One-dimensional hydrodynamic model generating a turbulent cascade

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.Comment: 5 pages, 4 figure

Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices

This paper deal with the motion of a polygonal ring of identical vortex points that are equally spaced at a line of latitude on a sphere with vortex points fixed at the both poles, which we call "N-ring". We give not only all the eigenvalues but also all the eigenvectors corresponding to them for the linearized steationary N-ring. Then, we also reduce the equations to those for a pair of two vortex points, when N is even. As a consequence of the mathematical and numerical studies of the reduced system, we obtain a transition of global periodic motions of the perturbed N-ring and the stability of these periodic motions

Non self-similar, partial and robust collapse of four point vortices on sphere

This paper gives numerical examples showing that non self-similar collapse can occur in the motion of four point vortices on a sphere. It is found when the $4$-vortex problem is integrable, in which the moment of vorticity vector is zero. The non self-similar collapse has significant properties. It is \textit{partial} in the sense that three of the four point vortices collapse to one point in finite time and the other one moves to the antipodal position to the collapse point. Moreover, it is \textit{robust} with respect to perturbation of the initial configuration as long as the system remains integrable. The non self-similar, robust and partial collapse of point vortices is a new phenomenon that has not yet been reported

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

Spot Dynamics of a Reaction-Diffusion System on the Surface of a Torus

Quasi-stationary states consisting of localized spots in a reaction-diffusion system are considered on the surface of a torus with major radius $R$ and minor radius $r$. Under the assumption that these localized spots persist stably, the evolution equation of the spot cores is derived analytically based on the higher-order matched asymptotic expansion with the analytic expression of the Green's function of the Laplace--Beltrami operator on the toroidal surface. Owing to the analytic representation, one can investigate the existence of equilibria with a single spot, two spots, and the ring configuration where $N$ localized spots are equally spaced along a latitudinal line with mathematical rigor. We show that localized spots at the innermost/outermost locations of the torus are equilibria for any aspect ratio $alpha=frac{R}{r}$. In addition, we find that there exists a range of the aspect ratio in which localized spots stay at a special location of the torus. The theoretical results and the linear stability of these spot equilibria are confirmed by solving the nonlinear evolution of the Brusselator reaction-diffusion model by numerical means. We also compare the spot dynamics with the point vortex dynamics, which is another model of spot structures

Statistical properties of point vortex equilibria on the sphere

We describe a Brownian ratchet scheme which we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a linear algebra problem $A\Gamma = 0$ where $A$ is a $N \times N(N − 1)/2$ non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant, and $\Gamma \in R^N$ is the (unit) vector of vortex strengths which must lie in the nullspace of $A$. Existence of an equilibrium is expressed by the condition $det($A^TA) = 0$, while uniqueness follows if$Rank(A) = N−1$. The singular value decomposition of$A$is used to calculate an optimal basis set for the nullspace, yielding all values of the vortex strengths for which the configuration is an equilibrium. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and after each random step we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the singular value drops below a predetermined convergence threshold, an equilibrium configuration is achieved and we find a basis set for the nullspace of A by calculating the right singular vectors corresponding to the singular values that are zero. For each$N = 4 \rightarrow 10$, we generate an ensemble of 1000 equilibrium configurations which we then use to calculate statistically averaged singular value distributions in order to obtain the averaged Shannon entropy and Frobenius norm of the collection. We show that the statistically averaged singular values produce an average Shannon entropy that closely follows a power-law scaling of the form$< S > \sim N^\beta$, where$\beta \sim 2/3$. We also show that the length of the conserved center-of-vorticity vector clusters at a value of one and the total vortex strength of the configurations cluster at the two extreme values ±1, indicating that the ensemble average produces a single vortex of unit strength which necessarily sits at the tip of the center-ofvorticity vector. The Hamiltonian energy averages to zero reflecting a relatively uniform distribution of points around the sphere, with vortex strengths of mixed sign