J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)249–252
000–000

Turbulence Spectrum and Transport Scaling
VLAD Madalina, SPINEANU Florin, MISGUICH Jacques1, REUSS Jean-Daniel1, BALESCU Radu2,
ITOH Kimitaka3 and ITOH Sanae4
National Institute for Laser, Plasma and Radiation Physics,
Association Euratom-MEC, P.O.Box MG-36, Magurele, Bucharest, Romania
1
Association Euratom-CEA sur la Fusion, CEA-Cadarache, 13108 Saint-Paul-Lez-Durance, France
2
Association Euratom-Etat Belge sur la Fusion, Université Libre de Bruxelles, Belgium
3National Institute for Fusion Science, Toki 509-5292, Japan
4
Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan(Received: 10 December 2003 / Accepted: 11 February 2004)

Abstract
The effect of the shape of the spectrum of an electrostatic turbulence on the scaling of the diffusion coefficient isstudied using the decorrelation trajectory method. We show that a strong influence appears at large Kubo numbers
due to trajectory trapping.
Keywords:
plasma turbulence, statistical approach, test particle transport
turbulence represented by an electrostatic potential φe(x, t),where x ≡ (x1, x2) are the Cartesian coordinates in the plane
perpendicular to B0. The test particle approach of the turbulent
transport relies on the following Langevin equation:

1. Introduction
The amplitudes and spectra of plasma turbulence aremeasured in a very large number of experiments on tokamak
devices. These experimental data can be used in a test particle
approach to determine the statistical properties of thetrajectories and in particular the transport coefficients. For
slowly varying or large amplitude turbulence a process ofdynamical trapping of the trajectories around the extrema of
the stochastic potential appears and strongly influences thetransport. Important progresses in the study of this nonlinear
process were recently obtained. New statistical methods weredeveloped [1,2] that permitted to determine the asymptotic
diffusion coefficient and also the Lagrangian correlation ofparticle velocity and the time dependent diffusion coefficient.
It was shown that the trapping process determines thedecrease of the diffusion coefficient and the change of its
scaling in the parameters of the stochastic field.
The aim of the present study is to determine the effectof the turbulence spectrum on the diffusion coefficient. It isalready known that in the quasilinear case the shape of thespectrum does not influence the asymptotic diffusion
coefficient D, which is determined only by the average wave
length and frequency. We show that in a turbulence with slow
time variation or large amplitude the shape of the spectrum
has a strong influence on the diffusion coefficient. This is anonlinear effect produced by trajectory trapping.

dx(t )= v [x(t ), t ],dtx(0) = 0(1)where x(t) represents the trajectory of the particle guidingcenter. The velocity v(x,z,t) is the E × B driftv(x, t ) ≡ −∇φ (x, t ) × ez

(2)

where ∇ is the gradient in the (x1, x2) plane and φ(x,t) =φe(x,t)/B0 is the electrostatic potential normalized with B0. The
electrostatic potential φ(x,t) is considered to be a stationary
and homogeneous Gaussian stochastic field, with zeroaverage. Such a stochastic field is completely determined by
the two-point Eulerian correlation function (EC), E(x,t) or
equivalently by the spectrum S(k,ω). These two quantities arerelated by the Fourier transformS (k , ω ) =1dx dt E (x, t ) exp (−i k ⋅ x − iω t ). (3)3 / 2 ∫∫(2π )The EC of the stochastic potential is defined by

E (x, t ) ≡ φ (x′, t′) φ (x′ + x, t′ + t ) .

(4)

The average 〈…〉 is the statistical average over the realizationsof φ(x,t), as is usually considered in theoretical studies
although it is actually experimentally obtained as a space andtime average over x′ and t ′ of the measured potential.

2. The model and the statistical method
We consider a constant confining magnetic field directedalong z axis B0 = B0 ez (slab geometry) and an electrostatic©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: madi@ifin.nipne.ro1249

VLAD M. et al., Turbulence Spectrum and Transport Scaling

percolation in stochastic landscapes [3] and yields γ = 0.7.
This value appears as a critical exponent valid for any EC ofthe potential that decays fast enough when x → ∞. We show
here that the shape of the EC can strongly influence thediffusion and in particular the value of the exponent γ . We
use a new statistical approach, the decorrelation trajectory
method, which is a semi-analytical approach developed in [1]and generalized to more complicated systems. In a recent
work [2] the results of this method concerning the LVC werevalidated by developing a more accurate approach.
The main idea in this method is to study the Langevinequation (1) in subensembles (S) of realizations of thestochastic field, which are determined by given values of thepotential and of the velocity in the starting point of thetrajectories. The potential and the velocity considered in sucha subensemble are Gaussian but non-stationary and nonhomogeneous, with space and time dependent averages. This
subensemble average Eulerian velocity determines an average
motion (the decorrelation trajectory) which is estimated byneglecting the fluctuations of the trajectories around theaverage trajectory. This approximation is analyzed in [1] and[2] where it is shown that it has the important property ofmaintaining the invariance of the average Lagrangian potentialin the case of static turbulence. The trapping process isessentially related with this invariance property of theevolution equation (1). The decorrelation trajectory method
eventually yields a set of trajectories with a weighting factor
for each one. It shows that the statistical properties of thecomplicated stochastic trajectories can be determined by
performing averages over these very simple and smooth
trajectories. They are determined from equations of motionthat have the same Hamiltonian structure as Eq. (1) but witha simple Hamiltonian function determined by the EC of thepotential.
A fast and precise code for determining the timedependent diffusion coefficient for given EC of the stochastic
potential was developed [2]. It can be used as a tool in theanalysis of the experimental data.

The statistical properties of the velocity components arecompletely determined by those of the potential; they arestationary and homogeneous Gaussian stochastic fields like
φ(x,t). The two-point Eulerian correlations of the velocity
components, Eij(x,t) ≡ 〈vi (x′,t′)vj (x′ + x′, t′ + t 〉 , are

Eij (x, t ) = −ε in ε jm∂ 2 E (x, t ),∂xn ∂xm(5)where ε12 = –ε21 = 1, ε11 = ε22 = 0. The EC of the velocity (5)evidences three parameters that characterize the (isotropic)
stochastic velocity field: the amplitude V = E11 (0, 0) , thecorrelation time τc and the correlation length λc, which are
the characteristic decay time and distance of this function.
These three parameters combine in a dimensionless Kubo
number

K=Vτc

λc

=τc
τ fl(6)

which is the ratio of τc to the average time of flight of theparticles over the correlation length, τ f l = λc /V.
Starting from this statistical description of the stochastic
potential, we will determine the Lagrangian velocity
correlation (LVC), defined by:

Lij (t ) ≡ vi [x(0), 0]vj [x(t ), t ] .

(7)

The mean square displacement and the running diffusion
coefficient are determined by this function:
txi2 (t ) = 2∫0 dτ Lii (τ ) (t − τ ),(8)tDi (t ) = ∫ dτ Lii (τ ),0(9)provided that the process is stationary.
For small Kubo numbers (quasilinear regime)corresponding to fast varying turbulence such that τ c << τ f l ,the results are well established: the diffusion coefficient does
not depend on the shape of the correlation and it is Dql =(λc2 /τ c)K2. For K > 1 ( τ c > τ f l ) the time variation of thestochastic potential is slow and the trajectories approximately
follow the contour lines of φ (x,t). This produces a trapping
effect: the trajectories are confined for long periods in smallregions. A typical trajectory shows an alternation of large
displacements and trapping events. The latter appear when theparticles are close to the maxima or minima of the potential
and consist of trajectory winding on almost closed small size
paths. The large displacements are produced when thetrajectories are at small absolute values of the potential. The
most important effect of trajectory trapping consists ofdecreasing the diffusion coefficient and of changing its
dependence on the Kubo number from the Bohm scaling [DB~ (λc2 /τc)K] to a trapping scaling [Dtr ~ (λc2 /τc)Kγ ] with γ < 1.
Simple heuristic arguments can be used to show that the valueof γ has to be in the range [0,1]. The lower limit γ = 0
corresponds to total trapping and the upper limit γ = 1
correspond to the absence of trapping.
The first estimation of γ is based on an analogy with the3. Results
We consider the following model for the Euleriancorrelation of the potential

⎛ x2⎞
c2
ε ( x ) = c1 exp ⎜ −⎟+2 α⎜ 2c0 ⎟⎝
⎠ 1+ x()(10)

where c0 , c1, c2 and α are parameters. This correlation isnormalized such that the amplitude of the velocity fluctuations
V 2 = –E11(0) = 1, which determines one of parameters, c2 =(1 – c1 /c0)/2α. This model of the EC permits to consider
several typical types of correlations. The first term is alocalized space-function while the second is an extended
function with algebraic behavior at large distance. The
variation of the parameter c1 between c0 and 0 determines acontinuous change of the shape of this EC from the pure
Gaussian (at c1 = c0), to a Gaussian with a tail, then to an2250

VLAD M. et al., Turbulence Spectrum and Transport ScalingFig. 1

The normalized diffusion coefficient F(K) = D(K)/DBobtained for the EC (10) with (c 1, c2) having the values(0, 0.5), (0.2, 0.4), (0.4, 0.3), (0.6, 0.2), (0.8, 0.1), (1, 0),from the upper to the lower curve; α =1 and c0 = 1.

Fig. 2The normalized diffusion coefficient obtained for analgebraic EC [c1 = 0 in Eq. (10)] with different values of
α. Dashed line corresponds to Gaussian EC.

Fig. 3The exponent γ in the scaling law of the diffusion
coefficient obtained for an algebraic EC [c1 = 0 in Eq.
(10)] as a function of α.

algebraic function with a perturbation around zero, which
decays and disappear at c1 = 0. The parameter α determines
the rate of decay of the EC. Its domain is α > –1 and it
defines two types of EC: short-range correlations for α > 0and long-range correlations for –1 < α < 0. In the first case
the EC decays to zero at large distances and in the secondcase the EC goes to minus infinity. This is possible because
actually the Eulerian correlation of the velocity has physicalsignificance and this correlation decays to zero in both cases.
In the first case the EC of the velocity is positive around zero
but at larger distances it has a region with negative values.
This shows that the stochastic potential is characterized byfinite size contour lines. In the second case the EC of thevelocity is everywhere positive and decays monotonously tozero. This corresponds to a potential with statisticallyimportant open contour lines.
The effect of trapping is represented by the function F(K)
= D(K)/DB , which appears in the solution obtained with thedecorrelation trajectory method [1] and is the asymptotic
diffusion coefficient normalized with the Bohm diffusion
coefficient. This function obtained for EC’s (10) with different
values of c1 is plotted in Fig. 1 as a function of K. One cansee that the scaling of this function is of power law type, F ~1/Kβ, when the EC contains only one of the two terms. The
exponents obtained for these two cases are different: β = 0.96
for the Gaussian and β = 0.4 for the algebraic EC with α = 1.
When the two terms in the EC (10) are present the scaling ismore complicated and depends on the value of c1 . At verylarge K an exponent close to β = 0.4 can be observed whileat K of the order 10 the exponent is closer to β = 0.96. A
large transition appears between these values. The shape ofthe EC also influences the constant that multiplies the K
dependent factor (significative distances appears between thecurves). In the quasilinear regime K < 1, the same result isobtained for all cases which shows that the shape of the ECdoes not influence the diffusion coefficient.
The function F(K) is represented in Fig. 2 for different
values of the parameter α in the EC (10) and for c1 = 0(without the Gaussian term). One can see that the decay rateof the EC with the distance strongly influences the diffusion
coefficient. The function scales as F(K) ~ 1/Kβ and β increases
when α increases. The trapping is stronger when the EC
decays rapidly (for large α) and as α increases the function
F(K) tends to the curve obtained for the Gaussian EC (dashedline). The diffusion coefficient obtained for the Gaussian ECis D ~ λc2 /τc K 0.04, thus it practically represents the limit of
total trapping Dt ~ λc2 /τc . For negative values of α, β is small
and it goes to zero when α → –1. In this limit F(K) saturates
at K > 1 which shows that the trapping is not present. Thus
the Bohm scaling of the diffusion coefficient appears forstochastic potentials with long-range correlations.
The exponent γ of the scaling of the diffusion coefficient(D ~ λc2 /τc Kγ ) is obtained from β as γ = 1 – β. Its dependence
on α, the decay rate of the EC, is presented in Fig. 3. We
have thus obtained a continuous variation of γ with α, fromthe value γ ≅ 1 for –1 < α < 0 to an asymptotic value which3251

VLAD M. et al., Turbulence Spectrum and Transport Scalingis γ = 0. At negative value of α the trapping is practicallyabsent and the transport is of Bohm type and when the EC islocalized (α → ∞) the trapping is total.

have obtained in the case of Gaussian and algebraic EC’spower law scalings in K for the diffusion coefficients. For thelatter case, we have determined the exponent γ of the diffusion
coefficient as function of the exponent α which describes theEC of the potential, Eq. (10). It is not a fixed value as in theestimation based on percolation theory [3] but a continuousfunction that decays from 1 to 0. For more complicated ECof the stochastic potential, the scaling of the diffusion
coefficient is not of power law type.

4. Conclusions
We have examined the effect of the shape of the Euleriancorrelation on the scaling law of the diffusion coefficient. We
have obtained analytical expressions for the time dependent
diffusion coefficient corresponding to given Euleriancorrelation of the stochastic potential, using the decorrelation
trajectory method. We have shown that in the nonlinear
regime K > 1, the scaling law of the diffusion coefficientdepends on the shape of the EC and not only on the global
parameters of the EC that define the Kubo number. A strong
influence has the space-dependence of the EC at large
distances, i.e. the small k components of the spectrum. Thiseffect is determined by the complicated process of dynamic
trajectory trapping in the structure of the stochastic field. We

References[1] M. Vlad, F. Spineanu, J.H. Misguich and R. Balescu,Phys. Rev. E 58, 7359 (1998).
[2] M. Vlad and F. Spineanu, “Trajectory structures andtransport”, Phys. Rev. E (2004), in print.
[3] M.B. Isichenko, Plasma Phys. Control. Fusion 33, 809(1991).

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