Principal Components of Electron Belt Variation

1. Principal Components of Electron Belt Variation Paul O’Brien

2. Analysis of Posterior Principal Components of a Statistical Reanalysis • A statistical model (TEM-2) was constructed from Polar, SCATHA, CRRES, and S3-3 data • The model describes flux in E, aeq, Lm coordinates in the Olson-Pfitzer Quiet field model • It was used as the “background model” for a reanalysis of HEO, ICO, and SAMPEX data for 1992-2007, at 6-hours steps • From that reanalysis, a “Gaussian Canonical” spatial correlation matrix was computed • The Principal Components are the eigenvectors of that matrix

3. Slot? (high energy, L~3) Atmospheric Loss Cone The green surface marks the zero crossing, and the red/blue surfaces mark ±1. (For the 1st PC, there’s no blue surface.) 1st PC represents globally correlated response. The peak response appears to be near 1 MeV at L~6

4. 2nd PC represents decoupling between high and low energies (with the node near 1 MeV)

5. 3rd PC represents decoupling between low L, low energy and high L, high energy

6. 4th PC decouples region near L~5 from rest of belt. Curious variation in the pitch-angle distribution: equatorially mirroring particles at L~4, ~300 keV are only weakly coupled to the rest of the distribution.

7. Solar Wind Driving of the PCs

8. Relationship of PCs to GOE >2 MeV Electron Flux No single PC captures the variability of the GOES data.

9. 1 2 3 4 5 6 7 8 9 10 11 PC-1 builds up slowly over several days PCs-4, 6 and 7 appear to have a significant response to the Dst effect

10. 1 2 3 4 5 6 7 8 9 10 11 PC-1 builds up slowly over several days PCs-4, 6 and 7 appear to have a significant response to the Dst effect

11. 1 2 3 4 5 6 7 8 9 10 11 PC-1 builds up slowly over several days None of the PCs strongly resembles the Dst effect. Several PCs look like they are driven by the solar wind.

12. Conclusions • The largest principal component is the unified global response • Based on the node structure of the first several components, the fastest diffusion is in pitch angle, then L, then energy: nodes are parallel to the coordinate(s) of faster diffusion • No single PC correlates with GOES >2 MeV flux • Solar wind and Dst effect responses are not concentrated in a single PC, but are spread over several • While the PCs of effectively reduce the dimensionality of the problem, they do not clearly delineate physical processes • We might do better with PCs of PSD in adiabatic coordinates

13. How to Determine Diffusion Coefficients from Reanalysis Results Paul O’Brien

14. We can do it • Diffusion is a linear time-evolution operator • The time-forward operator relates the spatial correlation function to the spatiotemporal correlation function of the system • A reanalysis can give us the spatial and spatiotemporal correlation functions we need to fully specify the time-forward operator (even if it’s not diffusion) under specified geomagnetic conditions q (e.g., Kp) • Subsequent linear operations can extract the diffusion coefficients from the linear operator

15. 1-D is Simple • In the 1-D case, we can directly invert the diffusion operator to obtain the diffusion coefficient from any one of the eigenvectors of Dq to within a constant offset. • Each eigenvector gives a distinct estimate of the diffusion coefficient, which should help estimate the error. • See V.3 “Quadrature (Spatial)” in Schulz and Lanzerotti • Yes, Virginia, everything you ever wanted to know about the radiation belts really is in S&L

16. Example in 1-D • Using the covariance and lag covariance of pitch angle distributions at L=4.5, 800 keV, we can infer the diffusion coefficient • This analysis uses the 2nd eigenmode only • The solution is fixed to 1/day at 40 degrees. • At “active” times, pitch angle scattering is relatively stronger at low pitch angles, and relatively weaker at higher pitch angles compared to quiet times. • EMIC waves, which are expected only at active times, might explain this difference. • The underlying reanalysis had only 7 grid points in pitch angle, so the blue and green curves are probably indistinguishable. • A better solution is possible if one simultaneously inverts multiple eigenvectors.

17. 3-D is harder • The diffusion operator is the sum of 6 operators (3 “diagonal” diffusion terms and 3 “off-diagonal” diffusion terms) • One way to tackle the problem is to set it up as an inversion on the first M>=6 eigenvectors • This is a “big” numerical problem. If there are N grid points, then we have to invert a 6Nx6N matrix • Another solution would be to set up a parameterized diffusion tensor, and then determine the (small number of) free parameters by a fitting procedure