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.