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SUMMARY:Vortex dynamics on the surface of a torus - Takashi Sakajo (Kyoto 
 University\; Kyoto University)
DTSTART:20191101T113000Z
DTEND:20191101T123000Z
UID:TALK133594@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 first review the mathematical fo
 rmulation of vortex dynamics with some applications to engineering and oce
 an problems. We then pay attention to vortex dynamics on the surface of a 
 torus. Although the flows on the surface of a torus is no longer a physica
 l relevance to real fluid flow phenomena\, it is theoretically interesting
  to observe whether the geometric nature of the torus\, i.e.\, a compact\,
  orientable 2D Riemannian manifold with non-constant curvature and one han
 dle\, yields different vortex dynamics that are not observed so far. The v
 ortex model is not only an intrinsic theoretical extension in the field of
  classical fluid mechanics\, but it would also be applicable to modern phy
 sics such as quantum mechanics and flows of superfluid films. Based on the
  model of point vortices\, where the vorticity distribution is given by di
 screte delta measures\, we investigate equilibrium states of point vortice
 s\, called vortex crystals\, moving in the longitudinal direction without 
 changing their relative configuration. Moreover\, we derive an analytic so
 lution of a modified Liouville equation on the toroidal surface\, where th
 e vorticity distribution is given by an exponential of the stream-function
 . The solution gives rise to a vortex crystal with quantized circulations 
 embedded in a continuous vorticity distribution in the plane\, which corre
 sponds to a model of shear flows in the plane known as Stuart vortex. A pa
 rt of the results presented in this talk is based on the joint works with 
 Mr. Yuuki Shimizu\, Kyoto University.
LOCATION:Seminar Room 1\, Newton Institute
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