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以及造成它们的不稳定性进行数值和理论研究。 以前的数值计算的开尔文和Rossby孤子将扩展到包括背景平均电流，引入严重的物理和数学困难的共振，微扰小因子，超渐近校正和临界纬度。 该项目的目标之一是更好地确定海洋和大气中非线性涡旋的性质，如热带不稳定涡旋（TIV），其中Rossby波嵌入平均剪切射流。 因此，将平均流量纳入解决方案将是工作的主要推动力。 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和径向基函数（RBF）谱方法将被用来离散他们。 由于径向基函数是一种没有规范网格的无网格方法，因此即使这些高梯度特征是弯曲的、非连续的或具有复杂拓扑结构的，也可以在锋区和关键层中密集地聚集网格点。 谱离散化产生一个大的非线性代数方程组系统。标准的参数或伪弧长延拓方法有很高的失败率，经常错过有趣的模式。 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。 目标是一个具有回溯和基于物理的选项的多元算法，当标准数字黑盒无效或非常慢时，它将取得胜利。 这些工具将用于计算大气中的大振幅斜压波。 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。 一个自然的结果，最近的工作TIV是调和线性不稳定性理论的五个范例和适用性的TIV成因。该项目中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、等离子体物理学和光学。 该项目将支持培训一名研究生，并将扩大代表性不足的本科生的参与。

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.