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

qq

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