Plasma Physics and Controlled Fusion

Plasma Physics and Controlled Fusion
EEC 231A Plasma Physics and Controlled Fusion N.C. Luhmann, Jr. Department of Electrical and Computer Engineering Fall 2015

Syllabus Quarter : Fall 2015
Course : EEC 231A – Plasma Physics and Controlled Fusion I Room : 2050 Academic Surge Time : Tuesday and Thursday, 8:00 AM - 9:20 AM Instructor : Professor N.C. Luhmann, Jr. Office : 3017 Kemper Hall Phone : (530) FAX : (530) Office Hours: Tuesday and Thursday 10:00 AM - 11:00 AM Website*: Final Exam: No final-just final project/report and homework *IMPORTANT: Please do not give out the password. The material has been posted solely for use in the 231 course sequence.

Fall 2015 Course Schedule Instruction Begins: Thursday, Sep 24
Instruction Ends: Friday, Dec 4 Quarter Ends: Friday, Dec 11 Veteran's Day Holiday: Wednesday, November 11 Thanksgiving Holiday: Thursday-Friday, November 26-27 20 Lectures No Final- Project/Report and Homework Course Projects and Homework: Due in class on indicated date to be posted on website

Expanded Schedule Nov.11: Veterans Day Nov.26: Thanksgiving Day
Thursday, 9/24 Tuesday, 9/29 Thursday, 10/1 Tuesday, 10/6 Thursday, 10/8: Homework #1 Due Tuesday, 10/13 Thursday, 10/15: Homework #2 Due Tuesday, 10/20 Thursday, 10/22: Homework #3 Due Tuesday, 10/27 Thursday, 10/29: Homework #4 Due Tuesday, 11/3 Thursday, 11/5: Homework #5 Due Tuesday, 11/10 Thursday, 11/12: Homework #6 Due Tuesday, 11/17 Thursday, 11/19: Homework #7 Due Tuesday, 11/24 Tuesday, 12/1: Homework #8 Due Thursday, 12/3 Friday, 12/4: Instruction Ends Quarter Ends: Friday, Dec 11 Nov.11: Veterans Day Nov.26: Thanksgiving Day

EEC231A – Plasma Physics and Controlled Fusion
3 units – Fall, 2015 Lecture: 3 hours Prerequisite: Graduate Standing in Engineering; consent of instructor Grading: Letter. Catalog Description: Equilibrium plasma properties; single particle motion; fluid equations; waves and instabilities in a fluid plasma; plasma kinetic theory

Expanded Course Description:
Plasma physics applications Particle motion in electromagnetic field; adiabatic invariants Fluid equations and diamagnetic drifts Debye shielding; plasma sheaths Maxwell’s equations in the plasma; the equivalent dielectric tensor Waves in cold and warm plasmas: CMA diagram; phase velocity surfaces; polarization and particle orbits; Fredericks and Stringer diagrams for low-frequency waves Electromagnetic waves: ordinary and extraordinary waves, Appleton-Hartree formula, microwave diagnostics. Alfven waves whistlers, e.m., cyclotron waves Electrostatic waves: Bohm-Gross waves, ion acoustic waves. two-ion hybrid waves, ion cyclotron waves Wave packets and group velocity in anisotropic media; resonance cones Diffusion in partially ionized gases Resistivity and diffusion in fully ionized gases; magnetic viscosity Magnetohydrodynamic (MHD) theory Single-fluid equations Kinetic theory; Vlasov equation and Landau damping Basic types of instabilities

Textbook: No required textbook- Instructor’s notes on class website*
Useful References: Francis F. Chen, Introduction to plasma physics and controlled fusion. Volume 1, Plasma physics, 2nd Edition Richard Fitzpatrick, Plasma Physics: An Introduction, August 1, 2014 by CRC Press, ISBN CAT# K20640 (see Texas Plasma) R.J. Goldston and P.H. Rutherford, Introduction to Plasma Physics, T.J.M. Boyd and J.J. Sanderson, The Physics of Plasmas *IMPORTANT: Please do not give out the password. The material has been posted solely for use in the 231 course sequence.

Supplementary Books The primary sources for the course will be the notes and pdf copies on the course website. The following are also useful. In particular, the books by Chen, Bellan, and Stix are highly recommended. 1. Krall and Trivelpiece, Principles of Plasma Physics (1973), McGraw-Hill. 2. Boyd and Sanderson, Plasma Dynamics (1969), Barnes & Noble. 3. Chen, Introduction to Plasma Physics (1974), Plenum Press 4. Stix, Waves in Plasmas (1992), American Institute of Physics, NY ISBN 5. Mikhailovsky, Theory of Plasma Instabilities (1974) NY: Consultant Bureau. 6. Heald and Wharton, Plasma Diagnostics with Microwaves (1965), John Wiley & Sons. 7. Bekefi, Radiation Processes in Plasmas (1966), John Wiley & Sons. 8. Bernstein and Trehan, Nuclear Fusion 1, 3 (1960). 9. Stringer, Plasma Physics 5, 89 (1963). 10. Allis, Buchsbaum and Bers, Waves in Anisotropic Plasmas (1963), MIT Press. 11. Davidson, Theory of Nonneutral Plasmas (1973), Benjamin Cummings/Addison Wesley Longman. 12. Griem, Methods of Experimental Physics, Vol. 9, Part A (1970), Academic Press

13. Plasma Physics, Ed. by B.E. Keen, Lecturers from the Culham Plasma Physics Summer
Schools (1974), London Institute of Physics. 14. Sagdeev and Galeev, Nonlinear Plasma Theory (1969), NY: Benjamin. 15. Kadomtsev, Plasma Turbulence (1965), Academic Press. 16. Simon and Thompson, Advances in Plasma Physics Vol. 6 (1976). [Parametric Instabilities References], Interscience. 17. Nishikawa, J. Phys. Soc. Japan 24, 916, 1152 (1968) 18. Allen and Phelps, “Waves and Microinstabilities in Plasmas-Linear Effects” Reports on Progress in Physics 40, 1305 (1977). 19. Franklin, “Microinstabilities in Plasmas-Nonlinear Effects,” Reports on Progress in Physics 40, 1369 (1977). 20. Porkolab and Chang, “Nonlinear Wave Effects in Laboratory Plasmas: A Comparison Between Theory and Experiment,” Review of Modern Physics 50, 745 (1978) 21. Luhmann, Laser Handbook Vol. 5, 455 (1985), North Holland-Physics Publishing (Amsterdam). 22. Luhmann, Rev of Sci. Instruments (Invited Article), “Instrumentation for Magnetically Confined Fusion Plasma Diagnostics,” 55, 279 (1984). 23. Luhmann, Millimeter and Submillimeter Wave Diagnostic Systems for Contemporary Fusion Experiments, “Diagnostics for Contemporary Fusion Experiments” 135, (1991), Editrice Compositori, Bologna). 24. P.M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press (6 April 2006)

25. Lackner and Lindinger, Plasma Physics and Controlled Fusion, Vol
25. Lackner and Lindinger, Plasma Physics and Controlled Fusion, Vol. 34, Number 13, December 1992 (1992), Pergamon Press. 26. Schmidt, Physics of High Temperature Plasmas An Introduction (1966), Academic Press. 27. Birdsall and Langdon, Plasma Physics via Computer Simulation (1985), McGraw-Hill, Inc.. 28. Teller, Fusion, Vol. 1, Part A (1981), Academic Press. 29. Teller, Fusion, Vol. 1, Part B (1981), Academic Press. 30. Button, Infrared and Millimeter Waves, Vol. 2 (1979), Academic Press. 31. Montgomery and Tidman, Plasma Kinetic Theory (1964), McGraw-Hill, Inc.. 32. Stix, The Theory of Plasma Waves (1962), McGraw-Hill, Inc.. 33. Northrop, The Adiabatic Motion of Charges Particles (1963), John Wiley & Sons, Inc.. 34. Chandrasekhar, Plasma Physics (1960), University of Chicago. 35. Cambel, Plasma Physics and Magnetofluidmechanics (1963), McGraw-Hill, Inc.. 36. Sitenko, Electromagnetic Fluctuations in Plasma (1967), Academic Press. 37. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (1967), Pergamon Press Ltd..

37. Kunkel, Plasma Physics in Theory and Application (1966), McGraw-Hill, Inc..
38. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (1964). 39. Vandenplas, Electron Waves and Resonances in Bounded Plasma (1968), John Wiley & Sons Ltd.. 40. Akhiezer, Collective Oscillations is a Plasma (1967), Pergamon Press Ltd.. 41. Ta-You Wu, Kinetic Equations of Gases and Plasmas (1966), Addison-Wesley. 42. Glasstone and Lovberg, Controlled Thermonuclear Reactions (1960), D. Van Nostrand Company, Inc.. 43. Rose and Clark, Jr., Plasmas and Controlled Fusion (1969). 44. Muraoka and Maeda, Laser-Aided Diagnostics of Plasmas &Gases (2001), IOP Publishing 45. Plasma Science, from Fundamental Research to Technological Applications (1995), National Academic Press. 46. Hutchison, Principles of Plasma Diagnostics (1987), Cambridge University Press. 47. White, Theory of Tokamak Plasmas (1989), Elsevier Science Publishers B.V.. 48. Longmire, Elementary Plasma Physics, Vol. IX (1963), John Wiley & Sons, Inc.. 49. Van Kampen and Felderhof, Theoretical Methods in Plasma Physics (1967), John Wiley & Sons, Inc.. 50. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Second Edition (1979), Plenum Press. 51. Trivelpiece, Slow-Wave Propagation in Plasma Waveguides (1967), San Francisco Press, Inc.. 52. Liboff, Introduction to the Theory of Kinetic of Equations (1969), John Wiley & Sons, Inc.. 53. Ichimaru, Basic Principles of Plasma Physics (1973), WA Benjamin, Inc..

54. Chu and Hendel, Feedback and Dynamic Control of Plasmas, Number 1 (1970), American
Institute of Physics, Inc.. 55. Delcroix, Plasma Physics (1965), John Wiley & Sons Ltd.. 56. Leontovich, Reviews of Plasma Physics, Vol. 5 (1970), Consultants Bureau. 57. Uman, Introduction to Plasma Physics (1964), McGraw-Hill, Inc.. 58. Holt and Haskell, Foundations of Plasma Dynamics (1965), Macmillan Company. 59. Davidson, Methods in Nonlinear Plasma Theory (1983), John Wiley & Sons, Inc.. 60. Sturrock, Plasma Physics (1994) Cambridge University Press. 61. Plasma Physics, by International Atomic Energy Agency (1965). 62. Leontovich, Reviews of Plasma Physics, Vol. 2 (1966), Consultants Bureau. 63. Swanson, Theory of Mode Conversion and Tunneling in Inhomogeneous Plasmas (1998) John Wiley & Sons, Inc.. 64. Hazeltine and Meiss, Plasma Confinement (1992), Addison-Wesley Publishing Company. 65. Scott, Gorini, and Sindoni, Diagnostics for Experimental Thermonuclear Fusion Reactors (1996), Plenum Press. 66. Physics of Alternative Magnetic Confinement Schemes, by ISPP-9, (1990), Editrice Composition. 67. Wakatani, Stellarator and Heliotron Devices (1998), Oxford University Press. 68. Brambilla, Kinetic Theory of Plasma Waves Homogeneous Plasma (1998), Clarendon Press. 69. Goldston and Rutherford, Introduction to Plasma Physics (1995), IOP Publishing Ltd.. 70. Brown, Basic Data of Plasma Physics, 1996 (1967), MIT Press.

71. Lindl, Inertial Confinement Fusion (1998), Springer-Verlag New York, Inc..
72. Seshadri, Fundamentals of Plasma Physics (1973), American Elsevier Publishing Company, Inc.. 73. Diagnostics for Contemporary Fusion Experiments, by ISPP-9, (1991), Editrice Compositori. 74. Kruer, The Physics of Laser Interactions (1988), Addison-Wesley Publishing Company, Inc. 75. Plasma 2010: An Assessment of and Outlook for Plasma Science , National Research Council, 2007 76. Frontiers in High Energy Density Physics: The X-Games of Contemporary Science , National Research Council, 2006 77. Fundamentals of Plasma Physics - 3rd edition, J. A. Bittencourt, Springer-Verlag New York, 2004 78. Plasma Physics and Fusion Energy, Jeffrey P. Freidberg, Cambridge University Press, 2007

H. J. de Blank

Typical temperatures and densities of astrophysical and laboratory plasmas.
H. J. de Blank

Plasma Parameters

Kip Thorne: T d e h n t i s - y p m a r u g w c , l f o b v and the densities and temperatures of specific examples of plasmas are indicated by dashed lines. as a nonrelativistic plasma. The boundaries of the plasma regime are marked with cross-hatching, Kelvin; and at the right kBT is plotted in units of electron volts. density ρ is plotted at the bottom. The temperature T is plotted at the left in units of degrees The number density of electrons n is plotted horizontally at the top, and the corresponding mass Blandford and Thorne, Applications of Classical Physics,

[From National Research Council Decadal Review, Plasma Science: From Fundamental Research
to Technological Applications (1995) [explanations added] .

Landmarks in the plasma universe
Boyd and Sanderson

Plasma Science: Advancing Knowledge in the National Interest (2007), National Research Council,
!

1.0 Preliminaries The Bellan textbook employs SI units. In addition, much of my notes employ CGS-Gaussian units so you need to be aware of the differences. In the following, we provide a brief overview of the important parameters associated with plasmas, together with a review of wave propagation phenomena in dispersive media.

Maxwell's Equations: CGS-Gaussian units 1. 2. 3. 4. where
1. 2. 3. 4. where for linear, isotropic media.

Maxwell's Equations: SI units

Saha Equilibrium The amount of ionization to be expected in a gas in thermal equilibrium is given by the Saha equation: where nn is the neutral density, ni is the ion density, T is the temperature in degrees Kelvin and Ui is the ionization energy of the gas. ni/nn is a balance between the rate of ionization (T dependent) and the rate of recombination (density dependent). As shown in the figure on the next slide, the fractional ionization is a very sensitive function of temperature and density. John Howard

(Ui=13.6 eV) as a function of temperature and number density.
Curve showing the dependence of the fractional ionization of a hydrogen (Ui=13.6 eV) as a function of temperature and number density. John Howard

Example: Saha Equation
Solving Saha equation

Example: Saha Equation (II)

Saha Continued

Characteristic Frequencies in a Plasma
Plasma frequency: The plasma frequency is given by where n0 is the particle density, e the charge, and m the mass. This is the frequency at which particles, displaced from their equilibrium positions, will oscillate due to the electrostatic coulomb restoring force. For electrons, we have: Thus, for a typical laboratory plasma with ne1012cm-3, we have fpe9 GHz. In the case of plasmas of magnetic fusion interest, this increases to GHz.

Plasma (Langmuir) oscillations
initial displacement of a bulk of electrons leads to charge separation, resulting electric field yields a restoring force, electron motion reverses, bulk of electrons becomes displaced in opposite direction, . . . (= angular plasma frequency of electrons).

Idealized Depiction of the Displacement of Electrons Relative to Ions Which Occurs During Plasma Oscillations

Plasma Oscillations

H. J. de Blank

Cyclotron frequency: The cyclotron frequency is given by ,
The cyclotron frequency is given by , where B is the magnetic field strength. This is the frequency at which charged particles gyrate about magnetic field lines. For electrons, we have . Again, focusing on plasmas relevant to magnetic fusion, this becomes GHz.

Uniform B field ― Cyclotron Gyration
Lorentz Force = centrifugal force Haimin Wang

Uniform B field ― Cyclotron Gyration (cont)
Frequency of Simple Harmonic Motion Haimin Wang

H. J. de Blank

Characteristic Lengths in a Plasma
There are a number of characteristic lengths associated with plasmas. Several of them are listed below. Debye length: The Debye length is given by This is the distance over which a stationary test charge will alter the electrostatic potential in a plasma. For example, electrons are attracted toward a test ion, and other ions are repelled. The charges within a Debye sphere centered about the test charge reorient themselves to shield more distant particles from the field. We can write this as and is cm for laboratory plasmas, and cm for fusion plasmas.

Debye Shielding and Debye Length
plasma shields out electric potentials, and its characteristic length is Debye length  o x + - plasma Poisson’s eqn Debye length and quasi-neutrality Boltzmann relation for electrons For Te=2eV and ne=2x1018m-3, D =7. 4 m Quasi-neutrality will be satisfied only when D << L Collective behavior means nD3 >> 1

Debye Length and Shielding
Plasma can shield out electric potential applied to it. If T=0 will attract same amount of ― from plasma. will attract same amount of + from plasma Shielding is perfect. If T 0 Thermal motion may make some local charge imbalance. The scale of this imbalance is called the Debye length.

Debye Shielding: Intuitive Approach
We have established that plasmas are neutral on timescales of order ω−1pe. An electron travelling at thermal speed vthe moves a distance Δx = vthe/ωpe in this time. This gives the distance scale over which any charge imbalance in the plasma is neutralized or shielded. The rigorous result is that free charges in the plasma are shielded out in a Debye length λD = vthe/(√2ωpe): where kB is the Boltzmann constant and we have anticipated a result relating vthe and the electron temperature Te for a plasma in thermal equilibrium. The Coulomb force between particles in a plasma is thus shielded by the mobility of free charges, and so is reduced in range from ∞ to ∼ λD. The hotter the particles, the more mobile they are and the greater is the range.

Debye Shielding: Intuitive Approach (cont)
When the density ne of electrons is high, the Debye length shrinks. The Debye shielding picture is valid provided there are enough particles in the charge cloud: A major consequence of this result is that plasma theory now becomes tractable—we can treat the plasma as a collection of independent fluid elements described by a distribution function which evolves under the influence of local forces and collisions. If the plasma dimension L is much larger than λD then charge perturbations or potentials are shielded out within the plasma and it remains quasi-neutral. However, because of the statistical nature of thermal motion, it is possible for small charge imbalances with associated potentials φ ∼ kBTe/e to arise spontaneously.

[Note: these equations give us the so-called plasma approximation]
The number of particles in a Debye sphere is nD3. We have a plasma if this number is large: [Note: these equations give us the so-called plasma approximation] Using the thermal speed , we see that Therefore, a particle traverses a Debye sphere in a time p-1: (characteristic time).

H. J. de Blank

Physics 142, Adrian Down

Larmor radius (gyroradius):
The Larmor radius rL is given by , where is the velocity perpendicular to the magnetic field B. This is the radius of the orbit of a particle about a magnetic field line. For electrons, we have: which is x10-3 cm for plasmas of magnetic fusion interest.

Electrons and ions spiral about the lines of force. The ions are
left-handed and electrons right. The magnetic field is taken out of the page

Gyro-motion in action

H. J. de Blank

M. Van Schoor

H. J. de Blank

M. Van Schoor

Density gradient scale length:
Plasma skin depth: The collisionless skin depth can be expressed as This is the attenuation length for a wave beyond cutoff. Density gradient scale length: In the case of an inhomogeneous plasma, the density gradient scale length is given by

Collisions Because of the long range Coulomb force, collisions between charged particles (in the sense that the particle momentum is significantly altered) are more likely the result of a large number of glancing encounters each of which slightly perturbs the particle trajectory. The plasma “collisionality” often refers to a dimensionless measure such as ν/ωT where ν is the actual collision frequency and ωT is the system transit frequency. An alternative and more intuitive measure is the ratio λmfp/L ∼ ωT /ν λmfp ≡ vth/ ν where defines the mean free path between collisions. A “collisionless” plasma satisfies the condition λmfp >> L. This condition can arise in hot plasmas for reason that the Coulomb collision frequency varies as whereupon where John Howard

Collision mean-free path:
The collision mean-free path mfp is related to the collision frequency ei through the relation Clearly, the mean-free-path measures the typical distance a particle travels between “collisions” (i.e., 90◦ scattering events). A collision-dominated, or collisional, plasma is simply one in which λmfp ≪ L, where L is the observation length-scale. The opposite limit of large mean-free path is said to correspond to a collisionless plasma.

Collision frequency: The electron-ion collision frequency is given by:
In the above, Z is the ion charge state, K is Boltzmann's constant, and ln  is the Coulomb collision parameter which is  10 for plasmas of interest. This assumes values of  106 sec-1 for typical laboratory plasmas, and 103 sec-1 for magnetic fusion plasmas.

Left: Schematic showing particle displacements in direct Coulomb
collisions between like species in a magnetized plasma. Right: Collisions between unlike particles effectively displace guiding centers. Because of the mass disparity, electrons bounce off almost stationary ions and execute a random walk of step length rL. In general, ions only move slightly, but very often. However, conservation of momentum implies that the diffusion rate for ions and electrons is the same (no charge separation, no electric fields).

Representative densities, temperatures and magnetic field
strengths together with derived plasma parameters in a variety of environments. Values are given to order of magnitude as all of these environments are quite inhomogeneous. Version K.pdf, March 28, 2007.

Some plasmas and their key parameters
Some plasmas and their key parameters. In all these examples the plasma parameter is large and the plasma is characterized by Coulomb interaction potentials that are weak compared with particle kinetic energies. John Howard

H. J. de Blank

Greg Hammett

H. J. de Blank

Three Criteria for the Existence of the Plasma State
Λ>> 1 (plasma parameter large-many particles in a Debye sphere)) 2. λD << L (small Debye length) 3. ωpeτen > 1 (low neutral collisionality)

Plasma physics is difficult – but why?
Combination of statistical physics and electromagnetism • Large variety of scales, from electrons to ions to fluids • A great variety of plasma descriptions must be mastered – single particle motion – Vlasov theory (electrons and ions described by distribution functions) – fluid descriptions (e.g., magnetohydrodynamics) – various hybrids of these • Collisions or their absence

Spatial/Temporal Scales in a Plasma
atomic mfp electron-ion mfp skin depth system size tearing length • Huge range of spatial and temporal scales • Overlap in scales often means strong (simplified) ordering not possible ion gyroradius debye length electron gyroradius Spatial Scales (m) 10-6 10-4 10-2 100 102 pulse length Inverse ion plasma frequency current diffusion inverse electron plasma frequency confinement ion gyroperiod Ion collision electron gyroperiod electron collision 10-10 10-5 100 105 Temporal Scales (s) Ravi Samtaney

Equations: A plasma is comprised of a collection of charged particles immersed in electromagnetic fields. We therefore require Maxwell's equations, together with an appropriate set of fluid or kinetic equations, to describe its behavior.

Maxwell's Equations: 1. where  = niqi+neqe with ne the electron number density (cm-3), and qe the electron charge (-e) where e = 4.8 x stat coulombs. 2. 3. where At this point we must choose a model to describe the plasma. For much of this course, we will use a simple picture in which the plasma is treated as two interpenetrating charged fluids, which are coupled through these charges. We will find that this is sufficient to describe the bulk of wave phenomena. To obtain effects such as Landau damping (collisionless damping), we will later employ a kinetic theory approach.

Plasma Equations - Two-Fluid Theory
Momentum equation: The momentum equation is given by with j=i,e (fully ionized, single ion species plasma). Here, is the pressure tensor for each fluid, the fluid velocity, and cj is the collision frequency for species j. In some cases, we will extend this to include additional ion species as well as neutral particles. The pressure tensor is represented, in general, by a 3 x 3 matrix. It is often useful to decompose into two tensors - one traceless and the other diagonal. We will often ignore the off-diagonal (shear) terms of the pressure tensor. For the case of a spherically symmetric (isotropic) plasma, we have P11=P22=P33=P, and This occurs, for example, if particle collisions are sufficiently frequent. In the presence of a magnetic field, one generally has

Continuity equation: Equation of State:
Particle conservation is described by the continuity equation, which is given by Equation of State: As you can see, the above set of equations is not closed. As we will see, we have sixteen equations for the eighteen unknowns ( ). We simply have used the first two moments of the Boltzmann equation. The next moment is the heat flow equation. A common practice is to truncate this infinite set of equations by replacing the heat flow equation with an equation of state which relates and n.

Common Assumptions 1. No collisions c = 0. 2. Isotropic plasma (no viscosity): , where Pj is a scalar. However, in the presence of a magnetic field, we will often have different pressure along, and perpendicular to, the field. Then 3. Simple equation of state: Then if we have an isothermal system  = 1 (infinite heat conductivity). For an adiabatic system (zero heat conductivity), where N is the number of degrees of freedom. Therefore,  = 5/3, 2, 3, depending upon the number of degrees of freedom. 4. Quasi-neutrality: nine for Z=1 (Znine, in general). This is true for dc and low frequency perturbations, but breaks down for high frequency electron waves due to ion inertia. This is valid only for large scale lengths compared to a Debye length. 5. Uniform plasma at rest and in equilibrium: [The subscript “0” will be explained below.]

Small Amplitude Oscillations (Linear Waves)
We assume that all quantities can be expressed as an equilibrium term and a small perturbation term: We will substitute these into Maxwell's equations and the fluid equations, and only retain the first order terms. We will neglect all second order terms. For , we will then neglect the term in the momentum equation. This will turn out not to be the case for an inhomogeneous plasma. Here, in equilibrium, we have neglecting For , we have However, for an inhomogeneous plasma, we get both and diamagnetic drifts.

Linearized Set of Equations
Making use of the small amplitude assumption, we can linearize the coupled fluid-Maxwell equations, as shown below. 1. Here, we have omitted the subscript “1” on perturbed quantities. 2. 3. 4. 5. 6.

But taking the divergence of Eq. (6), we have:
We see that we have 16 equations above. However, we have only 14 unknowns Two of the equations are redundant. To see this, take the divergence of Eq. 5. Then, taking /t of Eq. 3, we have: But taking the divergence of Eq. (6), we have: so and, from above, But this is just the difference of the electron and ion continuity equations;  Poisson's equation is superfluous. We can then omit for , and Poisson's equation. Alternatively, we will sometimes find it convenient to use Poisson's equation instead of one of the continuity equations.

Before proceeding with our study of plasma waves, let's quickly look at electromagnetic wave propagation in vacuum. In this case, , and we have: and so that Assuming a plane wave solution we have or which implies 2 = c2k2. This is the dispersion relation for electromagnetic waves in vacuum. Note that in the above we made use of The complexity of electromagnetic theory arises from the consideration of boundary conditions, waveguides, resonators, antennas, etc. Plasmas are a completely different story. A plasma is anisotropic in the presence of a magnetic field In addition, the particles can have resonant motions. Real plasmas are also inhomogeneous, as well as dissipative and nonlinear. Most of our work in this course will be concerned with plane wave analysis, since only a small number of boundary value problems have been studied to date.

Method of Attack There are two possible approaches to plasma wave analysis. The first is to assume a plane wave solution, and to obtain a general dispersion relation from our set of equations. We will then obtain a tensor dispersion relation. This is the approach of Stix. His work contains all the necessary information, and you can obtain the various waves by looking at limiting cases. The physical mechanism for each wave is not so obvious, however, in this approach. The second approach is to look at simple solutions first. This will give us the so-called principal modes, waves  and  to , and with or or Then, to find out what happens for other propagation directions or other frequencies, we will go back to the general theory. Before proceeding with derivations of the dispersion relations for various plasma waves, let us introduce some notations and a list of the various modes at which we'll be looking.

Notation: Parallel: the vector of the waves lies along .
Perpendicular: Longitudinal (Electrostatic): If the wave is longitudinal, vanishes, and Transverse (Electromagnetic): and finite. High, low or intermediate frequency: Compared to electron and ion plasma and cyclotron frequencies.

Bohm-Gross dispersion relation
Table of Waves Mode Dispersion Relation Characteristics Langmuir Oscillation: Langmuir Wave: Bohm-Gross dispersion relation Electromagnetic (Light) Wave: high frequency Ion Acoustic Wave: for low frequency

(have neglected ion dynamics, so will not see L-wave resonance)
Alfvén: low frequency Langmuir: Same as Electromagnetic (Light) Wave: high frequency (have neglected ion dynamics, so will not see L-wave resonance) Ion Acoustic: Same as case Electromagnetic: Electrostatic ion Cyclotron: where

Low frequency, magnetosonic
Extraordinary: Low frequency, magnetosonic Intermediate frequency, lower hybrid theory Upper hybrid frequency High frequency, upper hybrid

Plasma Physics and Controlled Fusion

Similar presentations

Presentation on theme: "Plasma Physics and Controlled Fusion"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Plasma Physics and Controlled Fusion

Similar presentations

Presentation on theme: "Plasma Physics and Controlled Fusion"— Presentation transcript:

Similar presentations

About project

Feedback