Effects of molecular ions on the collisional Rayleigh-Taylor instability: Nonlinear evolution

Authors


Abstract

[1] The nonlinear evolution of collisional Rayleigh-Taylor (CRT) instability in the F region of ionosphere is studied for two ions-electron (O+, NO+, and e) plasma. The enhancements in NO+ are found inside the O+ depletions. During linear phase of instability, NO+ concentration is mainly controlled by chemical process while during nonlinear phase, it is controlled by CRT induced transport process. In spite the dominant chemical loss during initial phase, significant NO+ ions could survive owing to their shielding by O+ depletions. The seeding of long wave perturbation, the O+ scale height below the base and prereversal enhancement play important role in supplying significant NO+ ions onto the topside which could possibly match the observations.

1. Introduction

[2] The large-scale plasma irregularities associated with equatorial spread F (ESF) phenomenon are generally believed to be generated by CRT instability [Haerendel, 1973]. In the presence of steep plasma density gradients in the bottomside of F region, the instability acts in such a way that regions of low plasma density below the F layer peak are transported to altitudes well above the F peak [Ossakow, 1981; Raghavarao et al., 1992] and seen as plasma depletions or bubbles which confirm the observations. The ion composition within the depletions has been the subject of a number of investigations. Typically, satellite mass spectroscopic observation [McClure et al., 1977] revealed that the ion composition can be vastly different inside and outside the depletions. Atomic O+ ions are depleted by as much as a factor of 103 to a concentration below that of the molecular NO+ ions. The NO+ ions were found to be in the topside and dominant in the O+ depleted region. Since O+ depletions on the topside are caused by nonlinear CRT instability mechanism, study of its evolution in presence of NO+ ions becomes important. The observation implies that the ions are efficiently transported from the altitude region where NO+ are the major ions. However, the ion composition measurements [Brinton et al., 1975; Narcisi and Szuszczewicz, 1981] and theoretically calculated profiles of NO+, O2+, and O+ [Anderson and Rusch, 1980] reveal that where NO+ become dominant ions in the bottomside, plasma scale height attains very large value. It inhibits the instability to grow therein and restricts the transport of plasma to higher altitudes. Therefore observations of bubble composition [McClure et al., 1977], on one hand, seem to imply that ions are efficiently transported from the altitude region where NO+ are the major ions. On the other hand, plasma scale height is very large in this altitude region which restricts the transport of plasma by CRT mechanism. This apparent contradiction needs to be examined more carefully.

[3] Until recently, the linear and nonlinear investigations of CRT instability were dealt by considering F region as one ion-electron (O+-e) plasma and ignoring the NO+ ion dynamics. Recent study of linear CRT instability by Sekar and Kherani [1999], however, shows that the introduction of NO+ as dominant ions along with the O+ dominant ions reduces the linear growth rate of CRT instability. Nevertheless, reduced growth rate is still larger than the damping rate. Present study deals with the nonlinear aspect of CRT instability in the presence of O+ and NO+ as major ionic species i.e; by considering the F region as two ions-electron plasma.

2. Numerical Simulation Model

[4] The nighttime equatorial ionosphere is considered in a slab geometry perpendicular to the magnetic field. Moreover instead of flux line integrated quantities, local values of physical quantities are used. This is a valid assumption in nighttime F region when E region conductivities become small. The positive unit vectors equation image, equation image, and equation image of the Cartesian coordinate system are directed along westward, upward, and northward respectively. The current continuity equation is solved for electrostatic potential while continuity equations for both the ions are solved for their corresponding densities.

equation image
equation image
equation image

where current density is given by

equation image

Here number densities of O+, NO+ ions and electrons are denoted by n1, n2, and ne, respectively, and ne = n1 + n2, while their velocities are denoted by equation image, equation image, and equation image. β and α are the reaction rates for ion-molecule exchange and dissociative recombination reactions respectively.

equation image

The steady state velocities in terms of forces are given by

equation image
equation image

where equation image, equation image, equation image, and νin are the electric field, Earth's magnetic field, neutral wind, and ion-neutral collision frequency, respectively, while Ω1,2 and m1,2 are the gyrofrequencies and masses of O+ and NO+ ions, respectively. The current density, from equation (4), can then be written as

equation image

In obtaining above expression, charge neutrality condition is used. With electrostatic perturbation (−∇ϕ) in electric field over ambient field equation image, equation (1) finally reduces to following equation:

equation image

which describes the spatial variation of polarization potential. Here Exo and Wy are the westward field and upward wind respectively. Temporal evolution of number densities of both the ions are obtained by solving the continuity equations separately:

equation image
equation image

where Eyo is the vertical electric field. For present investigation, only the effects of gravity and zonal electric field Exo are considered.

[5] Equations (9)–(11) are solved over region of 400 km east to 400 km west in zonal direction and over 182 to 532 km in vertical direction. The grid sizes are chosen to be uniform with values of Δx = 5 km and Δy = 2 km. As the problem of interest is to study the growth of the periodic perturbation in ion densities in zonal direction, periodic boundary conditions are imposed on both ni (i = 1,2) and ϕ in the zonal direction. In vertical direction, transmitive boundary conditions on number densities, equation image, and Neumann boundary conditions on ϕ, equation image, are imposed. Numerical scheme adopted by Sekar [1990] for one ion-electron plasma is modified for two ions-electron plasma in present investigation.

[6] Density profiles of both the ions and temporal variation of Exo are shown in Figure 1. The initial density perturbation is chosen of the form, n1,2(x, y) = n01,02(y)(1. − a cos[2πx/λ]), where equation image is the amplitude of initial perturbation defined as the fractional change in number density and λ is the wavelength of perturbation. Isodensity contours shown in Figures 2a and 3a, for O+ and NO+ ions respectively, represent initial perturbation. For O+ ions, starting from lower most contour with density value 103, density values of successive contours are 103.5, 104, 104.5, and 105 cm−3. For NO+ ions, starting from lower most contour with density value 1.5 × 103, density values of successive contours are 103, 5 × 102, and 102 cm−3. Similar choice is maintained throughout the investigation. Slight modulations, upwellings and downwellings, of contours in Figure 2a and 3a correspond to initial amplitude of perturbation, 5%, with respect to reference density values at different altitudes. The characteristics of these modulations are remarkably different for those O+ and NO+ contours which fall on altitudes above the base. Wherever upwellings are seen in O+ contours, downwellings in NO+ contours are noticed.

Figure 1.

Density profiles of both (a) ions and (b) temporal variation of ambient vertical plasma drift.

Figure 2.

Time evolution of O+ isodensity contours.

Figure 3.

Time evolution of NO+ isodensity contours.

[7] The linear analysis of CRT instability for two ions-electron plasma [Kherani, 2001] reveal that the growth rate is highly sensitive to density ratio n2/n1 and scale height ratio L2/L1 above the base where L1 and L2 are O+ and NO+ ion scale heights respectively. Moreover, simulation of CRT instability for one ion-electron plasma [Zalesak and Ossakow, 1980] reveal that the nonlinear growth depends on the wavelength of initial perturbation used in simulation. Following these features, various cases are considered in present investigation and are enlisted in Table 1. With the numerical simulation model discussed above, case 1 is studied as a first step toward the understanding of CRT instability with two ions. We present results in following section and discuss various features in subsequent section.

Table 1. Cases Considered for Simulation
CaseλLaboveaLbelowbO+-NO+ % at the base
  • a

    O+ scale height above base.

  • b

    O+ scale height below base.

Case 1200152050–50
Case 220052050–50
Case 3600152050–50
Case 42005750–50
Case 5200152085–15

3. Results Corresponding to Case 1

[8] In order to know the evolution of perturbation in O+, isodensity contours are plotted in Figures 2b–2f for different phases of evolution. Different phases chosen for investigation are 100, 300, 500, 700, and 1000 s which are written on the top of each plot. The growing upwelling and downwelling structures of contours are seen throughout the phases of evolution. During first 500 s (initial phase), structures are seen as slight modulations over initial isodensity contours. They grow slowly but not significantly to become prominent over initial density contours. After 500 s (later phase of evolution), these structures, however, grow rapidly and become prominent over initial density contours.

[9] The NO+ isodensity contours are plotted in Figures 3b–3f for similar time sequence. Interestingly, initially downwelling (upwelling) structures in Figure 3a above the base are now appeared as upwellings (downwellings) at 100 s and continue to grow during subsequent times. However, these structures grow rapidly and become prominent over initial density contours only after 500 s similar to O+ ions. More surprisingly, these contours are confined to smaller altitude range at 100 s than at 0 s. They continue to confine further during initial phase but not during later phase of evolution.

[10] In Figure 4a, altitude profiles of O+ ion density inside the upwelling structure are shown for similar choice of time sequence as earlier. During initial phase of evolution, profile is shifted to higher altitudes and no appreciable change is seen in the shape of profile as compared to initial altitude profile shown in Figure 1a.

Figure 4.

Time evolution of density variations of (a) O+ and (b) NO+ within the upwelling structures.

[11] However, at 700 s during later phase, O+ density profile becomes steepened above 400 km altitude compared to initial density profile. During subsequent time, profile becomes more and more steepened but the altitude, above which it occurs also shifted to higher altitudes.

[12] In Figure 4b, altitude profiles of NO+ ion density inside the upwelling structure are depicted. Unlike O+ case, its shape changes during initial phase also. At 100 s, density profile above 350 km altitude becomes steeper compare to initial density profile. At 300 s, it steeps further. At 500 s and beyond, however, profile becomes less and less steeper again. In Figure 5, density variations of O+ and NO+ ions along an assumed path is plotted. Throughout the ranges, NO+ density enhancements are collocated with O+ depletions.

Figure 5.

Density variations of both the ions at 1000 s along an assumed path.

4. Discussion

4.1. Upwelling Structures as O+ Depletions and NO+ Enhancements

[13] The slight upwellings of each contour in Figure 2a correspond to initial depletions in density with respect to the density value of corresponding contour. The time evolution of upwelling structures in Figure 2 confirms earlier numerical investigations [Scanapiecco and Ossakow, 1978; Ossakow, 1981] results. These upwelling structures are nothing but the O+ bubbles or depletions moving with large upward velocities. It is also evident that instability enters to nonlinear stage within 700 s.

[14] The initial downwellings of density contours lying above the base as depicted in Figure 3a correspond to depletion in NO+ density. The difference in initial morphology of O+ upwellings and NO+ downwellings is owing to their density gradients above the base which are positive and negative, respectively. Interestingly the downwelling structures change their morphology and become upwellings immediately within 100 s. Since NO+ number density decreases with altitude above the base, it implies that initially depleted NO+ region becomes enhanced region at 100 s. As time progress, an upwelling structure or enhanced region are stretched out to higher altitudes and seen as more and more enhanced region. The upwelling structures as depleted O+ and enhanced NO+ on topside is what Figure 5 and observation [McClure et al., 1977] reveal and reason was thought to be the CRT induced transport process [Anderson and Rusch, 1981; Narcisi and Szuszczewicz, 1981]. However, since CRT instability induced transport becomes efficient only after 500 s, it remained as a puzzle that what keeps NO+ ions to survive initially near F region peak where their life-time is much less than 500 s. Before attempting to explain this aspect, we would like to extract out some more information from Figure 4b. In Figure 4b, steepening of NO+ profile above 350 km altitude is seen during first 300 s. This feature is reflected as confinements of contours in Figure 3. Since the CRT instability is in linear phase during this time, any appreciable change in profile should be attributed to either linear phase of instability or some other process (if possible).

[15] However, for NO+, linear phase of instability is not the cause for altering the profile. If it was so then density had to increase at any altitude above 350 km altitude due to the uplift of NO+ ions by linear phase of CRT instability. However, what is noticed during first 300 s is opposite of it. Another process which may be important for NO+ is the chemical process and we will be discussing its effect shortly. After 300 s, profile begins to become less and less steep with the reduction of NO+ number density near 350 km altitude region. It means that the NO+ ions from lower altitudes are transported to higher altitudes significantly after 300 s. Thus the transport process induced by linear CRT instability becomes dominant after 300 s over the process which was more dominant during first 300 s. After 500 s, steepening reduces rapidly indicating the transition of CRT instability from linear to nonlinear phase. The upwelling structures of NO+ density contours seen in Figure 3 after 300 s is thus due to the advection of NO+ from lower altitudes to higher altitudes.

4.2. Transport Process Versus Chemical Process

[16] It is evident from foregoing discussion that NO+ ions exhibit different evolutionary aspects as compared to O+ ions. We feel that these aspects can be attributed to varying importance of different processes, chemical and transport, with time and altitude. For O+ ions, transport process dominates throughout the evolution. However, for NO+ ions, chemical processes are likely to play important role also. The evolution of NO+ ion number density follows continuity equation:

equation image

We define the rate of chemical process as ωc ∼ αne and rate of transport process as equation image. Here vpol and L2 are polarization velocity and NO+ scale height inside the depletion, respectively. The instantaneous value of scale height, L2, is inferred from Figure 4b. In Figure 6, velocity profiles inside the depletion are shown for different phases of evolution. The velocity is obtained using the equation, equation image, where equation image = −∇ϕ and ϕ is the solution of equation (8). For typical initial value of ne ∼ 103 cm−3, vpol ∼ 5 ms−1 at 350 km and value of L2 = 30 km, both ωc and ωt are order of 10−4 s−1. Thus both the rates are of comparable magnitude at 350 km altitude initially. However, as altitude increases, chemical rate dominates over transport rate since ne increases rapidly. At Fpeak, their values are 10−2 and 10−4 s−1 respectively. Thus its chemical process which decides evolution of number density initially for NO+ ions throughout the altitudes except near the base. Since chemical process is the dissociative recombination process through which NO+ ions are lost, more and more NO+ ions are lost as altitude increases. It causes the confinement of contours or steepening of its profile. The steepening of profile and increase in velocity at 100 s enforce ωt to be larger than ωc near the base.

equation image

Above the base, situation becomes more interesting. The depletion in O+ and so in the electron density continuously increases as CRT instability evolves. It reduces NO+ loss rate within the depletion as compared to background. However, effect is pronounced only above 380 km and after 300 s where considerable depletion is found. Thus, at 100 and 300 s, loss rate still remains larger than the transport rate at higher altitudes in spite the increase in transport rate. It again steepens the profile during 300–500 s. Meanwhile, vpol also increases with time. Both steepening in density profile and increase in vpol cause increase in transport rate during 300–500 s. At the same time, increasing electron depletion continuously reduces loss rate within it. At 500 s, degree of O+ depletion increases to 50% near 400 km resulting in 50% reduction in loss rate.

equation image

Hence both the rates become comparable even at 400 km altitude at 500 s. Since O+ density at Fpeak is not much differ from its value at 400 km we can safely say that both the rates attain comparable magnitude throughout the altitudes above the base. It ceases further steepening in NO+ profile as evident in Figure 4b after 300 s. After 500 s its the transport process which mainly decides evolution of NO+ density. In brief, though the chemical rate dominates initially above the base, the steepening of NO+ profile and increase in both polarization velocity and electron depletion make transport rate to rapidly overcome chemical rate. The continuity equation for NO+ ion density therefore can be approximated as

equation image

It is to be noted that even when chemical process dominates, NO+ enhancement will be found inside O+ depletion since the loss of NO+ ions inside the depletion is less than its loss into the ambient. This is the reason why we see NO+ upwelling in Figure 3 even during first 500 s. It is also obvious that the electron depletion acts as a shielding for NO+ ions which will be later supplied to the topside. Here we would like to mention that initial NO+ density profile is not in quasi chemical equilibrium which it is sought to be. We have run the simulation without any perturbation and found that its long way for profile to relax to equilibrium. Moreover, prereversal enhancement is responsible for presence of significant NO+ ions in bottomside F region. Since this is the time when ionosphere becomes susceptible for CRT instability, there is an ample possibility that NO+ profile may not find enough time to relax to chemical equilibrium before instability grows.

Figure 6.

Time evolution of velocity profile within the upwelling structures.

[17] In order to see effects of NO+ ions on the evolution of CRT instability, simulation with O+ alone is performed. The degree of depletion and upward velocities inside the depletion for this case is found to be larger than case 1. Such results are consistent with our linear analysis [Sekar and Kherani, 1999] which revealed that the growth rate in presence of NO+ ions reduces as compared to O+ ions alone.

4.3. Investigations With Other Cases

[18] So far, the nonlinear evolution of CRT instability for two ions-electron plasma is investigated under case 1. The investigation reveals that CRT induced transport process causes advection of both the ions into the topside, where NO+ density enhancements are collocated with O+ depletions consistent with observation [McClure et al., 1977]. However, in one of their ion-composition measurements inside the bubble, O+ ion density as low as 103 cm−3 and NO+ ion density as much as 103 cm−3 were found on topside. In Figure 7a, zonal density variations of both the ions at 460 km altitude are plotted. The O+ ions as low as 104 cm−3 and NO+ ions as much as 6 × 102 cm−3 are seen on topside. These findings are far too close to observation. Such rare observation indicates that both the ions must be transported from below the base where NO+ ions are dominant. Below the base, however, electron density scale height becomes very large and one suspect CRT instability not to work there. In order to resolve such discrepancy, zonal O+ and NO+ density variations at 340 km altitude just below the base is plotted in Figures 8a and 8b. Most importantly what we see is 70% enhancement in NO+ at 1200 s just below the base. The density fluctuations indicate considerable growth of instability below the base. Such growth is due to the nonlocal nature of instability which increases with wavelength of perturbation [Zalesak and Ossakow, 1980; Kherani et al., 2002] and could make instability to grow in the region where even negative gradients exist [Sekar et al., 1997]. However, we see that such growth becomes considerable only after 1000 s. By this time very large O+ scale height develops on bottomside owing to the motion of O+ depletion as seen in Figure 4a. It prevents the motion of plasma from lower altitudes. We have run the simulation for 1800 s and indeed found reduction in the transport of plasma onto the topside after 1200 s. In order to supply significant NO+ ions, thus such fluctuations must develop before the drastic change in O+ scale height. In this context, O+ scale height below the base becomes important. Our linear analysis [Kherani, 2001] reveal that below the base, linear growth rate mainly depends on O+ scale height. Small O+ scale height below the base can make growth faster there. Thus, in order to have close agreement, quantitatively, with the observation, one need to take proper background conditions such as density profiles and perturbation parameters mainly the wavelength. We now examine CRT instability by varying these parameters according to cases listed in Table 1. In Figure 7b, density variations along the zonal direction at 460 km altitude are depicted for both the ions. The simulation time is chosen to be 900 s which is the time of maximum upwelling for case 2. The comparison with Figure 7a reveal that the degree of depletion (enhancement) in O+ (NO+) densities are increased for case 2 compared to case 1. The NO+ ions as much as 103 cm−3 are found on the topside which were not seen for case 1. This feature, particularly, is consistent with satellite observation [McClure et al., 1977]. It is to be noted that NO+ scale height for both the cases is kept unchanged while O+ scale height is made 5 km for case 2. Moreover base of F region is shifted to higher altitude to maintain the same Fpeak altitude as for case 1. We feel that similar to the study [Ossakow et al., 1979] for single-ion case, for double-ion case both reduction in scale gradient and higher altitude shift of base could cause the degree of depletion and enhancement to be more for case 2 as compared to case 1.

Figure 7.

Zonal density variations of both the ions at 460 km altitude for (a) case 1, (b) case 2, (c) case 3, (d) case 4, and (e) case 5.

Figure 8.

Time evolution of zonal density variations of (a) O+ ions and (b) NO+ ions at 340 km altitude.

[19] The density variations along zonal direction are shown in Figure 7c corresponding to case 3. The increase in O+ depletion and NO+ enhancement compared to case 1 and 2 is noticed. Note that the initial O+ and NO+ density profiles are same as in case 1 but wavelength of perturbation is 600 km instead of 200 km. The numerical investigation by Zalesak and Ossakow [1980] for single O+ ion revealed that longer wavelength causes large depletion owing to the supply of O+ ions from much lower altitude as compared to shorter wavelength. We see that 6 × 103 O+ ions cm−3 are seen up to 460 km causing large depletion in O+ (Figure 7c). The O+ ions with density of order of 103 cm−3 were never seen at peak altitude region for case 1 and case 2.

[20] It was discussed above that O+ scale height below the base play an important role for CRT instability in two ions-electron plasma. This possibility is explored in case 4 where O+ scale height below and above the base are chosen as +7 and +5 km respectively. The zonal density variations at 460 km for this case are shown in Figure 7d. Interestingly, the findings of case 4 are very similar to case 3. More O+ ions from just below the base are transported to much higher altitude compared to case 2. The 6 × 103 O+ ions cm−3 are seen beyond 460 km which were never seen for case corresponding to shorter wavelength. Comparison of case 3 and case 4 implies that shorter wave perturbation with small scale height below and above the base can give results similar to long wave perturbation with large scale heights. The situation may arise where suitable combination of these parameters could give rise results similar to that of the rare observation [McClure et al., 1977].

[21] It was shown by Anderson and Rusch [1981] that the prereversal enhancement of electric field is responsible for supplying sufficient NO+ ions up to the base. In our investigation so far we have taken 50–50% O+ and NO+ ions at the base due to the prereversal enhancement. Under case 5, we have studied the situation where relative concentrations of O+ and NO+ ions are 85–15% at the base. The zonal density variations at 1000 s are plotted in Figure 7e show reduction in degree of NO+ enhancement as compared to case 1. The reduction in degree of NO+ enhancement is due to the availability of these ions on the bottomside which is highly reduced for case 5. Thus availability of NO+ ions near the base is also crucial for supplying significant NO+ ions onto the topside. The availability of NO+ ions highly depends on the ambient vertical drift [Anderson and Rusch, 1981]. Thus along with the suitable choice of density profiles and wavelength of perturbation, proper prereversal enhancement is also crucial for transport of significant NO+ ions onto the topside. Since these quantities are highly variable in nature, variability in occurrence of NO+ ions onto the topside [McClure et al., 1977; Narcisi and Szuszczewicz, 1981] is indeed expected.

5. Summary and Conclusion

[22] The investigation of CRT instability in the presence of NO+ ions is dealt with different background conditions like different seeding perturbations and varying density profiles. The results of the investigation are as follows:

  1. Both O+ and NO+ ions from lower heights where NO+ life time is short, are transported to higher altitude owing to the steep density gradient present in O+ ions. However, presence of NO+ ions slows down bubble motion and reduces the degree of depletion as compared to their values without NO+.
  2. Depletion in O+ and enhancement in NO+ ions are collocated throughout the altitudes above the base. It has close similarity with satellite observation [McClure et al., 1977]. The enhancement in NO+ is attributed to chemical effects during initial phase of evolution while attributed to transport process during later phase of evolution.
  3. Though NO+ concentration in initially dominated by chemical loss process, its shielding by electron depletion effectively makes them long-lived throughout the bottomside.
  4. Efficiency of advection depends on density gradients of both the ions, seeding perturbation of wavelength and nature of background electric field. Their day-to-day variabilities can be attributed to the day-to-day variabilities in occurrence of NO+ on topside as observed [McClure et al., 1977; Narcisi and Szuszczewicz, 1981].
  5. Longer wavelength can efficiently supply plasma compared to shorter wavelength from below the base where scale height is very large.
  6. In the presence of steep O+ density gradient below the base, shorter wavelength of perturbation also can supply enough plasma from this region.
  7. Prereversal enhancement in electric field play a crucial role for supplying sufficient NO+ ions up to the base which are then transported to higher altitudes by CRT instability.

Acknowledgments

[23] This work is supported by Department of Space, Government of India.

[24] Janet G. Luhmann thanks John Retterrer and another referee for their assistance in evaluating this paper.

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