J. Plasma Fusion Res. SERIES, Vol.2 (1999) 113-116

Theoretical Study of Inhomogeneous Plasma Turbulence
YAGI Masatoshix, ITOH Sanae-I., ITOH Kimitakar and FUKUYAMA Atsushi2
Research Institute for Applied Mechanics, Kyushu lJniversity, Kasuga 8l,6-8580

tNational Institute2Department of Nuclearfor Fusion Science, Toki, Gifu, 50g-52g2Engineering, Kyoto IJniversity, Kyoto, 606-8501 Japan(Received: 8 December 1998 / Accepted: 3 June 1999)

Abstract
A hierarchy of models, i,e., (l) a model with many degrees of freedom, (2) a model withintermediate degrees of freedom, (3) a model with a few degrees of freedom is considered to understandthe nature of turbulence. Results obtained from models of different levels are compared andcharacteristics of the interchange mode turbulence such as chaotic nature, cascade and statisticalexpression are discussed.

Keywords:
plasma turbulence, anomalous transport, nonlinear simulation, Lyapunov exponent

1. Introduction
Much work has been done in the research of

The nonlinear destabilization mechanism discussed
here originates from the anomalous electron viscosity
due to the energy cascades to short wavelength modes.

inhomogeneous plasma turbulence and associatedtransport. Recent research shows, however, nonlinear
instabilities play an important role in the transport
driven by the inhomogeneous plasma turbulence rather
than linear instabilities []. The analysis of subcriticalturbulence brings us a new insight for plasma
turbulence. As a typical example of the inhomogeneous
plasma turbulence, the interchange mode turbulence in asystem with the inhomogeneities of pressure andmagnetic field is considered. Based on the direct
simulation of the reduced MHD equations [2], it isThis mechanism also leads to the instability of thetrajectory in phase space. Positive Lyapunov exponents
and short correlation time are observed in this system. It

is a challenging problem to understand the relation
between the correlation time and the irreversibility due
to global transport. In the study of turbulence, a systemturbulent state is self-sustained and the fluctuation
amplitude is proportional to the power of the pressure

with high Reynolds number is important. However, it isdifficult to solve the equations in the inviscid limit evenby a super-computer. In order to understand the issues
above we consider a hierarchy of the models, i.e., (l) amodel with many degrees of freedom like direct
simulation t3l, (2) a model with intermediate degrees of
freedom such as a shell model t41, (3) a model with afew degrees of freedom such as the Lorenz model [5].
Results obtained from models of different levels arecompared and characteristics of the interchange mode

gradient. These results indicate that the strong

turbulence such as chaotic nature, cascade andturbulence is sustained by the inhomogeneities and theintermittency are discussed.

shown that (l) the nonlinear plasma turbulence develops
even in a linearly-stable region, (2) the threshold valueof the fluctuation amplitude for the nonlinear instability
is much lower than the stationary level (3) the strong

conventional method which is based on elaborate
evaluation ofthe linear srowth rate is insufficient.
*Corre sponding author' s e-mail : yagi @ post. riam.lqushu-u.ac.jp

113@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchYagi M. et al., Theoretical Study of Inhomogeneous Plasma Turbulence2. Model with Many Degrees of Freedom
We here adopt the three field reduced set ofamplitude is 3 times higher than that of the case without
electron nonlinearity in Eq. (2). An instantaneous

equations in a shear magnetic field with bad averaged
curvature. This model is relevant to study the nonlinear
dynamics and associated transport by the electrostatic

relation 6E n e2L', where ,1 represents the instantaneous
maximum Lyapunov exponent and 6E is given by

maximum Lyapunov exponent is evaluated from thecurrent diffusive interchange mode turbulence in asystem of magnetic hill. The equation of motion, Ohm's
law and the energy balance equation are given in au" =normalized form aswithol* =iky sxj - jak, p + P,v|,Q,!, l'',',*; ? (lv, 6@ l' *l o; l'. | 5o l')(1)

vl aO + ts@, vl Ql+tL.vi sdt
9dt

I i =-lft, sxQ + )",Y! i,Q)

=ik, sx6/ - jakrEp +trt,YatlQ,
I U; * I6O. jl + tO, Sjl = ift, sx6d +,t"Vl5;,o =- ik, Q +x,Y2Lp,(3)*{,otand

where d/dt = )ldt + t0, l , [, ] is the Poisson bracket
denoting the E x B nonlinearity. For the normalization,
the poloidal Alfven time and the electron skin depth areemployed (see Ref. [2] for details). Figure I shows thetemporal evolution of the growth of fluctuating pressure* lSQ.pl+td,6pl=ik, sx6@ +)(,Y216p.
9Uodt
Figure 2 shows the time evolution of 1. For comparison,
total energy defined byenergy (nearly equal to total energy), whichf ,.',*; ? v, Ql'+ l'demonstrates the nonlinear growth of fluctuations. We
choose the parameters l!" = X" = O.2, L, = 0.01 and s =E,=a = 0.5. The solid line is the result of the solution ofis also plotted. In the calculation, we renormalize theEqs. (1)-(3) and the electron nonlinearity in Eq. (2) is
kept. (Dashed line: the case without the nonlinearity inEq. (2).) In the small amplitude limit, the perturbations

amplitudes 60, 6/, 6p at each time step of A/ = 30. It isobserved that at t < 200, the behavior of the!,+ | o l'),obtained as order unity. This system describes thesubmarginal turbulence sustained by the nonlinear
instability which is associated with the strong normalcascade and the positive maximum exponent.

12102
100
L

?1 0-

8
4

5" 0't 106

0| "rinstantaneous maximum Lyapunov exponent is similar
to the fluctuation amplitude Q, j, p. However, at t > 200,
the behavior is different from that of the total energy.
The value of the maximum Lyapunov exponent isgrow following the linear growth rate. However, at thetime of t = 10, when the amplitude exceeds a certain
threshold value, the growth rate starts to increase. In thetime range of l0 < t < 60, the growth rate becomeslarger as the amplitude increases. This shows thenonlinear destabilization originated from the electrondynamics. During this phase, the normal cascade tohigher ft, mode develops. It is found that the saturation

1o{(l04080t

100120 160

1s0 200 250 3fi)t

Fig.2 Time evolution of the instantaneous maximum
Fig. 1 Fluctuating pressure energy versus time.

Lyapunov exponent.
114

Yagi M. et al., Theoretical Study of Inhomogeneous Plasma Turbulence

3. Model with Intermediate Degrees withsystem and the maximum Lyapunov exponent isFreedom

obtained as 2290. Figure 4 shows the scaling exponent

We extend a shell model by including the thermal(, for the structure function versus order p. The

instability terms (Rayleigh B6nard convection) [6].

structure functions are calculated by S!(n) = (lu,ln) andS!(n) = {e"lr), where the bracket represents the time

Normalized shell model is expressed in Fourier space as(4)average in the intervals with t = 0.2 - 10. The scaling
exponents are defined by (lu,lo) * k;l', and (lg,lo) *k;l?. fne circle and the triangle correspond to (i and(!, respectively. For comparison, the result obtained by
GOY model [8] is also plotted (denoted by the square).
The dashed line corresponds to K41 l9l: 6o = p/3.

(5)

Intermittency in shell models is the subject of muchcurrent work and is quite poorly understood at the moment. The scaling exponent for the structure function offluctuating velocity field agrees with the result of GOY
model, and is larger than that of fluctuating temperaturefield. They are not affected by the linear instability somuch, but reflect the characteristics of nonlinearity. Thissystem describes the strong turbulence with high:du, = i(o nui*t ui*z + b^u) , ui*, + c nui ,ui_r)+P,en-P,(u,de"? =,t e, (ui-.0)., - ui.,0)-,1^(ui,-20)-t+u)-,0i t)+ h, (ui*, 0),, + u),rqi*r)+g+Rour-k2r0nwhere * represents the complex conjugate, n = l, ..., N,
number ofeach shell, u,, the fluctuating velocity, gn, thefluctuating temperature, kn= C2n- I, the wave number,P,, the Prandtl number, Ro, the Rayleigh number. For

Rayleigh number, which contains the sufficient numbers

the boundary conditions, we assume uo= trN* r = 0 andof positive Lyapunov exponents, intermittency nature0o= 0N, r = 0 and the coefficients are given by

and the normal cascade [9].

k,-tan=kn, U,=-t,k,,^=;.

t,=-;.

knt

kn-zr^=-;.

4. Model with a Few Degrees of Freedom
We extend Lorenz model by introducing higherk.-,h,=-;.

Fourier harmonics which describe the shear flow effect.
Normalized five components model [0] is written as

and bt = bx = ct = c2= aN-r = aN = €t = €N = Et = 82=
hN-r = hx= O. Here the typical scale length and theV*=-xz+rX-Y'
ff =vw + P,(Y- X ).
V =xY - bZ, =- xw - yvv,#
# =- cxV -thermal diffusion time are used for normalization [7].
We use the following parameters: kt = C - l0-2, P, = l,Ro = 704, N = 20 for calculations. Figure 3 shows thedistribution of instantaneous Lyapunov exponents at t =l. It is found that first 12 Lyapunov exponents are
positive for the system with mode numbers N = 20,
where total Lyapunov exponents are given by 80 in thisfy,,W

3.5

104
0 100-l 104
130.^t-n't(p2104104
104
104t;

2.57t'2 104-3-4-5-6(6)

1.5
10.

1020 30 40 50 6070 80nFig.3 Distribution of instantaneous Lyapunovw.n 'g'c
7 I 9 105 6pFig. 4 Scaling exponent for the structure function versus
oroer.

exponents.

l15

Yagi M. et al., Theorctical Study of Inhomogeneous Plasma Turbulence

with4n2r= ---: -R", o, =---^------7n'+q"
lrc"\-,/+ q.
4n2 + a2
Tv=-:!f . P,, Yw=n'++- Pr- q"TE'+ q"Q2t-5

I

andtr -to-15-20^)Jq-.25

C= u,0414n'z+ q'z)

1000 2000 3m0 40mrwhere 4 indicates (normalized) wave number of cellarrangement. The Lorenz model is reproduced if we setFig. 5 Lyapunov exponents versus normalized RayleighV = W = 0. In simulations, the normalized Rayleighnumber is chosen to be an order parameter tocharacterize the system behavior and the otherparameters are fixed as q - Ic, b = 20, Tv = 10.0, c =number.

0.15 and P,= 4.O. Figure 5 shows Lyapunov exponents
versus norrnalized Rayleigh number. It is found that forthe parameter regime of r < 140, the similar behavior of
the Lyapunov exponent is observed to that of Lorenz
model. We observe the chaotic behavior and the limitcycle solution as well as the period-doubling process ofamplitude of energy or flux is predicted only by model(I), however, universal scaling law such as Kolmogorov

thesolution in the regime of 306 < r < 2200. For theregime of r > 2200 this model also falls into the limit
cycle. This fact indicates that a few numbers of Fouriermodes are not enough to describe the chaos in theregime with high Rayleigh number. However, thismodel is useful to study the transition to chaos fromwith a few degrees of freedom and that withlaminar state.

programme of Advanced Fusion Research Center,

inertia scaling law is also described by model (II).
Model (II) has an advantage to investigate multi-fractal
nature of turbulence, i.e., intermittency and transition
from chaos to turbulence. Inter-relation between a modelintermediate or many degrees of freedom should beinvestigated for future.
This work is partly supported by Grant-in Aid for
Scientific Research on Ministry of Education, Science,Sports and Culture Japan, by the collaboration
RIAM of Kyushu University.

5. Summary and Discussions
To study the interchange mode turbulence in thevarious range of Rayleigh number, we introduce
hierarchical model equations: (I) a model with manydegrees with freedom, (II) a model with intermediate
degrees of freedom, (III) a model with a few degrees offreedom. It is found that the nonlinear growth of
instability is characterized by normal cascade andReferencestll

K. Itoh et al., Plasma Phys. Control. Fusion 35, 543(1993).

t2l M. Yagi et c/., Phys. Plasmas 2, 4140 (1995).
[3] M. Yagi et a/., CHAOS 7, r98 (1997).
t4l M. Yamada and K. Ohkitani, J. Phys. Soc. Jpn. 56,4210 (1987).

not clear. It is left for a future work. The transition tot5l E.N. Lorenz, J. Atmos. Sci.20, 130 (1963).
t6l M. Yagi et a/., Chaos 9, issue 2 (1999) in press.
[7] P. Manneville, Dissipative Structures and WeakTurbulence, Academic Press Inc., P.55 (1990).
t8l M. Yamada and K. Ohkitani, J. Phys. Soc. Jpn.56,4210 (1987).

chaos is discussed based on the extended Lorenz model,

t9l A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 9

Various dynamical behaviors which are not described bythe Lorenz model are observed in the range of 306 < r <2200, however, for r > 22OO, this model is not relevant
to describe chaos. The absolute value of saturation(1941), 3r, 538 (r94r),32, 16 (r94r) and 31, 99

positive Lyapunov exponent. Intermittent nature isdiscussed by the higher order structure functions of ashell model. Intermittency is considered to be relatedwith the normal cascade [9]. At this point, the relation
between the submarginal instability and intermittency is(1941).
[10] T. Aoyagi et al., J. Phys. Soc. Jpn. 66,2689 (1997).

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