The parameterization presented in this paper is tested and integrated into the global climate model of Laboratoire de Météorologie Dynamique LMDZ. LMDZ is the atmospheric component of the IPSL coupled atmosphere-ocean model, IPSL-CM, used in particular for the CMIP exercises. Concerning boundary layer subgrid scale vertical transport, LMDZ combines a K-diffusion model based on a prognostic equation of the turbulent kinetic energy (TKE) with a mass flux parameterization . This so-called “thermal plume model” and its coupling to a bi-Gaussian cloud scheme is presented in Sect. , and  as it is an essential element of the proposed parameterization. Deep convection on the other hand is modeled by the Emanuel's scheme in interaction with an original parameterization of cold pools created by reevaporation of convective rainfall . The LMDZ6A configuration used here, developed for CMIP6, includes developments in many aspects of the model compared to previous CMIP5 configurations, as for example a modification of the thermal plume detrainment which led to a significant improvement of stratocumulus representation . The LMDZ model is a flexible tool which contains in particular a single column model (SCM) version. The SCM is extensively used for parameterization development, assesment and tuning based on 1D/LES comparisons.

The 1D/LES strategy is applied here with LES cases that embrace shallow convection scenes over earth and ocean as well as stratocumulus/cumulus transitions cases. These test cases were widely used in the development of the thermal plume model , the bi-Gaussian cloud scheme and the modified formulation of the thermal plume detrainment . They will be refered as IHOP/REF, ARMCU/REF, RICO/REF and SANDU/REF, FAST and SLOW and we will briefly recall the context of these different 1D cases.

IHOP comes from observations carried out during the International H₂O project on 14 June 2002 on the Great Plains . It represents an almost cloudless boundary layer.

The ARM case is derived from observations collected on 21 June 1997 at the Atmospheric Radiation Measurement site in Oklahoma, USA . It represents a typical diurnal cycle of the shallow convective boundary layer with appearance of a few cumuli clouds in the middle of the day.

The RICO case (Rain In Cumulus over the Ocean, ) is a case of shallow cumuli clouds over the ocean characterized by frequent precipitation.

Finally, the SANDU REF, FAST and SLOW cases are described in . They were built by compositing the large-scale conditions encountered along a set of individual Lagrangian 3 d trajectories performed for the northeastern Pacific during the summer months of 2006 and 2007. They are meant to represent oceanic boundary layers overhung by stratocumulus clouds which become thinner with a gradual transition to shallow cumulus. REF, FAST and SLOW refer to different configurations involving more or less rapid transitions.

LES reference simulations used in this study have been performed with the MESO-NH non-hydrostatic model for the IHOP, ARM and RICO cases. The LES used as a reference for the SANDU case is the one performed with the UCLA LES and described in .

As explained in , the calibration of the parameterizations of a GCM is a very sensitive step in climate modeling with the multiplication of sub-grid schemes and their associated free parameters. In LMDZ the calibration is done in successive stages from the scale of individual parameterization to the scale of the Earth's climate system as a whole. To facilitate this multi step and multi parameter tuning, automatic tuning techniques were developed based on the “history matching” approach, implemented in the “High-Tune explorer” or htexplo tool. The tool explores the model parameter space in successive waves and build an emulator of given metrics. Once this emulator is built, the parameter space is reduced at each wave according to the gap between the emulated values of the metrics and the target values of the LES. This tuning process on 1D simulations not only makes it possible to significantly reduce the parameter space but it also permits to focus and improve our understanding of the physical processes involved in the parameterizations and thus to adapt the underlying modeling work. More than a simple statistical calibration tool, it is therefore a real working support for the modeler. SCM tuning is then completed by one or several 3D waves.

Based on this approach, and using the previous mentioned 1D cases, were able to finely calibrate the key parameterizations on which we step on in the present work, the thermal plume parameterization and the cloud scheme. Here, we will reinvest the same tuning approach in order to obtain reliable and consistent comparison criteria with previous work.

Let ψ be a conservative variable, v the wind field and ρ the density. Using an Reynolds decomposition of the physical quantities ψ = ψ‾+ψ′ and in the case of non-viscous transport, we can write: 1∂ψ‾∂t=-v.grad(ψ‾)-divρw′ψ′‾ρ The first term on the right-hand side is the tendency due to large-scale advection, computed by the dynamical core of the model. The second term is the tendency due to turbulent transport. It is computed in LMDZ as the sum of a K-diffusive term and a mass flux term: 2ρw′ψ′‾=-Kρ∂ψ‾∂z+fψth‾-ψ‾

The K-diffusion term follows and includes a prognostic equation for the TKE. The mass flux transport relies on the so-called thermal plume model . It is activated only if the surface layer is unstable and relies on an original closure in Convective Availabale Potential Energy (CAPE) described by . It represents a population of thermal plumes through a single equivalent thermal plume with vertical velocity wth‾ and surface fraction α. The vertical variation of the convective mass flux f = ραwth‾, assumed stationary during one physics time step, is related to a lateral entrainment e and detrainment d through a mass conservation equation: 3∂f∂z=e-d The value of variable ψ within the thermal plume, noted ψth‾, is given by the conservation equation: 4∂fψth‾∂z=eψ‾-dψth‾+αρS, where S is a source term for variable ψ (S ≡ 0 for a conserved variable).

The entrainment rate ϵ = e/f depends positively upon the plume buoyancy B, according to the following equation: 5ε=max⁡0,B11+B1A1Bwth‾2-A2 where A1, B1, and A2 are tunable parameters.

The presence of parameters A1 and A2 in the computation of entrainment comes from the source term of the conservation equation for the vertical momentum (Eq. , with ψ = w). This source term reads S = A1B-A2wth‾2 and is the sum of a buoyancy and drag terms. Note that the plume fraction α is computed at each vertical level as f/(ρwth‾), f being computed from the mass conservation equation and wth‾ from this vertical momentum conservation equation.

For a given value of S, the source term in the vertical momentum conservation equation, parameter B1 (taking values in [0,1]) controls whether the plume velocity increases rapidly with no lateral entrainment (B1=0) or increases less rapidly with strong lateral entrainment of air from the environment, assumed at rest (large values of B1).

The detrainment rate, δ = d/f, is mainly positive in the regions where the buoyancy is negative and reads 6δ=max⁡0,-A1×B11+B1B*wth‾2+CQΔqt/qtwth‾/w020.5 with w0 = 1 m s⁻¹. Δqt is the contrast in humidity between the plume and its environment, and CQ is a tunable parameter. The CQ term was introduced to represent in part the enhancement of detrainment by the negative buoyancy subsequent to the evaporation of cloud water. The term is however active below the cloud as well. Note that a more physical parameterization of this CQ term is currently under development.

The computation of the buoyancy in the detrainment formulation reads: 7B*=gθv,th(z)-θvz+δzθv(z+δz), where θv is the virtual potential temperature. In this formula, the buoyancy is computed by comparing the plume internal properties at altitude z with the environment at altitude, z + δz, with δz = DZ×z (DZ being an adjustable parameter as well). This modification enabled the thermal plume model to simulate correctly the stratocumulus and transition cases, by making the plume “aware” of the top boundary layer inversion before reaching it, thus increasing artificially detrainment just below. As explained in , it may in part account for the fact that the air detrained at a given level has followed an overshoot and then a rapid downward motion, as occuring in fountains structures above stratocumulus or in so-called subsiding shells created around cumulus clouds by reevaporation of the detrained cloud water.

For the derivation of the variance model, below, we use the following Eq. (), obtained by combining Eqs. () and () without the diffusive term: 8∂ψ‾∂t=dρψth‾-ψ‾+fρ∂ψ‾∂z The first term on the right-hand is the tendency due to the thermal detrainment towards the environment, the second term is the tendency associated with the compensatory subsidence in the environment, the impact of which depends on the vertical gradient of the variable in the environment. Because the mass flux f is positive, a positive vertical gradient of the physical variable results in a positive tendency, as air with larger values of ψ is transported downward by subsidence. Here the fact that the tendency of the variable does not depend on entrainment is linked to the assumption that the entrained air is the average air of the environment. This hypothesis, which may be questioned, notably by investigating the impact of subsiding shells on the thermodynamic properties of the air in the vicinity of cumulus clouds , simplifies both mean and variance equation.

The LMDZ statistical cloud scheme is based on a bi-Gaussian PDF of saturation deficit in the presence of thermals . The saturation deficit is given by s = al(qt-qsat(Tl)) where Tl is the liquid temperature, qt the total specific humidity, qsat the total specific humidity at saturation and al is a factor depending on thermodynamic variables defined in and . The bi-Gaussian PDF of the deficit at saturation reads: 9PDF(s)=(1-α)Gs,senv‾,σenv+αGs,sth‾,σth, where G is a Gaussian function of the variable s with mean value senv and variance σenv2 in the environment and sth and σth2 inside the thermal plume, α being the fraction of the grid covered by the thermal plume. Because the fraction covered by thermal plumes is small (α ≪ 1) and the thermal plumes are moister than their environment (senv < sth), this distribution is positively skewed. We should notice that this PDF fails representing the negative asymmetry that may appear due to the presence of organised downward plumes within the compensatory subsidence. These plumes are however generally less organized than the ascending plumes forced by surface energy fluxes, and the asymmetry induced is also predominant near the surface, far from the boundary layer top where cumulus or stratocumulus clouds form.

With this choice of PDF, using the thermal plume fraction α to fix the weights of each Gaussian term, the variance and skewness of the saturation deficit are determined by the following equations being given the standard deviations σenv and σth and the mean values in both environment and thermals (the detailed derivation being given in Appendix A). 10V=ασth2+(1-α)σenv2+α(1-α)(s‾th-s‾env)211S=3α(1-α)s‾th-s‾envσth2-σenv2+α1-3α+2α2s‾th-s‾env3V32

Once the PDF is given, the cloud fraction cf of the mesh and the cloud water content qc can be computed as: 12cf=∫0∞PDF(s)ds13qc=∫0∞s⋅PDF(s)ds

If the mean values of the saturation deficit in the updrafts and in the environment are explicitly calculated in the GCM, it remains to determine the values of the standard deviations in order to close this cloud scheme.

In the current diagnostic version, the standard deviations of humidity in thermals and the environment are parameterized as follows: 14σth=BG2(α+0.01)γ2s‾th-s‾env+bq‾th and 15σenv=BG1αγ11-αs‾th-s‾env+bq‾env. BG1, BG2, γ1, γ2 and b being tunable parameters. This specific parameterization of the variance is based on a heuristic reasoning using the exchange surface and the specific humidity difference between the thermals and the environment . Our aim is to replace this multiparameter heuristic parameterization with a parameterization based on a well-established variance equation including all the air mass exchanges.

Equation () cannot apply to the variance of a variable because the variance is not conservative. Analytical equations for the variance evolution in the presence of detraining mass fluxes have been proposed by K05 with one single plume, or by for a general formalism with N plumes. However their implementation in the climate model requires knowledge of the humidity variance in thermals. Although this is not an insurmountable difficulty in our model where it is predicted, we have chosen to implement a more straightforward, but theoretically equivalent, methodology consisting in transporting q and q2 within the mass flux scheme. Let us first write the evolution of the variance of the total specific humidity in terms of q et q2: 16∂q′2‾∂t=∂q2‾∂t-∂q‾2∂t=∂q2‾∂t-2q‾∂q‾∂t.

To compute the convective variance tendency we only need knowledge of the mean total specific humidity q‾, its convective tendency ∂q‾∂t and the convective tendency of the total specific humidity square ∂q2‾∂t. q‾ and ∂q‾∂t are already calculated in the thermal plume model, therefore only ∂q2‾∂t remains to be estimated. Noting that the total specific humidity is treated as a conservative variable in the thermal plume transport model, we initially assumed that its square is also conservative. Indeed, the material derivative of a squared quantity is zero if the material derivative of this quantity is zero. We would then be justified in transporting the square of the total specific humidity as a conservative tracer, just like the total specific humidity itself. However, the square is not a purely conserved quantity within the thermal plume because air parcels entering with a different humidity will mix. To subsequently account for this mixing, we added a relaxation term to the thermal plume equation.

To compute Eq. () in our first conservative approach (without the relaxation term) we only have to call the plume transport routine for both total humidity and its square, instead of just the first one, with no need to implement explicit equations. The consistency of this approach can be checked by deriving the analytical development which joins and provides another perspective on the K05's formalism of variance transport. Applying the Eq. () both to q and q2, we obtain: 17∂q′2‾∂t=dρqth2‾-q2‾+fρ∂q2‾∂z-2q‾dρqth‾-q‾+fρ∂q‾∂z

Subsidence terms combine as follows: fρ∂q2‾∂z-2q‾fρ∂q‾∂z=fρ∂q′2‾∂z. Detrainment terms can also be rearranged: (qth2‾-q2‾)-2q‾(qth‾-q‾)=(qth‾-q‾)2+(qth′2‾-q′2‾). And the convective variance tendency finally reads: 18∂q′2‾∂t=dρqth‾-q‾2+qth′2‾-q′2‾+fρ∂q′2‾∂z

This approach leads to a similar equation that K05 derives from the mixing of two air masses and its impact on the humidity variance in the environment. This is not surprising because both approaches ultimately rely on the air mass conservation equations. In the present formalism we can underline the absence of the entrainment term, due to the already mentioned fact that we assumed that the entrained air had the average composition of the environment. We can interpret the different terms of the Eq. () as follows:

The first term represents the impact of the mean values difference between the thermal plume and the environment

This variance transport equation is limited so far to organized convective sources which are supposed to be dominant in the studied cases, it could subsequently include however the diffusive terms, computed with the same approach by calling the diffusive transport model for both q and q2. We could also include possible source/sink terms due to evaporation of precipitation for instance, or large-scale advection. This will be the subject of forthcoming publication. Here we place ourselves in a framework where the transport of variance is carried out by the organized convective terms and we only add a classical exponential dissipation term representing the small scale homogenization processes which will be discussed in Sect. . The present model finally reads: 19∂q′2‾∂t=dρ[(qth‾-q‾)2+(qth′2‾-q′2‾)]+fρ∂q′2‾∂z-q′2‾τ, where τ represents the relaxation time of the variance in the environment. In practice we can either implement this variance equation term by term or call the thermal transport equation for q and q2 as described before. However, the two methods lead to very different implementation of the variance model, each of which having its own interest. In the first method we explicitly calculate the different terms of the Eq. () which allows us to compare the relative importance of this terms, especially the one associated to the difference in humidity mean and variance. The second way is simpler to implement and can be transposed to other transport processes (diffusive transport or large-scale advection for example). In this case it is no longer necessary to extract all the physical quantities from the thermal model, only the tendencies of q2‾ and q‾ are necessary.

As highlighted by K05, an important question underlying this theoretical approach is the determination of humidity variance in updrafts: qth′2‾. In the diagnostic parameterization, it was prescribed using the humidity difference with the environment. However, the transport of q and q2 within the thermal model now gives us direct access to qth′2‾ = qth2‾-qth‾2.

With the same algebra as above, starting from the Eq. () applied to qth and qth2, we obtain the following expression for the vertical evolution of the variance in the thermal. 20∂qth′2‾∂z=ϵqth‾-q‾2+q′2‾-qth′2‾ when considering q2 as a conserved quantity. This expression can be completed by a relaxation term representing small-scale diffusion processes in the thermal. The advective-diffusive coupling leads to the introduction of a characteristic diffusion height z0 = wth‾τth in the thermal plume where wth‾ is the vertical advection and τth the diffusion timescale in the plume. 21∂qth′2‾∂z=ϵq‾-qth‾2+q′2‾-qth′2‾-qth′2‾wth‾τth

In practice, qth′2 can be computed applying the thermal plume model to q2 instead of q, adding the relaxation term in the computation, which requires to provide the value of qth obtained in the transport computation of q by the thermal plume model.

The Newton relaxation time τ which appears in the dissipation term is an important parameter of the variance calculation. It can be estimated as the relaxation time of the turbulent energy as was proposed in and . proposes to add to this small-scale relaxation a second dissipation process associated with slower large-scale two-dimensional (2D) dissipation due to horizontal wind shear instability. This second component would be present even in case of strong temperature stratification. Although it will be a part of our reflection when extending our work to the upper atmosphere, we neglect it here, as we only focus on shallow convective cases where the first component is expected to be dominant. The order of magnitude of the turbulence relaxation time is: 22τ=lTKE l being a mixing length of roughly 100 m . We obtain an order of magnitude of 100 to 1000 s for τ which is a fairly fast relaxation, especially compared to the 10 d estimated 2D relaxation . The prognostic model depends on a unique tunable parameter. We tested two options which led to small differences in the results either considering τ or l as the tunable parameter in Eq. (). In the second case, it was necessary, as in , to define an upper limit to the relaxation time as our TKE is likely to cancel out even below 3000 m. This prevents the relaxation terms from becoming too small and leading the variance to accumulate too strongly.

To replace the previous diagnostic model with a prognostic model based on humidity variance transport equations, it is first necessary to connect the humidity variance to the variance of the saturation deficit which drives the statistical cloud scheme. This well-known relationship can be written as: 23σ(s)2=al2q′2‾+αl2Tl′2‾-2αlq′‾Tl′‾ where αl = ∂qsat∂TTl, al was mentioned in Sect.  and Tl is the liquid temperature.

We see that a liquid temperature variance term appears in this equation as well as a cross-correlation term whose physical interpretations are described in . Several studies have analyzed the relative importance of humidity and temperature contributions to the variance . Although the temperature contribution is not completely negligible, it appears from these analyzes that order 1 is mostly dominated by the variability of specific humidity. In the context of this work, it seemed relevant to mainly concentrate on the specific humidity variability. Orders of magnitude of these variabilities can be estimated in LMDZ by simply diagnosing the differences in humidity and liquid temperature between the thermals and the environment. On this basis we obtain an order of magnitude of 10-4 for q′2‾ and closer to 10-5 for αl2Tl′2‾. The comparison in LES of the standard deviations of total specific humidity multiplied by al and the saturation deficit also show an excellent first-order agreement. In this work the hypothesis is made that the variance of s is controlled by the variance of the total specific humidity.

In the new approach, the diagnosed value of σenv is finally simply replaced by alq′2‾ and σth by alqth′2‾.

A long standing issue in parameterization development and improvement was the difficulty of retuning simultaneously the free parameters after a parameterization change, without which a physics improvement could be overled by a bad parameter tuning. This retuning is now made automatic thanks to the htexplo tool. The same tuning was applied here to the old model version and the new one with prognostic scheme. After a number of “waves”, we identify a number of configurations for both models, for which a series of metrics computed on SCM simulations of the IHOP, ARMCU, RICO and SANDU cases match the metrics values computed on LES to less than a given tolerance to error σ. We use the same metrics as in Sect. 5.1 of . These metrics are based on three model variables: potential temperature, humidity and cloud fraction. They are defined as temporal and spatial integrations over time and altitudes. For cloud fraction, three specific metrics are used: the first one is linked to the maximum cloud fraction on the vertical αcld,max, the second one represents an average altitude of clouds zcld,ave, and the last one corresponds to the altitude of the maximum cloud fraction zcld,max, computed as an averaged altitude weighted by the fourth power of the cloud fraction. Table  details all the different metrics used in this tuning (the three cloud metrics, and those concerning potential temperature θ and humidity qv).

The free parameters are also kept the same as in except for the diagnostic model parameter of the diagnostic variance scheme BG1 and BG2 (Eqs.  and ) which are replaced by parameters τ and τth in the prognostic scheme, allowed to vary between 100 and 2000 s. The other free parameters, common to both tuning exercises, are: (1) those involved in the modeling of the entrainment and detrainment rates of the thermal plume model (Eqs.  and ): A1, B1, B2, CQ and DZ and (2) the CLC and EVAP parameters are involved in the precipitation and rain re-evaporation model, CLC being associated with the critical incloud water from which precipitation is activated and EVAP being a free parameter of the precipitation flux equation which controls the fraction of precipitation that re-evaporates at a given altitude. The formalism of these parameterizations was introduced by the work of . Table  presents a summary of the different parameters to be tuned. More details on the parameterizations and free parameters can be found in .

At each tuning wave, LMDZ SCM-simulations are computed within the range of authorized parameters by the previous wave (NROY: not-ruled out yet). For each of this simulations, a score is assigned per metric, this score being defined from the difference between the target value of the metric (the average value of the LES) and the value for the given simulation, as well as from the accepted tolerance. The closer the score is to 0, the closer the simulation metric is to the target value, a score equal to 1 indicates that the difference between the simulated metric and the target value corresponds exactly to the tolerance which we are willing to accept. The simulations can thus be classified according to the obtained scores, either on an average score criterion or on a maximum criterion. Here we use the maximum criterion to classify the ten most efficient simulations in order to prevent large errors on certain metrics from being compensated by very good results on others.

Figure  compares the results obtained on all the metrics between the diagnostic and prognostic models after 20 waves of tuning. From bottom to top, the figure displays the scores of the 10 best simulations across the 11 studied metrics. The second row shows the average score for each simulation, and the top row shows the maximum score of each simulation, corresponding to the worst score obtained among all metrics. As mentioned previously, the simulations are ranked according to the order of their maximum scores. In particular we see that the ten best simulations achieve almost identical maximum scores in both cases, close to 0.75 which means that the worst metric is at a distance of 0.75 σ from the target value. Note however that this score is obtained for a somewhat reduced tolerance to error σ in the case of the prognostic case for the metrics relative to the cloud height, zcld,ave and zcld,max. This lower values were adjusted since the new scheme was globally producing a better adjustment of the cloud height.

It is important to underline that the tuning process does not aim to select an optimized configuration of the model but rather to highlight a range of parameters leading to acceptable results within the framework we have defined and the typical performance of the model in this range. In this context this tuning showed that the acceptable range of the relaxation time is 100 to 1300 s which is consistent with the previous qualitative considerations.

When using the TKE to define the relaxation time, the performance of the diagnostic scheme are comparable (results not shown) and the acceptable range of the mixing lengths shows up to be between 70 and 160 m.

In this section, we focus on the variance, skewness, and cloud fraction profiles, which constitute the main diagnostics examined in this study. Most other physical quantities are only weakly affected by the change in the variance parameterization, as shown in Sect. . The results for the almost cloud-free IHOP case – used to control the potential temperature profile, which remains largely unchanged by the new parameterization – are not discussed here. Figure  presents the time evolution of the vertical cloud fraction profile simulated by the LES (left column) for the ARMCU, RICO, and SANDU cases, together with the differences between the LMDZ SCM and the LES for the diagnostic (middle) and prognostic models (right). Overall, the cloud fraction simulated by the two models remains very similar, with the prognostic formulation leading to a slight reduction in biases. Even if biased, the representation of the cloud fraction has been much improved in comparison with general model performances in simulating the ARM cumulus case 20 years ago , in particular regarding the cloud base and cloud top. Also, the RICO and SANDU cases involve precipitation processes and radiation, to which the cloud fraction simulated by LES is also quite sensitive .

Figure  separates the total standard deviation into the standard deviation within thermals and within their environment, along with the corresponding difference in total specific humidity. Figure  shows the vertical profile of cloud fraction and the associated standard deviation of specific humidity for specific times of simulations, as well as the time evolution of the standard deviation averaged within the cloud layer. Figure  evaluates the representation of the third order moment and skewness.

While the cloud fraction and the standard deviation of total specific humidity are quite close between the two model versions and the LES (Fig. ), the prognostic model improves the representation of the standard deviation within thermals (Fig. ). It can be seen that in the diagnostic model, the standard deviation is almost proportional to the specific humidity difference between the thermals and the environment and one order of magnitude too large. This strongly impacts the third-order moment (Fig. ) through the first term of Eq. (). This major flaw is largely corrected by the prognostic model, although the standard deviation is there slightly underestimated. The third-order moment, which reflects the asymmetry of the humidity distribution, is now in better agreement with the LES. This demonstrates that an improved representation of the higher moments of the distribution profiles can be obtained by simply exploiting the transport of q and q2 within the thermals, with a tunable parameter that is easy to estimate using TKE. A possible flaw of the prognostic model is the lack of representation in the PDF of organized subsiding structures which tend to locally dry out the environment. Although subsidences are taken into account as sources of variance, they cannot negatively impact the third order moment because of the binary structure of our PDF with only thermal plume and environment. This is particularly visible in rare areas of negative skewness but it should not overshadow the quite accurate estimation of the third-order moments of the model.

Figure  further shows the contribution of the different terms of Eq. () in the variance transport. It shows the clear predominance of the first term of the variance transport equation, a term which represents the squared difference in humidity between thermals and environment. The two other terms, difference of variance and subsidence, appear to be an order of magnitude smaller so that they could be neglected if one wants to simplify the implementation of the explicit transport equation in the model. This result is specific to shallow convective cases, where the standard deviations of humidity, both in thermals and in the environment, are small compared to the mean values and their difference (see Fig. ). In contrast, in a deep convective case, such as that studied in K05, the ratio r = σqt is much higher, and one can therefore expect a significant impact from the variance difference term. K05 indeed shows that, in this case, the variance difference term is of the same order of magnitude as the squared humidity difference term. This point should not be overlooked when extending this study to deep convection cases. In that context, however, particular attention will also need to be paid to precipitating downdrafts, which can have a significant, and often negative, impact on the variance and skewness.

Accurately representing the vertical profile of cloud fraction during the stratocumulus-to-cumulus transition remains particularly challenging (SANDU case). showed that a carefully tuned modification of the detrainment parameterization represents an important first step toward simulating this transition. However, Fig.  shows that the cloud fraction still tends to remain saturated with the diagnostic model while it gradually fades in the LES. Moreover this cloud fraction is too thick at night in LMDZ especially from the second day of the simulation where it becomes significantly finer in the LES, leading to sometimes large differences in Fig. . Figures  and  show that both aspects are attenuated with the variance prognostic model, even though some notable differences still persist. Looking deeper at the variance profile, we can attribute this improvement to a better representation of the variance peak at the top of the clouds. Let's remind, indeed, that an increase in the variance of the statistical cloud scheme, in case of high cloud fraction, is likely to reduce the cloud fraction by increasing the fraction of air with lower humidity. This better representation of the variance peak can partly be attributed to the thermal plume detrainment peak in this area as we can see in Fig.  (in the middle bottom) even if this peak occurs a little too high. In fact, the detrainment is directly involved in the variance transport equation but was not taken into account in the diagnostic model which rather involves a term dependent on the thermal fraction (dashed red curve in the middle of Fig. ). This term vanishes to zero while the detrainment reaches its peak which explains the difference in behavior between the two models. We can notice, however, that the difference in humidity between thermals and environment also plays a crucial role at these altitudes (in black in Fig.  in the middle) but in the diagnostic model this term is multiplied by a factor which tends quickly to zero while in the prognostic model the multiplicative factor, which depends on the detrainment, cancels out higher and can even reach a local maximum at the top of the clouds.