Advanced plasma physics E-Book

Marino Dobrowolny

• Herausgeber: Youcanprint
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2019

Beschreibung

Il libro copre argomenti fondamentali nella fisica dei plasmi quali onde nei plasmi, instabilità e processi non lineari. Dopo una approfondita descrizione teorica vengono anche evidenziate osservazioni da satelliti di questi fenomeni in plasmi spaziali quali la ionosfera terrestre, la magnetosfera e il vento solare. L’ultima parte del libro tratta le interazioni di corpi carichi con un plasma. Nel caso di misure da satellite abbiamo a che fare con un campo magnetico esterno e col moto del satellite nel plasma, Non essendoci una teoria esatta in queste circostante vengono usati metodi approssimati che tuttavia conducono a delle stime credibili della corrente raccolta. Anche per questi argomenti le teorie presentate sono raffrontate a misure di satellite e in particolare ai risultati della missione TSS (Tethered Satellite System).

Leseprobe

Index

Introduction

Chapter 1: General properties of plasmas

1.1 Plasma as the fourth state of matter

1.2 Collective effects

1.3 Some fundamental plasma properties

1.3a Electron plasma oscillations

1.3b Debye shielding

1.4 Collisions in plasmas

Chapter 2: Charged particle orbits in external fields

2.1 Orbits in a constant and uniform magnetic field

2.2 Particle drift in uniform and constant electric and magnetic fields

2.3 The case of slowly varying fields

2.4 Polarization drift

2.5 Drift due to a magnetic field gradient

Chapter 3: Levels of description

3.1 Kinetic theory

3.2 Two fluid equations

3.3 One fluid equations

3.4 Magnetohydrodynamics

Chapter 4: Waves in plasmas from kinetic theory

4.1 The dielectric permittivity tensor

4.2 Linearization of the Vlasov problem

4.3 Derivation of the longitudinal dielectric permittivity

4.4 Landau damping

4.5 Dispersion relation for longitudinal electron waves

4.6 Ion effects on longitudinal waves

4.7 Dispersion relation for transverse electron waves

Chapter 5: Magnetohydrodynamic waves

5-1 Linearization of MHD equations

5-2 Dispersion relation for MHD waves

5-3 Alfvèn waves in the solar wind

5.3a The solar wind and the interplanetary magnetic field

5.3b Observations of Alfvèn waves in the solar wind

Chapter 6: Waves in a cold magnetized plasma

6.1 Derivation of a general dispersion relation

6.2 Propagation parallel to the magnetic field

6.3 Propagation perpendicular to the magnetic field

6.4 Waves at arbitrary angle with respect to the magnetic field: ion cyclotron wave

6.5 Waves at arbitrary angle with respect to the magnetic field: high frequency waves

Chapter 7: Current driven and drift wave instabilities

7.1 Stream instabilities

7.2 Bump in tail kinetic instability

7.3 Buneman instability

7.4 Ion sound wave instability

7.5 Drift waves

7.6 Drift wave kinetic instability

Chapter 8: Macroscopic instabilities

8.1 Kelvin Helmoltz instability in nature

8.2 Instability of tangential discontinuities in hydrodynamics

8.3 Instability of tangential discontinuities in a magnetized plasma

8.4 A general theory of the Kelvin Helmoltz instability in a magnetized plasma

8.5 Kelvin Helmoltz instability in high velocity solar wind streams

8.6 Rayleigh Taylor instability

Chapter 9: Magnetic reconnection

9.1 Resistive tearing mode

9.2 Model of a plane sheet

9.3 Equations for the linear stability problem

9.4 Particle orbits

9.5 Outline of the solution of the stability problem

9.6 Derivation of a dispersion relation

9.7 Evidence for reconnection in the geomagnetic tail

Chapter 10: Non linear waves

10.1 Non linear coherent waves

10.2 The Korteweg de Vries equation

10.3 The Burger equation

10.4 Ion acoustic solitons

10.5 Non linear Alfvèn waves

10.6 Trapping of particles in a wave

10.7 Exact non linear kinetic waves: the BGK solution

Chapter 11: Collisionless shock waves

11.1 Elementary considerations on shock wave formation

11.2 Shock waves from the method of characteristics

11.3 Shock waves in collisionless plasmas

11.4 The Earth bow shock

11.5 Collisionless mechanisms of dissipation

11.6 Anomalous resistivity

11.7 Anomalous viscosity

Chapter 12: Plasma turbulence: an introduction

12.1 Hydrodynamic turbulence

12.2 The Kolmogorov spectrum

12.3 Plasma turbulence

Chapter 13: Non linear wave particle interactions

13.1 An outline of the statistical theory of weak electrostatic turbulence

13.2 Basic equations of quasi linear theory

13.3 Application to the bump in tail instability

13.4 Resonance broadening

13.5 Non linear Landau damping

Chapter 14: Non linear wave-wave interactions

14.1 Non linear instability induced by mode mode coupling

13.1 Non linear coupling of three oscillators

13.2 Application to waves

13.3 Weak turbulence in a cold plasma

Chapter 15: Magnetohydrodynamic turbulence

15.1 Non linear interactions in incompressible MHD turbulence

15.2 Energy cascade and power spectra

15.3 Alfvenic turbulence in the solar wind

Chapter 16: Basic concepts in the interaction of charged bodies with plasmas

16.1 Debye shielding

16.2 Planar sheath in front of a negative plate: Bohm’s condition

16.3 Interpretation of the Bohm’s condition

16.4 Two scale theory: sheath and pre sheath

Chapter 17: Approximate theories of current collection by charged bodies

17.1 The old theory of Langmuir and Blodgett

17.2 I-V characteristics for the spherical diode

17.3 Spherical diode with an initial velocity for the emitted electrons

17.4 Application of the spherical diode theory to a probe in a plasma

17.5 Simplified theory of a spherical probe in an isotropic plasma

Chapter 18: Alpert’s kinetic theory

18.1 The general problem

18.2 Pre sheath: the quasi neutral region

18.3 The sheath problem

18.4 Current collection

18.5 Comparison with the spherical diode theory

18.6 Comparison with the simplified theory

Chapter 19: An introduction to charged probes in a magnetoplasma

19.1 Introduction

19.2 The theory of Parker and Murphy

19.3 The theory of J. Sanmartin

19.4 Effects of plasma turbulence

19.5 Conclusive remarks

Chapter 20: Ion flow around a moving charged probe

20.1 Introduction

20.2 Ionospheric aerodynamics

20.3 Studies of the compression region for high positive potentials

20.4 A model for the distribution of ions in the ram region

Chapter 21: Simplified theory of a moving probe in an unmagnetized plasma

21.1 Introduction

21.2 Solutions of Poisson’s equation

21.3 Solution in the ram region for a specific case

21.4 I-V characteristics

Chapter 22: Charging experiments in space: TSS data and their interpretation

22.1 Introduction

22.2 Electrical configuration and payloads of the TSS missions

22.3 Derivation of I-V characteristics through on board measurements

22.4 Results on I-V characteristics from the TSS-1R mission

22.5 A new theory of particle collection from highly charged bodies

ADVANCED PLASMA PHYSICS

Titolo | Advanced plasma physics

Autore | Marino Dobrowolny

ISBN | 9788831614344

© All rights reserved by the Author

No part of this book may

be reproduced without the

prior permission of the Author.

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Introduction

Plasma physics and the word “plasma” to denote a gas of charged particles has its origin with the works of Langmuir and collaborators in the 1920’s. In particular in 1929 Tonsk and Langmuir [1] provided the famous theory of plasma oscillations based on a uniform, neutral, zero temperature plasma. A small displacement of a slab of electrons from their equilibrium position gives rise through Coulomb forces to a restoring force, As a result one obtains harmonic oscillations at a frequency usually called the Langmuir frequency.

In the same years Langmuir and collaborators [2], in their studies of gas discharges, obtained fundamental results on the interaction of a body with a plasma. Those early results are still basic to theories of current collection by charged bodies in plasmas.

Since then plasma physics has grown to become a very large subject. Enormous progress has been reached in particular during the long search for thermonuclear fusion and in the investigations on laser fusion and plasma propulsion.

It was also realized that plasma is the most common state of matter in the Universe. Moving away from the Earth, we find plasmas in the ionosphere, the magnetosphere, interplanetary space, interstellar space and intergalactic space.

Satellites, with measurements in the near Earth space and interplanetary plasma, have provided a wealth of information about the processes taking place and in many cases this information has been unique in that these processes could not be simulated on Earth laboratories.

Given all this, we understand that the physics of plasmas is a very large subject that can hardly be covered in only one book. What you will find in this book is therefore not an exhaustive coverage but rather a selection of topics according to criteria which we are now going to explain.

We start with three chapters (Chapter 1, 2 and 3) where we outline the basic concepts which indeed define a plasma, introduce the basic results of particle orbits in electric and magnetic fields and give a survey of the possible levels of description when we study plasma processes.

We then go to the subjects of waves in plasmas (Chapters 4, 5 and 6), instabilities (Chapters 7, 8 and 9), non linear processes and turbulence (Chapters 10 to 15).

Each of these subjects, if treated completely, would require a book of its own and in fact there are books addressed solely to plasma waves or plasma turbulence. The treatment here must then necessarily be limited and the limitation was for the most part that of avoiding applications having to do with thermonuclear fusion as is the case for many instabilities and non linear processes leading to plasma diffusion across magnetic field lines.

We have choosen instead to describe mainly those plasma processes which can be observed in space through measurements from satellites giving also many examples of space data. Some of the phenomena we observe in space cannot not be reproduced in a laboratory and in these cases an understanding of the phenomena comes only from the space data. One important example is the observation of magnetoydrodynamic turbulence in the solar wind.

In the last part of the book (Chapters 16 to 22) we addressed the interaction of charged bodies with plasmas and there we devoted some time to illustrate the early work of Langmuir and collaborators. In the case of an isotropic plasma we have a complete theory describing current collection to a charged probe but even here the results of the early works can give useful (and much simpler) approximations.

In cases where we have a magnetic field in the plasma or the charged object (a satellite) is flowing in the plasma, as is the case for the measurements in space plasmas, it turns out that we do not have complete analytical theories. However, also for these cases, we are able to work out approximations which lead us to a reasonable understanding of the processes.

Finally, also for this part of the book, after outlining theories of current collection by charged probes, we resort to space experiments on charging to obtain data to be compared with theories.

Chapter 1 General properties of plasmas

1.1 Plasma as the fourth state of matter

We use the word plasma to describe a wide range of neutral substances which contain however free electrons and ionized atoms interacting through long range Coulomb forces.

Plasmas can be viewed as the fourth state of matter the other three being the solid, liquid and gaseous state. Solids, liquids and gases differ from one another because of the different strengths of the forces that keep together the constituent particles. These are very strong in solids, much weaker in liquids and almost absent in a gaseous state.

A substance is in one or another of these states according to its temperature. This determines the random kinetic energy of the constituents and the state is obtained by the equilibrium between the thermal energy and the binding forces between particles. If we heat a solid or a liquid, its atoms or molecules acquire more kinetic energy until that overcomes the potential energy which binds the particle constituents. At this point we obtain what is called a phase transition between solid and liquid state or liquid and gaseous state.

If we have a gas of molecules and continue to heat it, the molecules will dissociate into atoms, and, increasing the heating, the atoms will expel through collisions the outer electrons so that we end up with an ionized gas or a plasma. The transition from the gaseous to the plasma state is however obtained through a continuous process and is not a phase transition.

In the laboratory a plasma can be generated through the processes of photoionization or electrical discharge. In the photoionization one uses incident photons with energy greater than the ionization potential of the gas atoms. For example, the ionization potential energy for the external electrons of an Oxygen atom is 13.6 eV which can be obtained by ultraviolet radiation. The Earth’s ionosphere is a natural photo ionized plasma. On the other hand in a gas discharge we apply an electric field across the gas. If the gas is slightly ionized, the electric field will accelerate the free electrons to an energy sufficient to ionize neutral atoms.

We might think that the plasma state is a rare state of matter but just the opposite is true as in fact almost all the matter in the Universe is in the plasma state.

The plasmas found in nature and those created in the laboratory are characterized by parameters which vary by many orders of magnitude. For example, the electron density in intergalactic space is ; in the interstellar medium it is between and the latter value referring to interstellar clouds. In the solar corona N ranges from to , in interplanetary space from 1 to the latter value corresponding to strong corpuscular emissions from the Sun. In laboratory devices developed in the search for controlled thermonuclear fusion it is and in an electrical discharge .

When we go to the Earth’s ionosphere the matter is only partially ionized and an important parameter is the ratio between the electron density and the neutral density. This varies according to the altitude in the ionosphere.

Temperatures are also widely varying in natural plasmas (considered to be in equilibrium). In the Earth’s ionosphere T varies from 300K to 3000K. In the solar corona , in the interstellar gas and in the intergalactic gas . For thermonuclear devices

1.2 Collective effects

In a plasma the basic interactions between the charged particles are electromagnetic. Due to the long range of the electromagnetic forces, each charged particle in a plasma interacts simultaneously with a large number of other charged particles resulting in important collective effects. It is this feature, namely the existence of collective effects, that makes a plasma different from an ordinary fluid.

A charged particle has its own electric field and it is this which acts on other charged particles following Coulomb law. Furthermore, if a charged particle is moving, it has also an associated magnetic field which in turn produces forces on other charges.

We can indeed distinguish between weakly and strongly ionized plasmas on the basis of the possible interactions taking place. A weakly ionized plasma, i. e. a plasma which contains many neutrals, is characterized mainly by charge neutral interactions which occur (contrary to the Coulomb interactions) over distances of the order of the atoms diameter. On the contrary in a strongly or fully ionized plasma the multiple Coulomb interactions are dominant and they have a long range. It is mainly to these plasmas we will refer to in this book.

The dominance of collective effects leads us to a more quantitative definition of a plasma. This is obtained by recalling that a plasma is macroscopically neutral. This means that, in the absence of external fields, in a volume of the plasma sufficiently large as to contain many particles but sufficiently small compared to the lengths of variation of physical parameters such as density and temperature, the net electric charge is zero.

The neutrality is the result of a balance between the particle thermal energy, which tends to disturb charge neutrality, and the electrostatic energy which arises due to charge separation and tends to restore neutrality. The balance, as we will see, is obtained over a distance called the Debye length which is an important parameter of the plasma. This distance is given by

where Tand are electron temperature and density. The Debye length corresponds to the distance within which the field of an individual particle is felt by other charged particles inside the plasma. In other words, as it will be shown later, the Debye length is a characteristic distance over which the charges in a plasma collect around a given charge so as to screen its electric field.

A first criterion for the definition of a plasma, and therefore for the existence of collective effects, is given by

where L is a characteristic dimension for the variation of plasma parameters such as density and temperature.

Since the shielding effect is a consequence of the collective effects inside a Debye sphere, a second criterion is obtained by requiring the number of electrons in that sphere to be high, i. e.

1.3. Some fundamental plasma processes

Plasma dynamics can be investigated in a number of ways. We will go into that in Chapter 3 devoted to the possible levels of description. Here, on the other hand, we want to address a few fundamental plasma phenomena which can be described quite simply.

1.3a Electron plasma oscillations

The first phenomenon we consider is that of electron plasma oscillations. We have already said that, in equilibrium, a plasma tends to maintain a state of macroscopic neutrality. If, for some reason, we attempt to disturb this neutrality, the plasma reacts through electron oscillations which, on the average, maintain the electrical neutrality.

Suppose that a perturbation, in the form of a small negative charge is introduced in a small spherical region. The corresponding electric field will be radial and pointing towards the center of the region. Because of the electric field electrons will be moving radially outwards and, after some time, more electrons will leave the region than is necessary to maintain electrical neutrality. As a consequence an excess of positive charge is established within the region. Then the electrons will start moving radially inward.

This movement of electrons alternatively outwards and inwards is what constitutes an electron plasma oscillation. The total charge in the spherical region averaged over one period of these oscillations will be zero and it is in this way that electrical neutrality is maintained on the average. Notice that we have not talked about ions so far but this is legitimate as the frequency of the electron oscillations, as we will see, is high so that the motion of the ions which are are more massive than the electrons can in fact be neglected.

We will now describe mathematically these oscillations and, for the sake of simplicity, we will neglect electron thermal motions and electron pressure, We superimpose on the electron density a small perturbation by writing

with

Correspondingly we also assume that the electric field and the electron velocity are small perturbations so that we can use linearized equations. Regarding the electrons as a fluid, the linearized continuity and momentum equations for the electrons can be written as

(1.1)

(1.2)

Considering now singly charged ions, the overall charge density is

Therefore

(1.3)

Taking now the divergence of eq. (1.2) and using (1.1) for we obtain

and, substituting from eq.(1.3)

(1.4)

where

(1.5)

is called the electron plasma frequency. For the solution of eq. (1.4) we can write

so that varies harmonically in time with the plasma frequency. The same can be proved for all the other perturbed quantities. Furthermore for all the perturbations there is no change in phase from point to point which means that we do not have wave propagation or, in other words, the oscillations are stationary.

Another point is that the oscillations we are talking about are electrostatic which means that we do not have magnetic perturbations. To see this let us write the two Maxwell equations for and

For a harmonic time variation we obtain

The current density is given by

and from eq. (1.2) we obtain

Substituting in the equation we arrive at

having defined

For electron plasma oscillations it is so that the above equation reduces to

(1.6)

As the curl on any gradient function vanishes identically, we can then write that

(1.7)

where ψ is a magnetic scalar potential. Substituting (1.6) in (1.7) we obtain

However the only solution of this equation which is finite at infinity is

so that we have that

i.e. there is no magnetic field associated with the space charge oscillations.

In summary the electron plasma oscillations are stationary and electrostatic. They were first discovered by Tonsk and Langmuir [1] and are also called Langmuir oscillations. Notice that, if we include pressure gradient forces, the oscillations become propagating waves. We will see that in Chapter 4 where we will recover Langmuir waves from a kinetic treatment.

Finally, we give the values of the plasma frequency for various cosmic plasmas. In the solar corona , in the ionosphere , in interplanetary space , in intergalactic space .

1.3b Debye shielding

Debye shielding is another fundamental plasma phenomenon which can be explained quite simply. Let us focus on a given charge in the plasma. We will call it a test charge and suppose it is positive and amounts to +Q. Consider now a spherical coordinate system with its center at the position of the test charge. We want to determine the electrostatic potential which is established by this charge due to the combined effect of the test charge itself and of all the other charges of the plasma which are surrounding it.

As the positive space charge attracts electrons and repel ions, the electron and ion densities near the position of the space charge will be different whereas at larger distances from the origin the electrostatic potential will vanish and hence the electron and ion densities will be equal .

We can take maxwellian distributions for ions and electrons

having assumed that the electrons and the ions have the same temperature. The total charge density will then be given by

where δ is the Dirac delta function. Thus

From Maxwell equation

we arrive then at the differential equation

Suppose now that

Then the equation simplifies to

where is the Debye length which we have already introduced. Since the problem has spherical symmetry, the electrostatic potential will depend from r only (and not the orientation of r). Therefore the equation for the potential can be rewritten as

       (1.8)

To solve completely the problem let us first consider the case of an isolated charge +Q in free space. Then the electric field due to this charge will be directed radially outwards and will be given by

Consequently the electrostatic potential of this isolated charge will be given by

When the charge is in a plasma we can imagine that in the vicinity of the test charge the electrostatic potential will be close to . Therefore it is appropriate to seek the solution of eq. (1.8) in the form

where the function F(r) must be such that F(r)→1 when r→0 . Furthermore Φ(r) will have to vanish at infinity. The differential equation that we obtain for F(r) is

and has the general solution

(1.9)

This result is known as the Debye potential as it was first derived by Debye and Hunkel in their theory of electrolytes. It shows that Φ(r) becomes much less than the Coulomb potential when r exceeds the distance . (see Figure 1.1). Hence we can say, at least approximately, that a charged test particle in a plasma interacts effectively only with particles situated less than one Debye distance away and, therefore, the same test charge has scarce influence on particles at distances larger than one Debye length. Furthermore it can be shown that the test charge is neutralized by the charge distribution which surrounds it.

Typical values of the Debye length in various plasma environments are : in thermonuclear fusion machines; in the ionosphere; in the magnetosphere; in the solar wind.

Figure 1.1: Debye potential (solid line) versus Coulomb potential (dashed line)

1.4 Collisions in plasmas

Collisions in a plasma are quite different from those occurring in a neutral gas. This is due to the fact that the electrostatic forces between charged particles have a much longer range than the forces between neutral atoms. As a consequence, rather that close collisions modifying substantially the particle velocities, we have to consider distant encounters which, taken singly, produce only a very small effect. The cumulative effects of many of these collisions do lead however statistically to large deflection angles for the colliding particle and it is in relation to these that one introduces a collision frequency.

If we denote with the collision frequency for particles of species s with particles of species s’, the total collision frequency for species s will be given by

In general the species designation is important. For example the total collision frequency for electrons , in case of unit ionic charge and comparable temperatures of ions and electrons, is related to the collision frequency for the ions by

Notice that these collision frequencies measure the frequency with which a particle trajectory undergoes a major angular change due to Coulomb interactions with the other particles. Therefore, as single collisions imply small angular deflections, the collision frequency is not the inverse of the typical time between collisions. Rather it is the inverse of the typical time needed for enough collisions to occur that the particle trajectory is deviated by 90o.

Spitzer [3] has derived for the electron collision frequency

where

is the number of particles contained in a Debye sphere. It is for the solar chromosphere and for the solar wind.

It is clear that for high temperature diffuse plasmas the collision frequency will be small while the opposite is true for dense and low temperature plasmas. It turns out that most astrophysical plasmas fall in the first category. In particular this applies to the near Earth and solar wind plasmas which are the plasmas whose properties can be analyzed by means of measurements by satellites.

To decide if we can treat a plasma as collisionless or not (with respect to ordinary Coulomb collisions), we have to compare the above collision frequency with other typical frequencies which appear in the analysis of the various waves that a plasma can sustain (see Chapters 4, 5 and 6).

One of these frequencies that we have already encountered is the electron plasma frequency

Then we have the analogous frequency for the ions

In case the plasma has a magnetic field as it is the case for astrophysical plasmas, we further have the electron cyclotron frequency

which is the frequency of the circular motion of an electron around the magnetic field B, And obviously we have the analogous frequency for the ions

It turns out that for the space plasmas we mentioned before the electron collision frequency is much smaller than all the typical frequencies we have listed and, hence smaller than the frequencies of the oscillations that the plasma can sustain. Alternatively we can compare the mean free path for particle collisions defined by

with a typical electron velocity, with the wavelengths λ of plasma oscillations or the scale length L of variation of macroscopic quantities. For the astrophysical plasmas we want to refer to, we would find that

Hence these plasmas can be regarded as collisionless and this will be so for all the examples given in this book. On the other hand, we have to take into account that what we said so far refers to ordinary Coulomb collisions. There are other collisional effects arising from the interaction of waves with plasma particles. For example, anomalous collisions are occurring in presence of an ion acoustic turbulence (see Chapter 11). These anomalous collisions can be far more important than ordinary collisions and must in fact be taken into account in relation to structures observed in space plasmas such as shock waves and discontinuities.

Chapter 2 Charged particle orbits in external fields

Single particle motions in neutral gases are trivial: a particle moves on a straight line until it hits other particles or the walls of the container. In a neutral gas there is therefore no point of keeping track of the details of single particle motion and what suffices is a statistical averaging of these motions.

The situation is different in a plasma which can be considered collisionless in most astrophysical cases. There the details of single particle dynamics are not washed out by collisions and can influence the macroscopic behavior.

Furthermore the study of single particle dynamics has a direct relevance to the kinetic theory of plasmas (see Chapter 3). There one obtains an equation for the distribution of velocities of the charged particles which, in the collisionless case, is the Vlasov equation. And it is found that any function of the constants of particle motion is a valid solution of that equation.

For these reasons, and before going to illustrate various levels of description of the plasma dynamics, it is worthwhile to analyze single particle motions in prescribed fields and in particular electric and magnetic fields. What we will see is that it is possible obtain approximate solutions for the particle velocities in arbitrary electric and magnetic fields provided that the fields are slowly changing in space and time..

2.1 Orbits in a constant and uniform magnetic field

The Lorentz force is given by

and has no component in the direction of the magnetic field. The component of velocity parallel to the field will then be constant and so will be the corresponding energy

As for the perpendicular energy

it will also be a constant because a constant magnetic field cannot do any work on the particle.

In a coordinate system moving with the velocity , the orbit of the particle is a circle whose radius (called the Larmor radius) is obtained by equating the centrifugal force and the magnetic force. The result is

The corresponding angular frequency, which is called the cyclotron frequency, is given by

It is clear that charges of opposite sign traverse the orbit in opposite directions. The instantaneous center of gyration of the particle is called guiding center and moves along the lines of force with velocity . Therefore the orbit of the charged particle is an helix.

2.2 Particle drift in uniform and constant electric and magnetic fields

The starting point is Lorentz equation

(2.1)

We suppose the fields to be uniform and independent from time. Let us take the electric field in the y direction and the magnetic field in the z direction (see Figure 2.1)

Figure 2.1: Reference system for the static uniform fields considered

Consider now an ion starting at rest from the origin. As the magnetic force is perpendicular to it does not do any work and it is only the electrical force that can change the ion energy. At the ion kinetic energy is zero as we have supposed and the total energy (kinetic plus potential energy) is also zero. Furthermore as the fiels do not depend from time, this total energy will continue to stay zero during the particle motion.

In the initial phase of motion the ion is accelerated by the electric force while the magnetic force is small at low velocities. It therefore acquires a positive velocity in the y direction. Because of this velocity the ion starts to experience also a magnetic force

which turns its trajectory in the x direction and leads to the acquisition of a velocity . In turn this velocity generates a new component of magnetic force

and this is now in the negative y direction so that it counteracts the electric force. As a consequence the ion velocity decreases, goes to zero and then changes its sign. This means that the ions reach again but, as the velocity stays positive, they will have advanced in the x direction. The ions thus keep advancing in the y direction in a sequence of semi circles. This trajectory, which is indicated in Figure 2.2, is called a cycloid.

Figure 2.2: Trajectory of a positively charged particle with zero initial velocity in crossed electric and magnetic fields

If an ion starts with a finite rather than zero velocity, it will execute Larmor orbits which take the ion into regions of both positive and negative . However the ions will have a larger gyroradius in the orbit segment then in the segment where resulting again in an average drift to the right. On the other hand electron would have a larger gyroradius in the portion of the orbit but they rotate in the direction opposite to the ions so that they also drift towards the right. The actual trajectory is shown in Figure 2.3 and it is called a trocoidal trajectory.

Figure 2.3: Trajectory of a positively charged particle released with an initial velocity in the y direction

The magnitude of the steady drift can be calculated by assuming the existence of a constant perpendicular drift in the Lorentz equation (2.1) and the averaging that equation over the cyclotron motion. This leads to

and, solving for , we obtain the so called ExB drift

(2.2)

Notice that this is independent from the particle charge and mass and also from its initial velocity, One way of interpreting the ExB drift is that of recalling from special relativity that the electric field observed in a frame moving with velocity is

so that we can say that the particle drifts in such a way as to ensure that the electric field seen in its own frame is zero.

Finally, the ExB drift analysis can be generalized to describe the effect on a charged particle of any other force orthogonal to by simply making the replacement in Lorentz equation. Such a force will cause a drift

(2.3)

If is a gravitational force, this drift varies with particle polarity and therefore induces a current in the plasma. On the other hand the ExB drift just causes a bulk motion of the entire plasma and there is no current associated with it.

2.3 The case of slowly varying fields

WE now want to address charged particle motions in fields which are slowly varying both in space and time. More precisely these changes are subject to the following hypotheses:

1) The time variation is so slow that the fields can be considered as approximately constant during each cyclotron period

2) The spatial variations are so slow that the fields can be approximately regarded as uniform over the spatial extent of one Larmor orbit

The electric and magnetic fields will be related by Faraday’s law

Furthermore we will also suppose that

in which case we are able to discard relativistic effects.

The two hypotheses above imply that we can decompose the particle motions into a fast oscillatory component, which is the gyromotion, and a slow component obtained by averaging the gyromotion itself. Correspondingly we decompose the particle position and velocity into two terms

             (2.4)

Here “L” refers to the fast Larmor motion while “GC” refers to the slow motion of the guiding center. The magnetic field as seen from the particle can therefore be expanded as

where we keep only the first term in the expansion. The electric field can be expanded in a similar way. Now we can rewrite the Lorentz equation (2.1) as

On the other hand the gyromotion corresponds to the equation

and, if we subtract that from the previous equation, we remain with

If we now average this equation over one gyroperiod, the terms linear in the Larmor motion go to zero and we remain with one equation describing only slow quantities.

(2.5)

where with “< >” we denote the average over one gyroperiod.

We now decompose inti its components parallel and perpendiculat to the magnetic field

(2.6)

Then we have that

where is the unit vector in the magnetic field direction. The treatment has been quite general so far but now we want to introduce the hypotheses of a constant magnetic field direction. In this case eq. (2.5) reduces to1

(2.7)

The component of this equation parallel to the magnetic field is

(2.8)

and the component perpendicular to the field is

(2.9)

2.4 Polarization drift

Equation (2.9) can be put in the form

(2.10)

where