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