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II. Plasma Physics Fundamentals

II. Plasma Physics Fundamentals. 4. The Particle Picture 5. The Kinetic Theory 6. The Fluid Description of Plasmas 7. Waves in Plasma. 7. Waves in Plasmas. 7.1 Electrostatic Waves in Non-Magnetized Plasmas 7.2 Electrostatic Waves in Magnetized Plasmas.

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II. Plasma Physics Fundamentals

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  1. II. Plasma Physics Fundamentals 4. The Particle Picture 5. The Kinetic Theory 6. The Fluid Description of Plasmas 7. Waves in Plasma

  2. 7. Waves in Plasmas 7.1 Electrostatic Waves in Non-Magnetized Plasmas 7.2 Electrostatic Waves in Magnetized Plasmas

  3. 7.1 E.S. Waves in Non-Magnetized Plasmas 7.1.1 Wave fundamentals 7.1.2 Plasma Oscillations 7.1.3 Electron Plasma Waves 7.1.4 Sound waves 7.1.5 Ion Acoustic Waves

  4. 7.1.1 Wave Fundamentals • Any periodic motion of a fluid can be decomposed, through Fourier analysis, in a superposition of sinusoidal components, at different frequencies • Complex exponential notation is a convenient way to represent mathematically oscillating quantities: the physical quantity will be obtained by taking the real part • A sinusoidal plane wave can be represented as where f0 is the maximum amplitude, k is the propagation constant, or wave vector (k is the wavenumber) and w the angular frequency

  5. Wave Fundamentals (II) • If f0 is real then the wave amplitude is maximum (equal to f0) in r=0, t=0, therefore the phase angle of the wave is zero • A complex f0 can be used to represent a non zero phase angle: • A point of constant phase on the wave will travel along with the wave front • A constant phase on the wave implies

  6. Wave Fundamentals (III) • In one dimension it will be where vfis defined as the wave phase velocity • The wave can be then also expressed by • The phase velocity in a plasma can exceed the velocity of the light c, however an infinitely long wave train that maintains a constant velocity does not carry any information, so the relativity is not violated.

  7. Wave Fundamentals (IV) • A wave carries information only with some kind of modulation • An amplitude modulation is obtained for example by adding to waves of different frequencies (wave “beating”) • If a wave with phase velocity vfis formed by two waves with frequency separation 2Dw , both the two components must also travel at vf • The two components of the wave must then also have a difference in their propagation constant k equal to 2Dk

  8. Wave Fundamentals (V) • For the case of two wavebeating it can be written • By summing the two waves and expanding with trigonometric identities it is found • The first term of the r.h.s. is the modulating component (that does carry information) • The second term of the r.h.s. is just the “carrier” component of the wave (that does not carry information)

  9. Wave Fundamentals (VI) • The modulating component travels at the group velocity defined as • The group velocity can never exceed c

  10. 7.1.2 Electrostatic Plasma Oscillations • A displacement of electrons from their equilibrium position with respect a ion plasma background produces a restoring force due to the space-charge electrostatic field (no varying B field is considered) • Because the electron inertia this restoring force will cause electron oscillations as the electrons “overshoot” at their passage over the their equilibrium position

  11. Electrostatic Plasma Oscillations (II) • A purely electrostatic, one dimensional fluid problem without thermal motions (kBT=0) can be studied with the fluid electron equations of momentum and continuity written as • Unlike most fluid problems, here the Gauss’ theorem is needed, as the problem deals precisely with analyzing the result of a charge density fluctuation by taking into account the electron inertia

  12. Electrostatic Plasma Oscillations (III) • A simpler approach than the solution of the fluid problem can be followed • The Gauss’ theorem provides • For a small, one-dimensional, displacement of a layer of electrons of density ne = ni= n0 by a distance x, a space-charge electric field is generated equivalent to the one produced by a positive (due to the ions) surface charge s= qen0 x • The electric field produced by an infinite planar layer of surface charge s is

  13. Electrostatic Plasma Oscillations (IV) • Since each electron is subject to a force equal to -qeE the restoring force will be • The equation of the motion for each electron subject to a displacement x will be then which is a harmonic oscillator equation that describes electron plasma oscillations with characteristic frequency

  14. Electrostatic Plasma Oscillations (V) • The frequencywpe is called electron plasma frequency and depends only on the density • For a plasma with ne=1018 the plasma frequency is in the order of 9 GHz • Electron plasma oscillations have a frequency that does not depend on the propagation constant k, therefore the group velocitydw/dk is zero and no disturbance can propagate • From a physics standpoint this is explained by considering that in absence of collisions each electron will behave as an independent oscillator

  15. Electrostatic Plasma Oscillations (VI) • In a infinite system the electric fields due to the plasma oscillations do not propagate because the shielding produced by infinite charge layers • In a finite system fringe fields couple adjacent layers and the perturbation propagates • If the ion motion is also considered the plasma oscillations occur at a frequency where is the wpi ion plasma frequency defined as

  16. 7.1.3 Electron Plasma Waves • Thermal motions cause electron plasma oscillations to propagate: then they can be properly called (electrostatic ) electron plasma waves • By linearizing the fluid electron equation of motion with respect equilibrium quantities according to the frequency of the oscillations can be found as where

  17. Electron Plasma Waves (II) • Electron plasma waves have a group velocity equal to • In general a relation linking w and k for a wave is called dispersion relation • The slope of the dispersion relation on a w-k diagram gives the phase velocity of the wave

  18. 7.1.4 Sound Waves • For a neutral fluid like air, in absence of viscosity, the Navier-Stokes equation is • From the equation of state then • Continuity equation yields

  19. Sound Waves (II) • Linearization of the momentum and continuity equations for stationary equilibrium yield where mN is the neutral atom mass and cs is the sound speed. • For a neutral gas the sound waves are pressure waves propagating from one layer of particles to another one • The propagation of sound waves requires collisions among the neutrals

  20. 7.1.5 Ion (Acoustic) Waves • For a fully ionized plasma with low collisionality the same dispersion relation as for a neutral gas applies to the ion electrostatic oscillations and the resulting wave is called ion acoustic wave (or ion wave) • Unlike sound waves in a gas, ion acoustic waves in a plasma can occur also without collisions because ions are respondig to (long-range) plasma electric fields • A dispersion relation for ion waves can be derived by considering the fluid ion momentum equation • Because the low-frequency motion of the massive ions plasma quasi-neutrality can be assumed and Poisson equation is not used

  21. Ion (Acoustic) Waves (II) • Electrons can be described according to the Boltzmann relation, because they can be assumed always in regime conditions compared to the (slow) ion motion • Conventional linearization procedure leads to the dispersion relation where vs is the sound speed in a plasma • Since ions are subject to one-dimensional compression/rarefaction then gi=3. Electrons instead are isothermal, not adiabatic, and ge=1

  22. Ion (Acoustic) Waves (III) • The dispersion relation of ion waves is that of a constant velocity waves (the slope of the w-k diagram is a constant, phase velocity equals group velocity) • Electron plasma waves are constant frequency waves with a correction due to the thermal motions • Ion waves exists only when thermal motions (from electrons and/or ions) are present • Ion waves can occur even for zero ion temperature • Correction to the ion dispersion relation for k2lD2>>1 (where the quasi-neutrality is violated) lead to an asymptotic, constant frequency character for the ion waves

  23. 7.2 E.S. Waves in Magnetized Plasmas 7.2.1 Definitions for Waves in a Magnetic Field 7.2.2 Electron Oscillations Perpendicular to B 7.2.3 Ion Oscillations Perpendicular to B

  24. 7.2.1 Definitions for Waves in a Magnetic Field • The terms “parallel” and “perpendicular” are referred to the direction of k relative to an external, undisturbed magnetic field B0. • “Longitudinal” and “transverse” refer to the direction of k relative to the wave oscillating electric field E1. • If the wave oscillating magnetic field B1 equal to zero the wave is electrostatic, otherwise is electromagnetic • For e.m. waves or, for Fourier-transformed quantities

  25. Definitions for Waves in a Magnetic Field (II) • For a longitudinal wave then B1 also is equal to zero and the wave is electrostatic • Transverse waves are e.m. (B1 must be finite)

  26. 7.2.2 Electron Oscillations Perpendicular to B z wave fronts B0 k, E1 x • For longitudinal electron waves (e.s.) with no thermal motions in a fixed ion background the linearized fluid equations lead to a oscillation frequency that does not depend on k: where wh is named upper hybrid frequency

  27. Electron Oscillations Perpendicular to B (II) • The electron waves perpendicular to B are subjected to the Lorentz restoring force in addition to the electrostatic restoring force: then their frequency increase (like for any harmonic oscillator) • These waves do not have group velocity (because thermal motions have been neglected) • If B goes to zero, the usual plasma oscillations at the plasma frequency is recovered • If the plasma density goes to zero the frequency tends to the single particle Larmor frequency (since there is no more electrostatic restoring force)

  28. 7.2.3 Ion Oscillations Perpendicular to B z wave fronts B0 k, E1 x • Electrons cannot move along the x-axis to neutralize the ion charge variations due to the wave • In the plasma approximation and neglecting the electron mass the frequency for ion oscillations perpendicular to a magnetic field is obtained as

  29. Ion Oscillations Perpendicular to B (II) • The frequency wlh is named lower hybrid frequency and can be observed only for ion oscillation exactly perpendicular to the magnetic field (so that electrons cannot neutralize the ion motions) • For kalmost perpendicular to B the electrons can still move along the direction of the wave z electron motion wave fronts B0 k, E1 x

  30. Ion Oscillations Perpendicular to B (III) • For the case of k almost perpendicular to the magnetic field, the ion oscillations occur now at the ion cyclotron frequency: that, in the limit of zero ion temperature, can be written in terms of the ion sound speed as • In analogy to the case of electron waves in warm plasma, in this case the ion-acoustic oscillations are corrected to a higher frequency by the additional restoring effect due to the Lorentz force

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