1ESS200CPulsations and WavesLecture 182Magnetic Pulsations• The field lines of the Earth vibrate at different frequencies. The energy for these vibrations can come from external (exogenic) sources or internal (endogenic) sources.• Pc-1 waves (0.2 – 5s; 0.2 – 5 Hz) are produced by cyclotron resonance with ions.• Pc-2 waves (5 – 10s; 0.1 – 0.2 Hz) probably also produced by plasma resonance.• Pc-3 waves (10 – 45s; 22 - 100 mHz) produced by solar wind forcing of field-aligned resonance.• Pc-4 waves (45 - 150s; 7 - 22 mHz) produced by solar wind forcing and/or Kelvin Helmholtz instability.• Pc-5 waves (150 – 600s; 2 – 7 mHz) produced by Kelvin-Helmholtz instability or magnetopause oscillations.• Pi-1 waves (1 - 40s; 0.025 – 1 Hz) associated with downward field-aligned currents in auroral zone.• Pi-2 waves (40 – 150s; 2 – 25 mHz) produced by substorm triggered dynamics.3Field-Line Resonances• Proton beams moving back from the bow shock are unstable as they move through the incoming solar wind.• The waves produced are compressional and they push on the magnetopause, periodically generating compressional waves in the magnetosphere that can cross field lines.• Some field lines will resonate (standing wave) at these frequencies. Energy builds in the azimuthal direction of perturbation.• These resonances can be seen in ground magnetometers. They can be used to determine the mass content of the magnetic field line.4Maxwell’s Equations and Conservation Laws0))((00)(00=∇⋅+∂∂=×+=⋅∇=×∇×−∇=∂∂×+−∇=∇⋅+∂∂=⋅∇+∂∂γρμρρρputBuEBjBEtBBjpuutuut(continuity equation)(momentum equation)(Faraday’s law)(Ampere’s law)(B is divergenceless)(Ohm’s law)(conservation of specific entropy)5Linear Waves• Background quantities that can be large: B, ρ, p• Perturbed quantities that are small: b, δρ, δp, u E(=u x B), j(= x b/μ0)• Linearized equations become∇)()()(/)()(00FaradayButbmomentumBbptucontinuityut××∇=∂∂××∇+−∇=∂∂=⋅∇+∂∂μδρρδρ6One-Dimensional Cold Plasma Waves• For plane wave propagating in the x-direction oscillating quantities vary as expi(kx – ωt)• Then and and we can rewrite)()()()()/(ˆ)(000FaradayBxuxuBtbmomentumxbBxBbxtucontinuityxutxxxx∂∂−∂∂=∂∂∂∂+∂⋅∂−=∂∂=∂∂+∂∂μμρρδρikx−→∂∂ωit−→∂∂)(0)]([)(0]/))(ˆ([)(0][0FaradayBuuBkbimomentumbBBbxkuicontinuityukixxxx=−−=−⋅−=−ωμωρρωδρ7One-Dimensional Cold Plasma Waves• If we let B = (Bcosθ, 0, Bsinθ) and k = k where θ is angle between B and kwhere• Then the dispersion relations arexˆ0cossin]cos)/[(0]cos)/[(0cossin]sin)/[(22222222222=+−=−=+−xAzAyAzAxAuVuVkuVkuVuVkθθθωθωθθθω21)/(02ρμBVA=wavenalcompressioVkwaveAlfvenshearVkAA22222)/(cos)/(==ωθω8Wave Perturbations• In our mathematical development, we set k along x and the magnetic field in the x-zplane. If a wave is not compressional in this geometry, the velocity and magnetic field perturbations (u and b) must be along y. Ethen is along a direction perpendicular to Bin the ZY plane.• If the wave is compressional then the magnetic perturbation is along Z and j and Eare along y.• If we draw the waves in a coordinate system with B along Z with the wave vector in the x-zplane, 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 wavefor fast-mode waveBVAˆkVAˆ0/)(μbES×=9Waves 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 areθω2222cos/AVk =}]cos4)[({5.0/212222222222θωAsAsAsvcvccck −+±+=10Oscillations on Dipole Field Lines• Field lines are rooted in the conducting ionosphere and the conducting Earth and have natural resonating frequencies depending on the strength of the magnetic field, the plasma mass density and the length of the field line.• If the field line were straight and the density and field constant, the frequencies of resonance would be nB/2l(μ0ρ)1/2where n is the harmonic number, l is the length of the field line, B the number density and ρ the mass density.• Energy sources for these waves can be solar wind pressure variations or plasma anisotropies.• Mirror-mode grows when 1+β┴(1-β┴/βǁ)<0 where β is the ratio of plasma to magnetic pressure and ┴(ǁ) are the perpendicular (parallel) directions.11Ion Pickup and Ion Cyclotron Waves• 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 field, 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.12Waves in a Two-Fluid Plasma• Maxwell’s Laws• Conservation LawsPs= constant x (ns)γs= nsTs(polytropic law)where γsis the ratio of specific heats and Ts=kBTγs= 5/3 3D adiabatic= 2 2D adiabatic= 3 1D adiabatic= 1 isothermal= 0 isobaric)(),(),(),()(),(),()(0),()(/),(),(00002AmperetxEtxjtxBFaradayttxBtxEBofDivergencetxBPoissontxtxEt∂∂+=×∇∂∂−=×∇=⋅∇=⋅∇εμμερforcesmagneticandpressureexcludingvolumeunitperforcetheisFandnquqnjwheremomentummnFmnpBuEmquuutcontinuityunntsssqssssssssssssssssss′∑=∑=′=∇+×+−∇⋅+∂∂=⋅∇+∂∂,,)()()(0)(ρ13Waves in an UnmagnetizedPlasma• Assume ions are infinitely massive and geometry is one dimensional• Assume small perturbations, keeping only terms up to first order (linearization)• Taking time derivative of continuity equation and spatial derivative of others and substituting we