We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number
$\mathit{Pm}$
(and the limits thereof), with an emphasis on solution regularity. For
$\mathit{Pm}= 0$
, both
$\Vert \omega \Vert ^{2} $
and
$\Vert j\Vert ^{2} $
, where
$\omega $
and
$j$
are, respectively, the vorticity and current, are uniformly bounded. Furthermore,
$\Vert \boldsymbol{\nabla} j\Vert ^{2} $
is integrable over
$[0, \infty )$
. The uniform boundedness of
$\Vert \omega \Vert ^{2} $
implies that in the presence of vanishingly small viscosity
$\nu $
(i.e. in the limit
$\mathit{Pm}\rightarrow 0$
), the kinetic energy dissipation rate
$\nu \Vert \omega \Vert ^{2} $
vanishes for all times
$t$
, including
$t= \infty $
. Furthermore, for sufficiently small
$\mathit{Pm}$
, this rate decreases linearly with
$\mathit{Pm}$
. This linear behaviour of
$\nu \Vert \omega \Vert ^{2} $
is investigated and confirmed by high-resolution simulations with
$\mathit{Pm}$
in the range
$[1/ 64, 1] $
. Several criteria for solution regularity are established and numerically tested. As
$\mathit{Pm}$
is decreased from unity, the ratio
$\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $
is observed to increase relatively slowly. This, together with the integrability of
$\Vert \boldsymbol{\nabla} j\Vert ^{2} $
, suggests global regularity for
$\mathit{Pm}= 0$
. When
$\mathit{Pm}= \infty $
, global regularity is secured when either
$\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $
, where
$\boldsymbol{u}$
is the fluid velocity, or
$\Vert j\Vert _{\infty } / \Vert j\Vert $
is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range
$\mathit{Pm}\in [1, 64] $
show that
$\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $
varies slightly (with similar behaviour for
$\Vert j\Vert _{\infty } / \Vert j\Vert $
), thereby lending strong support for the possibility
$\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert \lt \infty $
in the limit
$\mathit{Pm}\rightarrow \infty $
. The peak of the magnetic energy dissipation rate
$\mu \Vert j\Vert ^{2} $
is observed to decrease rapidly as
$\mathit{Pm}$
is increased. This result suggests the possibility
$\Vert j\Vert ^{2} \lt \infty $
in the limit
$\mathit{Pm}\rightarrow \infty $
. We discuss further evidence for the boundedness of the ratios
$\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $
,
$\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $
and
$\Vert j\Vert _{\infty } / \Vert j\Vert $
in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.