Alex Lazarian Collaboration: Andrey Beresnyak

Astronomy Department, University of Wisconsin-Madison and Center for Magnetic Self-Organization (CMSO)

Imbalanced MHD Turbulence

Based on Beresnyak & Lazarian 2008, ApJ, 682, 1070 Beresnyak & Lazarian 2009, ApJ, 701, 000

It is difficult for weak wave to strongly cascade strong wave

Our model includes weak cascading for strong wave and strong cascading for weak wave.

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B

w+ w-

We use Elsasser variables

“Propagation balance” substitutes GS95 “critical balance” for imbalanced turbulence

Shearing rate

Propagation rate=

B

Anisotropic eddy

For the case of turbulence with more energy moving in one direction (imbalanced MHD turbulence) Beresnyak & Lazarian 08 proposed field wandering “propagation balance”.For balanced turbulence the angle of field wondering at perpendicular scale

GS95 critical balance:

~1

parallel scale corresponding to perpendicular scale is

which coincides with GS95 critical balance.

Application of the “propagation balance” allows to find the longitudinal scale of the strong wave

Strong wave with perpendicular scale induces strong cascading of the weak wave at the scale

Weak wave cascades strong weakly and the cascading involves interactions of the waves with the same parallel scale, as this is the feature of weak turbulence. The rms angle of emerging eddies is . Thus

Strong wave cascades from to

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New “propagation balance”

Strong cascadingof weak wave

Weak cascading of strong wave

shear rateenergy

Old critical balance

weakening factor

Asymptotic power-law solutions:

;

A model of strong imbalanced turbulence includes “balances” and rates of cascading

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dominant subdominant

Numerical simulations show difference in anisotropies

W+ W-

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different anisotropies!

Numerical data roughly agrees with our model

Our simulations require long times to achieve the steady state

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Energy spectra of w

's

wavevector1 110 10100 100

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Energy spectra of w

's

wavevector1 110 10100 100

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Imbalanced turbulence

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1. the anisotropies are different and the strong wave anisotropy is smaller;2. subdominant wave eddies are aligned with respect to the local field, while dominant wave eddies are aligned with respect to larger-scale field;3. the energy imbalance is higher than in the case when both waves are cascaded strongly, which suggests that dominant wave is cascaded weakly4. the inertial range of the dominant wave is shorter5. there is no “pinning” on dissipation scale, which suggest nonlocal cascading

Our model correctly reproduce the following features of simulations:

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Models with local cascading disagree with simulations

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B0=10 B0=1

balanced

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Imbalanced cascade is different from GS cascade

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Anisotropy calculated from SF's (mapping)

Balanced turbulence

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GS95 model assumes for strong turbulence which parallel scale is determined by uncertainty relation between cascading timescale and wave frequency: cas~1, i.e. ~vA/w.

For weak interaction ~const with cascading (and only waves with equal interact). Thus for strong wave ~const, but for weak wave is decreasing, cascade stops. Is stationary imbalanced turbulence possible?

w+ w-

w+ w-

Coexisting of weak and strong cascading is not trivial