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[1] A new technique is introduced for determining plasma fluxes using tomographic images of ionospheric electron density. Pairs of images are used to estimate the time derivative of the F-region electron density in the region of the northern equatorial anomaly at 120°E longitude. Characteristics of temporal variations are described, and two different methods are used to infer electron fluxes from the measured time derivaties. The height-dependent latitudinally-averaged vertical flux and a 2D “potential” flux are derived based on reasonable assumed boundary conditions. Examples illustrate that average vertical plasma fluxes in the local morning and evening hours are upward and downward, respectively, with peak magnitudes on the order of 10^{9} cm^{−2}s^{−1}. The 2D plasma fluxes in a meridional cross-section through the equatorial anomaly are calculated, and clearly illustrate the characteristic flow associated with the equatorial fountain effect.

[2] With the advent of satellite radiotomography (RT), many ionospheric images have been reconstructed that show interesting ionospheric structures with horizontal scales on the order of tens to hundreds of kilometers. The equatorial anomaly (EA) region is a region where such structures exist in abundance because of the fountain effect in response to the equatorial dynamo. It is therefore of interest to monitor and investigate this dynamic ionospheric region using the tomographic technique. For this purpose, a low-latitude ionospheric tomography network (LITN) was operated during the period 1994–1996. LITN consisted of a chain of six stations at locations spanning 14.6°N to 31.3°N (or 3.3°N to 19.7°N geomagnetically) at a longitude of 121 ± 1°E. Information about the stations, the data format, the processing techniques and some results has been published before [Huang et al., 1999; Andreeva et al., 2000, 2001; Yeh et al., 2001] and will not be repeated here. For this paper, we selected several sets of data to show the temporal variations of the ionospheric images and implied fluxes.

2. Temporal Variations of 2D Cross-Sections of Ionospheric Electron Density

[3] RT systems like LITN are used to image the electron density N(h, L, t) as a function of height h, varying from 200 km to 1000 km and latitude, L, over a span which depends on the length of the receiving chain. LITN spans about 17° in latitude or about 2,000 km in ground distance. The satellite takes 10 to 15 minutes to pass from one end of the tomographic receiving chain to the other end. In general, any ionospheric spatial variations less about 30–40 km will be smeared and cannot be accurately reconstructed. The satellite system NNSS consists, at various times, of 7 or more satellites in nearly polar orbits an an altitude of 1000 km. This provides many occasions where two successive satellite-passes are separated by a short interval of, say, thirty minutes or so. For large-scale ionospheric structures with slow temporal variations, particularly before sunrise or after sunset, images separated by about thirty minutes can be used to estimate the time derivative, , with little error. For this study data from nearly overhead satellite passages was employed so that the reconstruction plane was close to vertical (maximum elevation angle between 80–90 degrees). In all cases the images were derived from data measured by all 6 receivers in the LITN network and the experimental records were of excellent quality, without any breaks or jumps in the recorded dispersive doppler signal.

[4] We first present some examples of temporal variations of tomographic images derived using LITN data. Figures 1a, and 1b portray (in contours) two tomographic images on September 14, 1996 at 12:50 and 13:30 LT. The values of electron density are given in units of 10^{6} el/cm^{3} and the separation between contours is 0.1 · 10^{6} el/cm^{3}. In these images, the presence of the equatorial anomaly crest is apparent within geomagnetic latitude range 2°N–8°N (or within 14°N–20°N geographically as can be seen on the bottom scale of Figure 1). These and also other structural features existing in the region of equatorial anomaly region have been described previously [Andreeva et al., 2000].

[5] Shown in Figure 1c is the difference of the two images in Figures 1a, 1b in units of cm^{−3}s^{−1}. The derivative values vary within a range from −80 to +60 cm^{−3}s^{−1}. In the region of equatorial anomaly crest the derivative is positive, while just north of the crest it is negative. In other cases (not shown), derivative values vary within wide limits and reach magnitudes up to several hundred cm^{−3}s^{−1}. During a period of anomaly formation, we have found that the distribution of derivatives is often quite complicated, with a series of interlaced positive and negative striations.

[6] A typical example of time derivative distribution in the morning is shown in Figure 2a. An example of time derivative distribution in the evening ionosphere is depicted in Figure 2b. Derivatives in the morning are mostly positive and those in the evening are mostly negative except for cases of postsunset enhancement. Note that in most cases the distributions of electron density derivatives as well as distributions of density itself contain irregular structures with sizes of a few hundred km.

3. Height-Dependent Spatially-Averaged Vertical Flux Density

[7] As is well known, the electron density N satisfies an equation of continuity of the form

where is the velocity of electrons, P is the rate of production per unit volume, and L is the rate of loss per unit volume through chemical processes. In the upper F region ionosphere, production and loss terms are small and can be ignored.

[8] Define the flux density vector and rewrite (1) as

It is convenient to use a coordinate system with coordinates l (distance from the equator measured along earth's surface, positive to the north), h (height above earth's surface), and y (distance to the east, measured along earth's surface). The flux density vector will then have components denoted by {F_{l}, F_{y}, F_{h}}. We then integrate both sides of equation (2) over a volume, V, with incremental thickness, dy, along the y (east-west) direction and defined by lower and upper limits l_{0}, l_{1} in the north-south direction (defined by the limits of the reconstructed electron density image), and lower and upper altitudes h, h_{1}. The upper limit, h_{1}, corresponds to the upper limit of the reconstructed electron density image. The lower limit will be allowed to vary upward from the lowest reconstructed altitude h_{0}. Application of Gauss' theorem to the volume integral leads to a relationship between the flux through the bounding surface S of the volume V and the rate of change of the total number of electrons in the volume:

The surface integral on the left hand side of equation (3) includes 6 terms corresponding to each of the faces of the described volume. Recalling that the dimension of the volume in the y direction is incremental, dy, and defining the Jacobian J(h) = 1 + h/R which arises when integrating in (l, h) coordinates, both sides of equation (3) can be divided by dy to obtain:

The units of equation 4 are now electron flux per unit length, cm^{−1}s^{−1}. For simplicity, we will continue to refer to this quantity as “flux” in the next few sentences. We now assume that the flux through the upper boundary at h_{1}, typically located at 1000 km altitude, is much less than the flux through the lower boundary. Also, in the morning and evening hours, vertical fluxes in the anomaly region are expected to be significantly larger than horizontal fluxes so that the flux leaving the volume through the north and south faces is small compared to that through the bottom faces. We will also assume that the zonal gradient of the zonal flux density, is negligible. Then equation (4) simplifies to

which means that, under our approximations, the rate of increase of the electron content within the volume is accounted for by the upward flux through the bottom boundary.

[9] The spatially-averaged vertical flux density is defined to be the total flux divided by the length of the bottom surface, or

[10] The relation (6) defines the vertical flux averaged over a latitude range Δl = l_{1} − l_{0}. The average vertical flux density calculated for the two cases depicted in Figure 2 are shown in Figure 3. A positive value indicates that the plasma enters the region from below going upwards and vice versa. Thus, in the morning, the plasma enters the volume from below going upwards, and in the evening the average plasma flow is out of the volume in the downward direction.

[11] Among the cases that we have calculated, the largest vertical flux density observed in the morning was at h = 300 km on 10/27/94 and the largest evening flux density was at h = 300 km on 10/06/94. Typical maximum values are smaller, however, and amount to a few tenths of 10^{9} cm^{−2}s^{−1}. Figure 4 shows the average vertical flux densities calculated for October 6, 1994. Variations of the flux values and their height distributions are clearly seen in the Figure. In particular, during the evening interval 18:50 to 19:40LT a pronounced postsunset enhancement was observed with peak vertical flux density of about 1 · 10^{9} cm^{−2}s^{−1} at h = 300 km.

[12] It should be stressed that the calculations of average vertical flux densities are expected to be quite accurate since they are spatially integral quantities and noise is effectively filtered out. It should also be mentioned that in calculating the differences to approximate the time derivative ∂N/∂t one can choose also longer time intervals, even up to a few hours. In such cases, we must realize time-averaged flows are calculated. Of course, smearing occurs when the interval taken is too large.

4. Two-Dimensional Flows

[13] If the flux density is irrotational, i.e., , one can define a potential field Ψ such that . Then, equation (2) transforms into a classical Poisson equation for the potential function Ψ,

Assuming that the flux density vector is confined to the l, h plane containing the reconstructed electron density, horizontal F_{l} and vertical F_{h} components of the two-dimensional flux vector $\vec F = - \nabla \Psi$ can be expressed in terms of the function Ψ. Methods of solving the Poisson equation have been discussed extensively in books of mathematical physics. The boundary conditions used in this paper are F_{h} = F_{l} = 0 at the top boundary, F_{l} = 0 on the left boundary (this is equivalent to assuming the absence of interhemispheric flows, which may be valid for some, but not all, seasons), and ∂F_{h}/∂l = 0 on the left boundary. These are probably the simplest, yet still fairly realistic boundary conditions. While the Poisson equation (7) can be solved with other more complicated boundary conditions, accurate and reliable data is really lacking to point the way on how these simple boundary conditions should be modified. Perhaps model computations can provide some clues.

[14]Figure 5 shows examples of two dimensional flux density calculations in accordance with the described scheme for three cases for which the distributions of time derivatives are shown in Figures 1 and 2. Figure 5a is apparently a good illustration of the fountain effect, whereby plasma driven by crossed electric and magnetic fields squirts vertically upward from the magnetic equator, then turns to form the EA crest region. Figures 5b and 5c portray morning and evening flux density fields. The flows are mainly vertical, only minor variations of the flows around vertical caused by ionospheric irregularities are observed. This indicates that the spatially averaged vertical flux density calculated in section 3 for morning and evening cases is a reasonable approximation to the vertical flux density at any particular latitude within the LITN span. From these two-dimensional flows it is also possible to estimate the outflows through the boundaries. Recall that, in section 3, when computing the averaged vertical flux density, the flux through the North and South faces of the volume was ignored. Using the calculated 2D flow data we can now check the accuracy of this approximation on the North face. Using data shown in Figure 5, the flux through the North face is found to have a relative magnitude of 0.005 when compared with the flux through the bottom boundary for the morning case and a relative magnitude of 0.015 for the evening case, consistent with our original assumption that the flux through this boundary is negligible.

5. Discussion and Conclusion

[15] In the present paper time derivatives $\frac{\partial N}{\partial t}$ are derived from tomographic cross-sections obtained from close in time satellite passes during Autumn 1994 in the region of the equatorial anomaly. Typical examples of temporal variations are shown. Derivatives of the morning cross-sections are mostly positive and evening derivatives are negative except for postsunset enhancement cases. Within an interval of anomaly crest formation, changes in the sign of spatial distribution of time derivative take place which is associated with corresponding changes in spatial structure.

[16] Determination of time derivatives from experimental RT data allows the calculations of F-region plasma fluxes after taking into account some appropriate boundary conditions. If one wishes to extend the calculations to lower heights, then data about photochemical reactions would be needed. Such data can be supplied from model calculations and/or by another independent method. This can be done by restoring the terms P and L in equation (1). During morning and evening hours the vertical component of flux density noticeably exceeds the horizontal ones, therefore one can neglect the flux through the side boundaries compared to the flux through the lower boundary. The flux through the top boundary (about 1000 km) is also very small. For morning and evening hours a method was proposed for calculation of latitudinally averaged vertical flux densities. Typical maximum values of average vertical flux density are of the order of fractions of 10^{9} cm^{−2}s^{−1}. However, quite often they reach up to a few units.

[17] A technique was also proposed for calculation of two-dimensional potential flux densities from experimental RT data. Examples were presented that show two-dimensional plasma flows in a meridional cross-section in the EA region. The examples clearly illustrate quasi-vertical morning and evening flows and the fountain-effect near noontime, with the plasma squirting out of the equator and flowing to the EA crest region. Note that the method for calculating averaged vertical flux density and the method for calculating 2D potential flux densities from experimental RT data are proposed for the first time. These methods make it possible to carry out a comparison of the flows calculated from the models and those calculated from the RT experimental data. Indeed, such estimations have been recently made using magnetometer observations [Anderson et al., 2002]. In the future it seems promising to develop a combination of complementing approaches that can be checked and verified mutually.

Acknowledgments

[18] The material reported in this paper is based upon work supported by the U.S. National Science Foundation under grants ATM 97-13435, ATM 00-03418 and supplement, and by the Russian Foundation for Basic Research under grant 02-05-65350.