ESS 154/200C Lecture 19 Waves in Plasmas 2

Date Day Topic Instructor Due ESS 200C Space Plasma Physics ESS 154 Solar Terrestrial Physics M/W/F 10:00 – 11:15 AM Geology 4677 Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869) Date Day Topic Instructor Due 1/4 M A Brief History of Solar Terrestrial Physics CTR 1/6 W Upper Atmosphere / Ionosphere CTR 1/8 F The Sun: Core to Chromosphere CTR 1/11 M The Corona, Solar Cycle, Solar Activity Coronal Mass Ejections, and Flares CTR PS1 1/13 W The Solar Wind and Heliosphere, Part 1 CTR 1/15 F The Solar Wind and Heliosphere, Part 2 CTR 1/20 W Physics of Plasmas RJS PS2 1/22 F MHD including Waves RJS 1/25 M Solar Wind Interactions: Magnetized Planets YM PS3 1/27 W Solar Wind Interactions: Unmagnetized Planets YM 1/29 F Collisionless Shocks CTR 2/1 M Mid-Term PS4 2/3 W Solar Wind Magnetosphere Coupling I CTR 2/5 F Solar Wind Magnetosphere Coupling II; The Inner Magnetosphere I CTR 2/8 M The Inner Magnetosphere II CTR PS5 2/10 W Planetary Magnetospheres CTR 2/12 F The Auroral Ionosphere RJS 2/17 W Waves in Plasmas 1 RJS PS6 2/19 F Waves in Plasmas 2 RJS 2/26 F Review CTR/RJS PS7 – 2/22 2/29 M Final

MHD – Harmonic Perturbation If we let B = (Bcosθ, 0, Bsinθ) and k = k   where θ is angle between B and k, define Alfvén speed as

MHD – Harmonic Perturbation For cold plasma the dispersion relations are

Wave Perturbations In our mathematical development, we set k along x and the magnetic field in the x-z plane. If a wave is not compressional in this geometry, the velocity and magnetic field perturbations (u and b) must be along y (from y component of Faraday’s law). E is along a direction perpendicular to B in the ZY plane (as E=-vB). If the wave is compressional then the magnetic perturbation is along Z and j and E are along y. If we draw the waves in a coordinate system with B along Z with the wave vector in the x-z plane, then a non-compressive wave has its magnetic perturbation along Y. If we move the k vector into the Y-Z plane, the wave becomes compressional Energy flow is along Group velocity is for shear Alfven wave for fast-mode wave

Waves in Warm Plasmas In a warm plasma, a third mode appears called the slow mode. It is compressional but the field and thermal pressure fluctuations are in antiphase. The shear Alfven wave remains the same The fast and slow wave dispersion relations are

Topside Sounding Local cut-offs have zero delay X F-layer maximum Local cut-offs have zero delay Waves at higher frequencies propagate into the ionosphere Reflect at cut-off frequency Inversion techniques used to determine density profile Data from the Alouette spacecraft Horizontal axis – frequency (MHz) Vertical axis – virtual range

Harmonic Perturbation, first order quantities only: Governing Equations Harmonic Perturbation, first order quantities only: Faraday’s Law: Ampere’s Law: i.e.

Cartesian Coordinates Appleton-Hartree Assume: Then: Alternative formalism replaces j with the equivalent dielectric tensor

Polarized Coordinates Force Law: Therefore

Electrons only, parallel Therefore For parallel propagation modes split into R, L, and P modes

Electrons only, perpendicular O-mode: Therefore X-mode:

A-H Dispersion Relation

Appleton-Hartree

Quasi-longitudinal Approximation

Quasi-transverse Approximation

Appleton-Hartree Including Ions

Vlasov Equation (Collisionless Boltzmann): Liouville’s theorem: phase space density is constant along a particle trajectory, define Liouville operator L: Jeans’s theorem: Any phase distribution that satisfies the Vlasov equation is a function of the constants of the motion (ai):

Linearized Vlasov Equation The left hand side of this equation is the rate of change in f1 following an unperturbed particle trajectory. Formally Where the time integral is over the “past history” of the particle that passes through r, v at time t.

Wave Solutions Having obtained f1, the coupled Maxwell’s equations require either charge density (ρ1) [electrostatic] or current density (j1) [electromagnetic]

Harmonic perturbation Past history integral: Note that this form implicitly assumes that integrals converge at t = -

Landau Damping Parallel Propagation, electrostatic waves

Beam Generated Instabilities

Gyro-resonance Parallel Propagation, electromagnetic waves

Ion Pickup and Ion Gyro-Resonance If neutrals at rest are ionized in a flowing magnetized plasma, they are accelerated by the electric field associated with the flow so that they drift with the flowing plasma perpendicular to the field and form a ring (in velocity space) around the magnetic field. A wave grows parallel to the field resonating with the cyclotron motion. If the magnetic field is perpendicular to the flow, it is easy to visualize that the waves produced are not Doppler-shifted because they are moving perpendicular to the flow. If the magnetic field has a component parallel to the flow, the wave occurs at the frequency Doppler shifted from the ion gyro frequency by this component of the flow but the observer sees the wave near the gyro frequency because the observer is moving along the field line in the plasma flow.

Auroral Currents – Electron Gyro-resonance

Consequences of Low Density Because of low density electrons can be in direct gyro-resonance with faster-than light R-X mode waves

Resonance Condition Resonance condition must be modified to include relativistic corrections [Wu and Lee, 1979] Non-relativistic (R-mode gyro-resonance) : For non-relativistic gyro-resonance waves damped because of tail of distribution at large v Relativistic Resonance condition is an ellipse in velocity space:

Energy flow in the AKR Source Region