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Waves in plasmas

From Biocrawler, the free encyclopedia.

A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electrons and a single species of positive ions, but it may also contain multiple ion species including negative ions as well as neutral particles. Due to its electrical conductivity, a plasma couples to electric and magnetic fields. This complex of particles and fields supports a wide variety of waves.

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Terminology and classification

Waves in plasmas can be classified as electromagnetic or electrostatic according to whether or not there is an oscillating magnetic field. Applying Faraday's law of induction to plane waves, we find $\mathbf{k}\times\tilde{\mathbf{E}}=\omega\tilde{\mathbf{B}}$ , implying that an electrostatic wave must be purely logitudinal. An electromagnetic wave, in contrast, must have a transverse component, but may also be partially longitudinal.

Waves can be further classified by the oscillating species. In most plasmas of interest, the electron temperature is comparable to or larger than the ion temperature. This fact, coupled with the much small mass of the electron, implies that the electrons are much faster than the ions. An electron mode depends on the mass of the electrons, but the ions may be assumed to be infinitely massive, i.e. stationary. An ion mode depends on the ion mass, but the electrons are assumed to be massless and to redistribute themselves instantaneously according to the Boltzmann relation. Only rarely, e.g. in the lower hybrid oscillation, will a mode depend on both the electron and the ion mass.

The various modes can also be classified according to whether they propagate in an unmagnetized plasma or parallel, perpendicular, or oblique to the stationary magnetic field. Finally, for perpendicular electromagnetic electron waves, the perturbed electric field can be parallel or perpendicular to the stationary magnetic field.

Summary of elementary plasma waves
EM character	oscillating species	conditions	dispersion relation	name
electrostatic	electrons	$\vec B_0=0\ {\rm or}\ \vec k\\|\vec B_0$	$\omega^2=\omega_p^2+(3/2)k^2v_{th}^2$	plasma oscillation
	electrons	$\vec k\perp\vec B_0$	$\omega^2=\omega_p^2+\omega_c^2=\omega_h^2$	upper hybrid oscillation
	ions	$\vec B_0=0\ {\rm or}\ \vec k\\|\vec B_0$	$\omega^2=k^2v_s^2=k^2\frac{\gamma_eKT_e+\gamma_iKT_i}{M}$	ion acoustic wave
		$\vec k\perp\vec B_0$ (nearly)	$\omega^2=\Omega_c^2+k^2v_s^2$	electrostatic ion cyclotron wave
		$\vec k\perp\vec B_0$ (exactly)	$\omega^2=\omega_i^2=\Omega_c\omega_c$	lower hybrid oscillation
electromagnetic	electrons	$\vec B_0=0$	$\omega^2=\omega_p^2+k^2c^2$	light wave
		$\vec k\perp\vec B_0,\ \vec E_1\\|\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}$	O wave
		$\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\, \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}$	X wave
		$\vec k\\|\vec B_0$ (right circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}$	R wave (whistler mode)
		$\vec k\\|\vec B_0$ (left circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}$	L wave
	ions	$\vec B_0=0$		none
		$\vec k\\|\vec B_0$	$\omega^2=k^2v_A^2$	Alfvén wave
		$\vec k\perp\vec B_0$	$\frac{\omega^2}{k^2}=c^2\, \frac{v_s^2+v_A^2}{c^2+v_A^2}$	magnetosonic wave

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Electromagnetic electron waves

In an unmagnetized plasma, electromagnetic electron waves are simply light waves modified by the plasma. In the high frequency or low density limit, i.e. for $ω > > (4π n e e 2 / m e) 1 / 2$ or $n_e << m_e\omega^2\,/\,4\pi e^2$ , the speed is the speed of light in vacuum. As the density increases, the phase velocity increases and the group velocity decreases until the resonance at the plasma frequency.

The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction ck/ω (squared).

In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.

The O wave is the "ordinary" wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma. It is plane polarized with E₁||B₀. It has a cut-off at the plasma frequency.

The X wave is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with E₁||B₀) and partly longitudinal. As the density in increased, the phase velocity rises from c until the cut-off at ω_R is reached. As the density is further increased, the wave is evanescent until the the resonance at the upper hybrid frequency ω_h. Then it can propagate again until the second cut-off at ω_L. The cut-off frequencies are given by

$\omega_R = \frac{1}{2}\left[ \omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]$

$\omega_L = \frac{1}{2}\left[ -\omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]$

The R wave and the L wave are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ω_R (hence the designation of this frequency) and a resonance at ω_c. The L wave has a cut-off at ω_L and no resonance. R waves at frequencies below ω_c/2 are also known as whistler modes.

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