[1] The Naval Research Laboratory has recently developed a new two-dimensional code to study equatorial spread F (ESF): NRLESF2. The code uses an 8th order spatial interpolation scheme and the partial donor cell method. This allows the model to capture very sharp gradients over ∼ 4 grid cells and to assess the impact of numerical diffusion on the dynamics of ‘bubble’ evolution. Simulation results are presented that show new and complex ESF bubble dynamics: multiple bifurcations, secondary instabilities, density ‘bite-outs’ of over three orders of magnitude, and supersonic flows within low density channels (V ≃ few km/s). These results are consistent with radar and satellite observations, as well as optical images. It is also shown that numerical diffusion can inhibit bubble bifurcation and the development of small-scale structure.

[2] The equatorial ionosphere can become unstable to a Rayleigh-Taylor-like instability after sunset; this often leads to large scale electron density bubbles that can rise to high altitudes (∼1000 km) [Ossakow, 1981; Kelley, 1989]. Associated with these bubbles are smaller-scale, secondary fluid instabilities that can develop on the ‘walls’ of the bubble (G. Haerendel, preprint, 1974), as well as kinetic instabilities [Huba et al., 1978]. This phenomenon is generally known as equatorial spread F (ESF) because it ‘spreads’ the return radio signal from ionosondes [Booker and Wells, 1938]. It is a concern because ESF causes the scintillation of radio signals that can degrade and disrupt communications and navigation systems.

[3] Computer simulation models are required to understand the dynamic, nonlinear evolution of equatorial spread F. The breakthrough simulation study of ESF was performed by Scannapieco and Ossakow [1976] who demonstrated bubble rise through F-peak and into the topside of the F-region. Subsequently, extensive modeling studies were performed at NRL by Zalesak and co-workers [Ossakow et al., 1979; Zalesak and Ossakow, 1980; Zalesak et al., 1982]. More recently, Sekar and coworkers have performed a number of ESF modeling studies [Sekar, 2003, and references therein]. These numerical studies have shed light on a number of processes affecting the evolution of ESF: the role of a conducting E-layer, an inhomogeneous neutral wind, seeding conditions, molecular ions, and the interaction of multiple bubbles. Despite the progress made by these studies there are still a number of observations that have not been reported in simulation studies.

[4] In this Letter we present new results of the onset and evolution of ESF bubbles using a 2D simulation code (NRLESF2) recently developed at the Naval Research Laboratory. The simulation study shows for the first time multiple bifurcations, secondary instabilities, density ‘bite-outs’ of over three orders of magnitude, and supersonic flows within low density channels (V ≃ few km/s). These results are consistent with radar and satellite measurements, and all-sky optical images.

2. Numerical Model

[5] The Naval Research Laboratory has recently developed a new two-dimensional code to model equatorial spread F: NRLESF2. The basic equations solved in this model are [Zalesak et al., 1982]

where V_{⊥} = ∇Φ × e_{z}/Bc, ν_{in} is the ion-neutral collision frequency, V_{n} is the neutral wind, n is the electron density, g is gravity, x is the longitudinal direction (i.e., east-west direction), and y is altitude. We neglect the recombination loss term in equation (1) for simplicity and because it does not affect the growth rate of the Rayleigh-Taylor instability [Huba et al., 1996]. The code uses an Adams-Bashforth 2nd order temporal scheme and an 8th order spatial interpolation scheme. The plasma transport scheme is based on the distribution function method [Huba and Lyon, 1999] developed for 3D MHD simulation studies, and uses the partial donor cell method (PDM) for limiting the mass flux [Hain, 1987]. A feature of the PDM method in that numerical diffusion can be regulated through an adjustable parameter α. This parameter can be varied to assess the impact of diffusion of the evolution of the system. The potential equation (2) is solved using MUDPACK developed by John C. Adams (available at http://www.cisl.ucar.edu/css/software/mudpack/). An important feature of the code is that it can be run with very low numerical diffusion thereby permitting extremely sharp density gradients to form.

[6] The initial density profile is a Chapman layer with n(y) = n_{0} exp[1 − ξ − exp(−ξ)] + n_{1} where n_{0} = 10^{6} cm^{−3}, n_{1} = 10^{−6} cm^{−3}, ξ = (y − y_{0})/Δy, y_{0} = 438 km, and Δy = 70 km. This profile is perturbed with a single mode cosine function in x (i.e., east-west direction) with a 10% amplitude. The ion-neutral collision frequency is obtained from the empirical model NRLMSISE00 [Picone et al., 2002] and is ν_{in} ≃ ν_{0} exp(−y/L_{y}) where ν_{0} = 26.7 s^{−1} and L_{y} = 81.4 km. The simulation domain is 250 km in the x-direction (i.e., east-west), and from 250 km to 650 km in the y-direction (i.e., altitude). The mesh size is 192 × 384 so the grid resolution is Δx = 1.30 km and Δy = 1.04 km. We take gravity to be g = −9.8 m/s^{2} and for simplicity we do not impose a neutral wind V_{n} = 0. Finally, we show results for values of the PDM parameter α: α = 0.2 and 0.667; the numerical diffusion increases with increasing α.

3. Results

[7] In Figures 1 and 2we show gray-scale contour plots of the electron density at times t = 1575 s and 1905 s for the two cases α = 0.2 and 0.667. Also shown is the electron density profile as a function of altitude at t = 0 (dashed line). The electron density is normalized to n_{0} = 10^{6} cm^{−3}. As the system evolves, the F-region plasma layer falls because of gravity and the initial density profile is modified. At t = 1575 s the bottomside F layer has become unstable. For the low diffusion case (α = 0.2) the plasma bubble has bifurcated as it rises into the higher density F region. In contrast, for the larger diffusion case (α = 0.667) the bubble does not bifurcate. Subsequently at t = 1905 s, the low diffusion bubble has undergone small scale structuring on the ‘interior’ walls of the two, main bifurcated density structures. On the other hand, for the larger diffusion case, the plasma bubble does not structure and rapidly rises to the topside. This type of bubble behavior is consistent with previous simulation studies of a single layer plasma model [e.g., Zalesak et al., 1982; Sekar and Raghavarao, 1995].

[8] In Figure 3 we show the gray-scale contour plot of the electron density at time t = 2179 s for the low diffusion case α = 0.2. At this time the plasma disturbance has risen to ≃ 600 km and the plasma bubble has undergone multiple bifurcations in the topside F region. We also note that during this bifurcation process small-scale structures interact and merge. The initial bifurcated structure appears to be generated by a Rayleigh-Taylor-like instability that occurs when the bubble ‘flattens’ at the head as it rises through the ionosphere. A theory for this instability has been described by Hysell [2000] for an idealized bubble model. The subsequent small-scale structuring on the ‘interior’ walls of the bifurcated structures is consistent with a Rayleigh-Taylor-like instability because gravity and the density gradient are oppositely directed. In addition to gravity driving the instability, we note that there are also strong velocity shear flows at the edges of the bubbles that can affect the dynamics of the small scale structures. The role velocity shear plays in the development of the subsequent bifurcations needs further study. Finally we note that these results are consistent with observations: multiple bifurcated bubbles have been observed using radar measurements [Tsunoda et al., 1982] and in optical airglow images [Mendillo and Tyler, 1983; Makela and Kelley, 2003, 2005].

[9] An important point highlighted in Figures 1 and 2 is the impact of numerical diffusion on the simulated evolution of ESF. We demonstrated this by adjusting the PDM parameter α. We did not include a physical diffusion coefficient in the model equations for two reasons. First, physical diffusion is not included in previous ESF simulation studies, and second, the numerical diffusion is larger than the physical diffusion for the spatial grid considered here. The classical (i.e., collisional) cross-field diffusion coefficient is D_{C} = 2(ν_{en}/Ω_{e})(ckT/eB) [McDonald et al., 1981] and the turbulent (i.e., Bohm) diffusion coefficient is D_{B} = (1/16)(ckT/eB) [Huba, 2006]. In the F region D_{C} ≪ D_{B} because ν_{en} ≪ Ω_{e}; we estimate the diffusion coefficient to be D ≃ D_{B} ≃ 100 m^{2}/s for T = 1000 K and B = 0.5 G. The lifetime of a density gradient structure with L ≃ 5 km (i.e., L ≃ 4 Δx) is t ≃ 2.5 × 10^{5} s which is substantially longer than can be maintained numerically (as well as being physically meaningful). We conclude that a low numerical diffusion algorithm is needed to model the dynamics of ESF.

[10] In Figure 4 we plot the density as a function of east-west direction at time t = 2179 s at two altitudes: 370 km and 600 km. We find that the plasma density can drop over 3 orders-of-magnitude in plasma bubbles in the lower F-region. This result is consistent with observations from the Atmospheric Explorer satellite AE-C in which “large-scale (10 to >200 km) irregular bite-outs of up to three orders of magnitude” are seen in the magnitude of the ion concentration [McClure et al., 1977]. We also find that at high altitude (600 km) the plasma density can drop by a factor of 5 inside a plasma bubble. Again, this is consistent with AE-C observations [McClure et al., 1977] and ROCSAT-1 observations [Su et al., 2002; Le et al., 2003]. (We note that ROCSAT also observed even larger density bite-outs during a magnetic storm.) We also mention that ROCSAT-1 observed density enhancements (up to a factor of 2 – 3 above the ambient density) associated with ESF [Le et al., 2003]. At time t = 2167 s we observed a 17% increase in the plasma density at 600 km, just prior to the emergence of the density bubble. This is shown by the dashed line at 600 km in Figure 2. This is not as large some of the enhancements observed by ROCSAT-1 but nonetheless indicates that the physical process may be similar: the leading edge of a rising bubble produces a local density compression.

[11] In Figure 5 we plot the upward velocity V_{y} as a function of east-west direction at time t = 2179 s at two altitudes: 370 km and 600 km. We find that the velocity is as high as V_{y} ≃ 2 km/s at 370 km and is V_{y} ≃ 0.8 km/s at 600 km. The ion sound velocity for O^{+} at T ≃ 0.1 eV is C_{s} ≃ 1 km/s so these flows are supersonic in the lower F region. Equatorial bubbles have been observed rising at supersonic speeds from satellite [Aggson et al., 1992] and radar [Hysell et al., 1994] measurements. Two supersonic rising bubbles were reported by Aggson et al. [1992] from San Marco satellite data: the measured velocities were 1.7 km/s and 1.9 km/s. Hysell et al. [1994] reported an upwelling velocity ≃ 1.2 km/s using the CUPRI radar. To our knowledge, this is the first ESF simulation result showing supersonic flows. Additionally, ROCSAT-1 has observed upward flows V ≃ 0.4 km/s at an altitude of 600 km [Le et al., 2003] which is consistent with our simulation results. We also note there are downflows on either side of the plasma bubble at 600 km. This is indicative of vortices at the head of the rising bubble.

4. Summary

[12] We have presented new modeling results of equatorial spread F using a recently developed 2D code developed at the Naval Research Laboratory. In particular we have observed for the first time multiple bifurcations, secondary instabilities, density ‘bite-outs’ of over three orders of magnitude, and supersonic flows within low density channels (V ≃ few km/s). These results are consistent with radar and satellite measurements, and all-sky optical images. Furthermore, we have shown that using a low diffusion numerical scheme is critical to simulating ESF bifurcations and the development of smaller-scale structures.

[13] The model used in this Letter is very basic: it is predicated on a local description of the plasma and assumes the magnetic field lines are equipotentials. Needless to say, the model does not capture the full physics of ESF dynamics. However, it captures several aspects of the complex nonlinear behavior of rising plasma bubbles (noted above) that have not been reported by other modeling efforts to date. Alternative ESF models are based on flux-tube integrated quantities and include coupling to the E-region [Zalesak et al., 1982; Haerendel et al., 1992], and non-equipotential field line models that describe ballooning-mode type instabilities [Basu, 2002]. Basu [2002] provides a description of these different models in the linear regime and reports differences in the linear growth rates (a factor of 2 at most) between the various models. However, the focus of this paper is on the nonlinear dynamics of the system, not the linear phase.

[14] There are several areas we are currently investigating to improve the model as it applies to the equatorial ionosphere: inclusion of an E layer [Zalesak et al., 1982], adding ion inertia to better model supersonic flows and high altitude dynamics, and extending the model from 2D to 3D. Lastly, there are a number of future simulation studies planned to address critical ESF issues. For example, the ‘seed’ mechanism(s) for ESF bubbles is not entirely resolved. Two prominent triggering mechanisms are gravity waves [Hines, 1960; Kelley et al., 1981] and bottomside electrostatic turbulence associated with shear instabilities [Hysell and Kudeki, 2004; Hysell et al., 2005]. We intend to examine both of these mechanisms in a future paper.

Acknowledgments

[15] We thank S. Zalesak, S.L. Ossakow, D. Hysell, V. Eccles, and J. Makela for helpful discussions. This research has been supported by ONR.