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