Solitons and Wavepackets in the Ocean and Atmosphere and High-Order Numerical Algorithms

这项研究包括两个部分：(I)海洋和大气中非线性相干结构和波的数值和理论研究；(Ii)高阶数值算法，特别是谱方法的进一步发展。伪谱/牛顿/延拓数值方法将应用于三种相干结构：海洋中的赤道Kelvin波、海洋中的热带不稳定波(TIW)和大气中的大幅度斜压涡旋。斜率不连续的Kelvin角波的奇异性激发了提出的数值主题之一：混合？和有理Chebyshev重构以提高谱算法处理冲击、锋面和其他奇异性的能力。同时，为了解决相干结构的离散化非线性特征值问题，人们提出了计算切比雪夫多项式、勒让德多项式、张量积勒让德多项式或球谐多项式的根和代数族的新方法。这些地球物理问题，再加上对改进天气预报和气候模型“动力核心”的更广泛需求，促使了其他拟议的数值研究。一个是进一步的实验和理论，在谱元素代码中使用长球面基来代替通常的勒让德多项式。另一种是通过C8加窗和局部傅立叶基(Coifman-Meyer基)的思想，改进了区域光谱模式和数据分析方案与全球模式的融合。这通过密切相关的数学，扩展到在不规则区域上应用全局谱方法的基本问题：拟议的工作将建立在PI在这一领域的先前研究的基础上。这些主题，无论是地球物理的还是数值的，仍然充满问题。为什么开尔文波的椭圆形/角波/破碎分叉出现在这么多其他类型的波种中，包括普通的表面重力水波？这种分叉有什么共性？什么是非泛型？热带不稳定波为什么演变为准定常涡旋，类似于切变中的孤立子？为什么大气中的斜压涡旋是不稳定的，自聚焦成波包，间歇性而不是稳定的？具有高度不均匀网格的勒让德多项式真的是计算高阶p型、nite元素或谱元素的最佳方法吗？有理切比雪夫函数的变换和重建的组合能否很好地成功，如果其中一项对冲击和前沿只取得了部分成功？如何才能得到截断的球面调和或多维勒让德级数的水平曲线，而不经历转化为普通多元多项式的数值病态步骤？宽冲击开尔文波是被称为厄尔尼诺的海-气耦合振荡的主要海洋成分，其干旱和暴雨对全球影响很大。斜压不稳定是造成中纬度大范围天气的主要引擎。谱方法被广泛应用于科学和工程的几乎所有分支，从广义相对论中的重力波到地球中的地震波的模拟，再到汽车发动机的晃动；锋面、冲击和其他高梯度区域在大多数这些领域都是一个普遍存在的复杂问题：一个好的LTER/重建程序将具有非常广泛的适用性。傅里叶级数在天气气候的区域数据分析和模拟中具有许多技术优势，因此，对有限区域的光谱序列采用更好的混合和扩展方法将会产生直接的社会效益。多项式的x次方形式是出了名的病态；切比雪夫形式的求根方法在每个求根很重要的领域都是有价值的。在人类方面，研究生将接受地球物理和计算技术方面的广泛培训。一些本科生也将通过大学资助的项目参与研究。

ABSTRACTOCE-0451951The study has two components: (i) numerical and theoretical studies of nonlinear coherent structures and waves in the ocean and atmosphere and (ii) the further development of high-order numerical algorithms, especially spectral methods. Pseudospectral/Newton/continuation numerics will be applied to three species of coherent structures: equatorially-trapped Kelvin waves in the sea, Tropical Instability Waves (TIW's) in the ocean, and large-amplitude baroclinic vortices in the atmosphere. The singularity of the slope-discontinuous Kelvin corner wave motivates one of the proposed numerical topics: blending ?lters and rational Chebyshev reconstruction to improve the ability of spectral algorithms to cope with shocks, fronts, and other singularities. Also, the struggle to solve discretized nonlinear eigenvalue problems for coherent structures has led to new methods for computing the roots and algebraic varieties of polynomials expressed in Chebyshev, Legendre, tensor-product Legendre or spherical harmonic form. These geophysical problems together with broader need for improved "dynamical cores" for weather forecasting and climate modelling motivate the other proposed numerical studies. One is further experimentation and theory for a prolate spheroidal basis, instead of the usual Legendre polynomials, in spectral element codes. Another is improved blending of regional spectral models and data analysis schemes into global models through C8 windowing and local Fourier basis (Coifman-Meyer basis) ideas. This extends, through closely-related mathematics, to the fundamental problem of applying global spectral methods on irregular domains: the proposed work will build upon the PI's previous studies in this area.The topics, both geophysical and numerical, are still full of questions. Why does the Cnoidal/Corner Wave/Breaking bifurcation of the Kelvin wave occur in so many other kinds of wave species including ordinary surface gravity water waves? What is generic about this bifurcation? What is nongeneric? Why do Tropical Instability Waves evolve to quasi-steady translating vortices resembling solitons in shear? Why are baroclinic vortices in the atmosphere unstable, self-focusing into wavepackets, intermittent instead of steady? Are Legendre polynomials, with their highly nonuniform grid, really the best way to do high order p-type ?nite elements or spectral elements? Can a combination of ?ltering and reconstruction using rational Chebyshev functions succeed well where one or the other has succeeded only partially for shocks & fronts? How can one ?nd the level curves of a truncated spherical harmonic or multidimensional Legendre series without the numerically ill-conditioned step of converting to an ordinary multivariate polynomial?Broad ImpactKelvin waves are the main oceanic component of the coupled ocean-atmosphere oscillation known as El Nino, whose droughts and heavy rains have a large global impact. Baroclinic instability is the main engine of large-scale weather in the middle latitudes. Spectral methods are widely used in almost all branches of science and engineering from simulations of gravity waves in general relativity to seismic waves in the earth to sloshing ?ows in the engine of an automobile; fronts, shocks and other high-gradient regions are a ubiquitous complication in most of these ?elds: a good ?lter/recontruction procedure will have very wide applicability. There are many technical advantages to Fourier series for regional data analysis and modelling of the weather and climate; better blending and extension methods for limited-area spectral series will therefore have immediate societal bene?ts. The powers-of-x form of polynomials is notoriously ill-conditioned; root?nding methods for the Chebyshev form will be valuable in every ?eld where ?nding roots is important. On the human side, a graduate student will be trained broadly in both geophysics and computational techniques. Several undergraduates will also participate in research, through university-funded programs.