NASA Contractor Report 3787 

Theory and Modeling of 

Atmospheric Turbulence 

Part I: September 1, 1981 -August 31, 1982 

C. M. Tchen 

CONTRACT NASS-34622 

FEBRUARY 1984 

LOAN COPY: RETURN TO 

AFWL TECHNICAL LIBRARY 

KIRTLAND AFB, N.M. 87117 

https://ntrs.nasa.gov/search.jsp?R=19840010737 2020-02-21T19:11:24+00:00Z
- " ". -~ ~~ ~~ 

NASA Contractor Report 3787 

Theory and Modeling of 

Atmospheric Turbulence 

Part It September 1, 1981 -Azcgust 31, 1982 

C. M. Tchen 

Tbe City CoZZege Research Foundation 

New York, New York 

Prepared for 

George C. Marshall Space Flight Center 

under Contract NASS-34622 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Office 

TECH LIBRARY KAFB, NM 

1984 

FOREWORD 

Two approaches for the modeling of turbulence exist. One is to start 

from an open hierarchy of profiles and higher order correlations. The other 

is to formulate a closed sequence of transport processes: evolution of mean 

profiles, eddy transport coefficients (viscosity, diffusivity, and damping 

or amplification rate), and relaxation or memory loss for the approach of the 

transport coefficients to equilibrium. The former approach uses an ambiguous 

closure, or an arbitrary hypothesis of the length scale of mixing. The 

latter approach is based on a theory of transport and the analysis of the 

spectral structure of turbulence. 

The author follows the latter approach and develops a new kinetic method. 

This intact report includes research reported quarterly during the first year 

of effort, September 1981 through August 1982, with a second report to follow, 

which will include research for September 1982 through August 1983. 

ACKNOWLEDGEMENTS 

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. 

iii 

TABLE OF CONTENTS 

QUARTER ONE: Kinetic Basis of Cascade Transfer in Turbulence 

Page 

Abstract ............................ A-1 

I . Introduction ....................... A-2 

I1 . Micro-Dynamical State of Turbulence in the Configuration 

and Phase Spaces ........ : ............ 

I11 . The BBGK Hierarchy of Turbulence 

IV . The Scaling of Fluctuations ................ 

V . Kinetic Equation of Turbulence and Kinetic Theory of 

Transition ......................... 

VI . Transport Theory of Diffusivity .............. 

VI1 . Kinetic Basis of the Cascade Transfer ........... 

VI11 . Conclusions ........................ 

IX . Acknowledgment ...................... 

X . References ........................ 

A-6 

A-11 

A- 15 

A-20 

A-30 

A-35 

A- 40 

A-42 

A-43 

QUARTER TWO: Kinetic Theory of Turbulent Transfer with Double 

Memory Loss 

Abstract ............................ B-1 

I . Introduction ....................... B-2 

I1 . The Scaled Kinetic Equation of Turbulence ......... B-5 

I11 . The Lagrangian-Eulerian Transformation in the Phase Space. 

The Loss and Cutoff of Memory ............... B-9 

IV . Cascade Transfer with Double Memory Loss ......... B-16 

V . Eddy Viscosity and Collisionless Damping ......... B-21 

V 

I 
TABLE OF CONTENTS (cont) 

Page 

VI . Derivation of the Kilmogoroff Law from the Scaled Kinetic 

Equation ............................ B-28 

VI1 . Discussion .......................... B-31 

Acknowledgement ........................ B-35 

References .......................... B-36 

QUARTER THREE: Group Scaling Theory for the Enstrophy Transfer 

in Two-Dimensional Turbulence ........... 

Abstract .............................. C-1 

I . Introduction ......................... C-3 

I1 . Scaled Vorticity Equation of Turbulence ............ C-5 

I11 . Transport Phenomena in Turbulence ............... C-10 

IV . Double Memory Loss ...................... C-17 

V . Transfer Furnction ...................... C-19 

VI . Inertia Spectrum ....................... C-20 

VI1 . Discussions and Conclusions .................. C-21 

Appendix ........................... C-24 

Acknowledgement ......................... C-31 

References .......................... C-31 

QUARTER FOUR: A Group-Kinetic Theory of Turbulent Collective Collision 

Abstract .............................. D-1 

I . Introduction ......................... D-3 

I1 . Microdynamical State of Turbulence .............. D-8 

I11 . Group-Scaling Procedure .................... D-11 

IV . Turbulent Collisions and Memories ............... D-14 

vi 
TABLE OF CONTENTS (cont) 

Page 

V . Kinetic Equation of the Transition Probability .......... D-22 

VI . Determination of the Collisions by Means of the Transition 

Probability ........................... D-29 

VI1 . Shielding and Enhancement of the Collision ........... .. D-33 

VI11 . Summary and Discussions ..................... D-37 

Acknowledgment ......................... D-43 

References ........................... D-44 

vii 
CHAPTER A: QUARTER ONE 

Kinetic Basis of Cascade Transfer in Turbulence 

C. M. Tchen 

The Graduate Center and The City College of the City 

University of New York, N.Y. 10031 

ABSTRACT 

Among the transport functions which characterize t.he evolution 

of: a turbul-ent spectrum, the cascade transfer is the only function which describes the mode-coubling as the result of the 

nonlinear hydrodynamic state of turbulence. A kinetic theory 

combined with a scaling procedure is developed, to derive a 

kinetic equation of velocity distribution and the transition 

equations of path perturbations. These equations permit to 

formulate a transport theory of turbulence to investigate the 

eddy diffusivity, the eddy viscosity and the cascade transfer. 

Circumstances under which the transfer may appear in the form 
of a direct cascade or a reverse cascade are considered. 

I. INTRODUCTION 

The Navier-Stokes equation of motion, with a pressure 

gradient and a buoyancy as driving fields, can be used to describe the micro-dynamical state of turbulance and to serve as 

the basis of a statistical theory. But the nonlinearities in 

the inertia and the driving fields will confront us from the 

start of the analysis. Various statistical methods have been 

proposed in the literature but were not too successful in soling the closure of the hierarchy of equations. By changing the 

velocity function into a "particle" velocity as an independent 

random variable, the kinetic method is relieved of this diffiA-2 
culty. What remains is a nonlinear interaction between fluid 

and "particle", with a diffusivity as an integral operator to 

represent the memory in the non-Markoff process. 

The kinetic theory of turbulence presents the following 

three problems: (a) the kinetic equation of turbulence, (b) 

the transport theory of turbulence for the determination of 

the eddy diffusivity, (c) the kinetic basis of the cascade 

transfer. In the derivation of a kinetic equation, it is important not to fall into the Bogoliubov hierarchy. Its 

closure has not been as successful in fluids as in plasmas. 8 . 

This diffuculty may have prevented further progress in the 

kinetic theory of fluid turbulence. We introduce a "scaling 

procedure" to avoid this hierarchy and the involvement of a 

system of equations of singlet and pair distribution functions. ~ 

A- 3 
The eddy diffusivity which characterizes the kinetic equation 

is defined as the time integration of the Lagrangian correlation of field fluctuations. We develop a transport theory 

for the transformation of a Lagrangian correlation into an 

Enlerian correlation. This relationship, or the time-space 

transformation, has attracted many investigators by using the 

hypotheses of independance . and normality. The correction 

has also been evaluated. The problem referred to a diffusion model, i.e. to a micro-dynamical state without a driving 

field. This model would exclude the more realistic aspects of 

turbulence, as presented by the Navier-Stokes turbulence, the 

shear turbulence and the turbulent motions in a statified medium. We feel it important to find a generalization that incorporates the driving field. This generalization brings up 

A-4 
new difficulties. Our kinetic theory of turbulent transport 

helps in switching the driving f'ield to a new role of advection 

in the phase space and in prescribing probabilities of transition to replace the above hypotheses of independence and 

normality. 

In order to be qualified as a transport property, the 

diffusivity has to reachda statistical equalibrium within a 

finite time, called the relaxation time. This finite time is 

obtained, when a memory-loss can be found. The scaling procedure selects and organizes the necessary mechanism of memory-loss by the turbulent dissipations. 

The mode-coupling governs the transfer of modes across 

the spectrum in both directions toward the high as well as 

the low wavenumbers. It appears in the form of the moment of 

A- 5 

I 
the fluid-particle interaction, in'analogy with the nonlinear 

Landau damping in plasma turbulence. Formally, the fluid-particle interaction can be expressed by a series of high order 

moments, as we know that a velocity distribution could be expanded in this way. It is nevertheless an insufficient way Of representation. This indicates that, with these limitations, a 

pure continuum method on its own may not be successful in analyzing the mode-coupling, since it had failed in describing the 

Landau damping. 

11. MICRO-DYNAMICAL STATE OF TURBULENCE IN THE CONFIGURATION 

AND PHASE SPACES 

The hydrodynamic,al equations of the motion of an incompressible fluid are the equation of momentum 

A- 6 
and the equation of continuity 

where the evolution of the total fluid velocity &(t/x) 

" 

under 

the force 2 [%X) is governed by a differential operator . 6 

and 3 is the kinematic viscosity of the fluid. The force 

n 

E .l =-pvp - 

may be due to the gradient of 

f, or may take a more compl 

ture difference and the 

6 

pressure in a 

(3) 

fluid of density 

icate form involving the temperabuoyancy force as in a stratified medium. In all cases, the force can be written in the differential form 

A- 7 
or in the integral form 

where 

is Poisson's integral operator, such that 

Both forms result from applying the condition of incompressibility (lb) upon (la). Here and in the following the integration is understood to extend from -bo to w . 

The hydrodynamical equations (la) and (lb) 

can be considered as describing the micro-dynamical state 

of turbulence in the configuration space t,x. This can be c 

transformed into the phase space t,x,v, by introducing " a 

b - function 

A- 8 
which satisfies the equation of evolution 

with a differential operator 

in the phase space. The force can again be written in the integral form: 

with Poisson's integral operator 

and the source function 

in the phase space. 12 

A- 9 
The micro-dynamical state in the phase space (7) possesses 

certain important advantages over that in the configuration 

space (la) in eliminating the nonlinearities. This is done 

by changing the role of the velocity as a random function 

into the role as an independent random variable. By the same 

token, the inhomogeneous equation of micro-evolution in the 

configuration space is transformed into a homogeneous equation 

in the phase space. We shall fully utilize these advantages 

in developing our statistical theory of turbulence. Of courss, 

the micro-dynamics should still keep a nonlinear form in the 

phase space, as seen from . 3 hf in (7) . Fortunately the .- 

A 

treatment of this nonlinearity can be delayed by considering E - 

to be provisionally known. After we have closed the transport 

equations, we can pick up the nonlinear relationship (4a) which 

then will be in its uncomplicate form of a consititutive relation. 

A-10 
The equivalence between the two spaces (1) and (71, can be 

easily verified by taking the moments 

of (7). It is then seen that the equation (7) in the phase 

space actually corresponds to the simultaneous system of dynamical equations (la) and (lb) in the configuration space. 

111. THE BBGK HIERARCHY OF TURBULENCE 

If V is a stochastic variable by the stochastic behavior * 

Of cC(t,x) , the function (6) can be called a distribution 

/r 

function, denoted by 

retaining nevertheless its stochastic character, with a 

A-11 
micro - dynamical state (7). The function (12) and 

the evolution (7) contain fluctuations with all the minute 

details which are unnecessary in a statistical study. A 

coarse-graining procedure which eliminates all stochastic 

fluctuations is the ensemble average 

A .(> 

- 

called "global average". A fluctuating function subject to 

this average becomes deterministic, e.g. 

and the deviation from the average is the fluctuation 

or, in terms of the operator notations, 

where "1" is the "unit operator", is the average operator, 

A-12 
and is the fluctuation perator. 

Upon applying the average operator A' to the equation 

or 

when use is made of (9) and of the notation 

The functions (14) and (17) are called the singlet and the 

doublet distributions, respectively. 

In (16b) we see that the evolution of the singlet distribution depends on the doublet distribution, and that the 

latter evolution will expectedly depends on the triplet 

A-13 

I 
dis,tribution 

and the sequence continues to form the BBGK hierarchy. This 

hierarchy is well known in the kinetic theory of plasma turbulence, and is seen here to reappear in fluid turbulence 

with the difference that h(x,v) is defined by (lob) in an 

0- 

incompressible fluid, while A(x, v)= / in plasmas. 1-4,8 

-" 

The closure of the hierarchy constitutes a difficult 

problem. It has been attempted by finding a small parameter 

in plasmas, but such a parameter is lacking in fluids. This 

may explain why so little progress has been made in fluid 

turbulence by the kinetic approach. The analysis 0-f the spectral structure of turbulence requires the Fourier transformation of a correlation function, and therefore the second 

A-14 
moment of the doublet distribution. Thus this kinetic problem 

calls for the determination of both distribution functions. This 

obviously would pose a tremendous task. We shall devise 

a kinetic method, in which we can avoid the BBGK hierarchy and 

the doublet distribution function. To this end, we shall resort 

to the scaling procedure of fluctuations, which we shall describe in the following. 

IV. THE SCALING OF FLUCTUATIONS 

The BBGK hierarchy which begins with the singlet distribution (16) does not give any statistical information unless 

the doubiet distribution is known. This means that 

a minimum of two equations of the closed hierarchy will be 

A-15 
needed to describe the fluctuations statistically. In order 

to avoid the BBGK hierarchy completely, we shall scale the 

fluctuation 

into a macro-group &I"' and a micro-group e' . .c - 

The scaling is performed by writing the fluctuation operator 

x = A'")+ A/ 

A(" 1 into two components , A ' , which select the macrogroup and the micro-group, respectively, and are called 

"scaling operators". Since (15bZ is known as the "Reynolds 

decomposition" in turbulence, we may consider the scaling 

(20a) as a simple extension of this decomposition. 

A-16 
For the sake of convenience, we introduce 

A, = x + AC"/ 

so that 

The two groups E(OJ and E' have their durations of - u 

correlation rc (0) and T,' , with the inequalities 

stating that the micro-group E" is more random than the - 

macro-group .lo' . Their respective portions of the energy L 

are clearly separated by the wavenumber variable If in the 

spectral distribution S(k) of E - fluctuations, although - 

the individual groups and E' may overlap in their - 

Fourier compositions. Clearly, the two portions add to 

A-17 

I 
Similarly, we scale the velocity fluctuation 

L L 

into a macro-velocity Uco' and a micro-velocity U' of LC 

energies 

in the spectral distribution f(k) of U - fluctuations. c 

The two portions also add to 

The same scaling applies to the distribution function 

p(t x, v) in the superposition -c 

of a macro-distribution f"' and a micro-distribution {' . 

It is to be remarked that the scaling permits the 

A-18 
derivation of the spectral distributions by simply 

differenting (22a) and (24a) with respect to x , while this 

was not the case with the non-scaled energies (22b) and (24b). 

Thus our statistical problem amounts to finding a kinetic equation of the scaled singlet distribution only,instead 

of both the singlet and the doublet distributions in the unscaled treatment of the BBGK hierarchy. 

It is to be stipulated that the decomposition into the 

scaled groups (20) differs from the decomposition into the 

Fourier components or a group of such components. The latter 

decomposition is a mathematical transformation without adding 

any physical conceptto the treatment, while the former decomposition injects a scaling procedure of coarse-graining to 

the stochastic process by the operator A'") or the operator 

A-19 
A’=X-A(OJ . Consequently, the scaling procedure distinguishes between a macro-process representative of the evolution of a macro-field and a micro-process representative of 

the transport properties as shaped by the more random microfluctuations in the medium in which the macro-field is propagating. Well defined probabilities of transition, that are 

self-consistent with the once given micro-dynamical state of 

turbulence, will define the scaling operators A“ and A’ . 

V. KINETIC EQUATION OF TURBULENCE 

The micro-dynamical state of turbulence is described by 

the dynamical state (7) in the phase space. We apply the scaling 

operators p’” and A ‘ and decompose (7) into the system 

A-20 
which describes the evolution of the macro-distribution 

under the statistical effects of the more random micro-fluctuations E'!' , while the f ' fluctuations are ex- & 

cited by the gradients of the mean and the macro-distributions, 

according to (26 1. All operators including the scaling operators A'"), 4' and the differential operator L refer to all 

functions which follow them. 

If we disregard the group-coupling between f'O'and -f ' 

and merely express E' in terms of e' according to (9) by ubA' , we would again fall into the BBGK hierarchy in the form: 

with 

A-21 
by definition of the scaled doublet distribution function. 

In order to avoid this hierarchy which is equally undesirable, we call on the group-coupling between the two equations 

(26a) and (26b). For this purpose, we rewrite (26b) in the form 

by the use of (20cl . We integrate to find 

A { Tc x, v) E A "p (t, x, v) " " 

z =t at: 

along the trajectory of the evolution Of f?t.X,v) in the " 

phase space, and the right hand side is a driving field. The 

solution (28) is the time integration ofthe driving field in 

A-22 
the Lagrangian representation denoted by 

Here x(t-t) is the position along the trajectory at the - 

time t"C , when it is known that at time t, the position is 

x and the velocity is v . The trajectory is governed by: - - 

where V(tl) is an intermediate variable in - 

2 d x(tl) 

c 

= E(tl) . dt12 L 

The dynamical equations (31) Foverninr the trajectory are called 

the micro-dynamical equations of transition. The operator 

is called the propagator, or "Lagrangian operator". The last 

The stress in the phase space-4~ thus found to be 

A-23 
where 

is the eddy diffusivity. Hence, upon substituting the stress 

into the equation (26a) of macro-evolution, we obtain: 

In the evolution ofthe macro-distribution (34) of scale t , we have 

and a fortiori 

t 7) t,' 

from Fl), so that D' from (331 becomes a large-time diffusivity I 

4 

t- a3 

D' = A")L dt € yc x) U(t, t-t) I! '(t-t) , (37) - - -..c .5 

that asymptotically will take a deterministic form: 

A-24 
on account of the small correlation time (361 and the deterministic character (22a). Under this circumstance, we reduce 

(34) to the following kinetic equation 

It has the form of a Fokker-Planck equation, containing a 

memory as described by the diffusivity operator D' 

the group-coupling and the scaling procedure, the kinetic 

equation (391 is explicit in f"' , avoiding the BBGK hierarchy. 

KINETIC THEORY OF TRANSITION 

The transport theory for the determination of the diffusivity (38) requires the examination of the Lagrangian correlation 

A-25 
The trajectory in the phase space 

contains a fluctuating path p(,) in the retrograde tran- - 

sition, i.e. in a time interval "C , and is self-consistent 

with the same micro-dynamical state (31) as that of the kinetic 

equation of velocity distribution (392. 

A 

The fluctuating path t(-Z) is a random function, for u 

which we will develop a kinetic theory of transition. 

The first of the micro-dynamical equation of transition 

(31) can be written in the form 

called the "master equation of transition", where the velocity 

n v(-r) has been defined by (31) , and .4. 

A-26 
' is a probability of transition in the microscopic description. 

It is a function of the random variable 1 and has the 

moments : 

It is not difficult to verify, by means of the moments (44) 

that the master equation (42) will indeed reproduce the microdynamical equations (31) . 

We apply the scalings F, A. 4' to the master 

equation (42) and derive the following kinetic equations of 

transition 

" 

and the probability of transition in the micro-scale 

A-27 
where LA is a drift ve1ocit.y. The details of derivation, 

- 

4 

which are similar to those used in the derivation of the kinetic euuation of velocity. distribution, will not be repeated 

here. 

The coefficient of diffusion, as defined bv 

is calculated from the micro-dynamics (31) of the path fluctuation, written in the new form : 

We find the dispersion 

and the diffusion coefficient 

By scalings, we have: 

A-28 
The kinetic equation of transition., in the non-Markoff 

master equation (42) in self-consistency with the originally 

stated micro-dynamics of turbulence, in the form (11, (31), 

or (49), and therefore should not be confounded with the commonly known transition equation which was derived phenomenologically in the hypotheses of a Markoff process. l3 Note 

also that the kinetic equation of transition determines a 

retroqrade transition of nath ..I 

A-29 
VI. TRANSPORT THEORY OF DIFFUSIVITY 

Bv the Fourier decomnosition, we transform the formula 

(38) of diffusivity into 

with the use of (40) and (41). The details of the transformation are elementarv and have been omitted. For the sake 

of abbreviation, we have introduced the Fourier function 

and the fluctuatinq intensitv 

of spectral tensor 

When the Fourier transform is truncated within a lenqth interd 

val 2 , the truncation factor is Pshh) ,where 4 is 

A-30 
the number of dimensions. 

It is to be noted that the integral 

A. 

with a perturbed path A(-z), constitutes a Lagrangian corre- L. 

I 

lation of the E - field fluctuations, while the integral - 

without path perturbations is the Eulerian correlation. Hence 

the distinction and the relation between the two correlations 

reside in the path perturbations in the Fourier function 

c 

This function can be decomposed into two parts as 

A-31 
since /lo +A ’ , with 

For the analysis of the first part (56a), we calculate: 

from the Fourier transformation of the kinetic equations of 

transition (45) - (47), so that we find 

with the notations: 

A-32 
The second part (56b) contains the fluctuations already 

found in (571, and can thus be written in the form: 

The degenerated function 

can be considered as a time-integrated flux, so that, by combining (601 and (61), we obtain 

with 

in the transport of s’ of a path &)and a diffusivity 3 . Y 

-4 

4 

The results (58) and (62) for the two parts of the Fourier 

function (55b1 add to 

A-33 

I 
and yield a diffusivity which can be written in the form: 

This expresses that the diffusivity is endowed by the spectral 

tensor (drb')> and approaches its equilibrium at a relaxation ttime 

where and 5 ' have been defined by (59) and (63) , re9 

spectively. 

Note that, since the relaxation time (66) contains 3' 

in the integrand, through 5 ' and k/ from (63) , (59) and 

D 

(51b) , the form (65) is the integral equation for the determination of D' . .5. 

-u 

We conclude that the Lagrangian correlation of the 

' 

E- field fluctuations is expressed in terms of the spectral .L 

A-34 
function by the following formula: 

with 

from (64) - (66). 

VII. KINETIC BASIS OF THE CASCADE TRANSFER 

, We write the kinetic equation of velocity distribution 

(34) in the following explicit form: 

We take the moment of (68). The zeroth moment gives the 

equation of continuity 

A-35 
and the first moment gives the equation of momentum in the 

form: 

with a collisionless interaction 

The advection IO) - the production v(c M'?) I 

i'd 

(0) (0) 

d Ld 

the inertia A V (u.. LC . ) I the viscous damping Y I7 &. 2 (0) 

i 

and the driving field E'"' are all the regular terms recogC' 

nizable by isolating the macro-group from the hydrodynamical 

equation (la) which describes the micro-dynamical state of turbulence. The interaction is new, and can be verified to 

vanish if the memory should be absent. Since J" can be 

L' 

identified as a scaled Reynolds stress fromthe micro-flux of 

momentum, i.e. 

(71b) 

A-36 
the memory is indeed essential to turbulence. 

From the mixing-length hypothesis, the momentum flux of 

the gradient type, i.e. 

has been established, giving an interaction in the form: 

The transport coefficient K' is called the eddy viscosity. 

The comparison between the above two forms of interaction 

suggests a means of isolating this eddy viscosity, provided 

the transport of the gradient type (72) is valid. On the contrary, ,when the gradient transport is not valid, as is the 

case with certain stratified media, the kinetic interaction 

(70) could determine the correct type of transport appropriate 

to the problem. 

I 
describing the rate of change of the kinetic energy, as governed 

by the following transport functions: 

production 

coupling 

cascade 

transfer 

viscous 

dissipation 

flux transport 

Among these transport functions, only the mode-coupling 

function, or the cascade transfer function 

A-30 
possesses a structure that is characteristically kinetic, as 

involving the correlation 

between the fluid velocity &lo' and i 

(77) 

the singlet distribution 

$' . This coupling is analogous to the fluid-particle 

interaction which was considered to be the origin of the nonlinear Landau damping in plasma turbulence. 

Lagrangian correlation into an Eulerian correlation (or its 

Fourier transforml. We apply the same rule for the Lagrangian 

correlation (771, and obtain: 

where is defined by (67b). . 

Now when this Lagrangian correlation issubstituted into 

A-39 
(76), we find the transfer function in the form: 

Noting that Dy] is an operator of integrations - .c 

with respect to k' and Z , as prescribed by (65) and (66) a 

respectively, we can rewrite (79) in the explicit form, as 

follows: 

VIII- CONCLUSIONS 

By means of a scaling procedure, we have developed a kinetic 

equation of turbulence and a transport theory 

of eddy diffusivity. The transport theory is based 

upon the transformation of a Lagrangian correlation into an 

A-40 
Eulerian correlation by means of transition probabilities 

which we have found from a kinetic theory of transition. 

The transfer function describes the mode-coupling, or the 

cascade transfer across the spectrum. It appears in theaform 

of a fluid-particle,interaction, and is therefore a kinetic 

phenomenon like the Landau damping. A memory should exist 

in this interaction and is generated by the Lagrangian correlation between the fluid velocity and the "particle" velocity 

distribution. However, a memory-loss should be present for a 

collisionless dissipation to yield a finite relaxation time. 

A preliminary examination of the expression of the cascade 

transfer reveals that the cascade process can persist in the 

direction of high wavenumbers, so that either a transport along 

the gratient, or a collisionless dissipation in the explicit 

A-41 
form of a Landau damping can appear. With this distinction, 

an eddy viscosity or a rate'coefficient of Landau damping can 

be separately analyzed. In other circumstances, e.g. in the 

atmospheric turbulence with a stable stratification, a reverse 

cascade may appear, involving a transport counter to the gradi15 

ent. The reversal is associated with the "gap" phenomenon. 

An analytical expression of the cascade transfer of this kind 

will clarify the gap phenomenon and the concept of "negative 

viscosity". 

ACKNOWLEDGEMENT 

This work is performed under the grant NAS8-34622 from the 

National Acronautics and Space Administration. The hospitality 

of the Risa National Laboratory, Roskilde, Denmark, is deeply 

appreciated. 

A-42 
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- 

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A-43 
CHAPTER B: QUARTER TWO 

Kinetic Theory of Turbulent Transfer with Double Memory-Loss 

C.M. Ochen 

The Graduate Center and The City College of 

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

Abstract 

The transfer function governs the mode-coupling in 

strong turbulence. It is investigated here by means of a 

kinetic equation of turbulence, in consistency with the 

hydrodyxiamical system that describes the micro-dynamical 

state of turbulence. The equation of spectral balance is 

obtained by taking the moment of the scaled kinetic 

equation for the one-point distribution without the need 

of the two-point distribution. It is found that the transfer function is governed by two Lagrangian correlation 

functions and therefore by two memory-loss functions. 

The first correlation relates to two field fluctuations 

and finds the diffusivity as an integral operator. This 

perpetuates the memory to the second correlation which 

contains its own memory from the fluctuations of velocity 

and distribbtion of velocities. This correlation is known 

as the "wave-particle interaction" in plasmas. The competition between the two memory-loss functions and the 

cut-off of the memory form the essential basis for the 

closure of turbulence. We find two forms of transfer 

functions, for small and large scales, respectively. The 

B-1 
... 

first form belongs to the gradient transport and isolates 

an eddy viscosity. The second form belongs to a collisionless damping without gradient, and isolates a rate coefficient of collisionless damping. The transfer function 

for the large wavenumbers is applied to the derivation of 

the Kolmogoroff law of turbulence with the analytical determination of the numerical coefficient. 

I. INTRODUCTION 

The micro-dynamical state of fluid turbulence is 

usually described by a hydrodynamical system, for example 

the Navier-Stokes equation of motion. This equation is 

nonlinear in the velocity function and is inhomogeneous 

by having a driving field which may be the pressure 

gradient. On the other hand, in the kinetic representation 

the velocity is an independent variable and does not cause 

a nonlinearity, while the driving field forms a part of 

the advection and does not warrant an inhomogeneous 

equation. It is known that a homogeneous equation for the 

micro-dynamical state of turbulence can more explicitly 

give an analytical specification of the path perturbations, 

since the homogeneous equation contains a detailed differential operator of perturbations and therefore will yield 

by inversion an exact evolution operator. 

Unlike the kinetic theory of gases, the kinetic theory 

of the spectral structure of turbulence usually requires a

B-2 
two-point distribution function, since the spectral function is the Fourier transform of the two-point correlation function of velocity fluctuations. We see that such a 

procedurewould involve a hierarchy in the form of the 

Bogoliubov hierarchy . In order to confine ourselves to 

a single kinetic equation of one-point distribution and 

thus avoid the Bogoliubov hierarchy, weintroduce a procedure of scaling into a mean group and groups of macroand micro-fluctuations, repr-esenting three processes of 

transport: macro-evolution, micro-transport property, and 

relaxation8 ". These groups are in their increasing order 

of incoherence. The relaxation process, as the most random group, provides a memory-loss by path perturbations. 

Thus the closure lies in the memory-loss that is necessary for the transport property to reach its equilibrium, 

rather than-in closing the Reynolds stresses or other 

higher order transport functions. 

1-7 

Our first task is to derive a macro-kinetic equation, 

not through the closure of the Bogoliubov hierarchy, but 

by using a scaling procedure. The result is a macro-kinetic 

equation with a diffusivity serving as an integral operator, 

so that the memory can be extended to the distribution 

function that follows. Consequently, the kinetic equation 

takes the form of a mixture of the Fokker-Planck differential equation and the Boltzmann integral equation (Section 11) . 

B-3 
The kinetic equation of one-point macro-distribution 

can be transformed into a continuum representation by 

means of the moment method, and subsequently, upon multiplication by a macro-velocity, can derive an energy equation for the spectral balance of turbulence, bypassing 

the need for the two-point distribution function. Since 

the diffusivity acts as an operator, the distribution 

function that follows takes a Lagrangian form, and, together with the macro-velocity, will form a Lagrangian 

correlation. In this way, a transfer function is obtained, describing the mode-coupling across the spectrum, 

and is founded on two Lagrangian correlations: one from 

the field fluctuations for forming the diffusivity, and 

the other from the macro-distribution for describing the 

"particle-fluid'' interaction. Since each Lagrangian correlation presents a memory loss, the transfer function 

possesses two memory-loss functions, the analysis of 

which must be based on the Lagrangian-Eulerian transformation. The said transformation is in the phase-space and 

differs from an analogous for diffusion in the configuration space (Section 111). The competitive interplay of the two memory-loss functions provides us with a means of closure for the derivation of 

the transfer function in the form of a gradient transport 

or in the form of a collisionless damping.without a macroscopic gradient (Section IV). In this manner, we are able 

B-4 

. - . . . . . .. . . . . . . I 
to isolate an eddy viscosity and a rate of collisionless 

damping (Section V) . 

Finally, we test the applicability of our.kinetic 

method to the inertia turbulence. We derive analytically 

the Kolmogoroff law " and its numerical coefficient 

(Section VI) . 

11. THE SCALED KINETIC EQUATION OF TURBULENCE 

The hydrodynamical equations 

(a, + 6-v - vv2); = E, v-u = 0 , A 

"- - -4 re 

which describe the micro-dynamical state of turbulence, 

govern the total velocity as driven by a field E. This 

field may consist of the pressure gradient, the buoyancy, 

or any other fluctuating forces, including the random 

noise. 

- .* 

By the Reynolds decomposition, we can separate the 

total field 

into a mean field <E> I E and a fluctuation E. In an analogous way, we can decompose the field fluctuation 

L. c - 

B-5 
into a macro-field E(O) and a micro-field E', with unequal correlation times 

.* v 

in the increasing order of incoherence. The scaling can 

be made by means of the operators 

which select the. deterministic "global average", the macroand the micro-fluctuations, respectively. The macro-fluctuation.evolves in a medium possessing a transport property that is carried by the micro-field fluctuations. 

Finally, a relaxation is needed for the transport property 

to approach its equilibrium. This is made possible by a 

memory-loss. Thus the three transport processes of evolution, transport property, and relaxation form our main 

framework for the closure of turbulence. For the sake of 

convenience, we introduce an operator 

to represent an "accumalted macro-group" of fluctuating 

field 

and path 

B-6 
It is to be noted that the averages by A, A. are ensemble averages as specified by appropriate distribution 

functions. The groups as scaled by x, A (O'A', , only distinguish themselves by their degree of incoherence, and 

therefore may overlap in their spectral distributions 

with respect to frequency and wavenumber. 

The deterministic functions, which result from the 

global averages of the scaled fluctuating functions, inherit the same "rank" notations 

I 

of the originating scaled fluctuating functions. 

The hydrodynamical system of equations describing the 

micro-dynamical state of turbulence is usually complicated 

by the velocity function and the driving force which make 

the equations nonlinear and inhomogeneous. In order to 

avoid these difficulties, we adopt the kinetic approach. 

This transforms the velocity function into an independent 

variable, and incorporates the driving force into the advection in the phase space. This procedure renders the 

equations homogeneous and eliminates most nonlinearities, 

except the one representing the coupling between the driving 

field and the distribution function in the so-called "waveparticle" interaction. This last nonlinearity does not need 

B-7 
I 1111 1111 

our immediate attention. We may postpone after the closure 

and solve by an equation of state relating the fluctuations of field and velocity. These advantages will provide us with a simpler statistical basis of closure. 

As the point of departure, we write the following macrokinetic equation : 

8 

where ?(t,x,v) and f"'(t,x,v) are the mean and the macrodistributions, respectively, and 

*- " 

The eddy diffusivity is 

or, in the simplified form: 

where the Lagrangian representation of the micro-field is 

simply written as 

B-8 
and the Eulerian field at time t is written as 

since x(t) =x in E'(t,x(t)) = E'(t) . The propagator, or the 

Lagrangian evolution operator U(t,t-T) describes the evolution of the perturbed trajectory. The diffusivity may serve 

the role of an integral operator if it is written as D'{}. 

rr ,% .c L. + 

- 

111. THE LAGRANGIAN-EULERIAN TRANSFORMATION IN THE PHASE SPACE. 

THE LOSS AND CUTOFF OF THE MEMORY 

By means of the Fourier transformation in k-space, ?r 

the Lagrangian correlation can be written in the following 

form: 

It contains the spectral function 

giving the mean intensity 

where 

B-9 
is a factor of Fourier truncation within a length interval 

2X in d dimensions. The Fourier transformation has the 

Fourier kernels 

The frequency w = wk (k") is a function of k?' as determined 

by a certain dispersion relation, for example, reflecting 

k 

the interaction between internal gravity wave and turbulence . The perturbed path ( - T) forms a fluctuating 

orbit function e ik" ,,, . Since we can decompose the path 

15-17 

.y 

i into groups .y 

we find the corresponding "scaled orbit functions", or 

briefly, the "orbit components": 

On a separate occasion, we have developed a theory of 

transition probability for the orbital motion and determined the following correlation function to be 18-20. 

The first two factors on the right-hand side represent the 

factorization when the fluctuating functions 

B-10 
I 11.1 I 

are ,mutually independent I and h' is a correction factor 5 

when the hypothesis of independence does not hold. 

With the substitution of (10) into (61, we find that 

the Lagrangian correlation 

has, in the Fourier integral, a product of the spectral 

distribution <SI( k")) with the orbit function 

eThe components may form the groups 

representing the streaming by v,ii,z and the memory-loss 

by the macro- and micro-fluctuations, respectively. For 

- " 

the sake of convenience, we write the micro-component 

that incorporates the inter-dependence between the fluctuating intensity s' and the path fluctuations II'; also 

we write the accumulated macro-component as 

z Y 
The function M will be called the "memory-loss function". 

In particular, if the streaming predominates, we can 

degenerate (1 3a) into 

so that the Lagrangian correlation (12) is reduced to the 

Eulerian correlation by the relation: 

This is the well-known relation under the hypothesis of 

"frozen turbulence", and is valid in weak turbulence. In 

the general case where the turbulence is not frozen, we 

have the relationship 

between the two correlations by using W'I) as an integral 

operator. Note that 

E 

Q 

since M < -1. 
. 

In the derivation of a transport property or a transport function in equilibrium upon the r-integration of a 

Lagrangian correlation, it is obvious that the memory 

should not perpetuate indefinitely but should ultimately 

be cut off, as a requirement of the closure. We see that 

the memory function M consists of two components: The 

macro-component h (O) incorporates the strength and the decay of the memory. The micro-component h:, as belonging to 

the most random group at the tail of the memory-chain, has 

the function of cutting off the memory as the result of 

the diffusion by the micro-fluctuations. 

Recall that the diffusions byE- and E-fluctuations .y 

are governed by the dynamical equations of the orbital 

motions 

4 - 

A 

"LC 

4t I* (1 6a) 

and 

respectively. 

We perform the scalings and the ensemble averages by 

means of the scaled probabilities of transition 18,19 

The dynamical system (16b) gives the paths 

B-13 
and a macro-variance 

I 

belonging to the unmatured diffusion, or small-time diffusion. 

The mizro-variance can be written in the form 

belonging to the matured diffusion, or large-time diffusion with a transport coefficient 

Two forms of propagators U(T ,T - T') may be visualized, depending on the orbit evolution, whether it be perturbed by 

the u-fluctuations or by the E-fluctuations, giving an eddy diffusivity or an eddy viscosity, respectively. Alternatively, the micro-variance in the diffusion by g-fluctuations following (16b) is expected to deliver an eddy vis- 

'* IC 

cosity too, in an indirect way through D' Y and after the 

closure. For this reason, we shall assume a memory cut-off 

and a closure by (18a). A rough evaluation of K' could be 

made by relating it to the noise spectrum of the Brownian 

problem, or to the diffusion model (16a). But from a realistic viewpoint, K' must be related, in a self-consistent 

CI 

X 

X 

B-14 
way, to that same eddy viscosity which governs the mode 

transfer by small scale fluctuations (Section V). We adopt 

the latter viewpoint. 

The variances (17b) and (18a) determine the orbit components : 

with 

It is to be mentioned that, unlike h' which depends on 

the variance <a' ( - T)R' ( - TI> of one path R' ( - T) , the 

component 

M v CI 

contains a function 6' that depends on the micro-fluctuations of two paths, one is R' ( - T) and the other is w 

RL( - T) carrying the field intensity s', so that U 

N 

For small T, the two paths have still to grow, so khat 5' 

is small. For large T, the two paths are too far apart and 

B-15 
lose their correlation, so that 6' tends again to zero. 

It is for the intermediate T that 6' has a finite value, 

so that (20) may even become negative, yielding a negative portion of the Lagrangian correlation (15). 

Now we integrate the Lagrangian correlation with respect to T, to find the diffusivity in the phase space: 

This expression can also be used as an integral operator 

for integrations with respect to r and k". The memoryloss function M, as forming a part of M, governs the loss 

of memory for the diffusivity to approach its equilibrium. 

The time integral 

CI 

A 

will be called the relaxation time. 

IV. CASCADE TRANSFER WITH DOUBLE MEMORY-LOSS 

We take the moment of the kinetic equation (5) to 

obtain the following moment equation: 

with 

B-16 

.I,_. 
Upon multiplying (24) by ujO) and averaging, we obtain 

the energy equation: 

where the transport functions are as follows: 

production pC0& - {< lLT) E 1.; 

coupling 

transfer - (."? f0> 

r\rY 

dissipation 

The momentum equation and the energy equation are not 

purely hydrodynamical, since they contain the distribution 

function f (O? through J(.O) * and T'O). The function n. J'O) represents a collisionless interaction, and the transfer function T'O) represents a transfer between the two portions of 

the spectrum of macro- and micro-energies, respectively. 

This transfer originates from the "particle-fluid" interB-17 
action, as shown by the new Lagrangian correlation 

between the fluid velocity ui(O) and the distribution f (O) 

of the "particles". Thus the transfer function is more 

specifically 

from (25) (27) and (28). 

By applying the rule (121, we can transform the new 

Lagrangian correlation into 

with 

and get the transfer function in the form: 

This form can be made more explicit by inserting the expression (22) for D' , as follows: Y 

c 

B-18 
Here we have introduced for the sake of convenience: 

or, more briefly, 

where the dependence of i(k"), i(k') on -T,v, and of Y .c .c 

,(p!O) (k') on v may be considered to be understood.' l. cy ... By expanding the multiple derivatives a ,ar, we have: 

j 

with 

The second term on the right-hand side of (34) gives, 

after two successive integrations by parts: 

B-19 
with the obvious condition 

so that, after a substitution, (34) becomes 

with 

As far as their real values are concerned, 

N(O'(& h"), q:"[k'), i(k") and i(k') are even functions of I=" y 5 

U * 

k" and k' , so that the derivatives CI - 

will generate odd functions, and will not contribute to 

the integrations with respect to k" and k' in the infinite 

domain. Therefore they can be deleted from the expression 

- 4/ 

B-20 
of N'O) for the purpose of calculating T'O) from (32). jr 

The result 

is in the form of the moment of (pjo) as driven by the 

differentials 3.3 of the orbit functions. Hence the transI= 

fer function (32) takes the form 

and is seen to be controlled by two memory-loss functions 

M(k") and M(k') as imbedded in k(k") and i(k') . w 4 - r* 

V. EDDY VISCOSITY AND COLLISIONLESS DAMPING 

It is to be noted that the v-dependent orbit functions .. have their slowly varying component hi and their rapidly 

varying component h . Thus the differential will be 

mainly contributed by the differential ahv of the rapidly 

varying component in the manner 

V 

V 

We notice that the two orbit functions 

B-21 
compete in their roles of loss of memory for the approach 

of the transfer function (40) to equilibrium, under the 

scaling conditions 

Thus it suffices to select the function that is most effective in this role. On this basis, we shall calcu-late 

the asymptotic expressions by considering small and large 

scales. Since the path amplitudes E(o) ,E' and the transport coefficients Ki0) ,Ki are separated by k, the small 

and large values of k correspond to : 

3 S 

4'"' > A? ' 

and 

po)( 1' , 

respectively. 

Case (a). Large k 

For large k, or for even larger k", such that 

(43a) 

k u >> k ' 

(43b) 

(44 1 

in conjunction with (42) and (43a) , we can attribute the 

dominant role of memory-loss to the function 
which is independent of v, while leaving the role of 

streaming to the v-dependent orbit-function 

.* 

& 

The independence of (45) on v is seen from (18) and (19), 

after- having assumed h' -N 1 or <'EO . This approximation 

is legitimate, because in the integration (40) with respect to T, the role of h' in c( - T,~;V) is seen from 

(20) and (21) to be taken over by h' and ho for small and 

U 

5- 

5 " 

large T, respectively. Hence we reduce (39) and (40) 

and 

respectively'. Here 

is a modulation function having a streaming by hv and a 

memory-loss by hoh' . 

B-23 
1n.strong turbulence, the memory-loss predominates 

over the streaming, and renders the modulation function 

to become independent of v as in c 

Thus we obtain an eddy viscosity 

and transform (48) into 

The integrations herein involved are: 

and 

B-24 
Hence, by substitution into (52), we obtain the transfer 

func.tion 

7" 5 k 

in the form of the product of the eddy viscosit! 

(55) 

'Y 

the vorticity function R'O). Here, by isolating an eddy rj 

viscosity, we have found a cascade transfer of the gradient type. 

Case (b). Small k 

The small values of k imply an even smaller k'. This 

extremely small k' , as compared to k", suppresses all 

roles of h( - .r,k',v) before i( - T ,P:v), rendering 

" CN 

Thus we transform (39) into 

The factor k'!k'' emphasizes large values of k" so that 

Jr 

we can write 

as an approximation. When this function is substituted 

B-25 
into (57) and subsequently integrated with respect to T, 

following (401, the contribution from hv as a streaming 

is again negligible as compared to the memory-loss for 

large k" in strong turbulence. It follows a modulation 

function 

for the control of the transfer function (40) , yielding: 

Here the two integrals with respect to k' and kt' become 

separated into an energy 

F- " 

by (56) and (30b) , and a frequency 

called the rate of damping in the collisionless damping 

of the energy. Hence the transfer function becomes: 

B-26 
Since the transfer of energy is not of the gradient 

type, we cannot isolate an eddy vicosity as was the case 

of large k in (55). 

We may consider 

as integral operators which emphasize large and small 

values of k by their respective integrations 

and write 

In this manner, the two transfer functions (55) and (62) 

become 

and 

For large and small k, the operators (63) pick up 

B-27 
respectively, from the sum k' kt + kNk" as appearing in rj rj 

of the general transfer function (40). Thus the formula 

(40) is a general transfer function with the special forms 

(65) and (66). 

VI. DERIVATION OF THE KOLMOGOROFF LAW FROM THE SCALED KINETIC 

EQUATION 

Although the Kolmogoroff law' of the inertia turbu1enc.e has been derived by many investigators using diverse 

methods as seen in a recent review2' of the diagram technique" in conjunction with the direct interaction approxirnati~n~~, it is for the first time that the analytical 

derivation of the Kolmogoroff law and its numerical coefficient is obtained from the scaled kinetic equation of 

turbulence. Our kinetic method is generally valid for inertia and non-inertia turbulence. 

We assume an isotropic and homogeneous turbulence, 

and that the field 

E 3- j^Pp c 

comes from the pressure gradient Vp in an incompressible 

fluid of constant density p. The shear and the buoyancy 

are absent. The condition of incompressibility, as applied 

" 

B-28 
to the Navier-Stokes equation (1) gives the relation 

between E and u, and therefore also between their spectral 

distributions S (k) and F(k) . We have 

.c .5 

and find the relations (69 1 

where R'O) E (O) is the vorticity function, from (54), and 

the numerical coefficient is determined td be19t20 

Ri i 

The details of the calculation are omitted. 

In the inertia subrange, the energy balance in the 

spectral form is 13,14 

describing a constant cascade of energy transfer across the 

spectrum at the rate of energy dissipation E. Now the transB-29 
. . . 

fer function, as found in (551, can be written in the isotropic form: 

where 

is the trace of the eddy viscosity tensor (511, and the 

modulation function is given by(50), with wk= 0. 

From (9) , (18a) and (1 9) , we rewrite the modulation 

function (50) in the form: 

It contains the decay of the memory at a rate 

and the cut-off of the memory at a time uk-' such that 

The integral is evaluated by an interpolation, giving a 

modulation function 

B-30 
Hence we determine the eddy viscosity by the following integral equation: 

When the spectrum S(k) is converted into the spectrum 

F(k) by the relation (70), we find the solution of (731, 

giving the spectra 

of velocity, field and pressure fluctuations, respectively. 

The details of calculations are omitted. 

VI I. DISCUSSION 'P 

The micro-dynamical state of turbulence can be described by a nonlinear and stochastic hydrodynamical system 

which may be homogeneous or inhomogeneous. The Navier-Stokes 

system of equations is inhomogeneous, as in (l), or briefly 

as 

B-31 
The inhomogeneous system in the physical space may be converted into a homogeneous equation in the phase space, as 

Yr 

called the "master 

homogeneous system 

equation". It is equivalent to the in- 

(8 0) , provided 

Other homogeneous systems may be: 

(i) the Burgers equation 

(ii) the Korteweg-DeVries equation - 

(iii) the equation of geostrophic turbulence 

B-32 
From the conceptual viewpoint, being given the'microdynamical state of turbulence, the question may arise as 

to what treatment one should choose: the kinetic treatment 

or the hydrodynamic treatment. Since a homogeneous partial 

differential equation can describe the exact orbit dynamics 

by the differential perturbation operator L alone, there 

is an essential advantage to keep a homogeneous system, 

either in the hydrodynamic representation (83) or in the 

. 

kinetic representation (81). For this reason, we have 

chosen the kinetic method which, moreover, supresses all 

nonlinearities caused by the velocity since the velocity 

is an independent variable here. The only nonlinearity 

that subsists is from g=zfi in (81). This entails a Reynolds stress in the phase space, the evolution of which 

requires the knowledge of the Lagrangian correlation of 

E-field fluctuations and a theory of transport. The ap- A 

proach to equilibrium and the loss of memory constitute 

the basis for the closure of turbulence. 

There are two Lagrangian correlations involved: one

is the correlation of 5-field fluctuations as stated above, 

and the other is the correlation between the velocity fluctuations and the fluctuating distribution of velocities. 

The latter arises because our kinetic equation (5) of turbulence has a long memory. Since each Lagrangian correlation entails a memory, our kinetic theory of transport 

finds two memory-loss functions. Their competitive roles 

B-33 
I 

and their cut-off are the main issues of our analysis. 

The difficulties are minimized by introducing a scaling 

procedure which dispenses from our involvement wi.th the 

two-point distribution function. Then the scaled probabilities of transition determine the scaled orbit functions. The scaling permits the distinction between the 

macro- and micro-variances of path fluctuations which are 

associated with the unmatured (i.e. small time) and the 

matured (i.e. large time) properties of transport. Otherwise, the unscaled variance would appear as an external 

parameter. 

Our results indicate that the transfer function takes 

a gradient form for small scale turbulence, isolating an 

eddy viscosity, and a form of collisionless damping without gradient for large scale turbulence, isolating a rate 

of collisionless damping. 

We can write the equation of energy (26) in the form: 

with the transport functions: production P(O) , loss B , 

transfer T(O) and dissipation E'O). In statistical equilibrium the left-hand side is independent of k, so that, 

after a differentiation with respect to k, we have the 

spectral balance 

(0) 

B-34 
for scales larger than that of the inertia turbulence. 

Note that E ceases to be the governing parameter here. 

This means that 

) 0 in a net production (86a) 

- "(O) < 0 in a net loss. 

The change of signs in (86a) and (86b) refers to a direct 

cascade (i.e. a cascade transfer toward large wavenumbers) 

and to an inverse cascade (i.e. a cascade transfer toward 

small wavenumbers), respectively. The transfer function in 

the form of collisionless damping is preferred for the description of the inverse cascade. The arbitrary use of the 

transfer function in the form of gradient transport would 

result in a negative spectrum. 

ACKNOWLEDGEMENTS. This work was sponsored by a grant from 

the National Aeronautics and Space Administration. 

B-35 
REFERENCES 

1. A.S. Monin, Dokl. Akad. Nauk SSSR 177, 1036 (1967) 

[SOv. Phys. Doklady - 12, 1113 (1968m 

2. A.S. Monin, Priklad. Mat. Mekh. 31, 1057 (1967) 

[Appl. Math. and Mech. - 31, 1057 (1967)l. 

3. T.S. Lundgren, Phys. Fluids - 10, 969 ('1967). 

4. T.S. Lundgren, Phys. Fluids - 12, 485 (1969). 

5. F.R. Ulinich and B. Ya. Lyubimov, Zh. Eksp. Teor. 

Fiz. - 55, 951 (1968) [Sov. Phys. JETP - 28, 494 (196911. 

6. V.M. Ievlev, Mezh. Zhid. Gaza, No. 1 (1970). 

7. V.M. Ievlev, Dokl. Akad. Nauk SSSR 208, 1044 (1973) 

[Sov. Phys. Doklady - 18, 117 (1973)l. 

8. C.M. Tchen, Comptes Rendus Acad. Sci. Paris 286A, 

605 (1978). 

9. C.M. Tchen, "Kinetic Basis of Cascade Transfer in 

Turbulence", (manuscript) . 

10. T.S. Lundgren and Y.B. Pointin, Phys. Fluids 19, 

355 (1976). 

- 

11. J. Wefnstock, Phys. Fluids - 21, 887 (1978). 

12. J. Weinstock, Phys. Fluids - 19, 1702 (1976). 

13. A.N. Kolmogoroff, Compt. Rendus Acad. Sci. U.R.S.S. 

- 30, 301 (1941). 

14. W. Heisenberg, Z. Physik - 124, 628 (1948). 

15. J. Weinstock, J. Atmos. Sci. - 37, 1542 (1980). 

16. J. Weinstock, J. Atmos. Sci. - 35, 634 (1978). 

17. J. Weinstock, J. Geophys. Res. - 81, No.4, 633 (1976). 

18. C.M. Tchen, Versl. Ned. Akad. Wetensch., Afd. Natuurkunde, Amsterdam, - 53, 400 (1944). 

19. C.M. Tchen, "A Probability Theory of Retrograde 

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Applications, Greece, 1981. 

B-36 
20. C.M. Tchen, "A Kinetic Method of Turbulence". Manuscript accepted for publication in Proc. Int. Conf. 

of Fluid Dynamics and Applications, Greece, 1981. 

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(1965). 

- - 
CHAPTER C: QUARTER THREE 

Group-Scaling Theory for the Enstrophy Transfer 

in Two-Dimensional Turbulence 

C.M. Tchen 

The Graduate Center and The City College of 

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

ABSTRACT 

Since the detailed interactions between the individual modes contain too many minute details unnecessary for a statistical theory, we scale the modes into 

groups which can more easily decipher the governing 

transport processes and the statistical characteristics. 

An equation of vorticity transport describes the microdynamical state of two-dimensional, isotropic and homogeneous, geostrophic turbulence. By the group-scaling, 

we derive the equation of evolution of the macro-vorticity in the form of the Fokker-Planck equation with 

memory. The eddy diffusivity and the enstrophy transfer relax to equilibrium through the memory-loss by turbulent dispersions. The memory-loss is analyzed by developing a theory of probability of retrograde transition in the scaled form. We find &at the eddy diffusivity contains two time scales, characteristical of 

c-1 
the small-time scattering and the large-time diffusion. 

In the special cases, our formula can be degenerated to 

an empirical form suggested by Heisenberg when the scattering prevails, and to a form derived earlier by Tchen 

when the diffusion is dominant. In the inertia subrange, 

as governed by the enstrophy cascade, the spectral law 

FU = CE is derived with a numerical coefficient 

c = 2.59. 

2/3k-3 

5 

c-2 
I. INTRODUCTION 

The micro-dynamical state of turbulence can be described by the equations in the following forms: 

(a) a non-conservative (or inhomogeneous) equation 

in the physical space, 

(b) a conservative (or homogeneous) equation in the 

physical space,, 

(c) a conservative equation in the phase space. 

The form (a) contains a stirring force, which may be 

the pressure gradient, the buoyancy, or the random noise. 

The Navier-Stokes equation falls under the form (a). The 

two-dimensional geostrophic turbulence also belong to

the form (a), if the turbulence is driven by a random 

vorticity source, and will fall into the form (b) if such 

a source is absent. The one-dimensional Burgers equation 

and the Korteweg-DeVries equation also falls into the 

form (b). 

It is to be noted that the non-conservative equation 

(a) in the physical space can be converted into a conservative equation (c) in the phase space by regarding 

the stirring field as an advection. Therefore the problems of the conservative system should be the main objective of a statistical theory. With this emphasis in 

mind, we develop a statistical theory, using the scaling 

procedure to study the mode-couplings and the transport 

property, and relying upon the memory-loss to obtain the 

c-3 
, 

closure of turbulence. A kinetic theory' on the basis 

(c) has been developed by Tchen for the Navier-Stokes 

turbulence (a). Presently, we apply the scaling method 

to the two-dimensional, isotropic and homogeneous, geostrophic turbulence. 

In view of the many applications to the meso-scale 

flows in the atmosphere and the availability of numerical computations , the problem of geostrophic turbulence has attracted many investigators , see a review 

by Rhines' '. The geostrophic turbulence is characterized 

by an enstrophy transfer from large scales toward small 

2-7 

8-10 

scales, yielding a spectrum of velocity fluctuations 

following the k-3 law. Most theories analyzed the triad 

interactions 12-18 

The present theory simplifies the interactions by 

scaling a fluctuation into three groups, representing the 

evolution, the eddy transport and the relaxation. Our 

purpose is to derive analytically the enstrophy transfer 

function, the eddy transport coefficient and the spectral 

distribution. Upon a generalization to a kinetic method 

(c), as is equivalent to the non-conservative microdynamics (a.), we expect to find: 

(i) the spectral law k -5'3 in a direct cascade of 

energy transfer at large k, 

(ii) the spectral law k -5'3 in, an inverse cascade 

of energy transfer at small k, 

c-4 
(iii) the spectral law k-4 from an enstrophy transfer in a geostrophic turbulence that is driven by randomly distributed vorticity sources. These results will 

be reported on a separate occasion. 

11. SCALED VORTICITY EQUATION OF TURBULENCE 

The two-dimensional incompressible flow in the horizontal plane is governed by the following equations of 

motion: 

and 

v.LL =o A 

YHere 3 is the velocity, - 

is the scalar vorticity, 

is the differential operator, and v is the kinematic viscosity. By the condition (lb), the pressure gradient, 

the vertical buoyancy force, and the vertical Coriolis 

parameter are eliminated. The system (1) describes the 

c-5 
micro-dynamical state of geostrophic turbulence. 

The scaling may be made by means of the operator 

As0 

of global average and the fluctuation operator 

(4a) 

giving the deviation from the global averaqe. The fluctuation operator 

is subsequently divided into the operator A'O) of macrofluctuation and the operator A' of micro-fluctuation. 

For the sake of convenience, we introduce an operator 

giving an accumulated macro-group. Thus we have: 

and 

C-6 
The macro-fluctuation and the micro-fluctuation can instantaneously overlap in their wavenumber components, but 

their statistical functions are distinctly separated by a 

wavenumber variable k, and by their durations of correlation 

in the increasing degrees of incoherence. 

By applying A'O) and A' to (la), .we obtain the following scaled system: 

By a formal integration of (7b) and a substitution 

into (7a), we eliminate r,' to obtain the equation of 

evolution of the macro-vorticity: 

where 

is the eddy diffusivity, and U(t,t-T) is the propagator, 

c-7 
or the operator of evolution along an orbit that is perturbed according to (3). Without altering the value of 

the integral, the upper limit may be put to infinity, by 

the scale separation (61, giving a matured (i.e. largetime) diffusivity. It is to be noted that the diffusivity enters as an integral operatorK'0.in (81, so that 

this scaled vorticity equation takes the form of an integrodifferential equation, and not the customary Fokker-Planck 

differential equation. 

% 

When we multiply the vorticity equation (8) by 5 (O) 

and average, we get 

Here we have retained the kinematic viscosity in the differential operator L. 

In a homogeneous turbulence, the second term on the 

left-hand side vanishes, and the last term on the righthand side becomes 

C-8 
In an analogous way, we can write 

and find the first term on the right-hand side of (10) to 

be 

Here the diffusivity remains to be an integral operator 

with its time integration extended to the Lagrangian correlation 

The diffusivity is written in the form of a trace. The 

function E(O) is the rate of molecular dissipation of the 

enstrophy f<< (O)*>. The function T 5 

(O) represents the modecoupling and is called the "enstrophy transfer". 

5 

In terms of,the dissipation function and the transfer 

function, the equation of spectral balance (10) is reduced 

to the form: 

It represents the time decay of the mean vorticity or 

enstrophy 

c-9 
which has a spectral distribution F (k) of 5-fluctuations. 5 

111. TRANSPORT PHENOMENA IN TURBULENCE 

A. Lagrangian-Eulerian transformation in the scaled form 

The Lagrangian velocity follows a trajectory that is 

governed by the dynamical equation: 

or the orbit equation 

where the perturbed path 

'consists of a mean path Z(--T) and a path fluctuation - 

JL(-T). In the following calculations of the path perturbations, we shall neglect the effect of the kinematic 

.c 

viscosity v as compared to the larger effect from the 

eddy motions. 

By means of the Fourier transformation, we can write 

the Lagrangian and Eulerian correlations in the following 

c-10 
and 

respectively. Here ~'(k'') is a fluctuating intensity in 

the k-space, such that 

fu - 

- 

(4'(k1Q 5 7 (u'(k'7 &/(-hi y> ," .b - .*e 4r- 

(1 7a) 

is the spectral function, and 

is the factor of truncation, when the Fourier transformation is truncated within a length interval 2X in two 

dimensions, i.e. d= 2. The Fourier transformation has a 

c-11 
Fourier kernel 

with a frequency uk = uk(k“) that arises from a dispersion 

relation, e.g. as relating to an internal gravity wave. 

h 

19 

The integrations in (15)-(17) and in the following 

are understood to extend from -m to 03. 

The correlation in the integrand of (15) can be 

written as 

consisting of a mutually independent part that can be 

factorized as 

v 

T 

and a correction 

c-12 
accounting 

correction 

tering and 

for the mutual dependence. We find that this 

is not too important for the small-time scatthe large-time diffusion that are of interest 

in our problem. The mutually independent part contains 

an average 

that is calculated by the probabilities of transition in 

the scalings by A. and x. The derivation of the probability functions and the calculations of the averagere 

presented in Appendix A, finding 

A 

On the right-hand side appear the components of the orbit 

functions, or "orbit components". The mean path gives the 

component 

The fluctuations give the components 

C-13 

I 
in the unmatured and matured dispersions, respectively, 

of tibe scales m 

-1 

(O)-' and w6 , such that 

The details of the calculation are given in (A20)- (A24) 

of Appendix . The function FU(k") is the spectral distribution of velocity fluctuations, as described by (A19) 

in Appendix. 

If we lump the orbit components caused by the fluctuations into the function 

called the "memory-loss function", we can write the Lagrangian-Eulerian relationship in the form: 

This is a nonlinear 

contains W' through #uL * 

integral equation for WL (TI, since M 

h' , w' and (9). Note that the EulerY 

c 

K 

ian correlation is governed by the mean path Z(-T), while 

the Lagrangian correlation is governed by the mean path 

- 

C-14 
and the path fluctuation a(-.r). 

N 

6. Relaxation time 

Upon substituting (27) into (91, we find the diffusivity tensor and its trace in the form: 

It is seen that the diffusivity approaches its equilibrium 

in a relaxation time 

or, approximately, 

The approximation is based upon the interpolation. 

C-15 
C. Diffusivity 

The equation (28b) which determines the diffusivity 

k' is in reality an integral equation since the relaxation 

time G (k'') is also a function of F' through w' by (25b). 

As an approximation, we calculate the asymptotic 

values of K' by considering the following two special 

cases: 

K K' 

case (i) If GK is controlled by rn(O), we find 

with CK= 6 in two and three dimensions. 

case (ii) If GK is controlled by u' we find kt 

As an interpolation, we write: 

It is degenerated into 

(33) 

C-16 
IV. DOUBLE MEMORY-LOSS 

The first memory-loss function M(-T,~") appears in the 

Lagrangian correlation (27) which produces the diffusivity 

(32). The second memory-loss function arises from the 

second Lagrangian correlation (lld), with the same diffusivity in its role of an integral operator as appearing 

in the transfer function (llc). 

In order to derive the structure of the transfer function (llc) in isotropic turbulence, we first write the 

second correlation function as 

in analogy with (27). Here 

is the spectral intensity of $-fluctuations, such that the 

intensity 

41 

can be expressed in terms of F (k') by 5 

C-17 
Use is made of (lla). Note that the relationship 

between the two spectra follows directly from the definition (2). 

Now we apply the integral operator fC'{ } to (34) and f 

" 

write the transfer function in the following form 

according to the definition (llc), or in the alternate 

form 

when K'{ } is written out fully by means of (28b). It is 

seen that the transfer function approaches its equilibrium 

at a new relaxation time 

with two memories: 

C-18 
V. TRANSFER FUNCTION 

By the use of (13) and (17), the transfer function 

(37b) can be rewritten as: 

where, by definition (381, the relaxation time GT(k:k") 

sums up the contributions from the two competing memoryloss functions (39) under the conditions 

with the properties 

so that the role of memory-loss is taken over by M(-r,k"), 

i.e. 

Thus we can reduce (38) to 

G,k k '*) $ b '9 , (44) 

and separate the two integrations with respect to the wave-. 

numbers in (401, obtaining the degenerate form 

c-19 

I 
more briefly, 

in the notations (28b) and (36a). 

VI. INERTIA SPECTRUM 

In terms of the spectra FU(k) and F (k), we can write 5 

the spectral balance (12) in the form: 

Since the differential vanishes, i.e. 

in statistical equilibrium, the integral form should read 

where 

is a constant rate of dissipation of the enstrophy. A 

c-20 
fluctuating E would enhance the intermittency which we 

shall not consider at present. 

5 

The inertia subrange is described by the spectral 

balance 

by omitting the effect of molecular dissipation in the 

left member of (48). 

When we use the general formula (32) for k; and observe the relationship (36b) between the two fluctuations 

u and 5, we determine the spectra from (50). The results 

are : 

CI 

VII. DISCUSSIONS AND CONCLUSIONS 

The problem of turbulence starts with the nonlinear 

equation describing the micro-dynamical state of turbulence. This equation contains too many minute details 

which are too complicated and also unnecessary for the 

statistical theory of turbulence. Past theories either 

followed the kinetic method and closed at a high order 

distribution function, or the hydrodynamical method and 

c-21 
closed at a high order correlation. They analyzed the 

detailed interactions between individual modes. The 

present theory begins with the coarse-graining procedure 

by distinguishing between the fluc.tuations of the macrogroup and the fluctuations of the micro-group. It transforms the equation describing the micro-dynamical state 

into a scaled equation of turbulence, in which the macrofluctuations evolve in a medium of more random eddies belonging to the micro-group. In this way it analyzes the 

statistical couplings between groups of modes instead of 

the detailed couplings between individual modes. For the 

approach of the transport property to equilibrium, as a 

closure of turbulence, the memory-loss by turbulent dispersions is essential. We find a scattering by the smalltime dispersion and a diffusion by the large-time dispersion in the process of memory-loss and for the derivation 

of the eddy diffusivity. Our formula (32) of eddy diffusivity can be degenerated to the Heisenberg form (30) 

when the scattering prevails2', and to the form (31) when 

the diffusion is dominant. The latter form has been used 

as an empirical formula in modeling the atmospheric turbulence by Gisina21; it has also been derived analytically 

by Tchen22 from a cascade theory. The transfer of enstrophy is found to cascade down the spectrum toward the 

large wavenumbers. The spectral law F (k) =CE for 

the velocity fluctuations and the numerical coefficient 

2/3k-3 

5 5 

c-22 
e 

c=2.59, as derived in (51), agrees with the results from 

the triad interactions by Xraichnan! The nonstationarity 

of E: will cause a logarithmic factor due to the intermittency of turbulence, and will be considered on a sep5 

arate occasion. 

C-23 
APPENDIX. SCALING THEORY OF TRANSITION 

We distinguish between a direct transition and a .I 

retrograde transition of paths L?(T) and i(-~) in the positive time T and the negative time -T, respectively. The 

probabilities of transition have been investigated by 

T~hen*~, and were found to be governed by the Fokker- 

.y .c 

Planck equations. The derivations were based on a phenomenological ground. Here we shall re-examine the retrograde transition by a scaling procedure and on a basis 

that is consistent with the specified micro-dynamical state 

of turbulence, i.e. the orbit equation: 

1 

see (1 4b). 

The micro-dynamical state of the orbit can be described by the master equation 

upon the introduction of the &-function 

and the differential operator 

C-24 
I 

We may consider the independent variable 2 to be a random 

variable, so that P becomes a distribution function which 

- A 

satisfies the normalization condition 

and gives the moment 

It is easy to verify that the moment of (A2) reproduces 

the orbit equation (Al) identically. 

Now we apply the scaling operators A. and A' to (A2) 

and find the following coupled system of equations: 

.. We integrate (A6b) to get 

0 -LI 

and 

C-25 
where 

is a diffusivity and Ug is a propagator or evolution 

operator in the retrograde transition. The propagator 

U, may be considered as the inversion of the differential x. 

operator LR. 

When we substitute (A8) into (A6a), we derive the 

equation of the retrograde transition of the macro-group 

in the form of an integro-differential equation. 

In an analogous manner, we derive the equation of 

transition 

Here the governing diffusivity is 

as caused by the velocity fluctuations except u'. 

v 

C-26 

,...."". I -...-.,, _.... ......._ ... - 
We make a Fourier transformation of (A10) and (AI 1) 

by the formulas 

where K is an independent variable in the Fourier space, 

and d is the number of dimensions. We get 

4 

The solutions are: 

(A1 4a) 

(A1 4b) 

(A1 5a) 

with the following components of the orbit function: 

C-27 
(A1 6a) 

(A1 6b) 

(A1 6c) 

I 

Note that K is the diffusivity in a matured (i.e. c VI 

large-time T) dispersion, and K'O) is the diffusivity in 

an unmatured (i.e. small-time T) dispersion. In the 

latter case the orbit belongs to a free flight, so that 

e 

is locally stationary, and the trace in isotropic turbulence becomes 

the integration over a region of radius K gives 

$dv'= % -r kt in 2 dimensions 

= a m3 in 3 dimensions J 

C-28 
IHere x is the factor of Fourier truncation. In terms of 

the spectral function FU(~), such 

we find 

that 

in 2 dimensions (Alga) 

in 3 dimensions (A19b) 

in d dimensions (AZO) 

In conclusion, with the use of (A20) and the introduction of the frequencies m(O) and w;, such that 

and 

'"K 

= &/I?! K'z K2K' 

-- .v 

we find the orbit components to be 

(A2 1 b) 

C-29 
(A22b) 

Now the solutions (A151 can be applied to calculate 

and, subsequently 

- itc .j(-., FAo e rc - 

by the use of the formulas (A13). We obtain 

and hence 

(A23b) 

C-30 
ACKNOWLEDGEMENT 

This work was supported by the National Aeronautics 

and Space Administration. 

REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

C.M. Tchen, ''Kinetic approach to incompressible 

turbulence", C.R. Acad. Sci. Paris 286A, 605-607, 

(1978); "Kinetic approach to compressible turbulence'', Proc. Int. Symposium on Rarefied Gas 

Dynamics, Cannes. Published by Commissariat a 

1'Energie Atomique, Paris, 1979, ed. by R. Campargue, 

pp. 105-114. 

J.G. Charney, "Geostrophic turbulence". J. Atmos. 

Sci. - 28, 1087-1095 11971). 

R. Fjartoft, ''On the changes in the spectral distribution of kineticenergy for two-dimensional nondivergent flow". Tellus - 5, 225-230 (1953). 

D.K. Lilly, "Numerical simulation of two-dimensional 

turbulence". Phys. Fluids, Suppl. 11, 240-249 (1969) . 

D.K. Lilly, "Numerical simulation of two-dimensional 

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

(1969). 

- 

P.D. Thompson, "The equilibrium energy spectrum of 

randomly forced two-dimensional turbulence". J. Atmos. 

Sci. - 30, 1593-1598 (1973). 

K.S. Gage, "Evidence for a k-5'3 law inertial range 

in mesoscale two-dimensional turbulence". J. Atmos. 

Sci. - 36, 1950-1954 (1979). 

Ye.A. Novikov, "Spectral inequalities for two-dimensional turbulence". Izvestiya, Atmospheric and Oceanic 

Physics - 14, 474-476 (1978). 

C.E. Leith, "Diffusion approximation for two-dimensional turbulence". Phys. Fluids - 11, 671-673 (1968). 

C-31 
10. P.G. Saffman, "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). 

11. P.B. Rhines, "Geostrophic turbulence". Annual Review 

of Fluid Mechanics, vol. 11, Annual Reviews, Inc., 

401-441 (979). 

12. R.H. Kraichnan, "Inertia ranges in two-dimensional 

turbulence". Phys. Rev. - 10, 1417-1423 (1967). 

13. R.H. Kraichnan, "An almost-Markovian Galilean-invariant 

turbulence model. J. Fluid Mech. - 47,513-524 (1971). 

14. R.H. Kraichnan, "Inertia range transfer in two- and 

three-dimensional turbulence". J. Fluid Mech. 47, 

525-535 (1971). - 

15. J.R. Herring, "Statistical theory of quasi-geostrophic 

turbulence". J. Atmos. Sci. - 37, 969-977 (1980). 

16. A. Pouquet, M. Lesieur and J.C. Andre, "High Reynolds 

number simulation of two-dimensional turbulence using 

a stochastic model". J. Fluid Mech. - 72, 305-319 (1975). 

17. R. Salmon, G. Holloway and M.C. Hendershott, "The 

equilibrium statistical mechanics of simple quasigeostrophic models". J. Fluid Mech. - 75, 691-703 (1978). 

18. C. Basdevant, M. Lesieur and R. Sadourny, "Subgridscale modeling of enstrophy transfer in two-dimensional turbulence". J. Atmos. Sci. - 35, 1028-1042 (1978). 

19. J. Weinstock, "On the theory of turbulence in the 

buoyancy subrange of stably stratified flows". J. Atmos. 

Sci. - 35, 634-649 (1978). 

20. W. Heisenberg, "Zur statistische theorie der turbulenz". 

Zeit. Physik - 124, 628-657 (1948). 

21. F.A. Gisina, "Spectral characteristics of turbulence 

in a thermally stratified atmosphere". Izv. Atmos. 

Oceanic Phys. 5, no. 3, 247-257 (1969). [English translation: Bull. (Izv.) Acad. Sci. USSR, Atmos. Oceanic 

- 

Phys. - 5, 137-141 (1969). 

22. C.M. Tchen, "Repeated cascade theory of homogeneous 

turbulence". Phys. Fluids - 16, 13-30 (1 973) . 

23. C.M. Tchen, "Some mathematical relations which play a 

role in diffusion problems". Verslagen Ned. Akad. 

Wetensch. Afd. Natuurkunde - 53, 400-410 (1944). 

C-32 
CHAPTER D: QUARTER FOUR 

A Group-Kinetic Theory of Turbulent Collective Collision 

C . M. Tchen 

The City College and The Graduate School of The City University 

of New York, N. Y. 10031, U .S .A. 

AND J. H. Misguich 

Association Euratom-CEA, DLFartement de Physique du Plasma 

Centre d'Etudes Nuclgaires, 92260 Fontenay-aw-Roses, France 

ABSTRACT 

For the description of the microdynamical state of turbulence, a 

Liouville equation is taken that is equivalent to the basic hydrodynamical 

system of equations. The equation has the advantage of being homogeneous 

and contains less nonlinear terms. Our main objective is the derivation of 

the kinetic equation of turbulence which has a memory in the turbulent 

collision integral. We consider the basic pair-interaction, and the 

interaction between a fluctuation and the organized cluster of other 

fluctuations in the collection system, called the rcultiple interaction. 

By a group-scaling procedure, a fluctuation is decomposed into three groups 

to represent the three coupled transport processes of evolution, transport 

coefficient, and relaxation. By exploiting the property of quasi-stationarity 

at the different levels of degradation of coherence of the groups, we 

develop a transport theory with the closure by the memory-loss. The kinetic 

equation of the scaled singlet distribution is capable of investigating 

the spectrum of turbulence without the need of the knowledge of the pair 

distribution. 

The exact propagator describes the detailed trajectory in the phase 

space, and is fundamental to the Lagrangian-Eulerian transformation. We 

calculate the propagator and Its scaled groups by means of a probability of 

retrograde transition. Thus our derivation of the kinetic equation of the 

distribution involves a parallel development of the kinetic equations of 

the propagator and the transition probability. In this way, we can avoid the 
assumptions of independence and normality. 

Our result shows that the multiple interaction contributes to a 

shielding and an enhancement of the collision in weak turbulence and strong 

turbulence, respectively. The weak turbulence is dominated by the wave 

resonance, and the strong turbulence is dominated bythe diffusion. 

D-2 
I. INTPODUCTION 

The microdynamical state of turbulence can be described by the equation 

of evolution of the distribution function f(t,x,v) in the form: -" 

The differential operator 

the ballistic orbit that is represented by Jt + j. in the phase space " 

of position x and velocity v . * 

In plasma turbulence, (la) is called the Vl.asov equation, when E is the A 

- 

electrostatic field multiplied by the ratio of electric charge to mass. 

The equation can also be used to treat the fluid turbulence, when E 

A 

.. 

represents the pressure-gradient or other additional hydrodynamical forces. 

We decompose the, total distribution function 

A CI 

f=?+f 

into an average distribution 

by means of the averaging operator A , and a fluctuation 

- 

-a w 

f = Af. , 

by means of the fluctuation operator 

D-3 
The difference between A and ( ) lies in that the operator A , 

like any other operators, applies to all the functions which may follow, 

d -, 

while ( } is an ensemble average of the function or functions limited 

by the angular brackets. 

Thus by applying A to (la) , i.e. by premultiplying (la) by , - 

- 

we obtain the equation of evolution of f , as follows: 

The collective collision 

can be written in the form 

T =(.")r(r) . 

The collision operator 

has a diffusivity (5) which may serve as an operator to keep a memory. I - 

On the other hand, by applying to (la), we obtain the equation of 

evolution of f . This equation may take the following alternative forms: N 

where C is defined by (5), and 

D-4 
Our problem of the statistic,al theory of turbulence is to find the 

kinetic equation (4), with the turbulent collision (5) to be derived on the 

basis of one of the three equations (84 - (8c) of fluctuations. The three 

equations differ by their governing differentials 

>+x;, a +L , A t 3 +L , t t 

so that their integrations will have different evolution operators, as follows: 

U 

ftee ' 

A 

The operator A was introduced by Weinstock, 1-3 

'free is the free 

flight or ballistic propagator, describing a trajectory that is not 

perturbed by the E - field fluctuations, and U is the exact A A 

- 

propagator. 

The differential equation (8a) has the simplest source, but its 

A 

operator A is ccmplicate. On the other hand, the integration of the 

differential equation (8b) can be achieved by the simplest propagator 

'free of free flight, but the source is complicated by the nonlinear 

fluctu,ation C , as this will generate a hierarchy and requires a 

cl~sure.~ In the following we choose (8c), because the average source C 

is easier than the random source C , and the 'propagator U is simpler 

than the qperator A , and is governed by the well determined equation 

h. 

- 

-/ A 

+ 

of the exact trajectory in the form: 

or 

D-5 
The fluctuating source -L f gives a pair collision: N- 5 

that Is of the diffusLve type with an operator 

(13b) 

related to the diffusivity (D) . The two diffusivities are defined by: 

d 

% 

L 

Different levels of approximations have been made in the literature 

on the theories of turbulence in fluids and plasmas. As the lowest 

order, the quasi-1inea.r theory assumed a propagator of free flight 

and obtained a resonance denominator - k-v, where G) and k are the 

frequency and the wavenumber of a wave 

'free 

".. e 

interacting with the particle of velocity v. This resonance was 

fundamental to the Landau damping in the quasilincar theory. The next 

order approximation assumed an expansion around U in the renormalization 

5 

- 

theory. Interesting advances in theory of turbulence have made use of 

this assumption. 6Y7 

D-6 
With the group scaling, the derivation of the spectrum of turbulence 

can be Nade on the basis of the kinetic equation of the singlet distributiom 

without the pair distribution. Ve represent the three transport processes. 8 

of evolution, transport coefficient, and relaxation, by three scaled 

groups and analyse their interactions. The closure is obtained by the loss 

of memory in the processes and by exploiting the property of equasistationarity in the degrading coherence of groups. 

The paper is organized as follows. After a comparison between the 

hydrodynamical and the kinetic descriptions of the microdynamical state 

of turbulence (Sec. II), we introduce a group-scaling procedure for 

representing the three transport processes by the three groups (Sec. 111). 

We find that the kinetic equation of the macro-distribution has a turbulent 

collision integral with a memory. We investigate the pair interaction and 

the multiple interaction between a micro-fluctuation and the cluster of 

other micro-fluctuations that form a macro-group. Various forms of memory may 

develop : 

(i) First, the memory-appears in the collision that is controlled by 

the diffusivity acting. as an integral operator. Then, since the collective 

collision includes the pair collision and the multiple collision in the 

form of an integral equation, a memory ensues. 

(ii) The Lagrangian representation of a function is made by means of 

a propagator t'hat. is governed. by a kinetic eq,uation with a memory. 

Obviously, the memory group (i) which deals with the volution of 

the distribution function at the time instant t has a 'longer duration than 

the memory group (ii) which deals with the evolution of the propagator 

D-7 
....._ ....

for the shorter time-interval t-t' . We shall neglect the memory of the 

group (ii), by justifying from the property of quasi-stationarity 

between the groups (Sec. IV). We shall show that the propagator is related. 

to the probability of retrograde transition, and develop two parallel 

kinetic theories on the ,basis of the detailed dynamics of the trajectory 

(Sec. V). Finally, the collective collision is investigated, by means of a 

closure based upon the memory-loss (Sec. VI). A memory function appears and 

is controlled by the resonance function in weak turbulence and by the 

diffusion in strong turbulence, yielding two opposite outcomes: a 

shielded collision in weak turbulence, and an enhanced collision in 

strong turbulence (Sec. VII). 

11. PIICRODYNMIICAL S.TATE OF TURBULENCE ., 

The microdynamical state of turbulence can be described in two ways. 

First we can use the Navier-Stokes equations for the incompressible fluid: 

is the pressure gradient, and may even include other forces, such as 

buoyancy and random sources. 

D-8 
Alternatively, we can consider the microkinetic equation (la), 

rewritten in the form of the Liouville equation: 

[at+ if(t,x,v) ..... = . 

The evolution of the exact distribution function ?(t,x,v) is controlled by 

fhe differential operator 

” 

The microkinetic equation has been used as a basis of deriving the I 

I A 

BBGK equations by writing f(t,x,v) in the form of a summation of “ 

d - functions for N particles in the phase space. Presently, in 

order to secure the equivalence of the two descriptions, we write f 

in the form of a single d-function: 

8 

A 

h 

f(t,x,v) = STV - u(t,x>l . 

The density in the incompressible fluid is taken as unity here without 

loss of generality. The compressible fluid should have the density 

r(t,x) as a factor before the 6-function. Y 

By taking the moments, it can be demonstrated that the 

microkinetic description is transformed into the fluid description as 

represented by the Navier-Stokes equations. 

By the condition of incompressibility, we can write E in the following & 

two,forms: 

D-9 
n 4n 

One is nonlinear and the other is linear, so that the differential 

operator becomes 

with 

The Navier-Stokes equations (16) and the Liouville equation (18) 

may be called the primitive equations in the physical and phase spaces, 

respectively. 

The kinetic approach has several advantages. First, it transforms 

the inhomogeneous Navier-S tokes equation into tlle homogeneous equation 

(18) and rendered all nonlinear terms, as arising from the velocity 

function G(t,:) , into linear ones with the independent variable v , as 

in (18) and (21). The only nonlinear term kept is connected with E in 

Y * 

4 

u 

(18) for describing the mode-couplings. It can ultimately be treated by 

the linear equation of state (21). 

Secondly, the kinet-ic approach treats bot11 the wave-wave interaction and 

the wave-particle interaction. The latter is the mechanism of the linear 

and nonlinear Landau dampings or amplifications. This interaction is not 

explicit in the hydrodynamical approach. 

D-10 
Finally, should one propose a general kinetic theory that combines both 

the molecular motion and the turbulent motion, the microkinetic equation (18) 

can again serve as a basis, provided E includes both motions above, without 

a predetermined viscosity. Our kinetic theory that is based upon (18) with 

a differential operator (19) is concerned with the turbulent motion in a 

viscous' fluid medium. However, for the investigation of the turbulent collisions 

by eddies of size larger than the viscous cutoff, the effect of Y can be 

neglected. 

A 

- 

111. GROUP-SCALING PROCEDURE 

The decomposition of a fluctuating function fnto an average and a 

fluctuation is the usual practice in statistical methods. We denote the global 

average and the fluctuation by 

d n/ 

A and 

- 

A=1-A , 

respectively . 

Not all scales of fluctuations perform the same role in the three 

processes mentioned earlier: evolution, transport coefficient, and relaxation. 

For describing these processes and their couplings analytically, we 

represent them by three scales, using the three operators 

to form the three groups, or scaled fluctuations: 

They fluctuate and have their durations of correlation 

D-11 
, I, 

in the increasing degrees of incoherence. This is the property of 

degradation of group coherence, instituting a quasi-stationrity o f one 

group with respect to the other. The fields (24b) will also be called 

macro-, micro- and submicro-fields, respectively. 

The two-scale averaging procedure is obtained by further scaling A 

N 

in to 

A=AO+A' , N 

by using A(*) to find 

A' A = 0, AOAO = A' , A'A' = 0. 

Subsequently, the three-scale averaging procedure is obtained by another 

scaling of A' into 

' A' =A 

The averaging procedures of many scales may be denoted by 

- 

A E< > , A. =( }o , .A1 =<>1 , 

where is the operator of the global average, A. and AI are the 

operators of the accumulated averages: 

Ao=X+Ao , A1=A+Ao+A (1) 

D-12 
Although the groups Ao and A' may overlap in their wavenumbers 

instantaneously, their statistical functions, e.g. their spectral contents 

must lie in adjacent domains of wavenumbers, i .e. 

We conclude that the group-scaling procedure is a scaling into the 

three transport processes mentioned above. It can distinguish between the 

two diffusivities: the asymptotic diffusivity (Dl) and the non-asymptotic 

diffusivity (DO} , a distinction not present in the one-scale averaging 

procedure. 

- 

4 

" 

Recall that the spectral function, being the Fourier transform of the 

correlation function of velocities at two points, would, in the one-scale 

averaging, require a pair distribution function and its coupling with the 

singlet Gistribution function. In the present procedure of group-scaling, 

the singlet distribution function fo suffices. Indeed, fo gives u and 

((yo)2> , and thus derives the spectrum of u- fluctuation; by a differentiation 

with respect to k . 

0 

" 

" 

The degradation of group coherence (25) indicates a consecutive 

quasi-stationarity among the groups, and is an important property for 

classifying the interactions and placing the memory and the memory-loss. 

D-13 
IV. TURBULENT COLLISIONS AND MEMORIES 

A. Coll5sion in the Kinetic Equation of f 

0 

By scaling the Liouville equation (18) by means of Ad and A' , we get 

the equations of evolution of the macro-distribution 

and the micro-distribution in the form: 

or, equivalently, 

Here 

is the collective collision. The scheme falls into the framework that was 

described by (8c) and (8a). 

The differential equation (31a) can be integrated to give 

L 

D-14 
The lower limit of integration is set' to zero, and the initial value is 

ignored, since 

A. L'(t) f'(0) = 0 . (34) 

Here and in the following, the dependence on x or (x,v) is understood. ... " 

The propagator 

t A 

U(t, t') = exp - /dT" c(t'l) 1 

t' 

satisfies the differential equations (12). 

It is to be remarked that the differential equation (31a) for the 

evolution of f' has two sources written in the right hand side: The microsource -L'fo represents a coupling between the micro-operator L' and 

the macro-distribution fo and gives a fluctuating contribution to the 

solution, as seen from the first term in the right hand side of (33). The 

macro-source Co represents the collisional cluster in the medium in which f 

evolues. Although Co is quasi-stationary, it can give a random contribution 

through the operation A'U as seen from the second term in the right hand 

0 

Ir 

side. This term represents the effect of the quasi-stationary Co riding on 

the .random traj ec tory. 

In order to determine the collective collision, as defined by (32), we 

multiply (33) by L'(t) and average, obtaining: 

t 

Co(t> = Ao/dt' L'(t) ^V(t,t') L'(t') fo(t') 

J 

0 

D-15 
+ AoJ,"t' L' (t) U'(t,t') Co(t') . 

This expression consists of the pair collision 

and the multiple collision in the form: 

t 

dt' L'(t) U'(t,t') Co(t'> = - A. J' dt' Ho(t,t') Co(t'> 

0 

= - A. Ho * Co , 

Here Ho can be identified as the collective collision in the kinetic 

equation of Uo, i.e. 

f Lo(t) Uo(t,t') = - A L'(t) U' (t,t') f Ho(t,t') , (39) I 0 

as obtained upon scaling (12a) by Ao. Note that (12b) will not lead to 

the desired collision and is therefore not relevant here. 

As an option to (8a)-, we have the equation of evolution of the 

micro-distribution in the form 

(4 + A' i)f' = - L'fo , 

and obtain 

D-16 
by an integration and the use of the operator fl . It follows: 

A 

co - - - A. L'f' L 

I 

Here A A denotes the collision operator 0 

t 

AoA' = A. kt' L'(t) A (t, t') L'(t') 

with a diffusivi'ty 

J,' n 

Ao;' = A. dt' E'(t) /\(t,t') E'(t') . H * 

By replacing A by U in (42) - (44), we obtain the pair collision 

n * 

C 0 = A~A' { fo 1. , 

with a collision operator 

t 

AO&' = A. /o dt' L' (t) U(t,t') L' (t') A 

(44) 

D-17 
and a diffusivity 

AoB' = A. I dt' E'(t) G(t,t') E' (t') 

. 

Y 

U a 

-0 

By collecting the results (36) - (38), we obtain the following 

relation between the two collisions Co and in the form: 

with 

A. How Co' = A. dt' Ho(t,t') Co(t') . J,' (49) 

B. Collision in the Kinetic Equation of Uo 

The distribution function f and the propagator U have their evolutions 

A n 

of some resemblance. By repeating the calculations that have been made in 

Subsec. IVA, we obtain the collisions 

HO = AoA"{ Uo } 

= AoA'{ uo } 

of uo, as related by the formula: 

€Io = - JI0* H . (51) 

Here li0 and 11 are the collective collision and the pair collision 

in the equation of evolution of Uo , i.e. (39). The results (50a), (50b), 

and (51) are analogous to (42), (45) and (48), respectively. 

D-18 
C. Memories 

We summarise the results obtained above as follows: 

(i) Pr,imary memory in Co 

The kinetic equation of fo , in the form (30) and with the collision 

(42), carries a primary memory by the collision operator AoA.f j which can 

be written into two groups as follows: 

The second group forms a correlation A,(bYo{{ ). on its own, but does 

not have sufficient time to approach its equilibrium, by (25), and therefore 

will not contribute to a diffusive collision. Hence we can write 

as an approximation, reducing (42) - (49) into: 

(52b) 

I with 

D-19 
Hence the evolution 

with t-t' =Z carries a primary memory in cO from the fluctuations E' - 

of scale Tc , such that / 

by (25). 

(ii) Secondary memory 

On the other hand, if 

the kinetic equation 

in Ho 

we write U(t,t') = U(t ) with t-t' =T, 

has a collision integral governed by the collision operator 

decomposed into two parts. With the condition of quasi-stationarity 
, 

as a continuation of (56) from (25), the second integral has a .matured 

diffusivity, where we can put Z+m and obtain (~‘~T+Q$, while the 

first integral has an unmatured diffusivity. Since T?< 77 , the secondary 

memory carried by the collision integral Ho in the short time span T 

can be assumed negligible as compared with the primary memory. 

D. Resolution of the Integral Equations 

From the definition 

the Fourier transform of the convolution H * Co is - 

and is written as 

Here a change of variables T-c ’ has been made. The symbol ’= denotes 

a Fourier transformation. Hence we transform (53c) into: 

D-21 
V. KINETIC EQUATION OF THE TRANSITION PROBABILITY 

A. Path Dynamics 

The trajectory with which the propagator evolues is describe( 

the following differential equation of the path dynamics: 

or, identically, by the system 

with the conditions 

4 A 

x(t) = x, v(t) = v , I 4 - - 

which specify that at the instant of time t , 'the said trajectory 

passes by the phase point (x,v). 

" 

By integrating (63) and writing t' = t -x , we find the dynamical 

variables 

c 

/ 1 " 

and the displacements 

7- 
Now the path dynamics (63) that is driven by E has the displacements: 4 

* 

dt: L 

B. Retrograde Transition Probability 

The Lagrangian correspondent of an Eulerian function G(t,x) can - 

be written in the form 

and specifies the value of the function as observed by a fluid particle 

coming from the point f(t -T >' at the time t- , along the 

trajectory which at the time t passes the point x . The Lagrangian 

function can be written more conveniently as 

- 

- 

by means of the propagator U(t,t-t) . It is in reality a function 4 

of two states: 

Since ) (-T) is a random function, we can introduce a retrograde transition 

probability, written in the form: 

- 

D- 2.3 
or, briefly in the notation 

The first form expresses the state t,x as a condition and can be called 

a conditional probability. The second form implies a quasi-stationary 

transition, i.e. p varies more rapidly with -7g than with t,x 

Here 1 is an independent variable. 

... 

h 

.\ - c 

The concept of the retrograde transition probability has been introduced 

earlier and its equation of evolution was found to be of the Fokker-Planck 

type. More recently, by the use of this transition probability, the 

kinetic equation of turbulence was found. In the present Section, 

9 

10 

we shall devote to the relation between the retrograde transition probability 

and the propagator. 

n 

As the basis of the dynamics of p(-K l), we write the equation of 

.c 

the detailed evolution in the form 

satisfying the condition of normalization: 

D-24 
The differential operator is 

- 

In order for (71b) to describe the microdynamical state of the turbulent 

trajectory, we put 

The partial differential equation (71a) has its characteristical 

equations coinciding with (67). Evidently, by substituting (73) and 

integrating (71a) with respect to 1 , we will reproduce the basic 

dynamical equations (67). 

c 

The Fourier transforms of (71a) and (71b) are: 

and 

with 

and 

Here d=3 in the three dimensional space. The integration of (74a) 

with the condition (74b) gives the solution 

D-25 
It consists of a ballistic orbit function e-%'!' and a field-dependent 

orbit function, which is 

A 

y(-T,b) - = ~+p L kJ(-d, " 

if -E = 0. 

C. Relation Between the Propagator and the Transition Probability 

- 

The evolution of U is governed by a kinetic equation with a 

collision . It is to be noted that U(t,t') , being an evolution A 

operator, or propagator, describes the exact trajectory, and therefore 

is a functional of the path t(-T) during the interval of time from t 

to t-C in the retrograde transition. Hewe the propagator can be written 

A 

... 

as : 

or 

(76b) 

J- L - 

by the use of the transition probability "p<-t ,I ) . In an analogous manner, 

we have: 

- 

D-26 
In this way, the propagator is determined by the probability of transition 

A 

P(-d 1 Y or P(-d) 

- 

D. Lagrangian-Eulerian Transformation 

The Lagrangian function depends on two states, and can be written in 

three forms, by means of. a functional as in (68a), a propagator as in 

(G8b), and now a transition probability in the form: 

It admits a Fourier form: 

A 

U(t,t"r:) G(t-T)'= (ZX) p(-t,k) c(t-t ,IC), dA 

e 

by a Fourier transformation with respect to 1 . A time integration 

yields the convolution: 

" 

Hence the combined Fourier transformations with respe 

(794 

ct to both t and 

or 

D-27 
Use of (76b) has been made. 

We conclude that the transition probability helps in determining 

the propagator and the function to be operated upon, and thereby establishes 

the Lagrangian-Eulerian transformation. 

E. Kinetic Equation 

By scaling (71a) by x and x , we find the system: 

with the collision 

The procedure of determining the collision F(-C, R ) is the same -a. 

- 

as that yielded the collision H in the kinetic equation of U . We 

deduce 

- 

- c 

J=J - H+Y, 

- 

analytically, as was with (53c) and (53d). We have 

D-28 
with 

Now by neglecting the secondary memory, we can write the approximations: 

- - -c. 

c 

In the following, the determination of U and will be 

* 

performed through and p , so that we will have no more opportunity 

- 

of dealing with the kinetic equation (57) of the propagator. 

VI. DETERMINATION OF THE COLLISIONS BY MEANS OF THE TRANSITION PROBABILITY 

The result (79c) has transformed the convolution U Y G of the Lagrangian A 

form into its Fourier Eulerian correspondent. We shall apply this result 

to transform the convolution H Coy rewritten as 

- 

by definition (53a). We obtain: 

D-29 
- "" . 

.c c 

where 

is a dimensionless factor of memory, and 

is called the 'memory function", defining a life-time P1'3 for the 

memory. 

Rigorously speaking, (A> E " 

2 .hP' operates on the ballistic ." * 

orbit function (15), as well as on all the diffusivities which are 

embedded in is and Co . With the approximation that (D') varies 

slowly with v , as is true in strong turbulence, we can evaluate (85b) 

and get 

t: 

.% 

where, by definition (76b), we have 

D-30 
d- * (86b) = (2n) p(-d c I 

Now it leaves us the function p(-7J,k) , and we shall determine it 

- 

,. u 

by means of the Fourier transform of the kinetic equation of transition 

in the form 

from (80a). The collision is given by the expression 

as governed by the diffusivity (82c). The integration with the condition 

(74b) gives the probability 

which we substitute into (86b) to obtain 

Since (g) is caused by the field fluctuations 

D-31 
we can decompose the diffusivity into components, as follows: 

Within the available time spant and under the quasi-stationarity 

condition (58), the diffusivity (Kt (~+m)) reaches a matured development, 

leaving the non-asymptotic diffusivities of the order of magnitude: 

The negligible (K(l) (t 9 may be included under (KIT )) for brevity. 

The procedures indicated are to calculate the two diffusivities from 

the path dynamics (66b), to substitute the results into (89b), and 

subsequently into (853 I and (8%), to obtain P , o( andM . The calculations 

are lengthy, but can be simplified if we choose the highest power in Z 

for retaining the most rapid memory-loss. The results are collected 

- 

as follows: 

(a) results relating to (DI) 

D-32 
(b) results relating to (D') 

with " 

By applying these results, we transform (53b) and (53c) into: 

The last equation can be used to derive 

VII. SHIELDING AND ENHANCEMENT OF THE COLLISION 

With the approxima.tion of neglecting the secondary memory, as stated 

in (83), we can reduce (93) into the form 

rewritten as 

D-33 
by introducing the coefficient 

where 0;6 L%= are the real and imaginary parts of ', 

The result (95a) shows that the collecxive collision depends on the 

factor p , which in its Surn depends on the memory function . This 

function has a complicate integral (92). We shall calculate it 

approximately by means of an interpolation which correctly covers the 

three regions as dominated by (d , m , and 4)- k. v , separately. ,o 

9 --e 

For the sake of simplification of writing, we introduce the f.ollowing 

frequencies: 

with the ratios 

and the numerical coefftcients found to be: 

D-34 
Finally we find the results, as follows: 

We estimate that 7 is of the order of unity, and p is an index of the 

strength of turbulence. The asymptotic cases reduce the general formula 

into the following: 

(a) For weak turbulence, we have 5 >) , entailing d,<<a , we 

2 

find 

so that 

causing a shielded collision. 

(b) For strong turbulence, we have , entailing aL<< g,, , we 

find 

so that 

causing an enhanced collision. 

D-35 
The enhanced collision by the factor (3 can be evaluated for 

plasma turbulence and fluid turbulence. As an illustration, we consider 

the plasma turbulence in a st.rong magnetic field. Here the electric field 

fluctuation excites the velocity fluctuation. We found that both spectra 

possess the k'3 power law, that 3' is a constant quantity, and that 

the Lannor radius is a spectral cutoff. Since the E-fluctuation drops 

rapidly, the macro-electric energy ((5°)2> will cover the main body of the 

energy-containing portion of the spectrum to become approximately 

independent of k . These parameters determine a turbulent Reynolds number 

Re = ((EO)? a 1'2 

By definition (96a), we have 

obtaining 

and 

with 

(101a) 

For high intensity turbulence , R is large, and we reduce (101a) e 

D-36 

(101b) 
The collective collision is increased above the pair collision by 

an amount proportional to k-3'2. This means that the big eddies in a bath 

of smaller ones will be more apt to have an enhanced collective collision. 

An analogous increase of the effective diffusivity of a suspension of small 

particles in Brownian movements is well known. The application to fluid 

turbulence will be given at a later opportunity. 

VIII. .SUMMARY AND DISCUSSIONS 

The point of departure of the statistical theory of turbulence is the 

description of the microdynarnical state of turbulence, either by a. system of 

hydrodynamical equations in the t,x space, e.g. the Navier-Stokes 

equations (16), or by a consistent microkinetic equation in the t,xi,vi space, 

e.g. the Liouville equation in the form (18). Here by enlarging the 

i 

dimensionality from the passage of the t,xi space to the t,xi,vi space, we 

have eliminated the nonlinearity as connected with the velocity field, 

but have kept the nonlinearity as connected with the Ei-field from (21). 

If the Navier-Stokes equations are further decomposed into Fourier series, 

the multi-dimensional variable u , with fdz (i,k) , has the three 

o( 

directions i=1,2,3 and all the Fourier modes k running from-- tom . 

The distribution 

transforms the Navier-Stokes equation into a corresponding Liouville 

D-37 
equation of the form: 

(> + L)f t = 0, 

where 

is the differential operator, and 3 , are the coefficients. 

Aqv 

In this kinetic representation, the enlarged dimensionality by the 

Fourier transform renders L deterministic, so that the Liouville 

equation becomes linear. Since the linear equation does not distinguish 

between the average and the fluctuations, it does not lend to clearly 

express the coupling between the modes in afunctional form. An external 

fluctuation response assumption becomes necessary for establishing the 

physical and functional structure of L . For example, when L and f 

are decomposed into two components as 

L = (Ll0 + SL, f= (f)o +Sf, 

(103) can be transformed into 

An iteration along the direct interaction approximation was to assume a 

dissipative (L)o of the Fokker-Planck form. 

By contrast, the nonlinear Liouville quation (la) retains the mode-coupling

which are described by the equations of evolution (8a)-(8c). The form 

(8a) gives the collision in the most direct manner but requires the knowledge 

D-38 
L 

of the operator A whose physical structure and relation to U can only 

be obtained through other two equations, (8b) or (8c) . In the f om (8b) , the 

memory disappears in the ballistic orbit and reappears as a high order 

correlation 

A 

causing a hierarchy. Therefore we have chosen the form (8c) which does not 

present this difficulty. On the contrary it establishes the necessary 

relation betweenA and U , thanks to the transition probability. 

.. 4 

For the sake of simplification of discussion and abbreviation, we 

use the one-scale average by writing: 

- N 

f(t,X,V) = 7(1), f (t ,x,v) = 51) , (108a) -A " 

and 

(108~) 

from (22b). The time derivatives along the trajectories will be written as: 

h 4 

dt _~b~ +L , dt - 

- 

=3 +L . - 

t (108d) 

The average distribution and propagator, and their fl.uctuations, 

are governed by the following equations of evolution: 

a (109a) 

D-39 
AU 

dt f(1) = - f(lf) f(1) - C(1) rv 

and 

AY - 

dtU(l) - - ?(lf) v(1) - H(1), 

with the collisions: 

C(1) = -( F(lf) ?(1) > 

(109b) 

(110a) 

(110b) 

(llla) 

The evolution of f(1) is governed by a variable source coupling 

f and and a deterministic source -c which represents the cluster of 

Ir 

N 

fluctuations in the diffusive medium. Upon integrating (lO{)b), we get 

and subsequently upon multiplying by F(1') and averaging. we find 

the collision, as follows: 

or 

Here 

(113a) 

(113b) 

D-40 

i 
.is the pair collision, having an operator 

and -H*C 

" 

is the convolution of two collisions: 

In terms of the collision operator 

we can rewrite (113b) as 

We conclude that the variable source and the deterministic sources in (109b) 

yield a collective collision with two components: one is proportional to 

(z ) , and the other is proportional to <i)' , as re I.ated to the 

pair collision and multiple collision, respectively. The latter contributes 

to a shielding of the collision in a weak turbulence as gwerned by 

the resonance of the wave exp [-i( W - 5.y)T] , or to an enhancement 

of the collision in a strong turbulence as governed by the diffusion 

parameters of relaxation frequencies 0' and mo in the memory function. 

P 

Such opposite effects can find their analogies in the Balescu-Lenard equations 

in quiescent and in the effective diffusivity of a syspension of 

particles undergoing Brownian movement^,'^ respectively. More recently, 

a reversal of the effect of multipe collision in Brownian motion has 

also been found, dependent on the relative scales of the particles and 

the Brownian motion. 16-18 

D-41 
" 

.-. . 1.11 I 

We can give the following interpretation. A cluster of large size a may 

move with a high velocity and be scattered by the small scale turbulence, 

i.e. ka> 1. Since a part of the eddies will be displaced or shielded from 

participation in the collection interaction, the effective diffusivity 

is reduced. On the other hand, if ka< 1 , the cluster will participate 

together with the background eddies to contribute an enhanced collision. 

By the generalization to the two-scale averages, we obtain the kinetic 

equation of the distribution function fo with the collective collision 

'Co, as found in (30) and (48). This can derive the transfer function for 

the energy-cascade and determine the spectrum of turbulence without the 

intermediary of the pair distribution function (?(1') ;(l)) . 

10 

D-42 

-. 

\ 
ACKNOWLEDGMENT. The author C. M. Tchen wishes to thank the Dgpartement de 

Physique du Plasma, Centre d'Etudes Nuclbaires, for the hospitality which he 

I enjoyed during his visit. His effort in this work forms part of the 

research sponsored by the NASA Grant No. NAS8-34622. 

D-43 
REFERENCES 

'5. Weinstock, Phys . Fluids 12, 1045 (1969) . 

2J. H. Misguich and C. M. Tchen, "Equivalent Methods for Describing 

Quasi-Linear Turbulent Trajectories". Report EUR-CEA-FC-1157 (1982). 

3J. H. Misguich and C. M. Tchen, "A New Kinetic Description for Turbulent 

Collisions Including Mode-Coupling." Report EUR-CEA-FC-1158 (1982). 

4T. Boutros-Ghali and T.H. Dupree, Phys. Fluids - 24 (lo), 1839 (1981). 

5C. M. Tchen, Comptes Rendus Acad. Sci. Paris, 290B, 167 (1980). 

6T. H. Dupree, Phys. Fluids 9, 1773 (1966). 

7S. A. Orszag and R. H. Kraichnan, Phys. Fluids - 10, 1720 (1967). 

8C. M. Tchen, Phys. Fluids - 16, 13 (1973). 

'C. M. Tchen, Verslagen Ned. Akad. Wetensch. Afd. Natuurkunde 53, 400(1944). 

'OC. M. Tchen, "Kinetic Method of Turbulence". Preprint. 

"C. M. Tchen, Phys . Rev. - A8, 500 (1973) . 

12C. M. Tchen, Plasma Phys. - 18, 609 (1976). 

I3A. Lenard, Ann. Phys. (N.Y.) 2, 390 (1960) . 

- 

14R. Balescu, Phys . Fluids - 3, 52 (1960) . 

15J. M. Burgers, Proc. Acad. Sci. Amsterdam - 44, 1045, 1177 (1941). 

16A. R. Altenberger, J. Chem. Phys. - 70 (05), 1994(1979). 
I7W. B. Russel and A. B. Glendinning, J. Chem. Phys. 74(2), 928(1981). 

I8W. B. Russel, Ann. Rev. Fluid Mech. - 13, 425(1981). 

D-45 
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 3. RECIPIENT'S CATALOG NO. 

~ 

NASA - ~ 

. .. 

4. TITLE AND SUBTITLE 

~ .~ . . , , " 

5. REPORT DATE 

February 1984 

Theory and Modeling of Atmospheric Turbulence 

Part I: September 1, 1981-August 31, 1982 

6. PERFORMING ORGANIZATION CODE 

7. AUTHOR(S) 

11"". . 

48. PERFORMING ORGANIZATION REPOR r 7 

19. PERFORMING ORGANIZATION NAME AND ADDRESS 110. WORK UNIT NO. f 

I The City College Research Foundation 

138th Street and Convent Avenue 

New York. New York 10031 

"440 i 

1 1. CONTRACT OR GRANT NO. 

I NAS8-34622 I 

13. TYPE OF REPORT & PERIOD 

1 Contractor Report 12. SPONSORING AGENCY NAME AND ADDRESS 

National Aeronautics and Space Administration 

Washington, D. C. 20546 

31, 1982 

14. SPONSORING AGENCY CODE 1 

15. SUPPLEMENTARY Technical Monitors: George H. Fichtl and Margaret B. Alexander I 

Prepared for the Atmospheric Sciences Division, Systems Dynamics Laboratory 

Marshall Space Flight Center, Alabama 35812 . . " . Part I ~ of Final-Report -~ , ~ 

I 

16. ABSTRACT 

Among the transport functions which characterize the evolution of a turbule

spectrum, the cascade transfer is the only function which describes the mode-c

as the result of the nonlinear hydrodynamic state of turbulence. A kinetic theor

combined with a scaling procedure is developed. The transfer function governs the nonlintar mode-coupling in strong turbulence. It is investigated by a kinetic theory of 

turbulence. The master equation is consistent with the hydrodynamical system that 

cribes the micro-dynamical state of turbulence and possesses the advantages of being 

homogeneous and having fewer nonlinear terms. Since the detailed interactions betwe

the individual modes contain too many minute details for a statistical theory, 

modes are scaled into groups to decipher the governing transport processes and 

tical characteristics. An equation of vorticity transport describes the micro-dynam

cal state of two-dimensional, isotropic and homogeneous, geostrophic turbulence. By 

group-scaling, the equation of evolution of the macro-vorticity is derived in 

of the Fokker-Planck equation with memory. The micro-dynamical state of turbulenc

transformed into the Liouville equation to derive the kinetic equation of the 

distribution in turbulence. The collision integral contains a memory, which is an

lyzed by considering the pair collision and the multiple collision. For the inte

actions among the groups, two other kinetic equations are develped in parallel 

the propagator and the transition probability. 

17. KEV WORDS 

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118. DISTRIBUTION STATEMENT 

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I 

I Turbulence I Unclassified-Unlimited 

Atmospheric Boundary Layer 

Storm and Mesoscale Processes 

Global Scale Processes 

Atmospheric and Terrestrial Environ. Proc. STAR CATEGORY: 47