Long-term behaviour of the primitive equations with wind-driven boundary conditions: Convergence to the Ekman spiral

TL;DR

This paper proves that solutions of the 3D primitive equations with wind-driven boundary conditions exponentially converge to the Ekman spiral over time, establishing it as the system's unique equilibrium.

Contribution

It demonstrates the exponential convergence of primitive equations with wind-driven boundary conditions to the Ekman spiral, confirming its role as the system's unique long-term state.

Findings

Solutions converge exponentially to the Ekman spiral

Ekman spiral is the unique equilibrium

Long-term behaviour characterized by exponential decay

Abstract

In this article we investigate the long-term behaviour of the 3D incompressible, primitive equations with wind-driven boundary conditions and Coriolis force. We show that every solution converges exponentially fast to the Ekman spiral as $t \to + \infty$. In particular, this implies that the Ekman spiral is the unique equilibrium of the system.