The interaction of the magnetosphere with the ionosphere and atmosphere involves electromagnetic energy incident from the magnetosphere (for example as shown in Figure 1 on top left), which is output from an empirical Poynting flux model made from FAST satellite data (Cosgrove et al., 2014). The energy is dissipated by means of currents that flow through the ionospheric conductance, which arises because of the high rate of collisions with the neutral atmosphere, so that the currents cause heating and acceleration of the atmosphere. Various effects flow from this, with a particularly notable one being disruption of satellite orbits. Thus, it is important that we understand the physics that gives rise to the ionospheric conductance, which forms the inner boundary for magnetospheric modelling.

The ionospheric conductance has heretofore been calculated using a form of electrostatic theory that is very close to the textbook theory, where all time derivatives are set to zero and what remains of the equations of motion are applied to a boundary value problem. But since electrostatic theory is not valid in all cases, it is important to ask if we can show that it is valid for this ionospheric application, or, if not, to derive an electromagnetic calculation that can replace it.The transition between electromagnetics and electrostatics is addressed in the transmission line theory of electrical engineering. Consider the simple “lumped element” circuit shown in the top panel of Figure 3, consisting of a switched harmonic source with internal resistance, driving a capacitor. In order to be properly causal, this circuit is generally analyzed by taking the Laplace transform in time, which provides a solution as a sum of steady-state and transient terms. In many cases we are only interested in the steady-state part, which leaves the usual idea of a capacitive admittance operating in a harmonic circuit (Yin = iω0C, a positive imaginary number).

The terminology “lumped element” indicates that we are considering the capacitor to be very small, so that it doesn’t matter where the electrical connections are made to the parallel plates, and we can assume that the capacitor energizes everywhere all-at-once. But in reality when the switch is flipped, there is an electromagnetic signal that enters the capacitor on the side with the electrical leads, and then propagates across to the other edges, and bounces around until a steady state is reached. And depending on the size of the capacitor, there may be a portion of a wavelength inside the capacitor. When this happens the lumped element (electrostatic) analysis is too idealized to be of use.

To understand how this effect can be accommodated, consider the case of a capacitor that is long and thin, with the leads attached at the near end. Assuming that the signal cannot leak out and radiate away, this long and thin structure is a transmission line that is open-circuited at the far end. The signal propagates from the electrical leads to the far end where it reflects back, and then continues bouncing back and forth until a steady state is reached. Assuming there is only one propagating electromagnetic mode, the steady-state amounts to a superposition of two oppositely propagating waves, which are phased so that the current is zero at the open-circuited end.

From this description can be derived the well known formula for the steady-state input admittance seen by the source, which may be found, for example, in equation 3.88 from Collin (1966), and setting the load admittance to zero,where l is the length of the transmission line, Y0 is the characteristic admittance of the wave mode, and kz is the wavevector in the direction along the line (i.e., in the “parallel” or z direction). A schematic for the circuit with the open-circuited transmission line replacing the lumped-element capacitor is shown in the middle panel of Figure 3.

Since Y0 is usually a real number, the formula (1) provides that when the transmission line is short and the waves are not too lossy (i.e., kz is strongly real), then it does in fact function as a capacitor with admittance iY0kzl. But as the line gets longer the tangent function causes an oscillation between capacitive and inductive behavior, with near singularities where real(kz)l is a multiple of 90°. The singularities arise when the electric field of the reflected wave cancels that of the incident wave, where wave dissipation makes the cancelation imperfect, and is reflected in the imaginary part of kz. The famous “Smith chart” provides a graphical representation for lossless transmission lines that was a staple of microwave laboratories in the days before computers were widely available, which, by the way, was really not very long ago (bottom panel of Figure 3).

The ionosphere is not long and thin like this hypothetical capacitor. The capacitor was made thin to ensure that we do not question the coherence of the excitation produced by the electrical connections at the end. But as long as we stipulate that the excitation is coherent the capacitor can be made very wide, with electrical connections spread along its width. For example, the electrical connections could be phased so that they excite a simple plane wave, with some chosen transverse wavelength. In fact, assuming this very-wide geometry actually removes an approximation that we had swept under the rug, which is that to properly analyze the thin capacitor we should form a wavepacket in the transverse direction. If the capacitor is very wide, like, for example, the ionosphere, then there is no such approximation, and we can analyze one transverse wavelength at a time.

Thus, consider the gedankenexperiment shown in Figure 4, where the ionosphere is simplified to be a uniform slab of collisional plasma, with empty space below. The middle panel of Figure 3 is now the electromagnetic equivalent circuit for this simplified version of magnetosphere-ionosphere coupling, with the ionosphere represented by the open-circuited transmission line. The downward looking input admittance for an incident plane wave is given by the same transmission line formula (1), where Y0 and kz depend on the frequency and transverse wavelength of the plane wave.The real part of the input admittance is, of course, the ionospheric conductance. We can compare the input admittance to the well known electrostatic approximation, which is the field line integrated conductivity, σPl, where σP is the (zero frequency) Pedersen conductivity. Doing this we derive some preliminary criteria for electrostatic theory, where kz = 2π/λz − i/ldz, λz is the wavelength, and ldz is the dissipation scale length for the propagating electromagnetic mode.

To my knowledge, the last of the three criteria (2) was first derived by Cosgrove (2016), who named iY0kz the wave-Pedersen conductivity, since it replaces the usual (zero-frequency) Pedersen conductivity in an electromagnetic calculation of ionospheric conductance. Cosgrove (2016) also found that Y0 is strongly imaginary, while kz is strongly real, so that iY0kz is in fact strongly real, as expected for the ionospheric admittance. An important corollary comes from the tangent function dependence (1), which suggests the unexpected possibility that the ionospheric conductance could contain (near) singularities and change sign, if the parallel wavelength is ever comparable to the thickness of the ionosphere.Does this ever happen? Does iY0kz equal σP? Is there really only one propagating electromagnetic mode in the ionosphere? What happens when the ionosphere is vertically inhomogeneous? Stay tuned for the next episode. But I’ll give you a hint, the answer is not boring, just ask those referees who are recovering in the hospital— on second thought, don’t ask them (Cosgrove, 2022).

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*by Russell Bonner Cosgrove *

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*References*

*Collin, R. E. (1966), Foundations for Microwave Engineering, McGraw-Hill.*

*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.*

*Cosgrove, R. B., H. Bahcivan, S. Chen, R. J. Strangeway, J. Ortega, M. Alhassan, Y. Xu, M. V. Welie, J. Rehberger, S. Musielak, and N. Cahill (2014), Empirical model of poynting flux derived from fast data and a cusp signature, J. Geophys. Res., 119, 411–430, doi: https://doi.org/10.1002/2013JA019105.*

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