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

  • Chuong V. Tran (a1) (a2), Xinwei Yu (a1) (a2) and Luke A. K. Blackbourn (a1) (a2)
Abstract
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.

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The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence . The written permission of Cambridge University Press must be obtained for commercial re-use.
Corresponding author
Email address for correspondence: chuong@mcs.st-and.ac.uk
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Journal of Fluid Mechanics
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