Spectrum of MHD turbulence

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Spectrum of MHD turbulence
Stanislav Boldyrev
University of Chicago
(June 20, 2005)
Ref astro-ph/0503053 ApJ 626, L37, 2005
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Introduction Kolmogorov turbulence
Random flow of incompressible fluid
Reynolds number
v
L
ReLv/?gtgt1
?-viscosity
If there is no intermittency, then
and
Kolmogorov spectrum Kolmogorov 1941
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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.
, is constant for the
The energy flux,
Kolmogorov spectrum!
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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, ... 1960s
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Iroshnikov-Kraichnan spectrum
After interaction, shape of each packet changes,
but energy does not.
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Iroshnikov-Kraichnan spectrum
during one collision
number of collisions required to deform packet
considerably
?
?
Constant energy flux
Iroshnikov (1963) Kraichnan (1965)
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Goldreich-Sridhar theory
Anisotropy of eddies
?
B
L
Shear Alfvén waves dominate the cascade
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
Scaling of field-parallel and field-perpendicular
structure functions for different large-scale
magnetic fields.
Müller, Biskamp, Grappin PRE, 67, 066302, 2003
Weak field, B?0 Goldreich-Sridhar
(Kolmogorov) scaling
2
Strong field, Bgtgt?V Iroshnikov-Kraichnan
scaling
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New Model for MHD Turbulence
S.B., ApJ, 626, L37,2005
Analytic Introduction
Depletion of nonlinear interaction
Nonlinear interaction is depleted
Interaction time is increased
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,
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
As the scale decreases, ??0,
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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
?
?
?
In our theory, this angle is
Remarkably, we reproduced the reduction factor
in the original formula
The theory is self-consistent.
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Summary and Discussions
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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
• is EK . It is quite possible that
  spectrum is always EK ,
• but for weak large-scale field, the resolution
  of numerical
• simulations is not large enough to observe it.
  

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