This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniqueness of the minimiser of an energy functional arising from a model for atmospheric and oceanic ow. At each xed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury and Purser. Key words Rearrangement of functions, semigeostrophic, variational problems, generalised solution.

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In an inviscid and incompressible fluid, we prove the existence of nested co-axial vortex rings moving at the same speed.