Stability and bifurcation of 2D viscous primitive equations with full diffusion

TL;DR

This paper rigorously analyzes the stability and bifurcation behavior of 2D viscous primitive equations with full diffusion under thermal forcing, identifying critical thresholds for stability and the emergence of convection patterns.

Contribution

It provides a comprehensive stability analysis across thermal regimes and demonstrates the bifurcation leading to stable convection states near criticality.

Findings

Global nonlinear stability in subcritical regime

Asymptotic convergence at critical threshold

Nonlinear instability in supercritical regime

Abstract

This paper investigates the stability and bifurcation of the two-dimensional viscous primitive equations with full diffusion under thermal forcing. The system governs perturbations about a motionless basic state with a linear temperature profile in a periodic channel, where the temperature is fixed at $T_0$ and $T_1$ on the bottom and upper boundaries, respectively. Through a rigorous analysis of three distinct thermal regimes, we identify a critical temperature difference $T_c$ that fundamentally dictates the system's dynamical transitions. Our main contributions are fourfold. Firstly, in the subcritical case $T_0 - T_1 < T_c$, we use energy methods to establish the global nonlinear stability in $H^2$-norm, proving that perturbations decay exponentially. Secondly, precisely at the critical threshold $T_0 - T_1 = T_c$, we prove not only the nonlinear stability in $H^1$-norm but also the asymptotic convergence of all solutions to zero, leveraging spectral and dynamical systems theory. Finally, in the supercritical regime $T_0 - T_1 > T_c$, a bootstrap argument reveals that the basic state is nonlinearly unstable across all $L^p$-norms for $1 \leq p \leq \infty$. Finally, near the critical point, the dynamics are first reduced to a two-dimensional system on a center manifold. This reduced system then undergoes a supercritical bifurcation, generating a countable family of stable steady states that are organized into a local ring attractor. This work closes a significant gap in the stability analysis of the thermally driven primitive equations, establishing a rigorous mathematical foundation for understanding the formation of convection cells in large-scale geophysical flows.