Yuichi Yatsuyanagi

Simulations of diocotron instability using a special-purpose computer, MDGRAPE-2

The diocotron instability in a low-density non-neutral electron plasma is examined via numerical simulations. For the simulations, a current-vortex filament model and a special-purpose computer, MDGRAPE-2 are used. In the previous work, a simulation method based on the current-vortex filament model, which is called “current-vortex method,” is developed. It is assumed that electric current and vorticity have discontinuous filamentary distributions, and both point electric current and point vortex are confined in a filament, which is called “current-vortex filament.” In this paper, the current-vortex method with no electric current is applied to simulations of the non-neutral electron plasma. This is equivalent to the traditional point-vortex method. MDGRAPE-2 was originally designed for molecular dynamics simulations. It accelerates calculations of the Coulomb interactions, the van der Waals interactions and so on. It can also be used to accelerate calculations of the Biot–Savart integral. The diocotron mo...

Chaotic reconnection due to fast mixing of vortex-current filaments

We propose a new reconnection mechanism “chaotic reconnection”. A basic mechanism of the chaotic reconnection is examined by means of numerical simulations of collision between two vortex-current filaments. The term “reconnection” means a reconnection of the filaments. We conclude that thle chaotic process works to enhance the reconnection rate of the filaments. We shall propose a similar chaotic process as a candidate for the mechanism of the fast magnetic reconnection.

Filamentary magnetohydrodynamic simulation model, current-vortex method

A two-dimensional simulation model of the “magnetohydrodynamic (MHD)” vortex method, current-vortex method, is developed. The concept is based on the previously developed current-vortex filament model in three-dimensional space. It is assumed that electric current and vorticity have discontinuous filamentary (point) distributions on the two-dimensional plane, and both the point electric current and the point vortex are confined in a filament. In other words, they share the same point on the two-dimensional plane, which is called the “current-vortex filament.” The spatial profiles of the electric current and the vorticity are determined by the sum of such filaments. Time development equations for a filament are obtained by integrating the two-dimensional MHD equations around the filament. It is found that a special-purpose computer, MDGRAPE-2, is capable not only of molecular dynamics simulations but also of MHD simulations, because MDGRAPE-2 accelerates calculations of the Biot–Savart integral. The curren...

Numerical Simulations of the Vortex-Current Filaments Motion

Motion of the vortex-current filaments is examined via cutoff Biot-Savart numerical MHD simulations of single and double filaments. We have introduced a vortex-current filament by analogy with the vortex filament. The vortex-current filament consists of electric current density and vorticity which are parallel to the axis of the filament. Attention is needed to demonstrate the difference between vortex and vortex-current filaments. In this paper we present two numerical results of vortex-current filament motion, using cutoff Biot-Savart method. In the limit of no electric current, the two results are in good agreement with vortex filament systems. The first result is obtained for two circular ring filaments. Here we find there are some cases where the qualitative characteristics of the motion are unchanged by introducing electric current along the vorticity. On the other hand, the period of the mutual slipping motion of the filaments is changed. The second result is related to oscillation of an elliptic r...

Formation of current-vortex filaments

Time evolution of a system where electric current and vorticity coexist is demonstrated by two-dimensional magnetohydrodynamic simulations. Such a system has two characteristics. The first one is that once the electric current and the vorticity share the same position, they tend to move simultaneously, staying overlapped with each other. The second one is that the electric current and the vorticity evolve to create more overlapping regions where they coexist, even if their initial spatial distributions are not the same. During the evolution, the distributions of the electric current and the vorticity are fragmented, and the current-vortex filaments are formed. Once the current-vortex filaments are formed, they survive stably. This is due to the strong correlation between the electric current and the vorticity.

Statistical mechanical estimate of energy spectrum for N-point vortex systems

A statistical mechanical theory of the energy spectrum for two-dimensional N -point vortex systems is developed. The system in an infinite plane is considered. We focus our attention on the energy spectrum of the system “ in equilibrium ” and “ in a transient state ”. For like-sign point vortex systems in an infinite plane, we have succeeded in deriving a scaling law of the energy spectrum E ( k )∼ k -α for the intermediate k regime, via the 2-point correlation function R 2 ( r ). However, the applicability of the derived scaling law is limited by the validity of the asymptotic expansion used. By a direct numerical simulation, we obtain various powers α=2.11–3.19 for transient states. Using the Monte Carlo simulation, for equilibrium states we find that the energy spectrum in the intermediate regime does not obey a power law. It is concluded that for point vortex systems in an infinite plane, the scaling law of the energy spectrum in equilibrium and in the transient state depends on the system parameters,...

Dynamics of two-sign point vortices in positive and negative temperature states

The characteristic features of a two-sign point vortex system in positive and negative temperature states are examined by massive numerical simulations using MDGRAPE-2. The temperature is determined by a density of states for a microcanonical ensemble consisting of randomly generated 10 7 states. Since the density of states lias a single peak. the system has negative temperature states. The distributions of vortices in time-asymptotic equilibrium states in positive and negative temperature are obtained by time-development simulations. In positive temperature, both-sign vortices mix with each other and neutralize. In negative temperature, part of the vortices condense and form clumps exclusively consisting of the same-sign vortices, while the other part of the vortices distribute uniformly outside the clumps. It is found that the vortices inside the clumps gain energy and the vortices outside the clumps lose energy to keep the total energy constant. This suggests the common and essential role of the background vortices in the energy-conserving system that assists the formation of the clumps as well as the crystallization and generation of the symmetric configuration observed in the non-neutral plasma experiments.

Explicit Formula of Energy-Conserving Fokker–Planck Type Collision Term for Single Species Point Vortex Systems with Weak Mean Flow

This paper derives a kinetic equation for a two-dimensional single species point vortex system. We consider a situation (different from the ones considered previously) of weak mean flow where the time scale of the macroscopic motion is longer than the decorrelation time so that the trajectory of the point vortices can be approximated by a straight line on the decorrelation time scale. This may be the case when the number N of point vortices is not too large. Using a kinetic theory based on the Klimontovich formalism, we derive a collision term consisting of a diffusion term and a drift term, whose structure is similar to the Fokker–Planck equation. The collision term exhibits several important properties: (a) it includes a nonlocal effect; (b) it conserves the mean field energy; (c) it satisfies the H theorem; (d) its effect vanishes in each local equilibrium region with the same temperature. When the system reaches a global equilibrium state, the collision term completely converges to zero all over the s...

Self-organization mechanism in two-dimensional point vortex system

A kinetic equation for a two-dimensional double-species point vortex system is obtained. The obtained collision term consisting of a diffusion term and a drift term has several physically good properties, for example, conserving a mean field energy and satisfying H theorem. It should be emphasized that the drift term plays an essential role for self-organization in a point vortex system with negative temperature.

Mechanism of self-organization in point vortex system

A mechanism of the self-organization in an unbounded two-dimensional (2D) point vortex system is discussed. A kinetic equation for the system with positive and negative vortices is derived using the Klimontovich formalism. Similar to the Fokker–Planck collision term, the obtained collision term consists of a diffusion term and a drift term. It is revealed that the mechanism for the self-organization in the 2D point vortex system at negative absolute temperature is mainly provided by the drift term. Positive and negative vortices are driven toward opposite directions respectively by the drift term. As a result, well-known, two isolated clumps with positive and negative vortices, respectively, are formed as an equilibrium distribution. Regardless of the number of species of the vortices, either single- or double-sign, it is found that the collision term has following physically good properties: (i) when the system reaches a quasi-stationary state near the thermal equilibrium state with negative absolute temperature, the sign of dω/dψ is expected to be positive, where ω is the vorticity and ψ is the stream function. In this case, the diffusion term decreases the mean field energy, while the drift term increases it. As a whole, the total mean field energy is conserved. (ii) Similarly, the diffusion term increases the Boltzmann entropy, while the drift term decreases it. As a whole, the total entropy production rate is positive or zero (H theorem), which ensures that the system relaxes to the global thermal equilibrium state characterized by the zero entropy production.