25E10
Gyrokinetic Analysis of Phase-Space Interactionsin Two-Dimensional Electrostatic Turbulence
２次元静電乱流の位相空間相互作用に関するジャイロ運動論解析	 
T. Tatsuno*1,*2, G. G. Plunk*2
龍野	 智哉*1,*2,	 G. G. Plunk*2*1

Department of Communication Engineering and Informatics
The University of Electro-Communications
1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan
電気通信大学	 情報・通信工学科	  〒182-8585 東京都調布市調布が丘1-5-1*2

Institute for Research in Electronics and Applied Physics,
The University of Maryland, College Park, Maryland 20742, USA
Various local and nonlocal interactions may take place in the phase-space turbulence of magnetized weakly
collisional plasmas. Using electrostatic gyrokinetic simulations, it is shown that those interactions may becharacterized by the ratio between two collisionless invariants in two dimensions. Theoretically such avariety of interactions may be explained by generalizing the argument of Fjørtoft [Tellus 5 (1953)
225]. Practical application might include the control of large-scale fluctuations by means of the excitation of a particular small-scale perturbation due to the nonlocal interactions.

1. Introduction
turbulence [9]. The possibility of the nonlocal
Plasmas in fusion device, solar wind and others
interactions may be a potential problem inare usually collisionless or at most weakly colliconstructing a renormalized theory of turbulence.
sional. In such weakly collisional plasmas, turWe first introduce our model in Sec. 2 andbulence proceeds in phase space [1-8]. In magpresent the numerical results in Sec. 3.
netized plasmas, free streaming of particles parallel to the ambient field line brings about linear
2. Modelphase mixing, which ends up with the collisional
We use the normalized 2D electrostatic GKdissipation of entropy through creation ofequation∂gsmall-scale structures in parallel velocity [2].
(1)	 + φ R ,g = C R ,On the other hand, phase mixing perpendicular∂t
to the ambient field proceeds nonlinearly due towhere g is the gyro-average of the perturbed disthe small-scale fluctuations of electrostatic potribution function, φ is the electrostatic potential,
tential [3]. When gyrokinetic (GK) theory is• R is the gyro-average with the gyro-center posiapplied, the gyro-averaged potential differs€
among particles with different orbits, thus intro- € tion R fixed, {a,b} = zˆ × ∇a⋅ ∇b is the Poisson
ducing their decorrelation. In this case the vebracket, €and C is the collision operator. The
locity and position spaces are strongly coupled,electrostatic potential φ is obtained via theand as the turbulence proceeds, the entropy€casquasi-neutrality condition
cades in the velocity space as well as in the posi-€€g r dv = φ − φ R ,(2)€tion space [4-8].
r
In two dimensions (2D) there is another colliwhere • r is€ the gyro-average with the particle
sionless invariant in addition to the entropy,
position r fixed. Here we assumed zero rewhich is associated with the electrostatic potensponse electrons where electrons make no contributial [6,8,9]. As the 2D Navier-Stokes (NS) turtion€to the potential. This system of equations
bulence exhibits a dual cascade [10,11], 2D GK € possesses two collisionless invariants
also exhibits a dual cascade, where the difference€1
W =g 2 dRdv ,from the NS turbulence is that both velocity and2V
position spaces (or a phase space) need to be1
taken into account [9].
E=φ 2 + φ φ R r dr ,2V
In this presentation, we show that there arewhere W tends to cascade forward and E does
various routes of interactions in the phase-space{}

∫∫∫

∫

€

€

€

€

inversely.
Equations (1)-(2) are numericallysolved using an open-source, MPI-parallelized Fortran 95 code AstroGK [12].
3. Results
In order to characterize the velocity space structure we introduce the Hankel transform in the perpendicular velocity gˆ ( p) = J 0 ( pv⊥)g(v⊥)v⊥dv⊥ ,∫where J 0 is the Bessel function. Position spacestructure may be characterized by the wave number
k via the standard Fourier transform.
Figure 1 €shows the spectral density distribution€ W (k, p) = kp gˆ (k, p) 2 for three runs of the decay-€€€

ing-turbulence simulations made with varying initial ratio between W and E ( κ = W / E ).
While W corresponds to the spectral density over
the whole (k, p) plane, E is concentrated only
on the diagonal ( k = p ) as only those components€potential. Thus small€
contribute €the electrostatic€ κ corresponds to a state where spectral density is€ the diagonal, while large κ€concentratedalong€ to the other.
corresponds
In the middle case [moderate κ ; Fig. 1(c) and(d)], small-scale initial condition exhibits a typical€ in a difdual cascade where W cascades forward
fuse way and E cascades inversely along the2diagonal. When κ /k€
0 is smaller than unity [Fig.
1(a) and (b)], where k0 is the wave number of the€initial condition, the inverse cascade becomes non€local, which may be explained by the Kel€vin-Helmholtz
instability of the initial condition.
2€
When κ /k0 is much larger than unity [Fig. 1(e)and (f)], on the other hand, the large-scale diagonalcomponent is transferred to small scale directly as
the spectral diffusion of W reaches the diagonal.
€ This nonlocal transfer of E arises due to its conservation property during the evolution. Interestingly, in Fig. 1(f), the large-scale E is not suffi€ the spectral diffusion of W andcient to support€ density contains a dip along thethus the spectral
diagonal [Fig. 1(f)].
In the presentation, we€will express our results in€ explanation in
more detail including the theoreticalconjunction with the Fjørtoft’s theory [11].
Acknowledgments
This work is supported by the U.S. DOE Center
for Multiscale Plasma Dynamics, STFC, and the
Leverhulme Trust International Network for Magnetised Plasma Turbulence. Numerical simulations were performed at NCCS, NCSA, NERSC,NICS, and TACC.

Figure 1: Example spectral distributions. Diagonal( k = p ) is indicated by the dotted line.

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