1. Interaction of Shear Alfven Waves (SAW) with Trapped Energetic Protons in the Inner Radiation Belt X. Shao, K. Papadopoulos, A. S. Sharma Department of Physics and Astronomy, University of Maryland, College Park, MD, USA

2. Outline Proton-SAW Gyro-Resonant Condition Frequency selection for SAW-Proton resonance under inner belt condition Proton lifetime as a function of average SAW amplitude Ground-injected SAW power as a function of energy stored per unit shell volume

3. Proton-SAW Gyro-Resonant Condition Gyro-Resonant Condition: (non-relativistic proton, ω << Ω) SAW Dispersion Relation: Gyro-Resonant Condition for proton (v, ) with SAW:

4. Frequency Selection for Proton-SAW Resonance Proton Energy Frequency Range 30 MeV6-16 Hz 50 MeV5-15 Hz 100 MeV3.5-9.5Hz Frequency requirement for equatorial Proton-SAW resonance with at L=1.5 Frequency range ~ 5-15 Hz Broadband SAW:

5. Proton Lifetime Calculation I 1. Local Pitch Angle Diffusion Rate Proportional to Wave Energy 2. Bounce Averaged Diffusion Rate λ is the latitude and φ is the azimuthal angle (Wave energy trapped inside flux tube at φ)

6. Proton Lifetime Calculation II 3.Drift-Averaged Pitch Angle Diffusion Rate Pitch angle scattering amount is proportional to the stored SAW energy the proton experiences during its bounce-drift orbit.

7. Proton Lifetime Calculation III (Life Time) 4. Solve Pitch-Angle Diffusion Equation Use finite-difference to discretize Use iterative method to solve nonlinear eigen-value problem for lifetime Split temporal and pitch angle distribution

8. Local PAD Rate L = 1.5 for 30 MeV protons in presence of waves with f 0 = 13 Hz, Δf = 0.5 f 0 δB = 25 pT, (10 8 sec ~ 3 years) Alfven Velocity along L = 1.5 Field line From Global Core Plasma Model + Dipole Model Shift is due to increase of B Local Pitch-Angle Diffusion (PAD) Rate for Protons at L = 1.5 Loss Cone Earth Equator

9. Drift-Bounce Averaged PAD Rate  f/f=1/2, = 25 pT 6.5 Hz10 Hz 13 Hz Energy stored in SAW at L=1.5 and DL=.1 (volume = 3 x 10 20 m^3) with =25 pT is W= 75 kJ Loss Cone

10. Proton Lifetime f 1 = 6.5 Hzf 2 = 10 Hzf 3 = 13 Hz E = 30 MeV1688 days880 days595 days E = 50 MeV900 days586 days920 days E = 100 MeV580 days1032 days1600 days Df/f=1/2, Energy stored in SAW at L=1.5 and DL=.1 is W= 75 kJ Life time of (30-100 MeV) protons can be reduced to 1-3 years.

11. Injection of SAW Ionospheric Reflection Injection can be carried out at selected sites The remediation effects will be the same for global or sector injection as long as the total stored SAW resonance energy is the same. SAW is trapped inside the flux tube The loss of SAW mainly occurs at the ionospheric boundary.

12. Injection Power Requirement Injection power required to maintain 75 kJ at L=1.5 per.1 L width Typically, the required SAW injection power is ~ kW to reduce life time of (30-100 MeV) protons to 1-3 years. Wave Energy Evolution in Leaky Cavity R: Ionospheric Reflection Coefficient ΔT: Alfven Wave Travel Time R= 0.78 -0.95

13. Summary Estimates indicate that less than kWatt level of ULF injected into the L = 1.5-1.8 region is required to get interesting PRB removal lifetime (1-3 years).