J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Seesaw Mechanism in Turbulence-Suppression by Zonal Flows
Kimitaka ITOH1, Sanae-I. ITOH2, Masatoshi YAGI2, Atsushi FUKUYAMA3
1

2National Institute for Fusion Science, Toki, Gifu 509-5292, Japan
Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan3

Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan(Received: 1 September 2008 / Accepted: 20 November 2008)

Theory of non-local transport has been developed, taking an example of zonal flows (ZFs), the radial correlation
length of which is longer than those for microscopic fluctuations. ZF, which is driven by fluctuations at one radius,
can suppress fluctuations at distant radii. This mechanism induces new non-local interactions in turbulent
transport. That is, strong fluctuations at particular radius can suppress fluctuations at different radius, via
induction of ZFs. Stronger fluctuations suppress weaker fluctuations. This is called the seesaw mechanism via
ZFs.

Keywords: no-local transport, transient transport, zonal flows, turbulence suppression, drift wave turbulence,
nonlinear theory.

reduce the turbulence and turbulent transport. Because
the radial correlation length of ZF is longer than those formicroscopic fluctuations, which are inducing turbulent
transport, ZF, which is driven fluctuations at one radius,
can suppress fluctuations at distant radii. Thus, thefluctuations can exchange energy over the distance that ismuch longer than autocorrelation length of microscopicfluctuations. This mechanism induces new non-local
interactions in turbulent transport.
That is, strong
fluctuations at particular radius can suppress fluctuations
at different radius, via induction of ZFs. Stronger
fluctuations suppress weaker fluctuations. This is called
the seesaw mechanism via ZFs.
Owing to thismechanism, the turbulence transport is not determined bylocal parameters alone, but by parameters at far distance.
The transient response is much faster than the process
governed by diffusive processes.

1. Introduction
Research of non-local and transit transport is anurgent issue in modern plasma physics. Experiments
have reported the non-diffusive mechanisms in rapid
response of transport between distant radii (see, e.g.,
recent report [1]). Simulations have demonstrated thattransport barrier can be established while increasing
linear growth rate of local instabilities [2].
Theseobservations demonstrate that non-local interactions ofturbulent fluctuations that are responsible for anomaloustransport should have a key role in determining thetransport in confined, high-temperature plasmas.
Theory of non-local transport has been developed, based
upon the statistical theory of plasma turbulence [3]. In
this theoretical framework, global perturbations (whichmay be linearly stable) are shown to be excited by
microscopic turbulence. Such a global perturbations
have long radial correlation length and transmits thevariation by its phase velocity, not by diffusive process.
As a result of this, a change of microscopic turbulence at
one radius propagates across the plasma column much
faster than the expectation of diffusive response.
This new physics view awaits application tounderstand the non-local and rapid response in transport.
Example of such nonlinearly-driven, meso-scale
fluctuations is the zonal flow (ZF) [4]. ZFs growextracting energy from microscopic fluctuations so as to

2. ModelWe employ a predator-prey model of DW-ZF system.
Here, we notice that a zonal flow eigenmode (radialcorrelation length of which is chosen here as L) is excitedby a large number of drift waves (radial correlation lengthis much shorter than L). The local growth rate andnonlinear damping rate (through like-scale nonlinearities)
for drift waves vary in radius, and the scale length ofvariation can be of the order of L. In this case, thepredator-prey model takes a form

author’s e-mail: itoh@nifs.ac.jp

119©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

K. Itoh et al., Seesaw Mechanism in Turbulence-Suppression by Zonal Flows22�
�t I x = � x I – �2 I – � E I

 E=� E 1
t2L

 x = 0 +  1 –  0 x 2 .
L(1)

We take the case of  1 >  0 , and the opposite case isstraightforward.
Substituting Eq.(5) into Eqs.(3) and (4), we obtainthe following. Zonal flows are excited if collisional
damping is weak:

Ldx I – E

(2)

–Lwhere I x is the intensity of DW,  x is the local
growth rate, 2 I is the nonlinear damping rate of DW,
E is the intensity of ZF, � is a coupling coefficient
between DW and ZF and  is the linear damping rate ofZF. (Intensities and are normalized to the kinetic energy
density at the diamagnetic drift velocity.) Coefficients2 and � can also vary in radius. However, we takethem constant in radius, for the transparency of argument,
without loosing the essence of the problem.
Theintensity of zonal flow E is constant within thecorrelation length L.

1 � + 2� > 3 0�
2 1

1 � + 2� < 3 0�,
2 1ZFs are not excited, and bare drift waves occurs.
We consider the case of weak collisional damping
and ZF is excited. When the inhomogeneity of local
growth exceeds the threshold,This system describes that fluctuations at thelocation x are suppressed by zonal flow that is driven byfluctuations at the location x , x – x < L . Stationary

I = 1 � x –� E
�2if

1 � –� >3 0�2 1

From Eq.(1), we� x > �E(6)In opposite case,

3. Seesaw Mechanism

state is discussed in the following.
have(5)(7)

fluctuations are suppressed ( I = 0 ) in the region of weak
instabilityx < xc ,(3a)(8)

where the boundary xc satisfies the condition

andI=0if

� x < �E .

(3b)

1–2x
2xc 2
1 + c =3 
�
LL
� –�
1.

(9)

0Equation (2) provides

12L

In this case, fluctuations are suppressed completely in thedomain x < xc although the region is linearly unstableagainst linear drift mode.

Ldx I = �–L

if

E>0(4a)dx I < �.

(4b)

and

E=0if

12L

L–LThus, it describes the competition of fluctuations atdifferent locations through the interactions of zonal flows.
Consider the case that there two areas within a correlation
length of the zonal flow (i.e., drift waves are strongly
unstable in one area A, and are weakly unstable in theother area B). The level of zonal flow is enhanced byfluctuations in area A. Then the enhanced zonal flow
suppresses the fluctuations in area B. The enhanced
fluctuations in area A suppress fluctuations in the area Bthrough exciting zonal flows. This is called seesaw
mechanism in turbulent transport.
For explicit illustration, we take a parabolic modelfor the local growth rate, and take the origin at theminimum position (Fig.1):

(x)


1

0

-1.0

-0.5

0.0

0.5

x/L 1.0

Fig.1: Model of radial distribution of the local growthrate

120

K. Itoh et al., Seesaw Mechanism in Turbulence-Suppression by Zonal Flows

Profile of fluctuation intensity is shown in Fig.2 forthe cases of5I(x)1 � –� =6 0� (strong inhomogeneity of drive),2 1  = 6/410

31 � –� =2 0� (weak inhomogeneity of drive)2 12/201

and00.0 1 =  0 (no inhomogeneity).
It is shown clearly that the turbulence intensity in theregime of weaker instability is suppressed if the growthrate increases at distance.
In the case of strong
inhomogeneity, the fluctuations are completely
suppressed in the vicinity of the local minimum of thegrowth rate. In the moderate case, fluctuations arereduced in the entire region, but the suppression is notpredicted.
Figure 3 illustrates the profiles of theintensity of turbulent fluctuations (solid line), and zonal
flow (dashed line), together with the profile of linear
growth rate  x (dotted line). The competition offluctuations leads to the quench of fluctuations inunstable domain.
This mechanism naturally induces the fast responseat far distance. When the characteristics of turbulenceat one radius are modified at an instance, then theinfluence occurs in the region (width L) directly via
dynamics of zonal flows.
This transmission of
information does not need the diffusive propagation ofchange. Thus, fast transit response propagates acrossthe plasma column. The study of dynamical problem
will be discussed in separate article.

0.2

0.4

Fig.2:
Profile of� 1 – �0 –2 1 = 6  � – 1 ,0.6

0.8x/L 1.0

fluctuation intensity� 1 – �0 –2 1 = 2  � – 1forand1 = 0 .

  = 6/
18
70I(x)

65DW

43ZF

2
1
0
0.0

0.2

0.4

0.6

0.8x/L 1.0

Fig.3: Profile of fluctuation intensity, zonal flow
amplitude and local growth rate for the case of strong
inhomogeneity. The fluctuations near the minimum oflinear growth rate are quenched through competition withfluctuations in distant location.

4. Summary
Novel non-local interactions of microscopic
turbulence via zonal flow are investigated. Strong
fluctuations at particular radius can suppress fluctuations
at different radius, via induction of ZFs. This is called
the seesaw mechanism via ZFs.
Owing to thismechanism, the turbulence transport is not determinedby local parameters alone, but by parameters at fardistance. The threshold of turbulence suppression isformulated as a global condition.
The transient
response is much faster than the process governed bydiffusive processes.

References[1] S. Inagaki, et al.: Plasma Phys. Control. Fusion 48 A251
(2006)[2] M. Yagi, et al.: Plasma Phys. Contr. Fusion 48 A409 (2006)[3] S.-I. Itoh, K. Itoh: Plasma Phys. Control. Fusion 43 1055
(2001)[4] P. H. Diamond, S.-I. Itoh, K. Itoh and T.S. Hahm: Plasma
Phys. Control. Fusion 47 R35 (2005)

Acknowledgements
Discussions with P. H. Diamond, S. Inagaki, Y.
Nagashima, and A. Fujisawa are cordially acknowledged.
Work supported by the Grant-in-Aid for SpeciallyPromoted Research (16002005), by the Grant-in-Aid for
Scientific Research (19360418) and by collaboration
programme NIFS06KDAD005 and by collaboration
programme of RIAM Kyushu University.

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