3474
MONTHLY WEATHER REVIEW
VOLUME 135
Evaluation of Boundary Layer Similarity Theory for Stable Conditions in CASES-99
KYUNG-JA HA
Division of Earth Environmental System, Pusan National University, Busan, South Korea
YU-KYUNG HYUN
Policy Research Laboratory, National Institute of Meteorological Research, Korea Meteorological Administration, Seoul, South Korea
HYUN-MI OH
Division of Earth Environmental System, Pusan National University, Busan, South Korea
KYUNG-EAK KIM
Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, South Korea
LARRY MAHRT
COAS, Oregon State University, Corvallis, Oregon
(Manuscript received 6 September 2006, in final form 29 January 2007)
ABSTRACT
The Monin–Obukhov similarity theory and a generalized formulation of the mixing length for the stable
boundary layer are evaluated using the Cooperative Atmosphere–Surface Exchange Study-1999 (CASES99) data. The large-scale wind forcing is classified into weak, intermediate, and strong winds. Although the
stability parameter, z /L, is inversely dependent on the mean wind speed, the speed of the large-scale flow
includes independent influences on the flux–gradient relationship. The dimensionless mean wind shear is
found to obey existing stability functions when z /L is less than unity, particularly for the strong and
intermediate wind classes. For weak mean winds and/or strong stability (z /L ⬎ 1), this similarity theory
breaks down. Deviations from similarity theory are examined in terms of intermittency. A case study of a
weak-wind night indicates important modulation of the turbulence flux by mesoscale motions of unknown
origin.
1. Introduction
Monin–Obukhov (M–O) similarity is used to relate
the flux–gradient relationship in the surface layer to the
stability parameter, z /L, in stationary flow over homogeneous surfaces, where L is the Obukhov length. Actual surfaces are generally heterogeneous, to some degree. Even modest surface heterogeneity may influence
the weak turbulence in stable boundary layers.
Intermittency of the flux (Howell and Sun 1999;
Corresponding author address: Kyung-Ja Ha, Division of Earth
Environmental System, Pusan National University, Busan, 609735, South Korea.
E-mail: [email protected]
DOI: 10.1175/MWR3488.1
© 2007 American Meteorological Society
MWR3488
Coulter and Doran 2002; Salmond 2005; Acevedo et al.
2006) may also influence the time-averaged flux–
gradient relationship. The definition of the intermittency varies between studies. Normally, the nonstationarity of the turbulence is more complex than simple
on–off behavior (Nakamura and Mahrt 2005). Turbulence intermittency can be related to internal interactions between turbulent mixing and the mean shear
(Pardyjak et al. 2002; Fernando 2003), perhaps involving variation of the Richardson number about an equilibrium value. Or intermittency may be externally
forced by mesoscale motions, such as internal gravity
waves (Nappo 2002; Chimonas 2002) or horizontal meandering of the mean wind field (Anfossi et al. 2005). In
addition, downward transport of turbulence toward the
OCTOBER 2007
HA ET AL.
3475
FIG. 1. The 60-m tower instrumented with 7 levels of flux measurements.
surface (Ha and Mahrt 2001), and associated increase
of turbulence near the surface due to flux convergence
of the turbulence energy, is not included in the surface
layer similarity theory nor in the traditional concept of
boundary layers where the turbulence is controlled by
surface processes. The strongest turbulence may occur
above the surface inversion (Mahrt 1985; Ohya et al.
1997; Banta et al. 2002). Both internal and external
intermittency, as defined above, occur on scales larger
than individual eddies and thus contrast with finescale
intermittency associated with the substructure within
individual eddies.
For clear nights, the stratification is normally inverse
correlated with mean wind speed since weaker mean
winds correspond to smaller shear generation of turbulence and permit larger stratification. Here, we classify
the nocturnal boundary layer flows in terms of the
mean wind speed in addition to stability. Wind speed is
a dimensional quantity and therefore cannot provide
universal classification. From the point of view of turbulence similarity theory, the bulk Richardson number
is preferable to the mean wind speed since it is nondimensional. However, wind speed serves as a more external variable in contrast to z /L, which depends on the
turbulence itself. The mean wind speed does not inherit
errors in the estimation of vertical gradients required
for the Richardson number nor the ambiguity of defining the surface temperature. Banta et al. (2002) demonstrated that the speed of the low-level jet is a useful
overall predictor of properties of the nocturnal boundary layer. Anfossi et al. (2005) found that with mean
winds weaker than about 1.5 m s⫺1, the mean wind field
becomes more dominated by nonstationary meandering motions. As a result of such nonstationarity of the
mean flow and other processes noted above, the flux–
gradient relationship may depend on the speed of the
large-scale flow independent of the influence of stability.
2. Data
We analyze 5-min averaged data from a 60-m tower
taken during the Cooperative Atmosphere–Surface Exchange Study-1999 (CASES-99; Blumen 1999; Poulos
et al. 2002; Ha and Mahrt 2003). The observations were
made over relatively flat temperate grassland near
Leon, Kansas, in October 1999. The main site includes
a 60-m tower system (Fig. 1) with 8 levels of eddy correlation data based on Campbell sonic anemometers
(CSAT), 34 levels of thermocouple data from 0.23 to
58.10 m (Sun et al. 2002), the R. M. Young propeller
anemometer and wind vane data at 15, 25, 35, and 45 m,
and aspirated thermistor data at 5, 15, 25, 35, 45, and 55
m. Thermistor data are linearly interpolated to 5, 10, 20,
30, 40, 50, and 55 m in this study. The radiosondes were
also released at 270 m apart from the 60-m tower (e.g.,
Hyun et al. 2005). Turbulent fluxes and mean winds
were evaluated from fast response data at 8 levels (10,
20, 30, 40, 50, and 55 m) of the 60-m tower and 2 levels
(1.5 and 5 m) on a 10-m tower adjacent to the main
tower. The data were collected at 20 Hz. The data were
quality controlled following Vickers and Mahrt (1997).
3. Classification by wind speed
The data are divided into three different mean wind
speed regimes: strong-, intermediate-, and weak-wind
regimes. The three regimes based on the mean wind
speed at 5 m are Ûi ⱖ U ⫹ 0.55s, U ⫺ 0.55s ⱕ Ûi ⱕ
3476
MONTHLY WEATHER REVIEW
TABLE 1. Classifications of observed wind and calculated z /L at
5 m. U is an average for total 23 nights, Ûi is mean of each night
i, and s is std dev between the records for each night. Here, 䊉
denotes strong-wind cases, 䉺 denotes intermediate-wind cases 䊊
denotes weak-wind cases. Classifications of “S,” “I,” and “W”
mean strong-, intermediate-, and weak-wind cases, respectively.
Here, 䊉: U ⫹ 0.55s ⱕ Ûi, 䉺: U ⫺ 0.55s ⱕ Ûi ⬍ U ⫹ 0.55s,
䊊: Ûi ⬍ U ⫺ 0.55s A: 0 ⱕ z/L ⬍ 0.01, B: 0.01 ⱕ z/L ⬍ 0.1,
C: 0.1 ⱕ z/L ⬍ 1, D: 1 ⱕ z/L ⬍ 10
Wind at 5 m
Date
Magnitude
Shear
z /L at 5 m
Ûi
Class
5–6
6–7
7–8
8–9
9–10
10–11
11–12
12–13
13–14
14–15
15–16
16–17
17–18
18–19
19–20
20–21
21–22
22–23
23–24
24–25
25–26
26–27
27–28
䉺
䊉
䉺
䉺
䊊
䊉
䉺
—
䊊
䊉
䊉
䊉
䉺
䊊
䊊
䉺
䉺
䉺
—
䉺
䊊
䉺
䉺
䊉
䊉
䊉
䉺
䊊
䉺
䉺
—
䉺
—
䉺
䊉
䉺
䊊
䉺
䉺
䉺
䉺
—
䉺
䊊
䉺
䉺
D
B
C
C
D
B
C
C
D
B
B
A
D
D
D
D
D
D
D
C
D
C
C
2.3
5.4
2.2
2.3
1.2
4.1
3.2
—
1.9
6.1
4.9
7.3
2.1
1.3
1.6
2.4
2.5
2.7
—
4.0
1.5
3.1
3.4
I
S
I
I
W
S
I
—
W
S
S
S
I
W
W
I
I
I
—
I
W
I
I
U ⫹ 0.55s, and Ûi ⱕ U ⫺ 0.55s. Here, U is the mean
wind speed averaged over all nights, Ûi is the mean
wind speed of night i, and s is the standard deviation
(Table 1). Data are selected from 21 nights where the
flow remained in one regime during the entire night.
The strong, weak, and intermediate wind cases are (6–
7, 10–11, 14–15, 15–16, 16–17), (9–10, 13–14, 18–19, 19–
20, 25–26), and (5–6, 7–8, 8–9, 11–12, 17–18, 20–21, 21–
22, 22–23, 24–25, 26–27, 27–28), respectively. For the
strong-wind case, the 5-m average mean wind speed
exceeds 4.1 m s⫺1, while the mean wind is less than 2.1
m s⫺1 for the weak-wind nights. Classification in terms
of mean wind shear would have led to similar results.
The stability is expressed as z /L, where L is the
Obukhov length,
L⫽
⫺␪u 3
*
kgw⬘T ⬘␷
共1兲
Here, k is the von Kármán constant, g is the earth’s
gravitational acceleration, ␪ is the potential tempera-
VOLUME 135
ture, u is the friction velocity, and w⬘␪⬘ is the potential
*
temperature flux. In this study, L is evaluated from
fluxes at level z rather than from surface fluxes. The
ranges of z /L at 5 m are noted in the last column of
Table 1 based on the stability classes: A (0 ⬍ z /L ⬍
0.01), B (0.01 ⬍ z /L ⬍ 0.1), C (0.1 ⬍ z /L ⬍ 1), and D
(1 ⬍ z /L ⬍ 10). Stability is closely related to mean wind
speed. For example, the near neutral stability (class B)
generally occurs on strong-wind nights, while the weakwind night tends to be very stable (class D). Exceptions
occur. The use of 5-min fluxes leads to large scatter
associated with large random flux errors compared to
the 30-min or 1-h averages but leads to less capture of
nonstationarity of the mean flow and turbulence within
the flux calculation. Inclusion of nonstationarity by averaging the flux over longer time scales systematically
alters the flux–gradient relationship, examined in a
separate study. Use of 5-min averages instead of longer
averages reduces this bias at the cost of increased scatter.
4. Evaluation of similarity relationships
The flux–gradient relationship in the surface layer is
usually posed in terms of M–O similarity theory, which
relates nondimensional gradients to z /L. The nondimensional shear, ␾m, is defined as
␾m
冉冊 冉 冊
kz ⭸U
z
⫽
L
u ⭸z
*
共2兲
The nondimensional shear is often parameterized in
terms of the stability functions from Businger et al.
(1971) and Beljaars and Holtslag (1991), respectively:
␾m
␾m
冉冊
冉冊
z
z
⫽ 1 ⫹ 4.7 ,
L
L
冋
共3a兲
冉
z
z
z
⫽1⫹
a ⫹ b ⫻ e⫺d共zⲐL兲 1 ⫹ c ⫺ d
L
L
L
冊册
,
共3b兲
where a ⫽ 1, b ⫽ 0.667, c ⫽ 5, and d ⫽ 0.35. The
Beljaars–Holtslag formulation reduced the overestimation of the nondimensional gradients for very stable
conditions that occurred in Holtslag and De Bruin
(1988). The Businger formula was derived for z /L ⬍ 1.
Figure 2 shows the relationship between ␾m and z /L
(z ⫽ 5 m) for 6–7, 24–25, and 19–20 October. The vertical mean wind shear at the height of 5 m is obtained
by logarithmic finite differencing between 1.5 and 10 m.
For the weakest winds, the gradient depends on the
method of estimating gradients and no preferred
method emerges. The solid and dotted lines represent
the curves based on Eqs. (3a) and (3b) respectively.
OCTOBER 2007
HA ET AL.
3477
FIG. 2. The dependence of ␾m on z /L for strong- (6–7 Oct), intermediate- (24–25 Oct), and weak- (19–20 Oct)
wind cases at 5 m.
Note that most cases with z /L ⬎ 0 occur with weak
winds. For example, some near-neutral cases (small
z /L) occur with weak winds associated with small u
*
but very small heat flux. A few cases of small z /L occur
with large ␾m and large Richardson number. This behavior was not sensitive to choice of averaging time,
although sampling errors associated with the weak
fluxes may be important. We conclude that the stability
parameter z /L may not be an adequate indicator of the
stability for weak turbulence and strong stratification.
The nondimensional wind shear (Fig. 2) is well approximated by the empirical expression of Businger et
al. (1971) for the strong and intermediate wind classes.
In the weak-wind regime, the points are widely scattered even for small values of z /L. For large z /L, the
values of ␾m tend to occur between the two empirical
curves in Fig. 2. For z /L less than one, where all three
wind speed regimes contain data, the weak-wind regime exhibits much more scatter, presumably due to
the stronger intermittency (section 5a) and larger random flux errors. These results suggest that the mean
wind speed contains an independent influence not contained in the stability.
To estimate the contribution of self-correlation, we
construct randomizations of the original data based on
the method of Klipp and Mahrt (2004). With this
method, the original values of the mean shear and
fluxes are randomized such that the values of the mean
shear and fluxes for a given realization of the randomized data originate from different records. With no selfcorrelation, the correlation between ␾m and z /L for the
randomized data would be zero. The correlation between the nondimensional shear and z /L for the randomized data at 5 m is not significantly smaller than
that for the original data (Fig. 3), suggesting that linear
correlation cannot be confidently used as verification of
the degree of physical relationship between the nondimensional shear and z/L, even though the slope is significantly less for the randomized data.
5. Turbulence statistics
a. Intermittency
We evaluate a simple index of intermittency (turbulence variability; Mahrt et al. 1998), written as
FIG. 3. The relationship between ␾m and z /L for observed and randomized 5-m data for strong- (6–7 Oct),
intermediate- (24–25 Oct), and weak-wind cases (19–20 Oct).
3478
MONTHLY WEATHER REVIEW
VOLUME 135
FIG. 4. The dependence of z /L on the index of intermittency at (a) 5 and (b) 30 m.
FI ⬅
␴F
,
abs共F 兲
共4兲
where ␴F is the standard deviation of 5-min values of
the friction velocity within a 1-h period and F is the
friction velocity evaluated at 5 and 30 m. The 30-m level
is typically above the intense part of the surface inversion layer, where the local value of the momentum flux
is not representative of the surface momentum flux.
Here, z /L is used as an informal stability parameter
since 30 m is too high for the validity of M–O similarity
with significant stability. Figure 4 shows the intermittency index for u for the strong-wind (dark circles),
*
intermediate-wind (open circles), and weak-wind (gray
circles) nights. The intermittency index is small for
z /L ⬍ 0.1 and increases roughly linearly with z /L in
ln–ln coordinates (with large scatter) for z /L ⬎ 0.1. In
general, the intermittency index is greatest for the
weak-wind class. The intermittency, such as defined in
Eq. (4), partly explains the greater scatter in the relationship between ␾m and z /L for the weak-wind category (Fig. 2). The main qualitative features occur at
both the 5- and 30-m levels, but 30 m contains more
data with small z /L.
b. Deviation from the similarity prediction for the
flux–gradient relationship
For the present data (Fig. 5), the large intermittency
index generally corresponds to expected greater magnitude of the deviations from the fitted ␾m–z /L relationship, but such deviations of ␾m occur with either
sign with roughly the same probability. The intermittency leads to larger scatter but no detectable systematic changes. For the strong-wind regime, the intermittency index is generally below 0.3 and deviations from
similarity theory are small.
6. Dimensionless numbers
To further examine the dependence of turbulence on
stability, the relationship between z /L, the gradient Richardson number, and turbulent Prandtl number are
now examined for the strong-, intermediate-, and weakwind classes.
FIG. 5. The dependence of deviations from Businger’s relationship on the index of intermittency for (a) strong-,
(b) intermediate-, and (c) weak-wind cases at the height of 5 m.
OCTOBER 2007
HA ET AL.
3479
FIG. 6. The dependence of Ri on time for strong- (10–11 Oct), intermediate- (5–6 Oct), and weak- (18–19 Oct)
wind cases at 10 m.
a. Richardson number
The gradient Richardson number, Ri ⫽ (g/␪)[(⳵␪/⳵z)/
(⳵U/⳵z)2], in the surface layer is predicted to be proportional to z /L (Businger et al. 1971):
Ri ⫽ z ⲐL ⫻
共0.74 ⫹ 4.7z ⲐL兲
共1 ⫹ 4.7z ⲐL兲2
.
共5兲
The relationship was originally intended for z /L ⬍ 1.
For the selected nights at 10 m (Fig. 6), Ri remains less
than 0.1 throughout the night for the strong-wind case
but is more variable for the weak-wind regime with
short periods of very high values. We have chosen 10 m
for this part of the analysis because of more reliable
behavior of the temperature gradient estimates at this
level. With weak turbulence and very stable conditions,
10 m is probably above the surface layer.
During the weak-wind night, Ri decreases to values
below 0.25 during subperiods of “less weak” winds. The
occasional large values of Ri for the weak-wind case do
not necessarily occur for large values of z /L, and deviations from the M–O similarity relationship between
Ri and z /L are large for the weak-wind regime (Fig. 7).
Subperiods of large Richardson number sometimes correspond to small u but very weak heat fluxes and
*
therefore small values of z /L. Such small values incorrectly suggest near-neutral conditions. As a result, z /L
is an incomplete measure of stability.
Many of the observed values of Ri for the weak-wind
regime are greater than the M–O prediction (solid line,
Fig. 7) but still less than the relation of Ri ⫽ z /L (dotted line). Deviations from the M–O prediction are
much less for the intermediate and strong-wind regimes. Within the scatter of the data, Ri becomes independent of z /L for large values of z /L, suggesting z-less
turbulence; z /L should not be used as the stability parameter for these cases. For the weak-wind regime, one
could argue that Ri is independent of z /L for the entire
range of z /L (Fig. 7).
Figure 8 displays the eddy diffusivity for momentum
as a function of time (Figs. 8a–c) and z /L (Figs. 8d–f)
for strong- (16–17 October), intermediate- (22–23 October), and weak- (19–20 October) wind cases. The
eddy diffusivity decreases approximately exponentially
with increasing z /L for all three cases. The close relationship is partly due to the fact that the eddy diffusivity
FIG. 7. The dependence of Ri on z /L for strong- (10–11), intermediate- (5–6), and weak- (18–19) wind cases
at 10 m.
3480
MONTHLY WEATHER REVIEW
VOLUME 135
FIG. 8. The dependence of the eddy diffusivity for momentum on (a)–(c) time and (d)–(f) z /L for strong-,
intermediate-, and weak-wind cases at 10 m.
is proportional to the square of the friction velocity
while z /L is inversely proportional to the cube of the
friction velocity. The diffusivity Km is in general one or
more orders of magnitude smaller for weak winds compared to windier conditions (Fig. 8).
7. Mesoscale modulation during the weak-wind
night
b. Eddy Prandtl number
The eddy Prandtl number is defined as the ratio of
eddy diffusivity for momentum, Km, to that for heat,
K h:
Km ⫽
⫺w⬘u ⬘
⫺w⬘T ⬘␷
, Kh ⫽
,
⭸UⲐ⭸z
⭸␪Ⲑ⭸z
increasing z /L, which may also be due to selfcorrelation related to shared values of the heat and
momentum flux.
and
Pr ⫽
Km
.
Kh
共6a,b,c兲
Figure 9 shows the time evolution of the Prandtl number for the strong-, intermediate-, and weak-wind case
study days. For the strong- and intermediate-wind
nights, the eddy Prandtl number approaches values
near unity. Values of the Prandtl number much greater
than unity on the weak-wind night may be due to transport of momentum by nonlinear gravity waves. The
eddy Prandtl number increases with the Richardson
number for all three case studies although the large
self-correlation between the eddy Prandtl number and
Richardson number prevents definite physical conclusions. The eddy Prandtl number decreases slightly with
With weaker mean winds, mesoscale motions emerge
as an important influence on the wind and turbulence
fields. We find important modulation of the turbulent
flux on all of the weak-wind nights, although the time
scale of such modulation varies between nights. These
modulating motions are sometimes coherent across the
entire tower layer and sometimes confined to thin layers near the surface. The turbulence on 18–19 October
is modulated by wavelike modes with a period of
roughly 2 h (Fig. 10). This motion is superimposed on
other motions with a variety of time scales. The main
mesoscale mode is not a pure linear wave in that the
wind accelerations are often abrupt, assuming a microfront behavior. About 5 different mean wind maxima
can be identified, which generally lead to increased turbulence (Fig. 11) but less definable structure in the temperature field. Several of the events are coherent
throughout the entire tower layer, although the first
one appears to start at the lower levels. The last event
at 0515 LST is associated with cooling particularly in
OCTOBER 2007
HA ET AL.
3481
FIG. 9. (a)–(c) The time evolution of Pr for strong- (10–11 Oct), intermediate- (24–25 Oct), and weak- (19–20
Oct) wind cases at 10 m. (d)–(f) The relationship between Pr and Ri for the three wind cases. (g)–(i) The
relationship between Pr and z /L for the three wind cases.
the lowest 5 m and a change of mean wind direction
from westerly to northeasterly, apparently associated
with a density current.
The surface turbulence and fluxes are modulated by
these variations of the wind field (Fig. 11). There is
some evidence that the surface turbulence lags the accelerations at the top of the tower. However, the 2-m
turbulence is better related to the mean wind speed at
55 m than the noisier wind speed at 2 m (not shown).
The wind event at 2120 LST causes the largest increase
of turbulence during the night while the wind acceleration around midnight shows little enhancement of the
turbulence at any level within the tower layer.
During the stronger wind part of the mesoscale
mode, the standard deviation of vertical velocity (␴w)
increases by more than 100% (Fig. 11) while the surface
stress increases by about a factor of 5! Various computations of the bulk or gradient Ri are highly correlated
to the mean wind speed itself and do not lead to improved prediction of the turbulence. Unfortunately,
such important mesoscale modes are not adequately
captured by existing numerical models (study in
progress).
8. Mixing length
Above the surface layer, the flux–gradient relationship is often posed in terms of a mixing length. Mixing
length formulations for the stable conditions were examined based on Louis et al. (1981) shown in Eq. (8a)
and Ha and Mahrt (2001) shown in Eq. (8b), where
3482
MONTHLY WEATHER REVIEW
FIG. 10. One-minute averaged wind speed during the weak-wind
night of 18–19 Oct for the observational different levels.
Kh ⫽ ⫺
w⬘T ⬘␷
and
⭸␪Ⲑ⭸z
冏 冏
Kh ⫽ l h2
⭸U
,
⭸z
lh共Ri兲 ⫽ l0␾h共Ri兲 ⫽ l0关1 ⫹ 15Ri共1 ⫹ 5Ri兲1Ⲑ2兴⫺1Ⲑ2
冋
lh ⫽ l0 exp共c1Ri兲 ⫹
册
c2
,
Ri ⫹ c3
共7a,b兲
and
共8a,b兲
where c1 ⫽ ⫺8.5, c2 ⫽ 0.15, and c3 ⫽ 3.0. Figure 12
shows the mixing length as a function of Ri at 30 m. The
solid and dotted lines indicate the formulations in Eq.
(8a) and Eq. (8b), respectively.
The mixing length lh from Ha and Mahrt (2001) decreases more rapidly with Richardson number than that
of the Louis formulation. Usually the magnitude of the
mixing length is smallest in the early evening (gray
circle) and early morning (open triangle) periods. Compared with observations, the mixing length from Ha
and Mahrt (2001) shows better agreement than Louis et
al. (1981) for our data, although the purpose of the
Louis scheme is to parameterize fluxes over a relatively
large grid area with coarse vertical resolution. For the
strong-wind case, Ri lies within a narrow range centered about 0.2, while the mixing length values extend
over a broad range of values, suggesting that Ri is not
the only important influence on the mixing length. The
deviation of the mixing length from Eq. (8) is not systematically related to the intermittency index.
9. Conclusions
Surface layer similarity theory was evaluated for the
nocturnal boundary layer separately for different wind
regimes. While z /L is significantly inverse correlated
VOLUME 135
FIG. 11. The relationship between the std dev of vertical velocity
(␴w) at 2 m and the wind speed at 55 m during the weak-wind
night of 18–19 Oct.
with mean wind speed, some small values of z /L occur
for weak winds and some large values of z /L occur for
strong winds. In some cases of weak turbulence, z /L
was found to be a misleading indicator of the stability.
For example, weak turbulence, strong stratification,
and large Richardson number sometimes corresponded
to small near-neutral values of z /L.
Since the dependence of the flux–gradient relationship on stability is different for the different wind regimes, stability by itself is an incomplete predictor for
the flux–gradient relationship. The similarity theory is
least valid for weak-wind conditions. The somewhat independent role of the strength of the large-scale flow
may be related to shear generation of turbulence on the
underside of the low-level jet for stronger wind cases
and the increased influence of waves/meandering on
the turbulent flux–gradient relationship for the weakwind cases. However, a successful independent nondimensional parameter representing such influences
could not be identified.
For weak winds, meandering of the wind vector, density currents, and difficulties estimating weak fluxes all
influence the estimated flux–gradient relationship.
While intermittency greatly increased the scatter in the
flux–gradient relationship, intermittency did not lead to
significant systematic deviations from similarity theory.
The relationship between z /L and Ri is not well defined
for weak-wind conditions. The eddy Prandtl number
did not show a well-defined dependence on z /L. An
existing formulation of the mixing length based on Ri
rather than z /L performed well.
For a weak-wind case study night, mesoscale modes
of roughly 2-h periods strongly modulated the wind and
turbulence fields. The inability to predict such motions
would lead to large errors in the turbulence fluxes and
OCTOBER 2007
HA ET AL.
3483
FIG. 12. The dependence of the mixing length on Ri for strong- (14–15 Oct), intermediate- (27–28 Oct), and
weak- (19–20 Oct) wind cases at 30 m.
eddy diffusivity. Our investigation of other weak-wind
nights reveals that the wind and turbulence fields are
often strongly modulated by mesoscale modes of unknown origin.
Acknowledgments. We gratefully acknowledge the
helpful comments of two reviewers. This subject is supported by the Ministry of Environment as “the Ecotechnopia 21 project.” This subject is also supported by
the “Brain Korea 21 Project.”
REFERENCES
Acevedo, O. C., O. L. L. Moraes, G. A. Degrazia, and L. E. Medeiros, 2006: Intermittency and the exchange of scalars in the
nocturnal surface layer. Bound.-Layer Meteor., 119, 41–55.
Anfossi, D., D. Oettl, G. Degrazia, and A. Goulart, 2005: An
analysis of sonic anemometer observations in low wind speed
conditions. Bound.-Layer Meteor., 114, 179–203.
Banta, R. M., R. K. Newsome, J. K. Lundguist, Y. L. Pichugina,
R. L. Coulter, and L. Mahrt, 2002: Nocturnal low-level jet
characteristics over Kansas during CASES-99. Bound.-Layer
Meteor., 105, 221–252.
Beljaars, A. C., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327–341.
Blumen, W., cited 1999: CASES99 Field Catalog. [Available online at http://catalog.eol.ucar.edu/cases99/.]
Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971:
Flux–profile relationships in the atmospheric surface layer. J.
Atmos. Sci., 28, 181–189.
Chimonas, G., 2002: On internal gravity waves associated with the
stable boundary layer. Bound.-Layer Meteor., 102, 139–155.
Coulter, R. L., and J. C. Doran, 2002: Spatial and temporal occurrences of intermittent turbulence during CASES-99. Bound.Layer Meteor., 105, 329–349.
Fernando, H. J. S., 2003: Turbulence patches in a stratified shear
flow. Phys. Fluids, 15, 3164–3169.
Ha, K.-J., and L. Mahrt, 2001: Simple inclusion of z-less turbulence within and above the modeled nocturnal boundary
layer. Mon. Wea. Rev., 129, 2136–2143.
——, and L. Mahrt, 2003: Radiative and turbulent fluxes in the
nocturnal boundary layer. Tellus, 55A, 317–327.
Holtslag, A. A. M., and H. A. R. De Bruin, 1988: Applied modeling of the nighttime surface energy balance over land. J.
Appl. Meteor., 27, 689–704.
Howell, J., and J. Sun, 1999: Surface layer fluxes in stable conditions. Bound.-Layer Meteor., 90, 495–520.
Hyun, Y.-K., K.-E. Kim, and K.-J. Ha, 2005: A comparison of
methods to estimate the height of stable boundary layer over
a temperate grassland. Agric. For. Meteor., 132, 132–142.
Klipp, C. L., and L. Mahrt, 2004: Flux–gradient relationship, selfcorrelation and intermittency in the stable boundary layer.
Quart. J. Roy. Meteor. Soc., 130, 2087–2103.
Louis, J. F., M. Tiedtke, and J. F. Geleyn, 1981: A short history of
the operational PBL-parameterization at ECMWF. Proc.
Workshop on Planetary Boundary Layer Parameterization,
Reading, Berkshire, United Kingdom, ECMWF, 59–79.
Mahrt, L., 1985: Vertical structure and turbulence in the very
stable boundary layer. J. Atmos. Sci., 42, 2333–2349.
——, J. Sun, W. Blumen, T. Delany, and S. Oncley, 1998: Nocturnal boundary-layer regimes. Bound.-Layer Meteor., 88,
255–278.
Nakamura, R., and L. Mahrt, 2005: A study of intermittent turbulence with CASES-99 tower measurements. Bound.-Layer
Meteor., 114, 367–387.
Nappo, C. J., 2002: An Introduction to Atmospheric Gravity
Waves. Academic Press, 276 pp.
Ohya, Y., D. E. Neff, and R. N. Meroney, 1997: Turbulence structure in a stratified boundary layer under stable conditions.
Bound.-Layer Meteor., 83, 139–162.
Pardyjak, E., P. Monti, and H. Fernando, 2002: Flux Richardson
number measurements in stable atmospheric shear flows. J.
Fluid Mech., 459, 307–316.
Poulos, G. S., and Coauthors, 2002: CASES-99: A comprehensive
investigation of the stable nocturnal boundary layer. Bull.
Amer. Meteor. Soc., 83, 555–581.
Salmond, J. A., 2005: Wavelet analysis of intermittent turbulence
in a very stable nocturnal boundary layer: Implications for
the vertical mixing of ozone. Bound.-Layer Meteor., 114, 463–
488.
Sun, J., and Coauthors, 2002: Intermittent turbulence associated
with a density current passage in the stable boundary layer.
Bound.-Layer Meteor., 105, 199–219.
Vickers, D., and L. Mahrt, 1997: Quality control and flux sampling
problems for tower and aircraft data. J. Atmos. Oceanic Technol., 14, 512–526.