1. Introduction
This paper evaluates the ability of four planetary boundary layer (PBL) schemes to replicate observed clear-air convective boundary layer (CBL) depth h in the Advanced Research version of the Weather Research and Forecasting Model (ARW-WRF), through developing a suite of objective criteria, evaluating them using model virtual potential temperature Θυ profiles, and finally comparing them to observations, for four days in the 1997 Cooperative Atmosphere–Surface Exchange Study (CASES-97; LeMone et al. 2000). The CBL depth is an important tool in evaluating model representation of PBL evolution. Similarly, CBL depth determines the amount of dilution expected for trace gases emitted from the surface or entrained into the CBL top (e.g., Tucker et al. 2009), and is thus a vital parameter for idealized air pollution models. CBL growth destroys the convective “cap,” fixing the time at which the lower-CBL temperature becomes warm enough to overcome negative buoyancy to form deep convective clouds, making h a significant factor in convective initiation (e.g., Trier et al. 2011). Finally, h is an important parameter within the PBL schemes themselves.
The CBL depth results from the interplay of surface virtual temperature flux, shear-generated turbulence, and subsidence, making its counterpart in WRF reflect more than just the PBL scheme. The roles of surface and CBL processes are embodied in the focus on both surface and CBL schemes in the local land–atmosphere coupling studies recently described by Santanello et al. (2009, 2011). Others (e.g., Hu et al. 2010; LeMone et al. 2010a; Shin and Hong 2011; Svensson et al. 2012) have factored in the virtual potential temperature flux bias as well as the PBL scheme(s) in question. Shear-generated turbulence is handled by PBL schemes, but surface momentum flux (friction velocity u*) is used as input. However, the effect of its bias is likely relatively small for the buoyancy-driven CBLs considered here; moreover, comparisons with observations are difficult, because u* is a function of scale (e.g., Kustas et al. 2005; Strassberg et al. 2008). Uncertainty in both modeled and measured subsidence makes it difficult to evaluate its role.
Previous evaluations show that PBL schemes that parameterize vertical turbulent fluxes as proportional to the corresponding local vertical gradient tend to underestimate CBL depth; while those allowing for “nonlocal” transport, which account for fluxes by eddies stretching through the CBL, yield deeper (and sometimes too deep) values. Weisman et al. (2008) and Kain et al. (2005) note that the Yonsei University (YSU; Hong et al. 2006) nonlocal PBL scheme grows the CBL more aggressively than the Mellor–Yamada–Janjic (MYJ) local scheme (Janjic 2001) over the central United States, leading to too much drying out in the former case. Trier et al. (2011) show a similar result, but note that altering the surface fluxes can lead to almost as large differences. These studies focus on vertical profiles. As pointed out by Shin and Hong (2011), comparisons using default model h values are made difficult by different definitions for different PBL schemes. Selection of a universal h criterion provides a better basis for comparison. Indeed, the h criterion used by Hu et al. (2010) actually improved agreement with observations for both MYJ and YSU, although h from MYJ remained too low.
Our comparisons use model results on a 1-km horizontal grid, with the ultimate goals of looking at the impacts of local changes in land surface characteristics on CBL growth and resolving mesoscale (~10 km) motions. Such runs produce realistic-looking CBL circulations (e.g., LeMone et al. 2010b) that, along with the small grid spacing, violate the “horizontal homogeneity” assumptions implicit in the schemes themselves. In addition, the circulations as well as the PBL schemes produce fluxes, leading to a “competition” that only approximately represents reality. These issues are consistent with such finescale runs inhabiting Wyngaard’s (2004) “terra incognita.” However, WRF runs with grid spacing of 4 km or less are commonly done because of their success in replicating hurricanes (e.g., Davis et al. 2008), precipitating convection (Trier et al. 2008, 2010; Weiss et al. 2008; Weisman et al. 2008), urban air pollution (Bao et al. 2008), the impact of the land surface on mesoscale circulations (e.g., Case et al. 2008), and convective initiation (Trier et al. 2004).
Our definitions of CBL depth h follow the illustration in Fig. 1. In the figure, h1 is the top of the “mixed,” near-constant virtual potential temperature Θυ layer, and h2 is the base of the free atmosphere, rarely penetrated by boundary layer air over a region at a given time. In the transition or entrainment layer, between h1 and h2, lies the boundary layer top zi, where the normalized virtual temperature flux, 
Idealized CBL profiles, indicating definitions of CBL depth used in this paper. Idealized instantaneous top of CBL sketched in to show range of values on the bottom-right frame. TKE profile in botton-left frame based on Moeng and Sullivan (1994, their Fig. 9).
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The bottom-left frame shows the (large eddy simulation) LES-based turbulence kinetic energy (TKE) or 
The profiles represent data averaged over time and space for LES or observations; with statistical uniformity and a sufficient sample, they approach the ideal “ensemble average” that the PBL schemes are designed to replicate. At any instant, h varies in time and space, as illustrated by the schematic in the lower right frame. The scale of these variations can be up to several kilometers in the presence of CBL structures like Benard-like cells or horizontal convective rolls (for illustrations showing time–space variability in convective CBLs see Weckwerth et al. 1997; Cohn et al. 1998; Cohn and Angevine 2000; Weckwerth et al. 2004; Miao et al. 2006; Bennett et al. 2010; Traumner et al. 2011). Such structures, mimicked in WRF (e.g., Trier et al. 2004; Miao and Chen 2008; LeMone et al. 2010b), also vary in time and space, so it is necessary either to allow for these irregularities in interpretation, or more ideally, use averaged data for comparisons.
Armed with these definitions, we evaluate and select the best criteria for h diagnosis, apply them, and compare the results to observations in the following sections. Observations and methods of analysis are described in section 2; model runs, PBL schemes, and analysis of model results are described in section 3. We evaluate several h criteria, describe the impact of the resolved CBL eddies, and describe horizontal averaging in section 4. After discussing factors other than CBL schemes that influence h in WRF, we compare modeled h values to observations and assess the impact of horizontal averaging and grid spacing in section 5. We discuss and summarize the conclusions in section 6.
2. Observations
Figure 2 shows locations of the CASES-97 instrumentation, which provide an excellent dataset for evaluating the ability of the ARW-WRF to replicate the fair-weather diurnal cycle. OXF (Oxford), BEA (Beaumont), and WHI (Whitewater) denote 915-MHz radar wind profilers (RWP), operated by Argonne National Laboratory (wind and signal-to-noise ratio data are available online at http://gonzalo.er.anl.gov/ABLE/data_archive/). On five different fair-weather days, radiosondes were released every 90 min over the diurnal cycle, from the RWP sites, affording estimates of PBL depth (data are available online at http://www.eol.ucar.edu/projects/cases97). Aircraft sampled boundary layer fluxes along the three tracks shown in the figure. Surface fluxes were sampled at 10 different locations with winter-wheat, grassland, or bare-ground cover (numbered from 1–10 in the figure; we use 30-min data available online at http://www.eol.ucar.edu/isf/projects/cases97/asciiDownload30min.jsp). The S-band dual-polarization Doppler radar (S-Pol) and Weather Surveillance Radar-1988 Doppler (WSR-88D) radars, located around 37.55°N, 97.5°W, documented the clear-air convective structure. More details on the observations can be found in LeMone et al. (2000, 2002), and Yates et al. (2001).
Instrumentation for CASES-97, superposed on terrain (contour interval = 20 m). Numbers indicate surface-flux sites. At the vertices of the triangle lie 915-MHz radar wind profiler sites BEA (Beaumont), OXF (Oxford), and WHI (Whitewater), with Radio Acoustic Sounding Systems and minisodars with collocated radiosonde sites. Solid lines indicate flight tracks; square outlines the location of primary interest.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Soundings at Beaumont, Kansas, are the focus of this paper because of the uniformity of nearby vegetation (grassland), the distance from any major urban area, and the quality of the dataset. Surface fluxes for this study are based on sites 1 and 2, since they are grassland sites near Beaumont. Because virtual temperature flux so strongly influences h, it is important to note that the surface sensible and latent heat fluxes H and LE based on eddy correlation might be underestimated (e.g., Twine et al. 2000). Assuming that the other two terms in the budget—net radiation and flux into the soil—were accurate, Chen et al. (2003) used data for the whole experiment to find the H + LE for sites 1 and 2 are equal to an average of 77% of what would be required for balancing the surface energy budget; and LeMone et al. (2002) combined the data from sites 1–8 for individual days, finding that H + LE were 91%, 94%, and 85% of the values required to close the budget on 29 April, 10 May, and 20 May, respectively.
Table 1 summarizes conditions for the four days analyzed. Thin high clouds were observed on two of the days and the PBL was too dry (mixing ratio Q ~ 5 g kg−1) for appreciable low cloudiness, so the net radiation had a sinusoidal “clear sky” evolution on all four days. Winds ranged from light to moderate (Table 1), and inversions varied in strength (Fig. 3). Sonic temperature fluxes, a close approximation to virtual temperature fluxes, were similar for the four days and in the buoyancy-driven range associated with Moeng and Sullivan’s (1994) TKE profile in Fig. 1.
CASES-97 weather conditions around Beaumont, Kansas. Numbers apply to 1830 UTC (solar noon). Winds based on RWP and radiosonde profiles; surface values based on averages of sites 1 and 2 for 1815 and 1845 UTC.
Observed Θυ profiles (red) at Beaumont 1830 UTC (near solar noon) for the four days used in this study, along with profiles based on BouLac (green), MYJ (turquoise), QNSE (blue), and YSU (purple). CBL depth based on radar wind profiler (hRWP) included.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The determination of CBL depth exploits the strength of the sensors used.
For RWP, CBL depth hRWP is set equal to the height of the center of the gate (60 m for BEA) just below the maximum signal-to-noise ratio dropoff with height (Coulter and Holdridge 1998), suggesting that the values thus obtained lie within the lower transition layer. The hRWP (wind) values are based on ~10 total 1-min (5 min) samples over 50 min. The time stamp for the wind data is 10 min past the hour (the first 10 min being used for the Radio Acoustic Sounding System profiles); the average time for the wind data is 35 min past the hour.
For radiosondes, h1 and h2 were determined subjectively from Θ profiles, and from the height range across which the radiosonde vertical velocity (wsonde), a function of environmental turbulence, drops from about 5 to 3 m s−1 (Johansson and Bergstrom 2005). The apparent transition layer in the Θ profiles probably reflects the impact of the history of mixing along the balloon’s trajectory. The influence of larger-scale vertical air velocity fluctuations (associated with terrain or larger CBL eddies) can lead to scatter in wsonde-based h estimates.
3. ARW-WRF runs
a. The simulations
ARW-WRF version 3.2 simulations were run for the four days in Table 1, using the PBL schemes described in Table 2. Each simulation was run for 24 h, starting at 1200 UTC (0600 LST), using four two-way interacting nested grids with spacing of 27, 9, 3, and 1 km, respectively (Fig. 4). The vertical grid has 44 sigma levels, with the lowest half model level just below 5 m, 15 levels below 1 km, 22 levels at or below 2 km, and the top level at about 16 km. The 1-km inner grid was selected to include significant local land surface variability and to resolve mesoscale eddies (10 km or larger). Initial and boundary conditions for WRF are based on the 3-h North American Regional Reanalysis (NARR, see online at http://dss.ucar.edu/pub/narr/) data on a 32-km grid.
PBL schemes used in this study. K = eddy exchange coefficient, Lmix = length scale, and ɛ = dissipation. All are fed 
Domains used in the simulations, with MODIS land-use map and CASES-97 observing sites for innermost domain (D4). Land-use category numbers correspond to Table 3. Category 10, grasslands, is most common over the CASES-97 array, with croplands (category 12, mainly winter wheat) impinging from the west. Number of grid points in x and y directions are for D1 (127 107), D2 (235 223), D3 (229 175), and D4 (304 283).
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
In addition to the four PBL schemes, the physical parameterizations used include the Noah land surface model (Chen and Dudhia 2001a,b; Ek et al. 2003), the Rapid Radiative Transfer Model (RRTM) longwave parameterization scheme (Mlawer et al. 1997), the Dudhia (1989) shortwave radiation scheme, and the Lin et al. (1983) bulk microphysics scheme. Surface characteristics are based on the Moderate Resolution Imaging Spectroradiometer (MODIS) VEGPARM Table version 3.1.1, with surface roughness values z0 modified as shown in Table 3. Land-use types for the inner grid (Fig. 4) show that the Beaumont site is surrounded by grassland, consistent with the land cover at nearby surface flux sites 1 and 2.
Changes from Default MODIS VEGPARM 3.1.1.
Additional model runs were done to assess the impact of grid spacing; and one run was conducted to evaluate the impact of recent changes in YSU, as described in Shin et al. (2012). The YSU run uses WRF/YSU version 3.3.1 with the same settings as the primary runs. The impact of horizontal grid spacing was tested using sets of runs for 10 May with the outer three grids (27-, 9-, and 3-km spacing) but with no feedback from finer to coarser grids. The impact of vertical grid spacing was explored with one run using the MYJ scheme for 4–5 May with everything the same but 28 vertical levels.
b. The PBL schemes (for references see Table 2)
For convenience, we will refer to the three PBL schemes that use (1) as “TKE schemes.”
From Table 2, the scale Lmix for BouLac is determined from the Θυ profile using a parcel technique, while Lmix for QNSE and MYJ is intimately tied to e and CBL depth.
Each of the schemes is provided surface energy fluxes and u* from the Noah land surface model.
c. Analysis of model results
Hourly surface sensible and latent heat fluxes (H and LE) and vertical profiles of the wind components, temperature, mixing ratio, pressure, and the turbulence variables associated with the TKE schemes (TKE, Lmix, and KH) were extracted from the WRF runs for the grid square corresponding to Beaumont. Also, maps of W just above the PBL were extracted to assess impacts of subsidence.
The CBL profiles were objectively analyzed to determine h using the criteria in Table 4, moving upward from the lowest model level to the first height at which each criterion is met. There are three types: (i) those matching the Θυ at h to some near-surface level (criteria 1–3), (ii) those based on vertical gradients, including one (criterion 4) inspired by a method commonly used to subjectively identify the top of the well-mixed layer, and two based on Ri (criteria 4–7), and (iii) those based on a TKE threshold (criterion 8). Criterion 5 identifies the inflection point in the inversion layer [i.e., the lowest height at which Θυ,z reaches a maximum (Θυ,z)z = 0].
Criteria used to identify PBL depth.
We use (5) to evaluate Ri. Based on Vogelezang and Holtslag (1996), a correction of 
To filter out the effect of resolved CBL eddies on CBL profiles and hence h, 9- (25) point average profiles and surface data were computed from the 9 (25) grid squares forming a 3- (5) km square centered at Beaumont.
4. Evaluation of CBL criteria using modeled profiles
a. Evaluation of CBL depth criteria at a point
To evaluate the h criteria, we plotted the objectively determined CBL depths on the corresponding model Θυ profiles, as illustrated in Fig. 5. Then, using the definitions of Fig. 1, we sorted the objective estimates into categories (<h1 or “too low,” ~h1, ~zi, ~h2, >h2 or “too high,” or “erratic”), the last applying to a lack of consistency with the Θυ profiles. For example, a criterion that corresponds to h1 at one time, too low, at another time, and too high a third time, would be labeled erratic. Evaluations were done for all the 1-km data (all four schemes, all four days), and then checked by asking volunteers experienced in boundary layer meteorology to select the criteria that seemed to identify h most successfully. A second check was done by repeating the evaluation using the series of profiles for the 3- and 9-km grid runs. Since the model Θυ profiles often only had an unambiguous change in static stability at the top of the mixed layer, h1 was far easier to identify than h2 or zi. Plots of TKE profiles were included to understand the behavior of the TKE criterion.
Evaluation of CBL height criteria for MYJ at Beaumont, 4 May 1997. (top) Series of Θυ profiles with h based on the eight criteria indicated. (bottom) Corresponding profiles of TKE. Times are in UTC (CST + 6 h); Θυ (TKE) after 1400 UTC is incremented by 3 K (1 m2 s−2) each hour to separate profiles.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The results, shown in Fig. 6, show that the most consistent and successful criteria are based on the Θυ profile. The most successful criterion, which defines h as the height at which Θυ,z = 2 K km−1 (criterion 4) is consistent with mixed layer top h1 for all four schemes and all four days. This is not surprising: using the base of the inversion to estimate h is a common practice, as illustrated by the references in Table 4. Matching Θυ at h to level 1 (criterion 1) provides h values that vary between zi and h2, for QNSE and YSU, making it difficult to use as a universal estimate, with too high estimates based on this criterion for two days for MYJ. However, this criterion yields consistent estimates of h2 for all four days for BouLac. Unfortunately, this criterion is dependent on the height of the first grid level and therefore the vertical grid spacing. This problem is addressed by using Θυ at 0.05 h (criterion 3), but the h values, which vary between h1 and zi, are a function of PBL scheme and day. Moreover, the iterative process required to obtain h does not always converge, and h based on this criterion is too low on two days for the YSU scheme.
Evaluation of CBL-depth criteria based on one-point data.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The maximum-Θυ,z criterion 5 shows up as “high” primarily because of what appear to be too-high values in the afternoon; often (but not always) the morning values look like good estimates of zi. Figure 5 shows such behavior. Using MYJ with 28 vertical levels reveals h estimates that are uniformly too high (not shown). The layer Richardson number criterion 6 varied from around zi to too high, depending on the day and the PBL scheme, even though the threshold selected is lower than the 0.3 value used in the Community Climate System Model version 2 (Zeng et al. 2004).
There are two criteria labeled erratic: those based on thresholds of Riloc and TKE. The Riloc = 0.5 criterion 7 identified reliable h values for only two of the 16 cases. Figure 5 illustrates this erratic behavior. For much of the day, the h estimates based on this criterion look good; but the diagnosed CBL is far too shallow at 2000 and 2200 UTC. This follows from small vertical shear (not shown) and slightly stable stratification in the middle to upper CBL at 2000 and 2200 UTC. Similarly, the table shows mostly erratic behavior for the TKE = 0.2 m2 s−2 criterion 8. In Fig. 5, TKE = 0.2 m2 s−2 does a reasonable job until 1900 UTC, after which h appears to be underestimated at 2000 UTC (exaggerating the reduction in CBL depth at this time), and overestimated at 2100 UTC.
b. The impact of resolved CBL eddies
The PBL schemes, representing ensemble averages, assume negligible horizontal variation and hence should produce smooth CBL growth in fair-weather conditions. Instead, even the “successful” criteria can produce irregular growth at a given point. If we apply the 2 K km−1 criterion, one of the smoothest-varying of the eight criteria in Fig. 5, to other days and schemes, we find in Fig. 7 that “smooth” growth is rarely simulated. This to some degree mimics observations: our Fig. 1 and Fig. 5 in LeMone et al. (2002) suggest that departures of about 20%–25% from a smooth curve might be expected at a point, with larger fractional deviations relative to the mean CBL depth during the midmorning, when the CBL is growing rapidly (see also Bennett et al. 2010; Traumner et al. 2011). However, the hour-to-hour variability in h based on TKE and Riloc often tends to be much larger and thus looks unrealistic.
For the four days, simulated growth of the CBL for h1 equal to the height at which Θυ,z = 2 K km−1, for 1-km grid square corresponding to Beaumont.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The irregular growth of the CBL in Fig. 7 and the qualitative association of shallow TKE profiles with shallow and more stable CBL Θυ profiles in Fig. 5 suggest the influence of resolved CBL structures, which have repeatedly been documented in fine-grid WRF runs (e.g., Trier et al. 2004; Miao and Chen 2008; LeMone et al. 2010b). Their presence is confirmed in Fig. 8, which shows resolved vertical velocity W on the scale of a few kilometers in the mid-PBL not only for 4 May, but the other three days as well. Rampanelli et al. (2004) attribute them to the model responding to the unstable CBL, much as described in classic linear convection theory (e.g., Asai 1970).
For MYJ, mid-CBL (level 11, ~540 m) vertical velocity for the four days. The distance between the vertices of the profiler triangle is ~60 km.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Figure 9 shows the impact of W for the three TKE schemes, keeping in mind that the effects described reflect past time steps as well as the present.
For BouLac, MYJ, and QNSE, the relationship of vertical velocity W to TKE, Lmix, K or KH, and static stability Θυ,z, based on adjacent times with opposite W; W < 0 (dashed lines); W > 0 (solid lines, with vertical grid points superposed on top plot). Insets included for Θυ,z plots to emphasize stability changes. At 1-km grid square corresponding to Beaumont.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
For BouLac, W fortuitously switches from negative at 1800 UTC to positive at 1900 UTC, a half hour before and after solar noon, allowing us to compare two times with roughly equal surface virtual temperature flux. The large change in Lmix is related to a change in CBL depth (reflected in Θυ,z), because the air parcel energy approach (Table 2) typically makes Lmix equal to the vertical distance from the closer edge (top or bottom) of the CBL as defined by the Θυ profile. It follows from (1), (3), (4), and the Θυ profile, that TKE and K would be shallow as well. Here, we neglect the impact of shear, noting that its effect on the TKE budget is relatively small (e.g., Moeng and Sullivan 1994) for these buoyancy-driven CBLs (Table 1). From (4), the TKE and Lmix profiles reinforce one another to produce larger K for W > 0.
For MYJ, the winds are lighter than for the BouLac case (cf. 4 and 10 May in Table 1). Thus, TKE is even more strongly related to buoyancy production via (1), and hence, via (3), to Θυ,z, which is negative to 1200 m for W > 0 but positive above about 200 m for W < 0. The master length scale Lmix is related to both TKE and h (Table 2), both of which are suppressed for W < 0, leading [from (4)] to an extremely stunted KH profile.
For QNSE, there are again opposite W for the hours straddling solar noon in a buoyancy-driven CBL (Table 1). In this case, smaller W changes produce smaller but perceptible differences in TKE, Lmix, and KH. At 1800 UTC, W < 0 compresses and stabilizes the CBL, but only leads to slightly less TKE production than for W > 0.
The degree of W influence varies with the scheme. For MYJ, the complex relationship among TKE, Lmix and h makes KH extremely sensitive to static stability, and hence, to vertical velocity, which can result in wild fluctuations. Fortunately, these average out; for example, a large single-point KH value at 2100 UTC of ~2500 m2 s−1 on 10 May is hardly reflected in the corresponding 9-point horizontal average (335 m2 s−1) centered at Beaumont. Smaller sensitivity for QNSE is likely related to the smoothly varying functions of Ri used to obtain KM,H from their neutral values.
For YSU, the impact of resolved CBL eddies is relatively weak, since hourly W magnitudes do not exceed 20 cm s−1 at Beaumont. Moreover, KH is related to surface fluxes, the lowest model level, and hYSU via (6). For W < 0, the impacts on the Θυ profile are restricted to reducing h, slightly stabilizing the entrainment zone, and whatever feedback the compression of the superadiabatic layer has on the lowest model level and its interaction with the surface, with the reverse for W > 0.
c. Horizontal averaging to filter out CBL eddy impact
To mitigate the impact of resolved CBL eddies on h, we averaged the vertical profiles for the nine 1-km grid squares centered on Beaumont for all four days, and the 25 1-km grid squares centered on Beaumont for 10 May, the day that appeared to have the most irregular behavior. As seen in LeMone et al. (2010b), the W field for 10 May, illustrated in Fig. 10 and Fig. 8, is the most cellular of the four days, consistent with the weak winds (Table 1). Note the previously mentioned comparatively small W for YSU. This results from a more near-neutral CBL, which in turn reflects nonlocal fluxes; and, in the YSU version 3.2 used here, also an increase in heat flux out of the surface layer related to an adjustment of the Prandtl number.
As in Fig. 8, but for all four PBL schemes on 10 May.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Nine-point averaging smoothes CBL growth as defined by the top of the mixed layer (or where Θυ,z = 2 K km−1), as illustrated for MYJ in Fig. 11; however, noticeable temporal changes in stratification are not eliminated without 25-point averaging. Similarly, the TKE profiles require 25-point averaging to show reasonably smooth growth, particularly in the afternoon. Note that, after horizontal averaging, the TKE profiles above 100 m AGL more closely resemble their counterparts in Fig. 1 in both shape and depth, with TKE falling to free-atmosphere values above h1. BouLac and QNSE respond similarly to horizontal averaging. For all three PBL schemes, 25-point averaging reduces W to ~0.1 m s−1 or less.
For 1-km inner grid, comparison of 25-point (solid line) and 9-point (dotted line) averaged profiles of Θυ, TKE, and W, centered at Beaumont to point values at Beaumont (dashed line), for MYJ 10 May. In the figure, Θυ, TKE, and W are incremented by an additional 3 K, 1 m2 s−2, and 1 m s−1, respectively, for each hour after 1500 UTC.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Evaluations of objective h estimates based on 9-point-averaged TKE profiles appear in Fig. 12, along with a comparison of evaluations for 1-, 9-, and 25-point averages for 10 May. From Fig. 6, TKE-based h values are mostly erratic for single-point values; Fig. 12 shows only BouLac producing consistent h values on all four days with 9-point averaging. With 25-point averaging, however, diagnosed h values are more consistent with the Θυ profiles (i.e., fewer erratic judgments) for all three schemes. The 0.2 m2 s−2 threshold produces good h1 values for MYJ, while 0.101 m2 s−2 produces good h1 values for BouLac, and good zi values for QNSE. A lower threshold for BouLac (0.001 m2 s−2) also yielded zi estimates. In the table, it should be noted that too high reflects values that are just a bit over h2 at times.
Evaluation of TKE-based CBL depth as a function of horizontal averaging across 9 and 25 grid points forming square centered at Beaumont. “Free atmosphere” values of TKE for BouLac, MYJ, and QNSE are respectively 0.0001, 0.1, and 0.005 m2 s−2, respectively.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
5. Comparison to observations
a. Impact of confounding factors
In comparing model to observed CBL heights, factors other than the PBL schemes need to be considered. Among these are surface buoyancy fluxes, subsidence, horizontal advection, and Θυ stratification above the CBL. We discuss the first three here and consider the last factor in evaluation of the daily comparisons.
From LeMone et al. (2010a), a bias in surface buoyancy flux produces an h bias approximately equal to the square root of the ratio of the total modeled to total observed buoyancy flux since sunrise. The averages of the square roots of the ratios for the present cases (Table 5) range from 1.09 on 29 April to 1.11 for 10 May to 1.15 and 1.18 for 4 and 20 May, respectively. From the foregoing, these model overestimates of surface buoyancy flux could be artificial, offset by a likely underestimate in observed values. Moreover, allowing for hour-to-hour variability in the observations, the observed and modeled buoyancy flux curves have similar shapes. Thus, the modeled and observed fluxes are sufficiently close that we need not adjust for model surface-flux magnitude or phase differences.
Comparison of observed and modeled surface virtual temperature flux integrated from 1200 to 1850 UTC, 
The potential effect of Wsyn was evaluated by estimating its magnitude a number of ways using data for 10 May. We examined time series of four gridpoint vertical averages of W just above CBL top (1.5–2.1 km AGL), for the four PBL schemes, using the 1-km-grid 9-point averages, and the 3- and 9-km single-point values at Beaumont for the runs with no feedback from smaller scales. These showed all four schemes converging toward the same time series with the same time-averaged value (~−1 cm s−1) as the influence of the resolved CBL eddies decreased. This result, though not surprising, confirms that differences between modeled CBL depth resulting from use of different PBL schemes are not caused by differences in Wsyn. This value was confirmed by plotting maps of vertical velocity just above maximum CBL top, at level 22 (~1900 m).
It is beyond our capability, however, to estimate impact of Wsyn on comparisons of modeled to observed h, because rigorous estimates of Wsyn to the needed accuracy (<0.01 m s−1) are not available for WRF, and it is difficult to estimate subsidence from observations. For 10 May, above-CBL temperature budgets, sink rates for above-CBL features in the 90-min soundings during the day, and sink rates of the top of the RWP high signal-to-noise ratio field in the late afternoon, revealed Wsyn ~ 0.02–0.03 m s−1 at around 2 km AGL, consistent with NARR, but slightly greater over Beaumont than for the WRF simulations. Perhaps because of stronger winds and the associated impact of horizontal advection on sounding features, budgets and feature sink rates did not yield consistent Wsyn estimates at Beaumont for the other days.
The impact of horizontal advection is through resolved CBL eddies and larger-scale gradients. The impact of resolved CBL eddies is mitigated through time averaging (in the case of the RWP data) and spatial averaging (for model profiles). The impact of larger-scale gradients is likely small on these four fair-weather days. LeMone et al. (2000, their Fig. 13) shows small and fairly consistent h differences of the order of 100–200 m over 60 km for 4 May, 20 May, and 10 May; and LeMone et al. (2010a, their Fig. 9) shows a similar changes across the same region in 2002.
b. Comparisons for 1-km grid runs
For 4 May, Fig. 13 compares MYJ h based on the eight criteria to observations. From Fig. 6, the successful criteria (i.e., those correctly identifying h on model Θυ profiles) are (3) Θυ(h) = Θυ(0.05h) and (4) Θυ,z(h) = 2 K km−1; both reflect the h underestimate associated with MYJ. Figure 6 shows that h values based on criteria (1) [Θυ(h) = Θυ(Level 1)] and (2) [Θυ(h) = Θυ(Level 1) + 1 K] are too high for this day, so the apparent good matches in the figure through 2100 UTC are misleading. Similarly, the too-high max-Θυ,z and Ri(Level 1 − h) criteria lead to apparent good agreement with observations through 2100 UTC for the wrong reasons. The nine-point mean smoothes out the CBL growth curves, but not completely. As in the case of 10 May (Fig. 11), the nine-point smoothed Riloc- and TKE-based h curves are still quite irregular in the afternoon.
For MYJ on 4 May 1997 at Beaumont, comparison of observed h to values from MYJ, using the eight criteria of Fig. 6. For MYJ (black triangles), single-point values are represented by thin black lines and averages for the nine 1-km grid squares forming a 3-km square centered at Beaumont are represented by thick black lines. For observations, hRWP is represented by gray squares/lines. For radiosondes, downward-pointing triangles represent h1 and upward-pointing triangles represent h2, with solid gray lines for h based on Θ and dashed black lines for h based on sonde vertical velocity.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Figure 14 compares observed to model h using Θυ,z(h1) = 2 K km−1, based on average profiles for the nine 1-km grid squares centered at Beaumont. From Fig. 6, this criterion is a successful indicator of h1 for all four PBL schemes on all four days. As expected, BouLac, MYJ, and QNSE mostly underestimate h1, while YSU is closer to observed values on 29 April, 4 May, and 10 May. For 20 May, all the PBL schemes cross or intercept the observed h1 line, with YSU showing a high bias from 1700–1900 UTC, all but BouLac high at 1800 UTC, and the TKE schemes low for other parts of the day. Despite the smoothing, CBL growth retains some of the irregularity seen in Figs. 5, 11, and 13 for the TKE schemes, with the impact of resolved CBL eddies most evident in the afternoon. The midafternoon decrease in h1 for all four schemes on 4 May, when the sounding temperatures suggest a still-growing PBL, might result from errors in modeled subsidence. Alternatively, since wsonde-based h1 values appear comparable or higher than those based on Θ (perhaps due to local eddies lofting the balloon), the drop of wsonde-based h1 to below that from Θ at 2300 UTC could indicate that observed Θ profiles are starting to fail as a measure of CBL depth.
For Θυ,z = 2 K km−1 (criterion 4), h1 for the four schemes, for Θυ profiles at Beaumont, based on 9-point averages of 1-km grid profiles, as in Fig. 13. For observations, symbols as in Fig. 13 except for colors, with observed h based on sonde Θ in red; and h based on sonde vertical velocity in tan. For model: BouLac = Green; MYJ = turquoise; QNSE = blue; YSU = purple.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
As a consistency check, modeled and observed Θυ profiles at solar noon (1830 UTC) on the four days are compared in Fig. 3. There are clearly differences in Θυ magnitudes and vertical gradients above the CBL that likely reflect model initial conditions different from what was actually observed, especially for 29 April and 20 May, when both differ. Taking time-averaged hRWP as more representative than the instantaneous h1 based on the radiosonde Θυ profile, the TKE schemes underestimate the CBL depth for the two better-reproduced profiles on 4 and 10 May, and on 29 April, but they overestimate hRWP for 20 May. The YSU scheme replicates the CBL depth on 4 May, is slightly lower than hRWP for 29 April and 10 May, and overestimates hRWP on 20 May.
Figure 15 compares objectively determined model h to observations on all four days, for Θυ,z(h1) = 2 K km−1. The data are combined for the four days by first normalizing the h1 time series for each day by dividing the hourly model h1 on that day by the corresponding hRWP, then averaging the resulting time series, and finally multiplying the result by the corresponding 4-day-averaged hourly hRWP values. Though it includes data for 20 May, Fig. 15 shows that the TKE schemes produce too low a value, with YSU the closest to observations. As in Fig. 14, all PBL schemes appear too low starting at 2000 UTC.
Composite h1 based on Θυ,z = 2 K km−1, for all days and PBL schemes, with colors/symbols as in Fig. 14. To obtain the model curves, single-point model h1 is divided by the corresponding observed RWP h, averaged, and then multiplied by the four-day average RWP h. For comparison, average sonde-Θ-based h1 and h2 are plotted as well.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
With 25-point smoothing, TKE-based CBL depths increase more smoothly and become more reliable h indicators. Figure 16 shows the impact of the 25-point averages on h for the most successful TKE thresholds of Fig. 12 and compares them to h based on Θυ,z = 2 K km−1. The h curves based on 25-point-averaged TKE mostly correspond to the maximum height estimates based on the less-smoothed profiles. Furthermore, the 25-point TKE-based h curves are reasonably consistent with the three closely corresponding h1 curves based on the Θυ,z criterion.
CBL depths as a function of time, based on (left) Θυ,z = 2 K km−1 and (right) TKE, for single-point (dashed line), 9-point (dotted line), and 25-point (thick solid line) averages, for 1-km grids centered on Beaumont. Observations (gray symbols) included for comparison: from sonde Θ: triangle pointed up: h2, triangle pointed down h1, square: hRWP.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
c. Comparisons to results from coarser resolution
Figure 17 compares average Θυ, TKE, and W profiles for the runs with 3-km grids, 9-km grids, and horizontally averaged 1-km grids (25-point averages for the TKE schemes, 9-point averages for YSU). The averages are constructed from the four profiles centered on solar noon (1700, 1800, 1900, and 2000 UTC), by normalizing the heights z for each profile by its depth h1 (based on Θυ,z = 2 K km−1), interpolating the data to correspond to normalized heights at 0.02z/h1 intervals, averaging the four points at each normalized height, and then multiplying the heights by the corresponding time-averaged h1 value.
Impact of horizontal grid spacing on vertical profiles of Θυ, TKE, and W for 9-km (thick solid line), 3-km (dotted line), and 1-km grid spacing (25-pt average, dashed line) over Beaumont on 10 May, based on average of 4-hourly profiles centered on local solar noon, as described in text. Profiles of Θυ are offset by 1 K.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
The figure reveals interesting grid-scale-related similarities and differences in terms of stability, TKE, and the impact of resolved eddies. First, the 3-km grid still has resolved CBL eddies for all three TKE schemes, as revealed by the mean W profiles, with YSU values remaining small; while such eddies at 9 km are quite weak. As in the case of Fig. 9 for single-point 1-km data, W > 0 at 3 km probably accounts for the Θυ profiles being more unstable than for the 25-point averages of the 1-km grid data. Second, the 9-km grid Θυ profiles are more unstable than their counterparts based on the 25-point averaged 1-km data, for all three TKE schemes. This likely follows from the near absence of resolved CBL eddies, which at 1-km grid spacing transport heat from the lower CBL and make the average profile more near neutral. Apparently the parameterized fluxes are not sufficient make up for the lack of resolved eddies, a likely result of the lack of nonlocal fluxes in (3). Finally, the YSU Θυ profiles remain near neutral for all three grid spacings despite the lack of strong resolved CBL eddies, a result of including nonlocal fluxes in (3).
The behavior of the TKE profiles is consistent with the foregoing. At 3 km, larger TKE is consistent with W > 0 and more unstable Θυ profiles, just as at 1 km (Fig. 9). At 9 km, the larger parameterized TKE is consistent with the more unstable Θυ profiles, with the result being that the TKE scheme creates more subgrid-scale turbulence to make up for the absence of the resolved eddies.
Based on the criteria deemed useful for this day, Fig. 18 and Tables 6 and 7 mostly show small impact of horizontal grid spacing on h for 10 May, particularly given other sources of differences: 1) The 9-km grid point has an elevation about 30 m lower than the other grids. This offset increases the height relative to the ground for Θυ features above the CBL. Near the ground, there should not be much difference, and at intermediate heights, the offset should be between 0 and 30 m. 2) There is a ~3-km horizontal offset in the center of the grid squares, with the 9-km grid square the outlier, which seems to have minor impact. 3) Assigning the vertical gradient to midway between the Θυ grid points makes the exact position of h depend on the strength of the gradients above and below, which is typically less than the vertical grid spacing (of order 100 m at these heights, as shown in Fig. 3). 4) The h values in the tables are influenced by the method of finding average h (Tables 6 and 7). The 3-km grid values are not included in the tables because they are influenced by resolved eddies.
PBL depths based on Θυ,z = 2 K km−1 are fairly consistent for the three TKE schemes in Fig. 18 and Table 6. There are slight differences for YSU and BouLac, but they are less than the vertical grid spacing (Fig. 3).
TKE-based CBL depths are larger for the 9-km grid than the 1-km grid average for MYJ (h1) and QNSE (zi) in Fig. 18 and Table 7, but slightly smaller for BouLac. For the 3-km grid point, TKE-based CBL depth evolves more irregularly with time due to CBL eddies, especially for BouLac in the figure, with comparatively large depths between 1700 and 2000 UTC resulting from updrafts (vertical maxima 0.2–0.4 m s−1), and an anomalously shallow depth at 2200 UTC associated with a downdraft (vertical maximum 0.4 m s−1). It is encouraging, however, that the drop of TKE to background in Fig. 17 is not sensitive to grid spacing; and that the height at which this occurs is higher than h1, consistent with LES (see Fig. 1).
PBL depths based on level 1 Θυ values for the 9-km grid point have increases of the order of the elevation offset in Table 6. For YSU, zi is slightly deeper at coarser resolution but still less than the vertical grid spacing.
PBL depths based on 0.05-h Θυ values are larger for 9 km than for 1 km in Table 6. This follows from the greater CBL instability at 9 km, as illustrated by the vertical lines on the BouLac Θυ profiles in Fig. 17. This suggests that the level-1 criterion would be sensitive to horizontal grid spacing at coarser vertical grid spacing. We did not evaluate this criterion for YSU, since it was deemed too low for this day.
As in Fig. 16, but for 9-km grid (1 point, thick solid line), 3-km grid (1 point, dotted line), and 1-point grid (25 points, dashed line), centered on Beaumont.
Citation: Monthly Weather Review 141, 1; 10.1175/MWR-D-12-00106.1
Change of PBL height h with horizontal grid spacing based on Θυ profiles at 1700, 1800, 1900, and 2000 UTC, for Beaumont, 10 May.
For Beaumont, 10 May, average of PBL heights (m AGL) based on TKE profiles at 1700, 1800, 1900, and 2000 UTC.
6. Discussion and conclusions
Several criteria for determining CBL depth in WRF were evaluated against modeled Θυ profiles and then used to compare WRF-derived h from the BouLac, MYJ, QNSE, and YSU PBL schemes to observations on four different fair-weather days in CASES-97. Comparisons of h based on successful criteria to observations are consistent the well-documented low h bias for the TKE schemes, with more accurate to too-high h from the YSU scheme (Kain et al. 2005; Weisman et al. 2008; Hu et al. 2010). For BouLac, MYJ, and QNSE, the presence of resolved CBL eddies led to unrealistically irregular growth of h during the day, rendering some of the criteria useless without smoothing; their absence led to more unstable CBLs at coarser resolution, thus deepening the CBL according to some criteria. Furthermore, several of the criteria used were sensitive to PBL scheme and grid spacing either in the horizontal or vertical, suggesting that testing of selected criteria using model profiles is important once a model setup has been defined. Readers should be cautioned that the h metric should be used along with CBL profiles in PBL scheme evaluation, and that none of the comparisons discussed include low clouds or precipitation.
With these caveats in mind, we found the following for the present simulations. Depths are defined as in Fig. 1, with h1 the top of the well-mixed layer as defined by Θ or Θυ, h2 the top of the inversion layer, with zi somewhere in between.
The criterion Θυ,z(h) = 2 K km−1 is the most successful. The resulting CBL depth h1 varies more smoothly with time than those involving Riloc or TKE for the 1-km data even without averaging, and produces a close match to modeled mixed layer top h1 for all four schemes on all four days. Also, this criterion was relatively insensitive to grid spacing. For comparison, use of Θυ,z = 3 K km−1 (as in Liu and Liang 2010) yielded plausible values of h1 to zi, suggesting that Θυ,z = 4 K km−1 (as in Zeng et al. 2004) would be closer to zi for the present model runs.
Use of TKE at 1-km horizontal grid spacing is successful only after averaging the TKE profiles over a 5 km × 5 km square centered on Beaumont. Not surprisingly, different thresholds were required for different PBL schemes, with 0.2 m2 s−2 yielding good estimates of h1 for MYJ, 0.101 m2 s−2 yielding good estimates of h1 for BouLac, and 0.101 m2 s−2 yielding good zi estimates for QNSE. The TKE-based h estimates changed for BouLac, MYJ, and QNSE with horizontal grid spacing; however, the drop in TKE to “background” was relatively insensitive. Moreover, this background height is greater than h1, consistent with LES (Fig. 1).
The level-1 criteria suffer from a sensitivity to vertical grid spacing: with the dense vertical grid used here, the closeness of level 1 to the surface led to h estimates that ranged from zi to “too high.” The Θυ(level 1) + 1 criterion always produced too-high h values.
The 0.05h-criterion was designed to remove sensitivity to vertical grid spacing, but the iterative procedure used does not always converge, and the values thus obtained were “too low” for the YSU scheme on 2 days. Finally, h based on this criterion is larger at 9-km horizontal grid spacing than for the horizontally-averaged 1-km grid profiles using the TKE schemes, since the absence of CBL eddies for the larger grid spacing leads to a more superadiabatic lapse rate. It follows that a level-1 criterion with a coarse vertical grid (or a level-3 or -4 criterion with the grid used here) would run into the same problem.
The criterion Ri(Level 1 − h) = 0.2 yielded h values that tended to be too high. We tried Ri(Level 1 − h) = 0.3, the threshold used in the Community Climate System Model Version 2 (CCSM V2; Zeng et al. 2004). As expected, the resulting h values were even higher than the “too high” values using 0.2, suggesting a smaller threshold is appropriate. Given successful use of 0.3 in CCSM V2 (Zeng et a. 2004), our high bias probably reflects a sensitivity to vertical grid spacing.
For the 1-km grid, the criterion Riloc(h) = 0.5 is strongly dependent on the PBL scheme. For example, the greater CBL static stability for YSU sometimes translated into too-shallow a CBL using this criterion, and changes in stratification associated with W led to unreliable identification of depth for the TKE schemes; a larger threshold could help solve this problem. However, the increase in instability at coarser horizontal grids for the TKE schemes would also have an effect.
The maximum Θυ,z criterion did not work as well as hoped, often yielding h estimates that were too high. We would anticipate more success at higher vertical resolution, since increasing the number of vertical grid points from 28 to 44 improved its behavior.
Though the resolved eddies can look quite realistic, with patterns changing from cellular to linear with increasing winds, as pointed out in LeMone et al. (2010b), other aspects are artificial. For example, the Prandtl number modification in YSU version 3.2 that increases heat transport out of the surface layer led to more near-neutral conditions, much weaker resolved CBL eddies, and more linear convection with weak winds, compared to the other schemes here and to YSU version 2.1.2 in LeMone et al. even though the boundary layers are buoyancy driven in all cases. Moreover, stronger (but still linear) resolved CBL eddies appeared in a test run for 10 May with YSU/WRF version 3.3.1, which has been modified to reduce excess mixing as described in Shin et al. (2012). Also, not only do the TKE schemes fail to represent nonlocal transport, but they do not allow vertical TKE transport by the resolved flow, as observed in nature (e.g., LeMone 1976); none of the schemes accounts for horizontal turbulence transport.
While h is a useful metric for evaluating CBL schemes, profiles should also be considered. Comparing results from runs using WRF/YSU version 3.3.1 to the present runs indicated similar values of h [nearly identical using the Θυ(level 1)-criterion]. However, as pointed out by Shin et al. (2012), less-aggressive vertical mixing leads to more realistic vertical profiles.
The lack of clouds limits the conclusions of this study. However, some of the criteria can be easily generalized. For example, TKE would drop abruptly at cloud top. Also, one could replace Θυ in the Θυ,z criterion with the liquid-water potential temperature ΘL ≈ Θ − LQL/cp, where L is the latent heat of condensation, QL is the liquid-water mixing ratio, and cp is the specific heat at constant pressure. However, smoothing could be problematic in the presence of clouds. Indeed, as pointed out in LeMone et al. (2010b), the resolved CBL structures produce cloud fields that resemble observations. This, plus a potential role for such structures in convective initiation (e.g., Trier et al. 2004) and propagation (Trier et al. 1991) makes one hesitant to smooth them out for such applications.
Acknowledgments
We wish to acknowledge Chin-Hoh Moeng (NCAR), Jeff Weil (Cooperative Institute for Research in the Environmental Sciences, University of Colorado/NCAR), Jason Ching (Institute for the Environment, University of North Carolina), and Tim Glotfelty (Marine, Earth and Atmospheric Sciences, North Carolina State University), for poring over series of virtual potential temperature profiles to identify the objective criteria that best represented boundary layer depth. We also want to thank Prof. Julie Lundquist (University of Colorado, Boulder) for helpful suggestions. The comments by Hyeyum Hailey Shin (Yonsei University, South Korea) and the two anonymous reviewers provided useful input that removed errors and improved the paper. Song-You Hong (Yonsei University), Zavisa Janjic (NCEP), and Alberto Martelli (CIEMAT, Spain) were responsive to email requests, clarifying points about YSU, MYJ, and BouLac, respectively. Bonnie Slagel (NCAR) helped with some of the figures. Finally, we would like to acknowledge the NCAR Water System and STEP programs and the U.S. Air Force Weather Agency for their support.
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