The structure, stability and interaction of geophysical vortices

地球大气层和海洋中充满了漩涡结构或漩涡状的流体，这些环境受到稳定的垂直密度分层和行星自转的强烈影响。这些结构，或漩涡，存在于广泛的空间尺度，通常在巨大的雷诺数。一般来说，涡旋是行星大气的显著特征，它们被认为在塑造行星尺度环流方面发挥着核心作用。它们之间的相互作用可能非常复杂，直到今天，我们实际上还不知道这些相互作用如何对大气和海洋的整体运动做出贡献。该建议的主要目的是提供一个完整的描述涡旋的稳定性和涡旋的相互作用在地球物理流。行星旋转是重要的，当其相关的涡度是相当或大于相对垂直涡度。这可以用罗斯贝数Ro（相对于行星涡度的比值）来表示，与1相比，它很小。另一方面，当浮力频率大于水平涡度时，层结是重要的。同样地，这可以通过说弗劳德数Fr（水平涡度与浮力频率的比率）是小的来表示。当弗劳德数小于或等于罗斯比数时，控制方程组可以大大简化。然后，流动是很好的模拟的准地转（QG）方程。QG模型使基本的涡动力学的全面探索，从孤立的涡平衡和稳定性的三维QG湍流。我们现在理解了为什么QG模型中的涡旋往往是稳健的，它们如何对外部剪切和应变做出反应，是什么促使了同号和反号涡旋之间的强烈相互作用（位涡异常），以及湍流中涡旋种群的一般特性。重要的问题是：如何以及这些结果适用于有限的Ro和Fr？或者，当使用完整的运动方程时，QG结果是如何改变的？我们打算回答这些基本问题。当Ro和Fr不小时，出现两个新的特征。第一个是惯性重力波（IGW）的外观在超惯性频率，即在频率大于那些与旋涡运动。IGW通常引起与由涡旋引起的运动相比较弱的运动，因此IGW在许多情况下往往是次要的。有限Ro和Fr的另一个特点是增加了非地转运动的贡献，这在QG模型中是缺失的。非地转运动与高频IGW的不同之处在于，它们直接与涡旋相关联：它们是响应于瞬时位涡（PV）分布而产生的，并且对PV演变具有直接的平流影响。因此，非地转运动的保留对气流有重大影响，并意味着可能与QG动力学有很大的偏离。正是完整方程和QG近似之间的这一重要差异构成了所提出工作的基础。我们的目标是了解和量化这些影响的旋涡运动和旋涡稳定性的属性。

Vortical structures or swirling masses of fluid abound in the Earth's atmosphere and oceans, environments strongly influenced by both the stable vertical density stratification and the planetary rotation. These structures, or vortices, exist over a wide range of spatial scales and generally at huge Reynolds numbers. Vortices are conspicuous features of planetary atmospheres in general, and they are believed to play a central role in shaping planetary-scale circulations. Their interactions can be extraordinarily complex, and to this day we have virtually no understanding how these interactions contribute toward the collective motion of the atmosphere and the oceans as a whole. The main objectives of this proposal is to provide a complete description of vortex stability and vortex interactions in geophysical flows.The planetary rotation is important when its associated vorticity is comparable to or larger than the relative vertical vorticity. This can be expressed by saying that the Rossby number Ro, the ratio of relative to planetary vorticity, is small compared to 1. On the other hand, the stratification is important when the buoyancy frequency is larger than the horizontal vorticity. Likewise this can be expressed by saying that the Froude number Fr, the ratio of horizontal vorticity to buoyancy frequency, is small. When the Froude number is smaller than or comparable to the Rossby number, itself small, the system of governing equation can be greatly simplified. Then, the flow is well modelled by the `quasi-geostrophic' (QG) equations. The QG model has enabled a comprehensive exploration of basic vortex dynamics, from isolated vortex equilibria and stability to 3D QG turbulence. We now understand why vortices in the QG model tend to be robust, how they react to external shear and strain, what precipitates strong interactions between both like-signed and opposite-signed vortices (potential vorticity anomalies), as well as general properties of vortex populations in turbulence. The important question is: how well do these results apply to finite Ro and Fr? Or, how are the QG results altered when using the full equations of motion? We intend to answer these fundamental questions.When Ro and Fr are not small, two new features arise. The first is the appearance of Inertia-Gravity Waves (IGWs) at super-inertial frequencies, i.e. at frequencies larger than those associated with the vortical motion. The IGWs often induce weak motions compared to those induced by the vortices, and therefore IGWs tend to be of secondary importance in many circumstances. Another feature arising from finite Ro and Fr is the added contribution of ageostrophic motions, which are missing in the QG model. Ageostrophic motions differ from the high frequency IGWs in that they are directly associated with vortices: they are generated in response to the instantaneous potential vorticity (PV) distribution and have a direct advective effect on the PV evolution. The retention of ageostrophic motion thus has a significant impact on the flow and implies potentially large departures from QG dynamics. It is this important difference between the full equations and the QG approximation that underlies the proposed work. We aim to understand and quantify these effects on vortex motions and vortex stability properties.