Turbulent diffusion as a requirement for a f ( p ) p -5 spectrum
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Turbulent diffusion as a requirement for a f(p) p -5 spectrum J. A. le Roux & G. M. Webb IGPP, University of California, Riverside SHINE 2006
description
SHINE 2006. Turbulent diffusion as a requirement for a f ( p ) p -5 spectrum. J. A. le Roux & G. M. Webb IGPP, University of California, Riverside. EQUATION FOR ENERGETIC PARTICLE TRANSPORT IN RELATIVELY LARGE-SCALE COMPRESSIVE PLASMA FLUID TURBULENCE. Turbulent convection. - PowerPoint PPT Presentation
Transcript of Turbulent diffusion as a requirement for a f ( p ) p -5 spectrum
Turbulent diffusion as a requirement
for a f(p) p-5 spectrum
J. A. le Roux & G. M. Webb
IGPP, University of California, Riverside
SHINE 2006
2
p
fDp
ppz
fUp
ppz
ffCU
zt
fppeff 5
11
3
1 22
*32
*
EQUATION FOR ENERGETIC PARTICLE TRANSPORT IN RELATIVELY LARGE-SCALE COMPRESSIVE PLASMA
FLUID TURBULENCE
Turbulent convection)(
22
*eff
pc
c
tl
UUU
)( where)( eff
22 p
cteff
teff tUUf
)(2
22
2eff
pc
cpp t
l
UUpD
Non-linear diffusion expression where
T is turbulent diffusion
Turbulent stochastic acceleration
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Solve transport equation assuming: (i) Equal compressive wave intensities in all directions (U* = 0)(ii) turbulent diffusion dominates spatial diffusion by Alfven waves (t >> ) (iii) A steady state
045
41
9
4 42
2
2
2
p
fp
pp
U
z
f
t
tott
Equation to solve is:
tteff 94 9
4 tt
pp
UUp
UUpD
49
49
22
2
22
2
4
2
5493
where),(
22
2
ttot
sz
Us
pcepzf
S = 3 if 0S = 5 if 1/lc
Solution using separation of variables is:
A f(p) p-5 spectrum is the natural outcome of turbulent diffusion and stochastic acceleration by relatively
large-scale compressive plasma fluid turbulence
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Alternatively, assume: (i) Non-relativistic particles (ii) Equal compressive wave intensities in all directions (iii) A steady state (iv) integrate over p interval [p1, p2] after multiplying
transport equation with 4p2Ek - energy density form
0* U
2
1
2
1
42
5
2
5
4)(p
p
p
p
ppppcav
eff p
fD
m
p
p
fvDdpp
z
pE
z
94 teff
t
pp
UUpD
49
22
2
Diffusive energy flux through [p1,p2]
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Assume turbulent diffusion dominates- f(p) p-5
as shown before. Then one finds that:
05
22
1
26
p
pt
tot
p
fU
m
p
Thus a constant diffusive energy flux through interval [p1,p2] is a natural further consequence of turbulent diffusion and stochastic acceleration by relatively large-scale compressive plasma fluid turbulence.
