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摘要
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 parallel to omega parallel to(2) and parallel to j parallel to(2), where omega and j are, respectively, the vorticity and current, are uniformly bounded. Furthermore, parallel to del j parallel to(2) is integrable over [0, infinity). The uniform boundedness of parallel to omega parallel to(2) implies that in the presence of vanishingly small viscosity nu (i.e. in the limit Pm -%26gt; 0), the kinetic energy dissipation rate nu parallel to omega parallel to(2) vanishes for all times t, including t = infinity. Furthermore, for sufficiently small P m, this rate decreases linearly with Pm. This linear behaviour of nu parallel to omega parallel to(2) is investigated and confirmed by high-resolution simulations with P m in the range [1/64, 1]. Several criteria for solution regularity are established and numerically tested. As Pm is decreased from unity, the ratio parallel to omega parallel to(infinity) /parallel to omega parallel to is observed to increase relatively slowly. This, together with the integrability of parallel to del j parallel to(2), suggests global regularity for Pm = 0. When Pm = infinity, global regularity is secured when either parallel to del u parallel to(infinity)/parallel to omega parallel to(2), where u is the fluid velocity, or parallel to j parallel to(infinity)/parallel to j parallel to is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range Pm is an element of [1, 64] show that parallel to del u parallel to(infinity)/parallel to omega parallel to varies slightly (with similar behaviour for parallel to j parallel to(infinity)/parallel to j parallel to), thereby lending strong support for the possibility parallel to del u parallel to(infinity)/parallel to omega parallel to %26lt; infinity in the limit Pm -%26gt; infinity. The peak of the magnetic energy dissipation rate mu parallel to j parallel to(2) is observed to decrease rapidly as Pm is increased. This result suggests the possibility parallel to j parallel to(2) %26lt; infinity in the limit Pm -%26gt; infinity. We discuss further evidence for the boundedness of the ratios parallel to omega parallel to(infinity)/parallel to omega parallel to, parallel to del u parallel to(infinity)/parallel to omega parallel to and parallel to j parallel to(infinity)/parallel to j parallel to in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.
- 出版日期2013-6
