My understanding so far: given a small flow moving forward within a larger stationary body of water, the water ahead would pile up, creating hydrostatic pressure in all 2d directions (and thus acceleration), including to the sides. Wouldn't this side acceleration result in rotation, i.e. eddies or 2d vorticity?
But in my simulation code, no vorticity arises. (Of course, my code might be buggy.)
My understanding of the Shallow Water Equations and algorithm come from chapter 12 "Shallow Water" in Bridson's Fluid Simulation for Computer Graphics (whole chapter is available in the free preview)
Ideally, I'd like both a physical intuition and a mathematical understanding of this.