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Landau Damping

Landau Damping. Tim Larson Physics 312 March 2, 2005. Lev Davidovich Landau. Review. Linearized Vlasov & Poisson equations in 1D: These yield the dispersion relation. (no collision term!). Landau, J. Phys. (Moscow) 10 , 25 (1946). Review.

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Landau Damping

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  1. Landau Damping Tim Larson Physics 312 March 2, 2005 Lev Davidovich Landau

  2. Review • Linearized Vlasov & Poisson equations in 1D: • These yield the dispersion relation (no collision term!) Landau, J. Phys. (Moscow) 10, 25 (1946)

  3. Review • Landau stressed that this problem must be treated as an initial value problem. As such, k is assumed real. If  is then complex, its real part will give the oscillation frequency while its imaginary part will give the rate of damping (or growth). • Derived in the limit of high frequency and low damping, or equivalently • v = /k >> vthermal = (KT/m)1/2 Chen, Introduction to Plasma Physics (Plenum, New York, 1974)

  4. Points of Note • Purely mathematical result • Assumes no collisions • 15 years before the standard physical derivation given by Dawson • Doubts remained Dawson, Phys. Fluids4, 869-874 (1961)

  5. Experimental Confirmation • Cylindrical column of H plasma, length=230cm, radius=5.2cm • T=6.5eV or 9.6eV, n~2108cm-3 • rD≈1mm, mean free path ≈ 1000m for election-ion collisions, ≈40m for election-neutral collisions • Magnetic field B ≈183G • Boundary value problem, implies  real and k complex Malmberg & Wharton, Phys. Rev. Letters17, 175 (1966)

  6. Experimental Confirmation • First plot: upper curve is the log of received power. Lower is interferometer output. • Second plot: measured & calculated dispersion relations. Malmberg & Wharton, Phys. Rev. Letters17, 175 (1966)

  7. Experimental Confirmation • Im(k)/Re(k) vs. (v/vthermal)2 • For a Maxwellian distrubtion, Landau theory predicts Im(k)exp(-( v/vthermal)2) just as for Im(), in agreement with observations. Malmberg & Wharton, Phys. Rev. Letters17, 175 (1966) Chen, Introduction to Plasma Physics (Plenum, New York, 1974)

  8. Another Experiment • Almost same experiment, but different scale, done at Stanford. • T=0.18eV, n=6.9107 /cc • p/2 fitted to 68MHz Derfler & Simonen, J. Appl. Phys. 38, 5018 (1967)

  9. Another Experiment Dispersion Relations Derfler & Simonen, J. Appl. Phys. 38, 5018 (1967)

  10. Numerical Studies • Most straightforward approach is to simulate charged particles moving in an electrostatic field. • Damping is seen for some initial conditions, but not for others. Mixed behavior is also observed. • In this example, electric field amplitude is plotted against time. Decay by two orders of magnitude is seen at a rate differing from theory by less than 7%. The rise after t=45 is due to nonlinearities. Birdsall & Langdon, Plasma Physics via Computer Simulation, (McGraw-Hill,San Francisco, 1985)

  11. Phase Portraits • Phase portraits at • t= 2, 16, 30, 40, 90 • Initial velocities are • v=0.8, 0.9, 1.0 Birdsall & Langdon, Plasma Physics via Computer Simulation, (McGraw-Hill, San Francisco, 1985)

  12. Another Numerical Study Another approach is to solve the differential equations governing the plasma. These plots show electric field amplitude vs. time calculated from a phase mixing model for two different initial conditions. Brodin, Am. J. Phys. 65, 1 (1997)

  13. Interpretations • Conventional • Energy exchange with resonant particles • Validity contested • Phase Mixing & Thermal Spread • Oscillators dephase • Particles may have zero charge Sagan, Am. J. Phys. 62, 5 (1994)

  14. Interpretations • Resonant Diffusion • Relies on velocity-space diffusivity • Sometimes equated with conventional picture, but also allows for zero charge • Viscosity / Thermal Conduction • Complex transport coefficients can reproduce Landau damping Stubbe & Sukhorukov, Phys. Plasmas6, 2976 (1999) Puri, Phys. Plasmas7, 773 (2000)

  15. Thermal Spreading • Stubbe and Sukhorukov derive surprising result of Landau damping even in neutral gases. • Confirmed by earlier experiment by Meyer and Sessler measuring decay of sound waves in rarefied gas Stubbe & Sukhorukov, Phys. Plasmas6, 2976 (1999) Meyer & Sessler, Z. Phys. 149, 15 (1957)

  16. Physicist Deathmatch • Stubbe and Sukhorukov vs. Puri • Puri advocates resonant diffusion, claims it explains neutral gas Landau damping, rejects thermal spreading • S&S claim Puri’s treatment is equivalent to the conventional and uses incorrect energy relations • Paradigms differ Puri, Phys. Plasmas7, 773 (2000) Stubbe & Sukhorukov, Phys. Plasmas7, 775 (2000)

  17. Nonlinear Landau Damping • High enough amplitude waves can cause particle trapping • Collisions become non-negligible • Some authors claim BGK modes are steady state solutions, but this is contested. Electric field amplitude vs. time from numerical studies. Shown are (B/L)2=0.01, 6, 0.8, 1.0 Brodin, Phys. Rev. Letters78, 7 1997

  18. New & Crazy Applications • Particle beams (e.g. at the SLAC linac) • Superfluids • Condensed Matter • Quarks • Gravity Waves • Biological Systems (e.g. the synchronization of fireflies) Sagan, Am. J. Phys. 62, 5 (1994)

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