Principal Components of Electron Belt Variation Paul O’Brien

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

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

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

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

4th PC decouples region near L~5 from rest of belt 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.

Solar Wind Driving of the PCs

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

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

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

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

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

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

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.

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