The structure, stability and interaction of geophysical vortices

在地球的大气和海洋中有大量的涡旋结构或旋转的流体，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，存在于大范围的空间尺度和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，它们被认为在形成行星尺度的环流中起着核心作用。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是如何促成大气和海洋作为一个整体的集体运动的。本文的主要目的是对地球物理流动中的涡旋稳定性和涡旋相互作用提供一个完整的描述。当相关涡度与相对垂直涡度相当或大于相对垂直涡度时，行星自转是重要的。这可以表示为罗斯比数，相对于行星涡度的比率，比1小。另一方面，当浮力频率大于水平涡度时，分层是重要的。同样，这可以表示为弗劳德数Fr，即水平涡度与浮力频率之比很小。当弗劳德数小于或与本身较小的罗斯比数相当时，控制方程系统可以大大简化。然后，用“准地转”（QG）方程很好地模拟了这种流动。QG模型能够全面探索基本的涡旋动力学，从孤立的涡旋平衡和稳定性到三维QG湍流。我们现在了解了为什么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作出反应，是什么导致了同号和反号涡旋之间的强相互作用（潜在涡旋异常），以及湍流中涡旋种群的一般特性。重要的问题是：这些结果如何适用于有限的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都不小时，出现了两个新的特征。首先是惯性重力波（igw）在超惯性频率下的出现，即在频率大于与涡旋运动相关的频率上。相对于由涡旋引起的运动，igw往往引起较弱的运动，因此在许多情况下igw往往是次要的。由有限的Ro和Fr引起的另一个特征是地转运动的附加贡献，这在QG模型中是缺失的。地转运动与高频igw的不同之处在于它们与涡旋直接相关：它们是响应瞬时位涡（PV）分布而产生的，并对PV演变有直接的平流作用。因此，地转运动的保留对流动有重大影响，并意味着可能与QG动力学有很大的偏离。完整方程和QG近似之间的这个重要区别是所提出工作的基础。我们的目的是了解和量化这些对涡旋运动和涡旋稳定性的影响。

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.