Sensitivity of liquid clouds to homogenous freezing parameterizations

Water droplets in some clouds can supercool to temperatures where homogeneous ice nucleation becomes the dominant freezing mechanism. In many cloud resolving and mesoscale models, it is assumed that homogeneous ice nucleation in water droplets only occurs below some threshold temperature typically set at −40°C. However, laboratory measurements show that there is a finite rate of nucleation at warmer temperatures. In this study we use a parcel model with detailed microphysics to show that cloud properties can be sensitive to homogeneous ice nucleation as warm as −30°C. Thus, homogeneous ice nucleation may be more important for cloud development, precipitation rates, and key cloud radiative parameters than is often assumed. Furthermore, we show that cloud development is particularly sensitive to the temperature dependence of the nucleation rate. In order to better constrain the parameterization of homogeneous ice nucleation laboratory measurements are needed at both high (>−35°C) and low (<−38°C) temperatures. Key Points Homogeneous freezing may be significant as warm as −30°C Homogeneous freezing should not be represented by a threshold approximation There is a need for an improved parameterization of homogeneous ice nucleation


Introduction
Clouds play an integral role in the Earth's energy budget [Boucher et al., 2013] and can be sensitive to the presence of ice; however, an explicit understanding of ice formation processes and appropriate representation in models is currently lacking. Cloud droplets are frequently observed to supercool to temperatures approaching À35°C and even below [Choi et al., 2010;de Boer et al., 2011;Rosenfeld and Lensky, 1998;Rosenfeld and Woodley, 2000;Westbrook and Illingworth, 2011]. At these extreme supercooled temperatures, it is known that freezing can occur via the homogeneous nucleation of ice [Murray et al., 2010;Riechers et al., 2013], and this process therefore needs to be appropriately represented in cloud models.
Laboratory measurements show that homogeneous ice nucleation rates are strongly temperature dependent, and therefore, it has been commonly assumed that homogeneous freezing in cloud simulations may be approximated by a step function at a threshold temperature. A number of microphysics schemes currently use this approach with the threshold set at either À38 or À40°C [e.g., Forbes and Ahlgrimm, 2014;Kong and Yau, 1997;Lim and Hong, 2010;Morrison et al., 2005;Reisner et al., 1998;Thompson et al., 2008]. However, there is a finite but uncertain nucleation rate at temperatures warmer than these thresholds; hence, some schemes, albeit the minority, use parameterizations based on experimental measurements [e.g., Cotton and Field, 2002;Lynn et al., 2005;Milbrandt and Yau, 2005;Seifert and Beheng, 2006;Walko et al., 1995]; see Table S1 in the supporting information for a summary of microphysics schemes in the literature. Using a cloud resolving model, Fan et al. [2010] tested several existing heterogeneous and homogeneous freezing parameterizations including a relatively warm threshold of À36°C and found that deep convective anvil properties were sensitive to the different representations. In this paper we use a parcel model with detailed microphysics to show that supercooled liquid clouds formed at a range of updraft speeds are sensitive to the numerical representation of homogeneous freezing and that the commonly used threshold approximation of À40°C may be inappropriate.

Parameterizations
A summary of laboratory measurements of the homogeneous ice nucleation rate coefficient, J (nucleation events per unit volume and unit time for ice in pure water), as a function of temperature is shown in Figure 1. We neglect nucleation at droplet surfaces, since measurements indicate that it is not important for cloud-sized droplets [Duft and Leisner, 2004]. The data in Figure 1 show that J increases steeply over 5 orders of magnitude within~3°C, and there is a considerable spread in J values at all temperatures, along with a range in measured temperature (T) dependences (i.e., the gradient of J versus T).
HERBERT ET AL. Supporting Information: • Tables S1 and S2 and Figure S1 In order to assess the sensitivity of homogeneous ice nucleation in clouds to the uncertainty in the measured rate coefficients, we have used a number of parameterizations which are consistent with the data (see Figure 1). Pruppacher [1995] and Zobrist et al. [2007] used classical nucleation theory (CNT) to estimate values of J. These parameterizations are shown as solid, colored lines in Figure 1 and are individually referred to as J Prup and J Zob . A second set of parameterizations was developed here to represent sensitivity to the absolute value of J and the T dependence of J, referred to as J Prup-Low and J Zob-Shallow , respectively. The two parameterizations, shown as dashed colored lines in Figure 1, are based on J Prup and J Zob and are constrained to experimental measurements. The parameterizations to describe the J measurements are collectively referred to as J CNT ; the equations used for each parameterization are shown in Table S2 in the supporting information. Threshold freezing approximations have been used in previous modeling studies with temperatures of -35, -38, or À40°C (see Table S1 in the supporting information). For this study a value of À40°C was used in the threshold simulations due to its common use and is referred to as J Threshold ; the dotted green line in Figure 1 represents this function.

Description of Model
The simulations were performed using the Met Office Kinematic Driver (KiD) model. The KiD model described by Shipway and Hill [2012] is a one-or two-dimensional dynamical framework within which the dynamics and optional microphysical forcings are prescribed throughout the simulation. For the purpose of this study, the one-dimensional version was used with prescribed conditions so that a constant cooling rate was achieved. Only a single grid point was considered, and hydrometeor sinks were limited to precipitation, resulting in an idealized adiabatic parcel model simulating a trajectory through the atmosphere. The simulations were initialized under saturated conditions (relative humidity of 100% with respect to liquid water) at a temperature of À5°C and the parcel lifted along a saturated adiabat; the parcel therefore contained a population of water droplets soon after the simulation started. An additional set of simulations were initialized at À30°C to simulate a shallower cloud. Wright and Petters [2013], Vali [2014], and Herbert et al. [2014] have shown that the T dependence of the nucleation rate coefficient determines the stochastic time-dependent behavior of ice nucleation; therefore, the simulations were run under a range of cooling rates to additionally test the sensitivity of homogeneous freezing to the cooling rate. The Thompson two-moment bulk microphysics scheme was chosen from a number of existing embedded options in the KiD model; the scheme is one of several coupled to the Weather Research and Forecasting model and has been shown to be representative alongside other microphysics schemes within the KiD model framework [Shipway and Hill, 2012]. The scheme, described in full by Thompson et al. [2008], predicts cloud water, rain, cloud ice, graupel, and snow and includes a detailed treatment of in-cloud interactions between all hydrometeor species and water vapor. The size distributions of each hydrometeor species are represented by Marshall-Palmer distributions except for snow which is Figure 1. Homogeneous ice nucleation rate coefficient data determined from laboratory measurements (grey symbols and lines) and estimated using CNT following Pruppacher [1995] and Zobrist et al. [2007] (solid red and blue lines). Additional parameterizations adapted from the CNT-based lines and constrained by the measurements are shown as dashed red and blue lines. A threshold freezing approximation of À40°C is also included, shown as a temperature-independent step function (green dotted line). The data are taken from Benz et al. [2005], Butorin and Skripov [1972], Demott [1990] Table S2 in the supporting information.
Geophysical Research Letters 10.1002/2014GL062729 described using a combined exponential and gamma distribution. Cloud droplets, not described by the explicit activation of aerosols, are constrained to a concentration of 200 cm À3 , representing a relatively clean cloud. In each simulation, the primary production of ice was limited to homogeneous freezing of cloud and rain droplets only.
In the J Threshold simulations, all liquid water is converted into ice over a single time step once the threshold temperature is reached. For the J CNT simulations (J Prup , J Zob , J Prup-Low , and J Zob-Shallow ), the number of liquid droplets (both cloud and rain droplets) that freeze in each time step Δt is a function of T and droplet size and is calculated following where n liquid (r) is the number of droplets of radius r, V is the droplet volume, and J is the homogeneous ice nucleation rate coefficient at T. For each prescribed cooling rate, the simulations were run consecutively with each J(T) parameterization. In each set of simulations, an equivalent updraft speed, w, was calculated assuming a wet adiabatic lapse rate of À5.5°C km À1 , resulting in the range of 0.04 ≤ w ≤ 30 m s À1 . Figure 2 shows the one-dimensional evolution of the ice number concentration, cloud ice effective radius, snow mass mixing ratio, cloud ice fraction (defined as the ratio of ice mass to total cloud mass), and the total cloud mass mixing ratio as a function of parcel updraft speed. The first four columns in Figure 2 represent the J CNT simulations using J Prup , J Zob , J Prup-Low , and J Zob-Shallow , and the final column is using J Threshold , where all liquid water freezes at À40°C. The cloud liquid water content (LWC) prior to freezing can be inferred from the total cloud mass shown in Figure 2e at T = À30°C. The simulated LWC ranges from~0.4 to 2.0 g kg À1 (~0.5 to 3 g m À3 ) on increasing w from~0.1 to 20 m s À1 . This range is in agreement with LWC measurements from convective clouds made by Draginis [1958] and highly supercooled clouds by Rosenfeld and Woodley [2000]. In the analysis of the simulations, we define the first ice as a concentration of ≥1 m À3 ; however, it is worth noting that this small concentration increases by several orders of magnitude within a single degree. In all simulations, both cloud droplets and rain droplets are present and contribute to the glaciation of the cloud via homogeneous freezing.

Suitability of Threshold Approximation
A clear distinction between the J Threshold and J CNT simulations is evident in the evolution of all cloud variables. The first ice occurs >6°C warmer in the J CNT simulations with the most extreme case 10°C warmer than J Threshold , which corresponds to~1800 m difference in altitude. Due to the rapid freezing in the J Threshold simulations, the considerable differences observed are due to the increased amount of time for secondary processes, such as ice sedimentation, hydrometeor interactions, and depositional growth, to occur in the J CNT model runs. These combined processes affect both the size distribution of the ice particles and the total cloud mass mixing ratio, as seen in Figures 2b and 2e, suggesting a considerable impact on the cloud radiative properties of the evolving and glaciated cloud. The smaller ice particles may also increase the cloud lifetime, thus enhancing this impact. In these simulations, snow and graupel are minor constituents; however, they also demonstrate sensitivity to the representation of freezing, albeit indirectly, as shown in Figure 2c for snow. The simulations show a degree of cooling rate dependence, which is primarily due to the relationship between w and supersaturation, and the enhanced time-dependent processes at low w; this dependence is considerably enhanced in the J CNT simulations as evident in Figure 2d. At low w (~0.1 m s À1 ), the cloud fully glaciates between 3 and 5°C warmer than in the J Threshold simulation depending on the J CNT function used. A second set of simulations initialized at À30°C, thus simulating a shallower cloud, are included in Figure S1 in the supporting information. There is considerably less cloud LWC (<0.15 g kg À1 ) in these shallow clouds; however, the J CNT simulations continue to produce ice up to 8°C warmer than using J Threshold , and similarly, the evolution of other cloud properties is impacted. The differences between J Threshold and J CNT simulations clearly show that the threshold approximation is unable to represent the freezing behavior observed in simulations using the parameterizations constrained by laboratory data.

10.1002/2014GL062729
The results presented in Figure 2 should change the way we view homogeneous freezing. It is generally assumed that homogeneous freezing is a negligible process at temperatures above~À38°C when compared to heterogeneous ice nucleation; however, these results show that there may be considerable competition between the two nucleation modes. Observations and experiment-based extrapolations report atmospheric ice nuclei concentrations on the order of 10 2 to 10 5 m À3 at À34°C [DeMott et al., 2010;Murray et al., 2012]. Inspection of Figure 2a suggests that homogeneous freezing may provide between 10 and 10 6 m À3 ice Figure 2. Simulated one-dimensional evolution of cloud variables as a function of constant updraft speed using different homogeneous freezing representations. Variables include (a) cloud ice particle number concentration, (b) cloud ice particle effective radius (expression for the cross-section area weighted mean radius), (c) snow mass mixing ratio, (d) cloud ice fraction (ratio of ice mass mixing ratio to total hydrometeor mass mixing ratio), and (e) total cloud mass mixing ratio (all ice and water species). The first four columns demonstrate sensitivity of the cloud to the laboratory-constrained J(T) parameterizations, and the final column shows simulations using a threshold approximation of À40°C at which point all liquid instantly freezes.

10.1002/2014GL062729
particles at À34°C depending on the parameterization and updraft speed. Hence, homogeneous ice nucleation competes with heterogeneous ice nucleation at temperatures well above À38°C.

Sensitivity to Parameterization
Figure 2 demonstrates that the simulated cloud is sensitive to J CNT , in particular the T dependence of J. A systematic decrease in the absolute value of J from J Prup to J Prup-Low results in clouds which have approximately the same properties but are offset by~200 m in altitude, which demonstrates a relatively weak sensitivity. In contrast, a change in the T dependence from the steep J Zob to shallow J Zob-Shallow parameterization causes an onset of ice up to~3°C warmer. Unlike a change in the absolute value of J, a change in the T dependence does not have a simple linear effect on the cloud evolution. In the simulations, cloud evolution following ice onset proceeds more slowly in the J Zob-Shallow simulations than J Zob and allows more time for secondary processes and in-cloud interactions to occur before complete glaciation. The result is an increased ice depositional growth, leading to larger ice particles and increased cumulative sedimentation of ice. These changes may impact the radiative properties of the mixed phase and glaciated cloud as inferred from Figures 2b and 2e. As shown by Herbert et al. [2014], the T dependence of J controls the response of ice production to changes in cooling rate. This behavior can be seen in Figure 2d; the glaciation temperature (where cloud ice fraction = 1) is more dependent on w in the J Zob-Shallow simulations than any other J CNT . The result is a longer-lived mixed-phase regime which at low w (<0.1 m s À1 ) corresponds to additional time on the order of several hours.
In these simulations, the first ice ( Figure 2a) is observed at temperatures that correspond to small nucleation rate coefficients of approximately~1 cm À3 s À1 (see Figure 1). Hence, the simulated clouds are sensitive to J values that are more than 4 orders of magnitude below the range at which the majority of laboratory measurements have been made (J ≥ 10 4 cm À3 s À1 ). This paucity of measurements in this regime of J ≤ 10 4 cm À3 s À1 highlights a considerable gap in our current knowledge and the need for future laboratory studies to focus on homogeneous ice nucleation at higher temperatures. Figure 2 also shows that the scatter in laboratory measurements, as represented by subsequent model parameterizations, has implications for the evolution of simulated mixed-phase clouds and their microphysical properties. In these simulations, all J CNT parameterizations fall within the range of laboratory measurements of J and can be seen as conceivable representations; therefore, based on current measurements, it is not possible to determine which, if any, parameterization is more appropriate. Additional laboratory measurements at high and low temperatures would be required in order to establish the T dependence of J, a parameter which our study indicates is very important; such measurements would also help to define the absolute value of J.

Summary and Conclusions
In this study we used a parcel model with detailed cloud microphysics to show that the threshold freezing approximation, used in many microphysics schemes, is unable to suitably represent homogeneous freezing in liquid clouds. The evolution of cloud is sensitive to the finite rate of homogeneous nucleation well above À40°C, i.e., at temperatures where we traditionally assume only heterogeneous nucleation can produce ice in clouds. In some simulations, homogeneous freezing was active as warm as À30°C, which is considerably warmer (>8°C) than generally assumed to occur in clouds. A series of parameterizations based on CNT and constrained by the scatter in laboratory measurements was used to show that simulated clouds are sensitive to the chosen parameterization and therefore to the uncertainty in laboratory measurements. In particular, we found that the temperature dependence of the homogeneous ice nucleation rate coefficient is a key parameter for correctly determining the impact of homogeneous freezing on cloud properties. We recommend that future laboratory studies focus on nucleation at high (>À35°C) and low (<À38°C) temperatures in order to constrain a new parameterization.
The idealized simulations provide evidence for the unsuitability of using a threshold freezing temperature; however, it would be beneficial to extend this work to a three-dimensional spectral bin model where cloud scale interactions and feedbacks are included and explicitly represented. Moreover, inclusion of heterogeneous ice nucleation would also be required for understanding the competition between the two modes. Nevertheless, we recommend explicitly representing the temperature dependence of homogeneous ice nucleation rather than using a threshold approximation.