## Abstract

We study both theoretically and numerically two-dimensional

magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl

number $Pm$ (and the limits thereof), with an emphasis on solution

regularity. For $Pm=0$, both $\norm{\omega}^2$ and $\norm{j}^2$,

where $\omega$ and $j$ are, respectively, the vorticity and current,

are uniformly bounded. Furthermore, $\norm{\nabla j}^2$ is integrable

over $[0,\infty)$. The uniform boundedness of $\norm{\omega}^2$ implies

that in the presence of vanishingly small viscosity $\nu$ (i.e. in

the limit $Pm\to0$), the kinetic energy dissipation rate

$\nu\norm{\omega}^2$ vanishes for all times $t$, including $t=\infty$.

Furthermore, for sufficiently small $Pm$, this rate decreases linearly

with $Pm$. This linear behaviour of $\nu\norm{\omega}^2$ is investigated

and confirmed by high-resolution simulations with $Pm$ in the range

$[1/64,1]$. Several criteria for solution regularity are established

and numerically tested. As $Pm$ is decreased from unity, the ratio

$\norm{\omega}_\infty/\norm{\omega}$ is observed to increase relatively

slowly. This, together with the integrability of $\norm{\nabla j}^2$,

suggests global regularity for $Pm=0$. When $Pm=\infty$, global

regularity is secured when either $\norm{\nabla\u}_\infty/\norm{\omega}$,

where $\u$ is the fluid velocity, or $\norm{j}_\infty/\norm{j}$ is

bounded. The former is plausible given the presence of viscous effects

for this case. Numerical results over the range $Pm\in[1,64]$ show that

$\norm{\nabla\u}_\infty/\norm{\omega}$ varies slightly (with similar

behaviour for $\norm{j}_\infty/\norm{j}$), thereby lending strong

support for the possibility

$\norm{\nabla\u}_\infty/\norm{\omega}<\infty$ in the limit $Pm\to\infty$.

The peak of the magnetic energy dissipation rate $\mu\norm{j}^2$ is

observed to decrease rapidly as $Pm$ is increased. This result suggests

the possibility $\norm{j}^2<\infty$ in the limit $Pm\to\infty$. We

discuss further evidence for the boundedness of the ratios

$\norm{\omega}_\infty/\norm{\omega}$,

$\norm{\nabla\u}_\infty/\norm{\omega}$ and $\norm{j}_\infty/\norm{j}$

in conjunction with observation on the density of filamentary structures

in the vorticity, velocity gradient and current fields.

magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl

number $Pm$ (and the limits thereof), with an emphasis on solution

regularity. For $Pm=0$, both $\norm{\omega}^2$ and $\norm{j}^2$,

where $\omega$ and $j$ are, respectively, the vorticity and current,

are uniformly bounded. Furthermore, $\norm{\nabla j}^2$ is integrable

over $[0,\infty)$. The uniform boundedness of $\norm{\omega}^2$ implies

that in the presence of vanishingly small viscosity $\nu$ (i.e. in

the limit $Pm\to0$), the kinetic energy dissipation rate

$\nu\norm{\omega}^2$ vanishes for all times $t$, including $t=\infty$.

Furthermore, for sufficiently small $Pm$, this rate decreases linearly

with $Pm$. This linear behaviour of $\nu\norm{\omega}^2$ is investigated

and confirmed by high-resolution simulations with $Pm$ in the range

$[1/64,1]$. Several criteria for solution regularity are established

and numerically tested. As $Pm$ is decreased from unity, the ratio

$\norm{\omega}_\infty/\norm{\omega}$ is observed to increase relatively

slowly. This, together with the integrability of $\norm{\nabla j}^2$,

suggests global regularity for $Pm=0$. When $Pm=\infty$, global

regularity is secured when either $\norm{\nabla\u}_\infty/\norm{\omega}$,

where $\u$ is the fluid velocity, or $\norm{j}_\infty/\norm{j}$ is

bounded. The former is plausible given the presence of viscous effects

for this case. Numerical results over the range $Pm\in[1,64]$ show that

$\norm{\nabla\u}_\infty/\norm{\omega}$ varies slightly (with similar

behaviour for $\norm{j}_\infty/\norm{j}$), thereby lending strong

support for the possibility

$\norm{\nabla\u}_\infty/\norm{\omega}<\infty$ in the limit $Pm\to\infty$.

The peak of the magnetic energy dissipation rate $\mu\norm{j}^2$ is

observed to decrease rapidly as $Pm$ is increased. This result suggests

the possibility $\norm{j}^2<\infty$ in the limit $Pm\to\infty$. We

discuss further evidence for the boundedness of the ratios

$\norm{\omega}_\infty/\norm{\omega}$,

$\norm{\nabla\u}_\infty/\norm{\omega}$ and $\norm{j}_\infty/\norm{j}$

in conjunction with observation on the density of filamentary structures

in the vorticity, velocity gradient and current fields.

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

Pages (from-to) | 195-215 |

Number of pages | 21 |

Journal | Journal of Fluid Mechanics |

Volume | 725 |

Early online date | 14 May 2013 |

DOIs | |

Publication status | Published - Jun 2013 |

## Keywords

- Magnetohydrodynamic Turbulence
- Solution regularity