Originally published in:
Cosmic Electrodynamics 1 (1970) 67-89. All Rights Reserved Copyright © 1970 by D. Reidel Publishing Company, Dordrecht-Holland
The plasmapause, an abrupt drop in total electron density near L=4 (Carpenter, 1963; Gringauz, 1963), also divides the magnetosphere into two distinct regions. In addition it is found that this feature moves inward to L 3 during storm-times and relaxes toL 6 during quiet periods (Carpenter, 1967; Taylor et al., 1968; Chappell et al., 1969), reminiscent of the movement of the outer zone fluxes. With this in mind, it is tempting to consider the plasmasphere and the outer Van Allen belt as two physically distinct regions connected only indirectly by magnetospheric convection and diffusion across the plasmapause.
In the first part of this paper we review the motions of the plasmapause, the proton ring current and the low energy outer zone electron fluxes during a moderate magnetic storm to illustrate the relative location of these features. In Section 2 we investigate the electron morphology before and during another magnetic storm to emphasize that the outer edge of the inner zone, while strongly energy dependent, is relatively fixed during the storm-time enhancements in the outer zone. In Section 3 we reexamine the suggestion of Vernov et al. (1969), that the slot has moved in sympathy with the motion of the outer zone during the solar cycle and find that although the average location of the plasmapause and the average position of the outer zone maximum may have changed over the last 10 years, the profile of the inner edge of the slot has ostensibly remained fixed. Thus it is feasible to define an inner edge to the slot, which is primarily a function of electron energy.
In the final part of this paper we consider the natural cyclotron instability of the highly energetic electrons found within the plasmasphere and relate the observed persistent inner edge of the slot to the location where this interaction between the electrons and electro-magnetic VLF waves becomes ineffective. The role played by electrostatic waves in removing the energetic electrons will not be discussed here. Such emissions have been observed in the magnetosphere (Scarf et al., 1968) but they probably interact with lower energy particles.
Figure 1 shows the prestorm distribution of the thermal ion concentration, the proton flux, 31 keV < E < 49 keV, and the electron flux, E > 35 keV. The ion concentration and the proton flux were measured on OGO-3 in a highly eccentric orbit. The electron measurements were made at 90o pitch angles on Alouette 2 in a low polar orbit at an altitude ranging from 502 to 2982 km. Data from the Alouette orbit, closest in both local time and universal time to the OGO-3 pass, have been used. Unfortunately July 7 measurements were not presented by Taylor et al. (1968), and we have therefore used their prestorm July 3 ion data. The magnetic activity on July 3 and July 7 was, however, quite similar, K being 9 and 7 respectively on those two days, and from the studies of Carpenter (1967), Taylor et al . (1968) and Chappell et al. (1969) we can be fairly certain that the plasmapause was at a similar location on these two days. The important features of Figure 1 are that the plasmapause coincides roughly with the maximum of the quiet-time ring current and that the outer zone maximum of E > 35 keV electrons lies within the plasmasphere. Furthermore, there is a broad minimum in electron flux nearL 4.
The prestorm thermal ion concentration measured on July 3, 1966 by the OGO-3 experiment of Taylor et
al. (1968), the proton flux, 31 keV < E < 49 keV, measured on July 7, 1966 by the OGO-3 experiment of Frank
(1967) and the electron flux, E>35 keV, measured on July 7, 1966 by the Alouette-2 experiment of McDiarmid et
(1969) as a function ofL-value.|
Figure 2 shows similar measurements made on July 9 after the storm had begun. The scale of the proton intensity has been halved in this figure to accommodate the large increase in the ring current flux. The plasmapause, the ring current and the outer zone electron maximum have all clearly moved inwards during the main phase of this storm. Whereas the new plasmapause position may simply reflect the removal of the thermal particles beyond L= 3, the ring current and the outer zone electrons appear to be new particles, freshly accelerated or injected by the storm. It is also clear that these 'new' energetic particles appear just external to the plasmapause. In the case of the E > 35 keV electrons the stormtime injection has partially filled up the previously observed `slot' atL 4, moving the flux minimum in toL 3.
The same observations as Figure 1 made on July 9 during the main phase of the storm. The scale of
the proton flux has been reduced for this figure only.|
Measurements on July 11 during the recovery phase of the storm are shown in Figure 3. The plasmapause has now expanded and both the energetic electrons and protons have begun to decay. Swisher and Frank (1968) claim that the lifetime for the proton decay is consistent with that expected from charge exchange (Liemohn, 1961) with exospheric atomic hydrogen. It is however curious that the peak of the proton flux correlates well with the plasmapause during all phases of the July 8 storm. Clearly, since the number density of the ring current protons is much less than that of the plasmasphere, the charge exchange process itself cannot replace the prestorm thermal ion levels. Rather, the recovery of the plasmaspheric ion densities must originate from the ionosphere. Also, the neutral hydrogen concentration, and consequently the rate of charge exchange loss, should vary smoothly across the plasmapause, due to the fact that the convection electric fields, which presumably lead to the sharp drop in ion density (Nishida, 1966), do not affect the atomic hydrogen. It thus appears difficult to explain the rapid drop in proton flux just inside the plasmapause unless an additional process operates to exclude the ring current protons from the plasmasphere.
|Fig. 3. The same observations as Figure 1 made on July 11, during the recovery phase of the storm.|
The post-storm observations on July 13 are shown in Figure 4. The plasmapause has expanded still further and both the electrons and protons have continued to decay. It is particularly noticeable that while the proton maximum has moved out with the plasmapause, the outer zone electron maximum has remained roughly at the initial injection location and is now well within the plasmasphere. The electron decay, however, does appear to be restoring the original flux minimum at L=4 which was seen in Figure 1 In fact the only major difference between the prestorm measurements of Figure 1 and the post-storm measurements of Figure 4 occurs in the electron fluxes. The plasmapause has the same radius although the plasmaspheric density has not quite returned to its prestorm value, and the proton ring current is in the same position although the flux is about 50% greater.
|Fig. 4. The same observation as Figure 1 made on July 13 (post-storm).|
In summary, during a storm the plasmapause assumes a new position at lower radial values and a proton ring current builds up just external to the new plasmapause. The energetic electrons also appear just external to the plasmapause but not necessarily as close to it as the ring current. For electrons, with E > 35 keV, the new position of maximum flux can lie in the previous quiet-time minimum. After the storm the plasmapause expands and the ring current protons decay at a rate such that the maximum flux essentially keeps pace with the motion of the plasmapause. The electrons, however, decay much more slowly and the outer zone maximum created by the storm may lie several earth radii inside the quiet-time plasmapause. The net result is an injection of low energy electrons well into the plasmapause.
The division of the magnetosphere into two separate belts of radiation was discovered from the first deep space measurements of high energy particles (Van Allen and Frank, 1959). It seems fair to say that the early observations gave only a suggestion of the so called `slot,' leaving open the question of exactly which particles were responsible for this feature. It has since been established that the slot is due to a severe drop in the flux of high energy electrons. These particles have been intensively studied by several research groups and we now have detailed information on their flux, energy spectrum and pitch angle distribution. Since measurements of the higher electron energies at which the slot is best defined are not readily available for the July 1966 storm, we will examine the behavior of the electrons measured by Pfitzer et al. (1966) during another storm which occurred in September 1964.
Observationally, the slot is most pronounced for relativistic electrons during periods of low magnetic activity. At energies above 1 MeV the fluxes within a broad region centered about L = 2 to 3 are nearly always near or below the detectable limit. Typically, this entails a sharp drop in flux of more than two or three orders of magnitude from the peak values in the inner and outer belts. However, at energies below 100 keV, the slot is poorly defined and it tends to become completely filled in during magnetic storms. This was clearly illustrated for E > 35 keV electrons in the previous section. The location of the electron flux maximum in the inner zone, L, , and outer zone,L, , is shown in Figure 5 for the period just prior to the September 1964 storm. In each case the vertical bars indicate the energy range of the instrument. Also, indicated by the horizontal bars are the regions over which the flux fell below the detectable limits. One should especially notice that, whereas the location of the inner zone maximum depends only weakly on energy, there is a definite systematic increase in the L-value of the inner edge of the slot with decreasing energy. The outer edge of the slot exhibits similar, though more erratic, behavior and appears to mirror the location of the outer zone maximum. Figure 6 shows similar measurements taken 8 days after the storm. The inner zone fluxes are unchanged and the maxima have not moved. For the highest three energy channels the inner edge of the slot has also remained at the same location. However, at lower energies, electrons have been injected into the region previously devoid of particles, leading to an apparent inward motion of the flux minimum similar to that shown in Figure 2. Strictly one can no longer speak of a true slot in electron flux. Rather, the location of the flux minimum is now controlled by the inward injection of electrons originating in the outer zone.
|Fig. 5. The position of the maximum of the electron flux in the inner zone. LMAX,i, the maximum in the outer zone,L ,, and the electron slot as measured before the September 1964 storm by Pfitzer et al. (1966) at a variety of energies. The vertical bars indicate the energy range of each measurement. The ends of the horizontal bars denote the inner and outer edges of the slot in which the flux was below the detectable limit of the experiment.|
|Fig. 6. The same as Figure 5 except that the measurements were performed after the storm of September 1964. The lack of horizontal bars for the slot positions at the lowest two energies indicates that the electron flux did not fall below the detectable limit. The position indicated for the slot in these two cases is the location of the minimum flux.|
Even though the outer edge of the slot and the outer zone maximum have changed position in a complex manner, they appear, for a particular electron energy, to have moved in unison. At the highest two energies the motion is outwards; at the lowest three energies the apparent motion is inwards. Recalling the results of the previous section where it was seen that the E > 35 keV electron maximum lay within the plasmasphere at quiet-times, we can deduce from Figure 5 that the quiet-time outer zone maximum for the higher energies also lies within the plasmasphere. Similarly, since Figure 6 shows that the outer zone maximum moves outwards at the highest energies and inwards at the lowest energies during disturbed times, it appears that just after the onset of the storm the outer zone maxima at all energies will be external to the plasmasphere.
Figures 5 and 6 also indicate that the inner zone is relatively stable even during magnetically disturbed periods. This statement includes the outer edge of the inner zone or equivalently the inner edge of the slot. However, at the lower energies, particles are injected into or locally accelerated in the quiet-time slot region itself. Thus the outer zone maximum appears to move radially inwards and correspondingly the outer edge of the slot moves inwards to radial positions even lower than the previous inner edge of the slot. Nevertheless, if we keep in mind the physical nature of the apparent motion of the slot at low energies, it is clearly possible to define an inner edge profile that is only a function of energy. A theoretical explanation for the energy dependence of this stable profile will be explored in Section 4.
In the previous two sections we demonstrated that the distribution of particles in the magnetosphere changes significantly during geomagnetic disturbances. Generally the L-value of any specific morphological feature anticorrelates with the geomagnetic indices (Forbush et al. , 1962). A systematic movement of both the slot and the outer zone has also been noticed over the solar cycle (Frank and Van Allen, 1966; Vernov et al., 1969). In the present section we reconsider the question of this movement, asking whether the trend might not merely represent an enhancement in the average level of geomagnetic disturbance which itself is known to accompany increases in solar activity (Shapiro, 1969).
There has also been some confusion over the definition of the slot. Originally, this referred to the region of minimum counting rate of a detector such as a geiger tube which responds to both the inner zone protons, say E > 20 MeV, and the outer zone electrons, say E > 1 MeV. The data used by Frank and Van Allen (1966) and the majority of the observations summarized by Vernov et al. (1969) were of this form. More recently, however, magnetic spectrometers have been used successfully to distinguish between the electron and proton populations. Protons are generally distributed in a single zone which moves to higher L-values at lower energies (Davis and Williamson, 1966). In sharp contrast, electrons are found in two distinct zones. The severe drop in electron flux between the inner and outer zones is what we shall hereafter refer to as the slot.
Although observations of the Van Allen belts have been made since 1958, the task of determining possible motions over the solar cycle is not easy. In Section 2 we showed that the position of the slot and the outer zone maximum were clearly functions of energy. Thus in order to look for systematic changes, we should only compare similar energies. In addition, we should only include measurements that clearly reflect electron fluxes. In the slot region this requires strong discrimination against protons. A good instrument for this purpose is an electron spectrometer with magnetic selection of particles which also monitors the background counting rates. Finally, we should avoid measurements of the slot at very low altitudes which may be affected by the South Atlantic anomaly. For various reasons, primarily to obtain the largest sample of measurements, we shall restrict ourselves to electron energies around 1 MeV. Furthermore, we shall use geiger tube measurements only for observations of the outer zone.
The first measurement used in our study was made by Forbush et al. (1962) using a GM detector measuring electrons with E > l.l MeV flown on Explorer 7. They presentL, as a function of D. This is shown as position 1 in Figure 7 . The ends of the vertical bar show the most probable position of the maximum flux location for high and low D . The horizontal bar is at the most probable position for moderate DST and the ends of this bar give the time interval over which the measurements were taken.
|Fig. 7. Measurements made from 1960-1968 of the outer zone electron maximum (solid bars and dots), of the average plasmapause position (triangles) and the position of the slot (dashed bars). Only energetic electron measurements at energies near 1 MeV were used. See text for the references to each individual point.|
The second measurement of the outer zone used (points labelled 2), was made by Freeman (1964) on Explorer 12 using a GM detector sensitive to electrons with E > 1.6 MeV. Only nonstorm time points have been plotted.
The measurements labelled 3 are of electrons with E > 1.6 MeV. They were obtained by Owens and Frank (1968) using a GM detector on Explorer 14. The vertical bars represent the range of the positions of the outer zone maximum and the horizontal bars the time interval of the measurements. On occasion, the maximum occurred below L=4, the lower limit of reported observation, but only for short intervals of time. It was thus possible to calculate average positions which are indicated by the horizontal bars.
The measurements labelled 4 were obtained by Mihalov (1967) with an electron spectrometer (scintillator) in the energy range from 0.66 to 1.2 MeV on satellite 1964 45A in a polar orbit with an apogee of 3765 km. The spectrometer permitted an accurate definition of the slot as well as the outer zone maximum. The inner edge and outer edge of the slot are given by the ends of the vertical dashed line. The solid vertical line shows the range of L-values over which the peak in the outer belt occurred.
Points 5 are the measurements used in the previous section of Pfitzer et al. (1966) for the energy channel 0.69 to 1.7 MeV. Again the ends of the dashed lines are the inner and outer edges of the slot. The solid circles indicate the location of Lat various phases of the storm.
The measurements labelled 6 were made by Williams et al . (1968) with a solid state detector on Explorer 26 for electrons > 1 MeV. These measurements were contaminated by protons at low L-values but were entirely adequate for determining the outer zone electron maximum. The measurements plotted here represent the storm-time L-value of the outer zone maximum as a function of the maximum D of the storm. During storms with low D the new outer zone maximum appeared at the upper end of the vertical bar. During storms with high D, the new outer zone maximum appeared at the lower end of the vertical bar. The horizontal bar represents the period over which the measurements were made.
The measurements labelled 7, 8 and 9 are taken from unpublished data of A. Vampola (private communication). The instrument used in each case was a magnetic spectrometer. Points labelled 7 were obtained on satellite 1966-70A. The data have been corrected for the background proton fluxes, allowing excellent resolution of the slot. Each dashed vertical bar indicates the range over which the corrected counting rate of electrons with E=957+ 140 keV fell below 1 cm sec sterad keV . On September 6, 1966 the outer zone maximum moved into L 3. This was exceptional in that the geomagnetic K index had a daily sum of greater than 20 for the eight preceding days. In fact, on three of these days K > 44, indicating extreme disturbance. In contrast to the significant movement of the outer zone maximum over this period, one should note that the inner edge of the slot remained relatively fixed.
Points labelled 8 were obtained on April 9, 1968 on satellite 1968-26B and points labelled 9 were obtained on April 14, 1969 on satellite 1969-25C at 1.1 MeV with instrumentation very similar to that of satellite 1966-70A.
Since it was shown in Section 1 that the plasmapause often lies near the outer zone maximum we have also included in Figure 7 five measurements of the average plasmapause position taken over a period of five years.
Point a is the average plasmapause position, giving each day equal weight, taken from the whistler measurements of Carpenter (1966).
Point b was calculated from the Electron 2 and 4 satellite data of Bezrukikh (1968), which have been summarized by Gringauz (1969). It is not known whether this point is representative of the average over all local times. The point was plotted halfway between the launch date of the two satellites.
Point c was calculated by averaging the positions given by Taylor et al. (1965). This set of data was obtained from September 23 to December 10, 1964. It included both inbound and outbound passes of OGO-1 and thus a wide range of local times was sampled.
Point d was obtained from OGO-3 measurements by Taylor et al. (1968). Their data, however, were not representative of all local times since only inbound passes near midnight were used. In order to compensate for the known local time variations in the plasmapause location (Carpenter, 1966) we have reduced the mean plasmapause location by 8%.
Finally, points e were measured by the Lockheed ion mass spectrometer on OGO-5 (R. Chappell, personal communication). The inner point is the average position at which the density fell to 100 ions/cm and the outer point is the average position at which the density fell to 10 ions/cm. These points are derived from both inbound and outbound passes of OGO-5 over two months of operation and any normalization procedure to make these values representative of all local times would not significantly change these two points.
Examining Figure 7 , we find no dramatic solar cycle variations in the outer zone electron maximum. This is partially because this figure emphasizes ranges for the outer zone position. Figure 7 itself shows that the position of the outer zone maximum is strongly controlled by the strength of a magnetic storm. Presumably, if it were possible to accurately calculate, over the solar cycle, a 6 month running average of the outer zone position, there would be a systematic variation due to the 11 year modulation in geomagnetic activity (see for instance Shapiro (1969)). Point 1 on the other hand shows a different range for the outer zone maximum than the other points. This could be due to the fact that point 1 does not truly represent electrons as assumed or perhaps because storms in 1960 were much stronger than in later years.
Figure 7 also shows that the electron slot is quite stable over the solar cycle. This stability is especially apparent for the inner edge. In plotting the slot in Figure 7 we have used only individual points for the inner and outer edge rather than ranges. Changes would thus show up as a scatter in the positions of the inner and outer edge. While there is some moderate scatter in the position of the outer edge, very little scatter appears in the inner edge location. This conclusion agrees with the study of a single storm in Section 2.
Looking at the plasmapause positions in Figure 7 we do see a long term variation from 1963 to 1968. Here we have plotted real averages, not ranges or individual points. Since Carpenter (1967), Taylor et al. (1968), Gringauz (1969), and Chappell et al. (1969) show quite similar relationships between the plasmapause position and the K index and since the K index has a solar cycle variation (Shapiro, 1969), this apparent trend is quite possibly due only to the different average magnetic activity at the times of these observations. In summary, the position of the inner edge of the slot shows little if any solar cycle variation. The position of the outer edge shows some variability presumably in response to changes in the position of the outer zone maximum. The position of the outer zone itself may on the average change with solar activity but the variations appear to occur in the same range of L-values throughout the solar cycle. Throughout the period for which observations exist, the average location of the plasmapause occurs near the outer zone electron maximum. The average position of the plasmapause does vary but appears to be merely a function of the average magnetic activity.
The slot or gap in electron flux between the inner and outer radiation belts is generally attributed to an enhanced loss of electrons from this region of the magnetosphere. Most likely, this results from pitch angle scattering of electrons into the loss cone, thereby causing their precipitation into the atmosphere. Coulomb scattering of the high energy electrons with thermal particles can account for the observed electron lifetimes close to the earth but it fails completely for L > 1.25 (Walt, 1966). Alternatively, the electrons can be removed through a resonant interaction with waves oscillating near their Larmor frequency. Such whistler-mode waves are naturally generated during lightning discharges in the atmosphere and their subsequent escape into the magnetosphere (Helliwell, 1965) permits the resonant interaction with the Van Allen belt electrons to occur. The frequency spectrum of the ground detected whistlers led Dungey (1963) and Cornwall (1964) to hypothesize preferential electron dumping near L= 2 to 3.
Recent satellite observations of electromagnetic VLF waves deep within the magnetosphere, however, have shown these spherically generated whistlers to comprise only a minor portion of the total wave energy. Rather, the plasmasphere is almost continuously filled with a relatively broadband hiss (Russell et al., 1969; Dunckel and Helliwell, 1969). We show below that this emission is probably generated by cyclotron resonance with the Van Allen belt electrons themselves.
A necessary condition for this electromagnetic wave growth to occur is that the electron distribution function be anisotropic with more energy perpendicular than parallel to the geomagnetic field (Sagdeev and Shafronov, 1961; Andronov and Trakhtengerts, 1964). No thorough study of the pitch angle distribution in the inner magnetosphere has yet been reported. The existence of a loss cone along the geomagnetic field does however imply an anisotropy of the correct sign and such a distribution has now been observed (Katz, 1966; Parks, 1969). The instantaneous growth rate for this resonant interaction is proportional not only to the pitch angle anisotropy but also to the fractional number of particles near resonance. This, combined with the further condition that the radiation belts will only become truly unstable when the net wave growth exceeds all losses due to wave absorption and reflection, led Kennel and Petschek (1966) to the concept of a limit on the stably trapped particle flux.
As we shall show below, for a given energy electron, one expects the frequency of this natural emission to increase significantly with decreasing L. Using this fact, Tverskoy (1967) suggested that the slot was controlled by the degree to which these waves were absorbed in the ionosphere. As long as the wave frequency is well above the local proton gyrofrequency, wave absorption is known to decrease with decreasing frequency (Helliwell, 1965). At intermediate frequencies it may be possible for a significant fraction of the wave energy to be reflected back into the magnetosphere without suffering appreciable absorption. In terms of the theory for limiting stability of the radiation belts (Kennel and Petschek, 1966), a large reflection coefficient results in low stably trapped particle flux and vice versa. The slot should thus occur in the region of large wave reflection which in the above model occurs near L = 3.
There is however a basic error in the above argument. It occurs in Tverskoy's premise that waves are always guided by the geomagnetic field. In the absence of ducting this is more or less correct at high frequencies but becomes invalid whenever the wave frequency is below the local lower hybrid frequency of the propagating medium. Low frequency whistler mode waves can in fact propagate in a direction perpendicular to the magnetic field (Hines, 1957) enabling such waves generated near the equatorial plane to reflect and remain quasitrapped deep within the magnetosphere (Kimura, 1966; Thorne and Kennel, 1967). It is feasible that the near perfect magnetospheric reflection of these waves could account for the extremely low electron fluxes within the slot. The effective reflection coefficient, however, might still be reduced by the resonant damping of waves as they propagate through the magnetosphere (Kennel and Thorne, 1967; Thorne, 1968) but the observed features of the natural hiss band (Russell et al., 1969) suggesting this damping to be small.
Under our hypothesis the inner edge of the slot would occur at the L-value where magnetospheric reflection of the waves becomes ineffective, while the outer edge would probably be controlled by the balance between inward diffusion and loss. In the following subsection we review the essential features of this natural instability of the radiation belt electrons and incorporate the results of wave propagation studies to explain the general location of the slot.
The physical processes which combine to produce the slot could easily be obscured in a full analysis of the electron-whistler interaction. We shall therefore limit ourselves here to a simplified description which brings out the major features of the instability; the details of the calculation will be left for later presentation.
The Van Allen belt electrons will experience resonant scattering when the Doppler shifted wave frequency is equal to their cyclotron frequency. This condition can be expressed in the form
( 1-µ )= / (1)
where is the wave frequency, is the nonrelativistic gyrofrequency, = [1-(v/c) ]is the relativistic mass enhancement factor, and µ and are the components of the wave refractive index and the normalized electron velocity (v/c) along the geomagnetic field direction. Alternatively, (1) can be reexpressed in terms of the wave normal angle , the electron pitch angle , and the normalized wave frequency = / as
The observed frequency of the natural hiss band emission found within the plasmasphere is always well below both the plasma frequency and the electron gyrofrequency. Under this condition the whistler mode wave refractive index can be simply expressed as
where the dimensionless parameters
are functions of the local cold plasma medium. Along the magnetic equator, is typically between 10 and 100, although it can become comparable to unity outside of the plasmapause and at high geomagnetic latitudes. The value of depends strongly on frequency. It passes through zero when = at the lower hybrid frequency and for >> it is usually reasonable to take 1.
Combining (2) and (3) and dropping terms of order ( )2 yields the resonant wave frequency
For conditions in the magnetosphere (5) has a local minimum when = = 0 given simply by
The energy dependence of the emitted wave frequency is contained in the term
This is plotted in Figure 8 for = 1 and = 1, 10, 100 and . At high energies the quadratic term in (7) dominates and is then relatively independent of the cold plasma parameters.
In order to evaluate the resonant frequency at different locations in the magnetosphere one needs a model for values of the total electron density N and the magnetic field strength B. In Figure 9 we have used the usual dipole variation for and adopted the reference electron density profile given by Angerami and Carpenter (1966) to evaluate along the geomagnetic equator. Because we shall consider only resonant effects at quiet- times within the plasmasphere, no attempt has been made in Figure 9 to allow for the sharp drop in N at the plasmapause. It should also be noted that the ratio / varies from 3 to 20 over the L-range of 2 to 5, substantiating our assumption of large . Off the geomagnetic equator the electron density should remain roughly constant along field lines (diffusive equilibrium) whereas the magnetic field strength increases rapidly. The resonant frequency (6) which is proportional to B3 / N is thus an order of magnitude higher at 25o geomagnetic latitude, , than at the equator.
|Fig. 8. The scaling factor, , defined in Equation (7) as a function of energy for several values of , the square of the ratio of the electron plasma to electron gyrofrequency.|
It has been established by Kennel (1966) that the growth rate maximizes at =0 and in most cases the major contribution to the growth rate integral stems from particles with < 60o . Thus, at any given location, we can expect waves with frequency just above the minimum resonant frequency (6) to incur the most growth. The emission frequency will also increase with geomagnetic latitude. To compensate for this, and to allow for wave growth at non zero values of and we shall arbitrarily assume that the typical emission on a given field line occurs at ten times the minimum resonant frequency at the equator. Thus
where the L dependent parameter o (L) = 3 / 2 is evaluated from the equatorial values of Figure 9. The variation in this characteristic emission frequency with L is plotted in Figure 10. For each curve the energy factor was taken from the curve of Figure 8. This provides a good representation at energies above a few 100 keV. At lower energies, the curves in Figure 10 will require a slight downward adjustment whenever the characteristic value of is less than 100. This is important only for waves generated at high geomagnetic latitudes.
|Fig. 9. The equatorial parameters assumed in the calculations as a function ofL-value. The number density, N, and hence the electron plasma frequency, , has been modelled after the results of Angerami and Carpenter (1966). No plasmapause is included in this profile since our calculations only apply to the plasmasphere. The electron gyrofrequency, , has been calculated assuming a dipole field.|
Figure 10 merely indicates which wave frequencies are able to resonate with a given energy electron in the magnetosphere. Not all waves in the resonant band will grow at the same rate or to the same amplitude. Rather, one expects preferential growth of those frequencies which interact with electrons having the most intense flux and the largest pitch angle anisotropies (Kennel and Petschek , 1966). This whistler mode growth rate has already been evaluated by several authors and we refer those interested in the details of this calculation to the papers of Kennel and Petschek (1966), Kennel (1966) and Liemohn (1967). It suffices here to point out that the overall wave growth for each traverse across the equatorial plane is scaled to the energy flux of the electrons resonating with the VLF waves.
The emitted waves will propagate away from the equatorial region of growth, roughly following the magnetic field lines. In order for the natural instability of the radiation belts to develop, two conditions must apply. First some means must be found for returning the wave energy to the generation region, and secondly the growth incurred there must exceed all losses due to absorption and imperfect reflection. In other words, the system must lase. The wave growth, however, is rather small within the plasmasphere due to the extremely low fluxes of particles found in the region of the slot. One rarely expects the wave amplitudes to e-fold on each pass through the region of generation. It thus becomes mandatory to have a very efficient reflection process.
|Fig. 10. The characteristic resonant frequency for various energy electrons as a function of L-value. This has been taken as ten times the minimum resonant frequency at the equator. The lower hybrid resonance frequency at low altitudes occurs in the shaded band.|
At high frequencies, above 5 to 20 kHz, one must rely on reflection from the sharp density profile in the ionosphere. It seems very unlikely that this process can return a sufficient portion of the wave energy to the magnetosphere and we thus expect these high frequency waves to be lost by absorption in the ionosphere. This contention is substantiated by observations of VLF in the plasmasphere. The frequency of the natural hiss band emission is rarely above 3 kHz (Dunckel and Helliwell, 1969) showing that the high frequency components of the resonant wave band do not contribute to the instability.
Lower frequency waves, however, can undergo almost perfect reflection by an alternative mechanism. It has been shown by Kimura (1966) and Thorne and Kennel (1967) that low frequency whistler-mode waves propagating away from the equator tend to quickly increase the angle between their propagation vector and the ambient geomagnetic field. This results simply from an application of Snell's Law to waves moving into a region of lower refractive index. Near the equator, when > , the waves approximately follow the field lines even when is large. At high latitudes, however, when the wave frequency is less than the lower hybrid (hereafter referred to as LH) resonance frequency of the local plasma, it is possible for the wave group velocity to be oriented perpendicular to the field line. Further application of Snell's law in this region (Lyons and Thorne, 1969) shows that the waves which have attained large wave normal angles are forbidden from penetrating very far past the LH resonance point. They are thus forced to reflect and return to the equatorial plane remaining more or less on the same magnetic flux tube. Observational evidence for this magnetospheric reflection process, which is somewhat analogous to the total internal reflection found in optics, has already been presented by Smith and Angerami (1968).
A necessary condition for the above reflection process to occur is that the waves are at some point on their path below the LH resonance frequency. The latter tends to increase with latitude along a geomagnetic field line reaching a maximum of between 5 to 20 kHz at an altitude of 1000-2000 km above the earth's surface. To remain trapped within the magnetosphere, the waves must reflect well before this location. We have shaded this critical frequency of the LH band onto Figure 10. Along any given L-shell, it is clear that waves generated above this frequency are lost to the ionosphere. Waves of considerably lower frequency (produced by the higher energy electrons) may reflect, and thus take part in the natural instability of the Van Allen belts. From flux considerations one expects more rapid wave growth at the highest frequency taking part in the instability. This perhaps accounts for why the center frequency of the naturally occurring hiss band always lies between a few hundred Hz to few kHz, or just below the shaded LH band.
We can also use Figure 10 to predict the regions of electron loss. Clearly, the near perfect reflection of those waves generated at frequencies below the LH band would permit the natural hiss band to grow indefinitely. The only way to prevent this is through the complete removal of those particles which are responsible for generating or enhancing the wave amplitude. Simply stated, a perfect wave reflection must eventually lead to zero stably trapped flux (Kennel and Petschek, 1966). The extremely low quiet-time fluxes of electrons in the slot region are a clear indication of this complete flux depletion. By following the resonant frequency curves in Figure 10 up to the LH band we can predict the location at which this natural instability becomes ineffective for any given energy electron. We associate this point with the inner edge of the slot. The high, stable fluxes of electrons in the inner belt result simply from the fact that wave-particle interactions are very weak in this region.
One immediate question to be answered is the cause for the recovery of fluxes in the outer belt. It is known, however, that these outer belt electron fluxes are extremely variable and tend to become enhanced following bay activity. The outer edge of the slot probably occurs at the location where the rate of loss exactly cancels the injection or diffusion of these electrons into the plasmasphere.
In Figure 11 we have plotted the quiet- time location of the slot observed by Pfitzer et al. in September 1964 and by A. Vampola (private communication) in August 1966.
|Fig. 11. The calculated and observed location of the slot at quiet-times. The horizontal bars of the observations represent the inner and outer edges of the slot, and the vertical bars represent the energy range of the measurements. The horizontal extent of the solid bars representing the calculated inner edge of the slot indicates our imprecise knowledge of the lower hybrid frequency at the wave reflection point.|
A third set of measurements obtained in June 1968 by H. West (private communication) shows a similar trend and has therefore not been plotted. Each horizontal bar indicates the region over which the flux either fell below the detector's threshold or dropped to less than 1 electron cm sec sterad keV. In agreement with the results of Section 3 we notice that for a given electron energy, the inner edge of the slot has remained in the same location. There is, however, a definite broadening and shift of the slot towards lower L-values at higher energies. Also superimposed on Figure 11 are the theoretical predictions for the inner edge of the slot evaluated from the curves in Figure 10. The horizontal extent of each bar indicates our imprecise knowledge of the lower hybrid frequency at the wave reflection location. Nevertheless, the agreement between the theoretical and observed L-shell at which the inner zone fluxes begin to build up is sufficiently impressive to suggest that the physical processes described above are indeed responsible for determining the distribution of energetic electrons in the plasmasphere.
Although the thermal plasma and the energetic protons and electrons in the earth's magnetosphere have been studied separately for many years, the relative motions and locations of various features of the magnetospheric plasma have so far received little attention. Two storms have therefore been studied in this paper: one to locate the relative positions of the plasmapause, the proton ring current, and the outer electron zone before, during, and after a storm and the other to investigate the changes in the electron profiles at various energies before and after a storm.
The proton ring current was found to be created immediately outside the storm-time plasmapause. During the post-storm recovery, the plasmapause expanded to higher L -values and the protons decayed sufficiently rapidly so that the location of their flux maximum always coincided with the plasmapause. This rapid removal of protons primarily from within the dense region of the plasmasphere suggests that a loss mechanism in addition to charge exchange is operating. One likely candidate could be the resonantly unstable interaction with ion cyclotron waves (Cornwall, 1966) since conditions for wave growth are more favorable at high plasma densities.
Low energy electrons are similarly created external to but near the plasmapause. However, they decay much more slowly and as the plasmapause expands, the net effect of the storm is an injection of electrons deep into the plasmasphere. Although the outer edge of the electron slot is quite responsive to the changes in the position of the electron outer zone maximum, the inner edge, particularly at high electron energies, remains quite stable during the storm. Also, over the solar cycle we found that the outer zone electron flux maximum responded to changes in magnetic activity. A similar conclusion was drawn for the average plasmapause position. In contrast, the inner edge of the electron slot was found to remain at a roughly constant position. It is thus possible to define an inner edge profile for the slot purely in terms of electron energy.
In the final section of this paper we considered the loss of electrons from the region of the slot. By adopting the view that electrons are removed from the plasmasphere by the pitch angle diffusion associated with the natural instability of the radiation belts, it was shown that the inner edge of the observed slot agrees with the location where this interaction between the electrons and VLF waves becomes ineffective. On low L-shells the VLF waves generated by the electrons should penetrate the ionosphere and subsequently be absorbed. The observation by Dunckel and Helliwell (1969) of a reduction in the intensity of the plasmaspheric hiss within L = 2.5 is in qualitative agreement with the inner edge of the slot. Further within the magnetosphere, however, the waves produced by the instability remain trapped, permitting multiple amplification. This leads to a very low stably-trapped electron flux in the vicinity of the slot. The outer edge of the slot in our model was tentatively assigned to the location where the inward diffusion of electrons from the outer Van Allen zone exceeded the rate at which these particles could be lost.
Several features of the electron-VLF interaction were however omitted from our discussion for the purpose of clarity. The analysis is certainly invalid for energies lower than about 10 keV since the resonant frequency is then close to the electron gyrofrequency. Also, the electron energy resonant with a given wave frequency increases at higher geomagnetic latitudes. This will produce additional parasitic precipitation (Kennel, 1967) of the higher energy electrons whenever there is a strong instability at low energies. We have also restricted our attention to the quiet-time features of the electron morphology. It is known however, that the flux levels change appreciably during magnetic bay disturbances. Lower energy electrons are then injected well into the inner regions of the plasmasphere (Pfitzer and Winckler, 1968). Theoretically one expects electrons to be most unstable to the generation of VLF waves at high L-values. This seems to agree with the observed location at which the electrons are most rapidly depleted. While the present exploratory discussion leaves many unanswered points, it does predict the gross features of the energetic electron slot and introduces a new concept for our understanding of the Van Allen belt structure.
We are grateful to A. Vampola for both commenting on the final form of the manuscript and for allowing us to use his unpublished data. Valuable comments on our presentation of the paper were made by T. Farley and the study has benefitted from a number of discussions with H. West. The research was supported in part by the Academic Senate of the University of California at Los Angeles under Grant No.2525 and also by NASA grant NAS 5-9098.
Andronov, A. A. and Trakhtengerts, W.: 1964, Geomagnet. Aeron. 4, 181.
Angerami, J. J. and Carpenter, D. L.: 1966, J. Geophys. Res. 71, 711.
Bezrukikh, V. V.: 1968, submitted to Kosmich. Issled.
Carpenter, D. L.: 1963, J. Geophys. Res. 68, 1675.
Carpenter, D. L.: 1966, J. Geophys. Res. 71, 693.
Carpenter, D. L.: 1967, J. Geophys. Res. 72, 2969.
Chappell, C. R., Harris, K. K. and Sharp, G. W.: 1969, submitted to J. Geophys. Res.
Cornwall, J. M.: 1964, J. Geophys. Res. 69, 1251.
Cornwall, J. M.: 1966, J. Geophys. Res. 71, 2185.
Craven, J. D.: 1966, J. Geophys. Res. 71, 5643.
Davis, L. R. and Williamson, J. M.: 1966, Radiation Trapped in the Earth's Magnetic Field (ed. by Billy M. McCormac), Reidel Publishing Co., Dordrecht, Holland, p. 215.
Dunckel, N. and Helliwell, R. A.: 1969, submitted to J. Geophys. Res.
Dungey, J. W.: 1963, Planetary Space Sci. 11, 591.
Forbush, S. E., Pizella, G. and Venkatesan, D.: 1962, J. Geophys. Res. 67, 3651.
Frank, L. A.: 1967, J. Geophys. Res. 72, 3753.
Frank, L. A. and Van Allen, J. A.: 1966, J. Geophys. Res. 71, 2697.
Freeman, J. W., Jr.: 1964, J. Geophys. Res. 69, 1691.
Gringauz, K. I.: 1963, Planetary Space Sci. 11, 281.
Gringauz, K. I.: 1969, Rev. Geophys. 7, 339.
Helliwell, R. A.: 1965, Whistlers and Related Ionospheric Phenomena, Stanford University Press, California.
Hines, C. O.: 1957, J. Atmosph. Terrest. Phys. 11, 36.
Jelly, D. and Brice, N.: 1967, J. Geophys. Res. 72, 5919.
Katz, L.: 1966, Radiation Trapped in the Earth's Magnetic Field (ed. by Billy M. McCormac), Reidel Publishing Co., Dordrecht, Holland, p. 129.
Kennel, C. F.: 1966, Phys. Fluids 9, 2190.
Kennel, C. F.: 1967, Trans. Am. Geophys. Union 48, 180.
Kennel, C. and Petschek, H. E.: 1966, J. Geophys. Res. 71 1.
Kennel, C. F. and Thorne, R. M.: 1967, J. Geophys. Res. 72, 871.
Kimura, I.: 1966, Radio Sci. 1 (new series), 269.
Konradi, A.: 1968, J. Geophys. Res. 73 3449.
Liemohn, H. B.: 1961, J. Geophys. Res. 66, 3593.
Liemohn, H. B.: 1967, J. Geophys. Res. 72, 39.
Lyons, R. L. and Thorne, R. M.: 1969 (in preparation).
McDiarmid, I. B. and Burrows, J. R.: 1967, Can. J. Phys. 45, 2873.
McDiarmid, I. B., Burrows, J. R. and Wilson, M. D.: 1969, J. Geophys. Res. 74, 1749.
Mihalov, J. D.: 1967, J. Geophys. Res. 72 1081.
Nishida, A.: 1966, J. Geophys. Res. 71, 5669.
Owens, H. D. and Frank, L. A.: 1968, J. Geophys. Res. 73, 199.
Parks, G. K.: 1969, submitted to J. Geophys. Res.
Pfitzer, K.A. and Winckler, J.R.: 1968, J. Geophys. Res. 73, 5792.
Pfitzer, K.A., Kane, S. and Winckler, J.R.: 1966, Space Res. 6, 702.
Rao, C.S.R.: 1969, J. Geophys Res. 74, 794.
Rothwell, P. and Lynam, C.: 1969, Planetary Space Sci. 17, 447.
Russell, C.T., Holzer, R.E. and Smith, E.J.: 1969, J. Geophys. Res. 74, 755.
Sagdeev, R.Z. and Shafronov, V.D.: 1961, Soviet Phys. JETP 12, 130.
Scarf, F.L., Fredricks, R.W. and Crook, G.M.: 1968, J. Geophys. Res. 73, 1723.
Shapiro, R.: 1969, J. Geophys. Res. 74, 2356.
Smith, R.L. and Angerami, J.J.: 1968, J. Geophys. Res. 73, 1.
Swisher, R.L. and Frank, L.A.: 1968, J. Geophys. Res. 73, 5665.
Taylor, H.A., Jr., Brinton, H.C. and Smith, C.R.: 1965, J. Geophys. Res. 70, 5769.
Taylor, H.A., Jr., Brinton, H.C. and Pharo, M.W. III: 1968, J. Geophys. 73, 961.
Thorne, R.M.: 1968, J. Geophys. Res. 73, 4895.
Thorne, R.M. and Kennel, C.F.: 1967, J. Geophys. Res. 72, 857.
Tverskoy, B.A.: 1967, Geomagnet. Aeron. 7, 177.
Van Allen, J.A. and Frank, L.A.: 1959, Nature 183, 430.
Vernov, S.N. and Chudakov, A.E.: 1960, Soviet Physics Uspekhi 3, 230.
Vernov, S.N., Gorchakov, E.V., Kuznetsov, S.N., Logachev, Yu. I, Sosnevets, E.N. and Stopovsky, V.G.: 1969, Rev. Geophys. 7, 257.
Walt, M.: 1966, Radiation Trapped in the Earth's Magnetic Field (ed. by Billy M. McCormac), Reidel Publishing Co., Dordrecht, Holland, p. 337.
Williams, D.J. and Smith, A. M.: 1965, J. Geophys. Res. 70, 541.
Williams, D.J., Arens, J.R. and Lanzerotti, L.J.: 1968, J. Geophys. Res. 73, 5673.