An affine model of the dynamics of astrophysical discs
An affine model of the dynamics of astrophysical discs
复制标题
天体物理圆盘动力学的仿射模型
DOI：
10.1093/mnras/sty588
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发表时间：
2018
影响因子：
4.8
通讯作者：
G. Ogilvie
中科院分区：
文献类型：
--
作者：
G. Ogilvie
薄天体物理盘经常使用二维流体动力学方程建模。我们推导出这个模型的扩展，更准确地描述了在没有自重力，磁场和复杂的内部运动的情况下薄盘的行为。理想流体理论是直接从三维流体的汉密尔顿原理导出的，对变形梯度张量作了特定的近似。在投影到一个参考平面上后，我们用欧拉形式表示方程。圆盘被认为是一组流体柱，每个流体柱都能够进行与时间相关的仿射变换，包括三维中的平移和线性变换。因此，除了参考平面中通常的二维流体动力学之外，该理论还考虑了中平面的变形（如翘曲圆盘中发生的变形）以及伴随这种变形的内部剪切运动。它还允许通过水平流线周围的垂直重力场的变化或通过水平速度的发散在非圆盘中驱动的垂直膨胀。仿射模型的方程体现了能量和位涡守恒定律，即使是非平面圆盘。我们验证，他们重现正是三维翘曲和偏心盘的长期近似的线性理论。然而，仿射模型不依赖于任何长期或小振幅的假设，并应在更一般的情况下是有用的。
Thin astrophysical discs are very often modelled using the equations of two-dimensional hydrodynamics. We derive an extension of this model that describes more accurately the behaviour of a thin disc in the absence of self-gravity, magnetic fields and complex internal motions. The ideal fluid theory is derived directly from Hamilton's Principle for a three-dimensional fluid after making a specific approximation to the deformation gradient tensor. We express the equations in Eulerian form after projection on to a reference plane. The disc is thought of as a set of fluid columns, each of which is capable of a time-dependent affine transformation, consisting of a translation together with a linear transformation in three dimensions. Therefore, in addition to the usual two-dimensional hydrodynamics in the reference plane, the theory allows for a deformation of the midplane (as occurs in warped discs) and for the internal shearing motions that accompany such deformations. It also allows for the vertical expansions driven in non-circular discs by a variation of the vertical gravitational field around the horizontal streamlines, or by a divergence of the horizontal velocity. The equations of the affine model embody conservation laws for energy and potential vorticity, even for non-planar discs. We verify that they reproduce exactly the linear theories of three-dimensional warped and eccentric discs in a secular approximation. However, the affine model does not rely on any secular or small-amplitude assumptions and should be useful in more general circumstances.