1. Subionospheric VLF propagation
Prepared by Morris Cohen, Benjamin Cotts, Forrest Foust
Stanford University, Stanford, CA
IHY Workshop on
Advancing VLF through the Global AWESOME Network

2. Radio waves on the ionosphere
Magnetosphere
Ionosphere
Microwave
MF-HF Waves
LF Waves
Here is a simplified picture of the earth and the ionosphere. As you can see the propagation of waves are vastly different depending on the frequency of the wave. Microwaves for instance ( GHz) pass through the ionosphere relatively undisturbed and are therefore used for applications where transmission THROUGH the ionosphere is necessary. One example of this are GPS signals from the GPS satellite array operated by the US military. Though the small-scale variations in the ionosphere are responsible for the small inaccuracies (~10 m) of GPS locations, the overall signal is transmitted through the ionosphere extremely efficiently.
In contrast LF and VLF waves (< 300 kHz) are reflected very efficiently from the ionosphere. This means that instead of the wave being transmitted through the ionosphere, it is instead guided between the conducting layers of the earth and ionosphere, which is therefore known as the earth-ionosphere waveguide. The VLF waves transmitted by the Navy are in this category of wave and are used to communicate with submarines many thousands of kilometers away.
Atmosphere
Earth
N. Lehtinen

3. Ideal parallel-plate waveguide
Isotropic Ionosphere
Perfect Reflections
(reflection coefficient is 1)
Ionosphere
Reflection Height
Transmitter
Receiver
Earth
Perfect Reflections
Flat Earth 3

4. Basic waveguide analysis
Earth
Ionosphere
Propagation
Transmitter
Receiver
~80 km
Fields within waveguide
Modal components of propagating waves
Magnetic Field (B)
Transverse Electric (TE)
Transverse Magnetic (TM)
Transverse Electromagnetic (TEM)
Electric Field (E)
Vertical direction
Azimuthal direction
Radial direction (propagation)
Above ~1.8 kHz
Below ~1.8 kHz
All Frequencies 4

5. Typical Spectrogram
TE and TM
Weak TEM
TEM Wave

6. Typical conductivities
Nighttime Ionosphere
Daytime Ionosphere
 = 10-7 to 10-5 S/m
~80-90 km
~70-75 km
In this slide we show some typical “DC” conductivities of different materials. The unit used is are Siemens per meter. A larger conductivity indicates a better conductor, i.e. a conductor which will reflect more incident VLF energy
Salt water  = 4 S/m
Fresh water  = 10-2 S/m
Wet soil  = 10-3 to 10-2 S/m
Dry soil  = 10-4 to 10-2 S/m

7. Basic plasma conductivity
Electron response e- e- e- e- e- e- e- e- e- e- e- e- e- e-
Applied electric field e- e-
Polarization field e- e- e- e-
Applied electric field forced rearranging of electrons
Polarization opposes field, shields it from propagating further
Characteristic plasma response time ~ 1/p
p2 ~ Ne
Debye Shielding 7

8. Ionospheric Conductivityz
Electrons in motion forced to orbit magnetic field
Applied electric field can generate currents in other directions
Anisotropic conductivity
“Gyrofrequency” is a function of magnetic field and e- mass y x

9. Mode conversion Incident fields are “rotated” by electron response
TE and TM waves can be converted into each other
Reflected Wave
Incident Wave
Pure TM wave
Mixed TM and TE wave

10. Anisotropic Conductivity
Direction of wave incidence matters
Different reflection coefficients
Reflected Wave
Incident Wave
Incident Wave
Reflected Wave

11. Reflection coefficients
Sharp ionospheric boundary E E k k E k k E
Perpendicular incidence
Parallel incidence
Perpendicular reflection
Parallel reflection 11

12. Example 12

13. The Effect of Collisions
Electrons lose energy via collisions
Electron-neutral collisions are most prominent
Wave energy can be absorbed via collisions 13

14. Collisions and Magnetic Field
Lower D-region, collision frequency much higher than gyrofrequency
Higher altitudes, collisions rare, magnetic field dominates
Plasma frequency (electron density) increasing rapidly
Dominated by Magnetic Field
p= c
Dominated by Collisions 14

15. Ionospheric Parameters
Measures how strongly electron density affects wave propagation
Measures how strongly geomagnetic field affects wave propagation
Measures how strongly electron-neutral collisions affect wave propagation 15

16. Plasma Terms
X=1  = p
Plasma debye shielding fast enough to block wave
Z>>X, so collisions suppress the shielding
X=Z  c = p
Collision frequency weakens, Debye shielding wins out
VLF waves reflected
Daytime reflection ~65 km
Nightime reflection ~85 km
X=1
X=Z 16

17. Conductivity Tensor 17

18. Earth Ionospheric Changes Nighttime Ionosphere Daytime Ionosphere
Scattered Wave
Reflected Wave
~80-90 km
~70-75 km
Incident Wave
Mode conversion
Receiver
Transmitter
Earth 18

19. Refractive Index B Appleton-Hartree Equation Refractive index, n
Depends on , angle between wave and magnetic field
Depends on X, Y, and Z
For Collision-less plasma (such as magnetosphere) Z  0
When Collisions dominate (Y>>Z), Y can be ignored B 19

20. Ionospheric Absorption
“Helliwell” Absorption assumption
Normal incidence
Wavelength is much smaller than the size of any variation in the medium.
Loss () is proportional to the imaginary part of the refractive index
(in dB)
The lower ionosphere is assumed to be a lossy medium, which is what motivates the normal incidence assumption. In a lossy medium, the planes of constant phase will tend to be nearly parallel to the plane of incidence.
We’ve also assumed here that the wavelength is much smaller than any feature size. This allows us to use the so-called WKB approximation, which is a first-order approximation to the solution of an otherwise difficult differential equation. The details aren’t really important here.
What’s important is what we get from the assumption -- that losses are just proportional to the imaginary part of the refractive index, integrated over the path distance in the direction of propagation. This is exactly what we would do in a homogeneous medium, except here we’re allowing the medium to change slowly.
It’s not a perfect assumption, but is good as a first guess for incidence angles close to vertical.
A more accurate method would require finding the wavenormal angles as a wave progresses through the ionosphere (a combination raytracing/propagation approximation).
A much more accurate method would require finding not only the wavenormal, but all coupling, all modes, and all reflections at every interface as we go up in altitude. This is how so-called “full-wave” methods work, discussed briefly on the next slide. 20

21. Daytime vs. Nightime Higher reflection height at nightime
Nightime reflection
Daytime reflection
Higher reflection height at nightime
Absorption dominated by collisions at reflection height
Lower collisions  less attenuation at night 21

22. Attenuation by Frequency
From Barr et al. [2000] 22

23. References
K.G. Budden, The Wave-Guide Mode Theory of Wave Propagation, 1961, Prentice Hall
J.R. Wait, Electromagnetic Waves in Stratified Media, 1962, Pergamon Press.
J. Galejs, Terrestrial Propagation of Long Electromagnetic Waves,1972 Pergamon Press
R.A. Helliwell, Whistlers and Related Ionospheric Phenomena, 1965
R. Barr et al., ELF and VLF Radio Waves, J. Atmos. Sol.-Terr. Phy., Vol.2, , 2000. 23