126 research outputs found
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 -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 and minor radius . 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 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 . 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 where is a
non-normal configuration matrix obtained by requiring that all inter-vortical distances
on the sphere remain constant, and is the (unit) vector of vortex strengths
which must lie in the nullspace of . Existence of an equilibrium is expressed by the
condition A^TA) = 0Rank(A) = N−1AN = 4 \rightarrow 10< S > \sim N^\beta\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
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