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.
