To better understand the model integration, consider a cloud parameter ϕ. This represents any of the prognostic parameters used in this scheme, e.g., cloud ice qi or cloud liquid qc. The model then solves the equation ∂ϕ∂t+u⋅∇ϕ=∂ϕ∂tmicro+∂ϕ∂tvdiff+∂ϕ∂tconvection, where the left-hand side represents the resolved advection and the right-hand side the unresolved processes that need to be parameterized. The tendencies due to the microphysics routine are summarized here by ∂ϕ/∂tmicro. The tendencies ∂ϕ/∂tvdiff and ∂ϕ/∂tconvection are calculated by ECHAM6-HAM2's vertical diffusion and convection modules . Cloud microphysics modules not only include phase changes and aggregation processes but also the vertical advection of precipitation. For the prognostic treatment of precipitation, this aspect has strong implications for the accuracy and performance of the scheme. To this end, we separate the cloud processes from the prognostic advection of cloud ice. As will be elaborated carefully in Sect. , this allows us to employ a two-step reduction in the global model time step to (1) sufficiently resolve the computationally heavy cloud process calculations and (2) to achieve numerical stability for the vertical advection of cloud ice. This separation is shown in Fig.  by the local update boxes separating the calculation of cloud processes and sedimentation. In terms of the arbitrary cloud parameter ϕn at time step n, the workflow of a single microphysics sub-step can be expressed as ϕ′=ϕn+∂ϕn∂tcloudΔt,ϕn+1=ϕ′+∂ϕ′∂tsedΔt. The entire scheme then consists out of nout iterations of Eq. () where the calculation of the sedimentation tendency is done iteratively as well. This iterative process is shown in Fig. . The cloud process tendency is given by ∂ϕ∂tcloud=∂ϕ∂tiaccl+∂ϕ∂tislf+∂ϕ∂traccc+∂ϕ∂tcautr+∂ϕ∂te/s+∂ϕ∂tc/d+∂ϕ∂tact+∂ϕ∂tci+∂ϕ∂tmlt+∂ϕ∂tfrz. The individual terms are discussed in detail below. The abbreviations are as follows:

“iaccl”, accretion of liquid by ice

Equation () is in strong contrast to the original scheme. Due to the long time step of the global model, cloud processes have been calculated sequentially. This allowed us to represent a full cloud life cycle within one time step; from condensation/deposition to collisions and freezing/melting to evaporation/sublimation or precipitation formation. With the sub-stepping introduced to resolve the vertical advection of cloud ice, the new scheme also resolves the life cycle of rather short-lived clouds and hence does not need to introduce a specific order in which the cloud processes occur.

In practice, the tendencies for cirrus nucleation and deposition ∂ϕ/∂tci and cloud droplet activation ∂ϕ/∂tact are currently computed before the outer loop for two reasons. Both parameterizations are based on the time rate of change in supersaturation within an adiabatic parcel ascent. Particle formation depends on the maximal supersaturation that can be reached before condensational/depositional growth quickly depletes all supersaturation established by cooling. Such parameterizations are designed for global models where the time step is large enough, such that the entire process from parcel ascent to particle formation and subsequent depletion of supersaturation takes place within a single time step. With a reduced time step, this assumption does no longer hold. Furthermore, ECHAM6-HAM2 is designed in a modular way. Therefore, aerosol-related particle formation is not calculated within the cloud microphysics module but as part of the aerosol module HAM2 and a separate cirrus module. This approach allows us to choose the appropriate scheme for different applications and decouples the development of different modules by specifying a coupling interface.

In the following subsections the parameterizations used to calculate the individual terms in Eq. () are presented.

In this section we present an approach to find a compromise between computational efficiency and model accuracy. The goal is to increase model efficiency by distributing the workload between the two parts shown in Fig. : the computationally expensive outer loop with nout iterations calculating both cloud process rates and sedimentation and the computationally cheap inner loop with nin iterations calculating only the sedimentation of cloud ice. Sedimentation is calculated roughly 100 times faster than the cloud processes. Since the loops are nested, the number of calls to the sedimentation calculation will be ntot=nout×nin, reducing the time step in the sedimentation calculation and cloud processes to Δtin=Δt/ntot and Δtout=Δt/nout, respectively. To achieve numerical stability, we calculate the required number of sub-steps ntot online to achieve αk(Δtin)=vkΔtin/Δzk<1 for every level k. This requirement could be relaxed for an implicit scheme which is stable even for α>1. Since the outer loop is so much slower than the inner, the restriction to α<1 is not our main concern because it can easily be achieved by the inner loop. This leads to the following expression for ntot on N model levels: ntot=maxk∈1,…,Nαk(Δt).

The calculation of the process rates is not subject to the same restrictions as sedimentation. However, if we only use the inner loop (nout=1) to achieve numerical stability, we will not be able to represent the process rates acting on falling particles and impair model accuracy. On the other hand, if we set nin=1, we will have to achieve numerical stability solely by the expensive outer loop and impair computational efficiency.

We found a compromise between the two extremes by considering the trajectory of a falling ice crystals. For model accuracy, it is important that the cloud processes are calculated every time the ice crystals reach a new model level and are thus exposed to a new environment. This is equivalent to the requirement α(Δtout)<1. However, if fall speeds are very high and/or the level is very thin, the total rate of change in cloud ice will be dominated by sedimentation. Processes like sublimation and melting will not have enough time to significantly change the cloud ice content on those levels. Therefore, we neglect the last part of the trajectory where the ice only spends a fraction of a global model time step and calculate the number of outer iterations: nout=αmax=maxk∈Iout(x)αk(Δt) for the set Iout(x)={k|k∈{1,…,N}∧Σi=0kτk>x} for the time spent on level k, τk, a specified threshold time x and a model with N levels. Numerical stability is then achieved on all levels by using nin=ntot/nout inner iterations.

We illustrate our approach using the test case properly described in the next section. For the purpose of this section, the exact model setup is not important. Given the distribution of cloud ice in Fig. a and fall speed in Fig. b, we can calculate the time that cloud ice will spend in each level. Since the thickness of model levels varies from 90 m at the surface to 700 m at 8 km height and since gravitational size sorting and self-collection lead to the fastest fall speeds close to the surface, α varies from 57 at the surface to 1 at 8 km (Fig. c). At the same time the accumulated time spent below a certain height is only a fraction of a global time step (Fig. b).

The colored bars in Fig. c show different choices for the threshold time x in units of the global model time step Δt. Its value ranges from 0 (orange bar), meaning that the entire trajectory is considered, to 1 (brown bar), neglecting the part of the trajectory where ice does not spend at least one global model time step.

For the following estimation of the model error, we chose the strategy corresponding to the purple bar Iout(0.2). We neglect the last part of the trajectory where cloud ice spends less than 20 % of the global model time step. This choice represents a compromise between performance and accuracy; this will be further elaborated below.

To demonstrate the power of the sub-stepping described above, we employ a simple single-column setup with 31 vertical levels. At one single model level at around 8 km height, a constant ice source term with specified tendencies for all four ice moments is prescribed at every time step. The source terms are chosen such that the ice particles reach very high fall speeds, and thus sub-stepping is fundamentally important. The forcing is representative of hail formation and thus an extreme case that is probably not produced very often by the global model. However, it highlights the need for sub-stepping while the general conclusions are also true for smaller or dendritic particles.

We prescribe the tendencies for cloud ice with ∂qi/∂t=∂qrim/∂t=5×10-3g kg-1 s-1, ∂ni/∂t=1×103 kg-1 s-1 and ∂brim/∂t=∂qrim/∂tρrim-1, where we set the rime density to ρrim=900 kg m-3. This forcing implies a rime fraction Fr=1. The temperature profile is prescribed, constant in time and decreases linearly from 0 ∘C at the ground to -55 ∘C at 8 km. Relative humidity is set to 50 % with regard to liquid water. The relevant microphysical processes are sedimentation, sublimation and self-collection. The resulting ice profile undergoes a buildup phase until it reaches an equilibrium such that the source term is balanced by the precipitation sink.

Results from this idealized experiment are shown in Figs.  and . Since we are investigating numerical errors due to insufficient time resolution, the high-resolution run (T6; black, dashed line) with a time step of 6 s is regarded as the truth in the following analysis. We did not change the vertical resolution; therefore, CFL numbers α(Δt=6 s) are very small throughout the column and no sub-stepping needs to be applied in the T6 case. With the large CFL numbers for the global time step, errors in the sedimentation calculation are huge. The simulation without sub-stepping (NO) shows that the large errors in the sedimentation calculation lead to a numerical deceleration of sedimentation. This in turn strongly delays surface precipitation and leads to an accumulation of ice in the atmosphere; see Fig.  (orange lines). The opposite problem is encountered when only the inner loop is acting to reduce α (IN; purple line). While the error in the sedimentation calculation is reduced, the sequential treatment of cloud processes and sedimentation leads to an underestimation of sublimation and thus overestimation of surface precipitation.

The simulations OUT (green line) and FL (light blue line) reproduce the results of the high-resolution simulation T6 much better. The two simulations only differ by the fact that the FL simulation further reduces α by the additional sub-stepping of the sedimentation calculation to achieve α<1 throughout the column by the inner loop. This difference is most pronounced in the low, thin levels where α can still be very large even if it is reduced by the outer loop already. The effect of this can be seen by comparing the vertical profiles of the process rates of the two simulations in Fig. . In the lowest levels, the simulation OUT deviates from the reference T6, which leads to an overall error in the surface precipitation and the cloud ice profile. The simulation FL reaches very good agreement with the reference at all levels by ensuring α<1 and thus achieving numerical stability throughout the column.

The test case presented above allows us to use the optimization strategy to find a compromise between performance and accuracy. To assess performance, we measure the computation time of the cloud microphysics routine. Specifically the two parts cloud processes and sedimentation shown in Fig. . We then define the speedup as the ratio tT6/tSIM of the computation times t of any simulation SIM and the reference high-resolution simulation. Figure a shows the relative errors in surface precipitation and ice water path at equilibrium together with the speedup for each simulation. A new simulation OUT100 is introduced to provide a further benchmark. It uses only the outer loop to achieve α<1 throughout the column; i.e., it puts all the work into the expensive cloud processes iteration. This is equivalent to the strategy illustrated by the orange bar in Fig. c.

Figure a illustrates that the optimization strategy works as expected and a speedup of around 7 can be achieved by only considering part of the column to calculate the number of outer iterations. It also confirms the finding from the last section; using the inner loop is essential to reduce the error from more than 15 % in the OUT simulation to well below 5 % in the FL simulation while keeping the speedup almost identical. The simulation OUT100 achieves an almost exact match with the high-resolution simulation. However, it comes at the cost of drastically reducing model performance. Thus, the benefit of the sequential treatment of cloud processes and nested sub-stepping becomes clear.

Figure b adds more depth to the optimization strategy. By considering a range of different threshold times x, we can choose to trade accuracy for performance. By setting a high threshold time, we can achieve a speedup of up to 15 if we accept the larger error.

This analysis has been performed on a series of different test cases (varying particle size and density as well as varying temperature and relative humidity profiles; not shown), including one where ice melts to form rain. This last test is particularly interesting because the melting layer represents a sharp change in process rates from one level to the next. However, since the calculation of rain is largely independent of the sub-stepping (see the section below) and since melting is represented by a finite, physically based timescale, the optimization strategy did not lead to large errors. We chose to show a different test case here because we wanted to highlight the treatment of the sedimentation of cloud ice in the lowest levels where model levels are very thin.

While the values for the relative error vary roughly between 0 and 5 % depending on the test case and threshold values, the overall correspondence of relative error and speedup has been shown to be a robust result of our optimization strategy.

This section provides a closer look at the rain flux within the sub-stepping environment. A diagnostic treatment of precipitation is designed for very large time steps where the vertical movement of raindrops cannot be resolved. Since the new scheme in principle allows us to resolve falling hail particles, we are outside of the realm the rain flux scheme was originally designed for. Since this work focuses on the representation of cloud ice, we will not discuss potential improvements to the liquid phase that would benefit from the newly employed sub-stepping. The obvious improvement would be using prognostic rain as was done by . However, since their approach was different in terms of treating the cloud droplet and raindrop size distribution, a merging of these two approaches is beyond the scope of this paper but will be envisioned in future. Sticking with the rain flux approach, it is important to rule out any systematic biases of rain production associated with the sub-stepping employed for cloud ice.

To estimate the sensitivity of rain production to the number of outer iterations, we use a similar setup as for the ice sedimentation: a single-column simulation with an isothermal atmosphere at 20 ∘C and a relative humidity of 100 % throughout the column. A humidity tendency of 5×10-7 kg kg-1 s-1 is applied to the model levels between 16 and 26 (corresponding to 900 and 400 hPa in pressure levels). This forcing is representative of stratiform cloud formation in the global setup of the model and corresponds to a water column tendency of 2 mm h-1. For this experiment we fixed the cloud droplet number concentrations but vary their (constant) values from 50 to 1000 cm-3 to represent clouds with stronger and weaker rain production rates.

The simulations are run for 1 day with the humidity forcing active throughout the whole simulation. Every simulation is run once with nout=1, i.e., without sub-stepping affecting rain production, and once with nout=100, i.e., with a very large number of outer iterations. Figure  shows the vertically integrated rain production rate together with its constituents: rain enhancement by accretion of cloud droplets by raindrops and auto-conversion of cloud droplets to raindrops.

The first row in Fig.  indicates that different numbers of sub-steps and thus different time step lengths can lead to differences in the rain production rate. The simulation with one sub-step has a slightly delayed the onset of precipitation and tends to overshoot the total rain production by up to 10 % before reaching an equilibrium. For the simulations with the highest number concentrations and therefore weakest rain production rates, both the delay in onset and relative strength of overshooting is less pronounced. Eventually, every simulation reaches an equilibrium where the humidity input is balanced by the rain sink. This external constraint leads to vanishing differences in equilibrium rain production rates for different numbers of sub-steps. The second and third rows show the constituents of the total rain production rate. These rates show that there is no compensation of errors by the accretion and auto-conversion rates but rather that the differences are due to the overestimation of precipitation production by the linearized numerical integration method employed by the core model. The local update of qc and Nc by sub-stepping Eqs. () and () reduces the numerical error of the accretion and auto-conversion rates and prevents the overshooting of precipitation formation that can be seen in the nout=1 simulations. This claim is backed up by a simulation with high temporal resolution which is almost identical to the simulation with 100 sub-steps and therefore not shown. We conclude that sub-stepping is beneficial for the representation of the rain flux but the effect is small.

Varying the number of outer iterations from 1 to 100 is an extreme case. In the global model setup the number of outer iterations ranges on average from 5 in the tropics to 25 in midlatitudes. Since the model converges quickly with increasing number of outer iterations, the delayed onset and overshooting effects discussed here are an upper limit. We conclude that for our purpose a diagnostic scheme for liquid water is compatible with the prognostic treatment of cloud ice and no systematic biases are induced by the number of outer iterations used. This is important as the number of outer iterations is computed online and may vary between columns.

The standard version of ECHAM6-HAM2 is equipped with a two-moment scheme for both cloud liquid water and ice and diagnostic equations for snow and rain mass mixing ratios. For comparability, the new scheme can switch between calculating in-cloud and sedimentation tendencies based on the new single category and the original 2-category scheme. This way we are able to consider only the differences between the schemes that are due to the conversion of cloud ice to snow in the original scheme and the single-category approach while all compatible cloud processes (vapor deposition, melting and freezing) are identical.

In the following we will present results from three different microphysics schemes, summarized in Table . They are presented in the order similar to the evolution within ECHAM-HAM. The 2-category scheme treats ice and liquid water analogously by separating in-cloud and precipitation hydrometeors. While for liquid water this separation can be justified by the different scales on which growth by condensation and growth by collision–coalescence are efficient, the analogous argument does not hold for cloud ice; a perfect dendrite can reach a significant fall speed. This deficiency has been solved by including the mass flux divergence scheme to allow the in-cloud part of the ice population to fall . Here we call this scheme the 2.5-category scheme since the falling ice mass flux resembles a separate category. The problem with this is, that there is no physical distinction between falling ice and snow. Sublimation and melting are parameterized for both precipitation categories, but only for snow collisions with liquid water is included. This leads to an artificial competition between falling ice and snow that, depending on the cloud forcing, forms precipitation hydrometeors that can further grow by riming or are limited to sublimation and melting. Furthermore, other growth mechanisms (i.e., self-collection and vapor deposition) are neglected for both precipitation categories. The 1-category scheme is the logical successor in this line of development. By resolving the vertical advection of cloud ice, precipitation categories are no longer necessary. The spectrum of ice particles is represented, and one single set of cloud processes is parameterized for the entire ice hydrometeor population. With that, the conceptual problems of the previous schemes are solved. The cloud liquid and ice water contents of the three simulations in Fig.  highlight the differences between the schemes.

In the 2-category scheme simulation, cloud ice is not allowed to fall. In the single-column model, ice crystals are therefore restricted to the level they formed in, which are the levels where heterogeneous freezing of cloud droplets takes place. Since temperature decreases with increasing altitude, this process is most active at the cloud top. As soon as a sufficiently large number of ice crystals has formed, their accumulated depositional growth is able to quickly deplete the coexisting liquid water. The large mass transfer from the liquid to the ice phase grows the ice crystals to a size where conversion to snow is efficient. Those snow particles partly deplete the liquid cloud below the freezing levels by riming and subsequently sublimate.

The results from the 2.5-category scheme simulation show how the situation changes when the ice crystals themselves are allowed to fall. The ice crystals that formed at the cloud top do not accumulate in the levels of formation but spread throughout the cloud. An exponential tail of the ice crystal mass flux continually falls out of the cloud where it sublimates. Due to the steady removal of cloud ice, growth by vapor deposition is delayed (orange lines in Fig. a and b). The snow production rate is weak because ice sediments out of the cloud before it can efficiently grow to the snow threshold size. Consequently, the riming rate is virtually 0 since only collisions between snow flakes and cloud droplets are considered (purple line in Fig. c). The low riming and deposition rates lead to a mixed-phase cloud that is heavily dominated by liquid water (Fig. d). The challenge of treating the sedimentation of ice crystals diagnostically has been discussed in . Our results support the hypothesis that a diagnostic scheme likely overestimates sedimentation.

The 1-category scheme does not rely on a separate set of microphysical processes that are calculated only for diagnostic sedimentation categories like snow and falling ice in the 2.5-category scheme. Thus, there can be a competition between riming, vapor deposition and the removal by sedimentation. Residence times per level are resolved by the sub-stepping which allows us to compute physical processes on a theoretical basis instead of empirical relationships employed for the mass flux calculation for ice and snow.

These simulations nicely illustrate the theoretical considerations at the beginning of this section. The separation into multiple categories and the associated conversion rates strongly influence the resulting cloud structure, lifetime and phase fraction. A small snow threshold size leads to more snow and thus more riming, while for the threshold size of 100 µm used in the 2.5-category scheme, snow formation is almost completely inhibited by the steady removal of cloud ice by the mass flux divergence scheme. This sensitivity to a development choice that is not constrained by first principles is highly undesired. The problem is resolved by the prognostic, single category that does not have this degree of freedom and thus simulates the cloud in an objective, physically based manner.

The rime variables in the single-category scheme, i.e., the rimed ice mass mixing ratio and the rimed ice volume mixing ratio, determine the density and shape of the particles with heavily rimed particles being spherical and weakly rimed particles having dendritic geometry. As a result, particles with high rime fractions have a smaller projected area and thus a higher fall speed than their weakly rimed counterparts of the same mass. showed in a regional model that this adjustment of particle properties is crucial to correctly predict precipitation rates in a squall line simulation with strong convective updraft.

The ECHAM6-HAM2 GCM does not resolve those strong convective updrafts but parameterizes convection by the scheme. In its standard version, it employs very simplified microphysics which does not account for the coexistence of liquid water and ice. It is therefore questionable whether riming as such, and the resulting change in particle properties especially, is currently adequately represented in convective clouds.

From a purely stratiform cloud perspective, the effect of the particle properties on process rates is best illustrated by a seeder–feeder situation. Ice crystals form in the cirrus cloud and quickly grow to a few 100 µm in diameter in the highly supersaturated environment. The largest crystals sediment quickly and subsequently interact with a supercooled liquid cloud below by riming and the WBF process. Whether riming or vapor deposition dominates, strongly depends on the particle size. A few large particles will grow more strongly by riming while the growth of many small particles will be dominated by vapor deposition. Therefore, combining strong depositional growth in the cirrus regime with gravitational size sorting provides the optimal conditions for rime growth within the supercooled liquid cloud. In the following, we will explore this special case and shed light on the particle properties within the P3 parameterization.

The boundary and forcing profiles for the simulation are the same as in the last section (Fig. ) but with heterogeneous freezing in the mixed-phase cloud turned off. To investigate the sensitivity to the particle properties within the new scheme, we vary the effect that riming has on the mass-to-size relationship of the P3 scheme described in Sect. . Two simulations are done: the rime properties simulation uses the regular particle properties of the P3 scheme, and the dendritic properties simulation neglects the effect of riming on the mass-to-size relationship. With the notation from Sect. , this can be expressed by setting the rime fraction Fr=0, which results in Dgr=Dcr. The only remaining transition parameter then is Dth≡const, separating the small spherical ice regime from the dendrite regime.

Figure  shows a summary of the process rates and column-integrated water masses. We can see that neglecting the impact of riming on particle properties changes the thickness of the liquid cloud by roughly 10 %. This can be attributed to the different riming and deposition rates in the two simulations, which is ultimately a result of the slightly longer residence times within the mixed-phase cloud due to the smaller fall speeds in the dendritic properties simulation.

Figure  shows the particle properties for the two simulations. From the process rates (Fig. b and c), we can see that rime growth only exceeds growth by vapor deposition at about 7 h after the simulation start and quickly diminishes after that. This is due to the gravitational size sorting of the cirrus particles and the strong dependence on particle fall speed and diameter of the riming rate; large particles will reach the liquid cloud first and a significant amount of rimed ice forms upon impact with the cloud droplets. This is the time, where the two simulations differ most. While the particles from the rime properties simulation get slightly more spherical by riming and their fall speeds increase up to almost 2-fold, the particles from the dendritic properties simulation neglect the rime fraction for the fall velocity calculation. Therefore, these particles keep their dendritic shape and fall more slowly, which gives them more time to grow throughout the liquid cloud. The total ice particle density is largely dominated by the particle size. A fully rimed particle could reach a density of a several hundred kg m-3. However, the maximal rime fraction obtained here is roughly 60 %. Therefore, the particles do not reach densities higher than 100 kg m-3.

The idealized, purely stratiform and turbulence-free simulations shown in this section do not produce the conditions necessary to form heavily rimed particles even though the setup has explicitly been designed to create an environment that favors rime growth over depositional growth. It is therefore questionable whether the global setup is able to provide the forcing necessary to form heavily rimed particles at all. While this is subject to further investigation, the simulation shown here suggests that for stratiform clouds, the computational cost to solve prognostic equations to predict the rime fraction and rime density could be saved.

The single-category scheme is able to represent a wide range of particle properties representing small in-cloud ice crystals, larger dendrites and fast-falling structures like graupel and hail.

It is important to remember that the particle properties are parameterized by the particle size distribution, mass-to-size and mass-to-projected area relationships. Therefore, a predefined relationship between the four prognostic ice parameters and the particle properties exists, which is laid out in Sect. . We would like to stress the fact that it is the four predicted ice parameters (ice mass mixing ratio qi, rimed ice mass mixing ratio qrim, total ice number mixing ratio Ni and rimed ice volume Brim) that are prognostic and not the particle properties themselves.

This section provides a closer look at this peculiarity of a single-category scheme. We use a similar seeder–feeder simulation as in the last section but with a mixed-phase cloud instead of a pure liquid cloud as the feeder cloud. We will focus on three particular areas in the simulation, shown in Fig. a: (1) the mixed-phase cloud just before the cirrus particles impact, (2) the cirrus particles just above the mixed-phase cloud and (3) the resulting particles after impact.

From the corresponding particle size distributions in Fig. b we can see how a single category handles the addition of two ice masses with differing particle properties. The number concentration in the tail of the very large particles from the cirrus cloud (purple line) is lost by averaging even though it contributes almost 10 % to the total mass mixing ratio. This is because the total number concentration is heavily dominated by the mixed-phase cloud (orange line) with the cirrus cloud only making up a small fraction of about 1 % thereof. Since the microphysical process rates are calculated from the resulting particle size distribution (green line), particle collisions and the sedimentation sink are likely underestimated.

A solution to this issue has been proposed by by the use of multiple free categories. Ice of different origin will then be sorted according to its diagnosed properties and stored in separate variables, thus giving the properties themselves a prognostic flavor. However, adding prognostic variables for multiple free categories is computationally expensive. We argue that in the context of climate projections where we are interested in global and regional mean states of the atmosphere, the computational cost of this additional procedure likely outweighs the benefit from an improved representation of ice particle properties.

While a high-resolution model is able to produce different particle properties within a single cloud, the global model often represents a cloud with only a few grid boxes. Therefore, the regions where the use of multiple free categories could improve the model results are those where the seeder–feeder mechanism or convective anvils contribute significantly to the ice water path. How important these situations are in the global context in ECHAM6-HAM2 will have to be investigated in future studies. Then the benefit of multiple free categories can be revisited.