Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

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.