ESF is one of the oldest scientific topics in the equatorial aeronomy. ESF phenomena were observed mostly with ionosonde in early studies. An ionosonde can be viewed as an HF sweep frequency radar which receives signals backscattered from regions where the transmitted frequency is near the electron plasma frequency. Booker and Wells (1938) observed diffuse echoes from the F region of the equatorial ionosphere over a wide range of wave frequency and suggested that the diffuse echoes were caused by irregularities in the ionospheric F region. The ionogram traces showed a range of virtual heights as if the echoing region were spread over a range of altitudes, which is termed range spread F. At other times, the spread occurred only at the high frequency end, which is termed frequency spread F. In most studies with ionosonde, spread F was identified when diffuse traces occurred in the ionogram. Abdu et al. (1983) used range spread F index numbers, from 0 to 3, to specify the degree of spreading in the ionogram, with zero representing the absence of spread F, 1 and 2 representing weak and moderate degrees of range spreading, and 3 representing severe spreading to the extent of making the virtual height reading impossible.

Incoherent and coherent scatter radars have been used to study ionospheric irregularities and structures. Farley et al. (1970) performed a study of spread F using the radar at Jicamarca and found that strong scattering from field aligned irregularities (spread F irregularities) completely masked the incoherent scatter data. Backscattered echoes from spread F irregularities can extend from bottomside to topside F region, forming ionospheric plumes (Woodman and La Hoz 1976). In radar observations, spread F was often identified from backscattered echoes. Recently, Smith et al. (2016) introduced two metrics, “ESF occurrence index” and “ESF maximum height,” to describe ESF intensity. The first metric, ESF occurrence index, is defined as the percentage of bins between 1900 and 2400 LT and between 200 and 800 km altitude. The metric value is zero if ESF does not occur and 1 if ESF irregularities occur at all times between 1900 and 2400 LT and reach all altitudes in this time interval. The second metric is one based on the maximum height of the observed echoes, and the metric values range between 200 and 800 km or 0 (no ESF at all).

Measurements from low-Earth orbit satellites have been widely used to study ESF phenomena. Figure 1 shows an example of measurements made by the Communication/Navigation Outage Forecasting System (C/NOFS) satellite. In Fig. 1a, the blue line depicts the magnetic equator, the red line represents the latitude of C/NOFS, and the dashed magenta line, labeled on the right, represents the altitude (in km) of C/NOFS. The red line is very close to the blue line, indicating that C/NOFS flew almost along the magnetic equator. Figure 1b shows the ion density along the satellite track, and the irregular structures represent ESF irregularities.

Two quantitative definitions of ESF irregularities based on satellite measurements have been used in previous studies. One definition is the standard deviation of ion density variations in logarithmic scale divided by the mean of ion density in logarithmic scale, given by

$$ \sigma \left(\%\right)=100\times \frac{{\left[\frac{1}{10}{\sum}_{i=1}^{10}{\left(\log {N}_i-\log {N}_{0i}\right)}^2\right]}^{1/2}}{\frac{1}{10}{\sum}_{i=1}^{10}\log {N}_{0i}} $$

(1)

In the studies of Su et al. (2006, 2008) with measurements of the ROCSAT-1 satellite, the 1-s averaged data of the ion density were used. The 1-s data were linearly detrended in a 10-s segment (corresponding to ~ 70 km along the satellite track). *N*_{
i
} and *N*_{0i} in Eq. (1) are the measured ion density and the linearly fitted value at the *i*th data point, respectively. Kil et al. (2009), also using ROCSAT-1 data, used 100-s data (~ 700 km along the satellite track) instead of 10-s data, and the background ion density (*N*_{0i}) was represented by an 11-point smoothing curve. In the study of Huang et al. (2014), the ion density data were measured by the Planar Langmuir Probe (PLP) onboard the C/NOFS satellite. The original PLP measurement was made at 512 Hz. The ion density data used by Huang et al. (2014) were the 1-s averages over the 512 samples per second. In other words, the ion density data in the study of Huang et al. (2014) had temporal resolution of 1 s. The *σ* value is calculated over 10 data points (10 s). Each *N*_{
i
} is the ion density at the *i*th data point, and each *N*_{0i} is the average value over 60 s (corresponding to ~ 420 km along the satellite track) centered at the *i*th data point. Two earlier studies (Kil and Heelis 1998; McClure et al. 1998) used \( \sigma =\Delta {N}_i/\overline{N_{0i}} \) to define ESF but did not use the 10-point average. Occurrence of ESF is identified with *σ* > 0.3 or *σ* > 1%.

Another definition of ESF irregularities is the absolute perturbation of the ion density, given by

$$ \Delta N={\left[\frac{1}{10}{\sum \limits}_{i=1}^{10}{\left({N}_i-{N}_{0i}\right)}^2\right]}^{1/2} $$

(2)

In the studies of Huang et al. (2014) and Huang and Hairston (2015) using C/NOFS data, the method for the calculations of Δ*N* is the same as that for *σ*.

Both *σ* defined by Eq. (1) and Δ*N* defined by Eq. (2) can be used to characterize plasma density perturbations for ESF occurrence. When the background plasma density is high, especially in the evening sector during the periods of high solar radio flux, the global distributions of ESF occurrence based on the two definitions are similar. Significant difference in the global distributions occurs when the background plasma density is low, for example, in the postmidnight sector near June solstice at deep solar minimum (Huang et al. 2014). The global distribution of ESF occurrence based on Δ*N* with C/NOFS data at moderate solar activity is quite similar to that based on *σ* with ROCSAT-1 data (Kil et al. 2009).

Woodman and La Hoz (1976) proposed that the ionospheric plumes (ESF echoes) measured by the Jicamarca radar were caused by the plasma Rayleigh-Taylor instability. The numerical simulations of Scannapieco and Ossakow (1976) verified that the ionospheric plasma instability can indeed evolve into plasma bubbles. Sultan (1996) derived the linear growth rate of the Rayleigh-Taylor instability:

$$ \gamma =\frac{\Sigma_P^F}{\Sigma_P^E+{\Sigma}_P^F}\left({V}_P-{U}_L^P-\frac{g_e}{\nu_{\mathrm{eff}}^F}\right){K}^F-{R}_T $$

(3)

where \( {\varSigma}_P^E \) and \( {\varSigma}_P^F \) are the flux tube integrated E and F region Pedersen conductivities, *V*_{
P
} is the flux tube integrated plasm velocity perpendicular to the magnetic field, \( {U}_L^P \) is neutral wind in the geomagnetic meridional plane, *g*_{
e
} is the effective gravity, \( {\nu}_{\mathrm{eff}}^F \) is the effective F region collision frequency weighted by number density along the flux tube, *K*^{F} is the F region flux tube electron content height gradient, and *R*_{
T
} is the flux tube recombination rate.

As can be seen from Eq. (3), the vertical plasma drift contributes to the growth rate of the Rayleigh-Taylor instability through different processes. Besides the direct contribution by *V*_{
P
}, an upward plasma drift moves the F layer to higher altitudes and increases the growth rate by causing a decrease in \( {\nu}_{\mathrm{eff}}^F \) and in *R*_{
T
}. Other two factors, the neutral wind and the ion-neutral collision frequency, also affect the growth rate of the instability.

In Eq. (3), the term (\( -{g}_e/{\nu}_{\mathrm{eff}}^F \)) is positive because of the downward gravity. *K*^{F} is positive in the bottomside F region and negative in the topside F region. If we do not consider the plasma drift velocity, neutral wind, and recombination, the linear growth rate of the Rayleigh-Taylor instability is positive in the bottomside F region and negative in the topside F region. Farley et al. (1970) discussed the possible mechanisms for the generation of plasma irregularities in the equatorial topside F region and concluded that none of the linear theories of plasma instabilities (including the gravitational Rayleigh-Taylor instability) that had been put forward to explain the occurrence of ESF could explain the observations. The Rayleigh-Taylor instability must evolve into the nonlinear regime to reach the topside F region, as illustrated by Woodman and La Hoz (1976) and simulated by Scannapieco and Ossakow (1976).

The Rayleigh-Taylor instability must be initiated in the bottomside F region by some perturbations in the ionosphere or thermosphere before it grows into plasma bubbles or topside ESF irregularities. Two types of seeding perturbations have been proposed. One is atmospheric gravity wave, and the other is large-scale wave structure, a structure in the ionospheric plasma density. Shear flow in the lower F region of the equatorial ionosphere could be the source of large-scale waves or Kelvin Helmholtz instabilities (Hysell and Kudeki 2004). In numerical simulations, the initial condition is often a perturbation in the plasma density, with an amplitude of 1–5% (Scannapieco and Ossakow 1976; Zalesak et al. 1982), or a gravity wave. In the simulations of gravity wave seeding (Huang and Kelley 1996a, 1996b; Krall et al. 2013), the only initial perturbation for the Rayleigh-Taylor instability is a gravity wave, the first-order perturbation in the neutral velocity, and no initial perturbation in the plasma density is required. The gravity wave is to seed the Rayleigh-Taylor instability but does not contribute to the linear growth rate of the instability (see Huang and Kelley 1996a, 1996b for details).

Although initiation by a gravity wave seems likely, the gravity wave interaction cannot yield the large displacements observed without further amplification by the Rayleigh-Taylor instability, as suggested by Kelley et al. (1981, 1986). The simulations of Huang and Kelley (1996a, 1996b) and Krall et al. (2013) have demonstrated the formation of plasma bubbles seeded by gravity waves. Seeding by large-scale wave structure was proposed by Tsunoda (2005). Observational evidence of large-scale wave structure and its possible role in the generation of ESF irregularities was reported by Tsunoda and White (1981) and Thampi et al. (2009). Although ESF irregularities and plasma bubbles appear to be periodic sometimes (e.g., Huang et al. 2013), they are irregular and not periodic in most cases (e.g., Huang et al. 2012). Choi et al. (2017) analyzed C/NOFS data and found that a representative periodicity does not exist in the occurrence of bubbles and that the bubble spacing is generally irregular. Seeding perturbations for the Rayleigh-Taylor instability do not have to be periodic. Perturbations in the thermosphere and ionosphere, large scale or small scale, periodic or non-periodic, regular or irregular, can all initiate the Rayleigh-Taylor instability.