Polar cap plasma patch primary linear instability growth rates compared

Four primary plasma instability processes have been proposed in the literature to explain the generation of phase scintillation associated with polar cap plasma patches. These are the gradient drift, current convective, and Kelvin‐Helmholtz instabilities and a small‐scale “turbulence” process. In this paper the range of possible values of the linear growth rates for each of these processes is explored using Dynamics Explorer 2 satellite observations. It is found that the inertial turbulence instability is the dominant process, followed by inertial gradient drift, collisional turbulence, and collisional shortwave current convective instabilities. The other processes, such as Kelvin‐Helmholtz, collisional gradient drift, and inertial shortwave current convective instabilities, very rarely (<1% of the time) give rise to a growth rate exceeding 1/60, that is deemed to be significant (in publications) to give rise to GPS scintillation.


Introduction
Large regions of enhanced electron concentration drifting according to an E × B force from the auroral dayside ionosphere into the polar cap region of the ionosphere were theorized by Hill [1963] as an explanation for "sporadic F" observations via f o F 2 measurements in these areas. This theory was confirmed experimentally by Buchau et al. [1983] using optical all-sky camera images, and the structures (hundreds to thousands kilometer in scale) became generally known as polar cap plasma patches. They were soon observed to cause radio scintillation on radio signals traversing them [e.g., Weber et al., 1984]. Polar cap plasma patches are also strongly correlated with backscatter of high-frequency coherent scatter radar signals, which occurs when the signals meet electron concentration irregularities [Oksavik et al., 2006].
Irregularities cause scintillation through diffractive scattering [Hargreaves, 1992] or ionospheric lensing [Booker and Majidiahi, 1981] of the signal. Scintillation inducing irregularities is considered to form through a cascade of energy from longer wavelengths to shorter ones in a manner analogous to that which occurs in the generation of turbulence in neutral fluids [Kintner and Seyler, 1985]. This requires some initial process to create wave-like structures in the plasma where there are none to start with-a primary plasma instability [Kelley, 2009].
A plasma instability is any wave-like structure that can grow exponentially in amplitude from some initial perturbation that is assumed to preexist. In order for this to happen, some source of free energy-energy available to do work-is required [Mikhailovskii, 1992]. Such energy derives from a combination of the Earth's geomagnetic field and electric field mapped from the magnetosphere to the high-latitude ionosphere, (day to night) electron density gradients, and atmospheric gravity wave "seeding" [e.g., Hysell et al., 2014]. In this paper we consider only the linear regime, where the instability (consisting of alternating regions of higher and lower electron concentration) is assumed to grow exponentially in amplitude, at a particular growth rate. This linear regime is most likely to occur on the bottomside of the F region, with nonlinear processes dominating in the topside, particularly in the equatorial region [Hysell et al., 2014]. the formation of irregularities within the plasma patch. It should, therefore, be possible to calculate γ for each process and determine which instability is ultimately responsible for radio scintillation associated with plasma patches [e.g., Burston et al., 2009Burston et al., , 2010.
Doing this is complicated by the natural variation in the parameters that appear in the growth rate equations from one patch to the next or even for a single patch, both spatially and temporally. For example, the electric field (which appears in all the growth rate equations treated here) can be viewed as fluctuating in magnitude and direction around a quasi-DC component that in itself varies in magnitude and direction on a timescale of tens of minutes and spatial scale of hundreds of kilometers [Hanson et al., 1993], and the orientation of the field with respect to a gradient in electron concentration is always important. Hence, for any given instability the growth rate can take a range of values, depending on the range of values possible for the parameters involved. For this reason different instabilities may dominate in different circumstances that depend on the frequencies of occurrence and ranges of values of various solarterrestrial parameters that appear in growth rate equations.
With four instabilities under examination it would be expected that there would be four growth rate equations but for a single instability, there can in fact be more than one equation. This is because the growth rates may differ in the collisional and inertial plasma regimes and in the long and short perturbation wavelength assumptions. In the inertial regime only the bulk electric fields are important. In the collisional regime, the effects of individual particles' electric fields must be taken into account. The peak of the F region (at 300 km) of the ionosphere represents a transitional region from collisional to inertial regimes (delineated by where the collisional growth rate and ion-neutral collision rate become equal). Thus, it is necessary to consider both situations [e.g., Sojka et al., 1998]. Each instability is outlined below in its own section, with the appropriate growth rate equations given.

The Gradient Drift Instability
The gradient drift instability (GDI) requires a horizontal gradient in electron concentration, parallel to the electric field and a vertical magnetic field. Polar cap patches potentially meet all these requirements as the geomagnetic field is approximately vertical in the polar cap, the electric field has a large horizontal component, and the patch itself has gradients in all horizontal directions [e.g., Basu et al., 1990;Burston et al., 2009;Chaturvedi et al., 1994;Gondarenko and Guzdar, 1999, 2001, 2004a, 2004b, 2006a, 2006bGondarenko et al., 2003;2006b;Guzdar et al., 1998]. The growth rate in the collisional regime is given by where E is the electric field vector, B is the magnetic field vector, n is the electron concentration (Figure 1), and ∇n is the gradient of electron concentration per meter ( Figure 2). In the inertial regime the growth rate is given by (2) [Sojka et al., 1998] where v in is the ion-neutral collision frequency, given by where, in turn, r is the colliding particles' radius, assumed the same for all species and M is the colliding species' mass, also assumed the same for all species. T is the temperature, assumed the same for all species, and n n is the neutral concentration. The constant, k, is Boltzmann's constant [Kauzman, 1966]. This formula for the Journal of Geophysical Research: Space Physics 10.1002/2015JA021895 ion-neutral collision frequency is in fact that for the collision of two neutral species, and a more accurate version would take into account the small correction required for a singly ionized species colliding with a neutral species. The assumption that T is the same for all species is valid in quiet geomagnetic conditions only (Kp < 2), otherwise the collision rate will be larger than those used in this study. In the following, n n is taken as the density of monatomic oxygen ( Figure 3) and T is the O + ion temperature ( Figure 4) as these are the main constituents of the neutral and ionized atmosphere at F region heights.

The Current Convective Instability
The current convective instability may occur when the criteria for the GDI are met and current flows parallel to the magnetic field (z axis). Additionally, the initial perturbation must have a finite component along the direction of ∇n (y axis). For the CCI, there is a wavelength dependency as well as a difference between the collisional and inertial regimes, leading to four equations Ossakow, 1979, 1981;Huba, 1984;Kelley, 2009;Ossakow and Chaturvedi, 1979].
The growth rate for the CCI, collisional long wavelength regime, is given [e.g., Huba, 1984] by where q is the charge on the electron, k z and k y are the wave numbers in the z and y directions, respectively, n 1 and n 2 are the electron concentrations either side of a step-boundary perturbation (with n 2 > n 1 ), m e is the mass of an electron, m i is the mass of an ion, and v ei is the electron-ion collision frequency. The latter, in SI units, is given [e.g., Ichimaru, 1973] by  The growth rate for the CCI collisional short wavelength regime is given [e.g., Huba, 1984] by The growth rate for CCI inertial long wavelength regime is given [e.g., Huba, 1984] by CCI inertial short wavelength The short wavelength regime applies when kL << 1 and the long wavelength regime applies when kL >> 1, where k is the perturbation wave number and L = n/∇n is the electron concentration gradient scale length. L is given by Huba and Ossakow [1980] as 10-100 km. The perturbation wavelength on a plasma patch will be of the order of several tens of kilometer or less, making the short wavelength equations the more applicable.

Turbulence
The turbulence process was first put forward by Kelley and Kintner [1978] and further examined in Burston et al.
[2010] and Kelley [2009]. It takes into account the fact that the electric field fluctuates around the quasi-DC component indicated by E in the preceding equations. If it is assumed that these fluctuations are random in direction, then there is always a direction in which the instability can occur. The growth where k L is the smallest relevant wave number, in this case 2π/(scale of patch), or of order 2π/100,000 m. Substituting equation (9) into equation (1) gives the growth rate for the turbulent collisional regime as Substituting equation (9) into equation (2) gives the growth for the turbulent inertial regime as

Kelvin-Helmholtz Instability
The idea that primary ordinary Kelvin-Helmholtz waves structure plasma patches was examined observationally by Basu et al. [1990] and modeled by Gondarenko and Guzdar [2006a] who concluded that primary shear slows the development of the GDI. The idea was advanced again by Carlson et al. [2007] who, in contradiction of this earlier work, suggested it would accelerate the generation of irregularities. The Carlson hypothesis was further tested in a case study by Oksavik et al. [2012] where it was found not to be dominant over the GDI except in a narrow band of wavelengths. The growth rate used in these KHI studies of 2ΔU/L (where ΔU is the velocity change across a shear zone and L is the distance scale length) disguises the electric and geomagnetic field dependence and is therefore not appropriate for comparison with the equations already given in this paper. Instead, the equation used here is derived as a special case of the dispersion relation equation given in Mikhailovskii [1992] for ordinary Kelvin-Helmholtz waves.
where C 1 is the Alfven speed in zone 1, C 2 is the Alfven speed in zone 2, V 1 is the plasma velocity in zone 1, V 2 is the plasma velocity in zone 2, and k is the perturbation wave number. If n 1 and n 2 are the electron densities in zones 1 and 2, respectively, then the Alfven speed is given by Þ nj with j = 1 or j = 2 for the two different zones and m i and m e are the ion and electron masses, respectively.
Assuming that the patch (taken as zone 1) is moving and its surroundings (taken as zone 2) is stationary, then V 1 = V = |(E × B)/B 2 | and V 2 = 0. Hence, equation (12) becomes and a growth rate (rightmost term) given by which simplifies to with instability condition For unstable growth, there must be an imaginary component to ω, which implies a negative square root in equation (16).

10.1002/2015JA021895
Substituting V, C 2 1, and C 2 2 into equation (16) gives The instability condition becomes It is notable that the instability condition is independent of wave number, but larger wave numbers (shorter wavelengths) will grow faster.
The above four processes can be divided into two categories; those that are dependent on a specific geometrical relationship between the electric field and the electron concentration gradient and those that are not. The GDI, whether collisional or inertial, falls into the former category, whereas the others fall into the latter. Specifically, the GDI requires that the electric field be parallel to the gradient, which in turn means it will only operate on the trailing edge of a patch. (On the leading edge, the electric field would be antiparallel to the gradient.) At first glance it might be thought that this should apply to the turbulence process, too, but in fact, because it is assumed that the AC fluctuations in the electric field have no directional bias, all slopes are unstable approximately half the time.
It should be noted here that another possibility has been suggested for the structuring of plasma patches, particle precipitation during formation. Anisotropic precipitation of energetic particles from above the ionosphere would lead to localized regions of increased ionization and hence gradients in electron concentration within the plasma patch. By modifying n and ∇n in the patch and causing localized heating, this process would affect the subsequent growth rate of all plasma instabilities discussed here. This process would only occur during the formation of the patch and would cease once the patch left the region of the cusp [Oksavik et al., 2012;Walker et al., 1999]. Since it is not itself an instability process it cannot be directly investigated by the method outlined below.

Method
The possible magnitudes of the growth rate for the four processes discussed above (for both inertial and collisional regimes) are assessed below using some 550 days of data from the Dynamics Explorer 2 (DE2) satellite covering August 1981 to February 1983. Thus, the DE2 data are concentrated at the peak of solar cycle (#21) and therefore do not represent a whole solar cycle. Similarly, only data recorded by the satellite when at 300-400 km altitude and greater than 65°magnetic latitude (north and south) are used in order to constraint our study to the F 2 peak and polar cap regions. For illustration we do not separate the two hemispheres. However, it should be noted that there will be differences between hemispheres because of measurement times and the offset of the geographic from the geomagnetic and dip poles [e.g., Coley and Heelis, 1998]. The DE2 data were chosen as all of the parameters in the growth rates equations used are either directly observed by or can be calculated from DE2 observations, with a minimum in the way of approximation or assumption, except with regard to the electric field because of a partial instrument failure (discussed below) and the wave numbers of initial perturbations, the range of values of which are assumed. Full details of the instruments aboard the satellite can be found in Hoffman et al. [1981]. In the following, the relative magnitudes of growth rates for all the relevant processes are compared. This is done by randomly sampling the data for each parameter, which are then use to calculate all possible values of growth rate for each growth rate equation.
The values of v in and v ei used in the following were calculated from DE2 data by using equations (3) and (5), respectively. The values for T are taken as the (O + ) ion temperatures (whose histogram of occurrence is displayed in Figure 4), n n is the neutral monatomic oxygen concentration (histogram of occurrence displayed in Figure 3), and n = n i (i.e., the ion and electron concentrations are assumed equal, and histogram of occurrence is displayed in Figure 1). The full field strength B (histogram of occurrence displayed in Figure 5) is also approximated by the vertical magnetic field component (B z ). In the following only the magnitude of the growth rate is calculated (vector quantities are replaced by scalar ones in all cases) simplifying the equations slightly.

10.1002/2015JA021895
The electric field data come from the Vector Electric Field Instrument on board DE2. This consisted of three mutually perpendicular instruments measuring orthogonal electric field strength components, allowing recovery of the full electric field strength vector E [Heppner et al., 1978a[Heppner et al., , 1978b. Unfortunately, the z axis instrument did not deploy, and no z axis data were recorded. Because the axes were relative to the spacecraft, it is not possible to determine the components of the field in the Earth centered coordinate system without the z axis data. Hence, a method of approximating values of E = |E| had to be applied. Examination of the relevant data recorded in the x-y plane (spacecraft coordinates) showed that 50% of the time the values of each component E x and E y were the same order of magnitude and over 90% of the time E x and E y differed by no more than 1 order of magnitude (see Table 1). In the case of the AC field (see also Table 1) the E x and E y components also differed from each other by no more than 1 order of magnitude over 90% of the time. Since the spacecraft coordinate system continuously varied in relation to the Earth centered coordinate system, it was assumed that E z would also rarely differ from E x and E y by more than this. Hence, the approximation Given all of the above, the equation used for the collisional GDI growth rate, equation (1), becomes where ∇n was derived from DE2 Langmuir probe observations of electron concentration n (as displayed in Figure 1) in the following manner: First, only data recorded at 300-400 km altitude were admitted, in order to retain only F region altitudes. Second, only data at > 65°magnetic latitude were admitted (in either hemisphere), in order to retain only geographical regions where patches plausibly occur. The 65°magnetic latitude was chosen to capture patches during storms equatorward of the polar cap, as patches are more prevalent during such storm time conditions. Third, only gradients in the remaining data, indicative of a plasma patch, were admitted. Such gradients were defined as being at least 40% slopes over a distance of 140 (± 5 %) km, following [Coley and Heelis, 1995]. We believe that using this distance should exclude the smaller nonpatch structures within the auroral oval. Such gradients were calculated by dividing the electron concentration data into sequences covering 140 (± 5 %) km, performing a linear regression on them and taking the resulting straight line slope as |∇n|. Note that such sequences may overlap each other and that these slopes do not necessarily represent the steepest gradients possible on a given structure as they are only the slope in the direction of motion of the satellite. The histogram of |∇n| occurrence, as derived from DE2, is displayed in Figure 2.
The values of all other parameters used in this and all subsequent cases were those observed during the time periods when admissible gradients were present. Hence, the subset of the DE2 data used plausibly represents conditions within patches. No attempt has been made to independently verify the existence of patches at these times and places.
For the inertial GDI case, equation (2) becomes For the collisional, short wavelength CCI, equation (6) becomes For the inertial, short wavelength CCI, equation (8) becomes For the KHI, equation (18) becomes with n 1 and n 2 taken as the maximum and minimum figures in the electron concentration used to calculate |∇n|.
In equations 21-25, inclusive, the parameter, E, refers to the quasi-DC measurements of the electric field. For the two turbulence equations, the AC component is required. This was measured by DE2 in the range 4 Hz-1024 Hz across eight channels, so the integral over the measured region becomes X 2π=4 taking equation (20) into account for each E(k) before summation.
Equation (10), for the collisional turbulent growth rate, becomes For the inertial turbulent regime, equation (11) becomes The ideal approach to exploring the range of values of the growth rates for each process is to calculate a value of γ for every possible combination of the necessary input measurements (e.g., the magnetic and electric field data as summarized in Figures 5-7). However, given the subset of DE2 data selected this would become unmanageably large. Hence, in order to handle the number of output values of γ in a sensible time it was necessary to take representative samples of the full data sets of each input parameter. The method of simple random sampling was adopted, using the following equation to determine the minimum sample size, S, required where p = Z 2 /(4M 2 ), M is the desired margin of error expressed as a fractional percentage, P is the size of the population being sampled, and Z is a factor chosen dependent on the required confidence limit. The practical memory constraint led to the best possible sampling regime of a 95% confidence limit, corresponding to Z = 1.95, and a margin of error of 9% (M = 0.09) leading to a sample size of~116 for each parameter (as given in Table 2).

Results
In general, there are two sources of error relating to these results: instrument error and sampling error. The former applies to all the basic input measurements and introduces some level of uncertainty in their values; these are then combined when the calculation is made, generating an overall level of uncertainty that could be estimated from the individual uncertainties. The second arises since, instead of calculating gamma for every possible combination of input measurements (a practical impossibility), a representative sample of each input population has been used. Again, an overall uncertainty due to this sampling could be calculated and combined with the instrumental error. However, there is a third source of error relating specifically to the electric and magnetic fields. These have been taken as their full field magnitudes when, more rigorously, the horizontal and vertical components, respectively, should be used. This introduces an overestimating bias in Journal of Geophysical Research: Space Physics 10.1002/2015JA021895 both of these measurements. Finally, the approximation necessitated by the failure of the z axis Langmuir probe to deploy leads to an error in the electric field magnitude such that the values used can only be considered accurate to order of magnitude. This final source of error swamps the others. Hence, all values of γ presented below should be considered to be only order of magnitude accurate.
In [Carlson et al., 2007] evidence for a risetime (1/γ) of 60 s or less is presented for patches still in the cusp region. Hence, in the following the various growth rate mechanisms are compared to find which can give values for γ > 1/60 and under what circumstances. The percentage of derived growth rates (using the sample data) exceeding 1/6, 1/60, and 1/600 for each of the four processes is presented in Table 3. All processes except Kelvin-Helmholtz instabilities (KHI) are found capable of giving risetimes less than 60 s using the DE2 data. For collisional gradient drift and shortwave inertial current convective instabilities this occurs less than 1% of the time but occurs for the majority of the time for inertial turbulence instabilities and around 10% of the time for inertial gradient drift and collisional turbulence and shortwave current convective instabilities.

Conclusions
The derived distributions of possible linear growth rates for each mechanism show that only turbulence, inertial GDI, and shortwave collisional CCI would regularly give the risetimes of approximately 60 s or less observed in the cusp region during patch formation [e.g., Carlson et al., 2007, and references therein]. However, the relative importance of the various mechanisms could be significantly different in the nonlinear regime [Gondarenko and Guzdar, 1999, 2004a, 2004b, 2006a, 2006bGondarenko et al., 2003;Guzdar et al., 1998]. In this regard, the linear regime assumes the instability grows exponentially in amplitude, at the growth rate γ given by the above. While there are various mechanisms that can drive an instability, one possibility is an increasing velocity shear, δV (due to an exponentially increasing perturbation in electric field, δE = δE 0 e γt ), which over time could become large enough to drive secondary Kelvin-Helmholtz instabilities (KHI) that damp the growth of the instability, ending the linear growth regime. Based on equation (19), taking n 1 = n + δn and n 2 = n À δn (where n is the background electron density) and making the not unreasonable assumptions that n 2 ≫ δn 2 and m i + m e ≈ m i , this occurs for the ordinary KHI when t > t c where Using the sample mean values for B, E, and n, this would indicate that the linear regime would likely last for a duration of order 10/γ. This would indicate that grow rates greater than 1/6 would imply  that the mechanism becomes nonlinear within minutes, whereas growth rates less than 1/600 would take several hours for the mechanism to become unstable.
There are two limitations with regard to the method deployed here for comparing growth rates for candidate irregularity generating mechanisms associated with polar cap plasma patches that cannot easily be overcome. First, the data are concentrated at the peak of solar cycle 21 (late 1981 to early 1983) and therefore do not represent a whole solar cycle. Second, there is a tacit assumption that all the input variables in the growth rate equations are uncorrelated with all of the others. This may not be entirely accurate, especially with regard to the electric and magnetic fields and the electron concentration and its gradient.
Because the output values of growth rate are considered accurate only to order of magnitude, the relatively limited sampling of the populations is not considered a significant drawback. Nor is the use of full field values of the electric and magnetic fields or the approximation made in order to obtain values of electric field strength in the absence of z axis data.
It is remarkable that under the conditions selected as representative of patches, the AC component of the electric field is often larger than the quasi-DC component. This implies that the turbulence process can be a cause of scintillation on its own and that all slopes of a patch can, in the right circumstances, be sufficiently unstable to cause measurable phase scintillation. In the past it has been assumed that patches showing irregularities or scintillation throughout all regions, rather than just on the trailing edge had undergone the GDI to such an extent that the irregularities had penetrated throughout the patch having started forming only on the trailing edge. The present results show that the same situation could be obtained by the action of the turbulence process on all slopes, with irregularities forming on the entire circumference and working toward the center. In this situation the maximum irregularity amplitude need only be the radius of the patch, rather than the entire diameter. This mechanism would therefore appear at first glance to take approximately half the time to cause irregularities to form throughout an entire patch. In fact, because the electric field is fluctuating in magnitude and direction, the growth rate in any one particular direction should be halved. The time to reach complete instability is still accelerated, however.
The values of γ GDI and γ T obtained suggest that irregularities should be observed in association with most patches. This is not backed by the observational record on either count. The explanation arises from the use here of local, linear theory to obtain the growth rate equations used. Nonlocal, three-dimensional, Figure 7. Histogram of the AC electric field data (as summed over the measured range of 4 Hz-1024 Hz).  Gondarenko et al., 2003;Gondarenko and Guzdar, 1999, 2001, 2004a, 2004b, 2006a, 2006bGuzdar et al., 1998;Kelley, 2009]. It is possible that results may differ if nonlinear processes dominate.
This does not affect the main conclusion that the KHI is not important as it was negligible to start with. However, these results are based on the statistical analysis, and thus in this context the KHI is not the main player for the patch structuring. However, these results do not exclude the situation where the KHI dominates under conditions of the strong flow shears. Nor is the conclusion that turbulence is a process that must be taken into account affected, as its risetime will be reduced to a similar extent to that for the GDI, but not more so. If, taking the full complexities into account, none of the processes considered here can give a risetime less than 60 s, then the most probable explanation of the appearance of phase scintillation very rapidly, while the patch is still in the cusp, is that inhomogeneous precipitation of energetic particles into the patch as it is forming accelerates the action of the GDI and/or turbulence processes [Oksavik et al., 2012].
To completely understand what is happening, an extension of the work in Gondarenko et al. [2003], Gondarenko and Guzdar [1999, 2001, 2004a, 2004b, 2006a, 2006b, and Guzdar et al. [1998] must be made so that the effects of particle precipitation during formation are included, along with the effects of the AC electric field. Additionally, the patch geometry and initial electron concentration distribution should take account of recent observations in order to be fully realistic. (The assumption that patches are approximately circular throughout their lifetime has been demonstrated by Ionospheric Ray Tomography to be particularly unsound [e.g., Yin et al., 2009].) Further complications arise when compound processes are considered. For example, GDI and turbulence processes will be acting on the trailing edge of the patch at the same time. For such processes, the ratio γ GDI /γ T indicates which dominates. This is given by Hence, this can be obtained whenever simultaneous measurements of both the DC and AC components are available. Unless one or the other (or both) is negligible, the compound action of both will accelerate irregularity production compared to either one acting alone. This could lead to a situation where radar backscatter from irregularities is present throughout a patch but stronger from the trailing edge. The relative impact of turbulence versus  Figure 8. The percentage of derived growth rates exceeding the given growth rate for "turbulence" (Turb) and gradient drift (GDI) in both the inertial and collisional regimes. The inertial turbulent regime is most likely to exceed any given growth rate and is therefore the more important.
Journal of Geophysical Research: Space Physics 10.1002/2015JA021895 the GDI should also be more thoroughly examined by analysis of observational data based on equation (30) for each admissible electron concentration gradient. In this regard, it is interesting that comparing the quasi-DC and AC electric field strengths (Figures 6 and 7) shows that, under the conditions described for the selection of the data above, the summed AC field is usually stronger than the quasi-DC field and turbulence dominates (Figure 8).