Fully Three-Dimensional Stability Analyses and Evolution of Hurricanes and Tornadoes with Secondary Circulations

龙卷风和飓风是威胁美国人口的两种最具破坏性的自然现象。这些风暴的一个普遍特征是核心区有一个垂直排列的强烈涡旋。分析涡旋小扰动的线性化动力学的经典方法对强烈大气涡旋的行为有了实质性的洞察，有助于解释龙卷风中多个涡旋的形成和飓风中核心结构的快速变化等现象。然而，这些研究中的绝大多数都应用于基本的二维扰动到二维涡旋。在一些研究中已经考虑了三维扰动，但基本态涡旋在垂直方向上是均匀的。本研究将从两个方面推动大气强烈涡旋的研究。首先，经典的涡流稳定性分析方法将被扩展，以允许研究真正的三维涡流。运动方程将是三维滞弹性方程，这样既不需要流体静力、准地转或其他平衡近似。所有类型的强烈大气涡旋在垂直轴上并不是均匀的，但它们的风场和密度场往往随高度有相当大的变化。稳定性分析将考虑随高度任意变化的本征函数，以便首席研究人员能够在三维涡旋的核心中发现真正的三维本征模式，无论是不稳定的还是其他的。这种方法将基于数值模拟的结果和现有或未来的龙卷风和飓风的观测数据集，应用于轴对称的基本态涡旋。其次，将在龙卷风和飓风的发展和维持中发挥关键作用的二次环流纳入基本态流动。以往关于径向流入和垂直流出对这些涡稳定性的影响的研究仅限于高度理想化的涡旋。次级环流，特别是通过涡旋核心向上的垂直风的切变是否稳定或破坏涡旋的稳定，仍然是一个悬而未决的耐人寻味的问题，可能对理解龙卷风和飓风的核心动力学至关重要。对于稳定的涡旋，如发展中的热带气旋或“单胞”龙卷风涡旋，研究的自然扩展将是确定三维、非对称、非静力扰动的线性化演化及其对基本态涡旋的影响。这项研究的成功完成将有助于更好地理解和预测破坏性大气涡旋。

Tornadoes and hurricanes are two of the most destructive natural phenomena that threaten the U.S. population. A universal characteristic of these storms is a vertically aligned, intense vortex in the core region. The classical approach of analyzing the linearized dynamics of small perturbations to vortices has yielded substantial insight into the behavior of intense atmospheric vortices, helping to explain such phenomena as multiple vortex formation in tornadoes and rapid inner-core structure change in hurricanes. The vast majority of these studies, however, have been applied to essentially two-dimensional perturbations to two-dimensional vortices. In some studies, three-dimensional perturbations have been considered, but the basic-state vortices were homogenous in the vertical direction.This research will advance the study of intense atmospheric vortices in two ways. First, the classical methods of vortex stability analysis will be extended to allow for the study of truly three-dimensional vortices. The equations of motion will be the three-dimensional anelastic equations such that neither hydrostatic, quasi-geostrophic, or other balance approximations are required. Intense atmospheric vortices of all types are not homogenous along their vertical axis, but often have considerable variation of their wind and density fields with height. The stability analysis will allow for eigenfunctions that vary arbitrarily with height, so that the Principal Investigators can find truly three-dimensional eigenmodes, unstable or otherwise, in the cores of three-dimensional vortices. This approach will be applied to axisymmetric basic-state vortices, based on the results of numerical simulations and existing or future observational data sets of tornadoes and hurricanes.Secondly, the secondary circulation, which is known to play a critical role in the development and maintenance of tornadoes and hurricanes, will be incorporated into the basic state flow. Previous investigations on the influence of radial inflow and vertical outflow on the stability of these vortices have been limited to highly idealized vortices. Whether the secondary circulation, and in particular the shear of the vertical wind up through the core of the vortex, stabilizes or destabilizes the vortex remains an open and intriguing question and may be critical in understanding inner-core dynamics in tornadoes and hurricanes. For stable vortices, such as developing tropical cyclones or "one-celled" tornadic vortices, a natural extension of the research will be to determine the linearized evolution and impact on the basic-state vortex of three-dimensional, asymmetric, nonhydrostatic perturbations.Successful completion of this research will lead to better understanding and, potentially, better forecasts of destructive atmospheric vortices.