Introduction to Plasma Physics - Aalto · familiarize the reader with the main concepts and...

Introduction to Plasma Physics

Emilia Kilpua and Hannu Koskinen

HK, 11.12.2015

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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 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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ii 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 iii

9 Appendix 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

iv CONTENTS

Preface

This course is an introduction to basic concepts and methods of plasma physics. Itprovides basics for further studies of laboratory, fusion, space and astrophysical plasmaphenomena. 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 arediscussed: motion of charged particles in the electromagnetic field, kinetic plasma de-scription, macroscopic plasma quantities and equations, magnetohydrodynamics, Alfvenwaves, cold plasma waves, warm plasmas.

Most of these lectures deal with plasma in so high-temperatures that the plasmais practically fully ionized with only a small effect arising from neutral particles. Wewill also limit the discussion to non-relativistic plasmas with temperatures ranging fromabout a few eV to a few hundred keV. Quantum mechanical effects are also neglectedsince the interactions distances are usually much longer than the de Broglie wavelength.

Plasma physics is based on the main fields of classical physics: electrodynamics, me-chanics and statistical physics. These lectures require good understanding of bachelor-level basic physics and solid skills in undergraduate-level mathematical methods ofphysics (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 wego 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 physicslectures. However, as we will soon find out, this question is not trivial and it is difficultto give an exact definition of the plasma state. We refer at this point to a followingpractical description of plasma:

Plasma is quasi-neutral gas with so many free charges that collective electromagneticphenomena are important to its physical behavior.

Two key aspects of plasma can be found from this definition: 1) Due to the presenceof free charges plasma responds strongly to electromagnetic fields, and 2) in plasmacollective long-range interactions dominate. These characteristics of plasma lead to awide variety of interesting phenomena distinct from neutral gases, including collectiveshielding of individual charges, a large variety of new wave modes, and transfer of energyfrom 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 baryonicmatter in the universe is in the plasma state. Thus, plasma physics is a necessary tool inspace physics and in many astrophysical problems. What we mean by “space physics”needs a little explanation. Space physics investigates physical phenomena in space ofwhich 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 solarwind, and the magnetospheres, ionospheres, and upper atmospheres of the Earth and

3

4 CHAPTER 1. INTRODUCTION

other planets. Astrophysical plasma physics includes, in turn, the studies of plasmas andplasma processes farther in the universe, e.g., stars from the Sun-like objects to neutronstars, black hole accretion discs, and the interstellar medium.

Figure 1.1: Examples of plasmas near and far. Top row: Sun captured by ultravioletlight emitted by ionized helium atoms (Courtesy: SOHO/NASA), artist impression ofan 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. Inparticular, technological applications of plasma physics are numerous, including ther-monuclear fusion research, neon-lights, plasma displays, sterilizing of certain medicalproducts, and plasma processing of semiconductors and materials (e.g., etching andwelding). Visible examples of natural plasmas in the near-Earth environment are auro-ras and lightning.

1.2 Brief history of plasma physics

The word “plasma” originates from Greek where it means something molded. Britishscientist Sir William Crookes was the first to appreciate plasma as the fourth state ofmatter. He investigated the conduction of electricity in low pressure gases in electricaldischarge tubes (where air is ionized by applying a high voltage). In 1879 Crookespublished an article “On Radiant Matter” in The Popular Science Monthly where hestated: “So distinct are these phenomena from anything which occurs in air or gas at

https://archive.org/details/OnRadiantMatter

1.2. BRIEF HISTORY OF PLASMA PHYSICS 5

ordinary tension, that we are led to assume that we are here brought face to face withmatter in a fourth state or condition, a condition as far removed from the state of gasas a gas is from a liquid.”.

The term “plasma” was coined a few decades later by American physicists LewiTonks and Irving Langmuir. They conducted one of the first plasma experiments withelectric discharge tubes. These experiments already led to many important discoveriesconcerning the basic properties of plasma, such as the shielding of charge and plasmaoscillations.

A significant part of early plasma physics dealt with space and astronomical phe-nomena. A particular interest for space plasma physics in the early 1900s was radiobroadcasting. Radio waves are reflected from the ionosphere, the partially ionized layerof the upper atmosphere, which enables transfer of waves over long distances. Theefforts to understand radio communication led to the development of the theory howelectromagnetic waves propagate through non-uniform magnetized plasmas.

Figure 1.2: A few great minds of plasma physics. Top: William Crookes (Courtesy: Li-brary of congress), Hannes Alfven (Courtesy: Royal Institute of Technology, Stockholm),Irving Langmuir (Courtesy: IEEE).

Another main branch in early space plasma physics dealt with examining the con-nection between solar activity and the disturbances in the Earth’s magnetic field. Amost remarkable theory in the early 1900s in the field of solar–terrestrial studies wasSidney Chapman’s and Vincenzo Ferraro’s suggestion that magnetospheric storms arecaused when magnetized plasma clouds ejected from the Sun envelop the Earth’s mag-netosphere.

In the 1940s the Swedish scientist Hannes Alfven developed the formalism of mag-netohydrodynamic (MHD) theory. MHD treats plasma as a conductive fluid that cansupport magnetic fields. Over the years MHD has developed to one of the main tools ofplasma physics. Hannes Alfven has probably been the most influential individual in thehistory of plasma physics. His contributions to plasma physics are numerous including


theories describing the behavior of aurora, the radiation belts, now known as Van Allenbelts, the effect of magnetic storms on the Earth’s magnetic field and cosmic electrody-namics. Perhaps one of his best-known ideas is the theory of low-frequency magnetohy-drodynamic waves, now known as Alfven waves, in magnetized plasma. The basic modeof the Alfven wave propagates along magnetic field and it is the fundamental mode totransfer magnetic disturbances in plasma. Hannes Alfven was granted the Nobel prizein 1970 for his work on “fundamental work and discoveries in magnetohydro-dynamicswith fruitful applications in different parts of plasma physics”.

Radio astronomy started to develop in the 1930s when Karl Jansky observed radiowaves coming from the direction of the Milky Way. The Second World War broughtrapid developments in radio- and microwave technologies, which opened a new windowto the Universe at those radio frequencies that penetrate through the atmosphere. Partof the radiation is bremsstrahlung in hot astrophysical plasmas but it soon turned outthat all radio emissions could not be explained in this way. In the mid-1950s VitalyGinzburg argued that radio emissions from, e.g., the Crab nebula, i.e., the remainderof the supernova observed in 1054, must be synchrotron radiation by electrons gyratingin the strong magnetic field of the neutron star. This was an important milestone as itindicated the central role of the magnetic fields in cosmic plasma physics.

Modern plasma physics can be said to have originated after the Second World Warand 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. TheUS satellite Explorer 1 was launched a year later and its sole scientific instrument, aGeiger counter by James Van Allen, discovered radiation belts around the Earth. Thesebelts were later named Van Allen belts. Space exploration quickly expanded from thevicinity of the Earth further out in the heliosphere. The spacecraft have passed-by allplanets 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 evenreached 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 theSun. Numerical simulations have become an integral part in modern space research be-sides observations. The observational network in space physics is relatively sparse andsimulations are needed to fill this gap. In addition, simulations provide new physical in-sight to many space physics problems that can be tested by observations. Understandingof space plasma physics is also necessary for space technology, ranging from designing,manufacturing and testing scientific instruments to developing new propulsion systemsfor faster and more cost-effective space travel and exploration.

Plasma Astrophysics. As said earlier, almost all baryonic matter in the Universeis in plasma state. While the Sun is often considered to belong to the topics of spaceplasma physics, it is also a very typical star. The Sun is entirely in the plasma state anddetailed understanding of its plasma physics, e.g., the dynamo process creating its cycli-cally varying magnetic field, can readily be transferred to studies of other magneticallyactive 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 ofcosmic rays in astrophysical shock waves, emission of electromagnetic radiation fromradio frequencies to X- and gamma rays. It is clear that also in astrophysical contextsimulations using the most powerful computers today have become an essential tool.

Controlled Fusion Research. The early fusion experiments were conducted alreadyin 1930s but it was only after the Second World War when the interest in fusion researchreally sparked. The development of nuclear fission weapons raised interest also in fusionweapon technologies. It was proposed that fusion reaction could be controlled to makean effective reactor. Since then fusion research has quickly expanded as an importantinternational enterprise with several large experimental facilities being constructed withthe goal to develop a relatively clean and abundant energy source. The most prominentcurrent effort is the International Thermonuclear Experimental Reactor (ITER), whichis being constructed near Cadarache in the Southern France. When finished, ITERwill be the world’s largest tokamak nuclear fusion reactor. Much of the fusion researchnowadays is involved in studying how extremely hot plasma can be stabilized for longenough to attain sustained effective fusion and the tokamak geometry is the currentlyfavored approach in the large-scale devices. Another approach to controlled fusion is tocreate 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 scalesvary by many orders of magnitude. Plasmas exhibit diverse characteristics, their tem-peratures, densities and ionization degree can differ greatly as well as the importanceof collisions and electromagnetic forces to the behavior of plasma. Thus, different levelsof description are used to tackle different types of problems (see Figure 1.3). Differ-ent approaches can also provide alternative insights to understanding a given plasmaphenomenon.

Although collective behavior is a fundamental property of plasma, single particledescription (or an exact microphysical description) is the first step in understanding ofthe processes occurring in plasma. It is often a necessary approach, for example, whenstudying cosmic rays or energetic particles in the Van Allen radiation belts. In thisapproach 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., intime varying and curved magnetic fields) approximations or direct numerical calculationsare needed.

The next step is the kinetic theory. It is a statistical approach to average out in-dividual particle orbits and treat the motion of a large number of particles in form ofa distribution function. However, the detailed knowledge the particle distribution asa function of location and velocity is needed and in this sense kinetic theory is still


Figure 1.3: Levels of plasma descriptions.

microscopic. The core of the kinetic treatment is to determine the velocity distribu-tion functions and their evolution for each plasma species. From velocity distributionfunctions one can calculate macroscopic plasma variables, such as the bulk speed, tem-perature and density. The kinetic approach can deal with non-Maxwellian distributionsand 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 andspace. The evolution of these parameters are determined by means of macroscopic fluidapproach, the equations of which are analogous to the equations of hydrodynamics.However, the effects of electromagnetic fields on the charge particles and often differentbehavior of electrons and ions in a plasma make plasma fluid equations more complexthan hydrodynamic equations. In fluid description the velocity distributions of eachspecies 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 andcan be used to describe many plasma physical phenomena. Due to simplicity and com-putational effectiveness it is one of the main tools for global numerical simulations.Sometimes, a combination of different approaches are used. For examples, in hybridsimulations electrons can be described as a fluid and ions either as individual particlesor 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 thenext natural step from solid to liquid to gas, when the temperature is increased (Figure2.1). For example, when ice is heated its crystalline bonds are broken and it changes towater (liquid state). If more heat is added the molecular bindings break first, followedby independent H2O 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 intoelectrons and positive ions. At some point the fraction of the atoms that are ionizedbecomes large enough that the collective electromagnetic forces take over the behaviorof 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) andquark–quark bonds (energies > 175 MeV) resulting in quark-gluon plasma. Such anexotic plasma state dominated the universe just after the Big Bang and may exist in thecore of neutron stars. Experiments on CERN’s Large Hadron Collider are studying theproperties of quark–gluon plasma. However, this is beyond the scope of normal plasmaphysics 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 toa plasma, but a rough guideline is that already 0.1% degree of ionization typically givesclear plasma properties and 1% ionization means almost perfect conductivity. Thus,

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10 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS

plasma state is achieved after a remarkably small fraction of ionization. Partially ionizedplasmas 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, forexample, the solar wind, solar corona and magnetosphere. At the end of this Chapterwe will provide a more quantitative definition of a plasma state.

The degree of ionization for a gas in thermal equilibrium can be calculated fromSaha’s equation:

ninn

= 3× 1027T 3/2

niexp(−U/T ) , (2.1)

where ni is the ion number density ([ni] = m−3), nn the neutral number density, Ttemperature ([T ] = eV) and U the ionization energy ([U ] = eV, 1 eV ≈ 11604 K) , i.e. theenergy that is required to remove the outermost electron from the atom. From Saha’sequation 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 ofionization and recombination. This means that either the ionization source must becontinuous and strong enough, or the recombination rate must be low.

Contemplate: Why is the degree of ionization in Eq. 2.1 inversely proportional toion density? Using the literature find out the principle behind the derivation of Saha’sequation and determne the units of the factor 3 × 1027 (it is not dimensionless!). Youcan also try to calculate its value.

Apart from heating, ionization can be achieved by applying large local electric fieldsor by exposing the matter to ionizing radiation such as strong laser light, ultravioletlight, or X-rays. In fact, it is possible to produce plasma even from solid state. Anexample of low-temperature plasma sustained by solar EUV light and energetic particleprecipitation is the Earth’s ionosphere. Also the solar photosphere, that is the layerfrom which most of the solar irradiation emerges, is at a temperature of less than 6000 Kthat is well below the ionization energy of the photospheric gas. In that case the sourceof 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 overallplasma is quasi-neutral. This means that the positive and negative charges must haveapproximately equal charge densities:∑

s

ρqs =∑s

nsqs = 0 , (2.2)

where ρqs is the charge density ([ρqs] = C m−3), ns the number density ([ns] = m−3) andqs the electric charge of the species s. For plasma consisting of electrons and one singlycharged 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 elementarycharge (e = 1.6022 × 10−19 C). We see from Eq. 2.3 that if we in this case requirequasi-neutrality, electron and ion number densities must be equal, i.e., ne = ni.

A significant fraction of “free” electrons makes plasma electrically conductive. Infact, plasma is typically an exceptionally good conductor. When temperatures are highand densities low, collisions are rare, and thus, the resistivity is very small. If an electricfield is introduced in plasma, electrons quickly rearrange themselves and the electricfield is neutralized. As a consequence, no significant large-scale electric field can exist inthe rest frame of the unmagnetized plasma. The ability of plasma to shield out appliedelectric fields is one of its fundamental characteristics.

Contemplate: While the resistivity of plasma can be negligible, plasma is not a su-perconductor. Why?

Although plasma is neutral in large scales, deviations from charge neutrality candevelop in shorter scales. Let’ us now look more quantitatively a distance over whichquasi-neutrality is true. Suppose that a positive point charge qT is introduced into anotherwise quasi-neutral plasma. The “bare” Coulomb potential of the test charge isqT /4πε0r, where r is the distance from qT . Negative electrons are attracted to qT andthey 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 thethickness of the neutralizing electron cloud. The electrostatic potential φ can be derivedfrom the Poisson equation:

∇ ·E = −∇2φ = −ρtot(x)/ε0 , (2.4)

where ε0 is the permittivity of free space (≈ 8.8542× 10−12 A s V−1 m−1) and the chargedensity ρtot is composed from the contribution of qT (ρT ) and the polarization of thequasi-neutral plasma as a response to qT (ρpol):

ρtot(x) = ρT δ(x− xT ) + ρpol(x) . (2.5)


Here δ is Dirac’s delta function (recall its properties!). It ensures that the charge densityρT vanishes outside x = xT .

We assume that the plasma is sufficiently close to the thermal equilibrium so thatits density can be given by the Boltzmann distirbution:

ns = n0s exp (− qsφ

kBTs) , (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 beunbound. This means that we must require that the random thermal energy must bemuch greater than the average electrostatic energy. Thus, we can assume that qsφ kBTs and expand Eq. 2.6 as:

ns ' n0s(1−qsφ

kBTs+

1

2

q2sφ2

k2BT2s

+ ...) . (2.7)

The polarization charge density now becomes:

ρpol =∑s

nsqs ≈∑s

ns0qs0 −∑s

n0sq2s

kBTsφ = −

∑s

n0sq2s

kBTsφ , (2.8)

where∑sns0qs0 = 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 =qT

4πε0rexp (− r

λD) . (2.9)

The factor λD in Eq. 2.9 is called the Debye length:

λ2D = ε0∑s

kBTsn0sq2s

. (2.10)

When ions are much colder than electrons, the ion term can be dropped from the defi-nition of the Debye length.

Figure 2.3 shows how the shielded and “bare” Coulomb potentials of qT differ fromeach other. When the distance from qT is much smaller than λD the Coulomb potentialis 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 lengthis the distance over which significant charge separations (and electric fields) can occurin plasma. Intuitively, Debye length is the limit beyond which the thermal speed ofparticles 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 potentialand the shielded potential for two differentDebye lengths compared (λD1 > λD2).

Contemplate: How does the Debye-length change with density and temperature? Tryto give a physical explanation for this behavior

Table 2.2 shows some typical values for Debye length. Note that while in manyapplications/domains (tokamak, Earth’s ionosphere, solar corona) the charged regionsdo not exceed one millimeter, in some space plasmas (e.g., solar wind) Debye length canhave macroscopic values. See Exercise 2.4 for comparing typical sizes of the spacecraftand the Debye length of the medium they are measuring. Exercise 2.5 investigates theform 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 Debye length [m]

Solar core 10−11

Gas discharge tube 10−4

Tokamak 10−4

Ionosphere 10−3

Solar wind 10Interstellar medium 10Intergalactic medium 105

Using the Debye length we can formulate a more quantitative criterion for the ionizedgas to be in a plasma state. First, to guarantee the quasi-neutrality the plasma systemhas to have a size L of several Debye lengths:

λD L . (2.11)


Otherwise, significant charge separations can arise and the plasma becomes dominatedby the boundary effects.

Second, in order for the Debye length to be a statistically valid concept there haveto be enough particles within the Debye sphere. The number of particles in the Debyesphere is given by

ND = n× 4

3πλ3D (2.12)

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 chargedparticles dominate over binary interactions.

In many cases a parameter omitting the factor 4π/3 is used:

Λ = n0λ3D . (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 kineticenergy is larger than the Coulomb potential energy. This was an essential assumptionin the derivation of the Debye length.

2.3 Plasma frequency

Let’s now investigate the dynamic response of plasma to a small perturbation. Imaginethat a fraction of electrons are slightly displaced with respect to ions (Figure 2.4). Thecharge separation gives rise to an electric field that tries to restore the plasma quasi-neutrality. As a consequence, electrons are accelerated by the electric field back towardstheir original positions. Due to their inertia the electrons will overshoot and start tooscillate around the equilibrium position with a specific frequency. Electron oscillationsconvert continuously electrostatic energy to kinetic energy and back again keeping thetotal energy conserved. This kind of electron plasma oscillations were first observed byIrving Langmuir and Levy Tonks in a low pressure discharge tube filled with mercury va-por. 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 smallelectric field E1. We make the cold plasma approximation, i.e. assume there is nothermal motion. Ions are so heavy that they are practically unaffected, and we canconsider them as a fixed background. E1 is caused by a small perturbation n1 in theelectron density:

http://www.columbia.edu/~mem4/ap6101/

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∂t

+∇ · (neu) = 0 , (2.17)

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 orderterms (i.e. multiples of two small perturbation quantities, here n1u1) we get

∂n1∂t

+ n0∇ · u1 = 0 . (2.18)

The electric field causes a force F = qE1, and thus the equation of motion forelectrons is:

me∂u1

∂t= −eE1 , (2.19)

and E1 is determined from the Gauss law:

∇ ·E1 = −en1ε0

. (2.20)

Taking ∂/∂t of Eq 2.18 and using Eq 2.20 we obtain

∂2n1∂t2

+ (n0e

2

ε0me)n1 = 0 . (2.21)

Eq. 2.21 is the equation for a standing wave with the angular frequency called theelectron plasma frequency :

ω2pe =

n0e2

ε0me(2.22)


Plasma frequency gives the most fundamental time scale in plasma. Usually the termrefers to the electron plasma frequency. Ions are affected by the same electric field aselectrons, but due to their much larger mass their oscillation is much slower than theelectron oscillation (justifies our assumption of the fixed ion background). If the coldplasma approximation is relaxed (i.e. non-zero temperatures), the oscillation propagatesas 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 electricand magnetic forces is critical to understanding the behavior of plasma. A chargedparticle in a plasma moves under the influence of the Lorentz force and thus its equationof motion is:

mdv

dt= F = q(E + v ×B), (2.23)

where E is the electric field, B the magnetic field and v the velocity of the chargedparticle. Note that the electric and magnetic fields that are used to calculate the Lorentzforce arise from all particles in the plasma and include also external (applied) electricand magnetic fields. Thus, it is clear that calculation of the motion of a large numberof plasma particles is an immense problem.

Figure 2.5: Left: The electric field of the Lorentz force accelerates positive and negativecharges to opposite directions. Right: The magnetic part of the Lorentz force changesthe path of the particle. The direction of the bending can be inferred applying theright-hand rule to the vector product.

The electric field in the Lorentz force accelerates positive and negative charges inopposite directions (left-hand part of Figure 2.5). The magnetic part of the Lorentzforce 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 onthe charge (you can see this easily by calculating the power v ·F). It is often stated thattime-varying magnetic field is used to accelerate particles, but in fact, it is the inducedelectric 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 chargedparticles perform a circular motion about the magnetic field lines. The angular frequencyof this motion for species α is

ωcα =qαB

mα. (2.24)

The corresponding frequency is fcα = ωcα/2π. This “Larmor motion” is investigated indetail in Chapter 3. It is seen from above that due to their lower mass electrons spinmuch faster around the magnetic field than ions. Numerically for electrons and protons

fce (Hz) ≈ 28×B (nT)fcp (Hz) ≈ 1.5× 10−2 ·B (nT)

.

It is important for plasma physics that charged particles can move relatively freelyalong the magnetic field lines, but their motion perpendicular to the magnetic field ismuch more restricted. As a consequence, the magnetic field “binds” plasma particlestogether.

The Lorentz force per unit volume acting on charge density ρq and electric currentdensity J = ρqv is given by

f = ρqE + 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 accel-eration or heating of the particles, or vice versa. Which way the energy is transformedcan be found by considering the conservation law of the electromagnetic energy knownas the Poynting theorem.

Recall from your electrodynamics course that the energy densities of electric andmagnetic fields are given by

wE =1

2E ·D (2.26)

wM =1

2H ·B , (2.27)

where D = εE is the electric displacement field and H = B/µ the magnetic field in-tensity. ε is the permittivity and µ the permeability of the medium in consideration(in vacuum ε0 and µ0). Define the Poynting vector as S = E × H. From Maxwell’sequations it is an easy exercise to to derive

∇ · S = −E · J−E · ∂D

∂t−H · ∂B

∂t. (2.28)

The Poynting theorem is the integral of this expression over volume V

−∫V

J ·E d3r =

∫V∇ · S d3r +

∫V

∂

∂t(wE + wM ) d3r . (2.29)


The left-hand side is the work performed by the electromagnetic field per unit time (i.e.,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 ofelectromagnetic energy density. The last term on the right-hand side expresses the rateof change of the electromagnetic energy in volume V.

Assuming that there is no energy flux through the surface, the Poynting theoremstates that if J ·E > 0 the energy of the electromagnetic field in the volume V decreasesin 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 theinequality 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 wayparticles interact with each other. In a neutral gas particles interact primarily throughdirect binary collisions, where individual collisions lead to large deflections and can beconsidered “strong”. In plasma charged particles interact predominantly through theCoulomb force. These “Coulomb collisions” are long-range and each charged particleinteracts simultaneously with a large number of charged particles. The Coulomb collisionsare “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 thevery high temperature of the fully ionized plasmas makes them in many cases effectivelycollisionless.

Collisional plasmas can be divided into partially and fully ionized plasmas. Importantregions of fully ionized plasmas where collisions are frequent are the stellar interiors.Examples of partially ionized plasmas are stellar photospheres and the ionized layersof the upper atmospheres, the ionospheres, of planets. In weakly ionized plasma thedominant type of collisions are those between charged particles and neutral atoms andmolecules. When the ionization degree increases the Coulomb collisions between chargedparticles become dominant.

Often high temperature and tenuous plasmas are practically collisionless. Physicallythis 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 windthe effective mean free path between collisions is of the order the distance from the Sunto the Earth. It is important to understand that plasma being collisionless does not meanthat electromagnetic interactions between plasma particles would become negligible. Onthe 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 insteadof individual collisions.

2.5.1 Collisions with neutral particles

Charged particles interact with neutrals through direct collisions. A key quantity todetermine how often collisions occur is the effective cross-section σc that expresses thelikelihood of the interaction between two particles. For binary collisions σc is simplygiven as πd20, i.e. it is the cross-section of a neutral atom or molecule, whose radius isd0. If nn is the number density of neutral particles and 〈v〉 is the average speed of theparticles, we can calculate the collision frequency :

νn = nnπd20 〈v〉 (2.30)

and the average mean free path:

lmfp =〈v〉νn

=1

nnπd20. (2.31)

For binary collisions νn gives the frequency between individual collisions and the lmfpthe 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, orinelastic in which case the kinetic energy of a colliding particle is transferred to internalenergy of the neutral particle or molecular ion of the plasma. Inelastic collisions canthus lead to recombination, excitation, ionization, and charge exchange, which all areimportant processes in space plasmas. Auroras are examples of the excitation process ofneutral molecules in the Earth’s atmosphere due to precipitating high-energy electrons(Figure 2.7). Important charge-exchange processes are collisions where a high-energyproton collides with a slow atom. As a result of the charge-exchange a high-energyneutral atom (ENA) and a low-energy ion are formed:

p+fast + Hslow → HENA + p+

p+fast + Oslow → HENA + O+ . (2.32)

Examples of the first of these are the interactions of interstrellar hydrogen with solarwind protons and inner magnetospheric protons with the hydrogen geocorona aroundthe Earth. The latter process is important when the solar wind interacts with theatmospheres of unmagnetized planets Venus and Mars. ENA imaging provides a useful


Figure 2.7: Left) Precipitating electrons can excite the molecules into a higher energystate. Auroras form when exited atoms emit photons. Different colors of auroras dependon the energy levels of molecules present in the atmospheric layer the electrons can pen-etrate. Image Courtesy: Jouni Jussila. Right) Large ribbon of ENA emission detectedby 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 InterstellarBoundary Explorer (IBEX) spacecraft. The ribbon was found in 2009 and its origin haskept scientists puzzled. Currently, it is though that the ribbon is a reflection of particlesbouncing off a galactic magnetic field.

In plasmas collisions between neutrals and charged particles cannot happen too fre-quently. Otherwise, the behavior of the substance would be controlled by ordinaryhydrodynamic forces than by electromagnetic forces. If ω is the plasma frequency andτc 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 becauseeach charged plasma particle interacts with many far-away charges simultaneously, whilecloser encounters where the path would deflect significantly are much less common (seeExercise 2.7 to demonstrate this for a fully ionized plasma). However, the cumulativeeffect of many small Coulomb collisions can deviate the path significantly. For Coulombcollisions the collision frequency (νc) is the measure of the frequency with which theparticle trajectory is deviated by 90 due to many successive Coulomb interactions andlmfp is the distance traveled until such a deflection has accumulated. Here we give onlya very rough quantitative inspection of Coulomb collisions in plasma.

2.5. COLLECTIVE BEHAVIOR AND COLLISIONS 21

ion (at rest)

dc gc

electron

Figure 2.8: Coulomb collisions arelong-range interactions betweencharged particles and in plasmathey result typically only to asmall deflection in the particlepath.

Figure 2.8 shows an electron that is interacting with a positive ion. Since ions aremuch more massive than electrons we can assume that the ion is at rest. The path ofthe electron is a hyperbola, which far from the ion can be approximated by two straightlines at an angle γc.

The distance dc in Figure 2.8 is called the impact parameter and it describes theclosest approach distance between the electron and the ion. To estimate dc we investigatethe Coulomb force on the electron

Fc = − e2

4πε0d2c. (2.34)

The electron feels this force over the time τ = dc/ve, during which its momentum changesby the amount τ |Fc|:

|4(meve)| ≈e2

4πε0vedc. (2.35)

For large angle collisions (γc ' 90) this will be of the same order of magnitude as thetotal momentum of the electron meve. Thus, we get:

dc ≈e2

4πε0mev2e, (2.36)

and the approximation for the cross-section is:

σc = πd2c ≈e4

16πε20m2e 〈ve〉

4 , (2.37)

where ve has been replaced with the average speed 〈ve〉. The collision frequency becomes:

νei = neσc 〈ve〉 ≈nee

4

16πε20m2e 〈ve〉

3 . (2.38)

The average speed of the electrons can be replaced by their average thermal energykBTe = me 〈ve〉2 /2. Rewriting this using the plasma frequency we get:


νei ≈√

2

64π

ω4pe

ne

(kBTeme

)−3/2. (2.39)

Because ω4pe ∝ n2e, the collision frequency is proportional to the density and inversely

proportional to T 2/3.

This equation is only a rough approximation since most of the collisions are smallangle collisions. A correction for small angle deflections (not derived here) using theplasma parameter Λ = neλ

3D gives:

νei ≈ωpe32π

ln Λ

Λ. (2.40)

and the electron mean free path is:

lmfp =〈ve〉νe

=2ωpeλDνe

≈ 64πλDΛ

ln Λ. (2.41)

Contemplate: Why does the collision frequency increase with the decreasing temper-ature?

Electrons also collide with each other in plasma. For electron-electron collisions theCoulomb force is repulsive, and thus the colliding electron is deflected away from thetarget. Electron–electron collisions are more complex to deal with than electron–ioncollisions since we cannot assume anymore that the scattering electron is at rest. Sincethe Coulomb force is of the same order of magnitude, the deflection will be about thesame amount as for electron–ion collisions. In addition, ions collide with electrons andother ions. Due to much larger mass of an ion their momentum gain or loss in notsignificant when they interact with an electron. We can approximate:

νee ≈ νei (2.42)

νie ≈ (me/mi)νee

νii ≈√me/miνee

Contemplate: Explain why the electron collision frequency is much larger than theion–electron collision frequency.

2.6. PLASMA CONDUCTIVITY 23

2.6 Plasma conductivity

Because collisions change the momentum of the particles, they introduce a term corre-sponding to friction in the equation of motion (Eq. 2.23):

mdv

dt= q(E + v ×B)−mνc(v − u) , (2.43)

where νc is the collision frequency, irrespective whether the electron collisions occurbetween 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 whereall collision targets are at rest We can also assume that all electrons have the samevelocity ve, i.e. the plasma is cold (remember that the temperature arises from thevelocity 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 thecollisions balance each other, i.e., dv/dt = 0. Now the solution to Eq. 2.43 is:

E = −meνce

ve . (2.44)

Electrons carry the current density:

J = −eneve , (2.45)

Inserting Eq. 2.45 into Eq. 2.44 we obtain the relationship between the electriccurrent density and the electric field, i.e., Ohm’s law :

J =nee

2

meνcE (2.46)

and the conductivity is

σ =nee

2

meνc. (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 becold. What does this mean? We will return to this question when we discuss waves inthe cold plasma approxiamtion in Chapter 7.

For fully ionized plasma νc is the electron-ion collision frequency given by Eq. 2.40.


This yields the Spitzer resistivity :

η =me

nee2ωpe32π

ln Λ

Λ. (2.48)

Using the definitions of ωpe and λD one can write this as:

η =1

32πε20

e2m1/2e

(kBT )3/2ln Λ . (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 independentof the number of current carriers (electrons) as long as there are enough of them, whichis not always the case in low-density plasmas as we discuss in the next section in thecontext of magnetized plasma. A simple way to heat plasma is to pass an electric currentthrough it. However, according to Spitzer’s formula resistivity is inversely proportionalto the temperature. This means that when temperature increases, resistivity drops fastand plasma becomes such a good conductor that the Ohmic heating is not effectiveanymore.

Contemplate: Why does Spitzer’s resistivity not significantly depend on the electrondensity? 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 quan-tity because charged particles move in different ways perpendicular and parallel to themagnetic field. The equation of motion in a steady state situation can now be writtenas:

0 = −nee(E0 + ve ×B0)− νcmeneve . (2.50)

By computing the current as in Eq. 2.45 we can rewrite this equation as:

J = σ0E−σ0nee

J×B . (2.51)

To calculate J choose B = Bez. Using the electron gyrofrequency ωce (Eq. 2.24) wecan write:

Jx = σ0Ex −ωcνcJy

Jy = σ0Ey +ωcνcJx (2.52)

Jz = σ0Ez .

2.6. PLASMA CONDUCTIVITY 25

And thus:

Jx =ν2c

ν2c + ω2ce

σ0Ex −ωcνc

ν2c + ω2ce

σ0Ey

Jy =ν2c

ν2c + ω2ce

σ0Ey +ωcνc

ν2c + ω2ce

σ0Ex (2.53)

Jz = σ0Ez .

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):

−→σ =

σP −σH 0σH σP 00 0 σ‖

. (2.55)

The elements of the conductivity tensor, assuming for simplicity only one ion population,are given by:

σP =ν2c

ν2c + ω2ce

σ0

σH =ωceνc

ν2c + ω2ce

σ0 (2.56)

σ‖ = σ0 =nee

2

meνc.

The elements of the conductivity tensor depend both on the collision and gyro fre-quencies. σP is known as the Pedersen conductivity. It gives the conductivity in thedirection of the electric field E⊥ perpendicular to the magnetic field. The Hall conduc-tivity σH is the conductivity perpendicular to both the ambient magnetic and electricfields. The magnetic field-aligned conductivity σ‖ is the same as the classical collisionalconductivity in the absence of magnetic field. In collisionless plasmas it is typically sev-eral orders of magnitude larger than the perpendicular conductivities, meaning that theelectrons can quickly rearrange to cancel any electric field parallel to B and the electricfield in a plasma is typically perpendicular to the magnetic field. However, if there arenot enough current carriers available, finite E‖ can arise to accelerate the electrons toa 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 au-roras with potential drops of several kilovolts, which corresponds to the energy of theelectrons causing main auroral light.


2.7 Plasma definition: A summary

As a summary we will gather together three conditions that a gas must satisfy to be ina plasma state:

• Collective interactions dominate over binary interactions: There has to be enoughcharged particles within a Debye sphere, neλD 1

• Plasma is quasi-neutral: The size of the plasma system L has to be larger thanthe Debye’s length, λD L.

• Neutral collision frequency must be smaller than the collective inertial responsefrequency in plasma (i.e., the plasma frequency): ωτc > 1 (electromagnetic forcesdominate).

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 givenabove. 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

ninn

= 3× 1027T 3/2n−1i exp(−U/T ) ,

where ni is the number density of ions and nn of neutral atoms, T temperaturein eV, and U the ionization energy. Assume that the dominating species in theionosphere is O+ and their density 1011m−3 and temperature 0.3 eV. The ioniza-tion 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 0.5eV. You will notice that the ionization degree increases rapidly as a function oftemperature!

2. Derive the formula

ϕ =qT

4πε0rexp

(− r

λD

)for the screened potential qT of a test charge in a plasma with Boltzmann’s densitydistribution: nα(r) = nα0 exp(−qαϕ(r)/kBTα).Some hints: 1) Use e−x ' 1 − x when substituting the densities into Coloumb’slaw and make use of quasi-neutrality. 2) Make also use of spherical symmetry towrite

∇2ϕ =1

r2d

dr

(r2dϕ

dr

).

2.8. EXERCISES: BASIC DEFINITIONS AND PARAMETERS 27

After solving the differential equation require that the solution approaches theCoulomb 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 spacebetween 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 theplates is

φ =nq

2ε0(d2 − x2) .

(b) Show that for d > λD, the energy needed to transport a particle from a plateto 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 windand magnetosphere? (You can look this up on the Web.) Do spacecraft disturbthe medium they are trying to measure? What would you expect should happento a spacecraft passing through plasma?

5. A spherical conductor of radius a is immersed in a plasma and charged to a po-tential φ0. The electrons remain Maxwellian and move to form a Debye shield,but the ions are stationary during the time frame of the experiment. Assumingφ0e kBTe, derive an expression for the potential as a function of r in terms ofa, φ0 and λD. (Hint: Assume a solution of the form (exp(br))/r).)

6. Prove that g2/3 (g is the inverse of the plasma parameter Λ) is proportional tothe ratio of the average Coulomb potential energy between two electrons and theaverage kinetic energy of electrons.

Note: Since the plasma condition requires that Λ 1 this means that in plasma theaverage kinetic energy between electrons is much larger than the average potentialenergy. The result is intuitive since electrons are“free” in a plasma. The kineticenergy being much larger than the potential energy is also a central assumptionwhen deriving the Debye length, see Exercise 2.2.

7. Show that in a fully ionized plasma the frequency of small-angle collisions is muchlarger than the frequency of large-angle collisions. What plasma parameter givesthe 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 forceand we do not usually need to calculate the (very) complicated collision integral.

8. Derive the elements of the conductivity tensor

σ =

σP −σH 0σH σP 00 0 σ‖


starting from the equation of motion

E + ve ×B = −meνce

ve .

Sketch the electric field components in the plane perpendicular to the magneticfield.

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 envi-ronments.

10. Consider a spherical container of 1.5 m radius filled with completely ionized hy-drogen gas. Ions are are assumed cold.

(a) The electron density in the container is set to 4 × 1023 m−3, i.e., couple ofpercent of atmospheric number density on ground level. What should thetemperature be (upper and lower limits) so that the gas would behave like aplasma?

(b) If the temperature is set to 27C, what should be the electron density so thatthe plasma conditions are fulfilled?

(c) Let us mix some neutral particles to the gas, when the temperature is set to27C. The collision frequency nun between charged particles and neutrals is

< νn >= nnσn < 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 muchshould neutral gas be added so that the gas dynamics would be controlled bycollisions instead of collective electromagnetic interactions?

Explain also why in a) and b) the gas ceases to be plasma if the temperature anddensity are lower/higher than the obtained limits. It is helpful to revise how theDebye 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 influ-ence 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 col-lective plasma phenomena can be understood (even quantitatively) in terms of singleparticle motion. After all, plasma behaviour is based, ultimately, on the motions ofits constituent particles. Single particle description is a very useful approach in studiesof high energy particles in low density plasma where collisions are infrequent and theexternal magnetic and electric fields are much stronger than the fields generated by themotion of charged the particles themselves.

Our task in this chapter is to solve the equation of motion for a charged particle:

mdv

dt= q(E + v ×B) + Fnon−EM , (3.1)

where Fnon−EM govern non-electromagnetic forces.

We will start by investigating how a charged particle moves in the simplest magneticfield configuration, the static and uniform magnetic field in the absence of an electricfield. Then we will proceed to examine more complicated magnetic and electric fields, inparticular, to determine how the motion of charged particles is affected by spatial andtemporal 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, andthe magnetic field B is constant. Now the equation of motion (Eq. 3.1) of a chargedparticle is:

mdv

dt= q(v ×B) . (3.2)

29

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:

mdv

dt· v =

d

dt(mv2

2) = 0, (3.3)

which shows that the kinetic energy and the speed are both constant for a particle in astatic magnetic field.

Next we determine the trajectory of the particle. Let us choose the coordinate systemso that B = Bez. The components of Eq. 3.2 are:

mvx = qBvy

mvy = −qBvx (3.4)

mvz = 0 .

The velocity component parallel to the magnetic field (vz) is constant, i.e., the particlemoves at a constant speed along the magnetic field. This is because the v × B-forcehas no component parallel to the magnetic field. We concluded above (Eq. 3.3) thatthe total speed of the particle is also a constant, and hence, the absolute value of thevelocity perpendicular to the magnetic field (v⊥) must also be constant.

Taking the second time derivatives of the perpendicular velocity components gives:

vx = −ω2cvx

vy = −ω2cvy . (3.5)

Eq. 3.5 describes a simple harmonic oscillator at the Larmor (or cyclotron, or gyro)frequency :

ωc =qB

m. (3.6)

As the Larmor frequency is inversely proportional to the mass of the particle, elec-trons gyrate much faster than ions. For an electron ωc = 1.76×1011B, while for a protonωc = 9.58×107B, where the unit of the gyro frequency is rad s−1 and the magnetic fieldis given in Teslas.

Solving the spatial coordinates of the particle (Exercise 3.1) we see that it performsa 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

where v⊥ =√v2x + v2y is the particle velocity perpendicular to the magnetic field. The

Larmor radius is proportional to the mass of the particle and inversely proportional tothe magnetic field magnitude. Hence, electrons have much smaller Larmor radii thanions (see Exercise 3.2 for comparing Larmor frequencies and radii for particles in differentplasma 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 ofthis circular motion is called the guiding center (GC).

Figure 3.1: The trajectory of a charged particle in space in a homogeneous magneticfield is a helix.

Combining these motions we will see that the particle is gliding along the magneticfield while making a circle in the plane perpendicular to the magnetic field (i.e., its GCfollows 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:

tanα = v⊥/v‖ , (3.8)

and thusα = arcsin(v⊥/v) = arccos(v‖/v) . (3.9)

In many applications it is convenient to omit the relatively fast circular motion ofthe particle around the magnetic field as the main interest is to investigate the motionof the GC. Hannes Alfven was the first to introduce the idea of this guiding centerapproximation.

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 spaceand in time when compared to the Larmor motion:

rL/L 1 (3.10)

B−1∂B/∂t ωc , (3.11)


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 sensearound the magnetic field. We see from Eq. 3.7 that the Larmor radius is inverselyproportional to magnetic field magnitude (rL ∝ B−1), i.e., the stronger the magneticfield, the more tightly particles are “bound” to the magnetic field. We also see thatsince 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 thations lose more easily the guidance of the magnetic field than electrons.

Figure 3.2: Larmor orbits of positive ions and negativeelectrons in a magnetic field.

In the GCS gyrating charges form small current loops that are associated with themagnetic moment :

µ = πr2LI =1

2

q2r2LB

m=

1

2

mv2⊥B

=W⊥B

. (3.12)

The directions of the gyro motion of positive and negative charges is always such thatthe 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).

Figure 3.3: Charged particles which gyrate in amagnetized plasma form small current loops. Theassociated magnetic moment µ is always oppositethe magnetic field B.

I

µ

+

B

2LA rp=

Lr

3.2. MOTION IN CONSTANT PERPENDICULAR ELECTRIC ANDMAGNETIC FIELDS33

In vector form this is:µ =

q

2rL × v⊥ . (3.13)

When a large number of particles is present, the magnetization M is defined as themagnetic moment per unit volume:

M = −∑s

ns〈µs〉b , (3.14)

where < µs > is the average magnetic moment of the particle species s,

Magnetization contributes to the current density in plasma. Any circulation in themagnetization field gives rise to a magnetization current density that can be calculatedas:

JM = ∇×M . (3.15)

Note that the magnetization current density is often distinguished from the “true currentdensity” due to the motion of “free charges” in the medium.

3.2 Motion in constant perpendicular electric and mag-netic fields

Next we consider the particle motion in spatially uniform, static electric and magneticfields. The equation of motion 3.1 is now:

mdv

dt= q(E + v ×B) . (3.16)

The component of the equation along the magnetic field is:

mv‖ = qE‖ , (3.17)

where v‖ and E‖ are the velocity and electric field components parallel to the magneticfield. This equation describes acceleration along the magnetic field at a constant rate.

Due to high conductivity in the plasma free charges react quickly to qE‖. As aconsequence, the electric field component parallel to the magnetic field is typically closeto zero in plasma. However, as discussed in Section 2.6.2 “double layers” (i.e., sustainedparallel electric fields) can arise in some situations, e.g., if there are not enough chargecarriers to maintain the continuity of the electric current.

Let us then investigate the perpendicular component of Eq. 3.16.

mv⊥ = q(E⊥ + v⊥ ×B) . (3.18)

By choosing the frame-of-reference appropriately we can eliminate the electric fieldfrom Eq. 3.18 (remember that when conductor (e.g., plasma) moves in a magnetic


field, the observed electric field depends on the frame-of-reference). We can use thenon-relativisic Lorentz transformation (i.e., setting γ = 1):

E′

= E + vE ×B

B′

= B. (3.19)

Here E is the electric field in the non-moving frame (e.g., the spacecraft frame), E′

isthe electric field in the moving frame, which we take to be the plasma rest frame, andvE is the velocity of the frame-of-reference moving perpendicular to E and B (i.e., thedrift velocity of the GC). Since the electric field must vanish in the plasma rest-frameE

′= 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:

mdv

′⊥

dt= qv

′⊥ ×B , (3.21)

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 Larmormotion around the magnetic field and the drift of the GC at the velocity vE perpendicularto the magnetic and electric fields. The E × B-drift of electrons and ions is shown inFigure 3.4 in the special case when parallel velocity v‖ = 0. If the particle has a velocitycomponent parallel to the magnetic field the GC glides across the magnetic field and theactual path of the particle in 3-D space is a slanted helix. For a detailed derivation ofthe trajectory of a charged particle in constant and perpendicular electric and magneticfields see Exercise 3.5.

electron

ion

Figure 3.4: E × B-drift for electrons and ions in constant electric and magnetic fieldsfor the parallel velocity v‖ = 0.

It is important that the E ×B-drift speed does not depend on the mass, charge orthe velocity of the particle. This means that when many particles are present the whole

3.3. GENERAL DRIFT VELOCITY DUE TOA FORCE PERPENDICULAR TOMAGNETIC FIELD35

plasma drifts at the same speed. Hence, no electric current arises from the E×B-driftbecause there is no relative drift of electrons and positive ions. E×B-drift is also veryslow when compared to the particle gyro motion around the magnetic field. In Exercise3.6 the E × B-drift speed is calculated for an electron in the auroral ionosphere andcompared to its Larmor motion.

Contemplate: For a more complete physical picture consider the energy gain and lossof a particle during its gyro orbit around B under the influence of constant perpendicularE. This also explains why the drift is at the constant speed, although the effect of theforce F = qE is either accelerating or decelerating. See also Exercise 3.5.

3.3 General drift velocity due to a force perpendicular tomagnetic field

The particle motion in constant electric and magnetic fields can be generalized to driftsdue to a general constant external force perpendicular to the magnetic field:

dv⊥dt

=q

m(v⊥ ×B) +

F⊥m

. (3.22)

Assuming that the GC drift vD is caused by the force F⊥ we can make the transformationv⊥ = v′⊥ + vD:

dv′⊥dt

=q

m(v′⊥ ×B) +

q

m(vD ×B) +

F⊥m

. (3.23)

In the GCS two last terms have to cancel each other (see the previous section) and:

vD =F⊥ ×B

qB2. (3.24)

Note that this treatment assumes that F/qB c (otherwise the GC approximationis no longer valid).

A common example is the gravitational force F = mg, which causes the gravitationaldrift :

vg =m

q

g ×B

B2. (3.25)

Note that now vg depends on the sign of the particle’s charge. Hence the ions andelectrons will drift in the opposite directions and a net current density is produced. Thephysical reason for the gravitational drift is in the change in the Larmor radius as ionsand electrons gain and lose energy as they move in the gravitational field (i.e., due to


changes in v⊥). In most cases of interest the magnitude of the gravitational drift isnegligible to other GC drifts, but it has importance in some regions e.g., in the Earth’sionosphere and in the solar atmosphere.

It is also interesting to note that the gravitational drift is, in the same way as theE×B-drift, not in the direction of the gravitational force, but perpendicular to it. Figure3.5 illustrates a consequence of these two drifts by demonstrating what happens whena 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 electronsand ions creating a small electric field. Figure 3.5 shows that the direction of the electricfield is such that the associated E × B-drift is upwards where the layer has alreadymoved upward due to a ripple, and downward where the layer is already downward. Asa consequence, the ripple grows and result in an instability known as the Rayleigh-Taylorinstability .

vg (ions)

+++

---

-

++++

B plasma

E

E‰B

E‰B

E‰B

E--

E--

Fg=mg

+

vacuum

Figure 3.5: Illustration of how gravitational and electric drifts lead together to theRayleigh-Taylor instability.

We have now investigated particle motion in uniform magnetic and electric fieldsand found expressions for the GC drifts. In the following sections we will investigate theGC drifts in inhomogeneous fields, i.e., we allow either electric field or magnetic fieldto vary in space or in time. Now the equation of motion becomes too difficult to solveanalytically. We will generally assume that the changes in the fields are small whencompared 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 im-portant, as the response of plasma determines the properties of electromagnetic wavepropagation.

3.4. PARTICLE MOTION IN NON-UNIFORM ELECTRIC FIELDS 37

3.4.1 Spatially varying electric field

We now assume that the magnetic field is uniform and the electric field in the x-directionvarying sinusoidally in the y-direction:

E = E0 cos(ky)ex . (3.26)

Such an electric field can arise for example due to a wave motion. The equation ofmotion for the charged particle is now:

mdv

dt= q[E + v ×B] (3.27)

If electric field is weak we can use the undisturbed orbit to estimate Ex and average overa gyro cycle. We also consider the case of a small Larmor radius, krL 1. The resultis a small correction to the E×B-drift (Exercise 3.7):

vE =E×B

B2

(1− 1

4k2r2L

). (3.28)

The physical reason for this is that a charged particle with its GC near the maximumelectric 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 derivativeof E, and we can generalize:

vE =

(1− 1

4r2L∇2

)E×B

B2. (3.29)

Note that since electrons and ions have different Larmor radii, this drift is chargedependent.

3.4.2 Time varying electric fields

Let us next assume that the electric field is uniform in space, but it varies sinusoidallyin time:

E = E0 exp (iωt)ex . (3.30)

If changes are slow when compared to the particles gyro period (∂/∂t ωc) we findthe polarization drift (Exercise 3.8):

vP = − m

qB2

dvEdt×B =

1

ωcB

dE⊥dt

. (3.31)


The polarization drift separates particles with different charges and masses and givesrise to a polarization current (JP = nqvP ). Due to the large mass ratio between electronsand ions this current is carried mostly by the ions.

It is instructive to contemplate the physics behind the polarization drift. Assumethat 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 startsto feel the Lorentz force qv × B, and consequently starts to E × B-drift. If E willvary sinusoidally in time as assumed above, E will be reversed after some time and theparticle starts to E×B-drift in the opposite direction.

In the advanced space plasma physics course we will investigate also the cases whenthe rate of change in the electric field is at the same order as the gyro frequency of theparticle (ω ∼ ωc) and high-frequency electric fields (ω ωc). In the former case it isfound 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 directionof 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 themagnetic field vary in space. The basic motion is again the gyro motion, but nowthe particle experiences small field variations as it gyrates. As is seen from Figure 3.6the GC drift arises because the Larmor radius varies due to changes in the magneticfield magnitude in different regions. It is also obvious from the figure that the drift isperpendicular to both B and ∇B.

We assume that the magnetic field is only weakly inhomogeneous and can thus usethree-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 driftingparticle) can be calculated as an average over one gyro radius, by using the orbit in thehomogeneous 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 magnitudevaries 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:

dv‖dt

= − µm∇‖B . (3.34)

Perpendicular to the magnetic field one obtains from Eq. 3.24 the gradient drift :

vG =µ

qB2B× (∇B) =

W⊥qB3

B× (∇B) . (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 curvedwith a constant radius of curvature RC (positive inward). Now a drift arises from thecentrifugal force FC felt by the particle as it moves in the magnetic field:


FC = −mw2

‖RC

n , (3.36)

where w‖ is the particle’s speed along the magnetic field. In practice a sufficient accuracyis w‖ ' v‖. n is the unit vector in the direction of RC . FC is again perpendicular to themagnetic field and can be inserted to Eq. 3.24. We obtain:

vC = −mv2‖q

n×B

RCB2. (3.37)

Figure 3.7: A charged particle in a curved magnetic field.The radius of curvature is RC .

Now we need to express n in terms of B. If ds is a small displacement along themagnetic field, from Figure 3.7 we find that ds = RCdφ and db = ndφ. Dividing thesegives:

db

ds=

n

RC. (3.38)

Since d/ds denotes the derivative along the magnetic field, it can be replaced with (b·∇).Thus:

dB

ds= (b · ∇)B . (3.39)

We now obtain the expression for the curvature drift :

vC =mv2‖qB4

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 theparallel energy (explain why).

Assuming that there are no local currents (∇×B = 0), we can write Eq. 3.40 in asimilar form as the equation for the gradient drift:


vC =mv2‖qB3

B×∇B . (3.41)

Now it is possible to combine vG ja vC :

vGC =W⊥ + 2W‖

qB3B×∇B =

W

qBRC(1 + cos2 α)n× t , (3.42)

where t ‖ B ja n ‖ RC are unit vectors, and α is the pitch angle.

Contemplate: Investigate the differences between the E × B-drift and the combinedgradient and curvature drift (Eq. 3.42). For example, contemplate whether these driftsaffect primarily low or high-energy particles, do they give rise to an electric current andcan 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 themagnetic field cannot do work on a charged particles, but the electric field induced bya 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 writtenas 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⊥:

B

E

t

¶Ñ´ = -

¶

BE

Figure 3.8: A charged particle is accelerated by an electric field thatis induced by a time-varying magnetic field. If the time variations areslow when compared with the particle’s gyro motion, the magneticmoment of the particle stays constant.

dW⊥dt

= q(E · v⊥) . (3.43)

During one gyration the particle gains energy

4W⊥ = q

∫ 2π/ωc

0E · v⊥ dt . (3.44)

Assuming slow temporal changes we can replace the time integral by a line integral overa closed loop and use the Stokes law:

4W⊥ = q

∮C

E · dl = q

∫S

(∇×E) · dS = −q∫S

∂B

∂t· dS , (3.45)


where dS = n dS, n is the normal vector of the surface with the direction defined by thepositive circulation of the loop C. For small variations of the field ∂B/∂t→ ωc4B/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 Larmorradius of the particle is not changed significantly during one Larmor orbit. Thus, weobtain:

4W⊥ =1

2|q|ωcr2L4B = µ4B . (3.46)

On the other hand

4W⊥ = µ4B +B4µ (3.47)

and thus 4µ = 0. Hence, in a slowly time varying magnetic field the magnetic momentµ is conserved although the inductive electric field accelerates the particle throughoutits 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 mechanicsthe quantity related to a (nearly) periodic motion that stays constant when the systemchanges slowly, temporally or spatially, is called an adibatic invariant.

In Hamiltonian mechanics it is shown that if q and p are the canonical coordinateand momentum of the system and the motion is nearly periodic, the closed integral of pover one period in q

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 isthe magnetic vector potential and the canonical variables related to the motion perpen-dicular to the magnetic field are p⊥ ja rL. Using the Stokes theorem and assuming thatthe magnetic field and the particle’s perpendicular velocity do not change significantlyduring one Larmor gyration we obtain:

I =

∮p⊥ · drL =

∮mv⊥ · drL + q

∫S

(∇×A) · dS

=

∫ 2πrL

0mv⊥dl + q

∫S

B · dS (3.49)

= 2πmv⊥rL − |q|Bπr2L =2πm

|q|µ ,

i.e., the magnetic moment µ is an adiabatic invariant.


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,

Φ = Bπr2L =2πm

q2µ, (3.50)

is constant. When the magnetic field increases, the Larmor radius decreases and conse-quently the enclosed flux stays constant.

mirrorpoint, Bm

B

strong Bweak B

referencepoint, B0

Figure 3.9: A charged particle moving towards stronger magnetic field.

Now let us investigate a charged particle that is moving towards a stronger andstronger magnetic field (Figure 3.9). As the magnetic field increases the Larmor radiusof the particle gets smaller and smaller and its gyro frequency increases. We assume thatthe magnetic field changes so slowly that µ stays constant. Since µ = W⊥/B, it is clearthat to keep µ constant the perpendicular energy (W⊥), and hence the perpendicularvelocity (v⊥), of the particle increase with the increasing magnetic field. Because in theGC approximation the total kinetic energy is conserved, the parallel energy W‖ and theparallel velocity v‖ must decrease. See Exercise 3.10 for a demonstration how the parallelvelocity of a particle varies in a simple magnetic bottle configuration. W⊥ can increaseuntil W‖ → 0. In Section 3.5.1 we discussed that the magnetic field whose magnitudevaries in space causes the force F = −µ∇‖B. In this context this force is called themirror force and it slows down the GC motion and finally when all parallel energy hasvanished, it turns the particle around, in other words, the particle gets “mirrored”. Thephysical 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 magneticbottle is composed of two magnetic mirrors (not necessary of equal strengths) placedfacing each other. A charged particle can be trapped within the bottle. Magnetic bottleshave been used in various laboratory experiments to confine plasma (Figure 3.10), andthey are also found in natural plasmas. For example, the Earth’s dipole field forms a


huge magnetic bottle where charged particles bounce between the mirror points in thenorthern and southern hemisphere (Figure 3.11, see also Exercise 3.11). The high energyparticles trapped in this bottle form the Van Allen radiation belts. Another example ofa natural magnetic bottle is a solar coronal loop.

Figure 3.10: Two magnetic mirrors facing each other form a magnetic bottle. In practicalplasma confinement experiments much more complicated coil geometries are used toimprove the confinement conditions.

trapped particle

magnetic field line

mirror point

electronsions

North

South

Figure 3.11: The Earth’s dipole field forms a large magnetic bottle. The electronsgradient and curvature drift eastward and ions westward carrying a net westward currentaround the Earth.

For many practical purposes it is interesting to know which particles are mirroredand remain trapped in the magnetic bottle and which can escape from it (Exercise3.12). If a particle has too much parallel energy with respect to the maximum magneticfield magnitude in the converging field, it will escape from the bottle. For example, if aparticle 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 sinα.


Now the magnetic moment can be expressed as:

µ =mv2 sin2 α

2B. (3.51)

µ is assumed to be constant and on the other hand v2 ∝ W is also a constant. Thus,we find a relation between the pitch angle and the magnetic field magnitude at twolocations:

sin2 α1

sin2 α2=B1

B2. (3.52)

From the definition of the pitch angle we see that at the mirror point α → 90, asW‖ → 0. If B2 in Eq. 3.52 is the mirror field Bm (see Figure 3.9) then sinα2 = 1.Therefore, the strength of the mirror field Bm depends on the particle’s pitch angle atthe 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 mirrorfields, the particle will be trapped in the bottle if:

arcsin

√B0

Bm≤ α0 ≤ 180− arcsin

√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 particlebounces back and forth between the magnetic mirror, the bounce motion is nearly peri-odic. The bounce period τb is obtained from the formula:

τb = 2

∫ s′m

sm

ds

v‖(s)=

2

v

∫ s′m

sm

ds

(1−B(s)/Bm)1/2, (3.55)

where s is the arc length along the GC orbit and sm and s′m are the coordinates of the

mirror points. Note that the bounce period is defined over the whole bounce motionback and forth. The GC approximation is valid if τb ω−1c . Thus, the condition toconsider the bounce motion as nearly periodic is more restrictive than in the case ofLarmor motion:

τbdB/dt

B 1 . (3.56)

If this condition is fulfilled, there is an associated adiabatic invariant, which in plasmaphysics is called the second adiabatic invariant

J =

∮p‖ds , (3.57)

where p‖ = mv‖ for a non-relativistic particle.


To directly prove the invariance of J in a general case is a formidable task. Thecomplete 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 suffi-ciently symmetric, as e.g., in the quasi-dipolar planetary magnetic fields.

The corresponding third adiabatic invariant is the magnetic flux through the closedcontour defined by the GC drift:

Φ =

∮A · ds , (3.58)

where A is the vector potential of the magnetic field and ds is the arc element along thedrift path of the GC. The drift period τd has to fulfill τd τb τL. The invariant isweaker than µ and J because much slower changes in the field can break the invarianceof Φ.

In the Earth’s magnetosphere µ is often a good invariant. J is invariant for particlesthat spend at least some time in the magnetic bottle defined by the nearly dipolar field ofthe 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 mag-netic field.

The table below and Figure 3.12 present all three adiabatic invariants and associatedmotions.


Invariant Speed Time-Scale Validity

magnetic gyro motion gyro periodmoment µ v⊥ τL = 2π/ωc τ τL

longitudinal- parallel velocity bounce period τ τb τLinvariant J of GC w‖ τb and µ constant

flux invariant perpendicular velocity drift period τ τd τb τLΦ of GC w⊥ τd and µ and J constant

Every invariant has its characteristics energy conversion mechanism. First, let’sconsider a particle drifting across the magnetic field from the field B1 towards B2 withB2 > 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⊥2W⊥1

=B2

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∮ds to decrease. To compensate this, v‖, and thus, W‖ must increase. This is called

Fermi acceleration.

Enrico Fermi introduced this mechanism to explain the acceleration of cosmic raysto very high energies (107 − 1010 eV) in the magnetic fields of the universe. A typicalgalactic cosmic ray has wandered around in the galaxy for millions of years. The radiusof the Milky Way is of the order of 100 000 light years, and thus the particle has hada lot of time to “collide” with magnetic field structures in the galaxy that have a widerange of velocities. Note that in a given reference frame (e.g., ours) the particle eithergains or loses energy when it gets deflected by a magnetic structure (e.g., a mirror). As aresult, the velocity distribution of the seed population widens and finally some particlesend up at very high energies.

The modern version of Fermi acceleration, believed to be responsible for the acceler-ation of galactic cosmic rays, no longer relies on the conservation of the second adiabaticinvariant in a distribution of moving magnetic mirrors. Instead, particles are assumed tobe accelerated in shock waves generated in supernova explosions by a mechanism calleddiffusive shock acceleration. In this model, particles gain energy by repeatedly crossinga 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 staysconstant (i.e., third adiabatic invariant), the total energy can change if the drift shells are


compressed or expanded. As a summary, the characteristic energy conversion mechanismfor each invariant are

µ: W⊥ changes when Larmor radius (that is, |B|) changesJ : W‖ 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 complicatedthat numerical integration of the equation of motion is required. In this section we brieflydiscuss the motion in two simple but in practice very important geometries: the dipolefield 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 applicationof the orbit theory. Within the distances 2–7RE from the Earth’s center the dipoleis a reasonably good approximation of the geomagnetic field and all particles excepthigh-energy cosmic rays behave adiabatically as long as their orbits are not disturbedby collisions or time-varying electromagnetic fields.

In the following we use “geomagnetically” defined spherical coordinates. The dipolemoment ME is in the origin and points toward the south. Latitude (λ) is zero atthe equator and increases toward the north. Longitude (φ) increases toward the eastfrom a given reference longitude. The SI unit of ME is A m2. ME is often replacedby k0 = µ0ME/4π, which is also referred to as the dipole moment. The strength andorientation of the terrestrial dipole moment varies slowly and must be taken into accountin time scales of space climate. For our purposes sufficiently accurate approximationsare

ME = 8× 1022 A m2

k0 = 8× 1015 Wb m (SI : Wb = T m2)= 8× 1025 G cm3 (Gaussian units, G = 10−4 T)= 0.3 GR3

E (RE ' 6370 km)

The last (non-SI) expression is useful in practice because the dipole field on the surfaceof 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 shrunkinto a point at the origin. The source of a planetary or stellar magnetic field is actuallya finite, even large, region within the body giving rise to a whole sequence of highermultipoles. When moving away from the source the non-dipolar (quadrupole, octupole,etc.) contributions vanish faster than the dipole. Outside the source the field is apotential field (B = −∇Ψ). The potential for the dipole is

Ψ = −k0 · ∇1

r= −k0

sinλ

r2. (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

r3[3(k0 · er)er − k0] , (3.61)

from which

Br = −2k0r3

sinλ

Bλ =k0r3

cosλ (3.62)

Bφ = 0 .

The magnitude of the magnetic field is

B =k0r3

(1 + 3 sin2 λ)1/2 (3.63)

and the equation for the field line is

r = r0 cos2 λ , (3.64)

where r0 is the distance from the dipole to the point where the field line crosses thedipole equator. In dipole calculations we also need the length of the line element

ds = (dr2 + r2dλ2)1/2 = r0 cosλ(1 + 3 sin2 λ)1/2dλ . (3.65)

The geometric factor (1 + 3 sin2 λ)1/2 = (4 − 3 cos2 λ)1/2 appears frequently in dipoleexpressions.

Every dipole field line is uniquely determined by its (constant) longitude φ0 andthe distance r0. A useful quantity is the L parameter L = r0/RE . For a given L thecorresponding field line reaches the surface of the Earth at the latitude

λe = arccos1√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 =k0r30

(1 + 3 sin2 λ)1/2

cos6 λ. (3.67)

For the Earth, we findk0r30

=0.3

L3G =

3× 10−5

L3T . (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 considerabledeviations from this because the dipole is not quite in the center of the Earth, thesource is not a point, and the conductivity of the Earth is not uniform.


The guiding center approximation can be applied if the particle’s Larmor radius ismuch smaller than the curvature radius of the field defined by RC = |d2r/ds2|−1, whichfor a static dipole field is

RC(λ) =r03

cosλ(1 + 3 sin2 λ)3/2

2− cos2 λ. (3.69)

In terms of the particle’s rigidity mv⊥/|q|, we write

rL

∣∣∣∣∇⊥BB∣∣∣∣ =

mv⊥|q|RCB

∝ mv⊥|q|r0B

, (3.70)

and thus, the GC approximation is valid if

mv⊥|q| r0B . (3.71)

Contemplate: Rigidity is a widely used concept in cosmic ray studies. Considertwo otherwise identical cosmic ray particles, but with different momenta. Which one isaffected 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 bottlearound the Earth or magnetized planets are said to form trapped radiation. Let λm bethe mirror latitude of a trapped particle and let the subscript 0 refer to the equatorialplane. Then the equatorial pitch angle of the particle is

sin2 α0 =B0

B(λm)=

cos6 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 ifλe < λm, the particle hits the Earth before mirroring and is lost from the bottle. Inreality the loss takes place in the upper atmosphere at an altitude that depends on theparticle’s energy, i.e., on how far it can penetrate before it is lost by collisions. Thecritical 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 afunction of latitude see Exercise 3.15.

The conservation of the second adiabatic invariant requires that the bounce periodis much shorter than the variations in the magnetic field. For example, in the innermagnetosphere the bounce times of 1-keV electrons are a few seconds and of 1-keVprotons a few minutes. During magnetospheric disturbances typical time scales of thefield changes are minutes. Thus under such conditions J is a good invariant for electronsbut 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 tothe east.

Because ∇×B = 0, we find for vGC

vGC =W

qBRC(1 + cos2 α) (3.74)

=3mv2r20

2qk0

cos5 λ(1 + sin2 λ)

(1 + 3 sin2 λ)2

[2− sin2 α0

(1 + 3 sin2 λ)1/2

cos6 λ

].

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 neara current sheet. When two regions of oppositely directed magnetic fields are broughttogether, a sheet of current must arise according to Ampere’s law (∇ × B = µ0J) toaccount for the change in the magnetic field. An example of a current sheet in spaceplasmas is the tail current sheet in the Earth’s extended magnetotail, where a currentarises to separate the oppositely directed magnetic fields in the southern and northern taillobes. An even larger-scale current sheet is the “heliospheric current sheet” that extendsto the whole heliosphere and separates the opposite magnetic fields in the southern andnorthern 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 windand the Earth’s magnetosphere, and in fusion experiments.

Previously in this section we have generally assumed that the Larmor radius of acharged particle that gyrates in the magnetic field is small compared to the lengthscale of the field gradients. This assumption allowed us to use the GC approximationand Taylor expansion around the GC to estimate the particle’s orbit. However, neara thin current sheet there can be large field gradients over short spatial distances, andhence, neither the GC approximation nor the invariance of µ are no longer valid. Manyphysically interesting phenomena (instabilities, magnetic reconnection) arise near strongand thin current sheets where particle motion becomes chaotic and non-adiabatic.

The simplest model to describe the magnetotail current sheet is the Harris model forone- and two-dimensional configurations (Figure 3.13). In the two-dimensional Harriscurrent sheet the magnetic field is of the form

B = B0 tanh

(z

L

)ex +Bnez , (3.75)

where B0 and Bn are constant, Bn B0 and L is the characteristic thickness of thecurrent sheet. If Bn = 0, the field is one-dimensional. The magnetic field magnitude


changes from a constant value far away from the current sheet (B0) as a hyberbolictangent accross the sheet. The electric current points toward the positive y-axis and is

Jy =

(B0

µ0L

)sech2

(z

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 currentsheet the motion is normal Larmor motion, but in the vicinity of the current sheet themotion is more complicated. The monotonic motion in the ±y-direction is called Speisermotion. 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 Harriscurrent 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 changesfrom the current sheet outwards. (In the Advanced Space Plasma Physics course we willalso look the particle motion more quantitatively).

3.7. EXERCISE: SINGLE PARTICLE MOTION 53

3.7 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, derivesolution for the spatial coordinates of the particle in detail. Study how positivelyand negatively charged particles rotate in the magnetic field according to thissolution.

2. Derive numerical scaling formulas for the gyro frequency ωg and gyro radius rgusing electron mass, electron charge and km/s as scaling units (i.e., express particlemass, charge and velocity in these units) and calculate them (assuming v‖ = 0) for

(a) a 10-keV electron in the Earths magnetic field with B = 500 nT and plasmaelectron 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 correspondingplasmas.

3. A charged particle rotating in a magnetic field generates an electric current. Cal-culate the generated magnetic moment and show that the magnetic moment vec-tor antiparallel to the magnetic field vector B for both positively and negativelycharged 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 thatit follows a cycloid orbit:

x(t) =E0

B0

(t− 1

ωgsin(ωgt)

)

y(t) =E0

ωgB0

q

|q|(1− cos(ωgt))

Draw the orbits for a positive and a negative particle. Prove that the averagekinetic energy of the drift motion equals the average potential energy drop of theparticle in the electric field during half a cycloid orbit.


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. Assumethat the electric field is perpendicular to the magnetic field and has a magnitudeof 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 = B0ez and in-homogeneous electric field E = E0 cos(ky) ex, here B0 and E0 are constant andkrL << 1. Show that the drift speed is

vE =E×B

B2(1− k2r2L/4).

Give a physical interpretation for krL << 1 and compare its validity and driftspeeds for electrons and protons.

8. Starting from the equation of motion derive the polarization drift

vP =1

ωcB

dE⊥dt

.

9. Derive the forceF = −µ∇B .

on the guiding center of a charge (with magnetic moment µ) moving in an inho-mogeneous but straight magnetic field B. (Feel free to use text books in plasmaphysics.)

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, showthat a particle (mass m), which is mirroring between points −zm and zm, has alongitudinal velocity

v‖ =

√2µB0

m

√(zma0

)2

−(z

a0

)2

,

where µ is the magnetic moment. What is the particle velocity a the centre ofthe 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 particlesbounce between the two mirror points in the Earth’s dipole field. Pay attention tothe gyro motion around the mirror points. How does the gyro radius change alongthe orbit?

12. A group of charged particles with an isotropic velocity distribution is placed in amagnetic 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. Whatfraction of the particles is trapped? (Hint: Try to figure out what fraction of spacethe loss cone fills.)

3.7. EXERCISE: SINGLE PARTICLE MOTION 55

13. A proton with 1-keV kinetic energy and v‖ = 0 in a uniform magnetic field B =0.1 T is accelerated adiabatically as B is slowly increased to 1 T. The proton thenmakes an elastic collision with a heavy particle and changes direction so thatv⊥ = v‖. The magnetic field is then adiabatically decreased back to 0.1 T. Whatis the proton’s kinetic energy now?

14. Starting from the expressions for the components of the magnetic dipole fieldBr, Bλ and Bφ make a detailed derivation of the expression for the particle pitchangle at the equatorial plane as a function of its mirror latitude B(λm)

sin2 α0 =B0

B(λm)=

cos6 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 fieldline (from the equator towards Earth). Draw the loss cone size as a function oflatitude for L = 6. Study how the loss cone size varies between the equator andhigher latitudes and as the function of L.

16. Calculate the gradient-drift velocity of a 90-pitch angle particle in a dipolar mag-netic field. Start with equations

vD =F⊥ ×B

qB2

and F = −µ∇B. Before you start writing down equations, think very carefullywhat the pitch-angle assumption means.

Chapter 4

Kinetic Plasma Description

As the first step to understand plasma we studied how individual particles behave inelectric and magnetic fields. However, the definition of plasma requires that there hasto be a large number of particles within a Debye sphere and that the plasma system hasthe size of several Debye lengths (Eq. 2.2). In plasma charged particles that move inthe applied electric and magnetic fields generate their own fields. The computation ofthe motion of all plasma particles from Maxwell’s equations and the Lorentz force wouldbe an immense task. The kinetic plasma approach has its roots in statistical physics,representing the behaviour of a large collection of particles using distribution functionsin configuration and velocity space. In a case of plasma one needs to include Maxwell’sequations in the formulation of the theory.

Kinetic theory is one of the most challenging areas of plasma physics. Here we intro-duce the most central concepts only. Fortunately, the fluid description (see Section 1.3and Chapter 6) is sufficiently comprehensive to describe a large part of observed plasmaphenomena. However, the fluid approach loses the detailed information on distributionof the plasma constituents in the velocity space. For instance, the kinetic description isrelevant in situations where significant deviations from the local thermodynamic equi-librium arise and plasma particle species are non-Maxwellian. In particular in hot andtenuous plasmas there are not enough collisions to drive plasma towards Maxwelliandistribution. Kinetic theory must also be used when one considers phenomena occurringat short spatial (smaller than Debye length or Larmor radius) or temporal (faster thangyro or plasma frequency scales) scales. In addition, description of kinetic processes,such as instabilities and wave-particle interactions, require the knowledge of the velocityspace effects.

4.1 Distribution function

The dynamical state of a particle in a plasma at time t can be described by its position:

r = xex + yey + zez (4.1)

57

58 CHAPTER 4. KINETIC PLASMA DESCRIPTION

and velocity:v = vxex + vyey + vzez . (4.2)

Combining information of the particle’s position and velocity gives its location (r,v) ina 6-dimensional phase space (Figure 4.1). The infinidesimal volume element of phasespace is d3r d3v.

Figure 4.1: 6-dimensional phase space r

v

d3r

d3v

(r,v)

In statistical physics the single particle distribution function f(r,v, t) expresses thenumber density in a 6-dimensional phase space element at time t. Hence, f is a functionof seven independent variables. The units of number density in the configuration spaceis m−3 and in the velocity space is (m s−1)−3, thus the units of the distribution functionare [f ] =m−6 s3.

An example of a domain where kinetic processes prevail is the solar wind. The solarwind has low density (on average 5 cm−3), it is collisionless and is composed of differentparticle species. For examples, wave-particle interactions play an important role in thesolar wind and temperature anisotropies drive kinetic instabilities. Although large-scalesolar wind variations can be understood in terms of single-fluid approach, micro-scaleprocesses may affect the local solar wind properties. Figure 4.2 illustrates typical velocitydistribution functions (VDF) in the solar wind.

The distribution function needs to be normalized. The most intuitive normalizationis to require that the integration of the distribution function over the 6-dimensionalphase space volume V gives the total particle number N .∫

Vf(r,v, t) d3rd3v = N . (4.3)

The average density in spatial volume V is 〈n〉 = N/V . However, the density canusually vary with space and time, and thus, the particle number density is defined asthe zero order velocity moment of the distribution function

n(r, t) =

∫f(r,v, t) d3v . (4.4)

Note that in statistical and mathematical physics the distribution function is oftennormalized to 1. This is also a logical normalization, because then f(r,v, t) gives theprobability to find the particle at location r with velocity v at time t in the 6-dimensionalphase space (or if integrated over the whole phase space it states that the probability tofind 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 [Stevarket 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-Maxwellianstrahl that features an electron beam propagation along the magnetic field.

As an example let us investigate the Maxwellian velocity distribution function (seeAppendix 9.4):

f(v) = n

(m

2πkBT

)3/2

exp

(− mv2

2kBT

), (4.5)

where m is the mass of the particle and density n = 〈n〉 is assumed to be constant.Using the result: ∫ ∞

−∞exp(−x2) dx =

√π (4.6)

it is easy to show (Exercise 4.1) that the integral of the Maxwellian distribution over the3-dimensional velocity space gives n ([n] = m−3). The average and root-mean-squarevelocities for a Maxwellian velocity distribution are calculated in Exercise 4.2.

The definition of the particle density as an integral of the distribution function il-lustrates how macroscopical parameters can be expressed as velocity moments of thedistribution function: ∫

f d3v ;∫

vf d3v ;∫

vvf d3v .

Note that vv is a cartesian tensor, whose components are vivj . Velocity momentsdepend on time and space. Because in plasma different particle species have oftendifferent distribution functions we distinguish them using Greek subscripts.


The first-order moment yields the particle flux

Γα(r, t) =

∫vfα(r,v, t) d3v . (4.7)

Its SI units are (m−3)(m s−1) = m−2 s−1, which shows that the particle flux is thenumber of particles that traverse through a unit surface in a unit time. Dividing this byparticle density we get the average, or bulk, velocity at a given location:

Vα(r, t) =

∫vfα(r,v, t) d3v∫fα(r,v, t) d3v

, (4.8)

from which we can further determine the current density :

Jα(r, t) = qαΓα(r, t) = qαnαVα(r, t) . (4.9)

The second order moment gives parameters that are related to the square of thevelocity such as pressure and kinetic energy. In plasma physics pressure is a tensorquantity (particles are likely to have different velocities parallel and perpendicular tothe magnetic field) and it is defined to depend on how much the particle velocitiesdeviate from the average velocity Vα:

Pα(r, t) = mα

∫(v −Vα)(v −Vα)fα(r,v, t) d3v , (4.10)

which in a case if spherical symmetry reduces to Pα = pαI , where I is the unit tensorand pα the scalar pressure:

pα(r, t) =mα

3

∫(v −Vα)2fα(r,v, t) d3v = nαkBTα(r, t) . (4.11)

Here we have introduced the concept of temperature Tα. In the frame moving with thevelocity Vα, i.e., where Vα = 0, the temperature is given by:

3

2kBTα(r, t) =

mα

2

∫v2fα(r,v, t)d3v∫fα(r,v, t) d3v

, (4.12)

which for a Maxwellian distribution is the temperature of classical thermodynamics. Incollisionless plasmas equilibrium distributions may be far from Maxwellian and, conse-quently, temperature is a non-trivial concept. Temperature can be understood in termsof the width of the distribution function, but only for the Maxwellian distribution func-tion there is a unique level where to determine the width to correspond to the classicaldefinition of temperature.

The chain of moments continues to higher orders. The third order introduces theheat flux, i.e., temperature multiplied by velocity. Higher moments can be calculated,but do not have a simple physical interpretation. In plasma physics higher momentsthan the heat flux are seldom needed.

Contemplate: Write the distribution function of electrons that are all moving at thesame velocity V0 and the ions are all at rest. Write also equations for the electric currentand the pressure tensor

4.2. TIME EVOLUTION OF DISTRIBUTION FUNCTIONS 61

4.2 Time evolution of distribution functions

To determine how particle distribution functions evolve in space and time we need theappropriate equations of motion. We start by assuming that the number of particlesin the 6-dimensional phase space remains constant. We will investigate a small plasmaelement and follow its motion. Each point in the plasma element moves according toequations:

dr

dt= v ;

dv

dt=

F

m, (4.13)

where F describes the forces that influence the system. The number of particles withina volume V of a 6-dimensional phase space is:

N =

∫Vf(r,v, t) d3r d3v . (4.14)

Long-range forces affect in a similar way to all particles in the plasma element, butshort range forces, typically resulting from collisions, can scatter particles in and outfrom the phase space element. Here we consider only long-range forces. Figure 4.3illustrates the evolution of a phase space plasma element under the influence of long-range forces. All particles will be accelerated by the same force, and the phase spacedensity at time t2 will be the same as at time t1. Only if there are collisions the densitycan change.

plasma element

v

rr

v

t1 t2

Figure 4.3: A plasma element retains its density in 6-dimensional phase space as it movesunder the influence of long-range forces.

The conservation of particle number in volume V that moves with the particles isgiven by the continuity equation:

0 =∂N

∂t+∇u · (NU) =

∫V

(∂f

∂t+∇u · (fU)

)d3rd3v , (4.15)

where U = (x, v) = (v,F/m), and ∇u is the 6-dimensional gradient operator, whosecomponents are the the components of the gradients in the configuration and velocityspaces (∂/∂r, ∂/∂v) The first term on the left-hand side depends on the change in densityat each phase space point and the latter depends on how V changes with the motion so


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

∂t+∇u · (fU) = 0 . (4.16)

If the force F does not depend on the velocity (remember that U = (x, v)) we can writethe above equation as:

∂f

∂t+ v · ∂f

∂r+

F

m· ∂f∂v

= 0 . (4.17)

The Coulomb and gravitational forces do not depend on velocity, but the magneticpart of the Lorentz force does. However, fortunately:

∂

∂v· (v ×B) = 0 , (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 distributionfunction under the influence of long-range forces. This is called the Vlasov equation:

∂f

∂t+ v · ∂f

∂r+

q

m(E + v ×B) · ∂f

∂v= 0 . (4.19)

It was formulated by the Soviet theoretical physicist Anatoly Alexandrovich Vlasov inthe late 1930s.

In classical statistical physics the particle collisions are important and the equationcorresponding to the Vlasov equation is the Boltzmann equation:

∂f

∂t+ v · ∂f

∂r+

F

m· ∂f∂v

=

(∂f

∂t

)c, (4.20)

where the term on the right hand side describes the change of the distribution functiondue 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 interac-tions. In plasma physics Coulomb interactions are mainly long-range and weak. There-fore in plasma physics the average interactions between the particles are included in theBoltzmann equation through the external Lorentz force:

∂f

∂t+ v · ∂f

∂r+

q

m(E + v ×B) · ∂f

∂v=

(∂f

∂t

)c. (4.21)

4.3. SOLVING THE VLASOV EQUATION 63

The collision term includes only large-angle collisions between charged particles andthe possible collisions with neutrals, including charge-exchange processes. The generalcalculation of the Boltzmann collision term (∂f/∂t)c is a tedious task. Fortunately, hotand low density plasmas can often be considered collisionsless and the Vlasov equationis the appropriate approach.

The simplest situation taking account the collisions arises when the collisions occurpredominantly with neutrals. In this case the collision term can be approximated by theKrook model : (

∂f

∂t

)c

= −νc(f − f0) , (4.22)

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 time-independent plasma where collisions are taken into account using the Krook model.Taking into account the effect of long-range Coulomb interactions results in so-calledFokker-Planck equations. Their derivation is also a rather difficult task and beyond thescope 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 con-straint 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 tothe Vlasov equation. Furthermore, the force term in the Vlasov equation is nonlinear.Thus the Vlasov equation can be solved analytically only for small perturbations whenlinearization is possible.

We investigate here only the simplest case, where there are no background electric ormagnetic fields. Let us also assume 2-dimensional phase space (x, v), and that the plasmadistribution function is homogeneous and depends on speed only f0(v). We consider howthe plasma responds to a small perturbation. This corresponds to the setting in Section2.3 when we derived the formula for the plasma frequency. We consider again the electronmotion only and assume ions as a fixed background. Since plasma starts to oscillate, weassume that the perturbation will cause an electric field of the form of a plane wave:

E(x, t) = E exp[i(kx− ωt)] . (4.23)

If we denote the small perturbation to the distribution functioon by f1, the distri-bution function that enters to the Vlasov equation is f = f0 + f1 and

∂f

∂t+ v

∂f

∂x− e

mE∂f

∂v= 0 . (4.24)

In the case of small perturbations the linearization is possible, which means thatonly the first order terms of small perturbations will be considered. Since f0 is the


equilibrium solution of the Vlasov equation the sum of the zero order terms is triviallyzero. The first-order Vlasov equation thus becomes:

∂f1∂t

+ v∂f1∂x− e

mE∂f0∂v

= 0 . (4.25)

From Maxwell’s equations we only need:

ε0∇ ·E = ρ = −e∫f1 d

3v , (4.26)

which in the 1-dimensional case simplifies to the form:

ε0∂E

∂x= −e

∫ ∞−∞

f1 dv . (4.27)

Vlasov tried to solve these equations in the end of the 1930s by assuming that theperturbation of the distribution function also has the form of a plane wave:

f1(x, v, t) = f1(v) exp[i(kx− ωt)] , (4.28)

which is practically the same as using Fourier transformations in space and time. Withthis assumption the linearized Vlasov equation is reduced to:

−i(ω − kv)f1 −e

mE∂f0∂v

= 0 , (4.29)

with the solution:

f1 =ieE

m

∂f0/∂v

ω − kv. (4.30)

By inserting this to the Coulumb law (Eq. 4.27) we obtain:

ikε0E = −e∫ ∞−∞

f1dv = − ie2E

m

∫ ∞−∞

∂f0/∂v

ω − kvdv , (4.31)

from which we can cancel E.

If we know the equilibrium distribution f0 we can calculate the relation between thewave number and frequency related to the perturbation caused by the electric field,i.e., we have found the dispersion equation:

D(k, ω) ≡ 1 +e2

mε0

1

k

∫ ∞−∞

∂f0/∂v

ω − kvdv = 0 . (4.32)

4.3. SOLVING THE VLASOV EQUATION 65

Charge density ρ can be considered as the internal property of plasma and thus theMaxwell equation can be written as: ∇ ·D = 0, where D = ε0D(k, ω)E.

We can find the electron plasma wave we encountered in Section 2.3 by consideringthe dispersion relation at the long-wavelegth limit (ω kv). Now we can expand thedenominator in the integral:

1

ω − kv=

1

ω+kv

ω2+k2v2

ω3+k3v3

ω4+ ... (4.33)

For instance, by using the 1-dimensional Maxwellian distribution (for a complete deriva-tion see Exercise 4.4) :

f0(v) = n

(m

2πkBT

)1/2

exp

(− mv2

2kBT

)(4.34)

and taking into account only the leading terms, the dispersion equation reads:

1−ω2p

ω2

(1 +

3k2v2th2ω2

)= 0 , (4.35)

where vth =√

2kBT/m is the electron thermal speed .

Assuming infinite wavelength (k → 0) the solution is the standing plasma oscillationat the frequency ωp =

√ne2/(mε0) we found in Section 2.3. For finite wavelengths and

finite temperatures there is a small correction in the dispersion relation:

ω2 ≈ ω2p +

3

2k2v2th . (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 willderive the same dispersion relation starting from the warm plasma theory.

If the denominator in Eq. 4.32 cannot be expanded, the integral is not straightfor-ward to evaluate. If frequency ω is real, there is a singularity at v = ω/k along thepath of integration. In most situations frequencies are not real since waves in plasmaare typically either damped by collisions or amplified by some instability mechanism.Inserting ω = ωr + iωi in Eq. 4.30 we see that

f1 ∝ exp (−iωrt) exp (ωit). (4.37)

If ω is complex, the singularity is not along the real axis.


Vlasov did not find the correct way of dealing with the singularity. Lev Landaurealized in 1946 that because the perturbation must begin at some instant, the prob-lem can be treated as an initial value problem and, instead of a Fourier transform, aLaplace transform in the time domain can be applied. Once the initial transients of theperturbation have faded away, the asymptotic solution gives the intrinsic properties ofthe plasma, i.e., the dispersion equation. The exact Landau solution is complicated andtechnically 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 γthus depends on the form of the distribution function and it determines whether theperturbation

E = E0 exp[−i(ω + iγ)t] ∝ exp(γt) (4.38)

results in a growing or damped wave solution:

γ > 0: energy from wave to particles → growing wave (instability)γ < 0: energy from wave to particles → damped wave.

For instance, for the Maxwellian distribution γ is:

γ = −(π

8

)1/2 ωpek3λ3D

exp

(− 1

2k2λ2D− 3

2

)(4.39)

For the Maxwellian distribution γ is negative and the perturbation will damp (Figure4.4). This phenomenon is known as the Landau damping . The damping is a genuine col-lective effect characteristic for plasmas and important in describing how energy transfersfrom plasma particles to wave modes and vice versa. The particles that have a velocityclose to the phase velocity vph of the wave interact strongly with the wave (“resonance”)and can exchange energy.

f(v)

faster particles

slower particles

0vph

Figure 4.4: In a Maxwellian distribution, there are always more low than high energyparticles, and thus, wave loses more energy than gains back.

Landau’s original solution was not immediately accepted. The wave damping withoutenergy 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 analysisbut 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:

f0(v) = n

(m

2πkBT

)1/2

exp

(− mv2

2kBT

). (4.40)

It is a distribution towards which the gas thermalizes due to collisions. Note that thepower in the normalization factor depends on the degrees of freedom in the velocityspace. Each degree of freedom contributes a factor of 1/2. For a 3-dimensional isotropicdistribution the power is simply 3/2. Figure 4.5 illustrates an isotropic Maxwellianvelocity distribution function in the velocity space.

Figure 4.5: Isotropic Maxwellian velocity distribution function. The right-hand pictureshows contours of constant f in the velocity phase. The horizontal (vertical) axis showsthe velocity component perpendicular (parallel) to the magnetic field.

Many hot and tenuous plasmas are collisionless and cannot be described by a Maxwelliandistribution. However, in many cases the Maxwellian is a reasonable starting point. Forexample, the whole distribution may be moving with respect to the observer. If wedenote this velocity with V0 the 3-dimensional distribution function is

f(v) = n

(m

2πkBT

)3/2

exp

(−m(v −V0)

2

2kBT

). (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 theplasma. In particular, the magnetic field drives plasma towards anisotropy with differentproperties perpendicular and parallel to the magnetic field. For instance, assume thatparticles are trapped within a magnetic bottle and in the center of the bottle their


Figure 4.6: Velocity distribution function (left) and countours of constant f for aMaxwellian distribution that drifts perpendicular to the magnetic field at the velocityV0 (drifting Maxwellian).

distribution is Maxwellian. If the bottle is contracted, the mirror points move closertogether resulting in the increase of the pitch angles. This will flatten the isotropicdistribution in the direction parallel to the magnetic field and stretch it in the directionperpendicular to the magnetic field (parallel velocity of the particles will decrease, whileperpendicular velocities increase). The resulting anisotropic distribution is often calleda pancake distribution. In turn, if the bottle will be stretched the mirror points movefurther away and the distribution will be stretched parallel to the magnetic field to acigar-shaped distribution.

Anisotropic plasma can have a Maxwellian distribution both parallel and perpendic-ular to the magnetic field but with different temperatures T‖ ja T⊥. Now the distributionfunction will be

f(v⊥, v‖) =n

T⊥T1/2‖

(m

2πkB

)3/2

exp

(− mv2⊥

2kBT⊥−

mv2‖2kBT‖

). (4.42)

As the perpendicular velocity space is 2-dimensional, the normalization factor has thepower 2 × 1/2 = 1 for T⊥ (the width of the distribution is assumed to be the same inall perpendicular directions, the distribution is said to be gyrotropic), whereas there isonly one degree of freedom in the parallel direction.

Anisotropic plasma may also move accross the magnetic field, for example due to theE×B-drift or the gradient drift (Figure 4.7). Now the Maxwellian distribution reads:

f(v⊥, v‖) =n

T⊥T1/2‖

(m

2πkB

)3/2

exp

(−m(v⊥ − v0⊥)2

2kBT⊥−

mv2‖2kBT‖

). (4.43)

As discussed in Section 3.5.4 there are always some particles that can escape from themagnetic bottle. In the absence of a mechanism that would replenish the lost particlesthe distribution becomes a loss-cone distribution.

4.4. EXAMPLES OF DISTRIBUTION FUNCTIONS 69

Figure 4.7: Drifting pancakedistribution

Contemplate: Sketch a loss-cone distribution.

Another important special case is a field-aligned beam whose 3-dimensional distri-bution function is

f(v⊥, v‖) =n

T⊥T1/2‖

(m

2πkB

)3/2

exp

(− mv2⊥

2kBT⊥−m(v‖ − v0‖)2

2kBT‖

). (4.44)

It is often convenient to present the distribution function as a function of energyinstead of velocity. If all energy is kinetic, the energy is simply obtained from W =mv2/2. In the case the particles are in the external electric potential field U = −qϕ thetotal energy of particles is W = mv2/2 +U and the Maxwellian distribution function is

f(v) = n

(m

2πkBT

)3/2

exp

(− W

kBT

). (4.45)

This can be written as the energy distribution:

g(W ) = 4π

[2(W − U)

m3

]1/2f(v) . (4.46)

For derivation of the above form in a case U = 0 see Exercise 4.5. The normalizationfactor is determined by requiring that the integration of the energy distribution over allenergies gives the density.

A very important distribution function in space plasmas is the so-called kappa dis-tribution. Distribution functions are often nearly Maxwellian at low energies, but theydecrease more slowly at high energies. At higher energies the distribution is describedbetter by a power law than by an exponential decay of the Maxwell distribution. Sucha behaviour is not surprising if we remember that the Coulomb collisional frequencydecreases with increasing temperature as ∝ T−2/3 (see Section 2.5). Hence, it takeslonger time for fast particles to reach Maxwellian distribution than for slow particles.The kappa-distribution has the form (Figure 4.8):

fκ(W ) = n

(m

2πκW0

)3/2 Γ(κ+ 1)

Γ(κ− 1/2)

(1 +

W

κW0

)−(κ+1)

. (4.47)


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 ofmathematics. 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, theobserved 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 velocitycomponent normal to the surface. We define the differential flux of particles traversinga unit area per unit time, unit solid angle (in spherical coordinates the differential solidangle is dΩ = sin θdθdφ) and unit energy as J(W,Ω, α, r, t). The units of J are normallygiven 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 bykeV, MeV, or GeV. Thus it is important to pay attention to the correct factors of 10 indata displays!

Let us conclude this discussion by finding out how differential flux and distributionfunction are related to each other. We can write the number density in a differentialvelocity element (in spherical coordinates d3v = v2 dv dΩ) as dn = f(α, r, t) v2 dv dΩ.By multiplying this with v we obtain another expression for the differential fluxf(α, r, t) v3 dv dΩ. Comparing with our earlier definition of the differential flux we obtain:

J(W,Ω, r, t) dW dΩ = fv3 dv dΩ . (4.48)

Since dW = mv dv we can write the relationship between the differential flux andthe distribution function as:

J(W,Ω, r, t) =v2

mf . (4.49)

4.5. EXERCISES: KINETIC PLASMA DESCRIPTION 71

4.5 Exercises: Kinetic Plasma Description

1. Integrate the Maxwellian distribution function over three-dimensional velocityspace.

2. (a) Using the Maxwellian velocity distribution function

f(v) = n

(m

2πkBT

)3/2

exp

(− m

2πkBT

)calculate the average velocity 〈v〉

(b) Calculate also the root-mean-square velocity vrms =√〈v2〉. What is the

corresponding average kinetic energy 〈E〉 = m⟨v2⟩/2 ?

Hint: It is useful to remember that∫ ∞−∞

= exp(−x2

)dx =

√π.

3. Consider electrons in an unmagnetized (B = 0) homogeneous (∂/∂r = 0), time-independent (∂/∂t = 0) plasma in a weak constant electric field. Assume that theequilibrium distribution of the electrons is Maxwellian and take the collision intoaccount using the relaxation time approximation known also as the Krook model(

∂f

∂t

)c

= −νc(f − f0) .

Show that the conductivity of this plasma is given by σ =ne2

mνc.

4. Insert the one-dimensional Maxwellian into

D(k, ω) ≡ 1 +e2

mε0

1

k

∫ ∞−∞

∂f0/∂v

ω − kvdv = 0

and derive the dispersion equation for the Langmuir wave

5. Starting again from the simple Maxwellian velocity distribution show that theMaxwellian energy distribution (or Boltzmann distribution) becomes

f (W ) =2n√π

√W

(kBT )3exp

(W

kBT

).

6. Show that for large κ the kappa distribution approaches the Maxwellian distribu-tion.


7. Let us consider energetic particle measurements by satellites. The following differ-ential fluxes (particles/s keV) were obtained: 569000, 3850, 137, 4.52 correspondingto energies 30 keV, 80 keV, 240 keV and 800 keV.

(a) Plot the measurements as log(flux) vs. log(energy). What can you say aboutthe spectrum?

(b) Use the least squares method to fit a power-law spectrum of type f(E) =f0E

−γ to the data. What is the spectral index γ and f0 ?

(c) How many particles did the satellite measure altogether at the energy range50–200 keV?

Chapter 5

Macroscopic Plasma Equations

In the two previous chapters we have covered the microscopic and kinetic plasma de-scriptions. The Boltzmann and Vlasov equations derived in Chapter 4 can be consideredas the basic equations of plasma physics. In many cases it is not necessary to know theexact 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 velocitymoments of the Boltzmann and Vlasov equations. The resulting macroscopic variables,such as density, velocity and pressure, are functions of the position and time only. Thisis the fluid (or macroscopic, see Figure 1.3) plasma description. In the fluid theory, thetime evolution of macroscopic parameters is determined by means of fluid equations thatare analogous to, but generally more complicated than, the equations of hydrodynamics

There are different levels of fluid theories. Multifluid theories consider the plasmaparticle species independently. For example, the two-fluid model has separate equationsfor electron and ion fluids. The simplest and most important macroscopic model is calledmagnetohydrodynamics (MHD). MHD combines one-fluid (hydrodynamic and Lorentzforce) 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 physicsof neutral gases and fluids, and magnetic terms were added to the equations of hydro-dynamics. This led to the equations of MHD. Only later the equations of MHD werederived from the microscopic theory.

5.1 Macroscopic transport equations

For the needs of many applications we could start from the Vlasov equation, but retainingthe 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 velocitymoments of the Boltzmann equation for plasma particle species α:

∂fα∂t

+ v · ∂fα∂r

+qαmα

(E + v ×B) · ∂fα∂v

=

(∂fα∂t

)c. (5.1)

5.1.1 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 termis 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:

∫v · ∂fα

∂rd3v = ∇ ·

∫vfα d

3v = ∇ · (nαVα) , (5.2)

and we have found the equation of continuity

∂nα∂t

+∇ · (nαVα) = 0 . (5.3)

Continuity equations for charge or mass densities are obtained by multiplying Eq.5.3 by qα or mα, respectively:

∂ρmα∂t

+∇ · (ρmαVα) = 0 (5.4)

∂ρqα∂t

+∇ · Jα = 0 . (5.5)

The equation of continuity is an example of the general form of a conservation law

∂F

∂t+∇ ·G = 0 . (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 mαv and integrate over v (Exercise 5.1). This yields the momentumtransport equation, which is the macroscopic equation of motion.

5.1. MACROSCOPIC TRANSPORT EQUATIONS 75

nαmα∂Vα

∂t+ nαmαVα · ∇Vα − nαqα(E + Vα ×B) +∇ · Pα

= mα

∫v

(∂fα∂t

)cd3v . (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 repre-sent what we normally understand as pressure, while the off-diagonal elements representshearing or tension in the medium. The divergence of Pα contains information of inho-mogeneity and viscosity of the plasma. The Lorentz force term does not integrate tozero. The average electric and magnetic fields in the Boltzmann equation are determinedby both internal and external sources (ρext, Jext) and fulfill the Maxwell equations forthe average plasma properties.

∇ ·E =∑α

nαqαε0

+ρextε0

∇×B =1

c2∂E

∂t+ µ0

∑α

nαqαVα + µ0Jext . (5.8)

Because collisions transport momentum between different plasma populations, thecollision 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 relatedto the Krook model is

mα

∫v

(∂fα∂t

)cd3v = −

∑β

mαnα(Vα −Vβ) 〈ναβ〉 , (5.9)

where 〈ναβ〉 is the average collision frequency of particle species α with particles ofspecies β.

The momentum equation relates the fluid velocity to the density gradient and elec-tromagnetic 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 equationof hydrodynamics. In neutral hydrodynamics the only forces that are acting to fluid arethe 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 theenergy or heat transport equation (conservation law of energy). Integration is now quitea tedious process. We write here the equation in the form:


3

2nαkB

(∂Tα∂t

+ Vα · ∇Tα)

+ pα∇ ·Vα

= −∇ ·Hα − (P ′α · ∇) ·Vα +∂

∂t

(nαmαV

2α

2

)c

, (5.10)

where the isotropic part of the pressure pαI is written on the left-hand side and thenon-isotropic part P ′α on the right-hand side. The relation between the scalar pressurepα and temperature Tα is assumed to be that of an ideal gas pα = nαkBTα , The third-order term is Hα and it describes the heat flux. The energy equation states that thetemperature of plasma can increase due to compressive flow (∇ ·Vα < 0), dissipationdue to flow gradient induced stress ((P ′α · ∇) ·Vα), collisional energy exhange, and dueto divergence of the heat flux (sources or sinks).

Now we have macroscopic equations separately for each plasma species. In a realplasma several species co-exist. The simplest description of the real plasma consistsof electrons and protons (two-fluid model). The separate fluid components interactthrough collisions and electromagnetic interaction. Continuity equation and momentumtransport equations are valid separately for both fluids. In addition to electrons andprotons, there may be a variety of heavier ions, as well as neutral particles, which maycontribute 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 differenttemperatures or average velocities. Such situations often give rise to plasma instabilitiesto 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 Boltz-mann equation. This would lead to an equation with the fourth-order contribution, andso on. This is because the Boltzman equation includes both the zeroth and first ordervelocity. The chain of equations with increasing order of velocity (and with increasingcomplexity) must be truncated at some point to form a closed system of transport equa-tions. In many practical problems this is made in the second order, either by neglectingthe heat flux, or by substituting the energy equation by an equation of state. Here physi-cal insight is essential. Krall and Trivelpiece [1973] state this: “The fluid theory, thoughof great practical use, relies heavily on the cunning of its user”.

The simplest of closed system is the cold plasma model. It contains the conservationequations for mass and momentum and the related macroscopic variables are the densityand 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 bulkvelocity: 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 con-sidering, 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 thatplasma is isotropic, i.e., the diagonal elements of Pα are equal, and thus, the pressuretensor can be replaced by a scalar pressure. Thus, in the momentum transport equationthe term ∇ ·P degenerates to ∇p. The macroscopic variables appearing in this case arethe 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 = nkBT0 . (5.11)

In isothermal plasma changes of plasma parameters are so slow that the system hastime to thermalize during the time-scale of the change. In collisionless plasmas this istypically not a very good assumption.

The opposite limit is that the changes occur so fast that there is no heat exchangebetween the considered plasma element and its surroundings. Hence, we can set the theheat flux to zero (∇ ·H = 0). The resulting adiabatic equation of state can be derivedrather easily from the heat flux equation by assuming scalar pressure, using the densitycontinuation equation and writing d/dt = ∂/∂t+ V · ∇. The result is

3

2ndT

dt= T

dn

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 theadiabatic constant . In statistical mechanics it is shown that for monoatomic adiabaticgas γ = (f +2)/f , where f is the number of degrees of freedom. Thus, for 3-dimensionalideal gas γ = 5/3. In non-adiabatic cases γ 6= 5/3, e.g., for isothermal process γ = 1, forthe isobaric process γ = 0, and for the isometric (constant density) γ → ∞. Plasmasare not always isotropic. For instance, if a strong magnetic field is present or if thereare not enough collisions to maintain the isotropic velocity distribution the pressure isanisotropic. 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 willcombine these equations to a one-fluid theory called magnetohydrodynamics (MHD).


MHD is probably the most widely known plasma theory. In MHD the plasma is con-sidered as a single fluid in the centre-of-mass (CM) frame. This is a well-motivatedapproach in collision-dominated plasmas, where the collisions constrain the plasma par-ticles to follow each other closely and thermalize the distribution toward a Maxwellian,which makes the interpretation of velocity moments straightforward. MHD works alsoremarkably well in collisionless tenuous space plasmas, e.g., when studying the large-scale 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 ap-proximations. 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 completebecause the motion along the magnetic field is unconstrained in a homogeneous plasma.

Figure 5.1: The Earth’s magnetosphere simulated by the global magnetosphere-ionosphere simulation GUMICS-4. The colours indicate the plasma density. The smallinset shows the ionospheric conductivity from the electrostatic ionospheric module cou-pled 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) =∑α

nαmα ,


charge densityρq(r, t) =

∑α

nαqα = e(ni − ne) ,

macrospcopic velocity

V(r, t) =

∑α nαmαVα∑α nαmα

,

electric current densityJ(r, t) =

∑α

nαqαVα

and the pressure tensor in the CM frame

PCMα (r, t) = mα

∫(v −V)(v −V)fαd

3v ,

from which we get the total pressure

P(r, t) =∑α

PCMα (r, t) .

Summing the individual continuity and momentum transport equations over particlespecies yields the continuity equations

∂ρm∂t

+∇ · (ρmV) = 0 (5.14)

∂ρq∂t

+∇ · J = 0 (5.15)

and the momentum transport equation

ρm

(∂V

∂t+ V · ∇V

)= ρqE + J×B−∇ · P . (5.16)

The momentum equation corresponds to the Navier-Stokes equation of hydrodynam-ics where the viscosity terms are written explicitly (here they are hidden in the pressuregradient). At macroscopic level the deviations from charge neutrality are small and ρqEis usually negligible. The magnetic part of the Lorentz force J×B (often called Ampere’sforce) 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 writtenin the conservation form

∂

∂t

[ρm

(V 2

2+ w

)+B2

2µ0

]= −∇ ·H . (5.17)

Here w is the enthalpy that is related to the the internal free energy (per unit mass) ofthe plasma u by w = u + p/ρm. The RHS is the divergence of the heat flux vector H,


which is a third-order moment. After some reasonable approximations it can be writtenas

H =

(V 2

2+ u+

p+B2/µ0ρm

)ρmV − B

µ0

(V +

J

ne

)·B

− J×B

σµ0+

JB2

µ0ne+

meB

µ0ne2× ∂J

∂t. (5.18)

For derivation of the energy transport equation in adiabatic ideal MHD, see Exercise5.2. When integrated over a finite volume V the LHS of (5.17) describes the temporalchange of the energy of the MHD plasma in that volume and the RHS the the energyflux through the boundary ∂V and energy losses due to resistivity. Thus we have foundthe MHD equivalent of Poynting’s theorem of elementary electrodynamics.

In the case of MHD the third moment is usually neglected and an equation of stateis used to relate the changes in plasma pressure and density. MHD assumes Maxwelliandistribution and thus the pressure is isotropic. The adiabatic equation of state is writtenin the form:

d

dt(pρ−γm ) = 0 → pρ−γm = constant. (5.19)

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 plasmacurrent is the sum of all charged particle motions. In a single-fluid theory the currenttransport equation is derived by multiplying the momentum transport equations of eachparticle population by qα/mα and summing over all populations. This leads to a rathermessy expression including terms of different magnitudes and further approximationsare needed. Here we give the generalized Ohm’s law in the form that contains the mostimportant terms for space plasmas:

E + V ×B =J

σ+

1

neJ×B− 1

ne∇ · Pe +

me

ne2∂J

∂t. (5.20)

The terms that are proportional to me/mi and that contain the derivatives of the second-order terms VJ, JV and VV have been neglected. The collision integral has beenapproximated 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 5.3compares different terms in the generalized Ohm’s law during a magnetospheric substormin 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|.


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 framesare taken into account. Here the moving frame is attached to the fluid flow with thevelocity 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 tocompare the magnitude of the term V×B to the inertial, Hall (i.e., J×B) and resistiveterms. 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

τB me

ne2∂J

∂t=

me

ne2∂

∂t(

1

µ0∇×B) (5.23)

∼ me

ne21

τ

B

µL=

c

ω2p

B

τL

→ L c

ωp,

where ωp is the plasma frequency.

In a similar manner (try yourself!) we obtain the conditions to neglect the Hall term(∝ J×B):

L c

ωp, and τ 1

ωc. (5.24)

Finally the resistive term can be neglected if

V×B ∼ uB ηJ ∼ η 1

µ0

B

L(5.25)

⇔ u 1

µ0σL.


The ratio of the V ×B and ηJ terms is called the magnetic Reynold’s number :

Rm = µ0σuL . (5.26)

The resistive term can thus be neglected when the magnetic Reynold’s number islarge. This is indeed the case in many space plasmas that have large dimensions (L) andsmall resistivity. Note that the resistive term causes dissipation and converts magneticenergy to heat.

While the ideal MHD is a reasonable starting point, it is not at all clear that thenext term to take into account should be J/σ. In many space applications the Hall termJ×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 ac-tual inter-particle collisions, but which may lead to “effective” resistivity or viscosity atthe macroscopic level. Various wave–particle interactions and microscopic instabilitiestend to inhibit the current flow. Often the macroscopic effect of these processes looksanalogous to finite ν and is called anomalous resistivity .

5.4 Exercises: Macroscopic Plasma Equations

1. Derive the momentum equation

nαmα∂Vα

∂t+ nαmαVα · ∇V − nαqα (E + Vα ×B) +∇ · Pα

= mα

∫v

(∂fα∂t

)cd3v .

2. Derive the energy equation of adiabatic ideal MHD in the conservation form

∂

∂t(1

2ρmV

2 +p

γ − 1+B2

2µ0) +∇ · (1

2ρmV

2V +γ

γ − 1pV +

E×B

µ0) = 0.

Hints: Write the equation of state as d(pρ−γm )/dt and the conservation of mass asdρm/dt = −ρm∇ ·V to show that (γ − 1)V · ∇p = ∂p/∂t +∇ · (γpV). Then useideal Ohm’s law with Ampere’s and Faraday’s laws to show that

V · (J×B) = J · (V ×B) =∂

∂t

B2

2µ0+∇ · E×B

µ0.

5.4. EXERCISES: MACROSCOPIC PLASMA EQUATIONS 83

3. Investigate generalized Ohm’s law:

E + V ×B =J

σ+

1

neJ×B +

1

ne∇ ·Pe +

me

ne2∂J

∂t

during a substorm in the nightside magnetotail when following values have beenmeasured:

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, characteristictime scale ≈ 10 s and effective resistivity less than 1 s−1. Compare the magnitudesof various terms in Ohm’s law.

Chapter 6

Magnetohydrodynamics

Magnetohydrodynamics (MHD) describes electrically conducting fluids in the presenceof a magnetic field. This chapter deals with single-fluid MHD where different plasmacomponents (i.e., ions, electrons) move together and compose a single “fluid”. Theplasma is described by a single temperature, density and velocity. Single-fluid MHD is astrongly reduced theory but applies remarkably well in many situations. For instance, theplasma in the solar wind, Earth’s magnetosphere, solar corona, and in many interstellarregions can be treated with MHD. MHD has also vast applications in fusion plasmaresearch, in particular concerning the plasma confinement and stability. As described inthe previous chapter, MHD governs processes that are slow compared with the gyrationtime and on scales that are larger than the gyro (Larmor) radius.

We will discuss the general concepts of the MHD. We start by summarizing theMHD 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 discussthe importance and basic models of magnetic reconnection and finalize this chapter byinvestigating the MHD equilibrium.

6.1 MHD equations

The basic equations of MHD we derived in Chapter 5 are:

∂ρm∂t

+∇ · (ρmV) = 0 (6.1)

ρm

(∂

∂t+ V · ∇

)V +∇P − J×B = 0 (6.2)

E + V ×B = J/σ . (6.3)

85

86 CHAPTER 6. MAGNETOHYDRODYNAMICS

P = P0

(n

n0

)γ(6.4)

∇×E = −∂B

∂t(6.5)

∇×B = µ0J . (6.6)

The first four equations are the mass continuity equation, momentum equation, re-sistive Ohm’s law, and the adiabatic equation of state. We have used the basic MHDassumption that the temporal variations are so slow that the displacement current(ε0∂E/∂t) can be neglected in the Ampere–Maxwell law. The relationship betweenthe electric current and magnetic field is thus obtained from Ampere’s law ∇×B = µ0J.

As discussed earlier, the MHD equations are basically the combination of Navier-Stokes equations of fluid dynamics with Maxwell’s equations and Ampere’s force. It isimportant to note that in MHD the magnetic and velocity fields are taken as the primaryfields. The Maxwell equations and the MHD Ohm’s law give the current density andthe 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 usinvestigate 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

∂t= ∇× (V ×B− J/σ) . (6.8)

Remembering that ∇ · B = 0 and assuming that the conductivity is constant (fora case where the conductivity is not spatially homogeneous see Exercise 6.1) we obtainthe induction equation (one of the most important equations of plasma physics!):

∂B

∂t= ∇× (V ×B) +

1

µ0σ∇2B . (6.9)

From the induction equation we see that the magnetic field can change in time as aconsequence of two effects. We will investigate them separately.

6.2. MAGNETIC FIELD EVOLUTION 87

6.2.1 Diffusion

Assuming that the plasma is at rest (V = 0) the induction equation reduces to thediffusion equation:

∂B

∂t= Dm∇2B , (6.10)

where Dm = 1/(µ0σ) = η/µ0 is the diffusion coefficient. Thus, the magnetic field inplasma 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∇2B.

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 = µ0σL2B (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-handpart of Figure 6.1. A thin current sheet separates the regions of oppositely orientatedmagnetic fields. The magnetic field is chosen to be along the ±x-axis and the currentsheet normal is along the z-axis. In this case the diffusion equation becomes:

∂Bz∂t

=1

µ0σ

∂2Bz∂x2

. (6.13)


If at time t0 = 0 the current layer is infinitely thin (i.e., can be described by aδ-function), the magnetic field diffuses as:

Bz(x) = B0 erf

(µ0σ

4t

)1/2

x

, (6.14)

where

erf(u) =2√π

∫ u

0e−v

2dv . (6.15)

The magnetic field diffuses towards the current sheet and, as a consequence, thecurrent sheet broadens as the diffusion proceeds (right part of Figure 6.1). Oppositelydirected magnetic fields cancel each other decreasing the magnetic field gradient, andthus, slowing down the diffusion (Eq. 6.10). Physically, the magnetic energy is trans-formed to heat. This is called Joule (or Ohmic) heating . Increasing plasma pressurecompensates the decreasing plasma pressure.

6.2.2 Convection

If σ → ∞ (ideal MHD), the diffusion term becomes small and Eq. 6.9 reduces to aconvection equation:

∂B

∂t= ∇× (V ×B) . (6.16)

The convection equation describes how plasma flow and magnetic field are tied toeach other. In this case there is no diffusion of the magnetic field, but the plasmaand magnetic field “convect” (actually advect) together. It is commonly said that themagnetic 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 adimensionless 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, Vthe average plasma velocity perpendicular to the field, LB the characteristic length overwhich the magnetic field varies, and τd the diffusion time scale. Substituting ∂/∂t→ τand ∇ → L−1B , and neglecting directions, the induction equation reduces to

B

τ=V B

LB+B

τd. (6.17)

The ratio of the terms on the RHS (i.e., the ratio of the convection to diffusion) becomes

convection

diffusion=

V

LBτd = µ0σLBV (6.18)

This ratio corresponds to the dimensionless magnetic Reynold’s number (Rm) we en-countered in Section 5.3.3 while investigating the conditions under which the resistiveterm in the MHD Ohm’s law can be neglected. If Rm is large diffusion is slow and theconvection dominates. This corresponds to small resistivity and the ideal MHD limit.

Exercise 6.2 investigates diffusion times and Rm for a typical laboratory plasma andin the solar wind. Due to large spatial scales and high conductivities Rm is typicallyvery large in space and astrophysical plasmas. In the solar wind at the Earth orbit Rmis of the order of 1016−1017. This means that during the 150 million kilometres journeyfrom the Sun to the Earth the magnetic field in the solar wind diffuses only aboutone kilometre! Hence, diffusion is negligible in the solar wind. In turn, in laboratoryplasma spatial scales and conductivities are much smaller, and consequently, diffusiondominates.

6.3 Frozen-in condition

When the magnetic Reynolds number is very large, the magnetic field is “frozen-in” tothe plasma motion. As a consequence, two plasma elements that are initially magneti-cally unconnected cannot mix as long as the frozen-in condition applies. The frozen-inconcept was first brought to plasma physics by Hannes Alfven. Although very usefulthe frozen-in concept is often misunderstood and Alfven later denounced it as “pseudo-pedagogical”. The problem lies in picturing moving magnetic field lines. A magneticfield 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 plasmaelements are connected by a magnetic field line at time t, they are connected bya field line at all times. What is conserved is the magnetic connection between theplasma elements.

Let us test the frozen-in assumption by investigating two plasma elements under theassumption of the ideal MHD. We assume that two plasma elements are magnetically


connected at time t (Figure 6.3). This means that if we trace the magnetic field from oneplasma element, we end up at the other. Let the distance between the elements at timet 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 elementsis thus 4l + d(4l). In order the frozen-in concept to be valid we have to show that atthe time t + dt the plasma elements are still magnetically connected, i.e., we need toshow that d(4l×B)/dt = 0.

Figure 6.3: If ideal MHD assumption holds, two plasma elements that are magneticallyconnected (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 termin 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 tod(4l)

dt= 4u = (4l · ∇)u . (6.21)

Since we assume ideal MHD, let us investigate how the magnetic field changes in timestarting 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

dt=∂B

∂t+ (u · ∇)B = (B · ∇)u−B(∇ · u) . (6.23)

Now we can calculate d(4l×B)/dt:

d

dt(4l×B) =

d(4l)

dt×B +4l× dB

dt(6.24)

= [(4l · ∇)u]×B +4l× [(B · ∇)u−B(∇ · u)] .

6.3. FROZEN-IN CONDITION 91

Because we assumed at the beginning 4l to be parallel to B, 4l×B = 0, and the thirdterm on the RHS is zero. For the same reason 4l and B can be interchanged in the firstterm on the RHS. Thus the first and the second term are the same except for their signand we have

d

dt(4l×B) = 0 . (6.25)

Thus, we have obtained that 4l remains parallel to B and plasma elements that orig-inally are on a common field line remain on a common field line. This assumption isvalid as long as the ideal MHD approximation is valid.

B

S (t0+dt)

S (t0)

Figure 6.4: Magnetic flux through a surface moving with the plasma is conserved at theideal MHD limit.

An alternative way to express the frozen-in concept is to investigate the time varia-tions of the magnetic flux through a surface S (see Figure 6.3):

ΦS =

∫S

B · dS . (6.26)

The surface elements move with the plasma fluid velocity. By calculating the change ofthe magnetic flux from time t0 to t0 + dt it can be shown (Exercise 6.3) that

dΦS

dt=

d

dt

∫B · dS = 0 . (6.27)

Thus, assuming that the ideal MHD Ohm’s law applies, the magnetic flux through anyclosed 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 convectiondominates. In reality, the plasma has always some resistivity. However, the frozen-incondition applies if the characteristic time-scale of the process we are looking at is muchshorter than the diffusion time. This can also be seen from the magnetic Reynoldsnumber Rm = (V/LB)τB (see Eq. 6.18), where τB is the magnetic field diffusion time.


In space plasmas the first correction to the ideal MHD is often not the resistive termbut the Hall term J×B/(ne)

E + V ×B =1

neJ×B . (6.28)

This is expected to be the case, e.g., near current sheets separating magnetic fields ofdifferent strength and direction. In this Hall MHD the magnetic field becomes frozen-into the electron flow

E = −Ve ×B . (6.29)

As discussed earlier in Chapter 3 this is because due to their much smaller mass electronshave much smaller gyro radii and are tied more strongly to the magnetic field than theions.

6.4 MHD waves

MHD is a fluid theory and there are similar wave modes as in ordinary fluids (hydrody-namics). In addition, the presence of the magnetic field gives rise to new modes. In theMHD description we assume that the frequency of the wave is smaller than the charac-teristics frequencies in the plasma (gyro frequency and plasma frequency) and that thewavelengths are longer than microscopic plasma scales (Larmor radius). In addition, asdiscussed earlier, one of the basic assumptions of MHD is that the temporal changes areso slow that the displacement current can be neglected. As a consequence, MHD doesnot describe basic electromagnetic waves. However, this does not mean that electromag-netic waves could not propagate through the MHD plasma, rather MHD phenomena andhigh-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 frequencyof the wave, and thus determines how the wave travels in the medium. To derive thedispersion 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 thatis in a homogeneous background (applied) magnetic field. Thus, we can replace theresistive 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 andintroducing the speed of sound

vs =√γp/ρm =

√γkBT/m , (6.30)

6.4. MHD WAVES 93

where γ is the adiabatic constant. We obtain

∇p = v2s∇ρm . (6.31)

Use Ampere’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 ofequations

∂ρm∂t

+∇ · (ρmV) = 0 (6.32)

ρm∂V

∂t+ ρm(V · ∇)V = −v2s∇ρm + (∇×B)×B/µ0 (6.33)

∇× (V ×B) =∂B

∂t. (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. Asdiscussed 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 keepingonly the first order terms (the zeroth order terms automatically fulfil the equations andthe second order terms are assumed so small that we can neglect them). This leads tothe linearized equations:

∂ρm1

∂t+ ρm0(∇ ·V1) = 0 (6.38)

ρm0∂V1

∂t+ v2s∇ρm1 + B0 × (∇×B1)/µ0 = 0 (6.39)

∂B1

∂t−∇× (V1 ×B0) = 0 . (6.40)

Next, let us find equation for the velocity perturbation V1. By taking the timederivate of the linearized momentum equation (Eq. 6.39) we obtain

ρm0∂2V1

∂t2+ v2s∇

(∂ρm1

∂t

)+

B0

µ0×(∇× ∂B1

∂t

)= 0 . (6.41)

By using the linearized continuity equation (Eq. 6.38) ja the linearized Ampere’s law(Eq. 6.40) this can be written as

∂2V1

∂t2− v2s∇(∇ ·V1) + vA × ∇× [∇× (V1 × vA)] = 0 , (6.42)

where the vector vA is


vA =B0√µ0ρm0

. (6.43)

The magnitude of this velocity defines the Alfven 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 derivativesby algebraic operators (see Appendix 9.3)

∇· → ik · (6.45)

∇× → ik×∂/∂t → iω .

Eq. 6.42 simplifies to an algebraic equation

−ω2V1 + v2s(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

−ω2V1 + (v2s + v2A)(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 thatthe wave vector k is in the xz-plane. Denote the angle between k and B0 by θ. Figure6.5 summarizes the coordinate system.

Now we obtain

k = k(ex sin θ + ez cos θ) (6.49)

vA = vAez (6.50)

V1 = V1xex + V1yey + V1zez (6.51)

k · vA = kvA cos θ (6.52)

k ·V1 = k(V1x sin θ + V1z cos θ) (6.53)

vA ·V1 = vAV1z . (6.54)

6.4. MHD WAVES 95

z

B0

x

y

n,k

q

Figure 6.5: Coordinate system tostudy MHD waves

Inserting these to the dispersion equation (Eq 6.48) we obtain

−v2p + v2A + v2s sin2 θ 0 v2s sin θ cos θ

0 −v2p + v2A cos2 θ 0

v2s sin θ cos θ 0 −v2p + v2s cos2 θ

V1xV1yV1z

= 0 , (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 equatingthe determinant to zero.

Alfven wave

The y-component of the matrix Eq. 6.55 gives a linearly polarized wave mode with thephase speed

vp = vA cos θ , (6.56)

where vA is given by Eq. 6.43. This mode is called the Alfven wave.

It is seen from Eq. 6.56 that when the Alfven wave propagates along the backgroundmagnetic field (i.e. θ = 0) its phase speed is exactly vA. For oblique propagation thephase speed is less than vA. It is also clear from Eq. 6.56 that Alfven waves do notpropagate perpendicular to the magnetic field.

It is easy to verify that the eigenvector (0, V1y, 0) corresponds to the root v2p =v2A cos2 θ. Thus for Alfven 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

−iωρ1 + ρ0k ·V1 = 0 (6.57)


that Alfven wave is non-compressive, i.e., there are no density fluctuations. The plasmafluid motions are thus completely transverse signifying that the plasma elements oscillateperpendicular to the direction of propagation of the wave.

The wave magnetic field B1 can be calculated from the convection equation assumingharmonic temporal and spatial dependences

ωB1 + k× (V1 ×B0) = 0 (6.58)

→ B1 = − V1

ω/kB0 , (6.59)

i.e., the wave magnetic field is perpendicular to the background magnetic field B0. Figure6.6 demonstrates the propagation of an Alfven wave.

E1

V1

B0

k

B0

B1

Figure 6.6: Alfven wave propagating parallel to the magnetic field.

The Alfven wave is often called also as shear Alfven wave or non-compressionalAlfven wave.

The existence of the Alfven wave can be deduced also intuitively. The magnetic fieldline can be considered to behave like a tensed string. Transversal displacement of theelastic string generates a transverse wave that propagates along the string, in analogywith the Alfven wave propagating parallel to magnetic field lines.

6.4. MHD WAVES 97

Alfven 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. Ithas been suggested that Alfven waves could explain the heating of the outermost layer ofthe solar atmosphere, the corona (Figure 6.7). Understanding the properties of Alfvenwaves is also important for determining the stability, turbulence and heating in controlledfusion devices. The Alfven speed can differ greatly in space plasma, depending on thedensity and magnetic field magnitude in question. Typical Alfven speeds are calculatedin Exercise 6.4 in the Earth’s ionosphere, solar corona and interstellar gas cloud. It isalso instructive to contemplate whether the neutral mass density needs to be taken intoaccount when determing the Alfven speed (i.e., if plasma is not fully ionized, do neutralshave enough time to respond to the motion of the ions).

Figure 6.7: Japanese Hinode (“sunrise”) observations of fluctuating plasma that couldbe an indication of Alfven waves heating the corona. The presence of Alfven waves isdeduced by tracking the motions of coronal plasma.

Contemplate: The existence of Alfven waves was first suggested by Hannes Alfven(as the name hints!). In 1942 he noted that a new type of wave should be found inmagnetized plasmas that may be of importance to solar physics. It is instructive to lookAlfven’s original paper (click here) which also demonstrates his profound physical insight.

Fast and slow MHD (Alfven) waves.

The other solutions of the matrix Eq. 6.55 are obtain by setting the determinant of thecoefficients of V1x:n and V1z zero. The result is (Exercise 6.5)

v2p =1

2(v2s + v2A)± 1

2[(v2s + v2A)2 − 4v2sv

2A cos2 θ]1/2 . (6.60)

The solution with the larger phase speed is called the fast MHD wave and with thelower phase speed the slow MHD wave.

http://www.nature.com/physics/looking-back/alfven/index.html


The top panels of Figure 6.8 show the phase speeds of Alfven wave and the slow andfast MHD waves as a function of the angle between the wave vector and the backgroundmagnetic field (θ). The solution depends on the ratio between the Alfven speed vA andthe sound speed vs. Another way to illustrate wave properties is to use the wave normalsurfaces, see the bottom panels of Figure 6.8.

slow MHD

fast MHD

Alfvén

vA

vs

B0

(vA2+vS

2)1/2

slow MHD

fast MHD

Alfvén

vA

vs

B0

(vA2+vS

2)1/2

fast MHD

vA

vs

(vA2+vS

2)1/2

0 30° 60° 90°q

w/k

k||B0 k^B0

vA > vs

fast MHD

vA

vs

(vA2+vS

2)1/2

0 30° 60° 90°q

w/k

k||B0 k^B0

vA < vs

Figure 6.8: Top) Phase speeds as a function of the propagation angle and bottom) wavenormal surfaces for Alfven, fast and slow MHD waves. Cases with vA > vs and vA < vsare shown separately.

A wave normal surface shows the phase speed as a function of the angle betweenthe wave propagation direction and the magnetic field θ, i.e., it describes how the phasespeed varies with respect to the magnetic field direction. Actually Figure 6.8 shows 2Dcuts of the surfaces. Assuming that the system is gyrotropic, the surface is found byletting 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 pertur-bation V1 is in the xz-plane (Figure 6.5). Now k · V1 6= 0, and hence, the waves arecompressional and associated with density pertubations (Eq. 6.57).

Investigate first the propagation perpendicular to the magnetic field. Eq. 6.60 andFigure 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 =√v2A + v2s (6.61)

i.e., the phase speed depends both on the sound speed and the Alfven speed. Thiswave is called the magnetosonic wave. Magnetosonic speed defined by Eq. 6.61 is themaximum 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 Alfven speed canactually be larger than the speed of light (vA > c). In such cases the non-relativisticMHD approximation breaks down and the displacement current cannot be neglected.The modification to the dispersion relation for a mode propagating perpendicular to themagnetic field (θ = 90) including the displacement current is derived in Exercise 6.6.

ω2

k2=

v2s + v2A1 + v2A/c

2

Obviously this reduces to ω2/k2 = v2s +v2A for v2A c2, i.e., at the non-relativistic MHDlimit.

The eigenvector corresponding to this root is (0, 0, V1z) and from linerized continuityEq. 6.57 we obtain the density perturbation ρ1 = ρ0(V1z/vp). The linearized convectionequation (Eq. 6.58) gives

B1 =V1ω/k

B0 . (6.62)

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 thewave vector and the wave magnetic and electric fields are all perpendicular to eachother. However, mass flow and density fluctuate along the wave vector, and thus, themagnetosonic wave is longitudinal (Figure 6.9). It is also called “magnetoacoustic”,which derives from this property.


In the case vA vs the phase speed of the magnetosonic wave (Eq. 6.61) becomesvp ≈ vA, i.e., it approaches the Alfven speed. However, the wave is compressional and isoften called the compressional Alfven wave. Note that this situation corresponds to thecold-plasma (zero temperature) limit where vs goes to zero. From Eq. 6.60 it is clearthat 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 thecase where the magnetic field dominates (vA > vs) the dispersion equation gives for thefast MHD wave

vp = vA , (6.64)

i.e., the wave reduces to the Alfven 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 simplestdisturbance that can propagate in a collisional medium. The wave vector k is normal tothe pressure front and the restoring force is the pressure gradient. Because the magneticfield does not restrict the particle motion along the magnetic field in a plasma, soundwaves can propagate also in a magnetized plasma, see Figure 6.10.

Figure 6.10: Longitudinal sound wavepropagates along the magnetic field in acompressible and magnetized plasma. .

Figure 6.8 illustrates that in the case of parallel propagation either the fast or slowmode MHD wave reduces to the Alfven wave and the other one to the sound wavedepending on the ratio between the vA and vs. The fast MHD wave has always largerphase speed than the slow MHD wave.

A compressive MHD wave can steepen into a shock wave when the disturbance prop-agates faster than the characteristic speed of the medium. In space and fusion plasmasshocks may be produced by explosions (e.g., solar flares, supernovae, inertial confine-ment fusion), by a disturbance moving through a fluid with its speed exceeding the localcharacteristic information speed. Some examples are a coronal mass ejection movingthrough the solar wind faster than the local magnetosonic speed and the encounter ofsupersonic and super-Alfvenic fluid with a stationary object (e.g., the formation of theEarth’s bow shock).

6.5. MAGNETIC RECONNECTION 101

6.5 Magnetic reconnection

One of the most important plasma physical phenomena arises when the frozen-in con-dition breaks down. If the ideal MHD assumption holds, two initially (magnetically)separate plasma elements can never mix. For example, the Earth’s magnetosphere wouldalways stay closed to the solar wind. According to Eq. 6.9 the Reynolds number de-creases when the plasma flow speed or the length scale of field gradients decreases orwhen resistivity increases. When the Reynolds number becomes sufficiently small themagnetic field starts to diffuse. In a particle description the break-down of the frozen-incondition can be understood by the GC approximation becoming invalid, i.e., chargedparticles cease to follow the magnetic field. (As discussed earlier, this usually does nothappen simultaneously for electrons and ions.).

Figure 6.11: Reconnection between two plasma domains with oppositely oriented fieldsthat are flowing towards each other. Open arrows indicate the direction of the plasmaflow.

Figure 6.11 shows two ideal MHD plasma regions with oppositely oriented magneticfields flowing towards each other. Such situation arises for example at the interface be-tween the Earth’s magnetosphere and the solar wind when the interplanetary magneticfield is southward (at the nose of the magnetosphere the magnetospheric field pointsalways to the north). A thin current sheet forms between the regions introducing alarge magnetic field gradient. The exact microphysics that occurs in the thin currentsheet is not yet well-understood. However, if there are processes that increase resistiv-ity, diffusion can start leading to the reorganization of the plasma and magnetic field.Plasma elements that were originally in separate regions may now become magneticallyconnected.

The change of connection between the plasma elements is called magnetic reconnec-tion. The importance of reconnection lies in its ability to change the topology of themagnetic field and to convert magnetic energy to kinetic and thermal energy.

The concept of reconnection was first presented by Ronald Giovanelli in the 1940sto explain particle acceleration in solar flares. In a solar flare a huge amount of energyis released from the Sun in time-scales of only a few minutes. We cover here briefly themost elementary reconnection models.


Bz (x)

Jy (x)z

Ey

Vi

Vi

2l

x

+B0

-B0

Figure 6.12: 1-dimensional current sheet. y-directed electric field has been added thatbrings new plasma and magnetic flux towards the current sheet and maintains the bal-ance between diffusion and convection.

Diffusion in a 1-dimensional current sheet was treated in Section 6.2.1. To achieve asteady-state situation new magnetic flux and plasma have to be brought to the systemto replace the annihilated flux. This can be achieved by adding an electric field as shownin Figure 6.12. Outside the current sheet the ideal Ohm law gives Ey = ViB0, where Viis 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 andconvection. Assuming that the width of the current sheet is 2l Ampere’s law gives theelectric current Jy = B0/µ0l. Hence, the width of the current sheet can be written as:

2l =2

µ0σV. (6.66)

The scenario explained above is unphysical. What happans to the plasma that isbrought to the current sheet? The solution is to add an additional dimension as shownin Figure 6.13. This is the famous Sweet–Parker model formulated in the 1950s. In theSweet–Parker model the magnetic field annihilates in a finite domain called a diffusionregion (gray area in Figure 6.13). Plasma and the magnetic field flow away from theboundaries of the diffusion region. Figure 6.13 also illustrates that the plasma elementsthat were originally not magnetically connected (blue and red circles are at different fieldlines before entering the diffusion region) become connected after exiting the diffusionregion. In the outflow region the magnetic field is thus weaker and the plasma flow speedlarger than in the inflow region. It is important to note that the frozen-in conditionbreaks-down in the diffusion region but is valid outside. Hence, the diffusion region isthe region where the rearrangement of the magnetic field occurs.

The speed of reconnection, i.e., the reconnection rate, is typically expressed as theelectric field in the inflow region. It is an important quantity determining the inflowspeed. Estimates for the inflow and outflow speeds can be achieved by assuming incom-

6.5. MAGNETIC RECONNECTION 103

Vi

Vi

VoVo

E

E

Bi

Bi

inflow

inflow

outflowoutflow

2L

2l

Figure 6.13: Sweet-Parker reconnection. The diffusion region is shown by gray.

pressible flow (ρi = ρo = ρ), conservation of mass (ViL = Vol) and that all inflowingelectromagnetic energy transforms to kinetic energy.

Inflowing electromagnetic energy can be calculated from the inflowing Poynting flux:

|S| = |E×H| = EBiµ0

=ViB

2i

µ0. (6.67)

The mass that flows in a unit time to the diffusion region (ρVi) will be acceleratedto speed Vo. Hence, the change in energy in unit time and unit area is:

4W =1

2ρVi(V

2o − V 2

i ) . (6.68)

Equating the energy increase with the inflow energy flux and noting that Vo Vi gives

ViB2i

µ0=

1

2ρViV

2o (6.69)

⇒V 2o =

2B2i

µ0ρ= 2v2Ai . (6.70)

Thus the Alfven speed in the inflow region describes the speed of the outflowing plasma(under the used approximations within a factor of

√2). Using ViL = V0l 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 = µ0σvAiL is the Reynolds number calculated using the inflow Alfven speedknown as the Lundquist number. It is easy to show (Exercise 6.7) that half of theincoming magnetic energy is transformed to heat and the other half causes accelerationof particles. In space plasmas RmA is usually very large and thus the inflow and the


reconnection rate in the Sweet–Parker model is very slow. For a solar flare it would takedays to erupt, not minutes as the observations indicate.

The slow reconnection speed in the Sweet–Parker model can be traced to the propertythat all energy conversion occurs in a diffusion region whose length is much larger thanthe width of the outflow region. In 1964 Harry Petschek proposed that significantlyfaster reconnection rates can be obtained by introducing a vanishingly small diffusionregion. He added two slow mode shocks, i.e. slow MHD waves steepened to shocks, thatemanate from the diffusion region. The shocks deviate the plasma flow and magneticfield. The Petschek model is presented schematically in Figure 6.14.

Figure 6.14: Petschek fast reconnection. Two slow mode shocks (blue) emanate froma vanishingly small diffusion region and deviate the plasma flow (red lines) and themagnetic field (black lines).

Contemplate: Where does the energy conversion occur in the Petschek model? Wheredoes the magnetic field connectivity change?

The properties of MHD shocks are beyond our discussion here. They will be treatedin the course on space applications of plasma physics. However, a rigorous analysisshows that the inflow speed in the Petscheck model can be up to 10% of vAi, whichallows 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 astrophysicalphenomena and in fusion research. For example, solar prominences are huge structuresthat can remain stable up to several solar rotations before erupting and much of thefusion research deals with plasma confinement. We will start by investigating the MHDmomentum equation (Eq. 6.2). Assuming scalar pressure (∇ · P → ∇p) and time-independent (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 onsurfaces of constant pressure.

From the above equilibrium condition one can calculate the current perpendicular toB:

J⊥ =B×∇pB2

. (6.73)

This current is often called the diamagnetic current . It arises from the plasma pressuregradient. In the particle description the perpendicular current is the sum of all currentelements in the plasma and contains contributions from the magnetic drifts (gradient andcurvature drift related currents), polarization current, and the magnetization current.The magnetization current is caused by an inhomogeneous plasma density:

JM = ∇×M . (6.74)

Here the magnetization M is the density of magnetic moments µ (see Eq. 3.14). Figure6.15 shows the particle picture of the magnetization current. If the plasma density isnon-uniform, the gyration velocities do not sum to the zero, and hence, a net currentarises.

high density plasma

low density plasmacurrent

total

y

x

B

Figure 6.15: Single particle interpretation of the diamagnetic current.

Using Ampere’s law we can write the magnetic force in the form:

J×B = −∇(B2

2µ0

)+

1

µ0(B · ∇) B . (6.75)

The magnetic force consist of two separate terms. The first term on the RHS is thegradient of the magnetic energy density, i.e., of the magnetic pressure:

pB = B2/(2µ0) . (6.76)


The second term describes the tension force arising from the inhomogeneities of themagnetic field. This latter term can be divided into two components:

(B · ∇) B = Bd

ds(Bs) = B2 ds

ds+B

∂B

∂ss (6.77)

= B2 n

RC+ s

∂

∂s

(B2

2

),

where s is the unit vector along the magnetic field and RC is the radius of curvature. Itis now evident that:

1. The first term is anti-parallel to the radius of the curvature of field lines. Therelated 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 configura-tions. Sketching the magnetic field configurations helps to visualize how curvature andgradients 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 mo-mentum equation:

∇(p+

B2

2µ0

)=

1

µ0(B · ∇) B . (6.78)

See Exercise 6.9 for a demonstration of how in a simple magnetic field configurationBx = y and By = x the magnetic pressure and tension balance each other.

Assuming homogeneous magnetic field the sum of the magnetic and plasma pressuresis constant

∇(p+

B2

2µ0

)= 0 . (6.79)

The plasma beta

β =2µ0p

B2(6.80)

expresses the ratio of the plasma and magnetic pressures. It is one of the importantdimensionless parameters used to characterize plasmas.

An example of a MHD equilibrium configuration is the Harris current sheet we en-countered in Section 3.6.2. As discussed earlier, the Earth’s magnetotail, which canstay stable for long time periods, can be described by a Harris current sheet. In a1-dimensional Harris current sheet the magnetic field (assumed here to be in the z-direction) is given by:

B = B0 tanh

(z

L

)ey . (6.81)

6.6. MAGNETOHYDROSTATIC EQUILIBRIUM AND STABILITY 107

The pressure is given by

p = p0 cosh−2z

L, (6.82)

where p0 = B20/(2µ0)

As illustrated in Figure 6.16 (see also Exercise 6.10) the variations in the magneticfield and plasma pressure over the Harris current sheet balance each other.

Figure 6.16: Magnetic field and pressure variations in the Harris current sheet.

B

J

q r

z

B

J

q-pinch Z-pinch

Figure 6.17: Left) θ-pinch, and Right) Z-pinch

Other examples of 1-dimensional equilibrium configurations are θ- and Z-pinchesshown in Figure 6.17. In both cases it is convenient to use cylindrical coordinates. In aθ-pinch cylindrical coils drive an electric current and the magnetic field is axial, while in


a Z-pinch the electric current is axial and the magnetic field poloidal. The equilibriumconditions (∇p = −J×B) are (Exercise 6.11) :

d

dr

(p+

B2z

2µ0

)= 0 (6.83)

d

dr

(p+

B2θ

2µ0

)+B2θ

2µ0= 0 . (6.84)

6.7 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 plasmais zero. According to Eq. 6.75 in a force-free field the magnetic pressure gradient∇(B2/2µ0) is balanced by the magnetic tension force µ−10 (B · ∇) B. In real situationsthe force-free equilibrium is always an approximation, but often a very good one, tothe momentum equation. It is also evident from Eq. 6.85 that in a force-free field theelectric current flows along the magnetic field. Such currents are commonly called asfield-aligned currents (FAC).

Using Ampere’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 andthus difficult to solve.

The field-alignment of the electric current can be expressed as

∇×B = µ0J = 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

∇2B + α2B = 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 Lundquistin 1950 in terms of Bessel functions J0 and J1:

BR = 0 (6.91)

BA = B0J0

(α0r

r0

)BT = ±B0J1

(α0r

r0

), (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 heliceswhose pich angle increases from the axis (Figure 6.18). r is the radial distance from theflux rope axis, r0 is the radius of the flux rope and B0 is the maximum magnetic fieldmagnitude at the center of the flux rope (r = 0).

Figure 6.19: Left) Erupting coronal mass ejection whose structure is a magnetic fluxrope. Image taken by Solar Dynamic Observatory. Courtesy: NASA. Right) HubbleSpace Telescope image of a filamentary nebula (Dahlgren et al., 2007).

Flux ropes are common in space, astrophysical and fusion plasmas. The left-handpart of Figure 6.19 shows an erupting solar plasma cloud whose structure is often ap-proximated with a force-free flux rope. These plasma clouds maintain more or less theirintegrity while traveling away from the Sun to the orbit of the Earth and beyond and


they are the main drivers of severe magnetospheric disturbances. The right-hand partof Figure 6.19 shows a Hubble image of a planetary nebula. The substructures in thisnebula 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

∇2Ψ = 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 April2010. 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 originalsynoptic magnetogram. White areas indicate the maximum-strength positive flux andblack maximum-strength negative flux. Courtesy: NSO/GONG.

For example, the Sun’s magnetic field structure is often modelled by the so-calledPotential Field Source Surface (PFSS) model (Figure 6.20). The magnetic field is com-puted from the Laplace equation using spherical coordinates from the photosphere tothe “source surface”, nominally chosen to be at 2.5 Solar radii. At the source surfacethe Sun’s magnetic field is assumed to be purely radial. The inner boundary conditionsare obtained from solar magnetograms. Thus, PFFS assumes that there is no electriccurrent in the corona.

6.8. EXERCISES: MAGNETOHYDRODYNAMICS 111

6.8 Exercises: Magnetohydrodynamics

1. Derive the induction equation for the magnetic field in a case where the conduc-tivity 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 closedloop co-moving with plasma

dΦ

dt=

d

dt

∫B · dS = 0

is constant

4. Compute the Alfven speed in the following cases

(a) Earth’s ionosphere ne = 1011 m−3, B = 50µT, ions assumed to be mostlyO+.

(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 theionization degree is 1%.

5. Derive the phase speeds of the fast and slow Alfven waves(ω

k

)2

=1

2

(v2s + v2A

)± 1

2

[(v2s + v2A

)2− 4v2sv

2A cos2 θ

]1/2starting from the dispersion equation

−ω2V1 +(v2s + v2A

)(k ·V1) k

+ (k · vA) [(k · vA) V1 − (vA ·V1) k− (k ·V1) vA] .

6. Consider the propagation of Alfven waves taking the displacement current intoaccount. That is, start from the same equations as on the lectures but replaceAmpere’s law by

∇×B = µ0J +1

c2∂E

∂t.

Derive the dispersion equation for the mode propagating perpendicular to themagnetic field into the form

ω2

k2=

v2s + v2A1 + v2A/c

2.


7. Consider the Sweet–Parker reconnection model. Show that half of the incomingmagnetic energy is transformed to heat and the other half causes acceleration ofparticles.

8. Calculate the magnetic force J ×B for the following cases. Sketch also followingmagnetic field configurations and indicate the direction of magnetic forces in eachcase.

(a) B = xey

(b) B = ex + xey

(c) B = yex + xey

(d) B = reθ

9. Show that in the magnetic field configurationBx = y,By = x the magnetic pressureand tension balance each other. Show that if the configuration is stretched in they direction: Bx = y,By = α2x, where α2 > 1, this causes in certain regions a netforce 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 thatthe current density is

Jy(z) =B0

µ0hsech2

(z

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, thatMHD assumptions are valid and the electric current flows only inside the cylinderof radius R. Write the condition for the hydromagnetic equilibrium in cylindricalcoordinates. Calculate and plot the profiles of plasma pressure and the magneticfield, when the current is constant inside the cylinder. How would the resultchange, if the current would flow on the surface of the cylinder only?

Figure 6.21: Equilibriumpinch for Exercise 6.11

Βθ(r)

θ

rR

p=0p(r)=0

zJz(r)

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 andobserving wave emissions in plasma can give information on the plasma properties. Forexample, plasma density can be calculated easily from the plasma frequency and themagnetic field magnitude from the gyro frequency. In the previous chapter we inves-tigated MHD waves and considered frequencies well below the ion gyro and plasmafrequencies. When the frequency of the wave increases, one needs to take into accountthat the ion and electron dynamics become different, and hence, the one-fluid MHDdescription becomes invalid. This chapter investigates the wave propagation in the coldplasma limit. In reality the temperature of a plasma is never zero, but the temperatureeffects can be neglected if the wave propagates faster than the plasma thermal speed√

2kBT/m. As a consequence, cold plasma has zero pressure and there are no wavesrelated to pressure fluctuations, such as sound waves. At high frequencies, well abovethe ion gyro frequency, the ions can be considered as an immobile background as theycannot respond quickly enough to the wave. Note that we now consider much fasterfluctuations than in MHD, and thus, we need to take into account the displacementcurrent in Maxwell’s equations.

We start by deriving the general form of the dispersion equation. We proceed toinvestigate the waves that propagate exactly parallel or perpendicular to the magneticfield. 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 studyof general electromagnetic waves (see Appendix 9.3). To derive the cold plasma disper-sion equation we start from the density continuation equation, equation of motion and

113

114 CHAPTER 7. COLD PLASMA WAVES

Maxwell’s equations

∂n

∂t+∇ · (nV) = 0 (7.1)

m∂u

∂t= e(E + u×B) (7.2)

∇×E = −∂B

∂t(7.3)

∇×B− 1

c2∂E

∂t= µ0J = µ0

∑s

ensus = ~σ ·E (7.4)

∇ ·E =1

ε0

∑s

ens (7.5)

∇ ·B = 0 . (7.6)

In Eq. 7.4 we have used Ohm’s law with conductivity being a second rank tensor ~σ. sindexes 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∂t

+∇ · (n0u1) = 0 (7.11)

m∂u1

∂t= e(E1 + u1 ×B0) (7.12)

∇×E1 = −∂B1

∂t(7.13)

∇×B1 −1

c2∂E1

∂t= µ0J = µ0

∑s

ensu1s (7.14)

J = ~σ ·E1 (7.15)

∇ ·E1 =1

ε0

∑s

en1s (7.16)

∇ ·B1 = 0 . (7.17)

By taking the curl from Eq. 7.13 and using Eq. 7.14 we obtain

∇×∇E1 =1

c2∂2E1

∂t2− µ0

∂J

∂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’slaw (Eq. 7.14) to eliminate J from Eq. 7.18.

This leads to the homogeneous wave equation:

n× (n×E1) + ~K ·E1 = 0 , (7.19)

where n = ck/ω is the index of refraction and ~K is the dielectric tensor

~K = ~1− ~σ

iωε0. (7.20)

7.2 Wave propagation in non-magnetized plasma

Let us first consider a simple case where the background magnetic field B0 is zero. Thelinearized equation of motion now becomes

m∂u

∂t= eE . (7.21)

Note that here the subscript ”1” has been dropped for simplicity. Assuming harmonictime dependence this reduces to

m(−iω)u = eE . (7.22)

We use this equation to eliminate u from J =∑sensus and to obtain

J =∑s

n0e2

(−iω)msE , (7.23)

from which we can now identify the conductivity tensor

~σ = ~1∑s

n0e2

(−iω)ms. (7.24)

Using the definition of the plasma frequency and ωpe ωpi we obtain

i

ε0ω~σ = ~1

ω2p

ω(7.25)

and the homogeneous wave equation becomes

c2k× (k×E) = (ω2 − ω2p)E . (7.26)


Let us choose the wave vector k to be parallel to the z-axis. From Eq. 7.26 we nowobtain a matrix equation −c2k2 + ω2 − ω2

p 0 0

0 −c2k2 + ω2 − ω2p 0

0 0 ω2 − ω2p

ExEyEz

= 0 , (7.27)

from which we get the dispersion equation by setting the determinant of the matrix tozero

(−c2k2 + ω2 − ω2p)

2(ω2 − ω2p) = 0 . (7.28)

One of the roots is evidently ω = ωp. The electric field associated with this mode is inthe z-direction. Since we selected k to be in the z-direction, the electric field perturbationis parallel to the wave propagation. Hence, the wave is longitudinal : k · E1 6= 0 andk× E1 = 0. By assuming harmonic dependencies the linearized Gauss’s law (Eq. 7.16)becomes

ρ1 = iε0k ·E1 , (7.29)

and we see that the wave is associated with charge density fluctuations. In turn, assumingharmonic dependencies the linearized Faraday’s law (Eq. 7.13) becomes

ik×E1 = −iωB1 (7.30)

→ B1 =1

ωk×E1 ,

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 solutiondescribes oscillation at the plasma frequency we encountered in Section 2.3.

Another root of Eq. 7.28 is

ω2 = ω2p + c2k2 . (7.31)

The electric field has now components in the x and y-directions. Thus, the electricfield is perpendicular to k and from the linearized Gauss’s law we see that the wave isnon-compressional and transverse (the electric field perturbation is perpendicular to thedirection of propagation). The magnetic field associated with the wave is obtained fromEq. 7.30. The wave vector as a function of frequency is

k = ±1

c

√ω2 − ω2

p . (7.32)

Figure 7.1 displays the dispersion equation. At high-frequencies the solution ap-proaches the vacuum electromagnetic wave with ω = ck, i.e., its phase and group speedsapproach the speed of light. The interpretation is that the frequency of the wave be-comes so high that it does not interact with the plasma. In fact, it interacts but onlyvery weakly.

7.2. WAVE PROPAGATION IN NON-MAGNETIZED PLASMA 117

Figure 7.1: The dispersion equa-tion for an electromagnetic wavein a cold plasma where the back-ground 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 tofacilitate wave growth, this solution is unphysical. The solution with a positive imaginarypart makes the electric field to decay exponentially and the wave is said to cut-off at theplasma frequency. What happens physically is that when the wave frequency approachesthe plasma frequency the wave forces electrons to oscillate at the plasma frequency. Theoscillating electrons re-radiate the wave energy and the wave is reflected.

Figure 7.2: Reflection of electromagnetic waves from the ionosphere. Note that alsowaves with higher frequency than the maximum plasma frequency are affected by theplasma and refracted. The ionospheric density profile can be determined by sendingwaves at different frequencies and measuring the time the wave returns back. Such adevice 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 thusthe plasma frequency increases. As a response to the wave electric field electrons in the


ionosphere start to oscillate and they re-radiate the original energy. The total reflectionof 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 backgroundmagnetic field. The magnetic field introduces anisotropy, which leads to many new wavemodes. When a magnetic field is present, particles perform Larmor motion around themagnetic field lines. This introduces new possibilities to wave cut-offs and resonances. Inthis Section we first derive a general form of the dispersion equation and then proceedto investigate the so-called principal modes, i.e., waves that propagate either parallelor perpendicular to the magnetic field. Finally we will study wave propagation at anarbitrary angle.

7.3.1 Derivation of general dispersion equation

First, we need to determine the conductivity and dielectric tensors. Let the magneticfield to be in the z-direction. Assuming harmonic time dependencies and using the gyrofrequency ωcs = esB0/ms the linearized equation of motion (Eq. 7.12) can be nowwritten in the matrix form:

−iω −ωcs 0ωcs −iω 00 0 −iω

usxusyusz

=esms

ExEyEz

. (7.34)

Inversion of the matrix equation gives

usxusyusz

=esms

−iωω2cs − ω2

ωcsω2cs − ω2

0

− ωcsω2cs − ω2

−iωω2cs − ω2

0

0 0i

ω

ExEyEz

. (7.35)

Similarly as in the case with B0 = 0 the conductivity tensor is obtained by calculatingthe electric current J =

∑sesns0us1 = ~σ ·E.

~σ =∑ ns0e

2s

ms

−iωω2cs − ω2

ωcsω2cs − ω2

0

− ωcsω2cs − ω2

−iωω2cs − ω2

0

0 0i

ω

. (7.36)

7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 119

The dielectric tensor is now

K =

S −iD 0iD S 00 0 P

, (7.37)

where

S = 1−∑s

ω2ps

ω2 − ω2cs

(7.38)

D =∑s

ωcsω2ps

ω(ω2 − ω2cs)

(7.39)

P = 1−∑s

ω2ps

ω2. (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−∑s

ω2ps

ω2

(ω

ω + ωcs

)(7.41)

L = 1−∑s

ω2ps

ω2

(ω

ω − ωcs

). (7.42)

This is a useful division since electrons and ions respond in a different way to the wave.

z

B0

x

y

n,k

q

Figure 7.3: The choice of directions ofthe background magnetic field and theindex of refraction (wave vector) for awave propagating in a cold plasma.

Let us choose the magnetic field to be along the z-direction and the index of refractionn (and thus k) to be in the xz-plane (Figure 7.3). The homogeneous wave equation nowbecomes S − n2 cos2 θ −iD n2 cos θ sin θ

iD S − n2 0n2 cos θ sin θ 0 P − n2 sin2 θ

ExEyEz

= 0 . (7.43)

As before, we obtain the dispersion equation by setting the determinant to zero.

D(n, ω) = An4 −Bn2 +RLP = 0 , (7.44)


where

A = S sin2 θ + P cos2 θ (7.45)

B = RL sin2 θ + PS(1 + cos2 θ) . (7.46)

The dispersion equation can be modified to a useful form by solving for tan2 θ as afunction of n2

tan2 θ =−P (n2 −R)(n2 − L)

(Sn2 −RL)(n2 − P ). (7.47)

When a wave propagates through plasma, it may encounter regions of changingplasma frequency and gyro frequencies. In magnetized plasma we can find two cases inwhich 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 andbecomes evanescent. Physically, the wave is reflected and no energy is absorbed inthe 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 = −PS. (7.48)

7.3.2 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)

n2 = R (7.50)

n2 = L . (7.51)

The first root represents simple plasma oscillation (see Eq. 7.40), and thus, a non-propagating wave.


By inserting θ = 0 in Eq. 7.43 we can find the electric field eigenvector associatedwith 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 theroots n2 = R and n2 = L correspond to transverse waves. In the same way as in thecase of non-magnetized waves, we can confirm from the linearized Faraday’s law that thewaves are electromagnetic (i.e., they have non-zero magnetic field). From the linearizedGauss’s law we see that there are no charge density fluctuations.

We also see that the only difference between the eigenvectors corresponding to theroots n2 = R and n2 = L is the sign of the y-component. n2 = R corresponds to a wavethat rotates in the right-handed sense with respect to the magnetic field while n2 = L toa wave that rotates in the left-handed sense (confirm! In plasma physics the conventionof RH and LH is opposite to that used in optics). Hence, the solutions correspond tothe right-handed (R) and left-handed (L) polarized modes, respectively.

Right-handed mode

From 7.41 we obtain

n2R = R = 1−ω2pi

ω(ω + ωci)−

ω2pe

ω(ω − ωce). (7.54)

Consequently, the R-mode has a resonance (n → ∞) when the wave frequency ap-proaches the electron cyclotron frequency ω = ωce. This is because electrons rotatearound 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 (Exercise7.2)

ωR=0 ≈ωce2

[1 +

√1 + 4ω2

pe/ω2ce

]. (7.55)

The cut-off is divided to two branches depending on the density. At the low densitylimit (ωp ωc) the cut-off becomes

ωR=0 ≈ ωce(1 + ω2pe/ω

2ce) (7.56)

and at the high density (ωp ωc) limit

ωR=0 ≈ ωpe + ωce/2 . (7.57)


Left-handed mode

For the L-mode we obtain from 7.41

n2L = L = 1−ω2pi

ω(ω − ωci)−

ω2pe

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 ≈ωce2

[−1 +

√1 + 4ω2

pe/ω2ce

]. (7.59)

At low density limit we obtainωL=0 ≈ ω2

pe/ωce, (7.60)

and at the high density limit

ωL=0 ≈ ωpe − ωce/2 , (7.61)

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 andωL=0) the solution to the wave dispersion equation is called the free-space mode. Belowelectron and ion cyclotron frequencies the waves are called the cyclotron modes. At lowfrequencies (ω → 0) L- and R-modes merge and the dispersion becomes that of the shearAlfven wave n2 → c2/v2A we encountered in Section 6.4.2.

Faraday rotation

A linearly polarized plane wave can be expressed as a sum of left- and right-hand circu-larly polarized waves (R- and L-modes having equal amplitudes, E0). If we assume thatthe wave is linearly polarized along the x axis, and that the wave propagation (k) andthe background magnetic field (B0) are along the z-axis, we can write

E = E0[(eikRz + eikLz)ex + i(eikRz − eikLz)ey]e−iωt . (7.62)

The ratio of the Ex and Ey components is

ExEy

= cot

(kL − kR

2z

). (7.63)

Hence, due to different phase speeds of R- and L-modes the linearly polarized wave thatis 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 densityand magnetic field of the plasma. Considering frequencies above the plasma frequency


Figure 7.4: Wave modes propagating parallel to the magnetic field in the limit of highplasma 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

dz=

−e3

2m2eε0c ω

2neB0 (7.64)

and the total rotation from the source to the observer is

φ =−e3

2m2eε0c ω

2

∫ d

0neB · ds . (7.65)

The integral is calculated along the wave propagation path. The total rotation thusdepends on both the density and magnetic field of the medium. Exercise 7.3 appliesFaraday rotation to estimate the distance to a distant pulsar when the density of theinterstellar plasma is known.

Contemplate: Faraday rotation is an important tool in astronomy. Find an examplewhere Faraday rotation is used to obtain information on the physical properties of an


astronomical object. Pay attention to the fact that either density or magnetic field hasto be known from other methods.

Whistler waves

The investigation of dispersion characteristics of R-mode waves reveals an interestingfeature. Figure 7.4 shows that the R-mode propagates also in the region between electronand ion gyro frequencies. In this domain the dispersion equation can be approximatedas (Exercise 7.4):

k =ωpec

√ω

ωce, (7.66)

which gives the phase and group speeds

vp =ω

k=

c√ωce

ωpe

√ω (7.67)

vg =∂ω

∂k=

2c√ωce

ωpe

√ω . (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 wasobserved in telegraph lines during World War I. It took several decades before this phe-nomenon was explained. In 1953 L.R.O. Storey suggested that the sound was producedby waves that are propagating along the magnetic field lines from one hemisphere of theEarth to the other (Figure 7.5) and the whistling sound is the consequence of differentfrequencies arriving at different times. These whistler waves are produced by lightingstrokes 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


can be obtained from (see Eq. 7.68)

t(ω) =

∫ds

vg=

∫wpe(s)

2c√ωωce

ds , (7.69)

where ds is the line element along the magnetic field. Hence, lower frequencies arrive toan 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 homogeneouswave equation (Eq. 7.43) becomes S −iD 0

iD S − n2 00 0 P − n2

ExEyEz

= 0 (7.70)

and

tan2 θ =−P (n2 −R)(n2 − L)

(Sn2 −RL)(n2 − P )→∞ . (7.71)

The roots are now

n2 = P (7.72)

n2 =RL

S(7.73)

and the corresponding electric field eigenvectors are:

En2=P = (0, 0, E0) (7.74)

(7.75)

qEn2=RL/S = (iD

SE0, E0, 0) . (7.76)

Ordinary mode

The wave mode associated with the first root n2 = P is called the ordinary (O) mode.

n2O = P = 1−ω2pi

ω2−ω2pe

ω2≈ 1−

ω2pe

ω2. (7.77)

The electric field of the ordinary mode is along the background magnetic field, andthus, the wave vector k is perpendicular to wave electric field. The dispersion equationabove shows that the O-mode is not affected by the magnetic field (the particle motionis parallel to the magnetic field, and hence the magnetic part of the Lorentz for vanishesv × B0 = 0). Physically, the O-mode corresponds to the high-frequency transverseelectromagnetic wave. It is linearly polarized.

The O-mode has the cut-off (n2 → 0) when the wave frequency approaches theelectron plasma frequency, i.e., at ω = ωpe.


Extraordinary mode

The second mode is called the extraordinary mode. Now the wave electric field is per-pendicular to the background magnetic field (Eq. 7.75), and thus, the electric fieldhas components both parallel (longitudinal) and perpendicular (transverse) to the wavevector, see Figure 7.6. Hence, the X-mode has both electrostatic and electromagneticcharacteristics. The wave magnetic field can be calculated from the Faraday’s law. Ac-cording to Eq. 7.75 the X-mode is elliptically polarized. Figure 7.7 shows the solutionof n2 in the (ω, k)-space for ordinary and extraordinary modes.

z

B0

xy

E1

n,k

Ordinary (O) mode Extraordinary (X) mode

B1

z

1 1

1

w= ´B k E

B0

xy

E1n,k

B1

Figure 7.6: Magnetic field, electric field and the wave vector directions for O- and X-modes.

wUH

wLH

upper hybrid resonance

lower hybrid resonance

wpe

wX,R=0

wX,L=0

w

k

X-mode

O-mode

X-mode

magnetosonic mode

Figure 7.7: A plot of wave frequency as a function of wave number for ordinary andextrordinary modes.

The X-mode has two cut-offs when R = 0 and L = 0 (see Section 7.3.2). Calculationof resonances and cut-offs for extraordinary waves is a tedious task. The resonances


occur at S = 0 (see Eq. 7.38)

S = 1−ω2pe

ω2 − ω2ce

−ω2pi

ω2 − ω2ci

= 0 , (7.78)

and they are called hybrid resonances. The number of resonances depends on the ionspecies involved. Here we have assumed that the plasma consist of electrons and onepositive ion species. The resonance that occurs at the highest frequency, above both theelectron gyro and plasma frequencies, is called the upper hybrid frequency (ωUH). Thefrequencies near ωUH are so high that one can neglect the ion dynamics from Eq. 7.78and using ωpi ωpe the upper hybrid resonance is at

ω2UH ≈ ω2

pe + ω2ce . (7.79)

The lower hybrid resonance occurs between the electron and ion cyclotron frequen-cies. Assuming that ωce ωLH ωci one obtains the lower hybrid resonance at

ω2LH ≈

ω2ci + ω2

pi

1 + (ω2pe/ω

2ce)≈ ωceωci

(ω2pe + ωceωci

ω2pe + ω2

ce

). (7.80)

Further approximations are often made at the low and high density limits. In thehigh density limit ω2

c ω2p and we obtain

ωLH →√ωceωci . (7.81)

Both electrons and ions participate in the resonance oscillation. The lower hybrid res-onance at the high density limit is particularly important since the wave can be inresonance both with electrons and ions. This can facilitate the energy transfer betweenions and electron. For instance, in fusion devices these waves are in the microwaverange and microwave techniques are used to heat the plasma through the lower hybridresonance.

In the low density limit ω2p ω2

c 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 al-ready in Section 6.4.2. The cold plasma theory introduces a correction:

ω2

k2=

v2s + v2A1 + v2A/c

2, (7.83)

which guarantees that the group velocity of the wave remains below the speed of lighteven in a plasma where vA > c (in cold plasma vs is, of course, negligible).


7.3.4 Propagation in an oblique angle

Principal modes R, L, O ja X can be uniquely defined only when the wave propagatesexactly perpendicular or parallel to the magnetic field. Similar to MHD waves, coldplasma waves can propagate also at oblique angles to the magnetic field. It is possibleto draw the wave normal surface for each mode. However, the ratio of density andmagnetic 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 itswave normal surface and the surfaces may be traced in the (ω2

p/ω2, ωc/ω) space until

it disappears at the cut-off or at the resonance. As is seen from the figure cut-offs andresonance 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 elec-tric current due to polarization drift find the dielectric function for low frequenciesin the form

ε = ε0

(1 +

c2

v2A

).

2. Prove that for the right-hand polarized wave propagating parallel to the magneticfield the cut-off (nR → 0) occurs when

ωR=0 ≈ωce2

[1 +

√1 + 4ω2

pe/ω2ce

].

Show further that at the low density limit (ωp ωc) this reduces to

ωR=0 ≈ ωce(1 + ω2pe/ω

2ce)

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 ofplasma as

T = d/c+D/f2

where d is the distance to the source, f the frequency of the signal and D theso-called dispersion measure

D =e2

8π2ε0mec

d∫0

neds .

7.4. EXERCISES: COLD PLASMA WAVES 129

Consider a pulsar from which a signal at 100 MHz arrives 2 s later than the signalat 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 thewave assuming a linear polarization and 0.1-nT interstellar magnetic field.

4. Derive the dispersion equation for the whistler wave

k =ωpec

√ω

ωce.

Using this equation show that the group velocity of the whistler wave is

vg =∂ω

∂k=

2c√ωce

ωpe

√ω .

Compare the arrival times of the emitted low and high frequency waves.


P=0 (cut-off)

L=0 (cut-off)R=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. InMHD temperature appears through the equation of state, but always in relation to den-sity and pressure, while in the cold plasma theory we assumed that thermal effects canbe neglected. The inclusion of thermal effects introduces new wave modes in the plasmaand allows for free-energy that is necessary for the generation of plasma instabilities. Tofully describe warm plasma (wave modes, instabilities, etc.) one needs complex mathe-matical tools of kinetic theory. However, fluid description gives a simpler introductionto characteristics of warm plasmas. We now modify the fluid equations to take intoaccount the thermal effects. We add the pressure term to the equation motion

nαmαdVα

dt= enα (E + V ×B) +∇pα ,

which allows us to investigate temperature related phenomena in a number of specialcases. We first derive warm plasma dispersion equation and investigate two specialsolutions; the Langmuir wave and the ion sound wave. Then we proceed to a briefoverview 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 gradientof the electron pressure is included in the equation of motion, but we make a simplifiedassumption that the ion pressure gradient is zero, justified by their larger inertia. Usingthe adiabatic equation of state the pressure gradient ∇pe can be replaced by γp0n

−10 ∇ne1

(Exercise 8.1). Hence, the linearized continuity and momentum equations for electronsand ions are

131

132 CHAPTER 8. WARM PLASMA

∂ne1∂t

+ (∇ ·Ve1) = 0 (8.1)

∂ni1+

∂t+ (∇ ·Vi1) = 0 (8.2)

n0me∂Ve1

∂t= en0E1 −

γp0n0∇ne1 (8.3)

n0mi∂Vi1

∂t= en0E1 , (8.4)

respectively. We also need the Gauss law

∇ ·E1 = − e

ε0(ni1 − ne1) . (8.5)

Assuming again harmonic time and spatial dependencies, i.e., we are looking forplane wave solutions, we obtain

−iωne1 + in0k ·Ve1 = 0 (8.6)

−iωni1 + in0k ·Vi1 = 0 (8.7)

−iωmeVe1 = eE1 − iγp0n0

ne1k (8.8)

−iωmiVi1 = eE1 (8.9)

ik ·E1 =e

ε0(ni1 − ne1) . (8.10)

A brief calculation (Excercise 8.2) gives

(1−

ω2pi

ω2−

ω2pe

ω2 − k2(γkBTe/me)

)k ·E1 , (8.11)

where pe = nekBTe has been used to introduce the electron temperature. The expressionin 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 dispersionequation is

ω2 = ω2pe + k2(γkBTe/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 bederived assuming ions as an inmobile background (Exercise 8.3). The finite temperatureand associated thermal motions now allow electron plasma oscillation to propagate as awave (∂ω/∂k 6= 0). We notice also that the wave number depends on the frequency andhence the wave is dispersive. The dispersion equation is shown in the left-hand part ofFigure 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 smallcorrection to the cold plasma theory, we can assume that the temperature disturbancepropagates less than one wavelength during one plasma oscillation. This corresponds tothe long wave length limit (k2λ2De 1), i.e., the approach we used to solve the Vlasovequation in Chapter 4. Hence, the perturbation is assumed to be adiabatic. Sincehomogenous plasma without background fields is one-dimensional, the polytropic indexis γ = (d + 2)/d = 3, where d is the indicates the number of spatial directions, in thiscase d = 1. Using the relationship between the Debye length and thermal speed

λ2De =v2th,e2ω2

pe

(8.13)

we can write the dispersion equation in terms of the Debye length

ω2 = ω2pe(1 + 3k2λ2De) , (8.14)

which is the same as Eq. 4.36.

At the long wave length (small wave number) limit we can approximate

ω = ωpe

√1 + 3k2λ2De ≈ ωpe(1 +

3

2k2λ2De) . (8.15)

When frequencies are well below the electron plasma frequency (ω ωpe, the solutionto the warm plasma dispersion equation gives a new wave mode, the ion sound wave(Exercise 8.4)

ω =kcs√

1 + k2λ2De

, (8.16)


where we have introduced the ion sound speed

cs =√kBTe/mi. (8.17)

The dispersion equation is plotted in the right-hand part of Figure 8.1. Now the processhas been assumed isothermal (γ = 1), which is justified by the ions oscillating so slowlythat the electron temperature has time to relax over the oscillations. This mode can alsobe found from the Vlasov theory, when the solution is investigated from the appropriatefrequency domain.

At the limit of small wave number (k2λ2De 1) ω ≈ kcS , yielding the dispersionequation

ω

k=√kBTe/mi. (8.18)

At the limit of large wave numbers we obtain

ω =csλDe

=

√kBTemi

√n0e2

ε0kBTe=

√n0e2

miε0= ωpi , (8.19)

i.e., the wave frequency approaches the ion plasma frequency, see the right-hand part ofFigure 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 theelectron temperature, while in the denominator is the ion mass. Thus the electronsaccount for the pressure and ions for the inertia. If ion pressure would be taken intoaccount Te would be replaced by Te + γTi. However, Vlasov theory indicates that if theelectrons are not clearly warmer than ions, the ion waves are strongly damped. Notethat the ion acoustic wave can also propagate in collisionless plasma because chargedparticles interact due to long-range Coulomb forces. Electrons are highly mobile andthey quickly follow the ion motion to preserve the charge neutrality (remember that wehave assumed there to be no magnetic field!).

8.3 On plasma stability

Plasma reacts to a disturbance by starting to oscillate with a characteristic frequencyand wave length. Depending on the situation the oscillations may propagate, and growor damp. The growing oscillations can lead to a plasma instability. Instability requires asource of free energy, and hence, there are no instabilities in the cold plasma theory. Inaddition, the elementary approach to plasma physics often assumes an unperturbed statethat is in local thermodynamic equilibrium and particles can be described by Maxwellianvelocity 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 theform 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 distributionfunction. If there are no processes that would saturate the instability, the whole plasmasystem can explode. This happens both in space and laboratory plasmas. Solar flares andloss of plasma state in tokamaks are examples of large-scale plasma instabilities. One wayto categorize plasma instabilities is to divide them between microscopic and macroscopicinstabilities. A microscopic instability needs the kinetic approach and it depends onthe shape of the distribution function. A macroscopic instability is a configurationalinstability and can be described by macroscopic equations. We consider here only a fewsimple examples that can be understood either intuitively or that are straightforward tocalculate.

8.3.1 Z-pinch instability

Let us first investigate the equilibrium configuration of the Z-pinch from Chapter 6 (seeFigure 6.17) where plasma is confined by a toroidal magnetic field. The magnetic fieldarises from the electric current that is driven through the plasma. Figure 8.2 displayswhat may happen to an initially stable Z-pinch if the system is perturbed.

B2/2m0 decreases

B2/2m0 decreases

B2/2m0 increases

B2/2m0

increases

kink instability

sausageinstability

Figure 8.2: Kink and sausage instabilities

If the plasma tube is bent, the magnetic pressure will increase on the concave partof 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 itselfand the whole plasma can rise up from the equilibrium leading to the loss of the plasmastate. 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 decreasesin the nearby region. Larger magnetic field gradient tends to increase the compression


of the plasma and the whole plasma tube will break, if there is no mechanism to stopthe squeezing. This is called the sausage instability.

A common way to stabilize plasma is to wind a flux tube to a torus and drive anelectric current through the tube. This toroidal current creates a poloidal magneticfield around the torus. The superposition of toroidan and poloidal magnetic fields leadsto a spiral shaped magnetic field inside the torus. Such device is called tokamak, andit is nowadays the most common and important plasma confinement device in fusionexperiments (Figure 8.3). The growth rate of both sausage and kink instabilities can bestabilized in tokamaks, but driving large electric currents through the plasma may causekinetic instabilities related to changes in plasma distribution functions.

Figure 8.3: In a tokamak fusion reactor poloidal and toroidal electric currents create analmost 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 andinvestigate conditions that lead to growing wave perturbations. Let us investigate asimple example featuring two oppositely directed electron beams with different velocities.Essentially, this is a kinetic instability, a solution can also be found from macroscopictheory in the case the velocity difference between the beams is larger than their thermalmotion.

Let the densities of the electron beams be nα0 and nβ0 and the velocities Vα0 andVβ0. Assume that there is no background magnetic field and that ions are a fixedbackground, hence restricting the analysis to high frequency waves. We assume again asmall 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(nα0Vα1 + nα1Vα0) = 0 (8.20)

−iωnβ1 + ik(nβ0Vβ1 + nβ1Vβ0) = 0 . (8.21)

Assuming that beams are cold their linearized equation of motions become

−iωVα1 + ikVα0Vα1 = − e

meE1 (8.22)

−iωVβ1 + ikVβ0Vβ1 = − e

meE1 . (8.23)

Combining these with the Gauss law

ikE1 = − e

ε0(nα1 + nβ1) (8.24)

gives the dispersion equation

ikE1

(1−

ω2pα

(ω − kVα0)2−

ω2pβ

(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 dispersionequation has solutions with a positive imaginary part. Such a two-stream instabilitycan arise if the beam velocities differ from each other enough but not too much. Forderivation of the dispersion equation of even a simpler case with one-dimensional plasmawhere electrons flow with a constant velocity with respect to a stationary ion backgroundsee Exercise 8.5.

four real roots (stable) two real

roots (stable)

two complex roots (instability)

Figure 8.4: Left) The dispersion equation has four real roots. In this case the imaginarypart of the frequency is zero and plasma stable. Right) Two roots are complex. Oneof the complex roots has a positive imaginary part leading to growing of the waveperturbation and instable plasma.


8.3.3 On the stability of warm plasma

A general treatment of warm plasma instabilities requires kinetic approach. As discussedin Section 4.3, the perturbation will grow exponentially if the wave frequency is complexwith the imaginary part being positive (ωi > 0, see Eq. 4.37). Furthermore, the Vlasovtheory states that all monotonically decreasing distribution functions (∂f/∂v < 0) arestable. A positive slope (∂f/∂v > 0) makes the distribution potentially unstable, butdoes not guarantee instability.

unstable region

Figure 8.5: Example of a distribution that has a postive slope. If the wave moves withthe speed vph, it can be amplified at the expense of the energy of the faster movingbeam.

An example of a distribution with a positive slope is presented in Figure 8.5. Itfeatures a Maxwellian background (stable) with temperature T1 and a warm particlebeam (temperature T2) flowing along the z-axis. The number density of the beam isassumed to be smaller than the background density. Solving the Vlasov equation wouldindicate that unstable wave modes are found at the phase speeds that coincide with thepositive derivative of the distribution function. This so-called “gentle bump” instabilityrequires that the beam has sufficiently high speed when compared to the temperatureof the Maxwellian background. The instability is enhanced if the density of the bumpincreases, 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 treatmentof instabilities requires methods that are beyond the linear plasma theory and even touchthe boundaries of the current knowledge.

8.4. EXERCISES: WARM PLASMA 139

8.4 Exercises: Warm Plasma

1. Show that the equation of state p/ργm = constant leads to the following relationbetween first order (small) perturbations

p1 = p0γρm1

ρm0.

2. Derive the dispersion equation

1−ω2pi

ω2−

ω2pe

ω2 − k2(γkBTe/me)= 0

for non-magnetized electron-ion plasma by taking into account electron thermaleffects.

3. Consider isotropic plasma where the background magnetic and electric fields arezero, the ions are immobile, and the plasma is initially in equilibrium. Assume asmall disturbance in the electron mass density and derive the dispersion equationfor the Langmuir wave

ω2 = ω2pe + k2 (γkBTe/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 =kcs√

1 + k2λ2De

,

where cs =√kBTe/mi is the ion sound speed and λDe the electron Debye length.

5. Consider one-dimensional plasma where electrons flow with speed V0 with respectto a stationary ion background. Derive the dispersion equation for plasma oscilla-tions in the rest frame of the ions

(ω − kV0)2 = ω2p .

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

∇ · (fA) = (∇f) ·A + f(∇ ·A)

∇× (fA) = (∇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)−∇2A

The divergence theorem relates the volume integral of the divergence of a vector fieldA over a volume V to the surface integral of A over the surface S bounding the volumeV :

∫V

(∇ ·A) dV =

∮S

A · dS . (9.1)

The Stokes theorem relates the surface integral of the curl of a vector field A over asurface S to the line integral of A over its boundary ∂S:∫

S∇×A · dS =

∮∂S

A · dr (9.2)

141

142 CHAPTER 9. APPENDIX

9.2 Maxwell equations and useful concepts of electrody-namics

Although plasma is an electromagnetic medium the Maxwell equations are often writtenin the “vacuum” form

∇ ·E = ρ/ε0 Gauss’s law∇ ·B = 0 Gauss’s law for magnetism

∇×E = −∂B

∂tFaraday’s law

∇×B = µ0J +1

c2∂E

∂tAmpere-Maxwell’s law

(9.3)

Sources of the magnetic and electric fields, i.e. the charge density and the electriccurrent density, include all charges and currents and they can be determined from plasmadistribution functions (see Chapter 4). We call here B as magnetic field although rig-orously speaking B is the magnetic flux density. The magnetic field has the SI unitsV s m−2 = T, i.e., tesla. The SI units of the electric field E are V m−1, of the chargedensity ρ A s m−3 = C m−3 and the electric current density J, and A m−2. The naturalconstants appearing in Maxwell equations in SI units are

µ0 = 4π × 10−7 V s A−1 m−1 permeability of free spacec = 1/

√ε0µ0 = 299 792 458 m s−1 speed of light in vacuum

ε0 = (c2µ0)−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 “auxliar-ity fields” H, and D that are the magnetic field intensity, and electric displacement. Hand D and related to B and E through the constitutive relations:

D = ε0E + 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 electricfield and the magnetization to the magnetic field intensity

P = χeE (9.6)

M = χmH , (9.7)

where χe and χm are electric and magnetic susceptibilities determined by the propertiesof the medium. The electric susceptibility measures how easily a dielectric medium ispolarized when an external electric field is applied and the magnetic susceptibility givesthe 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 = εE (9.8)

H = B/µ , (9.9)

where ε = ε0 +χe, and µ = µ0(χm + 1) are the permittivity and permeability. Note thatthe permittivity and permeability are not necessarily constants, for instance, they canvary with the position, time and frequency. If the medium is anisotropic, permittivityand 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)

∇×H = Jf +∂D

∂t. (9.13)

The electric displacement D accounts for the effect of free charges ρf (∇·D = ρf ), whilethe sources of polarization P are bound charges (∇ ·P = ρb).

The total current J (see the vacuum form of the Ampere-Maxwell law) is the sumof the current produced by the flow of free charges Jf , and the magnetization andpolarization currents.

The polarization current is related to the change in polarization of the individualmolecules of the dielectric medium

JP =∂P

∂t. (9.14)

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 plasmaphenomena.

The magnetization current is associated to the circulation in the magnetization fieldM

JM = ∇×M (9.15)

The displacement current ε0∂E/∂t in the Ampere-Maxwell law arises from the gen-eration of magnetic fields by time-varying electric fields. This term is important as itallows 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 = +ε0 ∂E/∂t ,


from which one can easily derive the wave equations for E and H

∇2H− 1

c2∂2H

∂t2= 0

∇2E− 1

c2∂2E

∂t2= 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) , (9.17)

whereE0 wave amplitudeω = 2πf angular frequencyk = 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 constantphase.

Plane waves are often practical to present using the exponential form

E = E0ei(k·r−ωt)

B = B0ei(k·r−ωt) . (9.19)

If E0 and B0 are constants, the temporal and spatial dependencies of the electric andmagnetic fields are said to be harmonic. In such a case

∇ ·E0ei(k·r−ωt) = ik ·E0e

i(k·r−ωt) (9.20)

∇×E0ei(k·r−ωt) = ik×E0e

i(k·r−ωt) (9.21)

∂

∂tE0e

i(k·r−ωt) = −iωE0ei(k·r−ωt) , (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 ε and µ are constants thefollowing dependence between the wave number and wave frequency is

k =√µεω =

n

cω , (9.24)

9.3. BASIC CONCEPTS OF WAVE PROPAGATION 145

where we have defined the index of refraction

n =

√εµ

ε0µ0. (9.25)

The relationship between the angular frequency and the wave number is called thedispersion equation.

Dispersion equations are a central for studies of waves in plasmas. From a dispersionrelation one can determine characteristics of the wave propagation, for instance, thegroup and phase velocities.

The phase velocity is defined as the velocity at which the planes of constant phasemove

vp =ω

kn , (9.26)

where n is the unit wave normal vector. It is perpendicular to the surface of a constantphase, and thus parallel to the wave vector k and direction of wave propagation. If themedium is isotropic the wave propagation is in the same direction as the energy flux,defined in terms of the Poynting flux S = 1

2E × H∗, where the asterisk indicates thecomplex conjugate. In an anisotropic medium the electric field may have a componentparallel to the wave vector, and hence, the direction of energy flux is not necessary inthe 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 wavenumbers and frequencies.

The group velocity is the velocity at which the envelope of the wave packet propagates

vg = ∇kω(k) (9.27)

It is also the speed at which the energy propagates.


If the phase speed depends only on the physical properties of the medium and not onthe frequency of the wave, the medium is non-dispersive. In this case the group speed isequal to the phase speed and the wave packet maintains its shape as it propagates. Inturn, if the phase speed depends on the frequency, the wave package spreads out as thewaves 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 − iωε)E , (9.28)

where in the last equation the electric current has been written in terms of the electricfield 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 ‖ ez, E ‖ ex ja H ‖ ey gives

kEx = ωµHy

ikHy = −(σ − iωε)Ex , (9.29)

and the dispertion equation becomes

k2 = εµω2 + iσµω . (9.30)

Denoting k = |k|eiα we find

|k| =

√µω√ε2ω2 + σ2

α =1

2arctan(

σ

εω) . (9.31)

The electric field is thus

E = E0ex exp[i(|k|(cosα)z − ωt)] exp[−|k|(sinα)z] . (9.32)

To obtain a physical solution we need to choose phase α so that sinα > 0, i.e., the waveis damped (factor e−|k|(sinα)z) as it propagates through the medium, because we havenot assumed any source of free energy for wave growth.

The phase velocity is now

vp =ω

Re(k)=

ω

|k| cosα. (9.33)

The distance where the wave is damped by a factor of e is called the skin depth

δ =1

Im(k)=

1

|k| sinα. (9.34)

9.4. THE MAXWELLIAN DISTRIBUTION 147

9.4 The Maxwellian distribution

The Maxwellian (or Maxwell-Boltzmann) distribution describes how particle velocitiesare distributed in a gas. This distribution was originally derived by James Clerk Maxwellin the 1860s based on certain symmetries of the distribution function. A decade laterLudwig Boltzmann derived the Maxwelliann distribution both from the kinetic theoryand using the framework of statistical thermodynamics. The Maxwellian distribution isvalid for ideal gases where particles are in thermodynamic equilibrium i.e., all particlesof the same species can be described with the same temperature.

f(vx)

vx0

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

f(vx) = n

(m

2πkBT

)1/2

exp

(− mv2

2kBT

). (9.35)

Now f(vx) dvx gives the probability that a particle in a gas has a velocity component vxin the range vx + dvx. Particle velocities are randomly distributed around zero velocityand particles have an equal probability to propagate in the +x and −x directions. Eachof the velocity components may be treated independently, and thus, the distribution canbe generalized to three dimensiona by multiplying one-dimensional Maxwellians in eachthree directions. The resulting 3-dimensional Maxwellian velocity distribution is

f(v) = n

(m

2πkBT

)3/2

exp

(− mv2

2kBT

), (9.36)

where v =√v2x + v2y + v2z is the magnitude of the velocity vector v.

The spread of the velocities defines the thermal speed

vth =√

2kBT/m . (9.37)

The width of the distribution is controlled by the temperature, i.e., the higher thetemperature the broader the Maxwellian distribution.


The Maxwellian velocity distribution does not depend on the direction of v, only onits magnitude through v2. Hence, it is often more convenient to know the probabilityof particles to have their speeds in a certain range. In spherical coordinates d3v can bewritten as v2 sin θdθdφ. The distribution does not depend on the angular coordinates,and thus, the integration over angular coordinates gives 4π. The Maxwellian speeddistribution thus becomes

f(v) = 4πv2n

(m

2πkBT

)3/2

exp

(− mv2

2kBT

). (9.38)

Figure 9.3 shows that the shape of the distribution now differs from Gaussian, it is skewedtowards higher speeds. This is because there are more ways to achieve the higher speedswhen all three directions are considered (consider the how much volume each dv shellencloses when v increases).

f(v)

v0

<v>vrms

vp

Figure 9.3: Maxwellian speed distribution. Three different characteristics speeds areindicated.

Different types of characteristics speeds can be derived from the Maxwellian speeddistribution. Their relative locations are shown in Figure 9.3.

The most probable speed vp occurs at the highest point of the distribution. To calcu-late where f(v) has its maximum we set the derivative df(v)/dv to zero

df

dv= 4π

(m

2πkBT

)[2v exp

(−mv

2

2k

)− v2 mv

kBTexp

(−mv

2

2k

)]= 0 . (9.39)

This yields

vp =

√2kBT

m.

Thus, the thermal speed (velocity spread) equals to the most probable speed.

9.4. THE MAXWELLIAN DISTRIBUTION 149

The mean speed is obtained as the expectation value of f(v):

< v >=

∫ ∞0

f(v)v dv . (9.40)

Using the result ∫ ∞0

x3 exp−x2/a2 = a4/2

leads to

< v >=

√8kBT

πm. (9.41)

The root mean speed is defined as vrms =√< v2 >, where

< v2 >=

∫ ∞0

f(v)v2dv (9.42)

The result ∫ ∞0

x4 exp−x2/a2 =3√πa5

8

gives

vrms =

√3kBT

m. (9.43)

The thermodynamical equilibrium is typically reached through collisions. Many plas-mas are collisionless and are not in thermodynamical equilibrium. Hence, describing suchplasma with a Maxwellian distribution is no always sufficient. However, a Maxwell dis-tribution is often a good starting point. The collision frequency decreases when thetemperature increases (∝ T−3/2) and consequently it takes much longer for fast parti-cles to reach a Maxwellian distribution than for slow particles. Kappa distribution (seeSection 4.4) is an example of a case where the slow particle population can be describedwith a Maxwellian distribution, while the fast particles form a non-Maxwellian tail.

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