Takashi Sakajo

On a generalization of the Constantin–Lax–Majda equation

We present evidence on global existence of solutions of De Gregorios equation, based on numerical computation and a mathematical criterion analogous to the Beale-Kato-Majda theorem. Its meaning in the context of a generalized Constantin-Lax-Majda equation will be discussed. We then argue that the convection term can deplete solutions of blow-up.

On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term

We consider a one-dimensional model for the three-dimensional vorticity equation of incompressible and viscous fluids. This model is obtained by adding a generalized viscous diffusion term to the Constantin–Lax–Majda equation, which was introduced as a model for the three-dimensional Euler equation (Constantin P, Lax P D and Majda A 1985 A simple one-dimensional model for the three-dimensional vorticity equation Commun. Pure. Appl. Math. 38 715–24). It is shown in Sakajo T (2003 Blow-up solutions of the Constantin–Lax–Majda equation with a generalized viscosity term J. Math. Sci. Univ. Tokyo 10 187–207) that the solution of the model equation blows up in finite time for sufficiently small viscosity, however large a diffusion term it may have. In this paper, we discuss the existence of a unique global solution for large viscosity.

The N-vortex problem on a rotating sphere. III. Ring configurations coupled to a background field

We study the evolution of N-point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices is tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break up the structure. When the centre-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate) or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the centre-of-vorticity vector associated with the N-point vortices are no longer conserved as they are for the one-way coupled model described by Newton & Shokraneh. In the first case, energy is transferred to the ring, the length of the centre-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times, but ultimately breaks-up on a longer time-scale. In the latter case, energy is extracted from the ring, the length of the centre-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its centre-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the centre-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time.

An application of Draghicescu's fast summation method to vortex sheet motion

The fast summation method of Draghicescu is applied to computation of 2D vortex sheet motion. This paper reports our numerical experiments which show the effectiveness as well as difficulties of Draghicescus fast summation method in 2D vortex-sheet computations. For instance, the fast summation method is nearly thirty times faster than the direct summation method when we use 65536 = 2 16 vortex blobs. As a test problem, we re-examine Krasnys problem of computing a vortex sheet for a fairly long time.

The motion of three point vortices on a sphere

We consider the incompressible and inviscid flow on a sphere. The vorticity distributes as a point vortex. The governing equation for point vortices on a sphere is given by Bogomolov [3]. In the present paper, we study the motion of three point vortices. We prove that the motion is integrable Hamiltonian system and its solution never blows up in finite time. Prom the viewpoint of the configuration of three vortices, we classify the motion with assistance of the numerical computation.

Point vortex equilibria and optimal packings of circles on a sphere

We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N=3–12 and 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N=13–23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006 Proc. R. Soc. Lond. Ser. A 462, 3277–3299. (doi:10.1098/rspa.2006.1731)), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra, , where A is a N(N−1)/2×N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a non-empty nullspace. The answer is yes for N=3–9, 11–14, 16 and no for N=10, 15, 17–24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N-equal strength particles, , form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and ‘grow’ a packed state using a version of the ‘yin-yang’ algorithm of Longuet-Higgins (Longuet-Higgins 2008 Proc. R. Soc. A (doi:10.1098/rspa.2008.0219)). We also implement a version of our ‘Brownian ratchet’ algorithm to find new equilibria near the optimally packed state for N=10, 15, 17–24.

Equation of motion for point vortices in multiply connected circular domains

The paper gives the equation of motion for N point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail.

Efficient topological chaos embedded in the blinking vortex system

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.

Integrable four-vortex motion on sphere with zero moment of vorticity

We consider the motion of four vortex points on sphere, which defines a Hamiltonian dynamical system. When the moment of vorticity vector, which is a conserved quantity, is zero at the initial moment, the motion of the four vortex points is integrable. The present paper gives a description of the integrable system by reducing it to a three-vortex problem. At the same time, we discuss if the vortex points collide self-similarly in finite time.

Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet

We consider an axisymmetric, swirling vortex sheet in an inviscid and incompressible flow. Caflisch et al. pointed out that the vortex sheet acquired a singularity in finite time, but the property of the singularity was not revealed. In the present paper we show convincing numerical evidences of the singularity formation by applying the same numerical methods as what was used in the study of a two-dimensional (2D) vortex sheet. We find that the radial and axial components of the axisymmetric vortex sheet behave like the 2D singularity that has been observed in many vortex-sheet motions, while the azimuthal component of the sheet behaves differently. Furthermore, the singularity appears along the vortex line and the first derivative of the vortex sheet strength forms a cusp, while the known singularities are associated with the curvature along curves perpendicular to the vortex lines and the sheet strength has a cusp.