This paper establishes the equivalence of four definitions of two vector valued functions being rearrangements. Properties of the monotone rearrangement of a vector valued function are used to show existence and uniqueness of the minimizer of an energy functional arising from the semigeostrophic equations, a model for atmospheric and oceanic flow. At each fixed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury, and Purser [SIAM J. Appl. Math., 51 (1991). pp. 20-31].