We consider the temporal evolution of a slow downstream decrease in the velocity of a coastal current contained in the light upper layer of the ocean. The quasi-geostrophic model consists of two piecewise uniform potential vorticity regions separated horizontally by a free interface ("front") which intersects the vertical coastal wall in a "nose" region. As time increases, the slope of the front increases in this region, and the magnitude of the downstream convergence also increases, according to a nonlinear long-wave theory. At the time when this theory becomes invalid, the calculation is continued by numerical integration of the "contour dynamical" equations. This shows a continuation of the increase of the slope of the front near the nose, provided the total geostrophic transport is nonzero. (The case of zero transport is also discussed.) As time increases, a plume forms near the nose of the front, thereby transporting coastal water to very large offshore distances. It is suggested that this effect is responsible for some of the cold water plumes which extend to large distances from the coast of California. The cause of the small finite initial convergence (not implicit in our simple model) is attributed to differential upwelling or to a current instability.