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SUMMARY:Vortex dynamics on the surface of a torus - Takashi Sakajo (Kyoto 
 University\; Kyoto University)
DTSTART:20191101T113000Z
DTEND:20191101T123000Z
UID:TALK133591@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:As theoretical models of incompressible flows arising in engin
 eering and geophysical problems\, vortex dynamics is sometimes considered 
 on surfaces that have various geometric features such as multiply connecte
 d domains and spherical surfaces. The models are derived from the streamli
 ne-vorticity formulation of the Euler equations. In order to solve the mod
 el equations\, complex analysis and its computational techniques are effec
 tively utilized. In the present talk\,  we consider vortex dynamics on the
  surface of a torus. Although the flows on the surface of a torus is no lo
 nger a physical relevance to real fluid flow phenomena\, it is theoretical
 ly interesting to observe whether the geometric nature of the torus\, i.e.
 \, a compact\, orientable 2D Riemannian manifold with non-constant curvatu
 re and one handle\, yields different vortex dynamics that are not observed
  so far. The vortex model is not only an intrinsic theoretical extension i
 n the field of classical fluid mechanics\, but it would also be applicable
  to modern physics such as quantum mechanics and flows of superfluid films
 . Based on the model of point vortices\, where the vorticity distribution 
 is given by discrete delta measures\, we investigate equilibrium states of
  point vortices\, called vortex crystals\, moving in the longitudinal dire
 ction without changing their relative configuration. Moreover\, we derive 
 an analytic solution of a modified Liouville equation on the toroidal surf
 ace\, where the vorticity distribution is given by an exponential of the s
 tream-function. The solution gives rise to a vortex crystal with quantized
  circulations embedded in a continuous vorticity distribution in the plane
 \, which corresponds to a model of shear flows in the plane known as Stuar
 t vortex. A part of the results presented in this talk is based on the joi
 nt works with Mr. Yuuki Shimizu\, Kyoto University.
LOCATION:Seminar Room 1\, Newton Institute
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