论文标题
MHD湍流中吸引子维度的数学估计值
Mathematical estimates for the attractor dimension in MHD turbulence
论文作者
论文摘要
本工作的目的是得出严格的湍流MHD流量量（例如耗散量表的大小和各向异性），以及2D和3D状态之间的过渡。为此，我们计算了运动方程的吸引子维度的上限，这表明完全开发的流中存在模式的数量。该方法已经成功地用于得出2D和3D流体动力湍流的此类估计，这是耗散的$ \ Mathcal l_ \ Infty $ norm的函数，如\ cite {doering95}中。我们在这里解决了在3个空间方向上流动周期性的问题（空间周期$2πl$），该问题应用了永久性磁场。此外，耗散操作员的详细研究提供了有关流量结构的更多指示。在第2节中，我们回顾了动力学系统理论的工具以及在3D流体动力湍流的情况下它们导致的结果。第3节致力于研究一组模式，该模式最大程度地减少了与MHD湍流（粘性和Joule）总耗散相关的操作员的痕迹。最终，在第4节中得出了在强磁场下MHD湍流中吸引子维度和耗散量表的估计值，并将其与从启发式方面获得的结果进行了比较。
The aim of the present work is to derive rigorous estimates for turbulent MHD flow quantities such as the size and anisotropy of the dissipative scales, as well as the transition between 2D and 3D state. To this end, we calculate an upper bound for the attractor dimension of the motion equations, which indicates the number of modes present in the fully developed flow. This method has already been used successfully to derive such estimates for 2D and 3D hydrodynamic turbulence as a function of the $\mathcal L_\infty$ norm of the dissipation, as in \cite{doering95}. We tackle here the problem of a flow periodic in the 3 spatial directions (spatial period $2πL$), to which a permanent magnetic field is applied. In addition, the detailed study of the dissipation operator provides more indications about the structure of the flow. In section 2, we review the tools of the dynamical system theory as well as the results they have led to in the case of 3D hydrodynamic turbulence. Section 3 is devoted to the study of the set of modes which minimises the trace of the operator associated to the total dissipation in MHD turbulence (viscous and Joule). Eventually, the estimates for the attractor dimension and dissipative scales in MHD turbulence under strong magnetic field are derived in section 4 and compared to results obtained from heuristic considerations.
