Three Stratified Fluid Models: Benjamin-Ono, Tidal Resonance, and Quasi-Geostrophy.

[摘要] In this dissertation, we study three models arising in stratified fluid dynamics: the Benjamin-Ono equation, a two-layer tidal model, and the quasi-geostrophic equations. First, we compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles, under mild restrictions. For this class of initial conditions, we are able to obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient. We proceed to use these exact formulas to deduce precise asymptotics for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant on the eigenvalue. Second, we present a simplified analytical tidal model that demonstrates the influence of stratification on both large- and small-scale surface tidal elevations in a qualitatively similar manner as in previous global realistic-domain numerical simulations. Our analytical results demonstrate the potential for the presence of stratification and changes to stratification to alter the large-scale (barotropic) tide. We find that changes in stratification may change large-scale tidal elevations at the same order as small-scale tidal elevations if there is significant bottom topography and the damping acts primarily on the bottom layer. Third, we investigate the influence of bottom friction on quasi-geostrophic turbulence dynamics in a multi-level forced-dissipated model that includes buoyancy advection at the boundaries. Motivated by earlier studies of two-layer quasi-geostrophic turbulence, we study the influence of bottom friction on the horizontal scales and vertical structure of eddy kinetic energy with the aid of a ;;surface-aware’ modal decomposition.

[发布机构] University of Michigan

[效力级别] inverse scattering transform

[关键词] scattering data of the Benjamin-Ono equation;inverse scattering transform;small-dispersion limit;effect of stratification on tides;multi-level forced-dissipated quasi-geostrophic equations;Atmospheric;Oceanic and Space Sciences;Mathematics;Science;Applied and Interdisciplinary Mathematics