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

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

Proton-SAW Gyro-Resonant Condition

/ zzvk

Gyro-Resonant Condition:

zzvk (non-relativistic proton, ω << Ω)

SAW Dispersion Relation:

AzVk

Gyro-Resonant Condition for proton (v, ) with SAW:

v

Vv A

cos),(

Frequency Selection for Proton-SAW

Resonance

E

MVE A

2cos),(

2

degree 28L

Proton Energy

Frequency Range

30 MeV 6-16 Hz

50 MeV 5-15 Hz

100 MeV 3.5-9.5Hz

Frequency requirement for equatorial Proton-SAW resonance with at L=1.5

Frequency range ~ 5-15 Hz

2/,10 .,.

)/)(exp()(

00

220

ffHzfei

fffW

Broadband SAW:

Proton Lifetime Calculation I

1. Local Pitch Angle Diffusion Rate Proportional to Wave Energy

2. Bounce Averaged Diffusion Rate

)/)),((exp(),(

)(

)( 2202

2

vv

B

BD

2

1

)(cos)(cos

)cos(

)(2

1 72

M

M

dDS

DEE

B

dB

BvF

BS

B

M

eD

M

EEEE

B

0

722

2

)(cos)0(

)()(sin1),,(

)0(

)(

)(

1

)(cos)(

),0(

λ is the latitude and φ is the azimuthal angle

02 2/),0( BD

B(Wave energy trapped insideflux tube at φ)

Proton Lifetime Calculation II

0

22

2),0(

2

1

2

1

)/2(

1

B

dBF

dDV

rdD

VrD

BD

BD

D

3. Drift-Averaged Pitch Angle Diffusion Rate

Volume

EnergySAW

2 0

2

B

• Pitch angle scattering amount is proportional to the stored SAW energy the proton experiences during its bounce-drift orbit.

Proton Lifetime Calculation III

atmD

ffDS

St

f

0

0

0000

0000

0 )cos()sin()()cos()sin()(

1

),,()(),( 000 LEgtFtf

0000

00000 )cos()sin()()cos()sin()()()

11(

g

DSSgpatm

1)1

(

t

F

Fp(Life Time)

4. Solve Pitch-Angle Diffusion Equation

• Use finite-difference to discretize

• Use iterative method to solve nonlinear eigen-value problemfor lifetime

),,( ,0 iLEg

Split temporal and pitch angle distribution

Local PAD Rate

L = 1.5 for 30 MeV protons in presence of waves withf0 = 13 Hz, Δf = 0.5 f0

δB = 25 pT, (108 sec ~ 3 years)

Alfven Velocity along L = 1.5Field line

From Global Core Plasma Model+ Dipole Model

E

MVE A

2cos),(

2

Shift is due to increase of B

Local Pitch-Angle Diffusion (PAD) Rate for Protons at L = 1.5

Loss Cone

Earth Equator

Drift-Bounce Averaged PAD Rate

f/f=1/2 , <B>= 25 pT

6.5 Hz 10 Hz 13 Hz

• Energy stored in SAW at L=1.5 and DL=.1 (volume = 3 x 1020 m^3) with <B> =25 pT is

W= 75 kJ

Loss Cone

Loss Cone

Loss Cone

Proton Lifetime

f1= 6.5 Hz f2= 10 Hz f3= 13 Hz

E = 30 MeV 1688 days 880 days 595 days

E = 50 MeV 900 days 586 days 920 days

E = 100 MeV 580 days 1032 days 1600 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.

Injection of SAW

IonosphericReflection

• 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.

Injection Power Requirement

,AP

APR

Injection power required to maintain

75 kJ at L=1.5 per .1 L width

Rate Loss: Power, Injected: , PWPdt

dW

WT

RWP

ln

• Typically, the required SAW injection power is ~ kWto 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

S

SV

P

AA

55.0

06.0/1 0

R= 0.78 -0.95

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).