An analytical study of the frictional response of coastal currents and upwelling to wind stress

In this paper we present several theoretical results concerning currents forced by the wind in coastal regions, for a shallow sea and for a very deep sea. We investigate the time behavior and the spatial structure of the stream function and the momentum components, i.e., the onset, the transient, the asymptotic, and the periodic behavior of the currents and of the upwelling, forced by winds of different spatial and time structures. Results show that the intensity of the along coast jet initially grows linearly under a δ in time wind impulse, quadratically under a Heaviside in time wind impulse, and cubically under a linearly growing wind impulse. The asymptotic state is such that the intensity of the current vanishes if the wind impulse has a finite duration, while the intensity of the sea current has a final finite amplitude if the wind intensity goes to some finite value. If the wind stress is periodic in time, there is upwelling only when the period of the forcing is longer than a characteristic time scale, which is the sum of the inertial period and the friction e-folding time. Otherwise there are waves which propagate away from the region where the wind stress is acting. The spatial structure is such that the upwelling occurs in a horizontal region of the order of the Rossby deformation radius, corrected by the effect of friction (and by the effect of periodicity, when the wind stress is periodic in time). However, the horizontal gradient of the wind stress can be more important than the Rossby deformation radius in determining the horizontal extent of the upwelling region.