J. Plasma Fusion Res. SERIES, Vol.2 (1999) 343-346

Statistical Theory of Passive Scalar Advected by Turbulence:
An Application of the Lagrangian Direct-lnteraction
Approximation
GOTO Susumu and KIDA Shigeo
National Institute for Fusion Science, Toki, 509-5292 Japan(Received: 25 December 1998 / Accepted: 5 August 1999)

Abstract

A closure theory of turbulence so-called the Lagrangian direct-interaction approximation isformulated for a passive scalar field advected by isotropic turbulence. Solutions to the resultant closure
equations for the correlation functions of scalar and velocity fields are shown to be completely consistent

with well-known phenomenologies on small-scale statistics of turbulence.

Keyword:statistical theory of turbulence, direct-interaction approximation, passive scalar field

1. lntroduction
temperature, particle concentration, dye, smoke, etc,

The statistical properties of this system are different
depending on the Schmidt number s = v/r. It is
phenomenologically [-4] predicted that the passive

which is advected by turbulent velocity field andscalar spectral function @(k, r) defined by

We investigate a passive scalar field 0(x, t), e.9.,

diffused by molecular diffusion as@(k,t)=t*u,PdP=6xdt"_E'9
jctx j'I(1)ctx

Stokes equation,dr,du,*utdi =- t b -*"j+P d;,(i=r,2,3),
r'i -' Je)' dxdx, \'-k'sn followed by an exponentially decay for moderate s,

and the equation of continuity,

du,=0,=joxi@)

wherc Z(k, t) is the Fourier component of the one-timecorrelation function of the scalar field 0(r, 4 anO /OA
denotes a solid angle integration in the wavenumber
space, obeys different scaling laws depending on theSchmidt number in the large wavenumber range: Thespectrum consists of a scaling function proportional toThe velocity field ur(r, t) is governed by the Navier-a,f aoZr*,t1.

and the ft-r and the k't1/3 scaling ranges appear between

the ft-5l3 power law and the exponentially decaying(3)

ranges in the large and the small Schmidt number limits,

where t is the time, .r is the space coordinate, p is thepressure, p is the constant density, v is the kinematic

respectively (see (l 3)-( I 5) below).

viscosity of fluid, and r is the diffusion coefficient ofthe passive scalar. The summation convention is assumed for repeated subscripts.

analytically such a Schmidt number dependence of theThe main purpose of this paper is to show
statistics of passive scalar fields based upon theC o rre spondin g autho r' s e - mail : goto @ toki.the ory. nifs. ac.j p343

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchGOTO S. et dl., Statistical Theory of Passive Scalar Advected by Turbulencebasic equations (1)-(3) by developing the direct-

resents a nonlinear interaction between three Fourierinteraction approximation (DIA) t5,61.

modes i;(t), it{p) and il;(q).We call this nonlinear interaction appearing explicitly in the governing equation
the direct interaction between these three modes. Directinteractions between fii(kr) and the other modes are depicted schematically in Fig. 1. There is only a single direct interaction between an arbitrary pair of the Fouriermodes. This property, the sparseness of the nonlinear
couplings, arises from the constraint k + p + e = o in the2. Lagrangian DIA
2.1 Sparse nonlinear couplings
For simplicity of the analysis, we consider first themotion of a fluid confined in a periodic cube of side L,

which will be set at infinity ar the final stage offormulation. Then, we rewrite the Navier-Stokes
equation (2) in terms of the Fourier coefficients itlk, t)summation with respect to p and q on the right-handof the velocity field as= M'i-(k)side of (5), and is essential in the formulation of DIA.

!otl+ v*'l n,ro, ,,2.2 Assumptions

11 fij(- p, t) fr^(- q, t)

assumptions. [i] The correlation between a triplet ofFourier modes is assumed to originate mainly from their

The DIA is formulated under the following two(5)(k+ p+ q=6 1withdirect interaction. If we removed the direct interaction

M,^(kt=-L(24\'t Pti&\+k'(6)P''&\l'between three modes, then they would become
statistically independent of each other. [ii] Such an|2\LI lk^

artificial removal of only one direct interaction from thesystem might cause very small effects to the entire
statistics because there are a very large number ofinteractions in a high Reynolds-number system. The
second assumption has only to hold during the timescale of the auto-correlation function of the velocity

where Fu(t) = 6- - k k /k2. The right-hand side of (5)consists of a large number of terms, each of which rep-ttf')t (*r)

field. The above two assumptions may be justified forsystems with a large number of degrees of freedom [7].

mi(&r)

4(-&r - &s)

2.3 Langrangian fields
There have been proposed several DIA theories
depending on the choice of the statistical quantities forwhich a closed set of equations is constructed,

4(-kr - er)".g.,Kraichnan's original DIA [5] and the Lagrangian history
DIAs [8]. The present formulation, described in detail in

Ref. [6], is different from them in the sense that it isformulated with a set of two-time Lagrangian velocity
correlation and the Lagrangian response functions,4(-kr di(-&l - &2)

which are Galilean invariant. This Lagrangian statisticalquantities were first proposed by Kaneda [9]. Using tlle
Lagrangian position function V@, t lx', /') governed by

Fig. 1 Direct interactions between rj,(k,) and other
Fourier modes in the Navier-Stokes system, which
are expressed by triangles made by three Fouriermodes. We can see that there is only a single
direct interaction between Ai(krl and any othermodes. In this sense, this is a dynamical systemwith sparse nonlinear couplings. The DIA
assumptions are explained as follows: Although aremoval of the triad interaction shown with thicklines, for example, makes the three Fourier modes

dv+u,*0v
1{dl ' oxj =00)

with the initial condition ry(x, t I x', t') - d,(x - r'), wecan relate Lagrangian fields to Eulerian counterparts.
For instance, the Lagrangian velocity v iQ I x, t'), the velocity of a fluid particle at time t which passed at position r at time /'(< r), is expressed asuiki, Aikol andri;(-kr -ko) have no correlation,
the magnitude of the difference between the true

field and the fictitious field without this directr^v,(tlx'.t'1= | d'xu,(x,t)V@,tlx',t') (8)interaction remains much smaller than that of thetrue field.

J344

GOTO S. e, d/., Statistical Theory of Passive Scalar Advected by Turbulence

3. Solution to the Closure Equation
It is shown [0-12] that the closure equation (10)We apply the DIA assumptions to the Eulerian fields,
which have the sparse nonlinear couplings property, andderive a closed set of equations for the Lagrangian fieldcorrelation and response functions by the help of rela-

has solutions which are completely consistent withphenomenologies. It istions between the two kinds of fields such as (8).

@(k)=Cr7e-1t31'-5t3(13)2.4 Application to a passive scalar fieldin the inertial-advective range (k << min{ft1, k6},

In order to apply the Lagrangian DIA to a passive
scalar field, we introduce, similarly to (8), theObukhov [1] and Consin [2]),@(k)=CzXvlt2 e-tt2 k-l

Lagrangian passive scalar field,(14)

in the viscous-advective range (k6 << k << ks for s >> l,glx',fi=e{L)A3x0{x,t)tlt@,tlx',{) (9)JBatchelor [3]) and@A closed set of equations for the correlation and the re-(k) = C zX r-3 g2t3 P-t1tt(1s)sponse functions of the Fourier component of 0Q) ate
then derived under the two assumptions described inin the inertial-diffusive range (ft6 << ft << k6 for s << l,Batchelor, Howells & Townsend [4]. Here, the charac-s2.2.

teristic wavenumbers ftK (Kolmogorov wavenumber), ftg(Obukhov-Corrsin wavenumber) and fts (Batchelor

By a straightforward calculation [0] from the basic
equation (l) we can derive an evolution equation for thepassive scalar correlation function, while the responsefunction can be analytically solved. When the velocitywavenumber) are respectively defined byand the scalar fields ate statistically homogeneous, isotropic and stationary, the resultant clo-dpdq o(r, p, q) p^'' q

t'i-^'' [ [^,"[-''

ir*{q-lo&ntf"fo'Q(ip, tp't3ye*n(r7)kr= (el vK')t'o = stt2 kr(18)

The Lagrangian DIA we have described here is aquite simple closure theory applied to strong turbulence,
which is formulated under clear working assumptions.

i"'1 | + q2) t

The resultant closed system of equations has solutions, (10)

which are completely consistent with well-known
phenomenologies of small-scale statistics of turbulence

by Kolmogorov [3], Obukhov [], Corrsin [2],wavenumber, 6= @l(X,e'3t+usr+) is the nondimensional
spectral function,[- dkkz@(k)Batchelor [3] and Batchelor et al. [4].

There are many attempts to explain DIA byReynolds number expansions (calledrenormalized(l l)expansions) [4,15] or a diagram technique [6]. Even if

they yield a same set of closure equation, they do notseem to capture the essence of DIA [5]. The DIA is atheory for strong nonlinear systems, in which each oneof the strong nonlinear terms is treated as a perturbation
instead of the whole of the nonlinear terms as done inis the scalar dissipation rate, s is the energy dissipation
rate, ando(k, p, q)=(k+ p + q) (k+ p- q) (k- p+ q) (-k+ p+ q)4p'k'k"=(e | *zltta - fA 17*

4. Concluding Remarks[-r-'where [ = kl(ett4v-3t4; is the nondimensional

y =zr(16)andsure equation is written as

61i1 =

k*=(e / v3)"n ,(r2)other theories.

The function Q is the Lagrangian velocity correlation
function, which has already been known in the framework of the Lagrangian DIA for isotropic turbulence [6].
Thus, it is only the Schmidt number that characterizes
the resultant closure equation (10) for the passive scalarlllReferences
A.M. Obukhov, Isv. Geogr. Geophys. Ser. 13, 580949).

[2] S. Corrsin, J. Appl. Phys. 22, 469 (1951).
[3] G.K. Batchelor, J. Fluid Mech. 5, I l3 (1959).
[4] G.K. Batchelor, I.D. Howells and A.A. Townsend,spectrum.
345

GOTO S. e/ d/., Statistical Theory of Passive Scalar Advected by Turbulencell ll Y. Kaneda. Phys. Fluids 29,701 (1986).
|21T. Gotoh, Y. Kaneda and N. Bekki, J. Phys. Soc.

J. Fluid Mech. 5, 134 (1959).

t5l R.H. Kraichnan, J. Fluid Mech. 5, 497 (1959).
t6l S. Kida and S. Goto, J. Fluid Mech. 345, 307

Jpn. 57, 86 (1988).

(1997\.

ll3l A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30,[7] S. Goto and S. Kida, Physica D ll7, l9l (1998).
t8l R.H. Kraichnan, Phys. Fluids 8, 575 (1965),

301 (1941). English translation in Proc. R. Soc.
London, Ser. A 434, 9 (1991).
[l4] D.C. Leslie, Clarendon Press, Oxford, (1973).
ll5l R.H. Kraichnan, J. Fluid Mech. 83, 349 (1977).
l16l Jr. H.W. Wyld, Ann. Phys., N.Y. 14, 143 (1961).

erratum: Phys. Fluids 9, 1884 (1966).
t9l Y. Kaneda, J. Fluid Mech. 107, l3l (1981).

ll0l S. Goto and S. Kida. Phys. Fluids ll,1936 (1999).

346