INTERMITTENT BEHAVIOR IN NUMERICAL SOLUTIONS
TO A NONLINEAR SYSTEM OF EQUATIONS
by
Jon Alan Pollack
B.S.,
Yale University
(1976)
submitted in partial fulfillment
of the requirements for the
degree of
Master of Science
at the
Massachusetts Institute of Technology
(February,
Signature
Certified
of
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1979)
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Dept. of Meteorology, January, 1979
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Thesis Supervisor
Accepted
1 tT' Chairman, Department Committee
MIT LIfI
LIBRARIES
INTE1RITTEiT BEHAVIOR Ii
NiUMJIERICAL ;SOLUTIONS
2
TO A N Oi'NLIUAR SYSTEM OF EQUATIONS
by
Jon Alan Pollack
Submitted to the Department of ivieteorology
in January, 1979 in partial fulfillment of the requirements
for the Degree of Miaster of Science
ABSTRACT
The phenomenon of intermittency as observed in turbulent
flows is briefly discussed, and general properties exhibited
by intermittent systems are considered.
The flatness factor
and its use as a quantitative measure of the degree of
intermittency is discussed.
A set of nonlinear differential
equations similar in form to a system used to model twodimensional turbulence is presented.
analysis is performed.
A linear perturbation
Numerical integrations of the
equations are carried out for various values of the external
forcing.
Intermittent behavior is found on the smaller
scales of the system.
many types of solutions are found
and various properties of the system are investigated as
the value of the external forcing is varied.
Name and Title of Thesis Supervisor:
Edward N. Lorenz,
Professor of Meteorology
Table of Contents
Abstract
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Tab le of Content s ...
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Basic Equations
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Analysis
Linear
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Numerical Methods and Specification of Val,
VI.
25
"Low Forcing" Experiments
VII.
28
"Medium to High Forcing" Experiments
VIII.
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"Very High Forcing" Experiments .......
IX.
Experiments with Forcing in Band 3
XI.
"Random Forcing" Experiments
XII.
Final
Tables
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Figures
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Experiments with "Scale Independent Di
XIII.
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Remarks
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Bibliography .......
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Definitions and Statistical IMethods
IV.
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Qualitative Discussion of Intermittency ......
III.
X.
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Introduction
II.
V.
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Acknowledgements
I.
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Acknowledgements
The writer would like to express his appreciation to
Professor E.
N. Lorenz, Professor E.
fiollo-Christensen, and
Mr. Ronald Errico, all of the Department of iveteorology at
the Massachusetts Institute of Technology, for their time
and suggestions.
The writer would like to give special
thanks to his brother, Mr. Richard J. Pollack of the
Department of Entomology at Cornell University, who was of
invaluable assistance during the final preparation of many
of the figures.
I.
Introduction
Over the last three decades observations have indicated
that many processes associated with turbulence are intermittent
in nature.
The work of Batchelor and Townsend in the late
1940's presented evidence of spatial intermittency in the
fine structure of a turbulent velocity field.
Since the
viscous dissipation of turbulent kinetic energy occurs
mostly in this fine structure within small eddies, this
suggested that dissipation may be localized and distributed
in an intermittent manner throughout the fluid.
Sandborn (1959)
found that small-scale turbulence seemed to occur in discrete
lumps or bursts within a fluid rather than being uniformly
distributed in time or space.
Gordon (1974) found that the
transport of momentum in turbulent boundary layers is an
intermittent process in time and space and that much of the
Reynolds stresses in a fluid can be traced to a few, intermittent fluid motions.
Similarly, Heathershaw (1974) reported
that the production of Reynolds stress is intermittent and
occurs in bursts.
Gordon (1974) and Heathershaw (1974) both
indicate that the scale of intermittent phenomena observed
in the laboratory can be increased to scales of geophysical
interest, and Heathershaw (1974) presents some evidence of
intermittency in a geophysical boundary layer.
Observations then have inaicated that intermittency is
an important phenomenon which is a characteristic feature
of turbulent flows.
Experimental work has revealed that
turbulence is highly spotty and anisotropic, and that bursts
of turbulence, although occurring for only a small fraction
of the time, are responsible for the production of a large
portion of the observed stresses and fluxes within a fluid.
The effects of intermittency may present real problems
in predicting the future state of a fluid system.
iollo-
Christensen (1973) has written that attempts to do so by
solving the Navier-Stokes equations directly by numerical
means may be discouraged by the fact that "large scale flows
are characterized by sporadic concentrated bursts of activity."
Ramage (1976) has suggested that turbulence bursts represent
the most important mechanism for atmospheric change on all
spatial and temporal scales and that atmospheric predictability
is limited to that which can be achieved by statistical methods
since the onset of these turbulence bursts cannot be predicted
deterministically.
Mollo-Christensen (1973) has stated that both the
production of turbulent bursts and the subsequent dissipation
may be intermittent, and that the intermittency is due to
nonlinear interactions between different scales.
It is his
opinion that although there is an exchange of energy due to
weak interactions between scales with little separation, the
dominant processes are made up of nonlinear interactions
between a broad range of scales.
Kennedy and Corrsin (1961)
showed that intermittency does not occur in all nonlinear
random processes.
Some three dimensional simulations of
turbulence (e.g., Siggia and Patterson, 1977) have failed
to clearly demonstrate the presence of intermittency.
With
7
this in mind, an investigation was undertaken to see whether
intermittent behavior could be discovered in numerical
solutions to a system of relatively simple equations similar
in form to a set of low order equations which model two
dimensional turbulence (Lorenz, 1972) and which include local
nonlinear interactions between different scales.
II.
Qualitative Discussion of Intermittency
First, we will consider some general properties that are
exhibited by intermittent systems.
character of turbulent processes is
The lumpy or spotty
often revealed by an
observable intermittency in the time series of a signal from
a probe used to measure some property of the system.
Typically,
the time series is characterized by signals of relatively low
amplitude interrupted at times by high amplitude disturbances
corresponding to turbulent bursts.
Intermittency suggests
spottiness or lumpiness in time and space with activity
being concentrated in bursts with relatively calm or quiet
intervals in
between the bursts.
There are also certain statistical properties that an
intermittent signal can be expected to exhibit.
An intermit-
tent variable -will commonly have a high probability of being
found in a low amplitude state and in addition will take on
relatively high amplitudes corresponding to bursts.
W4hen
compared to a variable with a normal (Gaussian) probaoility
density, an intermittent variable will generally have a larger
than normal probability of being found near the mean, a larger
than normal probability of taking on values very far from the
mean, and consequently lower than normal probabilities at
intermediate values.
However, since these statistical prop-
erties give no information on the actual order of events, it
should be noted that if the time that the variable resides in
the low amplitude state does not occur in relatively
extended
periods, the signal may not appear to be intermittent.
Definitions and -StatisticalNethods
III.
Intermittency for some variable is often defined in terms
of an intermittency factor Y which measures the fraction of
time that a variable spends in a high amplitude or turbulent
state.
However, since it cannot in general be determined
a priori at what amplitude one should make a division between
high and low amplitude states, it would be a cumbersome
measure to use for our purposes.
It would require either that
a close examination be made of every numerical solution
individually before a suitable division, if any, could be
determined or that an accurate probability density of the
relevant variable be compiled when one would not know before
hand either the range of values that would be encountered or
the resolution that would be needed.
Furthermore, knowledge
of the intermittency factor would not really provide any
information on the intensity of intermittent bursts.
The
information it would provide might only give some indication
of their frequency of occurrence or their duration, and such
information might be somewhat sensitive to a subjective
decision of where to divide the high and low amplitude
states.
For example, for a variable whose characteristic
behavior is similar to that represented schematically in
Figure la, an intermittency factor could be defined with
relatively little ambiguity.
However, if the behavior was
more like that represented in Figure 1b, it would be far
more difficult to defend any particular division between
high and low amplitude states.
10
For these reasons another measure of intermittency which
depends upon high order statistics of a variable was employea.
Following the approach of Lumley (1970), we review the following definitions.
The distribution function
is defined so that
value of
S
is
.
defined by BCc)
j
Ac-7
c
=
.
The probability density function
d P(c
dc
. Bc)Ac
lies between
C
gives the
and CtC
The kth moment of a variable
,.
B(C)dc. The first
c
simply its mean, j
.
of a variable .
is equal to the probability that the
is less than c
probability that
as
P(c)
P
.
in the limit
is given by
moment of a variable
A
is
The kth central moment (moment about
the mean) is defined by subtracting out the contribution to
the integral due to the mean value of
(
(c )
Pcc =S(c-
,.
so that
B(c)dC
The second central moment of a variable J
is the variance,
while the square root of the variance gives the standard
deviation.
The third central moment when divided by the cube
of the standard deviation is referred to as the skewness,
while the fourth central moment when divided by the fourth
power of the standard deviation is known as the flatness
factor.
The flatness factor and skewness are both pure
numbers, independent of both the units used to express
and the choice of origin.
If instead of having a continuous history of the
variable
S,
we only have some of its values as sampled at
a finite number of points, we can rewrite some of the above
quantities in a discrete form that is more useful for
11
treating data.
If we have N
observations, the mean of
j
is
N
given by
by
J.
(
(
(j(-i)
The kth moment about the mean is given
so that
The standard deviation O0
gives an indication of the
width of the probability density function.
Skewness gives an
indication of the asymmetry of the probability density function
about the mean.
A~ variable whose probability density function
is symmetric about the mean will have a vanishing third
central moment and hence will have a skewness equal to zero.
In contrast, a variable whose probability density curve has
a long right (left) tail will tend to have a positive
(negative ) third moment due to the relatively large contributions that the large positive (negative) deviations make to
the sum of cubes, and the variable is said to be positively
(negatively) skewed or skewed to the right (left).
Since the fourth central moment depends more strongly
than the second central moment on large deviations from the
mean, the flatness factor gives an indication of the extent of
the tails of a probability density curve.
The probability
density curve of an intermittent random variable, when compared to that for a normally distributed variable, will tend
to have more values in the vicinity of the mean corresponding
to the low amplitude state and will also have substantially
longer tails corresponding to those extreme values in the
high amplitude state.
The values in the tails will contribute
quite strongly to the flatness factor since their fourth
power enters the sum involved in the fourth moment.
Since
the magnitude of the flatness factor reflects the extreme
values taken on by a variable, it seems that there might be
some motivation for using the magnitude of the flatness factor
in some way as a measure of intermittency.
A quantitative
measure that has been used for some time to indicate the
degree of intermittency in an intermittent variable is the
amount by which the flatness factor exceeds the value of 3.
A
flatness factor of
3
is characteristic of any variable with
a normal probability density.
The amount by which the
flatness factor exceeds the value of 3 has at times been
referred to as the "excess."
Before proceeding further, some properties of the
flatness factor will be discussed for the benefit of the
reader.
As mentioned previously, the flatness factor is a
nondimensional number which is independent of both the choice
For simplicity then, consider
of origin and scaling factors.
a variable
.
scaled so that
= 0
)
so that
where
0
and
Thus we see that
F
.
Then
F
=
F(= I
I*
.
Let
~_.
is always greater than or equal to 1.
For such a variable to have the minimum flatness factor of 1
it is required that (1J=
which means that
The initial assumption that
on values of !A
probability of finding I
is constant.
S
then requires that
T= 0
, where A
j
take
is some constant, and that the
in either state is equal.
Since
F
is independent of origin, this really only requires that
j
take on two distinct values and that it spend an equal
time in each state.
A square wave is a simple signal that
F
would produce a value of 1 for
The flatness factor for a variable will be influenced by
the shape of the signal.
By considering some simple symmetrical
curves the reader may gain a better feeling for the flatness
factor.
As mentioned earlier, a square wave has a flatness
factor of 1.
A signal made up of alternating positive and
negative peaks with the shape of half ellipses or semi-circles
has
F
= 1.2.
F
A sine curve has
triangular wave patterns have
F
=
=
Both sawtooth and
1.5.
These are all far
1.8.
less than 3, so one might ask what sort of simple symmetrical
curve could give a value greater than 3.
given.
Two examples will be
First, a curve of the form shown in Figure 2 has
F= ni3i+
which approaches n1-I
example is the curve
Sin
X
can be shown to approach
as n
which has
I
A second
gets large.
F=
,
ELYh * 2)!j~
--
which
as n) gets large.
The flatness factor is also influenced by the width of
peaks in relation to the duration of quiescent periods, so
that a relationship between flatness factor and the intermittency factor is suggested.
Batchelor (1953) pointed out
3
-
that F
for a variable which varies with a normal proba-
bility density during a fraction S
zero for the rest of the time 1-
of the time and which is
.
variable which takes on some amplitude
If we consider a
A
for a fraction
'
of the time and which is zero for the remainder of the time,
we find that
as
s->
0
*Since
F
-'-
3
which approaches
---L
(ort)
this two-state idealization seems a
reasonable first approximation to an intermittent variable,
this indicates that when a signal has only a few large
positive or negative spikes occupying a very small fraction
of the time, the flatness factor may be very high.
But as
pointed out by Gibson, Stegen, and Williams (1970) it may be
necessary to take extremely large sample sizes to measure
such a high flatness factor with any statistical significance.
In the flatness factor we have a quantitative, unambiguously defined, and objective measure that gives an indication
of the degree of intermittency in an intermittent variable.
However, as pointed out by Kuo and Corrsin (1971), some care
must still be exercised with this quantity since, although
an intermittent variable will most likely have a high flatness
factor, a high flatness factor in itself does not necessarily
mean that a variable is intermittent.
The flatness factor
at best can indicate the intensity of intermittency in a
variable that is known to be intermittent by other means.
Figure 3 comes from Kuo & Corrsin (1971) and gives a
good indication of the range of values of
F
that have been
obtained in turbulent flows by various researchers.
The
15
flatness factor for the first time derivative of velocity
fluctuations is plotted against the Reynolds number of the
flow.
Jhe various symbols refer to different studies which
include measurements made in the atmosphere, on the surface
of large bodies of water, and in the laboratory.
IV.
Basic Equations
The equations to be used for this study have a form quite
similar to the very low order model equations (VLOME) for
The VLOME are written as
turbulence of Lorenz (1972).
4
where
a.r]D
j=
,
is a coefficient of kinematic viscosity,
is the fundamental spatial period of the motion, 9i
is
an external forcing function which maintains the motion
against viscous effects, and CO is a constant.
These equations
are derived from the equations expressing the motion of a two
dimensional, homogeneous, incompressible, viscous fluid on an
infinite plane with external mechanical forcing.
After
requiring that the flow patterns have the same period in
both horizontal directions, Lorenz expressed the equations
-
in finite spectral form by expanding the vorticity field in
a double Fourier series, separated the dependent variables
into consecutive bands in wavenumber space and then constructed a low order model by keeping only a small number of
variables within each band.
The VLOMEViz
were then formed as a
special case by requiring that the flow pattern be unchanged
by a rotation of 900 about the origin.
In the VLOME the bands
were separated by half octaves in wavenumber space and only
one variable per band was retained.
jj
The remaining variable
describes the behavior in the jth band and is related
to the vorticity in that band.
In the very low order model,
the nonlinear interactions between different scales are all
local in wavenumber space--that is, they only involve
adjacent bands.
Another important property of the VLOMiE is
that they conserve energy and enstrophy in the absence of
forcing and dissipation.
Although the set of equations is
capable of representing motions on any scale, in practice
for computational purposes, the system is truncated by
setting ?j=o
for all
j
greater than some positive integer N
This corresponds to leaving unresolved those motions smaller
than a certain scale.
The equations that were used for this study can be written
YK
as
.YK<-aY,<-i- 3YK-1Yi<*t t Y.,, Y.
Y.
t3JN
While
which can be seen to be similar in form to the VLOMb.
as many as sixteen bands were initially included, the system
was reduced to seven bands for this study when it was found
that this was sufficient to produce intermittent behavior.
Since the bands are separated by half octaves, the seven
equations for
Y
y&(o,---)6))
govern the behavior in
bands 0 to
6 representing motions with wave numbers in the range of
k-
2
to
Q
.
-L
-
and wavelengths in the range of 21T-a
to 2f1- a
In this study, the external forcing is applied to only
one of the bands at any time.
Unless otherwise stated, the
forcing is applied to the band denoted by
roughly to a wavenumber of 2),
(corresponding
K=-
and we take 9K=
0
for K# 2.
Band 2 was chosen so as to keep the forcing on the large
scale side of the spectrum and to allow the forced band a
full set of adjacent bands with which to interact.
If this
system was scaled for the earth's atmosphere, then the
18
forcing could be roughly identified with wave numbers in the
range of 6 to 8 where the strongest abarotropic forcing
occurs.
Before proceeding any further, it should be emphasized
that we do not claim to be studying turbulence itself.
Rather, we will be examining the behavior and statistics of
a system of nonlinear equations similar in form to equations
which have been used to model certain properties of turbulence.
V.
Linear Analysis
The full system of equations is now written:
$Y.
yIYa-
+YY, -2AkY,
-3YY.
dY
{Ya. =Y.Y,
-
+- YK
3Y,
-
Al Y +
% = 2Y. -y3Ysy + 4YS - 2%Y 3
dt
+ Yy -A Y
2%% -3~YY4
6
dt3Y4Y
-a'AkYS
dy = ay4
dYY
4
Y5
-2%'.QY
The system possesses the steady state solution given by
YK=O For Ki 2- and
Ya=
Y
Small perturbations
about the steady solution are governed
(,..-,af)
initially
by the linearized equations:
~--A
-3,VY + 13
-
I
Tho only remaining ad justable parameters in the problem
are
A-
and oja
forcing
to grow.
gg
.
For a given
A
we may ask how large the
must be made before small perturbations will begin
Since the forcing does not appear in the perturbation
20
equations for
5
nd
t6
,
we no longer need to carry them
Actually, the linearized
along in the linear analysis.
indicate that perturbations in
equations for Va, 451 &nd V,
these bands will damp.
Our linearized system now reduces to
--A0
S_
(-3Y
dW.1
o0
Y
0
-A. Y
aY -f A-3Y
3
'
which we write schematically as
The critical condition for the stability of the solution
is the vanishing of a real part of an eigenvalue of 97L
The eigenvalues of 97L are the roots of the characteristic
equation which is obtained by equating the determinant
fll-21I
to zero, where I.
is the identity matrix.
The
resulting characteristic equation for this system is
1+-( 7A).'+(aoa AQ+ ~f)
A
+
*+ 36A2Y'+-18Y*) =0
If a value of 92 is found for which the eigenvalues of 77.
all have negative real parts and 9
is then continuously
increased, the characteristic equation may ultimately develop
either a real positive root or a pair of complex conjugate
roots with positive real parts.
In the first case the
characteristic equation must first develop a root equal to
zero, while in the second case it must first acquire roots
which are pure imaginary.
If
we take
gao
as a special case so that
Y= 0
,
the
roots of the characteristic equation are all negative:
'A= --A)-2.A,-8.Aand -A.
Niow we can proceecd in
looking for
the acquisition of zero or pure imaginary roots by the
characteristic equation as the forcing is increased.
The condition for a zero eigenvalue of
YL is that the
constant term in the characteristic equation vanish since it
is equal to the product of the roots.
-56AV+ 369A2Vt,9Y= O
Y
and
are both real
This requires that
A
which cannot be satisfied since
(we ignore the trivial
solution given by
=
=
0).
Therefore, if the
characteristic equation ever acquires roots with positive
real parts as the forcing is increased, the roots must be
complex conjugates, and the characteristic equation must
first obtain a pair of pure imaginary roots.
The condition
for a pair of pure imaginary roots is that the characteristic
equation have a factor of the form (11
where
a)
a
0
The characteristic equation has the form CqA*+C 3 a 3 t caAI
.
C 1A+c:o.
If it possesses imaginary roots, it must be expressible in the
+ 6o)= 0.
form (2ca)(haf+2b,2
Setting these two expressions
equal to each other, we find that it is required that
Cj = bz , c3 =b,,
ca.=
ba.+b.,
c=
&nd
a?b,,
c. = aob.
From these relations we obtain
a
=
~ CI
-
CI
CS
6
b
= CO
.
C*Cs
)c,CA=
C'
- C
=C
2. C 3
= cC3C
+ c,03
This last equation expresses the relationship that must be
satisfied by the coefficients in the characteristic equation
if it is to have pure imaginary roots.
Substitution of the
coefficients into this relation results in the following
equation which is quadratic in
2937Y*-
36
Y:
?/L90.A= 0
J666A/YI-
22
which has the solution
Y
Since
,
the critical value of the forcing
g2
at
which the eigenvalues of 9Z are pure imaginary is given by
If the forcing is any greater than this critical
-=
value, some of the eigenvalues of 77L would be expected to have
positive real parts, and small perturbations would begin to
grow with the result that the steady solution would no longer
be stable.
here:
Double precision values of qa
Q'=
26.2415564275619708
and Q
0 =
are given
5.12265130841071104
The linear analysis does not reveal what happens once the
perturbations have grown to amplitudes at which they can no
longer be considered small.
Ihis requires us to turn to
numerical integration of the equations to examine particular
time dependent solutions.
23
VI.
Numerical ±±ethods and Specification of Values
For numerically integrating the model equations forward
in time, the 4-cycle version of the N-cycle time-differencing
scheme of Lorenz (1971) was used.
fhe basic time step for
most of the work was taken to be At
=
0.10 units since
experiments showed that reducing it to smaller values did not
significantly alter the results.
For high values of the
forcing it was necessary to use a smaller time step
to avoid computational instability.
noted,
At
=
in order
However, unless otherwise
0.10.
For this study, the viscosity A. was fixed at 25 = 0.03125
units, and various properties of the system were investigated
as a funtion of the forcing.
It is not difficult to see that
the particular choice that is made for A
the nature of the results.
If we scale the governing
equations by letting Y=-kXK , divide by
define
Tzdft
should not change
a
and rewrite the equations in
,
and then
terms of
XK
and
T , we find that the new equations have the same form as
the original equations, except that k has been absorbed
into the time scale and the forcing has been scaled by 4
Effectively, this means that picking a particular value of4defines a dissipative time scale for the system and that
changing
ki is really equivalent to changing the forcing.
For this reason there is no point in varying both-k and
so we fix the value of -k and vary only the forcing.
The
forcing in the forced band was taken as a constant for each
experiment, although the particular value of the forcing
,
was varied from one experiment to the next.
For initial conditions in most of the experiments,
was taken equal to zero for
central bands
K
Y
K
Y1
= 0,1,5, and 6, while in the
was given the small value of 10~3 for
2, 3, and 4 so that the nonlinear interactions between
bands could begin.
In most cases a transient response was
observed; large amplitude disturbances would build up fairly
rapidly in all bands, but soon the effect of this build up
would pass.
In most cases, the equations were integrated
for 2000 time steps before any analysis or compilation of
statistics was begun to help insure that the effect of this
transient response would be minimal.
The computations were
performed in double precision arithmetic on an AiDAH.L 470
computer.
25
"Low Forcing" Experiments
VII.
ga
,everal runs were made with the forcing
in the
vicinity of the critical forcing derived in the section on
linear analysis in order to see if a change in behavior
occurs near this critical forcing.
A
=
WVith the choice of
0.03125, the critical forcing has a value of
approximately 0.020010.
Twenty runs were made with the
forcing ranging between 0.0175 and 0.0270.
conditions,
Ya,
was taken to be equal to
For initial
g3
,
while small
perturbations equal to 10~' were introduced in the other
bands.
This then in some respects was a numerical simulation
of the linear perturbation analysis.
The equations were
integrated for a total of 10,000 time steps during each run
and statistics were compiled during the last 5,000 steps.
-A summary of the computed flatness factors and standard
deviations appears in Tables la and 1b.
Shen
g, = 0.0200, all bands except 2 and 6 exhibit
flatness factors of 1.5 which would be consistent with the
presence of sinusoidal disturbances.
Zhis suggests that the
disturbances in these bands are not growing or damping but
rather are neutral.
At
%
=
0.0205, the flatness factors
for bands 0, 1, 3, and 4 have increased slightly.
,xamination
of the standard deviations at this forcing indicates that
there is a distinct difference in behavior between those
bands which were relevant for the later stages of the linear
analysis (0, 1, 3, and 4) and the other bands.
values of
F(Na)
The extreme
are not due to intermittent behavior.
Rather,
26
the value of
Y2
remains almost constant so that its
probability density curve is virtually a spike.
values of
F(Ya)
The high
are believed to be due to round off error
in the attempt to compute the ratio of two quantities both
of which are negligibly small.
Ys
and
YG
The small values of
'
for
suggest that their probability densities are also
highly peaked.
It seems that the linear analysis correctly
predicts the behavior near the critical forcing; disturbances
initially do not grow in bands 2,
5, and
6, and the numerical
results seem to indicate that disturbances begin to grow in
,> 0.0200 or at least that there is a
the other bands when
change in the behavior that occurs at about
Examination of the mean of
the relation
Yj=9Sgg
Yj
9,=
0.0200.
(not shown) indicates that
holds extremely well (to about 10 decimal
places) when 3a40. 0 2 0 0 .
-his is not merely due to the fact
that the system was started with the steady solution.
experiments, in which all the
Y1_ were
initially, produced the same result.
Other
set equal to 10-5
For
92 . 0.0200, this
relation still holds but with less precision until the
forcing reaches 0.0225.
For
9,> 0.0225,
Ya
begins to
decrease with increasing forcing and continues to decrease
until the forcing reaches 0.0265.
Apparently, by the time
the forcing reaches 0.0225, the disturbances are no longer
well described by the linearized equations.
For example,
disturbances in band 2 are no longer being damped as
evidenced by the oroadening of the probability density curve
for
Y that can be inferred from the rapid increases in f(Y)
27
and the drastic changes in
higher values of 92 in
detail.
F(Y2).
this range
The behavior for the
was
not examined in much
For example, the behavior of the system near
g2= 0.0260 where F(Yx)is about 1.5 for all
k except k = 6
would be interesting to investigate but was not of direct
interest here since no intermittency was indicated and since
the linear equations were no longer applicable.
A few other features can be notea about the behavior in
this range of low forcing.
For bands 0,
1,
3, 4, and 6, T(YK)
was virtually equal to the root mean square value of
YK--the
effect of the mean in the calculation of O(Y) was negligible.
However, this was not the case for bands 2 and 5.
5
Bands 2 and
also showed a much higher degree of skewness than the other
bands.
It is not surprising that band 2 would have different
statistical properties (a definite non-zero mean, for example)
since the forcing takes place in that band and is always
positive.
However, this is the first indication that the
behavior in band
5
is also exceptional.
28
VIII.
"1Medium to High Forcing" Experiments
Some indication of the behavior of the system was desired
over a fairly wide range of forcing substantially greater
than the "critical" forcing.
For this purpose, a large
number of experiments were conducted for which
YY
Initially,
between 0.025 and 3.7.
9, had
values
was taken to be 10-3
in bands 2, 3, and 4 and zero in the other bands.
Integrations
were carried out for 20,000 time steps with statistics
compiled during the final 18,000 steps.
Figures 4a, 4b, and
4c summarize some of the results of these experiments.
Flatness factors for each fK are plotted against the value
of the forcing in band 2.
indicates that
5 and
greater than 3.
Y6
Examination of these figures
have flatness factors significantly
Probability densities of Y( were computed
for several values of 9. in this range.
The probability
density curves had long tails to accompany the high flatness
factors thus indicating that intermittency was indeed being
observed.
Yq
also has
F > 3 over much of this range of
forcing, but we chose to focus our attention on the bands
representing the smaller scale motions since that is where
one would expect to observe most clearly any intermittency.
At low values of the forcing (less than about 0.1),
shows rapid variations with q. on all scales.
higher values of the forcing,
F
At slightly
seems to vary smoothly with
The resolution is not
the forcing in bands 0 through 4.
fine enough to determine whether
F
F
varies smoothly with
forcing in bands 5 and 6 where the intermittent behavior is
29
The plot of
observed.
2,
1,
in bands 0,
3,
F
seems to reach asymptotic values
F(Ys)
and possibly 4.
also seems to show
signs of approaching an asymptotic type of behavior with a
In contrast,
value of about 8.
F(%)
still shows a general
tendency to be growing with increased forcing, although its
rate of growth appears to be slowing.
It should be mentioned that certain features in the
flatness profiles can be recognized on many scales at certain
values of the forcing.
For example, at about
9a
=
0.425,
peaks in the flatness profiles can be seen in bands 0, 3, 4,
5, and
There is also a peak in band 1 although it is not
6.
apparent from the graph.
profile for
Yj
At the same forcing, the flatness
has a relative minimum.
can be observed at about
I = 0.8.
A
similar phenomenon
As further examples,
many of the flatness profiles exhib.it relative extrema in
the vicinity of
,=
1.4 and
9,= 2.6.
Perhaps these features
may indicate changes in regime that occur as the forcing is
increased.
Another curious feature is how the flatness profile of Y3
(and
Y
to some extent) seems to reflect in an opposite sense
the profile of
F(Y)from
%.=
0.20 to about
11 = 1.5.
Next, we decided to see what sort of behavior in band 6
was proaucing the high flatness factors observed there.
Figures
5
series of
0.5,
1.0,
through 10 are plots representing partial time
Y6
for values of the forcing
2.0,
and 4.0.
g
equal to 0.3, 0.4,
T' = 0 has no particular significance;
it merely indicates the starting time of the plot.
The most noticeable feature for 3
1.0 (Figures
5
through 8) is that
Y' is
=
0.3, 0.4, 0.5, and
periodic.
In this
range of forcing (0.3 - 1.0) it appears that we have a type
of behavior that we might call "periodic intermittency."
Another feature that can be noted by examining the
amplitude of the peaks of
YG
for different values of the
forcing is that the response of
6 to increased forcing is
decidedly nonlinear. Thus, we are assured that the system is
actually behaving nonlinearly.
The plots of
symmetry.
For g
Y6
=
can also be examined with respect to
0.3,
0.4 and 0.5, Y',
=
is asymmetric.-
However, for
exhibits a particular type of symmetry;
if one reflects the curve about the line of zero amplitude
and then displaces it to the right or to the left by the
Curiously, however,
proper amount the same curve is obtained.
comparison of other features of these curves such as their
shape and the number of lesser peaks seems to indicate that
the curve for
the curve for
at 3g. = 0.40 in some ways looks more like
91 = 0.30 than it
does for
0.
= 0.50.
Actually,
one could imagine the curve for 9, = 0.4 being slowly deformed
into either of these two other curves.
The appearance of
such a curve at intermediate steps during the deformation
might correspond to the actual appearance of
values of the forcing.
Y;
at intermediate
We will return to this possibility
later.
Another feature that was noted about these four curves
was that their periods were all fairly close.
The periods
31
for Y( for 9, = 0.3, 0.4, 0.5, and 1.0 were about 26.1, 25.7,
27.3, and 26.6 time units respectively.
25 plots of
Y, corresponding
An examination of
to forcings ranging from 0_ = 0.35
to 0.50 indicated that the average period for
Y4
over this
forcing range was about 2b.4 time units and that the departures
from this average period were always within 4 percent.
This
observation provided us with a valuable tool for examining
whether the flatness factor in band 6 varied smoothly with
the forcing over this range of forcing where periodic
intermittency was observed.
It was postulated that the
period for Y' in the range from
9a
= 0.30 to 1.0 (and
possibly over a wider range) was always close to some value
which we took as 26.6, so that much computer time could be
saved by compiling statistics over one period (approximately)
of 266 time steps instead of several thousand.
Since the
behavior in this range was observed to be both intermittent
and periodic, this approach was expected to give fairly
accurate results as long as both peaks (positive and negative)
However, if the period were slightly
were properly sampled.
shorter (or longer) than 26.6 units, we might be unlucky
and accidentally miss sampling part of a peak (or sample part
of an extra peak).
be inaccurate.
measure
F(Y)
and
If this occurred, then the results would
With this in mind, this approach was used to
F(Y )for
at intervals of 0.01.
100 values of
92
between 0.1 and 1.1
For each value of the forcing the
equations were integrated from the usual initial conditions
for 2000 time steps and then for an additional 266 steps
32
over which the statistics were compiled.
experiment for
and for
Y5
Y6
The results of this
are shown oy the solid dots in Figure 11a
in Figure 11b.
From Figure 11a, we see that above
91 = 0.20 the solid dots seem to describe a smooth curve
with the exception of those points corresponding to
0.38, 0.91, 0.92, and 0.93.
g,=
0.28,
Examination of the plot of 1'6
for 3a = 0.38 showed clearly that part of an extra peak had
been sampled.
This then was responsible for the apparent
departure of this point from the smooth curve.
,Asa further
check of the validity of this approach, some points from
Figure 4c representing statistics over 18,000 time steps
were plotted along with other specially selected points
whose statistics were compiled over 10,000 steps.
This group
of points are represented by the "x"'s on Figure 11a.
These
points generally show excellent agreement with our approximate
statistics and indicate that the curve of FUt)versus
, is
The points that apparently
indeed smooth in this range.
deviated from the smooth curve are shown to actually fall on
the smooth curve.
Also, we see that the estimated values
32
of F(YW)for values of
near 0.78 were too high and those
near 9. = 0.88 were probably slightly low.
method does not work below
Also, our short
ga.= 0.20 indicating that the
behavior in that range is probably different.
F(Y)
shows
a similar smooth variation with forcing.
Next, we decided to investigate what was responsible
for the decrease in F(Y)that occurs with increasing forcing
roughly between
ga
=
0.40 and
g2,=
0.50.
It was thought
33
that perhaps this decrease was due to a transition to another
type of behavior or regime.
In any case, the change in
is related to the probability density of
Y6
F(Y)
and should
therefore in some way be related to changes in the shape of
Figures 12a, 12b, and 12c show plots of
equal to 0.35, 0.40, 0.45, 0.50 and 0.55.
Yg
6;for values of
Each curve has
been normalized so that its minima and maxima have the same
amplitude.
In addition, the curves have been shifted so
that the first large peaks of each roughly coincide.
In
this way, it is easier to concentrate on the shape of the
curves.
The three figures represent part of a continuous
time series (with a little overlap) for each curve.
Originally, they were designed to be joined together, but due
to distortions at the edge of each page caused by the method
of reproduction there is no longer a neat match.
Nevertheless,
the figures are still adequate to serve our purposes.
At
first we will limit our discussion to the curves of
corresponding to
9a = 0.35, 0.40, 0.45, and 0.50.
YC,
These
four all exhibit the same type of symmetry that was mentioned
earlier.
This means that their odd moments should vanish so
that the expression for
F
reduces to F(Y;)=T
(fY
Over this range we can see two gradual changes that occur in
the shape of these curves.
First, there is a gradual
reduction and, in some cases, elimination of minor peaks
in relation to the main peaks.
Secondly, there is a
gradual broadening near the base of the major peaks which
indicates a more gradual build up to high amplitudes.
34
9a
Going from
=
0.35 to 0.40,
'4
Y6
1.99 (call this change R 2 ) and
of 4.81 (call this change R)
of R4 /(R 2)
=
increases by a factor
so that
F
changes by a factor
Using the normalized curves, we find
decreases by a factor of 0.906, reflecting the
Y
that
1.21.
increases by a factor of
6
reduction of the minor peaks, while
just barely
increases by a factor of 1.001, reflecting the fact that the
shape of the major peaks is unchanged.
increase in
F(Y6)
in
this range is
Apparently the
due to the reduction of
the smaller peaks in relation to the large peaks.
from
g
=
In going
0.40 to 0.50, the lower portion of the large peaks
broadens,while the smaller peaks are further reduced.
the unnormalized curves, we have R
that
F
Using
= 7.3 and R 2 = 2.9, so
is decreased by a factor of R4 /(R 2 )2 = 0.87. Using
the normalized curves we find that R 2 = 1.054 and R
= 0.97.
Apparently, in this range the broadening of the major peaks
near their bases adds significantly to the sum of squares
and more than makes up for the contributions lost by the
reduction of the minor peaks.
Therefore, the flatness factor
decreases.
We now consider the curve of
Y6
for
=
0.55.
This
curve is significantly different from the others; it lacks
the aforementioned symmetry, and its positive peak is bigger
than its negative peak.
sort of behavior.
This clearly represents a different
Examination of curves corresponding to
intermediate values of the forcing indicates that the change
to this type of behavior seems to occur around 9, = 0.54.
.No particular change in the profile of F(Y)
can be seen to
occur at 92 = 0.54, so that the idea that changes in
F
might
indicate regime changes is not supported here.
ie now return to a consideration of Figures 9 and 10
which are plots of sections of time series of Y6 for
and 4.0 respectively.
92. =
2.0
At these higher values of the forcing,
the periodic behavior seems to have disappeared.
ievertheless,
the appearance of these curves is not all that different
from that for
92
= 1.0 in some respects.
Large positive
peaks are always separated by large negative peaks, and, even
though the spacing between successive peaks is not constant
now, the average spacing between successive peaks is not
much different from what it was for lower values of the
forcing.
when
In an effort to determine if the curve for Y6
=
2.0 was possibly periodic with a very long period,
the equations were integrated for 200,000 time steps, and
after the initial 2000 steps the time between successive large
positive peaks was tabulated to see if any pattern could be
discovered.
No consistent pattern was discovered, but a
histogram of the time between successive large positive peaks
was assembled and is shown in Figure 13.
The range of values
is not very large; the shortest time between successive
positive peaks was about 253 time steps (25.3 units), while
the longest spacing observed was 284 time steps (28.4 units).
The average spacing for this sample of 740 was about 26.75
units which is quite close to some of the constant spacings
that had been observed with lower values of the forcing.
The reason for the distinct peaks in the histogram of the
It is possible that there is
spacings is not known for sure.
still a strong periodic component (or components) that keep
the range of values relatively narrow and also produce the
distinct peaks at some values.
A similar behavior was
observed at 9a = 4.0, but the spacing between successive
positive peaks showed a wider spread, ranging between 23.3
and 29.4 units.
'6 was periodic for 9.= 1.0
Since the behavior of
and apparently nonperiodic for
g. = 2.0, it seemed reasonable
that there should be a transition of some sort at some
intermediate value of the forcing.
To investigate this, the
equations were integrated for 50,000 time steps for
3.= 1.0,
1.1, 1.2, 1.3, 1.4, and 1.5, and the spacing between successive
large positive peaks and successive large negative peaks was
tabulated.
asymmetric.
For
=
1.0, the behavior of
periodic but
The minimum and maximum values were about
-0.1080 and 0.0772 respectively.
units.
Y, was
The period was about 26.64
The spacing between a maximum and the next minimum
was roughly 12.3 units, and the next maximum followed after
about 14.4 units.
For
3, =
1.1, the curve of
YC
was still
periodic and asymmetric.
Minima and maxima were about -0.0958
and 0.1022 respectively.
The period was about 26.88 units,
and the spacing was about 12.7 units from a maximum to a
minimum and about 14.0 units from a minimum to a maximum.
For
9, = 1.2, the behavior was still periodic and asymmetric
with minima and maxima of about -0.0911 and 0.1006.
The
37
period was about 26.425 units, and the spacing was about 13.1
units from maxima to minima and about 13.3 units from minima
to maxima.
For 32 = 1.3, the behavior was periodic and
symmetric.
The minima and maxima were about -0.1200 and
The period was about 25.63 units, and the spacing
0.1200.
between minima and maxima was about 12.8 units.
This return
to symmetric behavior was a surprise and raises the possibility that other similar transitions may occur in the range
g2
= 1.4,
the behavior
of forcings between 0.59 and 1.0.
At
was again periodic and symmetric.
The minima and maxima were
-0.1356 and 0.1356.
The period was about 27.376 units, and
the spacing between minima and maxima was about 13.7 units.
For
=
1.5, a new type of behavior was observed.
The
behavior was periodic and symmetric but was more complicated
The complete period was now
than that encountered earlier.
much longer, about 79.175 units, and involved three distinct
types of large peaks which appeared both above and below the
line of zero amplitude.
We denote these three types of
peaks as A, B, and C and attach a sign after them to indicate
whether the peak is positive or negative.
The magnitudes
of
peaks A, B, and C are about 0.1682, 0.1305, and 0.1564
respectively.
time in
If we define
going from X1to X2,
(X1 ,X2 ) as the separation in
then we can concisely describe
the interrelation of the large maxima and minima.
The order
of the peaks of course is cyclic and is given by A+, C-, 3+,
A-, 0+, 3-.
vie have S(A+,C-)
and S(B+,A-) = 12.05 units.
=
12.95,
S(C-,B+) = 14.5875,
Because of the symmetry of the
curve, this provides us with all the information on the
relation of the peaks.
The spacing between successive peaks
of the same sign for peaks
26.637
units respectively.
A,
3, and C are 27.5375, 25.0, and
Presumably the range from g,= 1.5
to 2.0 contains transitions to other types of behavior, but
this has not yet been investigated.
It should be noted that for
S
greater than about 0..2,
bands 2 and 5 both exhibited statistical behavior with respect
to the odd moments which sets these bands apart from the
others.
Y:,
Y.
and
Y5
consistently had positive means with
0.16 and Y5 somewhaT smaller.
Y5 was
The skewness of
consistently positive and usually had a value somewhere
between 1 and 2.
TLhe skewness of
Y
was consistently
negative with a value in the neighborhood of -0.2.
The odd
moments of the other bands exhibited no such consistent
behavior.
'e next investigated the possibility that the high values
of
F
in band 6 could be due to the truncation of the model at
that scale; perhaps if more bands were included the intermittent behavior would disappear.
To check this possibility,
the system was expanded to 10 bands and flatness factors for
each band were computed at several values of the forcing
between 0.25 and 4.0.
The equations were integrated for
12,000 time steps, and statistics were compiled during the
final 10,000 steps.
The results are summarized in Table 2.
Comparison of the results in Table 2 with those in Figures
4a, 4b, and 4c indicates that little has been changed in
39
bands 0 through 6 by mhe addition of the three new bands.
The intermittency we have observed in band 6 was not due to
the truncation of the model at that scale.
more dramatically how
grows smaller.
F
4e also observe
increases as the scale considered
40
"Very High Forcing" Experiments
IX.
rome indication of the behavior of YG at substantially
higher values of the forcing was desired.
The use of higher
forcing required a reduction in the time step to avoid
computational instability and meant that much more computer
time was needed to get more or less reliable statistics.
The equations
For this reason, only a few runs were made.
were integrated for 200,000 time steps which was close to
the maximum possible with the available time.
are summarized in Table
3, and
some values for lower
forcings are also included for comparison.
asymptotic behavior of
hold up through g
for
g
=
F
The results
The apparent
speculated about earlier seems to
12.5, but the flatness factors computed
=
25.0 seem to represent departures from this trend.
However, since the reduction in the time step meant that a
much shorter portion of the curve was being sampled, the
statistics for
g. =
25.0 may not be reliable.
Examination
of the spacing between successive large amplitude peaks for
these forcings indicates that the "regular" behavior that
had been observed up to
g,=
4.0 (in which each pair of
large peaks of one sign had a large peak of the other sign
in between) no longer seemed to hold at these higher
forcings.
In fact, it was not uncommon to have several
large peaks of one sign without any intervening large peaks
of the opposite sign.
41
A.
Experiments with Forcing in 1and 3
;s a variation on the previous work, the forcing was
moved from band 2 to band 3, and statistics over 198,000
Some of these results are
time steps were compiled.
The behavior of Y; is
summarized in Figures 14a and 14b.
again intermittent.
4
corresponding to
cxamination of partial time series of
3
=
0.3, o.4, o.5, 1.0, and 2.0 failed
to reveal any periodicity.
In fact, the occurrence of large
peaks was far more irregular than had been observed
previously when the forcing was in band 2.
The statistics
also did not seem to converge nearly as well as they had
when the forcing was in band 2.
Rather, the statistics
often seemed to wander about.
6
was highly skewed to the
right which supported earlier ooservations that those bands
removed by a factor of three from the forced band exhibited
exceptional behavior in their odd moments.
There were no
large amplitude negative peaks, and the spacing between
successive large positive peaks was extremely irregular,
ranging from 0.06 to over 850 units.
An interesting feature
of Figures 14a and 14b is the drop in F(YK) at 83
can be seen on many scales.
=
0.8 which
42
XI.
"Random Forcing" Experiments
In an attempt to determine whether some strong periodic
component was responsible for The characteristic periods of
roughly 25 to 28 units that were often observea, the following
experiment was devised.
The equations were integrated for
102,000 time steps with an average forcing 3,of 1.0.
The
forcing at each individual time step was taken either to be
0.0 or 2.0, and the particular forcing that was used was
determined randomly at each time step.
was no longer observed.
Periodic behavior
ihis feature may then depend on the
constancy of the forcing in time.
However, the spacings
between successive large maxima and between successive
large minima were tabulated, and a histogram of the spacings
between successive large maxima occurring during the final
100,000 time steps was assembled.
This appears as Figure 15.
iviany of the values are concentrated in a band centered on a
value which is only slightly higher than that which had been
observed earlier.
separate peaks.
;ithin this band there seem to be several
It seems reasonable that the band would
broaden due to the fact that the forcing is not constant in
time.
The complete absence of values between 41.0 and 50.0,
along with the presence of some values corresponding to
periods that are twice as long as the periods in the main
band, seems to argue rather strongly for the presence of one
or more periodic components.
The large peak at much lower
values was examined, and it was found that these values
almost always occurred when two large positive peaks were
43
not separated by a large anplitudue negative peak.
This would
probably occur when the forcing of 2.0 was on for several time
steps in succession not long after a large positive peak
had occurred.
44
XII.
Jxperiments with "Scale
Independent dissipation"
As a final variation, some runs were maae for a system
of equations in which the dissipation terms of the form
f
YV
were replaced by -4..Yx
so that the constant in
the dissipation term was the same for all the bands.
Other-
wise, the equations were identical to those used elsewhere.
The value of
A
= 0.125 so that the
was.taken to be 2
dissipation in the forced band would be equal to what it
had been in the earlier experiments.
These new equations
were integrated for 12,000 time steps with statistics
compiled during the final 10,000 steps for a number of
forcings between
ga
=
0.1 and
g;=
4.0.
The results appear
in Figure 16.
For values of the forcing between 0.2 and 0.6,
F(Y6 )
seems to increase steadily and reaches a value greater than 5.
However, there is a distinct arop by the time the forcing
reaches 0.7, and
F(Y
6)
seems to show no tendency toward
attaining high values of
F
as the forcing is increased
beyond this value within this range.
Incidentally, the
drop at 0.7 can be seen on all scales and might indicate
some large change in the nature of the solution.
These
results suggest that the higher dissipation we had in
the bands representing smaller scales is at least partly
responsible for the intermittent behavior observed in
those scales.
XIII.
Final Remarks
45
The existence of intermittency has been demonstrated in
the numerical solutions to a system of nonlinear differential
equations similar in form to a set that has been usea to
model some aspects of two-dimensional turbulence.
The system
has been shown to possess several different modes of behavior
which include:
a steady state solution;
symmetric periodic
solutions; asymmetric periodic solutions; nonperiodic solutions which seem to possess a fair amount of structure and a
strong periodic component; and fully nonperiodic solutions.
Although we cautioned earlier that we did not claim to
be studying any real physical system, some of the results
strongly suggest an affinity to observed processes.
The
intermittent peaks in the nonperiodic solutions are suggestive
of turbulent bursts in real flows.
The peaks observed on the
small scales could also be identified with the concentration
of vorticity in small regions which Batchelor (1953) points
out is a common tendency of hydrodynamical flows.
The transi-
tions between different modes of behavior as the external
forcing is changed is suggestive of any number of physical
processes in which changes of regime occur as some parameter
exceeds critical values.
Table la
ea
.0175
.OloO
.0105
.0190
.01)5
.0200
.0205
.0210
.0215
.0220
.0225
.0230
.0235
-u240
.0245
.0250
.0255
.0260
.0265
.0270
F(Y 0 )
5.5
4.5
3.5
2-5
1.0
1.5
1.8
2.5
3.5
4.5
5.0
3.5
2.4
1.9
1.7
1.6
1.5
1.5
1.5
1.5
F(Y 1 )
5.7
4.6
3.5
2.5
1.8
1.5
1.8
2.5
3.6
4.6
5.1
3.5
2.4
1.9
1.7
1.5
1.5
1.5
1.5
1.5
~1013
~10 13
.106
11.5
6.8
2.9
1.6
1.8
2.5
2.7
1.7
1.5
1-5
1.5
F(Y
n
)
-1012
F(Y 3 )
5.6
4.6
3.5
2.5
1.6
1.5
1.8
2.5
3.6
4.7
5.1
3.4
2.4
1.9
1.6
1.5
1.5
1.4
1.4
1.4
F(y
)
5.6
4.5
3.5
2.5
1.6
1.5
1.8
2-5
3.5
4.7
4.9
3.3
2.2
1.8
1.6
1.4
1.4
1.4
1.4
1.4
F(Y )
14.1
11.4
6.7
6.1
3.3
1.5
3.2
5.9
8.6
12.1
11.3
5.0
2.5
1.8
1.6
1.5
1.5
1-5
1.5
1.5
-
15-3
10.4
5.4
2.3
5.2
10.2
14.9
21.6
18.9
8.3
4.4
3.2
2.6
2.2
2.1
2.1
2.0
2.0
F(Y
2
4
-
012
1012
1013 -v1u 13 -',
C?7
8C-
7
901
_
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560T
o oZ
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T*?T
9 09
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'I*
117'I
T,6C
f7 99
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6o9q
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z6o1
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Pd7 'I
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V@9T
TCT
P *6
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51'0
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0*TT
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cg'?
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'I,0
1761
51ZI
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.()4
(
1)qP-T,
A x)
( ') )J
(OX),4
2
Table
F(Y 3 )
F(Y)
F(Y
3.03
1.92
3.85
6.05
10.5
1.39
3.12
1.96
4.32
7.26
14.1
2.01
1.38
3.01
2.00
4.05
8.09
19.1
0.025
1.95
1.37
2.89
2.11
3.81
7.97
21.8
12.5
0.025
1.97
1.35
2.89
2.17
4.11
7.85
19.9
25.0
0.025
2.07
1.46
3.90
3.07
7.83
13.2
29.6
25.0
0.0125
2.11
1.55
4.02-
3.29
8.55
14.1
27.6
g2
At
F(Y O )
F(Y)
F(Y
1.0
0.1
1.98
1.41
2.0
0.1
2.06
4.0
0.1
8.0
2
)
5
F(Y )
6
50
Figure la:
igure 10:
Curve witn well aefined int-ermittency factor
curve witiout a well defineu intermitcency factor
(x.+a'-x)
(x-x.)
X
-f
-(X.+dc'
.. (X-X.-aZ)
Figure 2:
Ourve with flatness fac tor
approaching n + 3/4 as n increases
50-
20
--
F 10
5
a
-
2
1
10
20
50
100
200
500
101
2x 103 5x10 3
104
as function of
Flatness factor of
F igure 3:
- eynolads number -R (from Cuo ana. Corrsin, 1971)
44444-
. ....
H"
4-4-44-4-4
I
o F (Y)
..............
x F (Yo)
T
m1m
Matflt
444 4444
-m-"
r
umii
I
I tr .. mrnt
d1ittttitHttt
-
41-
I
-+++H++
..... .
44-
i
4-444 t+4-444-1
1444m
111
niM
i~
m
M l
"_TT
tMMMM
1t-Iit
t
lmtjiltmmmttiti
i iTrT
l l~
4--4
4
TTT
TTII
H-H--++4-f~~+
III
iHt
4l
i
- 4ittti
4-4
If
t
ttf
3
-F(Y 3 )
+1
2
++
4jl
471
#
TV-
-~
rrllT4bw4TTnTl-1m' 4T~ThJ134
Li
t
4 4t~4
Ith
V
4-f
4
L-
l4
:-+
4
2.0
Flatness factor of Yk
figure 4a:
of forcing g2
(k = 0, 1,
4-0
3.0
2.5
2, 3)
as
function
2
I- I I IIIIT~
11U4411i1
I I~ II
I II
II IIrrIIrII
II IIrrII I
I Ii TrtrrIrrrIlI
.. . . . . .. . . . . . . . . . . ..
+-4-- -++444+
-
-H-H
t
frff--tH-Htt--Ht1itii--
I
..........
-- I
x F(YS)
0
-fr
4
F(Y+)
6
rllrllltl"-T"Itl,-m"-"-I-"t-l
'I
-L - - -----
4
Til
T1
44
I
w~-rn4T
tt
IIWThtiHI±±±itP H Html-fm
~It
4
4
+
-tmztfj~lt
-tH -*---M
iff
4-4-
Hi --H--H--iIF-H-H-
414T
Nil
ItI*IItI IIM
++HIH+HIE=
go.
1.0
±Figure 4b:
forcing g
2
15
2.0
2.5
Filatness factor of Yk (k = 4,
3.0
5)
as
TI-ti
fft44 {TF* itM4
3.5
function of
4.0
j
7-.7::
'"
T
17
..
;
15
....
.
...
-
-;
..-.
-
7'
A:r
F(Y~)
I
.........
*.t
-.
t
.. . ....
I
-
1 3
YF.-5
;
-,
5~
1
4.
I~I-T11
7 -t f
L
J
-
9
-
7-i;t
1
I~
O
v'ig -ure 4c:
1.0
..
1
~
1.5
-
-7
....
.....
.
2.0
iilatrie s fac-uor of Y
2.5
3.0
A.
4.0
as function of forc ing
(Ji
igure
5:
Time series of Y6 for g2 = 0.3
Figure 6:
Time series of Y
for g2 =
-,020
-.035
10
15
Figure 7:
20
25
30
35
40
Time series of Y
45
for
50
55
02
60
65
58
Y6
.11.10.
.09
.08
.07
.06
.05
.04.
,03.
.01
"'".11.
5
10
15
20
iigure o:
25
30
35
40
45
50
55
60
65
70
Time series of TC)'rtfor g2 =
75
80
85 90
59
S.15
60
Figure 9:
70
Time series of Y-C01-for g2 = 2.0
60
Figure 10: Time series
of Yo) for 4 2 = 4.0
ii
~ ~~
-
11 Y(F
)
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~
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r~1rrT.
1
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7
74
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.35
li;iure
of Y
12a:
"±ormalizeu" time series
for various values of forcin
Lacn curve
lauelleu witln value of'
.
5g
of for
Y
varo
va-u
- ftl~cf
~anKrv
1ue1~ ;i~;ASco
.35
AS
.45S
e igure 12c:
"Yormalizea" time series
of Y for various values of forcing2
,Laen curve
lauelleu with value of g2'
60
number of
occurrences
40
20
-
--
0
250
255
260
265
270
275
280
285
tirfme steps
histogram of number of time steps between
igure 13:
g2 - 2.0
successive large positive peaks of Yfor
( At = 0.1 time units)
-
I
I
F(Y0)
2-
1-
-.
-
3b
it
P
7
I
Si
I
:1:-I
2
7
t'
I
T
-
....
.....
I
fl'
I.
i:j
.....
F (Y)
-71 T
F(Ya)
1
F (Y3)
i
I
2
-1
T---1
2
-q
1
...
*.
-
..
E
I
I
2
...
..
-.
.S
1.0
I.
[..
-.
1.5
2.0
nLi~
J-:l. LIL J*'j
t
2.5
JJ
3.5
1.0
F
I
_J
1
4.0
,3
Figure
14a:
Flatness factor of Yk (k
function of forcing g
0,
1,
2,
4, 4)as
.I..
.,
14-13
t
-
11
1
p-
-:
;T
7
_4.
..- T I:
....
Kh-
J__
1 3
:Q
....
*~
*1~~~.. ....
17.
11.
,1.
111
-!
'T ,
-:
-I
T
0,
~ ~ ~.I7'-7 .....
--
..
xI
:
)F
5-
*
:
7a 7
71-
1
(Y
6-
.
....
T
;*-
7.YO
F
-
I-
... ..
n
H
I
12
7!7-
- - -:
II..II
7
-:
T11
[
-
4.
6
0' 5
2
-
..
0
7
7O
----
..
05
t
4
53
52
1L0
3,5
2.0
2.5
3.0
3.5
4,0
53
Figure 14b:
of forcink
i'laLness factor of Y
(k =,
6) as
function
number of
occurrences
40-
30-
20-
time sLeps
figure 15:
Histogram of number of Lime steps between
successive large positive peaks of Y with "ranuom forcing"
(At = 0.1 Lime units)
wnen T-2 = 1.0
3
x F (Y.)
IF-
--
Tx
-: -i
t
F(Y)
-
7:I
:*,-3
1-
2
F (Yx)
-
t
7
F
- F(Ye)
II
-
SF(Y3)
3-
-
ttI
I
I
F(Ys)
I *~..
2
56
F('4)
.
I
.
.
.
I
-
4--
2
-
0510
vFigure
16:
Flatness
15S
factor
of
2.0
yk
(k
2.
=0
to
-
t.
as
function
of' forcing9 g2 for experiments with "scale independent
dissipation",
3.5
-40
5l£LI0RAPHY
batchelor, 3.
K.,
The theory of homogeneous turbulence.
1953.
Cambridge, Cambridge University zPress, 194 pp.
Gibson, C. H.,
G. R. Stegen, and R. B. Williams, 1970:
Statistics of the fine structure of turbulent velocity
and temperature fields measured at high Reynolds
number.
Gordon,
J. Fluid
C. M. 9
ech.,
41,
Intermittent momentum transport in a
1974:
geophysical boundary layer.
rieathershaw, A. j.,
liature,
248,
153-167.
1974:
\ature, 24,
392-394.
"bursting" phenomena in the sea.
394-395.
Kennedy, j. &., and S.
Corrsin, 1961:
6pectral flatness
factor and "intermittency" in turbulence and non-linear
noise.
j. Fluid iviech.,
Kuo, A. Y.-S.,
10,
336.
and S. Corrsin, 1971:
Experiments on internal
intermittency and free-structure distribution functions
in fully turbulent fluid.
Lorenz,
. N.,
1971:
J. Fluid iviech.,
285-318.
An ii-cycle time-aifferencing scheme for
stepwise numerical integration.
644-648.
J,
ion. Weather Rev., 29,
Lorenz, E.
±.,
1972:
Low order models representing
realizations of turbulence.
Lumley, J. L.,
1970:
j. Fluid 7,ech.,
55, 545-5b3.
*Stochastic tools in turbulence.
New York, Academic Press, 194 pp.
MIollo-Christensen, E.,
turbulent flows.
Ramage, C. S.,
Bull.
1976:
Amer.
Sandborn, 3. A.,
1959:
D.,
Intermittency in large-scale
Ann. Rev. Fluid Mech., 3,
101-118.
trognosis for weather forecasting.
iieteor.
a boundary layer.
Siggia, E.
1973:
Soc.,
57, 4-10.
Intermittency of turbulent motion in
J. Fluid inech.,
and G. 6. Patterson Jr.,
6, 221.
1977:
Intermittency
effects in a numerical simulation of stationary threedimensional turbulence.' National Center for Atmospheric
Research, unpublished manuscript.