A comparison of the relative effect of the Earth's quasi‐DC and AC electric field on gradient drift waves in large‐scale plasma structures in the polar regions

Radio signals traversing polar cap plasma patches and other large‐scale plasma structures in polar regions are prone to scintillation. This implies that irregularities in electron concentration often form within such structures. The current standard theory of the formation of such irregularities is that the primary gradient drift instability drives a cascade from larger to smaller wavelengths that manifest as variations in electron concentration. The electric field can be described as the sum of a quasi‐DC and an AC component. While the effect of the quasi‐DC component has been extensively investigated in theory and by modeling, the contribution of the AC component has been largely neglected. This paper investigates the relative contributions of both components, using data from the Dynamics Explorer 2 satellite. It concludes that the contribution of the AC electric field to irregularity growth cannot be neglected. This has consequences for our understanding of large‐scale plasma structures in polar regions (and any associated radio scintillation) as the AC electric field component varies in all directions. Hence, its effect is not limited to the trailing edge of such structures, as it is for the quasi‐DC component. This raises the need for new experimental and modeling investigations of these phenomena.


Introduction
theorized that high-latitude spread F events, as recorded with analog ionosondes, were caused by plasma enhancements in the polar cap ionosphere convecting according to an E × B drift, caused by the combination of the Earth's magnetic field B and an electric field E mapped from the magnetosphere. Direct confirmation of this theory came with all sky images reported in Buchau et al. [1983], that showed just such enhancements, now generally known as polar cap plasma patches. A correlation between radio signal raypaths transecting polar cap plasma patches and UHF scintillation of the signal received at the ground was demonstrated soon after [Weber et al., 1984]. Scintillation associated with plasma patches has been found to occur across a wide range of radio frequencies including Global Navigation Satellite Systems (GNSS) [e.g., Jin et al., 2014].
Ionospheric scintillation is the phenomenon of rapid phase and/or amplitude fluctuations of a received signal, induced during passage of the signal through the ionosphere [Kintner et al., 2007]. For scintillation, the mechanism is propagation of the signal through "irregularities" in the ionospheric electron density. Such irregularities give rise to variations in the refractive index of the ionosphere that causes the scintillation [e.g., Hargreaves, 1992;Hunsucker and Hargreaves, 2002].
A number of explanations for the mechanism of irregularity formation within polar cap plasma patches and large-scale plasma structures in the polar regions have been proposed, including the current convective instability (CCI), first suggested in relation to auroral structures [Ossakow and Chaturvedi, 1979] and in plasma patches by Kelley [2009], the primary Kelvin-Helmholtz Instability (KHI) (possibly in conjunction with other processes) [e.g., Carlson et al., 2007;Gondarenko and Guzdar, 2006a;Oksavik et al., 2012], the gradient drift instability (GDI) [e.g., Basu et al., 1990;Burston et al., 2009;Chaturvedi et al., 1994;Gondarenko, and Guzdar, 2006b;Kivanc and Heelis, 1997;Weber et al., 1984] and a related "turbulent" process [Kelley and Kintner, 1978], which we term Kelley-Kintner-Turbulence (KKT). The term "turbulence" comes from magnetospheric electric fields that are turbulent in that they show continual variation at short time scales, and these fluctuations are also mapped to the ionosphere, exposing plasma to rapidly varying E × B drifts that can cause gradient drift instabilities, generating turbulent mixing of the plasma, if there is an electron concentration gradient present [Burston et al., 2010 Using Dynamics Explorer 2 satellite observations [Burston et al., 2016] investigated the range of possible values for the linear growth rates for each of these processes. In that study, we found that the inertial KKT instability gave rise to the largest growth rates followed by those for inertial gradient drift, collisional turbulence and collisional shortwave current convective instabilities. Growth rates for the primary KHI (in agreement with Oksavik et al. [2012]) were found to be far too small to be significant, as were the growth rates for the long wavelength CCI. These results, and similarities between the GDI and KKT, have inspired this follow-up study. In particular, the GDI requires an electric field perpendicular to a gradient in plasma concentration and a magnetic field perpendicular to both in order to occur. A preexisting perturbation in the plasma concentration gradient is also required. All these conditions can be met in the case of large-scale plasma structures in the polar regions since the geomagnetic field is approximately vertical, and there must be an appreciable electric field mapped from the magnetosphere to the ionosphere or the structure could not be convecting via an E × B drift force and hence would not exist. The growth rate for the GDI [Linson and Workman, 1970] is as follows: (collisional regime) and (inertial regime), where E is the electric field strength, B is the magnetic field strength, n is the free electron concentration, and ν in is the ion-neutral collision frequency [e.g., Sojka et al., 1998]. The subscripts c and i denote the collisional and inertial regimes, respectively. The cutoff between the two regimes is where the collisional rate and collisional regime growth rate become equal in value. In these equations E is usually considered to be the quasi-DC component of the magnetosphere-to-ionosphere mapped electric field. In other words, short time scale fluctuations in the direction and magnitude of the field are ignored. It is possible to derive equations that take account of the AC component of the field, separately, by replacing E with which represents the root-mean-square electric field strength due to the AC variations in the field with wavenumbers from k L to infinity, where k L is the smallest relevant wavenumber, in this case k L = 2π(∇n/n), where (n/∇n) is the scale of the structure. (1) and (2) yields

Substituting equation (3) into equations
and respectively. These ideas and equations (3)-(5) were presented in Kelley and Kintner [1978]. The subscript T is used for these growth rates, as these refer to Kelley-Kintner Turbulence.
The GDI and Kelley-Kintner Turbulence (KKT) give rise to growth rate equations that are similar in form. However, the differences between the electric fields driving them lead to the possibility of different effects. In equations (1) and (2) the fields are vector quantities and consequently the instability can only initiate on the trailing edge. Equations (4) and (5) are not vector relations, as the AC component of the electric field has no directional bias and fluctuates in all directions. Since there is also a gradient in electron concentration in all directions, this process is always viable in some direction and is not confined to the trailing edge of convecting plasma structures. As the only difference between the GDI and KKT equations is the electric field Journal of Geophysical Research: Space Physics 10.1002/2016JA022676 term, it is possible to assess which process likely dominates simply by dividing one by the other. In particular, both equation (1) ÷ equation (4) and equation (2) ÷ equation (5) yield which we redefine as E DC /E AC for simplicity, since henceforth only the relative magnitudes are important. It should be noted that two-or three-dimensional growth rate equations and nonlinear damping terms are unimportant here, since they will be identical for both the GDI and KKT cases except for the electric field values. Hence, E DC /E AC can be used as a theoretical measure of which process dominates at any given time.
For example, this has implications with regard to observations about the spatial structuring of polar patches. In particular, Kivanc and Heelis [1997] used Dynamics Explorer satellite data to classify patches into four types according to their internal irregularity structure distributions. These classes were (1) no structuring, (2) structuring only on the trailing edge, (3) structuring throughout the patch, and (4) structuring on the trailing and leading edges. Class one can be explained as having no instability operating or the instability operating for too short a time when the observations were made. Classes two and three can be explained by the GDI at earlier and later stages of growth, respectively. Class four can only be explained by the GDI if the direction of motion of the patch has altered radically, so that what was the trailing edge is now effectively the leading edge and vice versa. A simpler explanation would be that the KKT process is operating on all gradients at an appreciable rate. This could also explain the creation of fully structured patches (class three) at a rate faster than the GDI alone could produce as irregularities would only have to grow to the patch radius rather than the patch diameter for the case of the GDI alone.
A sufficient number of observations of E DC /E AC would therefore indicate statistically whether KKT is likely to be a significant irregularity generating process.

Method
The Dynamics Explorer 2 satellite (http://nssdc.gsfc.nasa.gov/nmc/experimentDisplay.do?id=1981-070B-09), launched in August 1981 and reentering in February 1983, flew in an elliptical, polar orbit, allowing it to make observations of the ionosphere above both polar caps. On board was the Vector Electric Field Instrument (VEFI) which consisted of three mutually perpendicular instruments capable of measuring electric field strength and allowing for recovery of the full vector electric field strength, E. This included separate measurements of both the quasi-DC and the AC components of the field up to 1024 Hz [Heppner et al., 1978a[Heppner et al., , 1978b.
In particular, the quasi-DC is the short duration component of the Earth's electric field < 4 Hz and the AC is the rest (4 Hz-1024 Hz). This quasi-DC component is stationary in time (over short time scales < 1 min) but is not stationary in space.
Since the interest here is the electric field within large-scale plasma structures in the polar regions, all measurements below 65°magnetic latitude (for each hemisphere) were excluded. Second, data above 400 km were excluded to eliminate data from far above the F peak. The retained data are all in the range 300-400 km, as DE2 never descended below 300 km during routine operation. Third, a method of identifying gradients in electron concentration was adopted as follows. Orbit segments with an electron concentration gradient of 40% or greater across the whole of a 140 km ± 5% orbit segment were retained. Figure 1 shows the electron concentration, the quasi-DC (x and y) components and the power spectra of the individual AC (x and y) components for an example orbit segment (meeting the above three criteria).
Because of the retention of only high-latitude data, the probable sources of gradients in electron concentration are only auroral structures and plasma patches. Since the steep gradients retained are calculated across 140 km orbit segments as a whole, not within them, it is unlikely that they are auroral in origin. This method was first used by Coley and Heelis [1995] and recently used by Burston et al. [2016] across a wider magnetic latitude. It should be noted that some retained orbit segments may overlap others. The data used to calculate the electron concentration gradients were taken from the DE2 Langmuir probe experiment. Strictly, as the gradient should be determined along its steepest direction rather than the satellite track this would tend to lead to an underestimate of the actual gradient.

10.1002/2016JA022676
The mean values of E DC and E AC were calculated for each orbit segment that met the criteria given above. However, there was a problem with the VEFI instrument; the z axis boom did not unfurl, and hence, no z axis data were recorded. This makes it impossible to directly calculate the full values of E DC and E AC . Because the axes refer to the spacecraft orientation it is also not possible to equate the x axis or y axis data with specific axes in the Earth-centered coordinate system. Hence, in order to progress, a method of approximating the full electric field had to be adopted. The method used is identical to that described in Burston et al. [2016] and uses where E x and E y are the x and y components of the electric field in spacecraft coordinates. The above applies equally to the quasi-DC and AC data and equation (7) was used for both cases. The mean values of E over each orbit segment of interest were calculated for both the quasi-DC data and the AC data, resulting in a pair of Figure 1. The electron density, DC electric field (x and y) components and the power spectra of the (x and y) components of the AC electric field for an example orbit segment that satisfies the three required criteria (magnetic latitude, height, and gradient).

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values, E DC and E AC for each orbit segment. The ratio of each of these pairs of means indicates which process, GDI or KKT, was dominant for each orbit segment.

Results
A total of 1148 DE2 satellite orbit segments met the criteria given in section 2. They were all recorded during solar maximum conditions using some 550 days of data covering August 1981 to February 1983. There are two sources of error in these measurements. The first is any systematic or random errors associated with the VEFI instrument itself. The second is associated with the technique (equation (7)) adopted to estimate the full electric field strength necessitated by the failure of the z axis component of VEFI. The work of Burston et al. [2016] estimate that the derived electric field values are generally correct to an order of magnitude. Hence, this dominates the intrinsic instrument error, given as 0.1 mV m À1 , which can be neglected. A third source of error is also generated by the partial instrument failure: Ideally, the locally horizontal components (in Earth-centered coordinates) of both E AC and E DC would be extracted from the data, as only these contribute to driving the GDI and KKT processes. This is impossible without the z axis data. The data used here are therefore overestimates of the actual driving electric fields but probably not to such an extent that the order-of-magnitude estimate is compromised. Comparing the orders of magnitude of the electric field  estimates is therefore valid and allows determination of whether E DC or E AC dominates for each orbit segment examined.
The results show that E AC is often similar or larger in magnitude than E DC , as illustrated in Figure 2, where the mean values of E DC and E AC for each orbital segment analyzed are plotted against each other as a scatter diagram. Note the difference in scales required for each axis. This is further illustrated in Figure 3, which shows a scatterplot of the ratio of each pair of means, E DC =E AC , for each orbit segment analyzed. In order to clarify how often the GDI is likely to dominate over KKT and vice versa the log ratios were binned by order of magnitude and plotted as a bar chart (Figure 4). A log ratio of order of magnitude < À1 implies E AC (KKT) dominance. A log ratio of order of magnitude >1 implies E DC (GDI) dominance. A log ratio of order of magnitude 0 implies that neither dominates the other.
In 34% of orbit segments analyzed show E AC and E DC to be of the same order of magnitude. Over 60% show clear dominance of E AC (KKT). Only 6% show clear dominance of E DC (GDI). Hence, in 94% (60% + 34%) of cases identified and analyzed by this method, it is likely that if irregularities formed within the patch KKT played a role in their development. Similarly, in 40% (34% + 6%) the DC component may also play such a role.

Conclusions
The AC component of the electric field is comparable to or larger than the quasi-DC component in 94% of the orbit segments found that likely traverse large-scale plasma structures in the polar regions. This fact both supports and greatly strengthens the conclusions of Burston et al. [2010], namely, that KKT can be as or more important than the GDI with regard to structuring of patches. The knowledge that KKT can often be as energetic a process as the GDI and can occur anywhere on the patch, rather than in a preferred location (trailing edge) affords a better explanation of the examples observed in Kivanc and Heelis [1997] of fully structured patches and patches with structure on the leading and trailing edges. In the latter case, there need not have been a complete reversal of the quasi-DC component, which seems an unlikely occurrence. Instead, KKT has occurred on the leading edge. In the former case, the KKT allows for an approximate halving of the time it would take to reach complete structuring as irregularities would move inward from all directions to the center of the patch, instead of the GDI having to grow across the whole patch diameter.
In order to fully understand irregularity growth in large-scale plasma structures in the polar regions and any associated radio scintillation, it is necessary to go beyond the previously referenced modeling work of Gondarenko and Guzdar [2006a] and include the effects of the AC fluctuations in the driving electric field. Addition of the effects of soft particle precipitation in the cusp during patch formation and use of realistic patch geometries derived from radar and ionospheric ray tomography studies would further enhance our understanding.
In addition, detailed knowledge of the power spectra of the electric field values with frequency may determine whether there is a preferential frequency driving the growth of KKT instabilities. In this regard, work by Mounir et al. [1991], albeit using data from the ARCAD-AUREOL-3 satellite, showed that the power spectrum of the electric field for individual plasma patches followed a power law decay with frequency (with index À1.4 to À1.8). This would indicate, if repeated in general (i.e., the index > À2), that the KKT growth rate may be driven by any AC frequency component and there is no preferential frequency range. The AC power spectra shown in Figure 1 for an example orbit segment tend to back this up. However, the AC electric field power spectra averaged over all orbit segments ( Figure 5) shows a steady decrease from 4 Hz to 256 Hz followed by a small increase up to the maximum observed frequency of 1024 Hz, possibly indicating that low and high frequency components are the more important and that more work is needed particularly if the high frequencies are significant.
In addition, in this study we have investigated only KKT and GDI, as these were found to give rise to the greatest growth rates using DE2 data [Burston et al., 2016] and differ only in their response to the quasi-DC or AC component. However, other instability modes such as CCI could also be present and may need also to be considered.