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Spectrum and small-scale
structures in MHD turbulence
Joanne Mason, CMSO/University of Chicago
Stanislav Boldyrev, CMSO/University of Madison at Wisconsin
Fausto Cattaneo, CMSO/University of Chicago
MHD Turbulence in the ISM
Kolmogorov~ k 5 / 3
• Statistical properties of MHD turbulence essential for
theoretical understanding of star-forming regions in the
ISM
best fit ~ k 3 / 2
• Pulsar signals exhibit scintillation  spectrum of the
interstellar electron density. Density fluctuations are a
tracer of the main turbulent energies.
• Phase structure function for PSR J0437-4715 and PSR
B0329+54 [1,2] yield a power law spectrum with
exponent different from Kolmogorov.
Taken from Shishov et al. [2]
[1] Smirnova et al. astro-ph/0603490.
[2] Shishov et al. A&A, 404, 557 (2003)
Incompressible MHD Turbulence
 t z   VA  z   z   z   P, z   v  b
Iroshnikov [1], Kraichnan [2]:
e.g. Maron & Goldreich [4]
..
• Isotropic
• Weak interactions:  l ~ lV A / vl
•E
• Confirm anisotropy but yield E k   k 3 / 2,
• Muller et al. [5] suggest anisotropic
spectrum depends on | B 0 |
k   k 3/ 2
parallel
perpendicular
zl
Goldreich & Sridhar [3]:
• Anisotropic: along
 k
5 / 3
~l
p
B0
• Critical balance: l / l ~ VA / vl
• E k
p
, l ~ l 2/3
B0  0, 5,10 (circles,
.. diamonds,triangles)
Taken from Muller et al [2].
• PSR J0437-4715 and PSR B0329+54
Dynamic alignment provides an explanation for these findings
[1] Iroshnikov. Soviet. Astron. 7, 566 (1964); [2] Kraichnan. Phys. Fluids, ..8, 1385 (1965); [3] Goldreich & Sridhar. ApJ,
438, 763 (1995); [4] Maron & Goldreich, ApJ, 554, 1175 (2001); [5] Muller et al. Phys. Rev. E, 67, 066302 (2003)
Theory of Polarization alignment
E


1
2
2
3
b

v
d
x

2
H C   v  b d 3 x
Decaying MHD turbulence:
• Free decaying MHD turbulence evolves towards the perfectly aligned
configuration b   v (Alfvenization effect [1-3]).
• Such configurations are very long-lived, being subject only to dissipation.
The nonlinear interaction terms ( z   z ) vanish for perfectly aligned
fluctuations.
Driven MHD turbulence:
• The energy cascade toward small scales must be maintained by the
nonlinear terms.
• Propose that the magnetic and velocity field fluctuations become aligned
within a scale dependent angle   .
• The turbulent eddies are locally anisotropic in the field perpendicular
plane.
[1] Dobrowolny et al. Phys. Rev. Lett. 45,144, (1980); [2] Grappin et al A&A,105,6 (1982); [3] Pouquet et al Phys. Rev. A, 33, 4266 (1986).
Alignment in Driven MHD turbulence

l ~ 2 /(3a )
• Scale dependent depletion of the nonlinear interaction.
The energy transfer time is increased   ~  / v 1 /   

• Assume fluctuations are aligned within a small angle
in the field perpendicular plane
0
v
b
• If
 ~ 3 /(3a )
   a / 3a  then constant energy flux 
E k   k
, E k   k
 5a  / 3a 
• Need to determine a a0, a1?
2
The value of a
•Conservation of cross helicity: minimize the total alignment  a=1, i.e.
 ~  ,
1/ 4
E k   k
3 / 2
,
l  1/ 2 ~  2 / 3
Testing the Theory: Numerical Results
• Moderate spatial resolution makes identification of the scaling law for the
energy spectrum difficult.
B x 
• However, angular alignment is realizable: ~
v   v  v   n n, n 
B x 
~
  sin    
~v   b
~
~v  b
~ 1/ 4
cos  
u b
ub
slope =0.25
  
k  10, B0  5, Re,m  800
scale, 
( B0  10, Re  Rm  800)
Testing the Theory: Exact relations
vL3 r    r  E k   k 5 / 3
4
5
• Isotropic magnetohydrodynamic turbulence (Politano & Pouquet [1])
• Hydrodynamic turbulence
z L w 2    w r
4
3
 z  v  b, w  v  b 
• Scale dependent dynamic alignment yields z L w 2  rvr3 ,
z L w 2 / w 2 ~  r
zL ~ v, w ~ v
[1] Politano & Pouquet, Geophys. Res. Lett., 25, 273 (1998)
Conclusions
..
• Magnetic and velocity field fluctuations become dynamically aligned
• Eddies are three-dimensionally anisotropic: ribbon-like dissipative structures rather
than filaments
• Perpendicular energy spectrum E k
 k
3 / 2
• Recover consistency with Politano & Pouquet relations
• Electron density fluctuations behave like a passive scalar expect energy spectrum
with exponent -3/2 and sheet-like eddy structure.
References
[1] Boldyrev, S. (2005) Astrophys. J. 626, L37.
[2] Boldyrev, S. (2006) Phys. Rev. Lett. 96, 115002.
[3] Mason, J., Cattaneo, F. & Boldyrev, S. Phys. Rev. Lett. submitted; astro-ph/0602382.
[4] Boldyrev, S., Mason, J. & Cattaneo, F. Phys. Rev. Lett. submitted; astro-ph/0605233.
Acknowledgement: This work is supported by the NSF Center for Magnetic Self-Organization in Laboratory and
Astrophysical Plasmas at the University of Chicago and the University of Wisconsin at Madison.
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