Spectrum of MHD turbulence
Stanislav Boldyrev
University of Chicago
Ref: astro-ph/0503053; ApJ 626, L37, 2005
(June 20, 2005)
2
Introduction: Kolmogorov turbulence
3/221
2
||21 ~ xxxVxV
213
||21 ~ xxxVxV
1xLv Re=Lv/η>>1
Reynolds number:
Random flow of incompressible fluid
η-viscosity
2x
If there is no intermittency, then:
3/522~~ kkVE kk
Kolmogorov spectrum [Kolmogorov 1941]
and
3
Kolmogorov energy cascade
kE
k kEnergy of an eddy of size is ;
it is transferred to a smaller-size eddy during time: k/1~
2/51~/~ kVV k
322 ~~ kVVE k
The energy flux, 3/5~/~ kEEJ k
3/522~~ kkVE kk
, is constant for the
Kolmogorov spectrum!
local energy flux
- “eddy turn-over” time.
fk
3/5k
4
MHD turbulence
dxBVE 22
V/~
No exact Kolmogorov relation. Phenomenology:
Energyis conserved, and cascades toward small scales.
•Is energy transfer time ?No, since dimensional arguments do not work!
•Need to investigate interaction of “eddies” in detail!
Non-dimensional parameter AVV / can enter the answer.
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]
5
Iroshnikov-Kraichnan spectrum
After interaction, shape of each packet changes, but energy does not.
wz
w z
6
Iroshnikov-Kraichnan spectrum
λ
λ
during one collision:
number of collisions required to deform packet considerably:
Constant energy flux:
[Iroshnikov (1963); Kraichnan (1965)]
7
Goldreich-Sridhar theoryAnisotropy of “eddies”
Lλ
Shear Alfvén wavesdominate the cascade:
B
┴ BCritical Balance
λL>>
[Goldreich & Sridhar (1995)]
8
Spectrum of MHD Turbulence in Numerics
[Müller & Biskamp, PRL 84 (2000) 475]
9
Goldreich-Sridhar Spectrum in Numerics
Cho & Vishniac, ApJ, 539, 273, 2000Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002
10
Strong Magnetic Filed, NumericsContradictions with Goldreich-Sridhar model
[Maron & Goldreich, ApJ 554, 1175, 2001]
Iroshnikov-Kraichnan scaling
11
Strong Magnetic Filed, NumericsContradictions with Goldreich-Sridhar model
Scaling of field-parallel and field-perpendicular structure functions for different large-scale magnetic fields.
[Müller, Biskamp, GrappinPRE, 67, 066302, 2003]
Weak field, B→0: Goldreich-Sridhar (Kolmogorov)scaling
2VB 2VB
B-parallel scaling
B-perp scaling
Strong field, B>>ρV : Iroshnikov-Kraichnan scaling
2
12
New Model for MHD Turbulence
Depletion of nonlinear interaction:
Nonlinear interaction is depleted
Interaction time is increased
A
1 2
This balances terms and in the MHD equations, as in the Goldreich-Sridhar picture, however, the geometric meaning is different.
1 2
[S.B., ApJ, 626, L37,2005]
For perturbation cannot propagate along the B-line faster than V , therefore, correlation length along the line is
Analytic Introduction
13
New Model for MHD TurbulenceAnalytic Introduction
Nonlinear interaction is depleted
Interaction time is increased
Constant energy flux,
3/2~ N 3/2~ l
Goldreich-Sridhar scaling corresponds to α=0:
2/1~ N
“Iroshnikov-Kraichnan” scaling is reproduced for α=1:
2/1~ l
Explainsnumerically observed scalings for strong B-field !
[Maron & Goldreich, ApJ 554, 1175, 2001]
[Müller, Biskamp, Grappin PRE, 67, 066302, 2003]
[S.B., ApJ, 626, L37,2005]
14
New Model for MHD TurbulenceGeometric Meaning
2/1~ l
S.B. (2005) “eddy”:
4/3~ line displacement:
3/2~ l
Goldreich-Sridhar 1995 “eddy”:
~line displacement:
As the scale decreases, λ→0,
turns into filament
turns into current sheet
agrees with numerics!
15
New Model for MHD Turbulence
4/3~
4/1~
Depletion of nonlinearity
S.B. (2005) “eddy”:
line displacement:
Remarkably, we reproduced the reduction factor in the original formula:
θ
In our “eddy”, w and z are aligned within small angle . One can check that: θ
In our theory, this angle is:
2/1~ l
The theory is self-consistent.
λ
16
Summary and Discussions
3. The spectrum of MHD turbulence may be non-universal.Alternatively, it may always be E~K , but in case 1, resolution of numerical simulations is not large enough to observe it.
┴-3/2
2VB 1. Weak large-scale field:
~dissipative structures: filaments
energy spectrum: E(K)~K ┴-5/3
3/2~ l[Goldreich & Sridhar’ 95]
2/1~ l4/3~
2VB 2. Strong large-scale field:
dissipative structures: current sheets
energy spectrum: E(K)~K ┴-3/2
4/1~ scale-dependentdynamic alignment
17
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 is E~K . 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.
┴-3/2 -3/2
┴
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