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Waves in plasmas

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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 k × E ~ = ω B ~ {\displaystyle \mathbf {k} \times {\tilde {\mathbf {E} }}=\omega {\tilde {\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.

Summary of elementary plasma waves
EM character oscillating species conditions dispersion relation name
electrostatic electrons B → 0 = 0   o r   k → ‖ B → 0 {\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}} ω 2 = ω p 2 + 3 k 2 v t h 2 {\displaystyle \omega ^{2}=\omega _{p}^{2}+3k^{2}v_{th}^{2}} plasma oscillation (or Langmuir wave)
k → ⊥ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0}} ω 2 = ω p 2 + ω c 2 = ω h 2 {\displaystyle \omega ^{2}=\omega _{p}^{2}+\omega _{c}^{2}=\omega _{h}^{2}} upper hybrid oscillation
ions B → 0 = 0   o r   k → ‖ B → 0 {\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}} ω 2 = k 2 v s 2 = k 2 γ e K T e + γ i K T i M {\displaystyle \omega ^{2}=k^{2}v_{s}^{2}=k^{2}{\frac {\gamma _{e}KT_{e}+\gamma _{i}KT_{i}}{M}}} ion acoustic wave
k → ⊥ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0}} (nearly) ω 2 = Ω c 2 + k 2 v s 2 {\displaystyle \omega ^{2}=\Omega _{c}^{2}+k^{2}v_{s}^{2}} electrostatic ion cyclotron wave
k → ⊥ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0}} (exactly) ω 2 = [ ( Ω c ω c ) − 1 + ω i − 2 ] − 1 {\displaystyle \omega ^{2}=^{-1}} lower hybrid oscillation
electromagnetic electrons B → 0 = 0 {\displaystyle {\vec {B}}_{0}=0} ω 2 = ω p 2 + k 2 c 2 {\displaystyle \omega ^{2}=\omega _{p}^{2}+k^{2}c^{2}} light wave
k → ⊥ B → 0 ,   E → 1 ‖ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\|{\vec {B}}_{0}} c 2 k 2 ω 2 = 1 − ω p 2 ω 2 {\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}} O wave
k → ⊥ B → 0 ,   E → 1 ⊥ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\perp {\vec {B}}_{0}} c 2 k 2 ω 2 = 1 − ω p 2 ω 2 ω 2 − ω p 2 ω 2 − ω h 2 {\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}\,{\frac {\omega ^{2}-\omega _{p}^{2}}{\omega ^{2}-\omega _{h}^{2}}}} X wave
k → ‖ B → 0 {\displaystyle {\vec {k}}\|{\vec {B}}_{0}} (right circ. pol.) c 2 k 2 ω 2 = 1 − ω p 2 / ω 2 1 − ( ω c / ω ) {\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1-(\omega _{c}/\omega )}}} R wave (whistler mode)
k → ‖ B → 0 {\displaystyle {\vec {k}}\|{\vec {B}}_{0}} (left circ. pol.) c 2 k 2 ω 2 = 1 − ω p 2 / ω 2 1 + ( ω c / ω ) {\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1+(\omega _{c}/\omega )}}} L wave
ions B → 0 = 0 {\displaystyle {\vec {B}}_{0}=0}   none
k → ‖ B → 0 {\displaystyle {\vec {k}}\|{\vec {B}}_{0}} ω 2 = k 2 v A 2 {\displaystyle \omega ^{2}=k^{2}v_{A}^{2}} Alfvén wave
k → ⊥ B → 0 {\displaystyle {\vec {k}}\perp {\vec {B}}_{0}} ω 2 k 2 = c 2 v s 2 + v A 2 c 2 + v A 2 {\displaystyle {\frac {\omega ^{2}}{k^{2}}}=c^{2}\,{\frac {v_{s}^{2}+v_{A}^{2}}{c^{2}+v_{A}^{2}}}} magnetosonic wave

ω {\displaystyle \omega } - wave frequency, k {\displaystyle k} - wave number, c {\displaystyle c} - speed of light, ω p {\displaystyle \omega _{p}} - plasma frequency, ω i {\displaystyle \omega _{i}} - ion plasma frequency, ω c {\displaystyle \omega _{c}} - electron gyrofrequency, Ω c {\displaystyle \Omega _{c}} - ion gyrofrequency, ω h {\displaystyle \omega _{h}} - upper hybrid frequency, v s {\displaystyle v_{s}} - plasma "sound" speed, v A {\displaystyle v_{A}} - plasma Alfvén speed

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

Bibliography

See also

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