LMDZ is the global atmospheric component of the IPSL-CM Earth System Model , historically and still actively involved in the CMIP exercises and named after the French climate institute where it is developed: the Institut Pierre‐Simon Laplace (IPSL). A set of papers describes the recent developments and the performances of the 6th version of LMDZ used for CMIP6 that we consider in this study and briefly describe hereafter.

The turbulent vertical flux ρw′ψ′‾ of a variable ψ in the boundary-layer is parameterized with an Eddy Diffusivity – Mass Flux approach and reads: 1ρw′ψ′‾=-Kψ∂ψ∂z+fth(ψth-ψ) where w′ and ψ′ are the turbulent fluctuations of the vertical velocity and of ψ respectively, the overline is the average operator in the Reynolds' equations system framework. The first term on the right hand side is the counter-gradient diffusion with Kψ the eddy diffusivity calculated with a TKE-l turbulent scheme. It is worth noting that the TKE-l scheme has recently been updated and we use here the new scheme from that exhibits better numerical properties as well as more robust and more easily tunable formulations of the different terms of the eddy diffusivity coefficients compared to the previous TKE-l scheme of the model . The second term on the right hand side of Eq. () is the transport of ψ by the so-called “Thermal Plume Model” , a mass-flux scheme that summarizes the mean behaviour of a population of shallow-convective plumes and rolls. Each atmospheric column is divided into a mean ascending updraft of mass flux fth and the “environment” i.e. the region with a compensating downdraft. ψth is the value of ψ within thermals. fth can be estimated from the continuity equation: 2∂fth∂z=eth-dth where eth and dth are the lateral entrainement (resp. detrainment) rate of air towards (resp. away from) the thermals. Further elements on eth and dth calculation are given in . Importantly, this mass-flux scheme only activates from the ground surface, when surface convective instability occurs. The combination of the LMDZ eddy-diffusivity and mass-flux schemes has proven successful in representing the structure of stratocumulus clouds, and in particular, the cloud-top dynamics .

The parameterization of clouds in LMDZ follows a purely macrophysical approach extensively described in . It is based on a statistical scheme that assumes a subgrid distribution of the total water vapour from which the cloud cover and the cloud total water specific content are diagnosed. In absence of deep or shallow convection from the surface – conditions under which surface-decoupled MPCs are found – the LMDZ cloud parameterization considers a subgrid Probability Density Function (PDF) F(q) of the total water content q. This PDF follows a log-normal distribution bounded by 0 (, right column in Fig. ). Assuming instantaneous adjustment to saturation, the cloud fraction αc and the cloud total water – sum of vapor and condensates – specific content qcld (kg kg⁻¹) read: 3αc=∫qlim∞F(q)dq4qcld=∫qlim∞qF(q)dq. The condensation threshold qlim is taken equal to the saturation specific humidity with respect to liquid qsl (resp. with respect to ice qsi) when T≥0 °C (resp. T<0 °C).

In presence of surface-based shallow convection, i.e. where the Thermal Plume Model is active, cloud fraction and specific cloud water content are calculated from a bi-gaussian PDF (left column in Fig. ) of the saturation deficit s. It is defined as s=al(q-qlim(Tl)) where Tl is the liquid temperature and al a thermodynamic function that depends upon temperature . Again, qlim is equal to the saturation specific humidity with respect to liquid qsl (resp. with respect to ice qsi) when T≥0 °C (resp. T<0 °C). One gaussian distribution is associated with the air properties within thermals, the other one with the properties of the air within the environment. Each of them is characterized by the mean saturation deficit (sth or senv) provided by the Thermal Plume Model and a standard deviation (σth or σenv). σth and σenv are determined by considering that the width of the distributions of s in thermals and the environment are mostly driven by the exchange of air between the plume and the environment. The dispersion of s in each of the two regions therefore increases when sth-senv increases (see and for details). Cloud fraction and cloud total water content are also computed assuming instantaneous adjustment to saturation in each of the two regions, the condensation threshold being s=0 which corresponds to saturation in each region.

Assuming that the specific content of vapour qv=qlim in all types of clouds, the mass content of cloud condensates is then partitioned between the liquid and ice phases by computing the fraction of cloud condensed water in the liquid‐phase xliq using a continuous function of temperature T : 5xliq(T)=qlql+qi=T-Tmin⁡Tmax⁡-Tmin⁡0.5, where ql and qi are the specific contents of liquid droplets and ice crystals respectively, Tmin⁡=-30 °C and Tmax⁡=0 °C.

The treatment of cold precipitation consists in diagnosing a vertical profile of precipitation flux assuming stationarity and equilibrium of the precipitation with the cloud field. Autoconversion of ice crystals is based on a sedimentation law and supercooled rain immediately freezes (see Sect. 2.7.3 of ). Importantly, the specific contents of snow and rain are not variables of the scheme as the latter computes fluxes. Following , the snow specific content can be estimated offline from the snow flux, assuming a fixed fall velocity vi for ice crystals.

In agreement with previous studies , showed that the temperature-based cloud phase partitioning is inappropriate to simulate the structure of polar MPCs, particularly because SLW droplets are often located near cloud top i.e. in the coldest part of the cloud. Subsequently, developed a phase partitioning parameterization based on the works of and that we briefly summarize here. The general rationale of the parameterization stems from the conceptual physical model of that predicts the SLW production in vertically moving air parcels containing – or not – pre-existing ice crystals. It starts from a slightly adapted version of the so-called linearized Squires' equation that predicts the evolution of supersaturation (here taken with respect to the ice phase Si) and reads: 6dSidt=-Siτp︸1+ai(w‾+w′)︸2

Term 1 is the relaxation of supersaturation due to vapour deposition upon pre-existing ice crystals, τp being the associated timescale which is a function of temperature and of the first moment of the ice crystals' size distribution. Term 2 is the source/sink term of supersaturation associated with vertical velocity that we arbitrarily decompose here into a large scale and resolved component w‾ and a turbulent subgrid one w′. Following , the w′ turbulent fluctuations are treated as a white noise process and Eq. () becomes a stochastic differential equation which admits a steady-state solution with the form of a Gaussian distribution f(Si) of variance σs2 and mean 〈Si〉. 7f(Si)=12πσs2exp⁡-(Si-〈Si〉)22σs28σs2=12ai2σw′2τdτp9〈Si〉=aiw‾τp. The timescale τp is a function of the first moment of the ice crystal size distribution M1. The latter is calculated assuming an exponential size distribution for ice crystals and using the pre-existing ice crystal mass concentration (at the beginning of the model physics time step). We further assume that the ice crystal number concentration follows the number of ice nucleating particles (INP) predicted by the parameterization. The latter depends upon temperature and the number concentration of aerosol particles with diameters larger than 0.5 µm (Naero5) that we fix in the simulations (see Sect. 2.2.2 of ).

The time-scale τd involved in Eq. () is the Lagrangian turbulent decorrelation time-scale which is the characteristic timescale over which a turbulent flow loses memory of its initial state. Assuming near-isotropic turbulence, the distribution of the turbulent vertical velocity of variance σw′2 is related to the turbulent kinetic energy e predicted by the TKE-l turbulent diffusion scheme: 10e=32σw′2 τd then reads: 11τd=43eεC0γτd where ε is the eddy dissipation rate, C0 is the Lagrangian structure-function experimental constant and γτd a tuning coefficient.

Assuming a liquid saturation adjustment process, the liquid cloud fraction αl and the specific amount of liquid ql are estimated by integrating the distribution from Siw the value of the supersaturation at the liquid water saturation point: 12αl=∫Siw∞f(Si)dSi13ql=∫Siw∞qsi(Si-Siw)f(Si)dSi. The liquid cloud fraction αl is bounded such that ql does not exceed the total cloud condensed water content qcld-qlim. qi is then estimated as the difference between the total cloud condensed water content and the estimated ql.

In the LMDZ cloud scheme, the condensation procedure is iterative and the condensation threshold qlim is adjusted at each iteration. In the mixed-phase temperature regime, following , made qlim depend on xliq at each iteration such that: 14qlim=xliqqsl+(1-xliq)qsi This enables more consistency between the liquid and ice water contents estimated from the cloud-phase partitioning scheme, and the cloud fraction αc and total cloud water content qcld estimated from the condensation scheme. The prediction of the cloud liquid ratio xliq is thus no longer a function of temperature but depends on subgrid turbulent activity and the properties of ice crystals. developed this parameterization for mid-level stratiform clouds and the authors kept the default temperature-dependent phase partitioning for convective boundary layer clouds namely for clouds formed through condensation using the bi-gaussian distribution of the saturation deficit. The rationale and hypotheses of 's parameterization – and in particular the use of steady-state solution which is – make it a priori designed to capture the phase of stratiform and surface-decoupled boundary-layer MPCs. However, some adaptations are required to make it suitable for surface-coupled MPCs forming at the top of convective boundary layers.

In convective boundary layer MPCs, in-cloud supersaturation and SLW production are strongly modulated by entrainment and detrainment to and from thermal plumes . Entrainment is the mixing of drier environmental air into ascending thermal plumes, leading to a dilution of moisture and a subsequent reduction in supersaturation within the plume. Importantly, detrainment transfers moister air from the plumes into the surrounding environment, increasing local moisture content and potentially generating supersaturation. This interaction between coherent turbulent updrafts and clouds in the environment should be distinguished from the local SLW generation associated with local, homogeneous, and nearly isotropic small-scale turbulent eddies considered in the previous parameterization. To account for this effect on clouds located in the environment surrounding convective thermal plumes, the 's parameterization has been modified as follows. In cases for which the “Thermal Plume Model” mass-flux scheme is active and for which clouds form using the bi-gaussian distribution of subgrid saturation deficit, we consider an adaptation of Eq. () with a 3rd “homogeneisation” term: 15dSidt=-Siτp︸1+ai(w‾+w′)︸2-Si-SEτE︸3, Such a term was originally introduced in to account for the exchange of vertically moving air parcels with their surrounding whose supersaturation is SE, with a typical timescale τE. It was not taken into account in in a first instance (see their motivations in their Sect. 2.2.1) but re-introduced here to account for air parcels mixing between clouds in the environment and the air in the thermals. SE is taken equal to the mean supersaturation within thermals (evaluated at the underlying layer such as it is not affected by the mixing with the environment at the same layer yet) and τE is taken inversely proportional to the magnitude of the net detrainment rate: 16τE-1=γEmax⁡(dth-eth,0.)ρ where ρ is the air density and γE a tuning coefficient.

Equations () and () are then adapted to take into account the new term: 17σs2=12ai2σw′2τd1τp+1τE18〈Si〉=aiw‾+SE/τE1τp+1τE

Figure  summarizes with schematics how the adapted LMDZ cloud scheme works when the new parameterization for phase partitioning is activated, considering separately surface-coupled – that is, when shallow convection is active – and surface-decoupled clouds.

The first case study focuses on a widespread single-layer cloud deck sampled during the Atmospheric Radiation Measurement Program's (ARM) Mixed-Phase Arctic Cloud Experiment (M-PACE; ) between 9 and 10 October 2004 along the North Slope of Alaska. Cloud formation was governed by strong sensible and latent surface heat fluxes during a cold-air outbreak event, where air of the ≈ 1500 m deep boundary layer was advected within a north-easterly flow from the coast of snow-covered Alaska to the open ocean surface under a persistent high pressure system. The case was built for a SCM and LES intercomparison exercise in . The simulation is 12 h long and the initial state provided by radiosoundings analysis at Utqiagvik (formerly known as Barrow), Alaska, corresponds to a convective boundary layer with purely liquid clouds topped by a 2 K temperature inversion. Surface temperature, turbulent sensible and latent heat fluxes are kept constant throughout the simulation. The advective forcings of the SCM are prescribed and consist in vertical profiles of temperature and humidity horizontal advection terms as well as that of vertical velocity that are constant in time. During the case study period, the cloud deck was sampled by 2 spiral flights of the North Dakota Citation aircraft. Here we focus on the observed liquid and ice water contents from the 1009 and 1010 flights and details on in-situ cloud probes's data process are provided in . From airborne and ground-based remote sensing data during the case study, estimate the liquid water path (LWP) to range between 110 and 210 g m⁻² and the ice water path (IWP) between 8 and 30 g m⁻². The observed ice crystals number concentration throughout the cloud layer ranges between 0.1 and 10 L⁻¹ with most values of around 1 L⁻¹ . The INP parameterization used in our cloud scheme with a default Naero5 value of 1 scm⁻³ provides an ice number concentration roughly between 0.1 and 0.6 L⁻¹ for the temperature range within the M-PACE cloud layer, namely between ≈ -15 and -10 °C. Those values are thus realistic but in the lower part of the measured range. Simulations with higher ice crystals number concentration will be investigated by varying Naero5. M-PACE simulations are run with the standard 10 min time step of the LMDZ SCM, but additional simulations using a shorter 1 min time step reveal an overall weak sensitivity of the results (not shown here). Given the relatively high SLW content in clouds, precipitating ice crystals during M-PACE mostly correspond to rimed particles, mostly with irregular, rosette and columnar shapes in the below-cloud precipitation layer, and with a mean mass diameter close to 2 mm . For this case, we therefore arbitrarily consider an ice crystals fall velocity vi=0.5 m s⁻¹ .

The second case is an 8 h long semi-idealized case based on observations from the Indirect and Semi-Direct Aerosol Campaign (ISDAC) and focuses on a single layer of stratocumulus cloud deck that formed over pack ice north of Utqiagvik, Alaska, on 26 April 2008. Clouds were located within a well-mixed layer decoupled from the sea-ice surface. It was set-up in for an intercomparison study of LES with bin or 2-moment bulk microphysics schemes. The initial profiles of temperature, moisture, and horizontal wind components are based on aircraft observations in the mixed layer and radiosoundings at Utqiagvik. The initial moisture profile contains a layer supersaturated with respect to liquid water between ≈ 700 and 800 m a.g.l. Surface turbulent sensible and latent heat fluxes are set to zero and a moderate subsidence of 0.4 cm s⁻¹ is prescribed above 800 m a.g.l. with a gradual decrease to 0 down to the surface. An important difference in the set-up compared to M-PACE is that the horizontal wind components, temperature and moisture profiles are nudged towards the initial profiles and nudging coefficients are specified to have the height dependency (see Appendix of ). At the end of the 8 h of simulation, the LES involved in the 's intercomparison exercise exhibit LWP and IWP values ranging roughly between 10 and 50 and 2 and 20 g m⁻² respectively, depending on the ice crystals' number concentration prescribed. Albeit initially built for LES, the almost same set-up is used here to force the LMDZ SCM. We however make two exceptions with respect to the 's protocol. First, we do not prescribe the ice nucleation rate and the ice number concentration as the number of ice crystals is diagnosed in the new cloud phase partitioning scheme. Nonetheless, the INP parameterization used with a default Naero5 value of 1 scm⁻³ provides an ice number concentration value at cloud-top temperature around 0.6 L⁻¹, which is fairly close to the approximate 1 L⁻¹ mean in-cloud ice crystal number concentrations estimated from multiple measurements during ISDAC . Second, we activate ice processes from the beginning of the run – and not 2 h later – as ice and liquid phases are treated jointly in the LMDZ cloud scheme. Note that LMDZ ISDAC simulations are run with a 1 min time step and not with the 10 min time step commonly used for the LMDZ SCM simulations. Justification for this choice is provided in Appendix . Precipitating ice crystals during ISDAC mostly correspond to unrimed and unaggregated dendrite crystals whose size rarely exceeds a few hundred µm. For this case, we therefore consider an ice crystals fall velocity vi=0.2 m s⁻¹ , which roughly corresponds to the mean-mass ice crystal velocity of the SAM LES presented in .

SCM simulations are run with the standard 95-level vertical grid of LMDZ with model layers' thickness increasing from ≈ 10 m near the surface to ≈ 100 m at 2000 m above ground level . Forcings for the MPACE and ISDAC single-column cases have been formatted to the international DEPHY-SCM standard (https://github.com/GdR-DEPHY/DEPHY-SCM/, last access 21 September 2025) so that they can be run by any SCM designed to read DEPHY-SCM inputs. Results of SCM simulations – especially LWP and IWP – will be compared to estimates (ranges of plausible values) from observations or reference LES simulations. Nonetheless, the ability of LMDZ to capture the overall structure of MPCs as well as realistic values of LWP and IWP necessarily depends on calibration choices, namely arbitrary choices for tuning parameter values.

This question is even more critical for the new parameterization of cloud phase partitioning that introduces new tuning parameters and that involves direct interactions between the cloud scheme, the shallow convection scheme and the vertical diffusion scheme.

The advantage of numerically-cheap and quick SCM simulations is that the parameteric sensitivity can be explored quite easily using Perturbed Parameter Ensemble (PPE) experiments, by randomly sampling the values of a given set of parameters within pre-determined ranges of plausible values. Then remains the question of the choice of parameters to include in the PPE, namely those for which we should assess the sensitivity. M-PACE and ISDAC boundary-layer clouds structure, water content and phase are expected to mostly depend on the calibration of the turbulent diffusion scheme's parameters, shallow convection scheme's parameters and cloud scheme's parameters including those controlling the subgrid distribution of water vapour (or saturation deficit), the autoconversion to precipitation and its evolution as well as the parameters of the cloud phase parameterization. Exploring with no a priori the full ranges of plausible values for the full set of possibly-impacting parameters is however misleading as parts of the explored parameters space, even if leading to reasonable LWP and IWP values for M-PACE and ISDAC clouds, may be climatically not viable – because leading to unrealistic climate states in 3D global simulation – and/or lead to unrealistic structures of other cloud types. To illustrate the sensitivity of the new cloud phase parameterization, we will therefore choose a model configuration in which all parameters of the shallow convection, turbulent diffusion, and cloud scheme that are not specific to ice and mixed-phase processes have been tuned on a series of warm boundary-layer cloud cases as in . Only 7 parameters related to cloud phase parameterization and ice precipitation will therefore be retained for the PPE exercise, together with Naero5 which controls the number of INPs. Table  provides the list of those parameters, their range of acceptable values and their default value. Note that using a toy-model version of the parameterization, show that the most critical parameters for the cloud phase partitioning in non shallow-convective boundary-layer MPCs are the parameter γτd controlling the Lagrangian turbulent decorrelation time-scale τd, the capacitance parameter of ice crystals C as well as the fraction of snowfall γs that is taken into account in the pre-existing ice crystal mass concentration for supersaturation relaxation term. Following , the number of simulations of the PPE is 10 times the number of parameters, namely 70.