24D07
Nonlinear processes of zonal flows near a plasma edge
プラズマ境界近傍における帯状流の非線形過程	 
Makoto Sasaki1,2, Kimitaka Itoh2,3, Sanae-I Itoh1,2 and Naohiro Kasuya2,3
佐々木真1,2, 伊藤公孝2,3，伊藤早苗1,2，糟谷直宏2,3
1

Research Institute for Applied Mechanics, Kyushu University, 6-1, Kasuga Fukuoka 816-8580, Japan1
九州大学応用力学研究所	  〒816-8580	  福岡県春日市春日公園6-1
2
Itoh Research Center for Plasma Turbulence, Kyushu University,
6-1, Kasuga Fukuoka 816-8580, Japan2
九州大学伊藤極限プラズマ研究連携センター	  〒816-8580	  福岡県春日市春日公園6-1
3
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan3核融合科学研究所	  〒509-5292	  岐阜県土岐市下石町322-6

The new mechanism of the mean flow generation is presented, investigating the system which consists ofturbulence, and stationary zonal flow (ZF) with considering the plasma edge effect. Based on the fluid
model, the set of coupling equations among turbulence, ZF, and mean flow is derived by evaluating theperpendicular Reynolds stress with the wave-kinetic equation. The model of the plasma edge is proposed:
the zero frequency components, ZF and mean flow, exchange the poloidal momentum with the flow in thescrape off layer (SOL) and becomes same with that in the SOL at the plasma edge. The effect of the plasmaedge is introduced in the coupling equations as the boundary condition. The finite mean flow is obtained,
whose value is comparable to the diamagnetic drift velocity.

1. Introduction
wave-kinetic equation. We consider the system thatSince the discovery of the high confinement
consists of turbulence and ZF near the plasma edge,
mode (H-mode) [1], the formation mechanism ofand the mean flow is shown to be excited. Thethe H-mode has been attracted much attention.
vorticity of the ZF is assumed to be the monotonic
The importance of a mean flow in the poloidal
plane wave. The vorticity of the zero frequency
direction near the plasma edge is widelycomponents is written as
recognized [2]. The strong radial electric field is(1),U = U Z e iq Z x + U Z* e −iq Z x + U MF
observed near the edge [3]. The generation
where qZ is the radial wavenumber of the ZF, andmechanism of the mean flow has been studied
U Z , U MF are the vorticity of the ZF and mean flowtheoretically [4]. The stationary zonal flow (ZF)
components, respectively. The complex conjugate
has been observed during the L-H transition [5]. €*It becomes more and more important to € of U Z is denoted by U Z . The set of the coupling
equations among the turbulence, ZF, and mean flow
understand the interaction among turbulence, ZFs€
is derived. We investigate the mean flow generationand mean flow.
in the case when the ZF is directly driven byIn this study, the new mechanism of mean flow
€turbulence. €The finite amplitude ZF in thegeneration is studied, analyzing the nonlinear
stationary state is obtained from the coupling
coupling between turbulence and ZF near theequation.
plasma edge. Due to the momentum exchange
The model of the plasma edge is proposed, andacross the plasma boundary, the net poloidal
theboundary condition for the coupling equation ismomentum can become finite in the plasma nearintroduced.
The zero frequency components canthe edge to generate a poloidal mean flow. In Sec.
exist independent of the topology of the magnetic
2, the model of the generation of mean flow nearfield. Actually, there is a flow with zero frequency
the plasma edge is described. The summary andin the scrape off layer (SOL) region, which isdiscussion are given in Sec. 3.
strongly prescribed by the particle and heattransport from the plasma. The zero frequency flow
2. Model of Mean Flow Generation
inside the plasma should be connected with that in
The model equations are based on the fluid
the SOL. Namely, they are equal at the edge byequations in the simple magnetic geometry (highexchanging their poloidal momentum. Therefore,
aspect ratio, circular plasma) [6]. The nonlinear
the boundary condition for the zero frequency
terms are focused on that from the perpendicularcomponents can be written as
Reynolds stress, which is evaluated with theU Z e iq Z a + U Z* e −iq Z a + U MF = U SOL€(2),
where the position of the edge is represented by
x = a , and U SOL is the vorticity of the flow in theSOL. Here, for the first step, we set U SOL = 0 , for€ the simplicity. By using the boundary condition Eq.
(2), the finite mean flow is expressed as€(3).
U MF = −2U Z cosqZ a€
The sign of the mean flow depends on the phase of
the ZF at the edge.
The physical mechanism of mean flow
generation
is explained. The ZF is the wave in€which the poloidal momentum is given andreceived between neighboring magnetic surfaces
due to the deformation of the turbulence vorticity.
When the ZF is not contiguous to the plasma edge,
the net poloidal momentum is zero. On the otherhand, when the ZF touches the edge, the exchangeof the poloidal momentum across the plasma edge
occurs, and the net momentum becomes finite.
Therefore the mean flow is generated. The process
is shown in Fig. 1. In the case of Fig. 1, the plasma
receives the positive poloidal momentum from theSOL, and the positive mean flow is realized.
The poloidal velocity of the mean flow isevaluated as

finite mean flow is generated by the momentum
exchange across the plasma edge, whose value iscomparable to the diamagnetic drift velocity.
The geodesic acoustic modes (GAMs) aredriven by turbulence when the safety factor
becomes large. Even when the ZF can not get theenergy from the turbulence directly, the finite ZF isdriven by the nonlinear coupling of the GAMs [7]in the presence of inhomogeneous turbulence, as isnear the edge. The finite mean flow is generated via
the same process with the ZF.

VMF
U~ 2kθ LZF ZF cosqZ a(4),Vdω*
where VMF , Vd are the poloidal velocity of the€mean flow and the diamagnetic drift velocity,respectively. The poloidal wavenumber of the€turbulence
is denoted by kθ , LZF is the radial€ spatial scale length of the ZF, and ω * is thefrequency of the drift wave. In the case ofthekθ LZF = 10, U ZF /ω * = 0.1, cosqZ a = 1 ,€ mean flow is V = 2V .
estimated value €of theMF
dThe generated mean flow is€ comparable to thediamagnetic drift velocity, which is not negligible.

3. Summary and Discussion€
The new mechanism of the mean flow generation
is presented, investigating the system that consistsof turbulence, and ZF with considering the plasmaedge effect. Based on the fluid model, the set of
coupling equations among turbulence, ZF, andmean flow is derived by evaluating the Reynolds
stress with the wave-kinetic equation. The model ofthe plasma edge is proposed: the zero frequency
components, ZF and mean flow, exchange thepoloidal momentum with the flow in the SOL sothat the value of the zero frequency components
becomes equal to the flow in the SOL at the edge.
The effect of the plasma edge is introduced in thecoupling equations as the boundary condition. TheFig. 1 Schematic of generation mechanism of themean flow. The radial structure of the poloidal velocity
with the zero frequency component, Vθ , is shown. Theorange dot line shows the last closed flux surface. The
red arrow in the upper figure shows the direction of thepoloidal momentum transport.

€Acknowledgments
This work is partly supported by a grant-in-aid
for scientific research of JSPS, Japan (21224014,22840034, 19360418) and by the collaboration
programs of NIFS (NIFS10KOAP023) and of theRIAM of Kyushu University and Asada Science
foundation.
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[7] M. Sasaki, et. al.: submitted to Nucl. Fusion (2011).