Nonlinear wave interactions in Rotating Shallow Water Equations and Applications

• Grant number: 24/01000-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2024
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Pedro da Silva Peixoto
• Grantee: Marco Antonio Dourado
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 21/06176-0 - Numerical methods for a new generation of weather and climate models, AP.PFPMCG.JP2

Abstract

One of the simplest models to describe geophysical fluid dynamics, that can yet capture complexfeatures of the atmosphere and ocean dynamics, is the so-called Rotating Shallow Water Equations(SWE). These equations consider the effects of the Earth rotation in a single layer of fluid, avoidingthe complications coming from vertical stratification. Furthermore, adding layers, and especially,considering the case with two layers, many phenomena in atmosphere and ocean can be modelled.The normal modes of the linearized set of SWE equations, i.e., its eigenfunctions, define 3different kinds of waves: inertia-gravity waves (East and West) and Rossby waves. Based on pre-vious knowledge about Shallow Water Equations and their normal modes using Fourier (SphericalHarmonics) spectral representation, this project aims initially to study the nonlinear interactionsbetween three linear waves (triads). We aim to research phenomena such as resonant and quasi-resonant interactions, and, also, the transfer of energy among the triads and how they can beuseful to understand the stability of Rossby modes.Furthermore, important atmospheric phenomena can be modeled by simplified models consid-ering only 4 or 5 nonlinearly interacting waves, which is a new perspective that has been studiedrecently and will be pursued in this project. For instance, effects such as the precession resonanceassociated with four waves, which promote a cascade process, and new studies point towards itsinfluence in generating low-frequency variability in the atmosphere and ocean.