Stratocumulus clouds can assume closed- or open-cell structures with strikingly different abilities to reflect solar radiation. While open cells have been linked to the presence of precipitation and low droplet concentrations, complete understanding of processes leading to their formation is lacking. Here we show that the structure of stratocumulus can be linked to two time scales: an updraft time scale (tup) and a rain initiation time scale (tr). When sufficient drizzle forms within updrafts (tr ≤ tup), cloud water in the outflow is depleted enough that an overcast cloud cannot be sustained. Using a simple parcel model, we relate these time scales to three observable parameters (droplet number concentration, cloud depth, and updraft speed) and derive a functional representation for the transition from closed to open cells. Eight well-documented observed and simulated cases of open- and closed-cell stratocumulus fit well into the classification based on our model.
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 Stratocumulus clouds, which are an important component of the Earth's climate system, can exhibit diverse structures, from solid decks to broken cloud fields, with strikingly different abilities to reflect solar radiation. Amazingly, these different cloud formations can occur within nearly the same environment [Wood and Hartmann, 2006], as evidenced by a common emergence of honeycomb-like “open” cells surrounded by overcast or “closed” cell areas. Despite considerable advances in understanding of processes governing the formation and maintenance of stratocumulus [Wood, 2012], there are no universal criteria for determining whether stratocumulus will adopt an open-cell or a closed-cell structure in a given environment [Feingold et al., 2010], which presents a challenge for climate models striving to predict the correct cloud properties [Wyant et al., 2010]. Open cells have been linked to the presence of precipitation and low droplet concentrations, but complete understanding of the many complex and interacting processes leading to their formation is lacking. For example, while all observed pockets of open cells (POCs) precipitate [Stevens et al., 2005], drizzle has been shown to be a necessary [Wood et al., 2008] but not a sufficient condition for POCs formation, as similar cloud base drizzle rates have been found in both cases [Wood et al., 2011]. Open cells have been found to exist only in regions with low droplet number concentrations (Nc), but a quantitative threshold has not been established, and in the Nc range of 30–100 cm−3, both open- and closed-cell cases are possible [Wood et al., 2008]. Detailed numerical cloud models are now able to replicate essential features of closed- and open-cell systems [Berner et al., 2011; Savic-Jovcic and Stevens, 2008; Wang and Feingold, 2009]. More importantly, these simulations are capable of mimicking the transition from a closed-cell structure to an open-cell structure simply by reducing Nc, suggesting that microphysics processes are of paramount importance [Feingold et al., 2010]. Similar to observations, however, the critical droplet concentration below which an overcast condition cannot be sustained appears to be case specific, and its relation to other parameters has not been identified. The goal of this study is to explore how basic dynamical, macrophysical, and microphysical properties affect the structure of the cloud layer and to derive a functional representation for the transition from closed to open cells.
2 Model Description
 We begin our analysis of the factors controlling the cloud structure with a simple conceptual model of stratocumulus, drawing heavily upon Kessler's seminal work on the continuity of water substance in the atmosphere [Kessler, 1969, 1995] and adapting it to the case of a well-mixed boundary layer. We consider a layer capped by a temperature inversion strong enough to prevent rising parcels from penetrating into the free troposphere. Liquid water is produced through condensation in updrafts and removed through evaporation in downdrafts and precipitation. In the absence of precipitation, condensation equals evaporation, and a horizontally uniform cloud layer is produced (Figure 1a). When precipitation is present, cloud water produced in the updrafts is partially depleted before reaching the downdrafts, which become thinner (Figure 1b). If precipitation removes enough liquid water, the downdrafts can become cloud free (Figure 1c). In the context of this conceptual model, we term overcast clouds as closed cells and broken clouds as open cells. Based on the above reasoning, we postulate that the overcast clouds cannot be maintained if precipitation forms in an ascending parcel before or soon after that parcel reaches the cloud top. This condition can be expressed in terms of an updraft time scale (tup) and a rain initiation time scale (tr), i.e., tr must be smaller than tup to produce open cells. Conceptually, we can view tup as an indicator of the intensity of mixing, which pushes the layer toward a vertically and horizontally uniform distribution of total (vapor + condensate) water, and tr as an indicator of how fast precipitation can disturb that uniformity. The resulting cloud structure reflects the balance between these two competing processes.
 To be of practical use, these time scales must be related to observable parameters. To do this, we turn to a simple parcel model. Being interested in precipitation initiation, we consider air parcels following an optimal trajectory for the fastest drizzle formation for a given cloud layer geometry (cloud depth H) and dynamics (updraft speed w). Thus, our parcels ascend with a constant velocity (w) from the cloud base to the cloud top, where w vanishes, and the parcel is moved away from the updraft by the divergent horizontal flow. The parcel's microphysics is described by four prognostic variables representing mass mixing ratios (q) and number concentrations (N) for cloud and rainwater (denoted by subscripts c and r, respectively) that evolve in time due to condensation, autoconversion (Au, the process of drizzle formation through collisions among cloud droplets) and accretion (Ac, drizzle growth via collection of cloud water). We use Au = cauqcaNc−b and Ac = cac (qc × qr)d, where a = 2.47, b = 1.79, d = 1.15, cau = 1350, cac = 67, qc and qr are in kilograms per kilogram, Nc is in cubic centimeters, and Au and Ac are in kilograms per kilogram per second [Khairoutdinov and Kogan, 2000]. Simulations are initialized at cloud base, with qc, qr, and Nr set to zero and Nc set to Nc0. The variable input parameters in the model are Nc0, H, and w (see the supporting information for a full model description).
3 Time Scale Analysis of Open and Closed Cells
 The time evolution of microphysical properties for two parcels representative of closed- and open-cell configurations (Figure 2) illustrates the time scale concept. The updraft time scale appears naturally as tup = H/w. Defining the precipitation initiation time scale tr is more intricate and requires setting a threshold value qr* such that qr* = qr(tr), which signifies the appearance of first rain. Guided by the qr evolution shown in Figure 2, we set qr* = 0.1 g kg−1 and later discuss the role of this threshold.
 To explore how the relationship between the two time scales depends on microphysics (Nc0), macrophysics (H), and dynamics (w) of the cloud layer, we perform over 3000 parcel model simulations covering a wide range of conditions: Nc0 varying between 5 and 1000 cm−3, H between 200 m and 2 km, and updraft speeds between 0.2 and 2 m s−1. Each triad (Nc0, H, w) uniquely defines a pair of time scales (tr, tup). Results for three representative updraft speeds (w = 0.2, 0.4, and 0.8 m s−1) are shown in Figure 3, where tr(Nc0,H) is shown by blue curves. For points above the diagonal, tr < tup, and drizzle is initiated within the updraft. For points below the diagonal, tr > tup, and drizzle can be initiated only after the parcel reaches the cloud top and resides there for a period of time, given by the difference tr(Nc0,H) − tup(H).
Table 1. Observed and Simulated Cases of Open and Closed Cells
 For any H, tr increases monotonically with increasing Nc0, as expected, since more numerous but smaller droplets delay rain formation. For any value of Nc0, tr decreases monotonically with increasing H. Since maximum cloud liquid water is proportional to H, this has a clear physical meaning of larger droplets being more efficient in forming drizzle. Once tr becomes smaller than tup, however, any further increase in cloud depth does not affect tr since drizzle is already initiated lower in the updraft. An interesting consequence of this is that for any (Nc0,w) pair, there is a maximum depth for closed cells, beyond which the cloud layer will always transition to an open-cell regime. For example, for w = 0.4 m s−1 (0.8 m s−1), overcast clouds cannot be deeper than 450 m (600 m) for Nc0 = 20 cm−3 or 800 m (1000 m) for Nc0 = 100 cm−3.
 The effect of updraft speed on cloud structure is subtler. Increasing updraft speed always accelerates precipitation formation (i.e., decreases tr) because a parcel's liquid water content increases faster in stronger updrafts. An important implication of this is that in clouds containing a spectrum of vertical motions, the strongest updrafts largely control rain initiation. Although tr is always shorter for faster updrafts (Nc0 and H being the same), tup also decreases for faster updrafts, and it is the ratio of the two that determines the structure. It turns out that for a given H, tup has stronger dependence on updraft speed than does tr. A comparison of Figures 3a–3c clearly illustrates this point. For example, for an 800 m cloud with Nc0 = 100 cm−3, cells would be open for w = 0.2 m s−1 and closed for w = 0.8 m s−1. Thus, slower updrafts are, in fact, more likely to break up the cloud layer for a given Nc0. For a given aerosol distribution, slower updrafts also produce fewer droplets (which further decreases tr); however, since activation fraction approaches unity for low aerosol concentrations, the effect of w through increasing tup may be more important.
 How do documented cases of open and closed cells fit into this conceptual framework? Despite the basic nature of input parameters required by our model, only few recent studies provide reliable estimates of Nc0, H, and w, particularly for the open-cell regime, in which horizontal averages are not representative of narrow cloudy updrafts. Eight cases from observations and simulations of open and closed cells from three field projects for which the needed parameters were reported are summarized in Table 1. Two cases come from field campaign observations over the Southeast (VAMOS Ocean-Cloud-Atmosphere-Land Study, VOCALS) and Northeast (Drizzle and Entrainment Cloud Study, DECS) Pacific. VOCALS Regional Experiment observations of overcast stratocumulus and pockets of open cells are from the 27–28 October 2008 case study [Wood et al., 2011]. The Drizzle and Entrainment Cloud Study (DECS) observations of a rift and surrounding overcast cloud were obtained from an instrumented aircraft on 16 July 1999 offshore of Monterey Bay, California [Sharon et al., 2006]. Two other pairs of cases are taken from large-eddy simulations (LES) of the VOCALS case described above [Berner et al., 2011] and a case from the Second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field campaign [Savic-Jovcic and Stevens, 2008].
 The ability of our simple model to correctly classify these cases is quite remarkable (Figure 3). For all open-cell cases, we find that rain is initiated in the updraft, i.e., tr/tup ≤ 1, in accord with the conceptual model. As expected, these cases are characterized by lower Nc0 and, in general, deeper clouds, with both Nc0 and H factors contributing to crossing the tr = tup barrier. For example, 800 m deep open cells observed during VOCALS (Figure 3b, red open circle) would likely remain open even if Nc increased from 20 to 70 cm−3, provided that w does not increase. Similarly, a 450 m deep overcast VOCALS cloud (Figure 3c, red filled circle) would approach but not necessarily cross the threshold to open cells if Nc decreased from 70 to 20 cm−3 and w stayed at 0.8 m s−1.
4 Criterion for the Transition to Open Cells
 Although the parcel model is still too complex to be solved analytically, a functional form for the transitional condition tr ≤ tup can be obtained by noting that in the updraft (and before rain initiation), qc change is dominated by condensation and qr evolution is controlled primarily by autoconversion (Figure 2). Approximating the condensation rate as γadw, where γad is the vertical gradient of adiabatic liquid water, we can integrate dqr/dt = Au from t = 0 to tr and then solve for tr (see the supporting information). Since the resulting analytical approximation does not account for accretion, it overestimates tr but provides a reasonable upper bound, particularly for Nc0 < 100 cm−3 and tr < 20 min (Figure 4), the parameter ranges where the majority of open-cell cases are expected to lie. The condition for the breakup of overcast clouds into open cells, which is defined as tr/tup ≤ 1, then becomes
where C = [(a + 1)qr*/(cauγada )]−1/(a+1).
 The functional form of equation (1) depends only on the parameters of the autoconversion rate formulation. The threshold qr* plays a moderate role in defining the critical value for the criterion [C ~ (qr*)−0.3] but, more importantly, does not alter its functional form. Although the value used here appears to be appropriate for the considered cases, analysis of additional observations can refine qr*. The derived condition can also be cast in terms of adiabatic liquid water path (LWPa) instead of H, since LWPa ~ H2 can be used in estimating tup. Average precipitation rate, on the other hand, may not be a good predictor for the transition because of its more convolved relationship with Nc and H and, consequently, with tr and tup.
 Criterion (1) is interesting because it links the qualitative change in cloud structure to quantitative changes in measurable cloud parameters. For the effects of Nc0 and H, which qualitatively have been identified previously, this expression provides an insightful quantitative refinement. Dependence H−1 is stronger than Nc0~0.5 (for a = 2.47 and b = 1.79) [Khairoutdinov and Kogan, 2000], but the range of the expected variability in Nc0 (a factor of 10, from 5 to 50 cm−3) is likely wider than in H (a factor of 3, from 300 to 1000 m), potentially balancing the real contributions from the two.
 The role of w in controlling the closed-to-open-cell transitions, being identified here for the first time, is noteworthy. Although the w dependence is the weakest of the three parameters (w~0.3 for a = 2.47), open cells are favored by slower updrafts. Thus, for a given intensity of mixing, layers with negatively skewed distribution of vertical velocity (i.e., with narrower downdrafts and wider but less vigorous updrafts) are more prone to formation of open cells than are layers with positive skewness of vertical velocity. In that sense, our result is consistent with preferential formation of open cells during late night [Wood et al., 2008], when radiative cooling drives the circulation more strongly from the cloud top, resulting in a negative skewness of the w distribution. If, however, updrafts tighten upon the formation of open cells, as one might expect, so does the criterion for their existence since the left-hand side of equation (1) increases. This provides for an interesting possibility that not all nascent open cells can survive the transformation.
 We have shown that the structure of stratocumulus can be linked to two time scales, namely, an updraft time scale (tup) and a rain initiation time scale (tr), which is manifested by the rainwater content in an ascending parcel exceeding a small prescribed threshold. When drizzle forms within updrafts (tr ≤ tup), cloud water in the outflow is depleted enough that an overcast cloud cannot be sustained. Using a simple parcel model, we relate these time scales to three observable parameters (droplet number concentration Nc, cloud depth H, and updraft speed w). Well-documented observed and simulated cases of open- and closed-cell clouds are shown to fit well into the model-derived classification for stratocumulus structure, based on these time scales. Finally, we derive a functional representation for the transition from closed to open cells, which not only helps to quantify the effects of microphysics (Nc) and macrophysics (H) in the formation of open cells but also brings out the previously overlooked role of cloud dynamics expressed through the updraft speed. The presented diagnostic approach compliments recent studies that have explored microphysical and thermodynamic controls on cloud, precipitation, and boundary layer structure in an interactive setup [e.g., Mechem et al., 2012; Wang et al., 2010]. The derived criterion for the transition to open cells provides valuable insights for processes governing the structure of boundary layer clouds and may serve as a basis for the much-needed refinements of their representations in global climate models.
 This research was supported by the U.S. National Oceanic and Atmospheric Administration (NOAA) Atmospheric Composition and Climate Program (NA10AANRG0083/56091). The Pacific Northwest National Laboratory (PNNL) is operated by Battelle Memorial Institute for the DOE under contract DE-AC05-76RLO1830. The authors are grateful to Robert Wood and an anonymous reviewer for helping to improve the clarity of this letter.
 The editor thanks Robert Wood and an anonymous reviewer for their assistance in evaluating this paper.