Stratified Rotating Boussinesq Equations in Geophysical Fluid Dynamics: Dynamic Bifurcation and Periodic Solutions

TL;DR

This paper analyzes the stability and bifurcation behavior of stratified rotating Boussinesq equations in geophysical fluid dynamics, revealing conditions for bifurcations and periodic solutions based on the Prandtl number.

Contribution

It provides a comprehensive bifurcation and stability analysis for these equations, including new results on Hopf bifurcation for different Prandtl number regimes.

Findings

Stability near the first critical Rayleigh number for Prandtl > 1

Existence of periodic solutions via Hopf bifurcation for Prandtl < 1

Development of a bifurcation theory for nonlinear dynamical systems

Abstract

The main objective of this article is to study the dynamics of the stratified rotating Boussinesq equations, which are a basic model in geophysical fluid dynamics. First, for the case where the Prandtl number is greater than one, a complete stability and bifurcation analysis near the first critical Rayleigh number is carried out. Second, for the case where the Prandtl number is smaller than one, the onset of the Hopf bifurcation near the first critical Rayleigh number is established, leading to the existence of nontrivial periodic solutions. The analysis is based on a newly developed bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) by two of the authors [16].