The basic idea for the new F23 parameterization is that INPCs have large variability, even on short timescales , which we represent by drawing a new INPC from the given distribution at a certain frequency – if immersion freezing occurs (i.e., T≤0 ∘C and cloud droplets are present). Drawing a new INPC mimics the evolution of the ice-active aerosol population, for example, by replenishment or time-dependent freezing at each grid point. The latter means that drawing at each time step does not necessarily reflect a change in aerosol population from one time step to another but a change in the present INP population due to the activation of more INPs with time. The main goal of this study is to test the concept of randomly drawing from a distribution of INPCs instead of applying a deterministic value (e.g., the median of observed INPCs), as is done in other freezing parameterization schemes.

We use the conceptional distribution of INPCs shown in Fig. , derived from the extensive data sets of a long-term time series of INPC temperature spectra measured in the maritime boundary layer at Cabo Verde , and widely dispersed ship-based observations . On average, the INPC field observations show striking consistency in terms of the temperature spectra's shape and variability over time and location. The same consistencies exist when compared with other long-term e.g., or composite INPC data sets e.g.,.

Based on the aforementioned data sets, we derived a function for the temperature-dependent, lognormally distributed INPC frequency as follows: 1Dμ,σ2=12π⋅σexp⁡-[ln⁡(a⋅INPC)-μ(T)]22σ2, with a=1 m3, INPC in per meters cubed (m-3), μ(T) the temperature-dependent mean, and σ2 the variance of the lognormal distribution (not the INPC itself). For the marine data sets, we find σ=1.37 and μ(T)=ln⁡(-(b⋅T)9×10-9), with b=1/(1 ∘C) and T given in degrees Celsius (∘C). By normalizing the distribution, a relative frequency distribution (RFD) as a function of temperature is obtained (Fig. ). This normalization is necessary, due to the discrete INPC field. Whenever an immersion freezing event occurs in the model (i.e., T≤0 ∘C and cloud droplets are present), an INPC value is drawn randomly from the RFD. That is, for two grid points with freezing events at the same temperature, the drawn INPC can differ by several orders of magnitude. For a large number of grid points, the relative frequency of the drawn INPC will follow a lognormal function, with the median INPC (red line in Fig. ) having the highest probability. For example, if all grid points were at a temperature of -16 ∘C, then the frequency of the drawn INPC would follow the distribution shown as a green curve in the lower-left inset of Fig. .

F23 assumes that all INPs are immersed in cloud droplets. To take into account the dynamic evolution of ice formation, the ice crystal number concentration (Ni) at a grid point is subtracted from the drawn INPC. This returns the number of newly frozen cloud droplets in a time step, i.e., hydrometeors moving from the “cloud droplet” class to the “ice crystal” class. Subtracting Ni is one approach to solving the need for time discretization when implementing a deterministic, time-independent scheme. Note that if other frozen hydrometeor species (snow and graupel) are represented in the model (not implemented in our setup), then the sum of their number concentrations should be subtracted from the drawn INPCs. No negative tendencies (INPC-Ni<0) are allowed in the scheme, since frozen cloud droplets will not melt due to a decrease in the INPCs. The change in the respective mixing ratios is calculated by multiplying the change in the number of frozen droplets by the average cloud droplet mass as follows: 2aΔNi=-ΔNc=max([INPC-Ni],0)2bΔQi=-ΔQc=ΔNiQc‾Nc‾. Here, Ni and Nc are the ice crystal and cloud droplet number concentrations per meters cubed (m-3). Qi and Qc are the ice crystal and cloud droplet mixing ratios (kg m-3), respectively, with the mean cloud droplet mixing ratio and number concentration denoted by the bar. The calculations are repeated at each time step, which means that INPs are only partially depleted through the subtraction of Ni. We interpret this as a way to imitate the time-dependent freezing and INP recycling, which has been shown to be crucial for realistic cloud development in LES for Arctic mixed-phase clouds (e.g., ).

The implemented INPC RFD field is discretized into bins of INPCs and temperature. The INPCs differ approximately by a factor of 2 (or Δlog⁡10(INPC)≈log⁡10(2)), while the temperature bins have a size of 1 ∘C. For this reason, we define temperature to the nearest degree when drawing from the INPC RFD, if not stated otherwise. Using the F23 immersion freezing scheme in MIMICA has approximately the same computational expense for the total simulation time as any other interactive ice-nucleation parameterization (for example, F62).

We use the well-established large-eddy simulation (LES) model MIMICA MISU MIT Cloud and Aerosol model;. For more information on the model, also see Appendix  or, e.g., or . The simulated case is based on a mixed-phase Arctic stratocumulus. The case study stratocumulus cloud was observed between 30 and 31 August 2008 during the ship-based ASCOS campaign Arctic Summer Cloud Ocean Study;. At that time, research vessel (R/V) Oden was drifting with an ice floe located at approximately 87∘ N. The atmospheric conditions were characterized by a high-pressure system, with large-scale subsidence in the free troposphere (for details, see ). This case has been used to study other microphysical cloud properties like dissipation , the influence of cloud condensation nuclei (CCN) hygroscopicity on cloud properties , secondary ice production , and the sustenance of an Arctic mixed-phase cloud, as well as in a model intercomparison study . We selected this case because it is an established case and because there are large uncertainties about the nature and concentration of INPs in the Arctic, which poses a challenge to modeling mixed-phase clouds in this region. Using other immersion freezing parameterizations in MIMICA, for example, an active site scheme, following , to simulate this case requires unrealistically active INPs in order to form ice. We focus on immersion freezing in this study because liquid-dependent ice nucleation is dominant in Arctic stratiform clouds . Contact freezing can be neglected, since very little interstitial aerosol was present in the case. Furthermore, these aerosols would need to collide with the supercooled cloud droplets in a very stable non-turbulent cloud, making contact freezing unlikely to occur. Deposition nucleation has the same limitation when it comes to interstitial aerosol and can additionally be neglected because of the temperature range which is not favorable for deposition ice nucleation. Secondary ice formation is not explicitly taken into account in this study, since we focus on primary ice formation. Simulations of the case with a diagnostic ice crystal number concentration (DIAG) instead of an interactive representation of ice nucleation assume a minimum ice crystal number concentration of 200 m-3 at grid points with a temperature below 0 ∘C, where there are sufficient cloud droplets. This means that at any given time step in the cloud, if Ni falls below 200 m-3 and T<0 ∘C, then ice crystals are produced to retain Ni=200 m-3, no matter the exact temperature below 0 ∘C. This approach is unspecific to the ice formation mechanisms. It combines immersion and contact freezing (due to the requirement of cloud droplets) and secondary ice processes.

The MIMICA simulations are initialized with the profiles of thermodynamic variables (e.g., potential temperature and pressure) and liquid cloud water measured at approximately 06:00 UTC on 31 August 2008. The initial ice or liquid potential temperature profiles are randomly perturbed in order for convection to develop more quickly. Consequently, any two simulations will yield different results, even if all parameters are held constant. A cloud layer was present between ca. 550 and 900 m a.g.l. (above ground level), capped by a temperature and humidity inversion, and decoupled from the surface (see , for profile details). The temperature within the cloud ranged from approximately -7 to -10 ∘C. The simulation setup follows . The domain covers a 96×96×128 grid, with a constant horizontal spacing of dx=dy=62.5 m (6 km × 6 km horizontal domain size). The vertical spacing is 7.5 m near the ground and in the cloud layer; between the surface and the cloud, it changes sinusoidally and reaches a maximum dz of 25 m, with a 1.7 km total vertical domain size. The time step is dynamic in order to satisfy the Courant–Friedrichs–Lewy (CFL) condition for the leapfrog time-integration method and ranges from ≈1–3 s to prevent numerical instabilities within the model. Our simulations cover 12 h, with the first 2 h utilized as a spinup period and subsequently omitted from the results. A large-scale steady state is maintained throughout the model runs, and the cloudy layer is present in the initial state of the simulations. However, the initial cloud is liquid, and cloud ice is only formed from the first model time step.

We excluded the hydrometeor categories of snow and graupel from all simulations, since it is known that MIMICA produces rather large volumes of graupel for this case , which dominates both the ice water path (IWP) and the number concentration of frozen hydrometeors. Because we are primarily interested in ice formation in our study, having ice crystals as the only frozen hydrometeors simplifies the analysis. For the same reason, snow and graupel were excluded previously from Arctic stratocumulus simulations conducted with MIMICA . Excluding snow and graupel means that no cold collection processes are active in our simulations; however, ice crystals can still grow by deposition, be transported, and precipitate. Aerosol is not represented prognostically in the model setup.

Observed liquid water path (LWP) and IWP were derived from measurements by a micrometer radiometer and millimeter cloud radar, respectively . Uncertainties for LWP are 25 g m-2, while for IWP they are approximately a factor of 2 .

The F23 and DIAG simulations differ only in how ice crystal formation is parameterized. Other related processes like deposition and sublimation are treated in the same manner, and the general setup follows Sect. . For F23, all parameters are set to the values described in Sect. , while DIAG ensures a minimum Ni of 200 m-3, where the temperature is below 0 ∘C, and there are sufficient cloud droplets.

LWP is substantially larger for F23 than for DIAG (Fig. a). IWP is larger for DIAG than for F23 (Fig. b) but does not compensate for the differences in LWP (the total mass of ice and liquid is still larger for F23). The large differences in LWP between F23 (INPC distribution) and DIAG (constant Ni) are due to enhanced freezing in DIAG, which leads to the increased evaporation of liquid droplets and deposition onto ice crystals via the Wegener–Bergeron–Findeisen process (WBF; see Fig. ).

To analyze the ice crystal number, Fig. c shows ICNB, the vertically integrated Ni (Fig. a). For DIAG, ICNB increases slightly from 1.88×105 to 1.96×105 m-2 over 10 h, while it increases dramatically throughout the F23 simulations from 3.2×104 to 1.0×105 m-2. Several primary processes cause the increase in the ice mass and number, namely (i) the vertical extent of the liquid cloud increases throughout the simulation (see dashed lines in Fig. ). (ii) The temperature within the cloud decreases throughout the simulation (see white contour lines in Fig. a and c). (iii) For F23, new ice crystals form whenever the drawn INPC is larger than the current Ni. Only process (i) is relevant for DIAG, since its requirements for ice formation are T<0 ∘C and the abundance of cloud droplets. For F23, all three processes are relevant, since it requires cloud droplets (i), the INPC RFD depends on the temperature (ii), and process (iii) is an inherent characteristic.

Figure d shows the time series of NIRS (i.e., incoming minus outgoing infrared radiation at the surface). NIRS is larger for F23 compared to DIAG. This can be expected, since LWP is larger for F23. The evolution over time is similar in both cases, with NIRS first increasing and then decreasing from hours 4–5. This pattern emerges from the combination of first increasing, then steady LWP (for F23, the LWP decreases slightly between hours 5 and 8, which causes a larger decrease in NIRS), and continuously decreasing temperature (see white lines in Fig. a and c), due to the radiative cooling of the cloud.

The profile of Ni for DIAG in Fig. a illustrates the principle of prescribing Ni diagnostically. For DIAG, within the cloud and ca. 100 m below it, Ni has the prescribed minimum value of 200 m-3. Down to the surface, the concentration is lower but constant, which is caused by steady sedimentation of ice crystals from above combined with sublimation from ice crystals (see Fig. b). Qi has the maximum values at the cloud bottom (Fig. b), which is commonly observed in Arctic mixed-phase clouds . Since no cold-phase collection processes are active in these simulations, the mass of single ice crystals can only grow by deposition. The total mass of ice is, however, also affected by transport processes like sedimentation. The vertical distribution of Ni and Qi in F23 is similar to DIAG (Fig. c and d). However, both Qi and Ni increase by larger rates throughout the simulation time for F23, which can also be seen in the IWP and ICNB values (Fig. b and c). Towards the end of the simulation time, F23 ice values are about half of DIAG. Even though ice nucleation depends on the temperature in F23, Ni is quite homogeneously distributed throughout the cloud (Fig. c). We explain this by the uniform stratification of the cloud and the subtraction of Ni from INPC (Eq. ) in the scheme, which homogenizes Ni vertically over time.

LWP as calculated in MIMICA is substantially higher than the 75th percentile of the observed values, irrespective of the treatment of ice nucleation (Fig. a). found this already in their model intercomparison study, where MIMICA was amongst the models with the highest LWP. The LWP exceeded the observations for simulations similar to ours, with no prognostic aerosol treatment but a simplified treatment of aerosol activation (note, however, that the MIMICA simulations in , included snow and graupel). MIMICA simulations, in which aerosol is modeled prognostically, yielded results closer to the observational range in the study by . The IWP for DIAG lies within the interquartile range of the observations (Fig. b), which was found in as well. The minimum Ni in DIAG was chosen in order to yield ice masses in the cloud similar to the observations. For the F23 simulations, IWP is lower than the 25th percentile of the observations. F23, however, only represents immersion freezing, leading to a lower ice mass. We can expect that multiplication processes were present in the observed cloud, which would have led to a higher IWP compared to model simulations that do not represent SIP. In fact, simulated the same case, using MIMICA and including a description of ice multiplication from breakup during ice–ice collisions. Their results show an increase in IWP by a factor of 2 to 3 when breakup is activated in the model. Adjusting our simulated IWP accordingly would result in values corresponding to the observed IWP for F23. LWP is decreased by approximately 25–35 g m-2 in simulations with ice multiplication in . This would bring our modeled LWP values closer to the observations, but LWP would still be higher than observed, due to simplified aerosol activation, as explained above.

One aspect that complicates the comparison of our simulations with observations is that we excluded snow and graupel. If snow and graupel were included in the simulations, then IWP might increase as riming converts liquid to frozen mass. On the other hand, this might also lead to faster precipitation and thus depletion of liquid or frozen water.

Note that our goal was to test F23 rather than include all the necessary processes in order to closely model the observations.

F62 calculates the number of cloud droplets that freeze to ice crystals from the temperature-dependent INPC (m-3) given in F62 (and shown by the black line in Fig. ), as follows: 3INPC(T)=0.02⋅exp⁡(-β⋅T), with β=0.6/(1 ∘C) and T in degrees Celsius (∘C). The changes in Ni, Nc, Qi, and Qc are calculated according to Eqs. () and (). Note that this parameterization is only strictly valid for -10>T>-30 ∘C, but we extrapolate it to the temperatures of the simulated cloud (dotted line in Fig. ). Analogously to F23, freezing happens where T≤0 ∘C and cloud droplets are present.

Figures b and show that very little ice is produced using the F62 ice-nucleation scheme. We explain this by the subtraction of Ni from INPC(T), since this only leads to considerable ice formation at the beginning of the simulation, when no ice is present yet or when the temperature decreases. This comparison between F23 and F62 illustrates the difficulty in simulating reasonable ice masses in warm mixed-phase clouds with conventional schemes that yield one INPC for one set of environmental conditions. Drawing from a distribution of INPCs according to F23 leads to substantially larger ice masses, and these might even be multiplied if, e.g., secondary ice processes would also be considered (see the discussion in Sect. ). In the future, it could be interesting to further analyze if the difference between F23 and F62 could be reduced by tuning F62 and how much it is related to the conceptual differences in the schemes.

To investigate the sensitivity of the simulated cloud to the characteristics of the F23 scheme, the following parameters (summarized in Table ) were varied: (i) the median and standard deviation of the INPC distribution (Sect.  and ); (ii) the size of the temperature bins (Sect. ); (iii) the frequency of drawing (Sect. ); and (iv) the resolution of the model domain (Sect. ). The sensitivity tests (i) mean that we cover a wider spectrum of INPC RFDs than the one defined in Sect.  and shown in Fig. .

The limitations and challenges of this study can be divided into three aspects, namely the limitations related to the parameterization itself, the data set it is based upon, and the limitations due to the modeling setup used to test the parameterization scheme.