AST 553. Plasma Waves and Instabilities. Course Outline. (Dated: December 4, 2018)

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1 AST 553. Plasma Waves and Instabilities Course Outline (Dated: December 4, 2018) I. INTRODUCTION Basic concepts Waves in plasmas as EM field oscillations Maxwell s equations, Gauss s laws as initial conditions Wave equation for the electric field Ohm s law, locality, concept of dispersion Conductivity σ(t, x) as a Green s function, causality Homogeneous stationary medium (HSM) Homogeneous waves with real frequencies Convolution theorem Electric field equation, dielectric tensor ɛ(ω, k) Dispersion relation, dispersion branches (curves, surfaces) Field representation in terms of eigenwaves Wave polarization, polarization vectors Wave packets with narrow spectrum in HSM Phase velocity and group velocity Homogeneous waves with complex frequencies Laplace transform, properties; inverse Laplace transform Solving the initial-value problem using the Laplace transform Landau s solution: assumptions, procedure, quasimodes Relation between ɛ(ω, k) and σ(ω, k) Finding a complex dispersion relation: summary

2 II. MONOCHROMATIC WAVES Waves in cold nonmagnetized plasma Hydrodynamic equations at zero pressure (without derivation) Dielectric tensor and dispersion relation Electromagnetic waves: nature, ω(k), n(ω), polarization Langmuir oscillations: nature, ω(k), n(ω), polarization Waves in cold magnetized plasma Dielectric tensor and dispersion relation in Stix s notation (S, D, R, L, P ) High- and low-frequency limits, Alfvén waves General dispersion relation Biquadratic equation for the refraction index, n Branches, slow and fast modes Equation for tan θ Cutoffs and resonances Electrostatic dispersion relation Cyclotron, lower-hybrid (LH), and upper-hybrid (UH) frequencies Perpendicular propagation O wave: dispersion, polarization, cutoff X wave: dispersion, cutoffs Parallel propagation Cutoffs, resonances, polarization L and R waves, whistlers Characteristic properties of magnetospheric whistlers Characteristic parameters of tokamak plasmas (Ω i, Ω e, Ω LH, Ω UH, ω pe /Ω e, ω pi /Ω i, c/v A ) Ad hoc thermal corrections Equation of state, limitations Nonmagnetized plasma Transverse and longitudinal susceptibility Warm species, Langmuir waves 2

3 Hot species: Debye shielding, ion sound Magnetized plasma Quasiparallel propagation (qualitatively) Quasiperpendicular propagation (qualitatively) Basics of the kinetic approach Probability distribution in different variables Canonical distribution, Liouville theorem Vlasov equation, linearized Vlasov equation (LVE) Longitudinal conductivity Initial-value problem via the Laplace transform, σ k (t) Phase mixing, decay of σ k (t) at large t Maxwellian plasma response, plasma dispersion function σ(ω, k) as a double integral, integration over t for Im ω > 0 General expression for χ(ω, k) at Im ω > 0 Plasma dispersion function at Im ω > 0 Landau rule, Landau contour Nyquist theorem, stability of distributions with one and two peaks Waves in warm nonmagnetized plasma Dielectric tensor, perpendicular and longitudinal dielectric functions Electromagnetic waves with ω/k c v T (continued as a homework) Dispersion relation of longitudinal waves: general and in Maxwellian plasma Langmuir waves in Maxwellian plasma: precise kinetic calculation Landau damping: physical mechanism, applicability of the linear theory Nonlinear saturation of Landau damping, BGK modes (basics) Inverse Landau damping, bump-on-tail instability (continued as a homework) Waves in warm magnetized plasma Dielectric tensor (derivation outline) Perpendicular propagation: X wave, O wave Bernstein waves as an example of essentially kinetic waves 3

4 General electrostatic approximation Electron Bernstein waves (EBW) in the electrostatic approximation (underdense and overdense plasmas) Electromagnetic dispersion relation at ω Ω e (coupling to X wave, Dnestrovskii-Kostomarov modes) Accessibility, CMA diagram, tokamak applications of EBW Ion Bernstein waves, lower-hybrid resonance (basics) Parallel propagation: L, R, and electrostatic waves (homework) III. QUASI-MONOCHROMATIC WAVES Geometrical-optics approximation of the dispersion operator Geometrical-optics parameter Weyl expansion of dispersion operators Equation for the electric-field envelope Mode conversion (continued as a homework) Geometrical optics in the adiabatic limit Scalar-wave model, equation for the mode amplitude Action density, action flux, action equation Local dispersion relation Consistency relation Ray equations, quantum analogy Calculating the action density along rays Wave energy and momentum Dissipation Energy and momentum equations in the presence of dissipation Dissipation power in magnetized plasma: general formula Specific mechanisms of wave dissipation: Landau damping (LD), properties Transit-time magnetic pumping (TTMP), properties Cyclotron damping (CD), properties Applicability of the linear theory: CD vs LD 4

5 IV. BROAD-SPECTRUM WAVES WITH BROAD SPECTRA Quasilinear theory Chirikov criterion Equations for the distribution function and he field Conservation laws: particles, energy, momentum Development of the bump-on-tail instability, effect of boundary conditions SEMINARS Seminar session 1 Speed of light as the limiting speed in cold plasma Electromagnetic surface waves in a plasma column Beam plasma instability (cold limit) Seminar session 2 QL and QT electron waves. Whistler waves: group velocity, angular dependence Plotting the dispersion curves numerically: n(θ), ω(k, θ), multiple ions Cold-plasma diagnostics: interferometry, reflectometry, Faraday rotation Seminar session 3 Low-frequency waves in warm magnetized plasmas (MHD model) Longitudinal conductivity in Lorentzian plasma, σ(t, x) Ion acoustic waves in Maxwellian plasma Seminar session 4 Two-stream instability, Nyquist diagrams Transverse waves, Weibel instability Dielectric tensor of warm magnetized plasma Seminar session 5 Parallel propagation in magnetic field, firehose instability, whistler damping 5

6 Evolution of GO waves: rays, action conservation, comparison with WKB Quasioptical (Schrödinger) approximation Seminar session 6 Mode conversion Cyclotron heating 6