## 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.

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 language | English |
---|---|

Article number | 066323 |

Number of pages | 9 |

Journal | Physical Review. E, Statistical, nonlinear, and soft matter physics |

Volume | 85 |

Issue number | 6 |

DOIs | |

Publication status | Published - 28 Jun 2012 |

## Keywords

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