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ESS 154/200C Lecture 19 Waves in Plasmas 2 1 ESS 200C Space Plasma Physics M/W/F 10:00 – 11:15 AM Instructors: C.T. Russell (Tel. x-53188; Office: Slichter 6869) R.J. Strangeway (Tel. x-66247; Office: Slichter 6869) • Date 1/4 1/6 1/8 1/11 1/13 1/15 1/20 1/22 1/25 1/27 1/29 2/1 2/3 2/5 • • 2/8 2/10 2/12 2/17 2/19 2/26 2/29 ESS 154 Solar Terrestrial Physics Geology 4677 Day Topic Instructor M A Brief History of Solar Terrestrial Physics CTR W Upper Atmosphere / Ionosphere CTR F The Sun: Core to Chromosphere CTR M The Corona, Solar Cycle, Solar Activity Coronal Mass Ejections, and Flares CTR PS1 W The Solar Wind and Heliosphere, Part 1 CTR F The Solar Wind and Heliosphere, Part 2 CTR W Physics of Plasmas RJS F MHD including Waves RJS M Solar Wind Interactions: Magnetized Planets YM W Solar Wind Interactions: Unmagnetized Planets YM F Collisionless Shocks CTR M Mid-Term W Solar Wind Magnetosphere Coupling I CTR F Solar Wind Magnetosphere Coupling II; The Inner Magnetosphere I CTR M The Inner Magnetosphere II CTR W Planetary Magnetospheres CTR F The Auroral Ionosphere RJS W Waves in Plasmas 1 RJS F Waves in Plasmas 2 RJS F Review CTR/RJS M Final Due PS2 PS3 PS4 PS5 PS6 PS7 – 2/22 MHD – Harmonic Perturbation wdr - k rux = 0 (continuity) wru - k( x̂(b × B + d p) - Bx b) / m 0 = 0 d p - cs2dr = 0 (momentum) (energy) w b + k(Bx u - ux B) = 0 (Faraday) • If we let B = (Bcosθ, 0, Bsinθ) and k = k x̂ where θ is angle 1 2 2 between B and k, define Alfvén speed as VA ( B / 0 ) (w k ) u - ( x̂(bz sinq + ( k w ) cs2 rux / B) - b cosq )B / rm0 = 0 (Faraday) (w k ) b + B(u cosq - ux ( x̂ cosq + ẑ sinq )) = 0 (cont., mom., energy) 3 MHD – Harmonic Perturbation (w k ) by + Buy cosq = 0 (w k ) bz + B(uz cosq - ux sinq ) = 0 (w k ) u - ( x̂(bz sinq + ( k w ) cs2 rux / B) - b cosq )B / rm0 = 0 2 \ (w k ) u + x̂(VA2 (uz sin q cosq - ux sin 2 q ) - cs2 ux ) + (w k ) b cosq B / rm 0 = 0 [(w / k)2 -VA2 sin 2 q - cs2 ]ux +VA2 sinq cosq uz = 0 [(w / k)2 -VA2 cos2 q ]uy = 0 [(w / k)2 -VA2 cos2 q ]uz +VA2 sin q cosq ux = 0 • For cold plasma the dispersion relations are ( /k) 2 VA2 cos2 ( /k) 2 VA2 shear Alfven wave: ux uz 0 compressional wave: uy 0 4 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=-vB). • 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 S ( E b) / 0 V A Bˆ for shear Alfven wave V A kˆ for fast-mode wave 5 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 2 / k 2 VA2 cos 2 • The fast and slow wave dispersion relations are 2 / k 2 0.5{cs2 vA2 [(cs2 vA2 )2 4cs2vA2 cos2 ] } 1 2 6 Topside Sounding Z O X F-layer maximum • Data from the Alouette spacecraft • Horizontal axis – frequency (MHz) • Vertical axis – virtual range • 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 7 Governing Equations Harmonic Perturbation, first order quantities only: ¶/¶t º –iw Ñ º ik Faraday’s Law: k ´ E = wb Ampere’s Law: k ´ b = –im 0 j– w2 E c i.e. 2 w k – 2 E – k(k×E) = iwm 0 j 2 c 8 Cartesian Coordinates Appleton-Hartree Assume: B 0 = B0z , Then: 2 w k – 2 E x – k ^k ||E z = iwm 0 jx 2 || k = (k ^, 0, k ||) c 2 w k – 2 E y = iwm 0 j y 2 c 2 w k – 2 E z – k ||k ^E x = iwm 0 jz 2 ^ c Alternative formalism replaces j with the equivalent dielectric tensor 9 Polarized Coordinates Force Law: –imwv = q(E + v´B0) –imwv x = q(E x + v yB0) –imwv y = q(E y – v xB0) q –iw(v x ± iv y) = m (E x ± iE y) – v yW ± iv xW [W = – qB0 /m] (± i)(± i)v y = – v y Therefore q –iwv± = m E± ± iWv± 10 Electrons only, parallel 2 w k – 2 E± = iwm0 j± 2 c 2 iw pe e0 E± j± = (w ± We ) Therefore 2 w pe m 2 =1w (w ± We ) w 2 E = –iwm j z 0 z 2 c w 2pe j z = i w e0E z w 2 = w 2pe For parallel propagation modes split into R, L, and P modes 11 Electrons only, perpendicular O-mode: w 2pe j z = i w e0E z 2 w k – 2 E z = iwm 0 jz 2 c Therefore X-mode: w 2pe m =1– 2 w 2 2 w k – 2 E y = iwm 0 j y 2 c w 2 E = –iwm j x 0 x 2 c 2 iw pe e0 E± j± = (w ± We ) 12 A-H Dispersion Relation w 2pe w 2 pe 2 2 1– 2 w w m2 = 1 – w 2pe We2 2 2 1 – 2 – 2 sin q ± G w w G= w 2pe W W 4 sin q +4 1 – 2 cos 2q w w w 4 e 4 2 e 2 2 13 Appleton-Hartree 14 Quasi-longitudinal Approximation w 2 pe W W 4 sin q << 4 1 – 2 cos 2q w w w 4 e 4 m2 = 1 – 2 2 w 2 pe w 2pe 2 2 1– 2 w w w 2pe We2 2 w 2pe We 2 1 – 2 – 2 sin q ± 2 w 1 – 2 cosq w w w w 2 pe sin q << 2 1 – 2 2 w w We2 2 e 2 m2 = 1 – w 2pe w2 W 1 ± we cosq 15 Quasi-transverse Approximation w 2 pe W W 4 sin q >> 4 1 – 2 cos 2q w w w 4 e 4 2 e 2 2 w 2 pe 1– 2 w m2 = 1 – w 2 pe 2 1 – 2 cos q w 16 Appleton-Hartree Including Ions 17 Vlasov Equation Vlasov Equation (Collisionless Boltzmann): f ν f a v f 0 t Liouville’s theorem: phase space density is constant along a particle trajectory, define Liouville operator L: L f ν a v , t Lf 0 Jeans’s theorem: Any phase distribution that satisfies the Vlasov equation is a function of the constants of the motion (ai): n Lf ( La i )f / a i i 1 18 Linearized Vlasov Equation ¶f1 q(v ´ B0 ) q + v × Ñf1 + × Ñv f1 = - (E1 + v ´ B1 )× Ñv f0 ¶t m m The left hand side of this equation is the rate of change in f1 following an unperturbed particle trajectory. Formally f1 (r, v, t) = - ò dt¢ t -¥ q (E1 + v ´ B 1 )× Ñv f0 m Where the time integral is over the “past history” of the particle that passes through r, v at time t. 19 Wave Solutions Having obtained f1, the coupled Maxwell’s equations require either charge density (ρ1) [electrostatic] or current density (j1) [electromagnetic] 1 (r, t ) q f dv 1 species j1 (r, t ) q νf dv 1 species 20 Harmonic perturbation Past history integral: i ( k v) f1 q q ( v B 0 ) v f1 [E1 (1 k v / ) kv E1 / ] v f 0 m m Note that this form implicitly assumes that integrals converge at t = - 21 Landau Damping Parallel Propagation, electrostatic waves i(w - k||v|| ) f1 - q q (v ´ B0 )× Ñv f1 = [E|| (1- k||v|| / w ) + k||v||E|| / w ]× Ñv f0 m m q2 ¶f / ¶v|| \ r = -i E|| ò dv|| 0 m w - k||v|| 22 Beam Generated Instabilities 23 Gyro-resonance Parallel Propagation, electromagnetic waves i(w - k||v|| ) f1 - q q (v ´ B0 )× Ñv f1 = [E^ (1- k||v|| / w )+ k||v × E^ / w ]× Ñv f0 m m éæ k||v|| ö ¶f0 k||v^ ¶f0 ù + ÷ êç1ú 2 è ø w ¶v w ¶v iq ë ^ || û \ jr = Er ò dvv^ 2m (w - k||v|| - We ) éæ k||v|| ö ¶f0 k||v^ ¶f0 ù + ÷ êç1ú 2 è ø w ¶v w ¶v iq ë ^ || û jl = El ò dvv^ 2m (w - k||v|| + We ) 24 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. 25 Auroral Currents – Electron Gyro-resonance 26 Consequences of Low Density Because of low density electrons can be in direct gyroresonance with faster-than light R-X mode waves 27 Resonance Condition Resonance condition must be modified to include relativistic corrections [Wu and Lee, 1979] Non-relativistic (R-mode gyro-resonance) : éæ k||v|| ö ¶f0 k||v^ ¶f0 ù + ÷ êç1ú 2 è ø w ¶v^ w ¶v|| û iq ë jr = Er ò dvv^ 2m (w - k||v|| - We ) For non-relativistic gyro-resonance waves damped because of tail of distribution at large v Relativistic Resonance condition is an ellipse in velocity space: k||v|| e (1 v 2 / c 2 ) 28 Energy flow in the AKR Source Region Electron Distribution in Density Cavity -1x105 -12.0 Upgoing to Magnetosphere -5x104 -13.4 2 3 Energy Flow Loss Cone -14.9 0 1 1. Acceleration by Electric Field 2. Mirroring by Magnetic Mirror 3. Diffusion through Auroral Kilometric Radiation 5x10 4 -16.3 Downgoing to Ionosphere 1x105 -1x10 5 -5x104 0 Parl. Velocity (km/s) 5x10 4 1x105 -17.8 29