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The spiral form of vortex breakdown observed in the numerical simulations of Ruith . (2003) is interpreted as the consequence of the development of a so-called nonlinear global mode originating in the convective/absolute transition of the instability in the lee of the vortex breakdown bubble.This local theory gives an excellent prediction of the precession frequency measured in the threedimensional DNS.

Motions of fluid particles advected by a 3D vortex filament are studied. As extensions of a circular vortex ring, which provides a basis of fluid transport by vortex, two types of motions, elliptic vortex ring and vortex soliton are considered. For an elliptic vortex ring, it is verified that the local induction equation (LIE) describes its motion properly if the aspect ratio is close to 1. Using the solution of LIE, particle motions are simulated for two different induction constants corresponding to a thin and a fat vortex for the circular vortex ring case, respectively. Intrinsic difficulty for calculating the finite transported volume by an elliptic vortex ring is suggested. For a vortex soliton, it is demonstrated that particle motions are confined in a torus near the kink of a soliton for a wide range of parameters that characterize the shape and the strength of the vortex soliton.

A new linear instability mechanism of curvature origin is established for a vortex ring. The curvature effect reduces O(2) × SO(2) symmetry of a circularcylindrical tube to O(2), and fuels a pair of Kelvin waves whose azimuthal wavenumbers on the core are separated by one. For Kelvin’s vortex ring, the growth rate and eigenfunctions are written out in closed form. In the inviscid case, the curvature effect dominates over the elliptically straining effect, but the former suffers from enhanced viscous damping. There are numerous excitable modes. As a first step toward an understanding of the route to a matured stage, we derive equations for weakly nonlinear evolution of amplitudes of the curvature instability. Our direct numerical simulation successfully captures the elliptical instability.

We consider the motion of identical vortex points on a sphere with two vortex points fixed at the both poles. The vortex points are spaced equally along a line of latitude, which is called the polygonal ring of vortex points or the -ring. Starting with the linear stability analysis, we investigate the unstable motion of the perturbed -ring; We give a brief summary in terms of the transition of unstable periodic motions for even -ring studied in a preceding paper (Sakajo 2004). Then we study an unstable non-trivial recurrent motion of a 3-ring as an example of complex dynamics of odd -rings.

We discuss the effect that the presence of a small viscosity has on the evolution of fields that are transported unchanged in the absence of viscosity. We employ a diffusive Lagrangian formulation and show that the Cauchy invariant, the helicity density, the Jacobian determinant, and the virtual velocity obey parabolic equations that are well-behaved as long as the diffusive transformations are invertible. We call such quantities . We showby numerical calculations that the loss of invertibility of the diffusive transformation can occur, and that the time scale on which it does can be short even when the viscosity is small. We present quantitative evidence relating the loss of invertibility to the physical phenomenon of vortex reconnection.

The analyticity strip method is used to trace complex singularities in direct numerical simulations of the Taylor-Green flows, performed with up to 2048 collocation points. No indication of finite-time real singularity is found. Simulations are also carried out beyond the time at which the truncated equations cease to approximate the original Euler equations. Kolmogorov-like turbulence is then obtained during an intermediate regime of the spontaneous relaxation of (time-reversible) spectrally-truncated Euler equations towards absolute equilibrium.

Helicity produced by nearly singular vortex interactions is shown to play a role in the ensuing development of turbulence. This might provide a link between turbulence and the dynamics of the three-dimensional Euler equations, where numerical evidence has suggested that there might be a singularity. Interactions between regions of oppositely signed helicity in both physical and Fourier space are shown to be associated with the transfer of energy to small scales and the formation of vortex tubes, both being properties of fully developed turbulence.

Superfluid turbulence has been one of the most important problems of superfluid hydrodynamics, in which quantized vortices play a significant role. The recent research on superfluid turbulence enters a new stage rather different from the old ones chiefly devoted to thermal counterflow. After describing the current motivation on these topics, we discuss our research which studies the energy spectrum both by the vortex filament model and the Gross-Pitaevskii model. Both energy spectra are consistent with the Kolmogorov law, which shows a close similarity between superfluid (quantum) and classical turbulence.

A study by computer simulation is reported of the behavior of a quantized vortex line at a very low temperature when there is continuous excitation of Kelvin waves with a low wave number. The energy of Kelvin wave is dissipated only at very high wave numbers. It was shown in previous report (Vinen . 2003) that nonlinear coupling leads to a net flow of energy to higher wave numbers and to the development of a simple spectrum of Kelvin waves. These results are likely to be relevant to the decay of turbulence in superfluid He at very low temperatures. To identify the wave number dependence of this spectrum more precisely, we improve the excitation and dissipation method. In this method, the operations of both excitation and dissipation are done in the Fourier space as contrasted with the previous method, whose operations were performed in the real space. The present results are consistent with our previous results not only on the wave number dependence but also on the absolute value. This means the spectrum that we have got is robust one.

The flow phase diagram predicted by Volovik (2003) is discussed based on available experimental data for He II and He-B superfluids. The effective temperature-dependent but scale-independent Reynolds number = 1/ ≡ (1−α′)/α, where α and α′ are the mutual friction parameters, and the superfluid Reynolds number characterizing the circulation of the superfluid component in units of the circulation quantum are used as the dynamic parameters.