开普勒环形剖面的浅水理论
A shallow-water theory for annular sections of Keplerian Disks
摘要：
一个缩放参数，导致浅水理论的非轴对称扰动薄开普勒磁盘的环形部分。发展一个理论结构，以帮助物理上理解已知的二维涡旋动力学与开普勒盘中三维涡旋动力学的关系。使用渐近标度参数曲张的开普勒磁盘的干扰被认为是在径向和垂直尺度上与磁盘的高度一致，而方位角尺度是完整的$2π$角的磁盘范围。缩放导致动态是径向地转和垂直流体静力。因此，位涡量出现，并显示出在拉格朗日意义上是守恒的。探讨了均匀位涡线性解，并证明了该理论包含通道流条件下平流旋转不稳定性的一个体现。线性化的解决方案，一个单一的缺陷上的无限域的开发和支持传播Rossby边缘波。线性非均匀位涡解也被开发，并显示出在某些方面是类似的严格二维无粘流的动力学。基于这一理论的框架下，参数的基础上的地球物理概念，提出支持的断言，平流旋转不稳定是在一个通用类的正压/斜压位涡不稳定。这种形式主义的扩展也提出了建议。本文所发展的渐近理论所得到的浅水方程为研究盘动力学开辟了一条新途径。
英文摘要：
A scaling argument is presented that leads to a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. To develop a theoretical construction that will aid in physically understanding the relationship of known two-dimensional vortex dynamics to their three-dimensional counterparts in Keplerian disks. Using asymptotic scaling arguments varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full $2π$ angular extent of the disk. The scalings lead to dynamics which are radially geostrophic and vertically hydrostatic. It follows that a potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain is developed and is shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and are shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. Based on the framework of this theory, arguments based on geophysical notions are presented to support the assertion that the strato-rotational instability is in a generic class of barotropic/baroclinic potential vorticity instabilities. Extensions of this formalism are also proposed. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach to studying disk dynamics.
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