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dc.contributor.authorTran, Chuong Van
dc.contributor.authorYu, Xinwei
dc.contributor.authorBlackbourn, Luke Austen Kazimierz
dc.identifier.citationTran , C V , Yu , X & Blackbourn , L A K 2013 , ' Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number ' Journal of Fluid Mechanics , vol 725 , pp. 195-215 . DOI: 10.1017/jfm.2013.193en
dc.identifier.otherPURE: 47133330
dc.identifier.otherPURE UUID: 3fc6541f-7d0a-47b3-a57e-b474a9bc060b
dc.descriptionLAKB was supported by an EPSRC post-graduate studentship.en
dc.description.abstractWe 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.en
dc.relation.ispartofJournal of Fluid Mechanicsen
dc.rights(c) Cambridge University Press 2013. 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.en
dc.subjectMagnetohydrodynamic Turbulenceen
dc.subjectSolution regularityen
dc.subjectQA Mathematicsen
dc.titleTwo-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl numberen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews. Applied Mathematicsen
dc.description.statusPeer revieweden

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