Bounds for the number of degrees of freedom of incompressible magnetohydrodynamic turbulence in two and three dimensions

Chuong Van Tran, Xinwei Yu

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We study incompressible magnetohydrodynamic turbulence in both two and
three dimensions, with an emphasis on the number of degrees of freedom
$N$. This number is estimated in terms of the magnetic Prandtl number
$\P$, kinetic Reynolds number $\Re$, and magnetic Reynolds number
$\Rm$. Here, $\Re$ and $\Rm$ are dynamic in nature, defined
in terms of the kinetic and magnetic energy dissipation rates (or
averages of the velocity and magnetic field gradients), viscosity and
magnetic diffusivity, and the system size. It is found that for the
two-dimensional case, $N$ satisfies $N\le\P\,\Re^{3/2}+\Rm^{3/2}$
for $\P>1$ and $N\le \Re^{3/2}+\P^{-1}\,\Rm^{3/2}$ for $\P\le1$.
In three dimensions, on the other hand, $N$ satisfies
$N\le\left(\P\,\Re^{3/2}+\Rm^{3/2}\right)^{3/2}$ for $\P>1$ and
$N\le\left(\Re^{3/2}+\P^{-1}\,\Rm^{3/2}\right)^{3/2}$ for
$\P\le1$. In the limit $\P\to0$, $\Re^{3/2}$ dominates
$\P^{-1}\,\Rm^{3/2}$, and the present estimate for $N$ appropriately
reduces to $\Re^{9/4}$ as in the case of usual Navier--Stokes
turbulence. For $\P\approx1$, our results imply the classical spectral
scaling of the energy inertial range and dissipation wave number
(in the form of upper bounds). These bounds are consistent with existing
predictions in literature for turbulence with or without Alfv\'en wave
effects. We discuss the possibility of solution regularity, with an
emphasis on the two-dimensional case in the absence of either one or
both of the dissipation terms.
Original languageEnglish
Article number066323
Number of pages9
JournalPhysical Review. E, Statistical, nonlinear, and soft matter physics
Volume85
Issue number6
DOIs
Publication statusPublished - 28 Jun 2012

Keywords

  • Dynamical Systems Theory, Magnetohydrodynamic Turbulence, Number of Degrees of Freedom

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