Pattern formation in the marginally unstable Ekman layer

Abstract

We consider the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress. Both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude $\lambda$. The fluid motion in a layer of finite depth as well as one of infinite depth is considered. The linear instability in the infinite depth case is known to depend on the direction of the applied stress for $\lambda \ne 90\dg$, but this dependence is weak for the stress-driven Ekman layer. By contrast, the weakly nonlinear motion exhibits for finite and infinite depths qualitatively different dynamics for different stress directions.

The problem is treated by the method of multiple scales. In the case of finite depth, this leads to the Davey–Hocking–Stewartson equation, an amplitude equation of complex Ginzburg–Landau type coupled to a Poisson equation. In the case of infinite depth, it leads to the anisotropic complex Ginzburg–Landau equation for the amplitude of the roll motion. Motions in both finite and infinite depth basins are explored by numerical simulation, and are shown to lead to chaotic dynamics for the modulation envelope in most cases. The statistics and the nature of the patterns produced in this motion are discussed.