A geometrical study of 3D incompressible Euler flows with Clebsch potentials

Koji Ohkitani, Department of Applied Mathematics, University of Sheffield

We consider a special class of flows which have Clebsch potentials to explore possibility of blow up of the incompressible 3D Euler equations. This is perhaps geometrically the simplest, but nevertheless nontrivial vortex. After briefly reviewing what Clebsch has done, we introduce a criterion for geometric non-degeneracy, which should be satisfied for a possible blowup. Some preliminary results of numerical simulations will be presented. We will consider two kinds of initial conditions: (1) a simple choice of Clebsch potentials, and (2) Kida's high symmetric flow. For (1), we test the above-mentioned criterion for the early stage evolution. For (2), we derive the expressions for initial Clebsch potentials and observe their evolution.

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