Spectrum and small-scalestructures 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 fortheoretical understanding of star-forming regions in theISM

• Pulsar signals exhibit scintillation spectrum of theinterstellar electron density. Density fluctuations are atracer of the main turbulent energies.

• Phase structure function for PSR J0437-4715 and PSRB0329+54 [1,2] yield a power law spectrum withexponent 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

Iroshnikov [1], Kraichnan [2]:

• Isotropic

• Weak interactions:

•

Goldreich & Sridhar [3]:

• Anisotropic: along

• Critical balance:

• ,

Dynamic alignment provides an explanation for these findings

• Confirm anisotropy but yield ,e.g. Maron & Goldreich [4]

• Muller et al. [5] suggest anisotropicspectrum depends on

• PSR J0437-4715 and PSR B0329+54

plzp

l

!" ~

parallel

perpendicular

)trianglesdiamonds,circles,(10,5,00 =B

Taken from Muller et al [2].

[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 alignedconfiguration (Alfvenization effect [1-3]).

• Such configurations are very long-lived, being subject only to dissipation.The nonlinear interaction terms ( ) vanish for perfectly alignedfluctuations.

Theory of Polarization alignment

Driven MHD turbulence:

• The energy cascade toward small scales must be maintained by thenonlinear terms.

• Propose that the magnetic and velocity field fluctuations become alignedwithin a scale dependent angle .

• The turbulent eddies are locally anisotropic in the field perpendicularplane.

[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).

• Assume fluctuations are aligned within a small anglein 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 α (α=0, α=1?)

b

v

Alignment in Driven MHD turbulence

0

!

l

)3/(3~ !" +

)3/(2~ !" +

•Conservation of cross helicity: minimize the total alignment α=1, i.e.

The value of α

• Moderate spatial resolution makes identification of the scaling law for theenergy spectrum difficult.

• However, angular alignment is realizable:

Testing the Theory: Numerical Results

( )!"

!,scale)800,10( 0 ===

meRRB

slope =0.25

( )bu

bu !="cos

800,5,10,0==>

meRBk

• Hydrodynamic turbulence

• Isotropic magnetohydrodynamic turbulence (Politano & Pouquet [1])

• Scale dependent dynamic alignment yields

Testing the Theory: Exact relations

vwvzL

!!"!! ~,~

[1] Politano & Pouquet, Geophys. Res. Lett., 25, 273 (1998)

• Magnetic and velocity field fluctuations become dynamically aligned

• Eddies are three-dimensionally anisotropic: ribbon-like dissipative structures ratherthan filaments

• Perpendicular energy spectrum

• Recover consistency with Politano & Pouquet relations

• Electron density fluctuations behave like a passive scalar expect energy spectrumwith 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 andAstrophysical Plasmas at the University of Chicago and the University of Wisconsin at Madison.

The End