Several recent simulation works in the non-ideal magnetohydrodynamic (MHD) formalism have shown the importance of ambipolar diffusion (AD) within the protoplanetary discs (PPDs) at large radii. In this study, we model the time evolution of a polytropic PPD in the presence of the AD. In this regard, the non-ideal MHD equations are investigated in the outer region of a PPD where the magnetic field evolution is dominated by the AD. The self-similar solution technique is used for a polytropic fluid including the self-gravity and viscosity. The ambipolar diffusivity and its derivative are crucial for the formulation of this study. Hence, this variable is scaled by an important factor, that is the Elsasser number. The self-similar equations are derived, and the semi-analytical and numerical solutions are presented for the isothermal and polytropic cases. The analytical approach enables us to know the asymptotic behaviour of the physical variables in a PPD, such as the angular momentum and magnetic field. Furthermore, the coupling/decoupling of magnetic field with the angular momentum was discussed analytically to find a corresponding model for the angular momentum loss at large radii of a PPD. Regarding this approach, we found that the magnetic braking induced by the AD at large radii has a high potential to loss the angular momentum even if the turbulent viscosity is not efficient. Also, the sign and values of vertical velocity strongly depends on the sign and values of radial field in the polytropic case.

中文翻译：

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原行星盘中双极性扩散的动力学效应

最近在非理想磁流体动力学（MHD）形式主义中的一些模拟工作表明，大半径的原行星盘（PPD）内双极扩散（AD）的重要性。在这项研究中，我们在AD的存在下模拟了多变PPD的时间演变。在这方面，在PPD的外部区域研究了非理想的MHD方程，在该区域中，磁场的发展主要由AD决定。自相似溶液技术用于包含自重和粘度的多相流体。双极性扩散率及其导数对于制定这项研究至关重要。因此，此变量的缩放比例很重要，即Elsasser数。推导了自相似方程，并给出了等温和多变情况的半解析和数值解。分析方法使我们能够了解PPD中物理变量的渐近行为，例如角动量和磁场。此外，通过分析讨论了磁场与角动量的耦合/去耦，从而找到了PPD大半径时角动量损失的相应模型。关于这种方法，我们发现，即使湍流粘度效率不高，大半径下由AD引起的磁制动也有可能损失角动量。而且，在多变情况下，垂直速度的符号和值在很大程度上取决于径向场的符号和值。分析地讨论了磁场与角动量的耦合/解耦，以找到相应的PPD大半径角动量损失模型。关于这种方法，我们发现，即使湍流粘度效率不高，大半径下由AD引起的磁制动也有可能损失角动量。而且，在多变情况下，垂直速度的符号和值在很大程度上取决于径向场的符号和值。分析地讨论了磁场与角动量的耦合/解耦，以找到相应的PPD大半径角动量损失模型。关于这种方法，我们发现，即使湍流粘度效率不高，大半径下由AD引起的磁制动也有可能损失角动量。而且，在多变情况下，垂直速度的符号和值在很大程度上取决于径向场的符号和值。