Vertical redistribution of moisture and aerosol in orographic mixed-phase clouds

. Orographic wave clouds offer a natural laboratory to investigate cloud microphysical processes and their representation in atmospheric models. Wave clouds impact the larger-scale ﬂow by the vertical redistribution of moisture and aerosol. Here we use detailed cloud microphysical observations from the ICE-L campaign to evaluate the recently developed Cloud Aerosol Interacting Microphysics (CASIM) module in the Met Ofﬁce Uniﬁed Model (UM) with a particular focus on different parameterisations for heterogeneous freezing. Modelled and observed thermodynamic and microphysical properties agree very 5 well (deviation of air temperature < 1 K , speciﬁc humidity < 0 . 2 gkg − 1 , vertical velocity < 1 ms − 1 , cloud droplet number concentration < 40cm − 3 ), with the exception of an overestimated total condensate content and a too long sedimentation tail. The accurate reproduction of the environmental thermodynamic and dynamical wave structure enables the model to reproduce the right cloud in the right place and at the right time. All heterogeneous freezing parameterisations except Atkinson et al. (2013) perform reasonably well, with the best agreement in terms of the temperature dependency of ice crystal number con- 10 centrations for the parameterisations of DeMott et al. (2010) and Tobo et al. (2013). The novel capabilities of CASIM allowed testing of the impact of assuming different soluble fractions on dust particles on immersion freezing, but this is found to only have a minor impact on hydrometeor mass and number concentrations. The simulations were further used to quantify the modiﬁcation of moisture and aerosol proﬁles by the wave cloud. The changes in both variables are on order of 15 % of their upstream values, but the modiﬁcations have very different vertical structures for the two variables. Using a large number of idealised simulations we investigate how the induced changes depend on the wave period ( 100 − 1800 s ), cloud-top temperature ( − 15 to − 50 ◦ C ) and cloud thickness ( 1 − 5 km ) and propose a conceptual model to describe these dependencies.

trast to thicker orographic clouds, the collision-coalescence process is less important and the interactions between air parcels travelling through the clouds at different altitudes is minimal. Also, their smaller horizontal and vertical extent implies that representative observations are obtained more easily. One particular question, for which observations in isolated mid-tropospheric mixed-phase wave clouds has been instrumental, is the glaciation of clouds. The formation of ice in all mixed-phase clouds, not only orographic wave clouds, plays a crucial role for the efficiency of precipitation formation (as already pointed out in early 60 studies by Bergeron (1935) and Findeisen (2015)) and the cloud optical properties (e.g. Joos et al., 2014;Vergara-Temprado et al., 2018).
In the atmosphere ice forms either via homogeneous freezing of solution droplets at temperature colder than about −35 • C or at warmer temperatures through the mediation of certain aerosol particles, which are called ice nucleating particles (INP).
Aircraft observations in orographic wave clouds have demonstrated the large increase in ice crystal number concentration due to the onset of homogeneous freezing at cold cloud top temperatures: For example, Heymsfield and Miloshevich (1993) showed that ice crystal concentrations of ∼ 60 cm −3 observed at temperatures colder than −35 • C in wave clouds over the central United States are consistent with box-model predictions assuming homogeneous freezing. Ice crystal concentrations at warmer temperatures were below the detection limit. Similarly, for wave clouds over Scandinavia Field et al. (2001) found homogeneous freezing to be dominant at temperatures colder than −35 • C, while ice at warmer temperatures was most likely 70 formed via immersion or contact nucleation, i.e. freezing mechanisms requiring INPs. Ice crystal number concentrations at these warmer temperatures has been observed to correlated with the presence of large aerosol particles (Baker and Lawson, 2006;Eidhammer et al., 2010) and chemical analysis of ice crystal residual found predominantly mineral dust with some contributions from organics and salts, which are known to be efficient INPs (e.g. Targino et al., 2006;Pratt et al., 2010). Depending on whether INPs are incorporated before or during the freezing event, different heterogeneous freezing mechanisms 75 are distinguished. In mixed-phase orographic clouds immersion freezing, i.e. INPs acting first as cloud condensation nuclei and later initiating the freezing of the cloud droplets, is likely the dominant freezing mechanism according to model-based (Hande and Hoose, 2017) analysis and comparison between parcel model simulations and observations (Field et al., 2001;Eidhammer et al., 2010). However, Cotton and Field (2002) could not completely reconcile box-model simulations using known freezing mechanisms with observations of hydrometeor number concentrations and mass mixing ratios. 80 The representation of heterogeneous freezing in numerical models relies on empirical relationships involving aerosol number concentrations and temperatures, because the fundamental processes of the ice nucleation process and those determining the efficiency of specific aerosol particles to act as INP are not yet understood. Several empirical formulation of heterogeneous (immersion) freezing have been proposed: Early parameterisations such as Meyers et al. (1992) are solely based on ambient air temperature, while later parameterisations additionally take into account the number concentration of large (> 0.5 µm) 85 aerosol particles (e.g. DeMott et al., 2010;Tobo et al., 2013;DeMott et al., 2015). The main difference between the latter parameterisations is the geographic regions, in which the underlaying observations were made, and hence they likely represent different chemical and/or mineralogical compositions of the INP population. Other recent parameterisations use estimates of the temperature-dependent number of active sites on specific materials and the surface area of the aerosol population to predict the number of INPs at a given temperature (Niemand et al., 2012;Atkinson et al., 2013). Again the main difference between 90 the parameterisations is the materials, for which the number of actives sites was determined. It is not clear how the different parameterisations affect cloud properties and whether the difference between the parameterisations can be directly assessed with observations of ice crystal number concentrations.
Previous work has demonstrated the usefulness of observations in orographic clouds to investigate cloud microphysical processes. However, detailed cloud-microphysical analysis in models was limited to box-, column or idealised two-dimensional -Is the numerical weather prediction model able to capture the thermodynamic conditions and wave cloud dynamics with 100 sufficient accuracy, i.e. i.e. the right cloud in the right place at the right time, to allow for a direct comparison of cloud microphysical properties between model and observations?
-Can observations of the vertical variation in ice crystal number concentration be used to assess the validity of different heterogeneous freezing parameterisations?
-How large is the modification of the water vapour and aerosol profiles by the wave cloud? How does the downward 105 transport of water vapour and aerosols depend on the upstream thermodynamic conditions? And under which conditions is the downward flux largest, i.e. can be best observed in future campaigns?
The analysis focusses on a wave cloud over the Central United States probed with the National Science Foundation (NSF) C-130 aircraft (17 th November 2007, RF03) during the ICE-L campaign Heymsfield et al. (2011);Pratt et al. (2010). Data from ICE-L has been used to investigate the relationship between upstream INP measurements and ice crystal number concen-110 trations (Eidhammer et al., 2010;Field et al., 2012), the depositional growth of ice crystals (Heymsfield et al., 2011) and to investigate the impact of using adaptive ice crystal habits in idealised model simulations (Dearden et al., 2012). The chemical analysis of cloud droplet and ice crystal residuals by Pratt et al. (2010) indicated that INPs active in the observed wave clouds are most likely mineral dust internally mixed with a significant salt component, as may be expected from aerosols emitted from playas in the Central United States.

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Details on the observations, models and their set-up are provided in the following section. In section 3 we present the comparison of observed wave cloud properties to the results from high-resolution simulations with the Met Office UM with a specific focus on the vertical gradient in ice crystal number concentration (sec. 3.3. A Lagrangian analysis of the simulations provides insight into the he modification of humidity and aerosol profiles by the wave cloud (section 4.1). The dependence of amplitude and shape of this modification on the gravity wave length and upstream thermodynamic conditions determining cloud top tem-120 perature and cloud thickness is assessed with additional idealised simulations in section 4.2. Finally, section 5 summarises the results and discusses implications for future aircraft observations in orographic wave clouds to constrain mixed-phase cloud microphysics.

Observational data from ICE-L
Detailed in-situ cloud microphysical observations in orographic wave clouds are available from the Ice in Clouds Experiment (ICE-L) conducted over the central US in November 2007(e.g. Eidhammer et al., 2010Heymsfield et al., 2011;Field et al., 2012). Various instruments onboard of the National Science Foundation (NSF) C-130 aircraft provide information of aerosol, cloud and ice populations in the observed mixed-phase clouds. Details on the instrumentation can be found in Heymsfield liquid water content, the 2D-C probe for ice water content and number concentrations, the CDP (cloud droplet probe) for cloud number concentrations, the tuneable diode laser hygrometer (TDL) for humidity measurements and aerosol size distributions 135 from the Ultra High Sensitivity Aerosol Spectrometer for size-resolved number concentrations (UHSAS). As in Field et al. (2012) we restrict 2D-C data to particles larger than 125 µm and correct the TDL humidity such that it is consistent with water saturation in the regions with a liquid water content (from the King liquid water probe) larger than 0.02 g m −3 . Further details on the data and its post-processing can be found in Field et al. (2012).

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We use the Unified Model (UM), the numerical weather prediction model developed by the MetOffice and used for operational forecasting in the UK, to conduct simulations of the wave cloud observed during research flight 3 of the ICE-L campaign (16 th November 2007). A global simulation (UM vn10.8, GA6 configuration, N512 resolution, Walters et al. (2017)) starting from the operational analysis at 12 UTC on 16 th November 2007 provides the initial and lateral boundary conditions for regional model simulations. Two regional nests are used, the first with a horizontal grid spacing of 1 km and the second with a grid 145 spacing of 250 m. Both nests are centred at the location of the observed wave cloud (42.116 • N, −105.1 • E). The analysis presented in this paper focusses on the innermost nest. In the vertical we use a stretched vertical coordinate system with 140 levels, which provides a vertical resolution of 130 − 200 m at the altitude of the observed cloud. Mass conservation is enforced in the regional simulations (Aranami et al., 2014(Aranami et al., , 2015 and sub-grid scale turbulent processes are represented with a 3D Smagorinsky-type turbulence scheme (Halliwell, 2015;Stratton et al., 2015). The cloud microphysics are represented 150 with the Cloud-AeroSol Interacting Microphysics (CASIM) module (see section 2.4). As we are particularly interested in the impact of ice nucleating particles (INPs) in the cloud we conduct sensitivity experiments with different heterogeneous ice nucleation parameterisations as well as different assumptions regarding the incorporation of ice nucleating particles (INP) into cloud droplets, which is pre-requisite for immersion freezing. The details of these sensitivity experiments are described in section 2.4. 155 5 https://doi.org/10.5194/acp-2019-940 Preprint. Discussion started: 20 January 2020 c Author(s) 2020. CC BY 4.0 License.

The KiD Model
For the analysis of a large set of wave clouds we conduct additional idealised simulations with the Kinematic Driver Model (KiD, Shipway and Hill (2012); Hill et al. (2015)). The KiD model uses prescribed dynamics to drive different microphysics modules and hence testing of different cloud microphysics and flow configurations in a relatively simple framework. Here, we conduct two-dimensional simulations of wave clouds with different horizontal wavelength (period T between 100 s and The upstream temperature profiles is given by a lapse rate of −8.104 · 10 −3 K m −1 and a surface temperature of 32.1 K. The initial pressure profile is computed using the hydrostatic approximation with a pressure of 886.2 hPa at 1000 m altitude (lowermost level). An initial profile of relative humidity is used with a relative humidity of 45 % below the moist layer, 70 % in 170 the moist layer and a linearly decreasing relative humidity above the moist layer with smooth transitions between the different layers: The initial profiles are based on the ICE-L case. However, we omit the vertical tilt of the orographic wave as well as the vertical gradient in maximum vertical velocity. Example cross-sections from the KiD simulations are shown in Fig. 9.

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Cloud microphysics are described by the CASIM module (section 2.4) as in the UM simulations. As in the UM simulations, the sensitivity to the heterogeneous freezing parameterisations as well as assumptions for the CCN activation of INP is tested as detailed in section 2.4. Together with the different settings for dynamic and thermodynamic conditions, we have a total of 45360 two-dimensional, idealised simulations.

The CASIM module 180
The Cloud-AeroSol Interacting Microphysics (CASIM) module is a recently developed double-moment cloud microphysics scheme for the UM (Shipway and Hill, 2012;Hill et al., 2015;Stevens et al., 2018;Miltenberger et al., 2018). Hydrometeors are represented by five different species, the size distribution of which is assumed to be a generalised gamma distribution with a fixed width. Hydrometeor mass and number of each hydrometeor species are computed prognostically. CASIM also includes prognostic mass and number of three soluble and one insoluble aerosol modes, for which log-normal distribution with a fixed 185 width are assumed. Additional tracers for aerosols incorporated into hydrometeors are available, which are transported in accordance with the hydrometeors, i.e. including sedimentation. The in-cloud aerosol tracers allow for an explicit representation of immersion freezing and to investigate the vertical transport of aerosol by hydrometeor sedimentation.
Key microphysical processes to be investigated in the mixed-phase clouds are activation of aerosols to cloud droplets, heterogeneous freezing, growth (sublimation) of ice crystals by vapour deposition, aggregation of ice crystals, and sedimentation of

Trajectory analysis
Kinematic air mass trajectories are computed to detect changes in specific humidity and aerosol number density due to sedimenting hydrometeors in the wave cloud. Trajectories are calculated with the Lagrangian Analysis Tool (Sprenger and Wernli,205 2015), which has been adapted to UM output, from the resolved wind-field at 5 min temporal resolution. For the KiD model, trajectories are calculated analytically based on the prescribed wind field (eq. 1).

Comparison of modelled cloud properties to observational data
On the 16 th November 2007 a wave cloud forming in the lee of the Medicine Bow National Forest was observed with three subsequent aircraft passes through the cloud at different altitudes. The average temperature of the three flight legs is −25 • C 210 (leg 1, z ≈ 6.9 km, ∼ 2040 UTC), −27.5 • C (leg 2, z ≈ 7.2 km, ∼ 2100 UTC) and −31 • C (leg 3, z ≈ 7.7 km, ∼ 2120 UTC).
The cloud had an along-flow extension of about 40 km and a vertical extension of at least 1 km. In the UM simulations a wave cloud of similar extent appears at the same location and roughly the same time (± 20 min). A horizontal cross-section of the modelled cloud at ∼ 7.2 km, i.e. the mean altitude of flight leg 2, is shown in Fig. 1 a together with the flight tracks. The modelled vertical cloud structure at 42.05 • N is shown in Fig. 1 b together with a projection of the aircraft legs on the plane of the cross-section. These plots already indicate that modelled cloud location and extent agree well with the observed cloud. In the remainder of this section we compare the observed and modelled cloud microphysical structure in more detail.

Thermodynamic conditions
The geometry of wave clouds is strongly controlled by the upstream humidity and temperature profile as well as the vertical velocity field. The upstream specific humidity is compared in Fig. 2 b. In general the model is somewhat more humid than observed at the 225 time and location of flight leg 3 with a deviation of about 0.2 g kg −1 , but agrees very well with the observed specific humidity at the other flight legs. The model suggests a quite large variability of the upstream specific humidity (roughly by a factor of 2) in the time window of the observations with a gradual moistening before 2100 UTC and a subsequent drying.
In Fig Using these hypothetical flight paths instead of the actual aircraft track eliminates the impact of slightly different horizontal wind direction in model simulations and the observed flow. The mean flow is from west to east, i.e. from left to right in these plots, and the cloud forms at the first peak in vertical velocity. The amplitude of the wave in terms of the vertical velocity is well 235 captured in the model at all three altitudes with maximum deviations of less than 1 m s −1 . The width of the positive vertical velocity peak is slightly larger in the model than in the observations and the peak occurs slightly further east. The secondary peaks in vertical velocity downstream of the main wave are less well captured, particularly at flight leg 3 (Fig. 3 a).
Overall, modelled flow pattern and thermodynamic conditions agree very well with observations (deviation of air temperature < 1 K, specific humidity < 0.2 g kg −1 , vertical velocity < 1 m s −1 . Hence the simulation can be used for an in-depth com- region of the flight legs is characterised by a relatively constant air temperature (variations < 0.5 K) and specific humidity (variations < 0.1 g kg −1 ) (Fig. 4 a-c, SI Fig. 1) in both the model and the observational data. The constant specific humidity reflects water saturated conditions given the observed constant in-cloud temperature. Consistent with the similar temperature in 250 model and observation, the in-cloud specific humidity is very similar in both data-sets. This is partly by design as the measured specific humidity was corrected such that the relative humidity is on average 100 % in regions with liquid water content larger than 0.02 g kg −1 (Heymsfield et al., 2011).
The deviations in the spatial distribution and amount of total condensate content between model and observations are larger than in all other variables considered so far (Fig. 4 d-f). In the upstream, updraft dominated cloud section, i.e. west of ∼ −105.1 • E, 255 the total condensate amount is clearly larger than in the observations. In most model runs as well as in the observational data there is little ice in this part of the cloud (Fig. 5) and hence the total condensate is controlled by the upstream humidity and the total lifting up to the considered point. Given the small deviations in upstream humidity between model and observations, the higher modelled total condensate values are likely due to the somewhat larger vertical velocities, which together with the similar horizontal wavelength result in larger vertical displacements of air parcels than in the observations (Fig. 3). Simula-260 tions using the A13 parameterisation have an even higher total condensate amount. In these simulations glaciation occurs very early (Fig. 5), hence the saturation pressure over ice is relevant for the equilibrium condensate amount and not the saturation   Fig. 4 a, b).
The CASIM microphysics explicitly considers the vertical transport of dust particles by hydrometeor sedimentation and there-355 fore allows us to quantify the downward transport of aerosol by the wave cloud. The Lagrangian change of aerosol content is shown in Fig. 8. The vertical structure is different to the moisture flux, with aerosol depletion only occurring at the very top of the cloud (above ∼ 9.7 km) and increases in aerosol number concentrations mainly towards cloud base. The profiles of the vertical dust transport are more sensitive to changes in the heterogeneous freezing parameterisation than the moisture flux, with larger changes also in the shape of the profiles. However, the differences are again smaller than the temporal variability.

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The treatment of the CCN activation of dust (using all dust for heterogeneous freezing or presenting activation assuming some soluble fraction on dust particles) has a much larger impact on the vertical aerosol transport than on the moisture flux. The resulting differences in the profile are on the same order of magnitude as the temporal variability (SI Fig. 4 c, d).
It would be interesting to constrain these downward fluxes with observational data, in particular given the uncertainties surrounding diameter-fallspeed relations often used in bulk models. The peak downward flux of about 0.1 g kg −1 is, however, 365 smaller than the temporal variation of the specific humidity (Fig. 2) during the average time a parcel needs to transit through the wave cloud (i.e. ∼ 30 min). As the aircraft data does not provide information on the temporal evolution of upstream humidity, it is not possible to use the aircraft data to constrain the vertical moisture flux. In addition, for such an assessment the construction of air parcel trajectories from the observed velocity field would be required. While this is in principle possible (e.g. Field et al., 2012), for the assessment of sedimentation fluxes the error in the upstream positions of air parcels would 370 need to be smaller than 500 m owing to the vertical gradient of upstream specific humidity. This is not feasible given the sparse observations of velocity (only sampled along flight legs) and the uncertainty in measured vertical velocity. However, detailed observations of the 3D velocity field for example with an on-board Lidar system and a better characterisation of the upstream and downstream humidity profiles, e.g. sampling in a quasi-Lagrangian manner, there is a potential for future field campaigns to constrain vertical sedimentation fluxes from wave clouds.

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Because wave clouds offer such an opportunity to detect sedimentation mediated fluxes of moisture and aerosol we assess in the following section how the amplitude of the sedimentation fluxes depend on the upstream thermodynamic conditions, which determine the cloud thickness and cloud top temperature, and on the horizontal wavelength of the gravity wave, that controls the horizontal extent of the cloud.

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The modification of moisture and aerosol profiles by hydrometeor sedimentation is investigated for the ICE-L case study in the previous section. However, the cloud-integrated sedimentation fluxes will vary for different wavelength, cloud top temperatures and cloud thicknesses and so will their impact on the vertical profiles of aerosol and moisture. To assess these dependencies, we use two-dimensional, idealised simulations with the KiD-model (section 2.3). Using an idealised model for this assessment allows us to vary the wavelength of the gravity wave, which would require changing the topography in the Unified Model.
In addition, we can carry out a large number of simulations sampling a large proportion of the relevant phase-space, which would not be possible with the UM due to the much larger computational costs. But we are able to link back to the case study by including the observed case in the phase space explored. Two exemplary realisations of wave cloud in the KiD-model are shown in Fig. 9 along with the profiles of Lagrangian changes in moisture (∆ Lagr q t ) and aerosol ∆ Lagr m du . As in the UM simulations, the profiles of moisture and aerosol changes have distinctly different shapes: While aerosol changes are concen-  Fig. 10 b shows the maximum difference between any two simulations with the same wave period, cloud top temperature and cloud thickness, but different heterogeneous freezing parameterisations. For the UM simulation the variability is about a factor 5 larger (colour-filled circle in Fig. 10 b), which is mainly due to the low values for the simulation with DM10 and = 0.01. The impact of the parameterisation choice is largest for cloud top temperatures just below the onset of homogeneous freezing (see e.g. Fig. 6). In this part of the parameter space ∆ Lagr q t varies by up to a factor 10 between A conceptual model of the sedimentation fluxes provides insight into the key variables controlling the modification of the moisture profile and may be used to represent these in models with a lower spatial resolution. Similar to previously proposed conceptional models for orographic precipitation Smith (1979); Smith and Barstad (2004); Seifert and Zängl (2010); Miltenberger et al. (2015), we chose an ansatz based on the consideration of the characteristic timescales of the cloud: The first term on the left side of the equation describes how much water is transferred from the gas-phase to frozen condensate due to depositional growth and freezing, while the second term describes the sedimentation of the condensate. Note that we ignore here the sedimentation of liquid cloud particles, which only has a minor impact compared to the impact of ice crystal sedimentation in the KiD simulations (not shown). The key variables are (i) the potential condensate G pot , which is the maxi-430 mum cloud condensate possible given thermodynamic constraints, initial humidity and vertical displacement, (ii) the in-cloud residence time τ ic , i.e. the time available for cloud microphysical processes, (iii) the timescale for depositional growth of ice hydrometeors τ dep and (iv) the timescale for sedimentation τ sedi . Note that in contrast to parcel-oriented formulations these timescales refer to the entire cloud and not to individual air parcels. Finally, G nuc denotes the condensate formed during ice crystal nucleation via homogeneous or heterogeneous freezing. A similar approach has been suggested by Seifert and Zängl 435 (2010) and Miltenberger et al. (2015) for describing the precipitation formation in warm-phase orographic clouds. As we show in the following, all parameters in equation 3 can be estimated from the upstream thermodynamic profiles and expected vertical displacement.
The potential condensate is the maximum condensate amount that would occur along a wave cloud trajectory if the air parcel's ice water content were in thermodynamic equilibrium. In warm-phase clouds the condensate amount in absence of sedimen-440 tation is often close to the potential condensate as a result of fairly small vapour deposition timescales (∼ 1 s), as e.g. used in saturation adjustment parameterisations. However, in mixed-and ice-phase clouds the potential condensate is typically not realised due to the longer timescales for depositional growth (in the order of 1000 s). G pot is not used as a measure of the condensate formed in the cloud, but as a "virtual" reservoir species from which condensate can be formed. Along air parcel trajectories G pot can be directly computed as the difference between the upstream specific humidity and the saturation pressure 445 over ice at the coldest point along the trajectory, if latent heating from phase-changes of water are neglected. Using trajectory data from the KiD experiments, the variation of G pot,Lagr with the wave period and cloud top temperature can be quantified ( Fig. 11 a). Further G pot can be computed from the wave amplitude A and the upstream temperature t 0 , specific humidity q v,0 and pressure p 0 profiles by assuming dry-adiabatic ascent of the parcel (lapse rate γ) and a hydrostatic balanced atmosphere: Another important cloud microphysical variable, that will be required for parameterising the characteristic timescales is the number of ice crystals in each cloud. To characterise the variability across the different clouds, we use only the maximum possible number of ice crystals n i, max formed either by homogeneous or heterogeneous freezing. In the Lagrangian data, this 455 is the integral of homogeneous and heterogeneous nucleation rates along the trajectory passing just below cloud top. Fig. 11 b shows that n i, max,Lagr depends strongly on cloud top temperature with a major increase around t ct ≈ −38 • C reflecting the transition to clouds dominated by homogeneous freezing. For clouds with colder cloud tops there is also a clear dependence on the time period of the wave clouds reflecting the interaction between the nucleation and growth of newly formed ice crystals (e.g. Kärcher et al., 2006). For the conceptual model, we find that using the heterogeneous parameterisation used in the 460 KiD model together with the minimum temperature expected from the maximum vertical displacement gives a reasonable estimate for temperatures warmer than −38 • C. For colder cloud-top temperatures, we use the homogeneous nucleation rate from the DM10 parameterisation (consistent with CASIM microphysics) and a correction factor depending on the wave period: 0.932 · log 1 0(T) + 0.228 for t ct < −42.5 • C and 1.48 · log 1 0(T) − 1.48 for t ct > −42.5 • C. Closely related to the ice crystal number is also the term G nuc describing the ice crystal mass formed by homogeneous or heterogeneous freezing.

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G nuc can be estimated from n i,max and a typical particle massq i , which can be directly obtain from the KiD simulations: q i = 10 −11.5 kg kg −1 (10 −9.6 kg kg −1 ) for clouds dominated by homogeneous (heterogeneous) freezing.
The in-cloud residence time τ ic describes the time available for condensate and precipitation formation (e.g. for warm clouds Miltenberger et al., 2015). Here, we define τ ic as time during which air parcels are super-saturated with respect to ice. This timescale τ ic,Lagr can be directly quantified from the KiD-model air mass trajectories (Fig. 12 a) or analytically calculated from 470 the prescribed wave flow and the upstream humidity profile: with η i,sat the vertical displacement required to reach ice saturation). The deviations between this estimate and the Lagrangian metric are less than 5 % (SI Fig. 5 b). From the resulting vertical profile of τ ic the largest timescale is selected (only considering cloudy parcels).

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The depositional timescale τ dep describes the characteristic timescale for the reduction of ice supersaturation for w = 0 m s and an ice crystal population characterised by the number concentration n i and mean ice particle diameter d i . The concept of describing depositional growth of ice crystals with a characteristic timescale τ dep is frequently used in literature and cloud microphysical parameterisations (e.g. Khvorostyanov, 1995): , c i the capacitance of the ice crystals, and f a ventilation factor. This concept needs to be extended to a single character-480 istic timescale for the entire cloud. To estimate this timescale we again utilise the KiD simulations. The cloud-scale deposition timescale can be estimated from the integrated deposition D and freezing rates G nuc as well as τ ic according to: τ dep,Lagr = τ ic,Lagr log(1 − D(G pot − G nuc ) −1 −1 . The resulting estimates are shown in Fig. 12 b. Immediately obvious is an inverse relation to the ice crystal number concentration, as expected from air parcel considerations, but this is not the sole determinant. In order to estimate τ dep from the a-priori known parameters, i.e. upstream profiles and vertical displacement, we determined the following least-square fits to the KiD model data (SI Fig. 5 c): if t ct ≥ −34.25 K 1.71 · 10 7 · n −0.764 The sub-division is necessary due to the fundamentally different behaviour in the parts of the parameter space dominated by homogeneous and heterogeneous freezing, respectively.
Finally, the sedimentation timescale needs to be determined, for which we use the same approach as for the deposition