Shock Acceleration at an Interplanetary Shock:A Focused Transport Approach

J. A. le Roux

Institute of Geophysics & Planetary Physics

University of California at Riverside

ct

f

p

f

dt

dp

p

f

dt

dpf

t

f

'

'

||

||

gg xV

EE

E

Eg

VbV

BV

VbV

||||||

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||

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1'

2

1'

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2

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pBB

vp

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dp

dt

dmB

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1. The Guiding center Kinetic Equation for f(x,p||,p’,t)

Transform position x to guiding center position xg (x = xg+rg)Transform transverse momentum to plasma frame p = p’

+ VE

Smallness parameter = m/q << 1 (small gyroradius, large gyrofrequency)Keep terms to order o

Average over gyrophase

where

Transform (p||, p’)(’, M)

ct

ff

dt

d

M

f

dt

dMf

t

f

'

'

gg1 xV

bbBBbVbV

VE

VbV

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g2

Eg1

||2||||

||

,'

,0

,

vdt

d

qB

m

Bq

M

q

Mv

t

BMq

dt

ddt

dM

v

where

Grad-B drift Curvature driftElectric field drift

Conservation of magnetic moment

To get focused transport equation, assume in guiding center kinetic equation E = -U B, whereby VE = U

Transform parallel momentum to moving frame (p|| = p’|| + mU||)

Transform (p’||,p’) (p’, ’ )

2. The Focused Transport Equation for f(x,p’,’,t)

Convection

PUI

iii

ij

iji

i

i

iii

ij

iji

i

i

i

i

iii

Qf

D

p

fpbE

p

q

dt

dUb

vx

Ubb

x

U

fbE

p

q

dt

dUbvx

Ubb

x

U

x

bv

x

fbvU

t

f

22

2

312

11

2

1

2231

2

1

Adiabatic energy changes

Parallel Diffusion

Mirroring/focusing

Advantages

Same mechanisms as standard cosmic-ray transport equation

No restriction on pitch-angle anisotropy - injection

Describe both (i) 1st order Fermi, and (ii) shock drift acceleration with or without scattering

Disadvantages

No cross-field diffusion

Gradient and curvature drift effect on spatial convection ignored

Magnetic moment conservation at shocks

v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-410-310-210-1100101102103104105106107108109101010111012

1 AU

downstreamupstream

SW Kappa distribution ( = 3) 1/r2

3. Results: (a) Proton spectra in SW frame

v/Ue

10-1 100 101 102 103f(

v)(s

3 /km

6 )10-310-210-11001011021031041051061071081091010101110121013

SW density = C

SW density 1/r2

0.7 AU

f(v) v-4.2

f(v) v-3.7

f(v) v-3.2

v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-410-310-210-1100101102103104105106107108109101010111012

__0.13 AU

---0.27 AU

___1 AU

_.._0.71 AU

…0.49 AU

SW Kappa distribution ( = 3) 1/r2

Shift in peak to lower energies

f(v) v-5.3

f(v) v-32

-1 0 1

f( )

0.1

1.0

-1 0 1

f( )

0.01

0.10

1.00

-1 0 1

f( )

0.10

1.00

100 keV 1 MeV

10 MeV

downstream

shock

upstream

(b) Anisotropies in SW frame at 1 AU

=82%

=25%

=9%

z(AU)

0.5 1.0 1.5

f(z)

10-2

10-1

100

101

102

103

104

100 MeV

10 MeV1 MeV

100 keV

(c) Spatial distributions across shock

t(hours)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

f(t)

10-2

10-1

100

101

102

100 MeV

10 MeV

10 MeV

100 keV

(d) Time distributions at 1 AU

v/Ue

10-1 100 101 102 103

f(v)

(s3 /

km6 )

10-310-210-11001011021031041051061071081091010101110121013

v/Ue

10-1 100 101 102 103f(

v)(s

3 /km

6 )10-310-210-11001011021031041051061071081091010101110121013

Strong shock (s=4) Weak shock (s=3.32.2)f(v) v-3.7

f(v) v-7.1

3. COMPARISON OF STRONG AND WEAK SHOCKS:

0.7 AU 0.7 AU

Spectrum harder than predicted by steady state DSA

Spectrum softer than predicted by steady state DSA

(a) Accelerated spectra

-1 0 1

f( )

10-2

10-1

100

(b) Anisotropies

-1 0 1

f( )

10-2

10-1

100

100 keV100 keV

Strong shock (s=4) Weak shock (s=3.32.2)

Pitch-angle anisotropy larger for a strong shock

k

IkkID

k

Ik

Iz

Uk

IkVU

z

z

IVU

t

I

kk

i

sh

Ash

Ash

)),((

)(

2

1

1D parallel Alfven wave transport equation at parallel shock solved

Convection

Convection in wave number space-Compression shortens wave length

Compression raises wave amplitude

Wave generation by SEP anisotropy

Wave-wave interaction – forward cascade in wave number space in inertial range

• Turbulence can be modeled as a nonlinear diffusion process in wavenumber space (Dkk)

• Ad-hoc turbulence model based on dimensional analysis

• Contrary to weak turbulence theory – assume that 3-wave coupling for Alfven waves is possible – imply wave resonance broadening – momentum and energy conservation in 3-wave coupling does not hold

Wave-wave interaction Model (Zhou & Matthaeus, 1990):

skk t

kD

2

• Kolmogorov cascade I(k) k-5/3

tNL << tA (U = VA >> VA ) or (B >> Bo)

Kraichnan cascade I(k) k-3/2

tA << tNL (U = VA << VA ) or (B << Bo)

U

ltt NLs

A

As V

ltt

Comments

k||re

101 102 103 104 105 106 107

I(k ||)

(eV

cm-2

)

1031041051061071081091010101110121013101410151016

downstream

at shock at 0.1 AU

upstream0.3 AU0.5 AU0.7 AU1 AU

Alfven wave Spectrum (Kolmogorov Cascade)

Need CK > 8 for cascading to work

Kolmogorov or Kraichnan Cascade?

Used Marty Lee (1983) quasi-linear theory extended to include nonlinear wave cascading via a diffusion term in wavenumber space.

Applied theory to Bastille Day and Halloween #1 CME events to calculate kmin for wave growth

Find that Kraichnan-type turbulence gives the correct kmin

Kolmogorov-type turbulence overestimates kmin by a factor 2-4

Kallenbach, Bamert, le Roux et al., 2008Bastille Day CME event

Halloween CME event #1

Vasquez et al. , 2007For cascade rates and proton heating rates to agree in SW – CK= 12 needed instead of the expected CK ~1

Kraichnan cascade rate is close to heating rate at 1 AU

A Weak Kinetic Turbulence Theory Approach to Wave Cascading?

))''()'()(()]''()(2)''()'(|[|')()(2)(

',',2

', kkkkIkIaakIkIadkkIkdt

kdIkkkkkk

Wave Kinetic Equation

Wave-wave interaction term a Boltzmann scattering term

When assume weak scattering, term can be put in Fokker-Planck form so that nonlinear diffusion coefficients for wave cascading Dkk can be derived

Linear wave-particle interaction

Nonlinear wave-wave interaction

Wave generation Wave decay