Generalized Geometry of 2D Incompressible Fluid Flows

We describe a family of generalized almost structures associated with a Monge-Ampère equation for a stream function of 2D incompressible fluid flows. Using an indefinite metric field constructed from a pair of $2$-forms related to the Monge-Ampère equation, we show the existence of generalized metric-compatible structures in our family of generalized structures. The integrability of isotropic structures on the level of Dirac structures and differential forms is discussed.