =
X
α
nα mα ,
5.3. MAGNETOHYDRODYNAMIC EQUATIONS
79
charge density
ρq (r, t) =
X
nα qα = e(ni − ne ) ,
α
macrospcopic velocity
P
α nα mα Vα
,
V(r, t) = P
α nα mα
electric current density
J(r, t) =
X
nα qα Vα
α
and the pressure tensor in the CM frame
PαCM (r, t) = mα
Z
(v − V)(v − V)fα d3 v ,
from which we get the total pressure
P(r, t) =
X
PαCM (r, t) .
α
Summing the individual continuity and momentum transport equations over particle
species yields the continuity equations
∂ρm
+ ∇ · (ρm V) = 0
∂t
∂ρq
+∇·J = 0
∂t
(5.14)
(5.15)
and the momentum transport equation
ρm
∂V
+ V · ∇V = ρq E + J × B − ∇ · P .
∂t
(5.16)
The momentum equation corresponds to the Navier-Stokes equation of hydrodynamics where the viscosity terms are written explicitly (here they are hidden in the pressure
gradient). At macroscopic level the deviations from charge neutrality are small and ρq E
is usually negligible. The magnetic part of the Lorentz force J×B (often called Ampère’s
force) is, however, essential in the theory of magnetic fluids.
The next equation in the velocity moment chain is the energy transport equation.
After some tedious but straightforward calculation the energy equation can be written
in the conservation form
"
∂
ρm
∂t
!
#
V2
B2
+w +
= −∇ · H .
2
2µ0
(5.17)
Here w is the enthalpy that is related to the the internal free energy (per unit mass) of
the plasma u by w = u + p/ρm . The RHS is the divergence of the heat flux vector H,
80
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
which is a third-order moment. After some reasonable approximations it can be written
as
H =
−
V2
p + B 2 /µ0
+u+
2
ρm
!
B
J
ρm V −
V+
µ0
ne
JB 2
J×B
me B
∂J
+
+
×
.
2
σµ0
µ0 ne µ0 ne
∂t
·B
(5.18)
For derivation of the energy transport equation in adiabatic ideal MHD, see Exercise
5.2. When integrated over a finite volume V the LHS of (5.17) describes the temporal
change of the energy of the MHD plasma in that volume and the RHS the the energy
flux through the boundary ∂V and energy losses due to resistivity. Thus we have found
the MHD equivalent of Poynting’s theorem of elementary electrodynamics.
In the case of MHD the third moment is usually neglected and an equation of state
is used to relate the changes in plasma pressure and density. MHD assumes Maxwellian
distribution and thus the pressure is isotropic. The adiabatic equation of state is written
in the form:
d
−γ
(pρ−γ
(5.19)
m ) = 0 → pρm = constant.
dt
5.3.2
Ohm’s law in MHD
It is also necessary to determine how the current density J depends on the electric field E.
Ohm’s law in a fluid description is a complicated issue. In the particle picture the plasma
current is the sum of all charged particle motions. In a single-fluid theory the current
transport equation is derived by multiplying the momentum transport equations of each
particle population by qα /mα and summing over all populations. This leads to a rather
messy expression including terms of different magnitudes and further approximations
are needed. Here we give the generalized Ohm’s law in the form that contains the most
important terms for space plasmas:
E+V×B=
J
1
1
me ∂J
+
J×B−
∇ · Pe + 2
.
σ ne
ne
ne ∂t
(5.20)
The terms that are proportional to me /mi and that contain the derivatives of the secondorder terms VJ, JV and VV have been neglected. The collision integral has been
approximated by a constant collision frequency ν, which using the conductivity σ =
ne2 /νme (Eq. 2.47) results in the first term on the RHS of Eq. 5.20. Exercise 5.3
compares different terms in the generalized Ohm’s law during a magnetospheric substorm
in the nightside magnetotail.
Assume further so slow temporal changes and large spatial gradient scales that |J×B|,
|∂J/∂t|, and |∇ · P| are all smaller than |V × B|.
5.3. MAGNETOHYDRODYNAMIC EQUATIONS
81
This leaves us with the standard form of Ohm’s law in MHD.
J = σ(E + V × B) ,
(5.21)
which is already familiar from elementary electrodynamics in cases when moving frames
are taken into account. Here the moving frame is attached to the fluid flow with the
velocity V.
5.3.3
Ideal MHD
If the conductivity is very large, we find Ohm’s law of the ideal MHD
E+V×B=0.
(5.22)
Let us investigate under what constraints the ideal MHD is valid. Hence, we need to
compare the magnitude of the term V × B to the inertial, Hall (i.e., J × B) and resistive
terms. Denote the characteristic length and time scales of the system with L and τ ,
respectively. First, the electron inertia term (∝ ∂J/∂t) can be neglected if:
V×B∼
L
me ∂J
B 2
τ
ne ∂t
me 1 B
∼ 2
ne τ µL
=
=
→L me ∂ 1
( ∇ × B)
ne2 ∂t µ0
c B
ωp2 τ L
c
,
ωp
(5.23)
where ωp is the plasma frequency.
In a similar manner (try yourself!) we obtain the conditions to neglect the Hall term
(∝ J × B):
c
1
, and τ .
(5.24)
L
ωp
ωc
Finally the resistive term can be neglected if
1 B
µ0 L
1
⇔u
.
µ0 σL
V × B ∼ uB ηJ ∼ η
(5.25)
82
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
The ratio of the V × B and ηJ terms is called the magnetic Reynold’s number :
Rm = µ0 σuL .
(5.26)
The resistive term can thus be neglected when the magnetic Reynold’s number is
large. This is indeed the case in many space plasmas that have large dimensions (L) and
small resistivity. Note that the resistive term causes dissipation and converts magnetic
energy to heat.
While the ideal MHD is a reasonable starting point, it is not at all clear that the
next term to take into account should be J/σ. In many space applications the Hall term
J × B/ne and the pressure term ∇ · P/ne are more important.
There are effects that originate at the microscopic level, which are not due to actual inter-particle collisions, but which may lead to “effective” resistivity or viscosity at
the macroscopic level. Various wave–particle interactions and microscopic instabilities
tend to inhibit the current flow. Often the macroscopic effect of these processes looks
analogous to finite ν and is called anomalous resistivity.
5.4
Exercises: Macroscopic Plasma Equations
1. Derive the momentum equation
nα m α
∂Vα
+ nα mα Vα · ∇V − nα qα (E + Vα × B) + ∇ · Pα
∂t
Z
= mα
v
∂fα
∂t
d3 v .
c
2. Derive the energy equation of adiabatic ideal MHD in the conservation form
∂ 1
p
B2
1
γ
E×B
( ρm V 2 +
+
) + ∇ · ( ρm V 2 V +
pV +
) = 0.
∂t 2
γ − 1 2µ0
2
γ−1
µ0
Hints: Write the equation of state as d(pρ−γ
m )/dt and the conservation of mass as
dρm /dt = −ρm ∇ · V to show that (γ − 1)V · ∇p = ∂p/∂t + ∇ · (γpV). Then use
ideal Ohm’s law with Ampere’s and Faraday’s laws to show that
V · (J × B) = J · (V × B) =
∂ B2
E×B
+∇·
.
∂t 2µ0
µ0
5.4. EXERCISES: MACROSCOPIC PLASMA EQUATIONS
83
3. Investigate generalized Ohm’s law:
E+V×B=
J
1
1
me ∂J
+
J×B+
∇ · Pe + 2
σ ne
ne
ne ∂t
during a substorm in the nightside magnetotail when following values have been
measured:
E ≈ 0.1 mV m−1 , V ≈ 100 km s−1 , B ≈ 1 nT
J ≈ 1 nA m−1 , n ≈ 1 cm−3 , Pe ≈ 0.1 nPa .
In these circumstances the characteristics scale length is L ≈ 104 km, characteristic
time scale ≈ 10 s and effective resistivity less than 1 s−1 . Compare the magnitudes
of various terms in Ohm’s law.
84
CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS
Chapter 6
Magnetohydrodynamics
Magnetohydrodynamics (MHD) describes electrically conducting fluids in the presence
of a magnetic field. This chapter deals with single-fluid MHD where different plasma
components (i.e., ions, electrons) move together and compose a single “fluid”. The
plasma is described by a single temperature, density and velocity. Single-fluid MHD is a
strongly reduced theory but applies remarkably well in many situations. For instance, the
plasma in the solar wind, Earth’s magnetosphere, solar corona, and in many interstellar
regions can be treated with MHD. MHD has also vast applications in fusion plasma
research, in particular concerning the plasma confinement and stability. As described in
the previous chapter, MHD governs processes that are slow compared with the gyration
time and on scales that are larger than the gyro (Larmor) radius.
We will discuss the general concepts of the MHD. We start by summarizing the
MHD equations derived in Chapter 5 and by investigating the magnetic field evolution.
We discuss magnetic field diffusion and the convection of plasma and magnetic field.
Then we proceed to investigate the wave modes found in MHD. We also shortly discuss
the importance and basic models of magnetic reconnection and finalize this chapter by
investigating the MHD equilibrium.
6.1
MHD equations
The basic equations of MHD we derived in Chapter 5 are:
ρm
∂ρm
+ ∇ · (ρm V) = 0
∂t
∂
+ V · ∇ V + ∇P − J × B = 0
∂t
E + V × B = J/σ .
85
(6.1)
(6.2)
(6.3)
86
CHAPTER 6. MAGNETOHYDRODYNAMICS
n
n0
∂B
∇×E = −
∂t
∇ × B = µ0 J .
P
= P0
γ
(6.4)
(6.5)
(6.6)
The first four equations are the mass continuity equation, momentum equation, resistive Ohm’s law, and the adiabatic equation of state. We have used the basic MHD
assumption that the temporal variations are so slow that the displacement current
(0 ∂E/∂t) can be neglected in the Ampère–Maxwell law. The relationship between
the electric current and magnetic field is thus obtained from Ampère’s law ∇ × B = µ0 J.
As discussed earlier, the MHD equations are basically the combination of NavierStokes equations of fluid dynamics with Maxwell’s equations and Ampère’s force. It is
important to note that in MHD the magnetic and velocity fields are taken as the primary
fields. The Maxwell equations and the MHD Ohm’s law give the current density and
the electric field in terms of the magnetic field.
6.2
Magnetic field evolution
MHD describes the magnetic field (B) and plasma motion (bulk speed V). Let us
investigate the relationship between V and B starting from the MHD Ohm’s law:
E + V × B = J/σ .
(6.7)
Taking the curl of this and applying Faraday’s law we obtain:
∂B
= ∇ × (V × B − J/σ) .
∂t
(6.8)
Remembering that ∇ · B = 0 and assuming that the conductivity is constant (for
a case where the conductivity is not spatially homogeneous see Exercise 6.1) we obtain
the induction equation (one of the most important equations of plasma physics!):
∂B
1
= ∇ × (V × B) +
∇2 B .
∂t
µ0 σ
(6.9)
From the induction equation we see that the magnetic field can change in time as a
consequence of two effects. We will investigate them separately.
6.2. MAGNETIC FIELD EVOLUTION
6.2.1
87
Diffusion
Assuming that the plasma is at rest (V = 0) the induction equation reduces to the
diffusion equation:
∂B
= Dm ∇2 B ,
∂t
(6.10)
where Dm = 1/(µ0 σ) = η/µ0 is the diffusion coefficient. Thus, the magnetic field in
plasma can evolve even in the absence of any plasma flow if the resistivity η is finite.
The magnetic field diffuses smoothing out spatial inhomogeneities described by the term
∇2 B.
The solution of the diffusion equation is of the form:
B = B0 exp(±t/τd ) ,
(6.11)
where the magnetic diffusion time τd is
τd = µ0 σL2B
(6.12)
and LB is the characteristic gradient scale length of the magnetic field.
Figure 6.1: Evolution of a one-dimensional current sheet due to magnetic diffusion.
(Assuming that no new flux is brought to the system).
As an example letus investigate the initial configuration presented in the left-hand
part of Figure 6.1. A thin current sheet separates the regions of oppositely orientated
magnetic fields. The magnetic field is chosen to be along the ±x-axis and the current
sheet normal is along the z-axis. In this case the diffusion equation becomes:
∂Bz
1 ∂ 2 Bz
=
.
∂t
µ0 σ ∂x2
(6.13)
88
CHAPTER 6. MAGNETOHYDRODYNAMICS
If at time t0 = 0 the current layer is infinitely thin (i.e., can be described by a
δ-function), the magnetic field diffuses as:
(
Bz (x) = B0 erf
where
2
erf(u) = √
π
µ0 σ
4t
Z u
1/2 )
x
2
e−v dv .
,
(6.14)
(6.15)
0
The magnetic field diffuses towards the current sheet and, as a consequence, the
current sheet broadens as the diffusion proceeds (right part of Figure 6.1). Oppositely
directed magnetic fields cancel each other decreasing the magnetic field gradient, and
thus, slowing down the diffusion (Eq. 6.10). Physically, the magnetic energy is transformed to heat. This is called Joule (or Ohmic) heating. Increasing plasma pressure
compensates the decreasing plasma pressure.
6.2.2
Convection
If σ → ∞ (ideal MHD), the diffusion term becomes small and Eq. 6.9 reduces to a
convection equation:
∂B
= ∇ × (V × B) .
∂t
(6.16)
The convection equation describes how plasma flow and magnetic field are tied to
each other. In this case there is no diffusion of the magnetic field, but the plasma
and magnetic field “convect” (actually advect) together. It is commonly said that the
magnetic field is frozen-in to the motion of the plasma (see Figure 6.2).
Figure 6.2: In a case of ideal MHD (σ → ∞)
magnetic field and plasma move together, i.e.
are frozen-in.
To determine whether convection or diffusion dominates, it is useful to introduce a
dimensionless parameter that is the ratio of the magnitude of convection and diffusion
6.3. FROZEN-IN CONDITION
89
terms. Let τ be the time scale of characteristic magnetic field temporal variations, V
the average plasma velocity perpendicular to the field, LB the characteristic length over
which the magnetic field varies, and τd the diffusion time scale. Substituting ∂/∂t → τ
and ∇ → L−1
B , and neglecting directions, the induction equation reduces to
B
B
VB
+
=
.
τ
LB
τd
(6.17)
The ratio of the terms on the RHS (i.e., the ratio of the convection to diffusion) becomes
convection
V
τd = µ0 σLB V
=
diffusion
LB
(6.18)
This ratio corresponds to the dimensionless magnetic Reynold’s number (Rm ) we encountered in Section 5.3.3 while investigating the conditions under which the resistive
term in the MHD Ohm’s law can be neglected. If Rm is large diffusion is slow and the
convection dominates. This corresponds to small resistivity and the ideal MHD limit.
Exercise 6.2 investigates diffusion times and Rm for a typical laboratory plasma and
in the solar wind. Due to large spatial scales and high conductivities Rm is typically
very large in space and astrophysical plasmas. In the solar wind at the Earth orbit Rm
is of the order of 1016 − 1017 . This means that during the 150 million kilometres journey
from the Sun to the Earth the magnetic field in the solar wind diffuses only about
one kilometre! Hence, diffusion is negligible in the solar wind. In turn, in laboratory
plasma spatial scales and conductivities are much smaller, and consequently, diffusion
dominates.
6.3
Frozen-in condition
When the magnetic Reynolds number is very large, the magnetic field is “frozen-in” to
the plasma motion. As a consequence, two plasma elements that are initially magnetically unconnected cannot mix as long as the frozen-in condition applies. The frozen-in
concept was first brought to plasma physics by Hannes Alfvén. Although very useful
the frozen-in concept is often misunderstood and Alfvén later denounced it as “pseudopedagogical”. The problem lies in picturing moving magnetic field lines. A magnetic
field line is just a mathematical abstraction and has no physical identity.
The correct way to express the frozen-in concept is to state that if two plasma
elements are connected by a magnetic field line at time t, they are connected by
a field line at all times. What is conserved is the magnetic connection between the
plasma elements.
Let us test the frozen-in assumption by investigating two plasma elements under the
assumption of the ideal MHD. We assume that two plasma elements are magnetically
90
CHAPTER 6. MAGNETOHYDRODYNAMICS
connected at time t (Figure 6.3). This means that if we trace the magnetic field from one
plasma element, we end up at the other. Let the distance between the elements at time
t be 4l. After time dt the plasma elements have moved distances u dt and (u + 4u) dt,
where u(r, t) is the plasma flow velocity. At time t+dt the distance between the elements
is thus 4l + d(4l). In order the frozen-in concept to be valid we have to show that at
the time t + dt the plasma elements are still magnetically connected, i.e., we need to
show that d(4l × B)/dt = 0.
Figure 6.3: If ideal MHD assumption holds, two plasma elements that are magnetically
connected (at the same field line) at time t stay magnetically connected at all times.
Let us begin by writing d(4l) in terms of the plasma flow velocity u. The first term
in the Taylor series of u is
4u = (4l · ∇)u .
(6.19)
From Fig. 6.3 we see that
4l + d(4l) = 4l + (u + 4u) dt − u dt ,
(6.20)
which leads to
d(4l)
= 4u = (4l · ∇)u .
(6.21)
dt
Since we assume ideal MHD, let us investigate how the magnetic field changes in time
starting from the convection equation:
∂B
∂t
= ∇ × (u × B)
= (B · ∇)u − (u · ∇)B − B(∇ · u) .
(6.22)
Here ∇ · B = 0 was used. In the frame of reference moving with the plasma
dB
∂B
=
+ (u · ∇)B = (B · ∇)u − B(∇ · u) .
dt
∂t
(6.23)
Now we can calculate d(4l × B)/dt:
d
d(4l)
dB
(4l × B) =
× B + 4l ×
dt
dt
dt
= [(4l · ∇)u] × B + 4l × [(B · ∇)u − B(∇ · u)] .
(6.24)
6.3. FROZEN-IN CONDITION
91
Because we assumed at the beginning 4l to be parallel to B, 4l × B = 0, and the third
term on the RHS is zero. For the same reason 4l and B can be interchanged in the first
term on the RHS. Thus the first and the second term are the same except for their sign
and we have
d
(4l × B) = 0 .
(6.25)
dt
Thus, we have obtained that 4l remains parallel to B and plasma elements that originally are on a common field line remain on a common field line. This assumption is
valid as long as the ideal MHD approximation is valid.
B
S (t0)
S (t0+dt)
Figure 6.4: Magnetic flux through a surface moving with the plasma is conserved at the
ideal MHD limit.
An alternative way to express the frozen-in concept is to investigate the time variations of the magnetic flux through a surface S (see Figure 6.3):
Z
ΦS =
B · dS .
(6.26)
S
The surface elements move with the plasma fluid velocity. By calculating the change of
the magnetic flux from time t0 to t0 + dt it can be shown (Exercise 6.3) that
dΦS
d
=
dt
dt
Z
B · dS = 0 .
(6.27)
Thus, assuming that the ideal MHD Ohm’s law applies, the magnetic flux through any
closed contour in the plasma, each element of which moves with the local plasma velocity,
is a conserved quantity.
The critical assumption when deriving the frozen-in theorem was the ideal Ohm’s law.
This requires that the E×B drift is faster than magnetic drifts, i.e., large-scale convection
dominates. In reality, the plasma has always some resistivity. However, the frozen-in
condition applies if the characteristic time-scale of the process we are looking at is much
shorter than the diffusion time. This can also be seen from the magnetic Reynolds
number Rm = (V /LB )τB (see Eq. 6.18), where τB is the magnetic field diffusion time.
92
CHAPTER 6. MAGNETOHYDRODYNAMICS
In space plasmas the first correction to the ideal MHD is often not the resistive term
but the Hall term J × B/(ne)
E+V×B=
1
J×B.
ne
(6.28)
This is expected to be the case, e.g., near current sheets separating magnetic fields of
different strength and direction. In this Hall MHD the magnetic field becomes frozen-in
to the electron flow
E = −Ve × B .
(6.29)
As discussed earlier in Chapter 3 this is because due to their much smaller mass electrons
have much smaller gyro radii and are tied more strongly to the magnetic field than the
ions.
6.4
MHD waves
MHD is a fluid theory and there are similar wave modes as in ordinary fluids (hydrodynamics). In addition, the presence of the magnetic field gives rise to new modes. In the
MHD description we assume that the frequency of the wave is smaller than the characteristics frequencies in the plasma (gyro frequency and plasma frequency) and that the
wavelengths are longer than microscopic plasma scales (Larmor radius). In addition, as
discussed earlier, one of the basic assumptions of MHD is that the temporal changes are
so slow that the displacement current can be neglected. As a consequence, MHD does
not describe basic electromagnetic waves. However, this does not mean that electromagnetic waves could not propagate through the MHD plasma, rather MHD phenomena and
high-frequency electromagnetic waves do not have a direct linkage.
6.4.1
MHD dispersion equation
Analyzing plasma waves properties requires the derivation of the dispersion equation.
The dispersion equation gives the relation between the wave number and the frequency
of the wave, and thus determines how the wave travels in the medium. To derive the
dispersion equation for MHD waves we start from the set of MHD equations (6.1–6.6)
given at the beginning of this chapter.
We consider here compressible, non-viscous, and perfectly conducting plasma that
is in a homogeneous background (applied) magnetic field. Thus, we can replace the
resistive MHD Ohm’s law (Eq. 6.3) by the ideal Ohm’s law E + V × B = 0.
Let us modify the adiabatic equation of state (Eq. 6.4) by taking its gradient and
introducing the speed of sound
vs =
q
γp/ρm =
q
γkB T /m ,
(6.30)
6.4. MHD WAVES
93
where γ is the adiabatic constant. We obtain
∇p = vs2 ∇ρm .
(6.31)
Use Ampère’s law to eliminate J and Eq. 6.31 to eliminate p from Eq. 6.2. Further,
the ideal Ohm’s law can be used to eliminate E from Eq. 6.5. We obtain the set of
equations
∂ρm
+ ∇ · (ρm V) = 0
∂t
∂V
ρm
+ ρm (V · ∇)V = −vs2 ∇ρm + (∇ × B) × B/µ0
∂t
∂B
∇ × (V × B) =
.
∂t
(6.32)
(6.33)
(6.34)
We assume the initial state to be in equilibrium where the density is constant ρm0 ,
the velocity is zero (V = 0), and the background magnetic field is constant B0 . As
discussed at the beginning of this chapter we consider here only small perturbations
(denoted by a subscript ”1”) to the initial equilibrium (subscript ”0”):
B(r, t) = B0 + B1 (r, t)
(6.35)
ρm (r, t) = ρm0 + ρm1 (r, t)
(6.36)
V(r, t) = V1 (r, t) .
(6.37)
We linearize the equations by inserting these to the equations 6.32–6.34 and keeping
only the first order terms (the zeroth order terms automatically fulfil the equations and
the second order terms are assumed so small that we can neglect them). This leads to
the linearized equations:
∂ρm1
+ ρm0 (∇ · V1 ) = 0
∂t
ρm0
∂V1
+ vs2 ∇ρm1 + B0 × (∇ × B1 )/µ0 = 0
∂t
∂B1
− ∇ × (V1 × B0 ) = 0 .
∂t
(6.38)
(6.39)
(6.40)
Next, let us find equation for the velocity perturbation V1 . By taking the time
derivate of the linearized momentum equation (Eq. 6.39) we obtain
∂ 2 V1
∂ρm1
ρm0
+ vs2 ∇
2
∂t
∂t
B0
∂B1
+
× ∇×
µ0
∂t
= 0.
(6.41)
By using the linearized continuity equation (Eq. 6.38) ja the linearized Ampère’s law
(Eq. 6.40) this can be written as
∂ 2 V1
− vs2 ∇(∇ · V1 ) + vA × {∇ × [∇ × (V1 × vA )]} = 0 ,
∂t2
where the vector vA is
(6.42)
94
CHAPTER 6. MAGNETOHYDRODYNAMICS
vA = √
B0
.
µ0 ρm0
(6.43)
The magnitude of this velocity defines the Alfvén speed .
Finally, let us try look for the solution assuming that it has the form of a plane wave
V1 (r, t) = V1 exp[i(k · r − ωt)] .
(6.44)
The temporal and spatial variations are now harmonic and we can replace the derivatives
by algebraic operators (see Appendix 9.3)
∇· → ik ·
(6.45)
∇× → ik ×
∂/∂t → iω .
Eq. 6.42 simplifies to an algebraic equation
−ω 2 V1 + vs2 (k · V1 )k − vA × {k × [k × (V1 × vA )]} = 0 .
(6.46)
Using a vector identity:
A × (B × C) = (A · C)B − (A · B)C
(6.47)
we obtain a useful form of the dispersion equation
2
−ω 2 V1 + (vs2 + vA
)(k · V1 )k
+(k · vA )[(k · vA )V1 − (vA · V1 )k − (k · V1 )vA ] = 0 .
(6.48)
From this we can find all MHD wave modes.
6.4.2
MHD wave modes
Select the z-axis to be parallel to background magnetic field B0 and the x-axis so that
the wave vector k is in the xz-plane. Denote the angle between k and B0 by θ. Figure
6.5 summarizes the coordinate system.
Now we obtain
k = k(êx sin θ + êz cos θ)
(6.49)
vA = vA êz
(6.50)
V1 = V1x êx + V1y êy + V1z êz
(6.51)
k · vA = kvA cos θ
(6.52)
k · V1 = k(V1x sin θ + V1z cos θ)
(6.53)
vA · V1 = vA V1z .
(6.54)
6.4. MHD WAVES
95
z
n,k
B0
q
x
y
Figure 6.5: Coordinate system to
study MHD waves
Inserting these to the dispersion equation (Eq 6.48) we obtain



2 + v 2 sin2 θ
−vp2 + vA
0
vs2 sin θ cos θ
V1x
s



2
2
2
0
−v
+
v
cos
θ
0
  V1y  = 0 ,

p
A
V1z
vs2 sin θ cos θ
0
−vp2 + vs2 cos2 θ
(6.55)
where vp = ω/k is the phase speed (see Appendix 9.3) of the wave.
There are three linearly independent non-trivial solutions that are found by equating
the determinant to zero.
Alfvén wave
The y-component of the matrix Eq. 6.55 gives a linearly polarized wave mode with the
phase speed
vp = vA cos θ ,
(6.56)
where vA is given by Eq. 6.43. This mode is called the Alfvén wave.
It is seen from Eq. 6.56 that when the Alfvén wave propagates along the background
magnetic field (i.e. θ = 0◦ ) its phase speed is exactly vA . For oblique propagation the
phase speed is less than vA . It is also clear from Eq. 6.56 that Alfvén waves do not
propagate perpendicular to the magnetic field.
It is easy to verify that the eigenvector (0, V1y , 0) corresponds to the root vp2 =
Thus for Alfvén wave the velocity perturbation V1 is in the y-direction, i.e.,
perpendicular both to the wave vector k and the background magnetic field B0 . Hence,
V1 · k = 0 and we see from the linearized continuity equation
2 cos2 θ.
vA
−iωρ1 + ρ0 k · V1 = 0
(6.57)
96
CHAPTER 6. MAGNETOHYDRODYNAMICS
that Alfvén wave is non-compressive, i.e., there are no density fluctuations. The plasma
fluid motions are thus completely transverse signifying that the plasma elements oscillate
perpendicular to the direction of propagation of the wave.
The wave magnetic field B1 can be calculated from the convection equation assuming
harmonic temporal and spatial dependences
ωB1 + k × (V1 × B0 ) = 0
→ B1 = −
V1
B0 ,
ω/k
(6.58)
(6.59)
i.e., the wave magnetic field is perpendicular to the background magnetic field B0 . Figure
6.6 demonstrates the propagation of an Alfveń wave.
k
B0
B1
V1
E1
B0
Figure 6.6: Alfvén wave propagating parallel to the magnetic field.
The Alfvén wave is often called also as shear Alfvén wave or non-compressional
Alfvén wave.
The existence of the Alfvén wave can be deduced also intuitively. The magnetic field
line can be considered to behave like a tensed string. Transversal displacement of the
elastic string generates a transverse wave that propagates along the string, in analogy
with the Alfvén wave propagating parallel to magnetic field lines.
6.4. MHD WAVES
97
Alfvén waves have been observed in the laboratory and in many space plasma regions,
for example in the solar wind, solar photosphere and in the Earth’s magnetosphere. It
has been suggested that Alfvén waves could explain the heating of the outermost layer of
the solar atmosphere, the corona (Figure 6.7). Understanding the properties of Alfvén
waves is also important for determining the stability, turbulence and heating in controlled
fusion devices. The Alfvén speed can differ greatly in space plasma, depending on the
density and magnetic field magnitude in question. Typical Alfvén speeds are calculated
in Exercise 6.4 in the Earth’s ionosphere, solar corona and interstellar gas cloud. It is
also instructive to contemplate whether the neutral mass density needs to be taken into
account when determing the Alfvén speed (i.e., if plasma is not fully ionized, do neutrals
have enough time to respond to the motion of the ions).
Figure 6.7: Japanese Hinode (“sunrise”) observations of fluctuating plasma that could
be an indication of Alfvén waves heating the corona. The presence of Alfvén waves is
deduced by tracking the motions of coronal plasma.
Contemplate: The existence of Alfvén waves was first suggested by Hannes Alfvén
(as the name hints!). In 1942 he noted that a new type of wave should be found in
magnetized plasmas that may be of importance to solar physics. It is instructive to look
Alfveń’s original paper (click here) which also demonstrates his profound physical insight.
Fast and slow MHD (Alfvén) waves.
The other solutions of the matrix Eq. 6.55 are obtain by setting the determinant of the
coefficients of V1x :n and V1z zero. The result is (Exercise 6.5)
1
1
2
2 2
2
vp2 = (vs2 + vA
) ± [(vs2 + vA
) − 4vs2 vA
cos2 θ]1/2 .
(6.60)
2
2
The solution with the larger phase speed is called the fast MHD wave and with the
lower phase speed the slow MHD wave.
98
CHAPTER 6. MAGNETOHYDRODYNAMICS
The top panels of Figure 6.8 show the phase speeds of Alfvén wave and the slow and
fast MHD waves as a function of the angle between the wave vector and the background
magnetic field (θ). The solution depends on the ratio between the Alfvén speed vA and
the sound speed vs . Another way to illustrate wave properties is to use the wave normal
surfaces, see the bottom panels of Figure 6.8.
w/k
vA > vs
w/k
fast MHD
(vA2+vS2)1/2
vA
vA < vs
fast MHD
(vA2+vS2)1/2
vs
vs
vA
0
30°
60°
q
90°
k||B0
k^B0
B0
vA
vs
0
30°
k||B0
90°
q
k^B0
B0
vs
vA
fast MHD
Alfvén
slow
MHD
60°
fast MHD
Alfvén
(vA2+vS2)1/2
slow
MHD
(vA2+vS2)1/2
Figure 6.8: Top) Phase speeds as a function of the propagation angle and bottom) wave
normal surfaces for Alfvén, fast and slow MHD waves. Cases with vA > vs and vA < vs
are shown separately.
A wave normal surface shows the phase speed as a function of the angle between
the wave propagation direction and the magnetic field θ, i.e., it describes how the phase
speed varies with respect to the magnetic field direction. Actually Figure 6.8 shows 2D
cuts of the surfaces. Assuming that the system is gyrotropic, the surface is found by
letting the wave normal curve rotate around the direction of the magnetic field.
At the oblique propagation angles for fast and slow MHD waves the velocity perturbation V1 is in the xz-plane (Figure 6.5). Now k · V1 6= 0, and hence, the waves are
compressional and associated with density pertubations (Eq. 6.57).
Investigate first the propagation perpendicular to the magnetic field. Eq. 6.60 and
Figure 6.8 show that when θ → 90◦ the phase speed of the slow MHD wave goes to zero.
The fast mode, in turn, can propagate to all directions. When θ → 90◦ the phase speed
6.4. MHD WAVES
99
of the fast mode reduces to
vp =
q
2 + v2
vA
s
(6.61)
i.e., the phase speed depends both on the sound speed and the Alfvén speed. This
wave is called the magnetosonic wave. Magnetosonic speed defined by Eq. 6.61 is the
maximum propagation speed of the MHD waves.
Figure 6.9: A fast magnetosonic wave propagates perpendicular to the magnetic field.
In a very tenuous plasma with large enough magnetic field the Alfvén speed can
actually be larger than the speed of light (vA > c). In such cases the non-relativistic
MHD approximation breaks down and the displacement current cannot be neglected.
The modification to the dispersion relation for a mode propagating perpendicular to the
magnetic field (θ = 90◦ ) including the displacement current is derived in Exercise 6.6.
2
ω2
vs2 + vA
=
2 /c2
k2
1 + vA
2 for v 2 c2 , i.e., at the non-relativistic MHD
Obviously this reduces to ω 2 /k 2 = vs2 + vA
A
limit.
The eigenvector corresponding to this root is (0, 0, V1z ) and from linerized continuity
Eq. 6.57 we obtain the density perturbation ρ1 = ρ0 (V1z /vp ). The linearized convection
equation (Eq. 6.58) gives
V1
B1 =
B0 .
(6.62)
ω/k
The electric field of the wave is obtained from the ideal Ohm’s law:
E1 = −V1 × B0 .
(6.63)
The magnetosonic wave is similar to the electromagnetic wave in the sense that the
wave vector and the wave magnetic and electric fields are all perpendicular to each
other. However, mass flow and density fluctuate along the wave vector, and thus, the
magnetosonic wave is longitudinal (Figure 6.9). It is also called “magnetoacoustic”,
which derives from this property.
100
CHAPTER 6. MAGNETOHYDRODYNAMICS
In the case vA vs the phase speed of the magnetosonic wave (Eq. 6.61) becomes
vp ≈ vA , i.e., it approaches the Alfvén speed. However, the wave is compressional and is
often called the compressional Alfvén wave. Note that this situation corresponds to the
cold-plasma (zero temperature) limit where vs goes to zero. From Eq. 6.60 it is clear
that in the cold plasma limit the slow mode MHD wave ceases to exist.
Next, let us investigate propagation parallel to the magnetic field (θ = 0◦ ). In the
case where the magnetic field dominates (vA > vs ) the dispersion equation gives for the
fast MHD wave
vp = vA ,
(6.64)
i.e., the wave reduces to the Alfveń wave.
In turn, for the slow MHD wave the phase speed becomes
vp = vs
(6.65)
i.e., it reduces to correspond the ordinary sound wave. A sound wave is the simplest
disturbance that can propagate in a collisional medium. The wave vector k is normal to
the pressure front and the restoring force is the pressure gradient. Because the magnetic
field does not restrict the particle motion along the magnetic field in a plasma, sound
waves can propagate also in a magnetized plasma, see Figure 6.10.
Figure 6.10: Longitudinal sound wave
propagates along the magnetic field in a
compressible and magnetized plasma.
.
Figure 6.8 illustrates that in the case of parallel propagation either the fast or slow
mode MHD wave reduces to the Alfvén wave and the other one to the sound wave
depending on the ratio between the vA and vs . The fast MHD wave has always larger
phase speed than the slow MHD wave.
A compressive MHD wave can steepen into a shock wave when the disturbance propagates faster than the characteristic speed of the medium. In space and fusion plasmas
shocks may be produced by explosions (e.g., solar flares, supernovae, inertial confinement fusion), by a disturbance moving through a fluid with its speed exceeding the local
characteristic information speed. Some examples are a coronal mass ejection moving
through the solar wind faster than the local magnetosonic speed and the encounter of
supersonic and super-Alfvénic fluid with a stationary object (e.g., the formation of the
Earth’s bow shock).
6.5. MAGNETIC RECONNECTION
6.5
101
Magnetic reconnection
One of the most important plasma physical phenomena arises when the frozen-in condition breaks down. If the ideal MHD assumption holds, two initially (magnetically)
separate plasma elements can never mix. For example, the Earth’s magnetosphere would
always stay closed to the solar wind. According to Eq. 6.9 the Reynolds number decreases when the plasma flow speed or the length scale of field gradients decreases or
when resistivity increases. When the Reynolds number becomes sufficiently small the
magnetic field starts to diffuse. In a particle description the break-down of the frozen-in
condition can be understood by the GC approximation becoming invalid, i.e., charged
particles cease to follow the magnetic field. (As discussed earlier, this usually does not
happen simultaneously for electrons and ions.).
Figure 6.11: Reconnection between two plasma domains with oppositely oriented fields
that are flowing towards each other. Open arrows indicate the direction of the plasma
flow.
Figure 6.11 shows two ideal MHD plasma regions with oppositely oriented magnetic
fields flowing towards each other. Such situation arises for example at the interface between the Earth’s magnetosphere and the solar wind when the interplanetary magnetic
field is southward (at the nose of the magnetosphere the magnetospheric field points
always to the north). A thin current sheet forms between the regions introducing a
large magnetic field gradient. The exact microphysics that occurs in the thin current
sheet is not yet well-understood. However, if there are processes that increase resistivity, diffusion can start leading to the reorganization of the plasma and magnetic field.
Plasma elements that were originally in separate regions may now become magnetically
connected.
The change of connection between the plasma elements is called magnetic reconnection. The importance of reconnection lies in its ability to change the topology of the
magnetic field and to convert magnetic energy to kinetic and thermal energy.
The concept of reconnection was first presented by Ronald Giovanelli in the 1940s
to explain particle acceleration in solar flares. In a solar flare a huge amount of energy
is released from the Sun in time-scales of only a few minutes. We cover here briefly the
most elementary reconnection models.
102
CHAPTER 6. MAGNETOHYDRODYNAMICS
x
+B0
Vi
Ey
Bz (x)
z
2l
Jy (x)
-B0
Vi
Figure 6.12: 1-dimensional current sheet. y-directed electric field has been added that
brings new plasma and magnetic flux towards the current sheet and maintains the balance between diffusion and convection.
Diffusion in a 1-dimensional current sheet was treated in Section 6.2.1. To achieve a
steady-state situation new magnetic flux and plasma have to be brought to the system
to replace the annihilated flux. This can be achieved by adding an electric field as shown
in Figure 6.12. Outside the current sheet the ideal Ohm law gives Ey = Vi B0 , where Vi
is the plasma inflow speed and B0 the magnetic field far away from the current sheet.
At the current sheet magnetic field is zero and the resistive Ohm’s law gives Ey = Jy /σ.
The width of the current sheet adjusts to maintain the balance between diffusion and
convection. Assuming that the width of the current sheet is 2l Amperè’s law gives the
electric current Jy = B0 /µ0 l. Hence, the width of the current sheet can be written as:
2l =
2
.
µ0 σV
(6.66)
The scenario explained above is unphysical. What happans to the plasma that is
brought to the current sheet? The solution is to add an additional dimension as shown
in Figure 6.13. This is the famous Sweet–Parker model formulated in the 1950s. In the
Sweet–Parker model the magnetic field annihilates in a finite domain called a diffusion
region (gray area in Figure 6.13). Plasma and the magnetic field flow away from the
boundaries of the diffusion region. Figure 6.13 also illustrates that the plasma elements
that were originally not magnetically connected (blue and red circles are at different field
lines before entering the diffusion region) become connected after exiting the diffusion
region. In the outflow region the magnetic field is thus weaker and the plasma flow speed
larger than in the inflow region. It is important to note that the frozen-in condition
breaks-down in the diffusion region but is valid outside. Hence, the diffusion region is
the region where the rearrangement of the magnetic field occurs.
The speed of reconnection, i.e., the reconnection rate, is typically expressed as the
electric field in the inflow region. It is an important quantity determining the inflow
speed. Estimates for the inflow and outflow speeds can be achieved by assuming incom-
6.5. MAGNETIC RECONNECTION
103
inflow
Bi
Vi
outflow
E
2l
Vo
E
Vo
outflow
Vi
2L
Bi
inflow
Figure 6.13: Sweet-Parker reconnection. The diffusion region is shown by gray.
pressible flow (ρi = ρo = ρ), conservation of mass (Vi L = Vo l) and that all inflowing
electromagnetic energy transforms to kinetic energy.
Inflowing electromagnetic energy can be calculated from the inflowing Poynting flux:
|S| = |E × H| =
Vi Bi2
EBi
=
.
µ0
µ0
(6.67)
The mass that flows in a unit time to the diffusion region (ρVi ) will be accelerated
to speed Vo . Hence, the change in energy in unit time and unit area is:
1
4W = ρVi (Vo2 − Vi2 ) .
2
(6.68)
Equating the energy increase with the inflow energy flux and noting that Vo Vi gives
Vi Bi2
1
= ρVi Vo2
µ0
2
⇒
Vo2 =
2Bi2
2
= 2vAi
.
µ0 ρ
(6.69)
(6.70)
Thus the Alfvén speed in the inflow region describes
√ the speed of the outflowing plasma
(under the used approximations within a factor of 2). Using Vi L = V0 l and the width
of the diffusion region from Eq. 6.66 the inflow speed is:
√
Vi = vAi ( 2/RmA )1/2 ,
(6.71)
where RmA = µ0 σvAi L is the Reynolds number calculated using the inflow Alfvén speed
known as the Lundquist number. It is easy to show (Exercise 6.7) that half of the
incoming magnetic energy is transformed to heat and the other half causes acceleration
of particles. In space plasmas RmA is usually very large and thus the inflow and the
104
CHAPTER 6. MAGNETOHYDRODYNAMICS
reconnection rate in the Sweet–Parker model is very slow. For a solar flare it would take
days to erupt, not minutes as the observations indicate.
The slow reconnection speed in the Sweet–Parker model can be traced to the property
that all energy conversion occurs in a diffusion region whose length is much larger than
the width of the outflow region. In 1964 Harry Petschek proposed that significantly
faster reconnection rates can be obtained by introducing a vanishingly small diffusion
region. He added two slow mode shocks, i.e. slow MHD waves steepened to shocks, that
emanate from the diffusion region. The shocks deviate the plasma flow and magnetic
field. The Petschek model is presented schematically in Figure 6.14.
Figure 6.14: Petschek fast reconnection. Two slow mode shocks (blue) emanate from
a vanishingly small diffusion region and deviate the plasma flow (red lines) and the
magnetic field (black lines).
Contemplate: Where does the energy conversion occur in the Petschek model? Where
does the magnetic field connectivity change?
The properties of MHD shocks are beyond our discussion here. They will be treated
in the course on space applications of plasma physics. However, a rigorous analysis
shows that the inflow speed in the Petscheck model can be up to 10% of vAi , which
allows much faster reconnection than in the Sweet–Parker model.
6.6
Magnetohydrostatic equilibrium and stability
MHD equilibrium structures are important for a number of space and astrophysical
phenomena and in fusion research. For example, solar prominences are huge structures
that can remain stable up to several solar rotations before erupting and much of the
fusion research deals with plasma confinement. We will start by investigating the MHD
momentum equation (Eq. 6.2). Assuming scalar pressure (∇ · P → ∇p) and timeindependent (d/dt = 0) equilibrium the momentum equation reduces to
J × B = ∇p .
(6.72)
6.6. MAGNETOHYDROSTATIC EQUILIBRIUM AND STABILITY
105
This means that the plasma pressure gradient and the Lorentz force must be in balance.
This equation gives B · ∇p = 0 and J · ∇p = 0 . Thus B and J are vector fields on
surfaces of constant pressure.
From the above equilibrium condition one can calculate the current perpendicular to
B:
B × ∇p
.
(6.73)
B2
This current is often called the diamagnetic current. It arises from the plasma pressure
gradient. In the particle description the perpendicular current is the sum of all current
elements in the plasma and contains contributions from the magnetic drifts (gradient and
curvature drift related currents), polarization current, and the magnetization current.
The magnetization current is caused by an inhomogeneous plasma density:
J⊥ =
JM = ∇ × M .
(6.74)
Here the magnetization M is the density of magnetic moments µ (see Eq. 3.14). Figure
6.15 shows the particle picture of the magnetization current. If the plasma density is
non-uniform, the gyration velocities do not sum to the zero, and hence, a net current
arises.
B
x
y
high density plasma
total
current
low density plasma
Figure 6.15: Single particle interpretation of the diamagnetic current.
Using Ampère’s law we can write the magnetic force in the form:
B2
J × B = −∇
2µ0
!
+
1
(B · ∇) B .
µ0
(6.75)
The magnetic force consist of two separate terms. The first term on the RHS is the
gradient of the magnetic energy density, i.e., of the magnetic pressure:
pB = B 2 /(2µ0 ) .
(6.76)
106
CHAPTER 6. MAGNETOHYDRODYNAMICS
The second term describes the tension force arising from the inhomogeneities of the
magnetic field. This latter term can be divided into two components:
(B · ∇) B = B
d
dŝ
∂B
(Bŝ) = B 2 + B
ŝ
ds
ds
∂s
!
∂ B2
2 n̂
+ ŝ
,
= B
RC
∂s 2
(6.77)
where ŝ is the unit vector along the magnetic field and RC is the radius of curvature. It
is now evident that:
1. The first term is anti-parallel to the radius of the curvature of field lines. The
related component of the force acts to reduce the stress in the field lines.
2. The second term is field aligned and cancels the field aligned component of ∇pB .
As a consequence only perpendicular component of ∇pB exerts force on the plasma.
In Excercise 6.8 the magnetic force is calculated for different magnetic field configurations. Sketching the magnetic field configurations helps to visualize how curvature and
gradients in the magnetic field are related to the direction of the magnetic force.
Hence, we obtain the condition for the MHD equilibrium dV/dt = 0 from the momentum equation:
!
B2
1
∇ p+
=
(B · ∇) B .
(6.78)
2µ0
µ0
See Exercise 6.9 for a demonstration of how in a simple magnetic field configuration
Bx = y and By = x the magnetic pressure and tension balance each other.
Assuming homogeneous magnetic field the sum of the magnetic and plasma pressures
is constant
!
B2
∇ p+
=0.
(6.79)
2µ0
The plasma beta
2µ0 p
(6.80)
B2
expresses the ratio of the plasma and magnetic pressures. It is one of the important
dimensionless parameters used to characterize plasmas.
β=
An example of a MHD equilibrium configuration is the Harris current sheet we encountered in Section 3.6.2. As discussed earlier, the Earth’s magnetotail, which can
stay stable for long time periods, can be described by a Harris current sheet. In a
1-dimensional Harris current sheet the magnetic field (assumed here to be in the zdirection) is given by:
z
ey .
(6.81)
B = B0 tanh
L
6.6. MAGNETOHYDROSTATIC EQUILIBRIUM AND STABILITY
The pressure is given by
p = p0 cosh−2
z
,
L
107
(6.82)
where p0 = B02 /(2µ0 )
As illustrated in Figure 6.16 (see also Exercise 6.10) the variations in the magnetic
field and plasma pressure over the Harris current sheet balance each other.
Figure 6.16: Magnetic field and pressure variations in the Harris current sheet.
q-pinch
Z-pinch
J
B
z
B
J
q
r
Figure 6.17: Left) θ-pinch, and Right) Z-pinch
Other examples of 1-dimensional equilibrium configurations are θ- and Z-pinches
shown in Figure 6.17. In both cases it is convenient to use cylindrical coordinates. In a
θ-pinch cylindrical coils drive an electric current and the magnetic field is axial, while in
108
CHAPTER 6. MAGNETOHYDRODYNAMICS
a Z-pinch the electric current is axial and the magnetic field poloidal. The equilibrium
conditions (∇p = −J × B) are (Exercise 6.11) :
d
dr
d
dr
6.7
B2
p+ z
2µ0
B2
p+ θ
2µ0
!
+
!
Bθ2
2µ0
= 0
(6.83)
= 0.
(6.84)
Force-free magnetic fields
If β 1 in magnetohydrostatic equilibrium, the pressure gradient is negligible and thus
J × B = 0.
(6.85)
Such configurations are called force-free fields because the magnetic force on the plasma
is zero. According to Eq. 6.75 in a force-free field the magnetic pressure gradient
∇(B 2 /2µ0 ) is balanced by the magnetic tension force µ−1
0 (B · ∇) B. In real situations
the force-free equilibrium is always an approximation, but often a very good one, to
the momentum equation. It is also evident from Eq. 6.85 that in a force-free field the
electric current flows along the magnetic field. Such currents are commonly called as
field-aligned currents (FAC).
Using Ampère’s law we can write Eq 6.85 as
(∇ × B) × B = 0 .
(6.86)
From this we see that the innocent-looking equation J × B = 0 is in fact non-linear and
thus difficult to solve.
The field-alignment of the electric current can be expressed as
∇ × B = µ0 J = α(r)B ,
(6.87)
where α is a function of position. Taking divergence of this we get
B · ∇α = 0 ,
(6.88)
i.e., α is constant along the magnetic field.
In the case α is constant everywhere, the equation
∇ × B = αB
(6.89)
is linear. Taking a curl of (6.89) we get the Helmholtz equation
∇2 B + α 2 B = 0 .
(6.90)
6.7. FORCE-FREE MAGNETIC FIELDS
109
Figure 6.18: Helical structure of a force-free magnetic field.
Solution to the Helmoltz equation in cylindrical symmetry was given by Lundquist
in 1950 in terms of Bessel functions J0 and J1 :
BR = 0
(6.91)
α0 r
= B0 J 0
r
0 α0 r
= ±B0 J1
,
r0
BA
BT
(6.92)
where BR , BA , and BT are radial, axial and tangential magnetic field components,
respectively. The solution is a magnetic flux rope where magnetic field lines form helices
whose pich angle increases from the axis (Figure 6.18). r is the radial distance from the
flux rope axis, r0 is the radius of the flux rope and B0 is the maximum magnetic field
magnitude at the center of the flux rope (r = 0).
Figure 6.19: Left) Erupting coronal mass ejection whose structure is a magnetic flux
rope. Image taken by Solar Dynamic Observatory. Courtesy: NASA. Right) Hubble
Space Telescope image of a filamentary nebula (Dahlgren et al., 2007).
Flux ropes are common in space, astrophysical and fusion plasmas. The left-hand
part of Figure 6.19 shows an erupting solar plasma cloud whose structure is often approximated with a force-free flux rope. These plasma clouds maintain more or less their
integrity while traveling away from the Sun to the orbit of the Earth and beyond and
110
CHAPTER 6. MAGNETOHYDRODYNAMICS
they are the main drivers of severe magnetospheric disturbances. The right-hand part
of Figure 6.19 shows a Hubble image of a planetary nebula. The substructures in this
nebula may be formed from magnetic flux ropes that are twisted around each other.
A special case of a force-free magnetic field is the current-free configuration ∇×B = 0.
Now the magnetic field can be expressed as the gradient of a scalar potential B = ∇Ψ.
Because ∇ · B = 0, the magnetic field can be found by solving the Laplace equation
∇2 Ψ = 0
(6.93)
with appropriate boundary conditions and using the methods of potential theory.
Figure 6.20: Potential Field Source Surface model of the Sun’s magnetic field on April
2010. Open positive (outward from the Sun) flux is in green, open negative flux in red,
and the tallest closed flux trajectories in blue. The fields are plotted over the original
synoptic magnetogram. White areas indicate the maximum-strength positive flux and
black maximum-strength negative flux. Courtesy: NSO/GONG.
For example, the Sun’s magnetic field structure is often modelled by the so-called
Potential Field Source Surface (PFSS) model (Figure 6.20). The magnetic field is computed from the Laplace equation using spherical coordinates from the photosphere to
the “source surface”, nominally chosen to be at 2.5 Solar radii. At the source surface
the Sun’s magnetic field is assumed to be purely radial. The inner boundary conditions
are obtained from solar magnetograms. Thus, PFFS assumes that there is no electric
current in the corona.
6.8. EXERCISES: MAGNETOHYDRODYNAMICS
6.8
111
Exercises: Magnetohydrodynamics
1. Derive the induction equation for the magnetic field in a case where the conductivity is not spatially homogeneous
2. Calculate the magnetic Reynolds number Rm and the diffusion time τd for
(a) a laboratory plasma where LB ≈ 0.1 m, V ≈ 103 m s−1 and σ ≈ 100 Ω−1 m−1
(b) solar wind where LB ≈ 10 solar radii, V ≈ 400 km s−1 and σ ≈ 3×104 Ω−1 m−1 .
3. Show that at the limit of large Reynolds number the magnetic flux through a closed
loop co-moving with plasma
dΦ
d
=
dt
dt
Z
B · dS = 0
is constant
4. Compute the Alfvén speed in the following cases
(a) Earth’s ionosphere ne = 1011 m−3 , B = 50 µT, ions assumed to be mostly
O+ .
(b) Solar corona: ne = 1014 m−3 , B = 50 mT, ions are protons
(c) Interstellar gas cloud: ne = 0.1 cm−3 , B = 0.1 nT, ions are protons and the
ionization degree is 1%.
5. Derive the phase speeds of the fast and slow Alfvén waves
2
ω
k
=
1/2
1 2
1 2
2
2
2
− 4vs2 vA
cos2 θ
vs + vA
±
vs2 + vA
2
2
starting from the dispersion equation
2
−ω 2 V1 + vs2 + vA
(k · V1 ) k
+ (k · vA ) [(k · vA ) V1 − (vA · V1 ) k − (k · V1 ) vA ] .
6. Consider the propagation of Alfvén waves taking the displacement current into
account. That is, start from the same equations as on the lectures but replace
Ampère’s law by
1 ∂E
∇ × B = µ0 J + 2
.
c ∂t
Derive the dispersion equation for the mode propagating perpendicular to the
magnetic field into the form
ω2
v2 + v2
= s 2 A2 .
2
k
1 + vA /c
112
CHAPTER 6. MAGNETOHYDRODYNAMICS
7. Consider the Sweet–Parker reconnection model. Show that half of the incoming
magnetic energy is transformed to heat and the other half causes acceleration of
particles.
8. Calculate the magnetic force J × B for the following cases. Sketch also following
magnetic field configurations and indicate the direction of magnetic forces in each
case.
(a) B = xey
(b) B = ex + xey
(c) B = yex + xey
(d) B = reθ
9. Show that in the magnetic field configuration Bx = y, By = x the magnetic pressure
and tension balance each other. Show that if the configuration is stretched in the
y direction: Bx = y, By = α2 x, where α2 > 1, this causes in certain regions a net
force toward the X-line and in other regions away from the X-line.
10. Show that the total pressure of the 1-dimensional Harris model is B/2µ0 and that
the current density is
B0
2 z
Jy (z) =
sech
.
µ0 h
h
Show futher that the model is in magnetohydrostatic equilibrium J × B = ∇p.
11. Consider the equilibrium pinch in Figure 6.21. Assume cylindrical symmetry, that
MHD assumptions are valid and the electric current flows only inside the cylinder
of radius R. Write the condition for the hydromagnetic equilibrium in cylindrical
coordinates. Calculate and plot the profiles of plasma pressure and the magnetic
field, when the current is constant inside the cylinder. How would the result
change, if the current would flow on the surface of the cylinder only?
z
J z (r)
p=0
p(r)=0
Βθ (r)
R
Figure 6.21: Equilibrium
pinch for Exercise 6.11
r
θ
Chapter 7
Cold plasma waves
Propagation of electromagnetic waves is one of the most important phenomena in plasma.
Characteristics of wave propagation are used in various ways to diagnose plasma and
observing wave emissions in plasma can give information on the plasma properties. For
example, plasma density can be calculated easily from the plasma frequency and the
magnetic field magnitude from the gyro frequency. In the previous chapter we investigated MHD waves and considered frequencies well below the ion gyro and plasma
frequencies. When the frequency of the wave increases, one needs to take into account
that the ion and electron dynamics become different, and hence, the one-fluid MHD
description becomes invalid. This chapter investigates the wave propagation in the cold
plasma limit. In reality the temperature of a plasma is never zero, but the temperature
effects
can be neglected if the wave propagates faster than the plasma thermal speed
p
2kB T /m. As a consequence, cold plasma has zero pressure and there are no waves
related to pressure fluctuations, such as sound waves. At high frequencies, well above
the ion gyro frequency, the ions can be considered as an immobile background as they
cannot respond quickly enough to the wave. Note that we now consider much faster
fluctuations than in MHD, and thus, we need to take into account the displacement
current in Maxwell’s equations.
We start by deriving the general form of the dispersion equation. We proceed to
investigate the waves that propagate exactly parallel or perpendicular to the magnetic
field. Finally, we briefly discuss arbitrary direction of propagation
7.1
General form of the dispersion equation
The treatment of waves in plasma at the cold plasma limit resembles closely the study
of general electromagnetic waves (see Appendix 9.3). To derive the cold plasma dispersion equation we start from the density continuation equation, equation of motion and
113
114
CHAPTER 7. COLD PLASMA WAVES
Maxwell’s equations
∂n
+ ∇ · (nV) = 0
∂t
∂u
= e(E + u × B)
m
∂t
∂B
∇×E = −
∂t
X
1 ∂E
∇×B− 2
= µ 0 J = µ0
ens us = ~σ · E
c ∂t
s
1 X
ens
∇·E =
0 s
∇ · B = 0.
(7.1)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
In Eq. 7.4 we have used Ohm’s law with conductivity being a second rank tensor ~σ . s
indexes all particle species that constitute the plasma.
Let us consider again a small perturbation (subscript ”1”) to the initial equilibrium
(subscript ”0”)
n = n0 + n1
(7.7)
u = u1
(7.8)
B = B0 + B1
(7.9)
E = E1 .
(7.10)
Insert these to Eqs. 7.1-7.6 and linearize
∂n1
+ ∇ · (n0 u1 ) = 0
∂t
∂u1
m
= e(E1 + u1 × B0 )
∂t
∂B1
∇ × E1 = −
∂t
X
1 ∂E1
∇ × B1 − 2
= µ 0 J = µ0
ens u1s
c ∂t
s
(7.11)
(7.12)
(7.13)
(7.14)
J = ~σ · E1
1 X
en1s
∇ · E1 =
0 s
(7.15)
∇ · B1 = 0 .
(7.17)
(7.16)
By taking the curl from Eq. 7.13 and using Eq. 7.14 we obtain
∇ × ∇E1 =
1 ∂ 2 E1
∂J
− µ0
.
c2 ∂t2
∂t
(7.18)
7.2. WAVE PROPAGATION IN NON-MAGNETIZED PLASMA
115
Assume harmonic time dependences (see Appendix 9.3) and use the linearized Ohm’s
law (Eq. 7.14) to eliminate J from Eq. 7.18.
This leads to the homogeneous wave equation:
~ · E1 = 0 ,
n × (n × E1 ) + K
(7.19)
~ is the dielectric tensor
where n = ck/ω is the index of refraction and K
~ = ~1 − ~σ .
K
iω0
7.2
(7.20)
Wave propagation in non-magnetized plasma
Let us first consider a simple case where the background magnetic field B0 is zero. The
linearized equation of motion now becomes
m
∂u
= eE .
∂t
(7.21)
Note that here the subscript ”1” has been dropped for simplicity. Assuming harmonic
time dependence this reduces to
m(−iω)u = eE .
We use this equation to eliminate u from J =
P
s
J=
X
s
(7.22)
ens us and to obtain
n0 e2
E,
(−iω)ms
(7.23)
from which we can now identify the conductivity tensor
~σ = ~1
X
s
n0 e 2
.
(−iω)ms
(7.24)
Using the definition of the plasma frequency and ωpe ωpi we obtain
ωp2
i
~σ = ~1
0 ω
ω
(7.25)
and the homogeneous wave equation becomes
c2 k × (k × E) = (ω 2 − ωp2 )E .
(7.26)
116
CHAPTER 7. COLD PLASMA WAVES
Let us choose the wave vector k to be parallel to the z-axis. From Eq. 7.26 we now
obtain a matrix equation



−c2 k 2 + ω 2 − ωp2
0
0
Ex



2 k2 + ω2 − ω2
0
−c
0
  Ey  = 0 ,

p
Ez
0
0
ω 2 − ωp2
(7.27)
from which we get the dispersion equation by setting the determinant of the matrix to
zero
(−c2 k 2 + ω 2 − ωp2 )2 (ω 2 − ωp2 ) = 0 .
(7.28)
One of the roots is evidently ω = ωp . The electric field associated with this mode is in
the z-direction. Since we selected k to be in the z-direction, the electric field perturbation
is parallel to the wave propagation. Hence, the wave is longitudinal : k · E1 6= 0 and
k × E1 = 0. By assuming harmonic dependencies the linearized Gauss’s law (Eq. 7.16)
becomes
ρ1 = i0 k · E1 ,
(7.29)
and we see that the wave is associated with charge density fluctuations. In turn, assuming
harmonic dependencies the linearized Faraday’s law (Eq. 7.13) becomes
ik × E1 = −iωB1
1
→ B1 =
k × E1 ,
ω
(7.30)
and we see that B1 = 0, i.e., the mode is electrostatic. The group speed vg = dω/dk
(see Appendix 9.3) is zero indicating that the wave does not propagate. This solution
describes oscillation at the plasma frequency we encountered in Section 2.3.
Another root of Eq. 7.28 is
ω 2 = ωp2 + c2 k 2 .
(7.31)
The electric field has now components in the x and y-directions. Thus, the electric
field is perpendicular to k and from the linearized Gauss’s law we see that the wave is
non-compressional and transverse (the electric field perturbation is perpendicular to the
direction of propagation). The magnetic field associated with the wave is obtained from
Eq. 7.30. The wave vector as a function of frequency is
k=±
1q 2
ω − ωp2 .
c
(7.32)
Figure 7.1 displays the dispersion equation. At high-frequencies the solution approaches the vacuum electromagnetic wave with ω = ck, i.e., its phase and group speeds
approach the speed of light. The interpretation is that the frequency of the wave becomes so high that it does not interact with the plasma. In fact, it interacts but only
very weakly.
7.2. WAVE PROPAGATION IN NON-MAGNETIZED PLASMA
117
Figure 7.1: The dispersion equation for an electromagnetic wave
in a cold plasma where the background magnetic field is zero.
The electric field of the plane wave is of the form
E = E exp[i(k · r − ωt)] .
(7.33)
When ω < ωp , the wavenumber k is purely imaginary. If the imaginary part is negative,
the wave electric field would grow exponentially. Since there is no energy in the plasma to
facilitate wave growth, this solution is unphysical. The solution with a positive imaginary
part makes the electric field to decay exponentially and the wave is said to cut-off at the
plasma frequency. What happens physically is that when the wave frequency approaches
the plasma frequency the wave forces electrons to oscillate at the plasma frequency. The
oscillating electrons re-radiate the wave energy and the wave is reflected.
Figure 7.2: Reflection of electromagnetic waves from the ionosphere. Note that also
waves with higher frequency than the maximum plasma frequency are affected by the
plasma and refracted. The ionospheric density profile can be determined by sending
waves at different frequencies and measuring the time the wave returns back. Such a
device is called ionosonde.
An example of the reflection arises when radio waves are sent to the ionosphere
(Figure 7.2). While the wave travels away from the Earth, the plasma density and thus
the plasma frequency increases. As a response to the wave electric field electrons in the
118
CHAPTER 7. COLD PLASMA WAVES
ionosphere start to oscillate and they re-radiate the original energy. The total reflection
of the wave occurs when the emitted frequency equals to the local plasma frequency.
7.3
Wave propagation in magnetized plasma
We now move to a more complicated situation and introduce a non-zero background
magnetic field. The magnetic field introduces anisotropy, which leads to many new wave
modes. When a magnetic field is present, particles perform Larmor motion around the
magnetic field lines. This introduces new possibilities to wave cut-offs and resonances. In
this Section we first derive a general form of the dispersion equation and then proceed
to investigate the so-called principal modes, i.e., waves that propagate either parallel
or perpendicular to the magnetic field. Finally we will study wave propagation at an
arbitrary angle.
7.3.1
Derivation of general dispersion equation
First, we need to determine the conductivity and dielectric tensors. Let the magnetic
field to be in the z-direction. Assuming harmonic time dependencies and using the gyro
frequency ωcs = es B0 /ms the linearized equation of motion (Eq. 7.12) can be now
written in the matrix form:





usx
−iω −ωcs
0
E
es  x 



0   usy  =
 ωcs −iω
 Ey  .
ms
0
0
−iω
usz
Ez
(7.34)
Inversion of the matrix equation gives




usx
es 



 usy  =
ms 

usz


−iω
2
ωcs − ω 2
ωcs
− 2
ωcs − ω 2
ωcs
2
ωcs − ω 2
−iω
2
ωcs − ω 2
0
0

0 

 Ex


0 
  Ey  .

i  Ez
(7.35)
ω
Similarly as in the case with B0 = 0 the conductivity tensor is obtained by calculating
P
the electric current J = es ns0 us1 = ~σ · E.
s


X ns0 e2 

s 
~σ =
ms 


−iω
2
ωcs − ω 2
ωcs
− 2
ωcs − ω 2
ωcs
2
ωcs − ω 2
−iω
2 − ω2
ωcs
0
0

0 


0 
.

i 
ω
(7.36)
7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA
119
The dielectric tensor is now


S −iD 0


S
0 ,
K =  iD
0
0
P
(7.37)
2
ωps
2
ω 2 − ωcs
(7.38)
where
S = 1−
X
s
X
D =
s
P
= 1−
2
ωcs ωps
2 )
ω(ω 2 − ωcs
2
X ωps
s
ω2
(7.39)
.
(7.40)
S and D are often decomposed into the left- (L) and right-handed (R) polarized modes:
S = (R + L)/2 and D = (R − L)/2, where
R = 1−
2 X ωps
s
L = 1−
ω2
2 X ωps
s
ω2
ω
ω + ωcs
ω
ω − ωcs
(7.41)
.
(7.42)
This is a useful division since electrons and ions respond in a different way to the wave.
z
n,k
B0
q
x
y
Figure 7.3: The choice of directions of
the background magnetic field and the
index of refraction (wave vector) for a
wave propagating in a cold plasma.
Let us choose the magnetic field to be along the z-direction and the index of refraction
n (and thus k) to be in the xz-plane (Figure 7.3). The homogeneous wave equation now
becomes



S − n2 cos2 θ −iD
n2 cos θ sin θ
Ex



iD
S − n2
0
(7.43)

  Ey  = 0 .
n2 cos θ sin θ
0
P − n2 sin2 θ
Ez
As before, we obtain the dispersion equation by setting the determinant to zero.
D(n, ω) = An4 − Bn2 + RLP = 0 ,
(7.44)
120
CHAPTER 7. COLD PLASMA WAVES
where
A = S sin2 θ + P cos2 θ
2
(7.45)
2
B = RL sin θ + P S(1 + cos θ) .
(7.46)
The dispersion equation can be modified to a useful form by solving for tan2 θ as a
function of n2
−P (n2 − R)(n2 − L)
.
(7.47)
tan2 θ =
(Sn2 − RL)(n2 − P )
When a wave propagates through plasma, it may encounter regions of changing
plasma frequency and gyro frequencies. In magnetized plasma we can find two cases in
which the wave ceases to propagate.
Cut-off occurs when n goes to zero. After the cut-off point n2 becomes negative,
and thus, n and k are imaginary. In such region the wave decays exponentially and
becomes evanescent. Physically, the wave is reflected and no energy is absorbed in
the plasma.
From Eq. 7.44 we see that cut-off occurs when P = 0, R = 0 or L = 0.
Resonance occurs when n2 approaches infinity (i.e., n and k approach infinity).
In resonance the wave energy is absorbed in the plasma and the wave is damped.
Resonance is an effective way to heat the plasma.
From Eq. 7.44 we see that resonance occurs when A = 0, i.e.,
tan θres = −
7.3.2
P
.
S
(7.48)
Propagation parallel to the magnetic field
We see from Eq. 7.47 that when the wave propagates exactly parallel to the magnetic,
i.e. θ = 0, the dispersion equation for cold plasma waves has three roots:
P
= 0
(7.49)
n
2
= R
(7.50)
n
2
= L.
(7.51)
The first root represents simple plasma oscillation (see Eq. 7.40), and thus, a nonpropagating wave.
7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA
121
By inserting θ = 0◦ in Eq. 7.43 we can find the electric field eigenvector associated
with the other two roots
En2 =R = (E0 , iE0 , 0)
(7.52)
En2 =L = (E0 , −iE0 , 0) .
(7.53)
Since we chose the magnetic field to be aligned with the z-direction, we see that the
roots n2 = R and n2 = L correspond to transverse waves. In the same way as in the
case of non-magnetized waves, we can confirm from the linearized Faraday’s law that the
waves are electromagnetic (i.e., they have non-zero magnetic field). From the linearized
Gauss’s law we see that there are no charge density fluctuations.
We also see that the only difference between the eigenvectors corresponding to the
roots n2 = R and n2 = L is the sign of the y-component. n2 = R corresponds to a wave
that rotates in the right-handed sense with respect to the magnetic field while n2 = L to
a wave that rotates in the left-handed sense (confirm! In plasma physics the convention
of RH and LH is opposite to that used in optics). Hence, the solutions correspond to
the right-handed (R) and left-handed (L) polarized modes, respectively.
Right-handed mode
From 7.41 we obtain
n2R = R = 1 −
2
2
ωpi
ωpe
−
.
ω(ω + ωci ) ω(ω − ωce )
(7.54)
Consequently, the R-mode has a resonance (n → ∞) when the wave frequency approaches the electron cyclotron frequency ω = ωce . This is because electrons rotate
around the magnetic field in the same sense as the electric field rotates in the R-mode.
Because ωpi ωpe and ωci ωce the R-mode cut-off (nR → 0) occurs when (Exercise
7.2)
q
i
ωce h
2 /ω 2 .
ωR=0 ≈
1 + 1 + 4ωpe
(7.55)
ce
2
The cut-off is divided to two branches depending on the density. At the low density
limit (ωp ωc ) the cut-off becomes
2
2
ωR=0 ≈ ωce (1 + ωpe
/ωce
)
(7.56)
and at the high density (ωp ωc ) limit
ωR=0 ≈ ωpe + ωce /2 .
(7.57)
122
CHAPTER 7. COLD PLASMA WAVES
Left-handed mode
For the L-mode we obtain from 7.41
n2L = L = 1 −
2
2
ωpi
ωpe
−
.
ω(ω − ωci ) ω(ω + ωce )
(7.58)
Hence, the resonance occurs now at the ion gyrofrequency ω = ωci .
Usually the ion motion is ignored when computing the L-mode cut-off
ωL=0 ≈
q
i
ωce h
2 /ω 2 .
−1 + 1 + 4ωpe
ce
2
(7.59)
At low density limit we obtain
2
ωL=0 ≈ ωpe
/ωce ,
(7.60)
ωL=0 ≈ ωpe − ωce /2 ,
(7.61)
and at the high density limit
i.e, both at low and high density limits ωR=0 = ωL=0 + ωce .
Figure 7.4 shows the solution of n2 in the (ω, k)-space for low- and high-density cases.
R- and L-modes are divided into two branches. Above the cut-off frequencies (ωR=0 and
ωL=0 ) the solution to the wave dispersion equation is called the free-space mode. Below
electron and ion cyclotron frequencies the waves are called the cyclotron modes. At low
frequencies (ω → 0) L- and R-modes merge and the dispersion becomes that of the shear
2 we encountered in Section 6.4.2.
Alfvén wave n2 → c2 /vA
Faraday rotation
A linearly polarized plane wave can be expressed as a sum of left- and right-hand circularly polarized waves (R- and L-modes having equal amplitudes, E0 ). If we assume that
the wave is linearly polarized along the x axis, and that the wave propagation (k) and
the background magnetic field (B0 ) are along the z-axis, we can write
E = E0 [(eikR z + eikL z )êx + i(eikR z − eikL z )êy ]e−iωt .
(7.62)
The ratio of the Ex and Ey components is
Ex
kL − kR
= cot
z
Ey
2
.
(7.63)
Hence, due to different phase speeds of R- and L-modes the linearly polarized wave that
is travelling along a magnetic field will experience the rotation of its plane of polarization.
This is called Faraday rotation. The magnitude of the rotation depends on the density
and magnetic field of the plasma. Considering frequencies above the plasma frequency
7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA
123
Figure 7.4: Wave modes propagating parallel to the magnetic field in the limit of high
plasma density (top), and low plasma density (bottom).
one can show that the rate of change in the rotation angle φ with the distance travelled
(assumed here to be in the z-direction) is
dφ
−e3
=
ne B0
dz
2m2e 0 c ω 2
(7.64)
and the total rotation from the source to the observer is
−e3
φ=
2m2e 0 c ω 2
Z d
ne B · ds .
(7.65)
0
The integral is calculated along the wave propagation path. The total rotation thus
depends on both the density and magnetic field of the medium. Exercise 7.3 applies
Faraday rotation to estimate the distance to a distant pulsar when the density of the
interstellar plasma is known.
Contemplate: Faraday rotation is an important tool in astronomy. Find an example
where Faraday rotation is used to obtain information on the physical properties of an
124
CHAPTER 7. COLD PLASMA WAVES
astronomical object. Pay attention to the fact that either density or magnetic field has
to be known from other methods.
Whistler waves
The investigation of dispersion characteristics of R-mode waves reveals an interesting
feature. Figure 7.4 shows that the R-mode propagates also in the region between electron
and ion gyro frequencies. In this domain the dispersion equation can be approximated
as (Exercise 7.4):
r
ωpe
ω
k=
,
(7.66)
c
ωce
which gives the phase and group speeds
ω
k
∂ω
=
∂k
vp =
=
vg
=
√
c ωce √
ω
ωpe
√
2c ωce √
ω.
ωpe
(7.67)
(7.68)
Thus both the phase and group speeds depend on the wave frequency.
This property of R-mode waves explains the puzzling “whistling” sound that was
observed in telegraph lines during World War I. It took several decades before this phenomenon was explained. In 1953 L.R.O. Storey suggested that the sound was produced
by waves that are propagating along the magnetic field lines from one hemisphere of the
Earth to the other (Figure 7.5) and the whistling sound is the consequence of different
frequencies arriving at different times. These whistler waves are produced by lighting
strokes that emit radio noise of broad frequency bands.
Figure 7.5: Whistler waves
The propagation time for a R-mode wave in in the frequency range ωci ω ωce
7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA
125
can be obtained from (see Eq. 7.68)
Z
t(ω) =
ds
=
vg
Z
wpe (s)
ds ,
√
2c ωωce
(7.69)
where ds is the line element along the magnetic field. Hence, lower frequencies arrive to
an observer after a longer time than the higher frequencies.
7.3.3
Propagation perpendicular to the magnetic field
When the wave propagates perpendicular θ = 90◦ to the magnetic field the homogeneous
wave equation (Eq. 7.43) becomes



S
−iD
0
Ex



0
 iD S − n2
  Ey  = 0
2
0
0
P −n
Ez
(7.70)
−P (n2 − R)(n2 − L)
→ ∞.
(Sn2 − RL)(n2 − P )
(7.71)
and
tan2 θ =
The roots are now
n2 = P
RL
n2 =
S
and the corresponding electric field eigenvectors are:
En2 =P
= (0, 0, E0 )
(7.72)
(7.73)
(7.74)
(7.75)
qEn2 =RL/S = (
iD
E0 , E0 , 0) .
S
(7.76)
Ordinary mode
The wave mode associated with the first root n2 = P is called the ordinary (O) mode.
n2O
2
2
2
ωpi
ωpe
ωpe
=P =1− 2 − 2 ≈1− 2 .
ω
ω
ω
(7.77)
The electric field of the ordinary mode is along the background magnetic field, and
thus, the wave vector k is perpendicular to wave electric field. The dispersion equation
above shows that the O-mode is not affected by the magnetic field (the particle motion
is parallel to the magnetic field, and hence the magnetic part of the Lorentz for vanishes
v × B0 = 0). Physically, the O-mode corresponds to the high-frequency transverse
electromagnetic wave. It is linearly polarized.
The O-mode has the cut-off (n2 → 0) when the wave frequency approaches the
electron plasma frequency, i.e., at ω = ωpe .
126
CHAPTER 7. COLD PLASMA WAVES
Extraordinary mode
The second mode is called the extraordinary mode. Now the wave electric field is perpendicular to the background magnetic field (Eq. 7.75), and thus, the electric field
has components both parallel (longitudinal) and perpendicular (transverse) to the wave
vector, see Figure 7.6. Hence, the X-mode has both electrostatic and electromagnetic
characteristics. The wave magnetic field can be calculated from the Faraday’s law. According to Eq. 7.75 the X-mode is elliptically polarized. Figure 7.7 shows the solution
of n2 in the (ω, k)-space for ordinary and extraordinary modes.
Ordinary (O) mode
Extraordinary (X) mode
z
z
E1
B0
B0
x
B1
x
y
n,k
n,k
B1 =
B1
y
E1
1
k ´ E1
w
Figure 7.6: Magnetic field, electric field and the wave vector directions for O- and Xmodes.
w
X-mode
wX,R=0
O-mode
wUH upper hybrid resonance
wpe
wX,L=0
X-mode
wLH lower hybrid resonance
magnetosonic mode
k
Figure 7.7: A plot of wave frequency as a function of wave number for ordinary and
extrordinary modes.
The X-mode has two cut-offs when R = 0 and L = 0 (see Section 7.3.2). Calculation
of resonances and cut-offs for extraordinary waves is a tedious task. The resonances
7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA
127
occur at S = 0 (see Eq. 7.38)
2
2
ωpi
ωpe
− 2
S =1− 2
2 = 0,
2
ω − ωce
ω − ωci
(7.78)
and they are called hybrid resonances. The number of resonances depends on the ion
species involved. Here we have assumed that the plasma consist of electrons and one
positive ion species. The resonance that occurs at the highest frequency, above both the
electron gyro and plasma frequencies, is called the upper hybrid frequency (ωU H ). The
frequencies near ωU H are so high that one can neglect the ion dynamics from Eq. 7.78
and using ωpi ωpe the upper hybrid resonance is at
2
2
ωU2 H ≈ ωpe
+ ωce
.
(7.79)
The lower hybrid resonance occurs between the electron and ion cyclotron frequencies. Assuming that ωce ωLH ωci one obtains the lower hybrid resonance at
2
ωLH
2 + ω2
ωci
pi
≈
≈ ωce ωci
2 /ω 2 )
1 + (ωpe
ce
2 +ω ω
ωpe
ce ci
2
2
ωpe + ωce
!
.
(7.80)
Further approximations are often made at the low and high density limits. In the
high density limit ωc2 ωp2 and we obtain
ωLH →
√
ωce ωci .
(7.81)
Both electrons and ions participate in the resonance oscillation. The lower hybrid resonance at the high density limit is particularly important since the wave can be in
resonance both with electrons and ions. This can facilitate the energy transfer between
ions and electron. For instance, in fusion devices these waves are in the microwave
range and microwave techniques are used to heat the plasma through the lower hybrid
resonance.
In the low density limit ωp2 ωc2 the lower hybrid resonance is at
ωLH → ωpi
(7.82)
Now only ions participate in the resonance oscillation.
For the low-frequency limit one obtains the magnetosonic mode we encountered already in Section 6.4.2. The cold plasma theory introduces a correction:
2
ω2
vs2 + vA
=
2 /c2 ,
k2
1 + vA
(7.83)
which guarantees that the group velocity of the wave remains below the speed of light
even in a plasma where vA > c (in cold plasma vs is, of course, negligible).
128
CHAPTER 7. COLD PLASMA WAVES
7.3.4
Propagation in an oblique angle
Principal modes R, L, O ja X can be uniquely defined only when the wave propagates
exactly perpendicular or parallel to the magnetic field. Similar to MHD waves, cold
plasma waves can propagate also at oblique angles to the magnetic field. It is possible
to draw the wave normal surface for each mode. However, the ratio of density and
magnetic field now varies and hence, the number of wave mode surfaces is much larger.
Figure 7.8 represents the characterization of waves using the CMA-diagram (Clemmow,
Mullaly, Allis). In the CMA-diagram a particular wave mode may be identified with its
wave normal surface and the surfaces may be traced in the (ωp2 /ω 2 , ωc /ω) space until
it disappears at the cut-off or at the resonance. As is seen from the figure cut-offs and
resonance define the ‘cages” where different wave modes are confined.
7.4
Exercises: Cold Plasma Waves
1. Consider a plasma consisting of free electrons and protons. Starting from the electric current due to polarization drift find the dielectric function for low frequencies
in the form
!
c2
= 0 1 + 2 .
vA
2. Prove that for the right-hand polarized wave propagating parallel to the magnetic
field the cut-off (nR → 0) occurs when
ωR=0 ≈
q
i
ωce h
2 /ω 2 .
1 + 1 + 4ωpe
ce
2
Show further that at the low density limit (ωp ωc ) this reduces to
2
2
ωR=0 ≈ ωce (1 + ωpe
/ωce
)
and at the high density (ωp ωc ) limit to
ωR=0 ≈ ωpe + ωce /2 .
3. The arrival time of a signal from a distant source depends on the dispersion of
plasma as
T = d/c + D/f 2
where d is the distance to the source, f the frequency of the signal and D the
so-called dispersion measure
e2
D= 2
8π 0 me c
Zd
ne ds .
0
7.4. EXERCISES: COLD PLASMA WAVES
129
Consider a pulsar from which a signal at 100 MHz arrives 2 s later than the signal
at 200 MHz. Assuming the density of the interstellar plasma to be 0.03 cm−3
calculate the distance to the pulsar. Calculate further the Faraday rotation of the
wave assuming a linear polarization and 0.1-nT interstellar magnetic field.
4. Derive the dispersion equation for the whistler wave
ωpe
k=
c
r
ω
.
ωce
Using this equation show that the group velocity of the whistler wave is
√
2c ωc e √
∂ω
ω.
vg =
=
∂k
ωpe
Compare the arrival times of the emitted low and high frequency waves.
130
CHAPTER 7. COLD PLASMA WAVES
P=0 (cut-off)
R=0
(cut-off)
L=0 (cut-off)
S=0
(res)
Figure 7.8: CMA diagram
Chapter 8
Warm plasma
In two previous chapters the temperature did not have an independent meaning. In
MHD temperature appears through the equation of state, but always in relation to density and pressure, while in the cold plasma theory we assumed that thermal effects can
be neglected. The inclusion of thermal effects introduces new wave modes in the plasma
and allows for free-energy that is necessary for the generation of plasma instabilities. To
fully describe warm plasma (wave modes, instabilities, etc.) one needs complex mathematical tools of kinetic theory. However, fluid description gives a simpler introduction
to characteristics of warm plasmas. We now modify the fluid equations to take into
account the thermal effects. We add the pressure term to the equation motion
nα mα
dVα
= enα (E + V × B) + ∇pα ,
dt
which allows us to investigate temperature related phenomena in a number of special
cases. We first derive warm plasma dispersion equation and investigate two special
solutions; the Langmuir wave and the ion sound wave. Then we proceed to a brief
overview of plasma instabilities.
8.1
Warm plasma dispersion equation
Now we include ions, but assume that there is no background electric or magnetic fields,
and that the plasma is homogeneous and initially at rest (density n0 , speed V0 = 0).
We assume a small initial perturbation, denoted again by subscript ”1”. The gradient
of the electron pressure is included in the equation of motion, but we make a simplified
assumption that the ion pressure gradient is zero, justified by their larger inertia. Using
the adiabatic equation of state the pressure gradient ∇pe can be replaced by γp0 n−1
0 ∇ne1
(Exercise 8.1). Hence, the linearized continuity and momentum equations for electrons
and ions are
131
132
CHAPTER 8. WARM PLASMA
∂ne1
+ (∇ · Ve1 )
∂t
∂ni1 +
+ (∇ · Vi1 )
∂t
∂Ve1
n0 me
∂t
∂Vi1
n0 m i
∂t
= 0
(8.1)
= 0
(8.2)
= en0 E1 −
γp0
∇ne1
n0
(8.3)
= en0 E1 ,
(8.4)
e
(ni1 − ne1 ) .
0
(8.5)
respectively. We also need the Gauss law
∇ · E1 = −
Assuming again harmonic time and spatial dependencies, i.e., we are looking for
plane wave solutions, we obtain
−iωne1 + in0 k · Ve1 = 0
(8.6)
−iωni1 + in0 k · Vi1 = 0
(8.7)
−iωme Ve1
−iωmi Vi1
e
ik · E1 = (ni1 − ne1 ) .
0
γp0
= eE1 − i
ne1 k
n0
= eE1
(8.8)
(8.9)
(8.10)
A brief calculation (Excercise 8.2) gives
2
2
ωpi
ωpe
k · E1 ,
1− 2 − 2
ω
ω − k 2 (γkB Te /me )
!
(8.11)
where pe = ne kB Te has been used to introduce the electron temperature. The expression
in the parenthesis is the dielectric function K(ω), in this case a scalar. The zeros of K(ω)
give the dispersion equation.
8.2
Langmuir wave and the ion sound wave
When frequencies are well above the ion plasma frequency (ω ωpi , and hence, the
(ωpi /ω)2 -term in Eq. 8.11) can be neglected) the solution to the warm plasma dispersion
equation is
2
ω 2 = ωpe
+ k 2 (γkB Te /me ) ,
(8.12)
8.2. LANGMUIR WAVE AND THE ION SOUND WAVE
133
i.e., we have again encountered the Langmuir wave. Note that this result can also be
derived assuming ions as an inmobile background (Exercise 8.3). The finite temperature
and associated thermal motions now allow electron plasma oscillation to propagate as a
wave (∂ω/∂k 6= 0). We notice also that the wave number depends on the frequency and
hence the wave is dispersive. The dispersion equation is shown in the left-hand part of
Figure 8.1. But what is the value for the polytropic index γ?
k
k
Figure 8.1: Solutions of the warm plasma dispersion equation. Left) Langmuir wave,
Right) ion sound wave
Considering that the inclusion of the temperature effects introduces only a small
correction to the cold plasma theory, we can assume that the temperature disturbance
propagates less than one wavelength during one plasma oscillation. This corresponds to
the long wave length limit (k 2 λ2De 1), i.e., the approach we used to solve the Vlasov
equation in Chapter 4. Hence, the perturbation is assumed to be adiabatic. Since
homogenous plasma without background fields is one-dimensional, the polytropic index
is γ = (d + 2)/d = 3, where d is the indicates the number of spatial directions, in this
case d = 1. Using the relationship between the Debye length and thermal speed
λ2De =
2
vth,e
2
2 ωpe
(8.13)
we can write the dispersion equation in terms of the Debye length
2
ω 2 = ωpe
(1 + 3k 2 λ2De ) ,
(8.14)
which is the same as Eq. 4.36.
At the long wave length (small wave number) limit we can approximate
q
3
ω = ωpe 1 + 3k 2 λ2De ≈ ωpe (1 + k 2 λ2De ) .
2
(8.15)
When frequencies are well below the electron plasma frequency (ω ωpe , the solution
to the warm plasma dispersion equation gives a new wave mode, the ion sound wave
(Exercise 8.4)
kcs
ω=q
,
(8.16)
1 + k 2 λ2De
134
CHAPTER 8. WARM PLASMA
where we have introduced the ion sound speed
cs =
q
kB Te /mi .
(8.17)
The dispersion equation is plotted in the right-hand part of Figure 8.1. Now the process
has been assumed isothermal (γ = 1), which is justified by the ions oscillating so slowly
that the electron temperature has time to relax over the oscillations. This mode can also
be found from the Vlasov theory, when the solution is investigated from the appropriate
frequency domain.
At the limit of small wave number (k 2 λ2De 1) ω ≈ kcS , yielding the dispersion
equation
ω q
(8.18)
= kB Te /mi .
k
At the limit of large wave numbers we obtain
cs
ω=
=
λDe
s
kB Te
mi
s
n0 e2
=
0 kB Te
s
n0 e 2
= ωpi ,
mi 0
(8.19)
i.e., the wave frequency approaches the ion plasma frequency, see the right-hand part of
Figure 8.1. Thus the ion sound wave has a resonance at the ion plasma frequency.
It is interesting to note that the numerator in the ion sound speed includes the
electron temperature, while in the denominator is the ion mass. Thus the electrons
account for the pressure and ions for the inertia. If ion pressure would be taken into
account Te would be replaced by Te + γTi . However, Vlasov theory indicates that if the
electrons are not clearly warmer than ions, the ion waves are strongly damped. Note
that the ion acoustic wave can also propagate in collisionless plasma because charged
particles interact due to long-range Coulomb forces. Electrons are highly mobile and
they quickly follow the ion motion to preserve the charge neutrality (remember that we
have assumed there to be no magnetic field!).
8.3
On plasma stability
Plasma reacts to a disturbance by starting to oscillate with a characteristic frequency
and wave length. Depending on the situation the oscillations may propagate, and grow
or damp. The growing oscillations can lead to a plasma instability. Instability requires a
source of free energy, and hence, there are no instabilities in the cold plasma theory. In
addition, the elementary approach to plasma physics often assumes an unperturbed state
that is in local thermodynamic equilibrium and particles can be described by Maxwellian
velocity distributions. Neither in that case is there free energy for waves to self-excite.
Free energy may be stored in the magnetic or plasma configuration, for example in the
form of magnetic tension or the relative streaming of plasma populations.
8.3. ON PLASMA STABILITY
135
Instability can be externally driven or result from the changes in plasma distribution
function. If there are no processes that would saturate the instability, the whole plasma
system can explode. This happens both in space and laboratory plasmas. Solar flares and
loss of plasma state in tokamaks are examples of large-scale plasma instabilities. One way
to categorize plasma instabilities is to divide them between microscopic and macroscopic
instabilities. A microscopic instability needs the kinetic approach and it depends on
the shape of the distribution function. A macroscopic instability is a configurational
instability and can be described by macroscopic equations. We consider here only a few
simple examples that can be understood either intuitively or that are straightforward to
calculate.
8.3.1
Z-pinch instability
Let us first investigate the equilibrium configuration of the Z-pinch from Chapter 6 (see
Figure 6.17) where plasma is confined by a toroidal magnetic field. The magnetic field
arises from the electric current that is driven through the plasma. Figure 8.2 displays
what may happen to an initially stable Z-pinch if the system is perturbed.
B2/2m0 decreases
B2/2m0
increases
kink
instability
B2/2m0 decreases
sausage
instability
B2/2m0 increases
Figure 8.2: Kink and sausage instabilities
If the plasma tube is bent, the magnetic pressure will increase on the concave part
of the bend (magnetic flux increases) and decrease on the convex part (flux decreases).
This creates a gradient in magnetic pressure, i.e., magnetic force, that strengthens itself
and the whole plasma can rise up from the equilibrium leading to the loss of the plasma
state. This is called kink instability.
The other instability shown in Figure 8.2 is related to the squeezing of the flux tube.
The magnetic pressure will increase at the part that is being compressed and decreases
in the nearby region. Larger magnetic field gradient tends to increase the compression
136
CHAPTER 8. WARM PLASMA
of the plasma and the whole plasma tube will break, if there is no mechanism to stop
the squeezing. This is called the sausage instability.
A common way to stabilize plasma is to wind a flux tube to a torus and drive an
electric current through the tube. This toroidal current creates a poloidal magnetic
field around the torus. The superposition of toroidan and poloidal magnetic fields leads
to a spiral shaped magnetic field inside the torus. Such device is called tokamak, and
it is nowadays the most common and important plasma confinement device in fusion
experiments (Figure 8.3). The growth rate of both sausage and kink instabilities can be
stabilized in tokamaks, but driving large electric currents through the plasma may cause
kinetic instabilities related to changes in plasma distribution functions.
Figure 8.3: In a tokamak fusion reactor poloidal and toroidal electric currents create an
almost force-free flux tube magnetic field configuration.
8.3.2
Two-stream instability
One way to investigate plasma instabilities is to derive the dispersion equation and
investigate conditions that lead to growing wave perturbations. Let us investigate a
simple example featuring two oppositely directed electron beams with different velocities.
Essentially, this is a kinetic instability, a solution can also be found from macroscopic
theory in the case the velocity difference between the beams is larger than their thermal
motion.
Let the densities of the electron beams be nα0 and nβ0 and the velocities Vα0 and
Vβ0 . Assume that there is no background magnetic field and that ions are a fixed
background, hence restricting the analysis to high frequency waves. We assume again a
small perturbation (allowing linearization) and investigate the plane wave solution. The
8.3. ON PLASMA STABILITY
137
linearized continuity equations for the two beams are
−iωnα1 + ik(nα0 Vα1 + nα1 Vα0 ) = 0
(8.20)
−iωnβ1 + ik(nβ0 Vβ1 + nβ1 Vβ0 ) = 0 .
(8.21)
Assuming that beams are cold their linearized equation of motions become
e
E1
me
e
E1 .
= −
me
−iωVα1 + ikVα0 Vα1 = −
(8.22)
−iωVβ1 + ikVβ0 Vβ1
(8.23)
Combining these with the Gauss law
ikE1 = −
e
(nα1 + nβ1 )
0
(8.24)
gives the dispersion equation
ikE1
2
2
ωpβ
ωpα
1−
−
(ω − kVα0 )2 (ω − kVβ0 )2
!
= 0.
(8.25)
The expression in the parenthesis is again K(ω). Figure 8.4 shows the plot of 1 − K(ω)
and a graphical representation for the roots of K(ω) = 0. Plasma is unstable if dispersion
equation has solutions with a positive imaginary part. Such a two-stream instability
can arise if the beam velocities differ from each other enough but not too much. For
derivation of the dispersion equation of even a simpler case with one-dimensional plasma
where electrons flow with a constant velocity with respect to a stationary ion background
see Exercise 8.5.
two complex
roots (instability)
four real
roots (stable)
two real
roots (stable)
Figure 8.4: Left) The dispersion equation has four real roots. In this case the imaginary
part of the frequency is zero and plasma stable. Right) Two roots are complex. One
of the complex roots has a positive imaginary part leading to growing of the wave
perturbation and instable plasma.
138
8.3.3
CHAPTER 8. WARM PLASMA
On the stability of warm plasma
A general treatment of warm plasma instabilities requires kinetic approach. As discussed
in Section 4.3, the perturbation will grow exponentially if the wave frequency is complex
with the imaginary part being positive (ωi > 0, see Eq. 4.37). Furthermore, the Vlasov
theory states that all monotonically decreasing distribution functions (∂f /∂v < 0) are
stable. A positive slope (∂f /∂v > 0) makes the distribution potentially unstable, but
does not guarantee instability.
unstable
region
Figure 8.5: Example of a distribution that has a postive slope. If the wave moves with
the speed vph , it can be amplified at the expense of the energy of the faster moving
beam.
An example of a distribution with a positive slope is presented in Figure 8.5. It
features a Maxwellian background (stable) with temperature T1 and a warm particle
beam (temperature T2 ) flowing along the z-axis. The number density of the beam is
assumed to be smaller than the background density. Solving the Vlasov equation would
indicate that unstable wave modes are found at the phase speeds that coincide with the
positive derivative of the distribution function. This so-called “gentle bump” instability
requires that the beam has sufficiently high speed when compared to the temperature
of the Maxwellian background. The instability is enhanced if the density of the bump
increases, it becomes colder (distribution narrower), or its speed increases.
Similar instabilities can arise due to relative motion between ion and electrons beams.
Plasma instabilities are a rich and varied field of plasma physics. A thorough treatment
of instabilities requires methods that are beyond the linear plasma theory and even touch
the boundaries of the current knowledge.
8.4. EXERCISES: WARM PLASMA
8.4
139
Exercises: Warm Plasma
1. Show that the equation of state p/ργm = constant leads to the following relation
between first order (small) perturbations
p1 = p0 γ
ρm1
.
ρm0
2. Derive the dispersion equation
1−
2
2
ωpi
ωpe
−
=0
ω2
ω 2 − k 2 (γkB Te /me )
for non-magnetized electron-ion plasma by taking into account electron thermal
effects.
3. Consider isotropic plasma where the background magnetic and electric fields are
zero, the ions are immobile, and the plasma is initially in equilibrium. Assume a
small disturbance in the electron mass density and derive the dispersion equation
for the Langmuir wave
2
ω 2 = ωpe
+ k 2 (γkB Te /me ) .
4. Assume that frequencies are well below the electron plasma frequency (ω ωpe .)
Starting from the general dispersion equation for warm plasma (Exercise 8.2),
derive the dispersion equation for the ion acoustic wave
ω=q
where cs =
p
kcs
1 + k 2 λ2De
,
kB Te /mi is the ion sound speed and λDe the electron Debye length.
5. Consider one-dimensional plasma where electrons flow with speed V0 with respect
to a stationary ion background. Derive the dispersion equation for plasma oscillations in the rest frame of the ions
(ω − kV0 )2 = ωp2 .
140
CHAPTER 8. WARM PLASMA
Chapter 9
Appendix
9.1
Useful vector identities and theorems
Some uselful vector identities are listed below:
A × (B × C) = B(A · C) + C(A · B)
(A × B) × C = B(A · C) + A(B · C)
∇ × ∇f
= 0
∇ · (∇ × A) = 0
∇ · (f A) = (∇f ) · A + f (∇ · A)
∇ × (f A) = (∇f ) × A + f (∇ × A)
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
∇ · (A · B) = (B · ∇)A + (A · ∇)B + B × (∇ × A) + A × (∇ × B)
∇ · (AB) = (A · ∇)B + B(∇ · A)
∇ × (A × B) = (B · ∇)A − (A · ∇)B − B(∇ · A) + A(∇ · B
∇ × (∇ × B) = ∇(∇ · A) − ∇2 A
The divergence theorem relates the volume integral of the divergence of a vector field
A over a volume V to the surface integral of A over the surface S bounding the volume
V:
Z
(∇ · A) dV =
V
I
A · dS .
(9.1)
S
The Stokes theorem relates the surface integral of the curl of a vector field A over a
surface S to the line integral of A over its boundary ∂S:
Z
∇ × A · dS =
S
I
∂S
141
A · dr
(9.2)
142
9.2
CHAPTER 9. APPENDIX
Maxwell equations and useful concepts of electrodynamics
Although plasma is an electromagnetic medium the Maxwell equations are often written
in the “vacuum” form
∇ · E = ρ/0
∇·B = 0
∂B
∇×E = −
∂t
1 ∂E
∇ × B = µ0 J + 2
c ∂t
Gauss’s law
Gauss’s law for magnetism
Faraday’s law
(9.3)
Ampère-Maxwell’s law
Sources of the magnetic and electric fields, i.e. the charge density and the electric
current density, include all charges and currents and they can be determined from plasma
distribution functions (see Chapter 4). We call here B as magnetic field although rigorously speaking B is the magnetic flux density. The magnetic field has the SI units
V s m−2 = T, i.e., tesla. The SI units of the electric field E are V m−1 , of the charge
density ρ A s m−3 = C m−3 and the electric current density J, and A m−2 . The natural
constants appearing in Maxwell equations in SI units are
µ0 = 4π × 10−7 V s A−1 m−1
permeability of free space
√
c
= 1/ 0 µ0 = 299 792 458 m s−1
speed of light in vacuum
0 = (c2 µ0 )−1 ≈ 8.854 × 10−12 A s V−1 m−1 permittivity of free space
In magnetic or dielectric media Maxwell equations are often written using the “auxliarity fields” H, and D that are the magnetic field intensity, and electric displacement. H
and D and related to B and E through the constitutive relations:
D = 0 E + P
(9.4)
H = B/µ0 − M ,
(9.5)
where P is the polarization density and M the magnetization.
In isotropic and linear medium the polarization density is proportional to the electric
field and the magnetization to the magnetic field intensity
P = χe E
M = χm H ,
(9.6)
(9.7)
where χe and χm are electric and magnetic susceptibilities determined by the properties
of the medium. The electric susceptibility measures how easily a dielectric medium is
polarized when an external electric field is applied and the magnetic susceptibility gives
the degree of magnetization of a material in response to an applied magnetic field.
9.3. BASIC CONCEPTS OF WAVE PROPAGATION
143
Now, the constitutive equations become
D = E
(9.8)
H = B/µ ,
(9.9)
where = 0 + χe , and µ = µ0 (χm + 1) are the permittivity and permeability. Note that
the permittivity and permeability are not necessarily constants, for instance, they can
vary with the position, time and frequency. If the medium is anisotropic, permittivity
and permeability are second rank tensors.
Using H and D the Maxwell equations can be written in the form
∇ · D = ρf
(9.10)
∇·B = 0
(9.11)
∇×E = −
∂B
∂t
(9.12)
∂D
.
(9.13)
∂t
The electric displacement D accounts for the effect of free charges ρf (∇ · D = ρf ), while
the sources of polarization P are bound charges (∇ · P = ρb ).
∇ × H = Jf +
The total current J (see the vacuum form of the Ampère-Maxwell law) is the sum
of the current produced by the flow of free charges Jf , and the magnetization and
polarization currents.
The polarization current is related to the change in polarization of the individual
molecules of the dielectric medium
∂P
JP =
.
(9.14)
∂t
Plasma consists of free charges so there is no unique way to determine the polarization.
However, the change in polarization and hence the polarization current are real plasma
phenomena.
The magnetization current is associated to the circulation in the magnetization field
M
JM = ∇ × M
(9.15)
The displacement current 0 ∂E/∂t in the Ampère-Maxwell law arises from the generation of magnetic fields by time-varying electric fields. This term is important as it
allows propagating electromagnetic wave solutions.
9.3
Basic concepts of wave propagation
In a vacuum (ρ = 0, J = 0) Maxwell equations read
∇ × E = −µ0 ∂H/∂t
∇ × H = +0 ∂E/∂t ,
144
CHAPTER 9. APPENDIX
from which one can easily derive the wave equations for E and H
1 ∂2H
c2 ∂t2
1 ∂2E
∇2 E − 2 2
c ∂t
∇2 H −
= 0
= 0.
(9.16)
The solution of the above equations is a wave propagating with the speed of light c.
An important special case is the plane wave represented by a sinusoidal function:
Ex (z, t) = E0 cos(kz − ωt) ,
where
(9.17)
E0
wave amplitude
ω = 2πf angular frequency
k = 2π/λ wave number .
In the vector form sine wave is
E(r, t) = E0 cos(k · r − ωt) ,
(9.18)
where k is the wave vector. It indicates the direction of motion of the planes of constant
phase.
Plane waves are often practical to present using the exponential form
E = E0 ei(k·r−ωt)
B = B0 ei(k·r−ωt) .
(9.19)
If E0 and B0 are constants, the temporal and spatial dependencies of the electric and
magnetic fields are said to be harmonic. In such a case
∇ · E0 ei(k·r−ωt) = ik · E0 ei(k·r−ωt)
i(k·r−ωt)
i(k·r−ωt)
∇ × E0 e
= ik × E0 e
∂
E0 ei(k·r−ωt) = −iωE0 ei(k·r−ωt) ,
∂t
(9.20)
(9.21)
(9.22)
and the Maxwell equations are transformed into an algebraic form
ik · D = ρ
k·B = 0
k × E = ωB
ik × H = Jf − iωD .
(9.23)
In a case of a medium for which ρf = 0, J = 0, and and µ are constants the
following dependence between the wave number and wave frequency is
k=
√
µω =
n
ω,
c
(9.24)
9.3. BASIC CONCEPTS OF WAVE PROPAGATION
145
where we have defined the index of refraction
r
n=
µ
.
0 µ0
(9.25)
The relationship between the angular frequency and the wave number is called the
dispersion equation.
Dispersion equations are a central for studies of waves in plasmas. From a dispersion
relation one can determine characteristics of the wave propagation, for instance, the
group and phase velocities.
The phase velocity is defined as the velocity at which the planes of constant phase
move
ω
vp = n̂ ,
(9.26)
k
where n̂ is the unit wave normal vector. It is perpendicular to the surface of a constant
phase, and thus parallel to the wave vector k and direction of wave propagation. If the
medium is isotropic the wave propagation is in the same direction as the energy flux,
defined in terms of the Poynting flux S = 21 E × H∗ , where the asterisk indicates the
complex conjugate. In an anisotropic medium the electric field may have a component
parallel to the wave vector, and hence, the direction of energy flux is not necessary in
the direction of wave propagation.
Another characteristic velocity associated with the propagation of the wave packet
(Figure 9.1).
Figure 9.1: A wave packet is a superposion of monochromatic waves with different wave
numbers and frequencies.
The group velocity is the velocity at which the envelope of the wave packet propagates
vg = ∇k ω(k)
It is also the speed at which the energy propagates.
(9.27)
146
CHAPTER 9. APPENDIX
If the phase speed depends only on the physical properties of the medium and not on
the frequency of the wave, the medium is non-dispersive. In this case the group speed is
equal to the phase speed and the wave packet maintains its shape as it propagates. In
turn, if the phase speed depends on the frequency, the wave package spreads out as the
waves of different frequencies travel at different speeds.
If the conductivity is finite, the Maxwell equations become
k·E = 0
k·H = 0
k × E = ωµH
ik × H = (σ − iω)E ,
(9.28)
where in the last equation the electric current has been written in terms of the electric
field using Ohm’s law J = σE. From this it is clear that k ⊥ E, k ⊥ H and E ⊥ H.
Such a wave is called transverse.
Choosing the coordinates as k k ez , E k ex ja H k ey gives
kEx = ωµHy
ikHy = −(σ − iω)Ex ,
(9.29)
and the dispertion equation becomes
k 2 = µω 2 + iσµω .
(9.30)
Denoting k = |k|eiα we find
|k| =
α =
q
p
µω 2 ω 2 + σ 2
σ
1
arctan( ) .
2
ω
(9.31)
The electric field is thus
E = E0 ex exp[i(|k|(cos α)z − ωt)] exp[−|k|(sin α)z] .
(9.32)
To obtain a physical solution we need to choose phase α so that sin α > 0, i.e., the wave
is damped (factor e−|k|(sin α)z ) as it propagates through the medium, because we have
not assumed any source of free energy for wave growth.
The phase velocity is now
vp =
ω
ω
=
.
Re(k)
|k| cos α
(9.33)
The distance where the wave is damped by a factor of e is called the skin depth
δ=
1
1
=
.
Im(k)
|k| sin α
(9.34)
9.4. THE MAXWELLIAN DISTRIBUTION
9.4
147
The Maxwellian distribution
The Maxwellian (or Maxwell-Boltzmann) distribution describes how particle velocities
are distributed in a gas. This distribution was originally derived by James Clerk Maxwell
in the 1860s based on certain symmetries of the distribution function. A decade later
Ludwig Boltzmann derived the Maxwelliann distribution both from the kinetic theory
and using the framework of statistical thermodynamics. The Maxwellian distribution is
valid for ideal gases where particles are in thermodynamic equilibrium i.e., all particles
of the same species can be described with the same temperature.
f(vx)
0
vx
Figure 9.2: The Maxwellian velocity distribution.
The one-dimensional Maxwellian velocity distribution (here given in the x-direction,
see Figure 9.2) has a simple Gaussian probability distribution form
m
f (vx ) = n
2πkB T
1/2
mv 2
exp −
2kB T
!
.
(9.35)
Now f (vx ) dvx gives the probability that a particle in a gas has a velocity component vx
in the range vx + dvx . Particle velocities are randomly distributed around zero velocity
and particles have an equal probability to propagate in the +x and −x directions. Each
of the velocity components may be treated independently, and thus, the distribution can
be generalized to three dimensiona by multiplying one-dimensional Maxwellians in each
three directions. The resulting 3-dimensional Maxwellian velocity distribution is
m
f (v) = n
2πkB T
where v =
q
3/2
mv 2
exp −
2kB T
!
,
(9.36)
vx2 + vy2 + vz2 is the magnitude of the velocity vector v.
The spread of the velocities defines the thermal speed
vth =
q
2kB T /m .
(9.37)
The width of the distribution is controlled by the temperature, i.e., the higher the
temperature the broader the Maxwellian distribution.
148
CHAPTER 9. APPENDIX
The Maxwellian velocity distribution does not depend on the direction of v, only on
its magnitude through v 2 . Hence, it is often more convenient to know the probability
of particles to have their speeds in a certain range. In spherical coordinates d3 v can be
written as v 2 sin θdθdφ. The distribution does not depend on the angular coordinates,
and thus, the integration over angular coordinates gives 4π. The Maxwellian speed
distribution thus becomes
3/2
m
f (v) = 4πv n
2πkB T
2
mv 2
exp −
2kB T
!
.
(9.38)
Figure 9.3 shows that the shape of the distribution now differs from Gaussian, it is skewed
towards higher speeds. This is because there are more ways to achieve the higher speeds
when all three directions are considered (consider the how much volume each dv shell
encloses when v increases).
f(v)
vp
<v>
vrms
0
v
Figure 9.3: Maxwellian speed distribution. Three different characteristics speeds are
indicated.
Different types of characteristics speeds can be derived from the Maxwellian speed
distribution. Their relative locations are shown in Figure 9.3.
The most probable speed vp occurs at the highest point of the distribution. To calculate where f (v) has its maximum we set the derivative df (v)/dv to zero
df
m
= 4π
dv
2πkB T
"
mv 2
2v exp −
2k
!
mv 2
mv
exp −
−v
kB T
2k
2
!#
= 0.
This yields
s
vp =
2kB T
.
m
Thus, the thermal speed (velocity spread) equals to the most probable speed.
(9.39)
9.4. THE MAXWELLIAN DISTRIBUTION
149
The mean speed is obtained as the expectation value of f (v):
Z ∞
f (v)v dv .
< v >=
(9.40)
0
Using the result
Z ∞
x3 exp −x2 /a2 = a4 /2
0
leads to
s
< v >=
The root mean speed is defined as vrms =
2
(9.41)
√
< v 2 >, where
Z ∞
< v >=
8kB T
.
πm
f (v)v 2 dv
(9.42)
0
The result
Z ∞
0
√
3 πa5
x exp −x /a =
8
4
2
2
gives
s
vrms =
3kB T
.
m
(9.43)
The thermodynamical equilibrium is typically reached through collisions. Many plasmas are collisionless and are not in thermodynamical equilibrium. Hence, describing such
plasma with a Maxwellian distribution is no always sufficient. However, a Maxwell distribution is often a good starting point. The collision frequency decreases when the
temperature increases (∝ T −3/2 ) and consequently it takes much longer for fast particles to reach a Maxwellian distribution than for slow particles. Kappa distribution (see
Section 4.4) is an example of a case where the slow particle population can be described
with a Maxwellian distribution, while the fast particles form a non-Maxwellian tail.