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NASA Contractor Report 38 17 

Theory and Modeling of 

Atmospheric Turbulence 

Purt IIt September I, 1982-August 31, 1983 

C. M. Chen 

The City CoZZege Research Foundation 

New York, New York 

Prepared for 

George C. Marshall Space Flight Center 

under Contract NAS8-34622 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1984 
FOREWORD 

The domain of statistical physics has extended from the traditional 

problems in quantum mechanics, the solid states and the kinetic theory of 

gases to other nonlinear problems in hydrodynamics, plasmas and optical 

systems, for which we need to consider the fluctuation effects. Conceptually, it has been found very helpful to view these statistical problems 

from a basic physical point of view which emphasizes their structural 

similarity. 

The study of stochastic systems has developed in two directions. First, 

the transition from a laminar state to a chaotic or turbulent state through a 

sequence of bifurcations shows a certain universality (see "Chaos and Universality," Nordita Selection, 1981). 

state of fully developed chaos or strong turbulence, the statistical methods 

used for their theoretical analysis find again a certain universality. 

proper understanding of the many basic phenomena in astrophysics, space, 

atmospherical and optical applications depends critically on our ability to 

analyze the turbulent characteristics and the collective processes in these 

nonlinear systems. The lack of a suitable methods of treatment and the difficulties encountered even for the simplest form of incompressible, homogeneous and isotropic turbulence, have hindered the theoretical development 

of strong turbulence. 

Secondly, when the system has reached a 

A 

From the physical point of view, a turbulent state is characterized by 

its transport properties: eddy diffusivity, eddy viscosity, coefficient of 

damping, or amplification. Their analytical determination requires a transport theory. The kinetic method is best suited for this purpose. The eddy 

transport coefficients, as induced by the fluctuations of small scales, govern 

the evolution of larger scales. This requirement of scaling leads to the concept of renormalization groups, from which we develop the group-kinetic method 

of turbulence. We have applied the method to problems of atmospheric turbulence. 

The group-kinetic method combines the advantages of the kinetic method and 

the group-scaling. The kinetic description has the advantage of transforming 

the system of hydrodynamical equations that govern 

into a master equation of lesser nonlinearity. The group-scaling enables the 

determination of the transport properties and the spectral structure by the 

one-point distribution function alone, without involving the two-point distribution. 

not by truncation of the infinite hierarchy of n-point distribution functions or 

correlation functions. 

a nonlinear stochastic system 

Our closure is obtained by a memory loss in the relaxation process and 

The six verbatim sections include research accomplished during the second 

year, September 1, 1982 through August 31, 1983, of this two-year contractual 

effort. 

iii 
ACKNOWLEDGmNTS 

The author wishes to acknowledge the support for this research of 

Mr. John S. Theon and Dr. Robert Curran of NASA Headquarters, Washington, 

D. C., and the technical direction of Dr. George H. Fichtl and Mrs. Margaret 

B. Alexander of the Atmospheric Sciences Division, Systems Dynamics Laboratory, 

Marshall Space Flight Center, Alabama. 

, 
TABLE OF CONTENTS 

SECTION 1: General Considerations on the Group-Kinetic Theory 

of Turbulence .................... 

I . Principle of the Group-Kinetic Theory .......... 

I1 . Spectral Distribution of Turbulence ........... 

I11 . Numerical Modeling .................... 

IV . Atmospheric Boundary Layer ................ 

SECTION 2: Group-Kinetic Theory of Two-Dimensional Geostrophic 

Turbulence ..................... 

I . Introduction ....................... 

I1 . Microdynamical State of Turbulence ............ 

111 . Group-Scaling Procedure ................. 

IV . Scaling of the Liouville Equations ............ 

V . Propagator and Probability of Retrograde Transition ... 

VI . Memory in the Collective Collision ............ 

VI1 . Spectral Balance ..................... 

VI11 . Statistical Equilibrium and Non-Equilibrium ....... 

IX . Spectral Distributions .................. 

X . Conclusion ........................ 

References ........................ 

Page 

1-1 

1-1 

1-2 

1-2 

1-3 

. 

2-1 

2-2 

2-3 

2-5 

2-7 

2-10 

2-14 

2-18 

2-23 

2-24 

2-29 

2-30 

SECTION 3: Equivalent Methods for Quasi-Linear Turbulent 

Trajectories .................... 3- 1 

Abstract ........................... 3-1 

I . Introduction ....................... 3-1 

I1 . Kinetic Quasi-Linear Equation .............. 3-2 

V 
SECTION 3: (continued) . 

Page 

I11 . Solution of the Asymptotic Quasi-Linear Equation: 

The Propagator Method .................... 3-14 

IV . Renormalized Quasi-Linear Kinetic Equation ......... 3-19 

V . Non-Asymptotic Quasi-Linear Kinetic Equation ........ 3-21 

VI . Solution of the Non-Asymptotic Quasi-Linear Equation: 

Green's Function ...................... 3-24 

Acknowledgments ....................... 3-29 

References ......................... 3-30 

SECTION 4: A New Kinetic Description €or Turbulent Collision 

Including Mode-Coupling ................ 4-1 

Abstract ............................ 4-1 

I . Introduction ........................ 4-2 

I1 . Turbulent Collisions and Higher Order Terms ......... 4-5 

I11 . Exact Kinetic Results .................... 4-10 

IV . Non-Asymptotic Treatment for Turbulent Collisions ...... 4-16 

V . Analysis of the Lagrangian Autocorrelation of Fields .... 4-22 

Appendix .......................... 4-28 

References ......................... 4-30 

SECTION 5: Spectral Structure of Turbulence in the Stable 

Atmospheric Boundary Layer ............... 5-1 

Abstract ............................ 5-1 

I . Introduction ........................ 5-2 

I1 . Basic Equations of Atmospheric Turbulence .......... 5-2 

I11 . Kinetic Foundation of the Pressure-Strain Correlation .... 5-8 

References ......................... 5-15 

vi 

J 
- Page 

SECTION 6: Kinetic Equation of Turbulence. ........... 6-1 

........................... 6- 1 

6-2 

Abstract 

I. Fluid Equations of Turbulence. .............. 

Master Equation for the Description of the Microdynamical 

State of Turbulence. ................... 

111. Kinetic Hierarchy of Turbulence. ............. 

11. 

6-5 

6-10 

References ........................ 6-14 

vii 
SECTION 1 

General Considerations on the Group-Kinetic Theory of Turbulence 

I. PRXNCIPLE OF THE GROUP-KINETIC THEORY 

The Navier-Stokes equation of motion with external 

forces is transformed into a master equation 

and self- consistent 

fi 

for the distribution function f(t,x,v) of velocity v , with the equivalence 

relation 

uu * 

A 

f(t,x,v) =p 5 - u(t,x)-) , \--- 

).u 

that is valid for a constant or variable density p, and for a fluctuating 

fluid velocity $(t,x). Here L is the differential operator for the perturbation 

of the trajectory in the phase space. 

0 

-w 

The total distribution function 

Iv 

consists of a mean distribution ?= (?} and a fluctuation f . The usual 

Fourier decomposition of a fluctuating function contains too many minute details, 

and a coarse-graining procedure is necessary. In analogy with the "renormalization 

Y rgroups", we decompose f 

of decreasing coherence, 

into groups 

f' , f" 

representative of the spectral evolution, eddy diffusivity, 

and relaxation, respectively. By formulating the relaxation as a functional of 

the diffusivity, we obtain a closure. 

The group-scaling has the advantage of determining the spectrum from the 

I 
1-2 

singlet distribution function alone, without involving the 2-point distribution 

function as is required in the conventional methods of statistical mechanics. 

The kinetic equation is derived for fo . It takes the generalized form 

of the 'Fokker-Planck equation, with the diffusivity D'(} as an integral 

operator to represent the memory and the collective behavior. It is converted 

into the hydrodynamic equation in the group form, from which we determine the 

eddy viscosity 

2 

K' F(k), parameters ] c * 

and calculate the spectral function F(k). The parameters include the Coriolis 

field and the length of stability from shear and buoyancy. 

11. SPECTRAL DISTRIBUTION OF TURBULENCE 

A. The application of the group- kinetic method to the Navier-Stokes 

equation of motion finds the direct and reverse cascades for the transfer across 

a spectrum, and derives the spectral distributions in inertia (k'5'31, shear 

(k-'), and geostrophic turbulence (k'3 and k 3. In boundary layer turbulence, 

the spectra of velocity fluctuations in separate directions 

and a spectral gap in the stable boundary layer is expected. 

-4 

can be treated, 

B. The extensions to the Zakharov equations, the nonlinear Schradinger 

equation and the Korteweg-deVries equation are important for analyzing the new 

dynamical characteristics and transport properties of soliton turbulence in fluids, 

plasmas and solid, and the self-focusing in optical turbulence. 

111. NUMERICAL MODELING 

- 

The kinetic equation for f and its conversion into fluid representation 

forms the basis for the modeling and prediction of turbulent profiles and 
1-3 

spectra in the turbulent boundary layer. Here the transport coefficients become 

explicit functions of t, x 

laws are analytically determined by our group- kinetic theory. The pressurevelocify correlation is also analytically derived. 

and the governing parameters, once the spectral - 

Our first aim in the modeling of turbulence is to provide an analytical 

basis for the Monin-Obukhoff similarity by determining the structure of the 

universal functions. Then, we can devise a pew numerical modeLing which will 

incorporates the effects of scales, the collective behavior and the memory. 

IV. ATMOSPHERIC BOUNDARY LAYER 

We can divide the turbulent atmospheric boundary layer into: 

A. The lower boundary layer, or surface layer 

B. The upper boundary layer which includes the free-convection layer, 

the mixed layer, and the Ekman layer. 

The 

and is characterized by the constant fluxes of momentum, temperature and 

surface layer extends from the ground up to a height of less than io0 meters, 

humidity. The main tasks are the prediction of profiles and spectra by analytically 

determining the universal functions that could not be determined by the empirical 

similarity theory of Monin and Obukhoff. 

The upper layer contains the additional effect of the Coriolis force, 

which enters into the differential operator in perturbing the trajectory. 

The above division of the atmospheric turbulence into two parts, as 

dependent on the absence qr presence of the Coriolis field, is analogous to 

the classification of plasma turbulence into two parts: the Langmuir 

turbulence (i.e. without magnetic field) and the magnetized plasma turbulence. 

J 
SECTION 2 

Group-Kinetic Theory of Two-Dimensional Geostrophic Turbulence 

C. M. Tchen 

The City College and The Graduate Center 

of The City University of New York, N. U. 10031 

ABSTRACT 

The two-dimensional geostrophic turbulence driven by a random 

force is investigated. On the basis of the Liouville equation which 

simulates the primitive hydrodynamical equations, we develop a groupkinetic theory of turbulence and derive the kinetic equation of the 

scaled singlet distribution. This distribution will suffice for the 

investigation of the spectrum of turbulence, without having to resort 

to the pair-distribution as was with the usual kinetic theories. The 

collision integral has a memory and describes the pair interaction and 

its enhancement by the multiple interaction. 

Our kinetic equation of turbulence is transformed into an 

equation of spectral balance in the equilibrium and non-equilibrium 

states, as governed by the direct cascade and the reverse cascade, 

respectively. The sequence of power laws 

are derived. 

k-3, k-4 of velocity fluctuations 

2- 1 
2-2 

I. INTRODUCTION 

The two-dimensional geostrophic turbulence has attracted much attention 

[Fjdrtoft, 1953; Charney, 1971; Rhines, 1979 . Theories based upon the 

diffusion approximation [ Leith, 1968 1, the spectral properties [Novikov 

1979 1, the direct interaction approximation [ Kraichnan, 1967, 1971a,b ; 

Pouquet et al., 1975; Salmon et al., 1978 1 , and numerical modeling and 

simulations [ Lilly, 1969a,b, Basdevant et al., 19783 have found a spectral 

driven by a random force [Saffman, 1971; Thompson, 19731 . Deviations from 

these laws have also been discussed [ Gage, 1979 1. The sequence of appearance 

of the spectra k-3 , 

cascade were analytically vague 

theory of the geostrophic turbulence driven by a random force. 

9 

k-3 . This spectrum changes into k-4 if the geostrophic turbulence is 

k”4 , and the dynamics of the direct and reverse 

in the lack of a generalized statistical 

Our investigation is divided into two parts. The first part develops a 

group-kinetic theory on the basis of our earlier scaling procedure 

1979 1 , and derives the kinetic equation of turbulence. Usually the analysis 

of the spectral structure requires either a pair-distribution function or a 

detailed Fourier decomposition. These are not needed here, because of the 

scaling procedure mentioned. Our kinetic approach has the advantage of not 

only including the random force in our homogeneous Liouville equation, but also 

of describing the interaction between the wave and the fluid particle, as 

characteristical of the large scale turbulence. The collective collision 

obtained will include the memory, the pair-collision and the multiple 

collision [Tchen and Misguich,l982] . The latter effect is essential and 

will be applied to the treatment of the reverse cascade that is often found 

with the large scale turbulence. 

Tchen, 1978, c 

With the kinetic foundation described above, the kinetic equation of 

turbulence is transformed into its hydrodynamical form, and is used to 

develop the spectral theory of geostrophic turbulence in the second part of 

this paper. The two parts are separately self-consistent and can be read 

independently. 
2-3 

11. MICRODYNAMICAL STATE OF TURBULENCE 

The Navier-Stokes equation of motion 

of an incompressible fluid satisfies the condition 

0.; = 0 0 

-.., 

By taking the curl, we obtain the vorticity 

h p = V%Lc 

+L * 

and transform (1) into the following vorticity equation of the two-dimensional 

geostrophic turbulencer 

Here we have 

The field 

/* a 4 

E - = c Ep + ,“x 

has two components. The component 

which is the gradient of pressure $ at a constant density f‘ 

present in the vorticity equation (3) , because 

, is not 
2-4 

I\ 4 

forms a finite vorticity source VxE . .!X .v. cx 

The other component 

The equations (l), (2) and (4) describe the microdynamical state of 

turbulence, and are used as the point of departure of our statistical 

treatment. In our transport theory by eddies larger than the viscous cutoff, 

the effect of the kinematic viscosity will be neglected, but will be 

restituted in the spectral balance where the viscosity gives a molecular 

dissipation. 

For the development of a kinetic method, we write the microkinetic 

equation in the form of the Liouville equation: 

[At +h] l(t,X,V) -II. - 0, 

with the differential operator 

A 

The detailed distribution function f is normalized to unity, as 

h 

dv f(t,x,v) = 1, 

and assumes the form 

in order to be 

equation of continuity (2). It is not difficult to show by taking the first 

two moments, that 

equations (1) and (2). The Liouville equation is homogeneous and has less 

nonlinear terms than does the Navier-Stokes equation, because the function 

u(t,x) of velocity of fluid is replaced by an independent variable v . We retain the only nonlinear term E *&f in order to describe the 

mode-couplings. 

consistent with the Navier-Stokes equation (1) and the 

the Liouville equation will reproduce the two hydrodynamical 

r 

*. 4A v-r 

-I 

J 
2-5 

111. GROUP-SCALING PROCEDURE 

The Liouville equation (9) contains too many minute details which are 

unncessary for a statistical treatment of turbulence. We apply a scaling 

procedure by dividing the detailed distribution 

h f=r+F (13) 

0u 

into a mean distribution 2 and a fluctuation f , and the fluctuation 

(14) tu 

f = fO+ f' 

- 

into a macro-group fo and a micro-group f' . The function f(t,x,v) 

gives the distribution of velocity in non-equilibrium. The macro-group fo 

calculates the spectrum of turbulence'without the need of developing a 

separate kinetic equation of the pair- distribution function (E f) . In the 

following, we develop a kinetic equation of the macro-group 

investigate the spectrum of turbulence. 

-1 

NN 

fo and 

The evolution of the macro-group is controlled by the transport 

property as shaped by the fluctuations of the micro-group, while the 

approach of the transport property to equilibrium is obtained by the 

relaxation. These three transport processes of evolution, transport 

property and relaxation are represented by the three groups 

fO, f' , f" 

of the distribution function, or by the three groups 

Eo, E' , E" v u - 

of the field. The interactions between the groups are described by a 

coupled system of equations. 

The groups have their time scales characterized by the correlation 

times 

x 
2-6 

For formulating the transport coefficients, the correlations, as given by 

the groups E'(t-t ), are integrated with respect to 77 and 

7; ' in the domains (0, t) , (0,t ), respectively, so that the evolution 

c'(7:- T'), .y 

times t 

following manner : 

,f, F 'can be ranged among the correlation times (15a) , in the 

The degradation of coherence (15) will constitute a property of 

quasi-stationarity of one group with respect to the other. 

The individual fluctuating groups 

0 

u 9 u' y u" Y ). * 

can be decomposed into Fourier components with overlapping wavenumbers. But the 

global averages e. g , 

are deterministic, and are separated by their adjacent wavenumber domains 

with k, k' varying from 0 to 60 . For the sake of convenience, we introduce the scaling operators 

- 

A , A" , A' A" (174 
2- 7 

IV. SCALING OF THE LIOWILLE EQUA'XONS 

By applying the operators A" and A' to scale the Liouville equation 

(9), we obtain the system of coupled equations: 

(b $)f' = - L'fo - c, . 

The second equation of the system may be equivalently replaced by 

(1, + A'Z)f' = - L'fo . 

Upon integrating (19) and (20), we have 

t 

ff = - A'[ dtG(t,t-C) L'(t--G) fo(t-t) 

JO 

t 

- A'/ d'l; $(t,t-T) Co(t--C'), 

0 

and 

dti(t,t-T) L' (t-T) { (t-T) f' = - A'it 0 

A A 

Here 

operators L and A'L, respectively. The initial value f' (0) has been 

neglected, since it cannot produce a finite correlation 

at large t , by (15). 

U andA are evolution operators as related to the differential fi h 

AOL' (t)f '(0) 

Subsequently, we prernultiply (21) and (22) by -AoL' to find the 

collision in the following two forms: 

C0 = Eo + Aok: L'(t) U'(t,t-T.) Co(t-t) 

p co - Ao Ho* Go , with H's -AoL1(t)U'(t,t-t) 
2-8 

co = A' A'[fo(t-T)) . 

The collective collision Co 

collision 

is seen to consist of two parts: The pair 

co= Aod'ffo(t-r)) , (25) 

fl 

from the correlation between two micro-fields 

to 4' The multiple collision, from the correlation AoL'U'Co between 

a micro-field and the cluster of the organized micro-fluctuations, is 

proportional to LQ . 

AOL'A'UL' , is propcrtional 

/2 

The governing collision operators 

t 

A'&' = AoJodC L' (t) A' ;( t, t-T) L ' ( t-t) 

are defined by the diffusivities 

n 

A'@' = A" Jo:, E'(t)A'U(t,t-C .ry )E'(t-t u ) ..L 

-a 

AoD' = A" /:T E'(t)A'A (t,t-T)E'(t-C r* ) .* u n 0 

which themselves may serve as operators. In (23) we have replaced 

AoL'(t)A'j(t,t-t)Co(t-T; ) by AoL'(t)U'(t,t-T)Co(t-C) 

without loss of generality. 

x 
2-9 

4 

The operator U(t,t-z), with t' = t-75 is governed by the Liouville 

equations : 

4 A 

The operator A is related to U and the relation has been developed 

by Weinstock [igw] . 

By scaling (28a) by means of A" we obtain the kinetic equation 

with the collisions 

as related by 

Ho yo- A" Hog Ho , 

in analogy with (23) - (25). 
2-10 

V. PROPAGATOR AND PROBABILITY OF RETROGRADE TRANSITION 

n 4 

The Liouville equations (9) and (28) which govern f and U have h the same differential operator L , and therefore the same characteristic 

equations governing the dynamical variables in the form 

or equivalently, 

with the conditions 

A 

x(t) = x, G(t) = v, (314 - .Le rc 

by the nature of the differential equation of the second degree, where 

the "initial conditions" in the retrograde transition from t to tt . By a change of variables 

5 and ,v are two independent variables. The values (31c) will be called 

h 

6 n 

x(t-t) = x - R(r) , v(t-t) = v - V(t), T=t-t', (324 h "-- ..r *.c 

we can write the characteristic equations in the integral form 

1c Jo .m 

- 0 

Note that ;(t-T) from (32a) can be rewritten as 

* 
2-11 

or as 

in the Lagrangian representation, with 

for the sake of brevity, The vertical bar in 

Lagrangian displacement x(t) was made during a time interval T: along 

the trajectory that passes by the point x, at the time instant t. 

The systems (31) and (32) are called the equations of the path 

dynamics in the differential form and the integral form, respectively. 

Both forms equally describe the trajectory of 

occupies the position x(t') at the time instant t' , provided 

the trajectory passes by the point x at the instant t while having 

a velocity v . 

[ denotes that the A t,z IM 

a fluid particle which 

A 

- ... r) 

It is to be stipulated that the essential function of the propagator 

A A 

U is to impose a Lagrangian representation of the function, say E as 

in (27a), in the form 

A 4 

E [t', G(tq)) I U(t,t')$(tl), .y 

w - 

following the trajectory that is determined by the differential 

dynamics (31). 

Alternatively, with the path dynamics (32) , the Lagrangian 

representation can be obtained in the integral form 

by using the probability density 

The integral relation is written for the purpose of transforming the 

Lagrangian function into the Eulerian function. 

n 
2-12 

The probability density satisfies the condition of normalization 

The density (34a) defines the probability 

for the fluid particle to occupies a position between 

at the time instant t-73 , provided it follows the trajectory which 

passes by the point x, at the time t. For the retrograde transition to be 

specified by the integral path dynamics of f(r), as given by (32), we 6. 

write [Tchen, 19441 

A p(t-Z, x-R 1 t,X) = Sfi-i+t-T, 1 t,X)l, 

or briefly 

The abbreviated form indicates a quasi-stationarity of $ 

space, so that the Liouville equation can be written in the form: 

in the t,x . 

with the differential operator 

It is not difficult to show by moments that (37a), together with (36), 

will reproduce the integral path dynamics with the initial velocity 

definition (12), v can be identified as *u(t,x) in the return to the 

A 

that is a random variable having a distribution f(t,x,v) . By z .-- n.- Y 

x 
2-13 

micro-hydrodynamical description. Thus the Lagrangian representation (33a) 

by means of the propagator is straight forwardly kinetic. On the 

other hand, the Lagrangian representation (33b) by means of the probability 

p(b,g) is valid in both the fluid and kinetic descriptions. In the fo 

the initial velocity is 

dissociated from the distribution f (t,x,v) In the latter, although the 

initial velocity 

probability density "p<t ,& ) , it is implicit in the path dynamics through 

the distribution ?(t,x,v), so that the same Liouville equation can be 

copsidered as an equation of a kinetic significance in the contracted 

dimensionality, i.e. in the sense that the governing path dynamics requires 

an initial velocity to be determined kinetically. The contracted dimensionality 

gives to the Liouville equation a simpler differential operator that is 

independent of 4 and therefore a simpler Fourier transformation. 

fi 

U(t,t') 

A 

u A 

u(t,x), and the Liouville equation (37a) becomes fi -* 

c ,- 

v is not directly involved as a variable in the * 

u 

"n. 

- 

In the following, we shall exploit this advantage of the contracted 

dimensionality by using the probability of transition in transfoming a 

Lagrangian function, e.g. (33a), into an Eulerian function, e.g. (33b). 
2-14 

VI. MEMORY IN THE COLLECTIVE COLLISION 

The problem of the derivation of the collective collision Co in 

terms of the pair collision c" involves the solution of the coupled 

integral equations (23) and (30), and the treatment of the Lagrangian 

functions associated with the propagator and its collision, from (29) 

and (30b) The problem is complicated. We shall make the simplification 

by separating the chain of correlations (15b) into two sections 

and 

7: 7 q > T'. 

referring to the processes of transport coefficient, i.e. diffusivity, 

and relaxation, i.e. propagator, respectively. The first section which 

has a correlation of long duration shall preserve its memory, while the 

correlation of the second section which has a short duration should not 

propagates its memory beyond the time spanz and exert an effect on the 

evolution of fo(t,x,v) , ...u 

With this memory-loss we simplify the equations of collisions (23) 

and (30b) into the system of equations 

which we can combine into 

C0 

It leaves the Lagrangian-Eulerian transformation by the method of the 

transition probability, as follows: 

(40) 
2-15 

We see the transformation 

from (33b), in which the kinetics of 

by the dynamics of 

as governed by (29) is replaced 

as governed by (37a). Note that 

are in the t,x,v space. On the other hand, in the T ,a space, p(c,g ) 

does not carry v , except in the path dynamics as an initial condition. 

..." U .c 

Y 

Further simplification is obtained by a Fourier transformation of 

(41) into 

- 

H + Co= (A'} (2?c)-$ p(w,k) Co(4k) vu = -%(d,k) * Co(cJ,k), bo (424 w 

with 

(3,k) = - (, A)7C ' lm dte i(d-k.v)t- - - Perk) (42b) * 

in d-dimensions, so that the integral equation (40) is transformed into the 

form 

or, by collecting Co, 

with 

- 

The probability function' 

differential equation of evolution: 

P( t ,j ) is determined by the following I 

that is scaled from the Liouville equation 

(43) 

(44) 
2-16 

with a fluctuating differential operator. We find the solution: 

1 

From the path dynamics (32), we find 

and that (@'(t)) 

evolution of F(r,l) from (451, 

Now we can calculate 6 from (42b), by substituting for 5 from (47). 

We find that 

is asymptotic and (m"0) is not asymptotic in the 

.I 

with 

is controlled by the memory-loss function 

which determines the life-time of memory in the form 

The characteristic frequencies are 

and the numerical coefficients are 

c =T/c' , &=(1/3)~'~ , co =(1/8)~/~ 

o( 

51a 

a 
2-17 

As seen from (43) and (44), the collective collision Co is 

related to the pair collision c" by a factor , which can be r 

as 

mitten 

in terms of the real and imaginary parts of & 

determination of (51) rests on the calculation of the memory integral 

(50a) and its decomposition into its real and imaginary parts. The 

calculation is made by an interpolation over the three regions dominated 

bYLt 

, by (49). Hence the 

and mo , separately. The results are given as follows: 

(a) For weak turbulence, i.e. 8 C.$: hb , we find a shielding 

in the collision by a factor 

0 

(b) For strong turbulence, i.e. 

enhancement in the collision by a factor 

%: ry , we find an 

(3 ?: 12.51, if no<< $' 

11 f 2.77 ($/m0)32 1, if mo >)a' 

3p 

We can approximate p by its constant asymptotic values which are : 

(a) weak turbulence , 

@I, 

0 (b) strong turbulence, /2 < "Jp', I* 

1, for &!'<$,' (large k) 

2.51, for$'&'(small k) . b 

(544 
2-18 

It entails the simplification 

of (43.). 

VII. SPECTRAL BALANCE 

The kinetic eqution (18) of fo , rewritten as 

(bt + AoL)fo = - Lo? f Co (56) 

has a collective collision which can be written as 

with a diffusivity found by (55b). - 

Consider now a homogeneous turbulence, i.e. with = 0, E = 0. .* * 

By taking the first moment of the kinetic equation (56), we obtain the 

following equation of macro-momentum: 

with a collision 

by (57). Subsequently,-we take the curl of (58) and (59), to obtain the 

following equation of macro-vorticity of the two-dimensional geostrophic 

turbulence: 

Finally, 

spectral 

upon multiplying (60) by go and averaging, we find the equation of 

balance in the form: .rr 
2-19 

with the following transport functions: 

cascade tranf er T" = ($9 VXQ") (624 

molecular dissipation E a t A!(PT")z) (62b) 

3; r* -A. 

5 ..L 

Note that the field 

0 E," = _EpO +,Ex 

consists of two parts: The first part has a zero vorticity by (a), and 

the second part, which is a random source, does not correlate with 5 . Thus the field bo 

the correlation (P-E") . But it governs the collision 

0 

does not appear in the spectral balance in the form of 

0 

*-- 

c r 

with 

through the diffusivities 

aD'> =(E)?/) .P (y> (634 

<Di)= rdT(Epl (t,xM%, t-t)Epl *cI (t4@$xp(Ex1 (t,x)&, t-C)EX' 4b (t-T) .(63d) 

By separating the collision into two parts as in (63a), we find the 

transfer function also separated into two parts, as follows: 

x z % 

> with do to 

0- * 

.1 

(644 

(64b) 

0 To = Tpo + Tx 

Tp 0 =(ro* vx Qpo) , TX =(CoeY~Qi> .. 

, r with 

-I.l.*. **d 

In the following, we shall follow the same method of treatment of 
2-20 

the transfer function as developed earlier 

we introduce the spectral functions Fu(k) 

density (sx'(k)) , such that 

by Tchen [1981] . As notations 

and F (k) , and the spectral 

5 

and the second spectral moments 

2 k 

R = 2 dk' kt2 F (k'), Ru = 2Ikdk' k' FU(k'), 

r 0 r 0 

with the relationship 

by definition (3) in d=2 dimensions. By omitting the details of calculation, 

we find the results as follows: 

(a) Cascade-transfer by the Ep - fluctuations 

The transfer function is obt2ined in the form: 

The coefficient /3 represents the effect of the multiple collision (54). 

The transfer function is seen to be governed by the eddy viscosity 

which has an asymptotic value 

-00 

;b 

for large k , and by the rate of damping 
2-21 

with a3 

(6 ') (K') h = [ _" d 

Fu(k") ] 

k 

U 

for small k . Note that (69a) is the solution of 

Q) 

The asymptotic formulas 

theory [Tchen, 1973, 19781 

(67) and (69a) agree with our earlier cascade 

and other dimensional theories 

pisenberg, 1948; Gisina, 19693 . (b) Cascade-transfer by EX - fluctuations h 

The transfer function is obtained in the form 

= /(Ki)R; , for large k 

TX I 1/3 ~.'(q*~, for small k , 

and is governed by the eddy viscosity 

for large lc, and the rate coefficient of damping 

for small k . Here the modulation function 

is characterized by the two frequencies as fqllows: 
2-22 

It is to be noted that the simplified formulas of all the transfer 

functions listed above are characteristically in the form of products of 

two functions. A general formula not in the form of a product has also 

been developed by Tchen (1981b), but is too complicated for the application 

to the present problem. 

We first consider the portion of the spectrum of large k . The small 

eddies are embedded in a gradient of big eddies and in their coupling with 

the bigger eddies, the small ones play the role of an eddy viscosity, 

to cause a loss of the bigger ones at the rate >t(p"t)< 0 . This transfer 

is in the pattern of a cascade from the big eddies to the smaller ones. 

It is of the gradient type and is called the direct cascade. Now we 

consider the portion of the spectrum of small 

no more gradient left in the medium in which they evolue, if the turbulent 

medium is isotropic. The coupling between the big and small eddies is 

not through the gradient transfer, but is of the type of the damping. The 

rate coefficients a:, fl.' are originated from the wave-particle interaction 

[Tchen, 19811 , and may become negative when k is sufficiently small. 

Then they give an amplification. When this happens, thb transfer functions 

reverse their roles from a loss into a production, i.e. 2t(<2> > 0 

Such a transfer mechanism is called the reverse cascade . An analogous 

phenomenon is known in plasmas as the Landau damping or amplification. 

k . There the big eddies have 

. 

If the demarcation wavenumber between the direct and reverse cascades 

is krev , the large and small values of k refer to 

krev and krev 9 (75) 

respectively. 

J 
2-23 

VIII. STATISTICAL EQUILIBRIUM AND NON-EQUILIBRIUM 

By differentiating the equation of spectral balance (61) with respect 

to k,, we obtain: 

The left hand side vanishes in a statistical equilibrium, yielding 

The constant of integration in the right hand side is determined by 

condition at k = bo , i.e. 

In addition, we note that 

T = 0, at k=O . P (79) 

For the inertial subrange in statistical equilibrium, we can write 

the spectral balance in the form 

by omitting the viscous dissipation 

subrange. Here f 

The statistical equilibrium requires that the two transfer functions 

find a net positive transfer to balance 

& by definition of the inertia 5 is a sink in the enstrophy transfer across the spectrum. 5 

. < 

On the other hand, if the net transfer is negative, ceases to be c 

a useful parameter, and the hypothesis of the statistical equilibrium 

becomes invalid. Consequently, we have to return to the original equation 

(61) for the spectral balance in non-equilibrium: 

by assuming a supply to the spectrum from an non-stationary source of larger scale. 

x 

L 
2-24 

Here we have again omitted the molecular dissipation. The negative net 

transfer, i.e. T 0, means an amplification of the enstrophy in time, 

and forms a reverse cascade toward the small wavenumbers in the Spectrum* 

t 

IX. SPECTRAT, DISTRIBUTIONS 

In the preceding Section, we have distinguished between a direct 

cascade and a reverse cascade, as characterized by a transfer of the gradient 

type at large k and by a transfer of the damping type at small k . We 

analyse the spectral distributions for these two cascades in the following 

lines. 

A. Direct Cascade 

We consider the joint enstrophy transfer by u and EX - fluctuations 

.L. 

u 

in the spectral balance (80), and rewrite it as 

with the use of (66a), (67), (70a) and (71). The eddy viscosity(pk') 

governs the transfer under the driving force exclusively, and the eddy 

viscosity (KU> governs the transfer without the driving force. 

of the following two separate equations of spectral balance: 

An approximate solution of (82) can be obtained by an interpolation 

k 

Ok 

2(rrL3ldIc1 kt2 F (k') - f 

5 5 

'E, 2{%3JdL' kf2 I 

0 

finding the asymptotic solution 

of (83a), with 
2-25 

<'ut}= bK 5 113 k-2 , bK= 2/C = 0.77 , 

and the asymptotic solution 

Note that, for obtaining the solution (85) of the equation of 

balance (83b), the eddy viscosity (71) has been written as 

= (s \ Jyktl 27 k" G(k"), for finite k, 

R 

where (s) is the trace of the spectral density tensor, i.e. 

As a white noise it has the form 

&(ttl)) = ( s ) , independent of k", for finite k" 

(88) = 0, for kt' =a) 

Also note that 

dk" = 2 k" dkl' . (89) - 

We have calculated G(k") from (73) and (74) for substitution into (86), to 

find(MX')as was written in (85b). The numerical coefficient 

determined from the equation: 

has been 
2-26 

Finally by substituting for(&') thus obtained, we have f 

as the solution of (83b). 

Now we return to the problem of the interpolation of the two 

asymptotic solutions (84a) and (85a), by incorporating the two eddy 

viscosities (84b) and (85b) in the equation (82) of the joint balance, 

which becomes : 

The equation (91) shows that the dominant eddy viscosities arein the form of 

(Eu') for k < -ko, and (%I) for k) ko 

Then we find the following formula of interpolatlon: 

1 with 

1 

'( l+k/k 0 ) 2 1+ k/ko 

and 

0.46 {s) -1/4 

(93) 

(94) 

(95) 

It is easy to verify that the general solut-on (93) -nclude the 

asymptotic solutions (84a) and (85a) for k < ko 

It is seen that the spectral law k-3 

in the sequence of increasing wavenumbers. 

and k > ko8 respectively. 

precedes the spectral law k'4 

B. Reverse Cascade 

For the reverse cascade which appears in the region of small wavenumbera, 
2-27 

we shall use the formulas (66b), (68) and (69), and write the spectral 

balance as 

or 

by a change of notations from (65b) and (65c). Here& is the rate of 

increase of 

F 

in non-equilibrium, and is assumed to be independent of 

balance which may include the supply from the synoptic scales. 

k in the global 

We divide both sides of (96b) by Ruo and differentiate, to obtain 

or 

where 

is seen to be a positive and dimensionless number in this subrange, 

since R" is a positive function which increases with 

by definition (68b). We can estimate bto be of the order of unity. 

k monotonically, b 

Finally we differentiate (99) with respect to k and find the 

spectrum 

where 

I 

P 
2-28 

and 

is estimated to be 

(102a) 

G=” 2, (102b) 

if we approximate 

Since the reverse cascade occurs at small k, we have taken the value 

(5442) for 13 . 

2/ /$yF>. Use of (68) and (69) has been made. lb II 

The solution (101) takes the following asymptotic expressions 

with 

k”3 , for k < ko 

-1/4 k-4, for k > ko, 

(103a) 

(103b) 

We note that the numerical coefficients (104) in the reverse cascade 

are smaller than their corresponding values (84a) and (85a) in the 

direct cascade. 
2-29 

X. CONCLUSION 

In the direct cascade, a critical wavenumber 

into two regions: The region at 

random force, and has a power law km4 . The region at k<ko has a 

negligible effect of the random force, and yields a power law k-3 In the 

reverse cascade, the same power laws repeat at a new critical wavenumber 

ko* ( < ko) , with their numerical coefficients modified by the multiple 

ko separates the spectrum 

k > ko shows a dominant role of the 

collision , and with different parameters, A bump appears in the transition 

from the direct cascade into the inverse cascade [Weinstock,l978 . 3 
2-30 

REFERENCES 

Basdevant 

modeling of enstrophy transfer in two-dimensional turbulence". J. Atmos. 

C, M. Lesieur and R. Sadourny (1978). 

Sci. 35, 1028-1042. 

Charney, J. G. (1971). "Geostrophic turbulence" J. Atmos. Sci. 28, 

1087-1095. 

Fjdrtoft, R (1953). "On the changes in the spectral distribution of 

kinetic energy for two-dimensional non-divergent flow". Tellus 2, 225-230. 

Gage, K.S. (1979)* "Evidence for a k a5I3 law inertial range in mesoscale 

two-dimensional turbulence", J. Atmos. Sci. 36, 1950-1954. 

Gisina, F. A. (1969). "Spectral characteristics of turbulence in a thermally 

stratified atmosphere". Izv. Atmos. Oceanic Phys. 2, no. 3, 247-257. 

kglish translation: Bull. (Izv) Acad. Sci. USSR, Atmos. Oceanic Phys. 

-9 5 137-1411 , 

Heisenberg, W. (1948). "Zur statistishce Theorie der Turbulenz". Zeit. 

Physika, 628-657. 

Herring, J. R. (1980). "Statistical theory of quasi-geostrophic turbulence". 

J. Atmos. Sci. 37, 969-977. 

Kraichnan, R. H. (1967). "Inertia ranges in two-dimensional turbulence". 

Phys. Rev. l0, 1417-1423. 

Kraichnan, R. H. (1971a) ."An almost-Markovian Galilean-invariant turbulence 

model". J. Fluid Mech. 47, 513-524. 

Kraichnan, R. H. (1971b). "Inertia range transfer in two- and threedimensional turbulence". J. Fluid Mech. 47, 525-535. 

a 
2-31 

Leith, C. E. (1968). "Diffusion approximation for two-dimensional 

turbulence". 

Lilly, D. K. (1969a) ."Numerical simulation of two-dimensional turbulence .I1 

Phys. Fluids, Suppl. 11, 240-249. 

Lilly, D.K. (1969b). "Numerical simulation of two-dimensional turbulence". 

J. Fluid Mech. 45, part 2, 395-415. 

Novikov, Ye A. (1978). "Spectral inequalities for two-dimensional 

turbulence: Izvestiya, Atmospheric and Oceanic Physics 2, 474-476. 

Pouquet A, M. Lesieur and J. C. And& (1975). "Hight Reynolds number 

simulation of two-dimensional turbulence using a stochastic model". 

J. Fluid Mech. 72, 305-319. 

Rhines, P. B (1979) . "Geostrophic turbulence". Annual Review of 

Fluid Mechanics, Vol. 11, Annual Reviews, Inc., 401-441. 

Saffman, P. G. (1971). "A note on the spectrum and decay of random 

two-dimensional vorticity distributions at large Reynolds number". 

Studies in Applied Mathematics, M, I. T. Press 50, no. 4, 377-383 (1971). 

Salmon R, G. Holloway and M, C. Hendershott (1978). "The equilibrium 

statistical mechanics of simple quasi-geostrophic models". J. Fluid Mech. 

-9 75 691-703. 

Tchen, C. M. (1944). "Some mathematical relations which play a role in 

diffusion problems". Verslagen Ned. Akad. Wetensch. Afd. Natuurkunde 53, 

400-410. 

Tchen, C. M. (1973). "Repeated cascade theory of homogeneous turbulence". 

Phys. Fluids 16, 13-30, 
2-32 

Tchen, C. M. (1978). "Kinetic approach to incompressible turbulence". 

Comptes rendus Acad. Sci. Paris 286 A, 605-607. 

Tchen, C. M. (1979). "Kinetic approach to compressible turbulence". Proc. 

Int. Symposium on Rarefied Gas Dynamics, Cannes, Commissariat 2 1'Energie 

Atomique, Paris, ed. by R, Campargue, 105-ll4. 

Tchen; C. M. (198la). "Group-scaling theory for the enstrophy transfer in 

two-dimensional turbulence'' Manuscript * 

Tchen, C.M. (1981b) "Kinetic theory of turbulent transfer with double 

memory-loss". Manuscript. I 

Tchen, C. M. and 3. H. Misguich (1982). "A group-kinetic theory of turbulent 

collective collision14, Manuscript. 

Thompson, P. D. (1973). "The equilibrium energy spectrum of randomly forced 

two-dimensional turbulence" . J. Atmos. Sci. 30, 1593-1598. 

Weinstock, J (1969). "Formulation of a statistical theory of strong plasma 

turbulence". Phys. Fluids g, 1045-1058. 

Weinstock, J. (1978). "On the theory of turbulence in the buoyancy subrange 

of stably stratified flows". J. Atmos. Sci. 35, 634-649. 
SECTION 3 

EQUIVALENT METUODS 

FOR QUASILINEAR TURBULENT TRAJECTORIES 

J. H. Misguich and C. M. Tchen* 

ABSTRACT. 

The propagator formalism is summarized in a self-consistent 

way for the asymptotic quasi-linear equation. 

A comparison is performed between the propagators and the 

Green's furictions in the case of the non-asymptotic quasi-linear 

equation. This allows to prove the equivalence of both kind of 

approximations used to describe perturbed trajectories of plasma 

turbulence. 

*This work investigates the detailed dynamics of the perturbed 

trajectory in turbulence. The manuscript is prepared for publication. 
3- 2 

Renormalization introduced in the microscopic 

theories of plasma and fluid turbulence essentially consists 

in taking into account the turbulent perturbation in particle 

trajectories. 

These turbulent modification (perturbed orbits) can 

of course be calculated by integrating the equations of motion 

in the fluctuating field. In a kinetic theory one uses however 

more compact and powerful 

the time-evolution of observable averages of all dynamical 

function by means of a unique entity: the propagator acting on 

phase variables x and x, or the Green function. 

tools which allow to directly describe 

- 

The aim of this work consists to prove the equivalence 

between two approximations which have been developed independently in these two foanalisms. This allows us to bridge the 

gap between different methods used by various authors. 

Turbulent modifications of the trajectories are 

described in the lowest approximation as quadratic function of 

the fluctuating field, and by considering all other trajectories 

as unperturbed by the turbulent field. This approximation has 

also been used in the quasi-linear equation for the distribution 

function and is thus referred to as the quasi-linear approxhation 

for the trajectories ( although it uses free trajectories as 

basic ones) . 

The turbulent trajectories so-obtained are then .used 

as a basic ingredient in the renormalized theory of plasma or 

fluid turbulence in the next approximation which is referred, to 

as the Renormalized Quasi-Linear approximation (RQLf 1,2,34,5,6, / 

This latter appears to be equivalent to the Direct Interaction 

Approximation (DIA) introduced in fluid turbulence /4,3/ . Thus 

the renormalized propagator appearing in this approximation ( in 

the weak-coupling limit) actually describes lowest order turbulent 

trajectories ( i. e. in the quasi-linear approximation 1 and 
3-3 

we will limit ourselves to this approximation for the particle 

trajectories. 

About the methods used to describe these trajectories, 

we have to remark that one has first integratedthe equations of 

motion /l/. WEINSTOCK /2/ has then introduced in this problen math~~~tizal 

operators'acting on the phase:wariables 2 and 

the role of which consists to describe the time-evolution of 

associated dynamical variables x(t) and v(t), and, consequently, 

(Liouville's theorem) to describe the time-evolution of distribution functions. In order to calculate the action of these 

propagators, WEINSTOCK had however to introduce particle 

trajectories which have been calculated by integrating the 

equations of motion. Some subtle effects , like that giving 

rise to an average displacement of the particles have been first 

missed by this method/3/. 

, the propagators /7/ 

Other unexpected effects, like the non-vanishing 

correlation betwesn particle velocity and Fosition, have been 

found due to the first explicit calculation of the turbulent 

propagator /8/. In Ref. 9 and 10 we have discussed the explicit 

relation between trajectories and the propagator for forward 

and backward propagation in time. 

Other authors have based the kinetic theory ori the 

Green functions /6/. We will compare here these methods and prove the equivalence between the approximations used. In spite of 

the global equivalence a few differences remain, namely concerning 

non-Markovian or memoPy effects. 
3-4 

The renormalized propagator and Green's function 

which willsbe derived are described by the quasi-linear 

equation which will be deduced here in a simple way. 

For electrostatic plasma turbulence, as well as 

for Navier-Stokes fluid turbulence, the starting point can be 

put in the form of the plasma Klimontovich equation 

( in the plasma case a ratio q/m is included in the electric 

field E, , with 9 the charge and m the mass of the particle). 

In the plasma case N(x,v,t) is the Klimontovich microscopic 

distribution 

ti 

4.2 

Here 5 and 1 are the phase space variables, while gi(t) and xi(t) 

are the dynamical (time-dependent) variables described by the 

exact equations of motion. The electric field is the microscopic 

one given in term of N by the Poisson equation /11,12/. 

The ensemble average introduced in statistical 

mechanics allows one to define the usual distribution function 

by -. F (~,y tk) = < N Cc,ytk)) 3. 3 

and the Vlasov mean field for inhomogeneous systems by (S(X_,t)) 

Fluctuations are defined as the difference with the 

average : 

F'=N -(N) 5.v 

E's E - <€> 
3-5 

The Klimontovich equation can be written in the compact form (at=;k 

'3,N =LN 

where L = +L' 

with 

Separation in average and fluctuating equations gites 

3.6 

3.504 

A e-. =. &. > 

describes the fluctuating part of the product. Let us remark 

that this last equation (x-5 1 also writes 

3.9 b. 

From the two equations (x.8) and (x.9) a general 

kinetic equation can be derived in the following way : equation 

in terms of f and the solution is 

substituted into (a.g) . A closed equation is then obtained 

for 7 , which still contains the initial fluctuation f'(b): 

this equation is called the general master equation. This 

equation, obtained by Weinstock from the Vlasov equation, is 

fully analoguous to the master equation obtained by PrigogineResibois in statistical mechanics, or to the one obtained by 

Xori or Zwanzig . 

(~9) is solved for f' 

In order to go to a kinetic equation, irreversibility 

has to be introduced in some way : it usualiy consists to take 

the limit of long tines(t-b) compared to microscopic elementary 

x 
3-6 

times (Eulek.ian correlation. time of fluctuating fields) and to 

neglect the term involving the initial fluctuation. 

One of us has developed with BALESCU a theory which 

shows that this general kinetic.equation can be obtained in 

a rigorous way for one ("kinetic") projection of the distribution 

function, due to the existence of two independent subdynamics 

/i3,14./. Tliis theory brings justification and necessary validity 

conditions for the simple derivation which is made here. 

The basic tool for solving equations (r.94b)is the 

propagator. We will first present the simple and trivial exemple 

of the free propagator associated with unperturbed particle 

trajectories. In such a simple case, the evolution equation 

reduces to 

The solution imnediatly writes 

Here the exponential operator 

which is solution of the szme equation 

allows us to describe the time-evolution of the distribution 

function as 

J 
3-7 

It is important to remark that this same 

describes particle trajectories: 

operator also 

and this allowsoneto check the "Liouville theorem" in its simple 

form 

This is the basic advantage of the propagators: the same 

mathematical tool allows one to describe the forward propagation 

of the distribution function in time: 

g(t).. u, Ct+t,t) xitw 

thus 

x_It+t)= [V,(ttt,t)]-4~ = U,(LjW~ 

Detailed demonstrations have been given in the general case /16/ 

In the case of equation T.3 one can define on one 

hand the propagator u(b,ko) describing the exact motion of the 

particles in the fluctuating field - E' or L' : 

71t u (t,to) = L IE,U UltjtO) g. Ao 

(u operates on both - X and V) - 

or, on the other hand, the Weinstock progagator associated with 

the homogeneous part of equation(L.g&) : 
3-8 

Here we have to stress the -fact that the fluctuation operator 3 

equally acts on L' and on A . The-solution of equations 

( IJ.~ 6)) can thus be written 

['&= A lLlb) f'(to) + A It yJ L' It') :It') E'.+c A 

The kinetic limit of long times can be justified 

by the subdynamics method /~.3,14/; it consists in neglecting 

the influence of the initial fluctuation f'/L) and in taking 

the asymptotic long time limit 3-w. In this case we have 

which describes the general mechanism of creation of fluctuations 

from the average function, by means of the fluctuating fields. 

Ud +u' 

we have 

Since we have 

d 
3-9 

the last term also writes as fo@a@S 

p.ns 

c ia7 

- l-al EkBy using the complete (but formal) solution(L.J2~) 

in 3.s we obtains the Weinstock master equation 

to 

the kinetic limit of which gives the general kinetic equation 

5 

j 

0 1 

n c 

ip 

: 

i I ?$t] =L(t)r[t) + 6% ( L'lt) ALL,t-%> L' hl) r(t-~) 3 E.% 

in an explicitly non-Markovian form (3@) *ends on f(t-3) 

earlier times). The second term in the r.h.s. represents the 

turbulent collision operator in its most general form. 

at 

The subdynamics method allows one to directly obtain 

the same equation in an apparently Markovian form ( including 

however all the non-Markovian effects : see h6/): 

These two forms are rigorously equivalent since the turbulent 

propagator v 

- 

is defined as the propagator of the kinetic 

equation itself: 

thus 

F.22 

E.t3 

Equation 3.22 remains a non-linear equation for the turbulent 

propagator. This dynamical non-linearity can be expressed also 

in the non-linear equation for the turbulent collision cperatcr 
- 

where v is a superooerator, non-linear in the operator G : 

This kind of operators have been studied in de 

The formal solution of (g.22) 

free propagator in the average field 

11 in Ref./l5/. 

can be written in terms of the 

in the form of the DYSON equation /18/ 

- 

where 6 is glven by E.ZY in terms of A and . The renomalihich is equivalent to the Direct 

(BIA) introduced by Kraichnan, 

PhLY zed quasi-linear approximati 

Interaction Approximation 

consists in approximating the Weinstock propagator A by the 

average turbulent propagator v . One obtains in this way: 

- 

where the RQL turbulent collision operator is defined by the 

following non-linear equation in terms of VRoL 

- 

: 

--------------- ............................................................ 1I.D. Kinetic equation ............................................ for turbulent "collisionless" Dlasmas 

Equation(%.%,2\) describes the kinFtic evolution of - 

the distribution function F=(N) : a priori it involves all 

collisional effects. It has however the same apparent form as 

the equation obtained usually by introducing fluctuations 

d 
3-11 

in the collisionless Vlasov equation , although such a procedure 

is not clearly justified usually. The equation obtained here is 

the actual justification of this usual procedure. Here the effect 

of individual particle collisions only appears lzter if one 

into account the specificity of the Rlimon'tovich equation : this 

appears in the binary correlations 

takes 

- 

flz = < N4($&k) N,(EL,Y,lt)) 

which involves the "self correlations" 

in addition to the usual ("distinct") correlations /19/ : 

For pratical purpose, it is sufficient to neglect self-correlations 

in order to describe the so-called "collisionless plasmas" from 

the Klimontovich equation. Correlations are then turbulent 

correlations, and the equations are identical to those obtained 

by the common procedure of the "fluctuating Vlasov equationl'but 

the justification is more clearly exhibited. 

Approximating the general kinetic equation x.20 or 

3.24 consists in approximating the propagators. The quasi-linear 

approximation consists in retaining in the turbulent collisions Z.29 

only free trajectories ( in the average field) determined byr:.zG 

One obtains 

3.32 
3-12 

Le. 

5.3s 

and, in the absence of any average field ( i.e. for an homogeneous 

ensemble) it reduces to 

-I t-to) y*y U,Lt,to) z e 

It is important to remark in the collision term c33 

that the propgator C)'fk-Z,t) takes into account ( here in an 

approximative way) the non-Markovian feature of the general 

kinetic equation. The effect of the propagator u"(b,k-t) in the 

electric field correlation consists to introduce a Lagrangian 

correlation, here taken along a free motion ( in a renormalized 

theory: alon the average motion). 

The explicit form of the quasi-linear equation can 

be easily obtained in the absence of an average field. We then 

have 

and 

Due to the obvious non-comnutation 

a gradient term appears(beside the usual diffusicn term in 

velocity space 1 which comes from the non-Ma'rkoyian feature of 

the kinetic equation : 

d 
where 

and 

This last non-Markovian term has been shown to be of importance 

in presence of a strong magnetic field /19./ . Here the fluctuation 

spectrum is given by the Eulerian correlation ( assumed homogeneous 

here for simplicity) : 
3-14 

The solution of equation 2-33 is written in terms 

of the turbulent propagator co c c 

iir A 

which is defined by the same quasi-linear equation: 

. The formal solution is a 7) and 2~ - '3 where '3, f - -I! -E 

time-ordered exponential operator: 

to 

/16/ 

where )(&is the operator of time-ordering which prescribes the 

ordering of the different factors coming from the expansion of 

the exponential, in the order of decreasing times from left to 

right. Non-commutation of the different terms in the exponent 

can be taken into account by means of an interaction representation. 

In a general way one can show /16/ that the solution 

of the equation 

9, U(t,t,) z Alt)+Bw-J UtLU rg -5 

a, uA(uo) = Aitr c),iu.) 

(where A and 3 do not commute) is given in terms of the 

(assumed elementary) propagator 

by the following formula ( see 'eq.3.10 in Ref ./16/) : 
3-15 

By choosing A:-$.! we obtain 

which can be put into the form 

The general form of the coefficients SI ,p has been 

given in eqs (5.14-18) in Ref. /1 /. In the simple case of 

a stationnary spectrum, 2 and E c do not depend on time. If 

we neglect in the exponent tens quadratic in the spectrum $ 

( i.e. the weak coupling approximation) the result can be written 

in the following siinple and tractable form: 

L 

.4%---. - .. -FiRrmwmiF3%5-F-Ed 
3-16 

The physical meaning of these various coefficients 

may be obtained in the following way. By integrating $P,g 19 

the exact equations of motion 

A(t) = wi 

+ (tl = E kl .IC 

up to the order Et- $ 

obtain : 

and by taking the long time limit, we 

We see that represents the turbulent contribution to the 

average displacement, the velocity, dependence of the coefficient 

represents an average acceleration or slowing down : these 

two terms are due to the velocity dependence of the diffusion 

coefficient. The 'term & represents the elementary process 

of the quasi-linear description: diffusion in velocity space . The term 6 represents the associated spatial diffusion ( CX'')*Z3 

an often neglected correlation between positizn and velocity. 

This whole set of terms have been discussed for the first time in 

Ref. /3 /. 

- 

- 

- 

iI represents 

the Dupree damping term) and the term 
3-17 

Inversely on can check that the propagator obtained 

in 3.AA allows one to recover results %.13.;tl ,coming from 

the equations of motion, by letting the obtained propagator acting 

on %=%/C) -- a\~d x:x[k] for instance: 

O((k-&,> = VO(t,t-B) = x,-t!! tp~~l~l a 

9.3.0. 

The obtained propagator thus takes correctly into account 

( and allows to describe) turbulent trajectories in the quasilinear, approximation , i.e. linearly in the spectrum. 

In this weak-coupling approximation and for a 

stationnary spectrum, we obtain the solution - of the quasi-linear 

equation in terms of the initial value F(x,y,o) as 

Let us introduce the Fourier transform by 

We thus have 

I 
3-18 

For later use, let us also introduce the Laplace 

transform 

JO 

We obtain 

In the simple case of a stationnary spectrum the 

solution OF 3.2 can be written 

and the solution q.27 can also be written formally as 

that will be compared later to the non-asymptotic case. 
3-19 

Renormalizing the quasi-linear equation essentially 

consists in taking into account the rbulent propagator in the 

collision term, instead of the free propagator. One thus have 

( see s.24 1: 

with * 

and 

'to E-. 3. 

This last equation is the RQL or DIA approximation of the DYSON 

equation 9.27. . It is equivalent to approximate the Weinstock 

propagator A by the turbulent propagator V . In equation 

$.A6 the present approximation is equivalent to keeping only 

the first term in the r.h.s.,(next term are thus responsible for 

"clumps" and other higher non-linear effects) . 

- 

The weak-coupling approximation of eq. q.2 for the 

propagator consists in considering in-this 5q;agion only the 

quasi-linear approximation of the collision term: 

This is the BOURRET 

It is equivalent to 

z.9 

approximation of the DYSON equation /17/. 

and the solution 
3-20 

has been developed explicitly in X.10-11 , 

The important phpical effects introduced this 

renormalized approximation is the main motivation of the present 

comparison between the techniques of propagator and Green's functions. 
3-21 

The asymptotic form of the general kinetic equation 

Zt or of the quasi-linear one g.32 has been obtained either 

by the projector technique and the subdynamics method, or by the 

usual arguments: 

F'(L0) -7 0 

&I --- P. A 

Some authors however consider the non-asymptotic equation obtained 

by simply neglecting the initial fluctuation in the general 

master equation. In spite of the fact that there do not exist, 

to the best of our knowledge, any rigorous justification of the 

consistency of such an equation, one can consider that the long 

'time limit may be taken afterwards; such an equation presents 

some advantage namely for introducing Laplace transforms. 

In the master equation p.Ag we simply take 

and limit ourselves to the quasi-linear approximation 

A(L,t() 3 U"/CIP) z. 3. 

We then obtain the non-asymptotic quasi-linear equation ( (E) =O) 

-3y.Z - 3, QV (Wt-3) 

- 

Jo 

where the integral kernel can be written in terns of the (Eulerian) 

We assume here an homogeneous spectrum ( S is x-independent). 
3-22 

In Fourier teansform ( 3.2) ) we have 

The so-obtained equation ( V.3 1 exhibits both 

non-Markovian ( F(k-3) 4 Tk) 1 and non-asymptotic ( k#* ) 

features. By neglecting these two effects, the turbulent collision 

ten would reduce to 

where 

“4. 

gives the usual velocity-diffusion term in the homogeneous case 

( k= 0). 

Let us introduce the Laplace transform of the 

complete equation ( V8? ) : 

and let us restrict ourselves to the case of a time-independent 

spectrum. Equation V.3 then writes: 

4 
where 

and 

such that 
By writing equation / Y.rz in the form 

one can define the associated Green function by the equation 

7 ~~Uu~~~~~?r$~.~:.*~~~~*.& 

The solution of equation 5.2 has been given by 

HORTON et al. / 5 / in an approximation which is equivalent to 

the weak-coupling renormalized approximation used in Section=, 

in the propagator formalism for the asymptotic equation. Our 

aim is to compare the approximations used in the two formalisms. 

Their solution writes [e?. 6.12 G %eF ./S/]'. 

This form actually assumes an isotropic tensor 

u.8.y = Hbt 

a ti This is equivalent tc neglecting the velocity dependence - =O 

as well for the dependence of H in b>-k,v as for its dependence 

in V, . 

av 

- 
3-25 

In the framework of these approximations, we will compare in 

Section@ wiWthe asymptotic solution obtained from the 

propagator 8.26. . 

Let us introduce a Fourier transform in velocity space: 

q. 6. 

The solution V1.3 can thus be written in the considered approxiby using 

d 
3-26 

One recognize here, in a scalar form, the spatial diffusion 

term -9 53 kz H 

( 9th 1 and also the crossed term o, H k corresponding 

to the correlations between position and velocity. 

(Dupree damping) , the velocity diffusion term 3 

Solution of eq. s.4 can be written 

L 

Here a propagator appears 

which is solution of 

and it generalizes to the non-asymptotic case the previous 

propagator ( for stationnary spectrum) : 
3-27 

It can be seen thag in the non-asymptotic case ( W1.q ) the 

non-Markovian contributions are taken into account in a much 

simpler way as compared to the asymptotic case ( F terms in V1.U ) 

This is due to the tensor m H/&k,,,v) -- : its approximate form for 

of the asymptotic case because 

b =k.V c is equivalent to-the Markovian approximation ( - =a) 

The knowledge o€ the expansion of the noncommuting operators in the exponent of ( VI. IS 1 : 

-&.$t &E 4k 2.i.p 

e 

- \I ko) = e. e e 

y:+ 

VI. r3 

we immediatly obtain here in the same approximation: 

Vf.20. 

d 
3-28 

When compared with the expression VI.41 obtained 

from the Green function, this last result has the advantage of 

taking into account the dependence of E upon both l.,k.,y 

and \r . The Fourier transform in velocity space, which 

has been introduced to integrate the Green function actually 

needed to qeglect such a dependence. Fort-this reason, one 

- 

- 

can say that the simple result obtained from 

from the non-asymptotic equation is equivalent 

mation 

the Green function 

to the approxiperformed in the result of the asymptotic equation if we futher 

neglect the non-Markovian contributions (i.e. Eq. q.26 where 

are given by z. 12-15 with F =0) . !!,PIX - ,5 5 

In conclusion, we have shown that the approximation 

used by HORTON et al. to calculate the Green function V1.Y in 

the case of the non-asymptotic quasi-linear equation is 

actually the same physical weak-coupling approximation used in 

the propagator formalism. However, the use of Green's functions 

seems to be more delicate if we would like to take into account 

- the tensorial feature of H, 

- but mainly its velocity-dependence ( and thus the non- 

- 

Markovian and non-asymptotic features). 

Neglecting these two effects is equivalent to. only 

consider the asymptotic and Markovian quasi-linear equation. On the 

- the tensorial feature of ,P and in the asymptotic case 

( or of H in the non-asymptotic case) 

- their velocity dependence (i.e. average displacements effects: 

see 8.n-lS 1 and non-asymptotic contributions in the case of the 

tensor H 

otherhan3,the propagator technique allows one to take into account 

- - 

= 

c 
3-29 

Acknowledgements : 

We want to thank Dr. Wendell HORTON for the discussions we had in Aspen I which have lead us to write the present 

comparison between the equivalent but so different techniques of 

propagators and Green's functions. 
3-30 

REFERSNCSS 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

T.H. DUPREE : Phys. Fluids 9 (1966) 1773. 

J. WEINSTOCX: Phys. Fluids - 12 (1969) 1045. 

J.H. MISGUICH, R. BALESCU: J. Plasma Phys. - 13 (1975) 385. 

D.F. DUBOIS, M. ESPEDAL: Plasma Phys. - 20 (1978) 1209. 

W. HORTON, Duk-In CHOI: Phys. Rep. 49 (1979) 273. 

J.A. KROMNES, R.G. XLEVA: Phys. Fluids _. 22 (1979) 2168. 

R. BALESCU : "Equilibrium and.Non-Equilibrium Statistical 

Mechanics", Wiley (1975) . J.H. MISGUICH, R. BALESCU : Phys. Lett. A - 48 (1974) 426. 

J.H. MISGUICH, R. BALESCU : Plasma Phys. 19 (1977) 611. 

J.H. MISGUICH, R. BALESCU : Plasma Phys. - 21 (1979) 749. 

Yu.L. KLIMONTOVICH : "The Statistical Theory of Non-Equilibrium 

Processes in a Plasma" Pergamon (1967) 

similitude trompeuse" , Note Interne DRFC 1218 ( Oct.1981). 

J.H. MISGUICH : "Equations de Vlasov et de Klimontovich: 

J.B. MISGUICH, R. BALESCU : Physica - 79 C (1975) 373 

R. BALESCU; J.H. MISGUICH : J. Plasma Phys. 13 (1975) 53. 

J.9. MISGUICH, R. SALESCU: Bull. C1. Sci., Acadgmie Roy. 

J.H. MISGUICH : Report EUR-CEA-FC 1020 (Nov. 1979). 

J.H. MISGUICH, R. BALESCU: Report EUR-CEA-FC 804 ( Apr.1976) 

J.H. MISGUICH, R. BALESCU : J. Plasma Phys. 19 (1978) 147. 

J.H. MISGUICH, R. BALESCU : J. of the Phys. SOC. Japan 

Belgique 61 (1975) 210. 

- 50 (1981) 1706. 

x 
SECTION 4 

A NEW KINETIC DESCRIPTION FOR TURBULENT COLLISION INCLUDING MODE-COUPLING 

J. H. MISGUICH and C. M. TCHEN 

ARSTRACT 

The micro-dynamical state of fluid turbulence is described by a 

hydrodynamical system. This is transformed into a master equation in a 

form analogous to the Vlasov equation for plasma turbulence. When the 

total distribution function is decomposed into a mean value and a 

fluctuation, the evolution of the mean distribution satisfies a transport 

equation , called the kinetic equation, and contains a turbulent collision 

that represents 

while the evolution of the fluctuation will form a transport equation for 

the turbulent collision. Themechanism of this collision is investigated. 

The report compares our theory with the clump theory of Dupree, the 

renormaliration theory of Misguich and Balescu and the direct interaction 

theory of Kraichnan. 

the statistical effect of the turbulent fluctuations, 

The paper is completed by Dr. Tchen at the City College of New York 

in collaboration with Dr. Misguish at the Centre d'Etudes Nucleaires, 

Fontenay-aux-Roses, France. 

The work is the result of cooperatioh befween Dr. Tchen at the City 

College of New York and Dr. Misguich at the Centre d'Etudes Nucleaires, 

Fontenay-aux-Roses, France. 
4- 2 

INTRODUCTION 

In plasma and fluid turbulence, the main average 

effect of the fluctuations consists to enhance (or replace) the 

role of particle collisions and to enhance consequently the 

transport processes ( turbulent diffusion, turbulent viscosity, 

a.s.0. .... ). The origin of such phenomena is generally referred 

to as "turbulent collisions". 

The starting point of theoretical descriptions are 

the Navier-Stokes equation for the macroscopic description of 

fluids, and the Klimontovich equation for the microscopic 

description of plasmas. The latter reduces to the Vlasov equation in the case of "collisionless" plasmas, which behave like 

an incompressible fluid in phase space. 

The most widely used among the theoretical descriptions of such processes are the Direct Interaction Approximation 

(DIA) introduced by Kraichnan in fluid turbulence, and the 

Renormalized Quasi-Linear (RQL) approximation introduced by 

Dupree and Weinstock in plasma turbulence. 

have been shown to be analoguous./l/ 

Both approximations 

Similar formalisms can thus be used to describe 

both kind of turbulence, and we will adopt here the language 

of plasma physics. The results can immediately be translated 

in the fluid language by replacing electric field fluctuations 

by pressure gradient fluctuations 

Turbulent collisions effects are described in RQL 

or DIA approximations by means of a Lagrangian correlation of 

fluctuating fields between two points whicb are separated in 

space and in time. These distances in space and in time are 

are actually related to each other by a complicated trajectory 

which involves all the dynamical problem; in RQL or DIA this 

trajectory reduces to an average diffusive trajectory, 

x 
4-3 

The purpose of the present work is to analyse the 

effects of higher mode coupling terms on this approximate 

description. Dupree for instance has introduced so-called 

'beta termsl'in plasma turbulence, the role of which is to 

ensure energy conservation to a higher degree of precision. 

Such terms have been proved to be important in drift wave 

turbulence /2/. 

The introduction of such higher order corrections 

has however been made in a rather intuitive manner : a simplified approach is presented in Section 11. Here we derive new 

results for turbulent collisions by using a general kinetic 

formulation of plasma turbulence (Section 111). Moreover we 

prove that these higher order corrections already appears 

from the non-linear dynamics, even when the non-linearity 

introduced by Poisson equation is not yet taken into account: 

they cannot thus be reduced to self-consistency effects only. 

Our main result (Eq. 111.14) consists to describe 

the deviation of the turbulent collision term from its RQL 

or DIA approximation in an exact way. This deviation involves 

a generalisation of the Beta terms introduced by Dupree, which 

includes here all higher order mode-coupling terms. A new 

approximation is proposed in Eq. (111.20) which goes beyond 

the RQL description; an approximate treatment of this equation 

is left for a future work. 

Another approach is presented in Section IV, which 

uses a simple non kinetic treatment of the Vlasov equation. 

Although the resulting expression involves four main contributions 

(Eq. IV.24) , we have been able to prove that in the kinetic 

regime an unexpected cancellation occurs between the last two 

contributions, and we recover the general result of the kinetic 

formulation. 

The consequences of this cancellation are examined 

in Section V where we focus our attention on a quantity CL 

defined as the infinite time integral of the Lagrangian 

autocorrelation of fluctuating fields. Exactly like in the 

classical case of Brownian motion, we demonstrate here that 

J 
4-4 

fn 

geneous and stationnary turbulence. Of course, when different 

scales can be introduced to describe the fluctuations, the 

Lagrangian correlation of the small-scale fluctuations remain 

inhomogeneous and non-stationnary, and this avoids the 

corresponding Ci to vanish in the asymptotic case. In such 

a scaling description of turbulence, Ci can be used as a bare 

description of the turbulent collision : higher order modecoupling terms then will introduce a dynamical shielding effect. 

This description will be developed in a subsequent work where 

an explicit calculation of the intrinsic or shielded turbulent 

collision,will be performed. 

an asymptotic description, CL goes to zero for an horno- 
4- 5 

11. TURBULENT COLLISIONS AND HIGHER ORDER TERMS 

............................ If, Schematical Introduction 

The recipe for the introduction of the higher order 

Beta terms can be summarized as follows. The Rlimontovich or 

Vlasov equation for the distribution function f(x,v,t) 

11.1 

with 

3 (2 =-% , q is the charge and m the mass of the particles, 

-. E in plasma is the electrostatic fluctuating field) can be 

separated as usual in fluctuating and average parts: 

11.2 

11.3 

with the usual notation c 

FZ <f>=RF 

and B-1-A denotes the fluctuating part of everything to its 

right. 

- 

The free propagator in the average field L is 

defined by 

11.4 

In the simple case of linear trajectories this reduces to 

11.5 9Jle (t,t*) = -X.F5_ U&, L) 

and the solution is a simple finite displacement operator in 

x-space: - ( t-b) y .E 

v, (th5 = e 

11.6 
4-6 

Solution .of (11.3) can be obtained as 

II.7b 

where time dependence is taken into account by the notation # 

which denotes the time convolution. This solution is formal 

since it is an integral equation which remainsto be solved. 

Substitution of (11.7) into (11.3) is equivalent 

to an iteration and gives 

2t~~S LF' +LI~ ~BL'v,~ L'F + BL'V, *BL'F' 

II.3b 

A simple handwaving argument can be applied in the last term 

in the r.h.s. : in OL'F' =,'f'- CL'F') the projector 

B means that the average has to be substracted from L'f' In 

the final calculation of average quantities the L' cannot 

be taken in average with f' : simple averages involving this 

last L' can only be taken: 

i) either with the other L' which yields 

ii) either with other fluctuating quantities which will appear 

to the right. This case can be schematized by an arrow 

4 

A 

L' uo SL' T') 

which indicatesthat L' is operated by uo but is excluded 

from the bracket average. 

The last term in (II.3b) thus involves at least the above two 

termsf and (II.3b) can be written: 

3~'~ LV'+C~^ +BL~u,+~~ t (L'u&)F'+ 4N+T> - 7 c 11.8 

Both underlined terms can be used to define an average renormalized propagator v and the solution 'writes. -0 
4-7 

When substituted into (11.2) this result allows us to write 

11.10 

In the r.h.s. the first term describes free motion in the 

average field, the second term a turbulent collision term 

(here in the weak-cdupbing RQL approximation). The third and 

4th terms are higher ordeE corrections to this approximation 

of the turbulent collisions. The complete expression for the 

collision appears to be in this handwaving derivation: 

~(c)~ 4 c F') = 4 L'v?~~c~)~ + (LTWS LW,+LJ): 

11.11 

Actually the last term ( equivalent to the Dupree Beta term /2/1 

is of the type 

4 L' L') < L' f') 

which involves the complete Cck) : Eq. 11.11 still remains an 

integral equation for e , which describe some kind of 

renormalization or shielding of the RQL-weak coupling approximation given by 

c O 0 4 L' V0*l')T 

The purpose of the present work is to elucidete the role of 

such higher-order implicit terms in the equation which defines 

the turbulent collisions. 

This above schematical introduction of the higher 

order terms can be made slightly more precise by considering 

the wave-vector dependence in Fourier transform. 

J 
4-8 

For the case of a turbulence which is homogeneous 

in average, we have a space-independent average function 

and 

and 

since we have no average Vlasov field. In this case (II.3a) 

writes 

11.12 

In order to perform the iteration, we write the formal 

solution for ?%-kt 

term, the asymptotic solution writes 

I . Neglecting as usual the initial value 

J-0. 11.14 

Let us consider the structure of this general expression. In 

the last term, the first B to the left prescribes the fluctuating part 03 what follows. (The integral IAt'' can be replaced 

by the discrete summation 5 with appropriate factor (2n/L)3 

which are ommited here for clarity). In a homogeneous system 

this implies 

i) k"= - k, which gives L',(V) \:,[e) 

ii) k"= -4' which gives L'-k, (k') \ Llk.') 

Kw 

k# 0 . In the same way the second B implies 

4 : two values of k" - appear to play a special role: 

9 

which is analoguous to tke Dupree Beta term 

i.e. a phase coherent term. 

- * 
4-9 

Neglecting higher non-coherent mode-coupling terms gives: 

1 

11.15 

Here the k' sumrnation is an approximate form for the complete 

mode-coupling term BL'F' in II.3a or 11.12 : 

Let us consider the structure of this equation: 

- The term LL'k, flk,) appears as a modification of the source - - 

term in 

tion of the average function f into an " effective " average 

function : 

: it has been interpreted by Dupree as a modifica- 

- The term < Lt, Lik,) appears as the weak-coupling approximation 

of the turbllen; collision term ( coherent part). 

In Ref. /3/ these both terms appear as 

1.e. a phase coherent term CFf' ( turbulent collisions) and 

the Beta term ( C4 1. This corresponds here to 

collisions 

the turbulent 

11.18 

and the Beta term 

The role of the term LL, L' 

not apparent . - k-g' F in Dupree's description is 

a 
4-10 

111. EXACT X INET I C RESULTS 

The appearance of turbulent collisions and higher 

order terms can actually be described more rigourously by 

using general expressions which have been obtained in the 

framework of our kinetic theory for plasma turbulence. 

........................ 

II1.A. General kinetic equation for turbulent ..................... Elasmas ------ 

In the kinetic regime of long times the exact 

solution of II.3a for the fluctuations can be written /4/ 

1 111.1 

I L 

This is a basic formula which describes the fluctuations in 

terms of the average distribution function. It holds in the 

kinetic regime of times long compared to the correlation time 

of electric field fluctuations; in such regime the influence 

of the 'nitial. fluctuation has been shown to vanish exactly /4/. 

Here A is the Weinstock propagator defined by the homogeneous 

part of Eq.(II.3a) : 

I11 * 2 

111.3 

111.4 

and where )(+is the time-ordering operator /5/ which prescribes 

that in the expansion of the exponential operator in power of 

its exponent, the various operators have to be time-ordered in 

order of decreasing times when t > to . 

J 
4-11 

C(kl : ds < L'lt) A (kit .7#) L' It.3)) VP-GA) 

z Ib * 

Solution 111.1 can be checked immediately: by using (III.2), 

Eq. (111.1) gives (I1.3a) (see 111.4). 

This simple exact solution allows us to write 

the average equation (11.2) as 

111.5 

equation t itself: 1 

i.e. 

111.7 

111.8 

This operator allows us to write the exact solution as 

111.9 

-

1II.B. -------Iteration -------- 

Let us now perform a kind of iteration of Eq.(III.l). 

A simple integral relation can be obtained between the Weinstock 

propagator A (111.2) and the turbulent propagator ( 111.7) : 

. .- 

J 
4-12 

i.e. 

+ :e t ~~,\-o)[~~'~t-s) - GIC-Q~J A 1t-0-t) 

1II.lOb 

1. A(L,L) =V lkhThis integral equation can be deduced from (111.2) by 

treating (B 1'1 G 1 as a perturbation: 

By substituting (111.10) into (111.1) we immediately obtain the following iterated formula for the fluctuating 

distribution function: 

J, '0 

111.11 

By substituting this exact solution in (11.2) we find a new 

expression for the turbulent collision term: 

Je 

which is given by 

111.12 

111.13 
4-13 

Introducing the variable S s t-B we have 

[02= j? = JTBJF =j'?JeZ 

and finally the result can be written as (8 +& 1: 

111.14 

The first term is nothing else than the RQL approximation of 

the collision term, which has been developed in plasma turbulence 

in analogy with the DIA approximation in fluid turbulence/6/.This 

approximation is obtained by replacing in the exact expression 

(111.6) the fluctuating trajectories described by the Weinstock 

propagator A by the average turbulent trajectories described 

by the turbulent propagator . The second term in (111.14) 

takes into account the deviation from the RQL description, by 

means of (BL'- G) . 

--------_ 1II.C. Higher -------_-------_-- order terms beyond ----RQL --- 

In order to make a bridge with the more intuitive 

calculation of Section 11, it is interesting to put this 

correction in another form. From (111.1) we have indeed: 

'0 

111.15 

Since 

we can write (111.14) as : 

111.16 
I 4-14 I 

JO 

111.17 

When compared with 11.11 the present exact result (111.17) can 

be analyzed as follows. The first term is the RQL approximation 

of the collision term, while the first term in 11.11 only 

represents the quasi-linear result ( I\ -tue 1 . The second 

term in (111.17) is a generalization of the second term of (11.11) 

i.e. a term involving at least three fluctuating fields; here 

A remains a fluctuating quantity which may introduce correlations between four fields, a.s.0. The third term in (111.17) 

is the general form of the Beta term obtained in (11.11). The 

present general formulation avoids using approximate treatments 

of averages like that denoted 3y the arrow in Eq.(II.ll) and 

avoids elimination of higher order mode-coupling effects. 

Moreover it allows to exhibit the implicit feature of this 

higher order Beta term: the deviation of G from the RQL approximation of G involves a Beta term which depends on G 

was given in terms of (L' F'> . . itself; this was already apparent in Eq. (11.11) where <L'F'> 

In summary, we have shown that the turbulent 

collision term involves a first lowest order approximation 

(QLin 11.11, RQL in 111.17) plus higher order terms which 

have been calculated. This latter correction involves an 

important , so-called Beta term, which actually depends on 

the complete collision term ( "implicit' terms 1. 

Two levels of approximation can be used to 

describe (111.14) . 
4-15 

lo. The RQL approximation for G- is solution of the 

non-linear equation (replace A by in 111.12) : 

2'. Higher order corrections can be obtained by replacing 

I\ by 7 in the iterated formula obtained in (111.141 

In this higher approximation we have I 

111.20 I 

This equation is the main tractable result of the present 

work, It represents a highly non-linear equation for the 

collision term of the average equation 

described in an approximation which goes beyond the RQL 

description. The calculation of an approximate solution for 

this new closed equation is left for future work. 
4-16 ' 

IV. NON-ASYMPTOTIC TREATMENT FOR TURBULENT COLLISIONS 

An integzra2.:- equation similar to (111.17) 

for the collision term of Eq.(11.2) has been obtained from 

the Vlasov equation directly /6/. We will show here that the 

physical content of the different terms appearsto be quite 

different. 

The fluctuating equation (11.3) can be written in 

the form 

' 4 L'F - 4L'F') 

IV. 1 

This f orm naturally introduces the exact propagator defined 

by 

IV. 2 

i.e. 

IV. 3 

Contrary to the Weinstock propagator, the exact propagator 

0 does not prescribe all intermadiate states to be fluctuating ones: allows l*transitionsn from an average *lstatel* to 

a fluctuating one, and vice-versa. The Weinstock propagator A 

(111.3) in turns describesthe irreducible propagation of 

fluctuating states /4/. 

Solution of (IV.1) can be written ( neglecting 

the initial fluctuation): 

i.e. 

IV.4 

a 
4-17 

t-to 

(L'FI) = Ut) f b. 4! L'lt) U[k,b) Li.cr-tl) fw 

- Id% (L'lt) u lt,W> ecw t-bo 

0 IV. 6 

Let us transform Eq.(IV.6) in order to introduce 

explicitly a first term which in the asymptotic limit (to 3-9,) 

gives the RQL approximation. For this, we simply decompose 

IV. 7 - 

where the average 

to the average turbulent propagator (111.8) in the limit: 

of the exact propagator actually tends 

of time intervals long compared to the correlation time of 

electric field fluctuations. Then (IV.6) becomes: 

IV. 8 

where the three contributions are given by 

I 1 

In order to build a bridge between the present 

treatment and the asymptotic results of Section 1x1, we have 

to calculate u' in terms of A and v For this, we have 

- 
4-18 

to derive from the definitions of the propagators some simple 

exact relations in the form of integral equations. 

Let us first derive the (integral) relation 

between u and u, : 

ut uo + U**L'U 

IV. 12 

which is readily obtained from (IV.2) and (11.4). We have to 

note that in such relations, the two propagators in the second 

term can be equally inverted since both formulas give the same 

iterated explicit solution : 

IV. 13 

- 

Let us now express u, in terms of \I and u, . From (111.71 and (11.4) we have: 

IV. 14 

Combining this with (IV.12) gives 

IV. 15 

Let us also express I) in terms of A . Eq.(III.2) can be 

written: 

which gives 

A= w - UxRL'A , IV.16 

Then IV. 15 gives 

IV. 17 

Here u, in the last term can be eliminated by 'using (IV.12 

and 16) : 
4-19 

and we finally obtain 

The fluctuating part of U can thus be written as 

i.e. 

i 

Iv. 20a 

br, explicitly: 

From the exact relation (IV.20) we can write the 

second contribution to the collision term (IV.10) in the form: 

(L'(t)[o+U'] L'(t-~))~~~-~)=c~ rl t Crrz 

IV.21 

with 
4-20 

IV.23 

The total collision term (IV.8) thus writes: 

IV.24 

Actually one can prove that, in the asymptotic limit, two 

last terms in (IV.24) exactly cancel (see Appendix) : 

IV. 25 

This implies that we have asymptotically: 

IV.26 

Now this result can be proved to be identical to 

Eq.(III.14) obtained in the asymptotic description. This 

identity can be proved as follows. From (IV.9) we have indeed 

which is indeed the first term of (111.14). On the other hand, 

we have from (IV.22): 

J, Jo 

IV. 28 

which is identical to the second term of (111.141. This achieves 

the demonstration of the equivalence of (IV.26) and (111.14). 

We have thus seen in the general result (IV.24) 

that in the asymptotic limit of a distant initial condition, 

last twD terms exactly cancel. This means that in (IV.6) the 

last implicit term is cancelled asymptotically by a part of 
4-21 

1 the contribution of u in the first ex icit and we 

remain with the asymptotic result given by the kinetic 

formulation (111.14). This remains a highly non-linear equation for the collision term, even in the approximation (111.20) 

where A has been replaced by . 
4-22 

V. ANALYSIS OF THE LAGRANGIAN AUTOCORRELATION OF FIELDS. 

Let us consider the exact relation (IV.6) which 

has been derived for the turbulent collision term from the 

Vlasov.or Xlimontovich equation: 

CCCL (L'F') = e& - dr (Ut) Ultrt-bj) CIW 

l-l0 v. 1 

where the first term eLis defined in terms of the Lagrangian 

correlation between two electric field fluctuations along the 

exact motion of the particle, which is described by the 

propagator U : 

v.2 

This autocorrelation of the electric field fluctuation can 

be visualized as the correlation between two points of the 

space-time (E,t) which are defined by the exact position of 

a particle which is at point x at time t and at a point 

x( t-3 I =U(k,h)< at time t-t ( see Fig.1). 

- 

- 

The time integral of this autocorrelation is fully similar 

to the well-known autocorrelation of fluctuating forces which 
4-23 

appears in the classical description of Brownian motion. The 

friction coefficient of the Langevin equation 

v. 3 

is related to the diffusion coefficient by the Eintein relation 

3: ki- - 

7- v.4 

arid is given by the time integral of the autocorrelation of 

the total fluctuating force F(t1 ( see p. 262 in /7/): 

v. 5 

Xere the upper integration limit? is larger than the 

correlationhkc kof the fluctuations, but smaller than the 

relaxation time : 

&(T 4xr=* 

V.6 7 

The point we want to stress here is the following: the exact 

autocorrelation has a strong peak near the origine (tsZe) 

v.7 

(see - Fig. 2 /8/) 

3 
4-24 

which makes that the total integral ( from zero to infinity) 

of the autocorrelation is actually zero: 

dt (F(t)._FIk+t)> - = 0 r v. a 

The finite integral however 

T (r) : J.' dt < f(tLFlkiG)) 

v. 9 

is a 5unction of C which reacksa quasi-plateau value at 

(see p. 261 in /7/) 

V.B. -----General result -_-_----__-_-- 

The point we want to show here is actually very 

similar. Let us consider Eq.W.1) and calculate the second 

term. From the exact equation for u we have: 

3,u: LU 

thus 

This can also be written 

v. 10 

v. 11 

v. 12 

v. 13 
4-25 

this gives an equation where fluctuating quantities have been 

eliminated: 

V.14 

By using partial integration 

we obtain 

V. 16 

BY 

This is an integral equation for e in terms of eL which 

is given by the time integral of the Lagrangian'autocorrelation 

of the fields. Here the integral operator has to be inverted 

in order to determine e in terms of e~ . An importantconclusion can nevertheless be reached for a particular 

homogeneous case. 

V. 18 

and 

v. 19 
4-26 

- I 

h the stationnary case v lLtt-t J=Vbjis t-independent, and 

c [I-%) becomes time-independent . We remain with 

JO - v.20 

For homogeneous systems, e and v are x-independent and 

( no average Vlasov field). We thus reach the 

result that for an homogeneous and stationnary turbulence 

the infinite time integral of the Lagrangian autocorrelation 

of fluctuating fields vanishes in the asymptotic description: 

1 =-%v 

1 i 

J v.21 I 

although the total turbulent collision term does not vanish. 

This result is very similar to Eq.W.8) for Brownian motion. 

This result implies that 

It is interesting to combine this result with (IV.25) : 

When reported in Eq.(IV.24) this gives 

v, 22 

V.23 

V, 24 

This means that in the asymptotic Analysis of the four terms 

appearing in (IV.8) for the collision term : 

I 
4-27 

1') last Contribution (IV.ll) goes to the complete result: (c 

3'),The total collision term is given by the result of the 

kinetic method developed in Section 111 : 

V. 27 

i.e. : i 

These equation (V.25,26,28) are three independent relations. 

The importance of the last result is that it allows us to 

describe the turbulent collision term by a modification of 

the RQL approximation. The contribution of G in (V.28) 

represents the equivalent of the Dupree Beta terms, here in 

a generalform. Their importance has been proved for energy 

conservation. We will come back later to this important point. 
AP~ENDIX 

4-28 

Let us prove the cancellation of cm and 5, in 

the asymptotic case. We calculate (IV.11) 

and use the exact asymptotic expression for c (111.121 : 

Let us modify the variables: 

A. 3 

% A. 6 
4-29 

(where Psk-k' ) . Substituting (A.6) into (A.5) gives 

u'(& ,\-t'+s) : d 8 ~lt,'t.e)lBL'/1-o).Gfl.e~ u 

t'-5 - 

1-3'*51 

~ A.7 

When sqbstituted into (A.4) this result allows us to express 

C$ as : ( change t' into t ) c 

A.8. 

On the other hand, cg 

by using 

given by (IV.23) can also be transformed 

2 

z-e 2 -zt 

j!B J!5 S=t.t'Jh J:l zj'slj;e 

A. 9 

and we obtain ( t'*S 1 : 

A. 10 

By comparing with (A.8) we have thus demontstrated the result 

(IV.25) : 

A.ll 
4-30 

REFERENCES 

1. J.- H. Misguich, R. Balescu, J. Plasma Phys. 13, 385 (1975). 

2. T. H. Dupree, D. J. Tetreaul;t, Phys, Fluids 21 

3. T. Boutros-GFhaIi, 7. H. Dupree, Phys. Fluids 24, 1839 (ZCBl). 

4. J. H. Misguich, R. Baleseu, Physica 79C 373 (1975). 

5.aJ. H. Misguich, R. Balescu, Bull. C. Sci. bead. Roy. Belg. 61, 210 (1970). 

6, J. H. Misguich, C. M, Tchen. Report. 

7. S. A. Rice, P. Gray: "The Statistical fleckanics of Simple Liquids", 

-' 425 (1978). 

-' 

Interscience 1965 . 
SECTION 5 

SPECTRAL STRUCTURE OF TURBULENCE 

IN THE STABLE ATMOSPHERIC BOUNDARY LAYER 

C.M. Tchen 

GRADUATE CENTER AND CITY COLLEGE 

OF THE CITY UNIVERSITY OF NEW YORK, N.Y. 10031, U.S.A. 

and 

S.E. Larsen, H. Pecselli, T. Milckelsen 

RIS0 NATIONAL LABORATORY, DK-4000 ROSKIDE, DENMARK 

ABSTRACT 

The hydrodynamical equations of turbulence are transformed into a master 

equation for the velocity distribution function. 

for the closure. 

A group-scaling is introduced 

The spectral balance for the velocity fluctuations of individual 

components shows that the scaled pressure-strain correlation and the cascadec 

transfer are two transport functions that play the most important roles. 

derive this eorrelation,.and find a power spectrum _klW3for the horizontal components, 

while the,spectrum for the vertical component drops rapidly by going to the larr,e 

scales. 

We 
I. TNTRODIICTTON 

5-2 

In a strongly stable boundary layer, the spectral distributions of the 

-3 large scale horizontal components take the power law k 

component decays rapidly toward the small k values. This spectral hump in the 

horizontal components are investigated here. 

of closure developed by Tchen . 

, while the vertical 

We use the group-kinetic method 

1 

Instead of the customary Fourier decomposition 

of a random function, we perform a group-scaling as a coarse-graining procedure. 

We replace the hydrodynamical equations of turbulence by a master equation for 

the velocity distribution, and expoit its advantage of being homogeneous and having 

less nonlinearity, since the turbulent velocity is now an independent 

variable. 

11. BASIC EQUATIONS OF ATMOSPHERIC TURBULENCE 

A. Microdynamical State 

The microdynamical state of atmospheric turbulence is governed by the 

equation of continuity and the hydrodynamical equations of Navier-Stokes, as 

follows: 

4 

The total function, as denoted by ( ), consists of an average quantity, as 

denoted by 

9 

N 

and a fluctuation, or a deviation from the average, as denoted by ( 1. The 

n 

7 

evolution of the velocity (A is controlled by the kinematic viscosit'y 3 and 

a 
5-3 

a field 

which is the sum of the adverse pressure gradient 

A n 

divided by a constant reference density f , and the buoyancy 0 

A 

The buoyancy is produced by the variable density J' 

acceleration 8 - IE tlie temperature T aii~ tile tiuinidity tiavc tticir 

mean values T, Q, 

under the gravitational 

/l 

-- constant reference values To , Qo , and fluctuations 

0, 4 , we can write, under the Boussinesq hypothesis, 

Po 

The temperature and the humidity satisfy the following equations of evolution: 

.I.- I 

with the molecular diffusivitylf . 

The system of equations (la), (lb), (5a) and (5b) describes the microdynami 

cal state of stratified turbulence. 

B. Group-Scaling of Fluctuations 

For the study of the spectral structure, wc will be interested in tltc 'I'liirliia1 , whose governing equations can be obtained by applying the 

and (51, where A E( ) is the 

n/ 

tionsh , 8 

fluctuation operator A s /-A 

averaging operator. These equations have terms proportional to the mean gradipnts 

.). 

?I I - 

to (1) 
5-4 

treated as constant parameters. 

temperature and the humidity into a single variable Y, called drift, such that 

For the sake of simplicity, we combine the 

A 

The constant coefficient y has the dimension of a frequency, and will always 

accompany the drift to form the buoyancy force. 

rv 

The differential equations which govern the fluctuating functions 5 and 

w N 

W can be obtained by applying the operator A to (1) and (5). These equations 

do not contain the mode-coupline,unless a Fourier transformation is made. 

- 

However, the Fourier form, like the original equations for the fluctuations, 

contain too many minute details. 

study. 

operators A and A , which compose A, in the form: 

A coarse-graining is necessary for a statistical 

For this reason, we introduce a group-scaling,.by means of the scaling 

0 N 

A" = An+ A', 

and the sequence continues by writing 

// 

A' = A(')+ A . 

Thus we have the macro-veloclty fluctuation 

o* 0 

A u=u , - -- 

the micro-velocity fluctuation 

,A ' 

A u=u , cthe submacro-velocity fluctuation 

and the submicro-velocity fluctuation 
5-5 

The goups ko , k', u 'have their correlation times - -4 

ranged .in the decreasing order of magnitude, indicating their decreasing 

coherence. 

0 

By the aid of A , we transform (la), (lb) and (5) into: 

It is not difficult to derive the following transport: equations oE eqergy 

in the group form: 

with the transport functions: 
5-6 

Thus we have the production functions fo and 

by mean drift gradient ywd , the buoyancy transport &id , the re-distrihuCion 

by wind shear V*kd and L 4 d 

y:H among the components d = 1, 2, 3, the cascade transfers To and -7-" d.( LV ' 0 

and the dissipations 

the repeating Roman indices does not apply to the Greek indices. 

equations have been approximated, by treating the mean gradients 

tdw and 2; . The summation rule which applies to. 

The transport 

c - -.., 

as constant parameters and by omitting the terms 

as arising from the inhomogeneity of turbulence. 

Unlike the non-scaled form, the energy equations of the macro-group contain 

the mode-couplings by the presence of the transfer functions. 

merit in these equations of being able to determine the spectral densities 

We also see the 

which are in reality the cumulative spec.tra1 distributions. 

C. Boundary Layer 

In boundary layers, the parameters (15) are restricted to the components 
5-7 

so that the energy equations (12) and (13) become reduced into the following 

particular form: 

13 zt (io,') = r,; +- e,* - 

The dissipation functions f ' 2 are negligible for large scale turbulence. 44 'w 

D. Kinetic Representation 

It can be shown that the hydrodynamical equations of turbulence (la) and 

(lb) are simply the zeroth and first moments of the master equation 

with the equivalence relation 

in the micro-dynamical state. The master equation has the advantage of being 

homogeneous, and its advective velocity does not cause a nonlinearity since V Is 

an independent variable. 

.. 

Like the hydrodynamical equations, the master equation can be submt tted 

to group-scalings, so that the evolution of 

controlled by the transport property. 

j!"c'.x,v) can be shown to be L..- 

x 
5-8 

called diffusivity, with the evolution operator A (t, t-T) 

property attains its equilibrium by a relaxation process, that is prescribed 

by fl(t,t-T) and achieved by the diffusion of the trajectory at a diffusivity 

. If the two diffusivities and 5 are structurally similar, a 

. This transport 

t // 

.., kz T & 

closure iq found. 

The analysis of the transport functions (18) will encounter many difficulties 

from the hydrodynamical approach. 

closure and the advantages of dealing with a homogeneous master equation of 

lesser nonlinearity in the kinetic approach will provide with a more powerEul 

tool €or the determination of the transport functions. 

It is hoped that the above relaxational 

111. KINETIC FOUNDATION OF THE PRESSURE-STRAIN CORRELATION 

Upon scaling the master equation to the macro-group, we have 

with the solution 1 

By taking the moment and differentiate, we find 

with 

(25a) 
5-9 

and the divergence-free condition: 

The 'differentiation by the operator v 

and (25b), and we have approximated the exact evolution operator 

approximate operator v, called the average propagator. Now we rewrite 

has yielded the two contributions (25a) P 

by the 

- 

these two terms in Fourier space, ;IS follows: A 

Here 

is an orbit function due to the unperturbed streaming, as opposed to the orbit 

-. due to the perturbed path ,!(-c) by turbulence. I 

The V-integration by parts gives 115 

Note that the effect of the orbit function! is small as compared to that of 

4 . The factor has the role of securing the divergence-free condition. 

V 

1. % 
5-10 

After the v- integration, we reduce (27) into: 

By definition (3a), the pressure I 

has a Fourier component 

9 

so that the pressure-strain correlation becomes 
5-11 

Here3 is a factor of Fourier truncation,'s0 

auto-correlation of Eo -field fluctuations, and PAC , 9: are diffusivity 

tensors from the cross-correlations. The. traces of the tensors do not possess 

indices. 

integrations with respect to z 

the divergence-f ree condition (26), and thar. 

is a diffusivity tensor from tlw 

,eh 0 

?* 

The diffusivities can play the role of integral operators in the 

and b9 . Note that (29) and (33) satisfy * 

is a boundary layer turbulence with the parameters (17). It leaves us the 

correlation (33b). Here the reduction of the diffusivity qb into aih and Do 

b,l ,c.:.L 

by (34a ) has,the benefit of confining ourselves to the velocity fluctuations 

, bo, by avoiding the pressure-field fluctuations. 

The diffusivity $O from the auto-correlation has been investigated earlier IJV 

2 

-% 

Tchen . It is found that for a strongly stable boundary layer, i.e., 

0 

the effect from 8' is negligible as compared to that from 7) . 

Brunt-Vais3la frequency. 

same method indicated above. 

which will be reported at a later opportunity, we expect that 

Here N is the 

The latter diffusivity can be investigated by the 

Without going into the details of the calculation 

a 
5-12 

The cascade transfer function is 

from our earlier work. The eddy viscosities are k and ' . Hence the 

spectral balance (18b) becomes 

For a steady state we differentiate (39) with respect to k, and obtain 

or 

by writing I 

The spectrum is thus found to be 

Y 

where 

is a factor without dimension. 

can be seen from the experiments in Pig. 1. 

power law 

The existence of the spectrum of power law 

'i'his spwtrim dirrc!rs from tlw 

suggested in the literature. The two spectra (42) and (44') are in the ratio 

J 
5-13 

4 fv-3 

\’ 

I I 1 I 1 t Ik 

I.-3 lo-’ IO’ 

Fig. 1. Normalized spectra Fll(k) , F22(k) in strongly 

stable turbulence. 
5-14 

for stable boundary layer. This small ratio is also verified by experiments. 

For a very stable boundary layer, where the conditon (45) is satisfied, the 

spectral distribution 

the redistribution and the cascade transfer, so that the power law k-3 is again 

valid. With the strong buoyancy, the spectrum G>O)decays rapidly by going ti:, 

(k) is similarly dominated by the balance between 

large k , without giving the opportunity for y033(k) to act efficient\>. 
1. 

2. C.M. Tchen, "Group-kinetic theory of pressure-strain correlation," (1983). 

3. H. R. Olesen, S.E. Larsen, and J. Hdjstrup, "Modelling velocity spectra 

C.M. Tchen, Chapter I in "Group-kinetic theory of tuburlence," (1983). 

in the lower part of the planetary boundary layer." 

Layer Meteorology. 

Submitted to Boundary- 
SECTION 6 

Kinetic Equation of Turbulence 

ABSTRACT 

The different types of turbulence in applications to the atmospheric 

and oceanic motions, the propagation of light, the solitary waves, and the 

plasmas have different governing equations. 

common Liouville form. 

method can be made available to treat the different types of turbulence. 

We shall bring them to a 

In this manner, we hope that a single statistical 

I 
6- 2 

I. FLUID EQUATIONS OF TURBULENCE 

Many basic phenomena in space, astrophysics, optics, atmospheric and 

oceanic sciences are nonlinear and random, i.e. they are in a turbulent 

state. 

our ability to analyze the turbulent behavior. 

A proper understanding of these phenomena depends critically on 

Although turbulence may 

appear in different forms and with different basic equations for the 

description of their microdynamical-states, there exists certain similarity. 

To exploit the analogy in the attempt of finding a common form of equation 

of microdynamical state, we first consider the incompressible and homogeneous 

turbulence, and write the governing Navier-Stokes equations, as follows: 

A 

Here U&x) is the velocity, p is the @onstant density 

in the incompressible fluid, 

the field E. 

which can be decomposed into a mean and a fluctuation. 

ity 9, which is a transport coefficient from the molecular dissipation, is 

neglig5bly small compared with its turbulent counter part, and can be 

ML 

is the pressure, whose gradient enters into 

) represents the total motion n F h 

The variable in the symbol ( w 

The kinematic viscosomitted in the analysis of the turbulent transport processes. 

gravity forms a buoyancy, and together with the Coriolis term may be included 

in the E field. 

The force of 

A 

4 

By applying the divergence-free condition (2) to the equation of momentum 

n A 

(l), we obtain the following relation between E and ct , in the form %a .I 
6 - 3 

called the “equation of state”. 

The equations of motion of an inviscid 

f 01 lows : 

compressible fluid are as 

(4) 

(5) 

By differentiating (5) with respect to t and eliminating 

we obtain the relation 

*A A 

u 

A 

‘d 

A 

The diffusion of a scalar Y($x).with chemical reaction E is governed (r 

by the following equation 

4 

where 3 is the diffusion coefficient, and U is the velocity field. 

The 

turbulence driven by a random field E. 

a vorticity 

.II. 

equations (7) are valid for the two-dimensional geostrophic A 

The geostrophic turbulence has .Iv 

with =: (3/6x ,$> ) in two-dimensions. 

The equation of propagation of laser light has the form: 

1 

Here is the light field propagation in the z-direction with an optical h 
6-4 

A 

wavenumber4 in a medium of fluctuating dielectric coefficient 2 . 

term --TI& I 

dielectric medium, causing the self-focusing of light. 

The 

represents the effect of the strong light intensity on the 

42 

192 The equation (8) is called the "nonlinear SchrGdinger equation" 

It also applies to the solitons in plasmas, the atmosphere and the oceans. 

Here U is the envelop of the waves. 

A 

The linearized form 

+ V, 2n u+k 24 (€-J)C=O, wfh x=-+- 2 h2 s ' (9) 

a)c= 7" 

3,4 is called the parabolic equation of propagation. 

A 

Although the equation (9) is linear in LL, the product of two random 

AA 

functions E U makes the problem statistically nonlinear. In the form (9), 

the propagation of light resembles the diffusion (7) of a scalar. 

Recently a great deal of effort on the nonlinear wave phenomena has 

5 

been made by using the Korteweg-de Vries (KdV) equation: 

where A is the coefficent of dispersion. 

adequately describes the long-time asymptotic behavior of wave motions in 

nonlinear dispersive systems, and that this asymptotic form can be 

It has been shown that this 

derived either from the nonlinear Schrodinger equation (8) or from the 

perturbations of the equations of wave motions. 
6-5 

-11. MASTER EOUATION FOR THE DESCRIPTION OF THE MICRODYNAMICAL STATE OFTURBULENCE 

Although the fluid turbulence obeys different forms of differential 

equations (1) - (7) for the description of the microdynamical state, their 

kinetic representation can be cast into a unified master squation, or 

6 Klimontovich equation: 

4 

Here $(tx,y) is the distribution of the random variable - v at time t n ' -.-- 

and 'position x , and L(t, X, V) is a differential operator which takes 

varied forms depending on the types of turbulence in consideration. 

These forms are found in the following. 

(i) For the inviscid incompressible turbulence as governed by (1) - 

(3), we have the differential operator 

the equation of state 

and the source 

The limits of integration are understood to extend to the whole 

available domain, unless otherwise specified. 

two representations, i.e. the fluid representation (1) - (3) on the one 

The equivalence between the 

hand and the kinetic representation (11) on the other, can be obtained by 

writing 
6-6 

With this equivalence, it is not difficult to transform the equation of 

state from the kinetic form (13) into the hydrodynamic form (3), and the 

master equation (11) into the hydrodynamic system (1) and (2) by means 

of moments 

(ii) For the compressible turbulence, as governed by (4) - (63, we 

have the same differential operator (12) and the same equation of state (13). 

The equivalence between the two representations is obtained by writing 

the distribution as 

and the source as 

By the equivalence relation (16), it is not difficul 

(17b) 

sform 

the kinetic representation with formulas (11) and (16) into the fluid 

representation with formulas (4) - (6). 

(iii) For the diffusion and the two-dimensional geostrophic turbulence, 

as governed 

differential operator 

by (7), the Liouville equation (11) remains valid, with a 

A 4 z L=U.V=3V , vand an equivalence condition 

(iv) For the light propagation, we can introduce 

to tra 
6-7 

and transform the SchrGdinger equations (8) and (9) into 

with 

in nonlinear propagation, and 

in l‘inear propagat ion. 

Note that (21) is already in a form analogous to the master equation 

so that further transformation into the kinetic representation becomes 

unnecessary. The same argument holds for the KdV equation (10). 

(v) The plasma turbulence is governed by (ll), now called the Vlasov 

A 

equation, with the differential operator (12). The E - field represents 

the electrostatic field with unit charge and mass.The equation of state in 

the form (13) remains valid, with the source 

A(X,V) = wp2 (23) 

ubv. 

where 0 is the constant plasma frequency. 

the normalization condition 

The distribution function has P 

where $($XI is the number-density, - 
6- 8 

z c 

c 

a, 

0 

rl 

5 

-2 

5 

7 

7 

c, 

Q) 

rl 

v) 

v) 

a, 

k 

F 

u 

d 

.rl 

a, 

0 

5 

9 

5 

8 

rl 

3 

P 

4J 

a, 

rl 

v) 

VI 

a, 

k 

Pi 

0 

a, 

0 

d 

a, 

rl 

7 

5 4J 

E! 

v) 

td 

4 

PI 

a, 

0 

rl 

8 

5 

!3 

7 

P 

c, 

0 

.A 

s 

k 

c, 

v) 

0 

a, 

M 

n n n n n 

U v 
6-9 

In Table 1, we have listed the different types of turbulence. Their 

governing equations are written in the master 

differential operators. 

distribution function by the equation of state in the form (13) through 

a sourceh . 

form with appropriate 

The self- consistent field is related to the 

This reduction of several types of turbulence into a unified 

master equation suggests that they can all be treated by a single 

statistical method. 

The description of the microdynamical state of turbulence by an equation 

of the master form presents the advantages that the equation is homogeneous, 

fi 

and that the replacement of the velocity function f.i.($x) by the independent - % - 

variable V prevents any terms connected kith the velocity from becoming 

nonlinear, as found in the differential, operators L and the sources (a) - (c). 

* A 

n 

a/ * The only surviving nonlinear term E. arises from the E -field. *k8 

Fortubat$ely this does not require our immediate attention, since it can be 

provisionally considered as a given random force until after the closure. 

Thereafter, E is determined by the equation of state with the proper source. 

A 

.H 

This nonlinear term describes the wave-particle interation 

explicitly, and therefore will enable the derivation of the nonlinear 

Landau damping in fluids in a direct manner. 

describe such an interaction only indirectly through a series of moments. 

The fluid representation may 

x 
6-10 

3. Kinetic Hierachy of Turbulence 

In the following lines we shall be concerned with the incompressible 

Navier-Stakes turbulence. The microdynamical state of turbulence is 

described in the kinetic representation by the master equation 

with a differential operator 

and a self- consistent field 

J 

so that the pressure F satisfies the Poisson equation 

by (13). The solution can be written in the form: 

We calculate 

by differentiating (29), and obtain 

* 

J 
6-11 

by introducing the integral operator 

The term (3l)yields a second order nonlinearity in the master equation. 

The master equation, which describes the microdynamical state of 

turbulence, contains fluctuations with all minute details which are 

unnecessary, if not impossible, in a statistical study. 

procedure is the "global ensemble average" 

A coarse-graining 

with the operator 

Ad J. 

Such an average is deterministic and may vary with $x .) or t '-'- x v The 

deviation from this average is obtained by the operator of fluctuation 

- A=. /-A, 

where "1" is the unit operator . 

,ry 

By applying the operators A , A to (25), we obtain: 

and 

or 

(34) 

(35) 

The operators operate on the functions which follow. 

- 

The equation (33) describes the evolution of the distribution /(t.x.y) 

in a turbulent medium which presents a turbulent collision c(t~ ' "/* /) 

J 
6-12 

rY* 

representing the statistical effects of the fluctuations f-4 upon r. Note 

that the collision, as defined by 

is the derivative of the Reynolds stress 

in the phase space. In this sense the equation (33) can be called the 

Reynolds equation in the phase space, and the equation of, evolution 

of the fluctuation, (34) or (35), is the Friedman equation, as obtained 

by the Reynolds decomposition 

By this decomposition, we can separate (32) into two components, as 

follows : 

where 

is the pair distribution function. Upon substituting (39) into the 

ensemble average of (32) and subsequently into (33), we get the following 

equation of evolution of 
6-13 

with 

me dependance of the collision upon the pair distribution function 

is expected. If we consider a source of the type (23), we reduce (41) to 

the BBGKY (Bogoliubov, Born, Green, Kirkwood and Yvon) hierarchy of plasmas 

Our general form (41) applies to other types of turbulence too. 7-12 

# 

The equation of evolution of f in (41) will be called the kinetic 

equation of turbulence. It remains to determine the collision as an explicit 

function of , i.e. in the form: 

- 

where ;e is a collision operator. 

x 
6-14 

REFERENCES 

1 S. S. Akhmanov, A. P. Sukhor_ukov, and R. V. Khokhlov, Zh. Eksp. 

Teor. Fiz. - 51, 1189 (1966) 1Sov. Phys.-JETP - 24, 797 (1967)l. 

V. E. Zakharov, Zh. Eksp. Teor. Fiz. - 53, 1735 (1967) [Sov. Phys.- 

JETP - 26, 994 (1967)j. 

V. I. Tatarskii,*Wave Propagation in a Turbulent Medium, translated 

by R. A. Silverman (McGraw-Hill, New York 1961). 

V. I. Tatarsnkii, The effects of the Turbulent Atmosphere on Wave 

Propagation (Translated by Israel Program for Scientific Translations; 

originally published in 1967). 

Technical Information Service, Springfield, Va. 1971) . 

D. J. Korteweg and G. de Vries, Phil. Mag. - 39, 422 (1895). 

2 

+ 

3 

@ 

4 

U.S. Department of Commerce, National 

5 

6 Yu L. Klimontovich, "The Statistical Theory of Non-Equilibrium Processes 

in a Plasmat1. Pergamon Press (1967). 

7 

8 

C. M. Tchen, Phys. Rev. - 114, 394 (1959). 

A. S. Monin, Dokl. Akad. Nauk SSSR - 177, 1036 (1967) [Sov. Phys.-Doklady 

- 12, 1113 (1968)l. 

9 A. S. Monin, Priklad. Mat. Mekh. - 31, 1057 (1967) [Appl. Math. and Mech. 

- 31, 1057 (1967)l. 

10 

11 

12 

T. S. Lundgren, Phys, Fluids 7 10, 969 (1967). 

T. S. Lundgren, Phys. Fluids - 12, 485 (1969). 

V. M. Ievlev, Dokl. Akad. Nauk 

Doklady - 18, 117 (1973)]. 

SSSR 7 208, 1044 (1973) [Sov. Phys.- 

J 
eling of Atmospheric Turbulence 

The City College Research Foundation 

138th Street and Convent Avenue 

New York, New York 10031 

National Aeronautics and Space Administration 

Washington, D. C. 20546 

Prepared for the Atmospheric Sciences Division Systems Dynamics Laboratory, 

Marshall Space Flight Center, Alabama 35812 

Part I1 of Two-Part Final Report 

16. ABSTRACT 

Two-dimensional geostrophic turbulence driven by a random force is investigated. 

Based on the Liouville equation, which simulates the primitive hydrodynamical equa- tions, a group-kinetic theory of turbulence is developed and the kinetic equation of 

the scaled singlet distribution is derived. This distribution will suffice for the 

investigation of the spectrum of turbulence. 

describes the pair interaction and its enhancemen The 

brium and non-equilibrium states. The propagator formalism is summarized in a selfconsistent way for the asymptotic quasi-linear equation. Comparison is made between 

the propagators and the Green's functions in the case of the non-asymptotic quasilinear equation to prove the equivalence of both kinds of approximations used to de- scribe perturbed trajectories of plasma turbulence. The microdynamical state of fluic 

turbulence is described by a hydrodynamical system and transformed into a master equation analogous to the Vlasov equation for plasma turbulence. 

bution function is decomposed into a mean value and a fluctuation, the evolution of tl 

mean distribution satisfies a transport equation, .i.e., ,kinetic equation, and contains 

a turbulent collision that represents the statistical effect of the turbulent fluctuation, while the evolution of the fluctuation will form a transport equation for the 

collision. The hydrodynamical equations of turbulence are transformed into a masterequation for the velocity distribution function. A group-scaling is introduced for 

the closure. 

components shows that the scaled pressure-strain correlation and the cascade transfer 

The collision integral has a memory and 

by the multiple interaction. 

kinetic equation is transformed into an equation of 4 spectral balance in the equiliWhen the total distriThe spectral balance for the velocity fluctuations of individual 

Turbulence 

Atmospheric boundary layer Unclassified-Unlimited 

Storm and mesoscale processes 

Global scale processes STAR Category: 47 

Atmospheric and terrestrial environmental 

processes 

For sale by National Technical Information Service, Springfield, Virginia 2 2 161 

NASA-Langley, 1984 

1