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- W2149274504 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 jVert ^{2} $ , where $omega $ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $Vert boldsymbol{nabla} jVert ^{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} jVert ^{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 jVert _{infty } / Vert jVert $ 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 jVert _{infty } / Vert jVert $ ), 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 jVert ^{2} $ is observed to decrease rapidly as $mathit{Pm}$ is increased. This result suggests the possibility $Vert jVert ^{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 jVert _{infty } / Vert jVert $ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields." @default.
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- W2149274504 date "2013-05-14" @default.
- W2149274504 modified "2023-10-18" @default.
- W2149274504 title "Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number" @default.
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- W2149274504 doi "https://doi.org/10.1017/jfm.2013.193" @default.
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