interaction of shear alfven waves (saw) with trapped energetic protons in the inner radiation belt...
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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