A Large-Eddy Simulation Study of Moist Convection Initiation over Heterogeneous Surface Fluxes

Song-Lak Kang Atmospheric Science Group, Texas Tech University, Lubbock, Texas, and National Center for Atmospheric Research, Boulder, Colorado

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George H. Bryan National Center for Atmospheric Research, Boulder, Colorado

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Abstract

This study uses large-eddy simulations to investigate processes of moist convection initiation (CI) over heterogeneous surface fluxes. Surface energy balance is imposed via a 180° phase lag of the surface moisture flux (relative to the sensible heat flux), such that the relatively warm surface is relatively dry (and the relatively cool surface is relatively wet). As shown in previous simulations, a mesoscale circulation forms in the presence of surface-flux heterogeneity, which coexists with turbulent fluctuations. The mesoscale convergence zone of this circulation develops over the relatively warm surface, and this is where clouds first form. Convection initiation occurs sooner as the amplitude of the heterogeneity increases, and as the surface moisture increases (i.e., Bowen ratio decreases). Shallow clouds initiate when boundary layer heights (zi) become greater than the lifting condensation level (LCL). Deep precipitating clouds initiate when the LCL and level of free convection (LFC) are roughly the same when averaged over the relatively warm surface, which is equivalent to the mean convective inhibition (CIN) becoming nearly zero. From the perspective of the entire (mesoscale) domain, cases with strongly heterogeneous surfaces have a wider distribution of both zi and LCL. Thus, a comparison of zi with LCL over a mesoscale area (i.e., within one mesoscale model grid box) may lead to misleading conclusions about CI and cloud-base height. It is also shown that as the amplitude of the surface-flux heterogeneity increases the mesoscale convergence zone becomes narrower and stronger. Furthermore, CI occurs earlier over relatively wet surfaces partly because turbulent eddies are more vigorous owing to slightly greater buoyancy.

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

Corresponding author address: Dr. Song-Lak Kang, Atmospheric Science Group, Department of Geosciences, Texas Tech University, Box 42101, Lubbock, TX 79409-2101. E-mail: song-lak.kang@ttu.edu

Abstract

This study uses large-eddy simulations to investigate processes of moist convection initiation (CI) over heterogeneous surface fluxes. Surface energy balance is imposed via a 180° phase lag of the surface moisture flux (relative to the sensible heat flux), such that the relatively warm surface is relatively dry (and the relatively cool surface is relatively wet). As shown in previous simulations, a mesoscale circulation forms in the presence of surface-flux heterogeneity, which coexists with turbulent fluctuations. The mesoscale convergence zone of this circulation develops over the relatively warm surface, and this is where clouds first form. Convection initiation occurs sooner as the amplitude of the heterogeneity increases, and as the surface moisture increases (i.e., Bowen ratio decreases). Shallow clouds initiate when boundary layer heights (zi) become greater than the lifting condensation level (LCL). Deep precipitating clouds initiate when the LCL and level of free convection (LFC) are roughly the same when averaged over the relatively warm surface, which is equivalent to the mean convective inhibition (CIN) becoming nearly zero. From the perspective of the entire (mesoscale) domain, cases with strongly heterogeneous surfaces have a wider distribution of both zi and LCL. Thus, a comparison of zi with LCL over a mesoscale area (i.e., within one mesoscale model grid box) may lead to misleading conclusions about CI and cloud-base height. It is also shown that as the amplitude of the surface-flux heterogeneity increases the mesoscale convergence zone becomes narrower and stronger. Furthermore, CI occurs earlier over relatively wet surfaces partly because turbulent eddies are more vigorous owing to slightly greater buoyancy.

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

Corresponding author address: Dr. Song-Lak Kang, Atmospheric Science Group, Department of Geosciences, Texas Tech University, Box 42101, Lubbock, TX 79409-2101. E-mail: song-lak.kang@ttu.edu

1. Introduction

Numerous studies have been performed to predict the location and timing of cloud formation and precipitation occurrence. However, accurately forecasting moist convection is still one of the most demanding and challenging issues in operational meteorology. With the advancement of measurement techniques, many detailed processes involved in moist convection have been illuminated recently (e.g., Fabry 2006; Wilson and Roberts 2006; Weckwerth et al. 2008). Also, with increased computer resources, various physical processes associated with moist convection can be investigated with a grid spacing that is fine enough to resolve energy-containing turbulent eddies (e.g., Bryan et al. 2003).

A failure in forecasting initiation of moist convection (CI) may result in an overall forecast error for convective-system structure and precipitation occurrence. There are many reasons why forecasts of CI may be incorrect, including errors in initial conditions, model errors (particularly for physical parameterizations), etc. This article focuses on small-scale structures and processes in the convective boundary layer (CBL). The CBL, by definition, is directly influenced by the earth’s surface, and the fluxes of heat from the surface are key to determining where and when CI first occurs. For example, in the CBL, clouds can form when warm moist low-level thermals penetrate into the capping inversion, the statically stable layer that caps the CBL. Thus, the intensity of the capping inversion can be a critical factor for determining where and when CI occurs (e.g., Mahrt 1977; Mahrt and Pierce 1980; Kalthoff et al. 2009). When moisture supply is limited in the CBL (e.g., over a surface characterized by large Bowen ratio), deep moist convection may still initiate under a strong capping inversion condition; several studies have shown that the strong inversion inhibits widespread development of cumulus cloud and can focus deep, moist convection in a local region (e.g., Mahrt 1977; Mahrt and Pierce 1980; Doswell and Bosart 2001).

In addition to the strength of the capping inversion, thermodynamic properties in the low-level CBL are closely related to CI. For example, Crook (1996) numerically examined the sensitivity of thunderstorm initiation to thermodynamic parameters of the CBL. He found that seemingly insignificant variation in temperature of 1 K and in moisture of 1 g kg−1 in the CBL could make a difference between deep convection and no convection in numerical simulations (using horizontal grid spacing of 2.5 km). The horizontal variability on the order of 1 K for temperature and 1 g kg−1 for water vapor mixing ratio has been observed in CBLs that include mesoscale fluctuations on scales larger than turbulent fluctuations (e.g., Weckwerth et al. 1996; Taylor et al. 2007).

Multiscale fluctuations of thermodynamic parameters have been reported in several studies (e.g., Trier et al. 2004; Fabry 2006; Weckwerth et al. 2008). Fabry (2006) analyzed data from radar, surface stations, soundings, and airborne in situ sensors during the International H2O Project (IHOP_2002). He found that on a scale of order 100 km, CI location is particularly sensitive to the variability in surface temperature; but, on a scale of order 10 km, CI location is extremely sensitive to the strength of upward vertical motions of convective scales in the boundary layer. He suggested that upward vertical motions magnify the effect of small-scale moisture variability and that moisture variability contributes similarly to the final erosion of convection inhibition (CIN). Weckwerth et al. (2008) also analyzed three-dimensional time-varying moisture fields collected by various instruments on 12 June during IHOP_2002. They suggested that mesoscale boundaries (e.g., drylines and outflow boundaries) enhance low-level moisture advection and convergence in the CI region, while persistent upward motions on the turbulence scale trigger convection. Mesoscale model-based studies also suggested that multiscale variations in land surface conditions could affect CI processes. For example, Trier et al. (2004) used a coupled mesoscale atmosphere–surface model and found that more surface sensible heat flux can result in a region of enhanced CBL height on a horizontal scale of order 100 km, and in the region with enhanced CBL height there can be organized circulations on a scale of order 10 km that directly initiate deep moist convection.

Spatial scales involved in CBL processes vary depending on thermodynamic and kinematic parameters in the boundary layer. In particular, horizontal scales of moisture fluctuations are likely different from those of temperature and vertical velocity. Couvreux et al. (2005) analyzed water vapor data collected by airborne water vapor differential-absorption lidar in a growing CBL over land during IHOP_2002; they found that the characteristic length scale of water vapor mixing ratio is somewhat larger than that of temperature and vertical velocity. This observation seems to be consistent with the idealized numerical simulation by Jonker et al. (1999); their large-eddy simulation (LES) of the boundary layer demonstrated that dynamically active scalars such as potential temperature have fluctuations of the order of the boundary layer depth (roughly 1 km) and remains this scale through 10 h of integration. However, passive scalars such as water vapor mixing ratio acquire mesoscale fluctuations even though the moisture field was horizontally homogeneous at the initial time. Earlier, Mahrt (1991) observed that amplitudes of 10-km-scale moisture fluctuations were an order of magnitude larger than amplitudes of turbulent fluctuations of temperature and vertical velocity in aircraft data, which were collected at 100–150 m above ground level (AGL). Similar results were found by LeMone and Meitin (1984). In Mahrt’s aircraft data, however, 10-km-scale fluctuations of temperature and vertical velocity are weaker than 1-km-scale turbulent fluctuations.

Spatial variations of surface heat and moisture fluxes are a significant and persistent cause of mesoscale fluctuations in the CBL. In particular on a scale of tens of kilometers or larger, surface heat and moisture fluxes satisfy surface energy balance (SEB) with net radiation and ground heat conduction (Mahrt 1991). The presence of mesoscale circulations induced by SEB heterogeneity is a common cause for the negative correlation between temperature and moisture on a scale of tens of kilometers or larger (e.g., Mahrt 1991; Mahrt et al. 1994; LeMone et al. 2002; Kang et al. 2007; Taylor et al. 2007). In addition to the negative correlation between temperature and moisture, mesoscale circulations produce near-surface convergence zones, which seem to be necessary to initiate deep moist convection (e.g., Hane et al. 1997; Gero et al. 2006; Kalthoff et al. 2009), at least in the absence of significant orography.

In this study we use large-eddy simulations to investigate the sensitivity of CI to the amplitudes of SEB heterogeneity on a scale of tens of kilometers under a zero background wind with a strong capping inversion of the CBL. We use a simple idealized framework with specified surface fluxes to obtain insight into the processes involved in CI. We focus primarily on the processes leading to CI over heterogeneous surface conditions, although in future studies we plan to investigate further the detailed CBL structures and processes, which might lead to better CBL parameterization for mesoscale models. The numerical model and numerical experiment setup are introduced in section 2. In section 3, we examine the processes involved in CI over the different SEB heterogeneity. In section 4, we decompose LES variables into mesoscale and turbulence components and analyze mesoscale and turbulence-scale structure. In section 5, we summarize the primary conclusions of this study.

2. Numerical experiments

We use the compressible nonhydrostatic numerical model of Bryan and Fritsch (2002). This model integrates the Navier–Stokes equations using third-order Runge–Kutta time differencing and fifth-order spatial derivatives for the advection terms following Wicker and Skamarock (2002). To minimize spurious oscillations in scalar concentrations, we use a weighted essentially nonoscillatory (WENO) advection scheme for scalar transport. The turbulence kinetic energy scheme of Deardorff (1980) is employed for a subfilter-scale (SFS) model.

The model domain and the prescription of surface heterogeneity are similar to Kang and Davis (2008) and Kang (2009). The model domain is 32 km in the x direction and 8 km in the y direction. The horizontal grid spacing is 100 m. The vertical grid spacing is 40 m up to 4000 m AGL, then linearly increases from 40 to 120 m between 4000 and 6000 m AGL, and then remains constant at 120 m up to the domain top of 10 800 m. The upper boundary is a flat, rigid wall with a Rayleigh damping layer (Durran and Klemp 1983) occupying 2.8 km beneath the model top. The lower boundary is also a flat, rigid surface. The surface momentum flux is derived from a simple surface drag parameterization (e.g., Stull 1988).

The mesoscale surface forcing is prescribed by sensible and latent heat fluxes that vary only in the x direction. We simulate the CBL in the middle of the day [~(1000–1400) local time (LT)] and to simplify analysis, we keep the surface heat and moisture fluxes fixed in time, assuming insignificant diurnal change in surface fluxes between late morning and early afternoon. The surface sensible heat flux variation is sinusoidal with mean value 〈F〉, amplitude A, and wavelength λ:
e1
Given the constraint of SEB, the surface moisture flux variation is prescribed with a 180° phase lag relative to the surface heat flux:
e2
(i.e., the sensible and latent heat fluxes are in antiphase). In this study, the SEB constraint means constant available energy H + LE, where H is the sensible heat flux and LE is the latent heat flux. Considering also net radiation Rnet and heat flux G into the ground, total SEB can be expressed as Rnet - G = H + LE, and thus Rnet - G is also assumed to be constant in this study. Above these spatial variations of H and LE (i.e., in the surface layer) temperature and moisture have negative correlation: the relatively warm surface is relatively dry and the relatively cool surface is relatively wet. These types of spatial variability of near-surface temperature and moisture can also be created by other atmospheric conditions, such as reduced vertical mixing on the cool side of a remnant outflow boundary, although these more complex processes are not studied herein.

For all simulations, the wavelength λ in (1) and (2) is chosen to be 32 km, considering that SEB is satisfied on a scale of tens of kilometers or larger (Mahrt 1991). The simulations performed with various values of 〈Fwθ〉, 〈Fwr〉, Awθ, and Awr are summarized in Table 1. In all simulations, 〈Fwθ〉 is set to 0.1950 K m s−1 [ρCpF〉 = 240 W m−2 where ρ is air density and Cp is specific heat at constant pressure]. In half of the simulations 〈Fwr〉 is 0.1325 g kg−1 m s−1 [ρLυFwr〉 = 400 W m−2 where Lυ is latent heat of vaporization], and in the other half of the simulations it is 0.0664 g kg−1 m s−1 (200 W m−2). We refer to the simulations with Awθ = Awr = 200 W m−2 as the strongly heterogeneous surface, and simulations with A = Awr = 50 W m−2 as the weakly heterogeneous surface. We also run simulations with A = Awr = 0 W m−2, which we refer to as homogeneous surfaces.

Table 1.

Experimental design parameters for the prescribed surface sensible heat and latent heat flux variation as defined in (1) and (2). Here ρ is air density, CP is specific heat at constant pressure, and Lυ is latent heat of vaporization.

Table 1.

For cases A200B06, A050B06, and A000B06, the domain-averaged Bowen ratio, ρCpFwθ〉/ρLυFwr〉, is 0.6. In these cases, the domain average of surface sensible heat flux is 240 W m−2 and the domain average of surface latent heat flux is 400 W m−2 (and thus total heat flux is 640 W m−2). We also run a set of simulations with the same distribution of sensible heat flux, but a lower domain-average latent heat flux of 200 W m−2 (and thus total heat flux is 440 W m−2). These cases—A200B12, A050B12, and A000B12—represent relatively drier surfaces that have a domain-averaged Bowen ratio of 1.2. The purpose of these drier-surface simulations is to test the generality of our results under different conditions, such as might occur in a different season, or perhaps in a different part of the world. Findell and Eltahir (2003) show that different mean surface conditions can affect the likelihood of CI.

Figure 1 shows the initial vertical profiles of temperature and moisture. This sounding is modified from Weisman and Klemp (1982). We include a CBL with a strong capping inversion and prescribe a somewhat drier free atmosphere than was originally used by Weisman and Klemp (1982). From the surface to 738 m AGL, the initial sounding has a potential temperature θ of 297.5 K and a water vapor mixing ratio rυ of 10.8 g kg−1. The initial capping inversion layer has a θ lapse rate of 3.5 K (24 m)−1 and an rυ lapse rate of 3 g kg−1 (24 m)−1 between 738 and 762 m AGL. In the initial sounding, the lifting condensation level (LCL) is 870 hPa (about 1.2 km AGL) and the level of free convection (LFC) is 690 hPa (about 3.1 km AGL). To initiate three-dimensional turbulent flows, random perturbations of 0.1 K are superimposed on the θ field at the lowest model level at the initial time.

Fig. 1.
Fig. 1.

Skew T–logp diagram of the temperature (solid) and moisture (dashed) profiles used as the initial condition for the numerical experiments. CAPE is 230 J kg−1, CIN is 170 J kg−1, the CBL height is 810 m, and the LCL is 1.2 km.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

The initial CBL height zi is 750 m. Throughout this paper, zi is determined by the level at which the vertical gradient of potential temperature is maximum [referred to as the “gradient method” by Sullivan et al. (1998)]. Also based on Sullivan et al. (1998), the potential temperature jump in the entrainment zone may be considered as a strong capping inversion that prevents large-scale folding of the interface between the CBL and the free atmosphere.

The lateral boundary conditions are periodic in both horizontal directions. All simulations are integrated for 6 h. However, our analysis focuses mostly on the 30-min period before moist convective initiation (roughly 2–4 h). For all the cases, the integration time step is 1 s and model output is saved every 100 s. The Coriolis acceleration and radiative tendencies (in the atmosphere) are neglected for simplicity and because of the short integration time. We use the Kessler (1969) microphysics scheme that considers only liquid water processes; ice microphysics can be reasonably neglected for the shallow clouds studied here, and because we focus on convective initiation (before formation of cold pools).

The warm patch is defined as the region where the sensible heat flux is above the domain average and the latent heat flux is below the domain average. The x dimension of the warm patch is x = −16 to 0 (Fig. 2). The cool patch is defined as the region where the sensible heat flux is below the domain average and the latent heat flux is above the domain average. The x dimension of the cool patch is x = 0 to +16 (Fig. 2). For analysis purposes, seven subdomains are considered, each with horizontal dimensions of 4 km × 8 km. As shown in Fig. 2, subdomains WL, WC, and WR are over the warm patch; subdomains CL, CC, and CR are over the cool patch; and subdomain MD is the middle region.

Fig. 2.
Fig. 2.

The one-dimensional sinusoidal surface sensible heat flux variation (scale on left) and surface latent heat flux variation (scale on right) prescribed for case A200B06. The available energy (net radiation − heat flux into the surface) is 640 W m−2. The domain-averaged surface sensible and latent heat fluxes are 240 and 400 W m−2, respectively (thin solid line). The x dimensions of the seven subdomains—WL, WC, WR, MD, CL, CC, and CR—are marked, which are defined starting from the subregion of x = −14 to −10 (and y = −4 to 4) ending to the subregion of x = 10 to 14 (and y = −4 to 4).

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

A summary of all specified surface fluxes is shown in Fig. 3. Although the total surface available energy ρcpFwθ + ρLυFwr is constant everywhere, the surface buoyancy flux (which drives generation of turbulent kinetic energy) is not constant. Specifically, varies sinusoidally, with maximum in the warm patch and minimum in the cool patch (Figs. 3d–f). Thus, a mesoscale circulation develops over the heterogeneous surfaces. The circulation has the same overall structure as that studied by Kang and Davis (2008), with ascent in the middle of the warm patch, descent in the middle of the cool patch, horizontal winds from the cool patch to the warm patch near the surface, and horizontal winds from the warm patch to the cool patch near the top of the CBL.

Fig. 3.
Fig. 3.

The distributions of (a)–(c) surface flux, where H is sensible heat flux and LE is latent heat flux; and (d)–(f) surface buoyancy flux.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

Some previous modeling studies of CBLs over mesoscale surface heterogeneity (e.g., Letzel and Raasch 2003; Kang and Davis 2008; Kang 2009) found that strongly heterogeneous surfaces can induce temporally fluctuating horizontal flows. Here, a similar temporal fluctuation occurs in some cases after ~3 h of integration. However, its onset is always after CI, and thus we do not analyze the temporal fluctuation in this study. Readers are referred to Kang and Davis (2008) and Kang (2009) for further analysis of the mesoscale circulation and its temporal fluctuation.

3. Moist convection initiation

In this section we focus on the development of clouds. We use the term “moist convection” to refer to clouds, whether or not they produce precipitation. We remind readers that the resolution used here is fine enough to resolve turbulent eddies (i.e., “dry convection”) in the CBL.

a. Timing of CI

Figure 4 presents the evolution of total liquid water mixing ratio that is integrated over the y and z directions. For all the heterogeneous cases (Figs. 4a,b,d,e), moist convection forms first over the center of the warm patch (the WC subdomain), which is consistent with some previous numerical model studies (e.g., Yan and Anthes 1988; Avissar and Liu 1996; Chagnon et al. 2004; van Heerwaarden and de Arellano 2008). In the horizontally homogeneous CBLs (Figs. 4c,f), moist convection is randomly scattered throughout the domain.

Fig. 4.
Fig. 4.

Time evolutions of cloud water (colors) and rainwater (contours) mixing ratios for cases (a) A200B06, (b) A050B06, (c) A000B06, (d) A200B12, (e) A050B12, and (f) A000B12. The values of the mixing ratios are summed over the y and z directions. The contour interval for rainwater mixing ratio is 1.0 × 103 g kg−1.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

Some previous studies found that moist convection initiates earlier over heterogeneous surfaces than homogeneous surfaces (e.g., Yan and Anthes 1988; Ek and Mahrt 1994; Avissar and Liu 1996; Chagnon et al. 2004; van Heerwaarden and de Arellano 2008). Consistent with these studies, we find that CI is earlier with the heterogeneous surfaces, and that CI occurs sooner as the amplitude of the SEB heterogeneity increases. For example, in case A200B06 (Fig. 4a), CI occurs at ~120 min over the center of the warm patch, which is about 60 min earlier than the weakly heterogeneous case (A050B06; Fig. 4b), and about 100 min earlier than the homogeneous case (A000B06; Fig. 4c).

For cases that have the same amplitude of flux heterogeneity, CI occurs earlier with the relatively wet surface. For example, considering the weakly heterogeneous simulations, the relatively wet surface case (A050B06; Fig. 4b) has CI at ~180 min, which is 50 min earlier than the relatively dry surface case (A050B12; Fig. 4e). So, in this case, relatively wetter surfaces produce moist convection more quickly. This result is expected, as the wet surfaces have greater surface available energy (cf. 640 W m−2 for the moist surfaces and 440 W m−2 for the dry surfaces). However, we note that the timing of CI is not solely a function of surface moisture; for example, clouds form in the weakly heterogeneous dry case (Fig. 4e) at roughly the same time as the homogeneous moist case (Fig. 4c), demonstrating that surface heterogeneity can be as important to CI as surface moisture. Nevertheless, the significant role of surface evaporation in CI has been suggested by many previous studies (e.g., Betts and Ball 1998; Clark et al. 2004; Alonge et al. 2007).

To identify the transition from shallow (nonprecipitating) convection into deep precipitating convection, we measure the time when moist convection grows to reach the height of 6 km by using cloud water and rainwater mixing ratios that are summed along the y direction. Within the numerical integration period of 6 h, both strongly heterogeneous surface cases (A200B06 and A200B12; Figs. 5a,d) and the weakly heterogeneous wet surface case (A050B06; Fig. 5b) develop deep precipitating convection. It takes less time to develop precipitating convection over the strongly heterogeneous surfaces than over the weakly heterogeneous surfaces, and over the wet surface than over the dry surface. Specifically, for the strongly heterogeneous wet surface case, moist convection grows to reach the height of 6 km at 230 min (A200B06; Fig. 5a), which is 45 min earlier than the strongly heterogeneous dry surface case (A200B12; Fig. 5d) and 115 min earlier than the weakly heterogeneous wet surface case (A050B06; Fig. 5b).

Fig. 5.
Fig. 5.

Vertical cross sections of cloud water (gray shades) and rainwater (contours) mixing ratio (a) at 230 min for case A200B06, (b) at 345 min for case A050B06, (c) at 360 min for case A000B06, (d) at 275 min for case A200B12, (e) at 360 min for case A050B12, and (f) at 360 min for case A000B12. The values of cloud water and rainwater mixing ratios are summed along the y direction. The contour interval for rainwater mixing ratio is 8 g kg−1 starting at 1 g kg−1.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

For the relatively dry surface cases, the amplitude of the flux heterogeneity seems to be a more critical factor for the development of precipitating convection. For case A200B12 (Fig. 5d), moist convection grows to reach the height of 6 km at 275 min. However, for case A050B12 (Fig. 5e), moist convection does not reach the height of 6 km within the 6-h integration period.

For the horizontally homogeneous CBLs, neither the relatively wet surface case (A000B06) nor the relatively dry surface case (A000B12) develops deep moist convection, although there are some subtle differences between these two cases. For example, for case A000B06 (Fig. 5c) moist convection reaches up to about 3 km AGL, but cloud formation barely occurs for case A000B12 (Fig. 5f).

b. Mechanisms for CI: CBL height and LCL

The comparison of CBL height zi with the LCL has been used previously to diagnose shallow cumulus onset time and cloud-base height (e.g., Wilde et al. 1985; Haiden 1997; Stull and Eloranta 1984; Craven et al. 2002; Siqueira et al. 2009). Here we note that all our simulations start with the same horizontally homogeneous atmosphere that has zi = 810 m and LCL = 1.2 km (Fig. 1), but with different surface conditions. At the time of initial cloud formation, we compute zi and LCL for every model grid point for all the cases (Fig. 6a). The LCL is computed using the method of Lawrence (2005) with pressure, temperature, and water vapor mixing ratio at the height of 140 m AGL (i.e., just above the surface layer).

Fig. 6.
Fig. 6.

Box-and-whisker plots of the CBL height (zi; dashed whiskers) and the LCL (solid whiskers) computed for each section at 120 min (a) for all cases and (b) for case A200B06. The box indicates the 75th and 25th quartile, and the symbol × within the box denotes the median value. The whiskers indicate the maximum and minimum values. Locations of subdomains are shown in Fig. 2.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

First, we compare CBL heights with LCLs over the whole domain. In Fig. 6a, we show box-and-whisker plots. For the strongly heterogeneous surface cases (A200B06 and A200B12), both zi and LCL have broader distributions than all the other cases. Also, minimum values of LCL are generally lower for these strongly heterogeneous surfaces. These results (specifically the higher values of zi and lower values of LCL) are consistent with development of more clouds for the strongly heterogeneous surfaces.

Given that the median value of zi is slightly lower for the strongly heterogeneous cases compared to the weakly heterogeneous surfaces (cf. A200 cases with A050 cases in Fig. 6a), one might expect moist convection to form at a lower level over the strongly heterogeneous surfaces. However, the comparison of zi with LCL for each subdomain of one case (A200B06; Fig. 6b) shows the opposite: moist convection actually forms at a higher level as surface heterogeneity increases. For the case in Fig. 6b, both zi and LCL tend to be higher over the warm patch, particularly over the subdomain WC.

The lower LCLs over the cool patch (Fig. 6b) are associated with less warming and more moistening of the surface layer. However, over the cool patch the local LCLs are still higher (on average) than the local CBL heights, and thus clouds do not develop over the cool patch. Clouds actually occur over the center of the warm patch (the WC subdomain) where most CBL heights are higher than the LCL. Thus, over surface-flux heterogeneity that induces mesoscale circulations, the comparison of mean values of zi and LCL over a mesoscale area (or within a mesoscale grid box) may not work well for estimating cloud-base heights. In fact, it has been reported that LCLs estimated with surface layer parameters are consistently lower than observed cloud-base heights (e.g., Stull and Eloranta 1984; Craven et al. 2002).

c. Mechanisms for CI: CIN and vertical velocity

CIN has been suggested as one of the most important factors for predicting convective development (e.g., Colby 1984; Crook 1996; Ziegler and Rasmussen 1998). We analyze CIN values also from two different perspectives: the whole domain and for each subdomain. CIN is computed for each grid point by using the method of Emanuel (1994) with the values of pressure, temperature, and water vapor mixing ratio at 140 m AGL at 120 min. Initially, CIN has a horizontally homogeneous value of 170 J kg−1, which is reduced to 0 J kg−1 over some portion of the domain (Fig. 7).

Fig. 7.
Fig. 7.

Box-and-whisker plots of the CIN (solid whiskers) and w (dotted whiskers) at 120 min for the various cases. CIN is estimated by using the method of Emanuel (1994) with the values of pressure, temperature, and water vapor mixing ratio at the height of 140 m AGL. The box indicates the 75th and 25th quartile, and the symbol × within the box denotes the median value. The whiskers indicate the maximum and minimum values.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

From the perspective of a mesoscale area, average CIN could be deficient as a method for forecasting CI because CIN values that are low enough to initiate moist convection may occur on only a small portion of the domain. For the strongly heterogeneous surface cases (cases beginning with A200), CIN distribution is somewhat broader than for the other cases. Furthermore, the relatively wet surface cases (cases ending with B06) have generally smaller CIN values than the relatively dry surface cases (ending with B12). Thus, based only on the CIN distributions shown in Fig. 7, one might forecast CI to occur earlier over the weakly heterogeneous wet surface (A050B06) than over the strongly heterogeneous dry surface (A200B12) because median CIN is lower. However, as presented in Fig. 4, CI actually occurs earlier over the strongly heterogeneous dry case (A200B12) than over the weakly heterogeneous moist case (A050B06).

Considering now distributions over the subdomains, we show in Fig. 8 the box-and-whisker plots at 120 min for the two cases. For case A200B12 (Fig. 8a), the median value of CIN over the warm patch (~40 J kg−1) is much less than the median value over the cool patch (~100 J kg−1). Furthermore, the minimum CIN is near 0 over the warm patch, but is >50 J kg−1 over the cool patch. Thus, CI occurs in a focused region (the WC subdomain) without much reduction of CIN in the surrounding areas. In comparison, for case A050B06 (Fig. 8b), the median values of CIN are comparable across all subdomains. The amplitude of the mesoscale circulation, which is analyzed further in section 4, is ultimately responsible for the mesoscale variation in CIN for these simulations.

Fig. 8.
Fig. 8.

As in Fig. 7, but for each subdomain at 120 min for cases (a) A200B12 and (b) A050B06. The subdomains are defined in Fig. 2.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

Like in many observational works (e.g., Arnott et al. 2006; Weiss et al. 2006; Weckwerth et al. 2008), maximum vertical velocity w is on the order of 1 m s−1 before CI, which cannot overcome the CIN value of 170 J kg−1 of the initial sounding; if using , then vertical velocity would need to exceed 18 m s−1. Hence, reduction of CIN is critical for CI. At 120 min, minimum CIN is reduced to near 0 J kg−1 in several cases (especially over the WC subdomain), and maximum vertical velocity increases to 6 m s−1 for all cases (Fig. 7), suggesting (incorrectly) that CI would occur for all cases. However, distributions over the subdomains must again be considered to explain these results. For example, for case A200B12 (Fig. 8a), over the center of the warm patch (the WC subdomain) the maximum vertical velocity is about 8 m s−1, which can nearly overcome the median CIN value of ~40 J kg−1. In contrast, over the center of the cool patch (the CC subdomain) the maximum vertical velocity is just 2 m s−1, which is not able to overcome the median CIN value of 100 J kg−1. For the weakly heterogeneous surface cases (e.g., case A050B06; Fig. 8b), the distributions of vertical velocity are not much different between the subdomains and median CIN has not been reduced enough for CI to occur. These results reinforce the conclusion that mesoscale averages can be misleading when diagnosing CI.

d. Mechanisms for CI: LFC

The processes involved in CI are summarized in Fig. 9, which shows zi, LCL, and LFC averaged over the WC subdomain for the time period leading up to CI. We use slightly different definitions of CI for this analysis because we now focus on the WC subdomain (where CI first occurs): CI of shallow (nonprecipitating) clouds is determined by the first occurrence of cloud water >1 g kg−1 averaged over the WC subdomain, and CI of deep (precipitating) clouds is determined by the first occurrence of average rainwater >1 g kg−1. For all cases, zi increases in time before the formation of deep convection. Convection initiation of shallow clouds occurs when zi becomes greater than the LCL (as discussed earlier). One might expect that CI of deep clouds occurs when zi exceeds the LFC, but the results in Fig. 9 show this is not the case. Deep convection actually occurs when the LFC and LCL become nearly the same; we note that CIN is zero when the LCL and LFC are the same.

Fig. 9.
Fig. 9.

Time series of LFC (green), LCL (red), and zi (black) averaged over subdomain WC for (a) A200B06 and (b) A050B06. The times of CI for shallow clouds and deep (precipitating) clouds are indicated by arrows.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

The rapid decrease of LFC between 120 and 150 min for the strongly heterogeneous case (Fig. 9a) is notable, especially because this does not occur in the weakly heterogeneous case (Fig. 9b). The same difference occurs in the relatively dry surface cases (not shown). This rapid decrease in LFC is attributable to the strong mesoscale ascent over the WC subdomain (associated with the mesoscale circulation), which cools and moistens the stable layer just above the CBL. The structure of this mesoscale circulation is analyzed further in the next section.

4. ABL structures and processes

a. Fourier spectra

This section looks more closely at the multiscale structure of temperature and moisture in the CBL before CI. We begin by analyzing spectra of potential temperature θ and water vapor mixing ratio rυ. Fourier spectra are calculated at 140 m AGL every 100 s for the 30-min period before CI. One-dimensional spectra are obtained by integrating two-dimensional spectral densities along the y direction. Then the spectra are temporally averaged over the 30-min period before CI for each case: 90–120 min for case A200B06, 150–180 min for case A050B06, and 190–220 min for case A000B06.

For all the heterogeneous cases, there is a spectral gap between the SEB heterogeneity scale of 32 km and the large-eddy scale of roughly 1 km for both θ and rυ (Fig. 10). In observed CBLs, such a spectral gap is not always present (e.g., Nicholls and LeMone 1980; Kang et al. 2007). A spectral gap might not exist because the local land surface may be homogeneous. Also, as shown by Kang and Davis (2008), it is possible for energy to cascade from the mesoscale to smaller scales and fill up the spectral gap.

Fig. 10.
Fig. 10.

One-dimensional spectra of (a) potential temperature and (b) water vapor mixing ratio at 140 m AGL during the time periods of 90–120 min for A200B06, 150–180 min for A050B06, and 190–220 min for A000B06. The vertical bars indicate one standard deviation of the spectra during the time period.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

We note that mesoscale moisture fluctuations exist in all cases, even in the homogeneous CBL (black curve in Fig. 10b); specifically, there are spectral density peaks near the 6-km scale. However, mesoscale temperature fluctuations are absent at these scales for the homogeneous simulation (black curve in Fig. 10a). As discussed in the introduction, Jonker et al. (1999) and de Roode et al. (2004) found a similar result; they suggested that the dominant length scale of mesoscale moisture fluctuations could depend on the ratio between the entrainment flux and surface flux of the passive scalar.

In the weakly heterogeneous CBL (A050B06; blue in Fig. 10b), the mesoscale moisture fluctuations exist on scales that are smaller than the imposed surface heterogeneity scale but larger than the predominant turbulence scale. In the strongly heterogeneous CBL (A200B06; red in Fig. 10b), the moisture fluctuations are insignificant. However, for this strongly heterogeneous case, the temporal standard deviations of moisture spectra (vertical bars in Fig. 10b) for the 30-min period before CI are much larger than those for cases A050B06 and A000B06. This result implies that moisture fluctuations are likely more transient in the strongly heterogeneous CBL. The same features are also observed in the spectra from the cases with the high Bowen ratio (not shown).

b. Scale decomposition

To decompose LES variables into mesoscale and turbulent components, we use a sharp-cutoff filter in Fourier space (e.g., Moeng and Wyngaard 1988; Kimmel et al. 2002; Wyngaard 2010). In this study, the surface heat flux heterogeneity is imposed only in the x direction, as shown in (1) and (2). The Coriolis acceleration, which changes the direction of the atmospheric motions induced by the surface-flux heterogeneity, is ignored. Thus, we assume that the mesoscale fluctuations associated with the SEB heterogeneity exist only in the x direction.

For the scale decomposition, a variable ϕ(x, y; z, t) is Fourier transformed:
e3
To obtain the mesoscale mean of ϕ, the Fourier-transformed variable is low-pass filtered with the cutoff wavenumber of κc:
e4
In contrast, the turbulent fluctuation of ϕT is obtained by high-pass filtering:
e5
Thus, a variable ϕ(x, y; z, t) is decomposed as follows:
e6
Furthermore, the total fluctuation is defined as the departure from a domain average, 〈ϕ〉(z, t):
e7
which can be further decomposed into mesoscale and turbulent fluctuations:
e8
where the mesoscale fluctuation is defined as the mesoscale mean minus the domain average:
e9

To denote additional averaging, such as averaging in time, we use the symbol { }. For example, we use {〈ϕ〉}P to represent the domain average of a variable ϕ that is also temporally averaged over the 30-min pre-CI period, P. As another example, {ϕM}y,P represents the mesoscale fluctuations that are spatially averaged over the y dimension and temporally averaged over the 30-min pre-CI period, P.

c. Vertical profiles of domain-averaged temperature and moisture

Figure 11 presents the vertical profiles of domain-averaged potential temperature and water vapor mixing ratio that are temporally averaged over the 30-min pre-CI period for all the cases. Horizontal bars in Fig. 11 indicate standard deviations. Considering the magnitudes of the standard deviations, the temporal changes of the vertical profiles are considered to be insignificant over this 30-min period.

Fig. 11.
Fig. 11.

Vertical profiles of domain-averaged (a) potential temperature and (b) water vapor mixing ratio, which are temporally averaged over the 30-min periods just before CI: 90–120 min for cases A200B06 and A200B12 (red lines), 150–180 min for cases A050B06 and A050B12 (blue lines), and 190–220 min for cases A000B06 and A000B12 (black lines). The solid lines are used for the wet surface cases (A200B06, A050B06, and A000B06) and the dotted lines are for the dry surface cases (A200B12, A050B12, and A000B12). The initial profiles are plotted with the gray lines.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

From the perspective of these domain-averaged statistics, the strongly heterogeneous cases have a somewhat more statically stable CBL structure (Fig. 11a), which has been shown in previous studies (e.g., Nicholls and LeMone 1980; Avissar and Schmidt 1998; Letzel and Raasch 2003; Kang and Davis 2008). In particular using aircraft observations, Nicholls and LeMone (1980) presented vertical profiles of leg-averaged potential temperature and specific humidity that steadily warm up and dry out with height from a low-level CBL, like the profiles of domain-averaged potential temperature and mixing ratio from the high-amplitude heterogeneity cases shown in Fig. 11. Nicholls and LeMone (1980) reported significant cumulus activity on the day of the particular profiles. Kang and Davis (2008) suggested that the increase of the overall CBL stability results from cold advection in the lower boundary layer and warm advection in the upper boundary layer owing to the surface-heterogeneity induced horizontal flows.

The slightly more stable CBL structure over the strongly heterogeneous surface keeps more moisture in a shallower layer, on average (Fig. 11b). Specifically, the domain-averaged water vapor mixing ratio below ~600 m increases by ~1 g kg−1 from the initial value. In contrast, the other cases have decreases in domain-averaged water vapor (with the exception of the low-amplitude heterogeneity case with the low-Bowen ratio, A050B06).

d. Mesoscale means and fluctuations

The mesoscale means are low-pass filtered with a cutoff wavelength of 4 km. The scale of 4 km has been used frequently to decompose a variable into mesoscale and turbulent components (e.g., Mahrt and Gibson 1992; Finkele et al. 1995; Kang et al. 2007; LeMone et al. 2007), although the choice of 4 km here is also based on the spectra of θ and rυ (Fig. 10). As defined in (9), mesoscale fluctuations are the deviations of mesoscale means from domain averages.

Figure 12 shows that the enhanced CBL heights over the warm patch become higher as the amplitude of heterogeneity increases. In comparison, the surrounding area has generally lower CBL heights as heterogeneity amplitude increases. (Hence, the box-and-whisker diagrams discussed earlier show broader distributions with strongly heterogeneous surfaces.) This result is consistent with the modeling study by van Heerwaarden and de Arellano (2008). Interestingly, the region where zi is higher than the LCL becomes narrower with larger amplitude; in fact, zi > LCL exists only over the center of the warm patch (subdomain WC) for case A200B06 (Fig. 12a), but zi > LCL exists over the entire warm patch (subdomains WL, WC, and WR) for case A050B06 (Fig. 12b). Thus, there is more focused region for CI over the center of the warm patch for case A200B06, which is consistent with the narrow region of clouds in Fig. 4a.

Fig. 12.
Fig. 12.

The mesoscale fluctuations of water vapor mixing ratio (g kg−1, colors), vertical velocity (m s−1, contours), and horizontal velocity (m s−1, vectors), which are spatially averaged over the y dimension and temporally averaged over the 30-min pre-CI period for cases (a) A200B06 (between 90 and 120 min) and (b) A050B06 (between 150 and 180 min). The thick white sold line represents the mesoscale means of CBL height (zi) and the thick white dotted line represents the mesoscale means of LCL. The contour intervals for vertical velocity are 0.04 m s−1.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

Figure 12 also shows the mesoscale fluctuations of horizontal and vertical velocities. Given that 〈u〉 ≈ 0 and 〈w〉 ≈ 0, the mesoscale fluctuations are equivalent to the mesoscale means. The magnitude of low-level convergence and midlevel vertical motion both increase with larger amplitude of the surface flux heterogeneity. In the pre-CI period, the magnitude of the simulated mesoscale vertical velocity is within the observed magnitude of vertical velocity under fair-weather conditions. For example, Mahrt et al. (1994) found vertical velocities with the magnitude of 0.3–0.6 m s−1 that were associated with inland-breeze fronts, Taylor et al. (2007) observed mesoscale vertical velocity on the order of 0.1 m s−1, and LeMone et al. (2002) inferred mesoscale vertical velocities on the order of 0.001 m s−1 from measurements of along-track convergence and divergence.

e. Differences between moist and dry surfaces

Here we show that the amount of moisture supplied by the surface also modifies mesoscale structure. Compared to the cases with relatively dry surfaces, the cases with relatively moist surfaces have stronger buoyant thermals. Figure 13a shows the (resolved scale) turbulence kinetic energy (TKE) for the 30-min period before CI. In the mid- and upper CBL, TKE is 10%–20% larger for the moist surfaces. This is because the additional moisture flux from the surfaces produces larger virtual potential temperature fluxes , as shown in Figs. 3e,f; specifically, at the surface, is ~5% larger in the warm patch for the moist surfaces. The resulting increase in TKE buoyancy tendency throughout the CBL (Fig. 13b) results in slightly more intense buoyant thermals.

Fig. 13.
Fig. 13.

Vertical profiles of (a) TKE and (b) the buoyancy tendency in the TKE budget. The profiles are calculated over the WC subdomain and are averaged over the 30-min time period before CI.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

The differences of mesoscale means of θ and rυ are presented in Fig. 14 for cases that differ only in Bowen ratio (i.e., they differ only in mean moisture flux from surface). The relatively moist surfaces have more cooling and moistening of the entrainment zone. For the strongly heterogeneous surfaces (Fig. 14a), the entrainment zone of the moister surface is cooler by 0.1–0.2 K and moister by 0.4–0.5 g kg−1, particularly over the warm patch (Fig. 14a); these differences in θ and rυ are consistent with the removal of CIN and the reduction of LFC. For the weakly heterogeneous cases (Fig. 14b), the entrainment zone of the moister surface is cooler by 0.1–0.3 K and moister by 0.5–1.0 g kg−1 over the whole domain. The greater cooling and moistening of the entrainment zone over the relatively moist surface is attributable to the stronger buoyant thermals that penetrate the capping inversion more vigorously, and is consistent with earlier CI.

Fig. 14.
Fig. 14.

Differences of mesoscale means of water vapor mixing ratio and potential temperature between the wet surfaces and the dry surfaces: (a) between A200B06 and A200B12, and (b) between A050B06 and A050B12. The mesoscale means are spatially averaged over the y dimension and temporally averaged over the 30-min pre-CI periods of the wet cases. The color intervals for water vapor mixing ratio are 0.01 g kg−1 in (a) and 0.02 g kg−1 in (b), and the contour intervals for potential temperature are 0.01 K. The solid white (orange) lines represent the CBL height for moist (dry) surfaces, and the dashed white (orange) lines represent the LCL for the moist (dry) surfaces.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-10-05037.1

Figure 14 also reveals a slightly warmer mixed layer over the relatively moist surfaces, which is somewhat surprising because the surface temperature flux profiles are identical. However, Sullivan et al. (1998) showed that CBLs have a downward plunge of warmer air pockets from the free atmosphere into the mixed layer, which then mix throughout the CBL; this is an essential process of warming the mixed layer. In our simulations, the warmer mixed layers in the moister surfaces are consistent with stronger engulfment of the overlaying warm free atmosphere by the more vigorous buoyant thermals. The warming effect difference between the moist and dry surfaces is larger over the cool patch than over the warm patch.

Last, the horizontal distribution of rυ is a function of the heterogeneity amplitude. Moisture in the CBL is more uniformly distributed in the strongly heterogeneous surface, with a difference range of 0.1 g kg−1 or larger (color shading Fig. 14a). In contrast, in the weakly heterogeneous surface, the moisture distribution shows local enhancement of rυ on a scale of ~1–2 km (alternating blue and green shades in Fig. 14b); these regions of enhanced rυ extend continuously from the surface layer to the entrainment zone. In the overlying entrainment zone, there are even larger rυ perturbations (of up to about 1 g kg−1; note pockets of yellow-to-orange shading in Fig. 12b). These moisture fluctuations, which are smaller than the imposed heterogeneity scale but larger than the turbulence scale, are likely associated with the peak in Fourier spectra noted in Fig. 10b.

5. Summary and conclusions

This study uses large-eddy simulation (LES) to investigate convective initiation (CI) over heterogeneous surface conditions under the constraint of constant surface available energy. For all the LES runs, the mean surface sensible and latent heat fluxes are constant, but the amplitude of sinusoidal surface flux heterogeneity vary between 50 and 200 W m−2 with the same heterogeneity wavelength of 32 km. In all the horizontally heterogeneous CBLs, CI occurs over the warm patch, where the surface sensible heat flux is above the domain average and the surface latent heat flux is below the domain average. With higher amplitude of the surface flux heterogeneity, CI occurs earlier in a more focused region over the center of the warm patch.

To check the generality of our results we also run a set of simulations with different average latent heat flux (i.e., different mean Bowen ratio). With the same amplitude of sensible heat flux heterogeneity, the relatively wet surface (lower Bowen ratio) yields earlier CI. Over weakly heterogeneous and homogeneous surfaces, surface latent heat flux is especially significant for determining the development of deep moist convection in these simulations; in contrast, over strongly heterogeneous surfaces, CI and its development into deep convection is less dependent on surface moisture. The earlier initiation for relatively wet surfaces is partially attributable to greater intensity of turbulent eddies, owing to slightly greater buoyancy.

Further investigation of the processes leading to CI shows that shallow clouds initiate when the CBL depth zi first exceeds the lifting condensation level (LCL). Deep (precipitating) clouds initiate at a later time when the average level of free convection (LFC) over the warm patch has decreased to roughly the same value as the LCL, which is equivalent to the average convective inhibition (CIN) becoming nearly zero.

With higher amplitude of surface flux heterogeneity, zi becomes relatively larger over the warm patch, and the region with these enhanced zi values becomes narrower, which results in a more focused CI region. Because of the focused CI region over strongly heterogeneous surfaces, the comparison of zi with LCL over a mesoscale area (or one mesoscale model grid box) may not be a useful method to predict CI. Specifically, over the center of the warm patch where CI actually occurs, the values of both zi and LCL are much higher than over surrounding areas. Furthermore, an examination of mean CIN over a mesoscale area may lead to an incorrect forecast for CI, because CIN is reduced only in a focused region.

Over weakly heterogeneous surfaces, there are moisture fluctuations on a scale smaller than the imposed heterogeneity scale but larger than turbulence scale (as shown in previous studies, such as Jonker et al. 1999). However, over strongly heterogeneous surfaces, moisture fluctuations on a scale larger than turbulence are generated only by the surface heterogeneity. Furthermore, over strongly heterogeneous surfaces, greater magnitude of mesoscale upward motion more significantly reduces the warming and drying effect by the entrainment of the free atmosphere into the CBL. In other words, the cooling and moistening associated with upward motion counteracts the entrainment.

These results suggest that NWP model performance of CI might depend on horizontal resolution, in particular over strongly heterogeneous land surface conditions. A significant error might be incurred if a numerical model has grid spacing that cannot explicitly resolve mesoscale circulations induced by varying surface conditions. The simulations herein have the same domain-average surface fluxes and the same initial conditions, but CI does not always occur, depending on the surface flux heterogeneity. Thus, we suggest that CBL parameterizations should be tested for various CBL regimes, focusing on strongly heterogeneous surfaces.

Acknowledgments

The Advanced Study Program of NCAR supported S.-L. Kang during this study. The authors thank Drs. Margaret LeMone and Stanley Trier for providing reviews of an earlier version of this manuscript.

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