# Plane waves in plasmas

Waves in plasmas can be classified as electromagnetic or electrostatic according to whether or not there is an oscillating magnetic field. A corresponding wiki article states:

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 longitudinal. An electromagnetic wave, in contrast, must have a transverse component, but may also be partially longitudinal.

Doesn't this only hold for plane waves which are not necessarily good approximations for waves in plasmas (e.g. for lower hybrid waves that have antennas near the plasma in fusion devices)? Is not the more accurate statement for longitudinal waves simply $\nabla \times \mathbf{E} = 0$? And $\langle \frac {\partial \mathbf{E}}{\partial t} \rangle = 0$ for electrostatic waves? They appear to both match in the special case of plane waves, but are plane waves appropriate representations for waves in plasmas?

Any comments on the validity of the above and further reading references on waves in plasmas that do not apply the plane wave approximation are appreciated.

• The magnetic fluctuations will either be linearly or elliptically polarized. In the former case, the wave is purely transverse and in the latter case, the wave can be approximated as planar owing to $\nabla \cdot \tilde{\mathbf{B}} = 0$. You are correct, the electric fields are generally not considered planar in this regard. You may find the following useful: physics.stackexchange.com/a/264526/59023 or physics.stackexchange.com/a/265731/59023 or physics.stackexchange.com/a/235549/59023. Commented Aug 9, 2018 at 21:04
• @honeste_vivere $\langle \nabla \cdot \mathbf{B} \rangle = 0$ only indicates the magnetic component (if it exists) of a wave is periodic in space and not necessarily planar (i.e. the amplitude of the real wave does not have to sinusoidal). Commented Aug 9, 2018 at 22:39
• @Mathew24 - Not necessarily. If we suppose the real part of the frequency is finite and the fluctuation is electromagnetic, then you can decompose the fluctuation in such a way to find $\mathbf{k} \cdot \mathbf{B} = 0$ for linear modes. If the mode is nonlinear or the imaginary part of $\mathbf{k}$ is non-zero, then I agree that we cannot assume $\mathbf{k} \cdot \mathbf{B} = 0$, i.e., we cannot assume planarity. Commented Aug 10, 2018 at 2:19
• @honeste_vivere Why would one assume linear modes being prevalent over nonlinear modes for plasma waves? Commented Aug 10, 2018 at 14:52
• Interestingly enough, even though many plasma waves have nonlinear properties, they still can maintain some of their linear properties. For instance, the amplitude of an observed mode may exceed a nonlinear threshold, but its time series profile can still look like a nice modulated sine wave and the frequency spectrogram can still be consistent with predictions from linear theory. The short answer is that one starts simple and if that fails, one adds a little complexity and iterates until the analysis can work. Commented Aug 10, 2018 at 18:13

Doesn't this only hold for plane waves which are not necessarily good approximations for waves in plasmas (e.g. for lower hybrid waves that have antennas near the plasma in fusion devices)?

I will address this in parts. First, the following applies to linear and quasi-linear – a first (and sometimes second) order correction to the linear approximation – approximations but cannot be generalized to fully nonlinear waves. Interestingly enough, even though many waves observed in space plasmas can be called nonlinear due to their amplitudes (or some other property), they often retain many of the linear properties predicted for the given mode [e.g., see examples in Giagkiozis et al., 2018; Wilson et al._, 2013, 2017].

Is not the more accurate statement for longitudinal waves simply $\nabla \times \mathbf{E} = 0$? And $\langle \tfrac{\partial \mathbf{E}}{\partial t} \rangle = 0$ for electrostatic waves?

Second, a linear electrostatic fluctuation satisfies $\mathbf{k} \times \mathbf{E} = 0$. This does not mean that they do not have their own displacement currents, i.e., electrostatic waves still have finite $\tfrac{\partial \mathbf{E}}{\partial t}$. It is really a statement that the wave vector, $\mathbf{k}$, is parallel to the fluctuating $\mathbf{E}$ [e.g., see example ion acoustic wave in Wilson et al._, 2010] not that $\tfrac{\partial \mathbf{E}}{\partial t} = 0$.

The nice thing about plasmas is they obey Maxwell's equations, thus the longitudinal part of any electromagnetic wave occurs in the electric field only owing to $\nabla \cdot \mathbf{B} = 0$. In the electrostatic case, $\mathbf{k}$ is along the fluctuating electric field and it is not a plane wave in the sense of the fields oscillating in a plane orthogonal to the direction of propagation.

They appear to both match in the special case of plane waves, but are plane waves appropriate representations for waves in plasmas?

Finally, yes there are significant limitations to the plane wave approximation. While the fluctuations seen plasmas may not satisfy all the assumptions for plane waves, this does not mean the approximation is invalid or cannot be used. For instance, we know the ideal gas law assumptions do not hold under most situations, but it is not an irrelevant approximation (it actually works frustratingly well in many situations).

Any comments on the validity of the above and further reading references on waves in plasmas that do not apply the plane wave approximation are appreciated.

Unfortunately, there is little that can be done with non-planar waves, i.e., those with non-stationary solutions or nonlinear properties. By nonlinear, I am specifically referring to fluctuations that have one or more of the following properties:

• those that cannot be approximated as a constant times $e^{i \left( \mathbf{k} \cdot \mathbf{x} - \omega t \right)}$;
• those with $A\left( \omega, \mathbf{k} \right)$, i.e., fluctuations where the amplitude depends upon frequency and/or wave vector; or
• those with $\Im\left[ \omega \right] \gg \Re\left[ \omega \right]$ that are observed in the plasma rest frame.

Bellan [2016] came up with a neat idea for finding $\mathbf{k}$ from single point measurements but even that, I think, is limited to a planar assumption.

I have listed several other references below on wave analysis in plasmas, some are observational applications and others are rigorous mathematical justifications for a given technique.

# References

• Bellan, P.M. "Revised single-spacecraft method for determining wave vector k and resolving space-time ambiguity," J. Geophys. Res. 121(9), pp. 8589–8599, doi:10.1002/2016JA022827, 2016.
• Giagkiozis, S., et al., "Statistical Study of the Properties of Magnetosheath Lion Roars," J. Geophys. Res. 123, doi:10.1029/2018JA025343, 2018.
• Kawano, H. and T. Higuchi "The bootstrap method in space physics: Error estimation for the minimum variance analysis," Geophys. Res. Lett. 22(3), pp. 307–310, doi:10.1029/94GL02969, 1995
• Khrabrov, A.V. and B.U.O. Sonnerup "Error estimates for minimum variance analysis," J. Geophys. Res. 103(A4), pp. 6641–6652, doi:10.1029/97JA03731, 1998.
• Means, J.D. "Use of the three-dimensional covariance matrix in analyzing the polarization properties of plane waves," J. Geophys. Res. 77(28), pp. 5551–5559, doi:10.1029/JA077i028p05551, 1972.
• Samson, J.C. and J.V. Olson "Some comments on the descriptions of the polarization states of waves," Geophys. J. 61(1), pp. 115–129, doi:10.1111/j.1365-246X.1980.tb04308.x, 1980.
• Wilson, L.B., et al., "Large‐amplitude electrostatic waves observed at a supercritical interplanetary shock," J. Geophys. Res. 115(A12), pp. A12104, doi:10.1029/2010JA015332, 2010.
• Wilson, L.B., et al., "Electromagnetic waves and electron anisotropies downstream of supercritical interplanetary shocks," J. Geophys. Res. 118(1), pp. 5–16, doi:10.1029/2012JA018167, 2013.
• Wilson, L.B., et al., "Revisiting the structure of low‐Mach number, low‐beta, quasi‐perpendicular shocks," J. Geophys. Res. 122(9), pp. 9115–9133, doi:10.1002/2017JA024352, 2017.