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Vortex Stretching by a Simple Hyperbolic Saddle

In: Applied and Industrial Mathematics, Venice—2, 1998

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  • Diego Cordoba

    (Princeton University, Department of Mathematics)

Abstract

We study solutions to the 3D Euler vorticity equation of the form $$\omega = \tilde \omega (x,t)\left( {\frac{{\partial t}}{{\partial {x_2}}},\frac{{\partial t}}{{\partial {x_1}}},0} \right) $$ in a neighborhood U. When the curvesf (x 1 x 2 t) = const are circles then these solutions are the well known axisymmetric 3D flow without swirl, and for this case there is no vortex stretching. If we assumef(x 1,x2, t) = const to be a set of curves that contain a simple hyperbolic saddle then vortex stretching may take place. We show that the angle of the saddle can not close faster than a double exponential in time and there is no breakdown. Similar results are obtain in two dimensional models.

Suggested Citation

  • Diego Cordoba, 2000. "Vortex Stretching by a Simple Hyperbolic Saddle," Springer Books, in: Renato Spigler (ed.), Applied and Industrial Mathematics, Venice—2, 1998, pages 229-234, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-4193-2_15
    DOI: 10.1007/978-94-011-4193-2_15
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