J. Plasma Fusion Res. SERIES, Vol.2 (1999) 93-96

A Theory of Lead-Time in Probabilistic Excitationof L/H Transition
TODA Shinichiro, ITOH Sanae-I., YAGI Masatoshi,
ITOH Kimirakar and FUKUYAMA Atsushi2
Research Institute for Applied Mechanics, Kyushu university, Kasuga 816-8580(Present Address: National Institute for Fusion Science, Toki, Gifu, 509-5292)

lNational Institutefor Fusion Science, Toki, Gifu, 509-5292

2Department of Nuclear Engineering,

Kyoto University, Kyoto, 606-8501 Japan(Received: 8 December 1998 / Accepted: 6 May 1999)

Abstrac't
A quantity of a lead-time, /1*6, is newly introduced to examine the probabilistic occurrence of the L/H transition. The lead-time is a time period during which a transition is likely to occur. We show that thelead-time has the statistical distribution as a function of the distance from critical parameter, e.g. ln" n"e I when the density is a key pararneter for transition. It has the dependence like /r"u6 n ln" - n 12 if the"obackground noise distribution is given as P (n") n ln"- n"nl-2.

Keywords:
probabilistic nature, L/H transition, lead-time, statistical variance, hysteresis

1. Introduction

probabilistic nature of occurrence. The cases in thepresence of the Gaussian noise [5] and the noise with apower law have been examined [6]. It has been foundTransition phenomena in plasmas (e.5., LIHtransition) are widely observed in various toroidal
confinement devices. A transition (bifurcation) isthat the transition occurs with a finite probability aroundthe original threshold condition if the noise is present.

modelled by the equation which includes a characteristicand cusp-type bifurcation. In the L/H transition study,
the model equation is tested [] to explain theexperimentally-observed dithering ELMs [2]. This study
clearly demonstrates that the high temperature plasma
has a transition nature. Further observation of repetitive
bifurcation was made on CHS device [3], and theimportance of the study of transition nature is more
widely recognized. Dynamics of transition in confined

The probability distribution around the threshold has
been obtained. Here, we introduce another concept,
which comes from the probability distribution, i.e., thelead-time. This statistical quantity is studied by the 0-D
dynamical equation of transition, with noise effect, inwhich a hysteresis exists in the model of the loss rate.

2. Zero-Dimensional Dynamical ModelEquations

plasmas has been discussed by use of the criticalcondition [4]. However, to understand the nature ofsudden onsets of L/H transitions, consideration of thebasic physics mechanism is still necessary. There exist
statistical variances in relevant variables. We haverecently explored a new aspoct of transition, that is, aWe here adopt two basic equations: One is thetemporal evolution of the plasma parameter. The other isthe dynamics of the loss rate that produces the hysteresisof the flux-gradient relation. We choose two@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research93

Toda S. et al., A Theory of Lead-Time in Probabilistic Excitation of L/H Transition

representative variables, i.e., the density and the loss
rate T in a layer with a finite width. The loss rate is
directly related to the turbulence level and the particle

the deviation from the mean fluctuation level and causes

diffusivity. The reduction to the 0-D model from the3. Probabilistic Excitation of L/H Transitiontransport equation has been discussed [7]. Thoughhighly simplified, the model form of the hysteresis has

We study the probabilistic nature which is causedby the statistical property of the turbulence. Temporal
evolution are investigated by solving eqs. (1) and (2).
The variation in a hysteresis characteristic is examined.
The variance of €o is taken account with s, =the variation in the hysteresis characteristic.

successfully applied to the investigation of the dynamicsof dithering ELMs (which cycles between L-mode andH-mode). The model equation used here takes the forms

9r=S-Tn,dt^l\o - ("))'. The parameters are chosen is as=0.5, b =(1)1.0 and 6= 0.01. Accordingly, Sr- 0.67, 52 - 1.21 and- 1.136. We set ,S = 1.0, i.e., 51 < S < Sz. We here
introduce the noise, the probabilistic function for eo
which obeys the power law, i.e., P(e") * le,l-2 in acertain region. The index of 2 is chosen as an example,
but this number is not far from the observation ofn"oand(*y=n-r+a(y-r)-b(y-r)3,-fA)

otwhere S is the particle influx into the layer, 6(= O@2plB?)) denotes the dynamical time difference betweenn and y, and the cubic equation a(Tb(y 1l3- l) -nonlinear simulation. In this example, the noise-

describes the shape of the hysteresis. Appropriate
normalization is used for n, y, and t.
The dynamical nature of the set of equations has
been studied. If all the coefficients (S, a, b, ('1 areconstant in time without noise, eqs. (1) and (2) predict
the stable stationary solutions or the dynamical solutionof a limit cycle. Stationary solutions are obtained for S <Sr (lower flux branch, H-mode), Sz < S (higher fluxbranch, L-mode) and a limit cycle appears for 51 < S <.S2, where S1 = (1 - "'l"tlUl {l + (2a13) T"tlUl and 52 0 + ",[a/3b) {l - (2a/3) "l"tZUt. The transition from Hmode to L-mode takes place at n"s = t + 2" ,["/Qb)/3,i.e., n"s is the threshold condition in the absence ofgenerator is used to obtain the fluctuation quantity eo inthe domain e"> | x 10-6 and in €o ( -l x l0{. In theregion 1.2 x l0+ < 1e,1 32.5 x 10-r, the relations P(eo)* le"l-2 and, P(e") = P(-€,) hold. We set P(e,) = P(1 x10-6) (= const.) for le"l < I x 10-6. The variance is
Q = 0.05.

Oscillations like a limit cycle with irregular bursts
are obtained. To show the probabilistic excitation of theplasma transition, the statistical distribution of thecritical density n", at which the transition from H-modeto L-mode takes place, is observed. The distributionfunction of n", P(n"), is obtained in Fig. 1. This
probability distribution clearly shows that the onset has
the statistical nature. The value of n" which takes themaximum probability nearly equals to n"6, where thetransition occurs in the absence of noise. In thisstatistical fluctuations, we observe 6nl6a = 0.37, wherenoise. We are here concerned this case as an example.

As the physics origin of the transport, theturbulence level is often discussed in terms of thestatistical averages. The turbulence level is also
associated with the statistical variance. For the confinedplasmas, which is generally far from thermal
equilibrium, the statistical variance is as important aquantity as the statistical average. The nonlinear

0. l0

simulation has shown a large temporal variation aroundthe average [8]. The experimental observation has

Qo-demonstrated that the statistical deviation from the meanp" o.os

value could be as large as the average itself [9]. Basedon these considerations, we have been considered thatparameters (5, a, b) are statistical variables and havefluctuation parts in time. In previous study [5,6], it has
been found that the variance in has a stronger effect onthe transition than that in S. Therefore, we here mainly
focus the effect of the statistical variances of a. We set a= as* E, and consider €, and €o as statistical variables,e.g., (€,> = 0 and (el) + O. The variance to comes from

8aab.

0.
tr"

3540

Fig. 1 The probability distribution for the density at theonset of the transition. Here, S = 1.0, ao = 0,5, b =1.0 and (= 0.01.

94

Toda S. et al.. A Theorv of Lead-Time in Probabilistic Excitation of L/H Transition

6n = 2",.u^ - zc,min and 6a = a",^u* - a",.in. The

So the model equation is given as

dN

dependence of log" P (2") on logel nc - n"s I is examined(3)- dr =P(n(t)).

in Fig. 2. The solid line indicates the relation P(n") n
ln" - n"ol-t.(A linear fitting in the region 1.2 x l}a <The partition rate N(t) is calculated as

fl c0 - fl c < 1.2 x l0-3 gives log" P(n") = Mo t M, log.ln"- n"6 | with Mo = -19 and M | = -1.99 and the correlationcoefficient R = 0.98. Scattering in the tail is statisticalnoise and we deduce in such a form like R = l-l45llrot

N(r)=r-l rg(ta{.

(4)

where we assume that the value of n(r = 0) is enough far

from n"s and the transition never occurs at t = 0, i.e.,N(t = 0) = l. Without the noise, the model equation is
simplified as -dN/dr = 6(r - t.), where 6(t - t-) is the
Dirac's delta function and, n(t - /.) = n"6. In this case,the transition always occurs at fl = flco. The rate N(t) isevaluated in the equation as N(r) = | - Jo6(t'- t-)dt'.

(1000 < ltot < 10000) when the number of observation oftransition, 1,o,, is increased. The distribution function isfound to obey the same power law P(n") n ln" - n,sl-2
as the generated noise. Analysis about other criticalparameters such as period and particle flux have been
done for the different type of noise [6]. The similar
result is also obtained in this studv.

The value of N(t) in the region I < /* tis unity and theone in the region t > t. is zero. The ratio N has thediscontinuity nature and ldN/dtl,=,- = -. In the presenceof the noises, the states N(t = 0) and N(t - -) are foundto be smoothly connected. This fact indicates that theparameter N is reflected by the probabilistic nature in

4. Analysis about the Lead-TimeThe finite probability of the transition below thecriterion n" I n"s allows one to estimate the effective
lead-time before the transition occurs. As in the previous

the LlH transition. The quantity N/ldN/drl is an
estimate of the effective time against the transitionphenomena and the change time for the state. We here
call this time 'lead-time': /1"u4 as shown in Fig. 3.
Without noises, /1"u0 is infinity and no transition takes
place in the region t < t.. At t = t*, t664 is zero and thetransition always happens. However, if the noise ispresent, the transition happens at n < n"s with a finite
probability of .6'*u P(n)dt after the time interval /r""4. In
this case investigated here, the lead-time is shown tohave the dependence such as tbud n lk" - nwhichsection, we here study the transition from the H-mode toL-mode. A time sequence of repeated transitions (totalnumber of 1,o,) in dithers is considered as a set of 1,o,observations of the transition events. In each temporal
evolution of transition, the initial condition is chosen as
the H-state, and the onset of H- to L-transition isobserved. The ratio that the plasma state remains the Hmode (the transition does not occur yet) is set N as themean value for the case of 1,o, times of the repeatedtransitions. Then the rate equation is given as -dN/dr ="612P(n(t))lrr, (= P"N), where ?,. is the time scale tohas inverse dependence for the background noise in ananalytic insight. The dependence of log"(N/ldN/dtl) on

complete one transition event after being triggered andP,, is the transition rate. The transition takes place withan expected rute qlP(n(t)). In this article, we choose
the normalized value ?r = I without lack of generality.

log"lz" - z"6l is examined in Fig.4. The solid line
indicates the relation (N/ldN/dtl) n ln, - r?.0 12. (A linear

fitting in the region 5.5 x 10r 3 n"o - n" < 2.5 x l0-3-23 103-3otr^-4?-s5-oL:2 l0' @rDigo @^i,nw-......Y........-sRb.....
Irro3i@z'-7I-8g-9_:@-otgl

Fig. 2 The dependence of log"P(n") on log" In" - n"o I isshown. The value of the slope for the solid line isL'jA

1.135n

1.t36

fJ37

Fig. 3 Lead Time (N/ldN/dtl) is shown as the functionof n".

-2.
95Toda S. et al.. A Theorv of Lead-Time in Probabilistic Excitation of [,/H Transition

gives log"(N/ | dN/dt l) = M z t M 3 log.ln - n col with M 2 =found to have the dependence of the inverse of that ofthe background noise field. The statistical study on theexperiments of transition will provide a uniqueinformation to understand the physics of the transition
as well as the nature of the turbulence in plasmas. Notethat we show the case for the one kind of noise in thispaper. Cases for other noises are left for future work.
One of authors (ST) acknowledges the support byJSPS Research Fellowships for Young Scientists, Thiswork is partly supported by Grant-in Aid for Scientific
Research on Ministry of Education, Science, Sports andCulture Japan and by the collaboration program ofAdvanced Fusion Research Center of Kvushu

2O and Mz = 2.01 and the correlation coefficient R =0.94. It is found that the numerical result shows thedependence trcuan ln"- n"e12 similar to the analytic one.

This fact is confirmed in the cases for other values of S.

5. Summary and Discussions
In this article, we study the probabilistic nature ofthe transition by 0-D model equation. The statisticalvariance for relevant parameters expresses theprobability of the transition based on the probabilisticview. The distribution of critical parameters is
confirmed to reflect the statistical property of thebackground turbulent field for the case P(e") * le"la.
Next a concept of lead-time is naturally introduced toUniversitv.

extend the probabilistic picture for the transition

References

phenomena. This time is the average time period, during

lll S.-I. ltoh, et al.,Phys. Rev. Lett. 67,2485 (1991).
t2l H. Zohm, Plasma Phys. Contr. Fusion. 3E, 105

which a transition is likelv to occur. The lead-time is(l996).
l3l A. Fujisawa, et al., Phys. Rev. Lett. 81, 2256
(l998).
[4] See a review, e.9., K. Itoh, S.-I. Itoh, Plasma Phys.

l09Contr. Fusion 38, I (1996).
t5l S.-I. ltoh, et a/., Plasma Phys. Contr. Fusion. 40,737 (1998\.
[6] S. Toda, et al., 1998 ICPP & 25th EPS Conf. on
Contr. Fusion and Plasma Physics, Praha, 29 June
- 3 July. ECAYol.22C, 1856 (1998).
[7] S.-I. ltoh, et a/., Nucl. Fusion, 33, 1445 (1993).
t8l M. Yagi, et cl., Phys. Plasmas, 2,4140 (1995).
t9l M. Endler, er a/., Nucl. Fusion, 35, 1307 (1995).

{8zEI7?654Fig.4 The dependence of log"(N/ldN/dtl) on log"ln" n, | . The value of the slope for the solid line is 2.

96