Radiation Belt Modeling Yuri Shprits 1 Collaborators: Binbin Ni 1, Yue Chen 2, Dmitri Kondrashov 1, Richard Thorne 1, Josef Koller 2, Reiner Friedel 2, Geoff Reeves 2, Michael Ghil 1, Tsugunobu Nagai 3 1 Department of Atmospheric and Oceanic Sciences, UCLA, Los Angeles, CA 2 Los Alamos National Lab, Los Alamos, NM 3 Tokyo Institute of Technology,Tokyo, Japan

Talk Outline Brief Introduction Dependence of the Quasi-linear Scattering Rates on the Wave-normal Distribution of Chorus Waves Versatile Electron Radiation Belt (VERB) Code UCLA-LANL Reanalysis Project

Dominant acceleration and loss mechanisms Losses 1)Plasmaspheric Hiss ( whistler mode waves) loss time on the scale of days 2)Chorus waves outside plasmapause provide fast losses on the scale of a day. 3)EMIC waves mostly in plumes on the dusk side very fast localized 4) Combined effect of losses to magnetopause and outward radial diffusion. Sources 1)Inward radial diffusion 2)Local acceleration due to chorus waves Review of modeling of losses and sources of relativistic electrons in the outer radiation belt JASTP (2008) Yuri Y. Shprits, Scot R. Elkington, Nigel P. Meredith, Dmitriy A. SubbotinReview of modeling of losses and sources of relativistic electrons in the outer radiation belt

Dependence of the Quasi-linear Scattering Rates on the Wave normal Distribution of Chorus Waves

Dependence of scattering rates on the wave- normal angle A  4 — B  2 +C=0 A=S sin 2  P cos 2   RL sin 2  PS (1+cos 2  C=PRL,  m =0.35 |  e | ;  =0.15 ; cut- offs at ±2 .; m  ° ;L=4.5

Assumed distributions of magnetic wave power for various models of the wave normal distribution: (red) - field aligned case, (blue) - oblique, and (black) - highly oblique.

Bounce-averaged pitch-angle (top row), mixed (middle row), and energy (bottom row) diffusion coefficients for energies of 10 keV (left column), 100 keV (middle column), and 1.0 MeV (right column). Solid lines show the results for “Fixed B” models, and dashed lines show the results for “Fixed E” models.

Pitch-angle (first three columns) and energy scattering (last three columns) rates of different resonance harmonics for electrons of 10 keV (top), 100 keV (middle), and 1 MeV (bottom) for Fixed B models. Scattering rates are computed for field- aligned, oblique, and highly oblique models of waves

Local pitch-angle (first 3 columns) and energy ( 4 th -6 th columns) diffusion coefficients for 10 keV electrons at the magnetic latitudes of 0°, 10°, 20°, and 30°.

Local pitch-angle (first 3 columns) and energy ( 4 th -6 th columns) diffusion coefficients for 1.0 MeV electrons at the magnetic latitudes of 0°, 10°, 20°, and 30°.

Pitch-angle and energy scattering rates of different resonance harmonics for electrons of 10 keV (top), 100 keV (middle), and 1 MeV (bottom). Scattering rates are computed for field-aligned, oblique, and highly oblique models of waves at L=6.5. Waves are assumed to be confined to 10° of the geomagnetic equator.

Discussion I Similar results are obtained by considering wave normal distributions predominantly field aligned, but with a variable width of the frequency distribution. This rather surprising and counterintuitive result is due to the phase mixing of the effect of oblique waves during the integration over the frequency range from lower to upper cut-offs and bounce-averaging of the local diffusion coefficients. The results are also insensitive to the assumed radial location and latitudinal distribution of the waves. Knowledge of the wave normal angle may play a much more important role if the frequency distribution is very narrow or the wave normal distribution is narrow. Anthropogenic VLF transmitters may have distinct frequency and wave normal angles. Scattering by such narrow-band whistler–mode waves may strongly depend on the wave normal distribution. In this study, we have considered highly oblique waves but still restricted the maximum wave normal angle far from the resonance cone. Currently statistical models of wave normal distributions of chorus waves are not available. The analysis presented in this study is clearly not applicable for magnetospherically reflected whistlers, which propagate very close to the resonance cone. Scattering rates for radiation belt electrons can be accurately determined from the knowledge of only magnetic or only electric field fluctuations unless the wave-normal angles are above 40°.

Summary I Latitudonal distribution of waves, frequency distribution of waves and background plasma density will play an important role in determining scattering rates. At 10 keV, 100 keV, and 1 MeV pitch-angle scattering affecting electrons near the edge of the loss cone occurs within 10°, 20° and 30° of the magnetic equator respectively. In fact dominant pitch-angle scattering for MeV electrons occurs at latitudes above 20° independent of the assumed wave- normal distribution. Energy scattering at 10 keV, 100 keV, and 1 MeV also dominated by waves within 10°, 20° and 30° of the magnetic equator respectively. The presented results show that knowledge of the wave normal distribution of chorus waves becomes important for determining the energy scattering rates at ~10 keV, for which Landau resonance dominates scattering at small pitch-angles MLT –averaging smoothes out non-systematic differences in diffusion coefficients. We also show that the knowledge of the latitudinal distribution of waves has a much more pronounced effect on scattering rates. Computations of the bounce-averaged diffusion coefficients of high energy electrons by chorus waves show that at least for high-energy particles, the wave-normal distribution of chorus waves is not a very important parameter as long as the wave normal angles do not exceed ~40°.

Versatile Electron Radiation Belt (VERB) Code

Fokker-Planck equation in terms of momentum and energy diffusion.

Two-grid approach used in the Versatile Electron Radiation Belt Code

Pitch-angle and energy diffusion coefficients

Simulations with VERB diffusion code. Phase Space Density at  =850 MeV/G ; K=0.025 G 0.5 R E Simulations with the UCLA 3D VERB code including radial diffusion, scattering by the upper band and lower band chorus, plasmaspheric hiss waves and hiss inside the plumes, and losses to the outer boundary.

Differential Electron Flux, E=1.0 MeV Simulations with the 3D VERB code including radial diffusion, scattering by the upper band and lower band chorus, plasmaspheric hiss waves and hiss inside the plumes, and losses to the outer boundary.

Simulations with constant and variable amplitudes of chorus waves

Peaks in Phase Space Density seen in Akebono and CRRES reanalysis

Simulations with various boundary conditions

Discussion II Diffusion with respect to any of the variables (pitch-angle, energy, or L) will change the gradients in PSD profiles with respect to two other variables. For example, pitch-angle scattering at lower L-shells may increase radial gradients and increase inward radial diffusive transport. If waves that produce pitch-angle scattering are in resonance with electrons only for a limited range of energies, pitch-angle diffusion will also change energy gradients and will affect energy diffusion. Radial diffusion will change the energy spectrum by accelerating or decelerating electrons and will also change the pitch-angle distribution. Energy diffusion may produce peaks in PSD, which will drive inward and outward radial diffusion. A peak in energy diffusion rates at a particular value of the equatorial pitch-angle will produce butterfly distributions, which will affect pitch-angle diffusion. Consequently, these scattering processes cannot be studied independently. The net effect of each loss or acceleration mechanism will depend on the initial conditions. While this study provides an initial assessment of the dominant acceleration and loss processes, more accurate modeling needs to be pursued. In particular, we ignore non-linear interactions, the effects of mixed diffusion, non-diffusive radial transport, electrons accelerated in the cusp region, steady convection below L = 7, MLT dependence of PSD, effect of magneto-sonic waves, and bounce resonances.

Summary II Simulations with the 3D VERB code using a realistic outer boundary and geomagnetic activity-dependent day-side and night-side chorus amplitudes show that radial diffusion, local acceleration by chorus waves, and losses by chorus, hiss, and EMIC waves may all play important roles in the dynamic evolution of the radiation belts. Our simulations show that combined acceleration by radial diffusion and energy diffusion can energize plasma sheet electrons (tens of keV) to MeV energies. Geomagnetic control of wave amplitudes plays a controlling role in determining the rate of local acceleration.

UCLA-LANL Reanalysis Project

Kp index Lifetime, days Phase Space Density Flux at E=1.0 MeV Time, days L-value Time, days L-value Idealized Radial Diffusion Simulations

Comparison of the radial diffusion model and observations, starting on 08/18/1990.

Make a prediction of the state of the system and error covariance matrix, using model dynamics Kalman Filter Compute Kalman gain and innovation vector Update state vector using innovation vector Compute updated error covariance matrix Assume initial state and data and model errors

Comparison of the model with data assimilation with Daily-averaged CRRES observations.

THEMIS inter-calibration

Comparison of Reanalysis obtained with polar- orbiting Akebono and nearly equatorial CRRES satelites

Comparison of Reanalysis obtained with polar- orbiting Akebono and nearly equatorial CRRES satellites

Reanalysis of the Radiation Belt Fluxes Using Various Magnetic Field Models

Errors associated with inaccuracies of the assumed magnetic field model

Comparison between UCLA Kalman filter approach and LANL ensemble Kalman filter

Remarkable Global Coherency

Innovation Vector

Summary III The dynamics of the PSD obtained with data assimilation using the CRRES and Akebono data compare favorably to each other. In particular, the reanalysis show favorable agreements in the locations and magnitudes of the PSD peaks and the magnitudes and radial extent of the dropouts. The reanalysis results show robust PSD peaks independent of the assumed magnetic field model. The PSD dropouts generally occur during the main phase of the storms, covering a large range of L shells, and are almost energy-independent, which indicates that electrons may be lost to the outer boundary of trapping and transported outward by radial diffusion, consistent with the observations and modeling of Shprits et al. [2006a]. Most of the strongest depletions of the radiation belts occur when there is a sudden increase in the solar wind dynamic pressure which indicates that electrons may be lost to magnetopause and diffuse outwards. The PSD enhancements during the storm recovery phases and formation of peaks in the radial profile of PSD, mostly within 4 ≤ L ≤ 5, cannot be produced by the variation in the outer boundary or systematic errors between the satellites providing data at high and low L shells, suggesting an in situ acceleration mechanism.

Data assimilation with synthetic data produced with a radial diffusion model with  Kp

Data assimilation with synthetic data produced with a radial diffusion model with  Kp

Innovation vector

Ni et al.,| 2009b

Simulations with VERB diffusion code. Phase Space Density at  =850 MeV/G ; K=0.025 G 0.5 R E Simulations with the UCLA 3D code including radial diffusion, scattering by the upper band and lower band chorus, plasmaspheric hiss waves and hiss inside the plumes, and losses to the outer boundary. [Shprits, Subbotin, and Ni, 2009]

Assumed distributions of magnetic wave power for various models of the wave normal distribution: (red) - close to field-aligned, (blue) - narrow, (cyan) - wide, and (black) -extremely wide.

Bounce-averaged pitch-angle (top row), mixed (middle row), and energy (bottom row) diffusion coefficients for energies of 10 keV (left column), 100 keV (middle column), and 1 MeV (right column) for the Fixed B models: (red) - close to field-aligned, (blue) - narrow, (cyan) - wide, and (black) - extremely wide.

Pitch-angle (first three columns) and energy scattering (last three columns) rates of different resonance harmonics for electrons of 10 keV (top), 100 keV (middle), and 1.0 MeV (bottom). Scattering rates are computed for different widths of a wave normal distribution