Shock Acceleration at an Interplanetary Shock:
A Focused Transport Approach
J. A. le Roux
Institute of Geophysics & Planetary Physics
University of California at Riverside
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
dp||
f
f
 Vg 

t
x g
dt

dp' 
f

p||
dt

f
 f 
 
p'   t  c
where
Vg  v||b  VE ,
dp||
dt
dp ' 
dt
 qE|| 



1 p'  v' 
dB
|| B  mVE 
,
2 B
dt
1 p||v ' 
1
 || B  p '    VE  b  ||VE 
2 B
2
Transform (p||, p’)(’, M)
f
f
dM f
d '
 Vg1 


t
x g
dt M
dt

f  f 
 
 '  t  c
where
Vg1  v||b  VE ,
dM
 0,
dt
d '
B
 qE  Vg2  M
,
dt 
t
Vg2  v||b  VE 
Conservation of
magnetic moment
M
  b ||  M B  2 B  mb  d v||b 
q
q
B
qB dt
Electric field drift
Grad-B drift
Curvature drift
2. The Focused Transport Equation for f(x,p’,’,t)
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’, ’ )
f
f
 U i  vbi 
t
xi
 bi
 f
U i
U i 2 dU i 2q
1
2
 1   v

 3bi b j
 bi

Ei bi 
2

x

x

x
v
dt
p
i
i
j

 
 1
 f
U
U i  dU i
1
q
  1   2  i  1  3 2 bi b j
 bi
  Ei bi  p
xi 2
x j v
dt
p
 2
 p

 
f 
 D
  QPUI
 
 
Convection
Mirroring/focusing
Adiabatic
energy changes
Parallel
Diffusion
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
3. Results:
(a) Proton spectra in SW frame
SW Kappa distribution ( = 3)  1/r2
1012
1011
1010
1098
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10-4
10
10-1
upstream
1 AU
f(v)  v-3.2
100
101
v/Ue
102
downstream
103
f(v)(s3/km6)
f(v)(s3/km6)
SW density = C
1013
1012
1011
1010
1098
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10
10-1
0.7 AU
f(v)  v-4.2
f(v)  v-3.7
100
101
v/Ue
SW density  1/r2
102
103
SW Kappa distribution ( = 3)  1/r2
---0.27 AU
…0.49 AU
_.._0.71 AU
___1 AU
f(v)(s3/km6)
__0.13 AU
1012
1011
1010
1098
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10-4
10
10-1
Shift in peak to lower energies
f(v)  v-5.3
100
101
v/Ue
102
f(v)  v-32
103
(b) Anisotropies in SW frame at 1 AU
downstream
1.00
1.0
upstream
0.10
=82%
=25%
f()
f()
shock
100 keV
1 MeV
0.01
-1
0
1

-1
0

1.00
f()
=9%
10 MeV
0.10
-1
0.1
0

1
1
(c) Spatial distributions across shock
104
103
100 keV
f(z)
102
101
100
10-1
100 MeV
10-2
0.5
1 MeV
1.0
z(AU)
1.5
10 MeV
(d) Time distributions at 1 AU
102
101
f(t)
10 MeV
100
10-1
10 MeV
10-2
100 keV
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t(hours)
100 MeV
3. COMPARISON OF STRONG AND WEAK SHOCKS:
(a) Accelerated spectra
f(v)  v-3.7
1013
1012
1011
1010
1098
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10
0.7 AU
10-1
100
Weak shock (s=3.32.2)
f(v)(s3/km6)
f(v)(s3/km6)
Strong shock (s=4)
101
102
v/Ue
Spectrum harder
than predicted by
steady state DSA
103
1013
1012
1011
1010
1098
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10
10-1
f(v)  v-7.1
0.7 AU
100
101
102
v/Ue
Spectrum softer than
predicted by steady
state DSA
103
(b) Anisotropies
Strong shock (s=4)
Weak shock (s=3.32.2)
100
f()
f()
100
10-1
10-1
100 keV
100 keV
10-2
-1
0

1
10-2
-1
0

Pitch-angle anisotropy larger
for a strong shock
1
1D parallel Alfven wave transport equation at parallel shock solved
I
t
 U sh  V A 
I
z

U sh  V A k I
z
k
 1 U sh 
 
I
 2 z 
  (ki ) I


 
I 
 Dkk ( I ( k ), k ) 
k 
k 
Convection
Convection in wave number spaceCompression shortens wave length
Compression raises wave amplitude
Wave generation by SEP anisotropy
Wave-wave interaction – forward
cascade in wave number space in
inertial range
Wave-wave interaction Model (Zhou & Matthaeus, 1990):
k2
Dkk 
ts
•
Kolmogorov cascade I(k)  k-5/3
tNL << tA
(U = VA >> VA ) or (B >> Bo)
t s  t NL 
l
U
Kraichnan cascade I(k)  k-3/2
tA << tNL
(U = VA << VA ) or (B << Bo)
ts  t A 
l
VA
Comments
•
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
I(k||)(eVcm-2)
Alfven wave Spectrum (Kolmogorov Cascade)
1016
1015
1014
1013
1012
1011
1010
109
108
107
106
105
104
103
downstream
at shock at 0.1 AU
upstream
0.3 AU
0.5 AU
0.7 AU
1 AU
101 102 103 104 105 106 107
k||re
Need CK > 8 for
cascading to work
Kolmogorov or Kraichnan Cascade?
Bastille Day CME event
Kallenbach, Bamert, le Roux et al., 2008
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
Halloween CME event #1
Vasquez et al. , 2007
For 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?
Nonlinear wave-wave
interaction
Wave Kinetic Equation
dI ( k )
 2 ( k ) I ( k )   dk '[| a k ,k ' |2 I ( k ' ) I ( k ' ' )  2a k ,k ' a k ',k I ( k ) I ( k ' ' )] ( ( k )   ( k ' )   ( k ' ' ))
dt
Linear waveparticle interaction
Wave generation
Wave decay
Wave-wave interaction term a Boltzmann scattering term
When assume weak scattering, term can be put in FokkerPlanck form so that nonlinear diffusion coefficients for
wave cascading Dkk can be derived