3a. Plane-wave reflectivity and path transmission
This article is part 2 of a two-part study; here, we briefly recap the
results of the first part, from JHB1. The work to follow entirely
depends on JHB1, and the reader should refer to that published article
for details beyond the brief recap here.
We rely on a numerical model of plane-wave reflection from a diffuse,
collisional, anisotropic D-layer [Jacobson et al. , 2010;Jacobson et al. , 2009; Jacobson et al. , 2012]. Our model
is a modernization of Pitteway’s groundbreaking treatment
[Piggott et al. , 1965; Pitteway , 1965]. We represent
the electron-neutral collision rate by an exponentially declining
function of altitude as is common in this field. For the electron
density, we use an exponentially increasing function of altitude, also
common in the field [see, e.g., Eq. 3.23, Section 3.2.3, inVolland , 1995]. See Table 1 for details.
Figure 1 summarizes the prediction of our plane-wave reflection model.
The vertical axis is the amplitude reflection coefficient, R ,
from the D-layer for a typical long-range-propagation angle of
incidence, in this case chosen as 85 deg. The reflection coefficient
shown has been averaged over all frequencies from 5 to 20 kHz. As shown
in JHB1, R varies continuously with solar zenith angle, but we
show the pure-day and pure-night extreme cases only. On the left of
Figure 1 is shown (a) the day-profile D-layer result, while on the right
is shown (b) the night-profile D-layer result (refer to Table 1 for
profile parameters). The abscissa is the wave magnetic propagation
azimuth. A separate curve is shown for each abs(dip angle), from 5 deg
(blue) to 85 deg (red), in steps of 5 deg. The curves for dip = 30 deg
and 45 deg are labeled in the night profile. For both (a) and (b), the
curve for dip = 45 deg is dashed. The horizontal black line marks the
nadir of the night-profile reflectivity level for dip = 45 deg.
How do we employ the single-reflection reflectivity from a plane-wave
model, in the context of long-range (”multi-hop”) propagation of
quasi-cylindrical waves in a spherical-shell waveguide? The article on
the first half of this project, JHB1, shows how this is done
heuristically but with satisfactory agreement with observations: First,
we correct the wave amplitude for the varying cross-sectional area of a
ray-bundle on the spherical Earth (see Eq. 7 in JHB1). Second, we rely
in JHB1 on a free parameter ”r ” , which is the effective number
of reflections per reference distance ρ0 = 1000 km (= 1
Mm). In JHB1 we demonstrated how comparison with observed received
electric-field amplitude resulted in a fit for r in the range 3
> r > 2.
Those two heuristics (correcting for the ray-bundle area, and invoking
an effective reflection-per-pathlength) were used in JHB1 to crudely
approximate long-range waveguide transmission in terms of the
single-hop, plane-wave reflectivity model. We define a ”logarithmic
reference transmission”, assuming perfect ground conductivity ,
along the Great Circle Path segment L i,m from VLF
emission point ”m” to sensor point ”i” :
\(ln(ref.\ transmission)\ =\ \ \frac{1}{\rho_{0}}\int_{0}^{L_{i,m}}{\ \ ln(R[Z_{i,m}}(t_{0}),\ \alpha_{i,m},\ I_{i,m}])ds_{i,m}\ +\ C(L_{i,m})\)Eq. (1)
where
L i,m = arcdistance along Great Circle Path from
lightning location m to station i
Z i,m(t) = time-dependent, location-dependent
solar zenith angle along path i,m
αi,m = location-dependent magnetic propagation azimuth
along path i,m
I i,m = location-dependent magnetic dip angle
along path i,m
R (Z i,m(t), αi,m,Ii ,m) local instantaneous
plane-wave reflectivity
ds i,m = differential path element along Great
Circle Path i,m
ρ0 = 1000 km
The term C(Li,m) in Eq. (1) is the geometrical
correction due to the variation of ray-bundle cross-sectional area. We
tabulate the correction, relative to its value at the reference distance
1000 km:
\(C(L_{i,m})\ =ln\ \{\sqrt{\frac{sin(\rho_{0}/R_{E})}{sin(L_{i,m}/R_{E})}}\text{\ \ }\}\)Eq. (2)
where RE is the Earth’s radius.
The logarithmic reference transmission (Eq. 1) must be multiplied by the
fitted parameter r to give an estimate of the actual logarithmic
path transmission assuming zero ground losses (see Eq. 9b in
JHB1). This r parameter was fitted to lie in the range of 2 to 3.
Physically, it is the number of hops per 1000 km reference distance,
subject to our model’s assumption of 85-deg angle-of-incidence.
Ignoring ground losses would be unacceptable if we were trying to
calculate absolute transmission in the waveguide. However, our
application involves examining the difference between day and night
conditions on the propagation anisotropy. The ground conductivity
effects are unchanged (on a given path) between day and night. Thus
modeling only D-layer losses is a satisfactory (though not perfect)
approach for our study of day-versus-night differences.
A further convenient simplification introduced in JHB1 is that we
actually solve for the log reflectivity ln(R) only for the two
extreme cases of pure day and pure night. Any intermediate case is
approximated by a linear combination of pure-day and pure-night, using a
smooth function of solar zenith angle (see Eqs. 10-11 in JHB1).This is done locally, at each point along the path integral in Eq.
(1), and for local solar zenith angle obtaining at the instant of the
lightning stroke. There is a crucial difference between, on the one
hand, making the linear combination locally (which we do), versus, on
the other hand, evaluating the path integral along the entire path both
for an artifactual day and an artifactual night case, then taking a
linear combination of those two results based on the proportion of the
path that is daylit. The approach latter would be clearly incorrect.