LMDZ is the General Circulation Model (GCM) used in this work. Developed in the 1970s at Laboratoire de Meteorologie Dynamique . It is based on simplified Navier-Stokes equations for fluid mechanics, as well as transport equations. It represents the second generation of a climate model initially described by . LMDZ is the atmospheric component of the IPSL coupled model. The latter is one of around twenty coupled models taking part regularly in the international coupled model intercomparison project (CMIP), the results of which are used in IPCC (Intergovernmental Panel on Climate Change) reports. We use here the LMDZ6A configuration of LMDZ designed for CMIP6 and described by .

LMDZ consists of two main parts, from a physical, mathematical and computational point of view. The first part, called “the dynamics”, concerns the numerical resolution of the atmospheric general circulation equations. This component manages horizontal exchanges between the model's grid cells. The second part, called “physics”, calculates the impact of radiation, small-scale processes (subgrid) and phase changes of water on dynamic variables via “physical parameterizations”. This “physical” part is made up of juxtaposed atmospheric columns, which do not interact with each other. Within each column, the variables are assumed to be statistically homogeneous in the horizontal plane.

The SCM version of LMDZ is built by extracting an atmospheric column from the GCM, incorporating all subgrid-scale parameterizations, and running it in a large-scale constrained environment. This approach has become central in the development and tuning of parameterizations of convection and associated clouds in several climate modeling groups . Parameterizations are often developed and evaluated within this single-column framework by comparing them with LES of the same atmospheric column. The SCM/LES approach was promoted in particular by GCSS (GEWEX Cloud Systems Study), a program aimed at improving the parameterization of cloud systems in climate models . A major advantage of the SCM is its low computational cost, which allows a large number of simulations, even on a laptop, making it particularly useful in the development phase, where extensive testing is required.

The role of convective parameterizations is to provide sources of heating Q1 and moistening Q2 following notations first introduced by to the conservation equations of potential temperature θ and specific humidity qv: CpDθDt=θT[QR+(Lv+fgLf)(c-e)]1-Cp1ρ∂ρw′θ′‾∂z=θT[QR+Q1],2DqvDt=e-c-1ρ∂ρw′qv′‾∂z=-Q2/Lv, where Cp is the heat capacity of dry air, QR is the radiative heating, c and e are condensation and evaporation rates, fg is the condensate ice fraction, Lv is the latent heat of vaporization, Lf the latent heat of fusion and -∂zρw′ϕ′‾/ρ (with ϕ=θ or qv) is the vertical convergence of the Reynold turbulent flux of ϕ accounting for the effect of the subgrid-scale turbulent or convective motions on the explicitly resolved large scale flow. Note that the θ/T factor on the right hand side of the first equation was missing in the equation of but not in the code of the parametrization. The convective parameterizations also often provide a source term Q3 for momentum but it is not involved in the coupling with cold pools described here.

Note that the equations above are simplified assuming that the ice fraction fg is unchanged by evaporation and condensation. Note also that Q2 is a sink of humidity expressed conventionally as a heating term with constant Lv.

The parameterization of turbulence, convection and clouds in LMDZ is based on a multi-scale, or object view, as described below.

The cold pool model has been entirely described in GL10. Here, we introduce the main equations and internal variables of the parameterization relevant for the rest of the paper.

The cold pool model represents a population of circular cold pools (or wakes) over an infinite plane, consistently with the way the Reynolds decomposition is used to separate, in GCMs, the 3D dynamical core from the 1D world of parameterizations of subgrid transport, assuming that the statistics of the turbulent or convective motions responsible for this transport are horizontally homogeneous on an infinite plane.

The centers of those cold pools are randomly distributed with a uniform number density Dwk (number of cold pools per unit area) assuming no overlap between two of them. Due to the complex life cycle of cold pools (including birth, death, collisions and merging), calculating their number density requires another parameterization. So far, the value of the cold pool number density is imposed. In LMDZ6A, it is fixed to a different value over ocean (10 cold pools over 100 km × 100 km) and over land (8 cold pools over 1000 km × 1000 km).

An important simplification of the model consists in assuming that, within a given column of the model and at a given time step, all the wakes have the same height, radius, and vertical profiles of thermodynamic variables which is equivalent to say that we are representing the population of cold pools through a mean effective cold pool. All those variables however evolve in time according to the cold pool physics, as explained below.

In practice, the model divides the space into two parts, the interior (wk) and exterior (or environment) of cold pools (ex) respectively. For temperature, humidity and vertical velocity, we introduce a wake anomaly δX=Xwk-Xex defined as the difference of its mean values in the two subdomains and X‾ the average over the horizontal domain.

The main driver of the parameterization is the cooling associated with convective unsaturated downdrafts. All this cooling, provided by the deep convection scheme, is applied to the interior of the cold pools thus creating a contrast in temperature δT between the interior and exterior of the cold pool (see Fig. ). Once the cold pools are initiated (which requires a prior activation of deep convection), they enter a positive feedback loop with deep convection. Cold pools reinforce deep convection both because the saturated drafts are fed by air coming from the environment of the cold pools (warmer near the surface than the average grid cell), and because of the lifting energy provided by the cold pool spreading as detailed in the following section. The temperature difference between the cold pools and their environment is controlled by competing effects: it increases due to cooling by convective downdrafts and decreases due to cold pool spreading and gravity wave damping. When deep convection stops, this temperature contrast decreases until the cold pool parameterization is switched off based on a combination of thresholds.

The top height of the cold pool is defined as the altitude hwk (and associated pressure pwk) where the temperature difference between (wk) and (ex) becomes zero. Below this level cold pools are cooler than their exterior. The boundary between the cold pool and the environment is considered to be infinitely thin, and at each point on this boundary, the cold pool spreads at a rate C. C is considered to be a random variable whose mean C* will give the rate at which the cold pool spreads. In the GL10 model, C* scales with the square root of the potential energy available in the cold pools, i.e the energy associated with their collapse, WAPE (Wake Available Potential Energy), given by: 3WAPE=g∫δρρ‾=-g∫0hwkδθvθv‾dz. So that: 4C*=k2WAPE, where ρ is the air density; θv is the virtual potential temperature.

Coefficient k in Eq. () should take a value between 0 and 1. Based on 3D CRM (Cloud Resolving Models) simulations, Lafore (2000) estimated this coefficient to 0.33 in the case of a linear structure such as squall line. This is the value retained in LMDZ6A.

When cold pools appear in a grid cell were they were not present before, their fractional cover σwk=πr2Dwk (where r is the cold pool radius) is set arbitrarily to 2 %, corresponding to an initial radius of about 2.5 km over ocean and 30 km over land with the values chosen for the wake number density in LMDZ6A. The cold pool radius then grows with rate r˙=C* so that the time evolution of the fractional area reads: 5∂tσwk=2πrC*Dwk=2C*πDwkσwk, This surface fraction increases over time. It is limited to 40 % of the domain. When this threshold is reached the growth of the cold pools is stopped and the radius remains constant.

It is assumed that below the top of cold pool (pwk), the vertical velocity profile associated with cold pools results solely from the spreading at the surface, without lateral entrainment (ewk=0) or detrainment (dwk=0) between the cold pool and its environment. Above this level, the subsidence induces a lateral convergence of air feeding the cold pool which can be reinforced by additional re-evaporation of rainfall below stratiform clouds. The shape of the vertical profile of the vertical p-velocity difference δω (where ω is the vertical p-velocity in Pa s⁻¹, while we will note w the vertical velocity in m s⁻¹) between the cold pool region and its environment is imposed as a piecewise linear function of pressure: δω increases linearly from zero at the surface up to a maximum value at pwk and then decreases linearly between pwk and a minimum pressure pupper corresponding to the upper bound of the cold pool model. The vertical subsidence which thus increases downward between pupper and pwk is fed by lateral entrainment ewk without detrainment so that ewk=σwk(1-σwk)∂pδω+∂tσwk. This lateral entrainment accounts for the horizontal component of the meso-scale circulation known to entrain dry and warm (in terms of potential temperature) air from low- or mid- tropospheric air into the cold pool (see Fig. ), in turn reducing the WAPE.

At pupper, the top of the cold pool model, δX cancels for all cold pool variables X. In GL10 model and in LMDZ6A, pupper was set to 600 hPa and there was also a nonzero velocity difference (δωcv) at pupper, accounting for the difference of the convective mass fluxes between (wk) and (ex). In the version used here, δωcv=0 above this level as we realized that the GL10 version was introducing a double counting of the vertical mass flux of unsaturated downdraft at pupper. This modification has however little effect on the parameterization results since the mass flux associated with downdrafts at pupper is small (result not shown).

As already said, it is the cooling associated with the re-evaporation of rain in unsaturated downdrafts that is the primary driver of cold pool development. This process is reflected in the model by assigning the heating term Q1unsat computed by the convective scheme to the interior of cold pools, while Q1sat acts on their environment. Consistent with this splitting, we assume that the saturated part of the convective scheme sees the profiles outside the cold pools, and the unsaturated downdrafts their interior. We further assume that thermal plumes are only active in the fraction of the horizontal surface located outside the cold pools. Thus the mass flux transport scheme is applied to any variable outside the cold pool ϕex=ϕ-σwkδϕ, and finally (θ/T)Q1th=-Cp(1-σwk)∂zρw′θex′‾ and Q2th=Lv(1-σwk)∂zρw′qvex′‾. The thermal plume model therefore induces a differential tendency that is opposite of the average tendency restricted to the environment. Ultimately, the contrast in convective tendencies (shallow and deep) between the cold pools and their environment reads: 6δQ1cv=Q1unsatσwk-Q1sat1-σwk-Q1th1-σwk,δQ2cv=Q2unsatσwk-Q2sat1-σwk-Q2th1-σwk.

It is the terms δQ1cv and δQ2cv that drive the time evolution of δθ and δqv given by: 7∂tδθ=θT(δQ1cv+δQ1wkCp)-ω‾∂pδθ-Kgwτgwδθ,∂tδqv=-δQ2cv+δQ2wkLv-ω‾∂pδqv.

This time evolution also includes a differential heating and moistening induced by the cold pools themselves of the air inside and outside the cold pools, under the effect of lateral air entrainment from the environment above pwk, subsidence inside the cold pools, and compensating ascent in the environment: 8θTδQ1wkCp=-ewkσwkδθ-δω∂pθ‾-(1-2σwk)δω∂pδθ,-δQ2wkLv=-ewkσwkδqv-δω∂pqv‾-(1-2σwk)δω∂pδqv.

The terms -ω‾∂pδ in Eq. () partially compensate for the fact that the contrasts δ are not transported by the dynamics until now. We therefore take into account, in the parameterizations, the vertical part of large-scale advection to partially compensate for this deficiency.

Finally, the last term, present only in the θ part of Eq. (), corresponds to the reduction in temperature contrasts by gravity waves with a coefficient specified as the ratio of an efficiency Kgw to a characteristic time 9τgw=σwk-(1-σwk)4NzDwk estimated as the time required for a wave with speed Nz (where N is the Brunt-Väisälä frequency and z is altitude) to travel a distance equal to the geometric mean of the cold pool size and the interval between cold pools.

Finally, the circulation induced in the cold pool parameterizations by the differential heating by convection (and cold pools themselves) contributes as well to the physics tendencies for temperature and humidity. The source terms only contain the vertical convergence of the Reynolds flux associated with mass flux transport (downward in the cold pool and upward in its environment) that reads ρw′ϕ′‾=1/gσwk(1-σwk)δωδϕ for any conserved variable ϕ so that: 10θTQ1wk=-Cpσwk(1-σwk)∂p(δωδθ),Q2wk=Lvσwk(1-σwk)∂p(δωδqv).

Finally, the cold pool model includes:

three prognostic variables, derived directly from the model equations: the profiles of δθ and δqv and σwk.

Triggering and closure formulations in LMDZ have been described in . Deep convection is triggered when the Available Lifting Energy (ALE) at cloud base exceeds the convective inhibition (CIN) threshold. This lifting energy is provided either by the thermal plume model (ALE_th) or by the cold pool parameterization (ALE_wk): 11max⁡(ALEth,ALEwk)>CIN. The intensity of the convection depends on the mass flux (Mb) at the cloud base. The closure consists in expressing Mb as a function of an available lifting power ALP, provided by the thermal plume (ALP_th) and cold pools (ALP_wk) parameterizations: 12Mb=ALPth+ALPwk(2Wb2+CIN), where 13Wb=wbsrf+wbmax1+ΔP(Ps-PLFC) is the vertical velocity at the level of free convection (LFC), ΔP=500 hPa and wb_srf and wb_max are model free parameters.

Concerning the boundary layer, the available lifting energy is taken as the maximum kinetic energy in the thermal plume below cloud base 14ALEth=12wth,max2. A notable improvement was introduced by , with the implementation of a statistical representation of the size distribution of cloudy thermal bases. In the new statistical triggering, deep convection is activated if both ALEth>|CIN| and at least one cumulus cloud within a grid cell exceeds a given size, specified by Strig (adjustable parameter). The available lifting power scales with wth,max3.

To calculate ALE_wk, the model assumes that the maximum speed (Cmax) on the cold pool contour will trigger convection. This is assumed to be proportional to the square root of WAPE, with a higher coefficient of proportionality than the one used for C* leading to the following relationship: 15Cmax=k′2WAPE, where k′=1.

The Available Lifting Energy associated with cold pools is thus expressed by the following relationship: 16ALEwk=12Cmax2.

Combining Eqs. () and (), one obtains: 17ALEwk=k′2WAPE, with k′=1, meaning that, in the cold pool model, the Available Lifting Energy associated with cold pools is equal to the collapse energy.

Each cold pool provides a power associated with its horizontal spreading. This power is calculated as the product of the kinetic energy supplied by the pool and the mass flux, evaluated over the entire contour of the cold pool (Lg=2πr), over the full height of the cold pool hwk, and using the spreading velocity C*. The power (ALP_wk,i) of an individual cool pool i is therefore: 18ALPwk,i=12ρC*3hwkLg. To obtain the average power (ALP_wk) over all cold pools in the domain, this individual power is multiplied by the cold pool number density Dwk: 19ALPwk=ALPwk,iDwk.

Since σwk = Dwkπr2, the lifting power ALP_wk reads: 20ALPwk=ϵρC*3hwkσwkDwkπ, where ϵ=0.25 is the lifting efficiency. This value means that 25 % of the power associated with cold pool spreading is available for deep convection. This rather arbitrary choice is tested hereafter.

Atmospheric Large Eddy Simulations (LES) are performed with non hydrostatic models, with a grid resolution fine enough to resolve the main structures (large eddies) that dominate the turbulent or convective transport. They provide an explicit and detailed representation of turbulent and convective motions within the boundary layer and of the associated clouds see e. g.among many others. LES were used as well for assessing parameterizations of deep convection, including cold pools that are represented explicitly in such simulations see e. g.. Note however that these simulations can become more dependent on the model parameterizations used when microphysics start to play a predominant role. One of the major strengths of LES lies in its ability to provide three-dimensional information not available from observations, making them an indispensable complement to the latter for understanding processes. In addition, LES can be used to validate the internal variables of parameterizations, enabling their physical realism to be assessed.

In the present study, we use the outputs of LES for two test cases, one over ocean and the other one over land. The test case over ocean was run with two different models, SAM and Meso-NH, with the exactly same setup in order to assess the uncertainty in LES, as it is known that RCE experiments are particularly sensitive to the microphysics scheme .

The LES over ocean were carried out in Radiative-Convective Equilibrium (RCE) mode see e. g.. In the RCE simulations used here, radiative computation is replaced by a constant cooling of -1.5 K per day, while the surface temperature is imposed at 300 K. Two LES of this RCE case are used here, one performed with the SAM model and the other one with Meso-NH . Both simulations cover an horizontal domain of 200 km × 200 km with resolution of 250 m. The lateral boundary conditions are cyclic for both models. The SAM RCE simulation ran for 44 d, a quasi steady-state regime being reached after about 40 d. SAM output are available every 3 h. The Meso-NH model was first run for 40 d at CRM (2.5 km) resolution and then restarted from the equilibrium reached at 250 m resolution for 10 d. Meso-NH outputs are available every 24 h.

The LES over land is issued from AMMA (African Monsoon Multidisciplinary Analysis) field campaign experiment . This case is derived from observations made on 10 July 2006 during which a relatively small, short-lived convective system formed over Niamey . This system, with a lifetime of around 6 h, was observed by various instruments (radar and atmospheric soundings), supplemented by satellite data. This case study represents a typical example of deep convection in the Sahel region . An LES of this continental case was carried out with the Meso-NH model over a 200 km × 200 km domain, with a horizontal resolution of 200 m. Lateral boundary conditions are cyclic and surface fluxes are imposed. Outputs are generated at a frequency of 30 min.

One advantage of RCE for parameterization development is that it targets a steady state regime. By contrasts, the AMMA case corresponds to a diurnal cycle of deep convection over land, for which the time matters, and for which errors in the phasing of the triggering of deep convection can alter comparisons.

In order to use LES for the assessment of the cold pool parameterization, the first challenge is to separate cold pools from their environment. Indeed, there is no a priori established framework for objectively identifying cold pools in observations and numerical models , and choices may depend in part on the physical picture one has of cold pools, and also, for the purpose at hand, on the picture underlying the parameterization. The first method for identifying cold pools proposed by was based on surface precipitation rates. In more recent studies, such as those by , the detection of cold pools is closer to a density current oriented detection, in which variations in temperature, pressure and wind are taken into account.

In the present study, the aim is not to isolate individual “cold pools objects”, but only to know whether a grid cell of the LES is inside or outside cold pools. Also the boundary conditions are idealized targeting the statistical homogeneity assumption that is at the basis of the Reynolds decomposition between dynamical core and physics parameterizations. In this idealized case with uniform surface temperature, cold pools can be identified using a threshold on the anomaly (after removing the domain average) of temperature at 10 m above surface, T10m, i.e. at the first model mid layer.

Figures  and show a horizontal moving average with a box of 3.25 km × 3.25 km of the divergence of the wind at 10 m above surface, V10m. From these maps, the centers and gust fronts of cold pools can be easily identified, corresponding respectively to the maximum and minimum of divergence values. Maxima of divergence of surface wind indicate the center of cold pools where cold air masses collapse. Precipitation is generally co-located with these divergence maxima (not shown). The fairly strong wind convergence observed around cold pools centers corresponds to the strong lift of air masses created upstream of the gust front at the cold pool's periphery.

Both the two LES of the RCE case and the LES of the AMMA case show cold pools clusters forming a common gust front. This can be explained by the fact that, during propagation, cold pools can merge to create a single, larger cold pool. We can also observe that wind convergence (and thus associated updrafts) is more intense where cold pools meet. This is in line with some studies that indicate that convection initiation on gust fronts is more efficient when two or more cold pools collide .

We superimpose on this map the T10m anomaly contours with different values to determine an optimal threshold for this anomaly. In the RCE case, the T10m anomaly at 0 K sometimes includes regions without cold pools centers, where divergence of surface wind is low (Fig. a and b) while anomaly contours at -0.2 and -0.4 K surround the centers of cold pools quite well. In the AMMA case, figure a clearly shows that the 0 K threshold is too high to identify cold pools. Fig. b, on the other hand, shows that the -1 K threshold follows gust fronts of cold pools better than the -0.5 K threshold. On the basis of these analyses, we retain the T10m anomaly thresholds at -0.2 and -1 K to identify cold pools in the RCE and AMMA cases respectively.

Once the threshold value is fixed for the T10m anomaly, we separate the full 3-dimensional LES domain between cold pool region (wk) and the rest of the domain (ex) form which we can compute the horizontal averages on each subdomain, Xwk inside cold pools and Xex outside, and then the cold pool anomaly δX=Xwk-Xex. This sampling allows to compute the vertical profiles of cold pools anomaly for temperature (δT), humidity (δqv) and vertical velocity (δw). Examples of temperature anomalies are shown in Fig. .

Note that we apply the same surface mask to the entire column to determine the vertical profiles. This simple vision of vertical cylinders is adopted to match the view underlying the parameterization but may be put into question in the presence of strongly tilted convection.

One can then compute the collapse energy (WAPE) of cold pools in the LES by integrating from the surface up to pwk, the virtual temperature anomaly, δθv (Eq. ). As suggested by , we take for pwk the pressure where the δT profile cancels out. This altitude is around 950 hPa (approximately 600 m) in the oceanic RCE case and around 800 hPa (approximately 2 km) in the AMMA case (Fig. ).

It is assumed in the parameterization that cold pools are identical disks of radius r. This assumption makes it easy to determine C* by the divergence theorem: 21∫∫divV10mdSwk=C*Lg.

After applying Eq. (), we obtain the following relation 22C*=divV10m‾SwkLg, where Swk is the area of cold pools.

Since Lg=2πr, σwk=Dwkπr2 and Swk=πr2, C* can read as a function of the mean divergence of wind at 10 m, the surface fraction (σwk) and the density (Dwk) of cold pools by the relation: 23C*=12divV10m‾σwkDwkπ.

The choice to calculate C* here from σwk rather than from the radius r is explained by the fact that it is easier to estimate σwk in the LES, since we did not use automatic cold-pool detection tools that would allow the radius of cold pools to be determined directly.

To apply this calculation of C* in the LES, we take the horizontal average of the surface wind divergence inside cold pools. The surface fraction (σwk) of cold pools calculated in the LES is 0.214 (Average over the available time steps between 5:00 PM and 10:00 PM) for the AMMA case and 0.253 (Average over the 24 time steps with the SAM model) for the RCE case. To determine Dwk, we manually counted the centers of cold pools visible on the surface wind divergence maps (Figs.  and ), as we did not use automated detection methods in this study that could generate their number automatically. We find a density, Dwk, of about 5 cold pools per 100 km × 100 km in the RCE case, and about 2.5 cold pools over the same domain in the AMMA case.

Finally we derive a direct estimation of the Available Lifting Energy (ALE_wk) and Power (ALP_wk) in the LES from a sampling of the vertical wind at cloud base.

To do this, we first determine an average cloud-base height at which we extract vertical velocities wb(x,y). This height corresponds to the altitude at which the average profile of condensed water reaches its first non-zero value. It is estimated at around 950 hPa on the two LES of the RCE case and at around 750 hPa for the LES of the AMMA case (cf. Fig. ).

We then separate the updrafts on gust fronts from those associated with thermal plumes. Since the updrafts on gust fronts are both stronger and more coherent horizontally than those associated with thermal plumes, we define a gust front mask based on a threshold applied to an horizontally moving average of the vertical velocity at cloud based wb, denoted as w̃b(x,y). Because the gust fronts are stronger in the AMMA case than in the RCE case, different choices were made for the size of the horizontal box of the moving average (1.25 km × 1.25 km for the RCE case and 2 km × 2 km for AMMA) and for the value of the vertical velocity threshold (0.6 m s⁻¹ for the RCE case and 2 m s⁻¹ for the AMMA case). Those values were retained after several tests so as to separate as effectively as possible the gust front form other ascents.

Figures  and overlays the updrafts within (red) and outside (green) gust fronts on maps of T10m anomaly (smoothed by applying a moving average with a box of 2.5 km × 2.5 km), for the RCE and AMMA cases respectively. The contours of the T10m anomalies used to identify cold pools (-0.2 K for RCE and -1 K for AMMA) are displayed as well. Visually, the gust fronts computed with w̃b(x,y) thresholds of 0.6 m s⁻¹ (RCE) and 2 m s⁻¹ (AMMA) align well with the contours of cold pools identified using these T10m anomaly thresholds. It also appears that most thermals are located in the environment of cold pools for both the RCE and AMMA cases. This retrospectively validates a choice made in LMDZ6A, where the effect of thermals was only computed outside cold pools.

Both ALE_wk and ALP_wk are computed from wb restricted to the gust front mask, noted wb,gust.

ALE_wk is estimated as the kinetic energy associated with the maximum value of wb,gust(x,y): 24ALEwk=max⁡(12wb,gust2).

ALP_wk represents the average updrafts power provided by all cold pools in the domain. It is calculated as the horizontal average of the cube of wb,gust times the surface fraction (σgust) covered by gust fronts: 25ALPwk=σgust12ρwb,gust‾3. The gust front mask is used to calculate σgust, which is 0.048 (LES SAM) for the RCE case and 0.045 for the AMMA case, for the times shown in Figs.  and . Characteristics of the cold pools estimated from the sampling are gathered on Table .

Physical parameterizations are defined by sets of mathematical equations designed to represent subgrid processes within a column of the model. The formulation of these equations is based both on a phenomenological understanding of the processes involved and on fundamental principles of physics. These parameterizations can be evaluated as a whole or in parts, by isolating certain equations or relationships between internal variables, or between internal variables and GCM state variables. LES offers the possibility of performing a priori validation and adjustment of these laws.

In the cold pool model, variables ALE_wk, ALP_wk and C* are determined from the collapse energy, WAPE (see Eqs. , and ). We compare in Table  the values obtained using the parameterization formulations (parameterized value P), based on WAPE deduced from δθv, with those obtained directly from resolved wind in the LES (sampled value S): the vertical speed at cloud base (wb) for ALE_wk and ALP_wk, and the mean divergence of wind at 10 m in cold pools for C*. These analyses are performed by averaging over the available time steps: 24 time steps for SAM and 7 for Meso-NH in the RCE case, and between 5:00 PM and 10:00 PM for the AMMA case.

The values of ALE_wk calculated by both methods are very close to each other. The largest error is an underestimation by about 30 % of the ALE_wk computed from WAPE compared to the wb,gust estimate. These results for the three LES confirm the validity of the hypothesis of equality between ALE_wk and WAPE assumed in the parameterization.

Table  shows that, C* values computed from the WAPE are systematically lower than those coming from the mean divergence of wind at 10 m in cold pools. This difference could be due to an underestimation of the coefficient k, imposed here at 0.33. With k=0.56, the calculation of C* based on the WAPE becomes comparable to those obtained from the mean divergence of wind at 10 m in cold pools (Table ). As discussed above, the value of 0.33 was retained following an oral communication by Lafore (2000). But other studies propose different values: estimate k at 0.68 based on CRM simulations of 2D squall grain, estimate it at 0.5 from observations of cold pools during the BAMEX experiment in the American Great Plains. Our results are thus compatible with the hypothesis of the model which postulates that the kinetic energy of cold pools results from the transformation of WAPE into kinetic energy with a coefficient k compatible with the published estimates.

Table  also shows that, for the three LES cases, the values of ALP_wk calculated with C* from WAPE are at least three times lower than those obtained from wb,gust. Two coefficients are involved in the calculation of ALP_wk with the parameterization formula: the coefficient k and the lifting efficiency ϵ, imposed respectively to 0.33 and 0.25. Using k=0.56 however in the calculation of C*, and keeping ϵ at is nominal value of 0.25 allows to reconcile the various estimates. This is compatible with the hypothesis of the parameterization according to which 25 % of the horizontal power provided by the cold pools during its propagation would be used to reinforce the intensity of the convection while a large part dissipates.

The analysis of the δT profiles in the LES confirms that cold pools are colder at the surface with temperatures increasing towards the top for the three LES. The cold pools are about three times deeper in AMMA (Fig. a) than for the RCE case (Fig. d). In the LES, the cold pool temperatures for the AMMA case (around 4 K) are lower than those of the RCE case (around 1.2 K). This is consistent with observations which indicate much colder pools over land than over the ocean. For the AMMA case in particular, observations reveal a temperature drop of approximately 5 K during the passage of the cold pool , a value fairly close to that of the LES. It should be noted however that the AMMA case corresponds to a relatively weak episode of continental convection. The δqv profiles indicate that at the surface, cold pools are wetter than the surrounding air in the RCE case and the AMMA case (Fig. b and e). In both cases, the excess of humidity within cold pools decreases with altitude up to the cold pools top. The humidity deficit above this level is due to the lateral entrainment of dry air from the mid troposphere and its subsidence into the cold pools (Fig. c and f). For the RCE case, this subsidence vanishes below 800 hPa (Fig. c), while for the AMMA case, it vanishes at a higher level, around 600 hPa (Fig. f).

The δT profiles simulated with LMDZ CTRL are qualitatively consistent with LES, with a cold pool top (where δT cancels) at about the right altitude. Cold pools simulated with LMDZ are however warmer than in the LES for the RCE case (Fig. a), and colder at the surface than the LES for the AMMA case (Fig. d). Consistently with LES, cold pools are also wetter at the surface and drier close to their top top (Fig. b and e). However the variations of δqv are much larger in LMDZ than in the corresponding LES. In particular, the cold pools are much too dry at their top in LMDZ. In both cases, cold pools are associated with subsidence. The height at which the subsidence of air masses into cold pools begins, fixed at 600 hPa in LMDZ CTRL, is too high compared to LES for the RCE case (Fig. e).

For the RCE case, the WAPE is significantly smaller in LMDZ CTRL than in the LES, with a difference of at least a factor of 2 (Table ). These low values of WAPE in LMDZ CTRL also translate into low ALE_wk values compared to LES (Table ). On the other hand, for the AMMA case, the WAPE simulated by the model, and consequently ALE_wk, are slightly higher than the values derived from the LES (Table ). The value of C* simulated by LMDZ CTRL is at least three times smaller than that of the LES in the RCE case and slightly lower for the AMMA case (Table ). ALP_wk is at least twenty times weaker in LMDZ CTRL than in the LES for all cases (Table ).

Various modifications of the cold pool parameterization are explored in the following section to try to correct the defects listed above.

We present here the impact of increasing the coefficient k from 0.33 to 0.56 (LMDZ V1 simulation) on the profiles of δT, δqv, δw as well as on the variables C*, WAPE, ALP_wk and ALE_wk (see Table ). In the RCE case, this modification significantly improves the profile of δw below pwk (Fig. c). It also allows for a better representation of the δw profiles below pwk in the AMMA case (Fig. f). These improvements are directly linked to the increase in C* for both cases (Table ), since the δw profile below pwk depends on the spreading of cold pools. The increase in C* could be associated with stronger air mass subsidence within the cold pool, which would contribute to a slight drying near the surface in both cases (Fig. b and e). For the AMMA case, this drying results in slightly drier cold pools at the surface in LMDZ V1 than in the LES, but they remain overall comparable to the latter. The increase in C* in both the RCE and AMMA cases also leads to a better representation of ALP_wk (an increase by at least a factor of 5 for both cases), even though this variable remains underestimated (Table ). We also note a warming effect from this modification in both the RCE and AMMA cases. The impact on the δT profiles in the AMMA and RCE cases is responsible for the decrease in the values of WAPE and ALE_wk for these two cases (Table ).

Figure  shows that the modification introduced in version V1 has a low impact on the θ and qv mean profiles (Fig. ), the black and blue curves being almost superimposed. Both CTRL and V1 simulations reproduce the θ profiles fairly well. Around 600 hPa, the temperature is too warm in the RCE case. Regarding humidity, a dry bias is present in the boundary layer in the RCE case, as well as between 800 and 400 hPa. For the AMMA case, there is a wet bias in the boundary layer and above 600 hPa, and a dry bias between 700 and 600 hPa. At the altitudes where these dry biases are found, notably between 800 and 400 hPa for the RCE case and between 700 and 600 hPa for the AMMA case, LMDZ produces almost no clouds, likely due to an overly dry atmosphere (not shown).

In the previous sections, we found that the altitude at which the subsidence of dry air above cold pools initiate is located around 800 hPa in the LES for the RCE case and around 600 hPa for the AMMA case, while in LMDZ, pupper is arbitrarily set to 600 hPa in the original version of the parameterization. In version V2, in addition to the change of the value of k from 0.33 to 0.56, we impose psrf-pupper=γwk,upper(psrf-pwk) where γwk,upper (fixed here to 3) is considered as a new free parameter in the following section.

Note that the numerical scheme used to estimate pwk was also modified compared to GL10 in order to solve instability issues.

The principle of the new scheme is to compute the height at which the vertical integral of δθ, from the surface to that level, reaches a fraction χ of the total integral from the surface to the first positive value of δθ. This fraction is taken slightly below 100 % to avoid fluctuations of pwk from one time step to the other, which happens when δθ is close to zero on a large range of altitudes around pwk as seen in Figs.  and . The new computation was activated already in CTRL and V1, not affecting the results significantly (figure not shown). The new computation becomes really important when pupper is deduced from pwk rather than being imposed.

Comparisons between LMDZ V2 simulations and LES show a better representation of the δqv profiles at the top of cold pools in both the RCE and AMMA cases (Fig. b and e). These results show that the dry bias at the top of the cold pool in the original version was due to advection of dry air from too high an altitude. This modification also reduces slightly the humidity of cold pools near surface in the RCE case, although they remain more humid than in the LES. Note finally that this modification has a very limited impact on the δT profiles.

Table  shows that the change in pupper weakly affects the variables WAPE, C*, ALE_wk and ALP_wk for these two cases.

Versions V2 does not modify much the mean vertical profiles except for a drying of the mid-troposphere in the RCE case (Fig. b), in a region where the CTRL simulation was already too dry. Although the time evolution of the mean profiles is the first target of physics parameterizations, we think however that the improvement of the internal variables is so strong for this modification that it should be adopted in the future in LMDZ.

As explained above, in the standard LMDZ configuration, thermals only interact with temperature and humidity profiles outside cold pools, inducing a differential heating in moistening (Eq. ). This choice was originally made to account for the fact that the atmosphere is more stable inside cold pools, and indeed the analysis above shows that the thermal plumes that reach cloud base are essentially located outside cold pools. Version V3 is identical to version V2, except that we consider that thermal plumes are active everywhere in the grid cell. Consistently, the terms Q1th and Q2th are removed from Eq. (). For the RCE case, this leads to a decrease in the surface humidity of cold pools, closer to the LES results (Fig. b). In the AMMA case, the effect is also present, although less pronounced (Fig. e). This result is expected because the vertical transport by thermals systematically dries the surface . This confirms the key role of boundary layer convection in regulating surface humidity on both continent and ocean , via the mixing of moist air with dry air above. One way to improve the representation of humidity anomaly profile without activating the thermal plumes everywhere would be to add a simple parameterization of shallow and cloud-free boundary layer convection (a simplified version of the thermal plume model) within the cold pool region.

In version V3, cold pools are colder than in version V2 in both the RCE and AMMA cases (Fig. ). In the RCE case, however, cold pools remain less cold than in the LES despite this effect. In the AMMA case, this cooling accentuates the overestimation of the cold anomaly. In both cases, this cooling leads to an increase in the WAPE, C*, ALE_wk, and ALP_wk variables (Table ). Version V3 does not introduce any notable changes, compared to V2, in the profiles of θ and qv for the RCE and AMMA cases.

The tests presented above show possible avenues for improving the cold pool parameterization. However, we see that the modifications do not sufficiently affect the mean profiles to reduce these biases significantly. All tests underestimate (for the RCE case) or overestimate (for the AMMA case) the cold temperature anomaly inside cold pools, as well as WAPE, ALE_wk, C*, and ALP_wk. We also observe systematic errors in the mean profiles, notably profiles that are much too dry for the RCE case.

In the GCM, these variables are not sensitive only to the parameters or formulation of the cold pool model. They are influenced by all other parameterizations, and in particular by the convection scheme to which the cold pool scheme is strongly coupled. In an attempt to see how modifications to other parameterizations could help reduce these biases, we performed automatic calibration simulations using the htexplo tool.

In practice, we decided to start from a tuning performed for the convective boundary layer by , for a configuration with 95 levels rather than 79 and using more recent versions of the codes than those used in the rest of the paper. This version indeed serves as the basis for preparing the future version of the climate model for the Fast-Trac part of the CMIP7 project, whose simulations are scheduled to begin in early 2026. We assume that the boundary layer parameters values have already been chosen to accurately represent the convective boundary layer and associated clouds, cumulus and stratocumulus. Regarding modifications to the cold pool model, those affecting the coefficient k and pupper are taken into account, as in the V2 configuration. Adjustments related to thermals (V3) are not considered here because they raise as many questions as they solve.

The tuning is performed using the htexplo tool as explained in Appendix . We choose metrics preferentially on the RCE case targeting the WAPE, ALP and the profiles of anomalies and mean variables. Indeed, we wish to avoid being overly dependent on errors in the phasing of the deep convective diurnal cycle. The adjustable parameters selected for the cold pool model are: k, γwk,upper, χ, and ϵ. Parameters involved in the deep convection scheme are also included: the minimum wb_srf and maximum wb_max vertical velocities at the base of the convective column; the fraction of the grid cell area in which precipitating downdrafts occur, σdesc; and the maximum precipitation efficiency in the Emmanuel scheme (EP_max). This efficiency is a maximum efficiency at the top of the convective columns. The difference 1-EPmax controls how much condensed water exits the convective clouds, and thus the moisture source in the upper atmosphere. Details on the metrics and parameter ranges chosen are given in Appendix .

The result of this tuning is the product of trial and error preliminary tuning experiments regarding the choice of parameters, their bounds, the metrics to adjust, and the associated tolerances. We finally perform 13 waves of tuning with the experimental setup described in Appendix A, and present here the 12 best configurations among all the runs performed during this tuning experiment. The best simulations are selected according to the 13 metrics involved in the tuning procedure (cf. Table ). For each metric we compute a score si=(mi-oi)/σi where mi is the metrics computed on the simulation, oi the target value (“observation”) and σi a tolerance to error. A combined metric is computed as the maximum of si over the 13 metrics: Si=maxisi. The best simulations are those with the smallest values of Si.

The analysis of the results for the RCE case reveals that the 12 bests configurations better simulate the targeted variables, particularly the mean humidity profile and the amplitude of the potential temperature deviations δθ. These deviations are more negative, consistent with a stronger WAPE and C* coefficient. Furthermore, the δqv profiles at the top of the cold pools, as well as the δw profiles, remain well represented across all 12 simulations.

By applying the parameters from the tuning performed on the RCE case to the AMMA case simulations, a significant reduction in the cold bias at the surface of cold pools is obtained (Fig. d), as well as an improvement in the mean humidity profiles across all 12 simulations. Only the best simulation (BEST TUNE) reproduces δqv and δw profiles consistent with the LES, the others tending to generate cold pools that are too dry at their top due to slightly overestimated δw profiles (Fig. e and f). The BEST TUNE simulation also provides a better representation of WAPE (Fig. a) and ALP_wk (Fig. b) in the AMMA case, with values very close to those from the LES at the time when cold pools reach their maximum development (Table ), despite a shift by about 3 h earlier in the afternoon.

It is well known that representing the diurnal cycle of convection remains a major challenge for GCMs. In LMDZ, a significant improvement in the representation of the diurnal cycle of precipitation was achieved thanks to the preconditioning of convection by thermals, as well as the role of cold pools in sustaining convection . Nevertheless, the delay in the timing of convective initiation remains a limitation. Intermediate tests have shown that tuning the parameter Strig, which defines the threshold cumulus size required to trigger deep convection, can delay convective onset. A work aiming at better representing the transition between shallow and deep convection is currently ongoing in the LMDZ team.

Several lessons can be learned about the range of the free parameters involved in the tuning when considering the best values (last column of Table ).

First the fraction of the grid cell assigned to unsaturated downdrafts, σdesc with a selected range of [0.042,0.048], is larger than the nominal value of 0.003, which contributes to first order to the increased intensity of cold pools, in particular for the RCE case. Concerning the parameters of the cold pool model itself, first the tuning of k from Eq. () confirms the findings of Sect. , with selected range of [0.56,0.57]. The value of γwk,upper that controls the height of the cold pool model is [3.6,4.1], compatible as well with the previous findings. With a range of [0.26,0.46], the tuning suggest that more than 25 % of the power associated with cold pool spreading is available for deep convection. Note finally that the χ parameter of the new numerical scheme for pwk computation is also strongly constraint by the tuning exercise (to a value of about 0.97). The values of parameters in the control simulation (initial values) and in the BEST TUNE simulation are presented in Table .

The htexplo tool retrieves automatically cold pool model parameter values close to those issued from direct analysis of LES. This highlights both the power of using such a tool for model development, avoiding a long phase of trial error in a multi-dimensional space, and the relevance of the model physics. Likewise, the range of values selected for σint leads to cold pool heights of about 600 m in the RCE case and 2 km in the AMMA case, in agreement with the expected orders of magnitude over ocean and over land, respectively.

One goal of the tuning is to intensify cold pools, particularly in the RCE case. The increase of σdesc is primarily responsible for strengthening cold pools in this case, through stronger evaporation.This increases confidence in the validity of the physics implemented in the cold pool model. In the AMMA case, the opposite effect (less colder cold pools) is explained by the rather large increase in C* (from 3 to 5 m s⁻¹), induced by the adjustment of k. This increase in C* accelerates the increase of the cold pools fractional area σwk, reducing the efficiency of evaporative cooling, which can no longer compensate for the dilution due to their rapid evolution. Although this mechanism is also present in the RCE case (with an increase in C* from about 1 to 2 m s⁻¹), it remains much less pronounced there, allowing the evaporation obtained with these σdesc values to effectively strengthen the cold pools.

We also consider that the increase of the difference 1-EPmax and σdesc played an important role in the improvement of the specific humidity profiles for the RCE and AMMA cases, although other parameters may also play a role. The increase of the difference 1-EPmax increases the amount of condensed water released at the top of the convective columns, thereby moistening the upper layers of the atmosphere. The value of EP_max was set to 0.999 in the control configuration. The increase in σdesc may both enhance humidity through enhanced evaporation and dry the boundary layer, for the AMMA case in particular, due to increased entrainment and downward advection of dry air from the free troposphere.

We also briefly examined the behavior of updrafts, given their strong coupling with downdrafts and cold pools. We note a decrease in undiluted adiabatic updrafts (which represent the theoretical capacity of the column to transport air and moisture) and a decrease in saturated updrafts in the tuned simulations (not shown). These effects are more pronounced in the AMMA case. The decrease in undiluted adiabatic updrafts would be linked to the drying of the boundary layer, itself associated with the increase in σdesc, as explained previously. This reduction in undiluted updrafts would also contribute to the decrease in saturated updrafts by limiting the moisture supply to the mixing with environmental air.