The 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 Impact Kelvin 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.
