On energetics and inertial-range scaling laws of two-dimensional magnetohydrodynamic turbulence

Luke Austen Kazimierz Blackbourn, Chuong Van Tran

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We study two-dimensional magnetohydrodynamic turbulence, with an
emphasis on its energetics and inertial range scaling laws. A detailed
spectral analysis shows that dynamo triads (those converting kinetic
into magnetic energy) are associated with a direct magnetic energy
flux while anti-dynamo triads (those converting magnetic into kinetic
energy) are associated with an inverse magnetic energy flux. As both
dynamo and anti-dynamo interacting triads are integral parts of the
direct energy transfer, the anti-dynamo inverse flux partially
neutralizes the dynamo direct flux, arguably resulting in relatively
weak direct energy transfer and giving rise to dynamo saturation. This
result is consistent with a qualitative prediction of energy transfer
reduction owing to Alfv\'en wave effects by the Iroshnikov--Kraichnan
theory (which was originally formulated for magnetohydrodynamic
turbulence in three dimensions). We numerically confirm the correlation
between dynamo action and direct magnetic energy flux and investigate
the applicability of quantitative aspects of the Iroshnikov--Kraichnan
theory to the present case, particularly its predictions of energy
equipartition and $k^{-3/2}$ spectra in the energy inertial range.
It is found that for turbulence satisfying the Kraichnan condition
of magnetic energy at large scales exceeding total energy in the
inertial range, the kinetic energy spectrum, which is significantly
shallower than $k^{-3/2}$, is shallower than its magnetic counterpart.
This result suggests no energy equipartition. The total energy spectrum
appears to depend on the energy composition of the turbulence but is
clearly shallower than $k^{-3/2}$ for $r\approx2$, even at moderate
resolutions. Here $r\approx2$ is the magnetic-to-kinetic energy ratio
during the stage when the turbulence can be considered fully developed.
The implication of the present findings is discussed in conjunction with
further numerical results on the dependence of the energy dissipation
rate on resolution.
Original languageEnglish
Pages (from-to)238-254
Number of pages17
JournalJournal of Fluid Mechanics
Early online date12 Jun 2012
Publication statusPublished - Jul 2012


  • Magnetohydrodynamic Turbulence


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