J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)260–262
000–000

Two-Dimensional Filamentary Mangnetohydrodynamics,
Current-Vortex Method
YATSUYANAGI Yuichi, HATORI Tadatsugu1, KATO Tomokazu2 and EBISUZAKI Toshikazu3
Kyoto University, Sakyo, Kyoto 606-8501, Japan
Kanagawa University, Hiratsuka, Kanagawa 256-1293, Japan
2
Waseda University, Shinjuku, Tokyo 169-8555, Japan
3
RIKEN, Wako, Saitama 351-0198, Japan

1(Received: 9 December 2003 / Accepted: 15 June 2004)

Abstract
A two-dimensional filamentary magnetohydrodynamic (MHD) simulation model, current-vortex method, isdiscussed. In the current-vortex method, a point electric current and a point vortex share the same position on thetwo-dimensional plane. Time evolution of an MHD system is traced by time evolution of the point electric currents
and the point vortices. A viscous effect due to the collisions between the particles may arise in such particle-basedsimulations. We demonstrate that the (instantaneous) viscosity depends on the number of the particles.
Keywords:
filamentary magnetohydrodynamics, vortex method, current-vortex filament, special-purpose computer,
MDGRAPE-2

1. Introduction

demonstrate the numerical results. In Sec. 4, we give
discussions and conclusions.

We have reported a two-dimensional filamentarymagnetohydrodynamic (MHD) simulation model, currentvortex method [1]. The concept is based on the traditionalvortex method in hydrodynamics. In the current-vortex
method, a point electric current and a point vortex share thesame position on the two-dimensional plane. Spatial profilesof the electric current and the vorticity are determined by thesum of such points. Time evolution of an MHD system istraced by time evolution of the magnetic and velocity fields
determined by the Biot-Savart integrals of the distributions of
the point electric currents and the point vortices. To accelerate
the calculations of the Biot-Savart integral, a special-purpose
computer for molecular dynamics simulations, MDGRAPE2, is used [2,3]. The current-vortex method is appropriate forsimulations of systems under the high magnetic Reynolds
number, because the spatial meshes are not necessary in thecurrent-vortex method. Only the positions of the point electric
currents and the point vortices are kept and traced in thesimulations.
It should be noted that the particles in the particle-based
simulations cause the collisions like ions and electrons. Such
collisions yield the viscosity. We demonstrate the viscosity
due to the collisions depends on the number of the particles(the point vortices).
In Sec. 2, we describe the basic equations and thecurrent-vortex method. In Sec. 3, we explain the numericalviscosity due to the collisions between the point vortices, and2. Filamentary magnetohydrodynamics
We use the two-dimensional ideal MHD equations,∂ω z= –(u ⋅ ∇)ω z + ( B ⋅ ∇) jz ,∂t(1)∂Az= –(u ⋅ ∇) Az ,∂t(2)∇ ⋅ u = 0,

(3)

E + u × B = 0,

(4)

B = – zˆ × ∇Az ,(5)ω z = zˆ ⋅ ∇ × u,

(6)zˆ
⋅ ∇ × B,
µ0

(7)

jz =

where B and u are the magnetic field and the velocity on thex-y plane, Az, jz and ωz are the z components of the magnetic
vector potential, the electric current density and the vorticity,respectively. The unit vector in z direction is denoted by ẑ.
The mass density is normalized to unity.
We assume that the electric current jz(r, t) and thevorticity ωz(r,t) have discontinuous distributions,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yyanagi@phys.h.kyoto-u.ac.jp1260

Yatsuyanagi Y. et al., Two-Dimensional Filamentary Mangnetohydrodynamics, Current-Vortex Method∑ J (t )δ (r – r (t )),ω ( r , t ) = ∑ Ω (t )δ ( r – r (t )),jz ( r , t ) =ii

electric resistivity and the kinetic viscosity. Thus, all theviscous effects in the simulations come from the numericalones. The following relation is used to determine the(instantaneous) numerical viscosity from the total kinetic
energy H and the enstrophy Γ,(8)izii(9)iwhere δ (r) is the two-dimensional Dirac delta function.
Notations Ji (t), Ωi (t) are the total electric current and thecirculation inside the i-th current-vortex filament pointed by
the position vector ri (t), respectively. They are confined ineach filament coaxially. Following the above expressions,magnetic field B(r, t) and velocity field u(r, t) are defined asfollows:∑u( r , t ) = ∑ Ω (t )∇G( r – r (t )) × zˆ.
B( r , t ) =Ji (t )∇G( r – ri (t )) × zˆ,dH= –ηΓ ,dt

(15)

H=

1 2u ( r , t )d r,2(16)Γ=1ω z2 ( r , t )d r,2(17)where(10)∫∫iii

To check the numerical viscous effect, the simulation
results of the diocotron instability in the magnetized pureelectron plasmas are used [7]. It is analytically shown thatthe two-dimensional equations of motion of low-density nonneutral electron plasmas with the guiding-center
approximation coincide with those of the two-dimensional
nonmagnetized fluids, i.e., the Euler equation. Thus, the time
evolution of the pure electron plasmas can be traced by thepoint vortex method. In this system, velocity u in energy H isdetermined by eq. (11). Enstrophy Γ is determined by(11)

iThe function G(r) is the two-dimensional Green function forthe Poisson equation. The right hand sides of eqs. (10) and(11) are the Biot-Savart integrals in the discretized form.
We rewrite the vorticity equation (1) and the magneticinduction equation (2) in terms of the filamentary
representations (8)–(11). Details are given in ref. [1]. The
obtained equations aredrk
J (t )= u( rk , t ) – k
B( rk , t ),
Ω k (t )dt(12)dΩ k (t )= 0,dt(13)d Jk (t )= 0.
dt(14)∫ (∇ ψ (r, t ) ) d r,
ψ ( r , t ) = ∑ Ω (t )G( r – r (t )),Γ=22ii(18)(19)

iwhere ψ (r, t) is stream function for the two-dimensional flow.
Note that the enstrophy is not well defined in the point vortex
system. Thus the enstrophy is determined by the stream
function ψ (r, t) that is well defined even in the point vortex
system. To evaluate the viscosity η numerically, we use eq.
(15) in the following form,

Equations (12), (13) and (14) are the equation of motion, theconservation of circulation and the conservation of total
electric current of the k-th filament, respectively.

η=3. Viscous effect
In this section, we discuss the viscous effect in particle
simulations. Here we limit ourselves to the purehydrodynamic case, i.e., Jk(t) = 0 in eq. (12) for simplicity. In
this limit, the current-vortex method exactly coincides withthe traditional point vortex method.
The point vortex method gives the analytically correct
solution to two-dimensional incompressible Euler equation[4]. On the other hand, in the limit of infinite number of point
vortices, the discontinuous vorticity field (9) converges to thecontinuous one. In this limit, the equations of the point vortex
method converge to the Euler equation for continuous fluids[5]. However, it appears that using an increased number ofvortices of decreased strength will not yield a converged
solution in numerical simulations [6]. This may be due to theviscosity that originates with the collisions between the point
vortices, even if there is no explicit viscous diffusion term in
the Euler equation. Such viscosity does not arise in the meshbased simulations for the continuous fluids, because there isno collisional process in the macroscopic fluid scale in theEuler equation.
In the model, we neglect the viscous parameters, i.e., theH (t ) – H ( 0 ),tΓ (t )(20)

becasue there are very small fluctuations in H(t) obtained by
the simulation results and it is difficult to evaluate theinstantaneous value of dH(t)/dt exactly.
Initial conditions are as follows. Initial ratio of the inner
to the outer radii of the annular electron distribution is 0.6.
There is no conducting wall around the electrons. We compare
the simulation results where total numbers of the pointvortices are 1.0 × 104, 4.0 × 104, 6.0 × 104 and 8.0 × 104. In
these simulations, the values of the circulation are equal. Timeevolutions of the instantaneous viscosity are plotted in Fig. 1.
In Fig. 1, linear growth stage is T = 30 – 90. In thisperiod, annular electron layer develops into clumps. After thelinear growth stage, the clumps are merged and the electron
distribution shows broad one. Then, the electrons aretransferred by the near-steady circular flow and there may beless collisional processes than the initial stage. Thus thephenomenon in the linear growth stage is more important thanthat in the nonlinear stage for us to reveal the mechanism of
the viscosity due to the collisions.
In the linear growth stage, the viscosity in the 1.0 × 104
2261

Yatsuyanagi Y. et al., Two-Dimensional Filamentary Mangnetohydrodynamics, Current-Vortex Method

4. Discussion and conclusion

Fig. 1

We have presented the simulation model where theelectric current and the vorticity are discretized in the samemanner as the traditional vortex method in hydrodynamics.
Time evolution of an MHD system is traced by time evolutionof the magnetic and velocity fields determined by the BiotSavart integrals of the distributions of the point electric
currents and the point vortices. To accelerate the calculationsof the integral, MDGRAPE-2 has been used. The model issuitable for the simulations of the high magnetic Reynolds
number.
We have demonstrated that the viscosity depends on thenumber of vortices. It seems likely that the value approaches
to zero at infinite number of point vortices, though it is not
yet demonstrated by the simulations. It is nessesary todetermine the critical value of the number of vortices wherethe viscosity due to the collisional effect becomes maximum.
We are now analytically evaluating the viscosity on thecurrent-vortex method.

Time evolutions of the instantaneous viscosity areplotted.

particles case is minimum, and the maximum viscosity
corresponds to the 2.0 × 104 particles case. In addition, during
all the simulation time, the 1.0 × 104 particles case shows thelowest viscosity. Thus we conclude that the viscosity is dueto the collisional process of the point vortices, even if there
is no explicit viscous diffusion term in the Euler equation.
On the other hand, the viscosity due to the collisions should
converge to zero for the continuous fluid, i.e., in the limit of
infinite number of vortices. In the 6.0 × 104 and 8.0 × 104
particles cases, the values of the viscosity is larger than the1.0 × 104 particles case, but smaller than the 2.0 × 104
particles case. Thus it is likely that the viscosity increases as
the number of vortices n increases upto a cirtain critical value,
say n0, and decreases for larger n (> n0). The value of n may
be between 1.0 × 104 and 6.0 × 104. Note that the viscosity
does not yet converge to zero in the simulation results. It isvery difficult to demonstrate the zero-viscosity result by thesimulation.

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[3] R. Susukita et al., Comput. Phys. Comm. 155, 115(2003).
[4] J.P. Christiansen, J. Comput. Phys. 135, 189 (1997).
[5] J. Goodman, T.Y. Hou and J. Lowengrub, Comm. Pure
Appl. Math. 43, 415 (1990).
[6] A. Leonard, J. Comput. Phys. 37, 289 (1980).
[7] Y. Yatsuyanagi et al., Phys. Plasmas 10, 3188 (2003).

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