J. Plasma Fusion Res. SERIES, Vol.4 (2001) 583_586

Boundary Gondition May Change Chaos
ITOH Sanae_I.*, KAWAI yoshinobul and yAGI Masatoshi
RIAM, Kyushu Llniversity, Kasuga g16-g5g0 Japan
Ilnterdisciplinary
Graduate school of Engineering Sciences, Kyushu IJniversity,
Kasuga g16-g5g0 Japan(Received: 5 December 2000 / Accepted: 27 August2001)

Abstract
Role of boundary condition for the appearance of chaos is examined.
Imposition of the boundary
condition is interpreted as the reduction of the system size z. Fora demonstration, Rayleigh-Benard

instability is considered and the shell model analysis is applied. It is shownthat the reduction of I

reduces the number of positive Lyapunov exponent of the system,
hence opens the route from theturbulence, to the chaos and to the limit cycle/fixed point.

Keywords:
chaos, ion-ion instability, Lyapunov exponent, Rayleigh-Benard

1. Introduction

number of existing modes reduces and the positive
Lyapunov number becomes small (say, unity), theThere have been reported various routes to turbu_
lent state in plasmas. Period doubling process due tosystem shows the chaotic behavior. And, the maximum
Lyapunov exponent becomes zero, only a limit cycle
solution exists. Imposition of the grid determines thenumber of unstable modes, from the fundamental tohigher harmonics, for given plasma parameters. The
gradual reduction of the distance between the grid andHopf bifurcation is a typical route which associates achaos. Experimentally, a plasma of unstable sysrem
often reaches the turbulent state without going throughthe chaotic state. It has been found that the ion_ion
instability in double plasma in the presence of ion beamcomponent directly enters into the turbulent state[l]. On

the central mesh cuts off the higher harmonics, so as toreduce the number of independent unstable modes. This

the contrary, when the bias grid is imposed at finite
distance from the central mesh, the discrete spectrum ofmechanism has an analogy to the inverse process ofperiod doubling (where the time period is replaced byunstable wave with an intermittent behavior was found.

This implies that a certain constraint upon the system

the space wavelength).

(e.g., boundary condition) reduces the number ofFor example, in the ion-beam system, the modes

degrees of freedom. Hence the phase diagram, on whichwith c1 < k)to < c2 can become unstable, where ft is thewave number, l,o is the Debye length, and numericalconstants satisfy cr - 0, being controlled by the ion_
neutral collisions, and c2 is determined by the ion_

the route to turbulence is determined, may also beaffected by the constraint.
In this paper, a working hypothesis is presented;

the boundary condition is imposed by the bias grid,
which makes the constraint for the number of unstable

Landau-damping, e.g., c2- | for the relevant parametersin [2]. Similar argument is applied to the Rayleigh_Benard instability, which corresponds to the flute modein plasmas. Here, the Rayleigh-Benard instability isconsidered, since the mode becomes unstable in a finite

modes in between the central mesh and the imposedgrid. If there are many unstable and independent modes,
and many positive Lyapunov numbers exist, the systemdevelops to the fully turbulent state. However. if thexCorresponding

instability. shell model

author's e-mail: s-iitoh@ riam.kyshu-u.acjp
s83@2001 by The Japan Society of plasma
Science and Nuclear Fusion Research

Itoh S.-I. et al., Boundary Condition May Change Chaos

where * represents the complex conjugate, n = 1,"", Nis the number of each shell, k, = cT"-t ' c is a numericalconstant, a, is the fluctuating velocity, pn is the fluctu-

region of k-space, can develop into turbulence, and issensitive to the constraint on the period of the system (inthe perpendicular direction to the magnetic field)'
In order to analyze the hypothesis of the basic
mechanisms theoretically, we examine the route fromthe turbulence to the chaos and to the limit cycle/fixedating temperature, Ru is the Rayleigh number' R" =hago;$l:r.t, and P. = v/r is the Prandtl number' To derivethese equations, k = k3 is assumed, and the normaliza-

tions are used: ftft --> k, krh-2 -+ t, v3h/r -+ u andlugh3lvr -+ p. The boundary conditions are given as

point. We adopt a shell model analysis, by which we
can extract all the turbulence, chaos and limit cycle/fixed point behavior t3l. The reduction of the systemsize is imposed, and the role of boundary condition on

It-t = lto = l/N+l = uN+z = 0 ald p-1 = Po= PN+t = PN+2 =0, which ensure no interaction between N-shell and thevacuum outside of N-shells. If we neglect the third termin RHS of (4) and (5), and add the force terms to them,respectively, then eq.(4) corresponds to the GOY model[5,6] and eq.(5) corresponds to passive scalar field

the appearance of chaos is shown below.

2. Model Equation
A simplified model, which describes Rayleigh-which involves with GOY model [7].
The linear stability analysis shows that the system
is unstable if R^> kX. The growth rate increases with theincrease of Ru and P,. The reduction of Ru value
corresponds to the reduction of the system size D' This
reduces the number of unstable modes in the shellmodel. The analysis on this shell model of Eqs'(4) andBenard turbulence in the fluid layer of height ft under
the gravity g is adopted. The temperature field is ? =Ze(x3) + 0, To@) = To - lxz and the density is p6 = p6(l+ op;3), where 0 is the fluctuating temperature, .r3 is thevertical distance from the bottom' cr is the thermal expansion coefficient and p6 is the density at the bottom'
The differential buoyancy is given by p6ag0.
Navier-Stokes equation for incompressible fluidand heat equation are written as(5) has been done and the turbulent , the chaotic and thelimit cycle nature has been reported in refs' [3'4]'(1)

V.y= 0,3. Analysis and Results
Here the nonlinear calculation is done for C = 1'0,
N = 13 (or N = 14) and a simple phase diagram isplotted on Ru P, plane (Fig.l). Heuristically, the higher$.0'Vir,=-fr+VV2v,+og96,, Q)* (, 'V; A= rV2 0+ Pv,6oPot-Ru number region above the line shows the turbulence'(3)and the limit cycle behavior with no positive Lyapunov

exponent is found below the line. An imposition ofwhere v is the kinematic viscosity, r is the heat diffusivity,6u is the Kronecker's delta and I = 3 represents
the direction of buoYancY'

Applying the Fourier transformation, we rewriteeqs.(l)-(3) into the shell-model equations (see [3'4] for

2.5 Wderivation). The nearest and the second nearest neighborcomplex component model is given as

turbulence

2.0 10
1.5 104

ff=i(o,u;.,u;,r-k,-ru)-rui,rt^-rr)-rui-')- P,klu^+ \p,

p =,\0.-,(,:-,1.0olff

limit cycle(4)
5.0 103t., - d., i^-,)0.0 ld0.0

1.0 2.o 3.0 4.O 5.0 6.0 7.0
Pr- o,- r(ui-, f^-, * "i-, f,- r)

Fig. 1 Phase-diagram on r9" - P, plane' The higher F"

number region above the line shows the+ k,(ui,, f,., - u:,, i,. r)\- R^un- k?, p,

gturbulence and the limit cycle behavior with nopositive Lyapunov exponent is found below the(s)

line.

584

Itoh S.-I. et al., Boundary Condition

May Change Chaos

l01r

rd

u.4

0.6

0.9

1 010tFig. 2 (a) The temporal evolutionof the total energy- - inthe case with P. = 2.0 and R" 2.0 xf O^.
=Fig. 3 (a) The temporal evolutionof the total energy- - inthe case with P, = 2.0 and R^ 1.0,

=f Or.

1018

10r6
I Or4

frl

1020

1012tgI Ors

1or0108l06
1or8104
102103Fig. 2 (b)power spectrum in the case with p.
.The= 2.0and r9" = 2.0 x IOa.

boundary condition can be simulated by
the reduction of
the value Ru. Cases with p, 2 for
2=and & _&= xlff

lOa are shown in Figs. 2 and 3, respectively.

temporal evolution of the total energyThedegrees of freedom with positive_exponent is positive in thedrops down toto the phase boundary
spectra are fitted- f-r.t, respectively, in theof 1000 </< 2500. On the other hand,the period
doubling cascade sach as | | 6, / lg,f f f / 4 f /2...., where /is the initial frequency, is observed
p. ginrange

the case of=[8]. This corresponds the fact that the fluctuation

appearance of theamplitude is very small and converges

to the limit cycle.
The intermittency is analyzed,
obtaining the scaling
exponent ofhigher order spectrum
function. For the casein Fig.a@), the intermittency, which
is observed by thethe number of

Lyapunov exponent.
The reduction of system size & is also

R" is kept constant and p. number

104. The maximum Lyapunov

region of 0.5 < p. < 7.5 and suddenlyas E(fl * J:-3'ts and E(fland the power spectrum are plotted.
Two frequency
spectra clearly show the difference
between theturbulence and the limit cycle. The

power spectrum in the case with p.
.The= 2.0and F" = 1,0 x lg'a.

in the diagram (Fig.l).
For P, - 2 and p, = 5, the power(6)

limit cycle demonstrates the reduction ofFig. 3 (b)zero at P. = 8, which corresponds

defined bv

E(t)=+1(l u.l' *l n,l')l03

considered if

is increased. Such acase is shown in Fig.4. The maximum
Lyapunov

deviation from the K4l prediction[9], is found fbr thecase of P, = 2 and, P, 5. However,=for p. > g, theintermittency disappears and thechaotic attractor does
not exist.

exponent vs P, number (a) and the power
spectra (b) for
different values of p, = 2.5 and 9 are plottej
for .l?, = 2 x585

May Change Chaos

Itoh S.-I. et al., Borrrrdary Condition

turbulent state rs

exponent. Well above the line' thein upper region',"uti"ed. Close to the phase boundary
might be
which
seen'
bewould
the chaotic behavior
study
Furtherscenario[10]'
related with Ruelle-Takens
positiveofnumber
theof the behavior of solution andmay clarify theLyapunov exponent across the boundary
the turbulence'
discrimination between the chaos andinstability can be
The role of system size on ion-ion
cold plasma apconsidered in a similar way' In theproximation, the parameter, (1 + (opb/op)2t3)3azniv*I] -300.0

o'

250.0
voo

200.0

dv

150.0
100.0
50.0

Reduction

0.0

o.o 2.O

4.0Pr

6.0 8'0

10'0

Lyapunovin the case of F"
numberPrandtl;;o;;;niJ"t"-ut(a) The log-log plot of the maximum
Fio.4' '"'= 2.0 x 10o.

Ru-number'fiIJ, may play the similar role to fromthe turbulence tooi the size I may open the route

point' Thesethe chaos and to the limit cycle/fixedwork'detailed analyses are left for our future

Acknowledgments

K' ItohAuthors would like to acknowledge Prof'
Grant-inbyfor discussions. Work is partly supported
Aid for Scientific Research of MoE Japan''loz2

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1oolor1oz

M' Buzzi'

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104

fof th.e total

Fig. a (b) The log-log plot of power spegtri.
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theshows the power spectrum for P':^2'0'

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thetodotted line corresponds

t6l K. Ohkitani and M' Yamada' Progr' Theor'
89.329 (1989).

t7l M.H. Jensen, G' Paladin and A' Vulpiani'
In this paper' we examined the role of boundary

Phys.

SSSR 30' 9iSf o.*. Kolmogorov, Dokl' Akad' Nauk 538 (1941)'

Imposition of thecondition for the appearance of chaos'the reduction oftheboundary condition is interpreted asof the systemsystem size. It is found that the reduction
associ,ir" ."du""* the number of degrees of freedom
Lyapunov exponent' Below theboundary in Fig.l, only the limit cycle

Phys.

Rev. A 45,7214 (1992)'(1971)'
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4. SummarY

ated with positiveof(lg4l),Dokl. Akad' Nauk SSSR 31'
Dokl. Akad. Nauk SSSR 32' 16
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solution is found'

Lyapunov
this fact shows that there is no positive586