CONCLUSIONS Simulation of Radiation Belt ULF OSCILLATIONS

DISCUSSION

The following features are noteworthy in the simulated GPS and geosynchronous flux levels (Plates 1 and 2):

1. There is an initial decrease in electron flux at L=4.2 due to buildup of the ring current, captured by the MHD simulations as an increased pressure in the inner magnetosphere due to enhanced convection during the period of southward IMF. The increased ring current causes a local decrease in total B which is more evident at geosynchronous orbit than at L = 4.2. Electrons move radially outward conserving the third adiabatic invariant in response to the decrease in local magnetic field strength due to build up of the ring current, which occurs on a time scale long compared to radiation belt electron drift periods [Schulz, 1997] (10.35 min at L=4.2 and 6.38 min at L = 6.6 at 1.6 MeV, using B₀=.305G at the Earth's equator). With conservation of their first invariant, energy decreases as does flux as electrons move radially outward. This decrease has been referred to as the Dst effect [Li et al., 1998; Kim and Chan, 1997], and is observed to be greater with increasing energy, as borne out in Plates 1 and 2, simply due to the steepness of the power law energy spectrum.

2. There is an abrupt rise in flux $\sim$ 09:00 UT, by half an order of magnitude in the most energetic simulated GPS channel (1.6 - 3.2 MeV), and a correspondingly smaller rise with decreasing energy, evident in Plate 1.

3. There is a dropout in flux at geosynchronous orbit in Plate 2 $\sim$ 02:00 - 03:00 UT on January 11, due to compression of the magnetopause inside geosynchronous orbit and recovery following arrival of the high density solar wind impulse at the magnetosphere [Reeves et al., 1998b].

The flux vs. energy and L plots (Plate 3) show that the flux peak moved inward from L = 5 to L = 4 on a timescale of several hours. Inward transport of the source population resulted in an increase in flux at higher energies as well, due to conservation of the first invariant. The spectrum continued to harden slightly after northward turning of the IMF at 17:30 UT, but there was no further inward radial transport of the flux peak, in particular associated with the high density solar wind pressure pulse at $\sim$01:00 UT on January 10.

The drift resonance between > 200 keV electrons and the torroidal oscillations apparent in the radial electric field component of the simulation data suggests the following coherent acceleration mechanism. Electron fluxes at geosynchronous were amplified by successive substorm injections throughout the period of steady southward IMF B_z beginning at 04:40 UT [Reeves et al., 1998b]. Electrons with the right drift phase are subjected to continuous acceleration by a radial electric field over a non-azimuthally symmetric drift path, given the electric field reversal on the timescale of half an electron drift period, as sketched in Figure 2.
  

Figure 2: Sketch of electron drift path and radial electric field orientation in a toroidal mode oscillation cycle. Solid arrows indicate electric field at t=0, and dashed as seen by an electron with wave frequency $\omega =\omega _D$, the electron drift frequency, starting at dusk for an m=2 azimuthal mode number.

Electrons with the opposite drift phase are continuously decelerated. Consider, for example, E_r with azimuthal mode number m=2, indicated by solid arrows in Figure 2. An electron with drift frequency $\omega _D=\omega$, the ULF wave frequency, drifting eastward from dusk (bottom) sees an electric field indicated by dashed arrows as it moves along its drift orbit. By the time it drifts around to dawn (top), half a drift period later, the electric field has reversed sign, but now dr/dt > 0 as it moves towards the dayside, vs. dr/dt < 0 as it moves towards the nightside, so such an electron experiences continuous - $\int E_rdr >$ 0, and is accelerated over its drift orbit. Examining an m=1 mode, there would be acceleration all the way around the drift path if the wave frequency is twice the drift frequency, with other resonance conditions possible, depending on the m-spectrum. An estimate of the magnitude of the acceleration has been made using the maximum radial electric field strength seen in the simulations, $\sim$ 8 mV/m, assuming a radial drift path of $\sim$ 1.5 R_E. A 100 keV electron at geosynchronous takes approximately one hour to drift around the earth and increase its energy to 200 keV for the assumed parameters, in another hour it increases its energy to 400 keV, and in two more hours it exceeds 1.6 MeV. Starting at 200 keV, its energy exceeds 1.6 MeV in less than 3 hours. These estimates, which assume continuous acceleration over a drift path, may be optimistic by factors of two, but are supported by the relativistic test particle results. Thus, it seems reasonanble to conclude that the observed increase in relativisitic electron flux in the simulations, and measured for the January 1997 event, can be explained in part by drift resonant acceleration in the radial electric field of torroidal eigenmodes which show enhanced power during the period of rise in electron fluxes seen by GPS.

The GPS observations suggest that the most rapid rise in flux at lower energies, 0.8 - 1.6 MeV, occurred around 11:00 UT,which coincided with the arrival of a moderate solar wind pressure pulse at the magnetosphere as seen by WIND, GEOTAIL and GOES spacecraft, as well as preliminary Dst [Li et al., 1998]. Figure 1 shows that ULF wave power in the same frequency range as seen in the power spectral analysis of the MHD simulation data (Plate 4) is also enhanced at this time. However, it is embedded in a pre-existing enhancement in ULF wave power in the same frequency range, following substorm activity which produced a peak in average K_p = 6 during the three hour intervals 06:00 - 09:00 and 09:00 - 12:00 UT. This activity interval is readily apparent in the Canopus magnetometer data as well [Baker et al., 1998], which provides a prelimary indication of longitudinal extent of the ULF oscillations in ground magnetometer data, under analysis. The MHD simulation data has been analyzed at local times shifted by 6, 12 and 18 hours relative to Plate 4, and similar but less well defined structure is apparent. Figure 3 shows a snapshot of the full electric field vectors in the equatorial plane of the simulations at $\sim$09:00 UT, and one can see a dominant m=2 azimuthal mode number, and large scale coherence of the toroidal oscillation superimposed on the convection electric field and magnetopause signatures.
  

CONCLUSIONS Simulation of Radiation Belt ULF OSCILLATIONS