Vertical Variations of Mixing Lengths under Neutral and Stable Conditions during CASES-99

Jielun Sun National Center for Atmospheric Research, Boulder, Colorado

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Abstract

An investigation on vertical variations of the mixing lengths for momentum and heat under neutral and stable conditions was conducted using the data collected from the Cooperative Atmosphere–Surface Exchange Study in 1999 (CASES-99). By comparing κz with the mixing lengths under neutral conditions calculated using the observations from CASES-99, the vertical layer where the Monin–Obukhov similarity theory (MOST) is valid was identified. Here κ is the von Kármán constant and z is the height above the ground. On average, MOST is approximately valid between 0.5 and 10 m. Above the layer, the observed mixing lengths under neutral conditions are smaller than the MOST κz and can be approximately described by Blackadar’s mixing length, κz/[1 + (κz/l)], with l = 15 m for up to z ~ 20 m for the mixing length for momentum and up to the highest observation height for the mixing length for heat. Above ~20 m, the mixing length for momentum approaches a constant. Both MOST κz and Blackadar’s formula systematically overestimate the mixing length for momentum above ~20 m, leading to overestimates of turbulence.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jielun Sun, National Center for Atmospheric Research, Boulder, CO 80307-3000. E-mail: jsun@ucar.edu

Abstract

An investigation on vertical variations of the mixing lengths for momentum and heat under neutral and stable conditions was conducted using the data collected from the Cooperative Atmosphere–Surface Exchange Study in 1999 (CASES-99). By comparing κz with the mixing lengths under neutral conditions calculated using the observations from CASES-99, the vertical layer where the Monin–Obukhov similarity theory (MOST) is valid was identified. Here κ is the von Kármán constant and z is the height above the ground. On average, MOST is approximately valid between 0.5 and 10 m. Above the layer, the observed mixing lengths under neutral conditions are smaller than the MOST κz and can be approximately described by Blackadar’s mixing length, κz/[1 + (κz/l)], with l = 15 m for up to z ~ 20 m for the mixing length for momentum and up to the highest observation height for the mixing length for heat. Above ~20 m, the mixing length for momentum approaches a constant. Both MOST κz and Blackadar’s formula systematically overestimate the mixing length for momentum above ~20 m, leading to overestimates of turbulence.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jielun Sun, National Center for Atmospheric Research, Boulder, CO 80307-3000. E-mail: jsun@ucar.edu

1. Introduction

Turbulence in the atmospheric boundary layer has been investigated based on knowledge gained from the wall and outer layers (where the effect of the wall shear on turbulence is small) under neutral conditions (e.g., Gibson and Launder 1978; Smedman 1988) (Fig. 1). Various names for the wall layer and the outer layer have been used in the literature. For example, the wall layer has also been called the surface layer (e.g., Tennekes and Lumley 1972). The outer layer has been called the free shear layer (e.g., Smedman 1988) or the outer Ekman layer (e.g., Blackadar and Tennekes 1968). Because the dominant length scale that influences turbulence varies between the wall and the outer layers, turbulence characteristics are significantly different between the two layers (e.g., Hunt et al. 1988). In this study, the layer that is directly influenced by wind shear at the ground and where the observation height above the ground is a critical length scale is called the wall layer; the layer above it is the outer layer (Fig. 1). Within the wall layer, the layer where the Monin–Obukhov (M–O) similarity theory (MOST) is valid is called the M–O layer. In the literature, the M–O layer is often called the inertial sublayer (Tennekes and Lumley 1972), the equilibrium layer (Townsend 1961), or the surface layer (e.g., Raupach et al. 1980; Garratt 1992). The layer below the M–O layer is referred to as the generalized roughness sublayer in this study.

Fig. 1.
Fig. 1.

Schematic of the various layers above the ground under neutral conditions.

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

a. Monin–Obukhov similarity theory and mixing lengths

Using the K theory, momentum τ and sensible heat fluxes H are parameterized as
e1
e2
where v is a wind vector, which consists of the zonal, meridional, and vertical components u, υ, and w; θ is the potential temperature; ρ is the air density; cp is the air specific heat at constant pressure; Km is the eddy viscosity; and Kh is the eddy diffusivity for heat. The overbar represents an average over a time window (i.e., the block average), the prime is the perturbation from the block average, and z is the height above the ground. Using Prandtl’s mixing length l, Km and Kh in (1) and (2) are related to l (e.g., Stull 1988; Garratt 1992) as follows:
e3
e4
where . Generally l is assumed to be the same for both Km and Kh. However, in this study, the mixing length for momentum lm and the mixing length for heat lh are distinguished. Using (3), (4), and and replacing the ls with lm and lh, (1) and (2) become
e5
e6
Monin and Obukhov [1954; English version can be found in Obukhov (2001), chapter 2.5] obtained a relationship between turbulence and mean variables based on the nondimensional analysis of the Buckingham π theory. MOST assumes constant momentum and heat fluxes, horizontally homogeneous and stationary flow, mean variables being functions of height only, and negligible effects of the Coriolis force. MOST describes momentum and sensible heat fluxes in a stratified flow as
e7
e8
In the above equations, ζz/L, is the Obukhov length, κ ~ 0.4 is the von Kármán constant, g is the gravitational constant, Φm and Φh are the M–O stability functions for momentum and heat, respectively, and T0 is a reference air temperature. The subscript 0 represents variables at the ground. In this study, the virtual potential temperature θυ approximately satisfies θυθ; therefore, θ instead of θυ, which is the formally correct temperature variable, is used in (6), (8), and L. MOST ignores the vertical variation of wind direction; thus, (more in section 2c). Using (5) and (6), (7) and (8) become
e9
e10
Because under neutral conditions (i.e., ζ = 0) Φm(0) = Φh(0) = 1 (commonly observed), (9) and (10) imply that both mixing lengths in neutral stratification are equal to κz in MOST. To generalize (9) and (10) to a layer deeper than the M–O layer, the dependence of lm and lh on height and stability can be expressed as
e11
e12
where lmN(z) and lhN(z) represent lm and lh under neutral conditions, respectively, and Fm and Fh are generalized stability functions. Therefore, the M–O layer is the layer where , , Fm = Φm, and Fh = Φh, where and are calculated using (5) and (6) with the observations under neutral conditions.

b. Self-correlation in MOST

A self-correlation problem can occur in a relationship between two variables if both variables depend on a common variable (Hicks 1978). Klipp and Mahrt (2004) pointed out that because of the appearance of u* on both sides of (7) and θ* on both sides of (8), there is a serious self-correlation problem in MOST. Andreas and Hicks (2002) and Baas et al. (2006) also discussed the self-correlation problem in data analyses. Mahrt (2007) predicted that the self-correlation in MOST would lead to Φm as a function of ζ1/3 for large ζ, which was observed by Grachev et al. (2005).

The self-correlation problem still exists if the stability parameter ζ is replaced by the gradient Richardson number, For example, if Φh defined in (8) is related to Ri, both Φh and Ri increase with . Therefore, the positive physical relationship between Φh and Ri (more in section 2d) can be contaminated by the self-correlation between Φh and Ri through their common dependence on . Different from Φh, physically Φm also increases with Ri, but the self-correlation between Φm and Ri leads to Φm decreasing with Ri through their common dependence on shear. Therefore, the self-correlation between Φm and Ri obscures the physical relationship between Φm and Ri. Because both (7) and (8) are needed to estimate sensible heat fluxes, the self-correlation problem cannot be completely avoided regardless of whether ζ or Ri is used as the stability parameter except under neutral conditions when Φm = Φh = 1.

c. Focus of this study

Because of the popularity of MOST in parameterizing turbulent fluxes for numerical models and difficulties in parameterizing turbulence in stable boundary layers (SBL) (e.g., Mahrt 1999), MOST needs to be further studied, especially for very stable situations. The Cooperative Atmosphere-Surface Exchange Study (CASES-99) conducted near Leon, Kansas, was designed to investigate stable boundary layers; therefore, its dataset would be ideal for examining the stability functions under stable conditions. However, before we do so, we need to know where the M–O layer was during CASES-99. This study focuses on the investigation of the vertical domain where MOST is valid by examining the height dependence of lm and lh using the CASES-99 dataset under neutral and stable conditions. The observation and the methodology for calculating essential variables used in this study are discussed in section 2. The height and stability dependence of lm and lh for the stability range from neutral to stable are examined in section 3. The main results are summarized in section 4.

2. Observations and essential calculated variables

a. Observations

The tower setup and the instruments deployed during CASES-99 were described by Poulos et al. (2002) and Sun et al. (2002). In this study, the measurements from the 60-m tower were used. Three-dimensional turbulence observations were obtained at nine levels (eight levels at a given time). The lowest sonic anemometer was moved from 1.5 to 0.5 m for the last one-third of the experiment period starting on 20 October 1999. The other turbulence observation levels were 5, 10, 20, 30, 40, 50, and 55 m. Together with the slow-response wind vanes at 4 levels (15, 25, 35, and 45 m), a total of 12 levels of the wind observation on the 60-m tower were used in this study. The slow-response wind observations were calibrated to the sonic anemometer measurements. The thermocouple temperature measurements at 34 levels (0.23, 0.67, 2.3, and every 1.8 m above 2.3 m) used in this study were calibrated with the aspirated temperatures at 4 levels (15, 25, 35, and 45 m) (Burns and Sun 2000). During CASES-99, the ground was covered mainly by ~0.10-m-tall grass (Sun et al. 2003). Since all the sonic anemometers were mounted on the booms pointing east, turbulent fluxes associated with wind from 270° ±60° could be distorted by the 60-m tower and were eliminated from the flux dataset. All the data except those mentioned above during the period between 1800 and 0600 LST over 18 days were used in this study.

Using the bulk formula with the observed momentum flux and wind under strong wind conditions, the roughness length for momentum, z0, is ~0.05 m. To avoid the ambiguity generated by using both displacement height and z0, the above z0 was calculated without a displacement height. The turbulent fluxes used in this study were calculated with mesoscale fluctuations, such as gravity waves, removed based on the method of Vickers and Mahrt (2003, 2006) at 10-min intervals (see their papers for details). No significant systematic errors are expected. The block average for obtaining mean variables was 10 min.

b. Wind and potential temperature profiles under neutral conditions

To examine lmN and lhN to find where MOST is valid, our nighttime dataset needs to include some neutral cases. Strong mixing over the entire 60-m layer often occurs when u* at 5 m is larger than 0.3 m s−1. Using this criterion, the Obukhov length, L (calculated using the turbulent fluxes at 5 m above the ground), is between 80 and 200 m with the median value of 150 m, which is within the normal range of L used for the neutral condition in the literature (e.g., Garratt 1992). A total of 33 wind profiles satisfy the above criterion.

Since the observed wind profiles under the neutral condition listed above are not perfect logarithmic profiles over the entire vertical observation range, the difference between the observed and fitted profiles depends on whether the entire observed wind profile or only a section of it is fitted to a logarithmic profile. If all 12 levels of the observed winds are used, the deviation of the observed winds from the fitted logarithmic wind profile is large below 10 m. On the other hand, if only the observed winds close to the ground are used to fit a logarithmic wind profile, the deviation of the observed winds from the logarithmic profile is large above 10 m. When the observed near-neutral wind at 5, 10, and 15 m is used to fit a logarithmic profile, the observed wind below 5 m and above 15 m is systematically stronger than the logarithmic wind, which is visible in Fig. 2a.

Fig. 2.
Fig. 2.

(a) Four examples of 10-min-averaged wind profiles under the neutral condition defined in section 2b for two periods when the lowest sonic anemometer was at 1.5 and 0.5 m. (b) The deviations of the observed neutral wind speeds from the logarithmic wind profiles that were fitted to the observed wind from 5 to 15 m. (c) The composite u* profiles for the two periods, where u* was calculated using the local momentum fluxes. For each time period, two u* profiles were composited: one for the neutral strong wind condition (marked with s), and one for the rest of the conditions (marked with w). The horizontal bars at the lowest observation level in (c) represent ±10% of the vertically averaged u* for each profile. The logarithmic wind profile fitted for each observed wind profile is marked by the solid line in the same color as the observations in (a).

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

With the fitting between 5 and 15 m, the difference between the fitted and the observed wind profiles is larger than 0.4 m s−1 below 5 m (Fig. 2b), but it is, in general, smaller than those if the entire or the bottom of the observed wind profile is fitted to a logarithmic profile. The positive deviation of our observed neutral wind profile from the logarithmic profile above ~15 m is qualitatively consistent with the wind observations presented by Kader and Yaglom (1978), who attributed the deviation to the increasing role of horizontal pressure gradients with height. The deviation close to and far away from the ground was also observed in neutral wind tunnels (e.g., Raupach et al. 1980; Kastner-Klein and Fedorovich 2002).

Using the same neutral condition, a similar deviation pattern occurs for the observed neutral potential temperature (Figs. 3a,b). If each observed potential temperature profile from 5.9 to 9.5 m is fitted to a logarithmic profile, the observed potential temperature is systematically larger than the logarithmic fitting below 5 m and above 15 m, and the deviation is larger than the measurement accuracy of 0.1°C. Similar to u* (Fig. 2c), the vertical variation of θ* is relatively small under strong mixing conditions (Fig. 3c). The depth of the best fitting layer for the observed temperature profile can be different for the observed wind profile.

Fig. 3.
Fig. 3.

As in Fig. 2 but for potential temperature. In (a) and (b), logarithmic potential temperature profile at each 10-min period was fitted to the observed thermocouple potential temperatures from 5.9 to 9.5 m; θ* in (c) was calculated using the local fluxes.

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

c. Calculation of vertical variations of wind and potential temperature

As mentioned in the introduction, MOST assumes no vertical variation of wind direction; therefore, the wind shear is the speed shear in MOST. In reality, the wind vector shear, that is, contributes to turbulence generation. Because momentum fluxes are correlated with the wind vector shear better than with the wind speed shear, the wind vector shear is used in this study. At 0.5 m, the wind speed shear is the same as the wind vector shear because of zero wind at z0 in both calculations. As a result of the Ekman rotation under weak winds (e.g., Grachev et al. 2005), the wind vector shear is larger than the speed shear by about 14%. Under neutral to near-neutral conditions, the difference between the two shears is negligible because strong turbulent mixing eliminates the vertical wind direction variation.

To calculate the wind shear at any of the flux observation levels in this study, the observed winds at three levels are used: the level where the wind shear is calculated, the one above, and the one below. The observed winds are then fitted to a log-linear wind profile as
e13
e14
where ua, ub, υa, and υb are the fitted coefficients. The wind shear is then
e15
The wind profile approaches the logarithmic one under neutral conditions, as expected. However, under weak wind conditions, the vertical variation of the observed winds is better described by a linear function of height than a logarithmic one.

Because of the high vertical resolution of the temperature observations, the central difference method with the thermocouple temperature observations above and below the calculation level is used to estimate the vertical temperature gradients. In calculating the vertical temperature gradients, the difference between using the logarithmic fitting and the central difference methods is small.

d. Stability parameters, MOST stability functions, and the critical Richardson number

The M–O stability functions, Φm and Φh, were originally proposed by Obukhov (1946) and Monin and Obukhov (1954). Later, Φm and Φh have been empirically developed using data collected from many field experiments (e.g., Foken 2006). Businger et al. (1971) and Dyer (1974) developed the stability functions [hereafter BD stability function; notice this is different from the Businger–Dyer stability function for unstable conditions referred to in Businger et al. (1971)] in stable stratified boundary layers for the stability range of 0 < ζ < 1. Over the years, the stability functions have been extended to ζ > 1 in the literature. For example, Holtslag and De Bruin (1988, hereafter HDB) extended Φm to ζ = 10 based on the work of Hicks (1976) and Carson and Richards (1978). Beljaars and Holtslag (1991, hereafter BH) developed an analytical Φh for ζ ≤ 10. In all these studies, a large ζ was obtained by using data with decreasing L under weak wind conditions. Based on large differences among various datasets presented in the literature, Högstrom (1990) concluded that MOST is unlikely valid for ζ > 0.5. As stability increases, turbulence becomes intermittent (e.g., Howell and Sun 1999; Sun et al. 2002, 2004) and nonstationary (Mahrt 2007). All these factors can affect turbulence parameterization.

To calculate Φm and Φh with nonturbulence variables, Louis (1979) and Louis et al. (1981) used Ri as the stability parameter. Both ζ and Ri can be influenced by their calculation methods and observation uncertainties (Padman and Jones 1985). In this study Ri is used and the consequent self-correlation problem for using Ri is discussed wherever it appears. Using the definition of Φm and Φh in (7) and (8), ζ and Ri are related as in the M–O layer (Fig. 4a). The BD Φm and Φh, the HDB Φm and BH Φh (hereinafter the HDBBH stability functions), and Φm and Φh formulated by Cheng and Brutsaert (2005, hereinafter CB) using the CASES-99 data are compared (Fig. 4). The BD stability functions are strongly constrained by the critical Richardson number, Ricr = 0.2, but the others are not.

Fig. 4.
Fig. 4.

Comparison of the relationship (a) between ζ and Ri within the M–O layer, (b) between Φm(Ri) and Ri, and (c) between Φh(Ri) and Ri using the BD and the CB stability functions, and the HDB stability function for momentum and the BH stability function for heat.

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

The existence and the value of Ricr in turbulence generation have been debated in the literature. The Ricr is traditionally defined as the Ri below which the flow would become turbulent. Since Richardson (1920) first proposed Ricr = 1, a range of Ricr values has been suggested. Assuming a constant wind shear and an exponentially decreasing air density with height, and using a linear, small-perturbation theory for stratified flow from laminar to turbulent, Taylor (1931) derived Ricr = 0.25. Later, Webb (1970) found Ricr = 0.2 based on atmospheric observations. Meanwhile, turbulence has been observed at Ri ≫ Ricr = 0.25 under various conditions (e.g., Kunkel and Walters 1982; Majda and Shefter 1998; Mahrt 1999; Monti et al. 2002). The latest research has suggested that 1) Ricr depends on how turbulence is generated, that is, whether flow transition is from laminar to turbulent or vice versa (e.g., Abarbanel et al. 1984; Strang and Fernando 2001; Fernando 2003; Troy and Koseff 2005) and 2) there is no universal Ricr (e.g., Canuto 2002; Galperin et al. 2007; Zilitinkevich et al. 2007). Since both HDBBH and CB stability functions level off with Ri, and the HDBBH stability functions compared well to observations (e.g., Howell and Sun 1999), the HDBBH stability functions are used in this study.

3. Determining the M–O layer

The critical components of MOST are two hypotheses: 1) lmN(z) = lhN(z) = κz and 2) the influence of the atmospheric stability on lm and lh can be described by Φm(Ri) and Φh(Ri) (Obukhov 2001). Under neutral conditions, the vertical integration of MOST leads to logarithmic wind and potential temperature profiles, which have been verified by observations in numerous studies; however, the logarithmic wind profile can be derived from different approaches. Von Kármán (1930) derived the logarithmic wind profile with the mixing length for momentum, which is defined as the ratio between wind shear and the second vertical derivative of wind with the assumption of a linear decrease of the Reynolds stress with height. Garratt (1992) demonstrated that the logarithmic wind profile can be derived by matching wind profiles in the outer and wall layers in a barotropic boundary layer. Therefore, the logarithmic wind profile is only a characteristic of the M–O layer under neutral conditions, but it is not a characteristic that defines the M–O layer.

a. Mixing length for momentum fluxes

The vertical variation of as a function of Ri is investigated at all the observation heights in this subsection (Fig. 5). Within the observed stability range, the calculated lm using the observed u* and wind shear is systematically smaller than the MOST prediction of at z = 0.5 m, z = 10 m, and z > 10 m. The above results imply that deviates from κz at those heights.

Fig. 5.
Fig. 5.

Comparison among lm estimated using the observations (black dots), the MOST formula, (green dots), and (red dots) as functions of Ri at z = 0.5–55 m. The function fm(z) is defined in (17) to describe the deviation of the observed neutral lm from κz. The HDB Φm(Ri) was used here and was calculated with the local momentum flux and local wind shear.

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

Because the neutral condition is associated with strong winds, which correspond to strong stress at night, is estimated by examining lm at large u*. Below ~10 m, lm monotonically increases with u* and becomes independent of u* at large u*. Therefore, below 10 m is determined as the maximum value of lm at large u*. As z increases, increases with u* and has no upper limit; above 10 m is estimated as the lm at the observed maximum u* at each level, which approximately equals the u* near the ground under neutral conditions as shown in Fig. 2c. Using with the HDB stability function, (red lines in Fig. 5) represents the mean value of at all the observed Ri much better than (green lines in Fig. 5) at all the observation heights. The better comparison also implies that Φm and Φh work approximately within our observation domain. Similarly, Garratt (1980) and Dellwik and Jensen (2005) also found that the mixing length for momentum under neutral conditions deviated from the MOST κz and the M–O stability functions were approximately valid outside of the M–O layer.

To investigate the vertical variation of , the errors that might influence the estimate need to be examined. Because of the better vertical resolution of the wind observation below 5 m in comparison with that above 5 m, and the additional condition of zero wind at z0, any significant error in the estimate close to the ground might be due to flux measurement errors. Since turbulence at 0.5 m is dominated by smaller eddies compared to those at 1.5 and 5 m, the wind measurement at 0.5 m is more likely to suffer high-frequency turbulence energy loss because of the relatively large pathlength of the sonic anemometer, which was 10 cm for all the sonic anemometers used during CASES-99. However, no significant differences in the cospectra of the vertical and horizontal winds between 0.5 and 5 m are observed based on the sonic anemometer measurements under the neutral condition. The momentum flux error due to the larger-than-10-Hz-frequency loss was estimated to be less than 15% based on limited comparisons between sonic anemometer and hot-film measurements at 1.5 m (D. Miller 2002, personal communication).

The choice of vector or speed shear does not contribute to any bias in the estimate because is estimated under the strong mixing condition. Because the wind shear at 5 m and above is calculated with a similar data resolution and the wind shear is found commonly to approach a constant value as z increases (e.g., Banta et al. 2006; Banta 2008), the observed large deviation of from κz above 5 m should not be caused by the vertical data resolution. Based on the observed scattering relationship between and u* and the possible high-frequency loss of the momentum flux at 0.5 m due to the pathlength averaging, the error bar for at each level is estimated in Fig. 6a.

Fig. 6.
Fig. 6.

(a) The comparison among , lmN = κz, and lmN = κz/[1 + (κz/l)] with l = 15 m. (b) The corresponding fm(z) = lmN/κz for the three lmN in (a). (c) The comparison among , lhN = κz, and lhN = κz/[1 + (κz/l)] with l = 15 m. (d) The corresponding fh(z) = lhN/κz for the three lhN in (c).

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

The vertical variation of shows that at 0.5 m is smaller than κz. The difference is attributed to the increasing deviation of the observed wind from the logarithmic wind profile toward the ground under strong winds while u* does not vary dramatically with height as shown in Fig. 2c. Between 1.5 and 5 m is approximately equal to κz, gradually becomes less than κz above 5 m, and approaches a constant value of ~6 ± 2 m above 20 m. The vertical variation of in Fig. 6a is qualitatively consistent with the laboratory results described in Klebanoff (1955) and van Driest (1956), the numerical results in Pope (2000, p. 307), and the field observations by Peña et al. (2010).

The neutral mixing length for momentum above the M–O layer was proposed by Blackadar (1962) as
e16
where l represents the mixing length far above the ground. With l = 15 m, (16) describes best for z less than 20 m except at z = 0.5 m. Above 20 m, (16) overestimates as approaches the constant value. The l value of 15 m obtained by best fitting lN to is different from the commonly used l = 40 m in numerical models (e.g., Beare et al. 2006; McCabe and Brown 2007). The height independence of above ~20 m implies that z is not a significant length scale there, which suggests that the layer below ~20 m is the wall layer, and above it, the outer layer. According to Townsend (1976), a self-preserving turbulent boundary layer requires that is independent of z, where Δ is a length scale. The observed constant above 20 m suggests the self-preserving character of the turbulence in the outer layer.
To describe the vertical deviation of from κz, a function,
e17
is introduced. If fm(z) = 1, equals κz, thus MOST is valid. Figure 6b shows that fm(z) ~ 1 above 0.5 m and below 10 m. The vertical variation of fm(z) suggests that MOST for momentum is valid somewhere between ~0.5 and ~10 m based on the consideration of the uncertainty of the estimate and the vertical resolution of the observations. The approximate constant above 20 m is shown in fm(z) as a function of 1/z.

b. Mixing length for heat fluxes

Similarly, the mixing length for heat, is calculated using the observed data, that is, . In general, is closer to the M–O value of than is to but is systematically smaller than the M–O value above 10 m (Fig. 7). The decrease of with Ri at each level is significantly influenced by the self-correlation because lh by definition decreases with increasing which contributes to the Ri increase. In contrast, the self-correlation does not impact the physical decrease of lm with increasing Ri because the wind shear in lm and Ri leads to an lm increase with Ri.

Fig. 7.
Fig. 7.

As in Fig. 5, but for lh. Here was calculated with the local θ* and The BH Φh was used.

Citation: Journal of Applied Meteorology and Climatology 50, 10; 10.1175/JAMC-D-10-05006.1

Applying the same method used in estimating to estimate (Fig. 6c), the difference between and κz is examined through a function,
e18
Similar to fm(z), fh(z) is close to 1 below 10 m and smaller than 1 above 10 m (Fig. 6d). With the uncertainty of the estimates, MOST for θ* [i.e., (8)] is approximately valid between the ground and somewhere below 10 m. With l = 15 m, (16) approximately describes from the ground up to the highest observation level, but is systematically larger than at the top two levels. The distinctive difference between fm(z) and fh(z) at 0.5 m, although the uncertainty of fh(z) is relatively large in comparison with that of fm(z), is due to the fact that strong mixing leads to the reduction of the vertical temperature gradient, but not of the wind shear. Because the magnitude of sensible heat fluxes is the product of u* and θ*, the validity of MOST requires both lmN and lhN to be equal to κz. Therefore, the M–O layer during CASES-99 is on average above 0.5 m and below 10 m.

4. Summary

The mixing lengths for momentum and heat were investigated using the CASES-99 data from the ground up to 60 m. Both mixing lengths under neutral conditions compare well to the MOST prediction of κz in the layer somewhere between 0.5 and 10 m, indicating that MOST is valid within the layer. Above the layer, both mixing lengths agree reasonably well with Blackadar’s mixing length, with l = 15 m up to ~20 m for the mixing length for momentum and up to almost the highest observation height for the mixing length for heat. Above ~20 m, the mixing length for momentum approaches a constant of 6 ± 2 m. Below 1.5 m, the mixing length for momentum is smaller than κz, but the mixing length for heat is approximately equal to κz. The difference between the two mixing lengths below 1.5 m is due to the increasing wind shear but reduced vertical temperature gradient under strong winds near the ground. With stable stratification, the stability functions developed for MOST can approximately represent the stability variation of the mixing lengths below and above the layer where MOST is valid, which is consistent with the literature.

MOST is the theoretical foundation for parameterization of turbulence using mean meteorological variables. However, MOST is only valid under the influence of the ground where the height above the ground is a control length scale. Above that height, Blackadar’s mixing length, which is a modification of the M–O mixing length of κz with a length scale of l, is commonly used in numerical models. The length scale l represents the mixing length far above the ground and is commonly tuned to be 40 m. Our results imply that using l = 40 m may overestimate turbulent mixing above 20 m, at least during CASES-99.

Acknowledgments

The author thanks Prof. Larry Mahrt at Oregon State University and Drs. Donald Lenschow and Peggy LeMone at NCAR for their valuable comments, and Dr. John Finnigan at CSIRO, Dr. Evgeni Fedorovich at the University of Oklahoma, and Dr. Zbigniew Sorbjan at Marquette University for their helpful discussions. The author also thanks Dean Vickers at Oregon State University for his calculation of turbulent fluxes, and all the anonymous reviewers’ constructive comments. The study is supported by U.S. Army Research Office MIPR3KNSFAR057. The University Corporation for Atmospheric Research manages the National Center for Atmospheric Research under sponsorship by the National Science Foundation. Any opinions, findings and conclusions, or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.

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  • Andreas, E. L, and B. B. Hicks, 2002: Comments on “Critical test of the validity of Monin–Obukhov similarity during convective conditions.” J. Atmos. Sci., 59, 26052607.

    • Search Google Scholar
    • Export Citation
  • Baas, P., G. J. Steeneveld, B. J. H. van de Wiel, and A. A. M. Holtslag, 2006: Exploring self-correlation in flux–gradient relationships for stably stratified conditions. J. Atmos. Sci., 63, 30453054.

    • Search Google Scholar
    • Export Citation
  • Banta, R. M., 2008: Stable-boundary-layer regimes from the perspective of the low-level jet. Acta Geophys., 56, 58 01558 087.

  • Banta, R. M., Y. L. Pichugina, and W. A. Brewer, 2006: Turbulent velocity-variance profiles in the stable boundary layer generated by a nocturnal low-level jet. J. Atmos. Sci., 63, 27002719.

    • Search Google Scholar
    • Export Citation
  • Beare, R. J., and Coauthors, 2006: An intercomparison of large-eddy simulations of the stable boundary layer. Bound.-Layer Meteor., 118, 247272.

    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327341.

    • Search Google Scholar
    • Export Citation
  • Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J. Geophys. Res., 67, 30953102.

    • Search Google Scholar
    • Export Citation
  • Blackadar, A. K., and H. Tennekes, 1968: Asymptotic similarity in neutral barotropic planetary boundary layers. J. Atmos. Sci., 25, 10151020.

    • Search Google Scholar
    • Export Citation
  • Burns, S. P., and J. Sun, 2000: Thermocouple temperature measurements from the CASES-99 main tower. Preprints, 14th Symp. on Boundary Layer and Turbulence, Snowmass, CO, Amer. Meteor. Soc., 358–361.

    • Search Google Scholar
    • Export Citation
  • 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, 181189.

    • Search Google Scholar
    • Export Citation
  • Canuto, V. M., 2002: Critical Richardson numbers and gravity waves. Astron. Astrophys., 384, 11191123.

  • Carson, D. J., and P. J. R. Richards, 1978: Modeling surface turbulent fluxes in stable conditions. Bound.-Layer Meteor., 14, 6781.

  • Cheng, Y., and W. Brutsaert, 2005: Flux-profile relationships for wind speed and temperature in the stable atmospheric boundary layer. Bound.-Layer Meteor., 114, 519538.

    • Search Google Scholar
    • Export Citation
  • Dellwik, E., and N. O. Jensen, 2005: Flux-profile relationships over a fetch limited beech forest. Bound.-Layer Meteor., 115, 179204.

    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372.

  • Fernando, H. J. S., 2003: Turbulent patches in a stratified shear flow. Phys. Fluids, 15, 31643169.

  • Foken, T., 2006: 50 years of the Monin-Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447.

  • Galperin, B., S. Sukoriansky, and P. S. Anderson, 2007: On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett., 8, 6569, doi:10.1002/asl.153.

    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., 1980: Surface influence upon vertical profiles in the atmospheric near-surface layer. Quart. J. Roy. Meteor. Soc., 106, 803819.

    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Gibson, M. M., and B. E. Launder, 1978: Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech., 86, 491511.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., C. W. Fairall, P. Ola, G. Persson, E. L Andreas, and P. S. Guest, 2005: Stable boundary-layer scaling regimes: The Sheba data. Bound.-Layer Meteor., 116, 201235.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1976: Wind profile relationships from the ‘Wangara’ experiment. Quart. J. Roy. Meteor. Soc., 102, 535551.

  • Hicks, B. B., 1978: Some limitations of dimensional analysis and power laws. Bound.-Layer Meteor., 14, 567569.

  • Högstrom, U., 1990: Analysis of turbulence structure in the surface layer with a modified similarity formulation for near neutral conditions. J. Atmos. Sci., 47, 19491972.

    • Search Google Scholar
    • Export Citation
  • Holtslag, A. A. M., and H. A. R. De Bruin, 1988: Applied modeling of nighttime surface energy balance over land. J. Appl. Meteor., 27, 689704.

    • Search Google Scholar
    • Export Citation
  • Howell, J. F., and J. Sun, 1999: Surface-layer fluxes in stable conditions. Bound.-Layer Meteor., 90, 495520.

  • Hunt, J. C. R., D. D. Strech, and R. E. Britter, 1988: Length scales in stable stratified turbulent flows and their use in turbulence models. Stable Stratified Flow and Dense Gas Dispersion, J. S. Puttock, Ed., Institute of Mathematics and Its Applications Conference Series, New Series No. 15, Clarendon Press, 285–321.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1978: Similarity treatment of moving-equilibrium turbulent boundary layers in adverse pressure gradients. J. Fluid Mech., 89, 305342.

    • Search Google Scholar
    • Export Citation
  • Kastner-Klein, P., and E. Fedorovich, 2002: Diffusion from a line source deployed in a homogeneous roughness layer: Interpretation of wind-tunnel measurements by means of simple mathematical models. Atmos. Environ., 36, 37093718.

    • Search Google Scholar
    • Export Citation
  • Klebanoff, P. S., 1955: Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Rep. NACA-TR-1247, National Bureau of Standards, 19 pp.

    • Search Google Scholar
    • Export Citation
  • Klipp, C. L., and L. Mahrt, 2004: Flux-gradient relationship, self-correlation and intermit-tency in the stable boundary layer. Quart. J. Roy. Meteor. Soc., 130, 20872103.

    • Search Google Scholar
    • Export Citation
  • Kunkel, K. E., and D. L. Walters, 1982: Intermittent turbulence in measurements of the temperature structure parameter under very stable conditions. Bound.-Layer Meteor., 22, 4960.

    • Search Google Scholar
    • Export Citation
  • Louis, J. F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17, 187202.

  • 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, United Kingdom, ECMWF, 59–79.

    • Search Google Scholar
    • Export Citation
  • Mahrt, L., 1999: Stratified atmospheric boundary layers. Bound.-Layer Meteor., 90, 375396.

  • Mahrt, L., 2007: The influence of nonstationarity on the turbulent flux-gradient relationship for stable stratification. Bound.-Layer Meteor., 125, 245264.

    • Search Google Scholar
    • Export Citation
  • Majda, A. J., and M. G. Shefter, 1998: Elementary stratified flows with instability at large Richardson number. J. Fluid Mech., 376, 319350.

    • Search Google Scholar
    • Export Citation
  • McCabe, A., and A. R. Brown, 2007: The role of surface heterogeneity in modeling the stable boundary layer. J. Fluid Mech., 122, 517534.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the atmosphere near the ground. Tr. Geofiz. Inst., Akad. Nauk SSSR, 24, 163187.

    • Search Google Scholar
    • Export Citation
  • Monti, P., H. J. S. Fernando, M. Princvac, W. C. Chan, T. A. Kowalewski, and E. R. Pardyjak, 2002: Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci., 59, 25132534.

    • Search Google Scholar
    • Export Citation
  • Obukhov, A. M., 1946: Turbulence in an atmosphere with a non-uniform temperature. Tr. Inst. Teor. Geofiz. Akad. Nauk SSSR,1, 95–115.

  • Obukhov, A. M., 2001: Turbulence and Atmospheric Dynamics. Center for Turbulence Research Monogr., Center for Turbulence Research, 514 pp.

    • Search Google Scholar
    • Export Citation
  • Padman, L., and I. S. F. Jones, 1985: Richardson number statistics in the seasonal thermocline. J. Phys. Oceanogr., 15, 844854.

  • Peña, A., S.-E. Gryning, and J. Mann, 2010: On the length-scale of the wind profile. Quart. J. Roy. Meteor. Soc., 136, 21192131.

  • Pope, S. B., 2000: Turbulent Flows. Cambridge University Press, 771 pp.

  • Poulos, G. S., and Coauthors, 2002: CASES-99: A comprehensive investigation of the stable nocturnal boundary layer. Bull. Amer. Meteor. Soc., 83, 555581.

    • Search Google Scholar
    • Export Citation
  • Raupach, M. R., A. S. Thom, and I. Edwards, 1980: A wind-tunnel study of turbulent flow close to regularly arrayed rough surfaces. Bound.-Layer Meteor., 18, 373397.

    • Search Google Scholar
    • Export Citation
  • Richardson, L. F., 1920: The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. London, A97, 354373.

  • Smedman, A.-S., 1988: Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Bound.-Layer Meteor., 44, 231253.

    • Search Google Scholar
    • Export Citation
  • Strang, E. J., and H. J. S. Fernando, 2001: Entrainment and mixing in stratified shear flows. J. Fluid Mech., 428, 349386.

  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, 666 pp.

  • Sun, J., and Coauthors, 2002: Intermittent turbulence associated with a density current passage in the stable boundary layer. Bound.-Layer Meteor., 105, 199219.

    • Search Google Scholar
    • Export Citation
  • Sun, J., S. P. Burns, A. C. Delany, S. P. Oncley, T. W. Horst, and D. H. Lenschow, 2003: Heat balance in nocturnal boundary layers during CASES-99. J. Appl. Meteor., 42, 16491666.

    • Search Google Scholar
    • Export Citation
  • Sun, J., and Coauthors, 2004: Atmospheric disturbances that generate intermittent turbulence in nocturnal boundary layers. Bound.-Layer Meteor., 110, 255279.

    • Search Google Scholar
    • Export Citation
  • Taylor, G. I., 1931: Effect of variation in density on the stability of superposed streams of fluid. Proc. Roy. Soc. London, 132, 499523.

    • Search Google Scholar
    • Export Citation
  • Tennekes, H., and H. L. Lumley, 1972: A First Course in Turbulence. MIT Press, 300 pp.

  • Townsend, A. A., 1961: Equilibrium layers and wall turbulence. J. Fluid Mech., 11, 97120.

  • Townsend, A. A., 1976: The Structure of Turbulent Shear Flow. Cambridge University Press, 429 pp.

  • Troy, C. D., and J. R. Koseff, 2005: The instability and breaking of long internal waves. J. Fluid Mech., 543, 107136.

  • van Driest, E. R., 1956: On turbulent flow near a wall. J. Aerosp. Sci., 23, 10071011.

  • Vickers, D., and L. Mahrt, 2003: The cospectral gap and turbulent flux calculations. J. Atmos. Oceanic Technol., 20, 660672.

  • Vickers, D., and L. Mahrt, 2006: A solution for flux contamination by mesoscale motions with very weak turbulence. Bound.-Layer Meteor., 118, 431447.

    • Search Google Scholar
    • Export Citation
  • von Kármán, T., 1930: Mechanische Ähnlichkeit und Turbulenz (Mechanical Similarity and Turbulence). Nachr Ges Wiss Göttingen Math Phys Klasse, 58 pp.

    • Search Google Scholar
    • Export Citation
  • Webb, E. K., 1970: Profile relationships: The log-linear range, and extension to strong stability. Quart. J. Roy. Meteor. Soc., 96, 6790.

    • Search Google Scholar
    • Export Citation
  • Zilitinkevich, S. S., T. Elperin, N. Kleeorin, and I. Rogachevskii, 2007: Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: Steady-state, homogeneous regimes. Bound.-Layer Meteor., 125, 167191.

    • Search Google Scholar
    • Export Citation
Save
  • Abarbanel, H. D., D. D. Holm, J. E. Marsden, and T. Ratiu, 1984: Richardson number criterion for the nonlinear stability of three-dimensional stratified flow. Phys. Rev. Lett., 52, 23522355.

    • Search Google Scholar
    • Export Citation
  • Andreas, E. L, and B. B. Hicks, 2002: Comments on “Critical test of the validity of Monin–Obukhov similarity during convective conditions.” J. Atmos. Sci., 59, 26052607.

    • Search Google Scholar
    • Export Citation
  • Baas, P., G. J. Steeneveld, B. J. H. van de Wiel, and A. A. M. Holtslag, 2006: Exploring self-correlation in flux–gradient relationships for stably stratified conditions. J. Atmos. Sci., 63, 30453054.

    • Search Google Scholar
    • Export Citation
  • Banta, R. M., 2008: Stable-boundary-layer regimes from the perspective of the low-level jet. Acta Geophys., 56, 58 01558 087.

  • Banta, R. M., Y. L. Pichugina, and W. A. Brewer, 2006: Turbulent velocity-variance profiles in the stable boundary layer generated by a nocturnal low-level jet. J. Atmos. Sci., 63, 27002719.

    • Search Google Scholar
    • Export Citation
  • Beare, R. J., and Coauthors, 2006: An intercomparison of large-eddy simulations of the stable boundary layer. Bound.-Layer Meteor., 118, 247272.

    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327341.

    • Search Google Scholar
    • Export Citation
  • Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J. Geophys. Res., 67, 30953102.

    • Search Google Scholar
    • Export Citation
  • Blackadar, A. K., and H. Tennekes, 1968: Asymptotic similarity in neutral barotropic planetary boundary layers. J. Atmos. Sci., 25, 10151020.

    • Search Google Scholar
    • Export Citation
  • Burns, S. P., and J. Sun, 2000: Thermocouple temperature measurements from the CASES-99 main tower. Preprints, 14th Symp. on Boundary Layer and Turbulence, Snowmass, CO, Amer. Meteor. Soc., 358–361.

    • Search Google Scholar
    • Export Citation
  • 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, 181189.

    • Search Google Scholar
    • Export Citation
  • Canuto, V. M., 2002: Critical Richardson numbers and gravity waves. Astron. Astrophys., 384, 11191123.

  • Carson, D. J., and P. J. R. Richards, 1978: Modeling surface turbulent fluxes in stable conditions. Bound.-Layer Meteor., 14, 6781.

  • Cheng, Y., and W. Brutsaert, 2005: Flux-profile relationships for wind speed and temperature in the stable atmospheric boundary layer. Bound.-Layer Meteor., 114, 519538.

    • Search Google Scholar
    • Export Citation
  • Dellwik, E., and N. O. Jensen, 2005: Flux-profile relationships over a fetch limited beech forest. Bound.-Layer Meteor., 115, 179204.

    • Search Google Scholar
    • Export Citation
  • Dyer, A. J., 1974: A review of flux-profile relationships. Bound.-Layer Meteor., 7, 363372.

  • Fernando, H. J. S., 2003: Turbulent patches in a stratified shear flow. Phys. Fluids, 15, 31643169.

  • Foken, T., 2006: 50 years of the Monin-Obukhov similarity theory. Bound.-Layer Meteor., 119, 431447.

  • Galperin, B., S. Sukoriansky, and P. S. Anderson, 2007: On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett., 8, 6569, doi:10.1002/asl.153.

    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., 1980: Surface influence upon vertical profiles in the atmospheric near-surface layer. Quart. J. Roy. Meteor. Soc., 106, 803819.

    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Gibson, M. M., and B. E. Launder, 1978: Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech., 86, 491511.

    • Search Google Scholar
    • Export Citation
  • Grachev, A. A., C. W. Fairall, P. Ola, G. Persson, E. L Andreas, and P. S. Guest, 2005: Stable boundary-layer scaling regimes: The Sheba data. Bound.-Layer Meteor., 116, 201235.

    • Search Google Scholar
    • Export Citation
  • Hicks, B. B., 1976: Wind profile relationships from the ‘Wangara’ experiment. Quart. J. Roy. Meteor. Soc., 102, 535551.

  • Hicks, B. B., 1978: Some limitations of dimensional analysis and power laws. Bound.-Layer Meteor., 14, 567569.

  • Högstrom, U., 1990: Analysis of turbulence structure in the surface layer with a modified similarity formulation for near neutral conditions. J. Atmos. Sci., 47, 19491972.

    • Search Google Scholar
    • Export Citation
  • Holtslag, A. A. M., and H. A. R. De Bruin, 1988: Applied modeling of nighttime surface energy balance over land. J. Appl. Meteor., 27, 689704.

    • Search Google Scholar
    • Export Citation
  • Howell, J. F., and J. Sun, 1999: Surface-layer fluxes in stable conditions. Bound.-Layer Meteor., 90, 495520.

  • Hunt, J. C. R., D. D. Strech, and R. E. Britter, 1988: Length scales in stable stratified turbulent flows and their use in turbulence models. Stable Stratified Flow and Dense Gas Dispersion, J. S. Puttock, Ed., Institute of Mathematics and Its Applications Conference Series, New Series No. 15, Clarendon Press, 285–321.

    • Search Google Scholar
    • Export Citation
  • Kader, B. A., and A. M. Yaglom, 1978: Similarity treatment of moving-equilibrium turbulent boundary layers in adverse pressure gradients. J. Fluid Mech., 89, 305342.

    • Search Google Scholar
    • Export Citation
  • Kastner-Klein, P., and E. Fedorovich, 2002: Diffusion from a line source deployed in a homogeneous roughness layer: Interpretation of wind-tunnel measurements by means of simple mathematical models. Atmos. Environ., 36, 37093718.

    • Search Google Scholar
    • Export Citation
  • Klebanoff, P. S., 1955: Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Rep. NACA-TR-1247, National Bureau of Standards, 19 pp.

    • Search Google Scholar
    • Export Citation
  • Klipp, C. L., and L. Mahrt, 2004: Flux-gradient relationship, self-correlation and intermit-tency in the stable boundary layer. Quart. J. Roy. Meteor. Soc., 130, 20872103.

    • Search Google Scholar
    • Export Citation
  • Kunkel, K. E., and D. L. Walters, 1982: Intermittent turbulence in measurements of the temperature structure parameter under very stable conditions. Bound.-Layer Meteor., 22, 4960.

    • Search Google Scholar
    • Export Citation
  • Louis, J. F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17, 187202.

  • 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, United Kingdom, ECMWF, 59–79.

    • Search Google Scholar
    • Export Citation
  • Mahrt, L., 1999: Stratified atmospheric boundary layers. Bound.-Layer Meteor., 90, 375396.

  • Mahrt, L., 2007: The influence of nonstationarity on the turbulent flux-gradient relationship for stable stratification. Bound.-Layer Meteor., 125, 245264.

    • Search Google Scholar
    • Export Citation
  • Majda, A. J., and M. G. Shefter, 1998: Elementary stratified flows with instability at large Richardson number. J. Fluid Mech., 376, 319350.

    • Search Google Scholar
    • Export Citation
  • McCabe, A., and A. R. Brown, 2007: The role of surface heterogeneity in modeling the stable boundary layer. J. Fluid Mech., 122, 517534.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the atmosphere near the ground. Tr. Geofiz. Inst., Akad. Nauk SSSR, 24, 163187.

    • Search Google Scholar
    • Export Citation
  • Monti, P., H. J. S. Fernando, M. Princvac, W. C. Chan, T. A. Kowalewski, and E. R. Pardyjak, 2002: Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci., 59, 25132534.

    • Search Google Scholar
    • Export Citation
  • Obukhov, A. M., 1946: Turbulence in an atmosphere with a non-uniform temperature. Tr. Inst. Teor. Geofiz. Akad. Nauk SSSR,1, 95–115.

  • Obukhov, A. M., 2001: Turbulence and Atmospheric Dynamics. Center for Turbulence Research Monogr., Center for Turbulence Research, 514 pp.

    • Search Google Scholar
    • Export Citation
  • Padman, L., and I. S. F. Jones, 1985: Richardson number statistics in the seasonal thermocline. J. Phys. Oceanogr., 15, 844854.

  • Peña, A., S.-E. Gryning, and J. Mann, 2010: On the length-scale of the wind profile. Quart. J. Roy. Meteor. Soc., 136, 21192131.

  • Pope, S. B., 2000: Turbulent Flows. Cambridge University Press, 771 pp.

  • Poulos, G. S., and Coauthors, 2002: CASES-99: A comprehensive investigation of the stable nocturnal boundary layer. Bull. Amer. Meteor. Soc., 83, 555581.

    • Search Google Scholar
    • Export Citation
  • Raupach, M. R., A. S. Thom, and I. Edwards, 1980: A wind-tunnel study of turbulent flow close to regularly arrayed rough surfaces. Bound.-Layer Meteor., 18, 373397.

    • Search Google Scholar
    • Export Citation
  • Richardson, L. F., 1920: The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. London, A97, 354373.

  • Smedman, A.-S., 1988: Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Bound.-Layer Meteor., 44, 231253.

    • Search Google Scholar
    • Export Citation
  • Strang, E. J., and H. J. S. Fernando, 2001: Entrainment and mixing in stratified shear flows. J. Fluid Mech., 428, 349386.

  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, 666 pp.

  • Sun, J., and Coauthors, 2002: Intermittent turbulence associated with a density current passage in the stable boundary layer. Bound.-Layer Meteor., 105, 199219.

    • Search Google Scholar
    • Export Citation
  • Sun, J., S. P. Burns, A. C. Delany, S. P. Oncley, T. W. Horst, and D. H. Lenschow, 2003: Heat balance in nocturnal boundary layers during CASES-99. J. Appl. Meteor., 42, 16491666.

    • Search Google Scholar
    • Export Citation
  • Sun, J., and Coauthors, 2004: Atmospheric disturbances that generate intermittent turbulence in nocturnal boundary layers. Bound.-Layer Meteor., 110, 255279.

    • Search Google Scholar
    • Export Citation
  • Taylor, G. I., 1931: Effect of variation in density on the stability of superposed streams of fluid. Proc. Roy. Soc. London, 132, 499523.

    • Search Google Scholar
    • Export Citation
  • Tennekes, H., and H. L. Lumley, 1972: A First Course in Turbulence. MIT Press, 300 pp.

  • Townsend, A. A., 1961: Equilibrium layers and wall turbulence. J. Fluid Mech., 11, 97120.

  • Townsend, A. A., 1976: The Structure of Turbulent Shear Flow. Cambridge University Press, 429 pp.

  • Troy, C. D., and J. R. Koseff, 2005: The instability and breaking of long internal waves. J. Fluid Mech., 543, 107136.

  • van Driest, E. R., 1956: On turbulent flow near a wall. J. Aerosp. Sci., 23, 10071011.

  • Vickers, D., and L. Mahrt, 2003: The cospectral gap and turbulent flux calculations. J. Atmos. Oceanic Technol., 20, 660672.

  • Vickers, D., and L. Mahrt, 2006: A solution for flux contamination by mesoscale motions with very weak turbulence. Bound.-Layer Meteor., 118, 431447.

    • Search Google Scholar
    • Export Citation
  • von Kármán, T., 1930: Mechanische Ähnlichkeit und Turbulenz (Mechanical Similarity and Turbulence). Nachr Ges Wiss Göttingen Math Phys Klasse, 58 pp.

    • Search Google Scholar
    • Export Citation
  • Webb, E. K., 1970: Profile relationships: The log-linear range, and extension to strong stability. Quart. J. Roy. Meteor. Soc., 96, 6790.

    • Search Google Scholar
    • Export Citation
  • Zilitinkevich, S. S., T. Elperin, N. Kleeorin, and I. Rogachevskii, 2007: Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: Steady-state, homogeneous regimes. Bound.-Layer Meteor., 125, 167191.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Schematic of the various layers above the ground under neutral conditions.

  • Fig. 2.

    (a) Four examples of 10-min-averaged wind profiles under the neutral condition defined in section 2b for two periods when the lowest sonic anemometer was at 1.5 and 0.5 m. (b) The deviations of the observed neutral wind speeds from the logarithmic wind profiles that were fitted to the observed wind from 5 to 15 m. (c) The composite u* profiles for the two periods, where u* was calculated using the local momentum fluxes. For each time period, two u* profiles were composited: one for the neutral strong wind condition (marked with s), and one for the rest of the conditions (marked with w). The horizontal bars at the lowest observation level in (c) represent ±10% of the vertically averaged u* for each profile. The logarithmic wind profile fitted for each observed wind profile is marked by the solid line in the same color as the observations in (a).

  • Fig. 3.

    As in Fig. 2 but for potential temperature. In (a) and (b), logarithmic potential temperature profile at each 10-min period was fitted to the observed thermocouple potential temperatures from 5.9 to 9.5 m; θ* in (c) was calculated using the local fluxes.

  • Fig. 4.

    Comparison of the relationship (a) between ζ and Ri within the M–O layer, (b) between Φm(Ri) and Ri, and (c) between Φh(Ri) and Ri using the BD and the CB stability functions, and the HDB stability function for momentum and the BH stability function for heat.

  • Fig. 5.

    Comparison among lm estimated using the observations (black dots), the MOST formula, (green dots), and (red dots) as functions of Ri at z = 0.5–55 m. The function fm(z) is defined in (17) to describe the deviation of the observed neutral lm from κz. The HDB Φm(Ri) was used here and was calculated with the local momentum flux and local wind shear.

  • Fig. 6.

    (a) The comparison among , lmN = κz, and lmN = κz/[1 + (κz/l)] with l = 15 m. (b) The corresponding fm(z) = lmN/κz for the three lmN in (a). (c) The comparison among , lhN = κz, and lhN = κz/[1 + (κz/l)] with l = 15 m. (d) The corresponding fh(z) = lhN/κz for the three lhN in (c).