And when you have got to that place, it’s time to appreciate that we now have three criteria for electrostatic theory that we should be able to test, if we can just figure out what is meant by these waves in an exact mathematical way. These criteria are given in equation (2) from Episode I, and for convenience let’s write them down here as well,

where σP is the usual (zero-frequency) Pedersen conductivity, l* *is the thickness of the gedanken ionosphere, kz = 2π/λz − i/ldz, λz is the parallel wavelength, ldz is the dissipation scale length, Y0 is the characteristic wave admittance, and we have named the composite quantity *i*Y0kz the wave-Pedersen conductivity, for the (single) propagating electromagnetic wave-mode. So how can we go forth and put some rigor to these quantities?

To begin with, this is physics, and so we will want some equations of motion. For these let’s take the Maxwell equations along with some order of the fluid equations for electrons and one species of ion. Taking the Fourier transform in space, these electromagnetic fluid equations may be expressed in matrix form as,

where the matrix *H*5 has been given in previous work, both for the 5-moment fluid equations (Figure A.1 of Cosgrove (2022)), and for the reduced set where the continuity and energy equations are omitted (Equation 7 of Cosgrove (2016)). `F⃗`(t) is a time-dependent vector containing the nonlinear terms. And most importantly, `X⃗` is a vector containing the physical quantities we want to solve for, such as electric field, magnetic field, electron velocity, ion velocity, electron density, and etc., depending on how many moments are retained in deriving the fluid equations from the kinetic equations.

As they are for electrostatic theory, the nonlinear terms will be dropped, in which case the solution may be written exactly as,

where`h⃗`

*j*and ω

*j*are the eigenvectors and eigenvalues of

*H*5, respectively,

`k⃗`is the Fourier transform

variable (the wavevector), and the

*a0j*are 16 arbitrary functions of

`k⃗`. The solution is easily verified by direct substitution, and this is something you can do. And if you wanted to set aside a couple of weeks you could also derive the matrix, except, I shouldn’t assume your engine is as weak as mine.

*Figure 1: Illustration of the calculation of dissipation scale length.*

The number 16 arises for the case of the 5-moment fluid equations, which together with the Maxwell equations comprise 16 scalar equations. If we were to omit, for example, the energy equations there would be only 14 modes, and this says something about the physical meaning of these modes. It is not quite right to equate these with the usual waves that are defined through physical approximations. Our waves are defined by their role in the exact solution (3) to the (linearized) equations of motion (2). Nevertheless, there is clearly a close relationship between them.

The solution (3) is the source-free initial value solution, which will decay to zero over time according to the time scales 1/*imag*(ω*j*). But as described in Episode I, what we actually want is the driven steady-state solution for a source that turns on, and then continues operating for a good while. Specifically we need the parallel wavelength (λjz), dissipation scale length (ldjz), and polarization vector (`P⃗`j) that describe the driven steady-state solution for each mode. We can obtain these (approximately) from the initial-value solution (3) by recognizing that the latter will describe the plasma evolution during any period when the source is turned off.

If the source that has been operating for a while were to suddenly turn off, the plasma would initially continue evolving in the same way, since it takes some time for the turn-off effect to propagate away from the source. So for example the plasma would continue oscillating at the source frequency, ω*0*, and the solution (3) must predict this. Since the frequency for each mode is ω*jr *(`k⃗`) = *real*(ω*j *(`k⃗`)), this means that ω*jr *(`k⃗`⊥, 2π/λjz) = ω*0*, where `k⃗`⊥ is the transverse wave vector determined by the source (Episode-I). So this allows us to solve for λjz, numerically at least.

Another quantity that must remain the same immediately after the source turn-off is the polarization vector, and so since the po- larization vector for each mode in equation (3) is the eigenvector `h⃗`*j *(`k⃗`), we can use our result for λjz to find the polarization vector as `P⃗`*j* = `h⃗`*j *(`k⃗`⊥, 2π/λjz).

Finally, to get the dissipation scale length, consider that if the transmitter were to turn on and then off again after transmitting for several cycles, then it would transmit a wave-packet such as is shown in the top panel of Figure 1 (one for each propagating mode). Using the usual understanding of wave-packet propagation, the wave-packets propagate with their group velocity, v*gjz* = −∂ω*jr*/∂kz, while decaying with time scale τ*j* = 1/*imag*(ω*j*), which is also illustrated in the panel. Thus if the transmitter were to do this repeatedly, that is, if it did not turn off, there would eventually result a signal that diminishes away from the antenna with the “dissipation scale length” ldjz = v*gjz*τ*j*, for each propagating mode. The bottom two panels of Figure 1 illustrate two instants in the temporal-ascent of the signal to steady-state.

*Figure 2: Illustration of magnetosphere-ionosphere coupling with electromagnetic ionosphere.*

Since a propagating wave can proceed in either of two opposing directions for any `k⃗`, equation (3) shows that the solution consists of a sum of up to 8 such “wave-modes,” which are paired modes with ω*kr* = − ω*jr*. The usual transmission line theory applies to the case where there is only one such wave-mode, and physical transmission lines are designed to ensure there is only one. But for our ionospheric application we do not have this luxury, and so we will want to examine the properties of the modes to find out which must be retained, and which can be discarded. Once the matrix *H*5 has been found, the eigenvalues and eigenvectors can be obtained using standard numerical tools, so that the properties of the modes can be determined.

We may discard modes that are not capable of propagating at the source frequency, that is, which cannot satisfy ω*jr *(`k⃗`⊥, kz) = ω*0* for any kz, based on typical ionospheric ranges for frequency and transverse scale (ω*0* and `k⃗`⊥). In analyzing the eigenvectors and eigenvalues, we find two “evanescent” modes with ω*jr *(`k⃗`⊥, kz) = 0, and we find the X-, O-, and Z-modes known from radio-frequency applications, which are way too high in frequency. Discarding these we are left with 4 wave-modes that can potentially contribute to ionospheric science, which we call the Whistler, Alfvén, Ion, and Thermal waves.

Of these, we may discard any that are not capable of transmitting energy over any significant distance, based on the credo that we should give electrostatic theory its best possible chance, and coupling into such modes would obviously prevent the electric field from mapping through the ionosphere. This failure-mode is embodied in the electrostatic criteria ldz ≫ l from equation (1). In analyzing the four we find that the Ion and Thermal waves have dissipation scale lengths (ldjz) less than 10 km, and generally much less. (Here it is important to note that we are only considering scales above a couple of hundred meters, and no conclusion is intended for smaller scales!) Therefore, there are only two wave-modes that need to be included in our transmission line for the ionosphere, the Whistler and Alfvén waves.

*Figure 3: Wave-Pedersen conductivity for the Alfvén and Whistler waves, compared to the usual (zero- frequency) Pedersen conductivity (λ⊥ = 100 km, **n*e* = 10 ^{11} m^{-3}*

*).*

*Figure 4: Panels a and b: Calculation of phase rotation and dissipation through the vertically-inhomogeneous ionosphere. Panel c: Total phase rotation versus transverse wavelength, for two different electron density profiles, and the same collision-frequency profiles as Figure 3.*

Now we are in a position to test the electrostatic criteria (1). Figure 3 shows the wave-Pedersen conductivity (*i*Y0kz) for both modes along with the usual (zero-frequency) Pedersen conductivity (σP), plotted versus altitude for a typical ionospheric-profile of collision-frequencies (profiles given in Cosgrove (2016), and for this example we take λ⊥ = 100 km and *n*e = 10^{11} m^{-3}). Neither mode agrees very well at all, and so *the third electrostatic criteria is not satisfied for either wave-mode*.

signal traversing the ionosphere. That is, using the numerical results obtained from setting ω

*jr*(

`k⃗`⊥, 2π/λjz) = ω

*0*, we integrate the inverse of λjz over altitude, from 400 km down to 100 km, while choosing the longest wavelength mode at each altitude. This “best case” calculation is illustrated in Panel a of Figure 4, where the vertical dashed-black line indicates the altitude for switching from the Alfvén to the Whistler wave.

This phase rotation is analogous to the argument for the tangent function that appears in the electrostatic criteria (1), except that the ionosphere is now vertically inhomogeneous. The results are summarized in panel c of Figure 4, by plotting the best-case phase rotation versus transverse wavelength for two different electron density (ne) profiles. It is found that the phase rotation can exceed 90°, even for transverse wavelengths as long as 100 km. Since tan 90° = ∞, *this represents a complete failure for the first of the electrostatic criteria*.

With respect to the second electrostatic criteria (ldz ≫ l), a similar analysis (Panel b of Figure 4) finds that this criteria is nearly satisfied, that is, it is satisfied by the Whistler wave, and with the exception of the lower E region at longer wavelengths, it is also satisfied by the Alfvén wave. This result gives rise to the idea that the Whistler wave dominates in the lower E region, and the Alfvén wave in the region above, such that there may be an important transitional effect.

Thus we have answered the first three questions posed in the last paragraph of Episode I, and in all three cases the answers speak against electrostatic theory in a very strong way. However, we might continue to wonder whether a more general form of electrostatic theory, such as electrostatic wave theory, might be sufficient to salvage the situation. This question turns out to be tied up with the question of the actual, vertical-inhomogeneity of the ionosphere, and of the interaction between the two modes that it causes. We will address both these questions in Episode III, and in the process we will recover the most rigorous model ever for the ionospheric conductance, which must needs be an electromagnetic model.

*by Russell Bonner Cosgrove*

*References*

*Cosgrove, R. B. (2016), Does a localized plasma disturbance in the ionosphere evolve to electrostatic equilibrium? Evidence to the contrary, J. Geophys. Res., 121, doi: https://doi.org/10.1002/2015JA021672.*

*Cosgrove, R. B. (2022), An electromagnetic calculation of ionospheric conductance that seems to override the field line integrated con**ductivity, Zenodo and ArXiv, doi: 10.48550/ARXIV.2211.10818, 10.5281/Zenodo.7416494.*

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