Even if a current occurs, the water is transparent, so without some help we won't be able to see anything. What we will do is spout red ink from the surface of the water, creating a thin red line in the water. By watching how the line of ink drifts, we can see how the water is flowing.
This photo is a view of the current from the side. Near the surface of the water, the current is pointing downwind, however just below that, you can see the current is
At a greater depth, there is almost no current, so the effect of the wind is only conveyed to a certain depth. The section that is affected by the wind is called the Ekman layer.