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Vortex dynamics on the surface of a torus

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CATW02 - Complex analysis in mathematical physics and applications

As theoretical models of incompressible flows arising in engineering and geophysical problems, vortex dynamics is sometimes considered on surfaces that have various geometric features such as multiply connected domains and spherical surfaces. The models are derived from the streamline-vorticity formulation of the Euler equations. In order to solve the model equations, complex analysis and its computational techniques are effectively utilized. In the present talk, we first review the mathematical formulation of vortex dynamics with some applications to engineering and ocean 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 physical 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 handle, yields different vortex dynamics that are not observed so far. The vortex model is not only an intrinsic theoretical extension in 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 direction without changing their relative configuration. Moreover, we derive an analytic solution of a modified Liouville equation on the toroidal surface, where the 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 corresponds to a model of shear flows in the plane known as Stuart vortex. A part of the results presented in this talk is based on the joint works with Mr. Yuuki Shimizu, Kyoto University.

This talk is part of the Isaac Newton Institute Seminar Series series.

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