Tran Chuong V, Yu Xinwei
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066323. doi: 10.1103/PhysRevE.85.066323. Epub 2012 Jun 28.
We study incompressible magnetohydrodynamic turbulence in both two and three dimensions, with an emphasis on the number of degrees of freedom N. This number is estimated in terms of the magnetic Prandtl number Pr, kinetic Reynolds number Re, and magnetic Reynolds number Rm. Here Re and Rm are dynamic in nature, defined in terms of the kinetic and magnetic energy dissipation rates (or averages of the velocity and magnetic field gradients), viscosity and magnetic diffusivity, and the system size. It is found that for the two-dimensional case, N satisfies N≤PrRe(3/2)+Rm(3/2) for Pr>1 and N≤Re(3/2)+Pr(-1)Rm(3/2) for Pr≤1. In three dimensions, on the other hand, N satisfies N≤(PrRe(3/2)+Rm(3/2))(3/2) for Pr>1 and N≤(Re(3/2)+Pr(-1)Rm(3/2))(3/2) for Pr≤1. In the limit Pr→0, Re(3/2) dominates Pr(-1)Rm(3/2), and the present estimate for N appropriately reduces to Re(9/4) as in the case of usual Navier-Stokes turbulence. For Pr≈1, our results imply the classical spectral scaling of the energy inertial range and dissipation wave number (in the form of upper bounds). These bounds are consistent with the existing predictions in the literature for turbulence with or without Alfvén wave effects. We discuss the possibility of solution regularity, with an emphasis on the two-dimensional case in the absence of either one or both of the dissipation terms.
我们研究二维和三维不可压缩磁流体动力学湍流，重点关注自由度的数量N。该数量是根据磁普朗特数Pr、动雷诺数Re和磁雷诺数Rm来估计的。这里，Re和Rm本质上是动态的，它们是根据动能和磁能耗散率（或速度和磁场梯度的平均值）、粘性和磁扩散率以及系统大小来定义的。研究发现，对于二维情况，当Pr>1时，N满足N≤PrRe(3/2)+Rm(3/2)；当Pr≤1时，N满足N≤Re(3/2)+Pr(-1)Rm(3/2)。另一方面，在三维情况下，当Pr>1时，N满足N≤(PrRe(3/2)+Rm(3/2))(3/2)；当Pr≤1时，N满足N≤(Re(3/2)+Pr(-1)Rm(3/2))(3/2)。在Pr→0的极限情况下，Re(3/2)主导Pr(-1)Rm(3/2)，此时N的当前估计值如在通常的纳维-斯托克斯湍流情况下一样适当地简化为Re(9/4)。对于Pr≈1，我们的结果暗示了能量惯性范围和耗散波数的经典谱标度（以上界的形式）。这些界与文献中关于有无阿尔文波效应的湍流的现有预测一致。我们讨论了解的正则性的可能性，重点是在没有一个或两个耗散项的二维情况下。
