1. Introduction
Cloud droplet temperature is an important parameter to consider while studying fundamental cloud microphysical and radiative processes. Cloud droplet temperature plays a role in determining the diffusional growth and decay rates of cloud droplets (Roach 1976; Srivastava and Coen 1992; Marquis and Harrington 2005), and is instrumental in specifying refractive indices of a droplet, which impacts radiative effects of clouds (Rowe et al. 2020). Supercooled cloud droplet temperature is also an important factor for ice formation in clouds, since homogeneous and certain heterogeneous ice-nucleation mechanisms require the presence of supercooled cloud droplets (Ansmann et al. 2009; de Boer et al. 2011), and the ice nucleation rates are highly temperature dependent (Wright and Petters 2013).
Multiple studies have shown that the presence of supercooled liquid water droplets is a prerequisite to ice formation in cold (<0°C) clouds, with contact or immersion freezing mechanisms dominant among the heterogeneous ice-nucleation pathways (Westbrook and Illingworth 2011; Field et al. 2012). Numerous qualitative and quantitative studies have reported the presence of supercooled liquid water droplets at the tops of a variety of cloud systems (Cunningham 1951; Hall 1957; Tricker 1971; Cooper and Vali 1981; Hobbs and Rangno 1985; Rauber and Grant 1986; Plummer et al. 2014; Zaremba et al. 2021; Hu et al. 2023a). For example, using airborne measurements within wintertime storms over the northern Colorado Rocky Mountains, Rauber and Grant (1986) observed liquid water layers in orographic cloud systems with tops as cold as −31°C. Plummer et al. (2014) observed supercooled water in cloud-top convective generating cells within continental cyclones between −31° and −11°C (almost ubiquitous at temperatures > −16°C). Zaremba et al. (2020, 2021) documented the presence of supercooled water at cloud tops over the temperature range −3° to −28°C, as well as ice in stratocumulus clouds with cloud-top temperatures > −5°C in cold sectors of extratropical cyclones over the Southern Ocean.
Ice formation often occurs near cloud boundaries, particularly cloud tops, where cloud-top generating cells are often found (e.g., Plummer et al. 2014, 2015). Evidence of mixing and entrainment/detrainment occurring between cloud-top generating cells has been observed by Plummer et al. (2014), including the presence of supercooled liquid water within and between the cells, due to their highly turbulent nature (Wang et al. 2020). Within these zones of entrainment and mixing in and around cloud tops and edges, as well as in downdrafts, the water droplets are exposed to sudden changes in relative humidity and can undergo rapid evaporation. While modeling microphysical processes in these scenarios, the traditional assumptions of cloud droplets being in thermal equilibrium with the environment, and the difference between the ambient temperature and that of the droplet being negligible (Pruppacher and Klett 1997, chapter 13), are likely to break down. Previous studies have speculated that evaporating cloud droplets at cloud edges can exist at a temperature much lower than that of the ambient environment, activating more ice-nucleating particles (INPs) than expected at ambient temperature (Mossop et al. 1968; Young 1974) and thus leading to enhanced ice formation. If, for example, a droplet temperature is lowered from −5° to −12°C during evaporation, the concentration of effective contact nuclei would increase from about 0.02 to 0.1 L−1 to possibly as high as 5 L−1 (Cooper 1980; Beard 1992). Several studies have also observed enhanced ice crystal concentrations near the evaporating regions of cumuliform, stratiform, and wave clouds (Langer et al. 1979; Hobbs and Rangno 1985; Rangno and Hobbs 1994; Field et al. 2001; Cotton and Field 2002; Baker and Lawson 2006).
Besides ice nucleation, improper specification of cloud droplet temperatures can impact cloud radiative effects by the erroneous specification of droplet refractive indices (Rowe et al. 2020) and substantial errors in the calculations of classical diffusional growth and decay rates (Roach 1976; Srivastava and Coen 1992; Marquis and Harrington 2005). Srivastava and Coen (1992) showed that large errors (about 10%–30% under certain conditions) in hydrometeor growth or decay estimation can occur if the traditional approach is used, which for simplicity eliminates the droplet temperature from the equations and is valid for only small temperature differences between the droplet and the ambient environment. Therefore, it is important to investigate the deviation in the droplet temperature from the ambient temperature under a wide range of environmental conditions.
Langmuir (1918), Topley and Whytlaw-Gray (1927), and Houghton (1933) were the earliest theoretical and laboratory studies of evaporation rates of stationary liquid water droplets. Houghton (1933) derived a theoretical expression for the rate of evaporation of water drops at rest based on the general evaporation equation of Jeffreys (1918). To explain the discrepancies in his results, he suspected that the drop, depending on its size and vapor density gradient, was actually cooling to a temperature considerably lower than the ambient environmental temperature, due to evaporation. The equilibrium temperature of a droplet is essentially determined by a balance between key heat transfer processes—evaporation, conduction, radiation, and convection—and as a result, might be different from the ambient temperature (Houghton 1933). The exact analytic solution to the problem at that point was, however, practically impossible because of the multiplicity of factors involved (Houghton 1933).
In their laboratory experiments using an advanced Schlieren optical method, Kinzer and Gunn (1951) measured a decrease in droplet temperature from 22.4° to 14.9°C within a few seconds for raindrops of 1.35 mm radii falling through subsaturated air. They compared their observation of this single scenario with theoretical results derived from the basic psychrometric equation, considering the effect of ventilation and environment upon the evaporation rates and the equilibrium temperatures of freely falling spherical raindrops. They concluded that the equilibrium temperature of a falling droplet is identical to the corresponding temperature of the ventilated wet bulb to within ±0.3°C. In their theoretical analysis, they assumed that the saturated vapor density at the droplet surface is a linear function of absolute temperature over a limited range. In experimental studies of evaporation rates of water droplets with radii between 20 and 600 μm falling through a wind tunnel at temperatures > 0°C, Beard and Pruppacher (1971) provided approximate time-independent formulations for droplet surface temperatures based on previous studies (Ranz 1956). However, they did not take into account the transient bulk droplet heating rate term balanced by the loss of latent heat and gain of heat by conduction. They assumed that the difference in ambient and droplet surface temperatures is much smaller than their average. Watts (1971, 1972) and Watts and Farhi (1975) calculated the relaxation time and equilibrium droplet temperatures for raindrops with radii between 100 and 3000 μm in above-freezing ambient temperatures, solving for temperature in space and time analytically. However, they did not extend their study to investigate the temperature evolution of evaporating supercooled cloud droplets. These studies made simplifying assumptions including a linear dependence of saturated vapor density on temperature. Watts (1971) demonstrated that the equilibrium temperature of a stationary droplet is approximately the same as that of the wet-bulb temperature (TWB) when the relative humidity (RH) is typically >70% (Fig. 4, Watts 1971). Watts (1971) mentioned one of the major limitations of his method is that the linearization is invalid when RH is very low, leading to overprediction of cooling of the droplet surface and thus overestimation of evaporation (Fig. 4, Watts 1971). Watts (1971) recommended that for low RH, the equilibrium temperatures should be read off TWB from a psychrometric chart. Srivastava and Coen (1992) solved for the steady-state growth/decay rates of a single, stationary droplet by providing iterative solutions to the explicit vapor and heat diffusion equations as well as by approximating saturation vapor–density differences as a quadratic function of temperature differences between the droplet and the environment. Their paper did not specify droplet sizes. Their study showed that large errors (∼10%–30% under particular conditions) can occur if the traditional growth equation, assuming a linear relationship between the saturation vapor density difference and the temperature difference between the droplet and the ambient environment, is used in growth or decay estimation of hydrometeors in numerical models.
A detailed investigation of the evolution of supercooled droplet temperatures in evaporating regions of the cloud, such as cloud edges and tops, is important as these droplets may be fundamental to the initiation of ice processes (Hobbs and Rangno 1985; Rauber 1987; Rauber and Tokay 1991). In this study, we explore the evolution of supercooled cloud droplet temperature in a subsaturated environment, numerically determining the steady-state droplet temperature (Tss), the time (tss) to reach Tss, and the time (tst) of droplet survival at Tss under varying conditions of RH, ambient temperature (T∞), and pressure (P), similar to what might occur at cloud top and cloud boundaries. We present a numerical analysis that includes the time-dependent droplet heating rate term and empirical relationships between saturation vapor density and temperature, thus eliminating some of the simplifying assumptions (viz., saturated vapor density at the droplet surface assumed to be a linear or quadratic function of the temperature difference between the droplet and ambient air, and exclusion of the time-dependent droplet heating rate term) appearing in previous studies (Kinzer and Gunn 1951; Watts 1971; Srivastava and Coen 1992). The current paper extends the analysis of Srivastava and Coen (1992) by including the time-dependent heating rate of a droplet, assuming empirical relationships between saturation vapor density and temperature, and also provides a comparison between the results. The novelty of this study lies in quantifying the time scales the droplet can exist at a lower temperature before evaporating into the atmosphere, as well as the potential impacts of lower droplet temperatures on the enhancement of ice nucleation at cloud boundaries, particularly cloud tops. To date, no such calculations have been done for supercooled cloud droplet temperatures and radii over a wide range of ambient temperatures, relative humidities, and pressures typically found in cloud tops and edges. We explore a broad range of the parameter space representing initial droplet characteristics and environmental conditions found in the atmosphere, especially focused on relatively higher temperatures (>−20°C), where ice nucleation might be enhanced via evaporative cooling of supercooled cloud droplets. We also assess the potential influence of radiation on droplet cooling, relative to evaporation, for a wide range of environments. Finally, we examine the potential impact of using the steady-state droplet temperature (Tss), instead of the usual approximation of the droplet temperature (Tr) to be the same as that of the environment (T∞), on the number concentrations of activated INPs in three commonly used ice nucleation parameterization schemes, for a range of specified environments.
2. Theoretical framework
The steady-state diffusional growth or evaporation of water droplets under various environmental conditions is described in several texts (e.g., Pruppacher and Klett 1997, chapter 13). A nucleated cloud droplet either grows or evaporates due to the diffusion of water vapor associated with the vapor density gradient between the droplet surface and the ambient environment. Heat transfer occurs at the drop–air interface due to differences in the temperatures of the droplet surface and the environment. Even if the droplet starts at the same temperature as that of the unsaturated environment, evaporation takes place at first due to the vapor density gradient, leading to the cooling of the droplet surface, and setting up a thermal gradient between the droplet and its environment. Usually, while calculating evaporation/condensation rates, the droplet is considered to be in thermal equilibrium with the environment, and the difference between the droplet temperature and that of the environment is assumed to be negligible. However, as discussed earlier, under certain scenarios, the validity and limitations of these assumptions need to be reconsidered, especially for supercooled cloud droplets exposed to rapid humidity changes near cloud boundaries. Under these conditions, there is a continuous exchange of vapor and heat between the droplets and the environment and heat and vapor diffusion equations need to be solved simultaneously to accurately estimate droplet heating/cooling and growth/decay rates. For evaporating droplets, the droplet temperature can even be much lower than the environmental temperature during this period.
Here, we consider the physical mass and heat transfer mechanisms of vapor and thermal heat diffusion in the framework of the evolution of an evaporating cloud droplet, including the impact of the time-dependent cooling on droplet temperature. We provide a numerical solution to the coupled system of equations solved using an iterative method instead of an approximate analytical solution, as provided by previous studies for raindrops (Kinzer and Gunn 1951; Watts 1971).
Conceptualized schematic of thermal and radial evolution of the idealized case of an evaporating cloud droplet within a cloud-top generating cell. The droplet with an initial radius (r0), and initial droplet temperature (
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
The equation used to evaluate L is a linear fit between −20° and 0°C to the data provided in List (1951).
The ODEINT solver in Python is used to numerically solve the system of ODEs, based on the Fortran library of ODEPACK, which uses a technique of switching automatically between the nonstiff Adams method and the stiff backward differentiation formula method to solve the equations more efficiently.
3. Results
a. Numerical results
Figures 2 and 3 depict the evolution of cloud droplet temperature Tr (red) and radius r (blue) in specific constant environments, for three different initial cloud droplet radii r0, and initial droplet temperatures
Thermal (red) and radial (blue) evolution of evaporating supercooled cloud droplets, with three different initial droplet radii (r0 = 10, 30, and 50 μm, marked by dashed, solid, dash–dotted lines, respectively), three different relative humidities (RH = 10%, 40%, 70%), four different pressures (P = 500, 600, 700, and 850 hPa), and ambient temperature T∞ = 268.15 K (−5°C). Droplet survival time tL is marked in seconds (blue) near their respective radius evolution curves (blue). Time to reach steady state tss is marked in seconds (red) along with the respective r0 values, and ΔT = T∞ − Tss is marked in kelvins (red) near the temperature evolution curve.
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
As in Fig. 2, but for an ambient temperature T∞ = 263.15 K.
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
As the cloud droplets evaporate within the subsaturated environment, their temperatures rapidly decrease due to evaporative cooling, eventually coming to a steady-state condition when the cooling is balanced by the conductive heating from the surrounding air. For an ambient temperature of 268.15 K, P = 500 hPa, and RH = 10%, Fig. 2a shows the droplets reach their steady-state temperature, Tss = 260.9 K, a reduction of nearly 7.3 K from the initial droplet temperature of 268.15 K. Irrespective of the initial drop size, the droplets always reach the same Tss for the same set of environmental conditions (RH, T∞, and P). The smaller the initial droplet radius (r0), the smaller the value of tss and the faster the droplet reaches Tss. For instance, a 10 μm droplet reaches Tss in ∼0.02 s while a 50 μm droplet requires ∼0.4 s. Comparing all the plots across all panels, it is observed that the time to reach steady state tss is not a strong function of environmental variables and depends primarily on droplet size. From Eq. (11),
The amount of droplet evaporative cooling from the ambient environment, ΔT = T∞ − Tss is independent of initial droplet size, but a strong function of RH as expected since drier conditions lead to stronger evaporative cooling (compare, for example, Figs. 2a, 2e, 2i). For instance, for P = 500 hPa, for RH = 10%, the droplets cool by almost 7.3 K to 260.9 K, whereas for RH = 70%, the amount of cooling is much smaller, ∼2.2 K. At P = 850 hPa, for RH = 10%, the droplets cool by 5.1 K (Fig. 2d), whereas for RH = 70%, the droplets cool by 1.6 K (Fig. 2l). The degree of cooling is a weaker function of P (compare, for example, Figs. 2a–d). For P = 500 hPa in Fig. 2a, the droplets cool by ∼7.3 K while for P = 850 hPa in Fig. 2d, the droplets cool by 5.1 K. The smaller decrease in droplet temperature at higher pressure levels is due to lower values of vapor diffusivity at higher atmospheric pressures. Thus, as P decreases with increasing altitude, the cooling of the evaporating droplets is enhanced so that the droplets reach even lower temperatures relative to the ambient environment.
The evaporating cloud droplet lifetime, defined as tL, is the time between the onset of evaporation and the time it reaches a radius of 0.7 μm. The 0.7 μm radius was chosen to avoid consideration of solution and curvature effects as noted earlier. The droplet radii decrease in size at different rates depending on the ambient subsaturation, pressure, temperature, and initial droplet size. The droplets reach Tss (when evaporative cooling is balanced by conductive warming), continuing to evaporate until they dissipate completely. Figure 2 also shows the temporal evolution of r in a subsaturated environment (in blue). As expected, the larger droplets survive longer compared to the smaller droplets for the same RH. For example, a 50 μm droplet survives for about 18.4 s after being introduced to an environment with RH = 10%, whereas a 10 μm droplet can survive for only 0.7 s under the same ambient conditions (Fig. 2a). Similar dependence between droplet size and droplet survival times can be seen for other RH, T∞, and P conditions. When the RH is 70%, the droplets survive longer due to weaker evaporation rates under more humid conditions (59.5 and 2.4 s for 50 and 10 μm droplets, respectively) (Fig. 2i). At higher P, for the same RH and T∞ conditions, the droplet survival times increase, with a larger increase for larger droplets. For example, for P = 500 hPa and RH = 40%, a 50 and 10 μm droplet survive for 28.7 and 1.2 s, respectively (Fig. 2e), whereas for P = 850 hPa, RH = 40%, the 50 and 10 μm droplet exist for 39.6 and 1.6 s, respectively (Fig. 2h). This is again due to the lower values of vapor diffusivity, leading to decreased evaporation rates at higher P.
Figure 3 shows similar relationships between the parameters of interest (Tss, tss, tL, ΔT), initial droplet characteristics, and environmental variables, but now for a lower initial ambient temperature of 263.15 K. Similar to cases shown in Fig. 2, the time to reach steady-state temperature depends primarily on initial droplet radius across all panels in Fig. 3, e.g., ∼0.02 s for 10 μm droplet, and ≤0.52 s for 50 μm droplet. As shown in Figs. 3a–d, as the P increases for RH = 10%, the amount of droplet evaporative cooling decreases from 5.6 K at P = 500 hPa (Fig. 3a) to 3.9 K for P = 850 hPa (Fig. 3d). For P = 500 hPa and a higher RH = 70%, the amount of cooling is 1.8 K (Fig. 3i).
Compared to T∞ = 268.15 K shown in Fig. 2, the amount of cooling for T∞ = 263.15 K in Fig. 3 is smaller, and the survival times of droplets are longer for all corresponding RH and P conditions. For instance, for RH = 10%, P = 500 hPa, the droplets in the lower T∞ environment cool to about 257.55 K, a reduction of about 5.6 K (Fig. 3a), as compared to a decrease of 7.3 K under the same RH and P conditions but with a higher T∞ (Fig. 2a). In general, the magnitude of the reduction in droplet temperature during evaporation is greater, and droplet survival time is smaller at higher ambient temperatures. This is due to greater values of the diffusion coefficient of water vapor (D) at higher ambient temperatures for a fixed RH, leading to enhanced net evaporative cooling and decay rates for the droplets.
Figure 4 shows the general relationship between the evaporating droplet temperature reduction, ΔT = T∞ − Tss (color shading) for a broad range of environmental conditions (RH, P, and T∞) and initial droplet radii (r0). As expected from Figs. 2 and 3, Tss and ΔT are not sensitive to r0 for a given environment. For the same r0, Tss and ΔT strongly depend on ambient RH and on T∞ and P for low RH. In contrast, there is little decrease in droplet temperatures at RH > 85% independent of T∞. The cloud droplets reach lower Tss for drier, higher ambient temperature, and lower ambient pressure environments. The strongest cooling of ∼10.3 K is observed for the driest ambient conditions (RH < 5%), lowest pressure (P = 500 hPa), and highest ambient temperature (T∞ ∼ 273.15 K) considered in this analysis. Hence, drier conditions, lower pressures, and higher ambient temperatures are most suitable for the greatest evaporative cooling of supercooled cloud droplets.
Shaded contour plot of the decrease in temperature (ΔT; in K), dashed blue contours are the time (tss; in s) to reach the steady-state temperature, and black contours are the time of survival (tst = tL − tss; in s) at the steady-state temperature Tss. The panels cover a range of relative humidities (RH = 1%–100%) and ambient temperatures (T∞ = 273.15–254.15 K; i.e., 0° to −19°C) of the environment for three different initial droplet radii (r0 = 10, 30, and 50 μm), and four different pressures (P = 500, 600, 700, and 850 hPa).
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
To get a sense of the potential implications of depressed cloud droplet temperatures for cloud microphysical processes such as ice nucleation, we need to quantitatively determine the time interval that droplets exist at lower temperatures before they evaporate completely (the cutoff radius for this study = 0.7 μm). Thus, we define a parameter for the time of droplet survival, tst, at the steady-state temperature, given by tst = tL − tss, where tL, as defined earlier, is the total time taken by the droplet to evaporate. As shown in Fig. 4, for the initial droplet radii of 10, 30, and 50 μm, values of tst increase for the larger droplets, as expected, for similar environments of P, RH, and T∞. For example, in Figs. 4a, 4e, and 4i, for values of RH and T∞ in the upper-left corner of the individual figures, 50 μm droplets survive at Tss ∼ 10 K lower than T∞ for at least 15 s before dissipation, whereas 30 μm droplets survive for between 5 and 7 s and 10 μm droplets for <1 s. At higher P, for the same RH and T∞, values of tst increase but the magnitude of droplet evaporative cooling is smaller. Thus, given a particular environment, even though smaller cloud droplets achieve the same lower steady-state temperatures as that of larger droplets as they are evaporating, the larger droplets exist for longer times at that steady-state temperature before they dissipate.
It is instructive to compare droplet lifetimes in the current study with those that would occur if droplet evaporative cooling were not considered. Table 1 provides a comparison between time to reach steady state (tss), and droplet lifetime (tL) as calculated from this study, and droplet lifetime (tc) as estimated with the classical approach of using T∞ as the steady-state temperature of the evaporating droplet surface, for initial droplet radii (r0 = 10, 30 and 50 μm), relative humidities (RH = 10%, 40%, 70%), and pressures (P = 500 and 850 hPa), and ambient temperature T∞ = 268.15 K (−5°C) and 263.15 K (−10°C). As shown in Table 1, tL is always greater than tc, since as the droplet begins to cool evaporatively, reduction in saturation vapor pressure at the droplet surface leads to a slower evaporation rate, as compared to that of the classical approach (which assumes that the difference between the temperature at the droplet surface and the ambient environment is negligible). The difference between the two lifetimes (Δt = tL − tc) increases with increasing r0, RH, P, and decreasing T∞. For example, in an ambient environment with RH = 10%, T∞ = 268.15 K, P = 500 hPa, Δt = 0.3, 3.0, and 8.5 s for r0 = 10, 30, and 50 μm, respectively, whereas for the same environment but with RH = 70%, Δt increases by at least 3.5 times to 1.2, 10.7, and 29.9 s for r0 = 10, 30, and 50 μm, respectively. For another environment, with RH = 40%, T∞ = 268.15 K, for P = 500 hPa, Δt = 0.6, 5.0, and 13.9 s for r0 = 10, 30, and 50 μm, respectively, while for P = 850 hPa, the respective values are 0.6, 5.1, and 14.4 s. Thus, from Table 1, the values of Δt are less sensitive to P and T∞ as compared to r0 and RH.
Comparison between different time scales (in s)—time to reach steady state (tss), droplet lifetimes (tL) as calculated from this study, and droplet lifetimes (tc) using the classical approach, for initial droplet radii (r0 = 10, 30, and 50 μm), relative humidities (RH = 10%, 40%, 70%), and pressures (P = 500 and 850 hPa), and ambient temperature T∞ = 268.15 K (−5°C) and 263.15 K (−10°C).
Figure 5 shows the relationship between the difference in droplet lifetimes, Δt = tL − tc and evaporating droplet temperature reduction, ΔT, for a broad range of environmental conditions (RH, P, and T∞) and initial droplet radii (r0). Figures 5a–l illustrate the general dependency of Δt on environmental conditions and initial droplet characteristics. For droplets as small as r0 = 10 μm, the difference between the two lifetimes is <10 s for all environments considered in this study (Figs. 5a–d). However, for larger droplets, the value of Δt strongly increases with r0 and RH. For environments with RH > 60%, Δt > 10 s for r0 = 30 μm (Figs. 5e–h), and >30 s for r0 = 50 μm (Figs. 5i–l). Even though the decrease in evaporating droplet temperatures under high RH conditions is typically ≤1 K, there occurs a large increase in droplet lifetimes. For example, for the largest droplets in this study (r0 = 50 μm) in environments with RH > 85%, Δt >100 s. Implications of these results on ice nucleation will be discussed in section 4.
Shaded contour plot of the decrease in droplet lifetime (Δt = tL − tc; in s) and black contours of the decrease in droplet temperature (ΔT; in K) for a range of relative humidities (RH = 1%–92%), and ambient temperatures (T∞ = 273.15–254.15 K; i.e., 0° to −19°C) of the environment for three different initial droplet radii (r0 = 10, 30, and 50 μm), and four different pressures (P = 500, 600, 700, and 850 hPa).
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
b. Comparison with thermodynamic wet-bulb temperatures
Comparison between thermodynamic wet-bulb temperatures and simulated droplet steady-state temperatures for two different ambient temperatures, three ambient relative humidities, and four ambient pressures.
c. Comparison with Srivastava and Coen (1992)
The results herein compare well with a previous study of evaporation of single, stationary hydrometeors by Srivastava and Coen (1992), who provided iteratively solved steady-state solutions, using saturation vapor pressure relations from Wexler (1976) to calculate saturation vapor density. In their analysis, they ignored the droplet heat storage and radiation terms in the droplet heat budget. The current study extends the analysis of Srivastava and Coen (1992) by including the time-dependent heating rate of the droplet, assuming empirical relationships between saturation vapor density and temperature using Lowe (1977). Our estimation of steady-state droplet temperatures, at corresponding temperatures, pressures, and relative humidities, agree well with their results (Fig. 1, Srivastava and Coen 1992) as shown in Table 3.
Comparison between temperature differences (ΔTsc and ΔTRRD) between an evaporating droplet and its environment as given by the numerical solutions from Srivastava and Coen (1992, or SC92) and this study (RRD), respectively, for four different ambient temperatures T∞ = 273.15, 268.15, 263.15, 258.15 K, two ambient relative humidities RH = 10% and 70%, and ambient pressure P = 600 hPa.
4. Potential implications for ice nucleation
As noted in the introduction, supercooled cloud droplet temperature and its time of survival in a subsaturated environment are important factors for ice formation in clouds, since homogeneous and certain heterogeneous ice-nucleation mechanisms require the presence of supercooled cloud droplets (Ansmann et al. 2009; de Boer et al. 2011). Ice nucleation rates are dependent on temperature (Wright and Petters 2013; Kanji et al. 2017) as well as time (Vali 1994). There have been two contrasting schools of thought on the issue of time dependence in the field of ice nucleation modeling—the time-independent “singular hypothesis,” also referred to as the deterministic model (Langham and Mason 1958) and the time-dependent “stochastic hypothesis,” also referred to as the classical nucleation theory (Bigg 1953). The singular hypothesis considers ice nucleation to be an instantaneous process where certain sites on the ice-nucleating particle activate instantly under specific conditions, and the ice crystal forms immediately. This approach ignores the time dependence of the process, resulting in a straightforward relationship between primary ice initiation and ambient conditions. On the other hand, according to the stochastic hypothesis, ice clusters in embryos form and vanish continually, with a temperature-dependent frequency, at the interface of immersed aerosol particles. Based on the results in section 3a, evaporative cooling of supercooled cloud droplets in subsaturated environments may lead to enhancement of ice nucleation, particularly near cloud boundaries, in two ways: (i) an instantaneous increase in activated INPs via a decrease in droplet temperature (supported by the singular hypothesis) and (ii) an increase in the supercooled droplet lifetime, within time scales associated with droplet freezing, before the droplets evaporate, which allows more time for nucleation events to occur (supported by the stochastic hypothesis).
One potential concern with the time-dependent evaporation enhancement mechanism of ice nucleation was that supercooled droplets might evaporate even before they can freeze (Phillips et al. 2007). Laboratory investigations of the time dependency of heterogeneous ice nucleation, especially for the ambient temperature ranges explored in this study, have been scarce. To our knowledge, no studies have reported nucleation rates at time scales < 1 s, or at temperatures higher than −14°C for supercooled water droplets. Vali (1994) concluded that the rate of freezing was dependent both on temperature and time from a small number of isothermal experiments with a period of 10–15 min, with temperatures between −16° and −21°C. Recent laboratory studies suggest weaker dependence on time, relative to temperature fluctuations for dust aerosol proxies (Wright and Petters 2013). Welti et al. (2012) reported a frozen fraction of droplets of 12% in 1 s at −30°C, and 82% in 1 s at −36°C for 800 nm kaolinite particles. Broadley et al. (2012) observed immersion freezing of droplets using optical microscopy, with an illite-rich powder called NX illite and found that 5 out of 50 droplets froze in 42.6 s, and 10 out of 50 droplets froze in 85.3 s at −28°C, while 6 out of 63 droplets froze in 70 s at −30°C (their Table 4 and Figs. 7a,b). Murray et al. (2012) summarized droplet freezing experiments with kaolinite and found a droplet freezing fraction of 12% in 250 s and 20% in 500 s for droplets with radii of 6.4 to 17.6 μm at −29°C. A more recent study by Jakobsson et al. (2022) experimentally observed the time dependency of nucleating aerosols. More relevant to this study, the frozen fraction ranged from 5% for rural continental aerosols at −14°C, 22% for mineral dust and combustion-dominated aerosols at −14°C and 40% for continental-polluted aerosols at −16°C, all of these at 1 s (their Figs. 9a,b). The current study shows that the evaporating droplets reach lower steady-state temperatures very quickly ∼0.03, ∼0.18, and ∼0.5 s for 10, 30, and 50 μm droplets, respectively. The droplet lifetimes at the steady-state temperatures typically ranged from ∼1 to 20 s for 10 μm, ∼5 to 200 s for 30 μm, ∼15 to 700 s for 50 μm initial size droplets, respectively. This study also shows that supercooled droplets, especially the larger ones under drier conditions, can survive at much lower steady-state temperatures for 5 s to over 10 min before evaporating. For example, from Fig. 2f, a droplet with r0 = 50 μm at an ambient temperature of −5°C, P = 600 hPa, and RH = 40% will cool by 4.2°C and survive for about 32 s at that steady-state temperature before dissipation. As shown in Table 1 and Fig. 5, the droplet lifetimes, as calculated from this analysis, are longer than those of evaporating droplets ignoring droplet cooling. This time difference allows more time for potential nucleation events to occur, especially for larger droplets in more humid environments, even if the decrease in droplet temperatures is typically lower. Thus, comparing droplet survival times obtained from this study with droplet freezing time scales available from experimental investigations, it appears plausible that droplet freezing events are likely to occur in the time window between the droplets reaching their steady-state temperature and complete evaporation.
Supercooled cloud droplets with large radii have been observed in cloud-top environments (Hobbs and Rangno 1985; Rauber and Grant 1986; Wang et al. 2020) and have been associated with aircraft icing (Rasmussen et al. 2002; Rauber et al. 2014). Evidence from field studies in cloud-top generating cells containing supercooled water droplets at temperatures ranging from −5° to −35°C clearly shows that a sufficient number of nucleation events occur in the cells within the time scales of the cells’ existence and that a plume of ice particles forms as ice particles grow and fall deeper into the cloud (Plummer et al. 2014, 2015; Wang et al. 2020; Zaremba et al. 2021). The current study suggests that droplet cooling by evaporation, and the increase in droplet lifetimes due to cooling at the droplet surface, may contribute to the rapid ice formation in these cells.
However, it is challenging to represent time-dependent stochastic nucleation in numerical models due to its complexity including many parameters for various components of INPs. If such statistics are not accurately modeled, active INPs can be overpredicted by several orders of magnitude at long times (Vali 1994; Vali and Snider 2015). It has also been observed that ice nucleation has a far stronger dependency on temperature than time (Wright and Petters 2013). Therefore, even though modern laboratory experiments have confirmed the existence of time dependence of INP activity, the singular hypothesis is still used as a simple approximation to the leading-order behavior of ice crystal initiation and is widely prevalent in numerical cloud models.
The results presented in section 3 are relevant for ice initiation in the presence of supercooled liquid water droplets via immersion freezing or contact nucleation, either through external contact (Young 1974) or an inside-out mechanism (Durant and Shaw 2005), as supported by the time-independent singular hypothesis. Contact freezing is thought to be more effective at ice production at higher temperatures as compared to the other modes of ice nucleation (Young 1974). Particles can encounter a droplet by Brownian motion, diffusiophoresis, and/or thermophoresis. Studies have shown that thermophoresis is more effective than diffusiophoresis for INPs with radii < 1 μm, with this process being most effective when droplets are evaporating in zones of high subsaturation (Slinn and Hales 1971; Carstens and Martin 1982). Hobbs and Rangno (1985) suggested that thermophoretically enhanced contact freezing nucleation in the evaporating regions of clouds might be the primary process responsible for the onset of rapid ice formation observed when significant numbers of cloud droplets are greater than 20 μm in diameter. Numerical results from our analysis show that larger evaporating cloud droplets (r0 = 30–50 μm) can exist at a steady-state temperature that can be several degrees lower than that of the ambient environment for about 5 s to over 10 min depending on the subsaturation of the environment before complete dissipation, thus potentially enhancing the probability of a contact nucleation event via thermophoresis.
An “inside-out” mode of contact nucleation called evaporation freezing was reported by Durant and Shaw (2005) from laboratory observations, wherein an insoluble INP immersed inside an evaporating supercooled water droplet comes in contact with the air–water interface and freezes it. As a potential explanation of this phenomenon, Baker and Lawson (2006) noted that as the droplets decrease in size during evaporation, the probability of a particle being near the droplet surface increases, potentially increasing the probability of inside-out contact nucleation. Hence, the number of potentially active INPs within droplets may increase in regions of cloud evaporation. Our work shows that droplets will also be at a lower steady-state temperature than the ambient environment, another factor favoring ice nucleation in evaporating regions of clouds. Hu et al. (2023b) showed from high-resolution WRF simulations of shallow, mixed-phase, postfrontal clouds over the Southern Ocean that the model drastically overpredicts supercooled liquid water compared to observations. They proposed a seeder–feeder mechanism by which cloud-top ice generation can seed ice growth through the cloud. Based on the results from the current study, it might be possible that the decrease in temperature felt by drops in this region could also play a role in ice formation in these clouds.
Over the last few decades, various parameterization schemes for heterogeneous ice nucleation have been developed for numerical cloud and climate models based on theory, laboratory measurements, and field observations (Fletcher 1962; Cooper 1986; Meyers et al. 1992; DeMott et al. 1998; Khvorostyanov and Curry 2000; Phillips et al. 2008). However, widely used INP parameterization schemes (Fletcher 1962; Cooper 1986; Meyers et al. 1992) in numerical cloud models underestimate the predicted concentrations of ice crystals, especially at higher temperatures (0° to −10°C) (e.g., Wallace and Hobbs 2006). This discrepancy can be due to inadequate parameterization of primary ice nucleation, secondary ice production, or both. In a more recent study, DeMott et al. (2010) aggregated data over 14 years from nine field experiments all over the globe and presented an improved INP parameterization scheme, which considers temperature as well as concentration of ambient aerosols (Na) with diameter > 0.5 μm. The common characteristic of all these schemes is the use of the ambient temperature to estimate activated INP, as opposed to the droplet temperature. The numerical simulations from this study show that evaporating droplets can exist at a lower temperature under certain scenarios, especially at lower relative humidity, ambient temperatures closer to 0°C, and lower-pressure environments. To investigate the maximum enhancement in activated INP concentrations due to evaporative cooling of supercooled water droplets in a prescribed environment, we assume that the activation in the parameterization schemes is related to the steady-state droplet temperature rather than the ambient temperature.
Figure 6 depicts that for the three commonly used schemes considered here, the maximum fractional increase in activated INP using Tss is higher for environments characterized by higher T∞ (i.e., closer to 0°C), lower RH, and lower P, due to enhanced droplet evaporative cooling. For the Fletcher scheme, the highest increase in activated INPs, by almost two orders of magnitude, is observed for P = 500 hPa, and almost 35 times for P = 850 hPa at very dry conditions, closer to 0°C (Figs. 6a,d). For T∞ < −5°C, the Fletcher scheme in general shows the greatest increase in the number concentration of activated INPs, especially for RH < 60%, compared to the Cooper and Demott schemes (Figs. 6e–l). The Cooper scheme shows lower increases overall with the highest increase by a factor of 15 for P = 500 hPa (factor of 7 for 850 hPa) at the highest of temperatures and driest of conditions considered here (Figs. 6e,h). For the Cooper scheme (Figs. 6e–h), there is a negligible increase in activated INPs for RH > 60%, whereas, for the Fletcher scheme (Figs. 6a–d), the same is true for RH > 80%. For the Demott scheme, the highest increase in activated INPs is observed for −2° < T∞ < 0°C and RH < 70%. A moderate increase by at least a factor of 5 occurs for T∞ as low as −8°C for drier scenarios, and for RH up to 90% for higher temperatures. There is a slight decrease in the number of activated INPs with increasing pressure (Figs. 6i–l).
Fractional increase in INP for three different parameterization schemes: (i) Fletcher (1962), (ii) Cooper (1986), and (iii) Demott et al. (2010), for a range of environmental relative humidities (RH = 1%–100%), and ambient temperatures (T∞ = 273.15–254.15 K; i.e., 0° to −19°C) for initial droplet radii, r0 = 30 μm and four different pressures (P = 500, 600, 700, and 850 hPa).
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
The spontaneous increase in activated INPs due to lower supercooled droplet temperatures can lead to enhancement of ice formation in certain regions of clouds, particularly at cloud tops and edges, especially at higher subfreezing ambient temperatures. Also, while modeling ice nucleation processes that require the presence of supercooled water droplets in subsaturated environments, a more accurate depiction of the evaporating cloud droplet temperature may partially address the discrepancy between INP concentrations and ice particle concentrations, since the concentration of activated INPs is sensitive to droplet temperature. Future modeling experiments, considering a population of freezing and evaporating droplets and considering the associated droplet survival time scales, latent heat exchange with their environments, and other feedbacks, are required to test the efficacy of the potential ice-nucleation enhancement mechanism through evaporation in a robust dynamical model setup.
5. Potential impacts of radiation
The contribution of radiative heating or cooling on droplet temperatures for environments close to saturation was assessed by Roach (1976) and Marquis and Harrington (2005). Marquis and Harrington (2005) demonstrated that the difference between the droplet temperature and that of the environment was ≤1 K for droplet radii ≤ 200 μm, neglecting shortwave heating (their Fig. 4a). They also reported that in nocturnal situations, the largest drizzle drops with radii ≥ 1000 μm could be as much as 3 K colder than the environment.
Values of ratio RER/EL (100%) for r0 = 10, 30, and 50 μm, T∞ = 273.15, 268.15, 263.15, and 254.15 K, RH = 10%, 20%, 70%, and 80%, and P = 500 and 850 hPa.
Figure 7 shows RER/EL (multiplied by 100%) as a function of T∞ and RH for all values considered in this study. The figure shows that for r0 = 10 μm and for both P values = 500 and 850 hPa, the radiative contribution is <10% for all T∞ and RH < 90%, and < 2% for most ambient conditions (Figs. 7a,d). At greater values of P, RER/EL is higher for all r0, due to reduced evaporative cooling of the droplet. Note that, as the evaporative cooling gets reduced at higher RH (particularly RH > 90%), radiation becomes more influential and should be included for investigations studying droplet temperatures in ambient environments close to and at saturation (e.g., as in Marquis and Harrington 2005). For larger r0, at the same RH and T∞, the effect of radiation starts increasing. For instance, for P = 850 hPa, T∞ = 267 K, and RH = 90%, the radiative contribution is slightly <20% and at 30% for r0 = 30 and 50 μm, respectively (Figs. 7e,f). For the typical subsaturated cloud-top environments and droplet sizes considered here, radiation at most plays a small role in altering the droplet temperatures.
Ratio RER/EL × 100%, for two different environmental pressures of P = 500 and 850 hPa and initial droplet radii r0 = 10, 30, and 50 μm.
Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-22-0239.1
6. Other processes
In this idealized numerical study, we have considered single, stationary, pure water droplets evaporating in an ambient environment, defined by prescribed temperature and moisture fields far away from a droplet, similar to the classical approach often used in the literature (Sedunov 1974; Rogers and Yau 1989; Srivastava and Coen 1992). Fukuta (1992) provided a correction to the ambient conditions at a radius similar to the mean distance between droplets (∼1 mm), leading only to minor modifications. Vaillancourt et al. (2001) compared two approaches to study droplet condensational growth rate—(i) a simple approach by solving the classical droplet growth equation described in Eqs. (11)–(15) of their paper and (ii) spatiotemporally varying temperature and water vapor distributions in the vicinity of the droplet. They found that the simple approach provided accurate time evolution of droplet growth rate for low cloud droplet concentrations (their Fig. A1). Thus, based on previous studies, the usage of steady-state expressions for the ambient environment provides a reasonable approximation of the evaporation rates of individual droplets, especially for low cloud droplet number concentrations.
In nature, a population of cloud droplets always exists in proximity to one another and grows or evaporates due to the presence of moisture and thermal gradients in their immediate vicinity. The volume around a droplet affected by these gradients has a radius of approximately 10 to 20 droplet radii (Vaillancourt et al. 2001; Grabowski and Wang 2013). For a droplet concentration of ∼100 cm−3, the mean interparticle distance, d ∼ 2100 μm. The droplet radii, considered in this study, range from 0.7 to 50 μm. For the largest considered droplet with a 50 μm initial radius, 20 droplet radii = 1000 μm, which is well below the calculated mean interparticle distance of ∼2100 μm for a droplet concentration of ∼100 cm−3. Typically, droplet concentrations in the cloud-top regions of orographic clouds, and stratiform clouds within cyclones, rarely exceed 100 cm−3 and are often closer to 40–50 cm−3, in which case the interparticle distance is even greater. A reasonable assumption under such cases can be to neglect the influence of other droplets and their immediate environments and assume a constant ambient temperature, pressure, and relative humidity, as done in this study. For a droplet number concentration of 50 cm−3, d = 2714 μm. Assuming D = 2.5 × 10−5 m2 s−1, for T∞ = 273.15 K and P = 850 hPa, the time required for diffusion to influence a length of 2.714 mm is about 0.3 s, which is ∼10tss for a 10 μm droplet, ∼2tss for a 30 μm droplet, and ∼tss for a 50 μm droplet. Now, assuming D = 4 × 10−5 m2 s−1, for T∞ = 263.15 K and P = 500 hPa, the time required for diffusion to influence a length of 2.714 mm, is about 0.18 s, which, generally, is ∼9tss for a 10 μm droplet, ∼1.1tss for a 30 μm droplet, and ∼0.5tss for a 50 μm droplet. So, for a population of larger droplets, say 50 μm, at lower pressures, there might be some interference from neighboring droplets. However, to determine its influence on droplet temperatures, high-resolution model simulations of coupled heat and mass transfer at the evaporating droplet interfaces, resolving the moisture and thermal gradients in the vicinity of the droplets, are required.
Since the droplets were assumed to have infinite thermal heat conductivity, the droplet temperatures are considered to be uniform, devoid of any thermal gradients that could give rise to internal drop circulation (Karbalaei et al. 2016). However, if one considers finite droplet heat conductivity, there is potential for even stronger cooling at the droplet interface, as evaporation is a surface phenomenon.
7. Conclusions
In this study, we examined the evolution of cloud droplet temperature and lifetime in a subsaturated environment by solving the fully coupled, time-dependent equations of mass and heat transfer between a stationary pure supercooled water droplet and its environment. We quantified the deviation of droplet temperature from that of the environment and the increase in droplet lifetime during droplet evaporation, for a range of initial droplet radii and temperatures and a range of atmospheric conditions, including ambient temperature, pressure, and relative humidity. This analysis aimed to identify environmental conditions and initial droplet characteristics that are more favorable for enhanced droplet cooling by evaporation. The motivation was to determine whether cloud droplet cooling by evaporation could potentially enhance the number concentration of activated INPs and amplify ice formation in cloud tops and edges.
The idealized numerical simulations show that an evaporating supercooled cloud droplet can exist at a temperature lower than that of the ambient atmosphere. The steady-state temperature of an evaporating droplet was shown to be within 0.75 K of the thermodynamic wet-bulb temperature. The degree of supercooling of the droplet increased for drier conditions, higher ambient temperatures, and lower pressures. Over a wide range of ambient relative humidity and temperature, the typical supercooling was on the order of 1–5 K. Larger evaporating cloud droplets (initial radii from 30 to 50 μm) can survive for times ranging from 5 s to over 10 min at these temperatures, allowing time for enhanced ice nucleation associated with lower droplet temperature. For very low RH, as might occur above stratiform cloud tops with cloud-top generating cells, calculations show droplet cooling can be even greater than 5 K (up to 10.3 K), especially for relatively higher ambient temperatures and lower pressures. Radiation was found to play a minor role in influencing droplet temperatures, except for larger droplets in environments with relative humidities closer to saturation.
The decrease in evaporating droplet temperature under high RH conditions (≳70%) is typically ≲1 K; however, the lower saturation vapor pressure at the droplet surface leads to a slower evaporation rate, resulting in an extended lifetime compared to what it would be if droplet cooling did not occur. For example, droplet lifetime will be increased by more than 100 s for larger droplets with r0 = 50 μm in environments with RH > 85%, allowing additional time for the possibility of an ice nucleation event.
This idealized study also explored the maximum increase in activated INPs that would occur based on widely used ice nucleation parameterization schemes if they were to use the steady-state cloud droplet temperature instead of the ambient temperature. The decrease in droplet temperatures during evaporation was shown to affect the predicted concentrations of activated INPs. The typical increase in activated INP concentrations was about a factor of 2–30 depending on ambient conditions, with the greatest increases (≥100) coincident with low RH, low P, and T∞ closer to 0°C.
This study recommends further consideration of the temperatures and lifetimes of supercooled cloud droplets in subsaturated environments, particularly when modeling heterogeneous ice nucleation processes that require supercooled liquid droplets since the concentration of activated INPs is sensitive to droplet temperature as well as lifetime. In INP parameterization schemes, the evaporating supercooled droplet temperature can be approximated by the thermodynamic wet-bulb temperature. The results from this study may also partially help reconcile discrepancies between observed ice particle concentrations and activated INPs, especially at relatively higher sub-0°C temperatures (between −20° and 0°C).
Acknowledgments.
This work was funded by the NASA Earth Venture Suborbital-3 (EVS-3) IMPACTS program under Grant 80NSSC19K0355 and the NASA CAMP2Ex program under Grant 80NSSC18K0144. This research was also supported by the National Science Foundation under Grant NSF AGS-2016106. We thank Tim Brice for providing more information about the thermodynamic wet-bulb temperature calculation procedure used by NWS. We also thank Dr. Zachary J. Lebo and two anonymous reviewers for their thorough reviews and comments to help improve the manuscript.
Data availability statement.
This analysis was theoretical and did not use any field data. The thermodynamic wet-bulb temperature data (Table 1) used in this study are calculated using the same procedure as publicly available from NWS wet-bulb temperature calculator at https://www.weather.gov/epz/wxcalc_rh.
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