## 1. Introduction

Numerous mesoscale modeling studies have suggested that land surface heterogeneity is quite significant in weather and climate forecasting. Weaver and Avissar (2001) for example, using a mesoscale atmospheric model, reported that specification of land surface heterogeneity was essential for accurate simulation of cloud development over the southern Great Plains of the United States. Ramos da Silva and Avissar (2006), also using a mesoscale model, documented that specification of land surface heterogeneity over the Amazon was essential to accurate simulation of rainfall. The fidelity of such mesoscale model results may depend on the parameterization of the atmospheric boundary layer (ABL). This study aims to advance our understanding of the physics of land–atmosphere interactions in a heterogeneous environment with a focus on the implications for parameterization of ABL processes in a mesoscale model.

Past research concerning interactions between land surface heterogeneity and the atmosphere is abundant. Studies using a mesoscale model with idealized, heterogeneous surface flux distribution on a scale of hundreds of kilometers have presented sea-breeze-like mesoscale circulations. In particular, these mesoscale model studies (e.g., Weaver 2004; Chen and Avissar 1994a, b) have suggested that the vertical heat and moisture fluxes directly associated with the mesoscale circulations (mesoscale fluxes) are often greater than the turbulent fluxes.

Some researchers, however, have disagreed with this result. Doran and Zhong (2002, 2000) and Zhong and Doran (1997, 1998) performed mesoscale model studies but with observation-based surface flux distributions instead of idealized conditions. They suggested that the mesoscale flux is not as significant as asserted in the previous numerical model studies. Doran and Zhong (2002, 2000) and Zhong and Doran (1997, 1998) reported that the magnitude of the vertical velocities generated by convergence and divergence is overestimated in previous research in which idealized surface flux conditions are used. Observational studies (Kang et al. 2007, hereafter KDL07; LeMone et al. 2002, 2007; Mahrt et al. 1994a, b) have supported the finding that typical mesoscale vertical velocities are weak. At least from a perspective of mesoscale vertical fluxes, the ABL structure observed over heterogeneous surfaces seems to differ from what is expected from the mesoscale model studies in which idealized surface flux conditions are used.

Many studies have focused on determining the scales at which land surface heterogeneity leads to organized mesoscale circulation in the ABL. Many of these studies have used a mesoscale model (e.g., Reen et al. 2006; Ramos Da Silva and Avissar 2006; Weaver and Avissar 2001; Chen and Avissar 1994,b). Theoretical studies (e.g., Dalu et al. 1996) have suggested that the optimal scale of surface heterogeneity needed to generate these mesoscale circulations is the local Rossby radius of deformation, which is estimated to be about 100 km for typical midlatitude atmospheric conditions (Pielke 2001). Baidya Roy et al. (2003), however, suggested that the atmosphere appears to respond the most to surface heterogeneity on scales of 1 to 100 km, with enhanced kinetic energy at a scale of 10–20 km. This finding suggests that additional study of land–ABL interactions at the scale of tens of kilometers is warranted. An alternative numerical approach to mesoscale modeling that can be used at this scale is large-eddy simulation (LES). LES can resolve both energy-containing turbulent eddies and the mesoscale eddies related to the surface heterogeneity, thus reducing the dependence, as compared to a mesoscale model, of the model results on the chosen subgrid parameterization.

Some studies using LES (Patton et al. 2005, hereafter PSM05; Letzel and Raasch 2003, hereafter LR03; Avissar and Schmidt 1998) have dealt with the effect of surface heterogeneity on a scale of tens of kilometers. PSM05 and LR03 simulated the structure of the ABL over a heterogeneous surface on a scale of tens of kilometers but their results appear to be inconsistent: quasi-steady (PSM05) and temporally oscillatory (LR03). LR03, specifying a high amplitude of 0.1–0.2 K m s^{−1} (about 120–150 W m^{−2}) over wavelengths of up to 40 km, reported that the generated mesoscale flow undergoes a temporal oscillation in the volume-averaged mesoscale and turbulent kinetic energies, which they call the kinetic perturbation energy. In contrast, PSM05, indirectly prescribing a low amplitude of 0.03 K m s^{−1} (about 40 W m^{−2}) over wavelengths of up to 30 km, observed no temporal oscillations in the kinetic perturbation energy. The structure of the ABL over a heterogeneous surface is likely divided into two regimes as a function of the amplitude of the surface heat flux variation: temporally oscillatory (LR03) or quasi-steady (PSM05).

One characteristic of the numerical model studies cited above is limited observational evaluation of their results. When observations have been used to evaluate the mesoscale model result, they are most often integrative measures (e.g., cloud or precipitation fields), which, although able to evaluate model performance over large regions and time spans, do not ensure that the details of surface-ABL interactions are simulated in a realistic fashion. Careful evaluation of the modeled versus observed surface energy balance or modeled versus observed boundary layer heights, winds, and mixing ratios, encompassing the mesoscale, are mostly absent from the literature focusing on land surface heterogeneity. The comparison of mesoscale model results with observations from the 1997 southern Great Plains experiment (SGP97; Jackson 1997) by Desai et al. (2006) and Reen et al. (2006) is one of the exceptions, although their work is limited to a case study. Collecting observations of both ABL structure and surface forcing over a mesoscale domain was one major goal of the International H_{2}O Project (IHOP_2002; Weckwerth et al. 2004; LeMone et al. 2007).

This study adds to the body of literature concerning LES of the convective ABL over heterogeneous land surfaces, focusing on how the convective ABL responds to surface heterogeneity in surface heat flux of varying amplitudes on spatial scales of 10–30 km. This study extends the studies of LR03 and PSM05 to elucidate in particular how this heterogeneity both alters convective ABL structure and modulates ABL turbulence. Although previous LES studies (e.g., PSM05; LR03; Avissar and Schmidt 1998) have encompassed this range of spatial scales and amplitudes of heterogeneity, our analyses are unique in two ways. First, we filter the simulated flow into mesoscale and microscale (turbulence) components. In this way we can address more directly how well existing turbulence parameterizations are likely to perform in mesoscale model simulations of the ABL over heterogeneous land surfaces. Second, we simulate the scale and amplitude of heterogeneity observed over a portion of the IHOP_2002 study region (KDL07), allowing a unique comparison between observed and simulated ABL characteristics. Our overarching goal is (i) to identify a domain, defined by model resolution and amplitude of the surface energy balance heterogeneity, within which mesoscale models that rely upon existing ABL turbulence parameterizations should function well and (ii) to describe the ABL structures that these models may fail to simulate outside of that domain.

## 2. Experiment setup

### a. Observational background

This numerical study is partially motivated by observations of the fair-weather midday ABL over a heterogeneous land surface obtained during IHOP_2002. The heterogeneous surface is a 60-km north–south oriented strip from approximately 36.4° to 37.0°N along 100.6°W. Although there was no sea-breeze-like mesoscale circulation observed, the ABL structures over the heterogeneous surface are obviously different from those over a homogenous surface (for detailed description, see KDL07). Repeated aircraft flights at approximately 65 m above ground level on five fair-weather days in May and June persistently showed that the surface heat flux increased linearly from south to north along a 16-km segment of this heterogeneous surface (36.47° to 36.62°N) (for details, see KDL07). Based on the measurements from two flux measurement sites at the ends of this segment, the range of sensible heat flux gradient between these two sites is estimated for fair-weather midday (1130–1430 LST) conditions (Table 1). Soundings of pressure, density, and potential temperature from a rawinsonde released over the Homestead site (36.55°N, 100.6°W) in the midst of this heterogeneous surface at about 1130 LST on 25 May are selected for the initial conditions for the LES runs. The initial potential temperature is constant (296 K) at altitudes less than 626 m and increases at the rate of 34.8 K km^{−1} between 626 and 806 m. Above 806 m, the potential temperature increases at the rate of 3.8 K km^{−1}.

### b. Surface heat flux variation

*F*〉 (

_{wθ}*z*= 0), amplitude

*A*, and wavelength

_{wθ}*λ:*

This sinusoidal variation of surface heat flux will likely generate sea-breeze-like circulations as schematically illustrated in Fig. 1 (e.g., LR03; Avissar and Schmidt 1988; Hadfield et al. 1991). Based on the potential temperature and wind velocity distributions along the *x*_{1} axis expected from this sinusoidal surface heat flux variation, the ABL is divided into two pairs of regions: (i) warmer (0 ≤ *x*_{1} < *λ*/2) and cooler regions (*λ*/2 ≤ *x*_{1} < *λ*) and (ii) middle (*λ*/4 ≤ *x*_{1} < 3*λ*/4) and edge regions (3*λ*/4 ≤ *x*_{1} < *λ* and 0 ≤ *x*_{1} < *λ*/4). In this study, *x*_{1} is the streamwise direction (corresponding to the south to north direction in our IHOP_2002 observational studies), and *x*_{2} is the crosswind (west–east) direction.

In (1), 〈*F _{wθ}*〉(

*z*= 0) is set at 0.20 K m s

^{−1}(250 W m

^{−2}), which is estimated based on the composite heat flux profile obtained from the measurements at the surface flux sites and aircraft over the 60-km flight track at midday (1130–1430 LST) on 25 May during IHOP_2002 (KDL07). The wavelength

*λ*in (1) is set at 32 km to emulate the slope between the monotonically varying surface heat flux variation observed over the 16-km transect between site 1 and site 2. To investigate the effect of the wavelength on our

*λ*= 32 km results, simulations with

*λ*= 16 km are also performed. We report on the simulation results with

*λ*= 16 km only when they differ significantly from the

*λ*= 32 km results. The amplitude

*A*in (1) is varied extensively from a minimum of 0 K m s

_{wθ}^{−1}to a maximum of 0.2 K m s

^{−1}. The minimum amplitude for a heterogeneous is 0.02 K m s

^{−1}(25 W m

^{−2}), approximately the average value of the observed sensible heat flux differences between the two flux tower sites (Table 1). This amplitude (0.02 K m s

^{−1}) is slightly lower than the amplitudes simulated by PSM05. The amplitude increases to 0.2 K m s

^{−1}(250 W m

^{−2}), which is the maximum value used in LR03 and Avissar and Schmidt (1998). The various combinations of

*A*and

_{wθ}*λ*that were simulated are summarized in Table 2.

### c. Model description and setup

This study utilizes the compressible nonhydrostatic numerical model of Bryan and Fritsch (2002) as an LES. The Bryan–Fritsch model, developed as a cloud-resolving model using LES techniques, has been used to solve many different nonlinear moist convection problems (e.g., Fanelli and Bannon 2005; James et al. 2006). In addition, from an LES perspective, this Bryan–Fritsch model has been applied to investigate the appropriate spatial resolution for the simulation of deep moist convection (Bryan et al. 2003). The utility of this Bryan–Fritsch model as an LES for the convective ABL study is demonstrated by producing the turbulence statistics characteristic of the ABL over a homogenous surface. The expected turbulence statistics are well known from both observational and numerical studies (e.g., Lenschow et al. 1980; Nieuwstadt et al. 1991). Some of these turbulence statistics will be presented in section 4, along with the results from numerical experiments for the ABLs over heterogeneous surfaces.

This Bryan–Fritsch model integrates the filtered compressible Navier–Stokes equation using third-order Runge–Kutta time differencing and fifth-order spatial derivatives for the advection terms. This coupled numerical scheme is found by Wicker and Skamarock (2002) to be the most accurate finite-difference solution for simulating a highly nonlinear flow. The turbulent kinetic energy scheme of Deardorff (1980) is employed for a subfilter-scale parameterization (Bryan and Fritsch 2002).

In an LES experiment the large, energy-containing turbulent eddies have to be explicitly computed. Given an effective resolution of 6Δ (where Δ is a grid mesh size; Bryan et al. 2003) and a typical size of energy-containing turbulent eddies [*l* ≈ 1.5 *z _{i}*, where

*z*is the ABL depth; Kaimal et al. (1972)], the grid mesh size has to be in the range of Δ < 0.25

_{i}*z*. Thus, considering the initial ABL depth of about 700 m (section 2a), the grid spacing is 100 m in the horizontal. Although this is relatively coarse resolution as compared to some state-of-the-art simulations of the ABL, our goal is to study the mesoscale organization of ABL convective turbulence. Therefore, we sacrifice finescale resolution in exchange for a large spatial domain, while retaining enough resolution to simulate well the dominant eddies in the convective ABL.

_{i}Because of the strong height dependence of the length scales of vertical velocity near the surface and increased dependence on the subgrid-scale parameterization, the fidelity of LES in the surface layer is questionable. Wyngaard et al. (1998) suggest the adoption of a high grid aspect ratio (horizontal dimension/vertical dimension). This high aspect ratio would produce a mean surface-exchange coefficient based on many surface-layer eddies. Thus, the grid spacing in the vertical is 10 m up to 100 m above the ground, linearly increases from 10 to 40 m between 100 and 1900 m above ground, and then remains constant at 40 m up to the model domain top of 3500 m.

The model domain is 32 km (or 16 km) in length in the *x*_{1} (here, south–north) direction and is 5 km in length in the *x*_{2} (here, west–east) direction. Without the Coriolis force, mesoscale flows by the surface forcing prescribed with (1) are likely to be generated only in the *x*_{1} direction, as shown in Fig. 1. Thus, in the *x*_{2} direction, it is assumed that there are only turbulent flows. In both horizontal directions the lateral boundary conditions are periodic. The vertical extent of the domain is 3.5 km. The upper boundary is a flat, rigid wall with a Rayleigh damping layer (Durran and Klemp 1983) occupying 1 km beneath the model top. The lower boundary is also a flat, rigid surface. Unlike the prescribed surface heat flux, the surface momentum flux is derived from a simple surface drag parameterization (Stull 1988).

The mesoscale surface heat flux variation is activated upon initiation of the simulation. Also, at this initial time, random perturbations of 0.1 K that initiate development of three-dimensional turbulent flows are superimposed on the potential temperature at the lowest atmospheric level. Considering that a model spin-up time is less than 1 h and that it is a midday ABL simulation (1130–1430 LST), the 4 h integration time is sufficient. Thus, we use the temporally constant surface heat flux distribution and neglect the Coriolis force. Occasionally we use results from model runs as long as 9 h to identify the temporal oscillations of surface heterogeneity–induced flows. However, the discussion is mostly based on the results from the model run between 1 h and 4 h. For all the cases, the integration time step is set to 1 s and the model output was saved every 100 s.

## 3. Horizontal averaging and wave-cutoff filter

Two types of averaging are applied to the LES results: one to study how the mean fields vary in relation to the surface heat flux distribution and a second to evaluate the structure of the ABL as a function of spatial scale. The temporally varying flows require filtering that differs from previous studies (e.g., PSM05; Avissar and Schmidt 1998) that used filtering that assumed that the mean and mesoscale ABL flows were steady with respect to time. In this study we define mean and mesoscale components of the flow that vary with time, similar to the work of LR03. To evaluate the impact of surface heterogeneity on microscale versus mesoscale flows, we go beyond LR03 and apply an additional wave-cutoff filter that produces low- and high-pass filtered components with no Fourier components in common. This allows us to examine components of the flow that would need to be resolved rather than parameterized with a mesoscale model.

### a. Horizontal averages

*z*and time

*t*, one of the variables from the LES is denoted by

*ϕ*(

*x*;

_{i}*z*,

*t*) where

*I*= 1, 2. The variables are horizontally refiltered aswhere

*G*(

*x*

_{i}−

*x*′

_{i}) is a horizontal spatial filter function and

*x*is limited to values of the form

_{i}*x*(

_{i}*n*) =

*nS*−

_{i}*S*/2 (

_{i}*n*= 1, 2, . . . ,

*L*/

_{i}*S*). Here,

_{i}*L*is the size of the model domain in the

_{i}*x*direction and

_{i}*S*is a filter scale in meters, which can be represented as

_{i}*S*=

_{i}*N*Δ

_{i}*x*(where

_{i}*N*is the number of LES grid points to be included in the filter scale and Δ

_{i}*x*is the LES grid mesh size, here 100 m). The range of

_{i}*S*is Δ

_{i}*x*≤

_{i}*S*≤

_{i}*L*. We use a filter function

_{i}*G*that is a simple spatial average defined as

*x*

_{1}direction; therefore, we expect a homogenous turbulent field in the

*x*

_{2}direction. Thus, the filter scale in the

*x*

_{2}direction is always the size of the model domain in that direction (

*S*

_{2}=

*L*

_{2}). However, to investigate the generated mesoscale flows and their influences on the turbulent flows, several different filter scales are used for the

*x*

_{1}direction. When

*S*

_{1}= Δ

*x*

_{1}, the refiltered variable is denoted by

*x*

_{1};

*z*,

*t*). With

*S*

_{1}=

*λ*/2, the variables averaged over the warmer, cooler, middle, and edge regions defined in section 2b are indicated by

^{W}(

*z*,

*t*),

^{C}(

*z*,

*t*),

^{M}(

*z*,

*t*), and

^{E}(

*z*,

*t*), respectively. When

*S*

_{1}=

*L*

_{1}, the domain-averaged variable is denoted by 〈

*ϕ*〉(

*z*,

*t*). From this horizontal domain average, the total fluctuation is defined as

### b. Wave-cutoff filter

*x*

_{1};

*z*,

*t*) (which represents a simple averaging over the homogeneous

*x*

_{2}direction) is refiltered again with a one-dimensional, low-pass wave-cutoff filter function

*G*′ in the

*x*

_{1}direction:where

*N*

_{1}=

*S*

_{1}/Δ

*x*

_{1}. In (5), the wave-cutoff filter

*G*′ iswhere

*j*is a location relative to the refiltered variable location. In (6),

*C*are the weighting coefficients at

^{j}*j*; these coefficients are determined by the choice of cutoff wavelength

*S*

_{1}(Tong et al. 1998). We use this low-pass wave-cutoff filter component [

*ϕ*] to decompose the LES variables into a low-pass filtered component and a residual:We shall refer to the first and second terms in the right-hand side of (7) as mesoscale and microscale (turbulent) components, respectively.

### c. Variance and vertical heat flux decomposition

*w*″ and

*θ*, and interscale components between

^{M}*w*and

^{M}*θ*″, respectively.

## 4. Results

### a. Mesoscale horizontal flows

Figure 2 shows that the surface heterogeneity–induced mesoscale horizontal flows can be temporally oscillating or quasi-steady depending on the amplitude *A _{wθ}* of surface heat flux variation. The mesoscale horizontal flows generated by high-amplitude surface heat flux variation are temporally oscillating, as was first found by LR03. For the wavelengths

*λ*of 32 and 16 km, surface heat flux amplitudes of 0.1 K m s

^{−1}or higher initiate the temporal oscillation. In contrast, for the amplitudes observed during IHOP_2002 (0.02–0.04 K m s

^{−1}; KDL07), the generated mesoscale flows are in a quasi-steady state. LR03 found no oscillations for amplitudes below 0.05 K m s

^{−1}at a wavelength of surface heat flux variation of 40 km or smaller. Also, PSM05, indirectly prescribing an amplitude of 0.03 K m s

^{−1}over wavelengths of up to 30 km, observed no temporal oscillations in the volume average of mesoscale and turbulent kinetic energies.

### b. Domain-averaged statistics

We examine domain-averaged vertical profiles of potential temperature. Some of these domain-averaged profiles have been presented in previous studies, and we review our numerical results to show this consistency. The temporal variability of these profiles, however, has not yet been documented. This is important for parameterization of the ABL within larger-scale models, because these parameterizations rely upon similarity theory developed for quasi-steady conditions. Our results show that the domain-averaged profiles often do not satisfy this assumption.

The vertical profiles are computed at each hour between 0–4 h (Fig. 3). Whereas the vertical profiles from a homogeneous ABL (BA000) demonstrate a typical well-mixed ABL structure, the profiles from the heterogeneous ABLs exhibit a somewhat statically stable ABL structure, with a warmer upper level and a cooler lower level. Previous studies (e.g., LR03; Avissar and Schmidt 1998) also have shown these statically stable ABL structures over heterogeneous surfaces. This increased static stability is likely the result of potential temperature advection by the generated mesoscale flow. From the perspective of the domain-averaged statistics, this heterogeneous ABL reduces buoyancy and decreases the intensity of turbulent vertical velocity throughout the model domain.

Figure 4a shows the vertical profiles of total vertical heat flux averaged over 1–4 h. In Fig. 4b, the temporal variability is estimated by the standard deviation of total vertical heat flux between 1–4 h. Here, standard deviation represents the distribution of profile values across the time. This standard deviation is normalized by the magnitude of the temporally averaged vertical heat flux. For all cases, the vertical profiles of total vertical heat flux averaged over 1–4 h appear to be linear. The temporal oscillations cause the temporal variability for the ABLs with higher *A _{wθ}* (BA150, BA200, and BA250) to be significantly larger than that for the homogenous ABL (BA000). In contrast, the temporal variability for the ABLs with lower

*A*(BA025 and BA050) is similar to that for BA000. This result implies that the domain-averaged heat flux profile is quasi-steady only for ABLs with lower

_{wθ}*A*.

_{wθ}In Fig. 4a, the ABLs with higher *A _{wθ}* have minimum values of heat flux near the ABL height that are less negative than those of the homogeneous ABL (BA000). This less-negative minimum kinematic heat flux has been shown in some previous LES results (e.g., Avissar and Schmidt 1998; PSM05). PSM05 suggested that this less-negative minimum value is caused by the spatial averaging of heat flux in a thicker capping inversion over a heterogeneous surface. We suggest that for the ABLs with higher

*A*, the temporal averaging of the heat flux may also cause the less-negative minimum value, considering the large temporal variability near the ABL height, as shown in Fig. 4b.

_{wθ}Figure 5 shows the vertical profiles of total variances of *θ*, *υ*, and *w* averaged over 1–4 h and the temporal variability of the total variances between 1–4 h. The vertical profiles from the homogeneous ABL (BA000) match the well-known typical convective ABL shapes (e.g., Lenschow et al. 1980; Nieuwstadt et al. 1991). Figures 5a–c (left-hand side) show mean variance profiles whose characteristics are consistent with past simulations of the convective ABL over one-dimensional, periodically heterogeneous surfaces (e.g., PSM05; Avissar and Schmidt 1998). These characteristics include increased *θ* and *υ* variances at the top and bottom of the ABL due to the generated mesoscale flows and decreased *w* variance throughout the ABL due to the increase in stability caused by these flows (Fig. 4). These characteristics intensify with increasing *A _{wθ}*. Variances of

*θ*are altered the most: lower

*A*cases (BA025 and BA050) show approximately 2–5 times the variance of of BA000 and ABLs with higher

_{wθ}*A*(BA150, BA200, and BA250) demonstrate approximately 20 times that of BA000 (Fig. 5a, left-hand side).

_{wθ}The temporal behavior of these variance profiles (Figs. 5a–c, right-hand side) reflects the oscillatory behavior of the mesoscale flows first documented by LR03. The degree to which the variance profiles are temporally variable differs, with *θ* variance profiles altered the most (temporal standard deviation in the variance can be as much as 75% of the mean variance) and *w* the least. The *θ* variance profiles are also noticeably enhanced even for the lower *A _{wθ}* cases (BA025 and BA050), whereas the

*w*variance profiles are not disturbed until the amplitude of heterogeneity is considerably larger (BA100 and higher

*A*). Observations of spectra over a similar heterogeneous surface by KDL07 are consistent with these findings.

_{wθ}### c. (Co)spectral analysis

*F*of

*θ*,

*υ*, and

*w*are obtained by integrating two-dimensional spectral densities

*ψ*along the

*x*

_{2}axis:where

*c*is

*θ*,

*υ*, or

*w*, and

*κ*(

_{i}*i*= 1, 2) is defined as the wavenumber in the

*x*direction. The energy spectra of

_{i}*θ*and

*w*presented in Figs. 6 and 7, respectively, are determined by computing one spectrum per 100 s and then temporally averaging over 1–4 h.

In the spectra of *θ* at 0.4 *z _{i}* (Fig. 6), mesoscale surface differential forcing appears to have given rise to microscale variances, as seen by the difference in spectral density between BA000 and other cases at approximately 1-km wavelength in Fig. 6. This result implies that, without considering this nonlinear transfer of variance from the mesoscale, the microscale variance would be underestimated in the ABL. The failure of the spectra normalized by mixed layer scaling to collapse into a single curve in prior observational studies (Kaimal et al. 1976; Young 1987) may be associated with this enhanced

*θ*variance at the microscale (about 1 km) in response to mesoscale forcing.

*θ*spectra, the

*w*spectra from the heterogeneous ABL show no spectral peak at the scale of surface heterogeneity (Fig. 7). From the zero-divergence continuity equation, the mesoscale vertical velocity, which is associated with convergence or divergence, can be estimated aswhere

*υ*is the generated mesoscale horizontal velocity. Note that the validity of this analysis should be limited to cases where the wavelength of surface heterogeneity is much greater than the ABL depth.

^{M}In Fig. 8, the *w* spectra from EA250 demonstrate relatively significant mesoscale peaks, whereas the *w* spectra from BA250 reveal no obvious mesoscale peaks. Our scaling argument (13) implies that shorter wavelength heterogeneity should produce, all others factors held equal, stronger mesoscale vertical velocity. Note also that the *w* spectra in Fig. 8 were computed using outputs averaged over 6–9 h instead of over 1–4 h (Figs. 6 and 7). During this period, the time-averaged ABL depth is 1600 m, almost 2 times the time-averaged ABL depth of 850 m over 1–4 h. Following (13), this higher ABL depth also favors a stronger mesoscale vertical velocity. In summary, our vertical velocity spectra appear to be consistent with those of LR03, showing suppression of spectra at the microscale in all cases and distinct mesoscale peaks when conditions [see Eq. (13)] favor a large mesoscale vertical velocity.

In Figs. 7 and 8, the inertial-range slopes are much steeper than the well-known Kolmogorov slope of −2/3. This steeper slopes are likely associated with the dissipation mechanism inherent in the integration scheme, working at scales smaller than the effective resolution of about 6Δ (Bryan et al. 2003; Skamarock 2004). Thus, these steeper slopes can be overcome by using higher resolution. However, we are limited in this respect by computational resources and the need to simulate large spatial scales. It does not necessarily follow that our simulation of the dominant ABL eddies is flawed: these spatial scales should be well resolved, as discussed previously.

Similar to spectra, one-dimensional cospectra are obtained by integrating two-dimensional cospectral densities along the *x*_{1} axis. In Fig. 9, the one-dimensional cospectra of *w* and *θ* from the heterogeneous ABLs exhibit two peaks. However, unlike the spectra, these cospectra show the mesoscale peaks at the scale of 10–20 km instead of at the prescribed heterogeneity scale. It should be also noticed that for the ABLs with higher *A _{wθ}* (BA150, BA200, and BA250), these mesoscale peaks become larger in magnitude than the microscale peaks, especially higher in the ABL. We will discuss the nature of this mesoscale flux in more detail in section 4e. Similar to the vertical velocity spectrum, the cospectra are suppressed at microscales, and the magnitude of the suppression increases as the amplitude of heterogeneity increases.

### d. Variance decomposition

It is clear from the simulations and past research that the statistics of the ABL are altered at the micro- and mesoscales by the presence of a heterogeneous surface. Next we wish to examine the micro- and mesoscale components separately to learn more about the nature of these changes and our ability to recover classic ABL statistics at the microscale by filtering out mesoscale features. If this can be done, we hypothesize that a mesoscale model with spatial resolution similar to the filter scale should be able to resolve the mesoscale ABL response to the surface heterogeneity and describe the ABL with traditional microscale parameterizations. If classic ABL profiles cannot be recovered, then it is unclear how a mesoscale model relying on traditional ABL parameterizations can simulate these ABLs accurately.

After applying the wave-cutoff filter (5), theoretically the wavelengths smaller than the cutoff wavelength should be completely removed. However, the smaller wavelengths are not in fact completely removed due to the imprecision of the spectral cutoff (Tong et al. 1998). That is to say, if the wave-cutoff filter is applied, theoretically 〈*ϕ ^{M}ϕ*″〉 = 0 in (10), but in fact 〈

*ϕ*″〉 =

^{M}ϕ*f*(

*λ*), where

_{c}*λ*is a cutoff wavelength. Thus, to identify a scale at which we can best segregate phenomena into mesoscale and microscale components, we have performed a variance decomposition (10) of

_{c}*θ*,

*υ*, and

*w*, with four different cutoff wavelengths of

*λ*/2,

*λ*/4,

*λ*/8, and

*λ*/16. Based on the contribution of the interscale component 〈

*ϕ*″〉 to total variance 〈

^{M}ϕ*ϕ*′

^{2}〉, we selected

*λ*=

_{c}*λ*/16(=2 km) as the scale which best decomposes the total variance into mesoscale and turbulent variances 〈

*ϕ*′

^{2}〉 ≈ 〈

*ϕ*

^{M}^{2}〉 + 〈

*ϕ*″

^{2}〉, leaving an interscale contribution of less than 5% for potential temperature and less than 1% for horizontal and vertical velocities.

Figure 10 shows the vertical profiles of turbulent variances averaged over 1–4 h and the temporal variability of these turbulent variances between 1–4 h. The vertical profiles of turbulent *θ* variances from the ABLs with lower *A _{wθ}* (BA025 and BA050) are nearly coincident with those from BA000. The temporal variability, seen in the rhs of Fig. 10a, indicates that BA025 and BA050 may satisfy the quasi-steady state condition. However, for the ABLs with higher

*A*(BA150, BA200, and BA250), the vertical profiles of the microscale

_{wθ}*θ*variances deviate somewhat from BA000. In addition, the temporal variability is much larger than in BA000, especially at the middle and upper levels. Although the microscale

*υ*variances from the ABLs with lower

*A*are somewhat larger than those from BA000, the shapes of the vertical profiles are similar to those from BA000. However, the microscale

_{wθ}*υ*variances from the ABLs with higher

*A*demonstrate values 2 times those from BA000 at 0.4

_{wθ}*z*–0.6

_{i}*z*. This seems to be associated with the larger transfer of mesoscale variance to the microscale at this middle level as compared to that at the lower and upper levels. In Fig. 10c, the profiles of microscale

_{i}*w*variance and their temporal variability are almost the same as those of total variances (Fig. 5c) because of the insignificant contributions of mesoscale variance to total variance.

### e. Decomposition of vertical heat flux

Figure 11 shows the time–height cross sections of the three terms in (11) for cases BA025 and BA250. The interscale term involving *w ^{M}* and

*θ*″ in (11) is disregarded due to the negligible contribution to total flux. In BA025, the turbulent vertical heat flux appears to be in a quasi-steady state (lhs of Fig. 11b). In contrast, in BA250, the turbulent vertical heat flux is not in a quasi-steady state. Approximately 30 min. after each time the interscale term 〈

*w*″

*θ*〉 and mesoscale term 〈

^{M}*w*〉 reach their maxima, the microscale vertical heat flux 〈

^{M}θ^{M}*w*″

*θ*″〉 is reduced in the middle- and upper-level ABL. Although at their maxima this interscale term becomes comparable to the microscale heat flux, the microscale heat flux is still the most significant contributor to total heat flux.

The significance of the interscale component 〈*w*″*θ ^{M}*〉, instead of mesoscale heat flux 〈

*w*〉, is related to the absence of mesoscale peaks in the vertical velocity spectra (Fig. 7). Also, this considerable interscale component is associated with the mesoscale peaks in the cospectra of vertical velocity and potential temperature (Fig. 9), located at a smaller scale than the imposed heterogeneity scale. The sea-breeze-like mesoscale circulation, which is accompanied by substantial mesoscale peaks in the vertical velocity spectra, is generated only when the amplitude of surface heat flux variation is the highest used in the experiment (0.2 K m s

^{M}θ^{M}^{−1}), the wavelength is the shorter of the two utilized (16 km), and the ABL depth grows to twice the initial depth (1600 m) (Fig. 8a). The relation between the mesoscale vertical velocity and ABL characteristics is consistent with a simple scale analysis of the zero-divergence continuity Eq. (13).

Figure 12a shows that the microscale heat fluxes from the ABLs with higher *A _{wθ}* are half of those from BA000, specifically at the middle levels of the ABL. The similar linear shapes of the vertical profiles of total heat flux as compared to that from BA000 in Fig. 4a result from the increased magnitude of the interscale component 〈

*w*″

*θ*〉 when the microscale heat fluxes are decreased. In Fig. 12b, the significant temporal variability of the microscale vertical heat flux in the ABLs with higher

^{M}*A*suggests that the microscale flux is substantially modulated as a function of time in the ABLs experiencing strong oscillations in the mesoscale flows.

_{wθ}## 5. Conclusions

As a function of the intensity of the heterogeneity, the fair-weather midday ABLs (1130–1430 UTC) over heterogeneous surfaces on a scale of tens of kilometers have significantly different structures: temporally oscillating (LR03) or quasi-steady (PSM05). These differing ABL responses arise as the result of large- versus small-amplitude surface heterogeneity, respectively. The existence or absence of this oscillatory behavior has a large impact on the resulting microscale structure of the ABL and governs the characteristics of the surface heterogeneity–induced mesoscale flows.

With an amplitude of heterogeneity in the range observed during IHOP_2002 (0.02–0.04 K m s^{−1}) the one-dimensional surface heat flux variation on a scale of tens of kilometers (32 km or 16 km in this study) generates quasi-steady mesoscale flows. Even with these quasi-steady flows, the structure of the ABL is disturbed from the homogeneous case. However, after filtering out mesoscale fluctuations from total fluctuations, the turbulent variances and vertical heat fluxes satisfy mixed-layer similarity profiles. This result is consistent with the observations of KDL07 on which the numerical experiment design of this study is based. The implication is that for the ABL, over weakly heterogeneous surfaces on a scale of tens of kilometers, mesoscale models of sufficient resolution that depend upon typical ABL parameterizations may suffice.

In the ABL with high amplitude of surface heat flux variation (0.10–0.20 K m s^{−1}; higher than the IHOP_2002 observation but still in the range used by previous LES studies), the generated mesoscale flow oscillates in time and the microscale structure of the ABL is nonstationary because of the oscillations. Mixed-layer similarity is violated for these conditions even when mesoscale fluctuations are filtered out. Given these findings, mesoscale models employing a typical ABL parameterization may fail to simulate the structure of the ABL over a strongly heterogeneous surface on a scale of tens of kilometers, even with a resolution high enough to resolve the mesoscale features. Past studies (LR03) showed that the oscillations, although robust with respect to model spin-up approaches, were disrupted by light winds blowing across the heterogeneous surface. To our knowledge these oscillations have not been observed, probably because of the large amplitude of heterogeneity required to generate them and perhaps also because of their vulnerability to mean wind. Thus, although we feel that the oscillatory behavior is something that could exist in the ABL, its practical significance may be limited. Research on the impact of large-amplitude heterogeneity on the microscale structure of the ABL in the presence of a mean wind may find more practical application than our more idealized results.

The significance of the nonlinear transfer of variance from the scale of the imposed mesoscale heterogeneity to other scales differs among atmospheric variables. Potential temperature variance increases at micro- and mesoscales, horizontal wind variance increases primary at the mesoscale and is not strongly affected at microscales, and vertical velocities are suppressed at the microscale but not strongly affected at mesoscales, all relative to the homogeneous case. Thus, the microscale variance of potential temperature (or any other scalar) may be underestimated and that of vertical velocity underestimated if this impact of mesoscale heterogeneity is not taken into account. Observations of spectra over a similar heterogeneous surface by KDL07) are consistent with these findings. These differing responses complicate ABL parameterization in heterogeneous environments.

We found that the vertical transport of heat by mesoscale vertical velocity is negligible compared with the transport by turbulent vertical velocity for the entire range of conditions we simulated. This result contradicts the conclusions of many studies using a mesoscale model with idealized surface flux conditions (e.g., Weaver 2004; Chen and Avissar 1994a, b), which suggests that mesoscale vertical heat flux (mesoscale vertical velocities transporting mesoscale gradients in potential temperature) is often greater than turbulent vertical heat flux. Our result is supported by some mesoscale model studies using surface-flux distributions based on observations (e.g., Zhong and Dora 1997, 1998) and observational studies (KDL07; LeMone et al. 2002; Mahrt et al. 1994a, b).

Although mesoscale heat flux is insignificant even in the ABL with a high amplitude of surface heat flux variation, the contribution of the interscale component (turbulent vertical velocities transporting mesoscale gradients in potential temperature) of the vertical heat flux to total heat flux becomes significant for high-amplitude heterogeneity (thus, oscillatory ABLs). This interscale component is periodic in these cases and is comparable to that of the turbulent vertical heat flux when the interscale heat flux reaches its maxima. This is expressed in the cospectra of vertical velocity and potential temperature as a decrease in power at the microscale and an increase at the mesoscale that becomes more pronounced as the amplitude of heterogeneity increases.

Mesoscale models will not resolve the turbulent vertical transport of mesoscale scalar fluctuations. Past mesoscale model studies that conclude that mesoscale vertical heat flux is significant when the amplitude of surface heat flux variation becomes high may be misrepresenting the physics of these environments. In addition, previous studies have shown that background wind (synoptic or larger-scale wind) may disrupt the organized mesoscale flows. Considering that no background wind was prescribed in this study, one could expect mesoscale vertical velocity to be further reduced and vertical transport due to mesoscale vertical velocity to be even smaller. Thus, we conclude that the transport of a scalar by turbulent vertical velocity is likely more significant than that by mesoscale vertical velocity, even in an ABL with high-amplitude surface heat flux variation.

The importance of mesoscale vertical velocities over a heterogeneous surface can be anticipated using the scaling of the conservation of mass equation presented in Eq. (13). Mesoscale vertical velocity increases in importance as the amplitude of the heterogeneity and boundary layer depth increase and the wavelength of heterogeneity decreases. Past observations and these numerical results imply that mesoscale vertical velocities in the ABL are typically quite small.

The total heat flux profile remains linear when the ABL is nonoscillatory. It is also linear for the oscillatory ABL when averaged over many oscillations. This contradicts some past work that used limited temporal averaging and found nonlinear flux profiles as a result. The modification of the shape of the linear profile (reported in the past and confirmed by our results) and the emergence of the oscillatory, interscale flux for high-amplitude heterogeneity present challenges for ABL parameterization in heterogeneous environments.

## Acknowledgments

This research was supported by the National Science Foundation through Grant ATM-0130349 and by the Department of Defense through Grant W911NF-06-1-0439. Computing resources for this study were provided by the National Center for Atmospheric Research, which is sponsored by the National Science Foundation. We gratefully acknowledge Dr. John C. Wyngaard for his suggestion on the use of wave-cutoff filter and a careful review of the original version of this manuscript. We thank George H. Bryan for his review of a draft of this manuscript as well as for providing the numerical model used in this study. We also thank Mark Kelly for his help with the use of the wave-cutoff filter, and Brian P. Reen and David P. Tyndall for their helpful discussion and comments on a draft of this manuscript.

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The mean and maximum values of the sensible heat flux difference between flux tower sites 1 and 2 for four of the 5 days* studied by KDL07, which presented the monotonically varying surface heat flux variation over the 16-km transect between sites 1 and 2 during the 5 days; *H*_{1} and *H*_{2} indicate the surface sensible heat fluxes measured at sites 1 and 2, respectively. These surface heat fluxes are the averaged values over 1130–1430 LST. The number in parentheses represents the value of the case for all the fair-weather days** during IHOP.

Experimental design parameters for the prescribed surface heat flux variation and model output mixed-layer scaling parameters. The domain-averaged value of surface heat flux is set at 〈*F _{wθ}*〉(

*z*= 0) = 0.20 K m s

^{−1}.