# Do the Shallow Water Equations produce 2d vorticity/eddies? Why/Why not?

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.

• If I recall, the fluid motion we call vorticity requires a 3 dimensional framework Commented Apr 28, 2017 at 13:14
• Perhaps because of vector cross products, Coriolis acceleration Commented Apr 28, 2017 at 13:15
• @docscience Yes, I think maybe "proper" vorticity is 3d only, but 2d vortices are produced by 2d Navier Stokes equation simulations (e.g. those swirling 2d smoke toys, or try wong's online webgl fluid simulation). To make them 3d, the 2d vortices can be thought of as having an axis along the 3rd dimension. Commented Apr 28, 2017 at 14:14
• I would tend to call the 2D rotation an 'eddy'. Commented Apr 28, 2017 at 17:14
• @docscience I agree, i actually almost wrote "eddies", I'll edit - title reads more cleanly now. Thanks. Commented Apr 29, 2017 at 2:12

And this definitely is true; They are even called as a "rolling waves" or "roll waves" because of this rotational character. Simply googling the latter even gives a lot of pictures for the issue. This linked publication "Dynamics of roll waves" also describes better the relation to equation $$c^2=gd$$ on page 182 in the question linked book "Fluid Simulation for Computer" as this is simply the Froude Number $$Fr=1$$.
At this speed there will be no rolling in the waves. The waves are created only because the external field (ie. gravity) produces a certain flow velocity $$V$$, which requires a certain flow depth $$y$$, and as $$Q=V*A$$ these all can be satisfied with only certain Flow $$Q$$, it means that the any Q less will produce wave a like flow, but without any rolling, as at this case the fluid particles are moving similarily like when you walk on conveyort belt. In reality this kind of flow is only possible in mild slops and low velocities, like the rain water flowing on the pavement, as with greater slopes and bigger water depth the velocities will quickly run above $$Fr=1$$
This bigger scale rolling starts then exactly at $$Fr=\sqrt3$$. So I expect your bug has been the simulation running below this limiting speed.