1. Introduction
Many features of tropical deep convection are accounted for by the quasi-equilibrium (QE) hypothesis. According to this hypothesis, deep convection responds very rapidly to changes in tropospheric stability because of large-scale circulation and radiative forcing, and the tropical troposphere is thus permanently close to a state of equilibrium. However, several authors have emphasized that an atmosphere in a permanent QE state would exhibit an exceedingly low variability at small and large scales (Neelin et al. 2008; Jones and Randall 2011; Raymond and Herman 2011). Using cloud-resolving model (CRM) simulations, Raymond and Herman (2011) showed that the response of deep convection to a perturbation was very fast (hours) only in the lower half of the troposphere and was much slower in the upper half. This points to the importance of the depth of moist convection and suggests that the QE hypothesis is valid in the region of the troposphere reached by cumulus clouds but not in the region reached only by deep convection. Cumulonimbus clouds efficiently warm the upper troposphere: when present, they bring the CAPE back to very low values in a matter of hours. However, they are short lived (about 30 min) and are present only as long as the triggering of new elements continues. It is then tempting to suppose, following Neelin et al. (2008) and Stechmann and Neelin (2011), that the main reason why deep convection departs from QE is that there are lapses of time where triggering of new convective cells does not occur and where the upper troposphere may drift freely away from QE.
Subcloud lifting processes and convective inhibition (CIN) are known to exert a strong control on deep convection onset and intensity, modulating the entropy flux from the boundary layer to the free troposphere (Emanuel et al. 1994). Mapes (2000) assumes that deep convective triggering occurs when turbulent kinetic energy in the boundary layer (the triggering energy) is sufficient to overcome CIN. With this picture in mind, the question of the occurrence and variability of moist convection in the tropics is strongly dependent on the departure of the troposphere from QE states and thus on the action of boundary layer processes on deep convection triggering. The present series of papers pursues of these ideas further and addresses the questions of deep convection triggering and its representation in climate models.
The QE hypothesis plays an important role in deep convection parameterizations since it makes it possible to express deep convection processes as a function of large-scale conditions. However, since departure from QE is a key factor for climate variability, convective parameterizations should not be bound too strongly by the QE hypothesis.
According to Jones and Randall (2011) (see also Xu et al. 1992), several methods have been used to drive the local atmospheric system away from QE. In a first approach (the superparameterization technique) a CRM is embedded within each general circulation model (GCM) grid cell, and the variability around QE is provided by the CRM internal variability [e.g., Plant and Craig (2008) emphasize the variability provided by CRMs for given large-scale conditions].
In a second approach (Palmer 2012) the tendencies computed by the physical parameterizations are perturbed randomly; the system is no longer driven toward QE but toward a target moving randomly around QE. We shall follow Neelin et al. (2008) and assume that movement away from QE occurs mainly when deep convection is not active. Consequently, determining the period of activity of deep convection is a key issue for representing climate variability.
In observations and in high-resolution simulations of moist convection, the triggering (or onset) of deep convection is the time when cumulus clouds reach the highest levels of the troposphere (i.e., congestus and cumulonimbus). Prior to this sharp transition, the convective boundary layer enters a transient regime (transition stage), during which cumulus clouds become gradually wider and deeper, but still remain in the low troposphere (Chaboureau et al. 2004; Guichard et al. 2004; Grabowski et al. 2006; Khairoutdinov and Randall 2006). Chaboureau et al. (2004) show that, during the transition phase, the updraft vertical velocities at cloud base are large enough to overcome the convective inhibition but that entrainment of exceedingly dry air limits the cloud vertical development. It is only when the lower free troposphere is moist enough that the sharp transition to deep convection occurs. Thus, they propose a two-step trigger in which stability and moisture are the two critical variables controlling the transition.
The objective is therefore to design multiple-step triggering that accounts for the evolving properties of the strongest boundary layer thermals applicable to any GCM that treats boundary layer structures independently from deep convection, for example, through the eddy diffusivity–mass flux approach. This type of scheme combines a diffusivity scheme, representing the small-scale turbulence, with a mass flux scheme representing the organized structures of the boundary layer (including the cumulus clouds).
The question of deep convection triggering is of particular interest over lands, where the boundary layer is, on average, higher than over the ocean (Medeiros et al. 2005) and is capped by a stronger inhibition layer. Thus, over land, the lifted parcel cannot reach its level of free convection (LFC) without some dynamical forcing, and the shallow and deep regimes are thus more distinct in space and time.
Actually, most current GCMs miss this transition phase and consequently represent the diurnal cycle of deep convection over land rather poorly (Grabowski et al. 2006; Yang and Slingo 2001; Guichard et al. 2004; Bechtold et al. 2004). According to Guichard et al. (2004), this is because the gradual moistening of the low free troposphere due to the detrainment at the top of cumulus clouds is not well represented in GCM parameterizations, so current GCMs cannot capture the succession of dry, shallow, and deep convection regimes.
Observations have shown that shallow cumulus clouds are the saturated part of thermals initiated at the surface and driven by buoyancy (LeMone and Pennell 1976). Here we define a thermal as a coherent structure rising from the surface to the top of the dry or cloudy boundary layer and carrying out most of the vertical transport of heat, moisture, and momentum. The thermal then divides into a subcloud layer and a cloudy layer. Deeper congestus and cumulonimbus clouds burst locally, overcoming an inhibition barrier, and are associated with precipitation and cold pools driven by the evaporation of rain under the cloud base.
In the current version of the Laboratoire de Météorologie Dynamique–Zoom (LMDZ) model, we treat shallow and deep convection separately. On the one hand, shallow convection is handled in a unified way with the boundary layer turbulence. This is done by combining a diffusive approach with a mass flux approach representing both dry and shallow convection. The so-called thermal plume model (Rio and Hourdin 2008) idealizes the effect of all dry and cloudy thermals contained in a model grid cell by considering a mean ascending dry or cumulus-topped thermal covering a fraction α of the grid cell and compensated by subsidence in the surrounding environment. In this way, shallow convection occurs when the ascending thermal condenses and no triggering criterion for shallow convection is required. On the other hand, deep convection and the associated precipitation and downdrafts are handled by the Emanuel episodic mixing and buoyancy sorting scheme (Emanuel 1991), coupled with a parameterization of cold pools driven by the evaporation of deep convective rain (Grandpeix and Lafore 2010).
Deep convection is first initiated if the dynamical lifting provided by boundary layer thermals is sufficient to overcome the convective inhibition. Once activated, deep convection is sustained by cold pools that provide an additional source of lifting. Then, the deep convection scheme is coupled with local lifting processes through two variables: the available lifting energy (ALE, expressed in joules per kilogram) and the available lifting power (ALP, expressed in watts per square meter). Convection triggering and closure are expressed in terms of ALE (convection is triggered when ALE > |CIN|) and ALP (cloud-base mass flux is proportional to ALP). In the LMDZ5B model, the lifting processes considered are (i) the boundary layer thermals (subscript BL) and (ii) the cold pools (subscript WK for wake) fed by unsaturated downdrafts resulting from the reevaporation of rain below cumulonimbus clouds. The ALE is the maximum of the lifting energies [ALE = max(ALEBL; ALEWK)] and ALP is the sum of the two lifting powers (ALP = ALPBL + ALPWK). The present paper is only concerned with deep convection triggering, that is, only with the ALE variable. Moreover, since we are specifically interested in convection initiation, only the lifting energy due to boundary layer thermals has to be considered (cold pools only act to maintain deep convection, after its onset). In the current version of the LMDZ5 GCM (LMDZ5B), the lifting energy is deduced from the maximum vertical velocity within the thermal:
The aim of this paper is to revisit the definition of ALEBL by identifying the key factors controlling the transition from shallow to deep convection. The final goal is to define a triggering criterion for deep convection from the properties of thermals associated with shallow convection.
Several studies using cloud-resolving models have been used to characterize this complex transition from shallow to deep convection and provide some insights into the variables that control deep convection triggering. While Chaboureau et al. (2004) proposes that deep convection starts when a variable called the normalized saturation deficit (NSD) at the cloud base reaches its minimum (as NSD is strongly linked to the cloud cover, triggering occurs when the cloud cover reaches a critical value), Wu et al. (2009) shows that the virtual temperature profile of the average cloud is a key factor, and Khairoutdinov and Randall (2006) and Grabowski et al. (2006) stress the importance of horizontal cloud size. Thus, several parameters seem to play key roles in deep convection triggering: at cloud base, the humidity of the troposphere, the cloud cover, and the size of individual clouds are significant, and above cloud base, the thermodynamic properties of cumulus clouds are important.
Here we tackle the problem of how deep convection triggering is represented in climate models. Using large-eddy simulation (LES) data in a continental case of transition from shallow to deep convection, we extract the statistical properties of the thermals at cloud base and propose a new computation of ALEBL. The goal is to propose a simple formulation of the triggering process, easily integrable in a GCM. This new formulation describes the whole transition process and in particular the episodic nature of the triggering.
Section 2 describes the theoretical framework, and section 3 describes the method. The cross-sectional spectrum of the thermals inside the domain is studied in section 4, and the vertical velocity spectrum inside the thermals is examined in section 5. The ALEBL computation is described in section 6. The triggering formulation is proposed in section 7, and some final comments are given in section 8.
2. Single versus spectral thermal approaches
In a typical GCM grid (L ≥ 100 km), the expected number of thermals can be very large. The “bulk thermal” approach may then be useful to predict their collective effect on heat and moisture transport. This approach considers a single (or average, or bulk) thermal of cross section Stot, covering a fractional area αtot (see Fig. 1). The vertical profiles of vertical velocity inside (i.e., in the ascending zone) and outside (i.e., in the surrounding environment) the bulk thermal are

(left) Side view and (right) top view of the (a),(c) single thermal vs the (b),(d) spectral approach to thermal modeling.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(left) Side view and (right) top view of the (a),(c) single thermal vs the (b),(d) spectral approach to thermal modeling.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
(left) Side view and (right) top view of the (a),(c) single thermal vs the (b),(d) spectral approach to thermal modeling.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
However, this approach is no longer useful when considering the size of the thermals. In such a case, a cross-sectional distribution has to be taken into consideration and requires a spectral approach. Figure 1 illustrates the differences between the single thermal and the statistical approaches.
Concerning shallow cumulus (topping boundary layer thermals), Neggers et al. (2003) and Rodts et al. (2003) studied the properties of the cloud field using aircraft measurements, satellite data, and large-eddy simulations. They showed that many distribution laws are possible fits for the cloud cross-sectional (size) spectrum over the domain; among them were the exponential law, the lognormal law, and some other power laws.






The internal fluctuations of vertical velocity in clouds may also be considered. Emanuel (1991) recalls that pioneering aircraft measurements have shown that in-cloud fluctuations exhibit a typical length scale of 100 m. From dual-Doppler cloud radar data analysis, Damiani et al. (2006) suggests a typical length scale of L = 200–600 m. Craig and Dörnbrack (2008) also give physical arguments supporting an L = 200–300-m length scale for variability. Malkus (1954) and Warner (1970) revealed that vertical velocity fluctuations were, at least, as large as the mean value across the cloud section.
Thus, studies suggest that both intrathermal (vertical velocity) and interthermal (cross section) fluctuations are important. Our aim is now to propose a corresponding theoretical representation of the boundary layer thermal plume field.
3. Data and methodology
a. Case description
The case investigated here is the African Monsoon Multidisciplinary Analyses (AMMA) case of 10 July 2006, where a small, short-lived convective cell developed over Niamey (Lothon et al. 2011). The whole transition was recorded by several ground-based instruments (radar, wind profiler, and atmospheric soundings) and completed by satellite data. This case study concerned a typical case of transition from shallow to deep convection over semiarid land with a high Bowen ratio (Bo ≈ 10) and associated with an elevated cloud base (zlcl ≈ 2.5 km). The structure of the boundary layer clouds evolved gradually from a “cloud street” organization (from morning to noon) to an isotropic structure composed of larger but more heterogeneous cells (from noon to midafternoon). Around 1540 LT, deep convective cells developed with associated cold pools. It was noted by Lothon et al. (2011) that the first convective cells developed over the largest horizontal cloud structures, which supports the relevance of the cloud-base cross section in describing the transition process and reinforces the hypotheses made in section 1. A modeling setup was developed to represent this case and a large-eddy simulation, able to represent the main observed features, was run (Couvreux et al. 2012).
b. The large-eddy simulation
The simulation uses the LES version of the Meso-NH nonhydrostatic model developed by Lafore et al. (1998). The domain is 100 × 100 × 20 km3, with a horizontal resolution of 200 m, a stretched grid on the vertical (from 50 to 250 m), and periodic lateral boundary conditions. The simulation lasts from 0600 to 1800 LT, at which time the cold pool generated by deep convection became too large relative to the domain. The lower boundary condition consists of imposed homogeneous surface latent and sensible heat fluxes. However, the observations showed a large positive surface temperature anomaly (around 5 K), over which the first cell developed (at 1540 LT). This heterogeneity is suspected of playing an important role in the triggering of deep convection (enhancing mesoscale circulation and breeze convergence over the hot spot; see Taylor et al. 2011). To simulate a similar onset of deep convection, a low-level moisture convergence is applied in the morning, linked to the monsoon flow, and a low-level ascent of 1.5 cm s−1 during the afternoon. With these conditions, the LES’s first cumulus appeared around 1100 LT and deep convection was triggered around 1630 LT. This simulation was evaluated against observations in Couvreux et al. (2012).
c. Data, definitions, and notations
1) Clouds and thermals
The material used in the present study comprises various fields extracted from the simulation every hour from 1200 to 1800 LT. In the LES, a column i is defined as cloudy if the liquid water content rc(i) ≥ 10−6 kg kg−1 in any vertical level k > kmin of a 200-m layer above kmin, where kmin is the lowest vertical level where the threshold rc = 10−6 kg kg−1 is reached in the LES domain. Within a cloudy column, the cloudy levels are those that verify rc(i, k) + ri(i, k) ≥ 10−6 kg kg−1, where ri(i, k) is the ice water content.
Adjacent cloudy columns (at cloudy levels) are then grouped to form individual clouds, described by (i) their cloud-base cross section si, (ii) their horizontal mean cloud-base altitude zlcl,i, and (iii) their horizontal mean cloud-top altitude ztop,i.
We also assume that each cloudy column corresponds to an individual draft that is grouped similar to the cloud to form a thermal originating from the surface and extending up to the cloud top ztop,i. Then, in the LES, for each cloud, we define a corresponding thermal, which is the ensemble composed of the subcloud and the cloudy part of the adjacent cloudy columns. The whole analysis is founded on cloud-base characteristics. Therefore, we discard thermals that will not reach the LCL, that is, the dry thermals.
2) Thermal field
The study domain corresponds to the extent of the horizontal area of the LES (Sd = 104 km2), in which Ntot thermals (and corresponding clouds) are present, covering an area Stot and a fractional area αtot
3) Thermal geometry
The geometry of a given thermal i is characterized by the altitudes zlcl,i and ztop,i of its cloud base and cloud top, respectively, and by its cross section si at cloud base. Since the LES horizontal resolution is 200 m, we arbitrarily assume, for simplicity, that the cross section of the elementary drafts is š = 4 × 104 m2. First, this length scale is consistent with the observational, high-resolution, and theoretical studies mentioned in section 2. Second, we will show in section 6 that this arbitrary parameter is of secondary importance.
A thermal i is then composed of ni adjacent drafts of cross section š underlying a cloud. The number of elementary drafts in a thermal i is noted
4) Vertical velocities










5) Mean properties





The arithmetic-mean cross section over the thermal population gives
Finally, the arithmetic-mean thermal average velocity over the thermal population is defined as
6) Data
For every thermal i we extract the cloud base zlcl,i and the cloud-top altitudes ztop,i. We then extract the following variables at cloud base: the thermal’s dimensionless cross section ni (or size in the following); the thermal’s cross-sectional average of vertical velocity
d. Method
Our final goal is to propose a new formulation of ALEBL, or in other words, to compute a maximum kinetic energy provided by the thermals, which has to be compared with CIN. Thus, the following LES analysis is aimed at finding the maximum value distribution for the thermal cross sections and for the thermal vertical velocities, so that ALEBL can be computed.
Three different types of errors are computed all along the study. The first category (see Fig. 2) represents the systematic error on the PDF [Nn(Δn)] estimate in each bin (Δn). The second category (see Figs. 3, 4) represents the fitting function error when using the least χ2 method. The last category (see Figs. 4, 6) gathers systematic errors, either on the arithmetical mean computation or on the systematic error on a function computed from several independent mean variables.

The N-normalized dimensionless cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

The N-normalized dimensionless cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
The N-normalized dimensionless cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(a) Time evolution of the N-normalized cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(a) Time evolution of the N-normalized cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
(a) Time evolution of the N-normalized cross-sectional distribution
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Time evolution of the average cross section of type-2 thermals at cloud base S2 (km2) from LES (solid) and from
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Time evolution of the average cross section of type-2 thermals at cloud base S2 (km2) from LES (solid) and from
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Time evolution of the average cross section of type-2 thermals at cloud base S2 (km2) from LES (solid) and from
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Our starting hypotheses are (i) two-step triggering [as suggested by Chaboureau et al. (2004)] and (ii) that the cloud-base cross section plays a crucial role in controlling deep convection triggering (see Lothon et al. 2011).
4. LES analysis: Distribution of maximum cross section at the cloud base
a. Cross-sectional spectrum: 
(s)

Mapes (2000), Khairoutdinov and Randall (2006), Rio et al. (2009), Grandpeix et al. (2010) and Del Genio and Wu (2010) have suggested that the subcloud layer processes play a key role in producing the dynamical forcing, which lifts the parcel from the surface layer to its LFC. In a conditionally unstable atmosphere, the LCL nearly corresponds to the top of the boundary layer and to the bottom of the CIN. We shall consider it as the most relevant level at which to represent the couplings between boundary layer processes and deep convection. Consequently, the present study focuses on the thermal properties at cloud base.













Category 1 gathers together a very large population of thermals topped by small cumulus clouds with cloud-base sizes (average value n = 3) essentially ranging from n = 1–40 drafts (only one type-1 thermal is expected to have a size larger than 40; see Fig. 2). Their depths fluctuate between 50 and 500 m (not shown).
Category 2 concerns small and intermediate thermals accounting for the distribution tail (i.e., the right branch of the N PDF plotted in Fig. 2) with a cloud-base area ranging from n = 1–160 drafts (see Fig. 2) and depths fluctuating between 50 and 2000 m (not shown). The remaining class of clouds (not shown) is not represented by the fitting function given in Eq. (5) and concerns deep convective clouds (appearing after 1630 LT in the LES).
b. Cross-sectional spectrum evolution
Figure 3a represents the N-normalized PDF evolution [defined in Eq. (5)] fitting the afternoon hours of the simulation (1200–1800 LT). The slope of the exponential distribution of type-2 thermals decreases with time, while it does not seem to vary appreciably for type-1 thermals.
Figures 3b and 3c give further details on the evolution of each cloud population. Figure 3b shows that N2 decreases throughout the transition period. It is less trivial to extract a trend for population 1, as the error bars are very large at 1200 (only small clouds are present) and 1300 LT. At those times, populations 1 and 2 more or less overlap. On the other hand, according to Fig. 3c, S2 increases from 1200 up to 1800 LT. In other words, the transition from shallow to deep convection gives rise to fewer but larger thermals, suggesting that the gradual drying and deepening of the boundary layer (Lothon et al. 2011; Couvreux et al. 2012) is associated with larger cloud bases and deeper cumulus. Since N2 and S2 tendencies are of opposite signs, the fractional coverage αtot [as suggested by Chaboureau et al. (2004) through the NSD] is a priori not the best proxy for describing the transition process. The average cross section seems more pertinent for the transition. This result reinforces the relevance of considering spectral thermals rather than a bulk thermal and of treating both the thermal population and the cloud-base mean cross section independently. We also noted that the ratio between the surface covered by type-1 thermals and the total surface covered by thermals
Since observations (Lothon et al. 2011) have suggested that the largest thermals are the key elements of the transition, we will study the statistical properties of the type-2 thermals only. However, since the cloud-base cross section is a variable that is absent from boundary layer parameterizations using the single thermal approach, we first need to establish empirical relationships between cloud-base cross section and vertical cloud development (that can be retrieved from any boundary layer parameterization).
c. Vertical versus horizontal scale of type-2 clouds
The mean horizontal length scale of type-2 clouds at cloud base is







The dependency of the thermal width at cloud base on the cloud-base altitude is consistent with the common hypothesis of a nearly constant aspect ratio (around 2, according to Rio and Hourdin 2008) for the boundary layer thermal structures. However, the issue of the potential mechanism(s) explaining the matching horizontal and vertical cumulus growth is trickier to address. The increase of cumulus buoyancy along the transition reported by Wu et al. (2009) may be associated with the decreasing lateral entrainment rate due to the increasing cloud width. Then, the entrainment process provides an explanation of how the cloud width increase causes the cloud-depth increase, but it does not reveal anything about how cloud depth feeds back onto cloud width. We suggest two potential mechanisms here, based on two diabatic processes, which may explain how cloud height influences cloud width.
The first one is diabatic cooling by rain evaporation. High-resolution simulations (Khairoutdinov and Randall 2006; Matheou et al. 2011; Boing et al. 2010) have shown that density currents induced by cumulus rain re-evaporation often appear before deep convection onset and play a key role in the transition from shallow to deep convection. They suppress convection in their core and favor it on their edges by lifting the surrounding unstable air, in particular where colliding density currents result in sparser but stronger updrafts. They tend to feed deeper and broader clouds as cold pools grow. The second mechanism is diabatic heating by condensation. Clark et al. (1986) assert that midlevel cumulus cloud heating can trigger gravity waves, which reflect on the tropopause and feed back onto the lower levels, selecting eddies having horizontal length scales comparable with the gravity wave spacing. Such a mechanism would operate a scale selection on thermal eddies and favor sparser and larger horizontal structures during the transition.
d. Maximum cross-sectional distribution: 
max(Smax)








The maximum value PDF

(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Therefore, from the PDF
5. LES analysis: Distribution of maximum vertical velocities of type-2 thermal
a. Method
In this section, the objective is to compute a statistical maximum velocity for the type-2 thermals. For that purpose, the whole LES simulation (i.e., from 1200 to 1800 LT) is grouped together in a single dataset of 9500 thermals. Then, only the thermals with sizes exceeding n = 40 drafts (i.e., diameters exceeding 1500 m) are kept to give a final dataset of 900 thermals, almost exclusively type 2. Finally, this dataset is divided into 10 samples sorted by increasing cross section.
For each sample k, characterized by its n range and composed of Ntot,k = 90 clouds, Table 1 shows the arithmetic means 〈⋅〉k over the clouds of various fields defined at cloud base in section 5c: (i) the average vertical velocity
Mean dynamical characteristics of the 10 thermal samples of category 2.


b. Vertical velocity moments
In an attempt to characterize the vertical velocity distribution inside the thermals, we first look at the sensitivity of the vertical velocity moments to the mean cross section of each sample. Figure 6 displays the pairs

(top to bottom) Scatterplots of sample mean
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(top to bottom) Scatterplots of sample mean
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
(top to bottom) Scatterplots of sample mean
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1







c. PDF of draft vertical velocities 









Normalized histogram of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Normalized histogram of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Normalized histogram of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Complementary comments
(i) Reference cross section š of the drafts
The fact that
(ii) Vertical velocity mean and standard deviation




d. Maximum vertical velocity distribution 









Figure 8a displays the sensitivity of the maximum-value PDF

(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
(a) The
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Of course, one can expect this particular draft to have a greater chance of being close to the thermals core, in which the rising parcels have less chances of mixing with surrounding dry air via lateral entrainment. Nevertheless, the present result shows that we can easily compute the maximum velocity in a thermal at the cumulus base, just by knowing its width.
This concordance between simulated and calculated maximums finally shows that the tail of the Gaussian distribution of the 200-m draft velocities

Histogram of the CCDF of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Histogram of the CCDF of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Histogram of the CCDF of
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
e. Summary
To sum up, the dynamical properties of the type-2 thermals are uniform over the thermal field. Since the cross-sectional spectrum for type-2 thermals is exponential, this result is somehow consistent with the exponential distribution for individual mass fluxes proposed by Plant and Craig (2008). Moreover, each thermal can be considered as being composed of independent drafts (i.e., with no spatial coherence) of typical length scale l ≈ 200 m, following a Gaussian distribution for the vertical velocities, in which the average is practically equivalent to the standard deviation. Finally, the Gaussian distribution effectively describes the maximum-value statistics, which mostly depend on the cloud-base cross section.
6. Statistical available lifting energy (ALEBL,stat)
a. ALEBL,stat computation






















Time series of ALEBL,stat (m2 s−2).
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Time series of ALEBL,stat (m2 s−2).
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Time series of ALEBL,stat (m2 s−2).
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
According to the LES, the morning-time large-scale inhibition is very high, and ALEBL,stat reaches the CIN (not shown) around 1300 LT. Therefore, since both observational (Lothon et al. 2011) and LES (Couvreux et al. 2012) data show that deep convection is triggered near 1600 LT, the dynamical threshold ALEBL,stat > |CIN| alone is not sufficient to describe the whole transition process.
b. Toward a new triggering formulation
Lothon et al. (2011) mentioned that, around 1200–1300 LT, the boundary layer moved from a regular, steady cloud-street organization to a more isotropic structure with bigger clouds. This could correspond to the beginning of the transition stage. Thus, although ALEBL > |CIN| is apparently not a pertinent threshold for deep-convection triggering, it may be relevant for describing the threshold for moving from a shallow cumulus regime to a transition regime. In the shallow cumulus regime, no clouds cross the inhibition layer. In the transition regime, many cumulus clouds have enough kinetic energy to overshoot the CIN, but are still too small to reach the high troposphere. So, we shall impose a complementary constraint on the size of the thermal to permit the triggering of deep convection.
7. Deep convection triggering formulation
In the current version of the LMDZ model, deep convection triggering by boundary layer thermals is exclusively based on the threshold condition ALEBL > |CIN|. As the associated thermal representation is deterministic, either no thermals trigger, or all the thermals trigger, deep convection. However, since a thermal spectrum is considered here, we can, a priori, expect to have both passive boundary layer cumulus clouds and overshooting clouds in a given domain. As already mentioned, the thermal size appears to be of primary importance in the triggering process; Lothon et al. (2011) noticed that the first deep convective cells occurred over a zone covered by the largest horizontal structures of the observed domain. Chaboureau et al. (2004) also stressed the existence of two-step triggering, in which a transition phase clearly appeared.
Hence, the main idea of the triggering formulation is that the thermal field must require (i) at least one thermal with a maximum kinetic energy exceeding the CIN, which means ALEBL,stat > |CIN| and (ii) a sufficient number of thermals having sizes that may potentially exceed a certain threshold value Strig. This threshold corresponds to an arbitrary limit after which the associated cloud no longer corresponds to a cumulus but to a congestus or a cumulonimbus cloud, whose top nearly reaches the freezing level (see section 3b in Part II for more details). It might be expected that the largest thermal size should grow gradually up to the time when it reaches this threshold.

















The point is now to determine what a reasonable estimation could be for τ. During a period Δt, the mean cumulus lifetime directly influences the correlation between the two consecutive scenes at t and t + Δt; thus, it could be reasonable, as a first approximation, to consider τ as the mean cumulus lifetime. Several LES studies have investigated this issue in various contexts (oceanic trade wind cumulus case, continental cumulus, etc.). Considering the studies by Zhao and Austin (2005), Heus et al. (2009), and Seifert and Heus (2013), the cumulus lifetime basically ranges from 1000 to 2000 s.
However, here τ represents a decorrelation time between two cumulus scenes only for type-2 thermals. We used 5-min interval snapshots from 1100 to 1530 LT in the LES to reach an estimation of this decorrelation time. We found an increase of the decorrelation time τ along the simulation, starting from τ = 1000 s between 1100 and 1200 LT and going to τ = 1800 s between 1400 and 1530 LT. Therefore, we shall consider the range 1000 < τ < 2000 s as a reference in Part II.
Looking back to Fig. 5, the distribution of Smax is broad, meaning that Smax may vary greatly around the median value
Recap: The three steps of the transition process
1) Preliminary condition
Only moist thermals are expected to trigger deep convection; consequently, the reference altitude for computing the thermals’ lifting energy (ALEBL) is taken at cloud base. Thus, the first necessary condition is that the boundary layer must be cloudy.
2) The dynamical threshold


3) The geometric threshold





Figure 11 illustrates the conceptual view of this formulation, from the first cloud to the triggering of deep convection. From this new formulation, a stochastic triggering parameterization is proposed in Part II.

Sketch of the transition from shallow to deep convection. Note that, in a numerical framework, a similar picture can be drawn by replacing τ by Δt, where Δt is the model’s time step.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1

Sketch of the transition from shallow to deep convection. Note that, in a numerical framework, a similar picture can be drawn by replacing τ by Δt, where Δt is the model’s time step.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
Sketch of the transition from shallow to deep convection. Note that, in a numerical framework, a similar picture can be drawn by replacing τ by Δt, where Δt is the model’s time step.
Citation: Journal of the Atmospheric Sciences 71, 2; 10.1175/JAS-D-12-0336.1
8. Discussion and conclusions
Here the transition process is described considering a statistical ensemble of thermals with intrathermal velocity fluctuations and interthermal cloud-base cross-sectional fluctuations. Data from a LES case of transition from shallow to deep convection over land provide the geometric and dynamical properties of the cloudy thermals at the cloud-base level during the transition from shallow to deep convection.
The thermal population is fitted by a sum of two exponential distribution laws corresponding to two types. During the transition time, type-2 thermals are less numerous, become wider, and feed deeper clouds. A linear relationship between type-2 thermal width at cloud base, cloud depth, and cloud-base altitude has been proposed and verified on the LES case. Moreover, type-2 thermals can be described as a sum of independent drafts with length scales of several hundreds of meters, having a velocity distribution
From the distribution of the maximum values of thermal sizes and draft velocities, a statistical maximum velocity
Thus, the present formulation proposes a three-step transition and consists of two consecutive thresholds, the first deterministic and the second stochastic. The first threshold is dynamic; it governs inhibition being exceeded by at least one thermal of the domain (i.e., ALEBL,stat > |CIN|). It represents the moment when shallow clouds start to overshoot the inhibition layer and reach their LFC, that is, the transition phase. The second threshold is geometric and rules deep convection triggering. Since deep convection tends to trigger where the largest horizontal structures occur, there is a threshold cloud-base cross section which has a certain probability of being exceeded at every independent cloud scene.
To sum up, the new triggering formulation (i) suggests a thermal size distribution, in which only the largest elements control the triggering; (ii) proposes a new computation of the thermal available lifting energy at the cumulus cloud base; (iii) allows the existence of a transition stage between shallow and deep regimes through a multistep process; and (iv) includes a stochastic component, to better mimic the episodic aspect of the onset of deep convection.
However, integrating such a formulation in a parameterization of deep convection triggering by boundary layer thermals is still a difficult task. The main difficulty is to retrieve a cross-sectional spectrum from the variables given by the boundary layer parameterization, which is single-thermal based in most cases. A triggering parameterization for the LMDZ model based on this formulation is proposed in Part II.
One may contest that this triggering formulation takes its inspiration from only one case study and so has little chance of being applicable in other situations. For this reason, the robustness of the formulation will be further investigated in Part II; the corresponding parameterization will be tested over various environmental conditions (continental and oceanic) and also in conditions favorable and unfavorable to triggering. It will be tested first in a single-column framework on different case studies and then in the global framework to estimate the added values in comparison to the standard approach in the full GCM.
Acknowledgments
The research leading to these results has been supported by both the French Department of Teaching and Research and the European Union, Seventh Framework Program (FP7/2011-2015) under Grant Agreement n282672. The authors thank Francoise Guichard and Jean-Philippe Lafore, from Météo-France (Toulouse), for their very helpful comments and for the numerous and enlightening discussions we had about deep convection issues. They also thank the three anonymous reviewers, whose meticulous analysis and enlightening suggestions made it possible to improve the quality of the manuscript.
APPENDIX
Maximum of a Large (≃100) Number of Random Variables with Identical PDFs








a. CCDF of the maximum







b. Inverse formula

















c. Case of a thermal cross section s
d. Case of vertical velocity 












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