# Equatorial Plasma Bubbles: A Review
## 3 Development of EPBs and Associated Irregularities: Theoretical Aspects
### Electrostatic Rayleigh-Taylor Instability
In the presence of a small sinusoidal perturbation in plasma density on the bottom side of the post-sunset equatorial F layer, a small perturbative electric field initially develops and the R-T instability starts to grow, as discussed in the last paragraph and shown in Figure 6. In order to determine the linear growth rate of the R-T instability in this initial stage, a set of equations has to be considered [54]. Only terms that are linear  
Figure 6: In this schematic figure, the shaded region represents the equatorial F layer, with higher plasma density, resting over a lower plasma density region (unshaded). The geomagnetic field is horizontal and northward (into the page). A gravitational current flows in the eastward direction. In the presence of a perturbation in plasma density on the bottom side of the F layer, perturbation electric fields \(\Delta\mathbf{E}\) develop. Motion of the plasma in the \(\Delta\mathbf{E}\ \times\ \mathbf{B}\) direction due to the perturbative electric fields is indicated by upward and downward arrows. in the density perturbation and electric field perturbation are retained in the continuity equation, which determines the temporal evolution of the density perturbation and the current divergence equation \(\
abla\bullet\textbf{J}~{}=~{}0\). These equations yield a dispersion relation for the wave-like perturbations in plasma density and electrostatic potential, and the linear growth rate \(\gamma\) of the electrostatic R-T instability is obtained as in [57; 58]:  
\[\gamma~{}=~{}\bigg{\{}-v_{in}~{}+~{}\Big{(}v_{in}^{2}~{}+~{}4g/L\Big{)}^{1/2} \bigg{\}}/~{}2 \tag{3}\]  
where \(g\) is the magnitude of the acceleration due to gravity and \(L~{}=~{}(\
abla N/N)~{}^{-~{}1}\) is the upward density gradient scale length. It can be seen from Equation (3) that, in the absence of collisions between ions and neutrals (\(v_{in}~{}=~{}0\)), the collisionless R-T growth rate is \(\gamma~{}=~{}(g/L)~{}^{1/2}\). On the other hand, when ion-neutral collision frequency is large enough to satisfy \(v_{in}~{}\gg~{}(g/L)~{}^{1/2}\), Equation (3) yields the R-T growth rate in the collisional limit:\(\gamma~{}=~{}g/~{}(L~{}v_{in})\)[54]. In the derivation of Equation (3) for the growth rate, the effects of an ambient zonal electric field and neutral wind, which are both generally present at the altitude of the base of the post-sunet equatorial F layer, have been ignored. Inclusion of these effects leads to generalized Rayleigh-Taylor (GRT) instability. Another complication is that the EPBs are aligned with the geomagnetic field, and the ionospheric parameters, which determine the growth rate of the R-T instability on the bottom side of the equatorial F layer, vary along a geomagnetic field line in both the hemispheres. In order to take this into account in a 2-D theory of R-T instability, [PERSON] [57] suggested that geomagnetic flux tube-integrated ionospheric parameters should be used. Following this suggestion, [PERSON] [59] derived a flux tube-integrated linear growth rate of a GRT instability, which includes the effects of an ambient zonal electric field **E** and neutral wind:  
\[\gamma_{RT}~{}=~{}\frac{\Sigma_{p}^{F}}{\Sigma_{p}^{F}~{}+~{}\Sigma_{p}^{E}} \left(V_{p}~{}-~{}U_{L}^{P}~{}-~{}\frac{g_{e}}{v_{eff}^{F}}\right)~{}K^{F}~{}- ~{}R_{T} \tag{4}\]  
Here, \(\Sigma_{p}^{F}\) and \(\Sigma_{p}^{E}\) are flux tube-integrated Pedersen conductivities of the F and E regions of the ionosphere along the geomagnetic field line connecting the equatorial F region with the conjugate E regions in both hemispheres. \(V_{P}\) is the vertical plasma drift at the dip equator due to an ambient zonal electric field **E**.