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 dc.contributor.author Tran, Chuong Van dc.contributor.author Yu, Xinwei dc.contributor.author Blackbourn, Luke Austen Kazimierz dc.date.accessioned 2013-05-20T16:01:02Z dc.date.available 2013-05-20T16:01:02Z dc.date.issued 2013-06 dc.identifier.citation Tran , 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.2013.193 en dc.identifier.issn 0022-1120 dc.identifier.other PURE: 47133330 dc.identifier.other PURE UUID: 3fc6541f-7d0a-47b3-a57e-b474a9bc060b dc.identifier.other WOS: 000319511200008 dc.identifier.other Scopus: 84880199193 dc.identifier.uri http://hdl.handle.net/10023/3539 dc.description LAKB was supported by an EPSRC post-graduate studentship. en dc.description.abstract 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. en dc.format.extent 21 en dc.language.iso eng dc.relation.ispartof Journal of Fluid Mechanics en 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.subject Magnetohydrodynamic Turbulence en dc.subject Solution regularity en dc.subject QA Mathematics en dc.subject.lcc QA en dc.title Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number en dc.type Journal article en dc.description.version Publisher PDF en dc.contributor.institution University of St Andrews. Applied Mathematics en dc.identifier.doi http://dx.doi.org/10.1017/jfm.2013.193 dc.description.status Peer reviewed en