1. 1 ESS200C Pulsations and Waves Lecture 17

2. 2 Magnetic 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.

3. 3 Field-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.

4. 4 Further Examples of Resonances in Magnetic Field data

5. 5 TOROIDALPOLOIDAL Sources of Various ULF Waves HUDSON ET AL. ANN. GEOPHYSICAE, 2004

6. 6 Distribution of Pc 5 Waves Measured in Magnetic Field Data HUDSON ET AL. ANN. GEOPHYSICAE, 2004

7. 7 Distribution of Pc 5 Waves Measured in Electric Field Data Pc4 Pc5 Poloidal Toroidal WENLON LIU ET AL. IN PREPARATION, 2009

8. 8 Maxwell’s Equations and Conservation Laws (continuity equation) (momentum equation) (Faraday’s law) (Ampere’s law) (B is divergenceless) (Ohm’s law) (conservation of specific entropy)

9. 9 Linear 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/μ 0 ) Linearized equations become

10. 10 One-Dimensional Cold Plasma Waves (Dropped Thermal Pressure) For plane wave propagating in the x-direction oscillating quantities vary as exp i(kx – ωt) Then and and we can rewrite

11. 11 One-Dimensional Cold Plasma Waves (Dropped Thermal Pressure) If we let B = (Bcosθ, 0, Bsinθ) and k = k where θ is angle between B and k where Then the dispersion relations are

12. 12 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

13. 13 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

14. 14 Oscillations 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(μ 0 ρ) 1/2 where 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 ┴ /β ǁ )<0 where β is the ratio of plasma to magnetic pressure and ┴ ( ǁ ) are the perpendicular (parallel) directions.

15. 15 Ion 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 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.

16. 16 Waves in a Two-Fluid Plasma Maxwell’s Laws Conservation Laws P s = constant x (n s ) γ s = n s T s (polytropic law) where γ s is the ratio of specific heats and T s =k B T Adiabatic approximation results in γ s =(N+2)/N where N=number of degrees of freedom γ s = 5/33D adiabatic = 22D adiabatic = 31D adiabatic = 1isothermal = 0isobaric

17. 17 Waves in an Unmagnetized Plasma 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 get

18. 18 Electrostatic Waves in an Unmagnetized Plasma: Alternate Approach Assume perturbations are plane parallel waves in 1D Substitute in continuity, momentum and Poisson equations For a solution the determinant must equal zero and Group velocity Here v g =0

19. 19 Electrostatic Waves in a Warm Magnetized Plasma In a 1-D adiabatic situation, pressure gradient is Our three equations become: Using the fact that the determinant must be zero, we obtain where v e =(2T 0 /m e ) 1/2 (thermal velocity of electrons) Rewriting we obtain dispersion relation of Langmuir waves The group velocity becomes approximately

20. 20 Electromagnetic Waves in an Unmagnetized Plasma Assume there is no unperturbed magnetic field and that k·E 1 = k·B 1 = 0 After some algebra ω 2 = ω p e 2 +k 2 c 2 Index of refraction n becomes n = c/v ph = ck/ω = (1- ω pe 2 /ω 2 ) 1/2 Group velocity is Where index of refraction goes to zero, is ω=ω pe, group velocity goes to zero and wave is reflected.