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Procedures used to produce the plots

The DREAM beta web service assimilates data from only a single GOES satellite. We use the NOAA-designated primary GOES satellite which, as of May 2010, is GOES-13. Future versions will assimilate available near-real-time data sources which might include multiple GOES satellites, LANL-GEO, GPS, the RBSP space weather broadcast, and others.

We fit a power law spectrum to the integral GOES energy channels which cover >0.8 MeV and >2 MeV. (Other GOES satellites have channels for >0.6 and >0.2 MeV.) We extrapolate the spectrum to arbitrarily high or low energies as needed. (More physically realistic procedures are certainly possible and errors due to extrapolation are a known limitation.)

The first adiabatic invariant, µ, is calculated from the fit spectrum and the model magnetic field. It is, of course, possible to use the measured GOES magnetic field but the beta version shown here does not. The second invariant, K, is also calculated using the model magnetic field. In calculating phase space density we assume (for the beta version) an isotropic pitch angle distribution.

We calculate the third invariant L* using the Tsyganenko 1989 model (T89) using the latest Kp values. One effect of using the T89 is a diurnal variation in the L* “position” of GOES. When GOES is on the day side it generally samples lower L* and when it is on the night side it samples higher L*. The specific values of L* also change with activity. At higher activity levels the “Dst effect” inflates the geomagnetic field with the result that lower L* values are sampled.

The plots of PSD show the L* value of GOES with a purple line.

Once we have calculated the magnetic invariants, (µ, K, L) we convert Flux (at fixed energy and pitch angle) to phase space density (PSD). It is actually the PSD values, not the flux values, that are assimilated in the DREAM model. This is because the physics model requires PSD and magnetic invariant coordinates.

The physics model in the DREAM beta web service is a simple radial diffusion model that uses the Brautigam & Albert [2000] formulation. It includes loss terms for the electron lifetime. Lifetimes are constants inside the plasmasphere and outside the last closed drift shell. Between the plasmapause and the magnetopause the lifetime is Kp dependent. (See Shprits et al., 2000.)

One assimilation is done for each µ-K pair. We currently calculate 25 separate assimilations for five values of µ and five values of K. The assimilations are computationally very fast and there is little trouble in scaling up to 100 or 1,000 assimilations for an operational version.

PSD is calculated for all bins from L* = 1-10. The model seldom produces appreciable PSD inside L*≈3. This is physically realistic and represents the point where inward radial diffusion is slow relative to the electron loss lifetime. We note that while the feature is realistic, the quantitative PSD values are not. We know this from other DREAM assimilation runs that use GPS data as well as geosynchronous data. The GPS observations at L*<5 significantly alter the assimilation results.

The magnetopause defines the limits of trapping for the radiation belts. The L* value of the last closed drift shell is also calculated using the T89 model. While the assimilation space extends to L* = 10, we set a very short electron lifetime outside the last closed drift shell. This creates a “ragged” outer boundary. We know from DREAM runs compared with near-equatorial POLAR observations between geosynchronous orbit and the magnetopause, that the PSD values in that region are generally quite good when realistic input spectra and pitch angle distributions are available.

The final step is to convert back from PSD to flux at fixed energy and pitch angle. At a given point in space different pitch angles have somewhat different L*. Similarly the conversion from µ to energy depends on the local magnetic field which is a function of both radius and local time. Therefore, converting back to flux is strictly possible only for a specified set of points in space. (E.g. along a spacecraft trajectory or a non-Keplerian set of points such as radial distance at fixed local time.) For computational simplicity the beta version here uses the (erroneous) assumption that L* = dipole L = R. Future versions will use the correct conversion procedure.

Conversion from K to equatorial pitch angle is linear. However, the range of pitch angle values in the final product is limited by the range of K values used in the assimilations. The conversion from µ to energy is proportional to the magnetic field strength and is L-dependent. Therefore, the range of L-shells for which flux can be calculated at a given energy and pitch angle is limited by the choice of the range of µ and K values used in the assimilations. As noted above there is no fundamental limit to the number and range of µ and K values that could be computed but, at some point, the extrapolation (in energy or pitch angle) from the omni-directional, integral-energy GOES measurements becomes physically unrealistic. The limitations on final DREAM output are illustrated in the figures below.

Figure 3:The relationship between the second invariant, K, and equatorial pitch angle α. Values are shown for L* = 6.6 but the relationship does not vary much as a function of L*. The current choice of limiting assimilations to K values between 0.0125 and 0.2 limits the range of equatorial pitch angles we can calculate to between about 50° and 80°. These limits are arbitrary and will be changed in future versions of DREAM.

Figure 4:The first invariant is proportional to energy divided by magnetic field strength (E/B) and therefore varies strongly as a function of L*. This plot shows the relationship between µ (Mu) and L* for a fixed energy of 2 MeV and a family of curves for different equatorial pitch angles.

Figure 5:This figure illustrates how a choice of fixed energy and pitch angle imposes limits on the L range in which fluxes can be calculated. This plot shows µ as a function of L* for an energy of 6 MeV. The red curve shows values for a 60° equatorial pitch angle. The heavy black lines show the range of µ values used in the DREAM beta web service (1,000-20,000). The intersection of the red curve with those lines defines the range of L-shells which in this example lie between about 3 and 6 RE.

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