1. Introduction
The land surface and the overlying atmosphere interact through a complex loop of feedback processes, which couple the energy and water cycles. The coupling between the land surface and the atmosphere is mediated by the state of the surface, which modifies the partitioning of the surface energy budget (SEB; Charney et al. 1975; Charney 1975; Delworth and Manabe 1989; 1993; Koster and Suarez 1994; Milly and Dunne 1994; Robock et al. 1995) and surface water budget (Rodríguez-Iturbe et al. 1999b; Rodríguez-Iturbe 2000; Laio et al. 2001; Porporato et al. 2004; Katul et al. 2007; Rigby and Porporato 2006) over time scales ranging from seconds–minutes to seasonal–interannual (Xue and Shukla 1993; Delire et al. 2004; Notaro 2008; Katul et al. 2007) and on spatial scales ranging from millimeters to hundreds of kilometers (Li and Avissar 1994; Rodríguez-Iturbe et al. 1995, 1999a, 2006; Avissar 1995; Plaza and Rogel 2000; Wheater et al. 2000; Western et al. 2002, 2004; Ronda et al. 2002; Skoien et al. 2003; Isham et al. 2005).
The variations in land surface properties modify the water and energy flux partitioning at the land surface, which affect the state of the overlying atmosphere (turbulence, heat, moisture, stability, clouds, precipitation, and dynamics) (Gentine et al. 2007, 2010, 2011b; Findell et al. 2011; Gentine et al. 2011a) and the near-surface turbulence, temperature, and moisture profiles (Businger et al. 1971). In turn, the change in the atmospheric state, mostly within the planetary boundary layer (PBL), affects the surface heat and moisture transport on possibly different spatial and temporal scales (Boers et al. 1995; Raupach and Finnigan 1995; Wulfmeyer 1999; Petenko and Bezverkhnii 1999; Sorbjan 2008; Paradisi et al. 2012; Katul et al. 1994a,b, 1999; Katul and Parlange 1995).
Boundary layer clouds exert an important radiative feedback onto the land surface through the decreased shortwave incoming solar radiation and increased longwave radiation (Mahrt 1991; Ek and Mahrt 1994) but also impact the large-scale dynamics (Bony and Emanuel 2005). The occurrence of boundary layer clouds is in turn related to the state of the land surface, especially on the surface turbulent heat flux partitioning, and, consequently, on the soil moisture state (Rabin et al. 1990; Ek and Holtslag 2004, hereafter EH04).
Shallow cumuli precondition the atmosphere for deep convection through moistening and heating of the free troposphere (Rio et al. 2009; Wu et al. 2009). Moist convection thus plays a prominent role in the radiative and hydrologic land–atmosphere coupling. The comprehension of the mechanisms triggering the onset of continental moist convection has nevertheless resisted a full theoretical treatment to date. This difficulty in the comprehension owes to the complicated and nonlinear boundary layer response to surface heat flux partitioning, which impacts both the rate of growth of the boundary layer as well as the condensation level in a nontrivial manner (van Heerwaarden and Guerau de Arellano 2008; van Heerwaarden et al. 2009).
Using a tendency equation of the relative humidity at the boundary layer top, Ek and Mahrt (1994), followed by EH04, demonstrated the role of direct surface moistening and boundary layer entrainment on the likelihood of cumulus onset. Huang and Margulis (2011) used this tendency equation to determine the evolution of the relative humidity at the top of the boundary layer in a series of large-eddy simulation experiments. Ek and Holtslag (2004) and Huang and Margulis (2011) highlighted the role of free-tropospheric stability on the occurrence of moist convection either over a dry or wet soil. Westra et al. (2012) used field observations to illustrate, for the first time, the increase in relative humidity at the boundary layer top over a drier soil, which we will refer to as a “dry soil advantage.” Other studies (Chagnon et al. 2004; Wang et al. 2009) hinted toward the existence of the dry advantage regime by satellite observation of shallow cloud occurrence over forested and deforested areas in the Amazon. A dry advantage regime was also documented through the sensitivity integration of a regional model under heat wave conditions in Europe (Stéfanon et al. 2013).
The relative humidity tendency equation used in EH04 is composed of an instantaneous surface evaporative term [evaporative fraction (EF)] and a “nonevaporative” boundary layer entrainment term. Although the evaporative fraction is often nearly constant during daytime (Gentine et al. 2007, 2011a), the boundary layer entrainment term is typically varying throughout the day. Since the nonevaporative term is not preserved throughout the course of the day, instantaneous observation of the boundary layer state using radio sounding, along with the tendency equation, is unable to clearly discriminate between the dry and wet soil advantage. On top of that, the boundary layer dynamics are themselves dependent on the evaporative fraction, so that the nonevaporative term is not really disentangled from the evaporative one. To resolve this issue, the boundary layer dynamics need to be taken into account in the relative humidity tendency equation.
The main objective of this work is to explicitly delineate the conditions leading to either dry or wet soil advantage in the triggering of moist convection based on surface or atmospheric observations that would be available anytime during the course of a day (free-tropospheric sounding, evaporative fraction). By moist convection we here refer to any type of convective boundary layer cloud formation, as well as to the triggering of deep convection. We will show that the use of conserved variables and the boundary layer dynamics permit us to objectively conclude whether moist convection will preferentially occur over a dry or moist surface, and we will highlight the role of the free-tropospheric moisture and temperature, which were missing in earlier results. This study will also show that the response is highly nonlinear in terms of surface evaporative fraction and free-tropospheric conditions.
The first part of the manuscript revisits the work of Ek and Mahrt (1994) and EH04 and provides a tractable diurnal solution of the relative humidity tendency at the boundary layer top in terms of conserved variables. The resolution includes the dynamics of the boundary layer using the analytical formulation of the boundary layer evolution of Porporato (2009, hereafter P09). The second part of the paper describes the diurnal cycle of relative humidity at the boundary layer top and the timing of boundary layer cloud onset over land as a function of surface and atmospheric conditions. The threshold between wet soil and dry soil advantages is explained in terms of dynamic and thermodynamic factors. In the last section, the surface and atmospheric conditions favoring the occurrence of stratocumulus, fair-weather cumuli, and shallow and deep cumulus are discussed, providing practical applications and new insights on the mechanisms at play in the formation of boundary layer clouds and convection over land.
2. Datasets
a. African Monsoon Multidisciplinary Analysis
To investigate the increase of relative humidity over either dry or wet surfaces, we use data from the African Monsoon Multidisciplinary Analysis (AMMA) field campaign. We here briefly describe the dataset used in this study. The reader is referred to Westra et al. (2012) for more details on the dataset. The observations used were gathered on 20, 22, 24, and 25 June 2006 at the AMMA site at Diori Hamani International Airport in Niamey, Niger (13.477°N, 2.175°E; 225 m above sea level). Figure 1 depicts the early morning radiosondes of the four days. The soundings were launched at 0733 UTC on 20 June, 0835 UTC on 22 June, 0834 UTC on 24 June, and 0833 UTC on 25 June.
Early morning soundings of (left) potential temperature, (middle) specific humidity, and (right) relative humidity over Niamey, Niger, during the AMMA field campaign.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
The date of 20 June 2006 was a hot, clear-sky day with temperatures of 27.3°C in the early morning (0600 UTC) and 38.2°C in the afternoon (1500 UTC), as observed with an eddy covariance system of the Atmospheric Radiation Measurement (ARM) Program mobile facility setup just outside the airport in Niamey. The relative humidity at the surface was similar to other days. It should be emphasized that the sounding of 20 June on Fig. 1 was taken an hour before the other soundings, so that the surface temperature appears lower and the relative humidity higher than in other soundings. After the rise of the boundary layer, the surface humidity conditions were similar in the different days. The surface was dry on 20 June, with a surface evaporative fraction of 0.1. The atmospheric mixed layer reached 1600 m at 1828 UTC. The early morning lapse rate of potential temperature between 1000 and 3000 m was 3 K km−1, as seen in Fig. 1. The inversion Bowen ratio, 
A mesoscale convective system occurred the night of 21 June, bringing 5 mm of rain. As a result, on 22 June the evaporative fraction was higher than on other days, reaching 0.3. The early morning lapse rate of potential temperature between 1000 and 3000 m was 2.75 K km−1, as seen in Fig. 1. The inversion Bowen ratio within this layer was 1.47. The mixed layer reached 1800 m at 1740 UTC. The temperature was higher that day, reaching 40°C at 1500 UTC, and the day was clear. On 24 and 25 June, the surface was dry again and the evaporative fraction reached 0.1. On 24 June the air temperature reached 40°C. The early morning lapse rate of potential temperature between 1000 and 3000 m was 2.95 K km−1, similar to previous days. It was extremely hot on 25 June, and the air temperature reached 44°C at 1400 UTC. On 24 June the mixed layer reached 1700 m at 1731 UTC, and a layer of shallow cumulus clouds developed in the morning. That day, the early morning sounding was moister than on 20 and 22 June. The inversion Bowen ratio between 1000 and 3000 m was much higher than in previous days and reached 1.8. On 25 June, the mixed layer was much higher than on other days and reached 2600 m at 1745 UTC. The high boundary layer was induced by the weak free-tropospheric stability (2.15 K km−1), as seen in Fig. 1. The inversion Bowen ratio was low (0.75) and the day was clear.
b. Cabauw
We use a second contrasting dataset located at Cabauw in the Netherlands to illustrate the onset of boundary layer clouds in more humid and colder conditions. The data were observed on 31 May 1978. The reader is referred to EH04 for details on the observations. Four radiosondes were launched from the Cabauw site during the morning of the study day, providing temperature and moisture profiles above the tower level. Additionally, the dataset is supplemented with information from radiosondes launched at De Bilt (about 25 km to the northeast) several times during the day, providing additional measurements of wind, temperature, and moisture. Sensible and latent heat fluxes were determined from profile and Bowen ratio methods. The 20-m air temperature reached 26°C at 1600 UTC, and the specific humidity was fairly constant between 7 and 8 g kg−1 during the course of the day. A fair-weather cumulus cloud layer developed at 1500 UTC. The surface was humid and the evaporative fraction was fairly constant during daylight hours at 0.8. The free-tropospheric potential temperature lapse rate was 3.7 K km−1, and the specific humidity lapse rate was −1.6 g kg−1 km−1. The inversion Bowen ratio was 0.93.
3. Relative humidity at mixed-layer top
a. Background
Mixed-layer model of the boundary layer. The inversion Bowen ratio is 
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
List of variables and parameters.
Our main goal here is to define the occurrence of cumulus onset and increase of RH(h) as a function of conserved variables that could be observed anytime throughout the day using early morning radio sounding, for instance. To do so, we will relate the boundary layer dynamics to the surface evaporative fraction using the conceptual model of the convective boundary layer of P09.
b. Impact of boundary layer dynamics
4. Tendency equation in terms of conserved variables
a. Derivation
Equation (13) demonstrates that the diurnal course of relative humidity at the boundary layer top is monotonic throughout the course of the day. The ratio A(t)/h(t) does not change sign during the course of the day and neither do the factors in the brackets of the rhs. If the PBL experiences moistening (drying), in terms of relative humidity, this moistening (drying) lasts until sunset. Once boundary layer clouds appear, the PBL humidity is mitigated by the cloud-base convective mass flux (Betts 1973; Arakawa 1974).
b. Timing of cloud onset
Equation (14) can be integrated analytically (not shown) in time, giving the evolution of RH(h) throughout the day. The time for which RH is equal to 100% is the time of cloud occurrence and can be found by inverting the integral of Eq. (14). Figure 3 depicts the time of cloud occurrence as a function of EF and the atmospheric stability (γθ), free-tropospheric relative humidity 
Time (hours from sunrise) of cloud occurrence as a function of evaporative fraction for a maximum available energy A0 = 500 W m−2 and for (left to right) varying values of free-tropospheric humidity and (top to bottom) free-tropospheric surface potential temperature. The free troposphere is more humid to the right and warmer to the bottom. Colored dots refer to the AMMA profiles. Black dot is Cabauw data.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
For visualization purposes, we only use a single variable to describe the humidity in the free troposphere 
Everything else held constant, the time of cloud occurrence increases with increased stability, since the latter reduces the boundary layer growth and the possibility to reach the LCL. The entrained warm air also increases the saturation specific humidity through the Clausius–Clapeyron relationship. As a consequence the relative humidity is decreased.
The time of cloud occurrence exhibits two opposite behaviors in response to EF. Under strong free-tropospheric stability (γθ > 4–5 K km−1), the time of cloud occurrence is reduced under rising EF. The moisture provided by the surface latent heat flux compensates for the dry air entrainment resulting in earlier higher relative humidity. Oppositely, under weak free-tropospheric stability (γθ < 4–5 K km−1), the time of moist convection onset rises with increased EF. Reduced free-tropospheric stability facilitates the growth of the boundary layer and the entrainment of free-tropospheric air. This dry and warm entrained air reduces the relative humidity through the reduction of the mixed-layer specific humidity (by moisture conservation) and through the rise of the saturation specific humidity (by heat conservation in the mixed layer and through the Clausius–Clapeyron relationship). At high EF values (above 0.6 to 0.8 depending on the surface free-tropospheric relative humidity and temperature), the time of cloud occurrence rises sharply in response to increasing EF and γθ. In those conditions there is significant control of the surface on the exact timing of cloud occurrence.
A moistened free troposphere (higher 
A warmer troposphere, as assessed by θf0, leads to a reduction of boundary layer cloud onset. The reduction in low-cloud cover is consistent with recent results of global warming scenarios in climate models (Brient and Bony 2013). Our results suggest that the reduction will be especially pronounced over moist surfaces (EF > 0.5) and under dry free-tropospheric conditions. It could thus be expected that the early dry season will show a reduction of cloud cover over land under global warming. Of course, our approach is only one-dimensional and does not include any horizontal circulations, which could alter the results presented here (Betts 2004).
Figure 4 shows the lifting condensation level (LCL) at the time of cloud occurrence. A strong sensitivity to the surface is observed in most cases beside at low EF values (left part of the plots). As EF increases, the PBL depth is reduced, and therefore, the LCL decreases as well at the time of occurrence. Significant sensitivity to surface conditions is observed at high EF values and under stronger free-tropospheric stability.
As in Fig. 3, but for cloud base (m) (LCL) at the time of cloud occurrence (hours from sunrise).
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
c. Moist and dry soil advantage regimes
One of the most important results of EH04 and Westra et al. (2012) is the somewhat counterintuitive realization that, in some cases drier soils may favor the occurrence of boundary layer clouds. These studies provided an important step forward in the understanding of the mechanisms controlling the likelihood of moist convection onset as a function of the surface and free-tropospheric state. They introduced the idea of different regimes of sensitivity to the soil wetness. Here, we will call them the “wet soil advantage” or “dry soil advantage” regimes. The exact limits between the two regimes remained unclear because the nonevaporative term of the tendency equation of EH04 varies throughout the course of the day and depends on the surface state through the boundary layer dynamic control. To describe the transition between the two regimes, we reinvestigate the sensitivity of the relative humidity at the mixed-layer top using Eq. (14). The sign change of 
Figure 5 depicts the separation between the two regimes, that is, where the sensitivity vanishes: 
Root of the sensitivity of the relative humidity at the mixed-layer top 
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
Wetter surfaces (higher EF) exhibit higher relative humidity under strong free-tropospheric stability (γθ) and over very wet surface (high EF), consistent with the observations of EH04 over Cabauw, the Netherlands (see below). EH04, Huang and Margulis (2011), and Westra et al. (2012) have already emphasized the role of the free-tropospheric stability on the relative humidity tendency.
Figure 5 demonstrates that the reference surface EF plays an important role in the control on RH(h): calculating the sensitivity of the relative humidity RH(h) to surface EF yields opposite results between low and high reference EF values, thus displaying an important nonlinear response. The wet soil advantage region increases (in terms of interval of EF) with decreasing free-tropospheric relative humidity 
In summary the wetter soil advantage regime is more likely under dry and cold free-tropospheric conditions and over wet surfaces (at large EF values). Conversely, drier soil advantage is more likely under weak stability, in a moist and warm free troposphere, and over low-EF surfaces (lower left part of Fig. 5). The important points to be underlined are that the reference EF value is important to determine the sign of the sensitivity since the sensitivity is highly nonlinear and that the humidity and reference temperature of the free troposphere play crucial roles in the evolution of the relative humidity at the top of the boundary layer.
For fixed free-tropospheric conditions and weak stability (γθ < 3.5–4 K km−1), the change of sign of 
Relative humidity increase over dry soil has recently been observed during the AMMA measurement campaign in the semiarid Sahel (Redelsperger et al. 2006). On 20 June 2006, relative humidity increased at the boundary layer top over a dry soil (EF ~ 0.1). The free troposphere was characterized by relatively weak stratification (γθ = 3.4 K km−1), medium-range relative humidity (qf0 = 14.36 g kg−1, 
In contrast to the dry soil advantage, EH04 illustrated observations of a wet soil advantage in Cabauw, the Netherlands, over 31 May 1978. The evaporative fraction was high (EF ~ 0.8), and the free-tropospheric conditions were reasonably warm (
Other recent observational studies showed that, in some conditions, moist convection is favored over dry soils. This includes the observations of more frequent deep convection development in the Sahel (Taylor et al. 2011, 2012) and of higher shallow cumulus cover in the Amazon (Chagnon et al. 2004; Wang et al. 2009). In the Sahel, drier and wetter patches of soil were compared. The patches had a spatial scale of 10–15 km and thus share the same synoptic-scale atmospheric forcing. Higher probability of triggering of convective storm is observed over drier patches. This is assumed to be induced by mesoscale breezes created by differential heating on the ground that creates convergence on the dry side (Taylor et al. 2011, 2012). Our results suggest that there could be a concurrent mechanism of boundary layer moistening over drier soils induced by the one-dimensional boundary layer dynamics described above, given the dryness of the Sahel (low EF) and the weak stratification of the monsoon period. This mechanism could precondition or help the triggering of convection by the mesoscale features. Consistent with the above results, in a recent regional model study, Stéfanon et al. (2013) observed that a drying of the soil induces an increase of shallow cumulus activity. This result was found under the specific conditions of European heat waves, with high temperatures, dry free troposphere, and weak stratification over the whole continent. The mesoscale circulation, in this case, was found to reinforce convection and precipitation over the mountains. On the other hand, the Amazon is characterized by high overall EF in the wet season. In case of very weak stratification (lower-right side of the diagram in Fig. 5), the system can bifurcate between dry and wet soil advantage regimes.
5. Dynamic versus thermodynamic moistening
a. Direct mixed-layer specific humidity increase
In contrast with the potential temperature, the boundary layer growth can either moisten or dry the boundary layer in terms of specific humidity (Mahrt 1991; Betts 1992). Since the mixed-layer specific humidity appears at the numerator of RH(h), it is enlightening to evaluate the regimes of increase or decrease in boundary layer specific humidity.
In the AMMA dataset, the early morning profile on 24 June 2006 exhibited the highest-inversion Bowen ratio (1.8), which corresponds to an EFc value of 0.44, still much higher than the observed EF value (0.1). Consequently, all of the AMMA days used here induced a specific humidity drying over drier surfaces. On the other hand, over the Cabauw site, the inversion Bowen ratio was 0.96, which corresponds to a critical EF of 0.59, lower than the observed EF value (0.8). Moister surfaces in this case led to a direct increase of the boundary layer specific humidity through the reduction in boundary layer height.
1) Limiting EFc cases: Dependence on entrainment parameter α
Here, we examine some limiting cases of the critical EFc. We first consider the case of pure encroachment, when the entrainment α vanishes. In this case EFc reaches 1/(1 + Binv). In other words, the critical surface Bowen ratio B0 exactly compensates for the inversion Bowen ratio Binv. When the surface Bowen ratio is higher (that is EF < EFc) the boundary layer dries up. The boundary layer growth induces the entry of dry free-tropospheric air into the boundary layer as an open system. The boundary layer moistens in the opposite case when EF > EFc since the surface moistening overcomes the entry of dry free-tropospheric air. When entrainment at the boundary layer top becomes very large, 
2) Limiting EFc cases: Dependence on inversion Bowen ratio
In the case of small-inversion Bowen ratio, which corresponds to either a weak stability or dry free-tropospheric profile, EFc approaches unity and the boundary layer always dries up. The small-inversion Bowen ratio induces rapid growth of the boundary layer and important drying associated with the large amount of entrained dry free-tropospheric air. The surface moistening cannot counterbalance the large quantity of entrained dry free-tropospheric air.
Large-inversion Bowen ratio, in turn, implies a small EFc. The strong stability induces a reduction of the dry air entrainment at the boundary layer top. A humid free-tropospheric profile prevents the reduction of the boundary layer specific humidity by entrainment. In all cases, insufficient free-tropospheric dry air is entrained within the boundary layer and cannot counteract the surface moistening.
It is important to stress that the boundary layer moistening in terms of specific humidity might be very different from the moistening in terms of relative humidity. The nonlinear response of the Clausius–Clapeyron relationship to temperature in the denominator of RH(h) might indeed reduce the relative humidity, even though the boundary layer experiences a specific humidity increase. In particular, the linear mixing of dry air from the free troposphere and dry air from the mixed layer may be saturated because of the convexity of the Clausius–Clapeyron relationship. In addition, the relative humidity at the mixed-layer top depends on temperature, which decreases as the boundary layer deepens. Consequently, a deeper boundary layer might be more likely to generate boundary layer clouds.
b. Factors controlling the relative humidity tendency
Ratio of factor 1 (direct moistening) to factor 2 (entrainment effect) on a log10 scale, as a function of EF and Binv. Values above 0 represent an advantage of the direct moistening over the free-tropospheric entrainment and drying. Colored dots refer to the AMMA profiles; colors as in Fig. 1. Black dot is Cabauw data.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
The dynamic control on the relative humidity tendency depends on the inversion Bowen ratio Binv and on the entrainment efficiency α. More efficient entrainment, such as in the case of intense shear (Pino et al. 2006), displaces EFsign toward unity. A critical value of Binv exists, 
The second part of the thermodynamic factor, which is always negative, corresponds to a relative humidity reduction induced by the air dilatation and cooling as the boundary layer deepens. This factor cannot explain the occurrence of moist convection over dry soils since it acts to reduce the relative humidity at the boundary layer top.
The ratio between the two terms of the thermodynamic factor is plotted in Fig. 7. The ratio is plotted on a logarithmic scale, and both factors are taken in absolute values. Interestingly, the departure between the dry and moist adiabatic is the dominant factor affecting the relative humidity tendency. The cooling of the actual temperature and dilatation at the boundary layer top is generally an order of magnitude smaller than the first term.
Ratio of the first term of the thermodynamic factor (departure between dry and moist adiabatic) to the second term (dilatation effect) on a log10 scale as a function of EF and free-tropospheric stability. The results are relatively insensitive to the moistening of the free troposphere and reference temperature. The crossover region between the preponderance of each factor is the bold black line. Colored dots refer to the AMMA profiles; colors as in Fig. 1. Black dot is Cabauw data.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
The departure between the dry and moist adiabatic is therefore the explanation for the occurrence of boundary layer clouds over very dry regions in the presence of very deep boundary layers, such as over the Sahara (Cuesta et al. 2008). In this case the cloud cover is generally composed of fair weather cumuli and the cloud cover is highly positively correlated with the depth of the boundary layer since the temperature change at the boundary layer top strongly affects the moist adiabatic but not the dry adiabatic. This effect is also visible on Fig. 1. The relative humidity increases almost linearly with height in the mixed layer. This quasi-linear behavior is confirmed by later sounding in the afternoon (not shown here). It is then evident that if the boundary layer sufficiently deepens, the relative humidity at the boundary layer top will reach 100%, even though the amount of specific humidity can be very small, like over the Sahara.
To differentiate regimes of dynamic or thermodynamic influence, the ratio of the two factors is plotted in Fig. 8. The ratio is plotted on a logarithmic scale and both factors are taken in absolute values. A positive value corresponds to a regime dominated by the dynamic factor. The dynamic factor is the main contributor of the relative humidity tendency above a diagonal going from (EF = 0, γθ = 4 K km−1) to (EF = 1, γθ = 1 K km−1), that is, in stable atmosphere with relatively wet surfaces. The thermodynamic factor is the main contributor below this diagonal, that is, under weak stability and dry soil. The AMMA data points belong to the region of thermodynamic influences, and therefore, drier surfaces can generate more clouds through the thermodynamic effect. On the other hand, the Cabauw site belongs to the region of dynamic influence, and increased cloud cover occurs over moister surface through the direct surface moistening. Surprisingly, the diagonal does not evolve much with the increase of the free-tropospheric temperature and relative humidity. In fact, the EF and the free-tropospheric stability are the main contributors to the growth of the boundary layer, as established by Eq. (10), and it is the depth of the boundary layer that principally determines the influence of the thermodynamic factor through the difference between the dry and moist adiabatic. An important consequence is that the changes in surface temperature are small compared to the changes in temperature induced by an extended boundary layer growth under weak stability and low EF. Indeed, a convective boundary layer has a nearly adiabatic profile, which corresponds to a temperature lapse rate of about 9.8 K km−1. Over deep boundary layers of a few kilometers, the temperature reduction at the boundary layer top can be of the order a several tens of degrees, which can be much larger than surface temperature difference between hot and cold regions.
Ratio of the dynamic to thermodynamic factors of the relative humidity tendency on a log10 scale for A0 = 500 W m−2. Gray line depicts the limit between dry vs wet soil advantage regimes. Bold black line depicts crossover between positive and negative values. Colored dots refer to the AMMA profiles; colors as in Fig. 1. Black dot is Cabauw data.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
6. Forced versus active moist convection, stratocumulus to deep convection transition
Even though the relative humidity tendency at the boundary layer top provides important insights onto the development of boundary layer clouds, it cannot discriminate between stratocumulus, shallow, or deep convection or between forced and active clouds (Stull 1988). Forced clouds refer to thermal plumes that condensate but are negatively buoyant above their LCL. Those plumes have reached their LCL but have not overcome their convective inhibition (CIN) and have thus not reached the level of free convection (LFC). Forced cloud cover can be either stratocumulus or forced shallow convection clouds. Conversely, active clouds have overcome the CIN and have reached their LFC. Active convection can be shallow or deep and generates a cloud base mass flux used as the boundary condition for a most convective scheme (Betts 1973; Arakawa 1974; Gentine et al. 2013a,b).
A useful indicator of the triggering of active convection versus forced convection is the difference between the mixed-layer equivalent potential 
The difference 
Difference (K) between the equivalent potential temperature of the mixed layer and the saturation equivalent potential temperature of the free troposphere just above the inversion (h+) at sunset for A0 = 500 W m−2. Negative (blue) regions denote stratocumulus occurrence, and positive (warm colors) regions depict the occurrence of shallow or deep convection. White areas denote regions without clouds. Black bold contour depicts the crossover between positive and negative regions.
Citation: Journal of Hydrometeorology 14, 5; 10.1175/JHM-D-12-0137.1
The dependence of the conditional instability to the surface EF is again nontrivial. Under drier free-tropospheric conditions, a rise in EF increases the occurrence of all clouds (less white contour), yet moister surfaces favor forced convection (stratocumulus and forced shallow convection). Under humid (
Figure 9 can be useful to make hypotheses on the effect of global warming on the occurrence of boundary layer clouds, assuming other variables remain constant (e.g., relative humidity, surface evaporative fraction, available energy at the surface, and free-tropospheric stability). Figure 8 indicates a reduction of the occurrence of clouds over land under global warming, especially over moist surfaces (high EF). The occurrence of deep convection, as defined by the regions in red in Fig. 9, increases as recently observed over land in climate model simulations (Wyant et al. 2012). Stratocumulus clouds also become less frequent over land under increased free-tropospheric temperature. In our conceptual model we do not account for the coupling with the large-scale circulations and the advection to/from the ocean (Betts and Ridgway 1988, 1989; Lintner et al. 2012). Therefore, the results have to be interpreted with caution.
7. Conclusions
The onset of moist convection is investigated using a mixed-layer model of the boundary layer. To facilitate an analytic development, a constant evaporative fraction is assumed. The sensitivity of the relative humidity at the boundary layer—defined by 
The dynamic factor comes from the interplay between the direct effect of the surface on the specific humidity of the PBL and the indirect effect via the boundary layer dynamics entraining drier and warmer free troposphere air. A dry surface reduces the specific humidity and thus tends to reduce the relative humidity. At the same time, however, it will cause the boundary layer to grow, entraining free atmospheric air. This air, if cold and dry enough, can cause the relative humidity to increase because of the Clausius–Clapeyron relationship. The prevalence of the direct or indirect effect is mainly controlled by the inversion Bowen ratio and by the entrainment efficiency. The dynamic regime pertains to boundary layers that are not too deep and reflects the typical mechanisms of boundary layer moistening and low-level cloud generation. This process is similar to the generation of low-level cloud over the ocean through the increase of the specific humidity. The dynamic regime dominates over wet surfaces and strong free-tropospheric stability and is favored under moister and colder free troposphere. In this regime boundary layer clouds and the relative humidity at the boundary layer top increase with a rise in surface evaporative fraction, that is, over moister surfaces.
The second thermodynamic factor concerns deep boundary layer regimes such as those observed over the Sahel and the Sahara. In this regime the departure between the moist and dry adiabatic induces an opposite response to surface moistening, and drier soils favor the generation of boundary layer clouds since drier surfaces increase the boundary layer depth. Warmer and drier free troposphere reduces the likelihood of cloud occurrence with the thermodynamic control.
The timing of cloud occurrence is closely related to the free-tropospheric conditions (stability, temperature, and humidity) and surface evaporative fraction. Moist convection onset occurs earlier over moist surfaces (large EF values) in regimes of strong free-tropospheric stability and cold environments. The opposite behavior is observed in regimes of weak free-tropospheric stability and warm environments. Not surprisingly boundary layer cloud occurrence is disfavored under dry and cold free troposphere but also under warm and moist free troposphere regimes.
Finally, we discuss the occurrence of active convection (shallow or deep) compared to forced convection (stratocumulus or fair-weather cumuli) as a function of conserved variables. Very wet surface conditions (evaporative fraction larger than 0.7) favor active convection under a wet atmosphere (e.g., Amazon). Drier, warm free-tropospheric conditions with weak stratification in turn favor active convection over dry soil. In the context of the one-dimensional model used in this paper, it is expected that deep convection will increase over land under global warming, especially over dry surfaces. On the other hand, the fraction of stratocumulus and shallow cumulus clouds is expected to decrease over humid surfaces.
Acknowledgments
This work has been carried out as part of grant NSF-AGS-1035986 of the National Science Foundation. The authors wish to thank Alan K. Betts for his comments on the manuscript as well as Françoise Guichard and Gert-Jan Steeneveld for providing the AMMA data used in this analysis.
APPENDIX A
Derivation of Critical EF
APPENDIX B
Dependence of c1 and c2 on Conserved Variables
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