## 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.

### a. Monin–Obukhov similarity theory and mixing lengths

*K*theory, momentum

*τ*and sensible heat fluxes

*H*are parameterized as

**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;

*c*is the air specific heat at constant pressure;

_{p}*K*is the eddy viscosity; and

_{m}*K*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

_{h}*z*is the height above the ground. Using Prandtl’s mixing length

*l*,

*K*and

_{m}*K*in (1) and (2) are related to

_{h}*l*(e.g., Stull 1988; Garratt 1992) as follows:

*l*is assumed to be the same for both

*K*and

_{m}*K*. However, in this study, the mixing length for momentum

_{h}*l*and the mixing length for heat

_{m}*l*are distinguished. Using (3), (4), and

_{h}*l*s with

*l*and

_{m}*l*, (1) and (2) become

_{h}*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

*ζ*≡

*z/L*,

*κ*~ 0.4 is the von Kármán constant,

*g*is the gravitational constant, Φ

*and Φ*

_{m}*are the M–O stability functions for momentum and heat, respectively, and*

_{h}*T*

_{0}is a reference air temperature. The subscript 0 represents variables at the ground. In this study, the virtual potential temperature

*θ*approximately satisfies

_{υ}*θ*≃

_{υ}*θ*; therefore,

*θ*instead of

*θ*, which is the formally correct temperature variable, is used in (6), (8), and

_{υ}*L.*MOST ignores the vertical variation of wind direction; thus,

*ζ*= 0) Φ

*(0) = Φ*

_{m}*(0) = 1 (commonly observed), (9) and (10) imply that both mixing lengths in neutral stratification are equal to*

_{h}*κz*in MOST. To generalize (9) and (10) to a layer deeper than the M–O layer, the dependence of

*l*and

_{m}*l*on height and stability can be expressed as

_{h}*l*

*(*

_{mN}*z*) and

*l*

_{hN}(

*z*) represent

*l*and

_{m}*l*under neutral conditions, respectively, and

_{h}*F*and

_{m}*F*are generalized stability functions. Therefore, the M–O layer is the layer where

_{h}*F*Φ

_{m}=*, and*

_{m}*F*= Φ

_{h}*, where*

_{h}### 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 Φ* _{m}* as a function of

*ζ*

^{1/3}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, * _{h}* defined in (8) is related to Ri

*,*both Φ

*and Ri increase with*

_{h}*and Ri (more in section 2d) can be contaminated by the self-correlation between Φ*

_{h}*and Ri through their common dependence on*

_{h}*, physically Φ*

_{h}*also increases with Ri, but the self-correlation between Φ*

_{m}*and Ri leads to Φ*

_{m}*decreasing with Ri through their common dependence on shear. Therefore, the self-correlation between Φ*

_{m}*and Ri obscures the physical relationship between Φ*

_{m}*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*

_{m}*ζ*or Ri is used as the stability parameter except under neutral conditions when Φ

*= Φ*

_{m}*= 1.*

_{h}### 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 *l _{m}* and

*l*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

_{h}*l*and

_{m}*l*for the stability range from neutral to stable are examined in section 3. The main results are summarized in section 4.

_{h}## 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, *z*_{0}, is ~0.05 m. To avoid the ambiguity generated by using both displacement height and *z*_{0}, the above *z*_{0} 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 *l _{mN}* and

*l*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

_{hN}*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.

(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

(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

(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.

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

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

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 *z*_{0} 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.

*u*,

_{a}*u*,

_{b}*υ*, and

_{a}*υ*are the fitted coefficients. The wind shear is then

_{b}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, Φ* _{m}* and Φ

*, were originally proposed by Obukhov (1946) and Monin and Obukhov (1954). Later, Φ*

_{h}*and Φ*

_{m}*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 <*

_{h}*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 Φ

*to*

_{m}*ζ*= 10 based on the work of Hicks (1976) and Carson and Richards (1978). Beljaars and Holtslag (1991, hereafter BH) developed an analytical Φ

*for*

_{h}*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 Φ* _{m}* and Φ

*with nonturbulence variables, Louis (1979) and Louis et al. (1981) used Ri as the stability parameter. Both*

_{h}*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 Φ

*and Φ*

_{m}*in (7) and (8),*

_{h}*ζ*and Ri are related as

*and Φ*

_{m}*, the HDB Φ*

_{h}*and BH Φ*

_{m}*(hereinafter the HDB–BH stability functions), and Φ*

_{h}*and Φ*

_{m}*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, Ri*

_{h}_{cr}

*=*0.2, but the others are not.

Comparison of the relationship (a) between *ζ* and Ri within the M–O layer, (b) between Φ* _{m}*(Ri) and Ri, and (c) between Φ

*(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.*

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

Comparison of the relationship (a) between *ζ* and Ri within the M–O layer, (b) between Φ* _{m}*(Ri) and Ri, and (c) between Φ

*(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.*

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

Comparison of the relationship (a) between *ζ* and Ri within the M–O layer, (b) between Φ* _{m}*(Ri) and Ri, and (c) between Φ

*(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.*

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

The existence and the value of Ri_{cr} in turbulence generation have been debated in the literature. The Ri_{cr} is traditionally defined as the Ri below which the flow would become turbulent. Since Richardson (1920) first proposed Ri_{cr} = 1, a range of Ri_{cr} 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 Ri_{cr} *=* 0.25. Later, Webb (1970) found Ri_{cr} *=* 0.2 based on atmospheric observations. Meanwhile, turbulence has been observed at Ri ≫ Ri_{cr} = 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) Ri_{cr} 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 Ri_{cr} (e.g., Canuto 2002; Galperin et al. 2007; Zilitinkevich et al. 2007). Since both HDB–BH and CB stability functions level off with Ri, and the HDB–BH stability functions compared well to observations (e.g., Howell and Sun 1999), the HDB–BH stability functions are used in this study.

## 3. Determining the M–O layer

The critical components of MOST are two hypotheses: 1) *l _{mN}*(

*z*) =

*l*(

_{hN}*z*) =

*κz*and 2) the influence of the atmospheric stability on

*l*and

_{m}*l*can be described by Φ

_{h}*(Ri) and Φ*

_{m}*(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.*

_{h}### a. Mixing length for momentum fluxes

The vertical variation of *l _{m}* using the observed

*u*

_{*}and wind shear

*z*= 0.5 m,

*z*= 10 m, and

*z*> 10 m. The above results imply that

*κz*at those heights.

Comparison among *l _{m}* estimated using the observations (black dots), the MOST formula,

*z*= 0.5–55 m. The function

*f*(

_{m}*z*) is defined in (17) to describe the deviation of the observed neutral

*l*from

_{m}*κz.*The HDB Φ

*(Ri) was used here and*

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

Comparison among *l _{m}* estimated using the observations (black dots), the MOST formula,

*z*= 0.5–55 m. The function

*f*(

_{m}*z*) is defined in (17) to describe the deviation of the observed neutral

*l*from

_{m}*κz.*The HDB Φ

*(Ri) was used here and*

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

Comparison among *l _{m}* estimated using the observations (black dots), the MOST formula,

*z*= 0.5–55 m. The function

*f*(

_{m}*z*) is defined in (17) to describe the deviation of the observed neutral

*l*from

_{m}*κz.*The HDB Φ

*(Ri) was used here and*

_{m}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, *l _{m}* at large

*u*

_{*}. Below ~10 m,

*l*monotonically increases with

_{m}*u*

_{*}and becomes independent of

*u*

_{*}at large

*u*

_{*}. Therefore,

*l*at large

_{m}*u*

_{*}. As

*z*increases,

*u*

_{*}and has no upper limit;

*l*at the observed maximum

_{m}*u*

_{*}at each level, which approximately equals the

*u*

_{*}near the ground under neutral conditions as shown in Fig. 2c. Using

*and Φ*

_{m}*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*

_{h}*κz*and the M–O stability functions were approximately valid outside of the M–O layer.

To investigate the vertical variation of *z*_{0}, any significant error in the

The choice of vector or speed shear does not contribute to any bias in the *z* increases (e.g., Banta et al. 2006; Banta 2008), the observed large deviation of *κz* above 5 m should not be caused by the vertical data resolution. Based on the observed scattering relationship between *u*_{*} and the possible high-frequency loss of the momentum flux at 0.5 m due to the pathlength averaging, the error bar for

(a) The comparison among *l _{mN}*

*= κz*, and

*l*

_{mN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (b) The corresponding

*f*(

_{m}*z*)

*= l*/

_{mN}*κz*for the three

*l*in (a). (c) The comparison among

_{mN}*l*

_{hN}*= κz*, and

*l*

_{hN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (d) The corresponding

*f*(

_{h}*z*)

*= l*/

_{hN}*κz*for the three

*l*

_{hN}in (c).

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

(a) The comparison among *l _{mN}*

*= κz*, and

*l*

_{mN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (b) The corresponding

*f*(

_{m}*z*)

*= l*/

_{mN}*κz*for the three

*l*in (a). (c) The comparison among

_{mN}*l*

_{hN}*= κz*, and

*l*

_{hN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (d) The corresponding

*f*(

_{h}*z*)

*= l*/

_{hN}*κz*for the three

*l*

_{hN}in (c).

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

(a) The comparison among *l _{mN}*

*= κz*, and

*l*

_{mN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (b) The corresponding

*f*(

_{m}*z*)

*= l*/

_{mN}*κz*for the three

*l*in (a). (c) The comparison among

_{mN}*l*

_{hN}*= κz*, and

*l*

_{hN}*= κz*/[1

*+*(

*κz*/

*l*

_{∞})] with

*l*

_{∞}= 15 m. (d) The corresponding

*f*(

_{h}*z*)

*= l*/

_{hN}*κz*for the three

*l*

_{hN}in (c).

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

The vertical variation of *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 *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

*l*

_{∞}represents the mixing length far above the ground. With

*l*

_{∞}= 15 m, (16) describes

*z*less than 20 m except at

*z =*0.5 m. Above 20 m, (16) overestimates

*l*

_{∞}value of 15 m obtained by best fitting

*l*to

_{N}*l*

_{∞}= 40 m in numerical models (e.g., Beare et al. 2006; McCabe and Brown 2007). The height independence of

*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

*z*, where Δ is a length scale. The observed constant

*κz*, a function,

*f*(

_{m}*z*) = 1,

*κz*, thus MOST is valid. Figure 6b shows that

*f*(

_{m}*z*) ~ 1 above 0.5 m and below 10 m. The vertical variation of

*f*(

_{m}*z*) suggests that MOST for momentum is valid somewhere between ~0.5 and ~10 m based on the consideration of the uncertainty of the

*f*(

_{m}*z*) as a function of 1/

*z*

*.*

### b. Mixing length for heat fluxes

Similarly, the mixing length for heat, *l _{h}* by definition decreases with increasing

*l*with increasing Ri because the wind shear in

_{m}*l*and Ri leads to an

_{m}*l*increase with Ri.

_{m}As in Fig. 5, but for *l _{h}*. Here

*θ*

_{*}and

*was used.*

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

As in Fig. 5, but for *l _{h}*. Here

*θ*

_{*}and

*was used.*

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

As in Fig. 5, but for *l _{h}*. Here

*θ*

_{*}and

*was used.*

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

*κz*is examined through a function,

*f*(

_{m}*z*),

*f*(

_{h}*z*) is close to 1 below 10 m and smaller than 1 above 10 m (Fig. 6d). With the uncertainty of the

*θ*

_{*}[i.e., (8)] is approximately valid between the ground and somewhere below 10 m. With

*l*

_{∞}= 15 m, (16) approximately describes

*f*(

_{m}*z*) and

*f*(

_{h}*z*) at 0.5 m, although the uncertainty of

*f*(

_{h}*z*) is relatively large in comparison with that of

*f*(

_{m}*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

*l*and

_{mN}*l*to be equal to

_{hN}*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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