I consider the flow of three-dimensional stratified rotating fluid with a free upper boundary. I show how the semi-geostrophic equations are derived as a limit of the Euler equations. Following earlier work of Benamou and Brnier, and Cullen and Gangbo, the equations are formulated in dual variables and the mapping to physical space is determined by optimal transportation using the energy as the cost function. I focus on the differences from the earlier work. These are the form of the energy, the definition of the space in which the energy is to be minimised, and the proof that the energy is strictly convex with respect to variations in the free upper boundary.