a focused transport approach to low energy ion acceleration
DESCRIPTION
SHINE 2006. A Focused Transport Approach to Low Energy Ion Acceleration. J. A. le Roux & G. M. Webb IGPP, University of California, Riverside. TRANSPORT THEORY. STANDARD FOCUSED TRANSPORT EQUATION – GYROPHASE AVERAGED BOLTZMANN EQUATION FOR GYROTROPIC DISTRIBUTION. - PowerPoint PPT PresentationTRANSCRIPT
A Focused Transport Approach to Low Energy Ion
Acceleration
J. A. le Roux & G. M. Webb
IGPP, University of California, Riverside
SHINE 2006
2
TRANSPORT THEORY
STANDARD FOCUSED TRANSPORT EQUATION – GYROPHASE AVERAGED BOLTZMANN EQUATION FOR
GYROTROPIC DISTRIBUTION
( )
scj
iji
i
i
i
i
j
iji
iiii
i
iii
t
f
p
f
x
Ubb
p
x
Upp
x
bv
p
f
x
Ubbp
dt
dUmbbqEp
x
bvbvU
t
f
⎟⎠
⎞⎜⎝
⎛∂
∂=
∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂
∂+
∂
∂−
∂
∂−
∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂
∂−−+
∂
∂+++
∂
∂
⊥
⊥⊥⊥
⊥⊥
'2
'
2
''
2
'
'''
2
''
||
||||||
Particle momentum p’ transformed to plasma flow frame where Ui =0
3
DRIFT EFFECTS IN FOCUSED TRANSPORT EQUATION
EVb +=+= ||||' vbvUdtdx
iii
Guiding center drift along BiElectric field drift
THE CONVECTION TERM
4
DRIFT EFFECTS IN FOCUSED TRANSPORT EQUATION
j
iji
iiii
i
i
x
Ubbp
dt
dUmbbqEp
x
bv
dt
dp
∂∂
−−+∂∂
= ⊥⊥
|||| ''
2''
( )⎥⎦
⎤⎢⎣
⎡×+•+∇−=⎟
⎠⎞
⎜⎝⎛ b
bbE ||||||||
2||2
1v
dt
d
Bq
mvqBMvmv
dt
d
PARALLEL MOMENTUM CHANGE TERM
Magnetic mirroring
Parallel guiding center drift
Contains curvature drift
5
DRIFT EFFECTS IN FOCUSED TRANSPORT EQUATION
TRANSVERSE MOMENTUM CHANGE TERM
4444 34444 21 j
iji
i
i
i
i
x
Ubb
p
x
Upp
x
bv
dt
dp
∂∂
+∂∂
−∂∂
−= ⊥⊥⊥⊥
2'
2'
'2''
||
( )4444444 84444444 76
⎥⎦
⎤⎢⎣
⎡×∇+
∇ו+
∂∂
+∇=⎟⎠
⎞⎜⎝
⎛⊥ ||2||||2'
2
1b
BE
q
M
B
B
q
Mq
t
BMBMvmv
dt
d
Magnetic mirroring
Gyration Grad-B drift Drift along Bi
6
CONSERVATION OF MAGNETIC MOMENT M
TRANSVERSE MOMENTUM CHANGE TERM CAN BE SHOWN TO GIVE:
)O( of if0 |||| εEEBM
dtdM bb ≈×∇•=
7
CONSTANTS OF MOTION AT PARALLEL SHOCK
Cvv
CUvv
==
=+=
⊥⊥ '
' ||||||
If E|| =0
CONSTANTS OF MOTION AT PERPENDICULAR SHOCK
CB
v
Cvv
=
==
⊥2
||||
'
'
If E|| =0 Agrees with shock drift theory
( ) )1('11'
' 221
22 −−+= sv
vμ
8
STANDARD FOCUSED TRANSPORT EQUATION INCLUDES:
STANDARD FOCUSED TRANSPORT EQUATION NEGLECTS:
gradient and curvature drift contribution to convection
Energy changes associated with part of acceleration drift (polarization drift)
Perpendicular diffusion – can be included by randomly varying field angle
dt
d
Bm
dt
dEk EVbE ו=
Convection along Bi and electric field drift
Energy changes associated with grad-B drift, curvature drift (part of acceleration drift), parallel drift (cross-shock potential), and compression of plasma flow along Bi
Magnetic mirroring, mirroring by cross-shock potential, and conservation of magnetic moment
Transport Theory consistent with Shock Drift Theory
TRANSPORT THEORY - SUMMARY
9
PARALLEL SHOCK: (i) Accelerated particle spectra (fluid frame)
v/Ue
10-1 100 101 102
f(v)
10-810-710-610-510-410-310-210-1100101102103
v/Ue
10-1 100 101 102
f(v)
10-810-710-610-510-410-310-210-1100101102103
Strong cross-shock potential
1 keV isotropic particle source
downstream downstream
No cross-shock potential
1st Reflection peak
1st transmission peak
10
PARALLEL SHOCK: (ii) Spatial variation across shock
z(AU)-0.02 -0.01 0.00 0.01 0.02
f(z)
100
z(AU)-0.02 -0.01 0.00 0.01 0.02
f(z)
100
10 keV
100 keV100 keV
10 keV
In fluid frame f(z) discontinuous across
shock
Discontinuity in f(z) enhanced by cross-
shock potential
Shock at z = 0
11
PARALLEL SHOCK: (iii) Anisotropies across shock (fluid frame)
μ-1 0 1
f(μ)
0.00.10.20.30.40.50.60.70.80.91.01.11.21.3
μ-1 0 1
f(μ)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
10 keV
100 keV
1 keV
upstream
10 keV
downstream
upstreamat shock
Reflection by cross-shock
potential
12
OBLIQUE SHOCK(BN = 45o): (i) Accelerated spectra (fluid frame)
v/Ue
10-1 100 101 102
f(v)
10-810-710-610-510-410-310-210-1100101102103
v/Ue
10-1 100 101 102
f(v)
10-810-710-610-510-410-310-210-1100101102103
Magnetic reflection
Magnetic reflection + cross-shock potential
reflection
Spectra harder than expected from
standard DSA theory
Hybrid simulationKucharek & Scholer
(1995)
13
z(AU)-0.02 -0.01 0.00 0.01 0.02
f(z)
100
OBLIQUE SHOCK(BN = 45o): (ii) Spatial variation across shock
10 keV
100 keV Magnetic reflection contributes substantially
towards discontinuity in f(z) across the shock at higher
energies
Standard assumption of f1 = f2
In DSA theory does not apply
14
μ-1 0 1
f(μ)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
OBLIQUE SHOCK (BN = 45o): (iii) Anisotropy across shock (fluid frame)
μ-1 0 1
f(μ)
0.00.10.20.30.40.50.60.70.80.91.01.11.21.3
upstream10 keV
100 keV
1 keV
upstream
at shock
downstream
10 keV
Magnetic reflection
Anisotropy enhanced by magnetic reflection
Anisotropy large at 100 keV – violates standard
DSA theory
15
v/Ue
10-1 100 101 102 103
f(v)(s
3/km
6)
10-810-710-610-510-410-310-210-1100101102103
v/Ue
100 101
f(v)(s
3/km
6)
10-1
100
QUASI-PERPENDICULAR SHOCKS (BN > 45o): Accelerated spectra (fluid frame)
BN = 70o BN = 89.4o
Only particles with E > 9 MeV (v/Ue > 95) can be reflected upstream
Only particles with v/Ue > 3 can be
reflected upstream
downstream downstream
16
NEARLY PERPENDICULAR SHOCK (Variable BN): Accelerated spectra (fluid frame)
ψ0 45 90 135 180 225 270 315 360
f(ψ)
0
50
100
150
200
250
Observed hourly averaged spiral
angles by Voyager 1 during 2004
v/Ue
10-1 100 101 102 103
f(v)(s
3/km
6)
10-1110-1010-910-810-710-610-510-410-310-210-1100101102103
Deviations from average BN lowers
threshold for particle reflection
17
SIMULATIONS: SUMMARY AND INTERPRETATION
•(1) SHOCK ACCELERATION RESULTS WITH FOCUSED TRANSPORT MODELDEVIATE FROM STANDARD DSA THEORY BECAUSE OF: Particle reflection at shock by field compression Particle reflection by cross-shock electric field (smaller effect) Particles are tied to field lines – have difficulty to go back upstream Particle momentum is in comoving frame
(2) THE MAIN DEVIATIONS FROM DSA THEORY AND SOLUTIONS ARE:Accelerated spectra is power law – but harder than predicted by DSA theory At low energies 2 prominent peaks in accelerated spectra downstream - DSA theory solution give smooth power lawSpatial distribution discontinuous in form of a spike across shock – even at higher energies – continuous distribution across shock is assumed in DSA theoryUpstream particle anisotropies large and field-aligned in direction away from shock even at higher energies - small anisotropies are assumed in DSA theory
(3) THE BASIC ACCELERATED SPECTRAL FEATURES PRODUCED BY FOCUSED TRANSPORT MODEL AGREE WITH MORE SOPHISTICATED PARTICLE CODES
(4) DISCONTINUOUS INTENSITY SPIKES, AND FIELD-ALIGNED UPSTREAM ANISOTROPIES PRODUCED BY FOCUSED TRANSPORT MODEL ARE PRESENT IN VOYAGER 1 OBSERVATIONS AT TERMINATION SHOCK
MAIN CONCLUSION: FOCUSED TRANSPORT WILL PROVIDE A MORE ACCURATE AND REALISTIC DESCRIPTION OF SEP ACCELERATION AT CME SHOCKS THAN STANDARD DSA THEORY
18
SIMULATIONS: PROBLEMS
(1) Particles are tied to field lines - if shock normal angle > 70O, particles have difficulty to achieve multiple shock encounters
(2) Can shock acceleration at a nearly perpendicular shock work by randomly varying the field angle without microscopic diffusion in focused transport model?
(3) The particle anisotropy at Voyager 1 peaks at some intermediate energy – focused transport model predicts an increase with decreasing energy – could indicate preacceleration should occur upstream of shock at lower energies – particle trapping upstream in non-linear self-generated waves possibly needed