A 1D Model for Nucleation of Ice From Aerosol Particles: An Application to a Mixed‐Phase Arctic Stratus Cloud Layer

Mixed‐phase clouds (MPCs) have been identified as significant contributors to uncertainties in climate projections, attributable to model representation of processes controlling the formation and loss of supercooled water droplets and ice particles from the atmosphere. Arctic MPCs are commonly widespread and long‐lived, with sustained ice crystal formation processes that challenge current understanding. This study examines the ice‐nucleating particle (INP) reservoir dynamics governing immersion‐mode heterogeneous freezing in an observed case of Arctic MPCs using a simplified 1D aerosol‐cloud model. The model setup includes prescribed dynamical forcings and thermodynamic profiles, and represents INPs as multicomponent and polydisperse particle size distributions. Diagnostic and prognostic approaches to immersion freezing parameterization are compared, including time‐independent (singular) number‐ and surface area‐based descriptions and a time‐dependent description following classical nucleation theory (CNT). The choice of freezing parameterization defines the size of the INP reservoir. The CNT‐based description yields an orders of magnitude larger INP reservoir than the singular parameterizations, which is the dominant factor for sustained ice crystal formation. The efficiency of the freezing process and cloud cooling are of secondary importance. A diagnostic treatment neglecting INP loss is only accurate when the INP reservoir size is large and INP depletion weak. Since a larger INP reservoir sustains ice crystal formation substantially longer, and ice water path scales with ice crystal concentrations for the conditions considered, resolving the source of differences in INP reservoir dynamics due to model implementation is a high priority for advancing climate model physics.

Supporting Information may be found in the online version of this article.Fridlind et al. (2012) conducted a large-eddy simulation (LES) case study of SHEBA observations evaluating the impact of diagnostic and prognostic treatments of INP budgets on long-lived MPC evolution.Table S1 in Supporting Information S1 gives representative SHEBA cloud conditions in a shallow cloud-topped PBL that is well-mixed from surface to cloud top (Fridlind & Ackerman, 2018;Fridlind et al., 2012).A unique feature of this SHEBA case study is that the PBL is supersaturated with respect to ice, S i , from cloud top down to the surface, such that any ice formed is growing at all elevations and the sublimation process can be neglected.In other words, once an ice crystal is formed from an INP, that INP ultimately will be removed by sedimentation to the surface with the ice crystal that it forms.The small ice crystal number concentrations, N i , and the absence of large drops or rimed ice also implies that secondary ice production is not active.Immersion, deposition and contact freezing were implemented with an instantaneous freezing treatment that was derived to be consistent with Continuous-Flow Diffusion Chamber (CFDC) measurements (i.e., not based on aerosol composition).The scheme predicts INP number concentrations available in all modes, N INP , as a function of environmental conditions and rates (e.g., presence in a drop or frequency of contact with a drop via combined phoretic forces), and a diagnostic treatment of INPs was also contrasted with a prognostic treatment (Appendix A).The major findings of the LES study were that a diagnostic INP treatment led to rapid cloud desiccation, in contrast to the observed long cloud lifetime.The prognostic treatment in the dominant immersion mode, on the other hand, resulted in rapid removal of INPs from the boundary layer with cloud-top entrainment being the only source sustaining N INP .Unrealistically high numbers of INPs being entrained from the air overlying the boundary layer were also needed to sustain the observed N i .Similar findings related to rapid INP depletion have been reported in other studies examining MPC evolution (Fan et al., 2009;Fridlind et al., 2007;Harrington & Olsson, 2001 Westbrook & Illingworth, 2013).This conundrum-sustained ice formation despite the expected rapid INP depletion-has led to the speculation that the presence of a slowly proceeding time-dependent immersion freezing process might reconcile the prognostic INP treatment with observations (Savre & Ekman, 2015;Westbrook & Illingworth, 2013;Yang et al., 2013).
Observed Arctic MPC events during ISDAC have been evaluated for continuous ice crystal formation using LES (Avramov et al., 2011;Fridlind & Ackerman, 2018;Savre & Ekman, 2015;Solomon et al., 2015;Yang et al., 2013).For example, Solomon et al. (2015), employing an instantaneous freezing mechanism (DeMott et al., 2010), found INP recycling to be important to sustaining simulated ice formation owing to sedimentation followed by sublimation of ice crystals that were subsequently mixed back into the cloud layer (not a feature of the SHEBA case study).Savre and Ekman (2015) applied a freezing probability that scales with a contact angle probability distribution function ("α-PDF") to emulate the varying freezing efficiencies among the same INP types (smaller contact angles represent more efficient INPs) (Hoose & Möhler, 2012;Lüönd et al., 2010;Niedermeier et al., 2011).In addition, this approach allows for each INP a time dependence of freezing.One can think of this as a hybrid model, where first a few particles out of the entire same particle population are selected to be activatable as INPs (i.e., not all particles can serve as INPs and potentially be activated) and then allow for a time dependence of freezing for each selected INP.This hybrid model approach has been applied previously to typically diverse particle-type populations (Vali & Snider, 2015;Wright et al., 2013;Wright & Petters, 2013), however, the hybrid model is in contrast to the classical nucleation theory (CNT) description where freezing is governed by a stochastic selection of INPs and associated time needed for activation (Alpert & Knopf, 2016;Knopf & Alpert, 2013;Knopf et al., 2002Knopf et al., , 2020;;Pruppacher & Klett, 1997;Zobrist et al., 2007).In the α-PDF scheme, as the most efficient INPs induced freezing, they are truncated from the α-PDF, leaving less efficient INPs available later in the simulations (Savre & Ekman, 2015).Results indicated that an evolving α-PDF supports longer-lasting in-cloud ice formation and that ice production is modulated by a combination of radiative cooling, INP entrainment, and INP scavenging in the three cases examined.
The two goals of this study are to present a new open-source column model for aerosol-modulated ice formation and to demonstrate its application to a simple, well-observed case study.We demonstrate how differing assumptions underlying commonly applied immersion freezing parameterizations (Alpert & Knopf, 2016;DeMott et al., 2015;Knopf et al., 2021;Niemand et al., 2012) impact the cloud-topped PBL's INP reservoir and associated ice crystal budgets in this case study of long-lived Arctic MPC and provide an open source modeling framework for future investigations.Hence, in this 1D model framework, the focus is to showcase the significant differences in N INP and N i that may result solely from differences in applied immersion freezing parameterizations when coupling and tracking of aerosol particles, INPs, and ice crystals is considered.A bottom-up approach is chosen where the aerosol types and associated particle size distribution (PSD) inform the availability of INPs in the model domain.The constraints in model complexity will allow us to examine unambiguously the leading effect of immersion freezing descriptions on N INP and N i using diagnostic and prognostic treatments.Furthering the 1D model approach to study the first-order impacts of differing freezing parameterizations on ice nucleation aims ultimately to guide implementation of greater complexity in larger scale models (GCMs) such as general circulation models that run simulations on the order of years and longer.Whereas here we limit aerosol and INP composition to dust and consider only three ice nucleation parameterizations in a single well-observed case study, future work will broaden applications of the 1D model to a wider parameter space of aerosol and meteorological conditions, and other freezing parameterizations.This initial study's focus is to examine the response in N INP and N i upon application of different freezing parameterizations using diagnostic and prognostic INP implementations while not claiming to explain N INP and N i observations from ambient measurements.The advantage of applying such a simplistic model is that it allows us to identify the effect of the different freezing parameterizations without introducing additional confounding process parameterizations that could compound and entangle this effect.It is our hope that this study motivates subsequent model studies evaluating different freezing parameterizations under diagnostic and prognostic INP treatments for different cloud types.
commonly applied immersion freezing parameterizations have been tested against ambient aerosol particles using a closure study approach (Knopf et al., 2021), which has been advocated for resolving discrepancies in the prediction of INP number concentrations (Burrows et al., 2022).That work included one CNT-based parameterization and two so-called singular parameterizations.The singular hypothesis states that ice nucleation occurs instantaneously, that is, only upon decreasing temperatures and reaching the temperature threshold for a specific particle that acts as INP (Levine, 1950).Thus, INPs are unique among a population of the same aerosol particles and are characterized by their activation temperature upon which freezing occurs instantaneously (Vali, 1971).In our model we prescribed one type of dust as aerosol.Following the singular hypothesis, only a small fraction of these dust particles can serve as immersion INP that are "activatable" within a given cloud layer, governed ultimately by the cloud top temperature (the minimum temperature in the cloud layer); following a CNT scheme, by contrast, all of the dust particles could be activated on a random base.Thus, the activatable INP constitute the reservoir of INP that can actually be activated within a given cloud layer.In a singular scheme, for instance, the INP reservoir increases significantly if the cloud top temperature decreases; by contrast, with the water activity (a w ) based immersion freezing model (ABIFM) CNT scheme, the INP reservoir itself does not change significantly when cloud top temperature decreases (all dust remain activatable and only few activatable INP are added to the reservoir), but rather the likelihood of freezing increases.As in the closure study, here we apply two singular parameterizations of mineral dust: an ice nucleation number-based (INN) approach (DeMott et al., 2015) and the ice nucleation active sites (INAS) approach (Niemand et al., 2012).Therefore, for a given temperature the number of maximum activatable INPs is determined, independent of the number of modeling time steps.We also apply one CNT parameterization approach, where ice nucleation, by definition, occurs randomly on particles of identical composition, and longer exposure to constant S i increases the freezing probability: namely, the water-ABIFM, which represents the temperature-and humidity-dependent freezing with two composition-dependent fitted parameters as given in Equation B7 (Alpert & Knopf, 2016;Knopf & Alpert, 2013;Knopf et al., 2020).Since nucleation occurs randomly in the CNT case, all particles in a composition class are considered as activatable INPs at a temperature-dependent rate (Knopf et al., 2020).Support for this consideration comes from studies that analyzed ambient particle populations and INPs micro-spectroscopically on an individual particle level (Knopf et al., 2018;Laskin et al., 2016).Those studies find that the INPs are chemically and morphologically not significantly different than particles that do not act as INPs (Alpert et al., 2022;China et al., 2017;Hiranuma et al., 2013;Knopf et al., 2014Knopf et al., , 2018;;Lata et al., 2021), consistent with a stochastic freezing process (i.e., one particle out of many same ones induced freezing).Consequently, in the ABIFM case, for each modeling time step, the number of activatable INPs is reassessed taking the entire activatable aerosol population into account (since special INPs are not assigned).This is in contrast to a singular immersion freezing approach, where a subpopulation of aerosol particles is determined as activatable INPs with their abundance dependent on temperature.This is also in contrast with hybrid immersion freezing parameterizations which include some kind of selection or specificity of the INPs followed by a time-dependent freezing description.ABIFM also encompasses immersion freezing at subsaturated conditions when the INP is immersed in a dilute aqueous solution such as a haze droplet (Knopf & Alpert, 2013).This latter case is not considered here since our aim is to compare to the singular freezing parameterizations, which only consider freezing at saturated conditions.Assigning which of the immersion freezing parameterizations is the most suitable to predict N INP is beyond this study; rather, we highlight the significant differences in the activatable INP reservoir and its impact on sustained ice crystal production.
Application of a singular-or CNT-based freezing description will yield differences in ice crystal formation in MPCs as illustrated conceptionally in Figure 1, which represents the development of a steady-state MPC within a mineral dust-laden atmosphere.As the cloud layer forms, we assume that all the dust particles are sufficiently hygroscopic to become promptly immersed in supercooled water via droplet activation at liquid-phase cloud base.In the ABIFM case (Figure 1a), where nucleation occurs stochastically among the droplets with immersed  1a stands in contrast to a singular interpretation of immersion freezing (Figure 1b).With a singular approach, the lowest temperature in the model domain determines the maximum number of activatable INPs, that is, the relatively sparse mineral dust particles that can serve as INPs at temperatures warmer than −20°C.Hence, the reservoir of INPs that can be activated within the cloud-topped PBL is orders of magnitude smaller than the mineral dust population, that is,  aer ≫ INP , illustrated by the sparse orange dots (activatable INPs) compared to black dots (mineral dust i.e., never available as INP given the minimum temperature in the domain in Figure 1b).This inequality likely holds for other hybrid immersion freezing parameterizations.As time elapses, activatable INPs are transported into the cloud layer, supporting additional freezing and sedimentation of ice crystals.As is evident, this progression leads to rapid depletion of N INP and N i as discovered in previous prognostic modeling studies (Fridlind et al., 2012 and references therein; Savre & Ekman, 2015).We note that aircraft-or ground-based in situ measurements would reveal a similarly dust-laden PBL in both scenarios at all times, consistent with observations (McFarquhar et al., 2011).However, below we show that the size and dynamics of the activatable INP reservoir are the dominating factors controlling the longevity of MPC ice formation, whereas the rate or efficiency of immersion freezing (i.e., a faster or slower ice nucleation process) and cloud cooling are of secondary importance for sustaining ice crystal formation.

1D Aerosol-Cloud Model
We use an aerosol-informed approach within a 1D modeling framework that is based on a widely simulated case that was observed over sea ice during the SHEBA campaign (Fridlind et al., 2012;Morrison et al., 2009;Zuidema et al., 2005) (see Table S1 in Supporting Information S1), which is selected because it represents minimal complexity from a process standpoint (Fridlind & Ackerman, 2018).This 1D aerosol-cloud model is sufficient to investigate the dependence of the INP reservoir dynamics, that is, the sensitivity of N INP and N i on immersion freezing parameterizations, on several leading factors identified in previous studies (Fridlind & Ackerman, 2018;Fridlind et al., 2012;Silber et al., 2021).Retaining a vertical model dimension rather than using a mixed-layer model allows examination of freezing processes under the controlling influence of temperature and relative humidity (RH), which vary weakly horizontally within the well-mixed PBL in this case.We note that droplets and ice may coexist at all elevations above liquid saturation in the 1D turbulent boundary layer, requiring no special treatment to represent the difference in ice saturation where droplets are growing versus evaporating (Bergeron, 1935;Findeisen, 1938;Wegener, 1911).All such elevations within the well-mixed PBL are roughly 20% supersaturated with respect to ice (Fridlind et al., 2012), rendering small differences in liquid saturation above cloud base negligible for ice formation and growth.This simplistic 1D aerosol-cloud model is sufficient to investigate the dependence of the INP reservoir dynamics on several leading factors identified in previous studies: PBL mixing time scale, cloud-top entrainment rate, and ice crystal sedimentation rate (Fridlind et al., 2012).We adopt the quasi-steady state LES baseline results reported by Fridlind et al. (2012) as the basis for a cloud-topped PBL case study, which has been shown to be similar to observations (Figure S1 and Table S2 in Supporting Information S1).Adopting a fixed 1D thermodynamic profile in this study (temperature, RH, liquid water content), adopted from the quasi-steady MPC state simulated by LES (Figure S1 in Supporting Information S1), is supported by the finding that, consistent with in-situ and radar observations, PBL N i yielded weak liquid water path depletion rates, attributable to weak precipitation rates (Silber et al., 2021).Lastly, we assume that each cloud droplet contains one mineral dust particle and that the dust is sufficiently hygroscopic to instantaneously activate as cloud droplet upon entering the cloud layer.

Governing Equations
The simplistic 1D aerosol-cloud model prognoses the INP and ice particle budgets using: In the model configuration used in this study, the INP and ice particle budgets are calculated at every time step using a time splitting approach that follows the order of the budget terms as in Equations 1 and 2. The implicit solution used here for S act applying the singular-based immersion freezing parameterizations follows: where δt is the time step length and τ act is an activation time scale set by default to 10 s (see Table S2 in Supporting Information S1).That value of τ act is chosen to reflect the time resolution of the CFDC instrument to observe INPs on which the singular parameterizations are based on.S act in Equation 3 follows the implicit Euler method (Hoffman, 2001).In this form the stepwise concentration change is proportional to , which approaches the explicit solution proportionality  −  act for δt ≪ τ act , but is less than 1 for all δt and τ act , thus guaranteeing that loss never exceeds the initial value.S act is treated explicitly for the case of the CNT-based immersion freezing parameterization (see Equation B10).
To prevent other numerical issues, S ent is computed implicitly as well, for the domain top height (z m ), representing the PBL top, by: where w e is the entrainment rate and   Imm INP,FT is the free-troposphere immersion freezing INP concentration equal to the initial domain INP size distribution.
The mixing terms S mix and  i mix are calculated in the following way: where τ mix is the PBL mixing time scale,   Imm INP is the PBL mean activatable INP number concentration,  i() is the PBL mean ice crystal number concentration.
Finally, the ice sedimentation term  i sed is set by: where v f is the number-weighted ice sedimentation rate (fall speed) set here to a constant value (see Table S2 in Supporting Information S1), and i ranges from 1 to m − 1 (index of layer below PBL top) instead of from 1 to m.Note that δz is always set to values larger than v f • δt to avoid potential numerical instability (CFL < 1).Here Euler's method proves adequate numerically since the sedimentation time scale is on the order of the turbulent mixing time scale, yielding weak vertical gradients in ice crystal number concentration.
As shown above, the only prognostic equations are for the number concentrations of activatable INPs and ice crystals.This simplified framework adopts fixed microphysical parameters such as ice crystal fall speed obtained from LES results which used a bin microphysics scheme.The 1D model is initialized with LES predicted temperature and fixed liquid water content as a function of height (see Figure S1 in Supporting Information S1).

Immersion Freezing Parameterizations
For Arctic MPCs considered in this study and that exist at temperatures greater than about −20°C, deposition ice nucleation, where ice forms from the supersaturated vapor phase onto the INP, is much less efficient than immersion freezing (Kanji et al., 2017;Knopf et al., 2018).Contact ice nucleation, where a supercooled water droplet collides with an INP, can be assumed to be negligible due to the weak aerosol scavenging by water droplets (Fridlind et al., 2012).Whereas potential secondary ice production processes (Korolev & Leisner, 2020) are generally thought to be active when droplets attain large sizes (Rangno & Hobbs, 2001), they are expected to play only a minor role in typically thin Arctic MPCs (Luke et al., 2021), consistent with a statistical predominance of low liquid water path and lack of drizzle (Silber et al., 2021).
For this sensitivity exercise we employ three immersion freezing parameterizations that have also been applied in a recent aerosol-ice formation closure study (Knopf et al., 2021).The three mineral dust immersion freezing parameterizations are based on the same laboratory cloud chamber experiments, and hence, all describe the experimental observations in the cloud chamber accurately (Alpert & Knopf, 2016;DeMott et al., 2015;Niemand et al., 2012).However, they represent the most contrasting features concerning underlying assumptions of the nucleation process and associated scaling behavior with respect to the aerosol population (i.e., number concentration and size) and activation time, thus, exerting, by design, a wide range of effects on N INP and N i when using diagnostic and prognostic treatments.Hybrid parameterizations of immersion freezing that fall in between the employed immersion freezing parameterizations are not discussed and considered beyond the scope of this study.
We (  ) can be derived (Niemand et al., 2012).ABIFM computes a heterogeneous ice nucleation rate coefficient, J het (∆a w ), (in m −2 s −1 ) (Alpert & Knopf, 2016;Knopf & Alpert, 2013;Knopf et al., 2020).Since the singular-based parameterizations are formulated to be valid when the INP is immersed in pure water droplets, in our model, immersion freezing by either singular-or CNT-based parameterizations only commences in the cloud layer, that is, at saturated conditions and presence of liquid water (Figure S1 in Supporting Information S1).Detailed information on the implementation of these three immersion freezing parameterizations is given in Appendix B.

Ice-Nucleating Particle Arrays
Application of a simplistic 1D aerosol-cloud model enables computationally efficient treatment of prognosticallyevolving INP reservoirs, which necessitates storing additional variables for each INP property (type, activation temperature, and size) throughout the simulation for the singular cases.The original LES model (Fridlind et al., 2012) included an instantaneous freezing parameterization, based simply on INP measurements rather than the aerosol population composition.We have added the INN and the more complex INAS and ABIFM immersion freezing parameterizations while allowing for implementation of multicomponent and polydisperse aerosol particle size distributions (PSDs).Table 1 displays the monodisperse and polydisperse mineral dust aerosol case studies we consider supporting the concept of the INP reservoir size governing the sustained ice crystal formation rate outlined in Figure 1.For the singular immersion freezing approaches, we construct INP arrays that store the number concentrations of INPs activatable at discrete temperatures.These INP arrays are depicted in Figure 2 and maximum activatable N INP , that is, the INP reservoir, are also given in Table 1.For the ABIFM approach, no INP arrays are required since the approach does not involve specifying mineral dust particles serving as INPs.In each simulation step, all mineral dust particles at a given S i are considered when determining N INP using J het (∆a w ).Hence, no additional variables need to be stored for the prognostic treatment using ABIFM.Thus, this CNT-based parameterization provides a computationally less demanding prognostic treatment, with potential relevance for application to more complex aerosol speciation models (Bauer et al., 2008;Liu et al., 2016) and within GCMs (Hoose et al., 2010;Liu et al., 2012).
In order to compare a time-dependent scheme with a singular scheme, an activation time must be adopted.Table 1 and Figure 2 include the ABIFM derived N INP for an activation time of 10 s, representing the ice nucleation activation times in INP instrumentation (DeMott et al., 2015).If not otherwise noted, for all case studies involving ABIFM, immersion freezing is only performed when the mineral dust is immersed in a supercooled droplet similar to the singular approaches, that is, we do not consider the freezing of haze particles for this comparison.Figure 2 shows that the singular approaches predict in most cases greater N INP compared to ABIFM (assuming 10 s activation time).However, as shown in the results section, these greater N INP are only briefly sustained in comparison to the typical lifetime of an Arctic MPC.

Simulating Changes in Immersion Freezing Efficiencies
To evaluate the impact of immersion freezing efficiencies on the INP reservoir and, in turn, ice crystal formation, we increase the freezing efficiency of INN, INAS, and ABIFM by a factor of 10 and 100 for all temperatures.We use the term freezing efficiency to avoid confusion since the singular parameterizations do not provide a freezing rate.
The enhancement of immersion freezing is demonstrated in Figure S2 in Supporting Information S1 for INAS density n s application and compared to K-Feldspar serving as INP (Atkinson et al., 2013).K-Feldspar represents the most efficient mineral dust INP.At the lowest temperatures of our simulation, increasing mineral dust n s by a factor 10 matches n s for K-Feldspar, and at the highest temperatures, mineral dust n s is about four orders of magnitude greater than expected for K-Feldspar.Increasing mineral dust n s by a factor 100 correspondingly increases n s further.
Clearly, this is an extreme case since pure K-Feldspar particles are not present in the atmosphere and the modified mineral dust n s represents orders of magnitude greater ice immersion freezing efficiency than those pure K-Feldspar particles for the considered temperature range.In other words, these cases present unrealistically fast immersion freezing for mineral dust particles, therefore, presenting an extreme test case.

Results
As outlined above, accurate analysis of the INP budget requires a prognostic treatment of INPs.However, in light of the common application of diagnostic INP treatments in models (Fridlind & Ackerman, 2018), we first briefly discuss this model configuration to contrast it with the results of the prognostic treatment.

Perils of Treating INP Diagnostically
When  For the diagnostic treatment, N INP in the singular parameterizations increases by more than a factor of 5,000 after 0.5 hr and by more than a factor of 100,000 after 10 hr, clearly indicating the significant "artificial" enhancement of INPs in the reservoir.Those numbers suggest that the effective INP reservoir is orders of magnitude greater than the actual physically present reservoir.The increase of the INP reservoir for the ABIFM application is not as pronounced, reaching maximum enhancement factors of less than 30 after 10 hr.Table 2 also shows the same enhancement factors for a prognostic treatment which are negligible.Evidently, a diagnostic INP treatment is not suitable to study INP reservoir dynamics and if applied, great care should be taken when interpreting cloud microphysical processes.From this we can infer that a diagnostic treatment of INPs may be a reasonable approach when the initial INP reservoir is large and INP depletion is low.However, as we have seen in the discussion of the conceptual approach, we expect any freezing parameterization that specifies INPs to yield a small INP reservoir, thus, leading to artificial enhancements of N INP .present in the simulation for given conditions.Table 1 Continued parameterizations.Activatable N INP for the singular and ABIFM approaches differ by several orders of magnitude due to the differences in the respective INP reservoirs that are available for activation (see also Figure 1).In the singular approaches within the cloud layer, INPs are promptly depleted according to their freezing temperatures (see also Table 1 and Figure   N i also differs for the singular and ABIFM approaches.These differing results can be entirely explained by the fundamentally different assumptions in the immersion freezing parameterizations.For the singular approach, ice crystal formation promptly commences in the cloud layer followed by sedimentation of ice crystals leading to enhanced N i below cloud.This initial burst of ice crystals decreases quickly over the following 9 hr due to the depletion of INPs (conceptually outlined in Figure 1b) similar to previous findings using an instantaneous freezing description (Fridlind et al., 2012).N i decreases with height due to the interplay of rapid freezing at the lowest temperatures and ice sedimentation.After about 5 hr, N i is nearly uniform across the domain, only sustained by entrained INPs.In the ABIFM approach, however, ice crystal formation draws from a much larger INP reservoir.Consequently, as soon as ice crystals form, a relatively steady N i is established that is maintained throughout the simulation (see also Figure 1a).In the cloud layer, a consistent N i gradient with height is visible, which results from the competition between freezing and sedimenting ice crystals.For the ABIFM approach, the role of entrainment at cloud-top is not as significant since much greater INP concentrations are present throughout the cloud, that is, the INP reservoir is much greater than the rate of addition of INPs by cloud top entrainment.Lastly, Figure 5 presents the ice crystal formation rate, dN i /dt, within the cloud layer.In the singular approaches, greater heights coincide with a greater dN i /dt that declines over time due to INP depletion.Only the top layer allows for continuous ice crystal formation due to INP entrainment.In the ABIFM approach, dN i /dt also exhibits a height (and temperature) dependence.However, over the course of the simulation, negligible change in dN i /dt is observed because INP depletion is so muted by the large reservoir.Case 2 with larger dust particles is depicted in Figure S4 in Supporting Information S1 and follows the same interpretation outlined in Figure 5.In this case, results for the INN case do not differ from Case 1, since particle size is not considered in this parameterization and the number concentrations are the same.

Physical Insights From Treating INP Prognostically
Figure 6 shows the PBL-averaged concentration of activatable N INP and N i for monodisperse dust cases 1 and 2 for the different immersion freezing parameterizations.The singular approaches display an initially large rise of N INP and N i which then rapidly decline due to the activation of INPs and subsequent ice crystal sedimentation.For the prognostic treatment using ABIFM, N INP and N i also decrease, but at a much slower rate.Table 2 presents the INP reservoir change for the prognostic cases 1 and 2. After 30 min and 5 hr, the singular descriptions, respectively,     3. Application of ABIFM for cases 1 and 2 results in τ INP and τ i being 2-3 orders of magnitude greater than for the singular freezing parameterizations, corroborating the effect of the ABIFM approach on sustaining ice crystal formation for these long-lived clouds.
The temporal evolution of dN i /dt and vertical distribution of N i are shown in Figure S5 in Supporting Information S1.The initial fast ice crystal formation rate by the singular descriptions lasts for less than a minute, then slows significantly, whereas dN i /dt only decreases by about 30% over 10 hr for the application of ABIFM.This contrasting behavior is also reflected in the temporal changes of N i in the cloud layer.After 10 s, the singular descriptions show a strong decline in N i with time, whereas ABIFM maintains almost constant N i throughout the cloud layer.From this we can conclude, as alluded to above, that a diagnostic treatment of each of these parameterizations (i.e., neglecting depletion of INP via activation) is only a good approximation when the INP reservoir is sufficiently large, and depletion is weak.

Prognostic Treatment of Polydisperse INP
We specify a more realistic ambient dust PSD (Table 1, Case 3) guided by field measurements during the ISDAC campaign (Earle et al., 2011;Savre & Ekman, 2015).Here the prognostic modeling results demonstrate that, independently of applied aerosol dust PSD the singular approaches lead to a rapid depletion of INPs and ice crystals whereas the ABIFM application demonstrates a substantially larger INP reservoir, and therefore, maintains significant N i throughout the simulation.Figure S6 in Supporting Information S1 shows that the presence of several thousands of dust particles per liter of air changes the scale of the examined parameters, mostly for the INAS and ABIFM cases.However, the trends in N INP , N i , and dN i /dt with height and simulation time are similar to the previously discussed cases of monodisperse dust populations where the application of ABIFM yields consistent N i and dN i /dt over the domain and simulated time.The domain-averaged N INP and N i (Figure 6) display similar trends as in the case of the monodisperse dust.Despite greater N aer , the singular parameterizations limit N INP and, thus, N i (see also Figure 2).This compensation leads to a similar depletion of ice crystals.In contrast, the application of ABIFM sustains greater ice crystal formation for these long-lived clouds.A similar result is obtained for the temporal evolution of dN i /dt and vertical distribution of N i (Figure S6 in Supporting Information S1), where singular parameterizations fail to replace ice crystals lost to sedimentation and ABIFM maintains almost constant N i throughout the cloud layer via continuous replacement.
Figure 7 demonstrates how the size-resolved INP and aerosol dust PSD in the middle of the cloud layer evolves.Application of singular freezing descriptions yields fewer N INP than for ABIFM, which is smaller than the total aerosol dust PSD by orders of magnitude.Therefore, the effect of INP depletion on the total aerosol dust PSD is ET AL.

10.1029/2023MS003663
16 of 26 negligible for those two singular approaches.As indicated in Figure 7, over the first 5 hr there is greater depletion of INPs that is also seen in Figures 7a  and 7b, respectively.The INN parameterization (Figure 7a) does not consider a particle size dependency and is valid for particles larger 500 nm.The size dependency of INAS is reflected by the fact that the fewer larger dust particles contribute an amount similar to the more numerous smaller particles to the INP population and, thus, respective depletion of activatable INPs (Figure 7b).Since for the application of ABIFM dN i /dt and N i are significantly and persistently greater than for the singular cases, the activation of INPs has a notable effect on the aerosol dust aerosol PSD (Figure 7c).It shows that the largest dust particles are becoming depleted, as expected from our understanding that greater particle surface area or larger particles yield increased freezing (Kanji et al., 2008;Knopf & Alpert, 2013;Knopf et al., 2018Knopf et al., , 2020;;Mason et al., 2016;Welti et al., 2009).Larger particles are also preferentially depleted in the singular approaches, owing to a size cut in the INN case, but negligibly relative to the total dust population compared with the ABIFM approach.

The Impact of Immersion Freezing Efficiency on INP Reservoir Dynamics
The results of our model analysis so far support the importance of the INP reservoir size in sustaining ice crystal formation over the long lifetimes of Arctic MPCs.However, how much does the ice nucleation efficiency (determined by the type of INP) impact continuous formation of ice crystals?Previous studies suggested that a time-dependent slower immersion freezing process could explain the sustained ice crystal formation (Fridlind et al., 2012;Savre & Ekman, 2015;Westbrook & Illingworth, 2013;Yang et al., 2013).If the efficiency of the freezing process is the determining factor of sustained ice particle production, this would imply that in the presence of a fast immersion freezing process, a prognostic treatment using ABIFM should yield results that approach or become equal to that from the singular descriptions.We examine this case by increasing the mineral dust immersion freezing efficiency for cases 1 and 2 by a factor of 10 and 100 (cases 4-7 in Table 1), applying unrealistically fast immersion freezing for mineral dust particles as outlined in Section 3.4, and therefore presenting an extreme test case.
The effect of enhanced immersion freezing efficiencies for singular and CNT approaches on N INP , N i , and dN i /dt with height and simulation time (not shown) follow the same trends as for cases 1 and 2. The domain-averaged N INP and N i for cases 4 and 5 and 6 and 7 are shown in Figure 8 and Figure S7    respectively.The singular approaches display an initial increase of N INP and N i that scale with the change in freezing efficiency, however, it is again followed by significant N INP and N i depletion.For example, INAS application in Case 7, that is, mineral dust particles 1.5 μm in size, and a factor of 100 increase in the freezing efficiency, yields the greatest N i of about 0.1 L −1 after 10 hr for a singular parameterization (Figure S7 in Supporting Information S1).τ INP and τ i for the singular approaches are the same as in previously discussed cases (Table S3 in Supporting Information S1) since the freezing efficiency was increased uniformly for all temperature bins.For Case 5, that is, mineral dust particles 1.5 μm in size, and a factor of 10 increase in the freezing rate, ABIFM application shows about 90% depletion of the INP reservoir over 10 hr.Despite this apparently large depletion, N i is maintained between 1 and 10 L −1 .Only for the most extreme Case 7, does ABIFM application result in N i decreasing to about 0.5 L −1 approaching τ INP and τ i values for the singular parameterizations while establishing overall greater N i .This analysis demonstrates that when applying ABIFM, as long as the INP reservoir is sufficiently large, even unrealistically fast freezing efficiencies do not yield the degree of N INP and N i depletion as encountered for the singular approaches.This in turn implies that the INP reservoir acts as the dominant factor maintaining sustained ice particle production.

The Impact of Cloud Cooling on INP Reservoir Dynamics
Continuous radiative cooling that is concentrated at cloud top and supports turbulent mixing of the PBL may lead to progressively declining PBL temperatures that can increase the numbers of available INPs because N INP increases exponentially with decreasing temperature (see Figure 2).To evaluate this scenario, we applied a cooling of the turbulently mixed PBL at a vertically uniform rate of 0.1 and 0.3 K hr −1 over the 10 hr simulation time, representing common and extreme cooling rates, respectively, to cases 1 and 2. The domain averaged N INP and N i are shown in Figure 9 and Figure S8 in Supporting Information S1.When cooling with 0.1 K hr −1 (Figure 9), the singular approaches show that the depletion of N i ceases at around 5 hr, but N i is still below 10 −2 and 10 −3 L −1 , only about a factor 10 more than without cloud cooling (Figure 6).By contrast, the application of ABIFM yields N i of 0.1-1 L −1 similar to the case without cloud cooling, yielding 2-3 orders of magnitude more N i compared to the application of singular freezing parameterizations.Extreme cloud cooling of 0.3 K hr −1 leads to an increase in N i after about 3 hr for the singular approaches but not reaching more than 0.03 L −1 .Application of ABIFM results in 0.4-2 L −1 , a factor of 10-100 more than for the singular approaches.Cloud cooling can increase N i for all applied parameterizations but is much more pronounced for the singular ones.For those cases, however, resulting N i is still low.ABIFM is not as sensitive to cloud cooling since the gain in activatable INPs due to the development of lower temperatures (Figure 2) is small compared to the size of the INP reservoir (all dust particles).Hence, cloud cooling has only a minor impact on N i when applying ABIFM.We conclude that a large INP reservoir shows little sensitivity toward addition of activatable INPs as a result of cloud cooling.
If solutes are present on mineral dust particles, an aqueous solution coating may form under subsaturated conditions.In equilibrium of the aqueous solution droplet or coating with the ambient humidity, RH equals condensed-phase a w by definition (Koop et al., 2000).ABIFM implicitly accounts for this scenario via non-zero heterogeneous ice nucleation rate coefficients for a w ≤ 1.We performed 1D model simulations for cases 1 and 2 applying ABIFM and consider immersion freezing to commence for subsaturated conditions at RH > 90%.Reducing a w below 1 results in a significant decrease of the nucleation rate (Knopf & Alpert, 2013).For these cases, 8% and 5% greater N i , respectively, are obtained after 10 hr simulation, which does not impact the conclusions of this study where we focus on in-droplet immersion freezing.However, such increases in N i highlight the potential importance of parameterization choices in accounting for immersion freezing under subsaturated conditions more generally.

The Impact of Coarse Mode INPs
The 1D aerosol-cloud model can be used to assess the impact of coarse mode particles serving as INPs.Field measurements have shown that the greatest values of N INP correlate with the largest particle sizes (Knopf et al., 2018(Knopf et al., , 2021(Knopf et al., , 2022;;Mason et al., 2016).Not shown here, when using the 1D model with ABIFM and mineral dust, 2 μm in size and N aer = 50 L −1 , N i larger than 0.5 L −1 over 10 hr can be sustained.This result points to the fact that a small number of coarse mode particles activatable as INPs can have a significant effect on N i in shallow Arctic MPCs.Sufficiently accurate and precise characterization of the ambient coarse-sized aerosol population

Conclusions and Discussion
Comparing three ice formation parameterizations based on the same laboratory measurements, the presented analysis using our 1D aerosol-cloud model indicates that the size of the cloud-topped PBL's activatable INP reservoir plays a dominant role in sustaining ice crystal formation in the class of long-lived MPCs considered.Furthermore, the type of immersion freezing parameterization adopted in the model defines the INP reservoir size.Using a prognostic treatment of INPs that activates via a singular freezing parameterization reproduces the rapid INP removal found in LES modeling studies (Fridlind & Ackerman, 2018;Fridlind et al., 2012, and references therein).By contrast, using a parameterization based on CNT, ice formation is sustained at a substantially higher rate without substantial INP depletion.The sensitivity analyses applying extreme immersion freezing efficiencies and cloud cooling rates, uncommon for these long-lived Arctic MPCs, demonstrate that those factors are of secondary importance for the maintaining of ice crystal formation and further emphasize the paramount role of the INP reservoir size.Furthermore, the presence of coarse mode particles serving as INPs, even when present in low concentrations, provides a sufficiently large INP reservoir, in the case of the ABIFM treatment, to sustain ice crystal formation over several hours.Where detailed LES studies informed by INP measurements have consistently underestimated N i compared with collocated in situ and remote sensing measurements (Fridlind et al., 2012;Silber et al., 2021), immersion freezing parameterizations that produce comparatively more crystals could improve overall model-observation agreement.The impact of the INP reservoir size on ice crystal number concentrations studied here is intended to provide a readily understood and easily tested foundation for approaching the full complexity of coupled processes in an actual cloud.A robust aerosol-INP closure approach (Burrows et al., 2022;Knopf et al., 2021) is required to quantify interpretation of immersion freezing parameterizations since differing nucleation physics assumptions lead to differing INP reservoir sizes.Future applications of the column model introduced here could help establish the degree of single-particle diversity required to reconcile singular and time-dependent approaches under these and differing conditions.
The three applied mineral dust parameterizations used here were derived from the same laboratory experiments, and all reproduce the same measurement data (Alpert & Knopf, 2016;DeMott et al., 2015;Niemand et al., 2012).However, application to ambient particles as previously shown (Knopf et al., 2021) and implementation in this 1D model yields very different N INP .Thus, although all INP descriptions might achieve satisfactory representation of the laboratory data, gross differences emerge when they are implemented under realistic MPC conditions.Because primary ice formation is a ubiquitous process in Arctic MPCs with low liquid water path (Silber et al., 2021) and a necessary preceding step to ice multiplication in clouds with higher liquid water path, more targeted laboratory and field studies are needed to better evaluate the best fidelity of immersion freezing parameterizations (Burrows et al., 2022;Knopf et al., 2020Knopf et al., , 2021)).
The 1D model results also demonstrate that treating INPs prognostically rather than diagnostically is important to simulating their impact on long-lived MPC properties when the activatable INP reservoir is small relative to their depletion.Prognostic treatment of INPs (coupled to aerosol and ice crystals, e.g., ensuring that both sources and sinks of rare activatable particles within a larger dust population are tracked) using singular approaches necessitates the setup and tracking of additional aerosol state variables, as demonstrated above.In this simple 1D model, this approach was readily tractable.However, when considering a 3D model that includes sources and sinks, and particle-particle interactions for multiple aerosol types (e.g., Bauer et al., 2008), keeping track of the activatable INPs will be computationally demanding, requiring additional variables dedicated only to INP activity.As demonstrated above, application of ABIFM does not require the setup of temperature-dependent INP arrays and corresponding tracking of additional variables to denote particle uniqueness.At a given computational time step, the number concentration and size of particles of same type at same S i are considered as activatable INPs, making this a computationally less demanding process.This also holds true when dealing with a more realistic, varying aerosol population, consisting, for example, mixtures of different mineral dusts, sea spray, and soot or organic particles.For each aerosol type that serves as a source of INPs, the application of ABIFM will not need the setup of INP-specific arrays, effectively representing a priori the lowest temperature previously experienced in an air mass, as compared to describing the aerosol population variety by singular approaches.This relative ease of implementing prognostic time-dependent parameterizations in global models with size-informed aero-modules (Bauer et al., 2008;Liu et al., 2016) is a strong motivation to better establish their accuracy versus singular parameterizations under typical MPC conditions.Considering that LES studies operate on the order of hours and GCMs on the order of years and longer, how to best formulate a prognostic treatment of INPs is crucial information.
Immersion freezing at liquid-subsaturated conditions can also occur.Of the three parameterizations employed here, it can be implemented as a process that is physically continuous across inactivated to activated aerosol conditions only in the ABIFM parameterization.The magnitude of its contribution will depend on the freezing rate of the specific INP type (Knopf & Alpert, 2013;Knopf et al., 2018), and the relative contribution of such INP activation would be greater than for the conditions studied here if a thinner cloud is present with a similarly humidified sub-cloud layer.In that case, ice crystal formation just above or below the liquid cloud boundaries could contribute a relatively greater fraction of layer ice formation.Immersion freezing at liquid-subsaturated conditions therefore warrants further examination in comparative studies of immersion freezing parameterizations.
Lastly, similar analyses of the activatable INP reservoir using the 1D model approach for more realistic aerosol conditions should hold relevance for other cloud types where environmental conditions supporting heterogeneous ice nucleation are sustained for relatively long periods of time.Supercooled mid-latitude altocumulus and synoptic cirrus present two such examples that overlap with but are not limited to Arctic latitudes.height and time).In the model configuration we use in this study, the T INP array has the first bin's lowest value set to the minimum domain temperature with a bin width of 0.1 K, while the following bins have an increasing width that follows a geometric progression with a bin width ratio between consecutive bins of 1.05.The last bin of the array is set such that its range includes a temperature of 268.15K (−5°C).For this parameterization we apply τ act = 10 s.

B.2. Singular (INAS)
The singular INAS-based parameterization of mineral dust particles acting as INPs is given by Niemand et al. (2012) as follows for a given particle size:

B.3. CNT (ABIFM)
The wateractivity-based immersion freezing model parameterizes the heterogeneous ice nucleation rate coefficient J het as a function of the water activity criterion, ∆a w (Knopf & Alpert, 2013).∆a w provides pairs of temperature T and water activity, a w .In equilibrium between the droplet's aqueous phase and water partial pressure, a w equals relative humidity (RH), also at subsaturated conditions.As such, ABIFM describes immersion freezing as a function of ambient T and RH.∆a w is derived from the ice melting temperature,   i w , and freezing temperature as a function of a w , in the following way: where a w (T) indicates the conditions during freezing (i.e., T and RH) and   i w ( ) is the a w corresponding to the ice melting temperature at a given T. For immersion freezing from water droplets (the case of this study), by definition a w (T) = 1.At subsaturated conditions representing non-activated aerosol, for which RH < 100%, also a w (T) < 1.
i w ( ) is given in Koop and Zobrist (2009).

•
A 1-D model informed by a largeeddy simulation allows detailed study of immersion ice-nucleating particles (INPs) • Stochastic immersion freezing yields a greater INP reservoir and more sustained ice formation than singular approaches • The efficiency of the freezing process and cloud cooling are of secondary importance for the sustenance of ice crystal formation Supporting Information:

Figure 1 .
Figure 1.Contrasting the stochastic (a) versus the time-independent (b) concept of the ice-nucleating particle (INP) reservoir governing the sustenance of ice crystal formation in shallow mixed-phase clouds over a sea-ice surface.(a) Application of the classical nucleation theory (CNT) derived water activity based immersion freezing model (ABIFM) that considers stochastic, time-dependent ice nucleation.All aerosol particles, here dust particles (black circles), are considered activatable ice-nucleating particles (INPs, orange circles) manifesting a large INP reservoir (N INP ).As soon as activatable INPs enter the cloud layer, they are engulfed by supercooled water and some INPs form ice crystals (N i ).Ice crystal sedimentation reflects the loss process of the INPs.Since the INP reservoir is large and its depletion by INP activation is negligible, continuous ice crystal formation is ensured throughout the cloud lifetime.(b) Application of the singular ice nucleation number based (INN) or singular ice-nucleation active sites (INAS) immersion freezing parameterizations that are non-stochastic and time-independent.For given minimum cloud temperatures, only a few activatable INPs out of all dust particles are available throughout the boundary layer.As soon as the activatable INPs enter the cloud layer and are engulfed in supercooled water, immersion freezing can commence.This initial INP activation is typically more pronounced than in the ABIFM case, leading initially to greater N i in the former case shown as a greater accumulation of ice crystals in (b) compared to (a).However, the few activatable INPs are quickly consumed by ice crystal formation due to the small INP reservoir present and ice crystal formation cannot be sustained over typical cloud lifetimes.Therefore, overall N INP and N i are significantly greater in the ABIFM case (a) compared to the singular case (b).
is the number concentration of activatable INP, N i is the number concentration of ice crystals, and z i is the height at grid cell index i counted from 1 (surface) to m (PBL top), the superscript Imm denotes the type of immersion freezing parameterizations used (INN, INAS, or ABIFM).  Imm * refers to additional variable dimensions depending on the parameterization used such that   Imm * =  for ABIFM, T INP for INN, and the combination of d and T INP for the INAS case, where T INP and d are the INP temperature and diameter arrays (see below), respectively.Note that the number concentration of activatable INPs is also denoted by N INP when applied immersion freezing parameterization is evident.The variables S act , S ent , and S mix are the INP activation, cloud-top entrainment, and turbulent mixing budget terms, and  i sed and  i mix are the ice sedimentation and mixing budget terms, respectively.In Equation 2, the nucleated ice particles (i.e., the ice crystals) do not preserve information on their associated INP and therefore the activation budget is summed over the full d array (in ABIFM), the T INP array (in INN), and over both the d and T INP arrays (in INAS).
Figure 2 Figure 2 treating INPs diagnostically N INP is updated in each model time step according to the predictions of the freezing parameterization using temperature and initial N aer .These INPs activate to form ice crystals, which ultimately sediment out at the surface.In this scenario, the loss of INPs will be compensated in the next time step by diagnosing again N INP .If N INP decreases due to activation to ice crystals, N INP − N i INPs will be added to correspond to predicted activatable N INP for initial N aer .INN does not consider size, and hence, yields the same N INP for the two dust cases examined.Figure 3 shows the evolution of N INP and N i over a model simulation time of 10 hr for cases 1 and 2 (monodisperse dust particles at 100 L −1 with 0.5 and 1.5 μm in size;

Figure 5
Figure5shows the prognostic INP treatment for Case 1, characterized by 100 L −1 monodisperse dust particles of 0.5 μm diameter, over 10 hr of simulation time when applying the different immersion freezing

Figure 2
Figure 2 2).A weak height (and temperature) dependence can be seen.Below cloud level, activatable INPs are present in the first few hours until effectively all INPs are mixed into the cloud layer where they activate to ice crystals.In the singular cases, over the duration of 10 hr the ice formation rate is greatest in the topmost layer due to entrainment of INPs (Figure 5).Activatable N INP are by contrast maintained in and below the cloud layer in the ABIFM approach.N INP below cloud lags compared to N INP within the cloud due to the assumed τ mix = 1,800 s.The cloud-top layer (model domain top) is affected by dust particle entrainment from the free troposphere, and consistently displays the largest N INP in the domain.The temporal evolution of

Figure 2 .
Figure 2. Ice-nucleating particle (INP) number concentration (N INP ) arrays for the monodisperse aerosol dust particles representing cases 1 and 2 and 4-7 in Table 1 where particles with 0.5 and 1.5 μm in diameter are given in left and right panels, respectively, and polydisperse aerosol dust particles representing Case 3. Red dashed, blue dash-dotted, and black lines represent singular ice nucleation number based (INN), singular ice nucleation active sites (INAS), and the classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM) parameterizations, respectively.For INN and INAS, at the lowest temperature, the shown INP number concentrations indicate the maximum activatable N INP , also termed INP reservoir, for given model conditions.For ABIFM a nucleation time of 10 s was applied where different nucleation times yield different N INP .Note the different y-axis scales.

Figure 3 .
Figure 3.The domain-averaged activatable ice-nucleating particle (INP) number concentrations (black lines), N INP , and ice crystal number concentrations (red lines), N i , for the applied diagnostic model simulations for cases 1 and 2 in Table 1 (monodisperse dust with d = 0.5 and 1.5 μm, respectively, and N dust = 100 L −1 ).N INP and N i are derived for three different immersion freezing parameterizations: Singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).
yield 40% and almost 90% depletion of the INP reservoir, explaining the small N i .After 10 hr, only about 5% of the INPs are left.In contrast, ABIFM application leads to a lesser depletion of INPs, ranging between 0.2% and 1.2% after 30 min with a maximum depletion of large INPs of 24% after 10 hr.This is also evident in the e-folding lifetimes for N INP , τ INP , and N i , τ i , given in Table

Figure 5 .
Figure 5. Evolution of the ice-nucleating particle (INP) concentration (N INP ; top), ice crystal number concentrations (N i ; middle) and ice crystal formation rate (dN i / dt; bottom) for prognostic model Case 1 monodisperse dust (d = 0.5 μm and N dust = 100 L −1 , see Table 1).Panels from left to right reflect the application of different immersion freezing parameterizations: singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).Black dashed lines indicate the cloud-base level.
Figure 5. Evolution of the ice-nucleating particle (INP) concentration (N INP ; top), ice crystal number concentrations (N i ; middle) and ice crystal formation rate (dN i / dt; bottom) for prognostic model Case 1 monodisperse dust (d = 0.5 μm and N dust = 100 L −1 , see Table 1).Panels from left to right reflect the application of different immersion freezing parameterizations: singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).Black dashed lines indicate the cloud-base level.

Figure 6 .
Figure 6.Domain-averaged activatable ice-nucleating particle (INP) number concentrations (black lines), N INP , and ice crystal number concentrations (red lines), N i , for the applied prognostic model simulations for cases 1 and 2 in Table 1 (monodisperse dust with d = 0.5 and 1.5 μm, respectively, and N dust = 100 L −1 ) and Case 3 (polydisperse dust).N INP and N i are derived for three different immersion freezing parameterizations: Singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).Note that the scale for N INP for Case 3 CNT(ABIFM) is different than all other panels.

Figure 7 .
Figure 7. Aerosol dust particle (N dust ) and ice-nucleating particle (N INP ) size distributions at 300 m (in-cloud) for the prognostic model polydisperse dust Case 3 (Table 1) are presented for (a) singular ice nucleation number based (INN), (b) singular ice-nucleation active sites (INAS), and (c) classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM) parameterizations.Solid black lines represent the initial N dust (at 0 hr simulation time).(a) and (b) Black dashed lines represent the maximum activatable N INP at 0 hr.Red dashed and blue dash-dotted lines represent activatable N INP at 5 and 10 hr simulation time, respectively.The INN parameterization is valid for particles larger than 500 nm.For this case, N INP are evolved using a bin-by-bin particle number weighted process based on the initial N dust .(c) Red dashed and blue dash-dotted lines represent N aer at 5 and 10 hr simulation time, respectively.

Figure 8 .
Figure 8.The domain-averaged activatable ice-nucleating particle (INP) number concentrations (black lines), N INP , and ice crystal number concentrations (red lines), N i , for the applied prognostic model simulations for cases 4 and 5 in Table 1 (monodisperse dust with d = 0.5 and 1.5 μm, respectively, and N dust = 100 L −1 and ice nucleation efficiencies enhanced by a factor 10). N INP and N i are derived for three different immersion freezing parameterizations: Singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).

Figure 9 .
Figure 9.The domain-averaged activatable ice-nucleating particle (INP) number concentrations (black lines), N INP , and ice crystal number concentrations (red lines), N i , for the applied prognostic model simulations for cases 1 and 2 in Table 1 (monodisperse dust with d = 0.5 and 1.5 μm, respectively, and N dust = 100 L −1 and applied radiatively driven cooling of the turbulently mixed planetary boundary layer at a vertically uniform rate of 0.1 K hr −1 (final minimum domain temperature is 1 K lower than in the standard case).N INP and N i are derived for three different immersion freezing parameterizations: Singular ice nucleation number based (INN), singular ice-nucleation active sites (INAS), and classical nucleation theory (CNT) water activity based immersion freezing model (ABIFM).
) = aer() ( 1 −  − aer () s (INP) ) (B3) and s(INP) =  (−0.517(INP−273.15)+8.934), (B4) where   sing (INAS) INP is the number concentration of activatable INPs in temperature bin T and diameter bin d.N aer (d) is the number concentration of aerosol particles with diameter d, A aer (d) is the individual particle surface area of a particle with diameter d, n s is the INAS density.In this application, N aer and A aer represent the number concentration of mineral dust particles, N dust , and corresponding surface area, A dust .Implemented in the model as: , , INP)( (, )).case on both d and T INP .Here, we set the T INP array as in the INN case, while the d array is set to have 35 with a minimum diameter of 0.01 μm and a bin-to-bin mass ratio of 2. For this parameterization we apply τ act = 10 s.
= aer(),(B8)    where N aer (d) is the number concentration of aerosol particles which represents the number concentration of mineral dust particles.Implemented in the model as: , ) = aer(, , )het( (), w()) 2  (B9) INP (illustrated as abundant orange dots in Figure 1a).Hence, the activatable INPs are distributed within the boundary layer.At each simulation time step, activatable INPs within the cloud layer will randomly activate to form ice crystals and sediment, thereby removing INPs from the cloud.Subsequently, mineral dust particles transported into the cloud layer from below and from above via vertical mixing within the PBL and cloud-top entrainment, respectively, augment INP activation in the next simulation time step.The INP reservoir remains orders of magnitude greater than the relatively few ice crystals that form, resulting in negligible depletion of N INP .Thus, as shown below, N i ≪ N INP , leading to negligible changes in the INP reservoir on a time scale of DeMott et al., 2015)ular, sometimes also termed deterministic, parameterizations and one freezing description based on CNT for mineral dust particles.INN yields the number concentration of activatable INPsDeMott et al., 2015).Note that this INN parameterization is only valid for mineral dust particles larger than 0.5 μm.When considering particle surface area, we apply the INAS density, n s (T), (in m −2 ), from which sing (INAS) INP

Table 1
Details of the Case Studies Considered With the Simplistic 1D Aerosol-Cloud Model Applying Singular Ice Nucleation Number Based (INN), Singular Ice-Nucleation Active Sites (INAS), and Classical Nucleation Theory (CNT) Water Activity Based Immersion Freezing Model (ABIFM) Parameterizations

Table 1
Figure 4 and Table 2 present the change in INP reservoir numbers for cases 1 and 2 as the ratio of INP number concentrations to the initially activatable INP number concentration, that is, ) and application of INN, INAS, and ABIFM.INAS accounts for particle surface area yielding about an order of magnitude greater N INP and N i for the larger dust particle scenario (Case 2).For the ABIFM approach, all 100 L −1 dust particles can serve as activatable INPs, thus, representing N INP .The resulting N i is limited to how many INPs can be activated during the 10 s simulation time step.As a result, N INP is much larger than N i .FigureS3in Supporting Information S1 reports the corresponding vertical distribution of ice crystals for the diagnostic treatment where INAS produces the largest N i in and below the cloud layer, followed by ABIFM and INN.Since we track the INPs, we can assess the impact on the model MPC system's INP reservoir.
in Supporting Information S1, INP for INN and INAS application reports two values, the first one accounting for the initial rapid decrease in N INP and the second for the general trend in declining N INP .

Table 3
Temporal Evolution of the Ice-Nucleating Particle (INP) Reservoir (N INP ) and Ice Crystal Number Concentration (Depicted in Figure 6) Expressed as the Respective e-Folding Lifetimes τ INP and τ i for the Prognostic Simulations Applying the Singular Ice Nucleation Number Based (INN), the Singular Ice Nucleation Active Sites (INAS), and the Classical Nucleation Theory (CNT) Water Activity Based Immersion Freezing Model (ABIFM) Parameterizations