# Rotating vortex patch numerics¶

In [1] we rigorously construct highly nonlinear rotating vortex patch solutions to the two-dimensional Euler equations. These are regions $$D$$ with unit vorticity in an otherwise-irrotational fluid that rotate with a constant angular velocity $$Ω$$. While the problem can be formulated as an integral equation for the boundary $$∂ D$$ of the patch, our proof also involves the flow outside the patch, which is determined by a stream function $$Ψ$$. We were unable to find any contour plots of $$Ψ$$ in the literature, and so we did our own numerics.

In the figures below, the yellow region is the vortex patch $$D$$, the black lines are streamlines in the moving frame (level curves of $$Ψ$$), the black dots are stagnation points, and the red lines are the singular streamlines which pass through a saddle point. See [1] for more information, and for a variety of other plots.