Statistical equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow w ater equations, a model of essentially 2D fluid Row with a free surface, is described. The model contains a competing acoustic turbulent direct energy cascade, and a 2D turbulent inverse energy cascade. It is shown. nonethele ss that, just as in the corresponding theory of the inviscid Euler equation , the infinite number of conserved quantities constrains the flow sufficien tly to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equa tions.