1. Introduction
The unified parameterization of turbulence and clouds in the planetary boundary layer (PBL) is one of the challenges in current numerical weather prediction (NWP) and general circulation models (GCMs; Siebesma et al. 2007; Golaz et al. 2002). The unified approach should ideally enable a better representation of interactions and transitions and eliminate compensating errors that arise from the separate modeling of turbulence and clouds.
One group of such unified parameterizations is based on the extension of the local downgradient parameterization with a nonlocal component that accounts for the transport caused by thermals. The nonlocal component can be introduced by a countergradient term (Deardorff 1972; Holtslag and Moeng 1991), which leads to upward turbulent heat transport in the presence of a slightly stable temperature gradient. A more sophisticated representation of thermals via mass flux is used in the eddy-diffusivity and mass-flux (EDMF) schemes (Siebesma et al. 2007; Neggers et al. 2009; Neggers 2009; Sušelj et al. 2012).
The other family of parameterization relies on a higher-order turbulent closure (HOC), where the higher-order moments enable the representation of nonlocal and countergradient transport. These parameterizations range from full third-order closure (e.g., Bougeault 1981) and schemes with prognostic equations for all second-order moments and a third-order moment for vertical velocity (e.g., Golaz et al. 2002) to closures for turbulence kinetic energy (TKE) together with prognostic scalar second-order moments (e.g., Nakanishi and Niino 2009) or scalar variances (e.g., Machulskaya and Mironov 2013).
The schemes with only prognostic TKE are able to simulate stratocumulus clouds (Duynkerke and Driedonks 1987; Bechtold et al. 1992) and, with proper treatment of cloudiness distribution and an advanced length-scale formulation, even trade wind cumulus (Bechtold et al. 1995; Bogenschutz and Krueger 2013).
In the present paper, we propose a new scheme that is based on prognostic equations for two turbulence energies. The additional prognostic turbulence energy is a compressed equivalent of the two scalar variances. In the closure hierarchy, this scheme is one step above schemes with only prognostic TKE.
Our scheme builds on the TKE scheme by Bašták Ďurán et al. (2014), which is used in the ALARO physical package of the ALADIN NWP model (Termonia et al. 2018; Wang et al. 2018) and is implemented in the Integrated Forecast System (IFS) model developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). We added a second prognostic turbulence energy by following the work of Zilitinkevich et al. (2007, 2008, 2013), who proposed the usage of two turbulence energies in the stable boundary layer. We extend their derivation to all atmospheric stabilities and include the influence of moisture.
The additional information from the second turbulence energy is used solely for the computation of the stability parameter, which thus obtains a prognostic character and an independence from local gradients. The scheme uses the downgradient approach, where the turbulent fluxes are proportional to the local gradients but the turbulent exchange coefficients are not strictly local. This enables transport across locally stable layers and deeper mixing, which eliminates the major problems associated with traditional downgradient schemes. A similar behavior was achieved in Bretherton and Park (2009) by extending their convective layers into stable regions and merging them if they overlap and by using averaged stability functions across each of these layers. Still, our scheme is unable to parameterize countergradient fluxes and is therefore less accurate than higher-order closure schemes. On the other hand, the two-energy scheme is simpler in design, and because the scheme is easier to implement, it is computationally cheaper and numerically stable for longer time steps.
The influence of moisture and phase changes of water on vertical transport plays an important role in the parameterization of PBL clouds. In HOC and TKE schemes, this effect is realized through a modification of the buoyancy terms or the Brunt–Väisälä frequency. The cloud fraction or the buoyancy terms can be computed by assuming a Gaussian distribution of temperature and moisture (Sommeria and Deardorff 1977). However, the distributions are skewed when shallow convection is present (Lewellen and Lewellen 2004) and need to be adjusted accordingly (Bechtold et al. 1995). A more advanced approach is to use a trivariate (temperature, moisture, and vertical velocity) probability density function (PDF; Golaz et al. 2002; Larson et al. 2012; Bogenschutz and Krueger 2013) for the direct computation of the buoyancy terms.
In our scheme, the buoyancy terms are computed as a cloud-dependent linear combination of covariances for temperature and moisture. The influence can then be represented via a modification of the stability parameter used in the stability dependency functions. We use the Gaussian formulation of Sommeria and Deardorff (1977) to determine the cloud fraction and compute the buoyancy terms according to Marquet and Geleyn (2013) with a skewness-dependent heuristic adjustment.
This paper is organized as follows. Section 2 gives a general overview of our TKE scheme for the dry and the moist case. In section 3, we describe the addition of a second turbulence energy and the corresponding stability parameter. Details about the turbulence energy solver are presented in section 4. The scheme is evaluated in section 5 in the single-column model (SCM) setup of the OpenIFS model and in section 6 in a long-term three-dimensional simulation of the IFS model. We finish with a summary and conclusions in section 7.
2. The TKE turbulence scheme
a. The dry TKE turbulence scheme
To demonstrate the impact of moisture on the turbulence scheme, we will first present the TKE scheme for the dry case, following closely Bašták Ďurán et al. (2014).
b. Length scale
The length scale L in the above expressions may be chosen independently from the stability functions and TKE. For our scheme, we will use a combination of two types of length scales.
c. Density fluctuation in moist case
Compared to the dry case, two instead of one (potential temperature) conservative thermodynamic variables are required to uniquely define an air parcel in the moist case (Betts 1973; Marquet 2011). This adds five more Reynolds-averaged Navier–Stokes (RANS) equations for second-order moments to the system (see, e.g., Stull 1988). The presence of water and its phase changes influence the density fluctuations and thus modify the buoyancy. We will accommodate these extensions and, analogous to the dry case, derive the relationships for turbulent vertical fluxes in the moist case.
Compared to Marquet and Geleyn (2013), the dimensionless quantity 
d. The moist TKE turbulence scheme
With the help of the symbolic computation system wxMaxima, the system of Eqs. (36)–(49) can be solved. The resulting stability functions 
As a consequence of the above, the derivation of the one-dimensional turbulent parameterization in Bašták Ďurán et al. (2014) can be applied also to moist case, where the flux Richardson number is substituted with its moist counterpart. The 
3. The two-energy turbulence scheme
Only a single stability parameter, namely, flux Richardson number 
- It can be computed from the gradients via the gradient Richardson number Ri:
- The
where
We will investigate a third option and express
The relationship Eq. (64) enables the computation of 
The two-energy scheme can be easily implemented into the TKE scheme, where the prognostic equation for TKE is already implemented. The second prognostic variable can be either 
In comparison to the downgradient prognostic model of Zilitinkevich et al. (2013), no assumption about stratification has been made in our scheme. This means our scheme can be used for both stable and unstable stratification.
4. Turbulence energy solver
An essential part of the turbulence scheme is the solver for the turbulence energies. The details of the TKE solver will be presented in the following subsection.
a. TKE solver
To ensure stability of vertical diffusion schemes, it is common practice to evaluate the turbulent exchange coefficients on half levels (i.e., between the levels where model variables are defined). Direct dependence of the exchange coefficients on TKE [see Eqs. (8) and (9)] implies that TKE would be best located on the half levels. Such a discretization however conflicts with the practical requirement to advect TKE consistently with the other prognostic variables located on the full levels. This issue can be solved either by putting TKE on the half levels and neglecting the advection of TKE or by using a full-level TKE and interpolating it to the half levels for the computation of the exchange coefficients. The former option works fine for scales coarser than 10 km and/or time steps longer than 5 min, where the advection of TKE contributes typically less than 1% to the total tendency. The latter option adds some implicit smoothing, which might not be desirable. Here, TKE is kept on full levels in order to ensure problem-free advection of TKE.
The simplified prognostic TKE equation [Eq. (67)] can be seen as a prognostic equation having only two sources: autodiffusion and relaxation. To prevent any potential stiffness issues caused by the source terms, an overimplicit treatment is adopted [following the conclusions of Kalnay and Kanamitsu (1988)]. The relaxation term is being discretized with the constant overimplicit parameter equal to 1.5. For the autodiffusion term, the overimplicit parameter β is set to the minimal value that ensures stability according to the method proposed by Bénard et al. (2000).
b. Modification for second turbulence energy 
As mentioned in section 3, we have chosen 
5. Idealized single-column model simulations
To evaluate our turbulence scheme, we implemented it into the ECMWF OpenIFS SCM.
The OpenIFS (Carver and Váňa 2017) model is the portable version of the IFS at ECMWF. The model is based on the IFS model cy40r1 (ECMWF 2014). All SCM simulations are performed with a 900-s time step using the two-time-level semi-Lagrangian scheme. The SCM runs with 91 atmospheric vertical levels (ECMWF 2017d), having 17 levels in the lowest 2 km.
We tested the SCM for three idealized cases, all developed by the Global Energy and Water Cycle Experiment (GEWEX): the continental cumulus case based on measurements over the Atmospheric Radiation Measurement (ARM) program Cloud and Radiation Testbed (CART) site in Oklahoma (Brown et al. 2002; Lenderink et al. 2004), the trade wind cumulus case based on observations from the Barbados Oceanographic and Meteorological Experiment (BOMEX; Siebesma et al. 2003), and the third test case is based on the GEWEX Atmospheric Boundary Layer Study (GABLS) project (Holtslag 2006; Beare et al. 2006; Cuxart et al. 2006).
This particular set of cases was chosen to analyze the turbulence scheme without interaction with other physical parameterizations. Although this is a limitation, the situations tested do include stable stratification (GABLS) and unstable stratification. The latter are two shallow convection cases: the quasi-steady state over ocean (BOMEX) and the diurnally driven case over land (ARM). Especially the capability of our scheme to represent moisture effects (i.e., without the need for a separate shallow convection scheme) is tested with the latter two cases.
Thus, the OpenIFS runs include only the turbulence scheme, and the remaining IFS physics are turned off. As a reference, we also run OpenIFS with the EDMF scheme and shallow convection parameterization (SCP; Bechtold et al. 2014).
The SCM results for the above cases were compared with results from the large-eddy simulation (LES) model MicroHH (van Heerwaarden et al. 2017). In the ARM and BOMEX simulations, MicroHH runs with moist thermodynamics and in the GABLS simulation with dry thermodynamics. Microphysical and precipitation processes are turned off for all cases. For the ARM case, the domain size is 6.4 km × 6.4 km in the horizontal and 4.4 km in the vertical with horizontal and vertical grid spacings of 100 and 34.375 m, respectively. For the BOMEX case, the domain size is 6.4 km × 6.4 km in the horizontal and 3.0 km in the vertical with horizontal and vertical grid spacings of 100 and 46.875 m, respectively. For the GABLS case, the domain size is 400 m × 400 m in the horizontal and 400 m in the vertical with horizontal and vertical grid spacings of 3.125 and 3.125 m, respectively. All the MicroHH simulations reproduced the results of the respective LES intercomparison studies.
The LES statistics were averaged over the whole horizontal domain in space and over one hour in time. All figures were generated with the Python matplotlib package (Hunter 2007).
a. Continental shallow cumulus
To outline the prognostic properties of the two-energy turbulence scheme, we start the analyses with the ARM case, which has a pronounced diurnal cycle with onset and decay of shallow convection.
Figure 1 shows the vertical profiles of selected variables for the LES and the SCM in four different configurations. Three configurations of the turbulence scheme (see section 2d) differ in the formulation of the flux Richardson number stability parameter: the TKE model based on gradients [TKEM-GR; see Eq. (51)], the TKE model from fluxes [TKEM-FL; see Eq. (53)], and the TKE model from the two turbulence energies [TKEM-TE; see Eq. (64)]. The fourth configuration uses the IFS implementation of the EDMF and a shallow convection parameterization, the IFS-EDMF+SCP. Please note that the internal cloud fractions C [Eq. (27)] and 
Comparison of vertical profiles of wind components, liquid water temperature, total specific content of water, cloud fraction, and turbulence cloud fraction after 10 and 12 h of integration for the ARM case. The IFS-EDMF+SCP stands for the IFS EMDF and SCP scheme. The internal cloud fractions C [Eq. (27)] and 
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
In terms of liquid water potential temperature, total specific content of water, and wind, the TKEM-TE performs best of all SCM configurations. The TKEM-GR and the TKEM-FL have in comparison a too-low PBL height due to too-weak mixing across the cloud top, and as a consequence, their PBL is too moist and too cold. The IFS-EDMF+SCP configuration is better in this respect thanks to the activity of the mass flux (both in the EDMF and in the SCP) but generates jagged profiles. Please note that the PBL height of the IFS-EDMF+SCP configuration is improved if the prognostic cloud scheme of the IFS (see ECMWF 2014) is activated.
The percentage of cloud fraction C and turbulence cloud fraction 
The evolution of the vertical turbulent fluxes—including the mass and convective fluxes for the IFS-EDMF+SCP—for heat, moisture, and wind components is presented in Figs. 2 and 3. The TKEM-GR and, even more, the TKEM-FL show an on–off behavior in the diurnal evolution of all displayed fluxes. This behavior can be also observed in the IFS configuration. Compared to that, the TKEM-TE configuration is more continuous in time, showing better agreement with the onset and decay of shallow convection. This positive feature can be attributed to the prognostic character of the stability parameter in the TKEM-TE approach.
As in Fig. 1, but for comparison of temporal evolution of vertical turbulent fluxes, including mass fluxes and convective fluxes for the IFS-EDMF+SCP, of liquid water potential temperature, and total specific content. Gray vertical lines indicate time, corresponding to the vertical profiles displayed in Figs. 1, 5, 6, and 7. Dotted areas mark regions with countergradient fluxes for the LES-ALL data.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
As in Fig. 2, but for turbulent fluxes of zonal and meridional wind component.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
The TKE and the flux Richardson directly influence turbulent fluxes via the turbulent exchange coefficient [see Eqs. (13) and (14)], so their characteristics can be used to assess the behavior of the turbulence scheme. The evolution of TKE and the flux Richardson number is depicted in Fig. 4. Additionally, profiles of the three types of turbulence energy (
As in Fig. 2, but for turbulence kinetic energy and flux Richardson number.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
(left) Vertical profiles of turbulence energies for 
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
As in Fig. 5, but after 12 h 30 min of integration.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
We can see that the TKEM-GR has a sharp transition between the mixing and nonmixing states because of its dependence on local gradients. The sharp transition leads to the accumulation of heat and moisture under the stable layer, which at some point results in the formation of clouds at the top of the PBL. The associated latent heat release enables turbulent mixing across the top of the PBL but only up to the next vertical level, where the new sharp transition is formed. The process repeats itself then on the next higher level, which results in the on–off behavior of the scheme.
The oscillating behavior of the TKEM-GR and the TKEM-FL is also caused by the direct link of the stability parameter to the local gradients or fluxes. If the local buoyancy flux exhibits a stable intermediate layer (see right in Figs. 5 and 6), the turbulent mixing in the PBL is interrupted until accumulation of heat and moisture from lower levels restarts it.
In comparison, the TKEM-TE has a smoother transition between stable and unstable layers and enables mixing across and into the locally stable layers (see right in Figs. 5 and 6). This is due to its link to the turbulence energies, which are vertically transported by the turbulence scheme and thus do not depend necessarily on the local gradients or the local turbulent fluxes (see left in Figs. 5 and 6). Thanks to that, the TKEM-TE is able to sustain continuous growth and later decline of the PBL.
In LES, the transport is caused mainly by intense transport in the active cells, which are not inhibited by local stability, and only the nonactive cells have a capped subcloud layer. Since the transport does not linearly depend on local stability, the LES appears to have turbulent transport across a locally stable layer in the domain average. This means that turbulent transport in the SCM grid should not depend only on local stability. This feature is mimicked in the TKEM-TE approach, which is confirmed by the good agreement of the TKE budget terms between the TKEM-TE and the LES. The comparison of TKE budget terms for one profile is displayed in Fig. 7.
As in Fig. 1, but comparison for TKE budget terms after 10 h of integration.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
b. Marine shallow cumulus
Next, we turn to BOMEX, representing a quasi-steady case of marine shallow cumulus. The case is a test whether our prognostic turbulence scheme is able to reach and hold a balance between turbulent mixing and a constant large-scale forcing.
The vertical profiles of selected variables are shown in Fig. 8. The TKEM-TE has the best agreement with the LES in this respect. The IFS-EDMF+SCP tends to overestimate and the TKEM-GR and the TKEM-FL underestimate the turbulent mixing in the upper part of the convective boundary layer.
As in Fig. 1, but for the BOMEX case after (left columns) 5 and (right columns) 21 h of integration and for mean cloud fractions instead of cloud fractions. The mean cloud fractions are computed from the start of the simulation until the specified time.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
The evolution of fluxes, presented in Fig. 9, shows that all the SCMs are able to hold a balanced state, which is accompanied by periodically rising thermals. The thermals in the TKEM-TE are too intense and less frequent than in the LES. This indicates that the speed of growth of the thermals is limited because of the cutoff formulation of the length scale above the PBL [see Eq. (18)]. The frequency and intensity of the thermals improves for smaller time steps (not shown). As in the ARM case, the penetration of the thermals is not deep enough for the TKEM-GR and the TKEM-FL. The IFS-EDMF+SCP has a good representation of the thermals, which are however too frequent in comparison with LES.
As in Fig. 2, but for the BOMEX case.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
c. GABLS case
To check the versatility of our turbulence scheme, we tested it also for the stable GABLS case. For the sake of brevity, we present only the vertical profiles of wind speed and potential temperature after 7 h 30 min of integration (see Fig. 10).
As in Fig. 1, but vertical profiles of wind speed and potential temperature for the GABLS case after 7 h 30 min of integration.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
All of the SCMs overestimate the turbulent mixing. Nevertheless, the profiles of potential temperature indicate the development of an elevated inversion within the SBL and generate a supergeostrophic jet peaking near the top of the boundary layer. These pronounced features in the LES are best modeled by the TKEM-FL and the TKEM-TE. The upper inversion in the IFS-EDMF+SCP has a smoother transition than the other SCMs.
The overestimation of mixing can be attributed to our particular choice of length scale, which might not be optimal for the conditions in GABLS case. Still, the results of our scheme are somewhat better than the operational IFS-EDMF+SCP scheme.
d. Countergradient fluxes
To evaluate the importance of the countergradient fluxes of heat and moisture in the shallow cumulus cases, we marked the countergradient regions in the LES data (see Figs. 1, 2, 8, and 9). The main countergradient egion is located above the PBL. The relatively small amplitude of the fluxes in this region indicates that their lack in the downgradient schemes does not lead to large errors. Another countergradient region can be found in PBL layers with near-constant liquid water potential temperature. The downgradient mixing in this region produces also a near-constant profile but enforces opposite signs of the vertical gradient and the turbulent flux. Altogether, we conclude that the downgradient approach is sufficiently accurate for the ARM and BOMEX cases. Similar observations regarding the downgradient schemes were made by Yamada (1977) for the Wangara case, Yamada and Mellor (1979) for the BOMEX case, and Bechtold et al. (1995) for the Rain in Cumulus over the Ocean (RICO) case.
6. Long-term 3D simulation
The SCM simulations allows us to study the turbulence scheme in idealized conditions independent from other parameterizations and the dynamics of the model, though the real challenge is to ensure that the turbulence scheme works well together with the other parts of the model and that it improves the forecast. We tested our turbulence scheme additionally in long-term three-dimensional global simulations. The aim of these tests was not an overall evaluation of its performance but the identification of possible weaknesses of the scheme, which might not be visible in the idealized simulations. Therefore, scores for selected variables only are presented.
We used for this the diagnostic tool of the ECMWF (for details, see Váňa et al. 2017; ECMWF 2017b), where a four-member ensemble simulation of the uncoupled global IFS model is run and compared to the model reanalyses and measurements for the 2000/01 time period.
We run the cy43r3 (ECMWF 2017c) version of the IFS model (i) with the IFS EDMF and SCP scheme (in figures marked as go7n) and (ii) the TKEM-TE turbulence scheme with two prognostic energies but without SCP (in figures marked as gugn). Apart from switching off the SCP, the following additional changes were made for the implementation of the TKEM-TE configuration into the full IFS model:
Two additional prognostic variables TKE and
Three more variables with inter–time step memory, but no advection (mainly for cost reasons), were introduced: turbulent heat flux, turbulent moisture flux, and the shear term in the TKE equation.
The turbulence scheme is computed with the same time step as the remaining physics and dynamics (i.e., no substepping is applied).
The existing coupling of physics and dynamics (see Wedi 1999) treats fast processes (i.e., turbulence and microphysics) by a first-order method while benefiting from a second-order averaging method along the semi-Lagrangian trajectory for the other processes. Thanks to the improved time consistency of the prognostic two-energy turbulence scheme, the second-order coupling can be now applied also to the tendencies from vertical diffusion. The remaining part of the fast-evolving processes requiring the first-order interface is then reduced to the tendencies of cloud water, cloud ice, and the two new turbulent energies only.
To stay consistent with the sequential time stepping and the present physics dynamics interface (Beljaars et al. 2004), the model variables seen by the physics (including now the variables used by the turbulence scheme) are updated by explicit dynamics and tendencies from other parameterizations that were already evaluated.
Finally, to stay consistent also with the cloud scheme, all the prognostic advectable quantities (prognostic cloud fraction, cloud ice and water, moisture, TKE,
The two-energy turbulence scheme was implemented in the same configuration as in the SCM simulations and without further adjustment of the IFS physics.
The evaluation of 2-m temperature and 2-m dewpoint temperature compared to ERA5 data (ECMWF 2017a) is displayed in Fig. 11. We can see that the impact of the turbulence scheme on the surface variables is comparable to the IFS-EDMF+SCP. As expected, a bigger impact of the turbulence scheme can be seen in Fig. 12, where the liquid water path compared to the Special Sensor Microwave Imager (SSM/I), version 6, dataset (see, e.g., O’Dell et al. 2008) and the total cloud cover (diagnosed outside the turbulence scheme) compared with CALIPSO data (see, e.g., Chepfer et al. 2008) are depicted.
An evaluation of (left) the 2-m temperature and (right) the 2-m dewpoint temperature for the IFS four-member ensemble long-term simulation against the ERA5 reference in the 2000–04 period. Comparison of differences for (top) the IFS-EDMF+SCP (marked as go7n) and (bottom) the TKEM-TE (marked as gugn) in the IFS cy34r3 configuration. The mean error (mean err) and the root-mean-square error (rmse) are computed for the annual cycle.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
As in Fig. 11, but for liquid water path with reference from (left) the SSM/I, version 6, dataset and (right) the total cloud cover with reference from CALIPSO data. The statistics for the total cloud cover are computed for latitudes between 50°N and 50°S.
Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0026.1
The TKEM-TE has a slightly better score than the IFS-EDMF+SCP for the liquid water path. The IFS-EDMF+SCP tends to underestimate the cloud cover and the TKEM-TE to overestimate the cloud cover. The low-level clouds especially are overestimated with the TKEM-TE. This agrees with the results from the SCM simulations, where the IFS-EDMF+SCP is drier than the LES and the TKEM-TE in the upper part of the PBL (see Figs. 1 and 8). Thus, a suitable adjustment of the IFS cloud diagnostics, which is calibrated for the IFS-EDMF+SCP, could partly compensate the overestimation of the cloud cover for the TKEM-TE scheme. This is confirmed by the good agreement of the internal cloud fraction of the turbulence scheme [Eq. (27)] with the LES data in Figs. 1 and 8. However, the two-energy turbulence scheme still seems to transport too much moisture into the upper part of the PBL for the shallow convection cases. As discussed in the previous section for BOMEX, we believe that an improvement of the length-scale formulation could mitigate this problem.
An increased influence of skewness in the relationship between the cloud fraction C and the turbulence cloud fraction 
7. Summary and conclusions
A turbulence scheme with two prognostic turbulence energies suitable for all atmospheric stabilities has been derived and presented. The influence of moisture and phase changes is represented via a modification of the buoyancy terms in the presence of clouds. Thanks to the assumed linear dependence of the buoyancy terms on the covariances of heat and moisture, this influence can be expressed via a modification of the stability parameter only. By using the “stability-dependent adjustment for turbulence energy modeling,” the stability parameter can be computed from the two prognostic turbulence energies. This way, the stability parameter gains independence from the local stability conditions and obtains a prognostic character. The scheme is still downgradient, and the turbulent fluxes are proportional to the local gradients of the diffused variables, but the exchange coefficients are not strictly local.
The performance of the two-energy scheme was tested and compared with a standard TKE scheme and the IFS EDMF scheme with shallow convection parameterization in three idealized SCM simulations. The test for the ARM case shows that our scheme is able to model shallow convection with a diurnal cycle. The turbulent fluxes are more continuous in space and time compared to the TKE scheme and result in deeper mixing in accordance with the LES results. The cloud fraction has acceptable strong amplitude, but the cloud layer is too thin. Similar conclusion can be drawn for the quasi-steady shallow cumulus case BOMEX. However, the thermals modeled by our scheme are too intense and too infrequent compared to the LES reference. This is due to the cutoff formulation of the length scale. The problem can be partly remedied with an increased temporal resolution. The scheme performs also satisfactorily for the stable stratification case GABLS but slightly overestimates turbulent mixing.
The long-term three-dimensional global simulations indicate that our turbulence scheme has reasonable behavior in a full atmospheric model even though no calibration of the full model has been undertaken. In fact, our scheme is able to outperform the default IFS configuration in some characteristics, for instance, in the 2-m temperature in the Arctic region. However, the two-energy scheme tends to overestimate the cloud cover, especially at low levels. This can be partly attributed to the specific adjustment of the cloud-cover diagnostics in the IFS model. Still, our turbulence scheme seems to transport too much moisture into the upper part of the PBL in the presence of shallow convection.
Besides the abovementioned tests made in the global model, the two-energy scheme has been implemented first in the ALARO canonical model configuration of the ALADIN system (Termonia et al. 2018) and is currently operationally exploited at several ALADIN NWP centers.
The aim of this paper was to present and evaluate a turbulence scheme with two prognostic turbulence energies, which is one step above the commonly used TKE schemes. We have shown that this approach has good potential. There are still possibilities for improvement, especially in the length-scale formulation and the cloud fraction/buoyancy terms computation. We plan to improve both of these aspects in future work. A promising approach for the length-scale formulation is usage of relaxation technique for dissipation rate of TKE proposed in Zilitinkevich et al. (2013) or the computation of the length scale according to Nakanishi (2001).
Furthermore, we are aware that the selected set of test cases is limited. We plan to extend our investigations to stratocumulus cases and cases with precipitation.
Acknowledgments
The authors thank D. Mironov, S. Zilitinkevich, P. Marquet, P. Bechtold, E. Machulskaya, J. Mašek, A. Schomburg, M. Hrastinski, and P. Smerkol, who helped to shape this paper with their constructive comments and discussions. Thanks also to M. Ahlgrimm and the OpenIFS team at ECMWF for their support related to the SCM experiments.
Ivan Bašták Ďurán and Juerg Schmidli were supported by the Hans Ertel Centre for Weather Research. This German research network of Universities, Research Institutes and the Deutscher Wetterdienst is funded by the Federal Ministry of Transport and Digital Infrastructure (BMVI).
Support of the ALADIN community, especially in its RC LACE component (ALARO physics project), was essential for the completion of the work. This work was supported by the Czech Ministry of Education, Youth and Sports within the National Sustainability Program I, Grant LO1415.
The LES data for estimation of cloud fraction were provided by D. Lewellen.
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