Evaluation of Boundary Layer Similarity Theory for Stable ...kgw T 1 Here, k is the von Kármán constant, g is the earth’s gravitational acceleration, is the potential tempera-ture,

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 stableboundary layer are evaluated using the Cooperative Atmosphere–Surface Exchange Study-1999 (CASES-99) data. The large-scale wind forcing is classified into weak, intermediate, and strong winds. Although thestability parameter, z /L, is inversely dependent on the mean wind speed, the speed of the large-scale flowincludes independent influences on the flux–gradient relationship. The dimensionless mean wind shear isfound to obey existing stability functions when z /L is less than unity, particularly for the strong andintermediate wind classes. For weak mean winds and/or strong stability (z /L � 1), this similarity theorybreaks down. Deviations from similarity theory are examined in terms of intermittency. A case study of aweak-wind night indicates important modulation of the turbulence flux by mesoscale motions of unknownorigin.

1. Introduction

Monin–Obukhov (M–O) similarity is used to relatethe flux–gradient relationship in the surface layer to thestability parameter, z/L, in stationary flow over homo-geneous surfaces, where L is the Obukhov length. Ac-tual surfaces are generally heterogeneous, to some de-gree. Even modest surface heterogeneity may influencethe weak turbulence in stable boundary layers.

Intermittency of the flux (Howell and Sun 1999;

Coulter and Doran 2002; Salmond 2005; Acevedo et al.2006) may also influence the time-averaged flux–gradient relationship. The definition of the intermit-tency varies between studies. Normally, the nonstation-arity of the turbulence is more complex than simpleon–off behavior (Nakamura and Mahrt 2005). Turbu-lence intermittency can be related to internal interac-tions between turbulent mixing and the mean shear(Pardyjak et al. 2002; Fernando 2003), perhaps involv-ing variation of the Richardson number about an equi-librium value. Or intermittency may be externallyforced by mesoscale motions, such as internal gravitywaves (Nappo 2002; Chimonas 2002) or horizontal me-andering of the mean wind field (Anfossi et al. 2005). Inaddition, downward transport of turbulence toward the

Corresponding author address: Kyung-Ja Ha, Division of EarthEnvironmental System, Pusan National University, Busan, 609-735, South Korea.E-mail: [email protected]

3474 M O N T H L Y W E A T H E R R E V I E W VOLUME 135

DOI: 10.1175/MWR3488.1

© 2007 American Meteorological Society

MWR3488

surface (Ha and Mahrt 2001), and associated increaseof turbulence near the surface due to flux convergenceof the turbulence energy, is not included in the surfacelayer similarity theory nor in the traditional concept ofboundary layers where the turbulence is controlled bysurface processes. The strongest turbulence may occurabove the surface inversion (Mahrt 1985; Ohya et al.1997; Banta et al. 2002). Both internal and externalintermittency, as defined above, occur on scales largerthan individual eddies and thus contrast with finescaleintermittency associated with the substructure withinindividual eddies.

For clear nights, the stratification is normally inversecorrelated with mean wind speed since weaker meanwinds correspond to smaller shear generation of turbu-lence and permit larger stratification. Here, we classifythe nocturnal boundary layer flows in terms of themean wind speed in addition to stability. Wind speed isa dimensional quantity and therefore cannot provideuniversal classification. From the point of view of tur-bulence similarity theory, the bulk Richardson numberis preferable to the mean wind speed since it is nondi-mensional. However, wind speed serves as a more ex-ternal variable in contrast to z/L, which depends on theturbulence itself. The mean wind speed does not inheriterrors in the estimation of vertical gradients requiredfor the Richardson number nor the ambiguity of defin-ing the surface temperature. Banta et al. (2002) dem-onstrated that the speed of the low-level jet is a usefuloverall predictor of properties of the nocturnal bound-ary layer. Anfossi et al. (2005) found that with meanwinds weaker than about 1.5 m s�1, the mean wind fieldbecomes more dominated by nonstationary meander-ing 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 thelarge-scale flow independent of the influence of stabil-ity.

2. Data

We analyze 5-min averaged data from a 60-m towertaken during the Cooperative Atmosphere–Surface Ex-change Study-1999 (CASES-99; Blumen 1999; Pouloset al. 2002; Ha and Mahrt 2003). The observations weremade over relatively flat temperate grassland nearLeon, Kansas, in October 1999. The main site includesa 60-m tower system (Fig. 1) with 8 levels of eddy cor-relation data based on Campbell sonic anemometers(CSAT), 34 levels of thermocouple data from 0.23 to58.10 m (Sun et al. 2002), the R. M. Young propelleranemometer and wind vane data at 15, 25, 35, and 45 m,and aspirated thermistor data at 5, 15, 25, 35, 45, and 55m. Thermistor data are linearly interpolated to 5, 10, 20,30, 40, 50, and 55 m in this study. The radiosondes werealso released at 270 m apart from the 60-m tower (e.g.,Hyun et al. 2005). Turbulent fluxes and mean windswere 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 maintower. The data were collected at 20 Hz. The data werequality controlled following Vickers and Mahrt (1997).

3. Classification by wind speed

The data are divided into three different mean windspeed regimes: strong-, intermediate-, and weak-windregimes. The three regimes based on the mean windspeed at 5 m are Ui � U � 0.55s, U � 0.55s � Ui �

FIG. 1. The 60-m tower instrumented with 7 levels of flux measurements.

OCTOBER 2007 H A E T A L . 3475

U � 0.55s, and Ui � U � 0.55s. Here, U is the meanwind speed averaged over all nights, Ui is the meanwind speed of night i, and s is the standard deviation(Table 1). Data are selected from 21 nights where theflow 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 thestrong-wind case, the 5-m average mean wind speedexceeds 4.1 m s�1, while the mean wind is less than 2.1m s�1 for the weak-wind nights. Classification in termsof mean wind shear would have led to similar results.

The stability is expressed as z/L, where L is theObukhov length,

L �

��u3

*

kgw�T ��1�

Here, k is the von Kármán constant, g is the earth’sgravitational acceleration, � is the potential tempera-

ture, u* is the friction velocity, and w�� is the potentialtemperature flux. In this study, L is evaluated fromfluxes at level z rather than from surface fluxes. Theranges of z/L at 5 m are noted in the last column ofTable 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 windspeed. For example, the near neutral stability (class B)generally occurs on strong-wind nights, while the weak-wind night tends to be very stable (class D). Exceptionsoccur. The use of 5-min fluxes leads to large scatterassociated with large random flux errors compared tothe 30-min or 1-h averages but leads to less capture ofnonstationarity of the mean flow and turbulence withinthe flux calculation. Inclusion of nonstationarity by av-eraging the flux over longer time scales systematicallyalters the flux–gradient relationship, examined in aseparate study. Use of 5-min averages instead of longeraverages reduces this bias at the cost of increased scat-ter.

4. Evaluation of similarity relationships

The flux–gradient relationship in the surface layer isusually posed in terms of M–O similarity theory, whichrelates nondimensional gradients to z/L. The nondi-mensional shear, m, is defined as

�m� z

L� �kz

u*��U

�z � �2�

The nondimensional shear is often parameterized interms of the stability functions from Businger et al.(1971) and Beljaars and Holtslag (1991), respectively:

�m� z

L� � 1 � 4.7z

L, �3a�

�m� z

L� � 1 �z

L�a � b � e�d�z�L��1 � c � dz

L��,

�3b�

where a � 1, b � 0.667, c � 5, and d � 0.35. TheBeljaars–Holtslag formulation reduced the overestima-tion of the nondimensional gradients for very stableconditions 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 ver-tical mean wind shear at the height of 5 m is obtainedby logarithmic finite differencing between 1.5 and 10 m.For the weakest winds, the gradient depends on themethod of estimating gradients and no preferredmethod emerges. The solid and dotted lines representthe curves based on Eqs. (3a) and (3b) respectively.

TABLE 1. Classifications of observed wind and calculated z /L at5 m. U is an average for total 23 nights, Ui is mean of each nighti, and s is std dev between the records for each night. Here, �

denotes strong-wind cases, � denotes intermediate-wind cases �

denotes weak-wind cases. Classifications of “S,” “I,” and “W”mean strong-, intermediate-, and weak-wind cases, respectively.Here, �: U � 0.55s � Ui, �: U � 0.55s � Ui U � 0.55s,�: Ui U � 0.55s A: 0 � z/L 0.01, B: 0.01 � z/L 0.1,C: 0.1 � z/L 1, D: 1 � z/L 10

Date

Wind at 5 m

z /L at 5 m Ui ClassMagnitude Shear

5–6 � � D 2.3 I6–7 � � B 5.4 S7–8 � � C 2.2 I8–9 � � C 2.3 I9–10 � � D 1.2 W

10–11 � � B 4.1 S11–12 � � C 3.2 I12–13 — — C — —13–14 � � D 1.9 W14–15 � — B 6.1 S15–16 � � B 4.9 S16–17 � � A 7.3 S17–18 � � D 2.1 I18–19 � � D 1.3 W19–20 � � D 1.6 W20–21 � � D 2.4 I21–22 � � D 2.5 I22–23 � � D 2.7 I23–24 — — D — —24–25 � � C 4.0 I25–26 � � D 1.5 W26–27 � � C 3.1 I27–28 � � C 3.4 I


Note that most cases with z/L � 0 occur with weakwinds. For example, some near-neutral cases (smallz/L) occur with weak winds associated with small u*but very small heat flux. A few cases of small z/L occurwith large m and large Richardson number. This be-havior was not sensitive to choice of averaging time,although sampling errors associated with the weakfluxes may be important. We conclude that the stabilityparameter z/L may not be an adequate indicator of thestability for weak turbulence and strong stratification.

The nondimensional wind shear (Fig. 2) is well ap-proximated by the empirical expression of Businger etal. (1971) for the strong and intermediate wind classes.In the weak-wind regime, the points are widely scat-tered even for small values of z/L. For large z/L, thevalues of m tend to occur between the two empiricalcurves in Fig. 2. For z/L less than one, where all threewind speed regimes contain data, the weak-wind re-gime exhibits much more scatter, presumably due tothe stronger intermittency (section 5a) and larger ran-dom flux errors. These results suggest that the meanwind speed contains an independent influence not con-tained in the stability.

To estimate the contribution of self-correlation, weconstruct randomizations of the original data based onthe method of Klipp and Mahrt (2004). With thismethod, the original values of the mean shear andfluxes are randomized such that the values of the meanshear and fluxes for a given realization of the random-ized data originate from different records. With no self-correlation, the correlation between m and z/L for therandomized data would be zero. The correlation be-tween the nondimensional shear and z/L for the ran-domized data at 5 m is not significantly smaller thanthat for the original data (Fig. 3), suggesting that linearcorrelation cannot be confidently used as verification ofthe degree of physical relationship between the nondi-mensional shear and z/L, even though the slope is sig-nificantly less for the randomized data.

5. Turbulence statistics

a. Intermittency

We evaluate a simple index of intermittency (turbu-lence 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).

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.


FI �F

abs�F �, �4�

where F is the standard deviation of 5-min values ofthe friction velocity within a 1-h period and F is thefriction velocity evaluated at 5 and 30 m. The 30-m levelis typically above the intense part of the surface inver-sion layer, where the local value of the momentum fluxis not representative of the surface momentum flux.Here, z/L is used as an informal stability parametersince 30 m is too high for the validity of M–O similaritywith significant stability. Figure 4 shows the intermit-tency index for u* for the strong-wind (dark circles),intermediate-wind (open circles), and weak-wind (graycircles) nights. The intermittency index is small forz/L 0.1 and increases roughly linearly with z/L inln–ln coordinates (with large scatter) for z/L � 0.1. Ingeneral, the intermittency index is greatest for theweak-wind class. The intermittency, such as defined inEq. (4), partly explains the greater scatter in the rela-tionship between m and z/L for the weak-wind cat-egory (Fig. 2). The main qualitative features occur at

both the 5- and 30-m levels, but 30 m contains moredata with small z/L.

b. Deviation from the similarity prediction for theflux–gradient relationship

For the present data (Fig. 5), the large intermittencyindex generally corresponds to expected greater mag-nitude of the deviations from the fitted m–z/L rela-tionship, but such deviations of m occur with eithersign with roughly the same probability. The intermit-tency leads to larger scatter but no detectable system-atic changes. For the strong-wind regime, the intermit-tency index is generally below 0.3 and deviations fromsimilarity theory are small.

6. Dimensionless numbers

To further examine the dependence of turbulence onstability, the relationship between z/L, the gradient Ri-chardson number, and turbulent Prandtl number arenow examined for the strong-, intermediate-, and weak-wind 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.

FIG. 4. The dependence of z /L on the index of intermittency at (a) 5 and (b) 30 m.


a. Richardson number

The gradient Richardson number, Ri � (g/�)[(��/�z)/(�U/�z)2], in the surface layer is predicted to be propor-tional to z/L (Businger et al. 1971):

Ri � z �L ��0.74 � 4.7z �L�

�1 � 4.7z �L�2 . �5�

The relationship was originally intended for z/L 1.For the selected nights at 10 m (Fig. 6), Ri remains lessthan 0.1 throughout the night for the strong-wind casebut is more variable for the weak-wind regime withshort periods of very high values. We have chosen 10 mfor this part of the analysis because of more reliablebehavior of the temperature gradient estimates at thislevel. With weak turbulence and very stable conditions,10 m is probably above the surface layer.

During the weak-wind night, Ri decreases to valuesbelow 0.25 during subperiods of “less weak” winds. Theoccasional large values of Ri for the weak-wind case donot necessarily occur for large values of z/L, and de-viations from the M–O similarity relationship betweenRi and z/L are large for the weak-wind regime (Fig. 7).

Subperiods of large Richardson number sometimes cor-respond to small u* but very weak heat fluxes andtherefore small values of z/L. Such small values incor-rectly suggest near-neutral conditions. As a result, z/Lis an incomplete measure of stability.

Many of the observed values of Ri for the weak-windregime are greater than the M–O prediction (solid line,Fig. 7) but still less than the relation of Ri � z/L (dot-ted line). Deviations from the M–O prediction aremuch less for the intermediate and strong-wind re-gimes. Within the scatter of the data, Ri becomes inde-pendent of z/L for large values of z/L, suggesting z-lessturbulence; z/L should not be used as the stability pa-rameter for these cases. For the weak-wind regime, onecould argue that Ri is independent of z/L for the entirerange of z/L (Fig. 7).

Figure 8 displays the eddy diffusivity for momentumas a function of time (Figs. 8a–c) and z/L (Figs. 8d–f)for strong- (16–17 October), intermediate- (22–23 Oc-tober), and weak- (19–20 October) wind cases. Theeddy diffusivity decreases approximately exponentiallywith increasing z/L for all three cases. The close rela-tionship 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 casesat 10 m.

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.


is proportional to the square of the friction velocitywhile z/L is inversely proportional to the cube of thefriction velocity. The diffusivity Km is in general one ormore orders of magnitude smaller for weak winds com-pared to windier conditions (Fig. 8).

b. Eddy Prandtl number

The eddy Prandtl number is defined as the ratio ofeddy diffusivity for momentum, Km, to that for heat,Kh:

Km ��w�u �

�U��z, Kh �

�w�T ��z

, and Pr �Km

Kh.

�6a,b,c�

Figure 9 shows the time evolution of the Prandtl num-ber for the strong-, intermediate-, and weak-wind casestudy days. For the strong- and intermediate-windnights, the eddy Prandtl number approaches valuesnear unity. Values of the Prandtl number much greaterthan unity on the weak-wind night may be due to trans-port of momentum by nonlinear gravity waves. Theeddy Prandtl number increases with the Richardsonnumber for all three case studies although the largeself-correlation between the eddy Prandtl number andRichardson number prevents definite physical conclu-sions. The eddy Prandtl number decreases slightly with

increasing z /L, which may also be due to self-correlation related to shared values of the heat andmomentum flux.

7. Mesoscale modulation during the weak-windnight

With weaker mean winds, mesoscale motions emergeas an important influence on the wind and turbulencefields. We find important modulation of the turbulentflux on all of the weak-wind nights, although the timescale of such modulation varies between nights. Thesemodulating motions are sometimes coherent across theentire tower layer and sometimes confined to thin lay-ers near the surface. The turbulence on 18–19 Octoberis modulated by wavelike modes with a period ofroughly 2 h (Fig. 10). This motion is superimposed onother motions with a variety of time scales. The mainmesoscale mode is not a pure linear wave in that thewind accelerations are often abrupt, assuming a micro-front behavior. About 5 different mean wind maximacan be identified, which generally lead to increased tur-bulence (Fig. 11) but less definable structure in the tem-perature field. Several of the events are coherentthroughout the entire tower layer, although the firstone appears to start at the lower levels. The last eventat 0515 LST is associated with cooling particularly in

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.


the lowest 5 m and a change of mean wind directionfrom westerly to northeasterly, apparently associatedwith a density current.

The surface turbulence and fluxes are modulated bythese variations of the wind field (Fig. 11). There issome evidence that the surface turbulence lags the ac-celerations at the top of the tower. However, the 2-mturbulence is better related to the mean wind speed at55 m than the noisier wind speed at 2 m (not shown).The wind event at 2120 LST causes the largest increaseof turbulence during the night while the wind accelera-tion around midnight shows little enhancement of theturbulence at any level within the tower layer.

During the stronger wind part of the mesoscalemode, the standard deviation of vertical velocity ( w)

increases by more than 100% (Fig. 11) while the surfacestress increases by about a factor of 5! Various compu-tations of the bulk or gradient Ri are highly correlatedto the mean wind speed itself and do not lead to im-proved prediction of the turbulence. Unfortunately,such important mesoscale modes are not adequatelycaptured by existing numerical models (study inprogress).

8. Mixing length

Above the surface layer, the flux–gradient relation-ship is often posed in terms of a mixing length. Mixinglength formulations for the stable conditions were ex-amined based on Louis et al. (1981) shown in Eq. (8a)and Ha and Mahrt (2001) shown in Eq. (8b), where

FIG. 9. (a)–(c) The time evolution of Pr for strong- (10–11 Oct), intermediate- (24–25 Oct), and weak- (19–20Oct) wind cases at 10 m. (d)–(f) The relationship between Pr and Ri for the three wind cases. (g)–(i) Therelationship between Pr and z /L for the three wind cases.


Kh � �w�T ��z

and Kh � lh2��U

�z�, �7a,b�

lh�Ri� � l0�h�Ri� � l0�1 � 15Ri�1 � 5Ri�1�2��1�2 and

lh � l0�exp�c1Ri� �c2

Ri � c3�, �8a,b�

where c1 � �8.5, c2 � 0.15, and c3 � 3.0. Figure 12shows the mixing length as a function of Ri at 30 m. Thesolid and dotted lines indicate the formulations in Eq.(8a) and Eq. (8b), respectively.

The mixing length lh from Ha and Mahrt (2001) de-creases more rapidly with Richardson number than thatof the Louis formulation. Usually the magnitude of themixing length is smallest in the early evening (graycircle) and early morning (open triangle) periods. Com-pared with observations, the mixing length from Haand Mahrt (2001) shows better agreement than Louis etal. (1981) for our data, although the purpose of theLouis scheme is to parameterize fluxes over a relativelylarge grid area with coarse vertical resolution. For thestrong-wind case, Ri lies within a narrow range cen-tered about 0.2, while the mixing length values extendover a broad range of values, suggesting that Ri is notthe only important influence on the mixing length. Thedeviation of the mixing length from Eq. (8) is not sys-tematically related to the intermittency index.

9. Conclusions

Surface layer similarity theory was evaluated for thenocturnal boundary layer separately for different windregimes. While z/L is significantly inverse correlated

with mean wind speed, some small values of z/L occurfor weak winds and some large values of z/L occur forstrong winds. In some cases of weak turbulence, z/Lwas found to be a misleading indicator of the stability.For example, weak turbulence, strong stratification,and large Richardson number sometimes correspondedto small near-neutral values of z/L.

Since the dependence of the flux–gradient relation-ship on stability is different for the different wind re-gimes, stability by itself is an incomplete predictor forthe flux–gradient relationship. The similarity theory isleast valid for weak-wind conditions. The somewhat in-dependent role of the strength of the large-scale flowmay be related to shear generation of turbulence on theunderside of the low-level jet for stronger wind casesand the increased influence of waves/meandering onthe turbulent flux–gradient relationship for the weak-wind cases. However, a successful independent nondi-mensional parameter representing such influencescould not be identified.

For weak winds, meandering of the wind vector, den-sity currents, and difficulties estimating weak fluxes allinfluence the estimated flux–gradient relationship.While intermittency greatly increased the scatter in theflux–gradient relationship, intermittency did not lead tosignificant systematic deviations from similarity theory.The relationship between z/L and Ri is not well definedfor weak-wind conditions. The eddy Prandtl numberdid not show a well-defined dependence on z/L. Anexisting formulation of the mixing length based on Rirather than z/L performed well.

For a weak-wind case study night, mesoscale modesof roughly 2-h periods strongly modulated the wind andturbulence fields. The inability to predict such motionswould lead to large errors in the turbulence fluxes and

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-windnight of 18–19 Oct.FIG. 10. One-minute averaged wind speed during the weak-wind

night of 18–19 Oct for the observational different levels.


eddy diffusivity. Our investigation of other weak-windnights reveals that the wind and turbulence fields areoften strongly modulated by mesoscale modes of un-known origin.

Acknowledgments. We gratefully acknowledge thehelpful comments of two reviewers. This subject is sup-ported by the Ministry of Environment as “the Eco-technopia 21 project.” This subject is also supported bythe “Brain Korea 21 Project.”

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FIG. 12. The dependence of the mixing length on Ri for strong- (14–15 Oct), intermediate- (27–28 Oct), andweak- (19–20 Oct) wind cases at 30 m.


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Evaluation of Boundary Layer Similarity Theory for Stable ...kgw T 1 Here, k is the von Kármán constant, g is the earth’s gravitational acceleration, is the potential tempera-ture,