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

Abstract

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.