Transcript Spectrum of MHD turbulence

Spectrum of MHD turbulence
Stanislav Boldyrev
University of Chicago
(June 20, 2005)
Ref: astro-ph/0503053; ApJ 626, L37, 2005
Introduction: Kolmogorov turbulence
Random flow of incompressible fluid
Reynolds number:
v L
Re=Lv/η>>1
x2
η-viscosity
V x1   V x2 ||3
x1
~ x1  x2 
If there is no intermittency, then:
V x1   V x2 ||2
~ x1  x2 
2/3
Kolmogorov spectrum
and
Ek ~ Vk k 2 ~ k 5 / 3
2
[Kolmogorov 1941]
2
Kolmogorov energy cascade
Ek
Ek ~ Vk k 2 ~ k 5 / 3
2
k
5 / 3
kf
k
k
2
3
Energy of an eddy of size  ~ 1/ k is E ~ V ~ Vk k ;
2
it is transferred to a smaller-size eddy during time:
  ~  / V ~ Vk1k 5 / 2
The energy flux, J 
- “eddy turn-over” time.
~ E /   ~ Ek k 5 /,3 is constant for the
Kolmogorov spectrum!
3
MHD turbulence
No exact Kolmogorov relation.
Energy


E   V 2  B 2 dx
•Is energy transfer time
Phenomenology:
is conserved, and cascades
toward small scales.
  ~  / V ?
Non-dimensional parameter
V / VA
No, since dimensional
arguments do not work!
can enter the answer.
•Need to investigate interaction of “eddies” in detail!
This is also the main problem in the theory of weak (wave) turbulence.
(waves is plasmas, water, solid states, liquid helium, etc…)
[Kadomtsev, Zakharov, ... 1960’s]
4
Iroshnikov-Kraichnan spectrum
z
w
w
z
After interaction, shape of each packet changes, but energy does not.
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Iroshnikov-Kraichnan spectrum
during one collision:
λ
number of collisions required to
deform packet considerably:
λ
Constant energy flux:
[Iroshnikov (1963); Kraichnan (1965)]
6
Goldreich-Sridhar theory
Anisotropy of “eddies”
λ
B
L
Shear Alfvén waves
dominate the cascade:
┴B
L>>λ
Critical Balance
[Goldreich & Sridhar (1995)]
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Spectrum of MHD Turbulence in Numerics
[Müller & Biskamp, PRL 84 (2000) 475]
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Goldreich-Sridhar Spectrum in Numerics
Cho & Vishniac, ApJ, 539, 273, 2000
Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002
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Strong Magnetic Filed, Numerics
Contradictions with Goldreich-Sridhar model
Iroshnikov-Kraichnan
scaling
[Maron & Goldreich, ApJ 554, 1175, 2001]
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Strong Magnetic Filed, Numerics
Contradictions with Goldreich-Sridhar model
B-parallel scaling
Scaling of field-parallel and
field-perpendicular structure
functions for different
large-scale magnetic fields.
B-perp scaling
B  V 2
B  V 2
2
Strong field, B>>ρV :
Iroshnikov-Kraichnan scaling
[Müller, Biskamp, Grappin
PRE, 67, 066302, 2003]
Weak field, B→0:
Goldreich-Sridhar (Kolmogorov)
scaling
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New Model for MHD Turbulence
Analytic Introduction [S.B., ApJ, 626, L37,2005]
Depletion of nonlinear interaction:
1
2
Nonlinear interaction
is depleted
Interaction time
is increased
For
perturbation cannot propagate along the B-line faster
than V , therefore, correlation length along the line is
A
This balances terms 1 and 2 in the MHD equations, as in the
Goldreich-Sridhar picture, however, the geometric meaning is different.
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New Model for MHD Turbulence
Analytic Introduction [S.B., ApJ, 626, L37,2005]
Nonlinear interaction
is depleted
Interaction time
is increased
Constant energy flux,
Goldreich-Sridhar scaling corresponds to α=0:
 N ~ 2 / 3 l ~ 2 / 3
“Iroshnikov-Kraichnan” scaling is reproduced for α=1:
 N ~ 1/ 2
l ~ 1/ 2
Explains
numerically
observed scalings
for strong B-field !
[Maron & Goldreich, ApJ 554, 1175, 2001]
[Müller, Biskamp, Grappin PRE, 67, 066302, 2003]
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New Model for MHD Turbulence
Geometric Meaning
S.B. (2005) “eddy”:
Goldreich-Sridhar 1995 “eddy”:
l ~ 2 / 3
line displacement:
l ~ 1/ 2
~
line displacement:
~ 3 / 4
As the scale decreases,
λ→0,
turns into filament
turns into current sheet
agrees with numerics!
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New Model for MHD Turbulence
Depletion of nonlinearity
S.B. (2005) “eddy”:
l ~ 1/ 2
In our “eddy”, w and z are aligned within
small angle θλ . One can check that:
line displacement:
~ 3 / 4
In our theory, this angle is:
~ 1/ 4
Remarkably, we reproduced the
reduction factor in the original formula:
The theory is self-consistent.
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Summary and Discussions
1. Weak large-scale field: B  V 2
l~
~
2/3
[Goldreich & Sridhar’ 95]
dissipative structures: filaments
-5/3
energy spectrum: E(K)~K┴
~ 1 / 4
2. Strong large-scale field: B  V 2
~ 3 / 4
l ~ 1/ 2
scale-dependent
dynamic alignment
dissipative structures: current sheets
energy spectrum: E(K)~K-3/2
┴
3. The spectrum of MHD turbulence may be non-universal.
-3/2
Alternatively, it may always be E~K┴ , but in case 1, resolution
of numerical simulations is not large enough to observe it.
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Conclusions
• Theory is proposed that explains contradiction between
Goldreich-Sridhar theory and numerical findings.
• In contrast with GS theory, we predict that turbulent eddies
are three-dimensionally anisotropic, and that dissipative
structures are current sheets.
• For strong large-scale magnetic field, the energy spectrum
-3/2
is E~K-3/2. It is quite possible that spectrum is always E~K┴
,
┴
but for weak large-scale field, the resolution of numerical
simulations is not large enough to observe it.
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