论文标题
小尺度和各向异性，低$ rm $ $
Small scales and anisotropy in low $Rm$ magnetohydrodynamic turbulence
论文作者
论文摘要
在本文中，我们通过得出代表该流量的吸引子的尺寸的严格上限来得出小尺度的尺寸和吸引子尺寸，以低$ $ $ $ $ $ $ $ $ $ $ $ $的磁流失动力湍流。为此，我们发现了与Navier-Stokes方程相关的Evolution Operator的任何$ n $维数量的最大增长率的上限。正如康斯坦丁等人所解释的。 （J. Fluid Mech。，1985），该最大值为零的$ n $的值是吸引子维度的上限。为了在3D期刊域的更精确的情况下使用此属性，我们被召集来计算$ n $模式的分布，从而最大程度地减少了总（粘性和焦点）耗散。事实证明，这组模式显示出MHD湍流的大多数众所周知的特性，以前是通过启发式考虑因素获得的，例如在强磁场下存在焦耳锥的存在。然后，在没有物理假设的情况下获得了小尺度和吸引子维度的寻求估计值，因为Hartmann和Reynolds数字的函数，并匹配了启发式结果的Hartmann数字依赖性。还建立了流量是三维和各向异性的必要条件（与纯粹的二维相反）。
In this paper, we derive estimates for size of the small scales and the attractor dimension in low $Rm$ magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any $n$-dimensional volume of the phase space by the evolution operator associated to the Navier-Stokes equations. As explained by Constantin et al. (J. Fluid Mech., 1985), The value of $n$ for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the more precise case of a 3D periodical domain, we are led to calculate the distribution of $n$ modes which minimises the total (viscous and Joule) dissipation. This set of modes turns out to exhibit most of the well known properties of MHD turbulence, previously obtained by heuristic considerations such as the existence of the Joule cone under strong magnetic field. The sought estimates for the small scales and attractor dimension are then obtained under no physical assumption as functions of the Hartmann and the Reynolds numbers and match the Hartmann number dependency of heuristic results. A necessary condition for the flow to be tridimensional and anisotropic (as opposed to purely two-dimensional) is also built.
