1. Introduction

The depth of the planetary boundary layer (PBL) and the intensity of the turbulence within it have a strong impact on the vertical and horizontal distribution of CO₂ in the atmosphere (Denning et al. 1995; Wofsy et al. 1988). During the daytime in summer the influence of photosynthetic uptake on the mixing ratio of CO₂ is diluted by deep convective turbulent mixing. The influence of respiration on the CO₂ mixing ratio at night is amplified near the surface by a shallow, stable boundary layer. The covariance between surface fluxes of CO₂ and the vigor of atmospheric mixing, which has been termed the “rectifier effect” (Denning et al. 1999; Law and Rayner 1999), has a strong seasonal character, with deeper convection during the daytime in summer when photosynthesis exceeds respiration (Denning et al. 1996). Observed distributions of CO₂ have been used to calculate spatial distributions of sources and sinks by inverse methods (e.g., Tans et al. 1990; Ciais et al. 1995; Francey et al. 1995; Fan et al. 1998). Since the rectifier effect influences the horizontal and vertical distributions of CO₂ in the atmosphere, it can lead to a serious bias in the calculated fluxes if not properly accounted for in inverse models (Denning et al. 1995). In order to study the diurnal and seasonal patterns of the rectifier effect, long-term, continuous observations of PBL dynamics and CO₂ mixing ratios over the continents are imperative.

Long-term, continuous observations of PBL structure were difficult or impossible until the recent development of robust boundary layer profiling radar and Radio-Acoustic Sounding System (RASS) (Ecklund et al. 1988). Most deployments of these systems to date have been too brief to capture seasonal information. For this study a radar profiler, RASS, and radiosonde system were deployed for the period from 15 March to 3 November 1998, near to a 447-m tall TV transmitter tower in northern Wisconsin. The tower was instrumented to measure continuously the turbulent flux profiles of latent and sensible heat, and flux and mixing ratio profiles of CO₂. Daytime convective PBL (mixed layer) depth measurements from the radar were verified against data from radiosondes.

Measurements of the vertical profile of CO₂ mixing ratio on the TV tower allowed us to study the evolution of the stable PBL at night. The stable layer is typically very shallow, usually less than 200 m, and therefore is not accessible to the profiler and RASS, which have a minimum altitude of 150 m above the ground with 60-m sampling interval.

Three to four days per month we were able to observe at night the height of the residual mixed layer from the previous day. These weather conditions were characterized by calm, fair-weather conditions, high surface pressure, and subsidence. From the rate of change of the residual-layer depth, we obtain an estimate of the subsidence rate, which was typically in the range 1–3 cm s⁻¹. These results provide a valuble and unique dataset to check subsidence estimates from weather prediction models. We also estimate the influence of subsidence on the structure of the mixed and stable layers.

2. Study site and measurements

The study site is located in Chequamegon National Forest in northern Wisconsin. The region is in a heavily forested zone of low relief. The tower is a 447-m tall television transmitter surrounded by a grassy clearing of about 180-m radius. The site, instrumentation, and flux calculation methodology have been described by Bakwin et al. (1998) and Berger et al. (2001). Three three-axis sonic anemometers at 30, 122, and 396 m above ground are used to measure turbulent winds and virtual potential temperature. Air from these three levels is drawn down tubes to a trailer where three LI-COR 6262 analyzers are used to determine CO₂ and water vapor mixing ratio fluctuations at 5 Hz for eddy covariance flux measurements. The lag times are approximately 16, 23, and 87 s (Berger et al. 2001). High-precision, 2-min mean CO₂ mixing ratios are sampled at six levels (11, 30, 76, 122, 244, and 396 m) by two LI-COR 6251 analyzers (Bakwin et al. 1998). Observations of net radiation, photosynthetically active radiation, and rainfall provide supporting meteorological data.

A National Center for Atmospheric Research (NCAR) Integrated Sounding System (ISS), which includes a radar profiler, a RASS, and a radiosonde system, was deployed about 8 km east of the tower from 15 March to 3 November 1998. The profiler is a sensitive 915-MHz Doppler radar that is designed to respond to fluctuations of the refractive index in clear air (Ecklund et al. 1988; White et al. 1991; Angevine et al. 1993, 1994a,c). The reflectivity measured by the profiler is related to the turbulence intensity, gradients of temperature and humidity, and particulates (Ottersten 1969; VanZandt et al. 1978; Wyngaard et al. 1980; White et al. 1991). The profiler can be used to measure the height of the mixed layer with a time resolution of 30 min or less, a vertical sampling of 60–100 m, a minimum height of 150 m, and a maximum height of 1500–3000 m depending on conditions (Angevine et al. 1994c). The RASS is an attachment to the profiler that measures temperature profiles up to a height of approximately 800 m above the ground by measuring the vertical propagation of an acoustic pulse (Angevine et al. 1994b). A detailed comparison of wind and temperature measurements from the tower and a similar profiler and RASS is given by Angevine et al. (1998). The ISS also includes a radiosonde system, and sondes were launched about once per week.

The depth of mixed layer can be derived from the signal-to-noise ratio (SNR) recorded by the profiler (Angevine et al. 1994c). The profiler SNR is related to the refractive index structure parameter, C_n², in clear air (White et al. 1991). This relationship is based on the assumptions that refractive index irregularities are in equilibrium with steady-state turbulence and that the radar wavelength lies in the inertial subrange of the turbulence (Ottersten 1969). Refractive index varies with both temperature and water vapor fluctuations (Wyngaard and LeMone 1980), however, in the mixed layer C_n² is dominated by water vapor. We extract the mixed layer depth, z_i, from the profiler SNR measurements by the method of Angevine et al. (1994c). First, we produce a contour plot of half-hour average range-corrected SNR from the profiler measurements for each day. Second, we define z_i as the median of the heights where SNR peaks occur over the period, as shown in Fig. 1.

Figure 2 shows the profile of potential temperature from a radiosonde launched at 1600 UTC 9 September 1998. UTC is 6 h ahead of local standard time. Here z_i is defined as the location of the sharpest change in potential temperature with height, which occurred at 940 m in this sounding. A 30-min average from the profiler at 1600 UTC gives an estimate for z_i of 850 m, in reasonable agreement with the radiosonde, which represents a point measurement. Turbulent fluctuations in z_i of ±200 m are common based on lidar observations of the convective PBL (Davis et al. 1997).

The comparison between measurements of z_i made by the radar profiler and from radiosondes launched during the deployment is shown in Fig. 3. The good agreement demonstrates that the z_i can be found accurately from the profiler SNR measurements. However, under unfavorable weather conditions such as precipitation or heavy clouds z_i cannot be estimated from the profiler SNR. Under these conditions the boundary layer is often not clearly defined (Stull 1988). In addition, the profiler is very sensitive to large cloud droplets and raindrops resulting in a high, relatively uniform SNR over the depth of the precipitation shaft.

Mixed layers shallower than 400 m, which typically occur in morning, are not well defined from the profiler SNR measurements. The CO₂ mixing ratio measurements from the tower (e.g., Fig. 4), however, can provide data when z_i is below 400 m. The top of the mixed layer is defined as the depth above ground to which the CO₂ mixing ratio is nearly constant provided that the net radiation is positive (warming the earth's surface).

Stable nocturnal boundary layers are more complicated than the daytime convective PBL. Mahrt et al. (1998) classified stable boundary layers into three different types: a very stable case with a thin, strongly stratified boundary layer; a deep, weakly stratified boundary layer; and an intermediate two-layer stratified boundary layer. It is possible to derive the height for the stable boundary layer from the tower CO₂ mixing ratio measurements because CO₂ is a very good indicator of the stratification. CO₂ released by microbial respiration at night builds up quickly in stable layers close to the ground, and the CO₂ mixing ratio is not altered by radiation like temperature or subject to saturation like water vapor. We define the top of the stable layer as the height at which CO₂ gradients first become very small. For example, as seen in Fig. 4, the heights of the stable layer are estimated to be 20.5, 53, and 183 m (i.e., half-way between adjacent measurement levels) during the periods of 0000–0300, 0400–1100, and 1200–1300 UTC, respectively. The stable layer, as defined here, typically grows over the course of the night as turbulent mixing from the earth's surface penetrates gradually upward through the stably stratified surface layer. This is consistent with the traditional view of the stable boundary layer (Stull 1988).

Horizontal advection may be important during the morning transition from stable to convective conditions (Yi et al. 2000) and could lead to erroneous identification of the stable layer top. However, for quantification of the depth of the stable layer we neglect cases when the virtual potential temperature flux is positive. As we will show, the CO₂ mixing ratio measurements at the tower allow us to estimate the depth of the stable layer for very stable and moderately stable (intermediate) conditions as defined by Mahrt et al. (1998) and Mahrt (1999), but not for the weakly stable conditions when the stable layer depth often exceeds 400 m. We refer to the very stable and intermediate cases collectively as the stable boundary layer.

Another feature that can be detected by the radar profiler is the top of the residual mixed layer from the previous day. The top of this residual layer is highlighted by the doted line in Fig. 1. A thin, strongly stratified stable layer also exists near the ground at 0100 UTC, clearly shown by the CO₂ mixing ratio profile (not shown for this day). The top of the residual layer was only observed under very clear and calm nighttime conditions, which typically occurred during periods of synoptic-scale subsidence. These conditions were encountered on three or four nights each month.

3. Results and discussion

a. Convective mixed layer

The monthly averaged diurnal cycles of z_i, net radiation, and sensible and latent heat fluxes are shown in Figs. 5 and 6. The maximum z_i occurs in May, corresponding with maximum sensible heat flux prior to full leaf-out, not maximum net radiation which occurs in July. The surface energy balance in July is maintained by a large latent heat flux due to transpiration. April is also characterized by deep, well-developed mixed layers due to generally large sensible heat fluxes. Here z_i depends on the time-integrated virtual potential temperature flux beginning after sunrise rather than on instantaneous virtual potential temperature flux.

In order to study the relationship between z_i and the cumulative virtual potential temperature flux we use the mixed layer model of Tennekes (1973):

View Expanded

where Θ_m is mixed layer mean virtual potential temperature, ΔΘ is the jump of Θ across the top of the mixed layer, () is virtual potential temperature flux, subscript s and i refer to the surface and z_i, respectively, γ denotes the lapse rate (= ∂Θ/∂z) above the top of the mixed layer, and w is the mean vertical velocity at z_i. The entrainment velocity is given by

w_edz_idtw(4)

Heating of the PBL air acts to decrease ΔΘ, while ΔΘ increases at the rate γw_e associated with entrainment. The subsidence effect was neglected in Tennekes' model because w is usually smaller than dz_i/dt and is difficult to observe directly. However, the terms associated with w are included here to give an understanding of how the subsiding motion affects mixed layer growth. The influence of shear on z_i is neglected. Mahrt and Lenschow (1976) found that shear contributions are important only for small z_i or weak stratification above z_i. Additional discussion regarding entrainment can be found in Tennekes (1973), Tennekes and Driedonks (1981), and Deardorff (1979).

If γ is taken to be constant, the solution of (1)–(3) is

View Expanded

where the subscript 0 refers to values at the time when ()_s changes sign from negative to positive. In order to determine z_i from (5) we reduce the unknowns by employing a simple relationship between the entrainment flux and the surface flux,

View Expanded

where c is an empirically determined constant. This linear relation is based on analysis of the turbulent kinetic energy budget. Although Zilitinkevich (1975) gave a more precise expression for c, we assume it to be constant here. Most of the published values of c lie between 0.1 and 0.3 (Stull 1976; Barr and Betts 1997; Davis et al. 1997) though some observations suggest values as large as 0.4–0.5 (Betts et al. 1992; Angevine et al. 1998). Combining (6) with (1)–(3) and assuming w = 0 yields

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Neglecting the terms in (7) related to Z_i0 is an excellent approximation because of the large powers of z_i. Thus, the relationship of ΔΘ to z_i becomes

View Expanded

It is also a good approximation to neglect the initial term on the right-hand side of (5) when z_i > 3Z_i0. Hence, substitution of (8) into (5) gives

View Expanded

Hence, z_i is proportional to the square root of the cumulative ()_s. However, we note that (9) is only valid during the period when the mixed layer is growing, (3) breaks down as w_e becomes zero (i.e., dz_i/dt = w). If subsidence is negligible (3) is valid only until the mixed layer reaches its maximum depth, otherwise, it is valid until the mixed layer reaches maximum depth and dz_i/dt becomes equal to w.

The last term in (9) is negative when w < 0 (subsidence). While w should be positive when the study site is in a low-pressure region, the top of the mixed layer is often difficult to define in this situation. Therefore, we focus here on a discussion of subsidence. In order to estimate the magnitude of the change in z_i caused by subsidence we assume w to be constant, which is reasonable over the course of a day since subsidence is a synoptic-scale phenomenon, and we drop the first term in (9) to obtain

z_isubsidencecwt.(10)

If we take w = −0.01 to −0.03 m s⁻¹, c = 0.2 and t = 8 h, then subsidence will cause the mixed layer depth to shrink by 350–1040 m. Therefore, a 20%–60% reduction of z_i could be caused by subsidence.

Figure 7 shows the relationship between z_i and

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along with a least squares linear fit to the data (r² = 0.98). Only data within the period when the mixed layer was growing were used. The linear fit is given by

z_iab(11)

where a = 97.1 m and b = 25.537 K^−1/2 m^1/2. This linear relationship between z_i and Γ is in good agreement with the theoretical prediction (9). The only approximations in (9) we have made are that γ and c are constants and the initial terms in (5) have been neglected. As seen from Fig. 5, 2 h is typically sufficient for the mixed layer to grow up more than 3Z_i0. A weak nonlinear relationship between z_i and Γ in Fig. 7 could result from a change in γ. The lapse rate can be estimated as γ ≈ 2(2c + 1)/b² in (K m⁻¹) by combining the linear term in (11) with (9) and by neglecting the subsidence term. If c = 0.2, then γ ≈ 4.3 K km⁻¹.

A more rigorous derivation of (10) suggested by Lanschow (2000, personal communication) can be found in the appendix. This derivation assumes constant flux divergence rather than constant w, which is a more reasonable assumption, especially when z_i is small.

A remaining question is: How can we determine, from the measurements of ()_s, when the mixed layer reaches its maximum depth? In general, the mixed layer does not stop growing as ()_s reaches its maximum value, but continues for some time depending on weather conditions and season. The frequency distribution of this time lag is shown in Fig. 8. Because of data selection, the weather conditions vary somewhat for the data in Fig. 8. Three hours is the dominant lag, and this case includes cloudy days in summertime and clear days in the months of March, April, and October. The weather conditions for 2-h lags are similar to 3-h lags. However, the weather is generally clear or partly cloudy on days when maximum z_i is reached 4 h after maximum ()_s. With clear skies in the summertime, the mixed layer can continue to grow for 5 or 6 h after the maximum ()_s since energy input is still substantial even with low sun angles. However, energy input often limits mixed layer growth during the spring and fall months, and during cloudy days in summer. We conclude that, to a reasonable approximation, z_i may be calculated integrating (11) to 2 h after ()_s reaches its maximum, after which the rate of growth slows significantly.

4. Concluding remarks

The depth of the mixed layer, z_i, has been derived by the combination of 915-MHz radar SNR and CO₂ mixing ratio measurements from a very tall tower. An empirical relationship between z_i and the square root of the cumulative surface virtual potential temperature flux,

View Expanded

is obtained (r² = 0.98) for this site. Insofar as the mixed layer model is valid, this result should apply to other locations. This model can be used to predict mean z_i during the period of the day when the mixed layer grows due to surface heating. There may be need for minor adjustment to the coefficients in (11) if the measuring height for ()_s is not at 30 m, as was used in this study.

The heights of nocturnal stable boundary layers were derived based on CO₂ mixing ratio measurements from the tall tower. The stable boundary layer heights typically increased over the course of the night. The weakly stable cases and windy nights were excluded because the height of boundary layer is greater than the tower height for those cases.

Subsidence has different influences on the evolution of the mixed, stable, and residual layers. Nighttime conditions when subsidence occurs generally have clear skies and strong radiative cooling that favor the development of a stable layer, trapping cold air near the ground. The divergence associated with subsidence suppresses growth of the stable layer somewhat. During the daytime, z_i depends on the competition between growth due to virtual potential temperature flux and reduction due to subsidence. The larger ()_s caused by the clear skies under conditions of subsidence favor increased growth of the mixed layer, but subsidence itself reduces z_i, as can be estimated by (10). We observed that a 10%–20% reduction in z_i could be caused by subsidence, based on comparison of Figs. 11 and 5, and excluding August and September. For August, a nearly 15% increase in z_i resulted from the fact that clear skies and drier air favored greater ()_s, and w was small (Table 1).

The residual layer was observed only on nights when the study site was under a synoptic high pressure system. We estimated the subsidence rate (w) as equal to the rate at which the residual layer top moved downward during the night (Table 1). Equation (10) should be valid for the residual layer as a result of (9) with ()_s = 0. We estimated the mean vertical velocities using (10), with c = 0.