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

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 $t\to +\mathrm{\infty }$. In particular, this implies that the Ekman spiral is the unique equilibrium of the system.