## 1. Introduction

This paper summarizes results from 5 nights of a 20-night study in May 2009 near a tall tower in a landscape with moderate terrain and mixed land use in the southeast United States. A focus was the vertical transport of trace gases—CO_{2}, H_{2}O, and perfluorocarbons—to four levels on the 328 m tower (2, 33, 68, and 328 m). Preliminary simulations with the SCIPUFF Lagrangian puff model (Sykes et al. 1999) found reasonable model-predicted perfluorocarbon concentrations at the lower three tower levels but one order of magnitude less than observed at 328 m, on both a weakly stable and a stable night.

The shortfall in simulated perfluorocarbon at 328 m was attributed to the assumption of a classical nocturnal boundary layer (NBL) in SCIPUFF where turbulence is assumed to decrease smoothly from high values near the surface to a small value at the NBL top (Stull 1988, chapter 9) and mixing length is parameterized. This model does not include turbulence variability or sources above the surface, i.e., from 1) wind shear due to low-level jets that can reach the surface (e.g., Banta et al. 2002; Burman et al. 2018), 2) top-down boundary layers in which turbulence kinetic energy (TKE) propagates downward from sources such as the low-level jet (Mahrt 2014), 3) topography and mixed land use that raises the effective lower boundary of the NBL from the surface to the displacement height, 4) uncertainty and variability in the mixing length—a key variable in vertical transport, 5) mesoscale variability near the nocturnal inversion, 6) small-scale time and space phenomena, e.g., gravity waves (Finnigan et al. 1984; Einaudi and Finnigan 1993), shear-flow instabilities (Newsom and Banta 2003), density currents (Blumen et al. 1999), turbulent bursting (Chimonas 1999), and sloping terrain (Wyngaard 2010).

Wind variability near the nocturnal inversion was associated by Reiter (1969) with the “nocturnal inversion jet” that should be called an “inversion wind maxima” since it has small horizontal shear. In this paper we will follow Reiter’s suggestion and distinguish between “low-level jets,” with well-defined cores between 200 and 1000 m above ground and speeds of 15–30 m s^{−1}, and the “inversion shear layer,” with poorly defined cores of less than 12 m s^{−1} and heights between 60 and 150 m above ground.

Simulation of vertical transport can be improved by using actual turbulence and mixing length data in place of parameterized profiles. Measurements available for this task in our study included four levels of winds, turbulence, and temperature on the tall tower, one sodar/radio acoustic sounding system (RASS) with 5 m vertical resolution, and a second sodar with 20 m resolution, both with 15-min-average data, and a network of 60 m towers. Vertical profiles of aerosol backscatter from a nearby lidar provided qualitative information on vertical mixing. Thus, accurate turbulence data were only available at the four levels of the tower and the 60 m towers with a thick layer between 68 and 328 m devoid of turbulence measurements. These sampling limitations suggested the need to calculate time–height profiles of turbulence and mixing length from temperature and winds measured by the towers and the sodars.

The use of temperature and wind gradients to determine TKE and fluxes is based on Monin–Obukhov scaling (MOS) in the surface layer and *z*-less scaling above the surface, which relate gradients to fluxes of heat, momentum, and chemicals. MOS is well established in the surface layer (e.g., Sorbjan 1989) but *z*-less scaling is more controversial above the surface layer (Nieuwstadt 1984; Basu et al. 2006). Moreover, both scalings are assumed to apply only to the inertial subrange (frequencies greater than ∼0.01 Hz) where energy is transferred to higher frequencies for dissipation, and not to low-frequency mesoscale or submesoscale eddies (e.g., Vickers and Mahrt 2006; Basu et al. 2006; Mortarini et al. 2018), which nevertheless can transport heat, momentum, and trace chemicals but without obeying MOS scaling.

The TKE and the mixing length can also be derived from the TKE equation since TKE in the NBL is controlled mainly by production via vertical wind shear, buoyancy loss, proportional to the vertical gradient of the potential temperature, and turbulent dissipation (Wyngaard 2010; Duarte et al. 2015). Recent studies have shown that the mixing length also depends on these two gradients through the TKE equation and the Richardson number (Grisogono 2010; Cheng and Canuto 1994). The turbulent dissipation term is of practical value when applying the TKE equation because unlike shear production and buoyancy loss, it is not dependent on vertical gradients but is a function of TKE itself (e.g., Brost and Wyngaard 1978).

The small-scale phenomena listed above also affect the mixing length and the TKE budget, e.g., through vertical TKE transport, or are not fully resolved by our instruments. We therefore treat them as anomalies in this study and estimate their importance through the 10 Hz sonic anemometer data collected at the four tower levels.

The objectives of this paper are to create a complete description of the 5 study nights, including temperature, winds, TKE, vertical mixing length, and Richardson number below 328 m from the mean temperature and wind distributions in order to better understand vertical mixing. These results are based on the TKE budget equation discussed in section 2 and the sonic anemometer and sodar measurements discussed in section 3. A primary goal is to examine coupling between levels and the role of wind shear and meso-*β* fronts. We will also use the top three tower levels to illustrate flux–gradient relationships and *z*-less scaling in greater detail (sections 3 and 4).

## 2. Methods

### a. Richardson number and boundary layer depth

*g*= acceleration of gravity (m s

^{−2}), Θ

*= the virtual potential temperature (*

_{υ}*K*),

*u*and

*υ*(m s

^{−1}) are the horizontal wind velocities, and

*, which replaces local derivatives with finite difference values, i.e.,*

_{B}*in the 0–33 m layer—weakly stable, stable, and very stable—and assume the ranges below apply to the layer:*

_{B}*z*= Δ

_{i}*z*= (

_{c}*z*– ground level), where Ri

_{i}*reaches a critical value Ri*

_{B}*.*

_{Bc}Straume (2001) suggest a value for Ri* _{Bc}* in the range of 0.2–0.5, depending on vertical resolution. A value of Ri

*of 0.4 was chosen in this study after comparing Δ*

_{c}*z*with the aerosol backscatter gradients.

_{c}As has been pointed out by previous investigators, the NBL height is often poorly defined. In this study we have defined the NBL height based on turbulence created in the surface layer, where Eq. (2) applies, and noted other situations with clearly distinguished sources of TKE aloft.

### b. Mixing length

*(Grisogono 2010). The total TKE and its vertical part TKE*

_{υ}*are defined, respectively, as*

_{υ}*u*′,

*υ*′, and

*w*′ are departures from a time average and

*r*is expressed in terms of a diffusion coefficient

*K*proportional to (TKE

*)*

_{υ}^{1/2}and Λ

*, i.e.,*

_{υ}*usually include a dependence on stability. Wyngaard (1988) recommended*

_{υ}*L*=

_{B}*σ*/

_{w}*N*= limiting scale of the energy containing eddies away from the surface,

*σ*= standard deviation of the vertical wind speed, and

_{w}*f*= Coriolis parameter (10

^{−4}s),

*λ*=

_{B}*λ*/Ri,

_{s}*λ*= 1 m, and

_{s}*κ*= von Kármán constant.

*z*-less scaling expressions for the mixing length that depend on TKE, wind shear, and the Richardson number. His second expression for the vertical length scale can be rewritten in terms of TKE and TKE

*as*

_{υ}_{0}= TKE

^{3/4}TKE

_{υ}^{−1/4}/|

*S*|,

*f*

_{2}=

*f*

_{2}(Ri/Pr) = (1 − Ri/Pr)

^{−1/2}

*S*= vertical wind shear, and Pr = Prandtl number = 0.8 + 5Ri (in stable conditions).

*from either the potential temperature or the CO*

_{υ}_{2}mixing ratio. For a quasi-conservative tracer with time-averaged mixing ratio

*σ*is the standard deviation of the tracer’s mixing ratio. We evaluated Eq. (8) from the potential temperature gradient and

_{r}*σ*

_{Θ}

*, the variance in the sonic anemometer virtual temperature. A second value for Λ*

_{υ}*can be found from the CO*

_{υ}_{2}mixing ratio measured by NOAA’s LI-COR 7000 at ground level via and samples obtained via tubes from each tower level, and concentration fluctuations measured by LI-COR 7500 analyzers at each level.

*r*is

*c*and

*ρ*are the CO

_{2}densities in g m

^{−3}. If we neglect pressure perturbations, then,

*σr*

_{vol}of the volume mixing ratio of CO

_{2}(

*r*

_{vol}= 29/44 ×

*r*

_{mass}).

### c. Vertical and time variation of TKE and Λ_{υ}

*for the 5 nights can be obtained with the sodar data and by interpolation between the four tall tower levels. Unfortunately, the large gap between the 68 and 328 m levels implies that much of the TKE and Λ*

_{υ}*structure between the levels will not be realized by interpolation alone. However, the good resolution in the Scintec sodar wind data (5 m) along with RASS temperatures suggests improved resolution is possible with the application of the TKE equation and Grisogono’s Λ*

_{υ}*relation [Eq. (7)]. Further details are given in appendix A. We note that TKE in the NBL can be simplified to include only shear production (first term), the buoyancy gain/loss (second term), and turbulent dissipation*

_{υ}*ε*(third term), while excluding pressure and turbulent transport (Kaimal and Finnigan 1994, chapter 1; Wyngaard 2010), i.e.,

## 3. Results and discussion

### a. Temperature and wind measurements

The primary instrument platform used in the experiment was a 328 m tall tower (33.40°N, 81.83°W) in moderately complex terrain with roughly equal parts of ∼2 km × 2 km forest and agricultural tracts in an elevation range of ∼100 m (Fig. 1). The estimated displacement height is 20 m and may be higher than inferred from the tall tower because of its location on elevated terrain (Fig. 1).

Description of instruments (*T* = temperature, *T _{d}* = dewpoint).

The tower has “A type” ATI three-dimensional sonic anemometers at 68 and 328 m, and an “Sx type” at 33 m. (Atmospheric Technology Inc., Longmont, Colorado), and, at each level, LI-COR 7500 CO_{2}/H_{2}O analyzers (LI-COR, Lincoln, Nebraska) with 10-Hz sampling. Slow-response pressure, thermistor-temperature, and capacitance-relative humidity sensors housed in passive solar radiation shields from Met One (Grants Pass, Oregon) were at each level. Radiation corrections were applied to the thermistor sensors but were less than 0.2°C at night because of steady winds above 33 m. These slow response sensors were used mainly for quality assurance of the sonic anemometer and LI-COR 7500 data.

A surface flux/weather station at the base of the tall tower included a Campbell Scientific CSAT-3 sonic anemometer (Campbell Scientific, Logan, Utah), and a LI-COR 7500 analyzer (20-Hz sampling).

The sonic anemometer speed was evaluated in a wind tunnel and a correction applied to all covariances containing the vertical wind component for the A-type instrument (appendix A). No correction was required for the Sx type instrument. The sonic anemometer temperature was calibrated on the tower with a side-by-side comparison against laboratory-calibrated temperature sensors to an accuracy of 0.2°C. This calibration yielded satisfactory results for vertical gradients except during the first few evening hours of 12 May when the boundary layer was well mixed and the temperature difference between 33 and 68 m was less than 1°C. A more precise correction for this night was obtained by assuming that the time in the early evening when the vertical heat flux at 33 m changed from positive to negative corresponds to [*θ _{υ}*(68 m) −

*θ*(33 m)] = 0 (

_{υ}*θ*is virtual potential temperature), which led to a reduction in the 33 m temperature of 0.05°C for the entire night of 12 May.

_{υ}During the study month the Remtech Model PA-2 sodar (PA-2, Remtech, Velizy-Villacoublay, France) was located ≈24 km east of the tall tower in a clearing ≈100 m across in a pine forest of 15 m tall trees (Fig. 1). The Remtech PA-2 sodar has a maximum vertical range up to 800 m, in 20 m increments. To reduce the effect of the forest the sodar data below 60 m were replaced with three levels (4, 18, 36 m) of bivane and cup anemometer data from the climatology tower located in an open area ≈500 m across, 8 km southwest of the sodar (“Cl Tow,” Fig. 1). The Scintec SFAS minisodar/RASS (Scintec, Tubingen, Germany) was located ≈8 km northeast of the tall tower in a pasture ≈400 m across (Fig. 1). The sodar/RASS has a maximum vertical range of 125–300 m in 5 m increments and has greater vertical resolution and sensitivity than the Remtech sodar.

The sodar winds were evaluated against nearby towers over several months. No correction was applied to the Scintec sodar wind data except for rejection of the topmost level. The sodar/RASS temperature data were averaged in the 50–100 and 100–150 m layers. The Remtech sodar *u* and *υ* components were found to be positively biased at wind speeds less than 1.5 m s^{−1}, i.e., there were relatively few occurrences of winds less than 1.5 m s^{−1}. A correction formula adjusted the distribution of sodar winds in the range −3 to +3 m s^{−1} to equal those at the *H* area tower without changing the ranks or signs.

The nine 61 m SRNL towers (identified by their respective “areas,” which are designated as A-area, B-area, etc.) were also used in the experiment (Fig. 1). Winds and turbulence were measured at 61 m above ground level with cup anemometers/bivanes (Met One, Grants Pass, Oregon), and aspirated platinum resistance temperature sensors. The D-area tower sensors are at 36 and 61 m above ground in a 25 m forest in the Savannah River flood plain.

The structure in Ri, TKE, and the mixing length Λ* _{υ}* in our study is mainly derived from sodar wind shear (5 m resolution) since our temperature sensors did not resolve vertical temperature variations as well as winds. Chimonas (1999) discussed layer structure in the stable boundary layer and pointed out temperature ramps in earlier work by Gossard et al. (1985) with scales of meters. The likely presence of comparable thin stable layers on our study nights would add fine structure (e.g., Balsley et al. 2003) not present in our results except in the sonic anemometer data.

Other compromises were also necessary because of limited data resolution. The main one was blending tall tower data with data from the minisodar 8 km removed. This was justified only after a careful comparison of the 33 and 68 m winds at the two locations revealed a high degree of spatial as well as vertical coherence as also reported by Burman et al. (2018). We also rely heavily on gradients between the 33 and 68 m levels of the tall tower, also justified by the good coherence between the two levels. Temperature profiles measured at the climatology tower (2, 18, 36, and 61 m) were found to better characterize vertical temperature structure in shallow inversions than the 2 and 33 m tall tower levels in part because of its more open exposure.

The choice of the appropriate separation for finite difference derivatives also depends on the scale of interest. Preliminary examination of sodar and tower data revealed that mesoscales tended to dominate the temperature and wind variation and therefore that derivatives should be chosen accordingly and to avoid sensor spacing so close that measurement error seriously degrades gradient values.

### b. CO_{2} and flux measurements

Carbon dioxide (CO_{2}) is a quasi-conservative tracer in the atmosphere with a lifetime from 5 to 200 years (Prentice et al. 2001). In the rural area where the experiment was carried out its vertical distribution is positive during the night, when respiration and decay of vegetation add CO_{2} to the atmosphere, and negative during the day when photosynthesis removes CO_{2}. Nearby industrial sources are limited except when the wind direction is from the city of Augusta, Georgia, to the west-northwest and occasionally from a power plant to the south. Other local CO_{2} sources are weak during the month of May because of modest heating and cooling requirements in the spring and light highway traffic. Previous studies (e.g., Kurzeja 2012) have shown little variation in CO_{2} fluxes with wind direction at the tower as expected from a mixture of 1-km-sized patches of forest and agriculture. Anomalous local CO_{2} sources can easily be detected in the CO_{2} time series, and none occurred during the 5-day study period.

Further insight into the value of CO_{2} as a tracer is given in appendix C which shows 10-Hz fluctuations in winds, temperature, CO_{2}, and H_{2}O during a 15-min period. A reasonable correlation between CO_{2} and temperature is seen and a better one between CO_{2} with H_{2}O. This suggests that CO_{2} is vertically stratified and without significant local sources.

NOAA’s Global Monitoring Division (GMD) collects four 30-s samples per hour of CO_{2} and CO via three air tubes that extend from 33, 68, to 328 m to a LI-COR 7000 continuous analyzer at the surface with a multiplexing system that switches to concentration standards (Andrews et al. 2014). High precision is achieved because the data streams are measured with the same instrument under uniform conditions. However, the limit of four samples per hour reduces the accuracy of averages.

Fluxes of heat, moisture, momentum, and carbon dioxide were measured at three elevations on the tall tower and at the surface station. The 15-min averages were calculated in the natural coordinate system (Lee et al. 2004), except that the coordinate rotation constants were found from a running 1-h average of covariance data rather than from each 15-min period to avoid rapid changes in the coordinate system and to preserve nonzero 15-min vertical wind data.

### c. Aerosol measurements

SRNL operates a Vaisala C31 ceilometer 16 km to the east-southeast of the tall tower (Chiswell and Parker 2011; Fig. 1). The ceilometer measures aerosol backscatter above 20 m with vertical resolution of 20 m. The gradient in aerosol backscatter is sensitive to the boundary layer height and rate of vertical mixing.

### d. Quality assurance of data

Data were quality assured by visual inspection of plots of 10-Hz data together with the slow response temperature, humidity, and NOAA GMD (LI-COR 7000) CO_{2} data The LI-COR 7500 analyzers on the tower were calibrated with the NOAA GMD CO_{2} sensors and the slow-response relative humidity sensors. No data were rejected or corrected, except as described above, for these 5 nights.

### e. Synoptic description

On the night of 10 May a weak cold front moved into the study area and stalled for the next 24 h with rain on the morning of 11 May before moving away. On 12 May high pressure to the north brought northeasterly winds before drifting eastward to the Atlantic Ocean on 14 May. The 850 hPa winds on 10–12 May were moderate from the west to northwest and decreased to light from the southwest on 13–14 May. The 1200 UTC 850 hPa winds (morning) were stronger than the 0000 UTC winds (evening) which indicates the presence of a low-level jet.

### f. General comments on the 5 nights

The nights of 10 and 13 May were mostly clear while 11–12 May were mostly cloudy; 14 May was partly cloudy and then clear; 10 and 11 May were included in the study because both were stable, the latter by virtue of very light winds rather than surface cooling; 14 May was included because of a persistent, quasi-steady low-level jet.

Sea breeze and other meso-*β* fronts are common nighttime events at the site (Kurzeja et al. 1991; Buckley and Kurzeja 1997). A dramatic example was seen at 0316 UTC 13 May in the ceilometer backscatter and radar images from the NWS NEXRAD Doppler radar in Columbia, South Carolina, 100 km northeast of the study area (not shown). The frontal speed was 10 km h^{−1} toward the northwest.

Based on the ranges given in Eq. (3), 12 and 14 May were weakly stable while the other nights were mostly stable (Fig. 2). The Richardson number varied significantly with time on the three stable nights, tended to increase during the last few hours of each night, and occasionally exceeded 0.7, i.e., very stable as at 0700–1100 UTC 10 May.

The bulk Richardson number (0–33 m) as a function of the UTC hour on the 5 nights. The dashed lines indicate the boundaries between weakly stable, stable, and very stable stability categories; see text for details.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

The bulk Richardson number (0–33 m) as a function of the UTC hour on the 5 nights. The dashed lines indicate the boundaries between weakly stable, stable, and very stable stability categories; see text for details.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

The bulk Richardson number (0–33 m) as a function of the UTC hour on the 5 nights. The dashed lines indicate the boundaries between weakly stable, stable, and very stable stability categories; see text for details.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

The virtual potential temperature Θ* _{υ}* at the four tower levels is shown in Fig. 3, corrected as described in section 2. On 10, 11, and 13 May the temperature difference between the surface and 328 m was a maximum the first few hours of the night, decreased to a minimum and then reached another maximum just before sunrise. On 12 and 14 May the temperatures at the lower three levels were within 1°C of each other with the 328 m temperature up to 2°C warmer.

Virtual potential temperature at four levels on the tall tower.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Virtual potential temperature at four levels on the tall tower.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Virtual potential temperature at four levels on the tall tower.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Figure 4 shows that the heat flux at the three tower levels was strongly negative at the beginning of 4 of the 5 nights and approached zero by sunrise. The vertical lines in Fig. 4 and succeeding figures between −0200 and +0200 UTC (2200–0200 LT) indicate wind obstructed by the tower. Factor-of-2 variations were embedded in the heat flux trends with pronounced oscillations on 13 and 14 May. The heat flux at 328 m was usually small and sometimes upward, except on the weakly stable night of 12 May when the fluxes at the three levels were comparable which implies a boundary layer height greater than 328 m. The large positive flux at 0400 UTC 13 May occurred during the sea breeze frontal passage.

Heat flux at the tall tower at 33 (solid), 68 (dotted), and 328 m (magenta) for (top) 10 to (bottom) 14 May. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Heat flux at the tall tower at 33 (solid), 68 (dotted), and 328 m (magenta) for (top) 10 to (bottom) 14 May. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Heat flux at the tall tower at 33 (solid), 68 (dotted), and 328 m (magenta) for (top) 10 to (bottom) 14 May. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

The values of Λ* _{υ}* were found in the 33–68 m layer and at 328 m with 68–328 m gradients. As mentioned in section 2b the NOAA LI-COR 7000 data are precise but the accuracy of 15-min averages is diminished by only four 30-s averages per hour. Variability was reduced by smoothing the CO

_{2}time series with a 1-h moving filter. These two approximations reduce the accuracy of the CO

_{2}-derived length compared with the potential temperature-derived value except when the latter gradient is small enough to be affected by measurement errors.

Mixing length calculations for the 5 nights are shown in Fig. 5 along with results of Brost and Wyngaard (1978) [Eq. (5)], Sorbjan (2014) [Eq. (6)], and Grisogono (2010) [Eq. (7)], the latter multiplied by a constant to yield the best fit to the data.

(a) Vertical mixing length between 33 and 68 m, derived from the potential temperature (dotted), from the CO_{2} mixing ratio (crosses), from Grisogono’s relation [Eq. (7)] multiplied by a constant (heavy solid), from Sorbjan’s relation [Eq. (6); thin solid], and from Brost and Wyngaard [Eq. (5); blue line]. (b) As in (a), but at 328 m.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Vertical mixing length between 33 and 68 m, derived from the potential temperature (dotted), from the CO_{2} mixing ratio (crosses), from Grisogono’s relation [Eq. (7)] multiplied by a constant (heavy solid), from Sorbjan’s relation [Eq. (6); thin solid], and from Brost and Wyngaard [Eq. (5); blue line]. (b) As in (a), but at 328 m.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Vertical mixing length between 33 and 68 m, derived from the potential temperature (dotted), from the CO_{2} mixing ratio (crosses), from Grisogono’s relation [Eq. (7)] multiplied by a constant (heavy solid), from Sorbjan’s relation [Eq. (6); thin solid], and from Brost and Wyngaard [Eq. (5); blue line]. (b) As in (a), but at 328 m.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

The Sorbjan results were calculated with a value of *λ _{s}* = 5.0 instead of Sorbjan’s somewhat arbitrary value of 1.0. Sorbjan’s expression is not shown at 328 m because it includes a stability correction designed to reduce the mixing length in the upper part of the NBL to a stably stratified free atmosphere. Note the reasonable agreement between the values derived from the potential temperature and from CO

_{2}in the 33–68 m layer except before 0000 UTC when the stability is near neutral. Values average approximately 10 m except on 12 and 14 May, the weakly turbulent nights when they are in the 20–40 m range. We attribute the smaller Λ

*values obtained with Sorbjan’s relationship Eq. (6) to elevated TKE on these nights not included in Eq. (6). We also note that Sorbjan’s Λ*

_{υ}*values do not capture observed temporal variations which we attribute to TKE variations. The Brost and Wyngaard curves are like those obtained with Grisogono, but the amplitudes of the latter are somewhat less and closer to the observations.*

_{υ}The good agreement in Λ* _{υ}* variations with time between Grisogono’s relation and Brost and Wyngaard (1978) argues in favor of a dependence of Λ

*on TKE, wind shear, and stability as discussed by Grisogono.*

_{υ}### g. Time–height variation of TKE and Λ_{υ}

Before discussing the time–height contours for the 5 nights we first show how the scheme worked for selected cases. Figure 6 shows results for 12 and 13 May. On 12 May the potential temperature and wind speed increased linearly with height to 328 m with embedded variable shear layers ∼20 m thick which were correlated with the Ri number and aerosol backscatter gradient, e.g., the well-mixed aerosol layers at 125 and 225 m correspond to minima in the Ri number and enhanced wind shear. The TKE (Λ* _{υ}*) maximizes (minimizes) in the high-shear layers in accordance with Eqs. (A1) and (7), respectively. The aerosol profile in Fig. 6a implies aerosol sources at the surface and above 300 m.

(a) Ri × 10 (solid line), wind speed (thin dotted line), potential temperature −287 K (asterisks), log_{10} of the aerosol backscatter (dashed line), and (b) TKE (thin line) and vertical length scale (thick line), at 0815 UTC 12 May. (c),(d) As in (a) and (b), but at 0815 UTC 13 May. The symbols are observed values. Note the different abscissa scale in (b) (0–60) compared with (a), (c), and (d) (0–20).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Ri × 10 (solid line), wind speed (thin dotted line), potential temperature −287 K (asterisks), log_{10} of the aerosol backscatter (dashed line), and (b) TKE (thin line) and vertical length scale (thick line), at 0815 UTC 12 May. (c),(d) As in (a) and (b), but at 0815 UTC 13 May. The symbols are observed values. Note the different abscissa scale in (b) (0–60) compared with (a), (c), and (d) (0–20).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Ri × 10 (solid line), wind speed (thin dotted line), potential temperature −287 K (asterisks), log_{10} of the aerosol backscatter (dashed line), and (b) TKE (thin line) and vertical length scale (thick line), at 0815 UTC 12 May. (c),(d) As in (a) and (b), but at 0815 UTC 13 May. The symbols are observed values. Note the different abscissa scale in (b) (0–60) compared with (a), (c), and (d) (0–20).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

On 13 May (Figs. 6c,d) the well-mixed aerosol layer between 70 and 180 m corresponds to strong wind shear, a weak gradient in potential temperature, and Ri numbers less than 0.3. The TKE and Λ* _{υ}* (Fig. 6d) are much smaller than on 12 May and correlated with each other which suggests a weak dependence on wind shear in contrast to 12 May.

Figures 7–11 show time–height contours calculated as described above. They reveal common features that distinguish our results from the classic NBL described: 1) oscillations with periods of 2–4 h on all the nights, especially on 14 May, 2) A TKE maximum near ∼0.3*z*/*z*_{i} rather than at the surface, 3) relatively high levels of TKE near and above the nominal *z _{i}* on all stable nights (10, 11, and 13 May), 4) weak and fragmented low-level jets (<15 m s

^{−1}) that were correlated with NBL variables at lower levels, 5) significant mesoscale wind shear in the 100–200 m layer with a tendency for stable layers above and below, 6) meso-

*β*fronts on each night occasionally extending to the surface, 7) factor-of-2–4 variations in the TKE and vertical mixing length at all elevations.

(a) Virtual potential temperature, (b) Scintec/tower winds, (c) Richardson number, (d) TKE, (e) Remtech sodar winds, (f) ceilometer aerosol backscatter, and (g) vertical mixing length on 10 May. The thick dotted line on the Ri plot is the boundary layer height calculated from the bulk Ri number.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Virtual potential temperature, (b) Scintec/tower winds, (c) Richardson number, (d) TKE, (e) Remtech sodar winds, (f) ceilometer aerosol backscatter, and (g) vertical mixing length on 10 May. The thick dotted line on the Ri plot is the boundary layer height calculated from the bulk Ri number.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) Virtual potential temperature, (b) Scintec/tower winds, (c) Richardson number, (d) TKE, (e) Remtech sodar winds, (f) ceilometer aerosol backscatter, and (g) vertical mixing length on 10 May. The thick dotted line on the Ri plot is the boundary layer height calculated from the bulk Ri number.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 11 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 11 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 11 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 12 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 12 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 12 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 13 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 13 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 13 May.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 14 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 14 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 7, but for 14 May. The dashed line in (f) is the estimated top of the NBL.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Pronounced oscillations are seen in all the variables, including the heat flux (Fig. 4) with periods of ∼2 h. Since they extend through the boundary layer, they fit the description of meso-*β*-scale events near the lower limits of 1 h and 20 km listed by Stull (1988, chapter 1). The most dramatic examples are the sea breeze on 13 May and the minisodar winds and Remtech winds on 14 May above the low-level jet core (400 m) shown in Fig. 11e, and also in the backscatter data, Fig. 11g. The TKE oscillations are similar to the oscillations in

On all nights the TKE maximum is generally between 30 and 50 m instead of near the surface which is probably due to ∼20 m variations in topography and vegetation near the tower. Hicks (2022) noticed a similar separation between the boundary layer flow and the surface using CASES data in which the flow above the surface retained upwind characteristics.

The tendency for elevated TKE near 30 m with periodic high TKE extending above the nominal *z _{i}* is illustrated in more detail in Fig. 12. Vertical profiles at seven intervals are shown on 10, 11, and 14 May, normalized by the square of the wind speed at 40 m and plotted against the height scaled with

*z*. Observed values are circles on the interpolated curves. On the weakly stable night of 14 May (Fig. 11c) the TKE decreases smoothly to 20% of its maximum value at ∼0.3 z/

_{i}*z*, whereas on 10 and 11 May (Figs. 7, 8a,b) significant TKE is seen above

_{i}*z*. It is unlikely that the boundary layer height calculated with Eq. (2) is too low on these two nights since it is confirmed by the surface inversion and the Ri number (Figs. 7, 8a,c). Instead, inspection of the minisodar plots indicates that the source of the TKE is weak jets and mesoscale shear between

_{i}*z*and 200 m.

_{i}TKE profiles between 0100 and 0800 UTC (a) 10, (b) 11, and (c) 14 May, normalized with the wind speed squared at 40 m, and plotted as a function of the height normalized with boundary layer height *z _{i}*. The symbols denote observed values.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

TKE profiles between 0100 and 0800 UTC (a) 10, (b) 11, and (c) 14 May, normalized with the wind speed squared at 40 m, and plotted as a function of the height normalized with boundary layer height *z _{i}*. The symbols denote observed values.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

TKE profiles between 0100 and 0800 UTC (a) 10, (b) 11, and (c) 14 May, normalized with the wind speed squared at 40 m, and plotted as a function of the height normalized with boundary layer height *z _{i}*. The symbols denote observed values.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Low-level jets were seen on 4 of the 5 nights but were weaker, transient, and lower than reported elsewhere (e.g., Berg et al. 2015; Whiteman et al. 1997) and tended to oscillate in strength. See, for example, the 250 m level at 0200–0600 UTC 10 May (Figs. 7b,e) and the layers above 75 m on 14 May (Figs. 11b,e). Not surprisingly, these oscillations lead to layering and local temporal maxima in the Ri number and other variables.

A tendency for clockwise rotation following the inertial oscillation was seen on all nights as well as vertical coherence with even weak low-level jets, e.g., at 0400–0600 UTC 12 May, when a weak meso-*β* front resulted in a minimum in TKE below 200 m.

Frontal zones are clearly visible on three of the nights—10, 11, and 13 May. On 10 May a weak low-level jet is seen between 100 and 400 m. A sea breeze is apparent on 13 May. On 11 May, however, northwesterly flow comes to a complete stop before resuming at 0700 UTC with an increase in TKE (Fig. 8d). It is tempting to associate these meso-*β* fronts with the description of microfronts in, e.g., Mahrt (2018). However, these fronts extend through the entire boundary layer and appear to be edges of air masses that shift laterally with the low-level jets.

The high TKE values above the surface suggests the possibility for “upside-down” boundary layers where energy is created aloft and transported downward (Mahrt and Vickers 2002). This possibility is examined by comparing the ratio of the TKE at 33 and 68 m (Fig. 13). A ratio greater than one implies a traditional boundary layer with the opposite for *z*-less and upside-down boundary layers. Traditional boundary layers are indicated on the stable nights of 10, 11, and 13 May, whereas no conclusion can be drawn for the other two nights because of a broad maximum in the 30–60 m range.

Times series of TKE(33 m)/TKE(68 m) (solid line) and TKE* _{υ}*(33 m)/TKE

*(68 m) (dotted line) on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure.*

_{υ}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Times series of TKE(33 m)/TKE(68 m) (solid line) and TKE* _{υ}*(33 m)/TKE

*(68 m) (dotted line) on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure.*

_{υ}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Times series of TKE(33 m)/TKE(68 m) (solid line) and TKE* _{υ}*(33 m)/TKE

*(68 m) (dotted line) on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure.*

_{υ}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Figures 7–11 show variations in the mixing length of 2–4, especially above the surface layer. These variations are not included in dispersion models and will affect vertical trace chemical transport.

In summary, the NBL presented in Figs. 7–11 differs from the classical NBL because it is predominantly time dependent and oscillatory with mesoscale fronts rather than controlled by surface stress. Mesoscale air masses are bounded by weak, low-level jets or inversion shear zones and persist for no more than 3–4 h. Below these active layers is a ∼30-m-deep surface layer affected by mixed land use, terrain variations and terrain slope.

### h. Level analysis

This section compares the Brost–Wyngaard (BW) mixing length [Eq. (5)] with the *θ _{υ}* mixing length in the 33–68 m layer. Figure 14 shows generally good agreement for mixing lengths less than 20 m but not above 20 m. The blue line shows the trend line for the BW relation on the weakly stable night of 12 May that is substantially steeper than the observations. A similar trend was noted on the other weakly stable night of 12 May (black dots).

Mixing length from the potential temperature vs the Brost–Wyngaard value [Eq. (5)] in the 33–68 m layer for the 5 nights in 15-min averages from 0000 to 1200 UTC. The blue line is a linear fit to the 12 May data.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Mixing length from the potential temperature vs the Brost–Wyngaard value [Eq. (5)] in the 33–68 m layer for the 5 nights in 15-min averages from 0000 to 1200 UTC. The blue line is a linear fit to the 12 May data.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Mixing length from the potential temperature vs the Brost–Wyngaard value [Eq. (5)] in the 33–68 m layer for the 5 nights in 15-min averages from 0000 to 1200 UTC. The blue line is a linear fit to the 12 May data.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Grisogono’s relationship is plotted against observations in Fig. 15 and is in good agreement with the *θ _{υ}* mixing length less than 14 m with no bias on the weakly stable nights of 12 and 14 May, but with more variability and outliers for mixing lengths greater than 15 m. Both Brost and Wyngaard’s and Grisogono’s expressions include a dependence on stability and TKE, but more analysis is needed to find the reasons for the outliers.

As in Fig. 14, but for Grisogono’s relation [Eq. (7)].

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 14, but for Grisogono’s relation [Eq. (7)].

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As in Fig. 14, but for Grisogono’s relation [Eq. (7)].

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

As noted in the introduction, MO and *z*-less scaling relate fluxes to gradients. The applicability of *z*-less scaling is supported indirectly by the good correspondence between TKE and shear production which suggests that TKE is in approximate balance between local production and loss—a requirement for *z*-less and MO scaling.

Further evidence for *z*-less scaling is in the ratio of *σ _{w}* to

*u*, the local value of the friction velocity,

_{L}The *z*-less scaling is often characterized by the asymptotic approach of *σ _{w}*/

*u*to a value of 1.4 with increasing stability using high-pass filtered (>0.01 Hz) data to exclude mesoscale phenomena (i.e., Nieuwstadt 1984). Basu et al. (2006) also calculated the ratio of

_{L}*σ*/

_{w}*U*from field observations, from laboratory experiments and LES to be 1.6, 1.5, and 1.4, respectively, using frequencies greater that 0.01 Hz. However, Wyngaard (1988) has noted, quoting Hunt (1985), that MO scaling with high-pass filtering ignores much low-frequency energy production in the NBL, and therefore may lack relevance.

_{L}Figure 16 shows the *σ _{w}*/

*u*ratio for the five nights at three levels along with Basu et al.’s results. Our mean values for the 33, 68, and 328 m levels are 1.63, 1.88, and 1.78, respectively, which are larger than Basu et al.’s measured value of 1.6 for his S3 stability class (

_{L}*z*/Λ = 0.25–0.5). The ratio at 33 m is smaller than at 68 or 328 m with a narrower variability range. This may be because the maximum TKE is close to 33 m for all days (Figs. 7–11).

Ratio of *σ _{w}*/

*u*at (a) 33, (b) 68, and (c) 328 m for the 5 nights and three levels. Basu et al.’s (2006) S3 stability class (

_{L}*z*/Λ = 0.25 − 0.5) field observations are shown on the right side of each figure. The center points of the bars (green lines) are the 50th percentiles, the blue lines are the 25th and 75th percentiles, and the ends are the 5th and 95th percentiles.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Ratio of *σ _{w}*/

*u*at (a) 33, (b) 68, and (c) 328 m for the 5 nights and three levels. Basu et al.’s (2006) S3 stability class (

_{L}*z*/Λ = 0.25 − 0.5) field observations are shown on the right side of each figure. The center points of the bars (green lines) are the 50th percentiles, the blue lines are the 25th and 75th percentiles, and the ends are the 5th and 95th percentiles.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Ratio of *σ _{w}*/

*u*at (a) 33, (b) 68, and (c) 328 m for the 5 nights and three levels. Basu et al.’s (2006) S3 stability class (

_{L}*z*/Λ = 0.25 − 0.5) field observations are shown on the right side of each figure. The center points of the bars (green lines) are the 50th percentiles, the blue lines are the 25th and 75th percentiles, and the ends are the 5th and 95th percentiles.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

A greater difference was seen in the variation of this ratio than found by Basu et al., i.e., a difference of 1.3, 2.3, and 2.1 in the (95th to 5th) percentile range at 33, 68, and 328 m levels, respectively, compared to Basu et al.’s value of ∼0.7, a factor of 2–3 greater. This result can be explained with Högström’s (1990) concept of inactive (nonturbulent) mesoscale eddies, which we can generalize to mean eddies of varying turbulent efficiencies.

Gravity waves were observed occasionally during the 5-night period, especially at 328 m, which we detected during our routine inspection of 15-min 10-Hz data. Appendix C shows a portion of a 15-min plot and a wavelet analysis of the same period. Interaction between gravity waves and the mean flow has been reported by many investigators (e.g., Finnigan et al. 1984; Einaudi and Finnigan 1993) and weak wave–mean flow interaction may partly explain the high ratios of TKE to *u _{L}* in Fig. 16c.

We can define high efficiency eddies (*σ _{w}*/

*u*< 1.5) as those with frequencies in the inertial subrange that convert a high percentage of TKE to momentum flux, while low efficiency eddies have high ratios of

_{L}*σ*/

_{w}*u*(>∼3), and a high fraction of inactive eddies. Gravity waves are assumed to be of variable efficiency (1.5 <

_{L}*σ*/

_{w}*u*< ∼3) because their conversion to turbulence depends on wave and mean flow properties (Finnigan et al. 1984). Thus, the eddies at 33 m are more “efficient” than the eddies at 68 and 328 m.

_{L}The anomalously high values of the *σ _{w}*/

*u*ratio at 68 m on 14 May (Fig. 16) that occurred between 0800 and 1200 UTC (Fig. 17b) deserve further attention. Figure 11c illustrates high stability at 68 m while Fig. 11d shows elevated levels of turbulence despite the weak vertical wind shear in the Scintec sodar. (Recall that the TKE at 68 m is measured rather than calculated.) On the other hand, Fig. 11e indicates a more vigorous low-level jet at 0800 UTC at the Remtech sodar, east of the tall tower. This suggests that the TKE at the tall tower may have been advected from near the Remtech sodar and that TKE is not in a quasi-steady balance between production and loss at this time.

_{L}Observed *σ _{w}* (thin solid) and

*u*(thick solid) at (a) 33, (b) 68, and (c) 328 m on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure. The circled period at 0315 UTC 11 May had weak turbulence and a gravity wave.

_{L}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Observed *σ _{w}* (thin solid) and

*u*(thick solid) at (a) 33, (b) 68, and (c) 328 m on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure. The circled period at 0315 UTC 11 May had weak turbulence and a gravity wave.

_{L}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Observed *σ _{w}* (thin solid) and

*u*(thick solid) at (a) 33, (b) 68, and (c) 328 m on the tall tower. The vertical lines on 12 May indicate wind shading by the tower structure. The circled period at 0315 UTC 11 May had weak turbulence and a gravity wave.

_{L}Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Further insight into ratio of *σ _{w}*/

*u*is from the period at 0315 UTC 11 May discussed in appendix C, when a gravity wave occurred along with very weak turbulence. This period, shown in Fig. 17 with a circle, is one with an elevated ratio of

_{L}*σ*/

_{w}*u*. Thus, it is an extreme example of the breakdown of

_{L}*z*-less scaling.

*shown in Figs. 7–11 and the 328 m sonic anemometer data. For simplicity dissipation loss was taken from Brost and Wyngaard (1978), i.e.,*

_{υ}*σ*/

_{w}*u*ratio seen in Fig. 16c.

_{L}(a) TKE terms in the 33–68 m layer for the 5 days, TKE (solid), 50 × TKE shear production (thin solid), 50 × TKE buoyancy sink (thin dashed), and 50 × dissipation loss (thick dot–dashed). (b) As in (a), but for the 300–328 m layer. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) TKE terms in the 33–68 m layer for the 5 days, TKE (solid), 50 × TKE shear production (thin solid), 50 × TKE buoyancy sink (thin dashed), and 50 × dissipation loss (thick dot–dashed). (b) As in (a), but for the 300–328 m layer. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

(a) TKE terms in the 33–68 m layer for the 5 days, TKE (solid), 50 × TKE shear production (thin solid), 50 × TKE buoyancy sink (thin dashed), and 50 × dissipation loss (thick dot–dashed). (b) As in (a), but for the 300–328 m layer. The vertical lines on 12 May indicate wind shading by the tower structure.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

## 4. Discussion

Our primary objectives in this paper were to determine realistic time–height profiles of TKE and the vertical mixing depth from the averaged distributions of temperature and wind from limited vertical profiles and to show that reasonable estimates of vertical fluxes of momentum and heat were also possible. We addressed the practical problem of the coarse vertical resolution of the sodars and towers by merging minisodar/RASS data with four levels of sonic anemometer and by showing that vertical wind shear was closely correlated with the TKE.

Previous research has relied on high-pass filtering to exclude “inactive” eddies that add to the TKE but otherwise do not interact with the mean flow or create a local flux. Our research has suggested that a longer averaging time (15 min) that includes a fraction of the mesoscale eddies and gravity waves, while modifying scaling constants and increasing variability, may nevertheless be more realistic and useful for parameterizations in models.

Other features of interest in our data were a TKE maximum usually between 30 and 50 m with occasional local maxima near and above the nominal boundary layer top, as well as oscillations in the wind, TKE, and heat flux with periods of ∼2 h.

The boundary layer depth *z _{i}* calculated from the bulk Ri number captured much of the temporal variation in vertically integrated TKE and was generally consistent with thermal gradients, and aerosol backscatter. It also was useful for scaling the vertical variation of TKE on weakly stable nights but not on stable nights. However, it should be used mainly as a reference since it changed with meso-

*β*frontal passages.

Our estimates of vertical mixing length were reasonable but were highly variable, especially at 328 m. The relationship derived by Grisogono (2010) was only marginally superior to the simpler expression of Brost and Wyngaard (1978) but could be substantially improved if outliers were explained and removed.

Stable boundary layers have been classified by Mahrt and Vickers (2002) into three types—traditional, *z*-less, and upside-down. In *z*-less boundary layers turbulence sources are present with no clear link to the surface, e.g., from gravity waves or the low-level jet. Upside-down boundary layers have an increase in TKE with height and a downward transport of TKE. Related to this concept are studies that have shown coherence between strong low-level jets and turbulence near the surface (e.g., Banta et al. 2003; Burman et al. 2018). Our paper does not address this topic in detail but, because of the relatively high TKE near the NBL top, does illustrate several related points. Although the low-level jets seen on our 5 nights are weak and short lived, continuity to the surface was apparent in all our derived variables, consistent with previous studies. However, in contrast, our study highlights much weaker shallow layers and meso-*β* fronts with similar properties to the low-level jet but over limited times above the surface layer. For example, the time–height TKE contours shown in Figs. 7–11 resemble plots shown in Banta et al. (2007, their Fig. 6). However, Banta et al. observed a maximum in TKE in the 10–40 m range instead of the 30–60 m range found in our study.

Another question is what proportion of TKE below ∼30 m is due to local generation from wind shear and what proportion is transported downward from above in an upside-down flux. If we require that TKE increase with height in an upside-down boundary layer then Fig. 13 implies that our 3 stable boundary nights are traditional, at least between 33 and 68 m, and possibly upside-down below 30 m.

## 5. Conclusions

This study examined winds and turbulence in and above the NBL on 5 successive nights with an array of towers, two sodars, and a ceilometer. A primary objective was the determination of accurate TKE and the vertical mixing length for input to transport models, motivated by preliminary simulations that found the models under predicted perfluorocarbon transport from the surface to 328 m by an order of magnitude for both weakly stable and stable nights.

The TKE equation along with a relation expressing the mixing length in terms of the TKE, wind shear and thermal stability were used with the sodar and tower data to create time–height contours of relevant variables in 15-min intervals. These plots together with detailed analysis at the tower levels (33, 68, and 328 m) yielded a different picture from the classical NBL in which elevated turbulence near the surface decreases to a minimum at the top of the boundary layer, interrupted by turbulent episodes on stable and very stable nights.

Our NBL in complex terrain and mixed land use is strongly time dependent and weakly height dependent with mesoscale air masses evolving with time and impacting each other forming meso-*β* fronts. At times these meso-*β* fronts are seen at the surface but more commonly remain above the surface layer, which in our area is near the displacement height of ∼20–30 m. The net impression is one of strong vertical coherence in the TKE and mixing length, with frequent detachment from the surface layer, which is different from the strong low-level jet cases in Burman et al. (2018) and Banta et al. (2003).

There are many causes of low-level jets (Stull 1988, chapter 12). Previous work has tended to assume one distinct low-level jet per night, but our results suggest mesoscale air masses form in different locations and different heights and cause weak and varying jets and the observed meso-*β* fronts. Our data also showed significant wind shear near the inversion height (100–150 m) with stable layers above and below.

The study provides indirect evidence that TKE behavior is at least approximately consistent with *z*-less scaling above the surface layer on stable nights even during nonstationary periods. One indication is that the TKE was explained well in terms of local wind shear production and dissipation loss. A second is the reasonably close match of the vertical velocity standard deviation to the local friction velocity scale.

The study also estimated the time and height variation of the vertical mixing length from the standard deviation and vertical gradient of the potential temperature and the CO_{2} mixing ratio and compared them with expressions of Brost and Wyngaard (1978), Sorbjan (2014), and Grisogono (2010). Grisogono’s *z*-less expression included a dependence on TKE and wind shear in addition to Ri that was consistent with our results.

An increase in the mixing length to a value of ∼20 m at 328 m was noted on the very stable nights and up to 50 m on weakly stable nights. Considerable variation in the length scale with time was observed, which implies that above 50 m, where the inverse dependence on *z* is weak, the TKE, wind shear, and the temperature gradient must be known to infer the length scale with confidence, and that simple representations of the length scale as a function of height and stability are approximate.

Implications for trace chemical transport are as follows: first, weakly stable nights are usually assumed to be well mixed, with a decrease in ground-level turbulence to the top of the boundary layer. Our results differ somewhat because the turbulence maxima occurred at 30–60 m and occasional stable layers were seen near 100 and 200 m. This implies that surface layer turbulence variables, e.g., the friction velocity or the Obukhov length, are likely to underestimate dispersion in the upper half of the boundary layer.

Second, turbulence on the stable nights was a weak function of height and often not capped by *z _{i}*. This behavior implies the need for vertical profiles of turbulence and TKE to adequately model mid- and long-range dispersion. The significant amount of TKE near and above the nominal boundary layer top and high mixing lengths on both weakly stable and stable nights may also explain the order of magnitude underprediction of the perfluorocarbon concentration at 328 m in our model simulations as mentioned in the introduction.

Mesoscale eddies may significantly enhance vertical transport especially on stable nights when vertical fluxes are otherwise weak. In addition, in our area complex terrain and land use with vertical scales of ∼20 m create a maze of overlapping flux footprints and a “blending height” (e.g., Mahrt 2000), where fluxes from the heterogeneous surface elements mix together.

The existing datasets offer several opportunities for future work. The most direct is a breakdown of tower variables and fluxes by wavelength to highlight the role of mesoscale events. A wavelet analysis (appendix C) is one option but a comparison of 5-min, 15-min, 30-min, and 1-h averages, already available, is an alternative picture. These data could be linked to the event description provided in this paper.

The validity of flux gradient relationships is important in mesoscale and global models and more information is needed for events with periods longer than minutes. The measured momentum and heat fluxes between 33 and 68 m can be used for this since the minisodar has 5 m vertical resolution and was close to the tall tower.

## Acknowledgments.

This document was prepared in conjunction with work accomplished under Contract DE-AC09-08SR22470 with the U.S. Department of Energy. Funding was provided by the DOE Office of Science Terrestrial Carbon Processes program. The authors thank the following individuals for assisting with the field studies. David Durden, Natchaya Pingintha (both now with NEON), and Natthaphol Lichaikul from the University of Georgia, John T. Hamilton Jr. and Ronald W. Johnson from the Savannah River National Laboratory, Arlyn Andrews, Jonathan Kofler, and Jonathan Williams from the NOAA/Global Monitoring Division, Erik Kabela, now with the Oak Ridge National Laboratory, and Borja Ruiz Reverter from Grupo de Física de la Atmósfera, Centro Andaluz de Medio Ambiente (CEAMA) Universidad de Granada, Granada, Spain. The authors thank Chuck Hunter of SRNL for reviewing the manuscript and for his support during his research and Mary Willoughby of UGA for help with the data repository. The authors are grateful to the reviewers for their helpful comments on the content and presentation.

## Data availability statement.

The data used in this study can be accessed from Athenaeum@UGA using the following links: https://esploro.libs.uga.edu/esploro/outputs/9949395383602959 for the Vaisala C31 ceilometer, https://esploro.libs.uga.edu/esploro/outputs/9949395383402959 for the Remtech sodar, https://esploro.libs.uga.edu/esploro/outputs/9949395383102959 for the tall tower meteorology, https://esploro.libs.uga.edu/esploro/outputs/9949395383202959 for the surface weather, and https://esploro.libs.uga.edu/esploro/outputs/9949396686702959 for the Scintec sodar/RASS.

## APPENDIX A

### Composite Profiles and Contour Plots

The temperature and wind contours were obtained by merging tall tower and Scintec sodar/RASS profiles for each 15-min period and then interpolating the temperature and winds to 5 m intervals with a cubic spline function. The *u* and *υ* curves were adjusted to approximate the vertical gradient of the Remtech *u* and *υ* components above 300 m under the assumption that differences between the tall tower and the Remtech sodar are small at these heights even though separated by ∼15 km.

*K*=

_{h}*K*/Pr and Pr is the Prandtl number.

_{m}*K*,

_{m}*C*is a constant, TKE

_{τ}_{υ}is the vertical component of TKE, and

*τ*is the dissipation time scale. We set

_{T}*τ*equal to the equilibrium time scale, i.e.,

_{T}*τ*≈

_{T}*τ*= Λ

_{TE}*/TKE*

_{υ}^{1/2}.

*K*= 2

_{m}*C*Λ

_{τ}*TKE*

_{υ}*TKE*

_{υ}^{−1/2}and TKE production is given by

*b*(

*z*) = TKE

*/TKE is the ratio of vertical to total TKE.*

_{υ}*c*(

*z*),

*are mutually dependent and the equations express one in terms of the other. In fact, Grisogono (2010) argued the mixing length is dependent on the TKE and wind shear as well as atmospheric stability for*

_{υ}*z*-less scaling. However, TKE and Λ

*are constrained by observed values and variances about the nighttime mean at 2, 33, 68, and 328 m and at 50 and 328 m, respectively.*

_{υ}*b*(

*z*),

*c*(

*z*), and the constants into a single function of height

*C*(

*z*). Then,

*is the adjusted TKE,*

_{a}*φ*is an empirical adjustment parameter, and the double overbar indicates an average over all times from 0400 to 1200 UTC (0000–0800 LT). The parameter

*φ*was found by comparing the variance in the measured and simulated TKE at 68 and 328 m and adjusting

*φ*to yield the best fit for each night.

*below 50 m were found from Eq. (6) by introducing an adjustable constant*

_{υ}*α*in Sorbjan’s expression to force a match between the calculated value at 50 m and the value derived from the potential temperature at 50 m (Fig. 3):

*(*

_{υ}*z*) was found with Eq. (7) using an exponential adjustment factor

*β*for the variance and a height-dependent multiplication factor

*d*(

*z*), i.e.,

*S* is the wind shear, and the double overbar is defined above. The exponent *β* was found as *φ* but at 50 and 328 m. Values for *φ* and *β* ranged between 0.75 and 0.9.

*were adjusted with the observations in a four-step process. First, preliminary time–height profiles of TKE and Λ*

_{υ}*and were obtained by linear interpolation between observed values at 68 and 328 m for TKE and at 50 and 328 m for Λ*

_{υ}*. These profiles showed no detail because wind shear was absent. The second step calculated TKE profiles from Eq. (A1) and found*

_{υ}*C*(50 m) and

*φ*(50 m) to match the observed 12-h time mean and variance. The process was repeated to find

*C*(328 m) and

*φ*(328 m). The function

*C*(

*z*) was found with linear interpolation while

*φ*(

*z*) was set equal to the average of the two levels, i.e.,

*. These profiles were adjusted as for TKE to find*

_{υa}*d*(

*z*),

*β*(50 m),

*β*(328 m), and

*and TKE profile by a parameter linearly dependent on height to force the profile to equal the observed values at 50, or 68, and 328 m at each time.*

_{υ}Matching nighttime means and variances of TKE and Λ* _{υ}* to the observations at the two levels is critical to ensure reasonable behavior of TKE and Λ

*between levels, i.e., we assume factors*

_{υ}*φ*and

*β*that fit the observed time-averages and variances at the two levels are also reasonable for layers in between.

## APPENDIX B

### Wind Tunnel Sonic Anemometer Calibration

The “A type” and “Sx type” ATI anemometers have difitaferent aerodynamic properties. Their accuracy was assessed in a wind tunnel, in 20° azimuth intervals, each over an elevation range from −60° to +60°, for wind speeds of 4 and 9.6 m s^{−1}. The average ratio of the Sx vertical velocity to the corresponding wind tunnel velocity was 0.99 compared with the A type ratio of 0.83. These factors were derived for the elevation range −20° to + 20°—typical of NBL turbulence. The factors were not strongly dependent on wind speed. The 0.83 factor was applied to all flux terms involving the vertical velocity. On the other hand, because both anemometers are relatively unobstructive to horizontal winds, no correction was necessary.

## APPENDIX C

### Gravity Wave Detection

This appendix outlines our method to detect gravity waves in the data. The first check was a visual inspection of the 10-Hz sonic anemometer data in 15-min intervals. These plots were generated to show bad or unusual data. Both 1-Hz and slow-response data were plotted. The fast-response data, sonic anemometer and Licor CO_{2}/H_{2}O analyzer data are shown in Fig. C1 at 328 m on 11 May. This particular 15-min period contained a large-amplitude gravity wave with a period of ∼4 min with very low high-frequency turbulence.

Plot of 10-Hz data at 0315–0330 UTC 11 May 2009. The data are normalized by subtracting the 15-min mean value and then dividing by the standard deviation over the 15-min period. The straight line is the mean while the positive and negative departures range from −5 to +5. The text to the right of each variable gives its mean value, standard deviation, number of corrected values, and sonic anemometer/Licor offset in tenths of a second.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Plot of 10-Hz data at 0315–0330 UTC 11 May 2009. The data are normalized by subtracting the 15-min mean value and then dividing by the standard deviation over the 15-min period. The straight line is the mean while the positive and negative departures range from −5 to +5. The text to the right of each variable gives its mean value, standard deviation, number of corrected values, and sonic anemometer/Licor offset in tenths of a second.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Plot of 10-Hz data at 0315–0330 UTC 11 May 2009. The data are normalized by subtracting the 15-min mean value and then dividing by the standard deviation over the 15-min period. The straight line is the mean while the positive and negative departures range from −5 to +5. The text to the right of each variable gives its mean value, standard deviation, number of corrected values, and sonic anemometer/Licor offset in tenths of a second.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

A gravity wave is clearly seen in the vertical velocity and the temperature between the third and twelfth minute, which are also out of phase indicating a negligible vertical heat transport, except during the growth and decay phases before the 1-min and after the 12-min marks. On the other hand, the phase relationships between *w* and *u* and *υ* suggest vertical momentum transport by the gravity wave.

Another view of this period is shown in the wavelet analysis (Torrence and Compo 1998) of Fig. C2 between 0000 and 0800 UTC 11 May. A large-amplitude wave with a period of ∼300 s (5 min) is shown at 0315–0330 UTC (circle and line segment). The figure also shows longer period but weaker waves with periods of up to 15 min between 0200 and 0400 UTC. A later period of gravity waves is shown near 0700 UTC, also within longer period waves.

Wavelet analysis of the vertical velocity variance at 328 m at 0000–0800 UTC 11 May. The line segment within the circle at 0315 UTC corresponds to the data in Fig. C1.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Wavelet analysis of the vertical velocity variance at 328 m at 0000–0800 UTC 11 May. The line segment within the circle at 0315 UTC corresponds to the data in Fig. C1.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

Wavelet analysis of the vertical velocity variance at 328 m at 0000–0800 UTC 11 May. The line segment within the circle at 0315 UTC corresponds to the data in Fig. C1.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-21-0268.1

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