Pattern formation in the oceanic Ekman-Langmuir layer

We have studied ocean surface patterns formed by the action of the wind and surface gravity waves. These surface patterns are due to roll motions in the fluid layer just below the surface and occur for wind speeds exceeding a threshold value. In the absence of surface waves, the wind stress balances the Earth's Coriolis acceleration to produce a classical boundary layer flow known as the "Ekman layer". When surface wave effects dominate the Coriolis acceleration, they are thought to be responsible for the formation of convective motions in the near surface layer known as "Langmuir circulations". We consider pattern formation in the Ekman layer, pattern formation aspects of Langmuir circulations, and pattern formation in environmental conditions for which both surface wave effects as well as the Ekman layer flow are relevant.; We have performed a linear stability analysis of the basic wind-generated current and have studied the dependence of the wind threshold value, the roll spacing, and the direction of the roll axes on nondimensional parameters describing the environmental conditions. In the northern hemisphere, the roll axes are typically at a small angle to the right of the wind direction and travel to the right (looking downwind).; We then derived amplitude equations describing the weakly nonlinear evolution of the roll amplitude near threshold. We have simulated these equations numerically with coefficients corresponding to a variety of environmental conditions. The simulated solutions are typically chaotic with exponentially decaying correlation functions. For some parameter choices we observed distinct spirals which appear to dominate the evolution of the amplitude. When surface gravity waves are absent, we have found qualitatively different behavior of the solutions for different wind directions. There is an interval of wind directions (north of east) for which solutions are regular while they are chaotic for the other directions.; We have also performed numerical computations of the rolls far above threshold and calculated their stability characteristics. We have found parameter regimes above threshold for which the rolls are stable and have characterized the instabilities bounding these regimes. Rolls tend to be unstable to longitudinal perturbations near threshold and to transverse perturbations far above threshold.; Finally, we have noted that traveling rolls need to have a certain strength to form distinct surface streaks. Close to threshold these streaks are fuzzy and may be difficult to observe.