The formation of fair-weather cumuli (FWC) has been analyzed in this study based on both a simple mixed layer model and a subset of the data collected from the Atmospheric Radiation Measurement (ARM) Program at the southern Great Plains (SGP) site. By analyzing conditions for the formation of FWC, the authors illustrate how different processes—such as the surface heat fluxes, the surface thermodynamic conditions, the entrainment processes at the boundary layer top, the vertical thermodynamic structure above the boundary layer, and large-scale subsidence—control the formation of clouds. The results of the analysis show that it is the highly nonlinear interaction among these factors that gives rise to the formation of FWC. For this reason, the occurrence of FWC may not simply follow changes in the surface conditions. The analysis indicates that the entrainment of moisture and surface processes play important roles in the formation of FWC, and the net effects of these processes can be evaluated by a parameter (l − β2)/B, where β2 is the ratio between the entrainment moisture flux and the surface moisture flux, and B is the extended Bowen ratio defined as the ratio of the surface buoyancy flux to the surface latent heat flux. The stratification above the inversion is another key parameter that influences cloud formation. The weaker the stability, the greater the potential for cloud formation. In most situations the net effect of subsidence is to reduce the relative humidity at the top of the mixed layer and thus is unfavorable for cloud formation, but the intensity of this reduction may vary depending on conditions of the boundary layer. In some specific conditions such as a moist boundary layer over an area with relatively small surface Bowen ratio, the net effect of subsidence on the relative humidity budget at the top of the mixed layer can be weak even though subsidence reduces the mixed layer depth substantially. In this study, some issues related to cloud onset and fractional cloudiness are also discussed based on the ARM SGP observational data.
It is well known that low-level clouds not only have a large impact on the boundary layer (BL) structure, but also strongly affect the earth's radiation budget (Albrecht 1981). How to represent these clouds appropriately in models is an important question for both climate simulations and weather forecast. Among all the low-level BL clouds, individual nonprecipitating shallow fair-weather cumuli (FWC) are the most difficult to treat since their lifetime and space scales are so short and small that it is impossible to resolve the cloud-related processes in large-scale models, or even mesoscale models. Instead, parameterizations have to be applied to include the cumulative effects of FWC on the resolvable scales in models. Thus, it is essential to understand how different processes and their interaction govern the formation and abundance of FWC.
Daytime FWC over land usually develop from convective thermals rooted in the lowest part of the BL (Stull 1985), which can be confirmed by observations that surface layer air can rise almost undiluted to the top of the mixed layer (ML; Crum et al. 1987). These undiluted thermals are most likely to form clouds. As a result, the actual cloud-base height should equal the lifting condensation level (LCL) of the surface layer air, and indeed this coincidence is often verified by many observations, such as Stull and Eloranta (1984, 1985) and Betts and Ball (1998). A strong link between the atmosphere and the underlying surface is also emphasized in many numerical simulations. The simulations done by Wetzel and Boone (1995) indicate that the BL cloud amount, the surface soil subgrid heterogeneity, and the liquid water budget are closely related to the surface energy balance serving as the central driving process for the entire system. The modification of the atmospheric BL structure due to land surface processes is also reported by Pielke et al. (1997).
Despite the strong feedback between the atmosphere and the underlying surface that has been suggested by previous studies, no simple relation between the formation of FWC and the condition of the underlying surface has been reported so far; instead, a complicated relation is often indicated by observations. Figure 1 gives an example of some cloud-related observations from the Atmospheric Radiation Measurement (ARM) field experiments. The rainfall, soil moisture, cloud fraction, and mean cloud-base height in June 1997 observed at the southern Great Plains (SGP) site are presented in this figure. The cloud cover is the percentage calculated from the ceilometer observations from 1200 UTC (0700 LST) to 2400 UTC (1900 LST). Since, in this study, only BL clouds are of concern, the high-level clouds observed by ceilometer are removed. Surprisingly, it appears that the occurrence and amount of FWC do not increase during the periods when there is high soil moisture, a condition that is usually believed to be a favorable condition for cloud formation, since high soil moisture is normally associated with a low LCL. Thus the observations strongly imply that in addition to the surface process, other physical processes may also play very important roles in the formation of FWC; but the relation among these processes is not well understood. Our current understanding of when and under what condition these clouds can form, and what controls the fractional cloudiness is incomplete.
Over the past decades many aspects of FWC have been widely studied from the structure of the trade wind cumulus BL to cloud parameterizations. Only a few studies really focus on the cloud formation problems. By deriving the tendency equation of the relative humidity at the top of the BL using a ML model, Ek and Mahrt (1994) examined the dependence of this tendency on soil moisture, surface fluxes, and the stratification above the BL. Since they did not fully apply the ML theory to this tendency equation and did not really consider the interaction among different processes, they failed to find the conditions for cloud formation and were unable to address the importance of some controlling parameters in the process of cloud formation. Thus, their results only answer part of the questions associated with cloud formation. Based on the relation between the LCL and the cloud base of cumuli, Wilde et al. (1985) studied the onset of cumuli, and found that cumulus clouds first form when the top of the entrainment zone, defined as the penetration region near the top of the BL, reaches the bottom of the LCL zone (the layer between the highest LCL and the lowest LCL in a certain area), and the cloud cover increases as more of the entrainment zone overlaps and extends above the LCL zone. But one important question that they did not address is under what conditions does the LCL zone overlap the entrainment zone. Consequently, why and how FWC can form under different situations are still unanswered questions.
To understand the basic physics underlying the formation of FWC, and possibly to develop a physically robust cloud parameterization scheme, many questions need to be answered. For example, how do surface processes interact with other processes to control the formation of FWC? What exact role do surface sensible and latent heat fluxes play in the cloud formation? To what extent can surface heat fluxes influence cloud formation? And how can the effects of surface fluxes on cloud formation be evaluated? The purpose of this study is to answer these questions. From a cloud parameterization perspective, such a study may also contribute to solving problems associated with cloud parameterization even though this work itself is not about parameterization.
The evolution of FWC may further be complicated by the mesoscale features or circulations induced by inhomogeneities of the underlying surface. It has been found that the development of the BL can vary significantly across an inhomogeneous area due to differential heating and cooling of the surface (Segal et al. 1988; Pinty et al. 1989; Pielke et al. 1991; Avissar and Chen 1993; Hong et al. 1995). A strong relationship between FWC and the inhomogeneous surface is confirmed by observations (Crum and Stull 1987; Schrieber et al. 1996). Using numerical simulations, however, Zhong and Doran (1997) argued that the formation of BL clouds is far more sensitive to ambient meteorological conditions than to spatially varying surface fluxes, even when surface fluxes vary significantly across the study domain. Thus the effects of variability in surface fluxes due to inhomogeneities on BL clouds need to be reexamined. But this is beyond the scope of this study. The purpose of this work is to illustrate how surface processes combined with the processes within and above the BL influence the formation of FWC. We will show that even for a homogeneous underlying surface, the formation of FWC is a complicated phenomenon and depends on the interaction among different BL processes. One good example of these is the trade wind cumuli where conditions at the surface are quite uniform, but a myriad of cloud structures are observed. The approach of this study will be both theoretical and empirical. In the next section we will present a description of the ARM observational data. Then based on simple ML theory and data analysis, a detailed discussion of the conditions needed for the formation of FWC will be presented in section 3. In section 4, we will discuss some issues related to cloud onset and fractional cloudiness. Finally, a summary discussion is presented.
2. Data processing
The data used in this study are from the ARM SGP Cloud and Radiation Testbed (CART) located on a 350 km × 450 km area centered at 36°36′N, 97°29′W in north-central Oklahoma. Detailed descriptions of the site and the experiment can be found in official reports released by the U.S. Department of Energy (DOE 1990, 1996), and many other publications, for example, Stokes and Schwartz (1994). The ARM SGP site includes many different observational facilities and supports a wide range of in situ and remote sensing observational systems. Many instrument clusters have been placed around the CART site; one central facility, which is heavily instrumented; and more than 30 other facilities called boundary, extended, and intermediate facilities, respectively. The analyses in this study are based on the observations made in 1996 and 1997. The following summarizes the data used in this analysis.
a. Surface data
Surface measurements are made in the SGP CART site using different observational systems. In this study only the data collected by the Surface Meteorological Observation System (SMOS), the Energy Balance Bowen Ratio (EBBR) System, and the Eddy Correlation (ECOR) Systems are used. The SMOS provides conventional in situ meteorological data, while the EBBR and ECOR are ground-based systems using in situ sensors to estimate the vertical fluxes of sensible and latent heat at the local surface. Fluxes estimated by the EBBR are from observations of net radiation, soil heat flow, and the vertical gradients of temperature and relative humidity using the Bowen ratio energy balance technique (Kanemasu et al. 1992). The ECOR calculates surface fluxes by the eddy-correlation technique, that is, by correlating the wind component with the sonic temperature and the water vapor density (which is obtained by an infrared hygrometer). For many years, the SMOS and the EBBR have collected almost complete data series that allow us to study the effect of surface processes on BL clouds at different timescales. The data collected by the ECOR are not as complete as those by the EBBR, but provide additional data sources on surface flux information. The surface flux data used in this study are mainly from the EBBR observations at the central facility. The observations from boundary, extended, and intermediate facilities also provide the spatial distribution information of different variables over the entire SGP site, which allows us to evaluate the effects of horizontal advection. The locations of different facilities at the ARM SGP site are shown in Fig. 2.
b. Sounding data
The upper-air system that supports the ARM SGP field experiment is called the Balloon-Borne Sounding System (BBSS). The BBSS provides vertical profiles of both the thermodynamic (temperature and relative humidity) and dynamic (wind speed and direction) state of the atmosphere. The BBSS is installed at the central facility and the four boundary facilities located near the mid-point of one side of the CART locale rectangle (see Fig. 2 for the locations). All the facilities use the same type of radiosonde. Under standard operations, five soundings (0530, 1130, 1430, 1730, 2030 UTC) are launched per day at the central facility, and one sounding (1730 UTC) per day at the boundary facilities. These soundings provide reliable high-resolution (about 10 m) information on the vertical structure of the BL. In this study, the thermodynamic structure within and above the BL is determined by the soundings launched at the central facility. Soundings from four boundary facilities are used to estimate subsidence and horizontal advection.
c. Cloud observation data
The ARM SGP field experiments provide several kinds of cloud and radiation observations. But since this paper focuses on the issues related to cloud formation, not the fine structure and microphysics of clouds themselves, only the data collected by the Belfort Laser Ceilometer (BLC) and the Micropulse lidar (MPL) are used in this study. Since the BLC or MPL may not operate for some periods due to technical reasons, to get relatively complete cloud information, we combine the data from both instruments, that is, clouds are considered existing whenever the BLC or MPL detects them. The BLC provides high temporal (30 s) and vertical (7.6 m) resolution of cloud-base information. The MPL is a ground-based optical remote sensing system designed primarily to determine the altitude of clouds overhead, but with a relatively coarse resolution. In this study, the time series of the BLC and MPL observations at the central facility are used to calculate the BL cloud cover fraction, which means we assume that Taylor's hypothesis is applicable to FWC. This may not be a very good assumption considering that large eddy structures such as thermals and plumes are not really “frozen” under some conditions. But since the spatial cloud coverage is not obtained from observations, the temporal one is an acceptable alternative when accuracy requirement is not a high priority. To evaluate how well this calculation represents the spatial fractional cloudiness, the calculated cloud cover fractions are compared with the observations from the Whole-Sky Imager (WSI). Since the WSI monitors the whole upper hemisphere including clouds at all levels, it usually overestimates the fraction of BL clouds. To make this comparison meaningful, we select the cases without high-level clouds. Figure 3 shows four examples of the comparison. For all these cases only low-level clouds were existing. Although it is difficult to provide a comprehensive evaluation of the WSI cloud fraction with the temporal cloud fraction calculated from the BLC, the case comparisons here at least indicate that the temporal fractional cloudiness estimated using the BLC and MPL data is a good alternative of spatial coverage under some circumstances. In fact, obtaining spatial coverage of FWC is a big challenge in boundary layer research. So far, no appropriate method can provide reliable cloud coverage data. The BLC and MPL observations at least provide an effective, economical, and objective way to get such information—unless a new instrument is being developed. As for this study, since we focus mainly on cloud formation issues, the fractional cloudiness is not critical to the results.
3. Formation of FWC
FWC are the products of the development of the convective BL (CBL) in which buoyancy is the dominant mechanism driving turbulent mixing. Such turbulence is not completely random, but often has recognizable structures such as thermals and plumes. The rise of a convective thermal is limited by an inversion layer that usually caps the ML. Whether FWC can form or not depends on the height that thermals can reach and the thermodynamic properties of thermals. Since the intense vertical mixing in the CBL tends to result in a layer with conserved variables such as the virtual potential temperature and specific humidity nearly constant with height, the bulk ML model is the simplest but most efficient representation of the CBL. One advantage of the ML model is that it avoids solving the fine structure of the CBL, but still retains the ensemble effects of the physical processes in the CBL. For this reason, we believe that it is appropriate to use an ML model to study the influence of BL processes on the formation of FWC. But the model will not be adequate to treat the cloud layer once clouds form, since we do not include the cloud radiative effects and phase changes in the model.
a. Mixed-layer model
The mean structure of the CBL can be represented in different complexity. The main difference between various versions of the ML model is how to deal with the structure of the inversion layer. In this study, we use a zero-order jump model that is illustrated in Fig. 4. In the zero-order jump model, the inversion layer is assumed to be infinitesimally thin. For cloud formation problems, this assumption is appropriate since the influence of the heat capacity and momentum deficit of this thin inversion layer is negligible compared with the contribution of the whole ML. For the clear CBL, the virtual potential temperature and moisture budget equations within the ML under horizontal homogeneous condition can be simplified as (Tennekes 1973)
where w′q′ and w′θ′υ are the kinetic moisture and buoyancy fluxes, respectively; the subscripts o and h refer to the surface and the level just below the top of the ML; h represents the depth of the ML. To first order, air density in the ML can be assumed constant, then the continuity equation can be written as
where E is the entrainment velocity and Wh is the mean vertical velocity at the top of the CBL. For the zero-order jump assumption, E can be represented as (Garratt 1992)
where Δθυ is the difference of the virtual potential temperature across the inversion, which is governed by
where γθ is the mean potential temperature lapse rate above the CBL, and we neglect the influence of moisture on the potential temperature above the CBL.
The entrainment fluxes at the top of the ML can be written as
where β1 and β2 are the proportionality factors, and their values represent the relative strength of the entrainment fluxes with respect to the corresponding surface fluxes (e.g., Betts 1973; Betts et al. 1990; Tennekes and Driedonks 1981).
Before we try to solve Eq. (6), first let us consider the growth of the CBL. Entrainment induced by the positive buoyancy always tends to deepen the CBL, while subsidence tries to reduce this growth. If other conditions are the same, then the maximum growth occurs when there is no subsidence. This is one extreme. The other extreme will be the case when the CBL stops growing if subsidence is strong enough to perfectly balance the entrainment growth. In some particular conditions, subsidence may be stronger than entrainment. However, if such a condition remains for a certain period of time, the CBL would collapse and invalidate the ML approach. Thus, any other situations will be somewhere in between these two extremes. For the first extreme, that is, no subsidence (Wh = 0), Eq. (6) degenerates into a homogeneous differential equation. It has a solution (Betts 1973)
To obtain Eq. (7), we have assumed that β1 is constant. We will discuss this assumption later. Equation (7) indicates that the strength of the inversion increases with the growth of the CBL. The strong inversion will hinder the entrainment growth as indicated by Eq. (3). Thus without this constraint, the CBL would never stop growing because there would be no mechanism to constrain entrainment.
For the second extreme, that is, entrainment perfectly balancing subsidence (E + Wh = 0), Eq. (6) becomes
With this relation, we can find the constant ML depth h in this case, which is
The similarity between Eqs. (9) and (7) ensures continuity when condition switches from one to the other. As we indicated before, since all the other situations should be bounded by these two extremes, it is reasonable to assume the general solution of Eq. (6) as
where α is a subsidence-dependent parameter with a value of 1 ≤ α ≤ 2. For Wh = 0, α = 2; and when ∂h/∂t = 0 (E + Wh = 0), α = 1.
This equation describes how the ML deepens under certain external forcings.
b. Boundary layer top entrainment
Entrainment is a very important process for the development of the CBL. In the zero-order jump model, the entrainment process is simplified and can be described by the entrainment rate E, which represents the mean CBL growth in the absence of subsidence, and the two dimensionless parameters β1 and β2, which represent the characteristics of the entrainment fluxes. Previous studies indicate that β1 can be taken as a constant, such as 0.2, although the value estimated from observations may vary from 0.1 to 0.5, or even larger in a shear flow (Ek and Mahrt 1994). In the previous section, to solve Eq. (6) we assumed that β1 is constant. If this assumption is not valid, then Eq. (11) and the subsequent Eq. (12), two equations that we rely on for the rest of the analyses, become questionable. Thus, we need to examine the validity of this assumption. Although no complete explanation of why β1 is constant has been given, some studies do provide good justification. For example, Stull (1976) argued that β1 is quasi-constant since a fixed amount of the energy input at the surface is used to entrain air from above, while the rest being dissipated. But this energetic argument is certainly not enough to prove a constant β1. Considering the difficulties of proving the assumption theoretically, using observational data to verify Eqs. (11) or (12) seems more realistic.
where δθυ and δh are the changes of the virtual potential temperature within the ML and the depth of the ML in a certain time interval δt. If β1 is a constant with the suggested value (0.2–0.5), the quantity (1 + β1)/(1 + αβ1) should also be an approximate constant since α varies between 1 and 2. Then, Eq. (13) describes a nearly linear relationship between the quantity δθυ/(δh − Whδt) and γθ. Thus, Eq. (13) provides an alternative way to verify the derived relations or to test whether β1 is a constant parameter as long as we know the values of δθυ/(δh − Whδt) and γθ. Unlike the conventional method for calculating β1, no flux data, which usually do not have a quality as high as that of conventional meteorological data, are needed in this method. In this study, the soundings launched at 1730 UTC (1230 LST) and 2030 UTC (1530 LST) at the central facility of the ARM SGP site in 1996 and 1997 are used for this calculation. Compared with δh and δθυ, γθ is relatively difficult to determine, because the thermodynamic structure above the CBL is often influenced by large-scale synoptic flows and cannot always be characterized by a constant γθ. To avoid the influence of this complexity on the accuracy of our calculation, only the soundings with simple thermodynamic structure (can be represented by a constant γθ) and with little horizontal advective influence are selected for this calculation. Figure 5 shows two examples of these soundings. In both cases the thermodynamic structure within and above the CBL is very simple and γθ can easily be determined. We also note that the profiles above the CBL show no substantial change from 1730 UTC (1230 LST) to 2030 UTC (1530 LST), which implies that horizontal advection is very weak for these two cases.
Since it is sometimes difficult to estimate the height of the CBL from the virtual potential temperature profiles, in this study we follow Mahrt (1976) and estimate the height of the CBL using the relative humidity profiles since the relative humidity profile combines the influence of decreasing moisture and increasing potential temperature with height to provide a sharper delineation of the CBL. In summary, the cases selected for this calculation must satisfy two requirements: 1) weak horizontal advection, and 2) simple stratification (represented by a constant γθ) above the CBL. For the first requirement, we first examine the potential temperature and specific humidity profiles above the CBL, and exclude those cases with substantial changes between 1730 UTC (1230 LST) and 2030 UTC (1530 LST). Second, we estimate the horizontal advective effects within the CBL using sounding data or surface observation data from different facilities distributed over the ARM SGP study area. A budget analysis (comparison among local change rate, horizontal advective rate, and the rate related to turbulent transport) is then done to determine which cases to use. Only those cases with horizontal advective rates less than 10% of the local change rates or turbulent transport rates are selected. Detailed descriptions and some examples of the budget analyses will be provided later. With these requirements, 24 cases are selected from 1996 and 1997 for this calculation.
Another difficulty is how to determine large-scale subsidence, since no direct observation is available for the ARM experiment. One method of estimating the mean vertical velocity is to integrate the continuity equation in the vertical to give
The horizontal divergence can be inferred from the horizontal wind fields. Fortunately, in the ARM experiment, soundings are launched at five different facilities at the same time every day (1730 UTC), which can be used to estimate the horizontal divergence. Detailed information on how to use wind field data to deduce the vertical velocity is given in Holton (1992). The calculation results are shown in Fig. 6. A nearly linear relation between δθυ/(δh − Whδt) and γθ is clearly indicated in the figure. This result is expected if β1 is constant. Therefore, the observations confirm that a constant β1 assumed in many studies is a fairly good approximation and Eq. (11) does describe a real connection between the strength of the inversion and the height of the CBL even though both vary in time.
Unlike β1, observations show that β2 is not a constant, it can either exceed unity, or become 0.5 or less (e.g., Mahrt 1991; Betts et al. 1990; Grant 1986; and others). But few of these studies explain why the behavior of β2 is different from that of β1. With this ML model, we can explore this issue a little further. Following Eq. (3), we may introduce a similar equation for the specific humidity
where Δq is the difference of the specific humidity across the inversion.
where L and Cp are the enthalpy of vaporization and the specific heat at constant pressure, respectively. Here, B is the extended Bowen ratio defined as B = (ρCpw′θ′υ0)/(ρLw′q′0). We also define the surface Bowen ratio Bo as Bo = (ρCpw′θ′0)/(ρLw′q′0). Using the definition of the virtual potential temperature, that is, θυ = θ(1 + 0.608q) ≈ θ + 0.608θq, it is easy to find the relation between B and Bo as
where θ is the mean potential temperature.
As indicated previously, for a certain stratification above the CBL, the strength of the inversion and the depth of the ML are constrained by Eq. (11). No such constraint, however, acts on the moisture budget, which means that the moisture difference across the inversion can be any value regardless of other variables. Equation (16) provides a way to estimate the value of β2 once the moisture jump across the inversion, the mean stratification above the ML, and the extended Bowen ratio are known.
The wide range of β2 can be inferred from the mean specific humidity profiles within the ML. Using Eqs. (1b) and (5b), it is easy to show that β2 > 1 corresponds to a decrease of the specific humidity with time, while β2 < 1 gives an increase of the specific humidity with time. Figure 7 shows four examples of the specific humidity profiles selected from the ARM soundings. It is very clear that the mean specific humidity within the ML can either increase or decrease with time, which implies that β2 can be either greater than or smaller than unity.
Obviously, the change of the mean specific humidity within the ML can also be affected by horizontal advection, but this is not the case for these four examples since observations indicate that the horizontal advection is very weak, which can be inferred from the overlapped profiles above the CBL from 1730 UTC (1230 LST) to 2030 UTC (1530 LST). Within the ML the horizontal advective effects can be estimated from the horizontal distributions of the specific humidity and wind fields over the SGP site shown in Fig. 8. Here, the specific humidity and wind are the averaged variables over the entire ML at 1730 UTC (1230 LST) from the sounding data at the central facility and the four boundary facilities. Based on a triangle-based cubic interpolation technique, we estimate the horizontal advective rate of the specific humidity for these cases. Table 1 shows the results of the budget analysis. We see that the calculated horizontal advective rates are very small compared with the local change rates or the rates associated with the turbulent transport. These four examples indicate that the entrainment moisture process is much more complicated than the entrainment process associated with buoyancy although they are related. The reason is probably because the strength of the inversion is constrained by Eq. (11), while no such constraint on the difference of moisture across the inversion exists. Thus, even with the same entrainment rate, the characteristics of the induced entrainment moisture flux can be very different from that of the induced entrainment buoyancy flux, which is usually a fixed portion of the surface buoyancy flux. For this reason, the entrainment moisture flux could be the key for understanding the formation of FWC.
c. Favorable conditions for cloud formation
To find out the conditions favorable for cloud formation, the most obvious approach is to consider how different processes control the variation of the thermodynamic fields at the top of the ML. Hence, it is useful to analyze the tendency equations of the related variables. With the results that we get from the previous analysis, the tendency equations of the virtual potential temperature and the specific humidity can be reorganized into a form that is easier to analyze. With Eqs. (1), (5), (12), and the definition of the extended Bowen ratio, it can be shown that
Next, we consider the tendency of the saturated specific humidity at the top of the ML (qsh). If we take the approximate form of qsh, that is, qsh = 0.622esh/Ph, then the tendency of qsh can be written as
where esh, Ph, and Th are the saturated water vapor pressure, pressure, and temperature at the top of the ML, respectively. Also, T0 and P0 are the surface temperature and pressure, Rυ and Rd are the gas constant of water vapor and dry air, g is the acceleration due to gravity, and γd is the adiabatic lapse rate. To get Eq. (20), we have used the Clausius–Clapeyron equation and the hydrostatic equation.
where we have used the definition of the virtual potential temperature, the relation L/RυT2h ≫ [g(θ − Th)]/RdγdθTh, and neglected [g(θ − Th)]/RdγdθTh in Eq. (21).
For the cloud-free CBL, qsh is greater then q. If (qsh − q) decreases with time, then cloud formation is possible. Thus, the necessary condition for cloud formation is
where C1 = (1/C2 + 0.608θ)Cp/L, and C2 = Lqsh/RυT2h. For typical values of Th from 273 to 310 K, C2 increases from 2.8 × 10−4 to 2.2 × 10−3 and C1 decreases from 1.5 to 0.26. Note that to obtain Eq. (23), we have applied the relation Lγd/RυTh ≫ g/Rd and neglected g/RdTh in Eq. (21).
The left-hand side term of Eq. (23) represents the ML warming induced by entrainment, an unfavorable effect to cloud formation. The first term on the right-hand side of Eq. (23) represents the growth of the CBL, a favorable effect to cloud formation, and the second term on the right-hand side of (23) describes the ML moistening/drying induced by entrainment depending on whether β2 is smaller or greater than unity. Thus, this term could either favor or hinder cloud formation. It is the net effect of these three processes that determines cloud formation. This may explain why there is no clear correlation between the occurrence of FWC and the surface hydrological conditions shown in Fig. 1 since the relative importance of these three processes may vary under different external forcings and ambient meteorological conditions.
1) Effect of γθ
With Eq. (23), we may explore how the stratification above the CBL influences the formation of FWC. To do so, we consider the following two specific cases.
- In the absence of subsidence (Wh = 0), Eq. (23) degenerates to Eq. (24), we have assumed β1 = 0.2. If γθ > 1.17γd, then the right-hand side of (24) is positive. Equation (24) is true only when β2 < 1, which means that for clouds to form the entrainment moisture process has to be weak enough so that there is a moisture convergence within the ML. Although Figs. 7a and 7d show two examples of moisture convergence within the ML, the statistics based on the ARM SGP observations indicate that this situation does not occur very often. On the other hand, if γθ < 1.17γd, then the right-hand side of (24) is negative, and in this case there is no particular restriction on β2. To form clouds, β2 could be either greater or less than unity as long as Eq. (24) is satisfied. Thus, even with a moisture divergence within the ML, clouds still can form as they did for 14 August 1996 and 6 July 1997, shown in Fig. 7. The ARM SGP observations show that FWC usually form under this condition. Thus, we may conclude that the weaker the stratification above the CBL, the greater the chance for cloud formation. Equation (24) also indicates that for a given γθ, the quantity (1 − βq)/B is the key parameter for cloud formation.
Since the entrainment rate E does not explicitly appear in (24), it would appear that E is irrelevant to cloud formation. But this is not correct since (24) is only a necessary but not a sufficient condition for cloud formation. If condition (24) is satisfied, E basically determines the time needed for clouds to form because this is the parameter representing the growth of the CBL. Clouds will form earlier for a larger E, but will take a longer time to form when E is smaller, and even may not form if E is sufficiently small.
- When subsidence perfectly balances entrainment (E + Wh = 0), with β1 = 0.2, Eq. (23) becomes
Equation (25) indicates that under this condition the formation of clouds is not related to the stratification above the CBL and only depends on the parameter [C1(1 − β2)]/B. For this particular case, a necessary and sufficient condition can be found if we consider this problem from the relative humidity point of view. It can be shown that the relative humidity at the top of the ML (rhh) can be written as (see appendix B)
where rh0 is the relative humidity in the surface layer, and C3 = Lγd/RυT20. Typical values of C3 are C3 ≈ (6.35 ± 0.78) × 10−4 m−1 with T0 from 273 to 310 K. Equation (26) indicates why the relative humidity profiles usually increase linearly with height at a similar rate (C3) within the ML as indicated by observations all over the world. Figure 5 gives two examples of relative humidity profiles.
A necessary and sufficient condition for cloud formation is that rhh has to be greater than unity, which gives
Equation (28) indicates that for FWC to form under this condition, the surface buoyancy flux has to be greater than a value determined by the surface relative humidity (rh0) and external conditions, such as Wh and γθ. For example, for a strong subsidence case (Wh = −0.02 m s−1) with a given CBL state and external forcing such as T0 = 300 K, rh0 = 60%, and γθ = 0.005°C m−1, the surface buoyancy flux has to be greater than approximately 100 W m−2 in order to form clouds. This value is possible in the real atmosphere.
2) Effects of subsidence
Investigating the effect of subsidence on cloud formation is not as easy as studying the effect of γθ. The reason is that the effect of γθ on cloud formation can be easily identified, since both (24) and (25) are not explicitly related to the entrainment rate E. However, when subsidence is important, no simplified relation such as Eqs. (24) and (25) can be obtained and we have to analyze the general condition Eq. (23) that involves both E and Wh. This is not easy since entrainment is a process that cannot be isolated from subsidence. Equation (12) indicates a complicated nonlinear interaction among subsidence, entrainment, and the depth of the ML. Thus, the best way to understand the effect of subsidence on cloud formation is to analyze the relative humidity tendency at the top of the ML directly.
The tendency of the relative humidity at the top of the ML (rhh) can be written as,
In Eq. (30), term (d) indicates that subsidence always reduces the tendency of the relative humidity at the top of the ML. This is the effect that can be readily seen since subsidence is directly related to the instantaneous time tendency of relative humidity. However, subsidence may also affect the relative humidity tendency through terms (a), (b), and (c) because of the nonlinear interaction among subsidence, entrainment, and the depth of the ML described by (12). According to Eq. (12), for a finite time, subsidence has an effect to lower the ML depth, which in turn increases the entrainment rate E. Thus, term (c) indicates that subsidence may lead to an increase in the relative humidity tendency at the top of the ML due to the enhancement of entrainment resulted from the accumulated or finite time effect of subsidence. As for term (b), since the reduced ML depth due to subsidence decreases its value (including the minus sign in front), subsidence has a negative effect on the relative humidity tendency at the top of the ML. Physically it means that the reduced h enhances the convergence of buoyancy flux within the ML to increase the temperature. The same applies to term (a) but with a more complicated situation since β2 can either be greater or less than unity. For β2 > 1, subsidence also reduces the relative humidity tendency due to the lowering of the ML height. Physically it means that whenever there is a moisture divergence in the ML, the decreased ML height due to subsidence enhances the divergence to further reduce the relative humidity tendency. For β2 < 1, things are just opposite. The reduced ML height due to subsidence increases the value of term (a) and thus tends to increase the relative humidity at the top of the CBL. In this case, we can say that subsidence enhances the moisture convergence in the ML. Therefore, whether subsidence has a net positive or negative effect on the relative humidity tendency at the top of the ML depends on how subsidence changes the value of these terms.
The first term on the right-hand side of (31) is the combination of terms (a), (b), and (c) in Eq. (30). To estimate the net influence of subsidence on the relative humidity tendency, Eq. (31) is differentiated with respect to subsidence to give
where we have used the approximation that the coefficients a1 and a2 are not sensitive to the change of subsidence, which can be easily verified by scaling analysis. Thus for the net effect of subsidence to increase the relative humidity tendency requires that
To see if this condition can occur in the real atmosphere, we need to first estimate the magnitude of δE/δWh. This can be done by solving Eq. (12) numerically for different Wh, and then calculating the quantity δE/δWh. In this calculation, we consider a strong surface forcing (ρCpw′θ′υ0 = 400 W m−2) case and a weak surface forcing (ρCpw′θ′υ0 = 30 W m−2) case with the same stratification (γθ = 7°C km−1). For each experiment, we chose three different values of subsidence. We also take β1 = 0.2, and leave α as a subsidence-dependent parameter. We note that α always appears in the form of (1 + αβ1), so the value of α should not influence our results too much. In this calculation, we simply assume a linear relation between α and Wh with α = 2 for Wh = 0 and α = 1 for a sufficiently large Wh such that δh/δt ≈ 0. Figure 9 shows the calculated results, which indicate that the range of δE/δWh varies from −0.3 for weak subsidence down to less than −0.6 for strong subsidence and weak surface buoyancy flux. This calculation gives us a basic idea how much the entrainment rate increases in response to increased subsidence.
Second we need to estimate the value of a2/a1. Although α1 and α2 involve many parameters, it is easy to see that a2 is almost a constant and a1 is sensitive to (1 − β2)/B and γθ. Figure 10 shows how a2/a1 varies with (1 − β2)/B and γθ. In this calculation, we take: θ ≈ Th = 290 K, q = 0.010 kg kg−1, β1 = 0.2, and α = 1.5. It is no surprise to see that for a wide range of (1 − β2)/B and γθ values, the estimated values of a2/a1 cannot satisfy Eq. (33) for most of the values of δE/δWh shown in Fig. 9. Thus, for most situations, the positive effect of subsidence on the budget of the relative humidity at the top of the CBL cannot compensate the negative effect of subsidence, which means that subsidence is unfavorable for cloud formation. However, when (1 − β2)/B and γθ both are sufficiently large, the value of a2/a1 may satisfy (33) in a situation with sufficiently strong subsidence and weak surface buoyancy fluxes such that the value of δE/δWh becomes sufficient small. In the real atmosphere, we may obtain large (1 − β2)/B in some specific conditions. For example, in a moistening boundary layer, β2 often becomes 0.5 or less (Grant 1986; Nicholls and Reading 1979). For over-sea Atlantic Trade-Wind Experiment (ATEX) and Barbados Oceanographic and Meteorological Experiment (BOMEX) cases, surface Bowen ratio Bo is around 0.06. A condition like this may give us a sufficiently large (1 − β2)/B. In this case, the overall effect of subsidence on cloud formation may be very weak, since most of the negative effects of subsidence are canceled by the positive effects.
3) Effect of Bowen ratio
Although individual surface Bowen ratios at certain times on certain days are not relevant to the formation of FWC since cloud formation is also related to the other physical processes within and above the CBL, Eq. (34) at least implies that small surface Bowen ratios should correspond to a greater chance of cloud formation. This result can be confirmed by the ARM SGP observations. Figure 11 shows the seasonal variations of the surface Bowen ratio in 1997 from 1730 UTC (1230 LST) to 2030 UTC (1530 LST) and the frequency of cloud formation during this period. For this calculation, only observational data collected at the central facility are used, and we have excluded days related to precipitation, fronts, and whenever the wind speed at the top of the BL exceeds 15 m s−1. Note that the concept of the frequency of cloud formation is different from the fractional cloudiness. The frequency of cloud formation defined here only accounts whether FWC exist regardless of cloud fraction. For example, if one cumulus case has a cloud cover of 30%, but another has a 10% cloud cover, both cases are considered FWC events in this study. If the estimated cloud cover of a specific case is less than 1%, then it is considered to be a clear-sky case. The interval to calculate cloud coverage in this calculation is 3 h from 1730 to 2030 UTC. To obtain Fig. 11, the surface flux data and cloud observation data are grouped into two seasons, namely, winter (December, January, and February) and summer (June, July, and August). The results indicate that the frequency of cloud formation varies with the averaged surface Bowen ratios. A smaller surface Bowen ratio corresponds to a greater chance of FWC, while larger surface Bowen ratios are associated with less chance of cloud formation. The seasonal cycle of the surface Bowen ratio is expected since the surface evaporation process is usually weak when surface temperature is low. This is consistent with the large surface Bowen ratios observed in the winter compared with the small Bowen ratios observed in the summer. However, we note that this relation is not very clear for the spring and fall seasons based on the data of 1997.
To show the overall relationship between the surface Bowen ratio and the chance of FWC, we also group the same data into clear and cloudy categories (Fig. 12). We see that cloudy category is associated with the smaller surface Bowen ratio.
d. A solution for the mixed-layer model system
Equation (23) is a necessary but not sufficient condition for cloud formation. In other words it only provides a potential for the CBL system to produce clouds. Obviously, for certain external forcings, cloud formation also depends on the initial state and the details of the development of the CBL. For certain initial conditions, Eqs. (12), (18), and (21) describe a closed CBL system, and can be solved numerically if we prescribe external forcings, such as the surface fluxes, the stratification above the CBL, and subsidence, and parameters β1, β2, and α. Among these variables, generally surface kinetic buoyancy flux w′θ′υ0, extended Bowen ratio B, and β2 are time dependent. However, to illustrate the basic physics underlying the formation of FWC, we may define time-averaged w′θ′υ0, B, and β2 (〈w′θ′υ0〉, 〈B〉, and 〈β2〉) so that they are independent of time. In this case, we can find the integrated form of Eqs. (12), (18), and (21) (see appendix C). With either differential or integrated system, we can analyze how the initial condition, external forcing, and the entrainment process influence cloud formation. Various cases will be examined in detail. For this simple model, we assume that clouds will form as soon as the relative humidity at the top of the CBL reaches 100%. But we note that in the real CBL clouds can form at a lower relative humidity. We will discuss this in section 4. The individual cases are considered as follows.
Case 1: No large-scale subsidence (Wh = 0). In section 3c we have already shown how the change of γθ affects cloud formation. By solving the closed system, we can investigate the influence of other factors as well. First we examine how (1 − 〈β2〉)/〈B〉, a key parameter representing the net effect of the surface forcing and the entrainment moisture process, influences cloud formation. To do so, we set the initial conditions and other parameters as β1 = 0.2, P0 = 1000 hPa, h0 = 100 m (initial ML height), θ0 = 300 K (initial potential temperature), rh0 = 70% (the initial surface layer relative humidity), which gives q0 = 0.0157 kg kg−1 (initial specific humidity), and ρCp〈w′θ′υ0〉 = 300 W m−2, and then investigate how the system responds to the change of (1 − 〈β2〉)/〈B〉. To illustrate the influence of γθ, we select four different values of γθ: 12°C km−1, 7°C km−1, 4°C km−1, and 2°C km−1. Figure 13 shows the results of the calculation. In this numerical experiment, we calculated the relative humidities at the top of the ML (rhh) for the designated values of different parameters, and then plotted them in the coordinate (1 − 〈β2〉)/〈B〉 versus the height of the ML (h) and the corresponding time. Thus the area in the figure where rhh < 1 indicates no cloud will form for the designated conditions, while the region where rhh ≥ 1 means it is possible for clouds to form but not necessary. Let us first consider the case of γθ = 12°C km−1 (Fig. 13a). Clouds may have difficulty forming in this case, although some combinations of parameters give rhh ≥ 1. For example, when (1 − 〈β2〉)/〈B〉 = 3.0, which is possible in the real atmosphere, the calculation shows that clouds could form since rhh reaches unity somewhere, but it will take an unrealisticly long time to get this point. As γθ decreases, the value of (1 − 〈β2〉)/〈B〉 at which clouds may form also decreases. When γθ decreases to 2°C km−1 (Fig. 13d), clouds can form at a reasonable height within reasonable time for a wide range of (1 − 〈β2〉)/〈B〉. This experiment indicates that the chance of cloud formation increases, and that cloud-base height decreases with the increase of (1 − 〈β2〉)/〈B〉 and decreasing γθ.
Case 2: The effect of the initial surface temperature on cloud formation is shown in Fig. 14. In this numerical experiment, we set γθ = 5°C km−1, and (1 − 〈β2〉)/〈B〉 was assigned to four different values (−0.4, −0.6, −0.8, and −1.0). The other parameters are the same, as in the previous experiment. We examine how the system responds to the change of the initial surface temperature, but with a fixed relative humidity at the surface (rh0 = 70%). We plot rhh in a coordinate of the initial surface temperature versus the height of the ML and the corresponding time. We see that for (1 − 〈β2〉)/〈B〉 = −1.0 (Fig. 14a), clouds cannot form when the initial surface temperature is low. Even for high temperature, it takes a long time to form clouds. When (1 − 〈β2〉)/〈B〉 increases to −0.4 (Fig. 14d), clouds can form for a wide range of the initial surface temperature. Thus, for certain γθ and (1 − 〈β2〉)/〈B〉, the higher the surface temperature, the easier it is to form clouds. This is why more FWC are observed in the summer than in the winter. The calculations once again indicate that the formation of FWC is very sensitive to the value of (1 − 〈β2〉)/〈B〉. For example, when θ0 = 290 K, clouds can still form with (1 − 〈β2〉)/〈B〉 = −0.6, but it is impossible when (1 − 〈β2〉)/〈B〉 decreases to −0.8.
Case 3: In section 3c, we have qualitatively analyzed how subsidence influences cloud formation. With the closed system, we can study this issue in detail. We consider two specific cases with different external forcings, but the same subsidence. Using these two examples, we will illustrate the net effect of subsidence may vary under different conditions. Table 2 shows the initial conditions and external parameters of these two cases. As we stated before, the condition of case (a) is similar to that of over-ocean cases with relatively low temperature, while case (b) may represent some conditions over land. We calculated the development of the ML with and without subsidence. The results are shown in Fig. 15. In case (a), clouds start to form after approximately 3.5 simulation hours. Subsidence hinders the growth of the ML, but the relative humidity tendencies at the top of the ML are about the same with or without subsidence. Subsidence does not delay the time at which clouds start to form. This is because most of the negative effects of subsidence on the relative humidity tendency at the top of the ML are canceled by the positive effects. But we note that the cloud-base height is reduced substantially by subsidence. However, situations are totally different in case (b). Although the height of the ML only reduces slightly, due to subsidence, subsidence does have a significant impact on the relative humidity at the top of the ML. Clouds may form in the absence of subsidence, but with subsidence it is almost impossible to form clouds. This experiment indicates that, even the same subsidence may result in different impact on cloud formation, simply because of different BL systems.
4. Fractional cloudiness
In section 3d, we assumed that the necessary and sufficient condition for clouds to form is that the relative humidity at the top of the ML (rhh) is 100%. But this condition may be too strict for the real atmosphere. This is because the ML model only represents the mean state of the CBL, and is unable to describe the fine structures of turbulence such as thermals and plumes that are actually related to the FWC. As indicated by Stull (1985), the formation of forced cumuli is merely the result of some convective thermals overshooting into the stable layer that caps the ML and rising above their LCL. Thus, to form FWC, rhh does not have to be 100%. Generally, the larger rhh, the greater potential of cloud formation. Then the question becomes, at what value of rhh is the onset of FWC possible? This question can be addressed with sufficient observational data. The long-term ARM SGP data provide a good data source for us to examine this question. A statistical relationship between rhh and the frequency of cloud formation and the fractional cloudiness can be obtained by analyzing these data. Figure 16 shows this relationship based on the ARM SGP observations in 1996 and 1997. The criterion for choosing days is the same as we stated before. We use the sounding data at 1730 UTC (1230 LST) and 2030 UTC (1530 LST) at the central facility to calculate rhh and then take the average. The frequency of cloud formation and the fractional cloudiness in this period are calculated from the BLC and MPL observations, as we stated before. The calculations indicate that both the frequency of cloud formation and the fractional cloudiness start to increase when rhh exceeds 80%, which is quite similar to the results of Ek and Mahrt (1991). But the cloud cover at rhh = 80%, of our statistics, seems a bit smaller than their results. Two reasons may be responsible for this difference. First, our average interval is much longer than theirs; second, the cloud cover determined in this study is temporal not spatial cloud cover. One thing that we should note is that the standard deviation of fractional cloudiness is so large for certain relative humidities that we speculate the relative humidity alone is not enough to parameterize the fractional cloudiness. One of the other most possible parameters controlling cloudiness of FWC could be the CBL stability, since the more unstable the BL, the stronger the overshooting will be. Following Stull (1994), in this study we used a parameter called “dimensionless mixed-layer Richardson number” to measure the stability of the CBL. But no clear connection between the fractional cloudiness and the stability of the CBL has been found based on the ARM data even though the statistics show a weak but highly scattered relation between them (not shown here). The result suggests that other parameters may also affect the fractional cloudiness. For example, our previous analysis indicates that the stratification above the BL and large-scale subsidence have a strong effect on the formation of FWC, it is not a surprise that they are also the most likely factors controlling the fractional cloudiness. But the relationship between them is not clear. Therefore, further study of this issue is needed.
5. Summary and discussion
The occurrence of FWC is one of the most common phenomena associated with the CBL. Intrinsically these clouds are often the visible tracers of the convective thermals rooted in the lowest part of the BL. Although the presence of these clouds has a large impact on modifying the structure of the BL and changing the earth's radiation budget, their formation has received relatively little attention. One reason for this shortcoming is because the importance of cloud formation is overshadowed by other issues, such as cloud maintenance and their influence on large-scale dynamics; the other is probably because of a lack of appropriate observational data. Since 1992, a tremendous volume of data has been collected from the ARM SGP CART site. A principal goal of ARM is to relate radiative properties to the composition of the atmosphere, including water vapor and clouds, and to develop parameterizations that can be used to accurately predict the radiative properties and to model the radiative interactions involving water vapor and clouds. Indeed, ARM provides a unique data source for these studies. Although the ARM data are being gathered in a way to achieve its main goal, there is no doubt that they can be used for other related studies as well. For example, this study shows that the long-term conventional meteorological observations and state-of-the-art remote sensing measurements from ARM provide an excellent data source to back up our analysis on the formation of FWC.
Observations from the ARM SGP site show that there is no direct relationship between the occurrence of FWC and the surface hydrological conditions, which suggests that FWC are the results of a complicated interaction between the surface process and the BL processes. To illustrate how different processes influence the formation of FWC, a detailed theoretical analysis is made based on a simple ML model. The reason that we choose this model as our major analysis tool is that the ML approach allows the relevant processes to be highlighted, thus providing an insight into the basic physics associated with cloud formation. Such an insight can provide for a focus on issues important for the successful simulation and parameterization in large-scale models. Further, the ML framework is useful in the analysis of observations.
Although the ML model itself is not new, for the first time the model is used to investigate systematically how the BL processes, external forcings, and their interactions affect the formation of FWC. This simplified approach provides many promising and interesting results. Some relations describing the formation of FWC resulting from this analysis may not be easy to obtain directly from a comprehensive model. The basic findings of this study are as follows.
The analysis indicates that there is a constraint between the strength of the capping inversion, the depth of the ML, and the above inversion stability despite that both the strength of the inversion and the depth of the ML vary in time. Large-scale subsidence affects this relation indirectly through reducing the depth of the ML. However, no such constraint can be found on the moisture field. Using the ARM sounding data, our analysis confirms that β1, the ratio between the entrainment buoyancy flux and the surface buoyancy flux, is quasi-constant; but β2, the ratio between the entrainment moisture flux and the surface moisture flux, depends on many factors, such as, the surface Bowen ratio and the difference of the specific humidity across the inversion. As a result, β2 has a wide range of values and can be either greater or less than unity, which means both moisture divergence and convergence can happen in the ML. For this reason, β2 is a parameter more sensitive to cloud formation than β1, as far as the entrainment fluxes are of concern. By analyzing the variability of the thermodynamic fields at the top of the ML, we derived the necessary condition for cloud formation. Basically this relation connects the three effects of entrainment on the development of the CBL, namely, warming, drying/moistening, and rising of the CBL. The relation also indicates that the net effect of the interaction between the surface process and the entrainment process on cloud formation can be evaluated by a parameter (1 − β2)/B.
The above inversion stability γθ is a very important external parameter that affects cloud formation. As γθ decreases it is usually easier to form clouds. For the case when the subsidence is negligible, if γθ < γd(1 + 2β1)/(1 + β1), then there is no restriction on the entrainment moisture process for cloud formation. If γθ > γd(1 + 2β1)/(1 + β1), then only when β2 < 1, will cloud formation be possible. It is also possible to form clouds in a state where subsidence perfectly balances the entrainment growth rate. But in this case, γθ is not directly relevant to the condition of cloud formation.
The influence of subsidence on cloud formation is related to the interaction between the surface process and the entrainment process. By analyzing the tendency of the relative humidity at the top of the ML, we illustrate that subsidence can affect the relative humidity budget at the top of the ML in two ways. First, subsidence directly reduces the instantaneous time tendency of relative humidity at the top of the ML, an unfavorable factor to cloud formation. Second, due to the nonlinear interaction among subsidence, entrainment, and the ML depth, the accumulated effects of subsidence can lead to an enhancement of entrainment, warming, and moistening/drying of the ML. Such accumulated effects of subsidence can either strengthen or weaken the instantaneous effect, and the magnitude of the enhancement or reduction may vary for different conditions. As a result, subsidence may act as a major obstacle to cloud formation in some conditions, while in the other conditions subsidence may not be that important in influencing cloud formation, even though it reduces the height of the ML. Overall, the net effect of subsidence is to reduce the relative humidity at the top of the ML and therefore is unfavorable for cloud formation in most situations. But for a moist BL with relatively small surface Bowen ratio, the negative effect of subsidence may be largely canceled by the positive effect. In this case, the net effect of subsidence on reducing the relative humidity at the top of the ML may be much smaller than expected.
Our study indicates that the frequency of cloud formation and the fractional cloudiness increase with the increase of the relative humidity at the top of the ML (rhh). Significant cumuli may form when rhh reaches 80%. The observations of the ARM SGP experiment also indicate that relative humidity may not be the only important parameter controlling the fractional cloudiness. To successfully parameterize cloud fraction, we need to consider other parameters as well. But no promising results are found based on this study.
The results from this analysis show that the simple ML model is very useful for the study of the formation of FWC. But due to the limitation of the model, the research is incomplete in the sense that some physical processes are oversimplified and some are neglected. For example: 1) the model only describes the mean state of the BL, so that it is unable to address problems associated with turbulence; 2) although wind shear is usually considered a minor factor for the development of the CBL, in some cases the shear across the BL may introduce extra turbulence and thus influence the entrainment process. We still have little knowledge about how wind shear influences the formation of FWC; 3) Our analysis indicates that the BL entrainment process plays a very important role in the formation of FWC, but from cloud fraction perspective, the entrainment process is treated too simply. Thus, no information about the fractional cloudiness can be obtained from the model itself. In spite of these deficiencies, the results from this study give us a basic picture of how different processes influence the formation of FWC and may provide some useful guides for future research. This study also provides the framework for effectively using datasets like those collected by the ARM program to do a comprehensive study of the factors that affect cloud formation and fractional cloudiness.
This research was supported by Department of Energy under Grant DEFG0297ER62337. The data used in this paper were obtained from the ARM Data Archive. We are grateful to the two anonymous reviewers, for their careful reading of the manuscript and their many constructive comments. Their helpful suggestions led to improvements in this paper.
Solution for Eq. (8)
Equation (8) can be reorganized as
It has a general solution:
Vertical Profile of Relative Humidity
The vertical gradient of the relative humidity within the ML can be written as
where we note ∂q/∂z = 0 and ∂T/∂z = −γd within the ML.
With qs ≈ 0.622 (es/P) ∂P/∂T = Pg/RdγdT, and the Clausius–Clapeyron equation, we can show that
Integrating Eq. (B3) from surface to the CBL height h gives
We now use the approximation
Eq. (B4) then becomes
Using a Taylor series expansion of the exponential function with a scaling analysis yields
Integrated Form of the System
Integrating Eq. (12), we may find, for Wh = 0,
and for Wh ≠ 0,
Integrating Eq. (18) yields
Integrating Eq. (C5) yields
Corresponding author address: Ping Zhu, MPO/RSMAS, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149-1098. Email: email@example.com