论文标题
低r m壁构成磁流失动力动力湍流的吸引子维度的边界
Bounds on the attractor dimension for low-R m wall-bound magnetohydrodynamic turbulence
论文作者
论文摘要
稳定的低$ R_M $ $ MHD湍流通过估计吸引子维度的估计。考虑了两个平行壁之间具有垂直磁场的平行壁之间的流动。该流量由其最大速度和磁场强度定义。鉴于相应的雷诺和哈特曼的数字，我们为吸引子的维度提供了上限，并找出必须选择哪些模式以实现该界限。将其与最小尺度和湍流各向异性大小的启发式估计值进行了比较。我们确定在非二维参数空间中分开不同模式集的边界，这些模式让人联想到在实际流动中观察到的三个重要的先前鉴定的过渡。第一个边界将模式的经典流体动力集与MHD集分开，其中各向异性采用``joule锥''的形式。第二个边界将3D MHD集与准2D MHD集分开。第三个分离集合，其中所有模式都表现出相同的边界层厚度左右，并且集合在集合中共存许多不同的``边界层模式''。然后，将定义这些边界的非二维关系与以各向同性和各向异性MHD湍流，3D和Quasi-2d MHD湍流以及湍流与层层层层层之间的过渡相提并论。除了这种3D方法外，我们还确定了使用2D MHD方程建模的强制性湍流流的上限，该方程在准2D MHD方向上相关。这种2D方法的优点是，虽然上限在三个维度上很松散，但在2D非线性项中存在最佳的上限。这使我们能够为准2D MHD流提供逼真的吸引子尺寸。
Steady Low $R_m$ MHD turbulence is investigated here through estimates of upper bounds for attractor dimension. A flow between two parallel walls with an imposed perpendicular magnetic field is considered. The flow is defined by its maximum velocity and the intensity of the magnetic field. Given the corresponding Reynolds and Hartmann numbers, we derive an upper bound for the dimension of the attractor and find out which modes must be chosen to achieve this bound. Thier, are compared to heuristics estimates for the size of the smallest scales and anisotropy of the turbulence. We identify boundaries separating different sets of modes in the space of non-dimensional parameters, which are reminiscent of three important previously identified transitions observed in the real flow. The first boundary separates classical hydrodynamic sets of modes from MHD sets where anisotropy takes the form of a ``Joule cone''. The second boundary separates 3D MHD sets from quasi-2D MHD sets. The third separates sets where all the modes exhibit the same boundary layer thickness or so, and sets where many different ``boundary layer modes'' co-exists in the set. The non-dimensional relations defining these boundaries are then compared to the heuristics known for the transition between isotropic and anisotropic MHD turbulence, 3D and quasi-2D MHD turbulence and that between a turbulent and a laminar Hartmann layer. In addition to this 3D approach, we also determine upper bounds for the dimension of forced turbulent flows modelled using a 2D MHD equation, relevant in the quasi-2D MHD regime. The advantage of this 2D approach is that, while upper bounds are quite loose in three dimensions, optimal upper bounds exist for the 2D nonlinear term. This allows us to derive realistic attractor dimensions for quasi-2D MHD flows.
