Forecasting low latitude radio scintillation with 3 D ionospheric plume models: 2. Scintillation calculation

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2008ja013840, 2010 Forecasting low latitude radio scintillation with 3 D ionospheric plume models: 2. Scintillation calculation J. M. Retterer 1 Received 20 October 2008; revised 23 September 2009; accepted 5 October 2009; published 19 March [1] A three dimensional model has been developed for the plasma plumes caused by interchange instabilities in the low latitude ionosphere to describe the structure and extent of the radio scintillation generated by turbulence around and within the plumes. With the inclusion of the processes that determine the transport of plasma parallel to the geomagnetic field lines as well as transverse to them, the model can predict the extent in latitude of the plumes and their scintillation. Diagnostics presented here include illustrations of the spectral density of the density irregularities that develop within the plumes. An extrapolation of the density irregularity spectrum down into the wavelength regime effective for radio wave scattering permits a phase screen calculation of the amplitude scintillation caused by the plumes. The scintillation produced by the model has much the same character as measurements of scintillation do in terms of the time and rate of onset of scintillation, duration, and latitudinal extent. Citation: Retterer, J. M. (2010), Forecasting low latitude radio scintillation with 3 D ionospheric plume models: 2. Scintillation calculation, J. Geophys. Res., 115,, doi: /2008ja Introduction [2] The upwellings of lower density plasma as a result of plasma instabilities that create plumes or bubbles in the equatorial ionosphere and powerful coherent reflection of radar signals [Woodman and La Hoz, 1976] from density irregularities are one of the most striking ionospheric phenomena. They commonly occur after sunset as the result of the prereversal enhancement [Eccles, 1998] in the upward plasma velocity lifting the F layer, but can also occur later at night as the result of electric fields associated with geomagnetic activity [Kelley and Retterer, 2008]. A practical consequence of the structures is the radio scintillation caused by the density irregularities of the plasma turbulence within them, because the resulting signal fades can cause outages of communication and navigation systems which rely on transionospheric radio propagation. The C/NOFS (Communication and Navigation Outage Forecasting System [de la Beaujardiere et al., 2004] satellite has been designed to study the plumes and monitor conditions within the ionosphere so that forecasts [Retterer, 2005] of their occurrence and effects can be made. [3] To investigate the structure and development of the equatorial plasma depletions in a realistic, time dependent ionospheric model, we have developed a new model [Retterer, 2009] to study the development of equatorial plasma depletions after sunset. Like the Naval Research Laboratory (NRL) models [alesak et al., 1982; Huba et al., 2008], the new model follows the temporal evolution 1 Space Vehicles Directorate, Air Force Research Laboratory, Hanscom AFB, Massachusetts, USA. This paper is not subject to U.S. copyright. Published in 2010 by the American Geophysical Union. of perturbations in the F region using the nonlinear continuity, momentum, and current conservation equations. The model relies upon detailed time dependent empirical and transport models to describe the background plasma and neutral thermospheric components accurately. One of the fundamental unanswered questions concerning the plumes has been: How far along a geomagnetic field line do the depletions and turbulence extend? By including parallel transport in a three dimensional treatment of the plumes, our new model addresses this question. [4] One of the remarkable characteristics of the plumes is the nearly universal spectrum of density irregularities found by in situ observations [Kelley and Livingston, 2003] over many different events, and the wide dynamical range of wavelengths over which this spectrum is close to a power law, from hundreds of kilometers down to meters. It is these density irregularities which cause the phase and amplitude scintillation of radio waves propagating through the plumes. Using the density structures predicted by the model, a phase screen calculation [Costa and Kelley, 1977] of the radio scintillation expected of the model will be performed to describe the structure and extent of the scintillation. This model has been developed to satisfy the needs of the C/NOFS program to predict the locations, times, extents, and strength of radio scintillation. 2. Scintillation Model [5] Radio scintillation is the term used to represent the random fluctuations in signal phase and amplitude that develop when the radio waves propagate through ionospheric electron density irregularities. A measure of the degree of scintillation in the strength of a signal is the quantity S 4 [Yeh and Liu, 1982], which describes the root mean 1of10

2 square fluctuations in signal intensity, normalized by the average signal intensity: S 2 4 ¼ hi 2 i hii 2 hii 2 [6] The most rigorous description of the effects of radio propagation through the ionospheric medium with density irregularities requires a solution of the wave equation with the irregular nature of the medium built into the description of the dielectric [Yeh and Liu, 1982]. With a statistical description of the density irregularities in the medium, a convenient analytical expression for S 4 can be derived with the weak scattering and thin screen approximations [Costa and Kelley, 1977]. This phase screen approximation expresses S 4 as an integral over Fourier wave number space of the spectral density of variations in density P N times a Fresnel factor: S4ps 2 ¼ 4 ð r eþ 2 dkt1 dk t2 L sc 2 2 ð1þ Fk ð t1 ; k t2 Þ P N ðk t1 ; k t2 ; k s ¼ 0Þ where r e is the classical electron radius, l is the radio signal wavelength, L sc is the extent along the signal propagation direction of the region with density fluctuations, and k s is the wave vector in the signal propagation direction while k t1, k t2 are two orthogonal wave vector components perpendicular to the signal propagation direction. The Fresnel factor for an infinitely thin scatter layer is " # Fk ð t1 ; k t2 Þ ¼ sin 2 k2 t1 þ k2 t2 kf 2 ð3þ with the parameter k 2 f ¼ 4 H s where H s is the distance between the observer and the position of the idealized phase changing screen imagined to contain the TEC irregularities. Although there are less limiting propagation models than the thin phase screen model available [Wernik et al., 1980; Knepp, 1983], its dependence on only the spectral information of the irregularities (i.e., no phase knowledge) permits the use of information from a plume simulation to estimate the strength of scintillation when extrapolation from the wave number regime of the simulation might be necessary. [7] How can we use the results of our simulations of lowlatitude plume formation [Retterer, 2009] to evaluate the strength of radio scintillation caused by the density irregularities produced by the plasma turbulence? The simulations give us, for a sequence of times, the plasma density as a function of position within a three dimensional volume near the geomagnetic equator. The density is tabulated on a cartesian computational grid defined in the geomagnetic equatorial plane. The vertical x grid is chosen to encompass the range of altitudes expected for the plumes, with apex altitudes typically from 200 km to 1200 km. The east west y grid is ð2þ ð4þ chosen to have a similar extent. The grid numbers points for x and y. Thus in the equatorial plane, the cell size is about 4 km in both dimensions perpendicular to the magnetic field. From each cell in the equatorial plane the coordinate in the third dimension extends along the field line, down to the points 90 km in altitude in each hemisphere. We will employ the Fourier transform of this spatial structure to provide us with estimates of the spectral densities needed for equation (2). By the Wiener Khinchin theorem [Blackman and Tukey, 1958], the spectral density P N (k) is the Fourier transform of the autocovariance function of density fluctuations, C N (x), which for a system of a finite volume can be written C N ðxþ ¼ 1 L 1 L 2 L 3 n x 0 n x 0 þ x d 3 x 0 where L i are the extents of the simulation volume in the three dimensions. Using the Fourier transform of the density nðkþ ¼ nðxþe ikx d 3 x ð6þ and some manipulation, the spectral density in the S 4 integral can be approximated for a system of finite volume as P N ðk t1 ; k t2 ; k s Þ ¼ jnk ð t1; k t2 ; k s Þj 2 L t1 L t2 L s where n(k t1, k t2, k s ) is the Fourier transform of the density from a function of position within the simulation volume to a function of the three wave number components. [8] Figure 1 shows the evolution of one of the spectral elements of density (the one dimensional Fourier spectrum component n(x, k y, g =0) 2 /L y ), at the geomagnetic equator (g = 0), with east west wavelength 20 km (k y =2p/20 [rad/km], although for convenience in this article and its plots we always give wave numbers without the 2p factor, i.e., as inverse wavelengths), as a function of time and altitude (x in the equatorial plane equals the apex altitude of the field line). This result is from a simulation of the Rayleigh Taylor instability in the low latitude ionosphere for the geomagnetically quiet case presented in [Retterer, 2009]: day of year 299, 2004 (25 October 2004) in the American sector (longitude 280 degrees). In Figure 1 we see at early times the linear growth phase, where the mode grows exponentially in time and rises with the F layer between 250 and 400 km. Then around LT = 20, the plume develops and very quickly rises in altitude, with enhanced spectral density of irregularities within it produced by nonlinear processes within the plasma. Numerous layers of intense spectral density are apparent within the highly structured plasma, which then subsequently fall with the ambient plasma and slowly decay with time. [9] Since k s is the wave vector component along the signal path, we see that n(k t, k s = 0) gives the Fourier analysis of variations of the signal path TEC: nðk t ; k s ¼ 0Þ ¼ ds d 2 tns; ð tþ e iktt ¼ d 2 te iktt ð5þ ð7þ ds nðs; tþ In fact, the spectral density P N (k t1, k t2, k s =0)equals P TEC (k t1, k t2 )/L s [Yeh and Liu, 1982], where P TEC (k t1, k t2 ) ð8þ 2of10

3 Figure 1. Spectral density of density fluctuations in the equatorial plane for one east west wave vector (wavelength 20 km), as a function of time and altitude. The quantity plotted is the one dimensional Fourier spectrum component ( n(x, k y,g =0) 2 /L y ), at the geomagnetic equator (g = 0), with east west wavelength 20 km (k y =2p/20 [rad/km]), as a function of local time and altitude x. is the spectral density of total electron content variations, approximated by P TEC ðk t1 ; k t2 Þ ¼ jtk ð t1; k t2 Þj 2 L t1 L t2 ð9þ where T(k t1, k t2 ) is the Fourier analysis of the total electron content integrated along the signal propagation direction. In this paper we consider only vertical signal incidence. Then to calculate T(k t1, k t2 ) we first need to tabulate the total electron content as a function of y, the east west grid coordinate, and an orthogonal cartesian coordinate, z, that varies in the geomagnetic latitude direction about a central latitude point,. The z grid is defined with a grid cell size equal to the y grid spacing, and with the same total extent, thus covering a few degrees in latitude range. The TEC tabulation is done by interpolating densities from the field line oriented plume grid into a regular altitude grid at each y and z, andthen integrating over altitude. For points off the geomagnetic equator, this altitude grid will require densities at altitudes beyond the altitude of the field line with largest apex altitude in the plume grid, so an extension of the density tabulation employing the structureless ambient density model is used to avoid discontinuities which would introduce undesirable high frequency power into the spectrum. From the tabulation of T(y, z, ), two dimensional Fourier analysis using the Fast Fourier Transform algorithm immediately leads to T(k y, k z, ) for equation (9). [10] Because the angular variation of the spectral density of the TEC fluctuations is integrated over in the S 4 formula, we can introduce a one dimensional spectral density that is a function only of the magnitude of the wave vector, k t : d PTEC i ð k t;þ ¼ k t 2 P TECðk t sin ; k t cos ; Þ ð10þ where is the angle of the wave vector in the transverse plane; tabulating this function in the simulation reduces some of the noise inherent in spectral estimation. (Note that we are not assuming that the spectral density of TEC fluctuations is isotropic. In fact, for example, at vertical signal incidence, the transverse direction with the most TEC fluctuation power near the equator should be the east west direction, while near the ends of the plume, both the eastwest and the north south directions will contribute significantly to the TEC fluctuation power of the plume.) Figure 2 shows the evolution of the vertical TEC spectral density at several latitude distances from the geomagnetic equator, as a function of local time and wave number (inverse wavelength in 1/km). The first feature to emerge is the 30 km 3of10

4 Figure 2. Spectral density of vertical TEC fluctuations as a function of the east west wave vector and time, for several latitudinal distances from the geomagnetic equator. wavelength ripple [Retterer, 2009] on the lower edge of the F layer that forms near the equator around 19 LT. Then, around 20 LT, the plume is formed, and a broadband spectrum is seen first only near the equator and then at latitudes out close to 20 degrees away. After a few hours, the decay of the strength of the irregularities becomes apparent. [11] Note that the one dimensional power spectrum of TEC fluctuations used for the scintillation calculation can be calculated approximately for a two dimensional plume simulation in the geomagnetic equatorial plane as described by appendix B of [Retterer, 2009] by assuming that the plume structures map along the geomagnetic field lines from the equatorial plane as described in that appendix. Although this includes only the east west variations, it does permit estimates of scintillation to be performed with the faster two dimensional plume algorithm, which can be important in an operational forecasting system. [12] In terms of the angular integrated spectrum, the integral for S 4ps at latitude becomes S4ps 2 ðþ¼4 ð r eþ 2 L sc dkt 2 Fk ð t; 0Þ PTEC i ð k t;þ ð11þ L s [13] The largest contributions to the integral will come from wavelengths where the Fresnel factor is close to unity, where k t 2 is p/2 k f 2 ; at a signal frequency of 250 MHz and with the distance H s equal to the height of the F layer (350 km), the wavelength for the maximum of the Fresnel factor is around 1 km. [14] These wavelengths are shorter than can be resolved in the grid of our simulation. Furthermore, if we wanted to evaluate the phase screen formula with the spectrum of TEC directly from a simulation, we would have to employ a grid cell size considerably smaller than 1 km, because unavoidable numerical inaccuracies due to the finite grid cell size distort the strength of modes near the shortest wavelengths of the system. This would come at a severe cost in execution time because the grids would have to be scaled up in cell numbers to provide the spatial coverage of the extent of the plumes, and the execution time of the solver for the electric potential elliptic equation increases as a high power of the number of grid cells (e.g., N 4 for the stabilized error propagation technique). In addition, the Courant condition for numerical stability, that a cell cannot be crossed in a single time step, would force us to reduce the time step at the same time, implying an increase in the number of steps required to integrate through the evolution of an instability. [15] To circumvent these problems, we employ a deferred approach [Retterer, 1979] to reach the limit of zero cell size, by examining the results with grids of fixed total size but increasing number of points (32, 64, 128, and 256 cells) in both coordinates perpendicular to the geomagnetic field to extrapolate to the limit of an infinitely fine grid. Figure 3 shows the one dimensional power spectra of the plasma density at one point in time for two dimensional runs with these grids. We see that as the grids get finer in spacing, the portion of the spectrum that is approximately a power law extends over wider ranges of k. An extrapolation to an infinitely fine grid suggests that the spectral density is a power law through many decades of k space. To apply this limiting procedure to a given case, we adopt the power law fitted to an intermediate range of wave numbers and apply it from that wave number regime down to the wave number regime of the Fresnel scale. This extrapolation of the spectrum assumes that the same physics applies through the range of wavelengths, which is not necessarily true. The observations in space, however, do show that the spectrum of density irregularities is close to a power law k 2 over several decades [Kelley and Livingston, 2003], down to wavelengths of tens of meters. The composite horizontal wave number spectrum determined by [Singh and Szuszczewicz, 1984] from a variety of observations is a power law in their medium wavelength regime (40 km down to 1 km), with a mean index of 1.4, and then a steeper power law in the intermediate wavelength regime (2 km down to 80 m), with a mean index of 2.5. Recent spectral fits to C/NOFS density observations by the Planar Langmuir Probe (PLP) instrument [Rodrigues et al., 2009] found similar shallow power laws for wavelengths longer than 60 m (spectral index around 1.5), and a break to a steeper power law shortward of that wavelength. Although the deviation from a power law of the spectra in Figure 3 also suggests a break to a steeper power law at shorter wavelengths, the fact that this break depends on grid resolution lead us to believe that this break is imposed by the unavoidable numerical inaccuracies of a discrete approximation to the equations of the model, and not the break observed in space, which is deter- 4of10

5 [Retterer, 2009] prior to plume formation, and the spectral line due to this structure distorts the estimates of spectral parameters around 19 LT. (The presence of a spectral line at higher wave number makes the spectrum appear to fall off more slowly with wave number, so the fitting algorithm chooses a smaller negative or even positive power law exponent.) The effect of the spectral line on the parameters of the vertical TEC spectrum power law fits, although weaker because of the integration over both the negative and positive excursions in a wavelength due to the tilt of the wave fronts, is also apparent here. In Figure 2 the spectral line due to these ripples can be seen beginning around 19 LT, between wave number 0.02 and 0.04 km 1, at dip latitudes 0 and 8 degrees; harmonics of the line are also faintly visible in the plot. Later, following plume formation, this spectral line broadens, the rest of the spectrum rises and engulfs Figure 3. Spectral density of density fluctuations in the equatorial plane as a function of the east west wave vector, at one instant of time (20.9 LT), for four simulations with differing grids: N = 32 (purple), N = 64 (blue), N = 128 (black), and N = 256 (red). mined by physical processes, and is expected at shorter wavelengths than can be resolved by the model, anyway. These observations give us some confidence in using the longer wavelength portion of the spectrum of irregularities in a simulation to set the magnitude of a power law spectrum that can be extrapolated with a power law down to the wavelengths relevant for the scattering calculation, at least for VHF signals. Basically, the simulation of the plume describes the evolution of the energy at longer wavelengths that is available for the cascade to shorter ones, but the cascade itself is described by a simple extrapolation procedure. Because of the break in the spectrum to a steeper power law at shorter wavelengths, using a single power law may overestimate the strength of scintillation at higher frequencies that have smaller Fresnel scales. A more elaborate model might use a combination of power laws, with a spectral break, to make the extrapolation down to shorter wavelengths. [16] Figure 4 gives the variation in the amplitude, S 0, and exponent, a, for the fitted power law, S 0 k a, for both the density and TEC at the equator for the day 299, 2004 case. Note that the spectrum is not expected to always be a power law. The initial spectrum will of course depend on the initial perturbation chosen for the simulation. It is only after plume formation (around 20 LT in this example) that the plasma enters a strongly nonlinear, fully turbulent state and the average spectrum smooths into a simple power law. Some measure of the validity of the power law fit is provided by the uncertainty in the fit of the power law parameters. The lighter curves in Figure 4 give an estimate of the range of possible values for the parameters, judged from the goodness of fit (on the basis of the correlation coefficient determined from the fit). Note that in this case, ripples with a wavelength of 30 km formed on the bottomside F layer Figure 4. Variation of the power law fitting parameters ((top) power law amplitude normalized by its peak over time; (bottom) exponent) for the spectral densities of density (black line) and vertical TEC (red line) as a function of time. The lighter curves give bounds on the fit of these parameters to the spectral densities. The larger range of uncertainty in the fits at earlier time confirms our expectation that the spectral density is not power law like until the turbulence reaches its full nonlinear state (after 20 LT). 5of10

6 Figure 5. Spectral density of vertical TEC fluctuations as a function of time and latitude. (top) Directly calculated spectral density for a long wavelength (50 km) mode. (bottom) Extrapolated spectral density at a wavelength of 1 km, which is near the wavelengths that give the largest contribution to the scintillation index S 4. the line, and the spectrum becomes power law like. Near the time of maximum power, the spectral exponent of the power law fit to the density spectrum is near 1.5, although the TEC spectrum has a steeper falloff (larger negative exponent, 2) because the vertical TEC integration smooths out mode structure on shorter scales. Following saturation of the instability, the power law amplitudes of both density and TEC fall as spectral modes decay; the exponents for both quantities increase in magnitude as well because shorterwavelength modes are decaying faster than longer wavelength modes. Figure 5 presents the spectral density of TEC fluctuations at a wavelength of 50 km as a function of time and latitude distance from the geomagnetic equator as directly determined from the simulation (Figure 5, top) along with the power law extrapolation to a wavelength of 1 km (Figure 5, bottom), which being near the Fresnel scale for 250 MHz radio signals, gives a sense of the strength of the spectral densities that will contribute most to the integral for S 4. The faster decay of the shorter wavelength modes is evident here, illustrating the formation of dead bubbles, the late evening large scale plasma depletions not associated with strong scintillation. [17] Recall that the phase screen formula is accurate only for weak scattering. For power law density irregularity spectra, comparison with full wave equation solutions [Dashen and Wang, 1993] shows that the phase screen formula does describe the resulting amplitude scintillation accurately when the irregularities are weak. When the irregularities 6of10

7 Figure 6. Scintillation strength, S 4, as a function of time and dip latitude for 250 MHz signal on a vertical signal path. are stronger, however, the full wave values of S 4 saturate at a value near unity. Physically, this results because negative fluctuations in intensity cannot exceed the signal intensity in magnitude, nor can positive fluctuations exceed the average intensity for long. To derive the actual strength of scintillation, S 4, we impose this saturation on the phase screen results, S 4ps, using a simple analytical formula: S 4 ¼ 1: exp S 4ps S4ps 2 : ð12þ The formula was obtained by visual examination of the [Dashen and Wang, 1993] results; the quadratic term in the exponential sharpens the turn from the linear to the saturated portions of the relation. It does not describe the strong focusing regime [Yakushkin, 1996] wherein for fluctuations of intermediate strength, the scintillation index peaks slightly above unity before asymptotically relaxing backto1inthesaturatedregimeofstrongerfluctuations. Applying this simple analytic transformation to obtain the strength of scintillation extends the usefulness of the phase screen as a measure of scintillation strength beyond the weak scattering limit in which it was derived. 3. Scintillation Maps [18] To construct a local map of the strength of scintillation requires carrying out the phase screen integral for the local bubble calculation, equation (11), with the saturation formula, equation (12), for every latitude and time. The ratio L sc /L s of path length containing scatterers to the size of the simulation needed for equation (11) is estimated using the cumulative distribution of RMS variation in density in the simulation; when scintillation is strong the ratio has values which are a reasonably large fraction of unity (for points on the equator, an idea of how this fraction varies can be seen from Figure 1 by judging the fraction of the simulation altitude range that is filled with stronger turbulence). The results for scintillation strength S 4 for vertical signal propagation at 250 MHz for the bubble calculation presented in the companion paper [Retterer, 2009] are shown in Figure 6 as a function of local time and dip latitude. We see the onset of scintillation near the geomagnetic equator, which then quickly spreads in latitude as the bubble rises. In subsequent hours the scintillation strength slowly decays and shrinks in latitude extent as the turbulence decays and falls with the motion of the ambient plasma in the late evening hours. The local time of onset is around 20:00; following that, strong scintillation lasts 4 5 h. [19] Once the scintillation producing irregularities are generated, they move longitudinally (usually eastward) with a speed that is bounded by that of the ambient plasma, typically around 100 m/s [see Retterer, 2009, Figure 7], equivalent to 3 per hour. Cross correlation analysis of spaced receiver observations of the scintillation structures [Bhattacharyya et al., 2001] also indicate drift speeds of this magnitude. The scintillation structures are expected to follow the latitude variation of the geomagnetic equator, i.e., their apex altitudes remain invariant, apart from the variations caused by E B drifts. Thus, once formed in a region, a region of scintillation can be tracked forward in time until it decays. [20] A regional map of scintillation at a given UT can be produced by mapping the local time variation of a plume calculated in the vicinity to nearby longitudes through the linear variation of local time with longitude at fixed UT, ignoring the longitudinal drifts because they are small on a global scale. In other words, within the region, the local time variation of the scintillation strength at every longitude is prescribed by a single plume calculation. The latitude variation is set by the combination of the dip latitude variation of the scintillation in the plume calculation and the variation of dip latitude with longitude. [21] A global map of scintillation for the C/NOFS system [Retterer, 2005] can then be produced by compositing these regional maps together. Because of the possibly short correlation distance of scintillation activity, independent plume development should be permitted every five or ten degrees in longitude, although correlation studies between the stations of the SCINDA network show that appreciable correlation can exist for stations as far apart in longitude as 30 degrees [Caton and Groves, 2006]. A sequence of global maps of scintillation throughout the day can be produced by superimposing regional maps produced with local plume calculations for the unstable regions at each time. In geomagnetically quiet times, only the postsunset region will likely experience the Rayleigh Taylor instability leading to plumes and scintillation, so this sequence of maps shows this region moving around the earth, following the sunset terminator line. [22] The most meaningful evaluation of the results comes from a comparison with measurements of S 4. For this, we present a comparison with data from the SCINDA [Groves et al., 1997] network. The SCINDA system consists of a set of ground based receivers that monitor the signal strength of radio transmissions from a set of satellite based transmitters and calculate the S 4 parameter from the fluctuations in signal intensity. On the day of our event, 25 October 2004, there were 3 SCINDA stations operating in South America: Ancon, Peru (latitude 12, longitude 283, diplatitude 0.8 ), Cuiaba, Brazil (latitude 15, longitude 304, diplatitide 6) and Antofagasta, Chile (latitude 24, longi- 7of10

8 Figure 7. Comparison of predicted scintillation (blue curves) with UHF SCINDA measurements (red curves) of amplitude scintillation at three South American stations. tude 290, diplatitude 11). They monitored satellite links to two geostationary communications satellites, Fleetsat 7 and Fleetsat 8, at a radio frequency near 260 MHz. None of the SCINDA links offers an exactly vertical geometry to match the assumption of our calculation. For the stations on the west coast (Ancon and Antofagasta) Fleetsat 7 offers the closest to vertical signal path, with zenith angles approximately 30 degrees and 44 degrees, respectively, for Ancon and Antofagasta, while the signal path to Fleetsat 8 was at much higher zenith angle. For Cuiaba, Fleetsat 8 presented a slightly smaller zenith angle (42 degrees) so we used that link in our comparisons. To correct for the difference from vertical signal incidence, a 1/cos c factor was introduced into the phase screen integral for S 4, where c is the zenith angle of the propagation path, although note that for very large zenith angles, the scattering turbulence may be remote from the observing station and occur at a local time more appropriate for the location where it is forming. [23] We have no measurements of the densities and drifts for the ambient plasma to determine whether the climatological characterizations of these quantities that we used were accurate; we can describe it as geomagnetically quiet day, in a scintillation season for the region. Figure 7 presents the S 4 scintillation index reported by the SCINDA stations at a 82 second cadence, along with the scintillation index predicted by the model. To apply one bubble calculation to these several stations at slightly different longitudes, the comparisons were made as a function of local time, by interpolating the predicted scintillation as a function of that variable. Note that each SCINDA system has an instrumental noise floor which set a minimum value of S 4 below which scintillation of geophysical origin cannot be detected. The Antofagasta data were processed to eliminate the worst of the nongeophysical spikes that contaminate the data by eliminating any points where S 4 was a factor of five or larger times the previous value. The results are that at all three stations, the model does well in predicting the onset time and duration of scintillation. At Ancon and Cuiaba (the lower dip latitude stations) S 4 saturates at a nearly constant level after onset until just before scintillation terminates, as the model predicts. At Antofagasta, however, there is structure in the SCINDA measurements which is not apparent in the model results. This is probably indicative of structure in the spacing of plumes or their bifurcated structure which the Fourier analysis of the east west structures, which used the whole simulation grid, integrates over. This is not an uncommon difference seen in comparisons with the SCINDA data: the model provides an envelope within which the actual S 4 varies. To address the issue would require perhaps a wavelet analysis of the structures to resolve their spatial structure instead of the Fourier analysis used here, but would also require more detailed knowledge of the initial conditions and seeding of the plumes than we can ever expect to know. Although the SCINDA study [Caton and Groves, 2006] showed some correlation between stations as far apart as 30 degrees, the distance between Cuiaba and the west coast stations is large enough that it is remarkable, and probably unusual, that the scintillation there correlated so well with the other stations to the west on this day. Along term study with knowledge of the conditions that trigger scintillation to establish the degree of correlation between the stations would be facilitated by the use of a model like the current one to relate the observations made at different latitudes and longitudes. [24] The scintillation at L band frequencies (1690 MHz) for this event is shown in Figure 8. The red curve gives the SCINDA measurement from the GOES 8 satellite to the Ancon station as a function of time; the instrumental noise floor has been subtracted (in quadrature). The solid blue curve gives the calculated S 4 index from the plume calculation when the power law exponent is fitted freely (as shown in Figure 4) to the TEC spectra. Because in situ measurements of density spectra [Singh and Szuszczewicz, 1984] suggest there is a break to a steeper spectrum at wavelengths shorter than 1 km, where the Fresnel scale for GHz radio signals falls, we imposed a steeper power law on the TEC spectra when the fitted power law was too shallow by limiting the power law exponent to be smaller 8of10

9 Figure 8. Comparison of predicted scintillation (blue curves) with L band SCINDA measurements (red curve) of amplitude scintillation at a South American station. The solid blue curve is the predicted S 4 if the spectral power law exponent is freely fit to the TEC spectra, while the dashed curve is the predicted scintillation if the exponent is not allowed to be larger than 2. than 2 (i.e., larger in absolute value) to produce the dashed curve that reproduces the amplitude of the GHz scintillation well. Both model curves represent the time of scintillation onset and duration fairly well. 4. Discussion [25] We have developed and presented here a model for predicting the strength of radio scintillation as a function of time, latitude, and longitude, given the drivers for the ionospheric structure: the plasma drift velocity, temperature, and the thermospheric parameters. This model offers the first physics based calculation of the latitudinal extent of the radio scintillation produced by equatorial plasma plumes. Although the model cannot encompass a first principle description of phenomena on all the scale lengths relevant for the generation of scintillation, the necessary extrapolations are based on observations and sound physical principles, and we believe that the model will be useful for operational predictions of scintillation and its impacts. [26] The first question to ask concerning the model is can it represent the morphology of scintillation properly? For the prereversal enhancement driven scintillation studied here, the time of scintillation onset (around 20:00 LT) and its duration (4 6 h) are about right in the model, and, apart from interplume structure, the temporal variation of scintillation in the model, e.g., the rate of onset at the beginning and rate of decay at the end, matched the observed variation. The next question is whether the model reproduces the climatology of scintillation [Aarons, 1993; Secan et al., 1995]. In the studies where we have compared the model climatology of plumes with the statistics of bubbles observed by the DMSP satellites [Huang et al., 2002], we [Retterer and Gentile, 2009] have also looked [Retterer, 2005] at the climatology of scintillation and established that this model does demonstrate at least the general features of the scintillation climatology, as a function of season, longitude, latitude, and phase of solar cycle. [27] The third question is whether the model reproduces the day to day variability of scintillation. This is a large question and addressing it has been hampered by the lack of long term data to characterize the ambient state of the ionosphere to determine its instability. A study of the data from the COPEX campaign using this model [Retterer et al., 2006], where an ionosonde at the geomagnetic equator provided vertical plasma drifts that have been analyzed for a period of a month, suggest that geomagnetic activity (even weak events) cause much of the variability of the plasma drifts, and using these drifts this model can explain most of the variability of occurrence of spread F. A more complete examination of this problem awaits better coverage of the variability of the ambient low latitude ionosphere, aided perhaps by the launch of the C/NOFS satellite [de la Beaujardiere et al., 2004]. [28] Acknowledgments. I thank O. de la Beaujardiere, B. Basu, T. Beach, and J. Huba for interesting discussions and thank K. Groves for providing the SCINDA data for the proof of concept demonstration. This research was partially supported by the Air Force Office of Scientific Research and the NASA Living with a Star and Heliospheric Space Theory programs. 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