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Unformatted text preview: 516 jOURNAL OF METEOROLOGY VOLUME 16 A GENERALIZATION OF THE MIXINGLENGTH CONCEPT1 By J. A. Businger University of Wisconsin3 (Original manuscript received 2 May 1958; revised manuscript received 7 April 1959) ABSTRACT A concept for the mixing length in diabatic conditions is introduced and elaborated. The basic idea is that
convective energy has effect on the mixing length but not on the size of the largest eddies. The theory devel
oped on this concept of the mixing length for the diabatic wind proﬁle gives satisfactory agreement with observations over a wide stability range. 1. Introduction In the last ten years, several papers dealing with
turbulent transfer in the lowest layer of the atmos
phere have been published. The progress made in that
period is mainly experimental. Several techniques of
collecting data were developed or improved. The two
expeditions to O'Neill, Nebraska in 1953 and 1956,
especially, resulted in a host of valuable data. Progress
was made also on the mode of representation. Several
studies, such as Deacon [7; 8]. Priestley [23], Monin
and Obukhov [18], Businger [4], Lettau [16], and
Inoue [12], were carried out to achieve in part a more
comprehensive representation of the data in order to
test several possible theories. Although unanimity
has not yet been reached on this point, the different
representations are more or less equivalent and in—
dicate the existence of a unique and universal relation
describing the turbulent structure of the surface layer
(e.g., the wind proﬁle in terms of a dimensionless
height as a function of stability). On purely theoretical
grounds, Batchelor [1] came to this same conclusion.
However, until the present no satisfactory theory has
been developed describing the observations in sufﬁ
cient detail. The theories presented by Lettau [14;
15], Businger [4; 5], Halstead [10], Ogura [19], and
Elliot [9] all introduce one or mere assumptions
which have an apparent lack of physical basis. This
situation is somewhat surprising because of the strong
limitations on the problem imposed by the general
assumption of a steady condition throughout the
surface layer—that is, that all time derivatives of
mean quantities are zero, and that the surface is
homogeneous and has a constant roughness parameter.
in sections 3 and 4 of this paper, a theory is developed
based on a model of turbulence which makes possible 1 The research reported in this article has been made possible
through support and sponsorship extended by the Geophysrcs
Research Directorate, Air Force Cambridge Research Center, under contract No. AF 19(604)—1384.
2 Present affiliation: University of Washington. the introduction of a generalized mixing length. The
agreement with observation, so farI is promising. Although the question of whether or not the coefﬁ—
cients of eddy diffusivity for momentum and heat
transport are equal is by no means settled, the equality
of the two is assumed in most of the above mentioned
theories and also in the present theory. It may be that
this assumption is not true, but it simpliﬁes the
theoretical approach considerably and provides a
result which eventually can be corrected because the
analysis does not depend on it in any fundamental
way. An empirical check of the equality of these
coeﬁicients is obtained by comparing the curvature of
the wind and temperature proﬁles from the O’Neill
data [27]. Comparison shows that these curvatures
are different, and therefore it is concluded that the
coefﬁcients must be different. It is very likely, how
ever, that the ratio of the coefﬁcients is a function of
the stability alone and does not affect the uniqueness
of the above mentioned universal relation. Another simpliﬁcation is made here with regard to
the zero displacement for the logarithmic proﬁle.
This zero displacement is not taken into consideration,
so the level 2 = 0 is assumed to be where the wind
velocity, according to the extrapolated logarithmic
proﬁle, is zero. Therefore, the theoretical results
cannot be trusted below a level of z = 1020, for instance.
For a more detailed discussion of this matter, reference
is made to Bjorgum [2]. 2. Dimensional characteristics and representation Since there is sufﬁcient experimental as well as
theoretical (Batchelor [1]) evidence that the Richard
son number, deﬁned as aé Oz
R1: / z gum/32y (1) (where 0 is potential temperature, at is horizontal mean
wind, g is acceleration of gravity and Tm is the mean OCTOBER 1959 J . absolute temperature), determines the structure of the
atmospheric turbulence in the surface layer, there
must exist a unique relation describing this structure.
Several studies [4; 7; 8; 12; 16; 18; 23] have been
carried out in order to give a concise and compre
hensive representation of this relation. Although most
representations are basically equivalent, because they
can be derived from each other, preference can be
given to some representations above others. From a
theoretical point of view, it is desirable to have a
representation which is easy and reliable to derive from
observations in order to be able to test the theory.
The ﬁrst representation of this kind is that of
Deacon [7], who plots a dimensionless parameter [3,
related to the curvature of the wind proﬁle, deﬁned as 26212/622 B = 612/62 (2) versus R1}. Although Deacon’s semiempirical general
ized wind proﬁle has not been found in theory, the [3
versus Ri relation is still of practical interest. In
recent studies, this relation has been used to compare
some theories with observation. A great advantage of
both the parameters 6 and Ri is that they can be
derived from wind and temperature measurements
without using the surface roughness. However, this
relation is difﬁcult to visualize in terms of the wind as
a function of height—that is, in a more integrated
form of the relation. Representations in this form are
given by Monin and Obukhov [18] and also by
Businger [5]. Monin and Obukhov represent the
universal relation using a dimensionless Wind differ
ence Y and a dimensionless height gr as 66 a 0
Y = we — Us) and r = kg—ﬁjf/aﬁui) (3) (where U = left/14* is the dimensionless wind, k is v.
Karman's constant, u* is the friction velocity, and 20
is the roughness parameter). The representation
Y = f (t) has the objection that it fails to show any
detail for nearly neutral proﬁles; that means it is
difﬁcult to ﬁnd the deviation from the neutral proﬁle,
as Lettau [17] has pointed out. This point is elab—
orated in section 4. Businger [5] used a representation
which is easy to interpret. Here, U was plotted as a function of 2/20 in a series of curves using the dimen
sionless stability number gee/62.20
5% = W
TmBﬁ/dz ' 14* as a parameter. Although this representation gives
good detail for the nearly neutral proﬁles, it is not as
concise as might be desired. A better representation can be introduced using a dimensionless velocity A. BUSINGER 517 difference, AU;
Z + Zn 20 AU=U—ln (4) versus the dimensionless height g’ or versus Ri. An
intermediate representation is given by Deacon [8].
He introduced a dimensionless coefﬁcient of eddy
transfer * K", 14* 17,3 — 2612/62 m and made a plot of 14%", versus Ri. In section (4), the u * . 0 n 
reelprocal value of Km 15 used as dimensionless Wind
gradient kzaii k
P=———=———. 14* 62 Km
In the following sections, both the representations 6 = f (Ki) (5) and
AU = h(R1§) (6)
will be used. 3. Model of turbulence under neutral conditions Turbulent structure under adiabatic conditions is
now well understood. Prandtl's [20] derivation of the
logarithmic law has found theoretical as Well as
experimental support. Therefore, the concept of the
mixing length must be regarded as a valuable tool,
with an exact deﬁnition in the neutral case, as was
pointed out by Businger [4]. An interesting fact,
furthermore, as shown by Inoue [11], is that the
logarithmic proﬁle is consistent with the similarity
hypothesis of isotropic turbulence when the mixing
length represents the size of the largest vertical eddies.
A consequence of this result is that in the equation of
energy decay, derived by Calder [6], and which under
steady adiabatic conditions can be written as d _._‘ at?
~—w’Eu’w’——D=0 (7)
62 62 ' (u’ and w’ are the horizontal and vertical components
of the turbulent wind respectively, E is the total
turbulent energy and D is the dissipation rate of
turbulent energy into heat), the term 6(—‘zv’~E_)/az,
representing the divergence of the turbulent energy,
is negligible in comparison to the two others. The rate of decay of turbulent energy 6 from one
wave number to the next is independent of the wave number and equal to the dissipation rate. This means
that
103
D = e ~ ka0 ~ — (8) 518 where 120 is the wave number of the largest vertical
eddies; 'w = (Wﬁ and l is the mixing length. The
equations (7) and (8), together with the equation for
the ﬂux of momentum pK— = — p We” (9) where the coefﬁcient of eddy transfer K is deﬁned by
K = wl, (10) deﬁne the logarithmic wind proﬁle, when the mixing
length 1 ~ (l/ko) ~ 2. The proportionality constant
which enters because of relation (8) is v. Karman’s
constant. It should be remarked that the mixing length as
introduced here is a Eulerian concept. Prandtl's
original concept of mixing length was “Lagrangian."
But, in his ﬁrst development where it was related to
u’w’, the transition to the Eulerian View was implied.
Therefore, the explicit expression obtained for the
mixing length in the neutral atmosphere, 1: k(z + 20), (11) is a Eulerian concept. (Note the analogy with the
kinetic gas theory 1) It is of interest now to present a different argument
in Prandtl’s original model. Consider an eddy with a
vertical velocity 10’ with respect to its surroundings
and a characteristic size 1’. The question is now how
far in the vertical will this eddy travel before it loses
its identity. That distance will then be the mixing
length for the eddy under consideration. To aHSWer
this question, we consider the energy of the eddy in
the z—direction, which will be proportional to pw’2l’3.
This energy dissipates at a rate proportional to the
eddy diffusivity of its surroundings K’, the gradient in
energy from the eddy to its surroundings, which is
proportional to w'2/l’ and the surface of the eddy
~ I”. The dissipation then is proportional to pK’w'2l’.
It may be assumed that K’ is proportional to the
 overall eddy diffusivity K. By the time all the energy
is consumed, the eddy has lost its identity and has
travelled its mixing length Z... In equation, pw’3l’3 wil/g
Z," N ——"‘—‘ ~ pK'w'w K ' (12) Averaging over all eddies, the lefthand side will give
the mixing length in the Lagrangian form lg, and the
righthand side will be proportional to 'wlo2/ K (where
lo is the predominant eddy size), provided the size
distribution of the eddies is independent of the in
tensity of the turbulence, which is implied in equa
tion (8). When we now consider all the eddies passing by a
point of observation, we may assume that on the JOURNAL OF METEOROLOGY VOLUME 16 average these eddies have travelled in the vertical
over a distance which is proportional to la. The mixing
length 1 obtained in this fashion is a Eulerian concept.
So we ﬁnally obtain, using (10), ‘10qu2 lo2
l N U N —‘ N “"
K I
01‘
l 1 1 (13)
w ._ ———— ~ Z
0 [30 This argument may seem to be superﬂuous, but it will
be shown in the next section that relation (12) provides
a link to a generalization of the mixinglength concept.
At the same time, it shows what assumptions are
made in the present theory. 4. Generalization of the model for diabatic conditions The previous section emphasized that there is con—
siderable agreement even in detail between Prandtl’s
model and the exact theory as well as the observations.
Therefore, it may be expected that a logical general—
ization of this model to diabatic conditions would
provide satisfactory results. Equation (7) is a special
case of the more general equation for diabatic con
ditions (see [6]). a __ 612 ag __.
— w’E —— u’w’ ——+——w’0’ — D = 0,
62 az Tm D = wakn (14) where a is a proportionality constant indicating the
efﬁciency with which density differences contribute
to the turbulent energy. It was Ogura [19] who
emphasized the usefulness of this equation in the
derivation of the diabatic wind proﬁle. Next, there is
the equation for the heat ﬂux (15) which is assumed to be similar to equation (9).
Further, there is some evidence from an analysis
of the power spectrum of the vertical wind component
that the turbulence behaves isotropic in the range of
high wave numbers [26] under all stability conditions.
This behavior must be related to the dissipating
structure which tends to be isotropic even in homo
geneous anisotropic turbulence, whereas the transition
from homogeneous anisotropic to isotropic turbulence
is very slow [28, p. 48]. The experimental evidence
indicates that the turbulent energy from friction as
well as from convection enters at the lowest wave
numbers and that equation (8) is also true for diabatic
conditions. The main difference between the diabatic
and adiabatic case lies in the way the frictional and
the convective energies enter into the turbulence. It OCTOBER 1959 J . may be said that the frictional energy enters the lowest
vertical wave numbers in an isotropic way, because the
horizontal turbulence extends to a lower frequency
range than the vertical turbulence. However, the
convective energy contributes only to the vertical
wind component of the lowest vertical wave number. Considering again an individual eddy, we see that
besides a dissipation of energy there is also a source of
energy proportional to fﬁr9’l““w’,
Tm
where 0' is the temperature difference between the eddy and its surroundings. In this case, instead of (12),
we ﬁnd wlllg
Oligo/ll2 '
Tm'w’ ln~ (16)
K— When averaging over all eddies and following a similar
line of reasoning as in deriving (13) and considering
that 0’ may be replaced by  1(69/62), we ﬁnd 1 ( oz’g 63 1 )“i
Tm 62 (how)2 . (17) It is easy to see that, in this more general case, 1 is no
longer proportional to 2 because l/ko ~ 2 is still valid.
The constant a’ is not the same as the a in equation
(14) because it contains also the constant of pro—
portionality between 1 / kg and 2. Under neutral condi~
tions, relation (17) should reduce to equation (11);
therefore, by replacing 1/ko by k(z + 20), we ﬁnd for
(17) that assuming that in this case a is the same in (14) and
(18). The equations (9), (10), (14), (15), and (18) now
deﬁne the complete structure of the atmospheric surface layer. From these relations, it is possible to
derive 3 R' e
"‘ i 3m + %[(aRi)2 +4<1  am} , 01f = ”—“i
1 — aR’L 2 (19) using (1) and (3). From this equation, the relation (5)
can be derived, considering that dlnRi
dlnf :3! by combining (1), (2), and (3), by differentiation of
(19). A consequence of the present theory is that for A. BUSINGER 519 R12 = — €10.43 = 4%, which is in agreement with Priest—
ley’s ﬁndings in the region of free convection [22]. From equation (19), the wind proﬁle or equation
(6) can also be derived. Writing the symbol for the
part of equation (19) in brackets and considering that
(see (1) and (3)) a U 1
6g“ _ Ri ’
it may be seen that
3 U P
f f or, when a function Q(§‘) = (1/§)(P — 1) is in
troduced, 6U 1 Ernst—FOG) and 5 r
U = In — —I— Q(§')d§ {0 (n (20) where
kgaé/aza) =k5 =——~——.
{0 " I;u*aa/az Thus, equation (6) takes the form f AU: r Q(§)d§ (21) which can be obtained by graphical integration. The
equations (20) and (21) indicate that U and AU are
not only functions of f but also of {0. So it is necessary
to use, in this case, a twoparameter representation.
However, in most cases i“ _z+20 f0 20 >>1 and, because Q(§‘) is a limited positive function, i‘ 5‘
d 2 d .
mac): foam: So equation (6) can be represented as f AU = f (and: (22)
provided 2» 20. Combining equation (22) with
equation (19), A U can be written also as a function of
Ri. Various representations of the universal function
as derived by the present theory are illustrated in
ﬁg. 1. It is now possible to compare equation (22) with
equation (3). Equation (3) can be written when 320 £25 JOURNAL OF METEOROLOGY VOLUME 16 FIG. 1. Various forms of the universal function as derived by the present theory. §>Oas
I
Y=hmr+j‘moa and when 3“ < 0 as
r Y=meuo+fema a» When g“ is very small, the absolute value of the integral
in the righthand side of these equations is small in
comparison with [In ZlfH. This explains, as was men
tioned in regard to equation (2), why detail is lost in
Monins and Obukhov’s representation. 5. Comparison with experimental data The theoretical relationship between [i and aRi as
given in ﬁg. 1 is compared with two groups of data—
i.e., a series of 260 hr of simultaneous wind and tem
perature proﬁles, measured at Shirley, New Jersey,
and a similar series of about230 hr measured at
O’Neill, Nebraska in 1953 and 1956. The O’Neill ob
servations were carried out by the Johns Hopkins and
MIT groups in 1953 and by the Texas A. and M.
group in 1956. The data reduction was carried out by
Lettau and the result was presented during the IUGG
at Toronto in 1957. The curvature [3 and the Richard
son number were divided in seven stability classes,
which are deﬁned in section 7.4.3 of [27]. For the
method of data reduction, applied reference is made to
[16]. The comparison of the theoretical curve with the
two groups of data is presented in ﬁgs. 2 and 3,
respectively. A reasonable ﬁt is obtained for a value a = 3 in ﬁg. 2. In ﬁg. 3, especially for nearly neutral
conditions, a value of a = 10 would ﬁt better. Also, the theoretical relation between AU and Ri
has been compared with experimental results. To obtain
’81], Z + 20
AU = —— —— ln
u* 20 from the proﬁles, it is necessary ﬁrst to evaluate both
u... and 20. The 20 was obtained by using the logarithmic
proﬁle under neutral conditions, and 14* was obtained
by assuming the logarithmic proﬁle to be valid to the
lowest level of wind observation. The data represented
in ﬁg. 4 are derived from 1953 observations by the
Johns Hopkins group for the 4m level and from 1956
observations by the Texas A. and M. group for the
8 and 16m levels, all at O’Neill, Nebraska. The total
was divided in ten classes of overall stability. In the
more stable classes, the determination of m was not
accurate because of the large deviation from the
logarithmic proﬁle. The theoretical curve shows also
in this case a reasonable ﬁt with the presented data
when or = 3. 6. Concluding remarks Although the present theory is not exact, a fairly
accurate description is obtained of the structure of the
turbulence in the atmospheric surface layer from
extremely unstable to stable conditions, provided the
steady state is maintained. In the very stable cases, the steadystate condition
is usually not .fulﬁlled because the transfer processes OCTOBER 1959 J . l.50 I25 FIELD BARE [0.0], on.
PLANTED WITH Rows or
050 SPINACH on com [Ami 3cm, V.V=4TO 8 «ma I =9T0 l2cm]
WIND 5 TEMPERATURE LEVELS:
a A v £1140, 80.50. AND 520 cm; 0 A v .180,l60,320,AND 640 cm, 025 l.00 ~0.75 '050 ~0.25 A. BUSINGER 521 0.0 0.25 Pl FIG. 2. Comparison of the theoretical 5 versus Ri relation for a = 3
with observations obtained at Shirley, New Jersey. [.50 O'NEILL
l953+l956 /3 versus Rt 0 2 =400 cm
Az=200cm
LIz . IOOcm 0.50 0.25 l.00 ‘ 0.75 0.50 ‘025 0.00.25
Ri FIG. 3. Comparison of the theoretical 6 versus Ri relation for a = 3 and a = 10
with observations obtained at O’Neill, Nebraska. become very slow. The result is a rather large scatter
of the observed points in this region. It is probably not possible to obtain a more precise
theory with Prandtl’s mixing length as a starting point. The constant of proportionality, :1, makes it
p0ssible to adjust the theory to the observations. It is,
of course, desirable to have a theory which predicts
this constant and agrees ...
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