Higher-order asymptotic theory and numerical analysis for three-dimensional dynamics of a vortex filament in a flw

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1. Three-dimensional motion of a vortex filament was investigated theoretically.i) A general formula for velocity of an axisymmetric vortex ring in a viscous fluid was obtained. Numerical computation was carried through for an infinitely thin core at the initial instant. Expansion of ring radius compares well with an experimental measurement.ii) Asymptotic development of the Biot-Savart integral and matched asymptotic expansions were extended to a high order, whereby the third-order correction to the speed of a vortex tube was obtained. Its relevance to the localized induction hierarchy was discussed.2. Three-dimensional instability of a vortex ring was calculated from the viewpoint of Hamiltonian spectrum theory. It was shown that a parametric resonance occurs between axisymmetric and bending modes due to the curvature effect.3. Direct numerical simulations of the Navier-Stokes equations using a highly accurate finite difference scheme were performed for generation and scattering of sound waves by vortices.i) An interaction of shock waves with a vortex ring was calculated and mechanism for sound generation was clarified.ii) Pressure fluctuations of small amplitude in a far region, generated by a head-on collision of vortex rings, was successfully computed.iii) With a numerical simulation, asymptotic theories for scattering of sound by Hill's spherical vortex were assessed.iv) A numerical simulation of the Gross-Petaevskii equation was performed for sound generation in the process of reconnection of quantized vortices.4. Using a model equation for MHD turbulence, the effect of an ordered structure in large-scale magnetic field upon scaling of characteristic time and intermittency was examined.