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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

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

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

• 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

• 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

• 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.

• 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

• 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)

• The modulating component travels at the group velocity defined as

• The group velocity can never exceed c

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• For a neutral fluid like air, in absence of viscosity, the Navier-Stokes equation is

• From the equation of state

then

• Continuity equation yields

• 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

• 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

• 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

• 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

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

• 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

• 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)

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

• 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)

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

• 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

• 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