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  • View in gallery

    Collection efficiency as a function of (top) aerosol radius at RH = 95%, and qa = qr = 0 and (bottom) relative humidity at droplet radius r = 25 μm, and qa = qr = 0. Dashed lines show the effect of charges with β = ±7. Prescribed conditions are Tair = 263 K and p = 800 hPa.

  • View in gallery

    Particle surface-area-normalized freezing efficiency from a selection of laboratory results. The solid black line is a best fit to all data [excluding Pitter and Pruppacher (1973)], and the dashed black line is a manually derived alternative fit of the same form as Eq. (11), but with different fit parameters. This line is used for sensitivity studies in section 3b.

  • View in gallery

    Two-dimensional histogram of the contact freezing rate as a function of (top) droplet and aerosol radii and (bottom) ambient temperature and RH. Prescribed conditions are p = 800 hPa and Na = Nr = 1 × 108 m−3.

  • View in gallery

    Comparison between INP number concentrations in the immersion, deposition, and contact modes. Prescribed conditions are p = 800 hPa, RHwater = 50%–101%, RHice = 100%–120%, r = 1–535 μm, Na = 1 × 108 m−3, Nr = 106–109 m−3, β = 0–7, and dt = 1 s. Contact freezing parameterization by Meyers et al. (1992) is also shown.

  • View in gallery

    Initial aerosol (black) size distribution, and mean cloud droplet (red solid) and rain droplet (red dashed) size distributions from Wet Dust run.

  • View in gallery

    (top) Sounding used to initialize the simulations. Wind speed is shown in knots (1 kt = 0.51 m s−1). The level of free convection and the equilibrium level are shown by the blue and red circles, respectively. CAPE is shown as the shaded region. (bottom) Orography used in the simulations, with Jülich (55.92°N, 6.35°E) in the center.

  • View in gallery

    Time series of mean vertically integrated cloud QC, QI, CDNC, and INC from the Wet Dust run.

  • View in gallery

    Mean horizontal cross section of contact INP number concentrations (m−3) at 4 h into the simulations.

  • View in gallery

    Mean horizontal cross section of ice crystal number concentrations (m−3) at 4 h into the simulations.

  • View in gallery

    Two-dimensional histograms showing density (bin count/sample count/bin area) for a range of values of temperature and droplet number concentrations, droplet radii, relative humidity, and contact INP number concentrations.

  • View in gallery

    Two-dimensional histograms showing density (bin count/sample count/bin area) for a range of values of contact INP number concentrations and droplet number concentrations, droplet radii, relative humidity, and temperature.

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Aerosol- and Droplet-Dependent Contact Freezing: Parameterization Development and Case Study

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  • 1 Institute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, Germany
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Abstract

A parameterization for contact freezing is presented that combines theoretical expressions for determining the collision efficiency with experimentally determined freezing efficiency results. The parameterization has dependencies on aerosol and cloud droplet physical properties, including electric charges, as well as ambient temperature and humidity. The highest freezing rate is obtained at large aerosol and large cloud droplet sizes, and at cold temperatures and low relative humidities, with typical dust aerosol and droplet properties. The number concentration of ice nucleating particles (INPs) in the contact freezing mode are generally lower than those in the immersion freezing or deposition nucleation mode; however, under certain conditions contact INP concentrations can exceed those of the other modes. The new parameterization is used in a high-resolution, semi-idealized simulation of a deep convective cloud, and a number of sensitivity studies are performed. Results indicate the greatest sensitivity is to the best-fit function to laboratory data. The simulations show that droplet properties and ambient relative humidity contribute significantly to contact freezing.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Luke B. Hande, luke.hande@kit.edu

This article is included in the Aerosol-Cloud-Precipitation-Climate Interaction Special Collection.

Abstract

A parameterization for contact freezing is presented that combines theoretical expressions for determining the collision efficiency with experimentally determined freezing efficiency results. The parameterization has dependencies on aerosol and cloud droplet physical properties, including electric charges, as well as ambient temperature and humidity. The highest freezing rate is obtained at large aerosol and large cloud droplet sizes, and at cold temperatures and low relative humidities, with typical dust aerosol and droplet properties. The number concentration of ice nucleating particles (INPs) in the contact freezing mode are generally lower than those in the immersion freezing or deposition nucleation mode; however, under certain conditions contact INP concentrations can exceed those of the other modes. The new parameterization is used in a high-resolution, semi-idealized simulation of a deep convective cloud, and a number of sensitivity studies are performed. Results indicate the greatest sensitivity is to the best-fit function to laboratory data. The simulations show that droplet properties and ambient relative humidity contribute significantly to contact freezing.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Luke B. Hande, luke.hande@kit.edu

This article is included in the Aerosol-Cloud-Precipitation-Climate Interaction Special Collection.

1. Introduction

Ice particles in mixed-phase clouds have a large impact on cloud lifetime, precipitation amount, and cloud radiative properties (Boucher et al. 2013; Lau and Wu 2003; Lohmann and Feichter 2005; Morrison et al. 2005), so correctly modeling ice formation processes is important for simulations performed on all spatial and temporal scales. Ice forms on aerosol particles, known as ice nucleating particles (INP), through four different processes: immersion freezing, condensation freezing, deposition nucleation, and contact freezing (Cantrell and Heymsfield 2005; Hoose and Möhler 2012; Murray et al. 2012; Vali et al. 2015). Immersion, condensation, and contact freezing require the presence of liquid water and are the primary ice formation mechanisms in mixed-phase clouds. There is only a subtle difference between immersion and condensation freezing, and as such they are often considered together. Deposition nucleation is active only above ice saturation and at colder temperatures compared to the other freezing modes.

The different ice nucleation processes depend primarily on environmental moisture and temperature, but particle properties (size and surface characteristics) and, in the case of contact freezing, droplet size and electrical charges, play a role. Hoose and Möhler (2012) state that the importance of particle size for contact freezing is not clear. In terms of particle type, recent studies suggest it has a significant influence on its ability to nucleate ice, with mineral dust being identified as a major contributor over a broad range of temperatures (Hoose and Möhler 2012; Murray et al. 2012; Atkinson et al. 2013). To account for the differences in nucleation ability of different aerosol species, parameterizations have been developed for immersion freezing and deposition nucleation (Niemand et al. 2012; Tobo et al. 2013; DeMott et al. 2010, 2014; Hiranuma et al. 2014; Steinke et al. 2015). However, a similar aerosol-dependent parameterization including dependencies on all the variables affecting nucleation has not yet been developed for contact freezing.

Developing such a parameterization remains a challenge because of a lack of quantitative experimental data systematically probing each of the dependencies for various aerosol species. Ladino et al. (2013) provide a review of recent contact freezing studies and highlight the need for more atmospherically relevant laboratory experiments. Parameterizations based on observations lack a sufficient amount of detail, often neglecting a complete description of aerosol and droplet physical properties (Young 1974; Meyers et al. 1992). For example, Phillips et al. (2008) includes dependencies on aerosol chemical properties but assumes the only difference between immersion and contact freezing is a constant shift in the freezing temperature. Diehl and Mitra (2015) include a detailed description of contact freezing, with a dependency on aerosol and droplet size; however, only between two and five aerosol sizes are considered for aerosols between 0.1 and 1 μm, depending on particle type.

Classical nucleation theory has been employed to describe contact freezing (Cooper 1974; Hoose et al. 2010; Wang et al. 2014); however, these parameterizations are not constrained by measurements. Electrical effects have not been included in earlier parameterizations, despite both experiments and theory indicating that they play a role (Wang et al. 1978; Ladino et al. 2013). These parameterizations are still commonly used in weather and climate models, or in some cases contact freezing is completely neglected.

Here we present a new parameterization for contact freezing, which combines a classical theoretical framework for calculating the collision efficiency with recent experimentally determined freezing efficiency measurements. In this way, the new parameterization includes dependencies on aerosol and cloud droplet properties, as well as ambient temperature and humidity, thus providing a new level of completeness. To perform sensitivity studies, the parameterization is applied to a semi-idealized simulation of a convective cloud, and the results are also compared against the commonly used Meyers et al. (1992) parameterization for contact freezing. Finally, the new parameterization is used to quantify the relative importance of the different variables affecting contact freezing.

2. Parameterization development

a. Theoretical determination of collision efficiency

Contact freezing is a two-step process: first, a collision between a supercooled droplet and an aerosol particle must take place, and, second, freezing of the droplet must result from the collision. The collision efficiency, which describes the efficiency with which interstitial aerosol particles collide with supercooled cloud droplets, is governed by many forces. Brownian motion dominates at small particle sizes, whereas inertial impaction and interception dominate at large particle sizes. The phoretic forces, thermophoresis and diffusiophoresis, occur when cloud droplets are either evaporating or growing by condensation. These all combine to determine the total collision efficiency.

The flux of aerosols to a collector drop falling at terminal velocity was determined by Wang et al. (1978). This model includes the effect of Brownian motion and the phoretic forces on the collision efficiency, which is given by
e1
where
e2
is the collision kernel (m3 s−1) due to phoretic forces and Brownian motion;
e3
is the sum of phoretic forces (kg m3 s−2); and
e4
and
e5
are the thermophoretic force (kg m s−2) and diffusiophoretic force (kg m s−2), respectively.

The governing equations presented above are also described in numerous other references (Martin et al. 1980; Pruppacher et al. 1998; Ladino et al. 2011). In such a model, hydrodynamic effects are treated very simply compared to the trajectory model (Grover et al. 1977), where the actual vapor density, temperature, and velocity fields are evaluated and used to determine the precise trajectory of an aerosol particle around a droplet. However, these simplifications have a negligible effect on the particle capture efficiency (Wang et al. 1978).

This model was extended to include the effects of inertial impaction, which dominates at larger aerosol sizes. According to Park et al. (2005), the collision efficiency due to inertial impaction is
e6
As a further extension, the effects of electrical charges on droplets and aerosols were parameterized through Wang et al. (2010). The collision efficiency due to electric charges is
e7
where K = 9 × 109 N m2 C−2,
e8
e9
are the mean charges (C) on the droplet and aerosol, respectively. The empirical parameter β (C m−2) is a coefficient that describes the amount of charge on the droplets and aerosols per unit surface area and can vary from between 0 (neutral) and 7 (highly electrified clouds). All the necessary equations for the undefined parameters above are presented in appendix A, and all symbols are defined in appendix B.
Assuming all particle–droplet collisions result in collection of the particle by the droplet, the collection efficiency (CE) is simply equal to the total collision efficiency:
e10
Figure 1 shows the collection efficiency for varying aerosol radii and ambient RH values. A prominent feature of Fig. 1 is the presence of the “Greenfield gap,” which is the transition region between the Brownian-motion-dominated small-aerosol size range and the inertial-impaction-dominated region at larger aerosol sizes. This region is where the collection efficiencies are lowest, and it is where the phoretic forces become important. The Greenfield gap has been experimentally confirmed, recently by Ladino et al. (2011) under laboratory conditions.
Fig. 1.
Fig. 1.

Collection efficiency as a function of (top) aerosol radius at RH = 95%, and qa = qr = 0 and (bottom) relative humidity at droplet radius r = 25 μm, and qa = qr = 0. Dashed lines show the effect of charges with β = ±7. Prescribed conditions are Tair = 263 K and p = 800 hPa.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Nagare et al. (2015) show that the expressions for thermophoresis provided by Wang et al. (1978) result in a collision efficiency that is almost independent of particle size, a different response compared to expressions provided by Andronache et al. (2006). Therefore, the collisions efficiencies shown in Fig. 1 could be an underestimation for submicron particles. Furthermore, Nagare et al. (2015) show a high sensitivity of the Wang et al. (1978) formulation of thermophoresis to temperature and note that this was formulated for warm rain processes, and its application to subzero temperatures may be problematic.

Other interesting features of Fig. 1 are the inverse relationships between the collection efficiency and the droplet radius and also the ambient RH. From Fig. 1, the increasing collection efficiency with decreasing droplet size is attributed to the increased contribution of Brownian motion to particle capture. Considering the trend with RH, the effect of diffusiophoresis is to create a force such that the aerosol particles move in the same direction as the vapor flux. For low-ambient-RH conditions, which corresponds to an evaporating droplet, this means a force away from the droplet and hence a decrease in the collection efficiency. However, thermophoresis is simultaneously present as a result of differences in droplet and air temperature, which create a force in the same direction as the temperature gradient and the opposite direction as diffusiophoresis. There is conflicting evidence as to whether thermophoresis or diffusiophoresis dominates (Santachiara et al. 2012); however, the above theoretical treatment implies enhanced collision efficiency in low-ambient-RH conditions. In the atmosphere, this would correspond to regions of dry-air entrainment into cloudy air. Note that the ambient RH has the strongest influence on the collection efficiency in the Greenfield gap: that is, aerosol particles between about 10−2 and 1 μm.

Figure 1 also indicates that electrically charged droplets and aerosols act to increase the collection efficiency, particularly within the Greenfield gap. The influence of the relative humidity on the collection efficiency is reduced when charged droplets and aerosols are present. An electric charge obtained with β = ±7 corresponds to a highly electrified cloud associated with thunderstorms (Wang et al. 2010); thus, this represents a realistic upper bound. Yair et al. (2010) provide a parameterization for the lightning potential index (LPI), based on the vertical velocity, and cloud liquid and ice water content. Although this cannot be directly incorporated into the parameterization presented here, it could be used to inform the choice of β in Eqs. (8) and (9). This could be done by assuming a linear scaling between LPI and β and, hence, provide a more realistic time-varying estimate of the collection efficiency as a result of electrical charges.

The collection efficiency for interactions between small droplets and small aerosols is dominated by Brownian motion, and this can give efficiencies greater than 1, as seen in Fig. 1 for r < 10 μm and a < 10−2 μm. It is therefore important to limit the maximum value of the collection efficiency to 1. Phoretic effects still influence the collection efficiency for droplet radii down to 6 μm, and below this droplet size the collection efficiency will always be unity for all aerosol sizes.

Diehl et al. (2006) show that interactions between small droplets and small aerosols contribute nearly 20 orders of magnitude less to the collection kernel, when compared to interactions between large droplets and large aerosols. This means the freezing rate for small droplet–small aerosol interactions would be vastly smaller than for large droplet–large aerosol interactions. The number concentration of smaller droplets is typically larger than the number concentration of large droplets, which would act to increase the freezing rate in this regime, but this is still insufficient to compensate for the difference in the collection kernel. Therefore, the impact of these interactions between small droplets and small aerosols on the contact freezing rate is small. This will be demonstrated in section 2c.

b. Laboratory results of freezing efficiency

Once an aerosol particle and cloud droplet collide, there is a certain probability that freezing of the droplet occurs. This second step in the process of contact freezing is quantified through the freezing efficiency, which depends on the ambient temperature, humidity, and the aerosol and droplet sizes. Recent experiments have attempted to quantify this in a systematic way, as summarized by Ladino et al. (2013).

Figure 2 shows a selection of results obtained from different laboratory instruments (Pitter and Pruppacher 1973; Svensson et al. 2009; Ladino et al. 2011; Hoffmann et al. 2013a,b; Niehaus et al. 2014; W. Cantrell 2015, personal communication), for dust particles with different mineralogies. Taken together, these data cover the complete temperature range over which contact freezing is thought to occur. There is considerable spread in the data, particularly at warmer temperatures. Nevertheless, the particle surface-area-normalized freezing efficiency (m−2) can be described by the temperature-dependent function:
e11
where A = 1.223 × 107, and B = −2.686 × 10−1, and were determined by means of a nonlinear least squares method, where the data are first approximated by a model, and the model parameters are refined through successive iterations that minimize the errors between the data and model. The reported freezing efficiency was normalized by the particle surface area assuming active sites at the particle surface are responsible for initiating contact freezing, similar to the ice nucleation active surface site (INAS) density approach for immersion freezing (Hoose and Möhler 2012). This approach follows Hiranuma et al. (2015).
Fig. 2.
Fig. 2.

Particle surface-area-normalized freezing efficiency from a selection of laboratory results. The solid black line is a best fit to all data [excluding Pitter and Pruppacher (1973)], and the dashed black line is a manually derived alternative fit of the same form as Eq. (11), but with different fit parameters. This line is used for sensitivity studies in section 3b.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Pitter and Pruppacher (1973) did not report the freezing efficiencies of their wind tunnel experiments; however, by assuming a certain number of collisions, as done by Ladino et al. (2011), the freezing efficiency can be estimated. These estimates were not used to define this function. At these warmer temperatures, the best-fit line is quite insensitive to the data used to define the temperature dependent function. Removing the Svensson et al. (2009) data results in an almost negligible change in the best-fit line, since the magnitudes of the errors between the data and the model are smaller than those at colder temperatures. A manually derived alternative fit is proposed in order to test the sensitivity of the parameterization to the freezing efficiency data. This alternative fit is shown as the dashed line in Fig. 2 and will be employed in the sensitivity studies of section 3b. To prevent unphysical high values, the freezing efficiency must have an upper limit of unity: that is, .

The freezing efficiency may have an additional dependency on droplet size, which is not considered in Eq. (11). Looking at the Ladino et al. (2011) data in Fig. 2, the effect of increasing the droplet size from 6.5 to 13 μm is to increase the freezing efficiency significantly. Since there are only two sizes considered in these experiments, extracting a relationship between the freezing efficiency and droplet size is prone to a significant amount of uncertainty. Therefore, it is of little use to attempt to parameterize this effect in the freezing efficiency until more data are available.

c. The contact freezing rate

The final step in defining the contact freezing rate is to combine the theoretically determined collection efficiency with the experimentally determined freezing efficiency results. The collection kernel (m3 s−1) first needs to be calculated from the collection efficiency, as shown below:
e12
where CE is given by Eq. (10), and Vr and Va are given by Eqs. (A16) and (A17), respectively. Finally, the contact freezing rate (m−3 s−1) is given by
e13
where Nr is the number concentration of droplets (m−3), nFE is given by Eq. (11) (m−2), Aa is the surface area of aerosol particles with radius a (m2), and Na is the number concentration of aerosol particles (m−3). The integration limits are given by the range of droplet and aerosol sizes over which the parameterization is valid: that is, r = 1–535 μm and a = 10−3–10 μm. The dust size distribution must be defined by the user. The droplet size distribution can be either calculated from model-diagnosed quantities, or, in the case where a bin microphysics scheme is used, the same bins can be employed in the integration over droplet sizes. Also, the total number concentration of INPs must not exceed whichever is smaller: the number concentration of aerosols or the number concentration of droplets.

Figure 3 shows 2D histograms of the contact freezing rate as a function of aerosol and droplet radii (top) and ambient temperature and RH (bottom). The highest freezing rates are obtained through this parameterization when large aerosol particles (≳0.3 μm) interact with large cloud droplets (≳30 μm). Indeed, at the very largest sizes, the frozen fraction is one. The reason for the high freezing rates at large aerosol and droplet sizes is twofold. First, at large aerosol and droplet sizes, the freezing efficiency approaches 1 (Fig. 1). Second, the collection kernel of Eq. (12) depends on the square of aerosol plus droplet radius.

Fig. 3.
Fig. 3.

Two-dimensional histogram of the contact freezing rate as a function of (top) droplet and aerosol radii and (bottom) ambient temperature and RH. Prescribed conditions are p = 800 hPa and Na = Nr = 1 × 108 m−3.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

The interactions between small aerosols (≲0.01 μm) and small droplets (≲10 μm) make a negligible contribution to the collection kernel, as also shown in Diehl et al. (2006). There is a small area of anomalously low freezing rates when aerosols and droplets of approximately the same size interact. This occurs when the terminal velocity of the droplets is approximately equal to the terminal velocity of the aerosol particles. It is offset to larger droplet sizes as a result of the difference in density between droplets and aerosols and results in a lower collision efficiency.

The bottom panel shows high freezing rates at cold temperatures and low-ambient-RH conditions, as a result of the increase in freezing efficiency with decreasing temperature and the increase in the collection efficiency with decreasing RH. At high values of RH and smaller droplet sizes, which are more relevant for in-cloud conditions, the phoretic forces have the strongest influence. At 255 K, a decrease in the RH of about 10% can increase the freezing rate by about an order of magnitude for 30-μm droplets. For large droplets, the contact freezing rate is mainly determined by temperature, meaning that large freezing rates can also occur in high-RH conditions. Figure 3 also indicates that, under the prescribed conditions, aerosol and droplet size has a larger influence on the freezing rate than ambient temperature and RH. The effects of electric charges were investigated in a similar manner, and found to be largely independent of ambient temperature and relative humidity.

To understand how the number concentration of ice nucleation events determined with the above parameterization compare to the other freezing modes and other contact parameterizations, Fig. 4 shows a comparison with the immersion freezing parameterization from Niemand et al. (2012) and deposition nucleation parameterization from Steinke et al. (2015). The Meyers et al. (1992) parameterization for contact freezing is also shown. The shaded areas show possible values based on the range of RHice values for deposition nucleation, and RHwater, r, Nr, and β for contact freezing. Here, a time step of 1 s is chosen in order to calculate contact number concentrations from the freezing rate given by Eq. (13). This is a typical microphysics time step in large-eddy simulations; however, numerical weather prediction models use a time step at least an order of magnitude larger, meaning an order-of-magnitude more contact INP concentrations than those shown in Fig. 4.

Fig. 4.
Fig. 4.

Comparison between INP number concentrations in the immersion, deposition, and contact modes. Prescribed conditions are p = 800 hPa, RHwater = 50%–101%, RHice = 100%–120%, r = 1–535 μm, Na = 1 × 108 m−3, Nr = 106–109 m−3, β = 0–7, and dt = 1 s. Contact freezing parameterization by Meyers et al. (1992) is also shown.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

For the most part, number concentrations of INPs in the contact mode are several orders of magnitude below those in the immersion or deposition mode. However, in conditions that correspond to low ambient RH, large droplet and aerosol sizes, and large number concentrations, INP concentrations produced in the contact mode can become greater than those produced in the other freezing modes, since the maximum values in the contact mode are greater than those in the other modes. When the prescribed aerosol radius is increased, contact freezing can become significantly larger than the other modes. There is also a very large spread in possible values of contact freezing INPs, covering up to 15 orders of magnitude. This highlights the significant influence that droplet properties, ambient RH, and electrical effects have on determining the number concentration of INPs in the contact mode. These number concentrations are mostly lower than those suggested by the Meyers et al. (1992) parameterization, except when the aerosol size becomes large.

3. Application in a simulation of a deep convective cloud

a. Model description

The nonhydrostatic regional-weather-forecasting Consortium for Small-Scale Modeling (COSMO) Model, version 5.01 (Schättler et al. 2008), was run at high resolutions capable of resolving energy-containing turbulence (Barthlott and Hoose 2015). To evaluate the above parameterization and perform sensitivity experiments, COSMO simulations were performed with high grid resolution of 100 m for a semi-idealized case of a convective cloud. In conjunction with the contact freezing parameterization presented above, the Niemand et al. (2012) parameterization for immersion freezing on desert dust particles, and the Steinke et al. (2015) parameterization for deposition ice nucleation on Arizona test dust were used. The two-moment cloud microphysics scheme of Seifert and Beheng (2006) was also used, which uses the supersaturation to define a power law, in order to calculate CCN concentrations that are representative of continental conditions. The parameterization is only applied to cloud droplets, and contact freezing of rain droplets is not considered. Since rain drops collect many particles through collision–coalescence, they are efficient in the immersion mode (Paukert et al. 2017).

A vertically constant, two-mode lognormal dust aerosol size distribution was used, covering particle sizes from 0.1 to 100 μm, which is based on observations from Jungfraujoch research station (M. Niemand 2015, personal communication) (mode 1: N = 0.015 × 106 m−3, μ = 1.355 × 10−6 m, σ = 1.443; mode 2: N = 0.000 01 × 106 m−3, μ = 8.518 × 10−6 m, σ = 1.358). The dust aerosols are not removed by precipitation in the model. Immersion freezing acts only on the immersed dust aerosols, and contact and deposition ice nucleation act on the interstitial dust. The droplet size distribution was calculated from the model-diagnosed cloud liquid water content and droplet number concentration in every grid box, assuming a modified gamma distribution, with parameters defined in Seifert and Beheng (2006) for droplets in the size range 1–535 μm.

The aerosol and droplet distributions were divided into 10 bins, over which the integration for the parameterizations was performed. The dust size distribution and the temporal and spatial mean of cloudy grid points for the droplet size distribution from one of the simulations, the Wet Dust run (Table 1), are shown in Fig. 5. The simulations were initialized from a sounding, with CAPE = 2801 J kg−1 indicated by the shaded region, and realistic topography was specified at each grid point, as shown in Fig. 6. The topography was chosen to represent the region near Jülich, in western Germany, with the mountains in the southwest of the domain. In this region, the measurement campaign High Definition Clouds and Precipitation for Advancing Climate Prediction [HD(CP)2] Observational Prototype Experiment (HOPE; http://hdcp2.zmaw.de) was conducted in spring 2013, for which realistic COSMO runs were performed by Barthlott and Hoose (2015) with up to 250-m horizontal grid spacing. Random temperature and wind fluctuations with amplitude 0.02 K and 0.02 m s−1, respectively, were added in the lowest 100 hPa of the atmosphere. A horizontal grid spacing of 100 m was used, with 100 vertical levels, and 600 × 600 grid cells horizontally. A time step of 2 s was used for the duration of the 9-h simulation.

Table 1.

Parameters investigated in the sensitivity study.

Table 1.
Fig. 5.
Fig. 5.

Initial aerosol (black) size distribution, and mean cloud droplet (red solid) and rain droplet (red dashed) size distributions from Wet Dust run.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Fig. 6.
Fig. 6.

(top) Sounding used to initialize the simulations. Wind speed is shown in knots (1 kt = 0.51 m s−1). The level of free convection and the equilibrium level are shown by the blue and red circles, respectively. CAPE is shown as the shaded region. (bottom) Orography used in the simulations, with Jülich (55.92°N, 6.35°E) in the center.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

A number of sensitivity experiments were performed to quantify the effect of various parameters on the contact INP number concentrations. These are detailed in Table 1. The parameters of interest in the sensitivity study are the ratio of immersed to interstitial dust aerosol ε, the temperature-dependent freezing efficiency fit, where “high” denotes the solid line and “low” denotes the dashed line in Fig. 2 and the effect of charges on aerosol particles and cloud droplets. An additional simulation is performed with the Meyers et al. (1992) contact freezing parameterization. The Meyers run represents the contact freezing parameterization from these authors, used along with the Niemand et al. (2012) and Steinke et al. (2015) parameterizations. Note that the Meyers simulation is independent of aerosol properties, so stating ε = 90:— only implies that there is 90% of the aerosol available for immersion freezing. Given the low hygroscopicity of uncoated dust (Ghan et al. 2001), the Dry Dust run would be more representative of atmospheric conditions with fresh dust particles that have not undergone aging. The Charge run represents an expected maximum in contact INP concentrations, whereas the Low nFE simulation represents the expected minimum.

b. Sensitivity studies

The evolution of the deep convective cloud is shown in Fig. 7. Here, the spatial mean vertically integrated time series of the cloud liquid content (QC), ice water content (QI), cloud droplet number concentration (CDNC), and ice number concentration (INC) from the Wet Dust run are shown. QC and CDNC show a peak at 4 h, and QI and INC increase to a maximum shortly after, between 4 and 5 h into the simulation, after which the cloud decays. In the last hour of the simulation, the cloud starts to deepen once again.

Fig. 7.
Fig. 7.

Time series of mean vertically integrated cloud QC, QI, CDNC, and INC from the Wet Dust run.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Figure 8 shows the mean horizontal cross section of the contact INP number concentrations from the five runs using the contact freezing parameterization presented here, at 4 h into the simulation, when the number concentrations are among the highest. These plots, as well as the subsequent domain-mean cross sections, are averaged over latitudes. By multiplying the contact freezing rate, given by Eq. (13), by the model time step (here 2 s), the number concentration of contact INP can be calculated.

Fig. 8.
Fig. 8.

Mean horizontal cross section of contact INP number concentrations (m−3) at 4 h into the simulations.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

The largest sensitivity is shown in the Low nFE simulation, where INP concentrations are three to four orders of magnitude lower than the other simulations. This implies that defining an appropriate best-fit function to the laboratory freezing efficiency data is critical. Another interesting difference is that the maximum INP concentrations in Low nFE are located at altitudes around 8 K colder than Dry Dust, as a result of the horizontal difference between the two fit functions in Fig. 2. The Wet Dust simulation shows lower maximum concentrations compared to the Dry Dust and Charge simulations; however, the differences between these three simulations are relatively small. The Meyers simulation contains contact INP number concentrations, which are significantly lower than the Wet Dust simulation. This should be expected since the input dust size is mostly greater than 1 μm, meaning there is an abundance of relatively larger particles that favor INP production via the parameterization presented here. Note that the Meyers et al. (1992) parameterization has no cold temperature limit; therefore, large INP concentrations are produced at very low temperatures.

The ice crystal number concentrations are shown in Fig. 9. Here, the number concentrations are lower than that of the total INP concentrations as a result of conversion of ice into other hydrometeors. The number concentration of INPs acts as the starting value for the ice crystal concentrations. As the simulation progresses, losses due to collision, riming, and evaporation, as well as gains due to freezing of rain and cloud droplets have the net effect of reducing ice crystal concentrations while producing snow, graupel, and hail. Ladino et al. (2017) present observations suggesting the opposite can occur: that is, that ice crystal concentrations are several orders of magnitude larger than INP number concentrations. This points to a missing ice multiplication process in the model. The largest sensitivities are in the simulations that have lower number concentrations of contact INP. There is a clear discontinuity in the number of ice crystals at 261 K for the Low nFE and Meyers simulations, where the immersion parameterization becomes active, and the concentrations at temperatures warmer than this are at least three orders of magnitude smaller.

Fig. 9.
Fig. 9.

Mean horizontal cross section of ice crystal number concentrations (m−3) at 4 h into the simulations.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

c. Dominant contributors to the contact freezing rate

A unique feature of the new contact freezing parameterization is the dependence on many new variables which affect the freezing rate, specifically the aerosol and droplet size and number concentrations, the relative humidity, and electrical charges. An analysis of the contribution of the droplet properties and the relative humidity to the contact INP concentrations is presented here. During the simulations, the dust aerosol properties and the electrical charges were constant and are therefore not investigated here. Figures 10 and 11 show 2D histograms of the droplet properties, relative humidity, and contact freezing INP concentrations, with the colors indicating the density (bin count/sample count/bin area). The droplet sizes were calculated from the cloud water content and cloud droplet number concentration in 10 logarithmic-spaced bins between 10−6 and 10−4 m. These histograms represent in-cloud conditions, defined as cloud water plus cloud ice mixing ratio greater than zero.

Fig. 10.
Fig. 10.

Two-dimensional histograms showing density (bin count/sample count/bin area) for a range of values of temperature and droplet number concentrations, droplet radii, relative humidity, and contact INP number concentrations.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Fig. 11.
Fig. 11.

Two-dimensional histograms showing density (bin count/sample count/bin area) for a range of values of contact INP number concentrations and droplet number concentrations, droplet radii, relative humidity, and temperature.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-16-0313.1

Figure 10 indicates that droplets are present in the cloud with concentrations fairly uniformly distributed from about 104 to 108 m−3, at temperatures warmer than 255 K. There is a preference for smaller droplets from between 1 and 10 μm, so much so that the largest droplet bins greater than about 20 μm are unoccupied. As expected, there are high values of relative humidity at warmer temperatures, and at temperatures relevant for contact freezing the in-cloud relative humidity can range from 20% to 100%. The sounding in Fig. 6 is quite dry in the midtroposphere. As such, there is a high occurrence of relative humidity values of 40%, which would correspond to cloud edge where one could see an enhancement in contact freezing in the lower-relative-humidity environment. The bottom-right panel indicates contact INPs are mostly present at low concentrations within the temperature range at which they are produced. However, contact INPs can sediment out or be transported in the updraft to colder temperatures.

Figure 11 shows that contact INPs are mostly present at lower concentrations. Higher contact INP number concentrations occur preferentially at larger droplet radii and over a larger range of droplet concentrations. The contact freezing events involving large droplets are very rare. Perhaps most interestingly, the relative humidity histogram indicates that contact INPs are present at lower relative humidities of about 40% almost as often as at higher relative humidities over 90%. This does not necessarily mean INPs are produced in the lower-relative-humidity areas almost as often as in the higher-relative-humidity areas, only that they are present here. Advection acts on these particles, possibly transporting them into an area of different environmental conditions. This does indicate that droplet properties and the ambient humidity play a significant role in contact freezing, and these variables should be included in parameterizations aiming for a sophisticated representation of contact freezing.

4. Summary and conclusions

By combining theoretical expressions for determining the collection efficiency with experimentally determined freezing efficiency results, a parameterization for ice nucleation in the contact mode has been developed. The parameterization includes dependencies on more variables than other parameterizations, most notably the surface area of aerosols and cloud droplets, making it particularly useful for detailed process and sensitivity studies. A decrease in the ambient RH of 50% results in an increase in the collision efficiency of about an order of magnitude. Highly charged droplets and aerosol have a similar effect on the collision efficiency, and the effect of RH is significantly diminished in the presence of charged droplets and aerosols. The size of droplets and aerosols has the largest influence on the collision efficiency. A dependence on temperature enters the parameterization through a best-fit function to laboratory observations of the freezing efficiency from numerous experiments.

Putting all of this together, the final parameterization produces the highest freezing rates when large aerosols interact with large droplets. Here, the freezing rate is more than 14 orders of magnitude larger than interactions between small aerosols and small droplets. Higher freezing rates are produced at lower-RH conditions, implying that, in areas of dry-air entrainment, the contact freezing rate would be enhanced. A comparison of this new parameterization to other freezing modes, as well as other contact parameterizations, suggests the number concentration of INPs produced from the parameterization presented here should be mostly lower than in other modes and parameterizations. However, under certain conditions, which correspond to large droplet and aerosol interactions in a low-RH environment with large number concentrations, the number of INPs could exceed those of the other freezing modes.

Semi-idealized, very-high-resolution simulations of a convective cloud were used to perform sensitivity studies and to evaluate the parameterization against another commonly used parameterization. The results indicate the largest sensitivity is to the best-fit function to laboratory data of the freezing efficiency. This underscores the importance of laboratory data to constrain ice nucleation parameterizations. Increasing the amount of interstitial aerosols resulted in a change in the contact INP number concentrations of a factor of 2. Using a realistic upper limit for the effect of charged droplets and aerosols resulted in only a slight increase in the contact INP concentrations.

The case study shows that in-cloud conditions are dominated by smaller droplets over a wide range of concentrations, and the relative humidity is often as low as 40%, presumably near the cloud edge. Contact freezing INPs are present most often in areas of high droplet concentrations and small droplet radii. The temperature and relative humidity do not have a large influence on where contact INPs are present in the cloud. This implies that contact freezing can be enhanced in low-relative-humidity environments near cloud edges, in contrast to the other freezing modes, which are not influenced by the relative humidity and are primarily dependent on temperature.

Acknowledgments

This work was funded by the Deutsche Forschungsgesellschaft through the research unit INUIT–2 (FOR 1525, HO4612/1–2). The authors would like to gratefully acknowledge Luis Ladino-Morino for sharing data and for advice on the determination of theoretical collision efficiency. Many thanks to Alexei Kiselev, Nadine Hoffmann, and Will Cantrell for sharing data.

APPENDIX A

Theoretical Expressions

The particle mobility (s kg−1) is given by Pruppacher et al. (1998) [Eq. (11.20)] as follows:
ea1
where
ea2
and
ea3
is the Knudsen number (unitless);
ea4
is the dynamic viscosity of air (kg m−1 s−1) (Ladino et al. 2011). The mean free path (m) is given by Ladino et al. (2011) as follows:
ea5
The particle diffusivity (m2 s−1) was taken from Pruppacher et al. (1998) [Eq. (11.21)]:
ea6
The thermal conductivity of moist air (J m−1 s−1 K−1) is
ea7
The mean ventilation coefficients for aerosol particles (unitless), and for mass (unitless) were taken from Ladino et al. (2011):
ea8
and
ea9
It is standard to assume that fh = fυ, where fh is the mean ventilation coefficient for heat transfer. Also
ea10
ea11
are the Schmidt numbers for collected particles (unitless) and for water vapor in air (unitless), respectively. To determine the terminal velocity of droplets in air, the Reynolds number must be computed:
ea12
with
ea13
and with the unitless number
ea14
where ln is the natural logarithm, and Cd is the drag coefficient, and, according to Eq. (10.142) of Pruppacher et al. (1998),
ea15
which is known as the Davies or Best number (unitless). Now the terminal velocity (m s−1) of droplets can be calculated through Eqs. (10.139) and (10.146) of Pruppacher et al. (1998):
ea16
The terminal velocity (m s−1) of aerosol particles is determined by applying Eq. (11.22) from Pruppacher et al. (1998):
ea17
Next, the droplet temperature and water vapor pressure must be computed. Equation (8.15) from Lamb and Verlinde (2011) provides a method to calculate the droplet temperature from the growth rate:
ea18
where
ea19
is the mass growth rate of the droplet, and
ea20
Here we have assumed the saturation ratio is unity, which is a valid approximation for the size of droplets considered in this parameterization. Using the droplet and air temperatures, Murphy and Koop (2005) can be used to calculate the saturated vapor pressure of the droplet psat(Tr) and air psat(Tair). Then, using the relative humidity, the actual vapor pressure of the droplet and air can be calculated as follows:
ea21
ea22
Following this, the water vapor density at the droplet surface and the water vapor density of air can be computed (Ladino et al. 2011):
ea23
ea24
The dimensionless Stokes number describes the inertial impaction properties of a particle:
ea25
The Cunningham slip correction factor can be computed by
ea26

APPENDIX B

List of Symbols

Table B1 shows the symbols used in this study, along with their definitions and units.

Table B1.

List of symbols.

Table B1.

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