Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. I. The Kelvin Modes
Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. I. The Kelvin Modes
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旋转多变体的非轴对称动态不稳定性。
DOI：
10.1086/305466
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
R. Durisen
中科院分区：
文献类型：
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作者：
J. Toman;J. Imamura;B. K. Pickett;R. Durisen
我们研究了旋转多面体的动力学不稳定性的线性政权使用近似拉格朗日技术和更精确的欧拉计划。我们认为非轴对称模式与方位角的依赖成比例的exp（IM），其中m是一个整数和IM是方位角，多面体具有广泛的压缩和角动量分布。我们确定了m = 2 - 4模式的稳定性极限，并找到了最不稳定的m模式的本征值和本征函数给定的平衡模型。在一定程度上，我们已经探索了参数空间，我们发现，不稳定性的开始是不是很敏感的可压缩性或角动量分布的多面体时，模型的参数为T/|W|.这里T是旋转动能，W是多面体的引力能。m = 2、3和4模式在T/|W|分别为0.26 - 0.28、0.29 - 0.32和0.32 - 0.35，与麦克劳林球状体的限值一致，在± 0.015（T/|W|.唯一的例外发生在我们测试的最可压缩的多边形，然后仅对于m = 4，其中不稳定性在T/|W| 0.37 - 0.39。增长最快的低m模的本征函数与麦克劳林椭球的本征函数相似，因为它们不显示大的垂直运动，仅弱依赖于z，并且随着椭球的赤道半径接近，振幅强烈增加。然而，多变本征函数在一个方面与麦克劳林本征函数有质的不同：它们形成了强大的旋臂。螺旋臂更强的可压缩多面体和多面体的角动量分布偏离显着的麦克劳林球体。然而，我们的近似拉格朗日方法，它明确假定非螺旋麦克劳林样的试验函数，产生合理的估计模式的周期和e-折叠时间的不稳定的m = 2模式，即使是高度可压缩和强烈的差分旋转多面体。本文中的线性分析和非线性流体动力学模拟之间的比较m = 2给出了很好的定量协议的特征函数，模式的速度，和e-折叠时间的动态不稳定模式。
We study the dynamic instabilities of rotating polytropes in the linear regime using an approximate Lagrangian technique and a more precise Eulerian scheme. We consider nonaxisymmetric modes with azimuthal dependence proportional to exp (imϕ), where m is an integer and ϕ is the azimuthal angle, for polytropes with a wide range of compressibilities and angular momentum distributions. We determine stability limits for the m = 2-4 modes and find the eigenvalue and eigenfunction of the most unstable m-mode for given equilibrium models. To the extent that we have explored parameter space, we find that the onset of instability is not very sensitive to the compressibility or angular momentum distribution of the polytrope when the models are parameterized by T/| W |. Here T is the rotational kinetic energy, and W is the gravitational energy of the polytrope. The m = 2, 3, and 4 modes become unstable at T/| W | ≈ 0.26-0.28, 0.29-0.32, and 0.32-0.35, respectively, limits consistent with those of the Maclaurin spheroids to within ±0.015 in T/| W |. The only exception to this occurs for the most compressible polytrope we test and then only for m = 4, where instability sets in at T/| W | ≈ 0.37-0.39. The eigenfunctions for the fastest growing low m-modes are similar to those of the Maclaurin spheroid eigenfunctions in that they do not show large vertical motions, are only weakly dependent on z, and increase strongly in amplitude as the equatorial radius of the spheroid is approached. The polytrope eigenfunctions are, however, qualitatively different from the Maclaurin eigenfunctions in one respect: they develop strong spiral arms. The spiral arms are stronger for more compressible polytropes and for polytropes whose angular momentum distributions deviate significantly from those of the Maclaurin spheroids. Nevertheless, our approximate Lagrangian method, which explicitly assumes nonspiral Maclaurin-like trial functions, yields reasonable estimates for the pattern periods and e-folding times of unstable m = 2 modes even for highly compressible and strongly differentially rotating polytropes. Comparisons for m = 2 between the linear analyses in this paper and nonlinear hydrodynamic simulations give excellent quantitative agreement in eigenfunctions, pattern speeds, and e-folding times for the dynamically unstable modes.