Following , we discriminate between different cloud types for convective transport predictions, as not all clouds transport ABL air towards the free troposphere. Forced clouds, related to air parcels that reach the lifting condensation level, are buoyantly too weak to reach the level of free convection. Consequently, forced clouds are neglected in this study. Clouds that reach the level of free convection are marked as active clouds, as the latent heat release increases the in-cloud buoyancy, thereby enhancing cloud growth. As a result, they affect the underlying atmosphere by venting. When the active clouds decouple from the ABL thermals, they lose their supply of energy and become passive. As a result, they do not contribute to the mass transfer anymore .

The part of the domain in which convective transport occurs is quantified by the area fraction of clouds, which is defined at each level independently . Note that we cannot use cloud cover, as this property is not locally determined but based on the vertically integrated liquid water path. Furthermore, we distinguish in the remainder of the paper between all clouds and cloud cores, i.e. active clouds, with subscripts c and cc, respectively. As a result, we can distinguish four indicators for cloud presence, namely, cloud cover (cc), cloud core cover (ccc), area fraction of clouds (ac) and area fraction of cloud cores (acc).

Mass transport can be approximated as the kinematic mass flux (M) multiplied with the spatial difference in the concentrations of chemical species at cloud base (ϕ) : w′ϕ′‾=M(ϕcc-ϕ‾(zb)), where ϕcc indicates the value in the cloud core, and ϕ‾(zb) indicates the domain-averaged value at cloud base.

The kinematic mass flux, M, is defined by the area fraction of cloud cores (acc), the difference between the cloud core vertical velocity (wcc) and the domain-averaged vertical velocity at cloud base (w‾(zb)) , through M=acc(wcc-w‾(zb)). In the remainder of this paper, we assume w‾(zb) to be zero. For models that run on a coarser grid resolution than the width of a cloud core, the variables of Eq. () cannot be resolved explicitly and therefore need to be parameterized. We start by parameterizing the area fraction of cloud cores (acc). NG06 approximated acc by the total area fraction of clouds (ac). The parameterization of ac is developed by CB95, which uses local variables that depend on temperature and moisture, and is expressed by ac=0.5+βarctan⁡(γ⋅Q1), where the constants β=0.36 and γ=1.55 represent a fit through the LES results of CB95. Q1 is calculated as Q1=s‾σs. Here, s denotes the saturation deficit in kg kg-1 and σs indicates the standard deviation of s. assumed for simplicity that Q1 can be represented as Q2=qt-qsσq, where qt and qs are, respectively, the total and saturation specific humidity, and σq is the spatial standard deviation of the specific humidity. Based on this work, NG06 applied this expression for acc, while the Q2 is replaced by Q3, which is further simplified to be applicable in a mixed-layer slab model, according to Q3=〈qt〉-qs|hσq|h. Here, 〈qt〉 is the total specific humidity averaged over the mixed layer (indicated by angle brackets) and qs|h and σq|h represents the respective values at the mixed-layer top. Although these adapted variables indeed coincidentally converted the expression for ac to a reasonable prediction for acc for the case evaluated by NG06, we demonstrate in Sect.  that this is not valid for all thermodynamic and dynamic conditions and that a different formulation should be applied.

As shown by , the cloud core vertical velocity can be scaled with the Deardorff convective velocity scale (w∗) . Building on this work, and showed for several ShCu cases that the inclusion of a prefactor improved this scaling: wcc≈0.84w∗, which will be further extended in this study. Furthermore, as shown by , the concentration of chemical species at the base of the active clouds can be parameterized as: ϕcc-ϕ‾(zb)≈-1.23(ϕ‾(zb)-〈ϕ〉).

The numerical model used in this study is DALES 4.0. This version contains several improvements over version 3.2 , including additional elements e.g. new landsurface submodels and the introduction of an anelastic approximation for density changes with height . In DALES, most (∼ 90 %) of the turbulent processes are solved explicitly in a convective ABL when run on a grid resolution of 100 m or less. As a result, only parameterizations for the smaller scale turbulent structures are needed, which makes it an adequate tool to use in our study. With the use of the Boussinesq approximation, the filtered Navier–Stokes equation is solved . Furthermore, DALES consists of no-slip boundary conditions at the bottom and periodic boundary conditions at the sides. At the top of the domain, a sponge layer is present which damps fluctuations caused by, e.g., convection waves.

In all cases, the horizontal grid resolution is set to 50m×50 m, which covers an area of 12km×12 km. A larger domain or increase in grid resolution proved not to be of significance. The vertical resolution and extent are case-dependent and are listed in Table . The direction of the wind is set in the x direction (u component), but differences are present in the velocities (Table ). Also the case-dependent surface kinematic heat and moisture fluxes are prescribed. Furthermore, the ABL top is defined as the height where the gradient of the virtual potential temperature (θv) exceeds 50 % of the maximum gradient in the vertical profile of θv .

Ten numerical experiments are run to simulate a range of ShCu and stratocumulus cases. Regarding the ShCu, 5 situations (TROFFEE, GoMACCS, SCMS, ARM, BOMEX) are selected. Additionally, TROFFEE+ and SCMS- consider an adapted wind velocity compared to the original TROFFEE and SCMS cases, respectively. The SCMScold case represents an adaptation on SCMS, where the initial vertical profile of θ is lowered by 2 K. This is done to represent a transition from stratocumulus to shallow cumulus, as discussed in Sect. . Regarding the stratocumulus, 2 situations (ATEX and DYCOMS-II) are analysed.

The ShCu simulations start in the early morning and are based on daytime convective conditions. The radiation is calculated as a function of time, depending on the geophysical location. The chemical mechanism applied in the ShCu cases is identical to that described in and contains 20 reactant species and three passive tracers. The latter are an emitted tracer (INERT; emission of 1 ppbms-1), an inert species that is initially only present in the ABL (BLS) and an inert species that is initially present in the free troposphere (FTS). To ensure that the reactions are fully resolved, the time step is forced to a maximum of 1 s. For all cases, the data are stored at a 1-min interval.

The stratocumulus experiments are solely performed to include representative data for the upper regime of the total cloud area fraction parameterization. Therefore, no chemical scheme is applied.

The temporal evolution of the total and active cloud area fraction is presented in Fig. . The cases TROFFEE and ARM are clearly affected by a different partitioning in sensible and latent heat fluxes, caused by the diurnal cycle in incoming solar radiation. This demonstrates that the initiation of ShCu formation is dependent on the surface forcing. As a result, the ShCu start to develop from mid-morning and diminish in the late afternoon. The GoMACCS and SCMS cases show different dynamics compared to TROFFEE or ARM, as ShCu start to develop in the early morning (06:30 and 07:00 LT, respectively). This can be explained by a high relative humidity in the initial profiles at the start of the day, therefore favouring cloud formation (not shown). The reason for these high values can be found in the geophysical location of these cases, which are close to the ocean, even though they are classified as continental cases. In contrast to the continental numerical experiments, the BOMEX case is characterized by a nearly constant surface forcing over the ocean and is therefore classified as a marine steady-state case. In the first half an hour, moisture and heat is building up in the ABL, which causes the sudden formation of ShCu around 05:30 LT. After 08:00 LT, the transport of energy is proportional to the supply of energy from the surface fluxes, so the temporal evolution of the area fraction of clouds and cores is in steady-state.

As is visible in Fig. , all continental ShCu cases show a time lag of 1 hour in the initiation of acc compared to ac. This can be explained by forced clouds, which are dominant during the first hour. This is also visible in Fig. a, where the ratio between ac and acc is shown. By focusing on the forced phase, it is visible that the area fraction of clouds increases with time, but that almost no active clouds are present. It is interesting to note that the dynamics in the BOMEX case are not comparable with the other cases, since it does not start in this forced phase, as mentioned earlier. In the next phase, the transition phase, the area fraction of clouds remains roughly equal, but the forced clouds are replaced by active clouds. During this process, the acc increases fast, indicating that the threshold for active ShCu growth is overcome. At the end of the transition phase, cloud venting affects the sub-cloud layer structures by redistributing thermals . As this transport of energy out of the sub-cloud layer affects the thermal structures, the area fraction of forced clouds decreases due to a decrease in the amount of thermals that reach the cloud layer. The area fraction of active clouds is not significantly affected by this process, while ac decreases, so that the accac ration increases. This process is clearly visible in the ARM and GoMACCS case (Fig. ). When the transport of energy is proportional to the increase in energy by the surface fluxes, we identify this period as the active phase. During the active phase, the ratio between ac and acc is roughly constant (ac=2.12acc), while both gradually decrease in time. In the final phase, the dissolving phase, the number of active clouds reduces rapidly due to the diminished surface forcing. In other words, the clouds decouple from the boundary layer thermals and are transformed into passive clouds. As such, the ratio between passive and active clouds increases (see Fig. a).

To only consider the clouds that enable vertical exchange, we perform a selection procedure based on the time period when the presence of acc is high. We show in Fig. b, that during this time period, ac and acc are coupled, but ac decreases faster by a factor of 2.12 (d=0.93). Here, d represents the index of agreement . This rough relationship is a valid first approximation, but one should note that the exact factor differs between conditions and that an independent parameterization of both components is needed, which will be derived in Sect. . To show the importance of this selection procedure, we compare our (selected) data with the data (no selection) from . Their relationship of 2.46 (d=0.77) is higher than ours, indicating that the effects of mass transport are underestimated, which decreases the accuracy of the mass transport parameterizations. Therefore, in the remainder of this paper, we use the selected data to evaluate the parameterizations and scaling for ShCu transport.

To asses the validity of the simplified statistical cloud area fraction parameterization (hereafter ac-parameterization) of NG06 (Eq. 12 therein) under different thermodynamic conditions, ten numerical experiments are performed to simulate a wide range of ShCu and stratocumulus cases (see Sect. ). As shown in Fig. a, the ac-parameterization of NG06 is not able to consistently represent the total cloud area fraction. Furthermore, by using Q3, no clear dependency is visible of the cloud area fraction on specific moisture conditions. An explanation for this could be found in the dependency of Q3 (Eq. ) to the volume of the ABL and the thickness of the transition zone, i.e. the region between cloud base and ABL. This dependency is only introduced in NG06, since Q1 in CB95 and Q2 are evaluated locally. Although NG06 simplified the expression for ac with the help of to reproduce the occurrence of active clouds with an atmospheric mixed-layer model, we show that this simplification can introduce significant errors depending on the evaluated case. Therefore, a revision of the ac-parameterization is needed such that the cloud area fraction can be reproduced for a wide range of atmospheric conditions. Furthermore, an independent representation of the active cloud area fraction, necessary for convective transport, is needed. In these analyses, we use the locally determined Q2 (Eq. ) as an indicator.

To include a wide range of boundary layer physics and cloud conditions between the ShCu and stratocumulus cases (i.e., between Q2=-2 and 1), two additional transition simulations, ATEX and SCMScold, are shown. ATEX represents a case where ShCu convection starts to develop, but an inversion causes the build up of moisture near the ABL top, resulting in a stratocumulus layer. Another approach is used for the SCMScold simulation, where the initial vertical profiles of θ were decreased by 2 K. As a result, the relative humidity is close to 100% near ABL top in the morning, thereby creating a stratocumulus layer. When the surface fluxes start to increase, convection starts to occur and the stratocumulus layer breaks. As is visible in Fig. b, a typical ShCu situation is present in the late afternoon which is comparable with the other ShCu cases and is captured by the revised ac-parameterization. As shown in Fig. b, using the proper index variable, Q2, results in a well-defined dependence of cloud area fraction. Furthermore, using this approach we can deduce an accurate parameterization for ac for all numerical experiments. By using the Levenberg–Marquardt algorithm for least square curve fitting, we find ac=0.5+αarctan⁡(βQ2+γ), where α=0.34 (0.002), β=1.85 (0.063) and γ=2.33 (0.111). The standard error of the parameter estimate (in parentheses) is calculated with the use of a covariance-matrix over the parameters. The residual standard error yields 0.036, which is calculated via the reduced chi-squared method. In ATEX and SCMScold, both the ShCu and stratocumulus regimes are generally captured well by Eq. (), while only the transition between this regime remains troublesome. This deviation is reflected in the relatively large residual standard error. Focusing solely on a cloud fraction lower than ac=0.3, i.e. ShCu cases, the residual standard error yields 0.007, as also shown in Fig. .

As mentioned before, acc cannot be parameterized by the expression for ac. This is confirmed by the ac= 2.12acc relation shown in Fig. . In Fig. , we show that a separate parameterization is needed for acc. We derive acc=0.292Q2-2, where the standard error in the parameter yields 0.001. The residual error yields 0.005. In addition to acc, we display the ac data (shaded) in Fig. , together with its parameterization, to demonstrate that ac and acc can be well-represented independently, but are not similar. As such, using ac to predict acc, as is currently assumed in the literature e.g., will lead to poor predictions of the active cloud area fraction. Furthermore, the simplified ac-parameterization of NG06 introduces inconsistencies depending on the evaluated case. Using Eqs. () and () removes these inconsistencies and is therefore essential to predict in-cloud transport and associated feedbacks accurately. Besides an improved representation of in-cloud transport in mixed-layer models, Eq. () is also relevant for global models that deal with the transport of atmospheric compounds other than water (e.g. the EMAC atmospheric chemistry-climate model; ), as the area fraction of active clouds is essential to calculate the correct vertical transport from a grid cell.

The cloud core vertical velocity, wcc, is the final component of the kinematic mass flux formulation (Eq. ). In this section, we evaluate the scaling of for various atmospheric conditions to complete the kinematic mass flux parameterization. showed that the wcc can be scaled with the Deardorff convective velocity scale (w∗). Building on this work, and found that a prefactor of 0.84 improved this scaling. Their analysis was based on four ShCu cases, where no selection of the data was applied to distinguish between active ShCu and forced/passive ShCu. Therefore, the presence of forced and passive clouds disturbs the scaling of wcc. As a result, their scaling value is lower due to the weaker vertical velocities related to forced clouds. By only taking the active phase into account, we find the following relation (Fig. ): wcc=0.91w∗ with d=0.90. Compared to the original prefactor, this increases the predicted kinematic mass flux (Eq. ) by ∼ 8 %. The high index of agreement shows that this relation is not affected much by different boundary layer dynamics and structures. However, as is visible in Fig. , the TROFFEE case is not as well-represented by the scaling. If we apply a fit through the TROFFEE data, we find a scaling factor of 0.85 (d=0.74), which is comparable to the result of who found 0.84 (d=0.94). The deviation of this case compared to other cases could be explained by a relative deep ABL depth (∼ 2 km). Combined with a strong surface forcing, w∗ increases strongly, while the wcc is not significantly affected. This results in a lower scaling constant.

Here, we briefly evaluate the kinematic mass flux prediction by comparing it to the observed kinematic mass flux in the simulations. In Fig. a it is shown that the parameterizations result in a good approximation for the observed M, even though it is overestimated for the TROFFEE case. We hypothesize that this latter exception is related to the deep (∼ 2 km) boundary layer, which only enables cloud growth for the most vigorous upward directed thermals with a high moisture level. When these thermals reach the LCL, the difference in moisture with its surrounding air is relatively high, leading to high values of σq and consequently a higher predicted acc than physically present. This hypothesis is supported by the small time span these active clouds are present during the most convective time of the day (Fig. ). To compare our prediction of the kinematic mass flux, based on separately parameterizing the cloud core area fraction and in-cloud vertical velocity, with the result one would obtain by making direct use of a proxy, we show the relation between the observed M and the scaled surface buoyancy flux in Fig. b. In this figure, the predicted M is equal to αw′θv′‾sθv‾s, where w′θv′‾s and θv‾s denote the surface buoyancy flux and domain-averaged surface virtual potential temperature, respectively, and α=142, based on linear regression. This prediction method could alternatively be used to roughly estimate the kinematic mass flux, however it is less accurate than the prediction obtained by combining the two separate parameterizations for acc and wcc. Again, the TROFFEE case is less well represented, which is most likely due to the specific atmospheric profile as mentioned before. In conclusion, the prediction of the kinematic mass flux is improved by making use of the parameterizations for its two main components (acc and wcc) compared to scaling with the surface buoyancy flux. Furthermore, as a result, we have information on both the in-cloud vertical velocity and the area fraction of active clouds, which is valuable information for atmospheric models that need parameterizations to represent convective transport of atmospheric compounds.

In this section we focus on the final component of the expression for convective transport of atmospheric compounds (Eq. ), namely the concentration of chemical species at cloud base, ϕcc-ϕ‾(zb). The parameterization, proposed by , showed that the concentrations of chemical species at the base of active ShCu can be predicted by Eq. () for a tropical case (TROFFEE). However, they stress that ABL dynamics could influence the parameterization. Therefore, we test the parameterization for all continental ShCu cases. The relation is illustrated for four chemical species (i.e., INERT, BLS, isoprene and CO) in Fig. , but the least squares regression is fit through all 24 evaluated chemical species. This yields: ϕcc-ϕ‾(zb)≈-1.18(ϕ‾(zb)-〈ϕ〉). For all relations, the index of agreement is 1.00.

We find similar results as but our constant is slightly less negative than their -1.23. Since we use the least squares method to find the optimum scaling constant, it means that compounds with the largest differences between ϕ(zb)-〈ϕ〉 affect the scaling constant the most. As shown in Fig. , this means that INERT has a dominating influence. Focusing on this compound (inset), we find that wind tends to increase the differences between species in the cloud core compared to their average at cloud base, while for a free convection situation the opposite is visible. As a result, the closure constant of Eq. () shifts slightly. This results in a slope of -1.17 in case of wind and a slope of -1.19 in case of free convection (not shown). We identify that dynamical segregation is occurring in the ABL, as shown for INERT in Fig.  and discussed by for a tropical case. Rising motions in the ABL transport high concentrations of the emitted species upwards, while lower concentrations of INERT are found in the downward motions. Therefore, higher concentrations of species are transported towards the free troposphere by cloud venting as would be expected compared to a well-mixed situation. Although the effects of chemical segregation are usually small for clear sky situations, they can be substantial for cloud-topped boundary layers due to cloud venting. As a result, the chemical parameterizations and scalings are affected. Furthermore, as is shown in Fig. , wind affects the distance and upward velocities in the thermals, resulting in less, but wider thermals in our domain. This affects the vertical transport of species and decreases this transport (max. 3.5 ms-1) compared to a free convection situation (max. 5.0 ms-1) where the thermals are narrower. As a result, the transport of chemical species to the cloud layer is less in the wind case, resulting in a smaller difference between ϕcc and ϕ‾(zb), which decreases the magnitude of the scaling constant of Eq. (). Next to an effect on convective transport, one has to note that dynamical segregation also modifies the mean reactivity in the ABL, as was shown by for clear sky conditions and in ShCu situations.