Introduction to Plasma Physics. Emilia Kilpua and Hannu Koskinen

Transcription

1 Introduction to Plasma Physics Emilia Kilpua and Hannu Koskinen HK,

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3 Contents 1 Introduction General definition and occurrence Brief history of plasma physics Levels of description Basic Definitions and Parameters Formation of the plasma Quasi-neutrality in plasma Plasma frequency Plasma response to electromagnetic fields Collective behavior and collisions Plasma conductivity Plasma definition: A summary Exercises: Basic Definitions and Parameters Single Particle Motion Motion in a static, uniform magnetic field Motion in constant perpendicular electric and magnetic fields General drift velocity due to a force perpendicular to magnetic field Particle motion in non-uniform electric fields Particle motion in non-uniform magnetic fields Examples of particle motion in simple geometries Exercise: Single Particle Motion i

4 ii CONTENTS 4 Kinetic Plasma Description Distribution function Time evolution of distribution functions Solving the Vlasov equation Examples of distribution functions Exercises: Kinetic Plasma Description Macroscopic Plasma Equations Macroscopic transport equations Equations of state Magnetohydrodynamic equations Exercises: Macroscopic Plasma Equations Magnetohydrodynamics MHD equations Magnetic field evolution Frozen-in condition MHD waves Magnetic reconnection Magnetohydrostatic equilibrium and stability Force-free magnetic fields Exercises: Magnetohydrodynamics Cold plasma waves General form of the dispersion equation Wave propagation in non-magnetized plasma Wave propagation in magnetized plasma Exercises: Cold Plasma Waves Warm plasma Warm plasma dispersion equation Langmuir wave and the ion sound wave On plasma stability Exercises: Warm Plasma

5 CONTENTS iii 9 Appendix Useful vector identities and theorems Maxwell equations and useful concepts of electrodynamics Basic concepts of wave propagation The Maxwellian distribution

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7 Preface This course is an introduction to basic concepts and methods of plasma physics. It provides basics for further studies of laboratory, fusion, space and astrophysical plasma phenomena. The most important goals of these lectures are to: familiarize the reader with the main concepts and phenomena of plasma physics give an overview of the importance and applicability of plasma physics teach the basic mathematical tools and approaches used in plasma physics After a brief introduction to fundamental plasma properties, following topics are discussed: motion of charged particles in the electromagnetic field, kinetic plasma description, macroscopic plasma quantities and equations, magnetohydrodynamics, Alfvén waves, cold plasma waves, warm plasmas. Most of these lectures deal with plasma in so high-temperatures that the plasma is practically fully ionized with only a small effect arising from neutral particles. We will also limit the discussion to non-relativistic plasmas with temperatures ranging from about a few ev to a few hundred kev. Quantum mechanical effects are also neglected since the interactions distances are usually much longer than the de Broglie wavelength. Plasma physics is based on the main fields of classical physics: electrodynamics, mechanics and statistical physics. These lectures require good understanding of bachelorlevel basic physics and solid skills in undergraduate-level mathematical methods of physics (e.g., vector calculus and analysis, Fourier analysis). 1

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9 Chapter 1 Introduction We start with a brief introduction to the idea of plasma as a state of matter before we go to the more technical treatment of plasma physics. 1.1 General definition and occurrence What is plasma? This is a natural question to ask at the beginning of plasma physics lectures. However, as we will soon find out, this question is not trivial and it is difficult to give an exact definition of the plasma state. We refer at this point to a following practical description of plasma: Plasma is quasi-neutral gas with so many free charges that collective electromagnetic phenomena are important to its physical behavior. Two key aspects of plasma can be found from this definition: 1) Due to the presence of free charges plasma responds strongly to electromagnetic fields, and 2) in plasma collective long-range interactions dominate. These characteristics of plasma lead to a wide variety of interesting phenomena distinct from neutral gases, including collective shielding of individual charges, a large variety of new wave modes, and transfer of energy from waves to particles (damping of oscillations) and vice versa (plasma instabilities). Plasma is ubiquitous in the universe. It is speculated that more than 99% of baryonic matter in the universe is in the plasma state. Thus, plasma physics is a necessary tool in space physics and in many astrophysical problems. What we mean by space physics needs a little explanation. Space physics investigates physical phenomena in space of which it is possible, at least in principle, to get detailed in-situ observational information. Thus its domain is mainly the solar system including the studies of the Sun, the solar wind, and the magnetospheres, ionospheres, and upper atmospheres of the Earth and 3

10 4 CHAPTER 1. INTRODUCTION other planets. Astrophysical plasma physics includes, in turn, the studies of plasmas and plasma processes farther in the universe, e.g., stars from the Sun-like objects to neutron stars, black hole accretion discs, and the interstellar medium. Figure 1.1: Examples of plasmas near and far. Top row: Sun captured by ultraviolet light emitted by ionized helium atoms (Courtesy: SOHO/NASA), artist impression of an active galactic nucleus (Courtesy: Alfred Kamajan), lightning (Courtesy: NOAA). Bottom row: plasma welding (Courtesy: Pro-Fusion), Joint European Torus (JET) fusion experiment (Courtesy: AFP/Getty Images), plasma thruster (Courtesy: NASA). Plasmas in our immediate environment are much less common, but they exist. In particular, technological applications of plasma physics are numerous, including thermonuclear fusion research, neon-lights, plasma displays, sterilizing of certain medical products, and plasma processing of semiconductors and materials (e.g., etching and welding). Visible examples of natural plasmas in the near-earth environment are auroras and lightning. 1.2 Brief history of plasma physics The word plasma originates from Greek where it means something molded. British scientist Sir William Crookes was the first to appreciate plasma as the fourth state of matter. He investigated the conduction of electricity in low pressure gases in electrical discharge tubes (where air is ionized by applying a high voltage). In 1879 Crookes published an article On Radiant Matter in The Popular Science Monthly where he stated: So distinct are these phenomena from anything which occurs in air or gas at

11 1.2. BRIEF HISTORY OF PLASMA PHYSICS 5 ordinary tension, that we are led to assume that we are here brought face to face with matter in a fourth state or condition, a condition as far removed from the state of gas as a gas is from a liquid.. The term plasma was coined a few decades later by American physicists Lewi Tonks and Irving Langmuir. They conducted one of the first plasma experiments with electric discharge tubes. These experiments already led to many important discoveries concerning the basic properties of plasma, such as the shielding of charge and plasma oscillations. A significant part of early plasma physics dealt with space and astronomical phenomena. A particular interest for space plasma physics in the early 1900s was radio broadcasting. Radio waves are reflected from the ionosphere, the partially ionized layer of the upper atmosphere, which enables transfer of waves over long distances. The efforts to understand radio communication led to the development of the theory how electromagnetic waves propagate through non-uniform magnetized plasmas. Figure 1.2: A few great minds of plasma physics. Top: William Crookes (Courtesy: Library of congress), Hannes Alfvén (Courtesy: Royal Institute of Technology, Stockholm), Irving Langmuir (Courtesy: IEEE). Another main branch in early space plasma physics dealt with examining the connection between solar activity and the disturbances in the Earth s magnetic field. A most remarkable theory in the early 1900s in the field of solar terrestrial studies was Sidney Chapman s and Vincenzo Ferraro s suggestion that magnetospheric storms are caused when magnetized plasma clouds ejected from the Sun envelop the Earth s magnetosphere. In the 1940s the Swedish scientist Hannes Alfvén developed the formalism of magnetohydrodynamic (MHD) theory. MHD treats plasma as a conductive fluid that can support magnetic fields. Over the years MHD has developed to one of the main tools of plasma physics. Hannes Alfvén has probably been the most influential individual in the history of plasma physics. His contributions to plasma physics are numerous including

12 6 CHAPTER 1. INTRODUCTION theories describing the behavior of aurora, the radiation belts, now known as Van Allen belts, the effect of magnetic storms on the Earth s magnetic field and cosmic electrodynamics. Perhaps one of his best-known ideas is the theory of low-frequency magnetohydrodynamic waves, now known as Alfvén waves, in magnetized plasma. The basic mode of the Alfvén wave propagates along magnetic field and it is the fundamental mode to transfer magnetic disturbances in plasma. Hannes Alfvén was granted the Nobel prize in 1970 for his work on fundamental work and discoveries in magnetohydro-dynamics with fruitful applications in different parts of plasma physics. Radio astronomy started to develop in the 1930s when Karl Jansky observed radio waves coming from the direction of the Milky Way. The Second World War brought rapid developments in radio- and microwave technologies, which opened a new window to the Universe at those radio frequencies that penetrate through the atmosphere. Part of the radiation is bremsstrahlung in hot astrophysical plasmas but it soon turned out that all radio emissions could not be explained in this way. In the mid-1950s Vitaly Ginzburg argued that radio emissions from, e.g., the Crab nebula, i.e., the remainder of the supernova observed in 1054, must be synchrotron radiation by electrons gyrating in the strong magnetic field of the neutron star. This was an important milestone as it indicated the central role of the magnetic fields in cosmic plasma physics. Modern plasma physics can be said to have originated after the Second World War and has expanded to several directions. A few main branches are briefly discussed below. Space Plasma Physics. The space age began with the launch of Sputnik in The US satellite Explorer 1 was launched a year later and its sole scientific instrument, a Geiger counter by James Van Allen, discovered radiation belts around the Earth. These belts were later named Van Allen belts. Space exploration quickly expanded from the vicinity of the Earth further out in the heliosphere. The spacecraft have passed-by all planets of our solar system. Some of them have become artificial satellites of Mercury, Venus, Mars, Jupiter and Saturn and carried landers with them. Mankind has now even reached beyond the solar system when Voyager 1 entered the interstellar space in The most detailed data is, however, obtained from the near-earth space and from the Sun. Numerical simulations have become an integral part in modern space research besides observations. The observational network in space physics is relatively sparse and simulations are needed to fill this gap. In addition, simulations provide new physical insight to many space physics problems that can be tested by observations. Understanding of space plasma physics is also necessary for space technology, ranging from designing, manufacturing and testing scientific instruments to developing new propulsion systems for faster and more cost-effective space travel and exploration. Plasma Astrophysics. As said earlier, almost all baryonic matter in the Universe is in plasma state. While the Sun is often considered to belong to the topics of space plasma physics, it is also a very typical star. The Sun is entirely in the plasma state and detailed understanding of its plasma physics, e.g., the dynamo process creating its cyclically varying magnetic field, can readily be transferred to studies of other magnetically active cool stars. Current topics of plasma astrophysics include neutron star magneto-

13 1.3. LEVELS OF DESCRIPTION 7 spheres, formation of astrophysical jets, accretion discs of black holes, acceleration of cosmic rays in astrophysical shock waves, emission of electromagnetic radiation from radio frequencies to X- and gamma rays. It is clear that also in astrophysical context simulations using the most powerful computers today have become an essential tool. Controlled Fusion Research. The early fusion experiments were conducted already in 1930s but it was only after the Second World War when the interest in fusion research really sparked. The development of nuclear fission weapons raised interest also in fusion weapon technologies. It was proposed that fusion reaction could be controlled to make an effective reactor. Since then fusion research has quickly expanded as an important international enterprise with several large experimental facilities being constructed with the goal to develop a relatively clean and abundant energy source. The most prominent current effort is the International Thermonuclear Experimental Reactor (ITER), which is being constructed near Cadarache in the Southern France. When finished, ITER will be the world s largest tokamak nuclear fusion reactor. Much of the fusion research nowadays is involved in studying how extremely hot plasma can be stabilized for long enough to attain sustained effective fusion and the tokamak geometry is the currently favored approach in the large-scale devices. Another approach to controlled fusion is to create the required hot and dense plasma state using intense lasers. 1.3 Levels of description Plasma processes are often extremely complicated and their spatial and temporal scales vary by many orders of magnitude. Plasmas exhibit diverse characteristics, their temperatures, densities and ionization degree can differ greatly as well as the importance of collisions and electromagnetic forces to the behavior of plasma. Thus, different levels of description are used to tackle different types of problems (see Figure 1.3). Different approaches can also provide alternative insights to understanding a given plasma phenomenon. Although collective behavior is a fundamental property of plasma, single particle description (or an exact microphysical description) is the first step in understanding of the processes occurring in plasma. It is often a necessary approach, for example, when studying cosmic rays or energetic particles in the Van Allen radiation belts. In this approach the task is to solve the equation of motion (F = ma) for a charged particle. Only in a few special cases the motion can be solved analytically and typically (e.g., in time varying and curved magnetic fields) approximations or direct numerical calculations are needed. The next step is the kinetic theory. It is a statistical approach to average out individual particle orbits and treat the motion of a large number of particles in form of a distribution function. However, the detailed knowledge the particle distribution as a function of location and velocity is needed and in this sense kinetic theory is still

14 8 CHAPTER 1. INTRODUCTION Figure 1.3: Levels of plasma descriptions. microscopic. The core of the kinetic treatment is to determine the velocity distribution functions and their evolution for each plasma species. From velocity distribution functions one can calculate macroscopic plasma variables, such as the bulk speed, temperature and density. The kinetic approach can deal with non-maxwellian distributions and it is often the required approach when studying plasma waves and instabilities. In many cases it is not necessary to know the exact evolution of distribution functions, but it is sufficient to determine how macroscopic plasma variables behave in time and space. The evolution of these parameters are determined by means of macroscopic fluid approach, the equations of which are analogous to the equations of hydrodynamics. However, the effects of electromagnetic fields on the charge particles and often different behavior of electrons and ions in a plasma make plasma fluid equations more complex than hydrodynamic equations. In fluid description the velocity distributions of each species are often implicitly assumed to be Maxwellian. The simplest description of plasma is the one-fluid or magnetohydrodynamic (MHD) theory. Although a very crude approximation, MHD is a widely applicable theory and can be used to describe many plasma physical phenomena. Due to simplicity and computational effectiveness it is one of the main tools for global numerical simulations. Sometimes, a combination of different approaches are used. For examples, in hybrid simulations electrons can be described as a fluid and ions either as individual particles or in terms of distribution functions.

15 Chapter 2 Basic Definitions and Parameters 2.1 Formation of the plasma Plasma is generally considered as the fourth state of matter because it arises as the next natural step from solid to liquid to gas, when the temperature is increased (Figure 2.1). For example, when ice is heated its crystalline bonds are broken and it changes to water (liquid state). If more heat is added the molecular bindings break first, followed by independent H 2 O molecules separating into hydrogen and oxygen atoms (gas state). In order to achieve plasma, even more heat has to be added to dissociate the atoms into electrons and positive ions. At some point the fraction of the atoms that are ionized becomes large enough that the collective electromagnetic forces take over the behavior of the system (plasma state). Figure 2.1: Plasma is considered as the fourth state of matter. Adding even more heat would finally break nuclear bonds (energies > MeV) and quark quark bonds (energies > 175 MeV) resulting in quark-gluon plasma. Such an exotic plasma state dominated the universe just after the Big Bang and may exist in the core of neutron stars. Experiments on CERN s Large Hadron Collider are studying the properties of quark gluon plasma. However, this is beyond the scope of normal plasma physics courses and will not be treated in these lectures. According to our practical plasma definition there has to be enough free charges. But how much is enough? There is no unique phase transition point when a gas turns to a plasma, but a rough guideline is that already 0.1% degree of ionization typically gives clear plasma properties and 1% ionization means almost perfect conductivity. Thus, 9

16 10 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS plasma state is achieved after a remarkably small fraction of ionization. Partially ionized plasmas can be found, for example, in ionospheres, neon-lights, and gas-discharge tubes. Examples of fully ionized plasmas include fusion plasmas and most of space plasmas, for example, the solar wind, solar corona and magnetosphere. At the end of this Chapter we will provide a more quantitative definition of a plasma state. The degree of ionization for a gas in thermal equilibrium can be calculated from Saha s equation: n i 27 T 3/2 = 3 10 exp( U/T ), (2.1) n n n i where n i is the ion number density ([n i ] = m 3 ), n n the neutral number density, T temperature ([T ] = ev) and U the ionization energy ([U] = ev, 1 ev K), i.e. the energy that is required to remove the outermost electron from the atom. From Saha s equation it is clear that the ionization degree increases rapidly with the temperature (Exercise 2.1). Note that to maintain the plasma state there has to be a balance of ionization and recombination. This means that either the ionization source must be continuous and strong enough, or the recombination rate must be low. Contemplate: Why is the degree of ionization in Eq. 2.1 inversely proportional to ion density? Using the literature find out the principle behind the derivation of Saha s equation and determne the units of the factor (it is not dimensionless!). You can also try to calculate its value. Apart from heating, ionization can be achieved by applying large local electric fields or by exposing the matter to ionizing radiation such as strong laser light, ultraviolet light, or X-rays. In fact, it is possible to produce plasma even from solid state. An example of low-temperature plasma sustained by solar EUV light and energetic particle precipitation is the Earth s ionosphere. Also the solar photosphere, that is the layer from which most of the solar irradiation emerges, is at a temperature of less than 6000 K that is well below the ionization energy of the photospheric gas. In that case the source of ionization is the heat coming from below the solar surface. 2.2 Quasi-neutrality in plasma Plasma consist of a mixture of positively and negatively charged particles, but overall plasma is quasi-neutral. This means that the positive and negative charges must have approximately equal charge densities: ρ qs = n s q s = 0, (2.2) s s where ρ qs is the charge density ([ρ qs ] = C m 3 ), n s the number density ([n s ] = m 3 ) and q s the electric charge of the species s. For plasma consisting of electrons and one singly charged ion species: n e ( e) + n i (+e) = e(n i n e ), (2.3)

17 2.2. QUASI-NEUTRALITY IN PLASMA 11 where n e is the electron number density, n i the ion number density and e the elementary charge (e = C). We see from Eq. 2.3 that if we in this case require quasi-neutrality, electron and ion number densities must be equal, i.e., n e = n i. A significant fraction of free electrons makes plasma electrically conductive. In fact, plasma is typically an exceptionally good conductor. When temperatures are high and densities low, collisions are rare, and thus, the resistivity is very small. If an electric field is introduced in plasma, electrons quickly rearrange themselves and the electric field is neutralized. As a consequence, no significant large-scale electric field can exist in the rest frame of the unmagnetized plasma. The ability of plasma to shield out applied electric fields is one of its fundamental characteristics. Contemplate: While the resistivity of plasma can be negligible, plasma is not a superconductor. Why? Although plasma is neutral in large scales, deviations from charge neutrality can develop in shorter scales. Let us now look more quantitatively a distance over which quasi-neutrality is true. Suppose that a positive point charge q T is introduced into an otherwise quasi-neutral plasma. The bare Coulomb potential of the test charge is q T /4πɛ 0 r, where r is the distance from q T. Negative electrons are attracted to q T and they form a neutralizing cloud around it (Figure 2.2) modifying its Coulomb potential. Figure 2.2: Debye shielding of a test charge q T. Let us compute the approximate form of the modified Coulomb potential and the thickness of the neutralizing electron cloud. The electrostatic potential φ can be derived from the Poisson equation: E = 2 φ = ρ tot (x)/ɛ 0, (2.4) where ɛ 0 is the permittivity of free space ( A s V 1 m 1 ) and the charge density ρ tot is composed from the contribution of q T (ρ T ) and the polarization of the quasi-neutral plasma as a response to q T (ρ pol ): ρ tot (x) = ρ T δ(x x T ) + ρ pol (x). (2.5)

18 12 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS Here δ is Dirac s delta function (recall its properties!). It ensures that the charge density ρ T vanishes outside x = x T. We assume that the plasma is sufficiently close to the thermal equilibrium so that its density can be given by the Boltzmann distirbution: n s = n 0s exp ( q sφ k B T s ), (2.6) where n 0s is the equilibrium number density in the absence of q T. For a gas to be in the plasma state the constituent electrons and ions must be unbound. This means that we must require that the random thermal energy must be much greater than the average electrostatic energy. Thus, we can assume that q s φ k B T s and expand Eq. 2.6 as: n s n 0s (1 q sφ + 1 qsφ 2 2 k B T s 2 kb 2 T s ). (2.7) The polarization charge density now becomes: ρ pol = s n s q s s n s0 q s0 s n 0s q 2 s k B T s φ = s n 0s q 2 s k B T s φ, (2.8) where n s0 q s0 = 0 due to quasi-neutrality, see Eq Inserting Eq. 2.8 into the s Poisson equation (Eq. 2.4) the potential turns out to be (Exercise 2.2): φ = The factor λ D in Eq. 2.9 is called the Debye length: q T r exp ( ). (2.9) 4πɛ 0 r λ D λ 2 D = ɛ 0 s k B T s n 0s qs 2. (2.10) When ions are much colder than electrons, the ion term can be dropped from the definition of the Debye length. Figure 2.3 shows how the shielded and bare Coulomb potentials of q T differ from each other. When the distance from q T is much smaller than λ D the Coulomb potential is recovered. For distances much larger than λ D the potential shows exponential decay, i.e. it decays much faster than the bare Coulomb potential. Thus, the Debye length is the distance over which significant charge separations (and electric fields) can occur in plasma. Intuitively, Debye length is the limit beyond which the thermal speed of particles is high enough to escape from the Coulomb potential of q T (see Exercise 2.3). Electric field due to q T is restricted within a sphere having the radius given by λ D.

19 2.2. QUASI-NEUTRALITY IN PLASMA 13 Figure 2.3: The bare Coulomb potential and the shielded potential for two different Debye lengths compared (λ D1 > λ D2 ). Contemplate: How does the Debye-length change with density and temperature? Try to give a physical explanation for this behavior Table 2.2 shows some typical values for Debye length. Note that while in many applications/domains (tokamak, Earth s ionosphere, solar corona) the charged regions do not exceed one millimeter, in some space plasmas (e.g., solar wind) Debye length can have macroscopic values. See Exercise 2.4 for comparing typical sizes of the spacecraft and the Debye length of the medium they are measuring. Exercise 2.5 investigates the form of the potential in the vicinity of a spherical conductor immersed in a plasma (e.g., a spherical electric probe measuring the properties of the solar wind). Table 2.1: Typical values of Debye length in different plasma environments. Plasma Solar core Gas discharge tube Tokamak Ionosphere Solar wind 10 Interstellar medium 10 Intergalactic medium 10 5 Debye length [m] Using the Debye length we can formulate a more quantitative criterion for the ionized gas to be in a plasma state. First, to guarantee the quasi-neutrality the plasma system has to have a size L of several Debye lengths: λ D L. (2.11)

20 14 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS Otherwise, significant charge separations can arise and the plasma becomes dominated by the boundary effects. Second, in order for the Debye length to be a statistically valid concept there have to be enough particles within the Debye sphere. The number of particles in the Debye sphere is given by and thus for the ionized gas to be plasma it is required that N D = n 4 3 πλ3 D (2.12) N D 1. (2.13) This criterion also guarantees that collective long-range interactions between charged particles dominate over binary interactions. In many cases a parameter omitting the factor 4π/3 is used: It is called the plasma parameter. Λ = n 0 λ 3 D. (2.14) In Exercise 2.6 the condition Λ 1 it used to prove that in a plasma the kinetic energy is larger than the Coulomb potential energy. This was an essential assumption in the derivation of the Debye length. 2.3 Plasma frequency Let s now investigate the dynamic response of plasma to a small perturbation. Imagine that a fraction of electrons are slightly displaced with respect to ions (Figure 2.4). The charge separation gives rise to an electric field that tries to restore the plasma quasineutrality. As a consequence, electrons are accelerated by the electric field back towards their original positions. Due to their inertia the electrons will overshoot and start to oscillate around the equilibrium position with a specific frequency. Electron oscillations convert continuously electrostatic energy to kinetic energy and back again keeping the total energy conserved. This kind of electron plasma oscillations were first observed by Irving Langmuir and Levy Tonks in a low pressure discharge tube filled with mercury vapor. Their original article can be found from mem4/ap6101/ Next, we derive the frequency of this electron oscillation as a response to a small electric field E 1. We make the cold plasma approximation, i.e. assume there is no thermal motion. Ions are so heavy that they are practically unaffected, and we can consider them as a fixed background. E 1 is caused by a small perturbation n 1 in the electron density:

21 2.3. PLASMA FREQUENCY 15 Figure 2.4: Electric field introduced to a plasma by a slight electron ion displacement. The continuity equation of the electron density is: n i = n 0 (2.15) n e = n 0 + n 1 (r, t). (2.16) n e t + (n eu) = 0, (2.17) where n e = n 0 + n 1 and the velocity caused by perturbation is u = u 1. Obviously, derivatives of the equilibrium quantities (here n 0 ) vanish and if we omit all second order terms (i.e. multiples of two small perturbation quantities, here n 1 u 1 ) we get n 1 t + n 0 u 1 = 0. (2.18) The electric field causes a force F = qe 1, and thus the equation of motion for electrons is: u 1 m e t = ee 1, (2.19) and E 1 is determined from the Gauss law: Taking / t of Eq 2.18 and using Eq 2.20 we obtain E 1 = en 1 ɛ 0. (2.20) 2 n 1 t 2 + ( n 0e 2 ɛ 0 m e )n 1 = 0. (2.21) Eq is the equation for a standing wave with the angular frequency called the electron plasma frequency: ω 2 pe = n 0e 2 ɛ 0 m e (2.22)

22 16 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS Plasma frequency gives the most fundamental time scale in plasma. Usually the term refers to the electron plasma frequency. Ions are affected by the same electric field as electrons, but due to their much larger mass their oscillation is much slower than the electron oscillation (justifies our assumption of the fixed ion background). If the cold plasma approximation is relaxed (i.e. non-zero temperatures), the oscillation propagates as a wave in a plasma as we will see later. 2.4 Plasma response to electromagnetic fields Plasmas respond strongly to electromagnetic fields, and therefore, the effect of electric and magnetic forces is critical to understanding the behavior of plasma. A charged particle in a plasma moves under the influence of the Lorentz force and thus its equation of motion is: m dv = F = q(e + v B), (2.23) dt where E is the electric field, B the magnetic field and v the velocity of the charged particle. Note that the electric and magnetic fields that are used to calculate the Lorentz force arise from all particles in the plasma and include also external (applied) electric and magnetic fields. Thus, it is clear that calculation of the motion of a large number of plasma particles is an immense problem. Figure 2.5: Left: The electric field of the Lorentz force accelerates positive and negative charges to opposite directions. Right: The magnetic part of the Lorentz force changes the path of the particle. The direction of the bending can be inferred applying the right-hand rule to the vector product. The electric field in the Lorentz force accelerates positive and negative charges in opposite directions (left-hand part of Figure 2.5). The magnetic part of the Lorentz force is always perpendicular to the particle s velocity (right-hand part of Figure 2.5). Thus, magnetic field can only change the path of the particle, but it cannot do work on the charge (you can see this easily by calculating the power v F). It is often stated that time-varying magnetic field is used to accelerate particles, but in fact, it is the induced electric field that is responsible for the acceleration.

23 2.4. PLASMA RESPONSE TO ELECTROMAGNETIC FIELDS 17 Figure 2.5 illustrates that in a static and homogeneous magnetic field the charged particles perform a circular motion about the magnetic field lines. The angular frequency of this motion for species α is ω cα = q αb m α. (2.24) The corresponding frequency is f cα = ω cα /2π. This Larmor motion is investigated in detail in Chapter 3. It is seen from above that due to their lower mass electrons spin much faster around the magnetic field than ions. Numerically for electrons and protons f ce (Hz) f cp (Hz) 28 B (nt) B (nt). It is important for plasma physics that charged particles can move relatively freely along the magnetic field lines, but their motion perpendicular to the magnetic field is much more restricted. As a consequence, the magnetic field binds plasma particles together. The Lorentz force per unit volume acting on charge density ρ q and electric current density J = ρ q v is given by f = ρ q E + J B. (2.25) Now v f = J E represents the power per unit volume acting on the moving charges. Depending on the sign of J E power is either extracted from the fields and used as acceleration or heating of the particles, or vice versa. Which way the energy is transformed can be found by considering the conservation law of the electromagnetic energy known as the Poynting theorem. Recall from your electrodynamics course that the energy densities of electric and magnetic fields are given by w E = 1 2 E D (2.26) w M = 1 2 H B, (2.27) where D = ɛe is the electric displacement field and H = B/µ the magnetic field intensity. ɛ is the permittivity and µ the permeability of the medium in consideration (in vacuum ɛ 0 and µ 0 ). Define the Poynting vector as S = E H. From Maxwell s equations it is an easy exercise to to derive S = E J E D t H B t. (2.28) The Poynting theorem is the integral of this expression over volume V J E d 3 r = S d 3 r + V V V t (w E + w M ) d 3 r. (2.29)

24 18 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS 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 of electromagnetic energy density. The last term on the right-hand side expresses the rate of change of the electromagnetic energy in volume V. Assuming that there is no energy flux through the surface, the Poynting theorem states that if J E > 0 the energy of the electromagnetic field in the volume V decreases in time, i.e., the energy is transferred to the particles. In the opposite case (J E < 0) the particles lose energy to the electromagnetic field. A rule-of-thumb is to interpret the inequality sign materialistically; if it is larger than, the matter gains. 2.5 Collective behavior and collisions One of the most distinct features separating plasmas from neutral gases is the way particles interact with each other. In a neutral gas particles interact primarily through direct binary collisions, where individual collisions lead to large deflections and can be considered strong. In plasma charged particles interact predominantly through the Coulomb force. These Coulomb collisions are long-range and each charged particle interacts simultaneously with a large number of charged particles. The Coulomb collisions are weak in the sense that vast majority of the collisions cause only minor deflections. This is the key to the collective behavior. Figure 2.6: Plasmas are divided to collisionless and collisional plasmas. Note that the very high temperature of the fully ionized plasmas makes them in many cases effectively collisionless. Collisional plasmas can be divided into partially and fully ionized plasmas. Important regions of fully ionized plasmas where collisions are frequent are the stellar interiors. Examples of partially ionized plasmas are stellar photospheres and the ionized layers of the upper atmospheres, the ionospheres, of planets. In weakly ionized plasma the dominant type of collisions are those between charged particles and neutral atoms and molecules. When the ionization degree increases the Coulomb collisions between charged particles become dominant. Often high temperature and tenuous plasmas are practically collisionless. Physically this means that the time between collisions, or the mean free path, becomes longer than

25 2.5. COLLECTIVE BEHAVIOR AND COLLISIONS 19 the temporal or spatial scales of problems under study. For example, in the solar wind the effective mean free path between collisions is of the order the distance from the Sun to the Earth. It is important to understand that plasma being collisionless does not mean that electromagnetic interactions between plasma particles would become negligible. On the contrary, they dominate the plasma behavior. At the collisionless limit it is, however, sufficient to consider the effect of average electromagnetic fields on the particles instead of individual collisions Collisions with neutral particles Charged particles interact with neutrals through direct collisions. A key quantity to determine how often collisions occur is the effective cross-section σ c that expresses the likelihood of the interaction between two particles. For binary collisions σ c is simply given as πd 2 0, i.e. it is the cross-section of a neutral atom or molecule, whose radius is d 0. If n n is the number density of neutral particles and v is the average speed of the particles, we can calculate the collision frequency: and the average mean free path: ν n = n n πd 2 0 v (2.30) l mfp = v = 1 ν n n n πd 2. (2.31) 0 For binary collisions ν n gives the frequency between individual collisions and the l mfp the average distance particles travel between two collisions. Note that binary collisions may be of very variable nature. They may be elastic, where two particles bounce off each other retaining their identities and energy states, or inelastic in which case the kinetic energy of a colliding particle is transferred to internal energy of the neutral particle or molecular ion of the plasma. Inelastic collisions can thus lead to recombination, excitation, ionization, and charge exchange, which all are important processes in space plasmas. Auroras are examples of the excitation process of neutral molecules in the Earth s atmosphere due to precipitating high-energy electrons (Figure 2.7). Important charge-exchange processes are collisions where a high-energy proton collides with a slow atom. As a result of the charge-exchange a high-energy neutral atom (ENA) and a low-energy ion are formed: p + fast + H slow H ENA + p + p + fast + O slow H ENA + O +. (2.32) Examples of the first of these are the interactions of interstrellar hydrogen with solar wind protons and inner magnetospheric protons with the hydrogen geocorona around the Earth. The latter process is important when the solar wind interacts with the atmospheres of unmagnetized planets Venus and Mars. ENA imaging provides a useful

26 20 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS Figure 2.7: Left) Precipitating electrons can excite the molecules into a higher energy state. Auroras form when exited atoms emit photons. Different colors of auroras depend on the energy levels of molecules present in the atmospheric layer the electrons can penetrate. Image Courtesy: Jouni Jussila. Right) Large ribbon of ENA emission detected by the IBEX satellite. Image Courtesy: NASA/IBEX/Heerikhuisen et al. tool to image the electric currents carried by ions around the Earth, Jupiter and Saturn. The right-hand panel of Figure 2.7 shows a huge ribbon imaged by the NASA Interstellar Boundary Explorer (IBEX) spacecraft. The ribbon was found in 2009 and its origin has kept scientists puzzled. Currently, it is though that the ribbon is a reflection of particles bouncing off a galactic magnetic field. In plasmas collisions between neutrals and charged particles cannot happen too frequently. Otherwise, the behavior of the substance would be controlled by ordinary hydrodynamic forces than by electromagnetic forces. If ω is the plasma frequency and τ c the mean time between collisions with neutrals, for plasma the condition must be fulfilled. ωτ c > 1 (2.33) Coulomb collisions between charged particles For Coulomb collisions the determination of the collision cross-section is a difficult task. Coulomb collisions are weak and they rarely result in large deflections. This is because each charged plasma particle interacts with many far-away charges simultaneously, while closer encounters where the path would deflect significantly are much less common (see Exercise 2.7 to demonstrate this for a fully ionized plasma). However, the cumulative effect of many small Coulomb collisions can deviate the path significantly. For Coulomb collisions the collision frequency (ν c ) is the measure of the frequency with which the particle trajectory is deviated by 90 due to many successive Coulomb interactions and l mfp is the distance traveled until such a deflection has accumulated. Here we give only a very rough quantitative inspection of Coulomb collisions in plasma.

27 2.5. COLLECTIVE BEHAVIOR AND COLLISIONS 21 electron d c ion (at rest) g c Figure 2.8: Coulomb collisions are long-range interactions between charged particles and in plasma they result typically only to a small deflection in the particle path. Figure 2.8 shows an electron that is interacting with a positive ion. Since ions are much more massive than electrons we can assume that the ion is at rest. The path of the electron is a hyperbola, which far from the ion can be approximated by two straight lines at an angle γ c. The distance d c in Figure 2.8 is called the impact parameter and it describes the closest approach distance between the electron and the ion. To estimate d c we investigate the Coulomb force on the electron e2 F c = 4πɛ 0 d 2 c. (2.34) The electron feels this force over the time τ = d c /v e, during which its momentum changes by the amount τ F c : e 2 (m e v e ). (2.35) 4πɛ 0 v e d c For large angle collisions (γ c 90 ) this will be of the same order of magnitude as the total momentum of the electron m e v e. Thus, we get: d c e 2 4πɛ 0 m e v 2 e, (2.36) and the approximation for the cross-section is: σ c = πd 2 c e 4 16πɛ 2 0 m2 e v e 4, (2.37) where v e has been replaced with the average speed v e. The collision frequency becomes: ν ei = n e σ c v e n e e 4 16πɛ 2 0 m2 e v e 3. (2.38) The average speed of the electrons can be replaced by their average thermal energy k B T e = m e v e 2 /2. Rewriting this using the plasma frequency we get:

28 22 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS ν ei 2 ωpe 4 64π n e ( ) kb T 3/2 e. (2.39) m e Because ω 4 pe n 2 e, 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 small angle collisions. A correction for small angle deflections (not derived here) using the plasma parameter Λ = n e λ 3 D gives: ν ei ω pe ln Λ 32π Λ. (2.40) and the electron mean free path is: l mfp = v e ν e = 2ω peλ D ν e 64πλ D Λ ln Λ. (2.41) Contemplate: Why does the collision frequency increase with the decreasing temperature? Electrons also collide with each other in plasma. For electron-electron collisions the Coulomb force is repulsive, and thus the colliding electron is deflected away from the target. Electron electron collisions are more complex to deal with than electron ion collisions since we cannot assume anymore that the scattering electron is at rest. Since the Coulomb force is of the same order of magnitude, the deflection will be about the same amount as for electron ion collisions. In addition, ions collide with electrons and other ions. Due to much larger mass of an ion their momentum gain or loss in not significant when they interact with an electron. We can approximate: ν ee ν ei (2.42) ν ie (m e /m i )ν ee ν ii m e /m i ν ee Contemplate: Explain why the electron collision frequency is much larger than the ion electron collision frequency.

29 2.6. PLASMA CONDUCTIVITY Plasma conductivity Because collisions change the momentum of the particles, they introduce a term corresponding to friction in the equation of motion (Eq. 2.23): m dv dt = q(e + v B) mν c(v u), (2.43) where ν c is the collision frequency, irrespective whether the electron collisions occur between neutrals or charged particles. u is the velocity of the collision targets Conductivity in non-magnetized plasma Let us first investigate non-magnetized plasma and choose the coordinate system where all collision targets are at rest We can also assume that all electrons have the same velocity v e, i.e. the plasma is cold (remember that the temperature arises from the velocity spread). Assume furthermore that the system has reached a static state, i.e., electrons have already been accelerated to the velocity where the Coulomb force and the collisions balance each other, i.e., dv/dt = 0. Now the solution to Eq is: Electrons carry the current density: E = m eν c v e. (2.44) e J = en e v e, (2.45) Inserting Eq into Eq we obtain the relationship between the electric current density and the electric field, i.e., Ohm s law: J = n ee 2 m e ν c E (2.46) and the conductivity is The inverse of conductivity σ is called resistivity (η). σ = n ee 2 m e ν c. (2.47) Contemplate: We have learned that that plasma is hot but here we assume it to be cold. What does this mean? We will return to this question when we discuss waves in the cold plasma approxiamtion in Chapter 7. For fully ionized plasma ν c is the electron-ion collision frequency given by Eq

30 24 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS This yields the Spitzer resistivity: η = m e ω pe ln Λ n e e 2 32π Λ. (2.48) Using the definitions of ω pe and λ D one can write this as: η = 1 32πɛ 2 0 e 2 m 1/2 e ln Λ. (2.49) (k B T ) 3/2 This shows that the resistivity has only a weak dependence on electron density through Λ. This means that if an electric field is applied to plasma the electric current is independent of the number of current carriers (electrons) as long as there are enough of them, which is not always the case in low-density plasmas as we discuss in the next section in the context of magnetized plasma. A simple way to heat plasma is to pass an electric current through it. However, according to Spitzer s formula resistivity is inversely proportional to the temperature. This means that when temperature increases, resistivity drops fast and plasma becomes such a good conductor that the Ohmic heating is not effective anymore. Contemplate: Why does Spitzer s resistivity not significantly depend on the electron density? And why is it inversely proportional to the temperature? Conductivity in magnetized plasma If a magnetic field is present in the plasma, the conductivity is generally a tensor quantity because charged particles move in different ways perpendicular and parallel to the magnetic field. The equation of motion in a steady state situation can now be written as: 0 = n e e(e 0 + v e B 0 ) ν c m e n e v e. (2.50) By computing the current as in Eq we can rewrite this equation as: J = σ 0 E σ 0 n e e J B. (2.51) To calculate J choose B = Be z. Using the electron gyrofrequency ω ce (Eq. 2.24) we can write: J x = σ 0 E x ω c ν c J y J y = σ 0 E y + ω c J x ν c (2.52) J z = σ 0 E z.

31 2.6. PLASMA CONDUCTIVITY 25 And thus: J x = νc 2 νc 2 + ωce 2 σ 0 E x ω cν c νc 2 + ωce 2 σ 0 E y ν 2 c J y = νc 2 + ωce 2 σ 0 E y + ω cν c νc 2 + ωce 2 σ 0 E x (2.53) J z = σ 0 E z. This set of equations describes a matrix equation between J and E where the conductivity tensor is (Exercise 2.8): σ = J = σ E, (2.54) σ P σ H 0 σ H σ P σ. (2.55) The elements of the conductivity tensor, assuming for simplicity only one ion population, are given by: ν 2 c σ 0 σ P = νc 2 + ωce 2 ω ce ν c σ H = νc 2 + ωce 2 σ 0 (2.56) σ = σ 0 = n ee 2. m e ν c The elements of the conductivity tensor depend both on the collision and gyro frequencies. σ P is known as the Pedersen conductivity. It gives the conductivity in the direction of the electric field E perpendicular to the magnetic field. The Hall conductivity σ H is the conductivity perpendicular to both the ambient magnetic and electric fields. The magnetic field-aligned conductivity σ is the same as the classical collisional conductivity in the absence of magnetic field. In collisionless plasmas it is typically several orders of magnitude larger than the perpendicular conductivities, meaning that the electrons can quickly rearrange to cancel any electric field parallel to B and the electric field in a plasma is typically perpendicular to the magnetic field. However, if there are not enough current carriers available, finite E can arise to accelerate the electrons to a large enough current. Such a structure is often described as an electric double layer. Parallel electric fields have been identified at a few thousand kilometers above the auroras with potential drops of several kilovolts, which corresponds to the energy of the electrons causing main auroral light.

32 26 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS 2.7 Plasma definition: A summary As a summary we will gather together three conditions that a gas must satisfy to be in a plasma state: Collective interactions dominate over binary interactions: There has to be enough charged particles within a Debye sphere, n e λ D 1 Plasma is quasi-neutral: The size of the plasma system L has to be larger than the Debye s length, λ D L. Neutral collision frequency must be smaller than the collective inertial response frequency in plasma (i.e., the plasma frequency): ωτ c > 1 (electromagnetic forces dominate). In Exercise 2.9 basic plasma parameters (plasma frequency, electron gyro frequency, Debye length and plasma parameter) are calculated and compared for different regimes, while in Exercise 2.10 the plasma state is investigated using the three conditions given above. These exercises demonstrates a wide range of conditions plasma may exist. 2.8 Exercises: Basic Definitions and Parameters 1. The degree of ionization is described by the Saha equation n i n n = T 3/2 n 1 i exp( U/T ), where n i is the number density of ions and n n of neutral atoms, T temperature in ev, and U the ionization energy. Assume that the dominating species in the ionosphere is O + and their density m 3 and temperature 0.3 ev. The ionization energy of oxygen is ev. What is the ionization degree of this plasma? Calculate the ionization degree also for temperatures of 0.1 ev, 0.2 ev, and 0.5 ev. You will notice that the ionization degree increases rapidly as a function of temperature! 2. Derive the formula ϕ = q ( T 4πɛ 0 r exp r ) λ D for the screened potential q T of a test charge in a plasma with Boltzmann s density distribution: n α (r) = n α0 exp( q α ϕ(r)/k B T α ). Some hints: 1) Use e x 1 x when substituting the densities into Coloumb s law and make use of quasi-neutrality. 2) Make also use of spherical symmetry to write 2 ϕ = 1 ( d r 2 r 2 dϕ ). dr dr

33 2.8. EXERCISES: BASIC DEFINITIONS AND PARAMETERS 27 After solving the differential equation require that the solution approaches the Coulomb potential of q T when r 0 and remains finite at all distances. Debye screening is considered as the most fundamental property of plasma. 3. An alternative derivation of Debye s length and further insight to its meaning: Consider two infinite parallel plates at x = ±d, set at potential φ = 0. The space between them is uniformly filled by a gas of density n particles of charge q. (a) Using Poisson s equation, show that the potential distribution between the plates is φ = nq 2ɛ 0 (d 2 x 2 ). (b) Show that for d > λ D, the energy needed to transport a particle from a plate to the midplane is greater than the average kinetic energy of particles 4. What is the size of a typical spacecraft used to measure plasmas in the solar wind and magnetosphere? (You can look this up on the Web.) Do spacecraft disturb the medium they are trying to measure? What would you expect should happen to a spacecraft passing through plasma? 5. A spherical conductor of radius a is immersed in a plasma and charged to a potential φ 0. The electrons remain Maxwellian and move to form a Debye shield, but the ions are stationary during the time frame of the experiment. Assuming φ 0 e k B T e, derive an expression for the potential as a function of r in terms of a, φ 0 and λ D. (Hint: Assume a solution of the form (exp(br))/r).) 6. Prove that g 2/3 (g is the inverse of the plasma parameter Λ) is proportional to the ratio of the average Coulomb potential energy between two electrons and the average kinetic energy of electrons. Note: Since the plasma condition requires that Λ 1 this means that in plasma the average kinetic energy between electrons is much larger than the average potential energy. The result is intuitive since electrons are free in a plasma. The kinetic energy being much larger than the potential energy is also a central assumption when deriving the Debye length, see Exercise Show that in a fully ionized plasma the frequency of small-angle collisions is much larger than the frequency of large-angle collisions. What plasma parameter gives the order of magnitude of the relation between the small- and large-angle collisions? This result allows us to describe the Coulomb collisions simply by the Lorentz force and we do not usually need to calculate the (very) complicated collision integral. 8. Derive the elements of the conductivity tensor σ = σ P σ H 0 σ H σ P σ

34 28 CHAPTER 2. BASIC DEFINITIONS AND PARAMETERS starting from the equation of motion E + v e B = m eν c v e. e Sketch the electric field components in the plane perpendicular to the magnetic field. 9. Calculate the electron plasma frequency, electron gyro frequency, Debye length, and plasma parameter for the following plasmas (note the units!) (a) Fusion device: T e 100 kev, n e cm 3, B 1 T (b) Ionosphere at 300 km altitude: T e 0.1 ev, n e 10 6 cm 3, B nt (c) Solar wind at 1 AU: T e 10 ev, n e 10 cm 3, B 5 nt (d) Core of the Sun: T e 1 kev, n e cm 3, no magnetic field (e) Neutron star environment: T e 100 kev, n e cm 3, B 10 8 T Contemplate different ranges these plasma parameters can have in different environments. 10. Consider a spherical container of 1.5 m radius filled with completely ionized hydrogen gas. Ions are are assumed cold. (a) The electron density in the container is set to m 3, i.e., couple of percent of atmospheric number density on ground level. What should the temperature be (upper and lower limits) so that the gas would behave like a plasma? (b) If the temperature is set to 27 C, what should be the electron density so that the plasma conditions are fulfilled? (c) Let us mix some neutral particles to the gas, when the temperature is set to 27 C. The collision frequency nu n between charged particles and neutrals is < ν n >= n n σ n < v >, where n n is neutral atom number density, collision cross-section σ n = m 2 and < v > is the average thermal velocity of ionized particles. How much should neutral gas be added so that the gas dynamics would be controlled by collisions instead of collective electromagnetic interactions? Explain also why in a) and b) the gas ceases to be plasma if the temperature and density are lower/higher than the obtained limits. It is helpful to revise how the Debye length and plasma parameter vary as a function of temperature and density!

35 Chapter 3 Single Particle Motion Plasma is composed of a large number of charged particles that move under the influence of electromagnetic fields. The electric and magnetic fields can be either external (applied fields) or generated by the charged particles themselves. A large part of collective plasma phenomena can be understood (even quantitatively) in terms of single particle motion. After all, plasma behaviour is based, ultimately, on the motions of its constituent particles. Single particle description is a very useful approach in studies of high energy particles in low density plasma where collisions are infrequent and the external magnetic and electric fields are much stronger than the fields generated by the motion of charged the particles themselves. Our task in this chapter is to solve the equation of motion for a charged particle: where F non EM govern non-electromagnetic forces. m dv dt = q(e + v B) + F non EM, (3.1) We will start by investigating how a charged particle moves in the simplest magnetic field configuration, the static and uniform magnetic field in the absence of an electric field. Then we will proceed to examine more complicated magnetic and electric fields, in particular, to determine how the motion of charged particles is affected by spatial and temporal field gradients. 3.1 Motion in a static, uniform magnetic field Assume first that the electric field E = 0, there are no non-electromagnetic forces, and the magnetic field B is constant. Now the equation of motion (Eq. 3.1) of a charged particle is: m dv dt = q(v B). (3.2) 29

36 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: m dv dt v = d dt (mv2 ) = 0, (3.3) 2 which shows that the kinetic energy and the speed are both constant for a particle in a static magnetic field. Next we determine the trajectory of the particle. Let us choose the coordinate system so that B = Bê z. The components of Eq. 3.2 are: m v x = qbv y m v y = qbv x (3.4) m v z = 0. The velocity component parallel to the magnetic field (v z ) is constant, i.e., the particle moves at a constant speed along the magnetic field. This is because the v B-force has no component parallel to the magnetic field. We concluded above (Eq. 3.3) that the total speed of the particle is also a constant, and hence, the absolute value of the velocity perpendicular to the magnetic field (v ) must also be constant. Taking the second time derivatives of the perpendicular velocity components gives: v x = ωc 2 v x v y = ωc 2 v y. (3.5) Eq. 3.5 describes a simple harmonic oscillator at the Larmor (or cyclotron, or gyro) frequency: ω c = qb m. (3.6) As the Larmor frequency is inversely proportional to the mass of the particle, electrons gyrate much faster than ions. For an electron ω c = B, while for a proton ω c = B, where the unit of the gyro frequency is rad s 1 and the magnetic field is given in Teslas. Solving the spatial coordinates of the particle (Exercise 3.1) we see that it performs a circular motion in the xy-plane: The radius of this gyro motion is Larmor (or cyclotron, or gyro radius): r L = mv q B, (3.7)

37 3.1. MOTION IN A STATIC, UNIFORM MAGNETIC FIELD 31 where v = vx 2 + vy 2 is the particle velocity perpendicular to the magnetic field. The Larmor radius is proportional to the mass of the particle and inversely proportional to the magnetic field magnitude. Hence, electrons have much smaller Larmor radii than ions (see Exercise 3.2 for comparing Larmor frequencies and radii for particles in different plasma regions). Therefore, the particle motion is divided into two components: 1. Linear motion along the magnetic field at a constant speed (v = constant) 2. Circular motion in the plane perpendicular to the magnetic field. The center of this circular motion is called the guiding center (GC). Figure 3.1: The trajectory of a charged particle in space in a homogeneous magnetic field is a helix. Combining these motions we will see that the particle is gliding along the magnetic field while making a circle in the plane perpendicular to the magnetic field (i.e., its GC follows the field line). The trajectory of a charged particle is thus a helix (Figure 3.1). The pitch angle of this helix is defined as: and thus tan α = v /v, (3.8) α = arcsin(v /v) = arccos(v /v). (3.9) In many applications it is convenient to omit the relatively fast circular motion of the particle around the magnetic field as the main interest is to investigate the motion of the GC. Hannes Alfvén was the first to introduce the idea of this guiding center approximation. The frame of reference where v = 0 is called the guiding center system (GCS). The GC approximation is valid if the applied magnetic field varies slowly in space and in time when compared to the Larmor motion: r L /L 1 (3.10) B 1 B/ t ω c, (3.11)

38 32 CHAPTER 3. SINGLE PARTICLE MOTION where L is the length scale of the inhomogeneity in the magnetic field. The ions gyrate in the left-handed sense and the electrons in the right-handed sense around the magnetic field. We see from Eq. 3.7 that the Larmor radius is inversely proportional to magnetic field magnitude (r L B 1 ), i.e., the stronger the magnetic field, the more tightly particles are bound to the magnetic field. We also see that since r L m the Larmor radius is much smaller for electrons than for ions (Figure 3.2). Thus, electrons are bound more tightly to magnetic fields than ions. This means that ions lose more easily the guidance of the magnetic field than electrons. Figure 3.2: Larmor orbits of positive ions and negative electrons in a magnetic field. In the GCS gyrating charges form small current loops that are associated with the magnetic moment: µ = πrli 2 = 1 q 2 rl 2 B 2 m = 1 mv 2 2 B = W B. (3.12) The directions of the gyro motion of positive and negative charges is always such that the magnetic moment is opposite to the externally imposed magnetic field (Exercise 3.3). This means that plasma particles tend to reduce the applied magnetic field, and therefore, plasma is diamagnetic (Figure 3.3). µ 2 A = pr L Figure 3.3: Charged particles which gyrate in a magnetized plasma form small current loops. The associated magnetic moment µ is always opposite the magnetic field B. I + B r L

39 3.2. MOTION IN CONSTANT PERPENDICULAR ELECTRIC AND MAGNETIC FIELDS33 In vector form this is: µ = q 2 r L v. (3.13) When a large number of particles is present, the magnetization M is defined as the magnetic moment per unit volume: M = s n s µ s ˆb, (3.14) where < µ s > is the average magnetic moment of the particle species s, Magnetization contributes to the current density in plasma. Any circulation in the magnetization field gives rise to a magnetization current density that can be calculated as: J M = M. (3.15) Note that the magnetization current density is often distinguished from the true current density due to the motion of free charges in the medium. 3.2 Motion in constant perpendicular electric and magnetic fields Next we consider the particle motion in spatially uniform, static electric and magnetic fields. The equation of motion 3.1 is now: m dv dt = q(e + v B). (3.16) The component of the equation along the magnetic field is: m v = qe, (3.17) where v and E are the velocity and electric field components parallel to the magnetic field. This equation describes acceleration along the magnetic field at a constant rate. Due to high conductivity in the plasma free charges react quickly to qe. As a consequence, the electric field component parallel to the magnetic field is typically close to zero in plasma. However, as discussed in Section double layers (i.e., sustained parallel electric fields) can arise in some situations, e.g., if there are not enough charge carriers to maintain the continuity of the electric current. Let us then investigate the perpendicular component of Eq m v = q(e + v B). (3.18) By choosing the frame-of-reference appropriately we can eliminate the electric field from Eq (remember that when conductor (e.g., plasma) moves in a magnetic

40 34 CHAPTER 3. SINGLE PARTICLE MOTION field, the observed electric field depends on the frame-of-reference). non-relativisic Lorentz transformation (i.e., setting γ = 1): We can use the E = E + v E B B = B. (3.19) Here E is the electric field in the non-moving frame (e.g., the spacecraft frame), E is the electric field in the moving frame, which we take to be the plasma rest frame, and v E is the velocity of the frame-of-reference moving perpendicular to E and B (i.e., the drift velocity of the GC). Since the electric field must vanish in the plasma rest-frame E = E + v E B = 0, which can be inverted as (see Exercise 3.4): v E = E B B 2. (3.20) This is called the E B-drift velocity. In the moving GC frame-of-reference the equation of motion is thus: m dv 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 Larmor motion around the magnetic field and the drift of the GC at the velocity v E perpendicular to the magnetic and electric fields. The E B-drift of electrons and ions is shown in Figure 3.4 in the special case when parallel velocity v = 0. If the particle has a velocity component parallel to the magnetic field the GC glides across the magnetic field and the actual path of the particle in 3-D space is a slanted helix. For a detailed derivation of the trajectory of a charged particle in constant and perpendicular electric and magnetic fields see Exercise 3.5. ion electron Figure 3.4: E B-drift for electrons and ions in constant electric and magnetic fields for the parallel velocity v = 0. It is important that the E B-drift speed does not depend on the mass, charge or the velocity of the particle. This means that when many particles are present the whole

41 3.3. GENERAL DRIFT VELOCITY DUE TO A FORCE PERPENDICULAR TO MAGNETIC FIELD35 plasma drifts at the same speed. Hence, no electric current arises from the E B-drift because there is no relative drift of electrons and positive ions. E B-drift is also very slow when compared to the particle gyro motion around the magnetic field. In Exercise 3.6 the E B-drift speed is calculated for an electron in the auroral ionosphere and compared to its Larmor motion. Contemplate: For a more complete physical picture consider the energy gain and loss of a particle during its gyro orbit around B under the influence of constant perpendicular E. This also explains why the drift is at the constant speed, although the effect of the force F = qe is either accelerating or decelerating. See also Exercise General drift velocity due to a force perpendicular to magnetic field The particle motion in constant electric and magnetic fields can be generalized to drifts due to a general constant external force perpendicular to the magnetic field: dv dt = q m (v B) + F m. (3.22) Assuming that the GC drift v D is caused by the force F we can make the transformation v = v + v D: dv dt = q m (v B) + q m (v D B) + F m. (3.23) In the GCS two last terms have to cancel each other (see the previous section) and: v D = F B qb 2. (3.24) Note that this treatment assumes that F/qB c (otherwise the GC approximation is no longer valid). A common example is the gravitational force F = mg, which causes the gravitational drift: v g = m q g B B 2. (3.25) Note that now v g depends on the sign of the particle s charge. Hence the ions and electrons will drift in the opposite directions and a net current density is produced. The physical reason for the gravitational drift is in the change in the Larmor radius as ions and electrons gain and lose energy as they move in the gravitational field (i.e., due to

42 36 CHAPTER 3. SINGLE PARTICLE MOTION changes in v ). In most cases of interest the magnitude of the gravitational drift is negligible to other GC drifts, but it has importance in some regions e.g., in the Earth s ionosphere and in the solar atmosphere. It is also interesting to note that the gravitational drift is, in the same way as the E B-drift, not in the direction of the gravitational force, but perpendicular to it. Figure 3.5 illustrates a consequence of these two drifts by demonstrating what happens when a wave-like ripple develops onto a horizontal surface between plasma (up) and vacuum (below). The gravitational drift is in the horizontal direction and it separates electrons and ions creating a small electric field. Figure 3.5 shows that the direction of the electric field is such that the associated E B-drift is upwards where the layer has already moved upward due to a ripple, and downward where the layer is already downward. As a consequence, the ripple grows and result in an instability known as the Rayleigh-Taylor instability. B E B F g =mg E + E E B v g (ions) E B E plasma vacuum Figure 3.5: Illustration of how gravitational and electric drifts lead together to the Rayleigh-Taylor instability. We have now investigated particle motion in uniform magnetic and electric fields and found expressions for the GC drifts. In the following sections we will investigate the GC drifts in inhomogeneous fields, i.e., we allow either electric field or magnetic field to vary in space or in time. Now the equation of motion becomes too difficult to solve analytically. We will generally assume that the changes in the fields are small when compared to the Larmor motion. 3.4 Particle motion in non-uniform electric fields Understanding the behavior of a charged particle in non-uniform electric fields is important, as the response of plasma determines the properties of electromagnetic wave propagation.

43 3.4. PARTICLE MOTION IN NON-UNIFORM ELECTRIC FIELDS Spatially varying electric field We now assume that the magnetic field is uniform and the electric field in the x-direction varying sinusoidally in the y-direction: E = E 0 cos(ky)ê x. (3.26) Such an electric field can arise for example due to a wave motion. motion for the charged particle is now: m dv dt The equation of = q[e + v B] (3.27) If electric field is weak we can use the undisturbed orbit to estimate E x and average over a gyro cycle. We also consider the case of a small Larmor radius, kr L 1. The result is a small correction to the E B-drift (Exercise 3.7): v E = E B B 2 ( 1 1 ) 4 k2 rl 2. (3.28) The physical reason for this is that a charged particle with its GC near the maximum electric field spends a significant time in a region of weaker electric field, and thus, experiences weaker E B-drift. The correction term depends on the second derivative of E, and we can generalize: v E = ( r2 L 2 ) E B B 2. (3.29) Note that since electrons and ions have different Larmor radii, this drift is charge dependent Time varying electric fields Let us next assume that the electric field is uniform in space, but it varies sinusoidally in time: E = E 0 exp (iωt)ê x. (3.30) If changes are slow when compared to the particles gyro period ( / t ω c ) we find the polarization drift (Exercise 3.8): v P = m dv E qb 2 dt B = 1 de ω c B dt. (3.31)

44 38 CHAPTER 3. SINGLE PARTICLE MOTION The polarization drift separates particles with different charges and masses and gives rise to a polarization current (J P = nqv P ). Due to the large mass ratio between electrons and ions this current is carried mostly by the ions. It is instructive to contemplate the physics behind the polarization drift. Assume that an ion is at rest in a magnetic field and that an electric field E is suddenly applied. The ion will start to move in the direction of E. While gaining speed the particle starts to feel the Lorentz force qv B, and consequently starts to E B-drift. If E will vary sinusoidally in time as assumed above, E will be reversed after some time and the particle starts to E B-drift in the opposite direction. In the advanced space plasma physics course we will investigate also the cases when the rate of change in the electric field is at the same order as the gyro frequency of the particle (ω ω c ) and high-frequency electric fields (ω ω c ). In the former case it is found that particles are in the resonance with the wave (see Chapter 7). 3.5 Particle motion in non-uniform magnetic fields Real magnetic fields can be homogeneous only locally. Both the strength and direction of the magnetic field varies and this gives rise to important drift motions Drift due to a magnetic field gradient Assume first that the magnetic field lines are straight, but allow the magnitude of the magnetic field vary in space. The basic motion is again the gyro motion, but now the particle experiences small field variations as it gyrates. As is seen from Figure 3.6 the GC drift arises because the Larmor radius varies due to changes in the magnetic field magnitude in different regions. It is also obvious from the figure that the drift is perpendicular to both B and B. We assume that the magnetic field is only weakly inhomogeneous and can thus use three-dimensional Taylor expansion near the GC (indicated by the subscript 0): B(r) B 0 + r ( B) (3.32) This expansion requires that r L /L 1, where L is the lenght scale of the field gradient. The Lorentz force at the GC (i.e., we consider the GC as if it were the drifting particle) can be calculated as an average over one gyro radius, by using the orbit in the homogeneous magnetic field from Section 3.1 and the Taylor expansion given above. After some calculations (Exercise 3.9) one obtains:

45 3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS 39 Figure 3.6: The Larmor radius of a particle gyrating in a magnetic field whose magnitude varies in space changes. This results in a GC drift perpendicular both to B and B. Figure shows the trajectory for a postively charged particle. F = µ B. (3.33) Parallel to the magnetic field this yields the acceleration: dv dt = µ m B. (3.34) Perpendicular to the magnetic field one obtains from Eq the gradient drift: v G = µ qb 2 B ( B) = W B ( B). (3.35) qb3 Thus v G depends both on the perpendicular energy and the charge of the particle. Thus, the gradient drift contributes to the net plasma current Drift due to a curved magnetic field Next, we assume that the density of magnetic field lines is constant but they are curved with a constant radius of curvature R C (positive inward). Now a drift arises from the centrifugal force F C felt by the particle as it moves in the magnetic field:

46 40 CHAPTER 3. SINGLE PARTICLE MOTION F C = mw2 R C ˆn, (3.36) where w is the particle s speed along the magnetic field. In practice a sufficient accuracy is w v. ˆn is the unit vector in the direction of R C. F C is again perpendicular to the magnetic field and can be inserted to Eq We obtain: v C = mv2 q ˆn B R C B 2. (3.37) Figure 3.7: A charged particle in a curved magnetic field. The radius of curvature is R C. Now we need to express ˆn in terms of B. If ds is a small displacement along the magnetic field, from Figure 3.7 we find that ds = R C dφ and dˆb = ˆndφ. Dividing these gives: dˆb ds = ˆn R C. (3.38) Since d/ds denotes the derivative along the magnetic field, it can be replaced with (ˆb ). Thus: db ds = (ˆb )B. (3.39) We now obtain the expression for the curvature drift: v C = mv2 B (B )B. (3.40) qb4 Similar to v G (Eq. 3.35), we see that v C depends on the charge of the particle, but while v G depends on the perpendicular energy, the curvature drift depends on the parallel energy (explain why). Assuming that there are no local currents ( B = 0), we can write Eq in a similar form as the equation for the gradient drift:

47 3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS 41 v C = mv2 B B. (3.41) qb3 Now it is possible to combine v G ja v C : v GC = W + 2W qb 3 B B = W qbr C (1 + cos 2 α)ˆn ˆt, (3.42) where ˆt B ja ˆn R C are unit vectors, and α is the pitch angle. Contemplate: Investigate the differences between the E B-drift and the combined gradient and curvature drift (Eq. 3.42). For example, contemplate whether these drifts affect primarily low or high-energy particles, do they give rise to an electric current and can they change the energy of the particle Drift due to a time varying magnetic field Next we allow the magnetic field vary in time. We discussed in Section 3.1 that the magnetic field cannot do work on a charged particles, but the electric field induced by a time variable magnetic field ( E = B/ t) can accelerate/decelerate particles (Figure 3.8). The particle s velocity perpendicular to the magnetic field can be written as v = dl/dt, where l is the length element of the path along the particle s trajectory. Let us take the scalar product of the equation of motion (Eq. 3.1 when F non EM = 0) with v : E B B Ñ E = - t Figure 3.8: A charged particle is accelerated by an electric field that is induced by a time-varying magnetic field. If the time variations are slow when compared with the particle s gyro motion, the magnetic moment of the particle stays constant. dw = q(e v ). (3.43) dt During one gyration the particle gains energy W = q 2π/ωc 0 E v dt. (3.44) Assuming slow temporal changes we can replace the time integral by a line integral over a closed loop and use the Stokes law: B W = q E dl = q ( E) ds = q ds, (3.45) C S S t

48 42 CHAPTER 3. SINGLE PARTICLE MOTION where ds = n ds, n is the normal vector of the surface with the direction defined by the positive circulation of the loop C. For small variations of the field B/ t ω c B/2π, where B is the amount by which the magnetic field changes during one Larmor orbit. Note that here the magnetic field changes are assumed to be so slow that the Larmor radius of the particle is not changed significantly during one Larmor orbit. Thus, we obtain: On the other hand W = 1 2 q ω cr 2 L B = µ B. (3.46) W = µ B + B µ (3.47) and thus µ = 0. Hence, in a slowly time varying magnetic field the magnetic moment µ is conserved although the inductive electric field accelerates the particle throughout its Larmor orbit Adiabatic invariants We found in the previous section that if the magnetic field is varying slowly in time, the magnetic moment µ of the charged particle stays constant. In statistical mechanics the quantity related to a (nearly) periodic motion that stays constant when the system changes slowly, temporally or spatially, is called an adibatic invariant. In Hamiltonian mechanics it is shown that if q and p are the canonical coordinate and momentum of the system and the motion is nearly periodic, the closed integral of p over one period in q I = p dq (3.48) is an adiabatic invariant. This statement requires a proof that we will not discuss here (see, e.g., classical mechanics textbooks by Goldstein or Landau and Lifshitz). The momentum of a particle in the electromagnetic field is p = mv +qa, where A is the magnetic vector potential and the canonical variables related to the motion perpendicular to the magnetic field are p ja r L. Using the Stokes theorem and assuming that the magnetic field and the particle s perpendicular velocity do not change significantly during one Larmor gyration we obtain: I = p dr L = mv dr L + q ( A) ds 2πrL = mv dl + q B ds (3.49) 0 S = 2πmv r L q BπrL 2 = 2πm µ, q i.e., the magnetic moment µ is an adiabatic invariant. S

49 3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS 43 When conserved µ is called the first adiabatic invariant in plasma physics. Also the magnetic flux that is enclosed by the particle with its Larmor motion, Φ = BπrL 2 = 2πm µ, (3.50) q2 is constant. When the magnetic field increases, the Larmor radius decreases and consequently the enclosed flux stays constant. reference point, B 0 mirror point, B m B weak B strong B Figure 3.9: A charged particle moving towards stronger magnetic field. Now let us investigate a charged particle that is moving towards a stronger and stronger magnetic field (Figure 3.9). As the magnetic field increases the Larmor radius of the particle gets smaller and smaller and its gyro frequency increases. We assume that the magnetic field changes so slowly that µ stays constant. Since µ = W /B, it is clear that to keep µ constant the perpendicular energy (W ), and hence the perpendicular velocity (v ), of the particle increase with the increasing magnetic field. Because in the GC approximation the total kinetic energy is conserved, the parallel energy W and the parallel velocity v must decrease. See Exercise 3.10 for a demonstration how the parallel velocity of a particle varies in a simple magnetic bottle configuration. W can increase until W 0. In Section we discussed that the magnetic field whose magnitude varies in space causes the force F = µ B. In this context this force is called the mirror force and it slows down the GC motion and finally when all parallel energy has vanished, it turns the particle around, in other words, the particle gets mirrored. The physical origin of the mirror force arises from the convergence of the magnetic field lines. The Lorentz force has a component opposite to the direction of convergence. The converging magnetic field regions have many important applications. A magnetic bottle is composed of two magnetic mirrors (not necessary of equal strengths) placed facing each other. A charged particle can be trapped within the bottle. Magnetic bottles have been used in various laboratory experiments to confine plasma (Figure 3.10), and they are also found in natural plasmas. For example, the Earth s dipole field forms a

50 44 CHAPTER 3. SINGLE PARTICLE MOTION huge magnetic bottle where charged particles bounce between the mirror points in the northern and southern hemisphere (Figure 3.11, see also Exercise 3.11). The high energy particles trapped in this bottle form the Van Allen radiation belts. Another example of a natural magnetic bottle is a solar coronal loop. Figure 3.10: Two magnetic mirrors facing each other form a magnetic bottle. In practical plasma confinement experiments much more complicated coil geometries are used to improve the confinement conditions. mirror point trapped particle North ions electrons magnetic field line South Figure 3.11: The Earth s dipole field forms a large magnetic bottle. The electrons gradient and curvature drift eastward and ions westward carrying a net westward current around the Earth. For many practical purposes it is interesting to know which particles are mirrored and remain trapped in the magnetic bottle and which can escape from it (Exercise 3.12). If a particle has too much parallel energy with respect to the maximum magnetic field magnitude in the converging field, it will escape from the bottle. For example, if a particle in the Van Allen radiation belt hits the Earth s atmosphere before it is reflected, it will be lost. Write the perpendicular velocity in terms of the pitch angle: v = v sin α.

51 3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS 45 Now the magnetic moment can be expressed as: µ = mv2 sin 2 α 2B. (3.51) µ is assumed to be constant and on the other hand v 2 W is also a constant. Thus, we find a relation between the pitch angle and the magnetic field magnitude at two locations: sin 2 α 1 sin 2 = B 1. (3.52) α 2 B 2 From the definition of the pitch angle we see that at the mirror point α 90, as W 0. If B 2 in Eq is the mirror field B m (see Figure 3.9) then sin α 2 = 1. Therefore, the strength of the mirror field B m depends on the particle s pitch angle at the reference point (subscript 0): sin 2 α 0 = B 0 /B m. (3.53) If B 0 is the weakest magnetic field in the bottle and B m is the weaker of the mirror fields, the particle will be trapped in the bottle if: B 0 B 0 arcsin α arcsin, (3.54) B m B m Otherwise the particle is lost from the bottle. It is said to be in the loss-cone. If the magnetic field does not change much during the time the charged particle bounces back and forth between the magnetic mirror, the bounce motion is nearly periodic. The bounce period τ b is obtained from the formula: s m τ b = 2 s m ds v (s) = 2 s m v s m ds, (3.55) 1/2 (1 B(s)/B m ) where s is the arc length along the GC orbit and s m and s m are the coordinates of the mirror points. Note that the bounce period is defined over the whole bounce motion back and forth. The GC approximation is valid if τ b ωc 1. Thus, the condition to consider the bounce motion as nearly periodic is more restrictive than in the case of Larmor motion: db/dt τ b B 1. (3.56) If this condition is fulfilled, there is an associated adiabatic invariant, which in plasma physics is called the second adiabatic invariant J = p ds, (3.57) where p = mv for a non-relativistic particle.

52 46 CHAPTER 3. SINGLE PARTICLE MOTION To directly prove the invariance of J in a general case is a formidable task. The complete proof is given by Northrop [1963]. The textbook by Goldston and Rutherford [1995] presents the proof for time-independent fields, which is long enough. Also the drift across the magnetic field may be nearly-periodic if the field is sufficiently symmetric, as e.g., in the quasi-dipolar planetary magnetic fields. The corresponding third adiabatic invariant is the magnetic flux through the closed contour defined by the GC drift: Φ = A ds, (3.58) where A is the vector potential of the magnetic field and ds is the arc element along the drift path of the GC. The drift period τ d has to fulfill τ d τ b τ L. The invariant is weaker than µ and J because much slower changes in the field can break the invariance of Φ. In the Earth s magnetosphere µ is often a good invariant. J is invariant for particles that spend at least some time in the magnetic bottle defined by the nearly dipolar field of the Earth. Φ is constant for energetic particles in the trapped radiation belts. However, any or all of the invariances can be broken by perturbations to the system. Figure 3.12: Three adiabatic invariants and corresponding motions in the Earth s magnetic field. The table below and Figure 3.12 present all three adiabatic invariants and associated motions.

53 3.5. PARTICLE MOTION IN NON-UNIFORM MAGNETIC FIELDS 47 Invariant Speed Time-Scale Validity magnetic gyro motion gyro period moment µ v τ L = 2π/ω c τ τ L longitudinal- parallel velocity bounce period τ τ b τ L invariant J of GC w τ b and µ constant flux invariant perpendicular velocity drift period τ τ d τ b τ L Φ of GC w τ d and µ and J constant Every invariant has its characteristics energy conversion mechanism. First, let s consider a particle drifting across the magnetic field from the field B 1 towards B 2 with B 2 > B 1 so that its magnetic moment µ (i.e. the first adiabatic invariant) is conserved. Such a drift can be caused e.g. by the E B drift. The conservation of µ leads to adiabatic heating: W 2 W 1 = B 2 B 1, (3.59) An example of adiabatic acceleration is given in Exercise 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 rays to very high energies ( ev) in the magnetic fields of the universe. A typical galactic cosmic ray has wandered around in the galaxy for millions of years. The radius of the Milky Way is of the order of light years, and thus the particle has had a lot of time to collide with magnetic field structures in the galaxy that have a wide range of velocities. Note that in a given reference frame (e.g., ours) the particle either gains or loses energy when it gets deflected by a magnetic structure (e.g., a mirror). As a result, the velocity distribution of the seed population widens and finally some particles end up at very high energies. The modern version of Fermi acceleration, believed to be responsible for the acceleration of galactic cosmic rays, no longer relies on the conservation of the second adiabatic invariant in a distribution of moving magnetic mirrors. Instead, particles are assumed to be accelerated in shock waves generated in supernova explosions by a mechanism called diffusive shock acceleration. In this model, particles gain energy by repeatedly crossing a single shock front from one side to the other (details not discussed in these lectures). Finally, if the magnetic flux through the closed contour particle s drift encloses stays constant (i.e., third adiabatic invariant), the total energy can change if the drift shells are

54 48 CHAPTER 3. SINGLE PARTICLE MOTION compressed or expanded. As a summary, the characteristic energy conversion mechanism for each invariant are µ: W changes when Larmor radius (that is, B ) changes J: 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 complicated that numerical integration of the equation of motion is required. In this section we briefly discuss the motion in two simple but in practice very important geometries: the dipole field and the field of a current sheet Motion in a dipole field Calculation of charged particle motion in the dipole field is an important application of the orbit theory. Within the distances 2 7 R E from the Earth s center the dipole is a reasonably good approximation of the geomagnetic field and all particles except high-energy cosmic rays behave adiabatically as long as their orbits are not disturbed by collisions or time-varying electromagnetic fields. In the following we use geomagnetically defined spherical coordinates. The dipole moment M E is in the origin and points toward the south. Latitude (λ) is zero at the equator and increases toward the north. Longitude (φ) increases toward the east from a given reference longitude. The SI unit of M E is A m 2. M E is often replaced by k 0 = µ 0 M E /4π, which is also referred to as the dipole moment. The strength and orientation of the terrestrial dipole moment varies slowly and must be taken into account in time scales of space climate. For our purposes sufficiently accurate approximations are M E = A m 2 k 0 = Wb m (SI : Wb = T m 2 ) = G cm 3 (Gaussian units, G = 10 4 T) = 0.3 G RE 3 (R E 6370 km) The last (non-si) expression is useful in practice because the dipole field on the surface of the Earth varies in the range G. The dipole field is an idealization where the source current is assumed to be shrunk into a point at the origin. The source of a planetary or stellar magnetic field is actually a finite, even large, region within the body giving rise to a whole sequence of higher multipoles. When moving away from the source the non-dipolar (quadrupole, octupole, etc.) contributions vanish faster than the dipole. Outside the source the field is a potential field (B = Ψ). The potential for the dipole is Ψ = k 0 1 r = k 0 sin λ r 2. (3.60)

55 3.6. EXAMPLES OF PARTICLE MOTION IN SIMPLE GEOMETRIES 49 It is a standard exercise in elementary electromagnetism to show that from which The magnitude of the magnetic field is and the equation for the field line is B = 1 r 3 [3(k 0 e r )e r k 0 ], (3.61) B r = 2k 0 r 3 sin λ B λ = k 0 cos λ (3.62) r3 B φ = 0. B = k 0 r 3 (1 + 3 sin2 λ) 1/2 (3.63) r = r 0 cos 2 λ, (3.64) where r 0 is the distance from the dipole to the point where the field line crosses the dipole equator. In dipole calculations we also need the length of the line element ds = (dr 2 + r 2 dλ 2 ) 1/2 = r 0 cos λ(1 + 3 sin 2 λ) 1/2 dλ. (3.65) The geometric factor (1 + 3 sin 2 λ) 1/2 = (4 3 cos 2 λ) 1/2 appears frequently in dipole expressions. Every dipole field line is uniquely determined by its (constant) longitude φ 0 and the distance r 0. A useful quantity is the L parameter L = r 0 /R E. For a given L the corresponding field line reaches the surface of the Earth at the latitude λ e = arccos 1 L. (3.66) The field magnitude along a given field line as a function of latitude is B(λ) = [B r (λ) 2 + B λ (λ) 2 ] 1/2 = k 0 r 3 0 (1 + 3 sin 2 λ) 1/2 cos 6 λ. (3.67) For the Earth, we find k 0 r0 3 = 0.3 L 3 G = L 3 T. (3.68) At the equator on the surface of the Earth the dipole field is 0.3 G, and at the poles 0.6 G (i.e., 30 and 60 µt), respectively. The observable geomagnetic field has considerable deviations from this because the dipole is not quite in the center of the Earth, the source is not a point, and the conductivity of the Earth is not uniform.

56 50 CHAPTER 3. SINGLE PARTICLE MOTION The guiding center approximation can be applied if the particle s Larmor radius is much smaller than the curvature radius of the field defined by R C = d 2 r/ds 2 1, which for a static dipole field is R C (λ) = r 0 3 cos λ(1 + 3 sin2 λ) 3/2 2 cos 2 λ. (3.69) In terms of the particle s rigidity mv / q, we write B r L B = mv q R C B mv q r 0 B, (3.70) and thus, the GC approximation is valid if mv q r 0 B. (3.71) Contemplate: Rigidity is a widely used concept in cosmic ray studies. Consider two otherwise identical cosmic ray particles, but with different momenta. Which one is affected more when travelling through a magnetic field, the one with more momentum, or one with less? And why? Write the rigidity in terms of the Larmor radius. The dipole field is a magnetic bottle and the energetic particles trapped in the bottle around the Earth or magnetized planets are said to form trapped radiation. Let λ m be the mirror latitude of a trapped particle and let the subscript 0 refer to the equatorial plane. Then the equatorial pitch angle of the particle is sin 2 α 0 = B 0 B(λ m ) = cos 6 λ m (1 + 3 sin 2. (3.72) λ) 1/2 This shows that the mirror latitude does not depend on L, but the mirror altitude does. If λ e is the latitude where the field line intersects the surface of the Earth and if λ e < λ m, the particle hits the Earth before mirroring and is lost from the bottle. In reality the loss takes place in the upper atmosphere at an altitude that depends on the particle s energy, i.e., on how far it can penetrate before it is lost by collisions. The critical pitch angle in the equatorial plane is (Exercise 3.14) sin 2 α 0l = L 3 (4 3/L) 1/2 = (4L 6 3L 5 ) 1/2. (3.73) The particle is in the loss-cone, if α 0 < α 0l. For the derivation of the loss cone size as a function of latitude see Exercise The conservation of the second adiabatic invariant requires that the bounce period is much shorter than the variations in the magnetic field. For example, in the inner magnetosphere the bounce times of 1-keV electrons are a few seconds and of 1-keV protons a few minutes. During magnetospheric disturbances typical time scales of the field changes are minutes. Thus under such conditions J is a good invariant for electrons but not for protons or heavier ions.

57 3.6. EXAMPLES OF PARTICLE MOTION IN SIMPLE GEOMETRIES 51 Both the gradient and curvature of the dipole field are directed toward the planet. In the dipole field of the Earth positively charged ions drift to the west and electrons to the east. Because B = 0, we find for v GC W v GC = (1 + cos 2 α) (3.74) qbr C = 3mv2 r0 2 cos 5 λ(1 + sin 2 [ λ) 2qk 0 (1 + 3 sin 2 λ) 2 2 sin 2 (1 + 3 sin 2 λ) 1/2 ] α 0 cos 6. 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 Particle motion in a current sheet geometry The single particle approach is also useful when describing charged particle motion near a current sheet. When two regions of oppositely directed magnetic fields are brought together, a sheet of current must arise according to Ampère s law ( B = µ 0 J) to account for the change in the magnetic field. An example of a current sheet in space plasmas is the tail current sheet in the Earth s extended magnetotail, where a current arises to separate the oppositely directed magnetic fields in the southern and northern tail lobes. An even larger-scale current sheet is the heliospheric current sheet that extends to the whole heliosphere and separates the opposite magnetic fields in the southern and northern solar hemispheres. Current sheets have also a key role in many solar phenomena (e.g., solar flares and coronal mass ejections), in the interaction between the solar wind and the Earth s magnetosphere, and in fusion experiments. Previously in this section we have generally assumed that the Larmor radius of a charged particle that gyrates in the magnetic field is small compared to the length scale of the field gradients. This assumption allowed us to use the GC approximation and Taylor expansion around the GC to estimate the particle s orbit. However, near a thin current sheet there can be large field gradients over short spatial distances, and hence, neither the GC approximation nor the invariance of µ are no longer valid. Many physically interesting phenomena (instabilities, magnetic reconnection) arise near strong and thin current sheets where particle motion becomes chaotic and non-adiabatic. The simplest model to describe the magnetotail current sheet is the Harris model for one- and two-dimensional configurations (Figure 3.13). In the two-dimensional Harris current sheet the magnetic field is of the form ( z B = B 0 tanh e x + B n e z, (3.75) L) where B 0 and B n are constant, B n B 0 and L is the characteristic thickness of the current sheet. If B n = 0, the field is one-dimensional. The magnetic field magnitude

58 52 CHAPTER 3. SINGLE PARTICLE MOTION changes from a constant value far away from the current sheet (B 0 ) as a hyberbolic tangent accross the sheet. The electric current points toward the positive y-axis and is J y = ( ) ( ) B0 z sech 2 µ 0 L L. (3.76) Figure 3.13: left) One-, and right) two-dimensional Harris current sheet. Examples of orbits near a current sheet are given in Figure Outside the current sheet the motion is normal Larmor motion, but in the vicinity of the current sheet the motion is more complicated. The monotonic motion in the ±y-direction is called Speiser motion. Particles in the Speiser motion carry most of the current in the current sheet. They do not conserve the magnetic moment. Figure 3.14: Trajectories of positively charged particles near the 1-dimensional Harris current sheet. Contemplate: Explain qualitatively the particle trajectories shown in Figure Consider the effect of gradient drift and note how density and magnetic field changes from the current sheet outwards. (In the Advanced Space Plasma Physics course we will also look the particle motion more quantitatively).

59 3.7. EXERCISE: SINGLE PARTICLE MOTION Exercise: Single Particle Motion 1. Consider the case with a uniform magnetic field with no background electric field. The Lorentz force on a charged particle is F = qv B. Starting from this, derive solution for the spatial coordinates of the particle in detail. Study how positively and negatively charged particles rotate in the magnetic field according to this solution. 2. Derive numerical scaling formulas for the gyro frequency ω g and gyro radius r g using electron mass, electron charge and km/s as scaling units (i.e., express particle mass, charge and velocity in these units) and calculate them (assuming v = 0) for (a) a 10-keV electron in the Earths magnetic field with B = 500 nt and plasma electron density n 0 = 100 cm 3 (b) a solar wind proton with bulk velocity 400 km 1, B = 5 nt and n 0 = 5 cm 3 (c) a 1-keV He + ion in a sunspot, where B = 0.05 T and n 0 = 109 cm 3. Compare the gyro frequencies with the plasma frequencies in the corresponding plasmas. 3. A charged particle rotating in a magnetic field generates an electric current. Calculate the generated magnetic moment and show that the magnetic moment vector antiparallel to the magnetic field vector B for both positively and negatively charged particles. Show also that the magnetic moment can be written as µ = W /B. 4. Show that the electric drift velocity v E = E B B 2 can be obtained from the Lorentz-transformed electric field E = E + v E B = Study a particle (mass m and charge q) in static and homogeneous electric E = E 0 e y and magnetic B = B 0 e z fields. If the particle is at rest at t = 0, show that it follows a cycloid orbit: x(t) = E ( 0 t 1 ) sin(ω g t) B 0 ω g y(t) = E 0 ω g B 0 q q (1 cos(ω gt)) Draw the orbits for a positive and a negative particle. Prove that the average kinetic energy of the drift motion equals the average potential energy drop of the particle in the electric field during half a cycloid orbit.

60 54 CHAPTER 3. SINGLE PARTICLE MOTION 6. Calculate Larmor radius, Larmor period and E B-drift speed for an electron (energy 0.1 ev) in auroral ionosphere where magnetic field is nt. Assume that the electric field is perpendicular to the magnetic field and has a magnitude of 20 mv m 1. How far does the electron drift during one Larmor period? 7. Consider a charged particle in a homogeneous magnetic field B = B 0 e z and inhomogeneous electric field E = E 0 cos(ky) e x, here B 0 and E 0 are constant and kr L << 1. Show that the drift speed is v E = E B B 2 (1 k 2 rl/4). 2 Give a physical interpretation for kr L << 1 and compare its validity and drift speeds for electrons and protons. 8. Starting from the equation of motion derive the polarization drift 9. Derive the force v P = 1 de. ω c B dt F = µ B. on the guiding center of a charge (with magnetic moment µ) moving in an inhomogeneous but straight magnetic field B. (Feel free to use text books in plasma physics.) 10. Let us study a magnetic bottle with B(z) = B 0 (1 + (z/a 0 ) 2 ) (sketch this field configuration!). Using conservation of energy and first adiabatic invariance, show that a particle (mass m), which is mirroring between points z m and z m, has a longitudinal velocity v = 2µB 0 m (zm a 0 ) 2 ( ) z 2, a0 where µ is the magnetic moment. What is the particle velocity a the centre of the bottle (z = 0, where B = B 0 ) and at the mirror points (z = z m and B = B 0 (1 + (z m /a 0 ) 2 ) )? 11. Draw a picture that shows how electron and ion orbits look like when the particles bounce between the two mirror points in the Earth s dipole field. Pay attention to the gyro motion around the mirror points. How does the gyro radius change along the orbit? 12. A group of charged particles with an isotropic velocity distribution is placed in a magnetic bottle with a mirror ratio of R m = B m /B 0 = 4. There are no collisions, so the particles in the loss cone simply escape and the rest remain trapped. What fraction of the particles is trapped? (Hint: Try to figure out what fraction of space the loss cone fills.)

61 3.7. EXERCISE: SINGLE PARTICLE MOTION 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 then makes an elastic collision with a heavy particle and changes direction so that v = v. The magnetic field is then adiabatically decreased back to 0.1 T. What is the proton s kinetic energy now? 14. Starting from the expressions for the components of the magnetic dipole field B r, B λ and B φ make a detailed derivation of the expression for the particle pitch angle at the equatorial plane as a function of its mirror latitude B(λ m ) sin 2 α 0 = B 0 B(λ m ) = cos 6 λ m (1 + 3 sin 2 λ m ) 1/2 and show that the loss-cone width in terms of the L-parameter is given by sin 2 α 0 = L 3 (4 3/L) 1/2 = (4L 6 3L 5 ) 1/ Derive the width of the loss cone as a function of latitude along a magnetic field line (from the equator towards Earth). Draw the loss cone size as a function of latitude for L = 6. Study how the loss cone size varies between the equator and higher latitudes and as the function of L. 16. Calculate the gradient-drift velocity of a 90 -pitch angle particle in a dipolar magnetic field. Start with equations v D = F B qb 2 and F = µ B. Before you start writing down equations, think very carefully what the pitch-angle assumption means.

62 56 CHAPTER 3. SINGLE PARTICLE MOTION

63 Chapter 4 Kinetic Plasma Description As the first step to understand plasma we studied how individual particles behave in electric and magnetic fields. However, the definition of plasma requires that there has to be a large number of particles within a Debye sphere and that the plasma system has the size of several Debye lengths (Eq. 2.2). In plasma charged particles that move in the applied electric and magnetic fields generate their own fields. The computation of the motion of all plasma particles from Maxwell s equations and the Lorentz force would be an immense task. The kinetic plasma approach has its roots in statistical physics, representing the behaviour of a large collection of particles using distribution functions in configuration and velocity space. In a case of plasma one needs to include Maxwell s equations in the formulation of the theory. Kinetic theory is one of the most challenging areas of plasma physics. Here we introduce the most central concepts only. Fortunately, the fluid description (see Section 1.3 and Chapter 6) is sufficiently comprehensive to describe a large part of observed plasma phenomena. However, the fluid approach loses the detailed information on distribution of the plasma constituents in the velocity space. For instance, the kinetic description is relevant in situations where significant deviations from the local thermodynamic equilibrium arise and plasma particle species are non-maxwellian. In particular in hot and tenuous plasmas there are not enough collisions to drive plasma towards Maxwellian distribution. Kinetic theory must also be used when one considers phenomena occurring at short spatial (smaller than Debye length or Larmor radius) or temporal (faster than gyro or plasma frequency scales) scales. In addition, description of kinetic processes, such as instabilities and wave-particle interactions, require the knowledge of the velocity space effects. 4.1 Distribution function The dynamical state of a particle in a plasma at time t can be described by its position: r = xê x + yê y + zê z (4.1) 57

64 58 CHAPTER 4. KINETIC PLASMA DESCRIPTION and velocity: v = v x ê x + v y ê y + v z ê z. (4.2) Combining information of the particle s position and velocity gives its location (r, v) in a 6-dimensional phase space (Figure 4.1). The infinidesimal volume element of phase space is d 3 r d 3 v. v d 3 v (r,v) Figure 4.1: 6-dimensional phase space d 3 r r In statistical physics the single particle distribution function f(r, v, t) expresses the number density in a 6-dimensional phase space element at time t. Hence, f is a function of seven independent variables. The units of number density in the configuration space is m 3 and in the velocity space is (m s 1 ) 3, thus the units of the distribution function are [f] =m 6 s 3. An example of a domain where kinetic processes prevail is the solar wind. The solar wind has low density (on average 5 cm 3 ), it is collisionless and is composed of different particle species. For examples, wave-particle interactions play an important role in the solar wind and temperature anisotropies drive kinetic instabilities. Although large-scale solar wind variations can be understood in terms of single-fluid approach, micro-scale processes may affect the local solar wind properties. Figure 4.2 illustrates typical velocity distribution functions (VDF) in the solar wind. The distribution function needs to be normalized. The most intuitive normalization is to require that the integration of the distribution function over the 6-dimensional phase space volume V gives the total particle number N. f(r, v, t) d 3 rd 3 v = N. (4.3) V The average density in spatial volume V is n = N/V. However, the density can usually vary with space and time, and thus, the particle number density is defined as the zero order velocity moment of the distribution function n(r, t) = f(r, v, t) d 3 v. (4.4) Note that in statistical and mathematical physics the distribution function is often normalized to 1. This is also a logical normalization, because then f(r, v, t) gives the probability to find the particle at location r with velocity v at time t in the 6-dimensional phase space (or if integrated over the whole phase space it states that the probability to find the particle somewhere in the phase space is 1).

65 4.1. DISTRIBUTION FUNCTION 59 Figure 4.2: Velocity distribution function (VDF) measured in the solar wind [Stevark et al., JGR, 2009]. VDF in the solar wind exhibits three components, 1) thermal core, 2) non-maxwellian halo with isotropic pitch angle distribution, and 3) non-maxwellian strahl that features an electron beam propagation along the magnetic field. As an example let us investigate the Maxwellian velocity distribution function (see Appendix 9.4): ( ) ) m 3/2 f(v) = n exp ( mv2, (4.5) 2πk B T 2k B T where m is the mass of the particle and density n = n is assumed to be constant. Using the result: exp( x 2 ) dx = π (4.6) it is easy to show (Exercise 4.1) that the integral of the Maxwellian distribution over the 3-dimensional velocity space gives n ([n] = m 3 ). The average and root-mean-square velocities for a Maxwellian velocity distribution are calculated in Exercise 4.2. The definition of the particle density as an integral of the distribution function illustrates how macroscopical parameters can be expressed as velocity moments of the distribution function: f d 3 v ; vf d 3 v ; vvf d 3 v. Note that vv is a cartesian tensor, whose components are v i v j. Velocity moments depend on time and space. Because in plasma different particle species have often different distribution functions we distinguish them using Greek subscripts.

66 60 CHAPTER 4. KINETIC PLASMA DESCRIPTION The first-order moment yields the particle flux Γ α (r, t) = vf α (r, v, t) d 3 v. (4.7) Its SI units are (m 3 )(m s 1 ) = m 2 s 1, which shows that the particle flux is the number of particles that traverse through a unit surface in a unit time. Dividing this by particle density we get the average, or bulk, velocity at a given location: vfα (r, v, t) d 3 v V α (r, t) = fα (r, v, t) d 3 v, (4.8) from which we can further determine the current density: J α (r, t) = q α Γ α (r, t) = q α n α V α (r, t). (4.9) The second order moment gives parameters that are related to the square of the velocity such as pressure and kinetic energy. In plasma physics pressure is a tensor quantity (particles are likely to have different velocities parallel and perpendicular to the magnetic field) and it is defined to depend on how much the particle velocities deviate from the average velocity V α : P α (r, t) = m α (v V α )(v V α )f α (r, v, t) d 3 v, (4.10) which in a case if spherical symmetry reduces to P α = p α I, where I is the unit tensor and p α the scalar pressure: p α (r, t) = m α (v V α ) 2 f α (r, v, t) d 3 v = n α k B T α (r, t). (4.11) 3 Here we have introduced the concept of temperature T α. In the frame moving with the velocity V α, i.e., where V α = 0, the temperature is given by: 3 2 k BT α (r, t) = m α v 2 f α (r, v, t)d 3 v 2 fα (r, v, t) d 3 v, (4.12) which for a Maxwellian distribution is the temperature of classical thermodynamics. In collisionless plasmas equilibrium distributions may be far from Maxwellian and, consequently, temperature is a non-trivial concept. Temperature can be understood in terms of the width of the distribution function, but only for the Maxwellian distribution function there is a unique level where to determine the width to correspond to the classical definition of temperature. The chain of moments continues to higher orders. The third order introduces the heat flux, i.e., temperature multiplied by velocity. Higher moments can be calculated, but do not have a simple physical interpretation. In plasma physics higher moments than the heat flux are seldom needed. Contemplate: Write the distribution function of electrons that are all moving at the same velocity V 0 and the ions are all at rest. Write also equations for the electric current and the pressure tensor

67 4.2. TIME EVOLUTION OF DISTRIBUTION FUNCTIONS Time evolution of distribution functions To determine how particle distribution functions evolve in space and time we need the appropriate equations of motion. We start by assuming that the number of particles in the 6-dimensional phase space remains constant. We will investigate a small plasma element and follow its motion. Each point in the plasma element moves according to equations: dr dt = v ; dv dt = F m, (4.13) where F describes the forces that influence the system. The number of particles within a volume V of a 6-dimensional phase space is: N = V f(r, v, t) d 3 r d 3 v. (4.14) Long-range forces affect in a similar way to all particles in the plasma element, but short range forces, typically resulting from collisions, can scatter particles in and out from the phase space element. Here we consider only long-range forces. Figure 4.3 illustrates the evolution of a phase space plasma element under the influence of longrange forces. All particles will be accelerated by the same force, and the phase space density at time t 2 will be the same as at time t 1. Only if there are collisions the density can change. v plasma element v t 1 r t 2 r Figure 4.3: A plasma element retains its density in 6-dimensional phase space as it moves under the influence of long-range forces. The conservation of particle number in volume V that moves with the particles is given by the continuity equation: 0 = N ( ) f t + u (NU) = V t + u (fu) d 3 rd 3 v, (4.15) where U = (ẋ, v) = (v, F/m), and u is the 6-dimensional gradient operator, whose components are the the components of the gradients in the configuration and velocity spaces ( / r, / v) The first term on the left-hand side depends on the change in density at each phase space point and the latter depends on how V changes with the motion so

68 62 CHAPTER 4. KINETIC PLASMA DESCRIPTION that the change of the volume does not change the total number of particles within it. As the conservation of particles has to apply for all phase space elements, we obtain: f t + u (fu) = 0. (4.16) If the force F does not depend on the velocity (remember that U = (ẋ, v)) we can write the 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 magnetic part of the Lorentz force does. However, fortunately: (v B) = 0, (4.18) v granting that Equation (4.17) applies also to the Lorentz force. Thus, we have arrived at an equation that describes the evolution of the distribution function under the influence of long-range forces. This is called the Vlasov equation: f t + v f r + q f (E + v B) m v = 0. (4.19) It was formulated by the Soviet theoretical physicist Anatoly Alexandrovich Vlasov in the late 1930s. In classical statistical physics the particle collisions are important and the equation corresponding to the Vlasov equation is the Boltzmann equation: f t + v f r + F m f ( ) f v =, (4.20) t where the term on the right hand side describes the change of the distribution function due to individual collisions. The Vlasov equation is sometimes called as the collisionless Boltzmann equation. Ludwig Boltzmann derived the collision term ( f/ t) c for strong short-range interactions. In plasma physics Coulomb interactions are mainly long-range and weak. Therefore in plasma physics the average interactions between the particles are included in the Boltzmann equation through the external Lorentz force: f t + v f r + q ( ) f f (E + v B) m v =. (4.21) t c c

69 4.3. SOLVING THE VLASOV EQUATION 63 The collision term includes only large-angle collisions between charged particles and the possible collisions with neutrals, including charge-exchange processes. The general calculation of the Boltzmann collision term ( f/ t) c is a tedious task. Fortunately, hot and low density plasmas can often be considered collisionsless and the Vlasov equation is the appropriate approach. The simplest situation taking account the collisions arises when the collisions occur predominantly with neutrals. In this case the collision term can be approximated by the Krook model: ( ) f = ν c (f f 0 ), (4.22) t c where f 0 is the equilibrium distribution and ν c is the constant average collision frequency. In Exercise 4.3 the conductivity is determined for unmagnetized, homogeneous and timeindependent plasma where collisions are taken into account using the Krook model. Taking into account the effect of long-range Coulomb interactions results in so-called Fokker-Planck equations. Their derivation is also a rather difficult task and beyond the scope of this book. 4.3 Solving the Vlasov equation The Vlasov equation is not easy to solve. It must, of course, be done under the constraint to fulfill Maxwell s equations because the source terms of Maxwell s equations (ρ, J) are determined by the distribution function, which, in turn, evolves according to the Vlasov equation. Furthermore, the force term in the Vlasov equation is nonlinear. Thus the Vlasov equation can be solved analytically only for small perturbations when linearization is possible. We investigate here only the simplest case, where there are no background electric or magnetic fields. Let us also assume 2-dimensional phase space (x, v), and that the plasma distribution function is homogeneous and depends on speed only f 0 (v). We consider how the plasma responds to a small perturbation. This corresponds to the setting in Section 2.3 when we derived the formula for the plasma frequency. We consider again the electron motion only and assume ions as a fixed background. Since plasma starts to oscillate, we assume that the perturbation will cause an electric field of the form of a plane wave: E(x, t) = Ê exp[i(kx ωt)]. (4.23) If we denote the small perturbation to the distribution functioon by f 1, the distribution function that enters to the Vlasov equation is f = f 0 + f 1 and f t + v f x e m E f v = 0. (4.24) In the case of small perturbations the linearization is possible, which means that only the first order terms of small perturbations will be considered. Since f 0 is the

70 64 CHAPTER 4. KINETIC PLASMA DESCRIPTION equilibrium solution of the Vlasov equation the sum of the zero order terms is trivially zero. The first-order Vlasov equation thus becomes: f 1 t + v f 1 x e m E f 0 v = 0. (4.25) From Maxwell s equations we only need: ɛ 0 E = ρ = e f 1 d 3 v, (4.26) which in the 1-dimensional case simplifies to the form: ɛ 0 E x = e f 1 dv. (4.27) Vlasov tried to solve these equations in the end of the 1930s by assuming that the perturbation of the distribution function also has the form of a plane wave: f 1 (x, v, t) = ˆf 1 (v) exp[i(kx ωt)], (4.28) which is practically the same as using Fourier transformations in space and time. With this assumption the linearized Vlasov equation is reduced to: with the solution: i(ω kv) ˆf 1 e mê f 0 v = 0, (4.29) ˆf 1 = ieê m By inserting this to the Coulumb law (Eq. 4.27) we obtain: ikɛ 0 Ê = e from which we can cancel Ê. ˆf 1 dv = ie2 Ê m f 0 / v ω kv. (4.30) f 0 / v dv, (4.31) ω kv If we know the equilibrium distribution f 0 we can calculate the relation between the wave number and frequency related to the perturbation caused by the electric field, i.e., we have found the dispersion equation: D(k, ω) 1 + e2 1 f 0 / v dv = 0. (4.32) mɛ 0 k ω kv

71 4.3. SOLVING THE VLASOV EQUATION 65 Charge density ρ can be considered as the internal property of plasma and thus the Maxwell equation can be written as: D = 0, where D = ɛ 0 D(k, ω)e. We can find the electron plasma wave we encountered in Section 2.3 by considering the dispersion relation at the long-wavelegth limit (ω kv). Now we can expand the denominator in the integral: 1 ω kv = 1 ω + kv ω 2 + k2 v 2 ω 3 + k3 v 3 ω (4.33) For instance, by using the 1-dimensional Maxwellian distribution (for a complete derivation see Exercise 4.4) : ( ) ) m 1/2 f 0 (v) = n exp ( mv2 2πk B T 2k B T and taking into account only the leading terms, the dispersion equation reads: where v th = 2k B T/m is the electron thermal speed. (4.34) ( 1 ω2 p ω k2 vth 2 ) 2ω 2 = 0, (4.35) Assuming infinite wavelength (k 0) the solution is the standing plasma oscillation at the frequency ω p = ne 2 /(mɛ 0 ) we found in Section 2.3. For finite wavelengths and finite temperatures there is a small correction in the dispersion relation: ω 2 ω 2 p k2 v 2 th. (4.36) The wave is now dispersive (see Appendix 9.3), i.e., propagates with different (finite) speeds at different frequencies. It is called the Langmuir wave. In Chapter 8 we will derive the same dispersion relation starting from the warm plasma theory. If the denominator in Eq cannot be expanded, the integral is not straightforward to evaluate. If frequency ω is real, there is a singularity at v = ω/k along the path of integration. In most situations frequencies are not real since waves in plasma are typically either damped by collisions or amplified by some instability mechanism. Inserting ω = ω r + iω i in Eq we see that f 1 exp ( iω r t) exp (ω i t). (4.37) If ω is complex, the singularity is not along the real axis.

72 66 CHAPTER 4. KINETIC PLASMA DESCRIPTION Vlasov did not find the correct way of dealing with the singularity. Lev Landau realized in 1946 that because the perturbation must begin at some instant, the problem can be treated as an initial value problem and, instead of a Fourier transform, a Laplace transform in the time domain can be applied. Once the initial transients of the perturbation have faded away, the asymptotic solution gives the intrinsic properties of the plasma, i.e., the dispersion equation. The exact Landau solution is complicated and technically beyond the scope of these lectures (for the full treatment see, e.g., Koskinen, 2011). The final result includes an imaginary part γ ( f/ v) v=ω/k. The value of γ thus depends on the form of the distribution function and it determines whether the perturbation E = E 0 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: ( ) ( π 1/2 ω pe γ = 8 k 3 λ 3 exp 1 D 2k 2 λ 2 3 ) D 2 (4.39) For the Maxwellian distribution γ is negative and the perturbation will damp (Figure 4.4). This phenomenon is known as the Landau damping. The damping is a genuine collective effect characteristic for plasmas and important in describing how energy transfers from plasma particles to wave modes and vice versa. The particles that have a velocity close to the phase velocity v ph of the wave interact strongly with the wave ( resonance ) and can exchange energy. f(v) slower particles faster particles 0 v ph Figure 4.4: In a Maxwellian distribution, there are always more low than high energy particles, and thus, wave loses more energy than gains back. Landau s original solution was not immediately accepted. The wave damping without energy dissipation by collisions has been one of the most astounding results of plasma

73 4.4. EXAMPLES OF DISTRIBUTION FUNCTIONS 67 physics. This unexpected result was discovered through a purely mathematical analysis but it was not experimentally verified in laboratories until the 1960s. 4.4 Examples of distribution functions We have previously considered primarily the one-dimensional Maxwellian distribution: ( ) ) m 1/2 f 0 (v) = n exp ( mv2. (4.40) 2πk B T 2k B T It is a distribution towards which the gas thermalizes due to collisions. Note that the power in the normalization factor depends on the degrees of freedom in the velocity space. Each degree of freedom contributes a factor of 1/2. For a 3-dimensional isotropic distribution the power is simply 3/2. Figure 4.5 illustrates an isotropic Maxwellian velocity distribution function in the velocity space. Figure 4.5: Isotropic Maxwellian velocity distribution function. The right-hand picture shows contours of constant f in the velocity phase. The horizontal (vertical) axis shows the velocity component perpendicular (parallel) to the magnetic field. Many hot and tenuous plasmas are collisionless and cannot be described by a Maxwellian distribution. However, in many cases the Maxwellian is a reasonable starting point. For example, the whole distribution may be moving with respect to the observer. If we denote this velocity with V 0 the 3-dimensional distribution function is ( m f(v) = n 2πk B T ) 3/2 exp ( m(v V 0) 2 ). (4.41) 2k B T This is called the drifting Maxwellian distribution (Figure 4.6). The magnetic field has a significant effect on how charged particles move in the plasma. In particular, the magnetic field drives plasma towards anisotropy with different properties perpendicular and parallel to the magnetic field. For instance, assume that particles are trapped within a magnetic bottle and in the center of the bottle their

74 68 CHAPTER 4. KINETIC PLASMA DESCRIPTION Figure 4.6: Velocity distribution function (left) and countours of constant f for a Maxwellian distribution that drifts perpendicular to the magnetic field at the velocity V 0 (drifting Maxwellian). distribution is Maxwellian. If the bottle is contracted, the mirror points move closer together resulting in the increase of the pitch angles. This will flatten the isotropic distribution in the direction parallel to the magnetic field and stretch it in the direction perpendicular to the magnetic field (parallel velocity of the particles will decrease, while perpendicular velocities increase). The resulting anisotropic distribution is often called a pancake distribution. In turn, if the bottle will be stretched the mirror points move further away and the distribution will be stretched parallel to the magnetic field to a cigar-shaped distribution. Anisotropic plasma can have a Maxwellian distribution both parallel and perpendicular to the magnetic field but with different temperatures T ja T. Now the distribution function will be ( ) ( ) n m 3/2 f(v, v ) = T T 1/2 exp mv2 mv2. (4.42) 2πk B 2k B T 2k B T As the perpendicular velocity space is 2-dimensional, the normalization factor has the power 2 1/2 = 1 for T (the width of the distribution is assumed to be the same in all perpendicular directions, the distribution is said to be gyrotropic), whereas there is only one degree of freedom in the parallel direction. Anisotropic plasma may also move accross the magnetic field, for example due to the E B-drift or the gradient drift (Figure 4.7). Now the Maxwellian distribution reads: f(v, v ) = n T T 1/2 ( m 2πk B ) 3/2 exp ( m(v v 0 ) 2 ) mv2. (4.43) 2k B T 2k B T As discussed in Section there are always some particles that can escape from the magnetic bottle. In the absence of a mechanism that would replenish the lost particles the distribution becomes a loss-cone distribution.

75 4.4. EXAMPLES OF DISTRIBUTION FUNCTIONS 69 Figure 4.7: Drifting pancake distribution Contemplate: Sketch a loss-cone distribution. Another important special case is a field-aligned beam whose 3-dimensional distribution function is ( ) ( n m 3/2 f(v, v ) = T T 1/2 exp mv2 m(v v 0 ) 2 ). (4.44) 2πk B 2k B T 2k B T It is often convenient to present the distribution function as a function of energy instead of velocity. If all energy is kinetic, the energy is simply obtained from W = mv 2 /2. In the case the particles are in the external electric potential field U = qϕ the total energy of particles is W = mv 2 /2 + U and the Maxwellian distribution function is This can be written as the energy distribution: ( ) m 3/2 ( f(v) = n exp W ). (4.45) 2πk B T k B T [ ] 2(W U) 1/2 g(w ) = 4π f(v). (4.46) m 3 For derivation of the above form in a case U = 0 see Exercise 4.5. The normalization factor is determined by requiring that the integration of the energy distribution over all energies gives the density. A very important distribution function in space plasmas is the so-called kappa distribution. Distribution functions are often nearly Maxwellian at low energies, but they decrease more slowly at high energies. At higher energies the distribution is described better by a power law than by an exponential decay of the Maxwell distribution. Such a behaviour is not surprising if we remember that the Coulomb collisional frequency decreases with increasing temperature as T 2/3 (see Section 2.5). Hence, it takes longer time for fast particles to reach Maxwellian distribution than for slow particles. The kappa-distribution has the form (Figure 4.8): ( ) m 3/2 ( Γ(κ + 1) f κ (W ) = n 1 + W ) (κ+1). (4.47) 2πκW 0 Γ(κ 1/2) κw 0

76 70 CHAPTER 4. KINETIC PLASMA DESCRIPTION Figure 4.8: Maxwell and Kappa distributions as a function of energy. Here is W 0 is the energy at the peak of the distribution and Γ is the gamma function of mathematics. When κ 1 the kappa distribution is close to the Maxwellian distribution (Exercise 4.6). When κ is smaller but > 1 the distribution has a high-energy tail. Velocity and energy distribution functions cannot be measured directly. Instead, the observed quantity is the particle flux to the detector (an example given in Exercise 4.7). Particle flux is defined as the number density of particles multiplied by the velocity component normal to the surface. We define the differential flux of particles traversing a unit area per unit time, unit solid angle (in spherical coordinates the differential solid angle is dω = sin θdθdφ) and unit energy as J(W, Ω, α, r, t). The units of J are normally given as (m 2 sr s ev) 1. Note that in literature cm is often used instead of m and, depending on the actual energy range considered, electron volts are often replaced by kev, MeV, or GeV. Thus it is important to pay attention to the correct factors of 10 in data displays! Let us conclude this discussion by finding out how differential flux and distribution function are related to each other. We can write the number density in a differential velocity element (in spherical coordinates d 3 v = v 2 dv dω) as dn = f(α, r, t) v 2 dv dω. By multiplying this with v we obtain another expression for the differential flux f(α, r, t) v 3 dv dω. Comparing with our earlier definition of the differential flux we obtain: J(W, Ω, r, t) dw dω = fv 3 dv dω. (4.48) Since dw = mv dv we can write the relationship between the differential flux and the distribution function as: J(W, Ω, r, t) = v2 m f. (4.49)

77 4.5. EXERCISES: KINETIC PLASMA DESCRIPTION Exercises: Kinetic Plasma Description 1. Integrate the Maxwellian distribution function over three-dimensional velocity space. 2. (a) Using the Maxwellian velocity distribution function calculate the average velocity v ( ) m 3/2 ( f(v) = n exp m ) 2πk B T 2πk B T (b) Calculate also the root-mean-square velocity v rms = v 2. What is the corresponding average kinetic energy E = m v 2 /2? Hint: It is useful to remember that ( = exp x 2) dx = π. 3. Consider electrons in an unmagnetized (B = 0) homogeneous ( / r = 0), timeindependent ( / t = 0) plasma in a weak constant electric field. Assume that the equilibrium distribution of the electrons is Maxwellian and take the collision into account using the relaxation time approximation known also as the Krook model ( ) f = ν c (f f 0 ). t c Show that the conductivity of this plasma is given by σ = ne2 mν c. 4. Insert the one-dimensional Maxwellian into D(k, ω) 1 + e2 1 f 0 / v mɛ 0 k ω kv dv = 0 and derive the dispersion equation for the Langmuir wave 5. Starting again from the simple Maxwellian velocity distribution show that the Maxwellian energy distribution (or Boltzmann distribution) becomes f (W ) = 2n ( ) W W π (k B T ) 3 exp. k B T 6. Show that for large κ the kappa distribution approaches the Maxwellian distribution.

78 72 CHAPTER 4. KINETIC PLASMA DESCRIPTION 7. Let us consider energetic particle measurements by satellites. The following differential fluxes (particles/s kev) were obtained: , 3850, 137, 4.52 corresponding to energies 30 kev, 80 kev, 240 kev and 800 kev. (a) Plot the measurements as log(flux) vs. log(energy). What can you say about the spectrum? (b) Use the least squares method to fit a power-law spectrum of type f(e) = f 0 E γ to the data. What is the spectral index γ and f 0? (c) How many particles did the satellite measure altogether at the energy range kev?

79 Chapter 5 Macroscopic Plasma Equations In the two previous chapters we have covered the microscopic and kinetic plasma descriptions. The Boltzmann and Vlasov equations derived in Chapter 4 can be considered as the basic equations of plasma physics. In many cases it is not necessary to know the exact evolution of the distribution function, rather we are interested in the macroscopic (and measurable) properties of the plasma (density, flow velocity, temperature, pressure, etc.) and their evolution in space and time. This can be achieved by taking velocity moments of the Boltzmann and Vlasov equations. The resulting macroscopic variables, such as density, velocity and pressure, are functions of the position and time only. This is the fluid (or macroscopic, see Figure 1.3) plasma description. In the fluid theory, the time evolution of macroscopic parameters is determined by means of fluid equations that are analogous to, but generally more complicated than, the equations of hydrodynamics There are different levels of fluid theories. Multifluid theories consider the plasma particle species independently. For example, the two-fluid model has separate equations for electron and ion fluids. The simplest and most important macroscopic model is called magnetohydrodynamics (MHD). MHD combines one-fluid (hydrodynamic and Lorentz force) effects and the Maxwell equations. Historically, the development did not proceed from microphysical to fluid theories. The development of plasma physics in the 1930s and 1940s started from the physics of neutral gases and fluids, and magnetic terms were added to the equations of hydrodynamics. This led to the equations of MHD. Only later the equations of MHD were derived from the microscopic theory. 5.1 Macroscopic transport equations For the needs of many applications we could start from the Vlasov equation, but retaining the collision term gives us a more complete macroscopic theory. When not needed, the 73

80 74 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS collision effects can be dropped at the macroscopic level. We start by taking the velocity moments of the Boltzmann equation for plasma particle species α: f α t + v f α r + q α (E + v B) f ( ) α m α v = fα. (5.1) t c Continuity equation (the zeroth moment) We first integrate Eq. 5.1 over the velocity space. For physical distributions f 0, as v, and the force term vanishes in the integration. If there are no ionizing, recombining, or charge-exchange collisions, the zero-order moment of the collision term is also zero. The integral of the first term of Eq. 5.1 yields the time derivative of density. The second term is of the first order in velocity. The integration gives: v f α r d3 v = vf α d 3 v = (n α V α ), (5.2) and we have found the equation of continuity n α t + (n α V α ) = 0. (5.3) Continuity equations for charge or mass densities are obtained by multiplying Eq. 5.3 by q α or m α, respectively: ρ mα t + (ρ mα V α ) = 0 (5.4) ρ qα t + J α = 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 Equation of motion (the first moment) Multiply Eq. 5.1 by m α v and integrate over v (Exercise 5.1). This yields the momentum transport equation, which is the macroscopic equation of motion.

81 5.1. MACROSCOPIC TRANSPORT EQUATIONS 75 n α m α V α t + n α m α V α V α n α q α (E + V α B) + P α ( ) fα = m α v d 3 v. (5.7) t c Equation of motion couples plasma velocity to number density. The term V α V α and the pressure tensor P α arise from the term with vv and moving the nabla-operator ( = / r) outside the integral. In the pressure tensor the diagonal elements represent what we normally understand as pressure, while the off-diagonal elements represent shearing or tension in the medium. The divergence of P α contains information of inhomogeneity and viscosity of the plasma. The Lorentz force term does not integrate to zero. The average electric and magnetic fields in the Boltzmann equation are determined by both internal and external sources (ρ ext, J ext ) and fulfill the Maxwell equations for the average plasma properties. E = α n α q α ɛ 0 B = 1 c 2 E t + µ 0 + ρ ext ɛ 0 α n α q α V α + µ 0 J ext. (5.8) Because collisions transport momentum between different plasma populations, the collision integral does not vanish, except for collisions between the same type of particles. The collision term is a complicated function of velocity. A useful approximation related to the Krook model is m α v ( ) fα t c d 3 v = β m α n α (V α V β ) ν αβ, (5.9) where ν αβ is the average collision frequency of particle species α with particles of species β. The momentum equation relates the fluid velocity to the density gradient and electromagnetic forces acting on the fluid element, but not on the single particles anymore. Note that the momentum equation has a close relationship to the Navier-Stokes equation of hydrodynamics. In neutral hydrodynamics the only forces that are acting to fluid are the pressure and viscous forces Energy equation (the second moment) Next, let us calculate the moments over vv. The second velocity moment yields the energy or heat transport equation (conservation law of energy). Integration is now quite a tedious process. We write here the equation in the form:

82 76 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS ( ) 3 2 n Tα αk B t + V α T α + p α V α = H α (P α ) V α + ( nα m α V 2 ) α t 2 c, (5.10) where the isotropic part of the pressure p α I is written on the left-hand side and the non-isotropic part P α on the right-hand side. The relation between the scalar pressure p α and temperature T α is assumed to be that of an ideal gas p α = n α k B T α, The thirdorder term is H α and it describes the heat flux. The energy equation states that the temperature of plasma can increase due to compressive flow ( V α < 0), dissipation due to flow gradient induced stress ((P α ) V α ), collisional energy exhange, and due to divergence of the heat flux (sources or sinks). Now we have macroscopic equations separately for each plasma species. In a real plasma several species co-exist. The simplest description of the real plasma consists of electrons and protons (two-fluid model). The separate fluid components interact through collisions and electromagnetic interaction. Continuity equation and momentum transport equations are valid separately for both fluids. In addition to electrons and protons, there may be a variety of heavier ions, as well as neutral particles, which may contribute to plasma dynamics through collisions, including charge-exchange processes. Sometimes it is also necessary to consider different species of the same type of particles; e.g., in the same spatial volume there may be two electron populations of widely different temperatures or average velocities. Such situations often give rise to plasma instabilities to be discussed in Chapter Equations of state The equation for the heat flux is found by taking the third velocity moment of the Boltzmann equation. This would lead to an equation with the fourth-order contribution, and so on. This is because the Boltzman equation includes both the zeroth and first order velocity. The chain of equations with increasing order of velocity (and with increasing complexity) must be truncated at some point to form a closed system of transport equations. In many practical problems this is made in the second order, either by neglecting the heat flux, or by substituting the energy equation by an equation of state. Here physical insight is essential. Krall and Trivelpiece [1973] state this: The fluid theory, though of great practical use, relies heavily on the cunning of its user. The simplest of closed system is the cold plasma model. It contains the conservation equations for mass and momentum and the related macroscopic variables are the density and bulk velocity. As the temperature is taken to be zero, the pressure tensor is zero.

83 5.3. MAGNETOHYDRODYNAMIC EQUATIONS 77 The particle distribution function becomes the delta function that is centred at the bulk velocity: f(r, v, t) = a δ(v V(r, t)), where a is the dimensional normalization factor. In the case of warm plasma the thermal effects can be taken into account by considering, e.g., isothermal or adiabatic approach. We assume non-viscous plasma, i.e., the non-diagonal elements of the pressure tensor are zero. Let us further assume that plasma is isotropic, i.e., the diagonal elements of P α are equal, and thus, the pressure tensor can be replaced by a scalar pressure. Thus, in the momentum transport equation the term P degenerates to p. The macroscopic variables appearing in this case are the number density n, the bulk velocity V, and the scalar pressure p. In isothermal plasma T = T 0 = constant and the equation of state is: p = nk B T 0. (5.11) In isothermal plasma changes of plasma parameters are so slow that the system has time to thermalize during the time-scale of the change. In collisionless plasmas this is typically not a very good assumption. The opposite limit is that the changes occur so fast that there is no heat exchange between the considered plasma element and its surroundings. Hence, we can set the the heat flux to zero ( H = 0). The resulting adiabatic equation of state can be derived rather easily from the heat flux equation by assuming scalar pressure, using the density continuation equation and writing d/dt = / t + V. The result is which gives the relations: 3 2 ndt dt = T dn dt, (5.12) T = T 0 ( n n 0 ) γ 1 ; p = p 0 ( n n 0 ) γ, (5.13) where γ = c p /c v is the polytropic index, which in the adiabatic case is known as the adiabatic constant. In statistical mechanics it is shown that for monoatomic adiabatic gas γ = (f + 2)/f, where f is the number of degrees of freedom. Thus, for 3-dimensional ideal gas γ = 5/3. In non-adiabatic cases γ 5/3, e.g., for isothermal process γ = 1, for the isobaric process γ = 0, and for the isometric (constant density) γ. Plasmas are not always isotropic. For instance, if a strong magnetic field is present or if there are not enough collisions to maintain the isotropic velocity distribution the pressure is anisotropic. In addition, pressure tensor does not even need to be diagonal. 5.3 Magnetohydrodynamic equations We have now derived macroscopic fluid equations for each plasma species. Next we will combine these equations to a one-fluid theory called magnetohydrodynamics (MHD).

84 78 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS MHD is probably the most widely known plasma theory. In MHD the plasma is considered as a single fluid in the centre-of-mass (CM) frame. This is a well-motivated approach in collision-dominated plasmas, where the collisions constrain the plasma particles to follow each other closely and thermalize the distribution toward a Maxwellian, which makes the interpretation of velocity moments straightforward. MHD works also remarkably well in collisionless tenuous space plasmas, e.g., when studying the largescale interaction of the solar wind, the magnetosphere and the ionosphere of the Earth (Figure 5.1). However, great care should be exercised both with interpretation and approximations. To some extent the electromagnetic forces take the role of collisions, e.g., constraining the motion across the magnetic field. This picture is, however, not complete because the motion along the magnetic field is unconstrained in a homogeneous plasma. Figure 5.1: The Earth s magnetosphere simulated by the global magnetosphereionosphere simulation GUMICS-4. The colours indicate the plasma density. The small inset shows the ionospheric conductivity from the electrostatic ionospheric module coupled to the MHD-based GUMICS-4. Courtesy: GUMICS team at FMI MHD transport equations The single-fluid variables are defined as: mass density ρ m (r, t) = α n α m α,

85 5.3. MAGNETOHYDRODYNAMIC EQUATIONS 79 charge density macrospcopic velocity ρ q (r, t) = α V(r, t) = n α q α = e(n i n e ), α n αm α V α α n αm α, electric current density J(r, t) = α n α q α V α and the pressure tensor in the CM frame from which we get the total pressure Pα CM (r, t) = m α (v V)(v V)f α d 3 v, P(r, t) = α P CM α (r, t). Summing the individual continuity and momentum transport equations over particle species yields the continuity equations ρ m t + (ρ m V) = 0 (5.14) ρ q t + J = 0 (5.15) and the momentum transport equation ρ m ( V t + V V ) = ρ q E + J B P. (5.16) The momentum equation corresponds to the Navier-Stokes equation of hydrodynamics where the viscosity terms are written explicitly (here they are hidden in the pressure gradient). At macroscopic level the deviations from charge neutrality are small and ρ q E is usually negligible. The magnetic part of the Lorentz force J B (often called Ampère s force) is, however, essential in the theory of magnetic fluids. The next equation in the velocity moment chain is the energy transport equation. After some tedious but straightforward calculation the energy equation can be written in the conservation form t [ ( ) V 2 ρ m 2 + w ] + B2 = H. (5.17) 2µ 0 Here w is the enthalpy that is related to the the internal free energy (per unit mass) of the plasma u by w = u + p/ρ m. The RHS is the divergence of the heat flux vector H,

86 80 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS which is a third-order moment. After some reasonable approximations it can be written as ( V 2 H = 2 + u + p + ) ( B2 /µ 0 ρ m V Bµ0 V + J ) B ρ m ne J B σµ 0 + JB2 µ 0 ne + m eb µ 0 ne 2 J t. (5.18) For derivation of the energy transport equation in adiabatic ideal MHD, see Exercise 5.2. When integrated over a finite volume V the LHS of (5.17) describes the temporal change of the energy of the MHD plasma in that volume and the RHS the the energy flux through the boundary V and energy losses due to resistivity. Thus we have found the MHD equivalent of Poynting s theorem of elementary electrodynamics. In the case of MHD the third moment is usually neglected and an equation of state is used to relate the changes in plasma pressure and density. MHD assumes Maxwellian distribution and thus the pressure is isotropic. The adiabatic equation of state is written in the form: d dt (pρ γ m ) = 0 pρ γ m = constant. (5.19) Ohm s law in MHD It is also necessary to determine how the current density J depends on the electric field E. Ohm s law in a fluid description is a complicated issue. In the particle picture the plasma current is the sum of all charged particle motions. In a single-fluid theory the current transport equation is derived by multiplying the momentum transport equations of each particle population by q α /m α and summing over all populations. This leads to a rather messy expression including terms of different magnitudes and further approximations are needed. Here we give the generalized Ohm s law in the form that contains the most important terms for space plasmas: E + V B = J σ + 1 ne J B 1 ne P e + m e ne 2 J t. (5.20) The terms that are proportional to m e /m i and that contain the derivatives of the secondorder terms VJ, JV and VV have been neglected. The collision integral has been approximated by a constant collision frequency ν, which using the conductivity σ = ne 2 /νm e (Eq. 2.47) results in the first term on the RHS of Eq Exercise 5.3 compares different terms in the generalized Ohm s law during a magnetospheric substorm in the nightside magnetotail. Assume further so slow temporal changes and large spatial gradient scales that J B, J/ t, and P are all smaller than V B.

87 5.3. MAGNETOHYDRODYNAMIC EQUATIONS 81 This leaves us with the standard form of Ohm s law in MHD. J = σ(e + V B), (5.21) which is already familiar from elementary electrodynamics in cases when moving frames are taken into account. Here the moving frame is attached to the fluid flow with the velocity V Ideal MHD If the conductivity is very large, we find Ohm s law of the ideal MHD E + V B = 0. (5.22) Let us investigate under what constraints the ideal MHD is valid. Hence, we need to compare the magnitude of the term V B to the inertial, Hall (i.e., J B) and resistive terms. Denote the characteristic length and time scales of the system with L and τ, respectively. First, the electron inertia term ( J/ t) can be neglected if: V B L τ B m e J ne 2 t where ω p is the plasma frequency. m e 1 ne 2 τ = m e ne 2 B µl = c ωp 2 L c ω p, t ( 1 µ 0 B) (5.23) In a similar manner (try yourself!) we obtain the conditions to neglect the Hall term ( J B): L c ω p, and τ 1 ω c. (5.24) Finally the resistive term can be neglected if B τl V B ub ηj η 1 B µ 0 L u 1 µ 0 σl. (5.25)

88 82 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS The ratio of the V B and ηj terms is called the magnetic Reynold s number: R m = µ 0 σul. (5.26) The resistive term can thus be neglected when the magnetic Reynold s number is large. This is indeed the case in many space plasmas that have large dimensions (L) and small resistivity. Note that the resistive term causes dissipation and converts magnetic energy to heat. While the ideal MHD is a reasonable starting point, it is not at all clear that the next term to take into account should be J/σ. In many space applications the Hall term J B/ne and the pressure term P/ne are more important. There are effects that originate at the microscopic level, which are not due to actual inter-particle collisions, but which may lead to effective resistivity or viscosity at the macroscopic level. Various wave particle interactions and microscopic instabilities tend to inhibit the current flow. Often the macroscopic effect of these processes looks analogous to finite ν and is called anomalous resistivity. 5.4 Exercises: Macroscopic Plasma Equations 1. Derive the momentum equation n α m α V α t + n α m α V α V n α q α (E + V α B) + P α = m α v ( ) fα d 3 v. t c 2. Derive the energy equation of adiabatic ideal MHD in the conservation form t (1 2 ρ mv 2 + p γ 1 + B2 ) + ( 1 2µ 0 2 ρ mv 2 V + γ γ 1 pv + E B ) = 0. µ 0 Hints: Write the equation of state as d(pρ γ m )/dt and the conservation of mass as dρ m /dt = ρ m V to show that (γ 1)V p = p/ t + (γpv). Then use ideal Ohm s law with Ampere s and Faraday s laws to show that B 2 V (J B) = J (V B) = + E B. t 2µ 0 µ 0

89 5.4. EXERCISES: MACROSCOPIC PLASMA EQUATIONS Investigate generalized Ohm s law: E + V B = J σ + 1 ne J B + 1 ne P e + m e J ne 2 t during a substorm in the nightside magnetotail when following values have been measured: E 0.1 mv m 1, V 100 km s 1, B 1 nt J 1 na m 1, n 1 cm 3, P e 0.1 npa. In these circumstances the characteristics scale length is L 10 4 km, characteristic time scale 10 s and effective resistivity less than 1 s 1. Compare the magnitudes of various terms in Ohm s law.

90 84 CHAPTER 5. MACROSCOPIC PLASMA EQUATIONS

91 Chapter 6 Magnetohydrodynamics Magnetohydrodynamics (MHD) describes electrically conducting fluids in the presence of a magnetic field. This chapter deals with single-fluid MHD where different plasma components (i.e., ions, electrons) move together and compose a single fluid. The plasma is described by a single temperature, density and velocity. Single-fluid MHD is a strongly reduced theory but applies remarkably well in many situations. For instance, the plasma in the solar wind, Earth s magnetosphere, solar corona, and in many interstellar regions can be treated with MHD. MHD has also vast applications in fusion plasma research, in particular concerning the plasma confinement and stability. As described in the previous chapter, MHD governs processes that are slow compared with the gyration time and on scales that are larger than the gyro (Larmor) radius. We will discuss the general concepts of the MHD. We start by summarizing the MHD equations derived in Chapter 5 and by investigating the magnetic field evolution. We discuss magnetic field diffusion and the convection of plasma and magnetic field. Then we proceed to investigate the wave modes found in MHD. We also shortly discuss the importance and basic models of magnetic reconnection and finalize this chapter by investigating the MHD equilibrium. 6.1 MHD equations The basic equations of MHD we derived in Chapter 5 are: ρ m t + (ρ m V) = 0 (6.1) ρ m ( t + V ) V + P J B = 0 (6.2) E + V B = J/σ. (6.3) 85

92 86 CHAPTER 6. MAGNETOHYDRODYNAMICS ( ) n γ P = P 0 (6.4) n 0 E = B (6.5) t B = µ 0 J. (6.6) The first four equations are the mass continuity equation, momentum equation, resistive Ohm s law, and the adiabatic equation of state. We have used the basic MHD assumption that the temporal variations are so slow that the displacement current (ɛ 0 E/ t) can be neglected in the Ampère Maxwell law. The relationship between the electric current and magnetic field is thus obtained from Ampère s law B = µ 0 J. As discussed earlier, the MHD equations are basically the combination of Navier- Stokes equations of fluid dynamics with Maxwell s equations and Ampère s force. It is important to note that in MHD the magnetic and velocity fields are taken as the primary fields. The Maxwell equations and the MHD Ohm s law give the current density and the electric field in terms of the magnetic field. 6.2 Magnetic field evolution MHD describes the magnetic field (B) and plasma motion (bulk speed V). Let us investigate the relationship between V and B starting from the MHD Ohm s law: E + V B = J/σ. (6.7) Taking the curl of this and applying Faraday s law we obtain: B t = (V B J/σ). (6.8) Remembering that B = 0 and assuming that the conductivity is constant (for a case where the conductivity is not spatially homogeneous see Exercise 6.1) we obtain the induction equation (one of the most important equations of plasma physics!): B t = (V B) + 1 µ 0 σ 2 B. (6.9) From the induction equation we see that the magnetic field can change in time as a consequence of two effects. We will investigate them separately.

93 6.2. MAGNETIC FIELD EVOLUTION Diffusion Assuming that the plasma is at rest (V = 0) the induction equation reduces to the diffusion equation: B t = D m 2 B, (6.10) where D m = 1/(µ 0 σ) = η/µ 0 is the diffusion coefficient. Thus, the magnetic field in plasma can evolve even in the absence of any plasma flow if the resistivity η is finite. The magnetic field diffuses smoothing out spatial inhomogeneities described by the term 2 B. The solution of the diffusion equation is of the form: where the magnetic diffusion time τ d is B = B 0 exp(±t/τ d ), (6.11) τ d = µ 0 σl 2 B (6.12) and L B is the characteristic gradient scale length of the magnetic field. Figure 6.1: Evolution of a one-dimensional current sheet due to magnetic diffusion. (Assuming that no new flux is brought to the system). As an example letus investigate the initial configuration presented in the left-hand part of Figure 6.1. A thin current sheet separates the regions of oppositely orientated magnetic fields. The magnetic field is chosen to be along the ±x-axis and the current sheet normal is along the z-axis. In this case the diffusion equation becomes: B z t = 1 2 B z µ 0 σ x 2. (6.13)

94 88 CHAPTER 6. MAGNETOHYDRODYNAMICS If at time t 0 = 0 the current layer is infinitely thin (i.e., can be described by a δ-function), the magnetic field diffuses as: { (µ0 ) σ 1/2 B z (x) = B 0 erf x}, (6.14) 4t where erf(u) = 2 u e v2 dv. (6.15) π 0 The magnetic field diffuses towards the current sheet and, as a consequence, the current sheet broadens as the diffusion proceeds (right part of Figure 6.1). Oppositely directed magnetic fields cancel each other decreasing the magnetic field gradient, and thus, slowing down the diffusion (Eq. 6.10). Physically, the magnetic energy is transformed to heat. This is called Joule (or Ohmic) heating. Increasing plasma pressure compensates the decreasing plasma pressure Convection If σ (ideal MHD), the diffusion term becomes small and Eq. convection equation: 6.9 reduces to a B t = (V B). (6.16) The convection equation describes how plasma flow and magnetic field are tied to each other. In this case there is no diffusion of the magnetic field, but the plasma and magnetic field convect (actually advect) together. It is commonly said that the magnetic field is frozen-in to the motion of the plasma (see Figure 6.2). Figure 6.2: In a case of ideal MHD (σ ) magnetic field and plasma move together, i.e. are frozen-in. To determine whether convection or diffusion dominates, it is useful to introduce a dimensionless parameter that is the ratio of the magnitude of convection and diffusion

95 6.3. FROZEN-IN CONDITION 89 terms. Let τ be the time scale of characteristic magnetic field temporal variations, V the average plasma velocity perpendicular to the field, L B the characteristic length over which the magnetic field varies, and τ d the diffusion time scale. Substituting / t τ and L 1 B, and neglecting directions, the induction equation reduces to B τ = V B L B + B τ d. (6.17) The ratio of the terms on the RHS (i.e., the ratio of the convection to diffusion) becomes convection diffusion = V L B τ d = µ 0 σl B V (6.18) This ratio corresponds to the dimensionless magnetic Reynold s number (R m ) we encountered in Section while investigating the conditions under which the resistive term in the MHD Ohm s law can be neglected. If R m is large diffusion is slow and the convection dominates. This corresponds to small resistivity and the ideal MHD limit. Exercise 6.2 investigates diffusion times and R m for a typical laboratory plasma and in the solar wind. Due to large spatial scales and high conductivities R m is typically very large in space and astrophysical plasmas. In the solar wind at the Earth orbit R m is of the order of This means that during the 150 million kilometres journey from the Sun to the Earth the magnetic field in the solar wind diffuses only about one kilometre! Hence, diffusion is negligible in the solar wind. In turn, in laboratory plasma spatial scales and conductivities are much smaller, and consequently, diffusion dominates. 6.3 Frozen-in condition When the magnetic Reynolds number is very large, the magnetic field is frozen-in to the plasma motion. As a consequence, two plasma elements that are initially magnetically unconnected cannot mix as long as the frozen-in condition applies. The frozen-in concept was first brought to plasma physics by Hannes Alfvén. Although very useful the frozen-in concept is often misunderstood and Alfvén later denounced it as pseudopedagogical. The problem lies in picturing moving magnetic field lines. A magnetic field line is just a mathematical abstraction and has no physical identity. The correct way to express the frozen-in concept is to state that if two plasma elements are connected by a magnetic field line at time t, they are connected by a field line at all times. What is conserved is the magnetic connection between the plasma elements. Let us test the frozen-in assumption by investigating two plasma elements under the assumption of the ideal MHD. We assume that two plasma elements are magnetically

96 90 CHAPTER 6. MAGNETOHYDRODYNAMICS connected at time t (Figure 6.3). This means that if we trace the magnetic field from one plasma element, we end up at the other. Let the distance between the elements at time t be l. After time dt the plasma elements have moved distances u dt and (u + u) dt, where u(r, t) is the plasma flow velocity. At time t+dt the distance between the elements is thus l + d( l). In order the frozen-in concept to be valid we have to show that at the time t + dt the plasma elements are still magnetically connected, i.e., we need to show that d( l B)/dt = 0. Figure 6.3: If ideal MHD assumption holds, two plasma elements that are magnetically connected (at the same field line) at time t stay magnetically connected at all times. Let us begin by writing d( l) in terms of the plasma flow velocity u. The first term in the Taylor series of u is u = ( l )u. (6.19) From Fig. 6.3 we see that l + d( l) = l + (u + u) dt u dt, (6.20) which leads to d( l) = u = ( l )u. (6.21) dt Since we assume ideal MHD, let us investigate how the magnetic field changes in time starting from the convection equation: B t = (u B) = (B )u (u )B B( u). (6.22) Here B = 0 was used. In the frame of reference moving with the plasma db dt = B + (u )B = (B )u B( u). (6.23) t Now we can calculate d( l B)/dt: d d( l) ( l B) = B + l db dt dt dt = [( l )u] B + l [(B )u B( u)]. (6.24)

97 6.3. FROZEN-IN CONDITION 91 Because we assumed at the beginning l to be parallel to B, l B = 0, and the third term on the RHS is zero. For the same reason l and B can be interchanged in the first term on the RHS. Thus the first and the second term are the same except for their sign and we have d ( l B) = 0. (6.25) dt Thus, we have obtained that l remains parallel to B and plasma elements that originally are on a common field line remain on a common field line. This assumption is valid as long as the ideal MHD approximation is valid. S (t 0 ) B S (t 0 +dt) Figure 6.4: Magnetic flux through a surface moving with the plasma is conserved at the ideal MHD limit. An alternative way to express the frozen-in concept is to investigate the time variations of the magnetic flux through a surface S (see Figure 6.3): Φ S = S B ds. (6.26) The surface elements move with the plasma fluid velocity. By calculating the change of the magnetic flux from time t 0 to t 0 + dt it can be shown (Exercise 6.3) that dφ S dt = d dt B ds = 0. (6.27) Thus, assuming that the ideal MHD Ohm s law applies, the magnetic flux through any closed contour in the plasma, each element of which moves with the local plasma velocity, is a conserved quantity. The critical assumption when deriving the frozen-in theorem was the ideal Ohm s law. This requires that the E B drift is faster than magnetic drifts, i.e., large-scale convection dominates. In reality, the plasma has always some resistivity. However, the frozen-in condition applies if the characteristic time-scale of the process we are looking at is much shorter than the diffusion time. This can also be seen from the magnetic Reynolds number R m = (V/L B )τ B (see Eq. 6.18), where τ B is the magnetic field diffusion time.

98 92 CHAPTER 6. MAGNETOHYDRODYNAMICS In space plasmas the first correction to the ideal MHD is often not the resistive term but the Hall term J B/(ne) E + V B = 1 ne J B. (6.28) This is expected to be the case, e.g., near current sheets separating magnetic fields of different strength and direction. In this Hall MHD the magnetic field becomes frozen-in to the electron flow E = V e B. (6.29) As discussed earlier in Chapter 3 this is because due to their much smaller mass electrons have much smaller gyro radii and are tied more strongly to the magnetic field than the ions. 6.4 MHD waves MHD is a fluid theory and there are similar wave modes as in ordinary fluids (hydrodynamics). In addition, the presence of the magnetic field gives rise to new modes. In the MHD description we assume that the frequency of the wave is smaller than the characteristics frequencies in the plasma (gyro frequency and plasma frequency) and that the wavelengths are longer than microscopic plasma scales (Larmor radius). In addition, as discussed earlier, one of the basic assumptions of MHD is that the temporal changes are so slow that the displacement current can be neglected. As a consequence, MHD does not describe basic electromagnetic waves. However, this does not mean that electromagnetic waves could not propagate through the MHD plasma, rather MHD phenomena and high-frequency electromagnetic waves do not have a direct linkage MHD dispersion equation Analyzing plasma waves properties requires the derivation of the dispersion equation. The dispersion equation gives the relation between the wave number and the frequency of the wave, and thus determines how the wave travels in the medium. To derive the dispersion equation for MHD waves we start from the set of MHD equations ( ) given at the beginning of this chapter. We consider here compressible, non-viscous, and perfectly conducting plasma that is in a homogeneous background (applied) magnetic field. Thus, we can replace the resistive MHD Ohm s law (Eq. 6.3) by the ideal Ohm s law E + V B = 0. Let us modify the adiabatic equation of state (Eq. 6.4) by taking its gradient and introducing the speed of sound v s = γp/ρ m = γk B T/m, (6.30)

99 6.4. MHD WAVES 93 where γ is the adiabatic constant. We obtain p = v 2 s ρ m. (6.31) Use Ampère s law to eliminate J and Eq to eliminate p from Eq Further, the ideal Ohm s law can be used to eliminate E from Eq We obtain the set of equations ρ m t + (ρ m V) = 0 (6.32) ρ m V t + ρ m(v )V = v 2 s ρ m + ( B) B/µ 0 (6.33) (V B) = B t. (6.34) We assume the initial state to be in equilibrium where the density is constant ρ m0, the velocity is zero (V = 0), and the background magnetic field is constant B 0. As discussed at the beginning of this chapter we consider here only small perturbations (denoted by a subscript 1 ) to the initial equilibrium (subscript 0 ): B(r, t) = B 0 + B 1 (r, t) (6.35) ρ m (r, t) = ρ m0 + ρ m1 (r, t) (6.36) V(r, t) = V 1 (r, t). (6.37) We linearize the equations by inserting these to the equations and keeping only the first order terms (the zeroth order terms automatically fulfil the equations and the second order terms are assumed so small that we can neglect them). This leads to the linearized equations: ρ m0 V 1 t ρ m1 t + ρ m0 ( V 1 ) = 0 (6.38) + v 2 s ρ m1 + B 0 ( B 1 )/µ 0 = 0 (6.39) B 1 t (V 1 B 0 ) = 0. (6.40) Next, let us find equation for the velocity perturbation V 1. By taking the time derivate of the linearized momentum equation (Eq. 6.39) we obtain 2 ( ) V 1 ρ m0 t 2 + vs 2 ρm1 + B ( 0 B ) 1 = 0. (6.41) t µ 0 t By using the linearized continuity equation (Eq. 6.38) ja the linearized Ampère s law (Eq. 6.40) this can be written as 2 V 1 t 2 v 2 s ( V 1 ) + v A { [ (V 1 v A )]} = 0, (6.42) where the vector v A is

100 94 CHAPTER 6. MAGNETOHYDRODYNAMICS The magnitude of this velocity defines the Alfvén speed. v A = B 0 µ0 ρ m0. (6.43) Finally, let us try look for the solution assuming that it has the form of a plane wave V 1 (r, t) = V 1 exp[i(k r ωt)]. (6.44) The temporal and spatial variations are now harmonic and we can replace the derivatives by algebraic operators (see Appendix 9.3) ik (6.45) ik / t iω. Eq simplifies to an algebraic equation Using a vector identity: ω 2 V 1 + v 2 s(k V 1 )k v A {k [k (V 1 v A )]} = 0. (6.46) A (B C) = (A C)B (A B)C (6.47) we obtain a useful form of the dispersion equation ω 2 V 1 + (v 2 s + v 2 A)(k V 1 )k +(k v A )[(k v A )V 1 (v A V 1 )k (k V 1 )v A ] = 0. (6.48) From this we can find all MHD wave modes MHD wave modes Select the z-axis to be parallel to background magnetic field B 0 and the x-axis so that the wave vector k is in the xz-plane. Denote the angle between k and B 0 by θ. Figure 6.5 summarizes the coordinate system. Now we obtain k = k(ê x sin θ + ê z cos θ) (6.49) v A = v A ê z (6.50) V 1 = V 1x ê x + V 1y ê y + V 1z ê z (6.51) k v A = kv A cos θ (6.52) k V 1 = k(v 1x sin θ + V 1z cos θ) (6.53) v A V 1 = v A V 1z. (6.54)

101 6.4. MHD WAVES 95 z n,k q B 0 x y Figure 6.5: Coordinate system to study MHD waves Inserting these to the dispersion equation (Eq 6.48) we obtain v 2 p + v 2 A + v2 s sin 2 θ 0 v 2 s sin θ cos θ 0 v 2 p + v 2 A cos2 θ 0 v 2 s sin θ cos θ 0 v 2 p + v 2 s cos 2 θ where v p = ω/k is the phase speed (see Appendix 9.3) of the wave. V 1x V 1y V 1z = 0, (6.55) There are three linearly independent non-trivial solutions that are found by equating the determinant to zero. Alfvén wave The y-component of the matrix Eq gives a linearly polarized wave mode with the phase speed v p = v A cos θ, (6.56) where v A is given by Eq This mode is called the Alfvén wave. It is seen from Eq that when the Alfvén wave propagates along the background magnetic field (i.e. θ = 0 ) its phase speed is exactly v A. For oblique propagation the phase speed is less than v A. It is also clear from Eq that Alfvén waves do not propagate perpendicular to the magnetic field. It is easy to verify that the eigenvector (0, V 1y, 0) corresponds to the root v 2 p = v 2 A cos2 θ. Thus for Alfvén wave the velocity perturbation V 1 is in the y-direction, i.e., perpendicular both to the wave vector k and the background magnetic field B 0. Hence, V 1 k = 0 and we see from the linearized continuity equation iωρ 1 + ρ 0 k V 1 = 0 (6.57)

102 96 CHAPTER 6. MAGNETOHYDRODYNAMICS that Alfvén wave is non-compressive, i.e., there are no density fluctuations. The plasma fluid motions are thus completely transverse signifying that the plasma elements oscillate perpendicular to the direction of propagation of the wave. The wave magnetic field B 1 can be calculated from the convection equation assuming harmonic temporal and spatial dependences ωb 1 + k (V 1 B 0 ) = 0 (6.58) B 1 = V 1 ω/k B 0, (6.59) i.e., the wave magnetic field is perpendicular to the background magnetic field B 0. Figure 6.6 demonstrates the propagation of an Alfveń wave. k B 0 V 1 B 1 E 1 B 0 Figure 6.6: Alfvén wave propagating parallel to the magnetic field. The Alfvén wave is often called also as shear Alfvén wave or non-compressional Alfvén wave. The existence of the Alfvén wave can be deduced also intuitively. The magnetic field line can be considered to behave like a tensed string. Transversal displacement of the elastic string generates a transverse wave that propagates along the string, in analogy with the Alfvén wave propagating parallel to magnetic field lines.

103 6.4. MHD WAVES 97 Alfvén waves have been observed in the laboratory and in many space plasma regions, for example in the solar wind, solar photosphere and in the Earth s magnetosphere. It has been suggested that Alfvén waves could explain the heating of the outermost layer of the solar atmosphere, the corona (Figure 6.7). Understanding the properties of Alfvén waves is also important for determining the stability, turbulence and heating in controlled fusion devices. The Alfvén speed can differ greatly in space plasma, depending on the density and magnetic field magnitude in question. Typical Alfvén speeds are calculated in Exercise 6.4 in the Earth s ionosphere, solar corona and interstellar gas cloud. It is also instructive to contemplate whether the neutral mass density needs to be taken into account when determing the Alfvén speed (i.e., if plasma is not fully ionized, do neutrals have enough time to respond to the motion of the ions). Figure 6.7: Japanese Hinode ( sunrise ) observations of fluctuating plasma that could be an indication of Alfvén waves heating the corona. The presence of Alfvén waves is deduced by tracking the motions of coronal plasma. Contemplate: The existence of Alfvén waves was first suggested by Hannes Alfvén (as the name hints!). In 1942 he noted that a new type of wave should be found in magnetized plasmas that may be of importance to solar physics. It is instructive to look Alfveń s original paper (click here) which also demonstrates his profound physical insight. Fast and slow MHD (Alfvén) waves. The other solutions of the matrix Eq are obtain by setting the determinant of the coefficients of V 1x :n and V 1z zero. The result is (Exercise 6.5) v 2 p = 1 2 (v2 s + v 2 A) ± 1 2 [(v2 s + v 2 A) 2 4v 2 sv 2 A cos 2 θ] 1/2. (6.60) The solution with the larger phase speed is called the fast MHD wave and with the lower phase speed the slow MHD wave.

104 98 CHAPTER 6. MAGNETOHYDRODYNAMICS The top panels of Figure 6.8 show the phase speeds of Alfvén wave and the slow and fast MHD waves as a function of the angle between the wave vector and the background magnetic field (θ). The solution depends on the ratio between the Alfvén speed v A and the sound speed v s. Another way to illustrate wave properties is to use the wave normal surfaces, see the bottom panels of Figure 6.8. w/k v A > v s w/k v A < v s (v A2 +v S2 ) 1/2 v A fast MHD (v A2 +v S2 ) 1/2 v s fast MHD v s v A q q k B 0 k^b 0 k B 0 k^b 0 B 0 v A v s fast MHD B 0 v s v A fast MHD slow MHD Alfvén (v A2 +v S2 ) 1/2 slow MHD Alfvén (v A2 +v S2 ) 1/2 Figure 6.8: Top) Phase speeds as a function of the propagation angle and bottom) wave normal surfaces for Alfvén, fast and slow MHD waves. Cases with v A > v s and v A < v s are shown separately. A wave normal surface shows the phase speed as a function of the angle between the wave propagation direction and the magnetic field θ, i.e., it describes how the phase speed varies with respect to the magnetic field direction. Actually Figure 6.8 shows 2D cuts of the surfaces. Assuming that the system is gyrotropic, the surface is found by letting the wave normal curve rotate around the direction of the magnetic field. At the oblique propagation angles for fast and slow MHD waves the velocity perturbation V 1 is in the xz-plane (Figure 6.5). Now k V 1 0, and hence, the waves are compressional and associated with density pertubations (Eq. 6.57). Investigate first the propagation perpendicular to the magnetic field. Eq and Figure 6.8 show that when θ 90 the phase speed of the slow MHD wave goes to zero. The fast mode, in turn, can propagate to all directions. When θ 90 the phase speed

105 6.4. MHD WAVES 99 of the fast mode reduces to v p = v 2 A + v2 s (6.61) i.e., the phase speed depends both on the sound speed and the Alfvén speed. This wave is called the magnetosonic wave. Magnetosonic speed defined by Eq is the maximum propagation speed of the MHD waves. Figure 6.9: A fast magnetosonic wave propagates perpendicular to the magnetic field. In a very tenuous plasma with large enough magnetic field the Alfvén speed can actually be larger than the speed of light (v A > c). In such cases the non-relativistic MHD approximation breaks down and the displacement current cannot be neglected. The modification to the dispersion relation for a mode propagating perpendicular to the magnetic field (θ = 90 ) including the displacement current is derived in Exercise 6.6. ω 2 k 2 = v2 s + v 2 A 1 + v 2 A /c2 Obviously this reduces to ω 2 /k 2 = v 2 s + v 2 A for v2 A c2, i.e., at the non-relativistic MHD limit. The eigenvector corresponding to this root is (0, 0, V 1z ) and from linerized continuity Eq we obtain the density perturbation ρ 1 = ρ 0 (V 1z /v p ). The linearized convection equation (Eq. 6.58) gives B 1 = V 1 ω/k B 0. (6.62) The electric field of the wave is obtained from the ideal Ohm s law: E 1 = V 1 B 0. (6.63) The magnetosonic wave is similar to the electromagnetic wave in the sense that the wave vector and the wave magnetic and electric fields are all perpendicular to each other. However, mass flow and density fluctuate along the wave vector, and thus, the magnetosonic wave is longitudinal (Figure 6.9). It is also called magnetoacoustic, which derives from this property.

106 100 CHAPTER 6. MAGNETOHYDRODYNAMICS In the case v A v s the phase speed of the magnetosonic wave (Eq. 6.61) becomes v p v A, i.e., it approaches the Alfvén speed. However, the wave is compressional and is often called the compressional Alfvén wave. Note that this situation corresponds to the cold-plasma (zero temperature) limit where v s goes to zero. From Eq it is clear that in the cold plasma limit the slow mode MHD wave ceases to exist. Next, let us investigate propagation parallel to the magnetic field (θ = 0 ). In the case where the magnetic field dominates (v A > v s ) the dispersion equation gives for the fast MHD wave v p = v A, (6.64) i.e., the wave reduces to the Alfveń wave. In turn, for the slow MHD wave the phase speed becomes v p = v s (6.65) i.e., it reduces to correspond the ordinary sound wave. A sound wave is the simplest disturbance that can propagate in a collisional medium. The wave vector k is normal to the pressure front and the restoring force is the pressure gradient. Because the magnetic field does not restrict the particle motion along the magnetic field in a plasma, sound waves can propagate also in a magnetized plasma, see Figure Figure 6.10: Longitudinal sound wave propagates along the magnetic field in a compressible and magnetized plasma.. Figure 6.8 illustrates that in the case of parallel propagation either the fast or slow mode MHD wave reduces to the Alfvén wave and the other one to the sound wave depending on the ratio between the v A and v s. The fast MHD wave has always larger phase speed than the slow MHD wave. A compressive MHD wave can steepen into a shock wave when the disturbance propagates faster than the characteristic speed of the medium. In space and fusion plasmas shocks may be produced by explosions (e.g., solar flares, supernovae, inertial confinement fusion), by a disturbance moving through a fluid with its speed exceeding the local characteristic information speed. Some examples are a coronal mass ejection moving through the solar wind faster than the local magnetosonic speed and the encounter of supersonic and super-alfvénic fluid with a stationary object (e.g., the formation of the Earth s bow shock).

107 6.5. MAGNETIC RECONNECTION Magnetic reconnection One of the most important plasma physical phenomena arises when the frozen-in condition breaks down. If the ideal MHD assumption holds, two initially (magnetically) separate plasma elements can never mix. For example, the Earth s magnetosphere would always stay closed to the solar wind. According to Eq. 6.9 the Reynolds number decreases when the plasma flow speed or the length scale of field gradients decreases or when resistivity increases. When the Reynolds number becomes sufficiently small the magnetic field starts to diffuse. In a particle description the break-down of the frozen-in condition can be understood by the GC approximation becoming invalid, i.e., charged particles cease to follow the magnetic field. (As discussed earlier, this usually does not happen simultaneously for electrons and ions.). Figure 6.11: Reconnection between two plasma domains with oppositely oriented fields that are flowing towards each other. Open arrows indicate the direction of the plasma flow. Figure 6.11 shows two ideal MHD plasma regions with oppositely oriented magnetic fields flowing towards each other. Such situation arises for example at the interface between the Earth s magnetosphere and the solar wind when the interplanetary magnetic field is southward (at the nose of the magnetosphere the magnetospheric field points always to the north). A thin current sheet forms between the regions introducing a large magnetic field gradient. The exact microphysics that occurs in the thin current sheet is not yet well-understood. However, if there are processes that increase resistivity, diffusion can start leading to the reorganization of the plasma and magnetic field. Plasma elements that were originally in separate regions may now become magnetically connected. The change of connection between the plasma elements is called magnetic reconnection. The importance of reconnection lies in its ability to change the topology of the magnetic field and to convert magnetic energy to kinetic and thermal energy. The concept of reconnection was first presented by Ronald Giovanelli in the 1940s to explain particle acceleration in solar flares. In a solar flare a huge amount of energy is released from the Sun in time-scales of only a few minutes. We cover here briefly the most elementary reconnection models.

108 102 CHAPTER 6. MAGNETOHYDRODYNAMICS 2l E y J y (x) +B 0 x V i B z (x) z -B 0 V i Figure 6.12: 1-dimensional current sheet. y-directed electric field has been added that brings new plasma and magnetic flux towards the current sheet and maintains the balance between diffusion and convection. Diffusion in a 1-dimensional current sheet was treated in Section To achieve a steady-state situation new magnetic flux and plasma have to be brought to the system to replace the annihilated flux. This can be achieved by adding an electric field as shown in Figure Outside the current sheet the ideal Ohm law gives E y = V i B 0, where V i is the plasma inflow speed and B 0 the magnetic field far away from the current sheet. At the current sheet magnetic field is zero and the resistive Ohm s law gives E y = J y /σ. The width of the current sheet adjusts to maintain the balance between diffusion and convection. Assuming that the width of the current sheet is 2l Amperè s law gives the electric current J y = B 0 /µ 0 l. Hence, the width of the current sheet can be written as: 2l = 2 µ 0 σv. (6.66) The scenario explained above is unphysical. What happans to the plasma that is brought to the current sheet? The solution is to add an additional dimension as shown in Figure This is the famous Sweet Parker model formulated in the 1950s. In the Sweet Parker model the magnetic field annihilates in a finite domain called a diffusion region (gray area in Figure 6.13). Plasma and the magnetic field flow away from the boundaries of the diffusion region. Figure 6.13 also illustrates that the plasma elements that were originally not magnetically connected (blue and red circles are at different field lines before entering the diffusion region) become connected after exiting the diffusion region. In the outflow region the magnetic field is thus weaker and the plasma flow speed larger than in the inflow region. It is important to note that the frozen-in condition breaks-down in the diffusion region but is valid outside. Hence, the diffusion region is the region where the rearrangement of the magnetic field occurs. The speed of reconnection, i.e., the reconnection rate, is typically expressed as the electric field in the inflow region. It is an important quantity determining the inflow speed. Estimates for the inflow and outflow speeds can be achieved by assuming incom-

109 6.5. MAGNETIC RECONNECTION 103 B i inflow V i E outflow V o 2l V o outflow E V i 2L inflow B i Figure 6.13: Sweet-Parker reconnection. The diffusion region is shown by gray. pressible flow (ρ i = ρ o = ρ), conservation of mass (V i L = V o l) and that all inflowing electromagnetic energy transforms to kinetic energy. Inflowing electromagnetic energy can be calculated from the inflowing Poynting flux: S = E H = EB i µ 0 = V ib 2 i µ 0. (6.67) The mass that flows in a unit time to the diffusion region (ρv i ) will be accelerated to speed V o. Hence, the change in energy in unit time and unit area is: W = 1 2 ρv i(v 2 o V 2 i ). (6.68) Equating the energy increase with the inflow energy flux and noting that V o V i gives V 2 o V i B 2 i µ 0 = 1 2 ρv iv 2 o (6.69) = 2B2 i µ 0 ρ = 2v2 Ai. (6.70) Thus the Alfvén speed in the inflow region describes the speed of the outflowing plasma (under the used approximations within a factor of 2). Using V i L = V 0 l and the width of the diffusion region from Eq the inflow speed is: V i = v Ai ( 2/R ma ) 1/2, (6.71) where R ma = µ 0 σv Ai L is the Reynolds number calculated using the inflow Alfvén speed known as the Lundquist number. It is easy to show (Exercise 6.7) that half of the incoming magnetic energy is transformed to heat and the other half causes acceleration of particles. In space plasmas R ma is usually very large and thus the inflow and the

110 104 CHAPTER 6. MAGNETOHYDRODYNAMICS reconnection rate in the Sweet Parker model is very slow. For a solar flare it would take days to erupt, not minutes as the observations indicate. The slow reconnection speed in the Sweet Parker model can be traced to the property that all energy conversion occurs in a diffusion region whose length is much larger than the width of the outflow region. In 1964 Harry Petschek proposed that significantly faster reconnection rates can be obtained by introducing a vanishingly small diffusion region. He added two slow mode shocks, i.e. slow MHD waves steepened to shocks, that emanate from the diffusion region. The shocks deviate the plasma flow and magnetic field. The Petschek model is presented schematically in Figure Figure 6.14: Petschek fast reconnection. Two slow mode shocks (blue) emanate from a vanishingly small diffusion region and deviate the plasma flow (red lines) and the magnetic field (black lines). Contemplate: Where does the energy conversion occur in the Petschek model? Where does the magnetic field connectivity change? The properties of MHD shocks are beyond our discussion here. They will be treated in the course on space applications of plasma physics. However, a rigorous analysis shows that the inflow speed in the Petscheck model can be up to 10% of v Ai, which allows much faster reconnection than in the Sweet Parker model. 6.6 Magnetohydrostatic equilibrium and stability MHD equilibrium structures are important for a number of space and astrophysical phenomena and in fusion research. For example, solar prominences are huge structures that can remain stable up to several solar rotations before erupting and much of the fusion research deals with plasma confinement. We will start by investigating the MHD momentum equation (Eq. 6.2). Assuming scalar pressure ( P p) and timeindependent (d/dt = 0) equilibrium the momentum equation reduces to J B = p. (6.72)

111 6.6. MAGNETOHYDROSTATIC EQUILIBRIUM AND STABILITY 105 This means that the plasma pressure gradient and the Lorentz force must be in balance. This equation gives B p = 0 and J p = 0. Thus B and J are vector fields on surfaces of constant pressure. B: From the above equilibrium condition one can calculate the current perpendicular to J = B p B 2. (6.73) This current is often called the diamagnetic current. It arises from the plasma pressure gradient. In the particle description the perpendicular current is the sum of all current elements in the plasma and contains contributions from the magnetic drifts (gradient and curvature drift related currents), polarization current, and the magnetization current. The magnetization current is caused by an inhomogeneous plasma density: J M = M. (6.74) Here the magnetization M is the density of magnetic moments µ (see Eq. 3.14). Figure 6.15 shows the particle picture of the magnetization current. If the plasma density is non-uniform, the gyration velocities do not sum to the zero, and hence, a net current arises. B x y total current high density plasma low density plasma Figure 6.15: Single particle interpretation of the diamagnetic current. Using Ampère s law we can write the magnetic force in the form: ( ) B 2 J B = + 1 (B ) B. (6.75) 2µ 0 µ 0 The magnetic force consist of two separate terms. The first term on the RHS is the gradient of the magnetic energy density, i.e., of the magnetic pressure: p B = B 2 /(2µ 0 ). (6.76)

112 106 CHAPTER 6. MAGNETOHYDRODYNAMICS The second term describes the tension force arising from the inhomogeneities of the magnetic field. This latter term can be divided into two components: (B ) B = B d ds dŝ (Bŝ) = B2 ds + B B s ŝ (6.77) = B 2 ˆn + ŝ ( ) B 2, R C s 2 where ŝ is the unit vector along the magnetic field and R C is the radius of curvature. It is now evident that: 1. The first term is anti-parallel to the radius of the curvature of field lines. The related component of the force acts to reduce the stress in the field lines. 2. The second term is field aligned and cancels the field aligned component of p B. As a consequence only perpendicular component of p B exerts force on the plasma. In Excercise 6.8 the magnetic force is calculated for different magnetic field configurations. Sketching the magnetic field configurations helps to visualize how curvature and gradients in the magnetic field are related to the direction of the magnetic force. Hence, we obtain the condition for the MHD equilibrium dv/dt = 0 from the momentum equation: ( ) p + B2 = 1 (B ) B. (6.78) 2µ 0 µ 0 See Exercise 6.9 for a demonstration of how in a simple magnetic field configuration B x = y and B y = x the magnetic pressure and tension balance each other. Assuming homogeneous magnetic field the sum of the magnetic and plasma pressures is constant ( ) p + B2 = 0. (6.79) 2µ 0 The plasma beta β = 2µ 0p B 2 (6.80) expresses the ratio of the plasma and magnetic pressures. It is one of the important dimensionless parameters used to characterize plasmas. An example of a MHD equilibrium configuration is the Harris current sheet we encountered in Section As discussed earlier, the Earth s magnetotail, which can stay stable for long time periods, can be described by a Harris current sheet. In a 1-dimensional Harris current sheet the magnetic field (assumed here to be in the z- direction) is given by: ( z B = B 0 tanh e y. (6.81) L)

113 6.6. MAGNETOHYDROSTATIC EQUILIBRIUM AND STABILITY 107 The pressure is given by p = p 0 cosh 2 z L, (6.82) where p 0 = B 2 0 /(2µ 0) As illustrated in Figure 6.16 (see also Exercise 6.10) the variations in the magnetic field and plasma pressure over the Harris current sheet balance each other. Figure 6.16: Magnetic field and pressure variations in the Harris current sheet. q-pinch B z Z-pinch J J q r B Figure 6.17: Left) θ-pinch, and Right) Z-pinch Other examples of 1-dimensional equilibrium configurations are θ- and Z-pinches shown in Figure In both cases it is convenient to use cylindrical coordinates. In a θ-pinch cylindrical coils drive an electric current and the magnetic field is axial, while in

114 108 CHAPTER 6. MAGNETOHYDRODYNAMICS a Z-pinch the electric current is axial and the magnetic field poloidal. The equilibrium conditions ( p = J B) are (Exercise 6.11) : ( d p + B2 z dr 2µ 0 ) ( d p + B2 θ dr 2µ 0 ) = 0 (6.83) + B2 θ 2µ 0 = 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 plasma is zero. According to Eq in a force-free field the magnetic pressure gradient (B 2 /2µ 0 ) is balanced by the magnetic tension force µ 1 0 (B ) B. In real situations the force-free equilibrium is always an approximation, but often a very good one, to the momentum equation. It is also evident from Eq that in a force-free field the electric current flows along the magnetic field. Such currents are commonly called as field-aligned currents (FAC). Using Ampère s law we can write Eq 6.85 as ( B) B = 0. (6.86) From this we see that the innocent-looking equation J B = 0 is in fact non-linear and thus difficult to solve. The field-alignment of the electric current can be expressed as B = µ 0 J = α(r)b, (6.87) where α is a function of position. Taking divergence of this we get i.e., α is constant along the magnetic field. In the case α is constant everywhere, the equation B α = 0, (6.88) B = αb (6.89) is linear. Taking a curl of (6.89) we get the Helmholtz equation 2 B + α 2 B = 0. (6.90)

115 6.7. FORCE-FREE MAGNETIC FIELDS 109 Figure 6.18: Helical structure of a force-free magnetic field. Solution to the Helmoltz equation in cylindrical symmetry was given by Lundquist in 1950 in terms of Bessel functions J 0 and J 1 : B R = 0 (6.91) ( ) α0 r B A = B 0 J 0 r ( 0 ) α0 r B T = ±B 0 J 1, (6.92) r 0 where B R, B A, and B T are radial, axial and tangential magnetic field components, respectively. The solution is a magnetic flux rope where magnetic field lines form helices whose pich angle increases from the axis (Figure 6.18). r is the radial distance from the flux rope axis, r 0 is the radius of the flux rope and B 0 is the maximum magnetic field magnitude at the center of the flux rope (r = 0). Figure 6.19: Left) Erupting coronal mass ejection whose structure is a magnetic flux rope. Image taken by Solar Dynamic Observatory. Courtesy: NASA. Right) Hubble Space Telescope image of a filamentary nebula (Dahlgren et al., 2007). Flux ropes are common in space, astrophysical and fusion plasmas. The left-hand part of Figure 6.19 shows an erupting solar plasma cloud whose structure is often approximated with a force-free flux rope. These plasma clouds maintain more or less their integrity while traveling away from the Sun to the orbit of the Earth and beyond and

116 110 CHAPTER 6. MAGNETOHYDRODYNAMICS they are the main drivers of severe magnetospheric disturbances. The right-hand part of Figure 6.19 shows a Hubble image of a planetary nebula. The substructures in this nebula may be formed from magnetic flux ropes that are twisted around each other. A special case of a force-free magnetic field is the current-free configuration B = 0. Now the magnetic field can be expressed as the gradient of a scalar potential B = Ψ. Because B = 0, the magnetic field can be found by solving the Laplace equation 2 Ψ = 0 (6.93) with appropriate boundary conditions and using the methods of potential theory. Figure 6.20: Potential Field Source Surface model of the Sun s magnetic field on April Open positive (outward from the Sun) flux is in green, open negative flux in red, and the tallest closed flux trajectories in blue. The fields are plotted over the original synoptic magnetogram. White areas indicate the maximum-strength positive flux and black maximum-strength negative flux. Courtesy: NSO/GONG. For example, the Sun s magnetic field structure is often modelled by the so-called Potential Field Source Surface (PFSS) model (Figure 6.20). The magnetic field is computed from the Laplace equation using spherical coordinates from the photosphere to the source surface, nominally chosen to be at 2.5 Solar radii. At the source surface the Sun s magnetic field is assumed to be purely radial. The inner boundary conditions are obtained from solar magnetograms. Thus, PFFS assumes that there is no electric current in the corona.

117 6.8. EXERCISES: MAGNETOHYDRODYNAMICS Exercises: Magnetohydrodynamics 1. Derive the induction equation for the magnetic field in a case where the conductivity is not spatially homogeneous 2. Calculate the magnetic Reynolds number R m and the diffusion time τ d for (a) a laboratory plasma where L B 0.1 m, V 10 3 m s 1 and σ 100 Ω 1 m 1 (b) solar wind where L B 10 solar radii, V 400 km s 1 and σ Ω 1 m Show that at the limit of large Reynolds number the magnetic flux through a closed loop co-moving with plasma is constant dφ dt = d dt B ds = 0 4. Compute the Alfvén speed in the following cases (a) Earth s ionosphere n e = m 3, B = 50 µt, ions assumed to be mostly O +. (b) Solar corona: n e = m 3, B = 50 mt, ions are protons (c) Interstellar gas cloud: n e = 0.1 cm 3, B = 0.1 nt, ions are protons and the ionization degree is 1%. 5. Derive the phase speeds of the fast and slow Alfvén waves ( ω k ) 2 = 1 2 ( ) vs 2 + va 2 ± 1 [ ( ] 2 1/2 vs 2 + v 2 2 A) 4v 2 s va 2 cos 2 θ starting from the dispersion equation ( ) ω 2 V 1 + vs 2 + va 2 (k V 1 ) k + (k v A ) [(k v A ) V 1 (v A V 1 ) k (k V 1 ) v A ]. 6. Consider the propagation of Alfvén waves taking the displacement current into account. That is, start from the same equations as on the lectures but replace Ampère s law by B = µ 0 J + 1 c 2 E t. Derive the dispersion equation for the mode propagating perpendicular to the magnetic field into the form ω 2 k 2 = v2 s + v 2 A 1 + v 2 A /c2.

118 112 CHAPTER 6. MAGNETOHYDRODYNAMICS 7. Consider the Sweet Parker reconnection model. Show that half of the incoming magnetic energy is transformed to heat and the other half causes acceleration of particles. 8. Calculate the magnetic force J B for the following cases. Sketch also following magnetic field configurations and indicate the direction of magnetic forces in each case. (a) B = xe y (b) B = e x + xe y (c) B = ye x + xe y (d) B = re θ 9. Show that in the magnetic field configuration B x = y, B y = x the magnetic pressure and tension balance each other. Show that if the configuration is stretched in the y direction: B x = y, B y = α 2 x, where α 2 > 1, this causes in certain regions a net force toward the X-line and in other regions away from the X-line. 10. Show that the total pressure of the 1-dimensional Harris model is B/2µ 0 and that the current density is J y (z) = B ( ) 0 z µ 0 h sech2. h Show futher that the model is in magnetohydrostatic equilibrium J B = p. 11. Consider the equilibrium pinch in Figure Assume cylindrical symmetry, that MHD assumptions are valid and the electric current flows only inside the cylinder of radius R. Write the condition for the hydromagnetic equilibrium in cylindrical coordinates. Calculate and plot the profiles of plasma pressure and the magnetic field, when the current is constant inside the cylinder. How would the result change, if the current would flow on the surface of the cylinder only? z J z (r) p(r)=0 p=0 Β θ (r) Figure 6.21: Equilibrium pinch for Exercise 6.11 R θ r

119 Chapter 7 Cold plasma waves Propagation of electromagnetic waves is one of the most important phenomena in plasma. Characteristics of wave propagation are used in various ways to diagnose plasma and observing wave emissions in plasma can give information on the plasma properties. For example, plasma density can be calculated easily from the plasma frequency and the magnetic field magnitude from the gyro frequency. In the previous chapter we investigated MHD waves and considered frequencies well below the ion gyro and plasma frequencies. When the frequency of the wave increases, one needs to take into account that the ion and electron dynamics become different, and hence, the one-fluid MHD description becomes invalid. This chapter investigates the wave propagation in the cold plasma limit. In reality the temperature of a plasma is never zero, but the temperature effects can be neglected if the wave propagates faster than the plasma thermal speed 2kB T/m. As a consequence, cold plasma has zero pressure and there are no waves related to pressure fluctuations, such as sound waves. At high frequencies, well above the ion gyro frequency, the ions can be considered as an immobile background as they cannot respond quickly enough to the wave. Note that we now consider much faster fluctuations than in MHD, and thus, we need to take into account the displacement current in Maxwell s equations. We start by deriving the general form of the dispersion equation. We proceed to investigate the waves that propagate exactly parallel or perpendicular to the magnetic field. Finally, we briefly discuss arbitrary direction of propagation 7.1 General form of the dispersion equation The treatment of waves in plasma at the cold plasma limit resembles closely the study of general electromagnetic waves (see Appendix 9.3). To derive the cold plasma dispersion equation we start from the density continuation equation, equation of motion and 113

120 114 CHAPTER 7. COLD PLASMA WAVES Maxwell s equations n t + (nv) = 0 (7.1) m u = e(e + u B) (7.2) t E = B (7.3) t B 1 E c 2 = µ 0 J = µ 0 en s u s = σ E (7.4) t s E = 1 en s (7.5) ɛ 0 s B = 0. (7.6) In Eq. 7.4 we have used Ohm s law with conductivity being a second rank tensor σ. s indexes all particle species that constitute the plasma. Let us consider again a small perturbation (subscript 1 ) to the initial equilibrium (subscript 0 ) Insert these to Eqs and linearize n = n 0 + n 1 (7.7) u = u 1 (7.8) B = B 0 + B 1 (7.9) E = E 1. (7.10) n 1 t + (n 0u 1 ) = 0 (7.11) m u 1 t = e(e 1 + u 1 B 0 ) (7.12) E 1 = B 1 (7.13) t B 1 1 E 1 c 2 = µ 0 J = µ 0 en s u 1s (7.14) t s J = σ E 1 (7.15) E 1 = 1 en 1s ɛ 0 (7.16) s B 1 = 0. (7.17) By taking the curl from Eq and using Eq we obtain E 1 = 1 c 2 2 E 1 t 2 µ 0 J t. (7.18)

121 7.2. WAVE PROPAGATION IN NON-MAGNETIZED PLASMA 115 Assume harmonic time dependences (see Appendix 9.3) and use the linearized Ohm s law (Eq. 7.14) to eliminate J from Eq This leads to the homogeneous wave equation: n (n E 1 ) + K E 1 = 0, (7.19) where n = ck/ω is the index of refraction and K is the dielectric tensor K = 1 σ iωɛ 0. (7.20) 7.2 Wave propagation in non-magnetized plasma Let us first consider a simple case where the background magnetic field B 0 is zero. The linearized equation of motion now becomes m u t = ee. (7.21) Note that here the subscript 1 has been dropped for simplicity. Assuming harmonic time dependence this reduces to m( iω)u = ee. (7.22) We use this equation to eliminate u from J = s en s u s and to obtain J = s n 0 e 2 ( iω)m s E, (7.23) from which we can now identify the conductivity tensor σ = 1 s n 0 e 2 ( iω)m s. (7.24) Using the definition of the plasma frequency and ω pe ω pi we obtain i ɛ 0 ω σ = 1 ω2 p ω (7.25) and the homogeneous wave equation becomes c 2 k (k E) = (ω 2 ω 2 p)e. (7.26)

122 116 CHAPTER 7. COLD PLASMA WAVES Let us choose the wave vector k to be parallel to the z-axis. From Eq we now obtain a matrix equation c 2 k 2 + ω 2 ωp c 2 k 2 + ω 2 ωp ω 2 ωp 2 E x E y E z = 0, (7.27) from which we get the dispersion equation by setting the determinant of the matrix to zero ( c 2 k 2 + ω 2 ω 2 p) 2 (ω 2 ω 2 p) = 0. (7.28) One of the roots is evidently ω = ω p. The electric field associated with this mode is in the z-direction. Since we selected k to be in the z-direction, the electric field perturbation is parallel to the wave propagation. Hence, the wave is longitudinal: k E 1 0 and k E 1 = 0. By assuming harmonic dependencies the linearized Gauss s law (Eq. 7.16) becomes ρ 1 = iɛ 0 k E 1, (7.29) and we see that the wave is associated with charge density fluctuations. In turn, assuming harmonic dependencies the linearized Faraday s law (Eq. 7.13) becomes ik E 1 = iωb 1 (7.30) B 1 = 1 ω k E 1, and we see that B 1 = 0, i.e., the mode is electrostatic. The group speed v g = dω/dk (see Appendix 9.3) is zero indicating that the wave does not propagate. This solution describes oscillation at the plasma frequency we encountered in Section 2.3. Another root of Eq is ω 2 = ω 2 p + c 2 k 2. (7.31) The electric field has now components in the x and y-directions. Thus, the electric field is perpendicular to k and from the linearized Gauss s law we see that the wave is non-compressional and transverse (the electric field perturbation is perpendicular to the direction of propagation). The magnetic field associated with the wave is obtained from Eq The wave vector as a function of frequency is k = ± 1 c ω 2 ω 2 p. (7.32) Figure 7.1 displays the dispersion equation. At high-frequencies the solution approaches the vacuum electromagnetic wave with ω = ck, i.e., its phase and group speeds approach the speed of light. The interpretation is that the frequency of the wave becomes so high that it does not interact with the plasma. In fact, it interacts but only very weakly.

123 7.2. WAVE PROPAGATION IN NON-MAGNETIZED PLASMA 117 Figure 7.1: The dispersion equation for an electromagnetic wave in a cold plasma where the background magnetic field is zero. The electric field of the plane wave is of the form E = E exp[i(k r ωt)]. (7.33) When ω < ω p, the wavenumber k is purely imaginary. If the imaginary part is negative, the wave electric field would grow exponentially. Since there is no energy in the plasma to facilitate wave growth, this solution is unphysical. The solution with a positive imaginary part makes the electric field to decay exponentially and the wave is said to cut-off at the plasma frequency. What happens physically is that when the wave frequency approaches the plasma frequency the wave forces electrons to oscillate at the plasma frequency. The oscillating electrons re-radiate the wave energy and the wave is reflected. Figure 7.2: Reflection of electromagnetic waves from the ionosphere. Note that also waves with higher frequency than the maximum plasma frequency are affected by the plasma and refracted. The ionospheric density profile can be determined by sending waves at different frequencies and measuring the time the wave returns back. Such a device is called ionosonde. An example of the reflection arises when radio waves are sent to the ionosphere (Figure 7.2). While the wave travels away from the Earth, the plasma density and thus the plasma frequency increases. As a response to the wave electric field electrons in the

124 118 CHAPTER 7. COLD PLASMA WAVES ionosphere start to oscillate and they re-radiate the original energy. The total reflection of the wave occurs when the emitted frequency equals to the local plasma frequency. 7.3 Wave propagation in magnetized plasma We now move to a more complicated situation and introduce a non-zero background magnetic field. The magnetic field introduces anisotropy, which leads to many new wave modes. When a magnetic field is present, particles perform Larmor motion around the magnetic field lines. This introduces new possibilities to wave cut-offs and resonances. In this Section we first derive a general form of the dispersion equation and then proceed to investigate the so-called principal modes, i.e., waves that propagate either parallel or perpendicular to the magnetic field. Finally we will study wave propagation at an arbitrary angle Derivation of general dispersion equation First, we need to determine the conductivity and dielectric tensors. Let the magnetic field to be in the z-direction. Assuming harmonic time dependencies and using the gyro frequency ω cs = e s B 0 /m s the linearized equation of motion (Eq. 7.12) can be now written in the matrix form: iω ω cs 0 ω cs iω iω Inversion of the matrix equation gives u sx u sy u sz = e s m s u sx u sy u sz = e s m s E x E y E z iω ω cs ωcs 2 ω 2 ωcs 2 ω 2 0 ω cs iω ωcs 2 ω 2 ωcs 2 ω 2 0 i 0 0 ω. (7.34) E x E y E z. (7.35) Similarly as in the case with B 0 = 0 the conductivity tensor is obtained by calculating the electric current J = e s n s0 u s1 = σ E. s σ = n s0 e 2 s m s ω cs iω ωcs 2 ω 2 ωcs 2 ω 2 0 ω cs iω ωcs 2 ω 2 ωcs 2 ω 2 0 i 0 0 ω. (7.36)

125 7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 119 The dielectric tensor is now K = S id 0 id S P, (7.37) where S = 1 s ω 2 ps ω 2 ω 2 cs (7.38) D = s ω cs ω 2 ps ω(ω 2 ω 2 cs) (7.39) P = 1 s ω 2 ps ω 2. (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 L = 1 s ω 2 ps ω 2 ω 2 ps ω 2 ( ω ) ω + ω cs (7.41) ( ) ω. ω ω cs (7.42) This is a useful division since electrons and ions respond in a different way to the wave. z n,k q B 0 x y Figure 7.3: The choice of directions of the background magnetic field and the index of refraction (wave vector) for a wave propagating in a cold plasma. Let us choose the magnetic field to be along the z-direction and the index of refraction n (and thus k) to be in the xz-plane (Figure 7.3). The homogeneous wave equation now becomes S n 2 cos 2 θ id n 2 cos θ sin θ id S n 2 0 n 2 cos θ sin θ 0 P n 2 sin 2 θ E x E y E z = 0. (7.43) As before, we obtain the dispersion equation by setting the determinant to zero. D(n, ω) = An 4 Bn 2 + RLP = 0, (7.44)

126 120 CHAPTER 7. COLD PLASMA WAVES where A = S sin 2 θ + P cos 2 θ (7.45) B = RL sin 2 θ + P S(1 + cos 2 θ). (7.46) The dispersion equation can be modified to a useful form by solving for tan 2 θ as a function of n 2 tan 2 θ = P (n2 R)(n 2 L) (Sn 2 RL)(n 2 P ). (7.47) When a wave propagates through plasma, it may encounter regions of changing plasma frequency and gyro frequencies. In magnetized plasma we can find two cases in which the wave ceases to propagate. Cut-off occurs when n goes to zero. After the cut-off point n 2 becomes negative, and thus, n and k are imaginary. In such region the wave decays exponentially and becomes evanescent. Physically, the wave is reflected and no energy is absorbed in the plasma. From Eq we see that cut-off occurs when P = 0, R = 0 or L = 0. Resonance occurs when n 2 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 we see that resonance occurs when A = 0, i.e., tan θ res = P S. (7.48) Propagation parallel to the magnetic field We see from Eq that when the wave propagates exactly parallel to the magnetic, i.e. θ = 0, the dispersion equation for cold plasma waves has three roots: P = 0 (7.49) n 2 = R (7.50) n 2 = L. (7.51) The first root represents simple plasma oscillation (see Eq. 7.40), and thus, a nonpropagating wave.

127 7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 121 By inserting θ = 0 in Eq we can find the electric field eigenvector associated with the other two roots E n 2 =R = (E 0, ie 0, 0) (7.52) E n 2 =L = (E 0, ie 0, 0). (7.53) Since we chose the magnetic field to be aligned with the z-direction, we see that the roots n 2 = R and n 2 = L correspond to transverse waves. In the same way as in the case of non-magnetized waves, we can confirm from the linearized Faraday s law that the waves are electromagnetic (i.e., they have non-zero magnetic field). From the linearized Gauss s law we see that there are no charge density fluctuations. We also see that the only difference between the eigenvectors corresponding to the roots n 2 = R and n 2 = L is the sign of the y-component. n 2 = R corresponds to a wave that rotates in the right-handed sense with respect to the magnetic field while n 2 = L to a wave that rotates in the left-handed sense (confirm! In plasma physics the convention of RH and LH is opposite to that used in optics). Hence, the solutions correspond to the right-handed (R) and left-handed (L) polarized modes, respectively. Right-handed mode From 7.41 we obtain n 2 R = R = 1 ωpi 2 ω(ω + ω ci ) ωpe 2 ω(ω ω ce ). (7.54) Consequently, the R-mode has a resonance (n ) when the wave frequency approaches the electron cyclotron frequency ω = ω ce. This is because electrons rotate around the magnetic field in the same sense as the electric field rotates in the R-mode. Because ω pi ω pe and ω ci ω ce the R-mode cut-off (n R 0) occurs when (Exercise 7.2) ω R=0 ω [ ] ce ω 2 pe/ω 2 ce 2. (7.55) The cut-off is divided to two branches depending on the density. At the low density limit (ω p ω c ) the cut-off becomes and at the high density (ω p ω c ) limit ω R=0 ω ce (1 + ω 2 pe/ω 2 ce) (7.56) ω R=0 ω pe + ω ce /2. (7.57)

128 122 CHAPTER 7. COLD PLASMA WAVES Left-handed mode For the L-mode we obtain from 7.41 n 2 L = L = 1 ωpi 2 ω(ω ω ci ) ωpe 2 ω(ω + ω ce ). (7.58) Hence, the resonance occurs now at the ion gyrofrequency ω = ω ci. Usually the ion motion is ignored when computing the L-mode cut-off At low density limit we obtain and at the high density limit ω L=0 ω ce 2 i.e, both at low and high density limits ω R=0 = ω L=0 + ω ce. [ ] ωpe/ω 2 ce 2. (7.59) ω L=0 ω 2 pe/ω ce, (7.60) ω L=0 ω pe ω ce /2, (7.61) Figure 7.4 shows the solution of n 2 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 (ω R=0 and ω L=0 ) the solution to the wave dispersion equation is called the free-space mode. Below electron and ion cyclotron frequencies the waves are called the cyclotron modes. At low frequencies (ω 0) L- and R-modes merge and the dispersion becomes that of the shear Alfvén wave n 2 c 2 /va 2 we encountered in Section Faraday rotation A linearly polarized plane wave can be expressed as a sum of left- and right-hand circularly polarized waves (R- and L-modes having equal amplitudes, E 0 ). If we assume that the wave is linearly polarized along the x axis, and that the wave propagation (k) and the background magnetic field (B 0 ) are along the z-axis, we can write E = E 0 [(e ik Rz + e ik Lz )ê x + i(e ik Rz e ik Lz )ê y ]e iωt. (7.62) The ratio of the E x and E y components is ( ) E x kl k R = cot z. (7.63) E y 2 Hence, due to different phase speeds of R- and L-modes the linearly polarized wave that is travelling along a magnetic field will experience the rotation of its plane of polarization. This is called Faraday rotation. The magnitude of the rotation depends on the density and magnetic field of the plasma. Considering frequencies above the plasma frequency

129 7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 123 Figure 7.4: Wave modes propagating parallel to the magnetic field in the limit of high plasma density (top), and low plasma density (bottom). one can show that the rate of change in the rotation angle φ with the distance travelled (assumed here to be in the z-direction) is and the total rotation from the source to the observer is φ = dφ dz = e 3 2m 2 eɛ 0 c ω 2 n eb 0 (7.64) e 3 2m 2 eɛ 0 c ω 2 d 0 n e B ds. (7.65) The integral is calculated along the wave propagation path. The total rotation thus depends on both the density and magnetic field of the medium. Exercise 7.3 applies Faraday rotation to estimate the distance to a distant pulsar when the density of the interstellar plasma is known. Contemplate: Faraday rotation is an important tool in astronomy. Find an example where Faraday rotation is used to obtain information on the physical properties of an

130 124 CHAPTER 7. COLD PLASMA WAVES astronomical object. Pay attention to the fact that either density or magnetic field has to be known from other methods. Whistler waves The investigation of dispersion characteristics of R-mode waves reveals an interesting feature. Figure 7.4 shows that the R-mode propagates also in the region between electron and ion gyro frequencies. In this domain the dispersion equation can be approximated as (Exercise 7.4): k = ω pe ω, (7.66) c ω ce which gives the phase and group speeds v p = ω k v g = ω k = c ω ce ω pe ω (7.67) = 2c ω ce ω pe 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 was observed in telegraph lines during World War I. It took several decades before this phenomenon was explained. In 1953 L.R.O. Storey suggested that the sound was produced by waves that are propagating along the magnetic field lines from one hemisphere of the Earth to the other (Figure 7.5) and the whistling sound is the consequence of different frequencies arriving at different times. These whistler waves are produced by lighting strokes that emit radio noise of broad frequency bands. Figure 7.5: Whistler waves The propagation time for a R-mode wave in in the frequency range ω ci ω ω ce

131 7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 125 can be obtained from (see Eq. 7.68) ds t(ω) = = v g w pe (s) 2c ωω ce ds, (7.69) where ds is the line element along the magnetic field. Hence, lower frequencies arrive to an observer after a longer time than the higher frequencies Propagation perpendicular to the magnetic field When the wave propagates perpendicular θ = 90 to the magnetic field the homogeneous wave equation (Eq. 7.43) becomes S id 0 E x id S n 2 0 E y = 0 (7.70) 0 0 P n 2 E z and The roots are now tan 2 θ = P (n2 R)(n 2 L) (Sn 2 RL)(n 2 P ). (7.71) n 2 = P (7.72) n 2 = RL (7.73) S and the corresponding electric field eigenvectors are: E n 2 =P = (0, 0, E 0 ) (7.74) (7.75) qe n 2 =RL/S = ( id S E 0, E 0, 0). (7.76) Ordinary mode The wave mode associated with the first root n 2 = P is called the ordinary (O) mode. n 2 O = P = 1 ω2 pi ω 2 ω2 pe ω 2 1 ω2 pe ω 2. (7.77) The electric field of the ordinary mode is along the background magnetic field, and thus, the wave vector k is perpendicular to wave electric field. The dispersion equation above shows that the O-mode is not affected by the magnetic field (the particle motion is parallel to the magnetic field, and hence the magnetic part of the Lorentz for vanishes v B 0 = 0). Physically, the O-mode corresponds to the high-frequency transverse electromagnetic wave. It is linearly polarized. The O-mode has the cut-off (n 2 0) when the wave frequency approaches the electron plasma frequency, i.e., at ω = ω pe.

132 126 CHAPTER 7. COLD PLASMA WAVES Extraordinary mode The second mode is called the extraordinary mode. Now the wave electric field is perpendicular to the background magnetic field (Eq. 7.75), and thus, the electric field has components both parallel (longitudinal) and perpendicular (transverse) to the wave vector, see Figure 7.6. Hence, the X-mode has both electrostatic and electromagnetic characteristics. The wave magnetic field can be calculated from the Faraday s law. According to Eq the X-mode is elliptically polarized. Figure 7.7 shows the solution of n 2 in the (ω, k)-space for ordinary and extraordinary modes. Ordinary (O) mode z Extraordinary (X) mode z B 0 E 1 B 0 B 1 B 1 n,k x y n,k x E 1 y B = 1 k E w 1 1 Figure 7.6: Magnetic field, electric field and the wave vector directions for O- and X- modes. w X-mode w X,R=0 w pe O-mode w UH upper hybrid resonance w X,L=0 X-mode w LH lower hybrid resonance magnetosonic mode k Figure 7.7: A plot of wave frequency as a function of wave number for ordinary and extrordinary modes. The X-mode has two cut-offs when R = 0 and L = 0 (see Section 7.3.2). Calculation of resonances and cut-offs for extraordinary waves is a tedious task. The resonances

133 7.3. WAVE PROPAGATION IN MAGNETIZED PLASMA 127 occur at S = 0 (see Eq. 7.38) ω2 pe S = 1 ω 2 ωce 2 ω2 pi ω 2 ωci 2 = 0, (7.78) and they are called hybrid resonances. The number of resonances depends on the ion species involved. Here we have assumed that the plasma consist of electrons and one positive ion species. The resonance that occurs at the highest frequency, above both the electron gyro and plasma frequencies, is called the upper hybrid frequency (ω UH ). The frequencies near ω UH are so high that one can neglect the ion dynamics from Eq and using ω pi ω pe the upper hybrid resonance is at ω 2 UH ω 2 pe + ω 2 ce. (7.79) The lower hybrid resonance occurs between the electron and ion cyclotron frequencies. Assuming that ω ce ω LH ω ci one obtains the lower hybrid resonance at ωlh 2 ω2 ci + ( ) ω2 pi ω (ωpe/ω 2 ce) 2 ω pe + ω ce ω ci ceω ci ωpe 2 + ωce 2. (7.80) Further approximations are often made at the low and high density limits. In the high density limit ω 2 c ω 2 p and we obtain ω LH ω ce ω ci. (7.81) Both electrons and ions participate in the resonance oscillation. The lower hybrid resonance at the high density limit is particularly important since the wave can be in resonance both with electrons and ions. This can facilitate the energy transfer between ions and electron. For instance, in fusion devices these waves are in the microwave range and microwave techniques are used to heat the plasma through the lower hybrid resonance. In the low density limit ω 2 p ω 2 c the lower hybrid resonance is at Now only ions participate in the resonance oscillation. ω LH ω pi (7.82) For the low-frequency limit one obtains the magnetosonic mode we encountered already in Section The cold plasma theory introduces a correction: ω 2 k 2 = v2 s + va va 2, (7.83) /c2 which guarantees that the group velocity of the wave remains below the speed of light even in a plasma where v A > c (in cold plasma v s is, of course, negligible).

134 128 CHAPTER 7. COLD PLASMA WAVES Propagation in an oblique angle Principal modes R, L, O ja X can be uniquely defined only when the wave propagates exactly perpendicular or parallel to the magnetic field. Similar to MHD waves, cold plasma waves can propagate also at oblique angles to the magnetic field. It is possible to draw the wave normal surface for each mode. However, the ratio of density and magnetic field now varies and hence, the number of wave mode surfaces is much larger. Figure 7.8 represents the characterization of waves using the CMA-diagram (Clemmow, Mullaly, Allis). In the CMA-diagram a particular wave mode may be identified with its wave normal surface and the surfaces may be traced in the (ωp/ω 2 2, ω c /ω) space until it disappears at the cut-off or at the resonance. As is seen from the figure cut-offs and resonance define the cages where different wave modes are confined. 7.4 Exercises: Cold Plasma Waves 1. Consider a plasma consisting of free electrons and protons. Starting from the electric current due to polarization drift find the dielectric function for low frequencies in the form ( ) ɛ = ɛ c2 va Prove that for the right-hand polarized wave propagating parallel to the magnetic field the cut-off (n R 0) occurs when ω R=0 ω [ ] ce ω 2 pe/ω 2 ce 2. Show further that at the low density limit (ω p ω c ) this reduces to ω R=0 ω ce (1 + ω 2 pe/ω 2 ce) and at the high density (ω p ω c ) limit to ω R=0 ω pe + ω ce /2. 3. The arrival time of a signal from a distant source depends on the dispersion of plasma as T = d/c + D/f 2 where d is the distance to the source, f the frequency of the signal and D the so-called dispersion measure D = e 2 d 8π 2 n e ds. ɛ 0 m e c 0

135 7.4. EXERCISES: COLD PLASMA WAVES 129 Consider a pulsar from which a signal at 100 MHz arrives 2 s later than the signal at 200 MHz. Assuming the density of the interstellar plasma to be 0.03 cm 3 calculate the distance to the pulsar. Calculate further the Faraday rotation of the wave assuming a linear polarization and 0.1-nT interstellar magnetic field. 4. Derive the dispersion equation for the whistler wave k = ω pe ω. c ω ce Using this equation show that the group velocity of the whistler wave is v g = ω k = 2c ωc e ω. ω pe Compare the arrival times of the emitted low and high frequency waves.

136 130 CHAPTER 7. COLD PLASMA WAVES P=0 (cut-off) R=0 (cut-off) L=0 (cut-off) S=0 (res) Figure 7.8: CMA diagram

137 Chapter 8 Warm plasma In two previous chapters the temperature did not have an independent meaning. In MHD temperature appears through the equation of state, but always in relation to density and pressure, while in the cold plasma theory we assumed that thermal effects can be neglected. The inclusion of thermal effects introduces new wave modes in the plasma and allows for free-energy that is necessary for the generation of plasma instabilities. To fully describe warm plasma (wave modes, instabilities, etc.) one needs complex mathematical tools of kinetic theory. However, fluid description gives a simpler introduction to characteristics of warm plasmas. We now modify the fluid equations to take into account the thermal effects. We add the pressure term to the equation motion n α m α dv α dt = en α (E + V B) + p α, which allows us to investigate temperature related phenomena in a number of special cases. We first derive warm plasma dispersion equation and investigate two special solutions; the Langmuir wave and the ion sound wave. Then we proceed to a brief overview of plasma instabilities. 8.1 Warm plasma dispersion equation Now we include ions, but assume that there is no background electric or magnetic fields, and that the plasma is homogeneous and initially at rest (density n 0, speed V 0 = 0). We assume a small initial perturbation, denoted again by subscript 1. The gradient of the electron pressure is included in the equation of motion, but we make a simplified assumption that the ion pressure gradient is zero, justified by their larger inertia. Using the adiabatic equation of state the pressure gradient p e can be replaced by γp 0 n 1 0 n e1 (Exercise 8.1). Hence, the linearized continuity and momentum equations for electrons and ions are 131

138 132 CHAPTER 8. WARM PLASMA n e1 + ( V e1 ) = 0 (8.1) t n i1 + + ( V i1 ) = 0 (8.2) t n 0 m e V e1 t n 0 m i V i1 t respectively. We also need the Gauss law = en 0 E 1 γp 0 n 0 n e1 (8.3) = en 0 E 1, (8.4) E 1 = e ɛ 0 (n i1 n e1 ). (8.5) Assuming again harmonic time and spatial dependencies, i.e., we are looking for plane wave solutions, we obtain iωn e1 + in 0 k V e1 = 0 (8.6) iωn i1 + in 0 k V i1 = 0 (8.7) iωm e V e1 = ee 1 i γp 0 n e1 k n 0 (8.8) iωm i V i1 = ee 1 (8.9) ik E 1 = e ɛ 0 (n i1 n e1 ). (8.10) A brief calculation (Excercise 8.2) gives ( 1 ω2 pi ω 2 ωpe 2 ) ω 2 k 2 k E 1, (8.11) (γk B T e /m e ) where p e = n e k B T e has been used to introduce the electron temperature. The expression in the parenthesis is the dielectric function K(ω), in this case a scalar. The zeros of K(ω) give the dispersion equation. 8.2 Langmuir wave and the ion sound wave When frequencies are well above the ion plasma frequency (ω ω pi, and hence, the (ω pi /ω) 2 -term in Eq. 8.11) can be neglected) the solution to the warm plasma dispersion equation is ω 2 = ω 2 pe + k 2 (γk B T e /m e ), (8.12)

139 8.2. LANGMUIR WAVE AND THE ION SOUND WAVE 133 i.e., we have again encountered the Langmuir wave. Note that this result can also be derived assuming ions as an inmobile background (Exercise 8.3). The finite temperature and associated thermal motions now allow electron plasma oscillation to propagate as a wave ( ω/ k 0). We notice also that the wave number depends on the frequency and hence the wave is dispersive. The dispersion equation is shown in the left-hand part of Figure 8.1. But what is the value for the polytropic index γ? k k Figure 8.1: Solutions of the warm plasma dispersion equation. Left) Langmuir wave, Right) ion sound wave Considering that the inclusion of the temperature effects introduces only a small correction to the cold plasma theory, we can assume that the temperature disturbance propagates less than one wavelength during one plasma oscillation. This corresponds to the long wave length limit (k 2 λ 2 De 1), i.e., the approach we used to solve the Vlasov equation in Chapter 4. Hence, the perturbation is assumed to be adiabatic. Since homogenous plasma without background fields is one-dimensional, the polytropic index is γ = (d + 2)/d = 3, where d is the indicates the number of spatial directions, in this case d = 1. Using the relationship between the Debye length and thermal speed λ 2 De = v2 th,e 2 ω 2 pe we can write the dispersion equation in terms of the Debye length which is the same as Eq (8.13) ω 2 = ω 2 pe(1 + 3k 2 λ 2 De), (8.14) At the long wave length (small wave number) limit we can approximate ω = ω pe 1 + 3k 2 λ 2 De ω pe( k2 λ 2 De). (8.15) When frequencies are well below the electron plasma frequency (ω ω pe, the solution to the warm plasma dispersion equation gives a new wave mode, the ion sound wave (Exercise 8.4) ω = kc s, (8.16) 1 + k 2 λ 2 De

140 134 CHAPTER 8. WARM PLASMA where we have introduced the ion sound speed c s = k B T e /m i. (8.17) The dispersion equation is plotted in the right-hand part of Figure 8.1. Now the process has been assumed isothermal (γ = 1), which is justified by the ions oscillating so slowly that the electron temperature has time to relax over the oscillations. This mode can also be found from the Vlasov theory, when the solution is investigated from the appropriate frequency domain. At the limit of small wave number (k 2 λ 2 De 1) ω kc S, yielding the dispersion equation ω k = k B T e /m i. (8.18) At the limit of large wave numbers we obtain ω = c s λ De = k B T e m i n 0 e 2 ɛ 0 k B T e = n 0 e 2 m i ɛ 0 = ω pi, (8.19) i.e., the wave frequency approaches the ion plasma frequency, see the right-hand part of Figure 8.1. Thus the ion sound wave has a resonance at the ion plasma frequency. It is interesting to note that the numerator in the ion sound speed includes the electron temperature, while in the denominator is the ion mass. Thus the electrons account for the pressure and ions for the inertia. If ion pressure would be taken into account T e would be replaced by T e + γt i. However, Vlasov theory indicates that if the electrons are not clearly warmer than ions, the ion waves are strongly damped. Note that the ion acoustic wave can also propagate in collisionless plasma because charged particles interact due to long-range Coulomb forces. Electrons are highly mobile and they quickly follow the ion motion to preserve the charge neutrality (remember that we have assumed there to be no magnetic field!). 8.3 On plasma stability Plasma reacts to a disturbance by starting to oscillate with a characteristic frequency and wave length. Depending on the situation the oscillations may propagate, and grow or damp. The growing oscillations can lead to a plasma instability. Instability requires a source of free energy, and hence, there are no instabilities in the cold plasma theory. In addition, the elementary approach to plasma physics often assumes an unperturbed state that is in local thermodynamic equilibrium and particles can be described by Maxwellian velocity distributions. Neither in that case is there free energy for waves to self-excite. Free energy may be stored in the magnetic or plasma configuration, for example in the form of magnetic tension or the relative streaming of plasma populations.

141 8.3. ON PLASMA STABILITY 135 Instability can be externally driven or result from the changes in plasma distribution function. If there are no processes that would saturate the instability, the whole plasma system can explode. This happens both in space and laboratory plasmas. Solar flares and loss of plasma state in tokamaks are examples of large-scale plasma instabilities. One way to categorize plasma instabilities is to divide them between microscopic and macroscopic instabilities. A microscopic instability needs the kinetic approach and it depends on the shape of the distribution function. A macroscopic instability is a configurational instability and can be described by macroscopic equations. We consider here only a few simple examples that can be understood either intuitively or that are straightforward to calculate Z-pinch instability Let us first investigate the equilibrium configuration of the Z-pinch from Chapter 6 (see Figure 6.17) where plasma is confined by a toroidal magnetic field. The magnetic field arises from the electric current that is driven through the plasma. Figure 8.2 displays what may happen to an initially stable Z-pinch if the system is perturbed. B 2 /2m 0 decreases B 2 /2m 0 increases kink instability B 2 /2m 0 decreases sausage instability B 2 /2m 0 increases Figure 8.2: Kink and sausage instabilities If the plasma tube is bent, the magnetic pressure will increase on the concave part of the bend (magnetic flux increases) and decrease on the convex part (flux decreases). This creates a gradient in magnetic pressure, i.e., magnetic force, that strengthens itself and the whole plasma can rise up from the equilibrium leading to the loss of the plasma state. This is called kink instability. The other instability shown in Figure 8.2 is related to the squeezing of the flux tube. The magnetic pressure will increase at the part that is being compressed and decreases in the nearby region. Larger magnetic field gradient tends to increase the compression

142 136 CHAPTER 8. WARM PLASMA of the plasma and the whole plasma tube will break, if there is no mechanism to stop the squeezing. This is called the sausage instability. A common way to stabilize plasma is to wind a flux tube to a torus and drive an electric current through the tube. This toroidal current creates a poloidal magnetic field around the torus. The superposition of toroidan and poloidal magnetic fields leads to a spiral shaped magnetic field inside the torus. Such device is called tokamak, and it is nowadays the most common and important plasma confinement device in fusion experiments (Figure 8.3). The growth rate of both sausage and kink instabilities can be stabilized in tokamaks, but driving large electric currents through the plasma may cause kinetic instabilities related to changes in plasma distribution functions. Figure 8.3: In a tokamak fusion reactor poloidal and toroidal electric currents create an almost force-free flux tube magnetic field configuration Two-stream instability One way to investigate plasma instabilities is to derive the dispersion equation and investigate conditions that lead to growing wave perturbations. Let us investigate a simple example featuring two oppositely directed electron beams with different velocities. Essentially, this is a kinetic instability, a solution can also be found from macroscopic theory in the case the velocity difference between the beams is larger than their thermal motion. Let the densities of the electron beams be n α0 and n β0 and the velocities V α0 and V β0. Assume that there is no background magnetic field and that ions are a fixed background, hence restricting the analysis to high frequency waves. We assume again a small perturbation (allowing linearization) and investigate the plane wave solution. The