25G11

Diffusion term implicitly appearing in the point vortex solution
for two-dimensional inviscid Euler equation
2 次元オイラー方程式における点渦解に現れる粘性Yuichi Yatsuyanagi and Tadatsugu Hatori1

八柳祐一 ，羽鳥尹承2

1

Shizuoka University, Faculty of Education, 836 Oya, Suruga-ku, Shizuoka 422-8529, Japan
2

National Institute for Fusion Schience, 322-6 Oroshi-cho, Toki, 509-5292 Gifu, Japan1

静岡大学教育学部 〒 422-8529 静岡市駿河区大谷 836
2
核融合科学研究所 〒 509-5292 土岐市下石町 322-6

Diffusion coefficient implicitly included in the point vortex solution for the two-dimensional inviscid Euler
equation is examined analytically. This diffusive effect arises from a discrete distribution of the vorticity.
The obtained diffusion coefficient includes a position correlation in addition to a time correlation. It can be
regarded as an extension of the well-known Green-Kubo formula.

1. Introduction
To explain a large scale structure formation, for example, great red spot on Jupiter, eddy at
Naruto, and typhoon, Onsager introduced a concept,"negative temperature" for the two-dimensional (2D)
point vortex system [1]. If absolute temperature ofa system is negative, there is more possible state at
higher energy than lower energy as the probability is
proportional to exp(-bH)) (H: system energy).
Much research effort has been devoted tounderstand the negative temperature state in thecontext of 2D turbulence [2-6]. One remarkable result may be an derivation of a mean field equation
for the point vortex system. This equation is calledsinh-Poisson equation [7]. Later, another member ofMontgomery group reported that a time asymptotic
distribution of 2D Navier-Stokes system at highReynolds number reached a state predicted by thesinh-Poisson equation. This implies that an equilibrium state for the viscous Navier-Stokes system issimilar to one for the inviscid point vortex systemand that the point vortex system may have a diffusive effect.
On the other hand, in the review article ofthe point vortex method, Leonard said that "It now
appears that using an increased number of point vortices of decreased strength will not yield a converged
solution. ... Ironically, best results with the point vortex method often are achieved by using only a few

vortices with a diffusive time integration scheme.[8]"This statement also implies a diffusive effect implicitly included in the point vortex system.
From these backgrounds, we have started an analytical estimate of the discussion coefficient for the 2D
point vortex system.
The organization of this paper is as follows.
First, we introduce the point vortex system as a microscopic solution. Second, the main result of thediffusion coefficient is presented. At last, we discuss
the result.
2. Euler Equation and Klimontovich Equation
The 2D Euler equation (vorticity equation)∂wz+ u ⋅ ∇wz = 0 			∂t
has an discrete solution:
wz (r , t ) = ∑ Wi d (r − ri (t )) 		
i(1)

(2)

where u(r,t) and wz(r,t) are the velocity field and thevorticity field. The solution (2) is called the point
vortex solution. A position and strength of the i-th
point vortex is given by ri = ri (xi , yi) and Wi. The
value of Wi is either W0 or -W0 where W0 is a positive constant. The vorticity field is discretized by theDirac delta function d(r - ri).
Usually, fluid equation that describes macroscopic phenomena has a macroscopic smooth solution. However, Eq. (2) is not a smooth solution. Thus

we consider the solution (2) is not a solution forthe macroscopic Euler equation but a solution for a
microscopic Euler equation whose form is identicalwith Eq. (1). To distinguish the microscopic equationfrom the macroscopic equation, we introduce a notation "hat". A variable with the hat means it microscopic.
∂wˆz(3)+ uˆ ⋅ ∇wˆz = 0∂tSimilar situation can be found in plasma
physics. Time evolution of an exact phase space densityfˆ(r , v, t ) = ∑ d(r − ri (t ))d(v − vi (t ))(4)
iis given by the Klimontovich(-Dupree) equationˆ∂fˆqˆ ) ⋅ ∂f = 0 .
(5)+ v ⋅ ∇fˆ + (Eˆ + v × B∂v∂tm
Taking an ensemble average of Eq. (5) yields a kinetic equation for the averaged phase space density,for example, Fokker-Planck equationq∂f∂   ∂f ∂f=⋅ D ⋅+ v ⋅ ∇f + (E + v × B ) ⋅ ,m∂v
∂v  ∂v ∂t(6)and in the inviscid limit, Vlasov equationq∂f∂f=0 .
+ v ⋅ ∇f + (E + v × B ) ⋅(7)m∂v
∂tNote that the Vlasov equation is the inviscid equation in approximation.
We consider that the microscopic Euler
equation that has the point vortex solution corresponds to the Klimontovich equation. The macroscopic, inviscid Euler equation corresponds to theVlasov equation. There is no corresponding equation
to the Fokker-Planck equation. Thus we derive it
analytically using the Klimontovich formalism [9].
3. Diffusion Coefficient for the Point Vortex SystemThe starting equation is the microscopic Euler equation (3). We assume each microscopic variable consists of an averaged macroscopic part and afluctuation.
wˆz (r , t ) = wˆz (r , t ) + dwˆz (r , t )= wz (r , t ) + dwˆz (r , t )(8)

Substituting Eq. (8) into Eq. (3) and averaging it, weobtain the following macroscopic Euler equation including a diffusion term in the right hand side:∂wz (r , t ) + ∇ ⋅ [u(r , t )wz (r , t )] = −∇ ⋅ duˆ(r , t )dwˆz (r , t )∂t(9)To evaluate Eq. (9), we introduce an linearized equation for the fluctuation.
∂dwˆz (r , t ) + u(r , t ) ⋅ ∇dwˆz (r , t ) = −duˆ(r , t ) ⋅ ∇wz (r , t )∂t(10)

This equation can be integrated:dwˆz (r , t ) = −∫t−∞d τduˆ(r − (t − τ )u(r , t ), τ ) ⋅ ∇(11)
wz (r , t )The resulting formula is given by∂wz (r , t ) + u(r , t ) ⋅ ∇wz (r , t ) = −∇ ⋅ (η ⋅ ∇wz )∂ttη = ∫ d τ duˆ(r , t ) duˆ(r − (t − τ )u, τ )

(12)

−∞

4. Discussion
The resulting equation is an extension of
the well-known Green-Kubo formula. In our result
position correlation due to the macroscopic flow u
is included in addition to time correlation. It may bepossible to evaluate the diffusion term numerically.
Acknowledgement
This work has been supported by KAKENHI(22740254).
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