The question of whether persistent ice crystal precipitation from supercooled layer clouds can be explained by time-dependent, stochastic ice nucleation is explored using an approximate, analytical model and a large-eddy simulation (LES) cloud model. The updraft velocity in the cloud defines an accumulation zone, where small ice particles cannot fall out until they are large enough, which will increase the residence time of ice particles in the cloud. Ice particles reach a quasi-steady state between growth by vapor deposition and fall speed at cloud base. The analytical model predicts that ice water content (wi) has a 2.5 power-law relationship with ice number concentration (ni). wi and ni from a LES cloud model with stochastic ice nucleation confirm the 2.5 power-law relationship, and initial indications of the scaling law are observed in data from the Indirect and Semi-Direct Aerosol Campaign. The prefactor of the power law is proportional to the ice nucleation rate and therefore provides a quantitative link to observations of ice microphysical properties.
 Long-lived mixed-phase clouds are frequently observed in the Arctic region [e.g., Shupe et al., 2006; Verlinde et al., 2007; McFarquhar et al., 2011] where they play an important role in the radiation balance [e.g., Lubin and Vogelmann, 2011]. Mixed-phase layer clouds with similar characteristics exist over many other regions of the earth as well [Westbrook and Illingworth, 2013]. Observations show that ice particles precipitate from these clouds nearly all the time [McFarquhar et al., 2011]. Recent experiments and modeling studies have led to great strides in understanding the complex and coupled radiation, dynamics, and microphysics of these clouds [Morrison et al., 2011], but why the mixed-phase clouds can exist for such a long time with steadily precipitating ice particles remains uncertain. One of the essential questions is how to replenish the ice nuclei that are quickly removed after activation and subsequent growth and sedimentation of ice crystals. It has been suggested that ice nuclei entrained from the top of the boundary layer might offset the ice particles lost at the cloud base [e.g., Avramov et al., 2011], and some tentative observational evidence for that has been found [Jackson et al.2012]. However, Fridlind et al.  concluded that ice nuclei concentrations above cloud top were too low to account for observed ice number concentrations in cloud, given realistic entrainment rates. Morrison et al.  suggested it might be the existence of low-efficiency contact ice nucleation that extends the mixed-phase cloud lifetime, while Fridlind et al.  showed that evaporation ice nucleation at cloud top may also help explain the persistence of mixed-phase clouds. Recently, Westbrook and Illingworth  argued that a time-dependent ice nucleation process in supercooled layer clouds would be a plausible explanation for the observed persistence of ice precipitation, and that concept can be taken as one of the motivations for this work.
 This letter provides an analytical model describing ice microphysical properties in a mixed-phase stratiform cloud, with emphasis on understanding the balanced state of ice nucleation, vapor growth, and sedimentation. The model is referred to as “minimalist” because we seek the minimum number of physical assumptions necessary to give a reasonable explanation for the presence of ice persisting over long time and also provide a reasonable estimate of the ice crystal mass and precipitation rate.
2 Model Description
 The model is based on the assumption of horizontally uniform and steady state conditions, in which the rate of formation of ice crystals within the cloud is balanced by the rate of removal of ice crystals through precipitation. For simplicity, we assume the cloud has uniform temperature and possesses a sufficient quantity of supercooled water so that the cloud humidity can be taken as saturated with respect to liquid water throughout. Ice crystals are assumed to nucleate at a uniform rate in the cloud and then to grow by water vapor deposition as they settle through the cloud. The cloud top is assumed to be closed. Steady state implies that the humidity within the cloud, the cloud droplet concentration, and the ice particle concentration do not change with time.
 To provide a context for this work, we draw on typical cloud properties observed for many hours on 26 April 2008 during the Indirect and Semi-Direct Aerosol Campaign (ISDAC) [McFarquhar et al., 2011; Ovchinnikov et al., 2011]. We set the cloud temperature at −10°C and thickness of the mixed-phase cloud at h=150 m. The ice equivalent diameter reached 1mm near the cloud base, but there was little signature of aggregation or riming due to the low liquid and ice water content. The liquid water profile is close to adiabatic.
 A central assumption in this model is that ice crystals form stochastically from plentiful ice nuclei, as suggested by recent laboratory work [e.g., Niedermeier et al., 2011; Welti et al., 2012]. Details of how the nucleation rates are distributed are not considered here, effectively neglecting any highly efficient ice nuclei as transients not relevant to the steady state. Ice crystals are assumed to originate from supercooled cloud droplets, with some fraction φ of the droplets containing ice nuclei. If the liquid cloud droplet concentration nw is homogeneous in the cloud, the number concentration of newly formed ice crystals in time Δt can be written as Δni=nwφ(1−e−Δt/τ), where τ is the characteristic time for heterogeneous ice nucleation (i.e., the inverse of the extensive nucleation rate). If τ>>Δt, Δni≈nwφΔt/τ, so we can define a volume ice formation rate , with units of m−3 s−1.
 Once nucleated, ice crystals grow and settle. The ice crystal radius ri increases with time due to vapor deposition growth at the rate approximated as [Lamb and Verlinde, 2011], where C is a shape factor, D is a modified diffusion coefficient accounting for heat transport and density, and si is the supersaturation with respect to ice (see supporting information). At constant temperature in the environment saturated with respect to liquid water, si is constant throughout the cloud, so if the initial radius of the crystal is ignored the result of time integration of the growth equation is .
 The ice particle terminal speed vi,t is assumed to have a power-law relationship with ice particle radius as , where b and k may depend on ice crystal habit and mass. For large crystals experiencing turbulent drag we can safely take the exponent to be k=1/2. If the deposition growth equation for ri is substituted into the ice terminal speed equation and then integrated over the distance from the point of nucleation to cloud base, we obtain equivalent ice crystal diameter as a function of height. Even for crystals nucleated at the cloud top, however, the resulting sizes do not exceed 200μm and cannot explain those observed in ISDAC (see supporting information). To obtain a realistic crystal size, it is necessary to consider growth in the presence of updrafts.
 Both observational data and model results show that updraft velocity in the cloud is maximum near the base and zero at the top of the cloud (see supporting information). For simplicity, we assume a linear decrease of updraft velocity with altitude: , where h is the cloud thickness and z is altitude above cloud base. Thus the ice fall speed under the influence of background velocity is vi=−b(2CDsit)k/2+ve(z). The result is a differential equation for the height of an ice crystal above the cloud base,
where P=v0/h and Q=b(2CDsi)k/2. The first term tends to increase fall speed due to depositional growth, and the second is the opposing updraft. The differential equation has the solution where c is a parameter depending on the initial condition: If the ice particle forms at cloud base, c=h, whereas if the ice particle forms at cloud top, c=0.
 Trajectories for growing crystals formed at cloud top and cloud base are shown in Figure 1. In order to compare with ISDAC observations, we take v0=0.3 ms−1, h=150 m, and , with r in meters and v in ms−1. Both particles stay in the cloud longer than 4000s (see Figure 1a), allowing large crystal size to be reached. Figure 1a also shows that both trajectories merge so that ice crystals tend to congregate as they approach the cloud base. We refer to this as reaching a quasi-steady state at the lower region of the cloud: no matter where ice particle forms it will have the similar size in the cloud base region, and the terminal speed of ice particles will be close to the background updraft speed. Quasi-steady state implies that the terms on the right side of equation (1) are nearly balanced such that dz/dt≈0, and therefore
The solution from equation (2) (black line in Figure 1) is slightly offset from the numerical solution of equation (1) at the base region but has very similar slope. The slope of these curves represents the ice crystal fall speed, so the velocity at cloud base can easily be obtained by differentiating equation (2), vi=(kQ/2P)t(k−2)/2=(k/2)(Q/v0)2/kh, where the second equality is obtained by eliminating t using equation (2) and taking z=0. At cloud base , so this becomes
where the second equality follows from the vapor growth rate equation. This result can be interpreted as the mean fall speed being proportional to the linear growth rate under quasi-steady state, i.e., the crystal only approaches cloud base at the rate at which it is able to grow by vapor deposition. Thus, in the quasi-steady regime the crystal growth times tend to converge to a single value, regardless of the initial location of the crystal nucleation event, and that time is much greater (and therefore the size much larger) than the time a growing crystal would take to fall through the depth of the cloud without an updraft.
 It should be noted that equation (3) is only satisfied in the quasi-steady region. To find when quasi-steady state can be expected at the cloud base region, different v0 are tested (see Figure 1b). It can be seen that quasi-steady state is valid only when v0 is larger than 0.2 ms−1. In addition, Figure1b shows that if v0 is larger than 0.25 ms−1, no matter where ice particles originate, they can reach diameters larger than 500μm, which is close to the observed value in ISDAC. We take this as observational support for the highly simplified picture of quasi-steady ice crystal growth at cloud base. Physically it means that cloud regions containing relatively large updraft velocities (comparable to terminal velocity of large ice particles) will suspend small ice particles, increasing their residence time in the ice-supersaturated cloud.
 Since in a quasi-steady state at cloud base all crystals have similar sizes and fall speeds (Figure 1), we can calculate the flux of ice particles out of the cloud as nivi and equate it with the column integrated nucleation rate
Combining equations (3) and (4), we get . Using the definition of ice water content to eliminate ri, we obtain a relationship between wi and ni,
where . This 5/2 power-law relationship between wi and ni is interesting because dilution or transport of ice crystals will tend to decrease wi and ni proportionally, i.e., they should follow a 1.0 power law. The prediction of a 2.5 power law is a result of ice particles being continuously formed due to the assumed stochastic ice nucleation process and smaller ice particles being held in the cloud by the updraft, with a concomitant increase in their residence time. This is consistent with the finding of Larson and Smith  that a 1.0 power law exists between wi and ni for an exponential size distribution of snow crystals and no mean updraft.
3 Comparison of Results With LES Cloud Model and ISDAC Observations
 In order to evaluate the plausibility of the minimalist model, we evaluate its predictions in the context of ISDAC cloud simulations and field observations. Two points should be mentioned as part of this comparison. First, the minimalist model is one dimensional, whereas the cloud simulations and observations contain all three dimensions. In the 1-D model, ice particles can only fall out of the cloud in the one column, while in 3-D, ice particles can separate horizontally due to dilution or transport and can fall out of downdraft regions more quickly. So we expect that G′ in equation (5) can be modified to , where γ is a 3-D correction parameter. Second, equation (5) describes the wi-ni relationship under the quasi-steady state conditions assumed in the model derivation, when updraft velocity exists in the cloud and linearly decreases with altitude. The extent to which the assumptions capture the essential physics is to be tested.
 Large-eddy simulations (LES) are performed using the System of Atmospheric Modeling (SAM) dynamical framework [Khairoutdinov and Randall, 2003] coupled with the Spectral Bin Microphysics (SBM) scheme [Khain et al., 2004] as described in Fan et al. . The simulation setup for the ISDAC case is similar to that used by Ovchinnikov et al. . The model's computational domain includes 64×64 columns and 160 vertical levels, using 50m grid spacing in both horizontal directions and 10m in the vertical. Size distributions for liquid and ice hydrometeors are predicted, each discretely represented by 33 size bins. For ice particle properties, the same relationship between ice crystal size and fall speed as in the minimalist model are used (more details can be seen in supporting information). Collision processes are not considered. Specification of cloud condensation nuclei follows Ovchinnikov et al. , producing a nearly constant droplet number concentration of around 200cm−3in the cloud. Ice particles are produced by prescribing a constant freezing probability of any droplet, regardless of its size, location, etc. Thus, a fraction (, where Δt is the model time step) of the droplet size distribution is converted to the ice size distribution every time step.
 To test whether wi and ni from the LES exhibit a 2.5 power law similar to equation (5), we analyze these variables from a time when the simulation has achieved a reasonably steady state (at 5h). We first select the columns containing an accumulation zone, where updraft velocity decreases with altitude in the cloud region, and then choose the corresponding wi and ni data at the base of the accumulation zone. Results are shown in Figures 2a and 2c. Each point represents the base of the accumulation zone for a single column in the LES. Colors in Figure 2a represent the updraft velocity at that point, while colors in Figure 2c represent the altitude of that point. Two power-law slopes clearly emerge in Figure 2c: One is the anticipated 2.5 slope, and the other is the 1.0 slope expected for transport and dilution. In addition, we note that there is no preferred updraft velocity and altitude for the data on the line with 2.5 slope, whereas for data on the line with 1.0 slope almost all points are at the top of the cloud. These columns actually contain only a small updraft at the cloud top, and the other part of the cloud is dominated by downdrafts. After removing these columns without a robust accumulation zone, we obtain a clear 2.5 power law between wi and ni (Figures 2b and 2c). Further analysis of horizontal layers in the LES are shown in the supporting information. Surprisingly, the 2.5 power law emerges not only in accumulation zones, but throughout the cloud, with the exception of just the cloud top region where entrainment and dilution is active and presumably the quasi-steady conditions are not reached.
 The LES observation of the 2.5 power law on horizontal layers allows us to consider whether there are similar indications in mixed-phase clouds sampled during ISDAC. A full analysis is not possible in the space limitations of this letter, but here we use 1s data from the 2DC and 2DP instruments taken during two horizontal in-cloud flight legs at around 700m and 800m on 26 April (the data set is discussed more fully by Fan et al. ). The data include ice particles larger than approximately 100μm in diameter, and both ni and wi are derived from the size distributions. Obtaining reliable measures of ni is especially challenging due to ice crystal shattering, but the data have been postprocessed to minimize such artifacts. Figure 3 shows the ISDAC data points plotted in log-log coordinates and lines with slope 1.0 and 2.5 for comparison. Despite the measurement challenges, this first analysis suggests that the observed (wi,ni) data lie within the bounds set by the two power laws and therefore to the plausibility of the minimalist model assumptions.
 Finally, we anticipate from equation (5) that the intercept of the 2.5-slope line will be sensitive to the nucleation rate, being proportional to . Figure 4 shows LES results for two ice nucleation rates: Blue points correspond to the LES with φ/τ=2×10−9 and red points to φ/τ=10−8. If we assume the correction parameter γ does not change much between these simulations, the shift of the 2.5 slope line will be due only to a change in the ice nucleation rate. The intercept shift predicted by the minimalist model is 1.5 log10(5)=1.05, which is very close to the best fitted line shift in Figure 4, 5.77−4.75=1.03. Indeed, the two γ values are 14.3 and 15.6 for low and high ice nucleation rates, respectively (shape factor C is set to be 1.0), so the assumption of constant γ is reasonable. This provides a compelling link between ice microphysical properties and the ice nucleation rate within the cloud, which may be used in future analysis of cloud observations.
 We have approached the problem of steady ice precipitation from long-lived mixed-phase clouds by assuming a steady state balance between new formation of ice particles due to a low stochastic ice nucleation rate throughout the cloud, and “quasi-steady” growth of ice particles by vapor deposition as they fall through updrafts and eventually out of the cloud base. A simple model based on this minimum number of assumptions is able to describe essential features of the ice microphysical properties of mixed-phase clouds. The model predicts a 2.5 power law between ice water content and ice number concentration in the cloud where updraft velocity decreases linearly with height. wi and ni from a LES cloud model with stochastic ice nucleation also follow the 2.5 power law, suggesting that the simple model can capture the properties of fully 3-D mixed-phase cloud based on the assumption of plentiful, low-efficiency ice nuclei and a stochastic ice nucleation process. Furthermore, a 2.5 power-law relationship between wi and ni is observed as an upper bound in ISDAC measurements, suggesting the assumptions of the minimalist model are plausible and motivating additional analysis of ISDAC and other mixed-phase cloud data sets. These observations open up the intriguing possibility that ice microphysical properties within supercooled layer clouds can be used to investigate the nature of the ice nucleation process and that similar models could serve as a bridge between complex field and cloud model observations and relatively idealized laboratory investigations of the time dependence of stochastic ice nucleation. The model also adds strength to the view that ice microphysics are tightly coupled with cloud dynamics and internal cloud variability, since the observed 2.5 power law is fundamentally tied to crystal growth in updrafts. Parameterizations based on these concepts are anticipated to be of value for larger-scale models in need of a physically based connection between cloud dynamics, ice nucleation, and the cloud microphysical properties that impact precipitation, cloud lifetime, and cloud optical properties.
 This research was supported by the DOE Office of Science as part of the Atmospheric System Research program, including through grant DE-SC0006949, and used data from the Atmospheric Radiation Measurement Climate Research Facility. We thank A. Korolev for providing the processed ice data from ISDAC. Simulations were performed using PNNL Institutional Computing at Pacific Northwest National Laboratory.
 The Editor thanks an anonymous reviewer for his/her assistance in evaluating this paper.