M. G. Deminov, A. T. Karpachev, and S. K. Annakuliev
Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Troitsk, Moscow Region, Russia
V. V. Afonin
Institute of Space Research, Moscow, Russia
The midlatitude ionospheric trough is the region of decreased electron density situated near the equatorial boundary of diffuse precipitation of electrons with energies of 0.5-1 keV (see, e.g., Rodger et al. ). During the main phase of a magnetic storm, one minimum in the latitudinal distribution of the ionospheric plasma electron density is generally noted in this region [Deminov et al., 1995a]. During this period the position of the minimum of the midlatitude ionospheric trough correlates with the magnitude of the magnetic field of the ring current DR better than it does with the Kp index [Deminov et al., 1995b].
During the magnetic storm recovery phase, the midlatitude ionospheric trough (MIT) structure becomes more complex. A qualitative analysis of sounding data from the Cosmos 900 satellite at altitudes of 400-500 km for nighttime has shown that during the recovery phase, the midlatitude ionospheric trough can be shown to consist of two troughs rather than one [Deminov et al., 1995a]. One of these troughs (MIT) tends to be situated near the diffuse precipitation boundary (DPB), while the other (the ring ionospheric trough (RIT)) seems to be related to the magnetospheric ring current [Deminov et al., 1995a]. None of the known empirical models of the position of the midlatitude ionospheric trough predicts the possibility of the simultaneous existence of two troughs clearly spaced in latitude (see, e.g., Rodger et al. ). Moreover, with two troughs existing at subauroral latitudes, the very term "position of the minimum of electron density of the midlatitude ionospheric trough" becomes ambiguous.
To overcome this contradiction, it is expedient to seek quantitative laws of variation of positions of the MIT and RIT as independent structures during the recovery phase of a magnetic storm. Knowledge of these laws is also essential for the identification of these troughs from experimental data, since the MIT is easily confused with the RIT, especially when only one of these troughs is present [Deminov et al., 1995a]. It is the goal of this paper to find out these laws on the basis of an analysis of the Cosmos 900 sounding data at altitudes of 430 50 km for nighttime in local winter and equinox at the recovery phase of 16 magnetic storms in 1978-1979.
Below we analyze the correlation of the invariant latitude FT(T) of the MIT minimum and the invariant latitude FR(T) of the RIT minimum as measured from the Cosmos 900 satellite at the universal time T with DR(T-T0 ) and Kp(T), where T0 is the characteristic time of delay of variation of FT or FR relative to DR. Time T0 is determined from the condition of the highest correlation of the position of the MIT or RIT with DR. The employed dependence of the magnetic field of the magnetospheric ring current DR on the Dst index and the solar wind parameters is that reported by Deminov et al. [1995b]:
where DR and Dst are measured in nanoteslas; P = 0.01nV2 is the solar wind pressure, where V is the solar wind velocity in kilometers per second and n is the density in cm -3. The coefficient k depends on the magnitude and sense of the interplanetary magnetic field vertical component Bz : k = 0.2, 0.25, and 0.3 at Bz > 1, 1 > Bz > -1, and Bz < -1, respectively. For analysis, we used the Cosmos 900 data on the ionospheric trough position, obtained from the sounding measurements of electron densities at altitudes of 430 50 km during nighttime (1800-0600 MLT) in local winter and equinox during 1977-1979. These data correspond to the recovery phases of 16 magnetic storms, for which data are available on the solar wind parameters [Cousens and King, 1986] necessary to calculate DR.
Analysis of the dependence of the correlations of FT(T) and FR(T) with DR(T - T0) on T0 for the entire aforementioned array of data shows that these correlations are at maximum at T0 = 1. Here and hereinafter time is measured in hours, and DR(T - T0) is in nanoteslas. For brevity, the times T and T0 are omitted; that is, expressions FT Kp and FR DR mean that FT(T) Kp(T) and FR(T) DR(T - T0), where T0 = 1.
The values of FT obtained from the Cosmos 900 data for the recovery phases of 16 magnetic storms correspond to the following ranges of variations of the magnetic activity indices: -4 > DR > -62, -19 > DR > -234, and -13 > DR > -234 for the local magnetic time intervals 1800-2200, 2200-0200, and 0200-0600 MLT, respectively. Therefore for the premidnight hours (1800-2200 MLT), the FT measurement data largely relate to a relatively late stage of the recovery phase of an intense magnetic storm. Because of this, the statistical characteristics of models for premidnight hours given below are approximate. In order to construct a model from these initial data on the MIT position, we will require the model being developed FT(DR, t, L) for the limiting cases of very low ( DR = 0) and high (e.g., DR = -500 ) magnetic activities not to contradict the model reported by Deminov et al. [1995b]:
Here and below, the invariant latitude F and geographic longitude L are measured in degrees, the local magnetic time t is counted from midnight ( t = MLT - 24 before midnight and t = MLT after midnight), the indices N and S correspond to F(L) for the northern and southern hemispheres, F0 = 65.4, and FL(0) = 41.4.
It should be noted that the formula for FT(0, t, L) was obtained on the basis of the Cosmos 900 data for the periods of low magnetic activity ( Kp 2) and the standard deviation of this formula from the experimental data is s = 2 [Deminov et al., 1992]. The formula for FT(-500, t, L) is an asymptotic approach to the empirical formula for FT during the main phase of a magnetic storm [Deminov et al., 1995b], which demonstrates a tendency of MIT shifting equatorward with magnetic activity increase. For example, for L = 36 in the near-midnight hours, FT = 55.6, 47.5, and 43.4 for DR = -50, -150, and -250. Moreover, the FT dependence on L is weakened with an increase of magnetic activity and almost completely disappears under DR < -50.
The empirical model of the MIT position at the storm recovery phase for the interval 1800-0600 MLT obtained on the basis of a statistical analysis of the Cosmos 900 data, the model standard deviation s in degrees of latitude, and the coefficient R of correlation of the model with the initial Cosmos 900 data is
Model (5) meets conditions (2): under none of the values of DR can the main ionospheric trough shift below the limiting latitude FL(t) or above the latitude FT(0, t, L). It follows from (5) that with decreasing magnetic activity during the magnetic storm recovery phase, the MIT moves poleward up to the high-latitude boundary FT(0, t, L). In this motion of the MIT toward high latitudes, the dependencies of FT(DR, t, L) on t and L become stronger.
One can obtain from formula (5) that FT = 56.3, 47.5, and 43.9 for DR = -50, -150, and -250, respectively, under L = 36 in the near-midnight hours. These values almost completely coincide with the values for the storm main phase presented above. Therefore formula (5) presented by Deminov et al. [1995b] for FT provides a FT variation continuously under a transition to the storm main phase.
A qualitative analysis has shown [Deminov et al., 1995a] that at the initial stage of the recovery phase of an intense storm, the MIT and RIT coincide; that is, they are looked upon as one trough moving toward higher latitudes. At a later stage, the velocities of motion of the MIT and RIT toward higher latitudes differ, and the high-latitude boundary of this motion proves to be far more equatorward for the RIT than it is for the MIT. During this period, positions of the MIT and RIT become different, and this difference is maximum at a later stage of the recovery phase of an intense storm. The empirical model of the RIT position at the storm recovery phase obtained on the basis of a statistical analysis of the Cosmos 900 data reflects these patterns:
where FR(0) = 56, DR0 = -100, FT(DR, t, L), and FT(DR0, t, L) are determined by equation (5). Statistical characteristics of model (6) are s = 0.9 and R = 0.96.
Formula (6) is correct if DRmin < -60, where DRmin is the lowest for the given storm value of DR, which is usually observed at the end of the storm main phase. For RIT to form, it is necessary to have the trough at the end of a magnetic storm equatorward from the FR(0). If this condition is not fulfilled, the RIT is not observed as a separate structure.
It follows from model (6) that at none of the values of DR can the ring ionospheric trough shift equatorward of FL(t) and poleward of FR(0). The high-latitude boundary FR(0) of the RIT poleward motion does not depend on local time and longitude. The MIT and RIT coincide at DR DR0. For the range DR0 < DR < -60, the difference between FT(DR, t, L) and FR(DR, t, L) normally does not exceed the value of deviation of the initial data from these models. That is why at the storm recovery phase, at DR < -60, the main and ring ionospheric troughs are normally looked upon as one trough moving toward higher latitudes. With magnetic activity further decreasing, the MIT and RIT clearly differ in both the region of localization and the character of dependence of this localization on magnetic activity: the MIT still moves toward higher latitudes, while the RIT stops near FR(0). During this period, FR(DR, t, L) does not practically depend on DR, t, and L, and the dependence of FT(DR, t, L) on t and L becomes stronger. In the limiting case DR = 0, the difference between positions of the MIT and RIT is determined by the boundaries FT(0, t, L) and FR(0) and averages to ~10o.
Figures 1, 2, and 3 illustrate visually typical dependencies of the trough position on the parameters, which were used in models (5) and (6). In Figures 1, 2, and 3, solid lines, dashed lines, solid cirkles, and open circles correspond to model (5), model (6), experimental data for MIT, and experimental data for RIT, respectively.
Figure 1 shows the dependency of the MIT and RIT positions on DR at fixed t = 0 and L = 36 according to models (5) and (6) together with the Cosmos 900 data reduced to these values of t and L. The procedure of reduction, for example, of measured values of the trough latitude FT(t, L) under fixed t, L and DR to t = 0 and L = 36 is
where FT(DR, t, L) and FT(DR, 0, 36) are determined by model (5). It is worth noting that the FN(L) and FS(L) functions coincide under L = 36. The values of correlation coefficients R and standard deviations s presented above for equations (5) and (6) correspond to the data in Figure 1.
Figure 2 shows the dependencies of the MIT and RIT positions on t for the fixed L = 36 and DR = -20 or -100 according to models (5) and (6) together with the Cosmos 900 data reduced to these values of L and DR. It can be seen that RIT actually does not depend on LT ( R = 0.1 and s = 0.9 ). The MIT dependency on LT is the most significant under low magnetic activity: R = 0.80 and s = 1.5 under DR = -20 and R = 0.68 and s = 1.5 under DR = -150.
Figure 3 shows the dependencies of the MIT and RIT positions on geographic longitude L under the fixed values of t = 0 and DR = -20 for the northern and southern hemispheres together with the Cosmos 900 data reduced to these values of t and DR. It can be seen that RIT actually does not depend on L ( R = 0.1 and s = 0.9 for the northern hemisphere and R = 0.07 and s = 1 for the southern hemisphere). The MIT dependency on L is well manifested for the southern hemisphere (R = 0.74 and s = 1.2) and is much less pronounced for the northern hemisphere (R = 0.27, s = 1.7). Thus the FT dependence on L could not have been taken into account. However, this dependency provides a reasonable agreement with the experimental data under transition from the recovery phase of a magnetic storm to quiet conditions; so the dependency is left in the model.
Figures 1, 2, and 3 illustrate one more characteristic feature: the higher the magnetic activity level, the more accurate is model (5). Under high magnetic activity the FT dependency on DR dominates, because the FT dependencies on t and L are significantly weakened. As a result, model (5) for the recovery phase of a storm is more accurate than the MIT position model for quiet conditions: s = 2.0 for FT = FT(0, t, L) under Kp < 2 and s = 1.5 for FT = FT(0, t, L). Figure 1 also demonstrates that it is impossible to provide the above indicated accuracy under DR > -60 without dividing the troughs into MIT and RIT.
Models (5) and (6) make it possible to predict localization of the MIT and RIT at the storm recovery phase and from experimental data to distinguish the MIT from the RIT. However, because of the initial data set's not being large enough, these models are not capable of quantitatively predicting the probability of occurrence of the MIT and RIT at DR < -60. Therefore we will resort to qualitative characteristics of these probabilities. Analysis shows that at the recovery phase of an intense storm at DR > -60, it is rare that the MIT and RIT are observed simultaneously as two troughs clearly spaced in latitude. During the premidnight hours the MIT is normally more clearly pronounced than the RIT, and frequently the RIT fails to be present in the experimental data at all. For the postmidnight hours, the opposite tendency prevails: at -30 > DR > -60 the RIT alone is often noted, but the MIT appears as an additional trough at lower magnetic activity. During the premidnight hours, at -30 > DR > -60, the probabilities of occurrence of the MIT or RIT do not differ significantly, and two troughs clearly spaced in latitude can be noted simultaneously. Additional analysis shows that for a weak magnetic storm, the probability of occurrence of the RIT decreases. Occurrence of the RIT at the recovery phase seems to require the trough to shift equatorward of FR(0) by the end of the storm main phase. It should be noted that the RIT sometimes occurred 1-2 days after the storm ended, including during the evening hours.
The Kp index is more often used as an indicator of magnetic activity for the position of the ionospheric trough at subauroral latitudes (see, e.g., Rodger et al. ). To estimate the effectiveness of using the Kp index in determining the dependence of the MIT position on magnetic activity, we will use the same Cosmos 900 data array as that used in constructing model (5).
The empirical model of the MIT position at the storm recovery phase for the interval 1800-0600 MLT, obtained from statistical analysis of these data, has the following form:
where F(t) and F(L) are determined by relationships (3) and (4). This model is seen to be less accurate than model (5). Statistical characteristics of models (5) and (7) depend on local time (see Table 1). The Cosmos 900 data for 1800-2200 MLT largely refer to a relatively later stage of the recovery phase of an intense magnetic storm. As is seen from Table 1, models (5) and (7) feature the same accuracy for these conditions. For premidnight and postmidnight hours, model (5) is far more accurate than model (7).
Practically all of the previously constructed empirical models of position of the electron density minimum at subauroral latitudes are based on data arrays, each of which includes both quiet conditions and all the storm phases. Analysis shows that, for such data sets, the relative number of cases when the RIT is noted instead of the MIT is generally small. Therefore it can be assumed that the empirical models thus constructed correspond to the MIT. A nearly complete set of such models has been constructed using, basically, the Cosmos 900 data for nighttime conditions in local winter 1978-1979 [Deminov et al., 1992]. For an altitude of 430 km, the model of Deminov et al.  can be presented in the form
The accuracy of this model is typical of the accuracy of known models of this type. For example, at F(L) = 0, equation (8) practically does not differ from the most frequently used model of Kohnlein and Raitt . Table 1 lists the statistical characteristics of model (8) for the same Cosmos 900 data set on the MIT position that was used in determining relationships (5) and (7). The standard deviation related accuracy of model (5) for premidnight and postmidnight hours is seen to be about 3 times that of model (8). For premidnight hours, the difference in accuracy of models (5), (7), and (8) is slight. Allowing for corrections to the model of Kohnlein and Raitt  associated with the interplanetary magnetic field [Ben'kova et al., 1989] does not alter the situation. Statistical characteristics of the model of Ben'kova et al.  from the same Cosmos 900 data on the MIT position are s = 3.6 and R = 0.88 for the entire interval 1800-0600 MLT, s = 3.6 and R = 0.85 for 2200-0200 MLT, and s = 4.2 and R = 0.86 for 0200-0600 MLT.
Thus for premidnight and postmidnight hours, the DR index is a better indicator, compared with the Kp index, of magnetic activity for the MIT position at the recovery phase of an intense magnetic storm. For these conditions, the accuracy of model (5) is about 2 and 3 times that of model (7) and that of known models, model (8) included, respectively. For premidnight hours (1800-2200 MLT), at a relatively later stage of the recovery phase of an intense magnetic storm, the difference in accuracy of (5) and (7) is insignificant.
Analysis shows that the use of the Kp index in constructing an empirical model of the RIT position at the storm recovery phase leads to relatively great errors. The coefficient of correlation of such a model with the Cosmos 900 data does not exceed 0.82, and it would be impractical to introduce Kp as an indicator of magnetic activity for the RIT position. Model (6) seems to be the first empirical model of the RIT position.
It should be noted that model (5) is based on satellite data for local winter and equinox. Use of the model for local summer conditions requires special analysis because dependencies of the MIT position on magnetic activity level may evidently be different for local winter and summer.
We now consider the possible causes of dynamics of the midlatitude ionospheric trough at the recovery phase of a magnetic storm, which follow from models (5) and (6).
By the end of the main phase of an intense magnetic storm, the midlatitude trough, the boundary of diffuse electron precipitations, the plasmapause at the equatorial plane of the magnetosphere, and the maximum of the ring current ion energy density are all found at the lowest latitudes ( L shells) but, it seems, never cross the equatorial boundary FL(t). In this context, a significant contribution to the ring current energy density is made by the O + ions, especially for E < 30 keV [Hamilton et al., 1988]. At the recovery phase of a magnetic storm, all these structures shift toward higher latitudes, to their values specific to quiet conditions.
For intense storms, two stages of the recovery phase can be isolated. The first, fast stage with a characteristic time of 5-9 hours seems to be determined by the energetic O + ion losses of the ring current, while the second, far slower stage is determined by the energetic H + ions [Hamilton et al., 1988]. By the end of the first stage, all the aforementioned structures seem to be found near L = 3, which corresponds to F = 55. By that time, the vertical distribution of the electron density in the equatorial plane of the magnetosphere appears like two steps: in addition to the internal plasmapause near L = 3, an external plasmapause forms at higher L shells (see a review by Singh and Horwitz ). The value L = 3 corresponds to the median position of the internal plasmapause [Singh and Horwitz, 1992], the maximum density of the residual ring current ion energy [Hamilton et al., 1988; Lui et al., 1987], and practically coincides with the boundary latitude FR(0) for the RIT. This means that with a subsequent decrease in magnetic activity, the internal plasmapause, the magnetospheric ring current, and the RIT remain near L = 3. Because of this, at the second stage of the storm recovery phase, the MIT and RIT clearly distinguish themselves. At this stage, out of all the aforementioned structures, only the MIT, the DPB, and possibly the external plasmapause continue their motion toward higher values of L.
Comparison of the aforementioned structures suggests that the latitude of the RIT minimum is found near the internal plasmapause, but at somewhat smaller L shells. Formation of this minimum at altitudes of 350-450 km seems to relate to the following chain of processes: Coulomb interactions of the ring current ions with the surrounding electrons within the plasmasphere, where a major role is played by the O + ions with energies below 20 keV [Kozyra et al., 1987]; heating of the surrounding electrons in the region of this interaction; formation of a peak of the electron temperature and the subauroral red arcs at ionospheric altitudes through heat transfer along the L shells from the region of heating [Kozyra et al., 1987]; and an increase in the coefficient of recombination of ionospheric electrons at F region altitudes through vibrationally excited species, such as N 2* and O 2* in the region of the electron temperature peak (see, e.g., Rodger et al. ) and, as a result, formation of the RIT. An additional cause of the decrease in density of the ionospheric ions of the F2 layer can be the sequence of the processes: precipitation of energetic O + ions from the ring current, heating of the thermosphere, increase of the molecular components in the thermosphere composition at F2 layer altitudes, and increase in the recombination rate of the ionospheric electrons [Fuller-Rowell et al., 1990]. These processes seem to be the main causes of formation of the RIT. The close relationship of the RIT with the magnetospheric ring current justifies the name of this trough. It also follows from this relationship that, at the storm recovery phase, the invariant latitude of the maximum of the rate of heating of the ionospheric plasma in the ring current region is described by equation (6).
As was noted above, the RIT stops moving at the second, relatively slow stage of the storm recovery phase, whereas the MIT and the DPB continue to move toward higher latitudes to their values characteristic of quiet conditions. The variation in the flux of precipitating electrons (with energies of about 1 keV) with latitude is one of the main causes of formation of the polar wall of the MIT [Gal'perin et al., 1990]. For a pronounced MIT to form at this period seems to require that the DPB be pronounced as well; that is, an abrupt change in the precipitating electron flux with latitude should take place. Normally, the DPB is far more pronounced in premidnight hours than in postmidnight hours [Gal'perin et al., 1990]. Because of this, the MIT may not be noted in postmidnight hours. The relationship between the MIT and DPB is also tracked in the character of the dependence of the position of these structures on magnetic activity. In particular, it follows from the DPB model of Gussenhoven et al.  and model (8) for the MIT that correlation of these structures with Kp is maximum for the interval 1800-2100 MLT. For premidnight and postmidnight hours, the DR index is a better indicator, compared with Kp, of magnetic activity for the MIT position at the recovery phase of an intense magnetic storm (see Table 1). The close relationship between the MIT and DPB suggests that the same law is specific to the DPB as well.
At the second, relatively slower stage of the magnetic storm recovery phase, relaxation of DR is far delayed in relation to the decrease of Kp. As is seen from Table 1, in transition from dusk to premidnight hours, the accuracy of model (5) increases, whereas the accuracy of models (7) and (8) decreases. This means that relaxation of the MIT position to a value characteristic of quiet conditions proceeds faster in dusk than in premidnight hours. Such a relationship seems to hold for the DPB as well. This effect appears to be associated with the fact that the electron precipitation that forms the DPB occurs virtually locally in the coordinate system L shell-longitude because of the low electric field of the magnetospheric convection at subauroral latitudes at the storm recovery phase. As a result, the DPB position at a fixed MLT is determined not only by the characteristic lifetime of these electrons, but also by the DPB position in the previous sector of MLT because of the Earth's rotation. The MIT and RIT are found more poleward in dusk hours than in premidnight hours. As a result, relaxation of the DPB and MIT positions to values characteristic of quiet conditions will proceed faster in premidnight than near-midnight hours. Formally, this leads to a relatively higher correlation of the MIT and DPB positions with Kp in premidnight hours compared with postmidnight hours.
1. An empirical model of the dynamics of the position of the MIT minimum at altitudes of 430 50 km at the recovery phase of an intense magnetic storm is constructed for the nighttime during local winter and equinox. The accuracy of this model with respect to standard deviations for premidnight and postmidnight hours is 3 times that of the known models. This increased accuracy is attained by isolation of the magnetic storm recovery phase, separation of the MIT from the RIT, and introduction of a magnetic activity indicator for the MIT position: the ring current magnetic field DR, which is a better index than Kp. In dusk hours at a later stage of the recovery phase, the DR index is not much more effective to use in comparison with Kp because of the relatively fast relaxation of the MIT position to the undisturbed value during these hours. Much the same dependence of the model accuracy on local time seems to be characteristic of the DPB as well. The relatively fast relaxation of the MIT and DPB positions to the undisturbed value in premidnight hours may be associated with the effect of the Earth's rotation, that is, with the dependence of the position of these structures at a fixed MLT on their position in the previous MLT sector.
2. For the same geophysical conditions, an empirical model of the RIT minimum position is constructed, wherein the DR index is the magnetic activity indicator. This model seems to be the first empirical model of the RIT minimum position. A qualitative analysis shows that the RIT is closely associated with the magnetospheric ring current. It follows from this relationship that the model of the RIT position can be used for determining the invariant latitude of the maximum of the rate of heating of the ionospheric plasma in the magnetospheric ring current region. The Kp index is impractical to use for the description of variation of the RIT position.
3. The empirical models constructed make it possible to predict localization of the MIT and RIT at the storm recovery phase. They are also essential for identification of these troughs from experimental data, since the MIT is easy to confuse with the RIT, especially when only one of these troughs is noted.
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