When the waves settle down, the surface of the water forms a curve called a parabola. As the box spins, the water tends to continue moving in a straight line tangent to the circle. However, the box restrains the water and forces it to keep moving in a circle. The water near the edge of the box goes around in one large circle in the same time that the water near the center goes around in a small circle. That means the water near the edge travels faster than the water near the center. The faster an object moves in a circle, the larger the force necessary to hold it in the circle. This force is called the centripetal force.
The surface of a body of water in equilibrium is always perpendicular to the net forces on the water. The diagram below (click to enlarge) shows the forces on the water in relation to the tilt or slope of the water surface.
The diagram shows that the tilt or slope of the water surface indicates the size of the force holding the water in its circular path. The flat bottom of the parabola shows that little force is needed to hold the water there in its circular path, while the steep outer regions show that a large force is required in those areas.
The horizontal component of the buoyancy provides the centripetal force.
You can prove to yourself that the water forms a parabola. A parabola has the equation y = x2. Draw a parabola on a piece of graph paper and tape the paper to one side of your rectangular box so that you can look through the box and see the paper. Then rotate the box until you find the speed at which the bottom of the parabola you drew matches up with the lowest part of the water surface. (Note that the X and Y axes of this graph must have the same scale.) The water surface should exactly match the curve of the parabola drawing at every other point.