Coherent Structures, Vortices and Waves in Jets and Instabilities

拟议的研究是对海洋和大气中的非线性相干结构和波及其造成的不稳定性进行数值和理论研究。以前对Kelvin和Rossby孤子的数值计算将扩展到包括背景平均流，这将引入共振、微扰小因子、超渐近校正和临界纬度等严重的物理和数学困难。该项目的目标之一将是更好地确定海洋和大气中非线性涡旋的性质，例如热带不稳定涡旋(TIV)，平均切变喷流中嵌入了Rossby波。因此，将平均流量纳入解决方案将是这项工作的主要主旨。孤子和其他非线性行波是非线性微分方程本征值问题的解。谱Galerkin谱方法和径向基函数(RBF)谱方法都将用于离散它们。由于径向基函数法是一种无网格方法，没有标准网格，即使这些高梯度特征是弯曲的、丝状的或具有复杂的拓扑结构，也可以在前锋区和关键层密集地聚集网格点。谱离散产生了一个大型的非线性代数方程组。标准的参数或伪弧长延拓方法失败率很高，经常遗漏有趣的模式。其中一个数值任务是开发基于物理的替代方案，将共振构建到求解器中。目标是一种具有回溯和基于物理的选项的多算法，当标准的数值黑盒无效或非常慢时，该算法将获胜。这些工具将用于计算大气中的大幅度斜压波。即使这些是弱不稳定的，也应该有可能理解局域相干结构的观测结果。最近关于TIV的工作的一个自然结果是调和了线性不稳定理论的五种范式以及每一种范式对TIV起源的适用性。该项目将开发的分析和数值方法将适用于海洋学和气象学以外的许多领域，包括量子力学、等离子体物理和光学。该项目将支持一名研究生的培训，并将扩大未占名额的本科生的参与范围。

The proposed research is to perform numerical and theoretical studies of nonlinear coherent structures and waves in the ocean and atmosphere, and the instabilities that make them. Previous numerical computations of Kelvin and Rossby solitons will be extended to include background mean currents, which introduce the severe physical and mathematical difficulties of resonances, perturbative small divisors, hyperasymptotic corrections and critical latitudes. One of the aims of the project will be to determine better the nature of nonlinear vortices in the ocean and atmosphere, such as Tropical Instability Vortices (TIV), with are Rossby waves embedded in mean sheared jets. Thus, incorporating mean flows into the solutions will be a major thrust of the work. Solitons and other nonlinear traveling waves are solutions to nonlinear differential equation eigenvalue problems. Both spectral Galerkin and radial basis function (RBF) spectral methods will be used to discretize them. Because RBFs are a meshless method without a canonical grid, it will be possible to cluster grid points densely in the frontal zones and critical layers even when these high gradient features are curving, filamentary, or otherwise have a complicated topology. The spectral discretizations generate a large system of nonlinear algebraic equations. Standard parameter-or-pseudoarclength continuation methods have a high failure rate, often missing interesting modes. One numerical task is to develop physics-based alternatives that build the resonances into the solver. The goal is a polyalgorithm with backtracking and physics-based options that will triumph when standard numerical black boxes are ineffective or very slow. These tools will be applied to compute large amplitude baroclinic waves in the atmosphere. Even though these are weakly unstable, it should be possible to understand observations of localized coherent structures. A natural outgrowth of recent work on TIVs is to reconcile the five paradigms of linear instability theory and the applicability of each to TIV genesis. The analytical and numerical methods that will be developed in the project will have applicability in a number of fields beyond oceanography and meteorology, including quantum mechanics, plasma physics, and optics. The project will support the training of a graduate student and will broaden participation by under-represented undergraduate students.