Point-vortex integrability and long-time 2D Euler

Klas Modin

 slides.com/kmodin

Are there low dim, near integrable stable invariant manifolds?

1. Small vorticity formations merge to larger
2. Well-separated blobs interact by blob-vortex dynamics (BVD)
3. Dynamics is not integrable \(\Rightarrow\) blobs continue to merge
4. \(k\)-BVD integrable \(\Rightarrow\) quasi-periodicity prevents further mixing

Known about integrability

Point vortex dynamics on \(\mathbb{S}^2\)

• 3-PVD is integrable (Sakajo, 1999)
• 4-PVD is integrable for vanishing momentum (Sakajo, 2007)
• 4-PVD non-integrable in general (Bagrets & Bagrets, 1997)

Symplectic reduction theory:

only \(SO(3)\) symmetry needed in proof

Blob vortex dynamics on \(\mathbb{S}^2\)

• 3-BVD is integrable
• 4-BVD is integrable for vanishing angular momentum

Vanishing angular momentum