Spectrum and small-scale structures in MHD turbulence Joanne Mason, CMSO/University of Chicago

1. 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

2. Taken from Shishov et al. [2] MHD Turbulence in the ISM • 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. [1] Smirnova et al. astro-ph/0603490. [2] Shishov et al. A&A, 404, 557 (2003)

3. 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 • Iroshnikov [1], Kraichnan [2]: • Isotropic • Weak interactions: parallel perpendicular • Goldreich & Sridhar [3]: • Anisotropic: along • Critical balance: • , 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) Incompressible MHD Turbulence Dynamic alignment provides an explanation for these findings

4. 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. Theory of Polarization alignment • 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. [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).

5. Alignment in Driven MHD turbulence • 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 a (a=0, a=1?)

6. The value of a • Conservation of cross helicity: minimize the total alignment  a=1, i.e.

7. slope =0.25 Testing the Theory: Numerical Results • Moderate spatial resolution makes identification of the scaling law for the energy spectrum difficult. • However, angular alignment is realizable:

8. Testing the Theory: Exact relations • Hydrodynamic turbulence • Isotropic magnetohydrodynamic turbulence (Politano & Pouquet [1]) • Scale dependent dynamic alignment yields [1] Politano & Pouquet, Geophys. Res. Lett., 25, 273 (1998)

9. Conclusions • 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. 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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