Nonlinear Wave and Plasma Structures in the Auroral and Subauroral Geospace

By Evgeny Mishin and Anatoly Streltsov

Nonlinear Wave and Plasma Structures in the Auroral and Subauroral Geospace presents a comprehensive examination of the self-consistent processes leading to multiscale electromagnetic and plasma structures in the magnetosphere and ionosphere near the plasmapause, particularly in the auroral and subauroral geospace. It utilizes simulations and a large number of relevant in situ measurements conducted by the most recent satellite missions, as well as ground-based optical and radar observations to verify the conclusions and analysis. Including several case studies of observations related to prominent geospacer events, the book also provides experimental and numerical results throughout the chapters to further enhance understanding of how the same physical mechanisms produce different phenomena at different regions of the near-Earth space environment.

Additionally, the comprehensive description of mechanisms responsible for space weather effects will give readers a broad foundation of wave and particle processes in the near-Earth magnetosphere. As such, Nonlinear Wave and Plasma Structures in the Auroral and Subauroral Geospace Nonlinear Wave and Plasma Structures in the Auroral and Subauroral Geospace is a cutting-edge reference for space physicists looking to better understand plasma physics in geospace.

• Presents a unified approach to wave and particle phenomena occurring in the auroral and subauroral geospace
• Summarizes the most current theoretical concepts related to the generation of the large-scale electric field near the plasmapause by flows of hot plasma from the reconnection site
• Includes case studies of the observations related to the most “famous events during the last 20 years as well as a large number of experimental and numerical results illustrated throughout the text

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 3, 2021
• ISBN: 9780128209318

Evgeny Mishin

Dr. E. V. Mishin received his Ph.D. in Plasma Physics (1974) and D.Sc. in Radiophysics (1985) from the Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Academy of Sciences (IZMIRAN), Troitsk, Moscow, RF. He worked at IZMIRAN as Junior Scientific Researcher, Senior Scientific Researcher, and Head Laboratory (1974-1993), was a Scholar and Visiting Professor at the Max-Planck Institute for Aeronomy, Lindau, Germany (1993-1999), Visiting Scholar at the MIT Haystack Observatory, Westford, MA (1999-2001), Senior Research Scientist at Boston College, Chestnut Hill, MA (2001-2008), and Senior Research Physicist at the Air Force Research Laboratory since 2008. Dr. Mishin is an expert in active space experiments with intense particle and electromagnetic beams and nonlinear plasma effects in the auroral and subauroral geospace. He is a Fellow of the American Physical Society.

Book preview

Nonlinear Wave and Plasma Structures in the Auroral and Subauroral Geospace

Evgeny V. Mishin

Air Force Research Laboratory, Space Vehicles Directorate, Albuquerque, New Mexico, United States

Anatoly V. Streltsov

Embry-Riddle Aeronautical University, Department of Physical Sciences, Daytona Beach, Florida, United States

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1. Introduction: near-Earth space environment

1.1. Bow shock

1.2. Magnetosheath

1.3. Polar cusps

1.4. Magnetosphere

1.5. Ionosphere

1.6. Electric currents

1.7. Aurora and auroral oval

1.8. Magnetosphere-ionosphere (MI) coupling

Chapter 2. Plasma waves and instabilities

Chapter 2.1. Plasma waves

Chapter 2.2. Plasma instabilities

Chapter 2.3. Nonlinear interactions

Chapter 3. Auroral geospace

Chapter 3.1. Earthbound injections in the magnetotail

Chapter 3.2. Substorms

Chapter 3.3. Multiscale aurora: structure and dynamics

Chapter 3.4. Alfvénic aurora

Chapter 4. Nonlinear effects in natural and artificial aurora

Chapter 4.1. Energetic electron impact on the upper atmosphere

Chapter 4.2. Artificial aurora

Chapter 4.3. Theory of Artificial and Enhanced Aurora

Chapter 4.4. E/F-region turbulent heating

Chapter 5. Subauroral geospace

Chapter 5.1. Subauroral flows

Chapter 5.2. Subauroral ULF wave structures

Chapter 5.3. Subauroral arcs

Chapter 5.4. Generation and dynamics of subauroral VLF whistlers

Index

Copyright

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Preface

From smart watches and cell phones to GPS satellites, humanity increasingly relies on the ability of modern gadgets to exchange information via electromagnetic waves. The quality of such communication links, however, depends on the propagation characteristics of the near-Earth plasma environment, which is frequently not at equilibrium. In particular, plasma irregularities occurring in the ionosphere during space storms cause severe distortion of wave trajectories, leading to the interruption of communication and navigation. Enhanced fluxes of MeV, so-called killer electrons forming the Earth’s radiation belts, strike spacecraft, thereby shortening their lifetime. These phenomena represent just a fraction of the Sun-driven space weather affecting technological systems in near-Earth space. The self-explanatory term space weather remains in use since the space era began.

Humankind will probably never have control of space weather. The main practical purpose driving research in this field is, therefore, to identify possible effects of space storms, to understand and to predict storm development from the Sun to the ground, and then to develop methods to mitigate those effects. The Sun is the ultimate energy source of space weather. Ground-based and onboard sensors monitor the Sun to detect the onset of coronal mass ejections (CMEs), when billions of tons of the hot solar plasma are ejected into interplanetary space and create a shock wave moving from the Sun at a speed greatly exceeding that of an undisturbed solar wind.

CMEs are manifested by sharply intensified optical (solar flare), radio, and X-ray radiation. The radiation covers the distance between the Sun and the Earth (1 AU ≈ 150 million kilometers) in about 500 seconds, whereas the shocks approach the Earth in about 1–2 days. The shock arrival at the magnetosphere’s boundary, the magnetopause, leads to an abrupt compression of the magnetosphere, indicated by a so-called storm sudden commencement (SSC) in the geomagnetic field. Another global manifestation is the so-called shock aurora caused by electrons precipitating from the disturbed magnetosphere into the lower atmosphere. Fig. 0.1 provides an impressive illustration of the space weather concept.

Figure 0.1 Artist's not-to-scale illustration of a Space Weather event: Coronal Mass Ejection manifested by a solar flare (white light) creates a shock wave moving toward the Earth's magnetosphere. 

Source: NASA www.sunearthplan.net/3/inter.

A more continuous effect results from a so-called magnetic cloud—a twisted, magnetic structure carried by the perturbed solar wind—which interacts with the Earth’s magnetic field. This interaction, via a complex process of reconnection or merging at the magnetopause, leads to the electromagnetic and plasma energy inflow and a strong distortion of the geomagnetic field. The interaction is particularly strong at polar latitudes because of this region’s magnetic connection to the solar wind. This process results in the intensification of a global electric field and current system accompanied by particle energization and precipitation into the atmosphere, creating aurorae even at mid-latitudes during major space storms.

The electromagnetic and particle energy flowing into the magnetosphere makes the magnetosphere–ionosphere system unstable. Various plasma instabilities lead to energy release and the ultimate relaxation of the system to a new equilibrium state after the magnetic cloud passes on. Instabilities in the outer magnetosphere, such as reconnection at the magnetopause and in the magnetotail, maintain the global energy balance by injecting electromagnetic and particle fluxes into the inner magnetosphere. In turn, instabilities in the inner magnetosphere generate electromagnetic waves and plasma turbulence that enhances energy dissipation in a global electric circuit, energizes plasma, and makes particles trapped in the geomagnetic field precipitate.

The goal of this book is to consistently apply the methods of plasma theory to selected geospace phenomena that are critical for the understanding of electromagnetic and plasma disturbances in the near-Earth plasma environment at auroral and subauroral latitudes. We intend to demonstrate that the underlying physics of many of these phenomena is the same in the different parts of the strongly coupled magnetosphere–ionosphere system. Nonetheless, the same classical plasma instabilities and nonlinear interactions occurring at different latitudes can lead to different observational phenomena, depending on the background parameters and the disturbance magnitude. We illustrate the results of the theoretical analysis with a large number of observations made over the past 40 years, starting with the first active experiments conducted in space and ending with the most recent observations from the ground sensors and satellites.

This book consists of five parts. In Part 1, we provide a general description of the basic structural elements of the near-Earth space environment for readers unfamiliar with the subject. Part 2 describes the linear theory of plasma waves, basic plasma instabilities, and nonlinear wave–particle and wave–wave interactions. Part 3 outlines processes occurring in the auroral geospace during substorms, as well as the spatial and temporal characteristics of different types of aurorae and their physical sources. Part 4 contains a survey of the classical aurora resulted from the collisional impact of energetic electrons on the atmosphere, describes observations of noncollisional auroral features, artificial aurora experiments, the theory of nonlinear electron beam–plasma interactions applied to aurorae, and plasma heating effects in the auroral E region. Part 5 describes the basic features and processes occurring during auroral substorms in the subauroral geospace, including subauroral–auroral boundary processes, subauroral flow channels and arcs, as well as the generation and dynamics of VLF whistler waves.

As a rule, we will not dwell on theoretical details by giving all the basic derivations and concepts; rather we will provide just enough background for understanding theoretical and experimental results in subsequent chapters. Nonetheless, the requirement for this text is a reasonable familiarity with the contents of a typical graduate physics or engineering curriculum, including classical mechanics, vector algebra, Maxwell’s equations, and calculus. Inquisitive readers can find further details and rigorous derivations in comprehensive reviews and textbooks listed in Recommended Reading.

This book is not just a summary of research results; rather it is a presentation of the basic issues of auroral and subauroral plasma physics, together with illustrations of simple models wherever appropriate. The material covered could serve as a good foundation on which an undergraduate or graduate student could build an understanding of the past and present research in this field. We have already used some parts of this book in undergraduate and graduate Space Plasma Physics courses at Embry-Riddle Aeronautical University. For the experienced researcher, we hope that this book is a useful presentation of the status quo in the field. Concerning references, we believe that the historical aspects of the field are subordinate to a proper grasp of the underlying physics described in most recent books and review papers.

This book is a result of 40+ years of the authors’ active research, started in the USSR and continued in the US, in cooperation with many colleagues whose hard work led to the development of this discipline. We are grateful to all of them. We would especially like to acknowledge and thank Roald Sagdeev, who introduced the many fascinating ideas constituting the core of nonlinear plasma physics, as well as Albert (Alec) Galeev, Gerhard Haerendel, Tor Hagfors, Vitaly Shapiro, Dennis Papadopoulos, Goran Marklund, Larry Lyons, John Foster, Bill Burke, and Vilen Mishin. Our special thanks go to Charlotte Johnson for her able assistance in editing this manuscript and making it more readable for people unfamiliar with plasma instabilities.

We acknowledge the continuous financial support of our activity by the Air Force Office of Scientific Research. And last, but not least, our deep appreciation to our wives, Luba Mishin and Natalia Streltsov, for their constant encouragement during the writing of this book and for bearing with us through its successful completion.

Chapter 1: Introduction

near-Earth space environment

Abstract

The near-Earth space can be defined as the region where the solar wind magnetic field and plasma interact with the magnetic field and plasma supplied by the Earth. It starts at the distance ∼10–12 RE toward the Sun and extends to >100 RE in the direction from the Sun. It consists of several large, distinctive regions, which, in turn, contain smaller regions with different parameters of the plasma and the magnetic field, and as a result, with different dominant wave and particle processes. In that sense, the near-Earth space is similar to a Russian Matryoshka doll or Chinese Box, where smaller and smaller objects are nested inside the larger ones.

Keywords

Auroral oval; Birkeland currents; Ionosphere; Magnetosphere; Near-Earth space

The near-Earth space can be defined as the region where the solar wind magnetic field and plasma interact with the magnetic field and plasma supplied by the Earth. It starts at the distance ∼10–12 R E toward the Sun and extends to >100 R E in the direction from the Sun. It consists of several large, distinctive regions, which, in turn, contain smaller regions with different parameters of the plasma and the magnetic field, and as a result, with different dominant wave and particle processes. In that sense, the near-Earth space is similar to a Russian Matryoshka doll or Chinese Boxes, where smaller objects are nested inside the larger ones.

Fig. 1.1 depicts five main regions in the near-Earth space: (1) Bow Shock, (2) Magnetosheath, (3) Cusps, (4) Magnetosphere, and (5) Ionosphere. Each of these regions has its own subregions, with different parameters of the plasma and the magnetic field. For example, the integral parts of the Magnetosphere are the Magnetotail, Plasmasheet, and Plasmasphere. In the Magnetotail, we distinguish the Mantle, Lobes, and Current sheet. In the plasmasphere, which contains mainly a dense cold plasma, corotating with the Earth, two energetic particle populations deserve special attention. One of these populations is radiation belt particles, and the other is ring current particles.

Let us discuss these regions in some detail.

1.1. Bow shock

This is a region at a distance ≈10–12 R E from the center of the Earth toward the Sun. Here, the shock wave associated with the supersonic solar wind plasma around the obstacle (Earth's magnetosphere) is formed. Indeed, the average velocity of the solar wind near the Earth (at distance 1 AU from the Sun) is u sw   =   400   km/s; the electron temperature T e   ≈   10⁵   K; and the ion temperature T i   ≈   10⁴   K. A typical value of the sound speed in the solar wind plasma is c s ∼ 40   km/s, so that the sonic Mach number M s   =   u sw /c s   ≈   10. Therefore, the solar wind is supersonic near the Earth's orbit, and the shock wave appears near the location where the supersonic flow interacts with the Earth's magnetosphere. This is the bow shock.

Figure 1.1 Near-Earth space environment and magnetospheric currents. 

From the internet.

1.2. Magnetosheath

This is a region between the bow shock and the Earth's magnetosphere. Here, the transition from the interplanetary magnetic field carried by the solar wind to the magnetic field generated inside the Earth occurs. The plasma in the magnetosheath is the postshock solar wind plasma of the density n   ≈   5   cm −³ and T e   ≈   T e   ≈   10   eV.

1.3. Polar cusps

These are two narrow, funnel-like regions in the Northern and Southern hemispheres where the solar wind plasma can penetrate to low altitudes up to the ionosphere due to the dipole geometry of the Earth's magnetic field (Smith and Lockwood, 1996).

1.4. Magnetosphere

This is a part of the near-Earth space with primarily the magnetic dipole field although disguised by the interaction with the solar wind (e.g., Bagenal, 1985). The magnetosphere is normally divided into the dayside magnetosphere and the magnetotail. The boundary separating the magnetosphere from the magnetosheath is called a magnetopause. Technically speaking, this is the boundary around the dayside magnetosphere and the magnetotail, but more often, it is used to specify the boundary between the magnetosheath and the dayside magnetosphere (Paschmann, 1979; Russel, 1981).

One of the main parameters describing the dayside magnetopause is the so-called standoff distance, which defines the distance from the center of the Earth to the subsolar point where the magnetopause is supposed to be. This distance is calculated from a pressure balance between the dynamic pressure in the solar wind, , and the magnetic field pressure inside the magnetosphere, . Basically, it is assumed that the magnetic field in the solar wind is weak (usually, B SW   ≈   4–8   nT near the Earth), and the pressure in the solar wind is due to the dynamic pressure only. Inside the magnetosphere, the plasma is relatively cold, stationary, and diluted, and the pressure there is due to the magnetic field only. If we assume that the magnetic field inside the dayside magnetosphere is dipole, then the magnitude of this field in the equatorial plane is , where r is the geocentric distance, R E   =   6371.2   km is the radius of the Earth, and B E   =   3.2   ×   10 −⁵   T. In this case, the equation defining the distance to the magnetopause in the subsolar point from the center of the Earth, , is

(1.1)

For typical parameters of the solar wind, u sw   =   400   km/s, m i   =   m p   =   1.67   ×   10 −²⁷   kg, and n   =   6   cm −³, this distance is , which is less than the average distance to the magnetopause observed by satellites in the subsolar region. This value can be corrected by considering effects from the Chapman–Ferraro currents flowing around the dayside magnetosphere. These currents will be described shortly in this chapter. Here, we just note that they originate from the magnetic curvature and gradient drift motion of the particles in the solar wind facing the strong magnetic field on the magnetosphere. The resulting current flows in the ecliptic plane from dawn to dusk producing a magnetic field, which increases the magnetic field inside the magnetosphere and decreases it outside the magnetopause.

If the magnitude of the magnetic field produced by the Chapman–Ferraro current is equal to the magnitude of the field outside the magnetopause, then it will cancel the magnetic field outside and double it inside. In this case, to calculate the distance to the magnetopause, one should use in Eq. (1.1) instead of , and the resulting expression for is

(1.2)

Now, for the same typical parameters of the solar wind, and this value corresponds to the observations.

1.4.1. Magnetotail

Plasma Mantle . This is the region of the magnetosphere adjacent to the magnetopause. Here, the plasma density is   ≈   0.1–1.0   cm −³ and T e   ≈   100   eV.

Tail Lobes . Northern and southern lobes of the magnetotail extend downstream from the Earth to >200 R E . Magnetic field lines in the lobes are nearly parallel to each other and the strength of the magnetic field is   ≈   20   nT. The plasma density here is very low, ≈ 0.01   cm −³, T e   ≈   100   eV, and T i   ≈   1   keV. This region of the magnetotail maps along the magnetic field to the polar cap and provides a spatially homogeneous polar rain of electrons with energies of a few hundred eV into the ionosphere.

Plasmasheet. This is a central part of the magnetosphere. In the nightside magnetosphere, it separates two tail lobes. The magnetic field in the nightside of the plasmasheet is weaker than in the lobes. The field is supposed to be near zero in the most central part of the nightside plasmasheet where the reconnection occurs. In the part of the plasmasheet closer to the Earth, the magnetic field lines are closed, and this region maps by the magnetic field to the auroral oval in the high-latitude ionosphere. The average plasma density in the plasmasheet is   ≈   0.3–1.0   cm −³, T e   ≈   0.5–1.0   keV, and T i   ≈   3.0–6.0   keV. The magnetic field here is weaker than in the lobes and the plasma is denser than the plasma in the lobes.

Boundary Layers. Two additional subregions in the magnetotail deserving special attention are the plasmasheet boundary layer and the low-latitude boundary layer (LLBL). They are shown in Fig. 1.1. These layers represent narrow transition regions, where parameters of the plasma change significantly over a relatively short distance leading to strong transverse gradients in plasma density, temperature, and velocity. These gradients cause the development of the hydrodynamics instabilities (e.g., Kelvin–Helmholtz instability in LLBL), which affect the electromagnetic dynamics of the magnetosphere.

1.4.2. Plasmasphere

The plasmasphere consists of a torus of relatively cold and dense plasma of ionospheric origin corotating with the Earth. Here, n   >   100   cm −³ and T e   ≈   T i   ≈   1   eV. The period of the plasmasphere rotation around the Earth is   ≈   26   h or ∼10% longer than the period of the Earth's rotation. The plasmasphere is bounded in the radial direction by a sharp, well-defined boundary called the plasmapause. During quiet geomagnetic conditions, the plasmapause locates on the magnetic field lines that map down to ≈60 degrees magnetic latitude. The characteristic scale size of the plasmapause in the radial direction can be in the range 0.01–0.1 R E , and the plasma density changes over this distance from <10   cm −³ outside the plasmasphere to >100   cm −³ inside (Carpenter and Anderson, 1992; Lui and Hamilton, 1992).

The plasmasphere also contains several populations of energetic particles which normally are considered separately. These populations include particles forming radiation belts and carrying ring current.

1.4.3. Radiation belts

The first population consists of low-density energetic electrons and ions forming so-called radiation belts around the Earth. The electrons with energies >0.5   MeV accelerated in the plasmasheet, form an outer radiation belt in the region between L   =   2.4 and L   =   6 magnetic shells. Protons with energies >100   MeV, originating from the decay of neutrons produced in the atmosphere by cosmic rays, form the inner radiation belt near L   =   1.5. The two radiation belts are separated by the gap or slot region near L   =   2.0.

The density of the energetic particles in both radiation belts is 10 −⁴–10 −⁶ of the total density, but because of their high energy, these particles create a real danger for satellites and humans operating in space. They also can generate some type of plasma waves (for example, VLF whistler-mode waves) via cyclotron wave–particle interactions.

1.4.4. Ring current

The second population comprises of ions with a typical energy of ≈50–70   keV. The mechanism producing the ring current is the magnetic field curvature and gradient drift, as is the case of the currents on the dayside magnetosphere. However, the ring current ions are energized in the magnetotail and move along an almost parallel magnetic field in the plasmasheet until they reach the region of the strong dipole magnetic field in the plasmasphere. Then due to the gradient and curvature of the magnetic field ions start drifting westward and electrons eastward in the ecliptic plane around the Earth. The net current flows in the westward direction in the equatorial plane. During storm time, substantial fraction of the ring current ions comes from the ionosphere.

1.5. Ionosphere

The ionosphere is a partially ionized gas occupying the range of altitudes from 80 to 2000   km above the Earth. Some books suggest considering this gas as a plasma and some do not. The reason for that discrepancy is that the density of the neutral atmospheric particles in the main regions of the ionosphere (<400   km) is 100–1000 times higher than the density of the charged particles, which makes collisions with neutrals and electrochemical reactions between different species very important participants of the ionospheric processes. Excellent review of physics and chemistry of the ionosphere is given by Schunk (1983), Schunk and Nagy (2004), and Kelley (2009).

The midlatitude ionosphere is mostly produced by the photoionization of the neutral atmosphere by EUV and X-ray radiation from the Sun. Two other important production mechanisms are (1) the impact ionization of neutrals by superthermal electrons and (2) charge exchange. The dissociative and radiative recombination balances the ionization and creates a dynamically stable configuration of charged particles with some averaged values of the main parameters.

Tables 1.1–1.3 list some of the major reactions used in photochemical models, with the reaction rate coefficients from Grubbs et al. (2018), unless noted. As common, we denote the excited state of nitrogen, N( ²D), as N# and τ i,n   =   T i,n (K)/300.

Table 1.1

Table 1.2

Table 1.3

1.5.1. Ionospheric regions

The classical daytime midlatitude ionosphere consists of four regions: D, E, F 1, and F 2, as shown in Fig. 1.2A:

• The D region occupies the altitude range 70–90km. A typical electron density here is ∼10²–10³ cm−³. The main ion species are NO+ and O2+. The main sources of ionization are solar Lyman-α, galactic X-rays, and galactic cosmic rays. This region practically disappears during the nighttime.

• The E region with the peak electron density 1–2×10⁵cm−³ is between 95 and 140km. The main ion species are O2+ and NO+. The main sources of the ionization are solar Lyman-β, soft X-rays, and UV Continuum.

• The F1 region occurs at the altitudes 140–200km. The electron density is 10⁵–10⁶cm−³. The main ion species are O+ and NO+. The main sources of the ionization are solar He II and UV Continuum (100–800Å). This region also disappears during nighttime.

• The F2 region is the region with the ionospheric density peak between 200 and 400km. The peak density in this region is 5×10⁵–5×10⁶cm−³. The main ion species are O+ and N+. The main sources of the ionization are solar He II and UV Continuum (100–800 Å).

These ionospheric parameters are typical for middle latitudes only. At high latitudes, the ionosphere strongly depends on particle precipitation from the plasmasheet, and hence, on the geomagnetic conditions. The nighttime ionosphere between the F 2 and E regions features the so-called valley, which depth and location varies with latitude (Fig. 1.2B).

Figure 1.2 (A) The average structure of the ionosphere. (B) The electron density in the nighttime ionosphere at various geographic latitudes. 

Adapted from (A) Jursa, A.S., 1985, Handbook of Geophysics and Space Environment, AFRL, National Technical Information Service, Springfield, VA 22161. (B) Titheridge, J., 2003. Ionization below the night F2 layer—a global model. J. Atm. Solar-Terr. Phys. 65, 1035–1052. https://doi.org/10.1016/S1364-6826(03)00136-6.

Because the main energy source for the ionospheric production is the radiation from the Sun, the ionosphere demonstrates a strong temporal variability depending on the position and intensity of the Sun. Thus, the density of the ionosphere at noon is more than 10 times larger than the density in the same location during local midnight. The daytime density of the ionosphere during the solar maxima can be 10 times more than the daytime density during the solar minima. Moreover, the nighttime F 2-region density during the solar maxima can be more than the daytime density during the solar minima.

The amount of solar radiation used to ionize the neutrals also depends on the latitude, and therefore, the ionosphere at low and middle latitudes is denser than the ionosphere at high latitudes. At the same time, at high latitudes (particularly, in the auroral zone), the precipitation of energetic electrons from the plasmasheet is an important source of the ionization. This source strongly depends on the geomagnetic activity in the magnetotail (e.g., substorms). As a result, the ionospheric density can be very high in the auroral zone and change significantly over relatively short time intervals and spatial scales.

1.5.2. Ionospheric conductivities

One very important distinction of the dynamics of the ionospheric plasma below 400   km altitude from the dynamics of the magnetospheric plasma is that the ionospheric plasma is embedded into the dense neutral gas. Collisions between neutral and charged particles significantly affect the electromagnetic processes occurring in the ionosphere. In particular, collisions provide a finite conductivity of the ionospheric plasma, connecting currents and the electric field in the ionosphere.

Expressions for the ionospheric conductivity are derived in a straightforward way from the equations of motion for charged particles

(1.3)

Here, index s indicates the species of the charged particles (e for electrons and i for ions), is the collision frequency between species s and neutrals, and is the velocity of the neutrals. Let us consider electrons moving without acceleration parallel to the magnetic field or without any magnetic field . Also let us assume that neutrals are stationary . In this case, Eq. (1.3) gives

(1.4)

Here, is the parallel electron conductivity. The total parallel conductivity, , includes a contribution from electrons and ions. In the plasma consisting of electrons and one species of ions only, it is

(1.5)

If the plasma consists of multiple ion species with different masses and charges, then the parallel conductivity is

(1.6)

Here, index i marks different ion species and .

If one will consider electrons moving without acceleration under some angle to the background magnetic field, and assume that neutrals are stationary , then Eq. (1.3) gives

(1.7)

It is convenient to analyze Eq. (1.7) by introducing an orthogonal coordinate system with the z axis aligned with the ambient magnetic field B. In this case, components of Eq. (1.7) become

(1.8)

or

(1.9)

Here,

(1.10)

The relations between the electric field and the total current carried by the electrons and multiple ion species can be obtained in a similar way:

(1.11)

Here, is given by Eq. (1.6), is called Pedersen conductivity, and is called Hall conductivity

(1.12)

The relation between the current and the electric field in the ionosphere in the matrix form is

(1.13)

The Pedersen conductivity is responsible for the Pedersen currents flowing in the ionosphere in the direction of the electric field. This current is carried mostly by ions. It causes dissipation of the electric field energy in the ionosphere and the ionospheric heating. The Hall conductivity is responsible for the Hall current flowing in the ionosphere in the direction perpendicular to the electric field and mostly carried by electrons.

Both conductivities result from the fact that collisions with neutrals demagnetize ions in the ionosphere, and they start to move in the direction of the electric field instead of participating in the drift. Electrons remain magnetized, and they continue to move perpendicular to with the velocity of the electric drift. Thus, collisions effectively separate electrons from ions, the ions carry Pedersen current in the direction of the electric field, and the electrons carry Hall currents in the direction perpendicular to .

The Hall and Pedersen currents arise from the peculiarities of the electric drift motion in the collisional media. They both depend on the orientation of the background magnetic and electric field relative to each other. These fields are oriented differently at high and low latitudes. At high latitudes, the magnetic field has a large angle with the ionosphere and with the electric field produced in the ionosphere. At low latitudes, the magnetic field in the south–north direction is parallel to the ionosphere and, if there is an electric field in the east–west direction in the ionosphere, then the drift pushes electrons in the vertical direction and creates a vertical component of the electric field. By considering the contribution from this field, one can get the relation between the east–west electric field and current in the ionosphere, , where is called Cowling conductivity.

Fig. 1.3 shows typical profiles of , , and with an altitude reproduced from Jursa (1985). It should be noticed here that all three conductivities are proportional to the plasma density in the ionosphere, particularly in the D and E regions, as well as on the temperature of electrons and ions. This fact has been used in many active ionospheric experiments based on changing ionospheric conductivity by heating electrons in the D and E regions with HF waves produced by powerful ground transmitters, like the High-frequency Active Auroral Research Program (HAARP) facility in Gakona, Alaska.

1.6. Electric currents

Interactions between the plasma and magnetic field carried by the solar wind with plasma and the magnetic field of the Earth's origin distort the dipole geometry of the Earth's magnetic field, and these distortions of the magnetic field generate a system of electric currents threading different near-Earth space regions. The interactive visualization of the currents in the near-Earth space is shown on the website http://meted.ucar.edu/hao/aurora/txt/x_m_3_1.php. These currents include dayside magnetosphere or Chapman–Ferraro currents, nightside magnetosphere or tail currents, cross-tail or neutral sheet current, ring current, field-aligned or Birkeland currents, and the ionospheric currents, as depicted in Figs. 1.1 and 1.4. Let us consider those in some detail.

• Dayside Magnetosphere or Chapman–Ferraro Currents. These currents occur on the dayside magnetopause. They are carried by the solar wind particles experiencing the magnetic field curvature and gradient drifts. Both these drifts cause the motion of the electrons and ions in opposite directions and produce electric current flowing in the eastward direction in the ecliptic plane on the magnetopause. This current increases the Earth's magnetic field inside the magnetopause and decreases it outside. This effect is consistent with a simple physical picture of solar wind compressing the magnetosphere on the dayside and increasing the magnetic field inside.

Figure 1.3  Example of distribution of Pedersen (σ P ), Hall (σ H ), and Cowling (σ C ) conductivities with altitude. In general, these conductivities depend on the plasma density and temperature. 

Adapted from Jursa, A.S., 1985, Handbook of Geophysics and Space Environment, AFRL, National Technical Information Service, Springfield, VA 22161.

• Nightside Magnetosphere or Tail Current. This is a system of two solenoid-like currents flowing around the magnetotail. The currents are the result of the geometry of the magnetic field in the tail, which is described by almost uniform and almost parallel magnetic field lines. Because the magnetic field in the southern magnetosphere is pointing from the Earth, and in the northern magnetosphere, it is pointing toward the Earth; the currents around the southern and northern parts of the tail should flow in opposite directions.

• Cross Tail or Neutral Sheet Current. This current flows across the tail through the neutral sheet providing the closure of the northern and southern tail currents.

Field-Aligned or Birkeland Currents. Field-aligned currents, named after their discoverer Birkeland currents, are different from other currents in the magnetosphere in several ways. First, they are carried mostly by the electrons traveling along the ambient magnetic field and originated from polarization charges at plasma boundaries and often driven by a parallel voltage between the ionosphere and equatorial magnetosphere (e.g., Arnoldy, 1974). There exist several possible mechanisms producing potential drops with different spatial characteristics and temporal behavior (Baumjohann, 1982; Lyons, 1992). It is common to distinguish large-scale, quasi-stationary Region 1 and 2 currents (Iijima and Potemra, 1978) and small-scale currents carried by Alfvén waves.

Figure 1.4 (A) A schematic illustration of the global magnetospheric current system in the Northern Hemisphere: Region 1 and 2 currents, the magnetopause (Chapman–Ferraro, black), partial ring current (black dashed), and the Pedersen currents (green). Red/blue lines indicate upward/downward current regions in the polar region and Region 1 and 2 currents. (B) A global view of Region 1 and Region 2 currents. (C) Ionospheric closure of the field-aligned currents. 

Adapted from (A) Carter, J., Milan, S., Coxon, J., Walach, M.-T., Anderson, B., 2016. Average field-aligned current configuration parameterized by solar wind conditions. J. Geophys. Res. Space Phys. 121, 1294–1307. https://doi.org/10.1002/2015JA021567. (B, C) Pictures from the website http://meted.ucar.edu/hao/aurora/txt/x_m_3_1.php.

o Large-Scale, Region 1 and Region 2 Currents. The current system connecting the magnetopause (and the solar wind) with the polar ionosphere is called the Region 1 current system. The poleward boundary of R1 currents coincides with the polar cap boundary. This region is≈100–200km wide and the current density of this current is≈1μA/m². The current system connecting the inner boundary of the plasmasheet with the equatorward part of the auroral ionosphere and the ring current with the subauroral ionosphere is called the Region 2 current system.

o Small-Scale Alfvénic Currents. Ultra-low-frequency shear Alfvén waves, generated in the magnetosphere by coupling between shear and fast MHD waves or wave–particle interactions, or by the different sources in the ionosphere, carry field-aligned currents playing an important role in the exchange of the mass, energy, and momentum between the ionosphere and the magnetosphere. These current systems have transverse sizes in the ionosphere ≈10–100km and oscillate with frequencies 0.5–100mHz. Satellite and ground-based observations suggest that these currents are closely related to the bright, discrete auroral arcs, and other nonluminous wave and plasma phenomena in the auroral and subauroral ionosphere.

Ionospheric Currents. The Pedersen and Hall currents are two main currents in the lower ionosphere. They are localized in the ionospheric D and E regions, where the corresponding Hall and Pedersen conductivities maximize. Fig. 1.4C show a schematic plot of these currents in the ionosphere.

o Pedersen Current. The Pedersen current flows in the direction of the electric field in the ionosphere. It is carried mainly by the bulk ions due to ion-neutral collisions that demagnetize ions. That is, collisions disrupt ion gyrorotation around the magnetic field thus making ions move in the direction of the electric field instead of drift.

o Hall Current. The Hall current flows in the direction of the drift. It is carried mainly by the bulk electrons in the altitude range where ions are demagnetized but electrons remain magnetized.

1.7. Aurora and auroral oval

Aurora, known as polar or northern lights (aurora borealis) or southern lights (aurora Australis), is a natural airglow in the Earth's sky. As auroras were formerly thought to be the first light of dawn, the name Aurora came from the Latin word for dawn, morning light, while Borealis was coined by Galileo in 1619 from the Roman goddess of the dawn and the Greek name for the north wind (Siscoe, 1986). Auroral emission is produced when fluxes of energetic electrons and protons precipitate along the magnetic field into the upper atmosphere below ∼130   km. The region of the most frequent occurrence of aurorae is the auroral or Feldstein oval (Feldstein, 2016). Fig. 1.5 presents examples of aurora and a snapshot of the auroral oval taken from the Polar satellite over the Northern Hemisphere. Clearly, aurorae fill in a continuous, oval-shape pattern around the geomagnetic pole replicating the shape of the Earth's magnetosphere: compressed on the dayside and stretched on the nightside. The oval maps into the plasma domains of the Earth's magnetosphere with precipitating ≤20   keV electron fluxes. The auroral oval is a natural system of reference for description of rapidly changing phenomena in the geospace.

1.8. Magnetosphere-ionosphere (MI) coupling

Magnetosphere–ionosphere coupling includes many different subjects in the global study of the near-Earth space physics, with many different complex and complicated phenomena to explore. This coupling includes various linear and nonlinear mechanisms providing the exchange of energy, mass, and momentum between the ionosphere and the magnetosphere. These mechanisms work in the same spatial domain but different geomagnetic conditions, on different spatial scales, and with different timeframes. Several very different mechanisms can produce very similar observational effects, and the same physical mechanism can produce very different observational effects under different conditions. Fig. 1.6 (courtesy of Joe Grebowsky, NASA Goddard Space Flight Center) illustrates geophysical processes manifesting coupling between different regions in the terrestrial magnetosphere, ionosphere, and atmosphere.

Figure 1.5 (Top) Examples of auroral displays: (A) Corona and (B) rayed arcs (https://en.wikipedia.org/wiki/Aurora). (Bottom) Ultraviolet (UV) image from the Polar satellite over the Northern Hemisphere. 

(Top) From Mishin, E., 2019. Artificial Aurora experiments and application to natural aurora. Front. Astron. Space Sci. 6, 14. https://doi.org/10.3389/fspas.2019.00014. (Bottom) From http://eiger.physics.uiowa.edu/∼vis/examples.

Ultra-low-frequency (ULF) shear Alfvén waves and field-aligned currents carried by those are the main participants in the electromagnetic coupling between the ionosphere and magnetosphere in the auroral and subauroral zones. One of the main mechanisms demonstrating the importance of magnetosphere–ionosphere coupling for understanding the origin and dynamics of intense ULF waves, currents, and density structures is the active feedback from the density disturbances in the ionosphere on the structure and amplitude of the magnetospheric, field-aligned currents causing these disturbances.

The basic idea of this mechanism is that the ULF field-aligned current interacting with the ionosphere changes the ionospheric conductivity by precipitating or removing electrons in the E region, and these variations in the conductivity feedback on the structure and amplitude of the incident current. First, variations in the density change the reflection of the ULF waves from the conducting bottom of the ionosphere. Second, if the large-scale electric field exists in the ionosphere, the density variations change the Joule dissipation of this field and generate some additional field-aligned current contributing to the current reflecting from this location.

Figure 1.6 Schematic plot illustrating coupling between different regions in the terrestrial magnetosphere, ionosphere, and atmosphere. 

Courtesy of Joe Grebowsky, NASA Goddard Space Flight Center.

If the ULF Alfvén wave, carrying the field-aligned current, is trapped in some resonator cavity in the magnetosphere, then the ionospheric feedback can work in a constructive way and increase the amplitude of the wave and the density disturbances in the E region, which will lead to the development of the ionospheric feedback instability (IFI) suggested first by Atkinson (1970).

IFI has been extensively studied in the global magnetospheric resonator, formed by the entire magnetic flux tube with both boundaries in the ionospheric E-region and the ionospheric Alfvén resonator formed by the E region and a strong gradient in the Alfvén velocity at the altitude 0.5–1.0 R E . It will be considered in this book in more detail in Chapter 3.4.

References

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2. Atkinson G. Auroral arcs: result of the interaction of a dynamic magnetosphere with the ionosphere.  J. Geophys. Res.  1970;75:4746.

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4. Baumjohann W. Ionospheric and field-aligned current systems in the auroral zone: a concise review.  Adv. Space Res.  1982;2(10):55.

5. Barth C, Lu G, Roble R. Joule heating and nitric oxide in the thermosphere.  J. Geophys. Res.  2009;114:A05301. doi: 10.1029/2008JA013765.

6. Carpenter D, Anderson R. An ISEE/Whistler model of equatorial electron density in the magnetosphere.  J. Geophys. Res.  1992;97:1097–1108.

7. Carter J, Milan S, Coxon J, Walach M.-T, Anderson B. Average field-aligned current configuration parameterized by solar wind conditions.  J. Geophys. Res. Space Phys.  2016;121:1294–1307. doi: 10.1002/2015JA021567.

8. Duff J, Dothe H, Sharma R. A first-principles model of spectrally resolved 5.3 mm nitric oxide emission from aurorally dosed nighttime high-altitude terrestrial thermosphere.  Geophys. Res. Lett.  2005;32:L17108. doi: 10.1029/2005GL023124.

9. Feldstein Y. The discovery and the first studies of the auroral oval: a review.  Geomagn. Aeron. Engl. Transl.  2016;56:129–142. doi: 10.1134/S0016793216020043.

10. Grubbs G, Michell R, Samara M, Hampton D, Hecht J, Solomon S, Jahn J.M. A comparative study of spectral auroral intensity predictions from multiple electron transport models.  J. Geophys. Res.: Space Phys.  2018;123:993–1005. doi: 10.1002/2017JA025026.

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14. Lui A, Hamilton D. Radial profiles of quiet time magnetospheric parameters.  J. Geophys. Res.  1992;97:19325.

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16. Mishin E. Artificial Aurora experiments and application to natural aurora.  Front. Astron. Space Sci.  2019;6:14. doi: 10.3389/fspas.2019.00014.

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Chapter 2: Plasma waves and instabilities

Chapter 2.1

Plasma waves

Abstract

This chapter gives a brief synopsis of the linear theory of plasma waves in a uniform plasma, consecutively moving from simple to complex. We start with general definitions of plasma electrodynamics, dielectric permittivity, plane waves, wave–particle resonances, and end up considering wave propagation in nonuniform plasmas and collisionless wave damping. As a rule, we do not dwell on theoretical details giving only the basic derivations and concepts, just enough for understanding theoretical and experimental results in subsequent chapters. Nonetheless, the requirement for this text is a reasonable familiarity with the contents of a typical undergraduate physics or engineering curriculum, including classical mechanics, vector algebra, Maxwell's equations, and calculus. Interested readers can find further details and rigorous derivations in comprehensive reviews and textbooks in Recommended Reading.

Keywords: Collisionless wave damping, Dispersion equations, Drifts motion, Fluid theory, Kinetic approach, Plane waves, Plasma resonances

This chapter gives a brief synopsis of the linear theory of plasma waves, consecutively moving from simple to complex. We start with general definitions of plasma electrodynamics, dielectric permittivity, plane waves, wave–particle resonances, and end up considering wave propagation in nonuniform plasmas and collisionless wave damping. As a rule, we do not dwell on theoretical details giving only the basic derivations and concepts, just enough for understanding theoretical and experimental results in subsequent chapters. Nonetheless, the requirement for this text is a reasonable familiarity with the contents of a typical undergraduate physics or engineering curriculum, including classical mechanics, vector algebra, and Maxwell's and differential equations. Interested readers can find further details and rigorous derivations in comprehensive reviews and textbooks in Recommended Reading.

2.1.1. Background

2.1.1.1. Plasma: the fourth state of matter

Conventionally, a plasma is an ionized gas whose properties are determined by the collective interaction of particles via the long-range Coulomb force, . Here is the elementary charge, is distance, and is the permittivity of vacuum. Charged particles in motion generate electromagnetic fields that affect motion of other particles, thereby creating a fast-remote response to local perturbations. Because of collective behavior of large particle ensembles, plasmas are similar to solids, while interaction in fluids occurs only between neighbors and in gases via accidental collisions. Thence, a macroscopic description of plasmas, the magnetohydrodynamic model, is similar to that of liquid metals. A symbiotic relationship between plasma particles and electromagnetic fields supports a great variety of collective motions—plasma waves. What distinguishes plasma from the other states of matter is that the waves direct the plasma state.

The collective interaction implies that the interaction volume contains a large number of charged particles. The distance beyond which electric charges are shielded is the Debye length or radius, . It is the maximum distance of charge separation over which electrons of the density, , can spontaneously move apart from ions. At larger distances, polarization electric fields maintain the plasma quasineutrality, , with no net charge in unit volume (the subscript "e and i" denotes electrons and ions, respectively). The Debye radius is found as follows. Assume that electrons in the plane layer move away from ions over the distance, . Poisson's equation, , determines the emerging electric field: . The resulting electron potential energy is . As the potential energy of electrons due to accidental charge separation cannot exceed their thermal energy, , the sought-for separation distance is . The characteristic lifetime of charge separation, , determines the Langmuir or plasma frequency, .

So, the number of particles in the Debye sphere, , should be large: . Another condition, which is important for weakly ionized plasmas, is a secondary role of electron collisions. That is, the mean free path of thermal electrons, , should be greater than . In fully ionized plasmas with the electron–ion collision frequency, , this condition is satisfied automatically at . However, in a weakly ionized plasma, the electron–neutral collision frequency, , can be dominant. In this case, the condition or defines the critical density of neutral particles, N n∗, which separates plasmas from ionized gases.

Further, unless noted, we use the density, mass, electric (magnetic) fields, and frequency, , in cm −³, kilograms, V/m (Tesla), and Hz, respectively. Temperatures are in electron volts (1   eV   =   11,605   K) to avoid repeated writing of Boltzmann's constant. Useful expressions in these units are kHz, m/s, and m. For simplicity, we assume that plasma consists of electrons and one singly charged ion species. The electron-to-ion mass ratio, , is about and in the F- and E-region ionosphere, respectively.

Henceforth, tensors are denoted by the accent, and vectors by the bold face, , where is the unit vector along the axis. For brevity, sometimes the dot or scalar product of two vectors, , will be shown simply as or . The sign, ∗, means matrix multiplication: or, omitting the summation sign, . Here the indices, , denote the mutually orthogonal axes of an arbitrary system of coordinates, . We use a standard notation: the nabla, , and Laplacian, . Usually, the overline, , means averaging over the wave or gyration period, while an angle bracket denotes averaging over the ensemble of particles.

2.1.1.2. Maxwell's equations and dielectric permittivity

Maxwell's equations connect electric ( ) and magnetic ( ) fields with the electric charge density, ρ, and current, , in continuous media. In SI units, the Maxwell system reads

(2.1.1)

Here and is the in free space permeability and speed of light. Substituting Poisson's equation (2.1.1b) into the dot product of with Ampere's law (2.1.1a) yields the continuity equation

(2.1.2)

Polarization charges emerge in a conducting medium under action of applied electric fields. The charge density, ρ, can be expressed as , with the dipole moment of unit volume or the polarization vector, . Here the summation goes over all charged j-particles in unit volume with displacements, . As follows from the continuity equation (2.1.2), the electric current in unit volume, , is simply . For weak fields,

connects to through a linear integral relation (see Landau and Lifshitz, 1960).

(2.1.3)

where the dielectric response function, , in an anisotropic medium is a tensor.

For a monochromatic wave, , the relation (2.1.3) reduces to , with the electric susceptibility tensor

(2.1.4)

As a result, we have the relation between and the electric field (Ohm's law) via the conductivity tensor, :

(2.1.5)

The electric induction, , connects to the electric field via

(2.1.6)

Here we denote the dielectric permittivity tensor as . The dot product of with Eq. (2.1.6) gives the well-known equality, .

The relations (2.1.5) and (2.1.6) determine the dispersion law for monochromatic waves

(2.1.7)

Here is the identity tensor or, in index notation, , where is the Kronecker delta: if or 0 if .

Applying the cross product of to both sides of Faraday's law (2.1.1c) with the aid of Ampere's law (2.1.1a) and equality (2.1.6) we arrive at the wave equation in the space–time domain

(2.1.8)

2.1.1.3. Plane waves

In the linear approximation in the wave amplitude, wave and plasma perturbations in an unbounded uniform medium can be represented as a superposition of plane waves:

(2.1.9)

with angular frequencies, , and wave vectors, k. Here is the complex amplitude with the phase, ψ. The exponential notation means that the real part should be taken as the measurable quantity, or

where is the complex conjugate of . As , using instead of with implies . Here means the averaging of a function, , over the wave period.

The phase velocity, , is the speed of a point of the constant phase,

(2.1.10)

which may exceed the speed of light. In any event, there is no violation of the theory of relativity, since plane waves do not carry information, unless being modulated. Modulations travel at the group velocity

(2.1.11)

As follows from the Fourier representation (2.1.9), an electrostatic wave, , where is the electric potential, has the longitudinal ("l) polarization, . However, electromagnetic, , waves with mainly the transverse polarization (t"), , may also have the longitudinal component.

2.1.1.4. Dispersion and wave equations

With the aid of the identity, , the wave equation (2.1.8) can be presented as

(2.1.12)

Although it is valid for both, "t and l," waves, it is more convenient for potential waves to substitute into the dot product of with Eq. (2.1.8) to arrive at

(2.1.13)

For plane waves, Eqs. (2.1.12) and (2.1.13) become a system of algebraic equations

(2.1.14)

Here is the refraction index. For potential waves, .

In order to have nontrivial solutions of Eq. (2.1.14), the determinant of the matrix, ), and the total of must vanish. The resulting dispersion relation for the wave normal modes (eigenmodes) reads

(2.1.15)

Here (indicated by the wavy accent) is the scalar dielectric permittivity. For potential waves, the contribution of each species to can be easily obtained from Poisson's equation as follows. The perturbed charge density is . Then, from the continuity equation (2.1.2) and dispersion law (2.1.7) we have

(2.1.15c)

We need the particles' response to the applied field in order to determine the dielectric tensor. For that, it is necessary to determine motion of particles under the action of the electromagnetic force. In general, the kinetic theory is most accurate for this task, whereas in some specific cases thermal motion is unimportant and plasmas can be treated as fluids using hydrodynamic equations.

2.1.1.5. Hydrodynamic approach

In the fluid or hydrodynamic approach the plasma response to applied fields is insensitive to their individual velocities, . Thus, they can be replaced by the mean velocity, , for each species

Here is the number of particles and .

Motion of nonrelativistic particles with the electric charge, q, and mass, m, under action of external electric, , and magnetic, , fields obeys the Lorentz force

(2.1.16)

The total derivative, , is taken at the position of the particle, . Averaging over the ensemble of particles replaces , by , taken in the reference frame of the fluid element (parcel) moving at the average speed, .

In the laboratory frame, the variation, , of any function, , during the time interval, , comprises two parts. The first one is the time variation at the given point, . The second part is the spatial variation, , between two points at the distance, , traveled by the fluid parcel during the interval, . Their total divided by gives the total or convective derivative (the Euler equation), that is, the partial time derivative   +   advective derivative

(2.1.17)

Accounting for the kinetic pressure, , gravity, , and the frictional force, , leads to the two-fluid hydrodynamic equations of motion for electrons ( ) and ions ( )

(2.1.18)

Here is the mean transport collision frequency obtained by averaging the collisional integral on the r.h.s. of the kinetic equation (2.1.21) over the j species. It determines the mean free path:

Finally, in the absence of the source (e.g., ionization) and loss (e.g., recombination) of charged particles, the continuity equation for the electron or ion fluid reads

(2.1.19)

The set of equations (2.1.18) and (2.1.19) describes the response of the electron and ion fluids. Together with Maxwell's equations and equations of state, they constitute the basis of the two-fluid magnetohydrodynamic (MHD) theory. It can be easily generalized to the multifluid MHD model by adding continuity and momentum equations for multiple ion species. If the difference between the ion and electron fluids has an insignificant effect, a plasma can be treated as a magnetized conducting fluid in terms of the one-fluid MHD model (see Section 2.1.6). The MHD approach satisfactorily describes large-scale, slow plasma processes, such as ultra-low frequency (ULF) waves ubiquitous in space plasmas.

2.1.1.6. Kinetic approach

In general, the plasma response depends on the distribution of plasma particles over velocities. The distribution function (DF), , defines the density of particles with velocities between and at position and time or at each point (r, v) in phase-space

Hereafter, stands for and for . In general, the DF is a function of the particle integrals of motion, i.e., variables that remain constant along the particle trajectory. In thermal equilibrium with the temperature, T, the distribution of each species is a Maxwellian distribution function (MDF)

(2.1.20)

Here is the kinetic energy of nonrelativistic particles. In a magnetized plasma, the temperatures along and across the magnetic field may differ, i.e., . In this case, a typical anisotropic distribution function, , is a bi-Maxwellian distribution:

Averaging over the ensemble of particles is achieved by multiplying by the DF and integrating over velocities:

For example, the average kinetic energy parallel and perpendicular to B in a bi-Maxwellian plasma is and , respectively. In cold ( ) plasmas, reduces to , where is Dirac's delta function. In this case, there is no distinction between individual particles, and the fluid theory is accurate. In warm plasmas, the fluid approach may also work under certain conditions on the wave motion (see Section 2.1.4).

The DF variation under the influence of electromagnetic fields in a uniform plasma is described by the kinetic (Vlasov) equation

(2.1.21)

Here is the collisional integral. The physical meaning of the left-hand side (hereafter, l.h.s.) of Eq. (2.1.21) becomes clear in the frame of particles moving in phase-space

It is simply the continuity (Liouville) equation for the phase density of particles along the six-dimensional phase-space trajectory.

A remark is in order concerning the procedure of taking moments of the kinetic equation. That means multiplying the Vlasov equation by various powers of v and then integrating over velocity. This procedure creates a system of differential equations for the mean quantities—the density, mean velocity, temperature, etc. However, the resulting system of differential equations is not complete because the advective derivative, , always yields a higher moment, which requires its own equation. For example, the continuity equation (zero order moment) includes the first-order moment—the mean velocity. The momentum equation (first order) includes the second-order moment via the pressure gradient, etc. In order to truncate this sequence, sometimes it is sufficient to assume either the adiabatic (no heat exchange) or the isothermal (fast heat exchange) equation of state and ignore the heat transport equation. Otherwise, the closure should be considered depending on the process under consideration (see, e.g., Braginskii, 1965).

Next, we describe some typical cases of motion of individual (test) particles necessary for derivation of the dielectric properties of plasmas (see more details in, e.g., Sivukhin (1965) and Bellan (2008)).

2.1.2. Drift in static magnetic fields

2.1.2.1. Cyclotron rotation

We consider first a uniform and stationary magnetic field, , parallel to the axis, . In the absence of the electric field, , Eq. (2.1.16) becomes

(2.1.22)

Here , with the angular cyclotron (gyro) frequency, .

As follows from the equation of motion (2.1.22), the magnetic field does no work on charged particles, so their kinetic energy and speed remain constant. Indeed, , while the perpendicular energy, , is conserved as . Thus, particles move on helical trajectories evenly