2b. Prior modeling of east-west asymmetry of VLF propagation
The reflection of radio waves from the underside of the ionosphere
became an active area of research during the 1950s [see the historical
references given by, e.g., Barber and Crombie , 1959; Wait
and Spies , 1960; Wait and Spies , 1964; Yabroff , 1957].
The problem was nontrivial due to the anisotropy of the dielectric,
associated with the gyration of charged particles about the geomagnetic
field. This was especially true for VLF waves, whose height of
reflection occurs in the lowermost ionosphere, namely the D-layer. The
strong electron-neutral collision rate in the D-layer further
complicates models of VLF reflection. The models needed to address
practical challenges, e.g.:
(a) What is the VLF reflectivity?
(b) How does it depend on solar zenith angle?
(c) How does the reflectivity depend on angle-of-incidence?
(d) How does the reflectivity depend on local propagation magnetic
azimuth (reckoned clockwise from local magnetic North) and on local
magnetic declination (”dip angle”)?
(e) How does the reflectivity depend on electron-neutral collision rate?
Starting late in the 1950’s, sharp-boundary treatments of the
collisional, anisotropic VLF reflection process were set up analytically
and solved numerically with newly available digital computers
[Barber and Crombie , 1959; Wait and Spies , 1960;Wait and Spies , 1964; Yabroff , 1957]. The first
numerical model of an arbitrarily-layered (rather than just a sharp
boundary) D-layer [Piggott et al. , 1965; Pitteway ,
1965] followed quickly, although its physical implications appear to
have been only slowly appreciated. The Pitteway model for the
continuously varying D-layer solved the Maxwell Equations for the
altitude-dependent, anisotropic, and complex susceptibility tensor. All
of the sharp-boundary models, as well as the Pitteway model, dealt with
the elementary reflection of an incident plane wave.
Such plane-wave models are excellent for providing insights on ”process”
questions, such as those cited in the previous paragraph. However, for
long-range ”multi-hop” propagation, it is more efficient, though less
heuristically instructive, to cast the problem in terms of waveguide
modes in the spherical-shell Earth-ionosphere waveguide (EIWG). The
modes are akin to cylindrical waves from a point source within a
parallel-plane waveguide, except that the waveguide elements are
(approximately) concentric spherical surfaces [see the
illuminating tutorial by Cummer , 2000]. A waveguide model
provides a point-to-point complete description of the VLF transmission
along any given Great Circle path. This includes all portions of the
path. The first portion consists of 3-dimensional expansion of the
wavefield into a hemisphere. The next portion takes account of the first
ionospheric reflection, which effectively is a transition to
spherical-shell EIWG propagation. This transition needs many
higher-order modes to describe the wavefield, because at such a short
range (e.g, < 1000 km) a broad range of plane-wave ”angles of
incidence” are at play [Cummer , 2000]. Ultimately, however,
at longer range the waveguide modes simplify. For a vertical-dipole
source near ground level, and a vertical-dipole receiver also near
ground level, the modes simplify at large distances to the fundamental
Transverse Magnetic (TM) mode. Thus the transmission characteristics
vary from 3-dimensional expansion into a hemisphere, to a single 2-D,
fundamental TM mode in the waveguide.
The waveguide approach was perfected in the Long Range Propagation
Capability, or LWPC [Pappert and Ferguson , 1986] suite of
computer codes developed by the United States Navy. The LWPC includes an
atlas of Earth-surface conductivity. The user can select a D-layer
model, usually exponential profiles of electron density and of
electron-neutral collision rate. The LWPC contains ”everything” in one
master code suite. LWPC uses just an approximation of the D-layer
electron-density profile, but that is justified by the impossibility of
knowing any better profile at any given instant.
One adverse side-effect of its end-to-end completeness is that the LWPC
blurs (to the LWPC user) the role of local parameters, such as
solar magnetic propagation azimuth and local magnetic dip angle. These
vary along the path, but the LWPC’s end-to-end approach path-integrates
over their local variations, and all the user sees is the result of the
path integration. Thus, despite its completeness, premiere accuracy, and
reliability, the LWPC is not pedagogically illuminating for exploring
individual local processes in isolation.