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Unformatted text preview: Jim}: 1955 J . W. HUTCHINGS 263 TURBULENCE THEORY APPLEED Ti} LARGESCALE
ATMOSPHERIC PHENGMENA By J. W. Hwtchings New Zealand Meteorological Service (Manuscript received 5 August 1954) ABSTRACT An outline is given of a preliminary attempt to apply the methods which have proved serviceable in the
treatment of smallscale turbulence to the wider problem of largescale turbulence in the atmosphere. In
particular, application is made of theoretical results giving the form of the correlation function for velocity
ﬂuctuations in the inertial range according to the theory of Kolmogoroif. It appears that some of these
results may be Valid over a considerably greater range of eddy sizes than might at ﬁrst have been antici
pated. Some attempt is also made to apply these methods to the determination of the correlation functions
for largerscale ﬂuctuations of pressure and temperature in the atmosphere. Actual data taken at the earth’s
surface and in the free atmosphere are used to test the validity of these deductions. 1. Introduction During the last few years, there has occurred a
renewed interest in some of the fundamental problems
of turbulence, and many important results have been
obtained regarding the turbulent ﬂows encountered in
aerodynamic and hydraulic laboratories. These inves
tigations have naturally been concerned with turbu
lence on a relatively small scale, the scale being
speciﬁed in advance by the dimensions of the appa
ratus used. Much of this work can be directly extended
to the investigation of the turbulent ﬁelds occurring
naturally in the atmosphere, and in this way many
important properties of atmospheric turbulence have
been obtained. This renewed interest in smallscale turbulence may
in some measure be traced to the fruitful concepts ﬁrst
put forward by Kolmogoroff (1941) and later inde
pendently set out and ampliﬁed by von Weizsacker
(1948) and Heisenberg (1948). The hypothesis ad
vanced by Kolmogoroff enabled speciﬁc quantitative
predictions to be made about the values of many of
the important statistical parameters which charac
terize an arbitrary turbulent ﬂow, and made possible
a comparison of theory with observation. This circum—
stance has contributed much to the recent progress in
this ﬁeld. In meteorology, these advances in smallscale tur
bulence theory have been accompanied by a gradually
increasing awareness of the importance of scale (Priest
ley and Sheppard, 1952) in dealing with atmospheric
problems. There has been, too, a growing acceptance
of the view that the atmosphere is essentially a large
scale quasihorizontal turbulent ﬂowﬁeld and that
many fundamental meteorological problems are ca
pable of being formulated and interpreted by means
of turbulence theory. This View was ﬁrst put forward by Defant (1921) for the surface phenomena of the
temperate zones of the earth, but has since found
wider application in a great variety of investigations
concerning transport phenomena in the atmosphere.
Indeed, it seems as if one of the most fruitful ﬁelds for
the application of modern ideas on turbulence lies in
the upper atmosphere, away from the disturbing effects
of orography and free from the marked dissipative
effects of smallscale turbulence in the boundary layer
near the surface of the earth. ' Nevertheless, up to the present, there has been little
or no attempt to make speciﬁc and numerical applica
tion of the results obtained in the domain of small—
scale turbulence to the wider problem of large—scale
quasihorizontal atmospheric turbulence. On this scale,
the general circulation of the atmosphere itself must
be considered as the statistical mean ﬂow. This article
represents a preliminary attempt to treat the large
scale phenomena of the atmosphere along lines stem—
ming from modern research in the theory of turbulence
and may serve the following purposes: 1. To show how recent developments in smallscale turbulence
theory can be applied to a much wider range of turbulence ele
ments than might at ﬁrst have been thought possible; 2. To draw the attention of more meteorologists to recent
developments in turbulence theory, and to suggest that much
already existing statistical material can be given a new interpre
tation along these lines; and 3. To indicate the need for further extensions of turbulence
theory in order to approach some of the important problems of
dynamic meteorology — for example, the problem of geostrophic
departures and the question of the transfer of energy between
atmospheric motions of different scales. 2. The internal structure of a turbulent ﬂuid Batchelor (1953) has recently given a comprehen—
sive discussion of the modern approach to the theory
of homogeneous and, in most applications, isotropic 264 turbulence. The structure of a turbulent ﬂuid will
therefore be discussed here only brieﬂy. According to the View put forward by Kolmogoroff,
any fully developed turbulent ﬂow may be thought of
as a superposition of pulsations or velocity ﬂuctuations
of different magnitudes, varying from what might be
called changes in the mean 'ﬂow down to the very
smallest ﬂuctuations belonging to the state of laminar
ﬂow. For any ﬂuid ﬂow, it is a well established prin—
ciple that when the Reynolds number exceeds a certain
critical value the ﬂow becomes unstable for small dis
turbances, so that a superimposed ﬂow in the nature
of a pulsation or ﬂuctuation with characteristic dimen
sions and velocity is generated. Kolmogoroff’s View is
based on the application of this principle to each
superimposed ﬂow, so that with increasing Reynolds
number each superimposed ﬂow in turn becomes 'un
stable and further disturbances are generated. Thus,
at sufﬁciently high Reynolds numbers, ﬂuctuations of
all magnitudes are present. This expresses the well
established intuitive idea that the larger eddies of any
turbulent motion break up into a series (or cascade)
of smaller eddies, which in turn break up, i.e., pass
their kinetic energy on to eddies of even smaller scale.
Kolmogoroff further supposed that, as the energy
passes down the system of eddies, the direct inﬂuence
of the larger energy—containing eddies is gradually lost,
so that all eddies smaller than a certain critical size
have statistical properties which to a large extent are
random. It follows, then, that as long as we deal only
with eddies smaller than this critical size, the turbu
lence may be considered approximately homogeneous
and isotropic. In Kolmogoroff’s terminology, we may
say that any ﬁeld of turbulence is locally isotropic and
the range of eddy sizes over which thisrcondition is
fulﬁlled may be called the range of local isotropy. Since almost all of the dissipation of energy by
viscosity takes place in the smaller eddies, and since
these must adjust themselves so as to convert all the
energy they receive into heat, it is evident that the
statistical state of the small eddies must depend
entirely on two parameters: 1. The average rate at which energy is handed on to them from
larger eddies (which is approximately identical with the total
average rate of energy dissipation by viscosity, e); and 2. The kinematic viscosity, :1, which determines the rate at
which the small eddies can convert kinetic energy into heat. As the Reynolds number increases, smaller and
smaller eddies are brought into existence; and at very
high Reynolds numbers, the larger eddies of the range
of local isotropy will contribute very little to the dissi
pation of energy by viscosity. There will thus exist at
the largeeddy end of the range of local isotropy a
subrange in which the statistical properties of the
eddies are independent of v and depend only on c.
This range of eddies is dominated purely by inertial JOURNAL OF METEOROLOGY VOLUME 12 forces and may be called the inertial range. In this
article, we will be concerned almost entirely with con
ditions in the inertial range. 3. Correlation functions in the inertial range As we are considering conditions in the inertial range
only, we may consider a ﬁeld of turbulence which is
approximately isotropic and statistically steady in
time. It then proves convenient to introduce the longi
tudinal velocity correlationfunction f (r) and the lat
eral velocity correlationfunction g(r) for two points a
distance r apart in any direction. With use of the bar
symbol to denote averages over a sufficiently long
period of time, these functions are deﬁned by u.(A)_ 1MB) 2
“12 f0) = un(A$)t_:2ln(B), where up(A), 14,,(B), 14,,(A) and un (B) are the turbu
lent components of velocity at the two points A and
B, respectively, measured parallel (sufﬁx p) and nor
mal (sufﬁx n) to the vector separation AB. In isotropic
turbulence, 14—1,? = u_,,2 = 142. The introduction of the equation of continuity for
an incompressible fluid leads, as was shown by von
Karman and Howarth (1938), to the relation g = f
+ %r (if/61' between the functions f and g. If the tur
bulence is taken to be isotropic in two dimensions
only, the same reasoning as employed by von Karman
and Howarth leads to the relation g = f + r (if/6r
between the functions f and g. These functions, f and
g, may be called Eulerian space correlationfunctions
and either may be denoted by R(r). In the inertial range, R(r) must depend solely on e,
and dimensional analysis shows that the form of R(r)
is then given by u2:1 — R(r)] ~ eiri. Introducing the
above relations between f and g, we ﬁnd that, a being
a constant, f (r) , and g(r)= 1 — mg, and g(r) = 1 — garg or
= 1 — gaﬁ, according as the turbulence is taken to be isotropic
in three Or two dimensions. The ratio (1 — f) / (1 — g)
is thus 0.75 or 0.60, according as the turbulence is
three or twodimensionally isotropic. In wind—tunnel turbulence, the turbulent ﬂuctua
tions are normally only a few per cent of the mean
speed of ﬂow parallel to the sides of the tunnel, and
it is a plausible assumption to consider that the turbu
lent ﬁeld is simply carried downstream past a single
observation point at a constant rate U equal to the
mean speed of ﬂow. The space interval r then corre
sponds to a time interval 7 = r/ U, and it is possible
and practically very convenient to deﬁne longitudinal
and lateral velocity correlationfunctions f ( r) and g(r)
by using values of the velocity components parallel JUNE 1955 J . and normal to the mean ﬂow at a single point in space
but separated by an interval of time (lag interval) 7.
These functions may be called Eulerian time correla
tionfunctions, and either may be denoted by R(r).
In these circumstances, a change of variable shows
that R(r) and R(r) have the same form and that all
the above relations are true when r is substituted for r.
In other circumstances, this assumption of the equiva
lence of space and time correlations is much more
questionable: but it will appear that in the particular
case of largescale atmospheric turbulence, the above
assumption represents at least a crude approximation
to actually occurring conditions. 4. Application to largescale atmospheric phenomena In recent years, the worldwide extension of the
regular network of aerological soundings has demon
strated that at middle and higher levels the moving
cyclonic and anticyclonic vortices, which are such a
prominent feature of the circulation of the lower
atmosphere in temperate latitudes, are replaced by a
largescale system of troughs and ridges in a strong
and extensive westerly ﬂow. Average annual or seasonal
charts show that, with the exception of minor or local
asymmetries, the mean ﬂow at upper levels over large
portions of both hemispheres is very nearly zonal, the
whole circulation being dominated by vast cyclonic
vortices located near the poles. It thus seems possible
to consider many of the familiar features observed on
daily or twicedaily upper—level synoptic charts as
elements in a huge zonal turbulent ﬂow. These turbu
lence elements will consist of motions with all different
lengthscales, ranging from the large slowmoving
troughs and ridges, through the lesser and quicker
moving synoptic systems, down to the smallest ﬂuc
tuations that can be revealed by the density of the
aerological net used in constructing the charts. In
these circumstances, the mean ﬂow will correspond to
an average of the individual (once, twice or even four
times daily) patterns taken over a long period of time.
The time of averaging will thus be measured in months
or even years. This may be compared with averaging
times of the order of minutes commonly used in wind
tunnel and smallscale atmospheric research. This (largely kinematical) view of the problem may
be regarded as an extension of the ideas of Defant
(1921) to the atmospheric circulation at higher levels
and as an application of the philosophical viewpoint,
ﬁrst put forward by Richardson (1930) in connection
with diffusion, that turbulence is essentially a compen
sation for smoothing and that, as a consequence, the
kind of turbulence encountered is greatly dependent
on the kind of smoothing (or averaging) employed.
But these views are no doubt oversimpliﬁed; and to
arrive at a more realistic conception of the applica
bility of turbulence theory to the largescale atmos W. HUTCHINGS 265 pheric phenomena, it is necessary to consider some of
the more important aspects in which well deﬁned
differences exist between large and smallscale turbu
lent motions. The most important of these considera
tions appear to be: 1. The use of the continuity equation for an incompressible
ﬂuid; this implies certain restrictions on the corresponding large
scale atmospheric ﬂow; 2. The extent to which largescale turbulence may be consid
ered sufﬁciently homogeneous and isotropic; 3. The inﬂuence of the Coriolis force on motions of larger
scales; 4. Uncertainties regarding the sources of energy of the large
scale velocity ﬂuctuations observed in the atmosphere; and 5. The assumption of the equivalence of space and time corre
lationfunctions. This last consideration is not a difficulty in principle
but is of some importance from the point of view of
the data normally available for practical calculation
of correlation functions. In the following subsections,
these ﬁve questions will be brieﬂy discussed. Assumption of incompresstbility.—Recent theoreti
cal studies (Charney, 1948) have tended to show that
the largescale motions of the atmosphere at middle
levels may be regarded as approximately horizontal
and nondivergent. From a kinematic point of View,
it would thus appear probable that many of the results
obtained for the turbulent motion of a homogeneous
incompressible ﬂuid could be applied without serious
modiﬁcation to the problems of middlelevel large
scale atmospheric turbulence. It would also seem
probable that the SOOmb level would be suitable for
such applications. Homogeneity and isotropy in large scale turbulence.—~
It has already been emphasized that the application
of Kolmogoroff’s hypothesis‘involves only the less
stringent condition of local isotropy and does not
require that the complete turbulent ﬁeld be accu
rately homogeneous and isotropic. Some observational
evidence regarding the extent to which the complete
largescale turbulent ﬁeld in the atmosphere may be
regarded as homogeneous and isotropic may be ob
tained from the work of Brooks et at (1950). Although
only a limited amount of data was discussed, much
evidence was put forward by Brooks et al to show that
in the free atmosphere, away from the effects of orog
raphy, the distribution of the vector deviations of the
instantaneous wind vectors from the longterm mean
vector wind is approximately symmetrical and circu
lar. These results may be interpreted as indicating a
certain measure of isotropy in actuallyoccurring large
scale atmospheric turbulence. The symmetrical circu
lar distribution implies that the largescale zonal
turbulent component of the wind, u, and similarly the
meridional turbulent component, 2), are normally dis
tributed with equal standard deviations 0'“ and a,”
respectively. Some independent computations of a“ 266 and 0,, together with the average westerly (U) and
southerly (V) components of the wind at the 30,000ft
level are given in table 1 for Auckland, New Zealand
(36°47’S, 174°38’E), and for Nandi, Fiji (17°45’S,
177°27’E). It will be seen that the relative magnitudes
of a“ and 0., vary somewhat from month to month,
but that at Auckland there does not appear to be any
marked systematic difference between them, so that
over a long period of time it might be expected that
there would exist at the level an approximate equality
between a“ and en. Conditions at N andi are less cer—
tain, and more computations would be necessary to
determine if the apparent systematic differences be
tween au and (7,, would persist over a long period of
time. In this article, no further use is made of data
for N andi. Batchelor (1953) has shown theoretically that, for
smallscale turbulence in a homogeneous incompres
sible ﬂuid, the effect of pressure forces on the turbulent
motion is to transfer kinetic energy between the differ
ent directional components of the motion and thus
lead to an approximate equality of turbulent kinetic
energy in each direction. In the atmosphere, the ap
proximate equality of 0.. and (7,, may doubtless be
attributed at least in part to the same process. Nevertheless, the results of Brooks et al (1950) imply
that u and v are uncorrelated, thus leading (Hutchings,
1952) to zero meridional ﬂux of absolute angular mo
mentum across any parallel of latitude. Actually, the
theoretical necessity for a poleward ﬂux of absolute
angular momentum over most of the earth is well
recognized, and practical calculations using observed
wind data have amply demonstrated the reality of the
ﬂux and have also determined its approximate magni
tude. From these results, we may infer the magnitudes
of the correlation coefﬁcients between 14 and v at
various levels in the atmosphere. In general, they
appear to be less than 0.2 at most middlelatitude
stations but may approach 0.3 at the higher levels for TABLE 1. Standard deviations and mean wind components (kn),
30,000 ft, Auckland and Nandi. Cu, an U V
Auckland
July 1950 39.4 25.2 25.0 14.1
July 1951 30.6 30.6 36.0 —0.3
July 1952 35.8 31.8 47.6 —0.6
July 1953 31.4 31.6 47.7 ——11.2
July 1950—1953 35.6 31.2 39.1 0.5
June 1952 31.8 36.4 73.3 —6.6
July 1952 35.8 31.8 47.6 —0.6
August 1952 26.0 21.9 21.1 2.6
June, July, Aug. 1952 38.0 30.6 46.6 ——1.5
Nandi
July 1950 18.2 21.1 35.5 5.8
July 1951 16.2 19.8 35.2 5.1
July 1952 9.7 14.5 28.8 —6.3
July 1950—1952 15.2 19.7 33.2 1.5 JOURNAL OF METEOROLOGY VOLUME 12 stations near 30 deg lat, across which parallel the
maximum transport of absolute angular momentum
takes place. It would thus appear likely that the re
sults for Auckland given in table 1 represent the
greatest departures from isotropy likely to be experi
enced in the atmosphere. With regard to the amount of homogeneity to be
expected in largescale atmospheric turbulence, we
may obtain some information from the charts of
standard vectordeviation given by Brooks et al (1950).
It would appear from these charts that the homo
geneity of large—scale turbulence varies considerably
from level to level in the atmosphere and also varies
somewhat with geographical location and season of
the year. In general, largescale turbulence appears
to be most homogeneous at and above the ZOOmb
level. From the above considerations, we may conclude
that, on the average, largescale turbulence in the
atmosphere is neither theoretically nor practically ex
actly homogeneous and isotropic; but at certain atmos
pheric levels and for areas which are not too large, the
requirements for homogeneous isotropic turbulence
will be approximately fulﬁlled. . Inﬂuence of the Coriolis force—As is well known, all
atmospheric motions whose periodic times are of the
order of half a pendulum day are strongly inﬂuenced
by the Coriolis force arising from the earth’s rotation.
Again, the variation of this force with latitude is
known to exert an appreciable controlling inﬂuence on
the largescale ﬂow patterns observed in the atmos
phere. However, the Coriolis force, although in the
nature of an inertial force, depends linearly on the
velocity and would not therefore be expected to con
tribute directly to the statistical decay of turbulent
energy. As pointed out by Batchelor (1953), the effect
of nonlinear inertial forces is to spread the turbulent
energy over an increasing range of wavenumbers. It
may thus be anticipated that many of the results
obtained for smallscale turbulence, particularly those
of a kinematic nature, may be applied directly to the
largerscale problem. On the other hand, it would
appear likely that important modiﬁcations in small
scale theory will be necessary before the results will
be fully applicable to largescale ﬂuctuations of dy
namical quantities such as atmospheric pressure. Supply of energy to atmospheric motions.—Many
basic questions regarding the sources of energy of the
velocity ﬂuctuations observed in the free atmosphere
cannot as yet be answered with any certainty. In
smallscale turbulence observed in a windtunnel and
in mechanically produced turbulence in the boundary
layer, there is no doubt that kinetic energy is com
municated directly to the mean ﬂow and larger eddies
by what may be regarded as an external agency. This energy is then passed down the system of smaller and
smaller eddies and ﬁnally dissipated irreversibly into JUNE 1955 J . molecular motion (heat) by the smallest turbulence
elements. In the free atmosphere, however, it is not certain
that the largescale atmospheric disturbances derive
any major part of their kinetic energy from the mean
ﬂow, and indeed perhaps even the reverse is true.
Very probably both processes take place. At times
there is a transfer of energy from the mean ﬂow to
feed a young and growing disturbance, while in other
circumstances the mean ﬂow may be regenerated from
time to time by the growth of particular largescale
disturbances which are able to transform most efﬁ
ciently gravitational and thermal energy into kinetic
energy. Whether, in a statistical sense, one type of
transfer process can be said to be dominant is not at
present known. In the same way, it is not at present
known if and to what extent the smallerscale atmos
pheric motions may be regarded as deriving all or part
of their kinetic energy from motions on a larger scale.
Nevertheless, from a fundamental point of View it is
necessary to have some information regarding the
energy sources present in largescale turbulent ﬂows,
because without this knowledge it is not possible to
determine beforehand the range of eddies to which
Kolmogoroff's hypothesis applies or even to determine
if such a range exists. Our present lack of knowledge
thus prevents a strictly deductive approach, and it is
necessary to proceed in an indirect fashion. Equivalence of space and time correlations.——The basic
condition for the equivalence of space and time corre
lationfunctions is that each turbulence element, as it
moves past the point of observation, should undergo
only relatively unimportant internal changes. This
condition is probably fulﬁlled when, as in the wind
tunnel, the magnitudes of the turbulent ﬂuctuations
are very small compared with the mean velocity of
ﬂow. The comparative smallness of the ﬂuctuations
no doubt also ensures that the translational speed of
the turbulence elements approximates c105ely the mean
velocity of ﬂow. But the above basic condition is also
capable of being fulﬁlled in cases Where, as in large
scale atmospheric turbulence, the turbulent ﬂuctua
tions of velocity are of the same order of magnitude
as the mean speed of ﬂow and the translational speed
of the turbulence elements differs considerably from
that of the mean ﬂow. The important condition that
needs to be fulﬁlled is that the zonal advection of the
velocity ﬁeld should be more important than the effect
of development in determining velocity changes at a
ﬁxed point. Although there are undoubtedly occasions
on which this is not true, synoptic experience with
upperlevel charts tends to show that, on the average,
this assumption may be a reasonable approximation
to actually occurring conditions. Summary—From the above review of the difﬁculties
likely to be experienced in the application of small W. HUTCHINGS 267 scale turbulence theory to atmospheric phenomena on
a larger scale, it appears that many simple and direct
applications would be likely to give useful results but
that it is not possible to say a priori that all the con
ditions for its strict application are fulﬁlled. Some
doubt exists as to the full effect of the Coriolis force
on turbulence of this scale, but the greatest uncer
tainty involved would seem to lie in our present lack
of knowledge regarding the supply of energy to atmos
pheric disturbances of different scales. In view of these uncertainties, it seems clear that it
would be more proﬁtable to proceed in an inductive
manner by making direct application of the theory to
large—scale turbulence and then deciding, by a com
parison of the relevant theory with observational re
sults, whether or not there does in fact exist a range of
atmospheric motions to which the theory may be
reasonably applied. Indeed, such a procedure might
yield indirect knowledge regarding the processes which
govern the transfer of energy between atmospheric
motions of different scales and might also provide
information regarding the dimensions of atmospheric
motions beyond which the geostrophic (Coriolis) con
trol plays a dominant part. 5. Correlation functions in the free atmosphere The most fundamental statistical functions charac
terizing the ﬁeld of turbulent ﬂuctuations of a par
ticular physical quantity are the power spectral
function for the quantity considered and the associated
correlation function. This latter function is connected
with the spectral function by means of the Fourier
transform relations. Of the two kinds of function, the
correlation function is the more easily computed from
observed data. This section gives details of actually
computed correlation functions for several variables of
interest in largescale atmospheric turbulence. Velocity ﬂuctuations.—In the atmospheric case, the
simplest functions that can be considered are the two
dimensional longitudinal and lateral correlation func
tions, f(T) and g(r), respectively. These may be de
ﬁned by f(7) =@, and g(r) =@’ we) M) where u and 21 represent, respectively, the westerly
and southerly components of the turbulent wind
velocity, and T represents the variable time lag em
ployed. The bar symbol here represents a mean value
taken over a sufﬁciently long period of time, while the
turbulent components are represented by the devia
tions of the instantaneous wind components from their
respective means. When the conditions for the equivalence of space
and time correlation functions are satisﬁed, we may, in
the inertial range, assume that [1 — 13(1)] ~ 1}. 268 There may be circumstances in which the above con
ditions are not satisﬁed, and in this case some infor
mation on the probable form of R(r) may be obtained
from a recent paper by Ogura (1953). Using a simple
onedimensional model of small—scale turbulence sug
gested by the results of von Weizsacker (1948), Ogura
shows that, in the range where the Eulerian space
correlationfunction R(r) is given by [1 — R(r)] N r},
the Eulerian time correlationfunction R(r) may be
expressed in the form [1 — R(r)] ~ 7’”, m being a
complicated function of U (JO—l, where U is the veloc
ity of the mean ﬂow and (JV gives a measure of the
magnitudes of the turbulent velocity components in
the direction of the mean ﬂow. The value of m lies
between 2/3 and 1, the minimum value (m = 2/3)
occurring when U >> (JV and the maximum value
(m = 1) when U << (EV. It is at present not clear
how far this simpliﬁed model of smallscale turbulence
may be applied on a larger scale, but from these re
sults it is to be expected that there will always be some
inaccuracy in assuming the equivalence of space and
time correlationfunctions, even when allowance is
made for the regeneration and decay of the turbulence
elements as is done in the model investigated by
Ogura. Nevertheless, if we attempt to apply these
results to largescale atmospheric turbulence, it is clear
that we must interpret U as .a mean translational
speed for the largescale atmospheric disturbances and
not 'as a mean velocity of ﬂow. We then know that
above the 500mb level U is approximately constant
with height, while the intensity of turbulence as meas
ured by (ﬁaﬁ increases considerably. It might then
be expected that, if R(r) is identiﬁed with R(r), values 6 I2 /8 24
1' HRS. JO J‘ 4! FIG. 1. Longitudinal (westerly) (A) and lateral (southerly) (B)
correlation functions at 500 mb, Larkhill. Circles denote observed
values, full lines theoretical linear relation, slope 2/3. JOURNAL OF METEOROLOGY VOLUME 12 TABLE 2. Correlation functions, f and g; 500 mb, Larkhill,
December 1944—February 1945. 0'“ = 29.8, m; = 21.4 kn.
Time lag, 7:
(hr) 6 12 18 24 30 36 48
f: 0.904 0.817 0.740 0.685 0.643 0.607 0.517
g: 0.741 0.566 0.413 0.315 0.235 0.228 0.077
(1—f)/(1—g): 0.37 0.42 0.44 0.46 0.47 0.51 0.52 of m near the theoretically expected value of 2 / 3 would
be found at lower levels while those at higher levels
would deviate towards the value m = 1. The calcula
tions in this section have therefore been made at two
atmospheric levels, the 500mb level (where we might
expect values of m near 2/3) and the 30,000—ft level
(at which values of m nearer 1 might be expected).
The SOO—mb data used in this section consist of
radar upperwind determinations at 6hr intervals for
Larkhill, England (51°12’N, 1°48’W), during the win
ter period from December 1944 to February 1945.
These data were taken from the Daily weather report ' of the Meteorological Ofﬁce, London. The computed correlation functions, f and g, for this period are shown
in table 2. . In ﬁg. 1, the quantities l—f and 1—g have been
plotted against 7', both on logarithmic scales, and com
pared with the theoretically expected linear relation ~
with slope 2/ 3. The agreement is clearly not perfect;
but in view of the inaccuracy involved in substituting
f(r) and g(1) for f(r) and g(r), respectively, it is
perhaps as good as could be expected with the data at
hand. The results tend to indicate the probable appli
cability of Kolmogoroff’s hypothesis to a wider range
of phenomena than might at ﬁrst have been thought
possible. It will be noted that in the above data the
condition of complete isotropy (as judged by the ap
parent inequality of 0",, and 0,) is not particularly well
fulﬁlled. If f and g have the form appropriate to the
inertial range, the ratio (1 —f)/(1—g) should have
the value 0.6; it may be seen from table 2 that this
requirement is also not well fulﬁlled by these data. The data for 30,000 ft consist of radar upper—wind
determinations at 6—hr intervals at Auckland, New
Zealand (36°47’S, 174°38’ E) for the winter period
from June to August 1952. These data were obtained
from the original records of the New Zealand Meteoro
logical Service. The computed correlation functions,
f and g, for this period are shown in table 3. Compu
tations were not continued beyond the 24—hr lag
interval. TABLE 3. Correlation functions, f and g; 30,000 ft, Auckland,
June—August 1952. 11.. = 38.0, a"; = 30.6 kn. ‘ Time lag, 7:
(hr) 6 12 18 24
f: 0.88 0.79 0.68 0.59
g: 0.78 0.61 0.43 0.31
(1—f)/(1—g): 0.55 0.54 0.56 0.59 JUNE 1955 J . The data in table 3 are plotted in ﬁg. 2, in the same
way as before. It will be seen that an approximately
linear relation is given by the plotted points up to the
24hr lag interval, and that the slopes of the two lines
are identical. The value of the slope is approximately
0.88, which differs considerably from the 2/3 slope
found in the 500mb data from Larkhill. This devia
tion of the slope towards the value 1 is, however, in
good agreement with the tendency to be expected if
the model of small—scale turbulence proposed by
Ogura (1953) is taken to be applicable to turbulence
elements on a larger scale. If a relation of the form
[1 —— R(r)] ~ 1’" is assumed, the reasoning of section
3 shows that the ratio (1 — f)/ (1 — g) should have
the value 1/(m + 1), and for m = 0.88 this value is
0.53. It will be seen from table 3 that the computed
values of the ratio (1 — f)/(1 — g) lie close to this
value. Nevertheless, too much emphasis should not
be placed on this result, as the data at the two levels
in question have been obtained at different stations.
For a full investigation, it would of course be desirable
to study the variation of the index m with height by
using data from different levels at the same station,
and then repeating the procedure for several stations
in different wind regimes. This course was not followed
here, owing to the fact that the computed correlation
coefﬁcients for 30,000 ft at Auckland were already
available and that it seemed more important to gain
some experience with the form of the correlation func
tions at higher latitudes than to investigate the details
of the variation of the index m with height. This latter
investigation would gain more signiﬁcance if attempted lRK) 6 l? I! 24
T HRS. W. HUTCHINGS 269 in conjunction with the computation of both space
and time correlations. Pressure ﬂuctuations—One of the most important
features of small—scale turbulent ﬂow is the existence
of turbulent ﬂuctuations in ﬂuid pressure accompany
ing the turbulent variations in velocity. In smallscale
ﬂow, it seems very plausible that these pressure ﬂuc
tuations (p) are connected with the corresponding
ﬂuctuations in velocity through the Bernoulli equa
tion, and that we might write formally p = %p62, Where
p is the density and c is the magnitude of the total
turbulent velocity. Dimensional reasoning then shows
that the correlation function h(r) for pressure ﬂuctua
tions has the form [1 —— h(r)] ~ r4/3. In the same way
as before, we may also write [1  ~ 74/3. In the case of largescale atmospheric turbulence,
it is well known that the Coriolis force almost exactly
balances the pressure gradient for motions whose
period approaches half a pendulum day, so that it
might be expected that the 74/3 law would cease to
hold beyond a certain value of 1, this value being quite
independent of the actual limits of r determined by
the inertial range. It might be, however, that in certain
circumstances, for example in low latitudes or in a
region where pressure changes are strongly inﬂuenced
by orographic or topographic effects, the 74/3 law might
hold over a range considerably greater than that indi
cated by considerations of geostrophic control. Owing to instrumental and other difﬁculties, accu l 2 J 6 E
T lIRS. Id 24 $354.2“ 6072 90/21? FIG. 2. Longitudinal (westerly) (A) and lateral (southerly) (B)
correlation functions at 30,000 ft, Auckland. Circles denote ob
served values, full lines to ﬁt observed data, slopes 0.88. FIG. 3. Autocorrelation function, surfacepressure ﬂuctuations,
Buenos Aires, July 1943. Circles denote observed values, full line
theoretical linear relation, slope 4/3. 270 rate data regarding shortperiod pressure ﬂuctuations
in the free atmosphere are difﬁcult to obtain. There is
more information available regarding surfacepressure
changes, but even here published computations of lag
correlationcoefﬁcients have usually been available
only for large lag intervals of one day or more. Very
detailed computations (with lag intervals down to 1 hr
over part of the range) have, however, been published
by Dedebant et al (1953) for the surface—pressure ﬂuc
tuations at Buenos Aires during July 1943. As this
station is situated on the lee side of the Andes cor
dillera, it seems well suited for investigating whether
or not anything resembling the theoretically antici
pated 4/3 law can be found in atmospheric data. With
the values of h(r) taken from the above publication,
ﬁg. 3 shows the values of [1 — h(r)] plotted against
the corresponding values of 7, both on logarithmic
scales. It will be seen that a linear relation gives a
very satisfactory ﬁt to the observations up to a lag
interval of 60 hr, but beyond that a systematic devia
tion from the linear relation can be observed. The
slope of the line drawn corresponds to the theoretical
value of 4/3. Another station for which detailed com
putations of lag correlation coefﬁcients have been
carried out is Wellington, New Zealand (41°17’S,
174°46’E). As long as we are concerned with pertur
bations whose periods are less than half a pendulum
day at the latitude of Wellington, it seems likely that
the associated pressure changes would be of the Ber
noulli type and that these smallperiod ﬂuctuations
might approximate the 4/3 law. The relevant points
for the pressure fluctuations at Wellington for the
period 1 January 1941 to 31 December 1942 are shown
in ﬁg. 4, and it may be seen that an approximately
linear law is followed up to a lag interval of about
18 hr, but that beyond this systematic deviations
appear. The value for the slope of the line shown in 0"
02 0'/ v / R00 0 05 0025 00/! J 6 .9 lé‘ IJ' [82/24 3036‘ 46 1‘ HRS. FI_G. 4. Autocorrelation function, circles surfacepressure ﬂuc
tuatlons at Wellington, crosses SOOmb geopotential ﬂuctuations
at Larkhill. Full lines to ﬁt data, slopes 1.5 and 1.0, respectively. JOURNAL OF METEOROLOGY VOLUME 12 ﬁg. 4 is 1.5, which is only a very rough approximation
to the expected value of 1.33. It is interesting to note
that the length of a half pendulum day in the latitude
of Wellington is approximately 18 hr. Some attempt has also been made to examine data
from the free atmosphere, with use of the 6—hr values
of the SOOmb geopotential at Larkhill for the period
from December 1944 to February 1945. This is the
same period as was used for the velocity correlations,
and the data have been taken from the Daily weather
report of the Meteorological Ofﬁce, London. The rele
vant points are also plotted in ﬁg. 4 and seem to indi
cate an approximately linear law up to about the 24—hr
lag interval. The length of the half pendulum day at
Larkhill is about 16 hr. The indicated slope is 1.0,
which represents a considerable deviation from the
theoretical value. The reasons for this deviation are
not entirely clear. Part of the deviation is no doubt
due to the use of time correlations in place of space
correlations, but it should also be emphasized that the
determination of the correlation function for pressure
ﬂuctuations in the free atmosphere by means of the
Bernoulli equation does not have a ﬁrm justiﬁcation
even when only ﬂuctuations of small period are con
sidered. Pressure changes in the free atmosphere are
the result of very complicated processes, and it is
probable that the Bernoulli equation is applicable only
in special circumstances. Temperature ﬂuctuations.—If the grid of a wind
tunnel is heated, turbulent ﬂuctuations of temperature
occur downstream. In contrast to the phenomenon of
ﬂuid pressure, there is no clear dynamical basis for
dealing with ﬂuctuations in temperature, and some
special assumption appears to be necessary. For small—
scale turbulence, Obukhoff (1949) has proposed that
the ﬂuctuations in temperature for a homogeneous
incompressible ﬂuid should, in the inertial range, 5 I? /8 24
1' HRS. J0 36 42 46 FIG. 5. Autocorrelation function, SOOmb temperature, Lark hi’lll. Circles denote observed values, full line to ﬁt data, slope
0. . JUNE 1955 J . follow the same law as the ﬂuctuations in velocity,
i.e., the 2/3 law should apply. There appears to be only a very faint analogy be
tween these small—scale temperature ﬂuctuations and
those occurring on the larger scale, but it was never
theless thought worthwhile to compare some practical
calculations with the above theoretical prediction. To
reduce the error due to the use of time correlations
instead of space correlations, data from the SOOmb
level in preference to the higher levels were used for
these computations. The 500mb temperature data for
Larkhill, taken at 6hr intervals, have been used over
the same period as was used for wind and pressure
correlations, that is, December 1944 to February 1945.
The relevant points are shown in ﬁg. 5 and show an
approximately linear region of slope 0.7 from the 12hr
to the 48hr lag interval. Computations were not con
tinued beyond the 48hr lag interval. Although the
data are not sufﬁcient to allow deﬁnite conclusions to
be drawn, these results are not inconsistent with the existence of a 2/3 law for the temperature ﬂuctuations
at this level. 6. General appreciation The work represents a ﬁrst attempt to apply some
of the results obtained in smallscale turbulence theory
to the general problem of atmospheric turbulence on
a larger scale. The conclusions must therefore of neces
sity be preliminary and tentative. Much further work
will be necessary before the exact nature of the results
can be ascertained and the conditions for their validity
established. Many of the necessary reﬁnements to the
present crude approach will be possible only by further
reﬁnement and clariﬁcation‘of the basic ideas of tur
bulence theory itself. The theory of turbulence is at
present in a very rudimentary state. In future prac
tical work in the atmosphere, more attention must be
given to the determination of actual space correlations
instead of time correlations. Only in this way will it
be possible to avoid one of the basic difﬁculties in the
treatment given in this article. Nevertheless it would seem, from the preliminary
results obtained so far, that the application of turbu
lence methods to the interpretation of largescale
atmospheric phenomena is well worth further investi
gation on a more rigorous and extensive basis. W. HUTCHINGS 271 Acknowledgments.—Grateful acknowledgment is
made to Dr. J. F. Gabites of the New Zealand Mete
orological Service for details of the standard deviations
of the 30,000—ft wind components at Auckland and
Nandi, as well as for the velocity correlationcoefﬁ—
cients at 30,000 ft for Auckland. Thanks are also due
to the Director, Applied Mathematics Laboratory,
Wellington, for details of the pressure correlations at
Wellington, and to Miss E. Farkas for assistance with
the computations in this article. The work is pub— lished by permission of the Director, New Zealand
Meteorological Service. REFERENCES Batchelor, G. K., 1953: The theory of homogeneous turbulence.
London, Cambridge Univ. Press, 197 pp. Brooks, C. E. P., C. S. Durst, N. Carruthers, D. Dewar, and J. S.
Sawyer, 1950: Upper winds over the world. Geophys.
Mem., No.85, 150 pp. Charney, J. G., 1948: On the scale of atmospheric motions.
Geofys. Publ., 17, No. 2, 17 pp. Dedebant, G., R. Di Maio, and E. A. M. Machado, 1953: Las
funciones aleatorias y su aplicacién a la meteorologia.
Meteoros, 3, 140—173. Defant, A., 1921: Die Zirkulation der Atmosphare in den ge
ma'ssigten Breiten der Erde. Geograf. Amt, 3, 209—266. Heisenberg, W., 1948: Zur statistischen Theorie der Turbulenz.
Z. Phys., 124, 628—657. Hutchings, J. W., 1952: A note on the distribution of freeair
wind vectors about their mean. Quart. J. r. meteor. $05.,
78, 105—106. Karman, T. von, and L. Howarth, 1938: On the statistical theory
of isotropic turbulence. Prue. ray. Soc. London, A, 164,
192—215. Kolmogoroff, A., 1941: The local structure of turbulence in in
compressible viscous ﬂuids for very large Reynolds num
bers. C. R. Acad. Sci. U.R.S.S., 30, 301—305. Obukhoff, A. M., 1949: Structure of the temperature ﬁeld in
turbulent ﬂow. Izv. Akad. Nauk S.S.S.R., Ser. Geogr.
Geoﬁz., 13, 58—69. Ogura, Y., 1953: The relation between the space and time corre»
lation functions in a turbulent ﬂow. I. meteor. Soc. Jap.,
31, 355—369. Priestley, C. H. B., and P. A. Sheppard, 1952: Turbulence and
transfer processes in the atmosphere. Quart. J. r. meteor.
$00., 78, 488—529; Richardson, L. F., and J. A. Gaunt, 1930: Diffusion as a com—
pensation for smoothing. Mem. r. meteor. Soc., 3, 171—175. Weizsacker, C. F. von, 1948: Das Spectrum der Turbulenz bei
grossen Reynoldsschen Zahlen. Z. Phys., 124, 614—627. ...
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