This section first describes similarities between simulation results and observations of coherent echoes reported in the literature, then connects results to a more general theory of coupled FBI/GDI growth than that presented in §4.2.
4.5.1 Connection with observations
This work not only represents the first kinetic simulations of coupled FBI/GDI but also lends insight to observations of E-region plasma irregularities observed by radars.
95 95
0.0 0.2 0.4 0.6 0.8 1.0
P(Vph)/max(PE12)
-800 -600 -400 -200 0 200
Vph [m/s]
0.0 0.2 0.4 0.6 0.8 1.0
Normalized P(Vph)
E
0z= 6 mV/m!
E
0z= 9 mV/m!
E
0z= 12 mV/m!
= 15±1º = 2 m#
(a)!
= 2 m!
= 3 m!
= 8 m!
= 15±1º 0z = 12 mV/m#
(b)!
Figure 4·8: Phase-velocity spectra at ✓ = 15 : (a) Power in 2-m waves normalized to the spectrum of the 12-mV/m run; (b) Self-normalized power in 2-m, 3-m, and 8-m waves for the 12-mV/m run.
Figure 4·8: Phase-velocity spectra at ✓ = 15 : (a) Power in 2-m waves normalized to the spectrum of the 12-mV/m run; (b) Self-normalized power in 2-m, 3-m, and 8-m waves for the 12-mV/m run.
Patra et al. (2005) reported east-west asymmetries in Type II irregularities observed with an 18-MHz radar located near the magnetic equator, and attributed the asym-metry to the tilt in kilometer-scale primary waves at E-region altitudes. Hysell et al. (2007) connected east-west asymmetries with up-down Type I asymmetries observed with a 50-MHz radar and noted that the depleted phases (i.e. troughs) of kilometer-scale primary waves should have larger electric fields than the correspond-ing enhanced phases (i.e. crests), leadcorrespond-ing to observations of larger line-of-sight drifts and preponderance of Type I echoes in westward-aligned beams. The density results presented in Figure 4·2, while not directly comparable to kilometer-scale processes, are consistent with those observations and the total electric field results presented in Figure 4·3 account for the development of Type-I irregularities within the de-pleted region westward of a large-scale wave. In that region, the positions of density troughs and crests modifies the electrostatic potential in a manner that enhances the polarization electric field. This adds to the background and ambipolar electric fields within the density trough between the two large-scale density crests. These results also support the conclusion by Ronchi et al. (1991) that long wavelength activity affects the characteristics of short wavelength two-stream irregularities. In the work presented here, long wavelength activity creates the electrostatic potential field that drives short-wavelength two-stream irregularities within the density trough between long wavelength waves Sudan et al. (1973). It is worth noting again that Figure 4·7c shows a thin band of relatively high normalized power near Vph ≈ −425 m/s for 0◦ < θ < 30◦, suggesting that two-stream irregularities have a constant Vph over this range. This result is consistent with early claims that the phase speed of Type-I irregularities is constant with zenith angle (Cohen and Bowles, 1967). Furthermore, hVphi (the white line) never exceeds ±Cs ≈ 350 m/s, suggesting that the mean phase speed saturates at Cs. This claim is also consistent with observations (Sudan, 1983).
4.5.2 Dispersion relation
An analysis of instability growth in these simulations must account for magnetized electrons and unmagnetized ions with arbitrary wavevector in the presence of a 2-D background gradient. Sudan et al. (1973) derived the two-fluid dispersion relation for an isothermal, electrostatic, quasi-neutral plasma with a strictly vertical back-ground gradient and static horizontal backback-ground magnetic field. The appendix of Fejer et al. (1975b) shows the derivation of a similar two-fluid dispersion relation, al-lowing for plasma production and an arbitrary wavevector. Sudan (1983) developed a nonlinear theory of Type II irregularities from which he obtained a linear disper-sion relation similar to that given by Sudan et al. (1973). Dimant and Oppenheim (2011b) derived a fluid dispersion relation for the combined FBI/GDI with arbitrary magnetization, gradients, and wavevector, including production and recombination effects. Makarevich (2016) presents a general dispersion relation for E- and F-region instabilities that makes no assumptions about altitude, wavevector, or background density gradient.
Equation A29 with equations A34 and A35 in Dimant and Oppenheim (2011b), under the additional assumptions kk = 0 and κi 1, yield a local linear growth rate appropriate to the present work:
ωi(k) = ψ⊥
1 + ψ⊥
1
νi ω2r− k2Cs2 − Ωe
k× ˆb
· G νek⊥2 ωr
, (4.4)
The symbols ψ⊥, νi, νe, Ωe, ωr, and Cs have the same meanings as in equations 4.2;
ˆb is a unit vector parallel to the magnetic field (−ˆy in the present geometry) and G ≡ n−10 ∇n0. Note that some of the notation used here differs from that used in Dimant and Oppenheim (2011b) for the sake of consistency.
Figure 4·9 shows ωi(k) from equation 4.4 evaluated numerically for 2-m, 3-m,
and 8-m waves propagating at θ = 15◦, given the initial density and total electric field in each run. In the calculation of ωr(k) for ωi(k), Ve0 includes the Hall drift and the diamagnetic drift. Panels (a), (b), & (c) show that ωi(k) is non-positive everywhere for 2-m waves when E0z = 6 mV/m, and becomes increasingly positive with increasing E0z, as expected. The location of ωi(k) > 0 is not exactly cospatial with the peak in FB irregularities in Figure 4·2c, which coincide with a more localized region centered on the peak in ET. The reason is two-fold: First, 2-m waves develop quickly along the entire positive vertical density gradient of the background wave and are less severely damped near the central trough than longer-wavelength waves.
Nonlinear wave interaction along the density gradient produces cascading features composed of a range of wavelengths from a few to tens of meters, effectively washing out the 2-m waves. Second, the preceding fluid analysis does not capture the fact that the kinetic FB growth rate peaks at a few meters. In the region of enhanced electric field, the true growth rate (i.e. including kinetic effects) will be higher for waves with wavelengths of a few meters.
There are also trends for fixed E0z and varying wavelength. Panels (a), (d), &
(g) show that, in the run with E0z = 6 mV/m, ωi(λ = 8 m) > ωi(λ = 3 m) >
ωi(λ = 2 m) with ωi(λ = 2 m) ≤ 0. This is consistent with Figure 4·2a, in which long-wavelength gradient-drift turbulence grows along the positive vertical density gradient of the background wave. For E0z = 9 mV/m, panels (b), (e), & (h) show that ωi(λ = 8 m) ≈ ωi(λ = 3 m) ≈ ωi(λ = 2 m) > 0 along the positive gradient near the central region but ωi(λ = 8 m) > ωi(λ = 3 m) > ωi(λ = 2 m) ≈ 0 away from the center. This is consistent with the increased growth of meter-scale waves in the central region and the predominance of longer wavelengths near the edges, but does not exactly predict the smallest-scale wave growth in the central trough for the reasons described above. For E0z = 12 mV/m, panels (c), (f), & (i) show
Growth Rate at = 15º!
0128256384 Zonal [m]0128256384 Zonal [m] -60-40-200204060 γ [s−1 ]
(a)! (b)!
(d)! (e)!
(c)!
(f)!
(g)! (h)! (i)!
Figure 4·9: Local linear growth from equation 4.4 for waves traveling at θ = 15◦ from +ˆx. Rows (top to bottom): λ = 2 m, λ = 3 m, and λ = 8 m. Columns (left to right): E0z = 6 mV/m, E0z = 9 mV/m, and E0z = 12 mV/m.
that ωi(λ = 2 m) > ωi(λ = 3 m) > ωi(λ = 8 m) > 0 along the positive gradient.
Again, the prediction made by the fluid growth rate is consistent with wave growth along the positive density gradient but does not predict meter-scale FB turbulence in the central trough. The reader may benefit from comparing Figure 4·9 to the supplemental movies of relative perturbed density for each simulation run.