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
• Statistical properties of MHD turbulence essential for theoretical understanding of star-forming regions in the ISM
• 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.
MHD Turbulence in the ISM
[1] Smirnova et al. astro-ph/0603490. [2] Shishov et al. A&A, 404, 557 (2003)
Taken from Shishov et al. [2]
2/3~fitbest k
3/5~Kolmogorov k
Incompressible MHD Turbulence
bvzzzzVz ,PAt
Iroshnikov [1], Kraichnan [2]:
• Isotropic
• Weak interactions:
• 2/3 kkElAl vlV /~
Goldreich & Sridhar [3]:
• Anisotropic: along
• Critical balance:
• , 3/5 kkE
lA vVll /~/
3/2~ ll
0B
Dynamic alignment provides an explanation for these findings
• Confirm anisotropy but yield , e.g. Maron & Goldreich [4]
• Muller et al. [5] suggest anisotropic spectrum depends on
• PSR J0437-4715 and PSR B0329+54
2/3 kkE
plzp
l ~
parallel
perpendicular
)trianglesdiamonds,circles,(10,5,00 BTaken from Muller et al [2].
..
|| 0B
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[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)
..
Decaying MHD turbulence:
• Free decaying MHD turbulence evolves towards the perfectly aligned configuration (Alfvenization effect [1-3]).
• Such configurations are very long-lived, being subject only to dissipation. The nonlinear interaction terms ( ) vanish for perfectly aligned fluctuations.
vb
Theory of Polarization alignment
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.
xdvbE 322
2
1 xdH C 3bv
[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).
zz
• Assume fluctuations are aligned within a small angle in the field perpendicular plane
• Scale dependent depletion of the nonlinear interaction. The energy transfer time is increased
• If then constant energy flux
• Need to determine
3/
bv
Alignment in Driven MHD turbulence
0
23/5 , kkEkkE
/1/~ v
l
)3/(3~
)3/(2~
•Conservation of cross helicity: minimize the total alignment =1, i.e.
The value of
,~ 4/1 ,2/3 kkE
3/22/1 ~ l
• Moderate spatial resolution makes identification of the scaling law for the energy spectrum difficult.
• However, angular alignment is realizable:
Testing the Theory: Numerical Results
4/1~~~
~~sin
bv
bv
,scale)800,10( 0 me RRB
slope =0.25
bu
bucos
800,5,10 ,0 meRBk
xx
B
Bnnnvvv ,~
• Hydrodynamic turbulence
• Isotropic magnetohydrodynamic turbulence (Politano & Pouquet [1])
• Scale dependent dynamic alignment yields
Testing the Theory: Exact relations
rz wL
3
42 w bvwbvz ,
,32rrL vz w
3/53
5
4 kkErvL r
vwvzL ~,~
[1] Politano & Pouquet, Geophys. Res. Lett., 25, 273 (1998)
rLz ~/ 22 ww
• Magnetic and velocity field fluctuations become dynamically aligned
• Eddies are three-dimensionally anisotropic: ribbon-like dissipative structures rather than filaments
• Perpendicular energy spectrum
• 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.
Conclusions
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.
2/3 kkE
The End