1. Spectrum of MHD turbulence
Stanislav Boldyrev
University of Chicago
(June 20, 2005)
Ref: astro-ph/ ; ApJ 626, L37, 2005

2. Introduction: Kolmogorov turbulence
Random flow of incompressible fluid
Reynolds number: v L
Re=Lv/η>>1
η-viscosity
If there is no intermittency, then:
and
Kolmogorov spectrum [Kolmogorov 1941]

3. Kolmogorov energy cascade
local energy flux
Energy of an eddy of size is ;
it is transferred to a smaller-size eddy during time:
- “eddy turn-over” time.
The energy flux,
, is constant for the
Kolmogorov spectrum!

4. MHD turbulence ? No exact Kolmogorov relation. Phenomenology:
is conserved, and cascades
toward small scales.
Energy
No, since dimensional
arguments do not work! ?
Is energy transfer time
Non-dimensional parameter
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, ’s]

5. Iroshnikov-Kraichnan spectrumw z w z
After interaction, shape of each packet changes, but energy does not.

6. Iroshnikov-Kraichnan spectrum
during one collision:
number of collisions required to
deform packet considerably: λ λ
Constant energy flux:
[Iroshnikov (1963); Kraichnan (1965)]

7. Goldreich-Sridhar theory
Anisotropy of “eddies” B λ L λ
L>>
Shear Alfvén waves
dominate the cascade: ┴ B
Critical Balance
[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, 2000
Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002

10. Strong Magnetic Filed, Numerics
Contradictions with Goldreich-Sridhar model
Iroshnikov-Kraichnan
scaling
[Maron & Goldreich, ApJ 554, 1175, 2001]

11. Strong Magnetic Filed, Numerics
Contradictions with Goldreich-Sridhar model
B-parallel scaling
B-perp scaling
Scaling of field-parallel and
field-perpendicular structure
functions for different
large-scale magnetic fields.
[Müller, Biskamp, Grappin
PRE, 67, , 2003]
Weak field, B→0:
Goldreich-Sridhar (Kolmogorov)
scaling 2
Strong field, B>>ρV :
Iroshnikov-Kraichnan scaling

12. 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 and in the MHD equations, as in the
Goldreich-Sridhar picture, however, the geometric meaning is different. 1 2

13. 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:
Explains
numerically
observed scalings
for strong B-field !
“Iroshnikov-Kraichnan” scaling is reproduced for
α=1:
[Maron & Goldreich, ApJ 554, 1175, 2001]
[Müller, Biskamp, Grappin PRE, 67, , 2003]

14. New Model for MHD Turbulence
Geometric Meaning
Goldreich-Sridhar 1995 “eddy”:
line displacement:
S.B. (2005) “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
Depletion of nonlinearity
S.B. (2005) “eddy”:
line displacement:
In our “eddy”, w and z are aligned within small angle One can check that: θ λ θ
In our theory, this angle is:
Remarkably, we reproduced the
reduction factor in the original formula:
The theory is self-consistent.

16. Summary and Discussions
1. Weak large-scale field:
dissipative structures: filaments
energy spectrum: E(K)~K ┴
-5/3
[Goldreich & Sridhar’ 95]
2. Strong large-scale field:
dissipative structures: current sheets
energy spectrum: E(K)~K ┴
-3/2
scale-dependent
dynamic alignment
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

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 ┴ ┴