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.

[1] (1,2)

Z Hassainia, N Masmoudi, and MH Wheeler, Global bifurcation of rotating vortex patches, to appear in Comm. Pure Appl. Math. arxiv (36 pages)

Symmetry \(m=3\)

Symmetry \(m=5\)