waves in plasmas

{{Short description|Concept in physics}}

{{No footnotes|date=December 2024}}

In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. 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 wave phenomena.

The electromagnetic fields in a plasma are assumed to have two parts, one static/equilibrium part and one oscillating/perturbation part. 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 $\backslash mathbf\{k\}\; \backslash times\; \backslash tilde\{\backslash mathbf\{E\}\}\; =\; \backslash omega\; \backslash tilde\{\backslash mathbf\{B\}\}$, implying that an electrostatic wave must be purely longitudinal. 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 smaller mass of the electron, implies that the electrons move 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.

class="wikitable" |+ Summary of elementary plasma waves
EM character	oscillating species	conditions	dispersion relation	name
rowspan = 5 | electrostatic	rowspan = 2 | electrons	$\backslash mathbf\; B\_0=0$ or $\backslash mathbf\; k\; \backslash parallel\; \backslash mathbf\; B\_0$	$\backslash omega^2\; =\; \backslash omega\_p^2\; +\; 3\; k^2\; v\_\backslash text\{th\}^2$	plasma oscillation (or Langmuir wave)
$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$	$\backslash omega^2=\backslash omega\_p^2\; +\; \backslash omega\_c^2=\backslash omega\_h^2$	upper hybrid oscillation
rowspan = 3 | ions	$\backslash mathbf\; B\_0=0$ or $\backslash mathbf\; k\; \backslash parallel\; \backslash mathbf\; B\_0$	$\backslash omega^2\; =\; k^2\; v\_s^2\; =\; k^2\; \backslash frac\{\backslash gamma\_e\; k\_\backslash text\{B\}\; T\_e\; +\; \backslash gamma\_i\; k\_\backslash text\{B\}\; T\_i\}\{M\}$	ion acoustic wave
$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$ (nearly)	$\backslash omega^2=\backslash Omega\_c^2\; +\; k^2v\_s^2$	electrostatic ion cyclotron wave
$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$ (exactly)	$\backslash omega^2\; =\; \{\backslash left[\{\backslash left(\backslash Omega\_c\; \backslash omega\_c\backslash right)\}^\{-1\}\; +\; \backslash omega\_i^\{-2\}\backslash right]\}^\{-1\}$	lower hybrid oscillation
rowspan = 8 | electromagnetic	rowspan = 5 | electrons	$\backslash mathbf\; B\_0\; =\; 0$	$\backslash omega^2\; =\; \backslash omega\_p^2\; +\; k^2c^2$	light wave
{{nowrap|$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$,}} $\backslash mathbf\; E\_1\; \backslash parallel\; \backslash mathbf\; B\_0$	$\backslash frac\{c^2k^2\}\{\backslash omega^2\}\; =\; 1\; -\; \backslash frac\{\backslash omega\_p^2\}\{\backslash omega^2\}$	O wave
{{nowrap|$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$,}} $\backslash mathbf\; E\_1\backslash perp\backslash mathbf\; B\_0$	$\backslash frac\{c^2k^2\}\{\backslash omega^2\}\; =\; 1\; -\; \backslash frac\{\backslash omega\_p^2\}\{\backslash omega^2\}\backslash ,$ \frac{\omega^2 - \omega_p^2}{\omega^2-\omega_h^2}	X wave
$\backslash mathbf\; k\; \backslash parallel\; \backslash mathbf\; B\_0$ {{nowrap|(right circ. pol.)}}	$\backslash frac\{c^2k^2\}\{\backslash omega^2\}=1-\backslash frac\{\backslash omega\_p^2/\backslash omega^2\}\{1-(\backslash omega\_c/\backslash omega)\}$	R wave (whistler mode)
$\backslash mathbf\; k\; \backslash parallel\; \backslash mathbf\; B\_0$ {{nowrap|(left circ. pol.)}}	$\backslash frac\{c^2k^2\}\{\backslash omega^2\}=1-\backslash frac\{\backslash omega\_p^2/\backslash omega^2\}\{1\; +\; (\backslash omega\_c/\backslash omega)\}$	L wave
rowspan = 3 | ions	$\backslash mathbf\; B\_0=0$		none
$\backslash mathbf\; k\; \backslash parallel\; \backslash mathbf\; B\_0$	$\backslash omega^2\; =\; k^2\; v\_A^2$	Alfvén wave
$\backslash mathbf\; k\backslash perp\backslash mathbf\; B\_0$	$\backslash frac\{\backslash omega^2\}\{k^2\}\; =\; c^2\backslash ,$ \frac{v_s^2 + v_A^2}{c^2 + v_A^2}	magnetosonic wave

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• $\backslash omega$ - wave frequency,
• $k$ - wave number,
• $c$ - speed of light,
• $\backslash omega\_p$ - plasma frequency,
• $\backslash omega\_i$ - ion plasma frequency,
• $\backslash omega\_c$ - electron gyrofrequency,
• $\backslash Omega\_c$ - ion gyrofrequency,
• $\backslash omega\_h$ - upper hybrid frequency,
• $v\_s$ - plasma "sound" speed,
• $v\_A$ - plasma Alfvén speed

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(The subscript 0 denotes the static part of the electric or magnetic field, and the subscript 1 denotes the oscillating part.)