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

这项研究有两个组成部分：(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/延拓数值将应用于三种相干结构：海洋中的赤道捕获开尔文波、海洋中的热带不稳定波和大气中的大振幅斜压涡。坡面不连续开尔文角波的奇异性激发了提出的数值主题之一：混合？字母和有理切比雪夫重建，以提高频谱算法应对冲击、前沿和其他奇点的能力。此外，解决相干结构的离散非线性特征值问题的努力导致了计算以Chebyshev， Legendre，张量积Legendre或球调和形式表示的多项式的根和代数变体的新方法。这些地球物理问题，加上对天气预报和气候模拟改进“动力核”的更广泛需求，激发了其他提出的数值研究。一个是进一步的实验和理论，为一个长球基，而不是通常的勒让德多项式，在谱元代码。另一个改进是通过C8窗口和局部傅立叶基（Coifman-Meyer基）思想将区域光谱模型和数据分析方案混合到全球模型中。通过密切相关的数学，扩展到在不规则域上应用全局谱方法的基本问题：提议的工作将建立在PI先前在该领域的研究基础上。这些课题，无论是地球物理的还是数值的，仍然充满了疑问。为什么开尔文波的余弦/角波/破断分岔会出现在许多其他种类的波中，包括普通的表面重力水波？这种分岔有什么共性？什么是非通用的？为什么热带不稳定波在切变中演变成类似孤子的准稳定平移涡？为什么大气中的斜压涡旋不稳定，自聚焦成波包，是间歇性的而不是稳定的？具有高度非均匀网格的勒让德多项式，真的是求解高阶p型的最好方法吗？黑色元素还是光谱元素？可以结合吗？使用有理切比雪夫函数的过滤和重建很成功，其中一个或另一个仅部分成功的冲击前线？怎么可能呢？截断球面调和级数或多维勒让德级数的等高线，而不需要将其转换为普通多元多项式的数值病态步骤？广泛的影响开尔文波是被称为厄尔尼诺的海洋-大气耦合振荡的主要海洋成分，其干旱和暴雨对全球有很大的影响。斜压不稳定是中纬度地区大尺度天气的主要动力。光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模拟到地球上的地震波，再到晃动？发动机内的发动机；锋面、冲击和其他高梯度区域是其中普遍存在的复杂因素。埃尔兹：好？重建程序将具有非常广泛的适用性。傅里叶级数在区域数据分析和天气和气候建模方面有许多技术优势；因此，对有限区域光谱序列进行更好的混合和扩展方法将具有直接的社会效益。众所周知，多项式的x次幂形式是病态的；根?切比雪夫形式的计算方法在任何情况下都是有价值的。在哪里？找根很重要。在人类方面，研究生将在地球物理学和计算技术方面接受广泛的训练。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.