Editorial Type:
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Article Type:
Research Article

Aerosol Replenishment and Cloud Morphology: A VOCALS Example

Qingfang Jiang Naval Research Laboratory, Monterey, California

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Shouping Wang Naval Research Laboratory, Monterey, California

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Print Publication:
01 Jan 2014
DOI:
https://doi.org/10.1175/JAS-D-13-0128.1
Page(s):
300–311
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Abstract

The impact of aerosol replenishment in a stratocumulus-topped boundary layer (STBL) on cloud morphology and albedo is examined using a large-eddy simulation (LES) model in conjunction with a prey–predator-type dynamical model following Koren and Feingold. In both the LES and the prey–predator models, the aerosol replenishment is represented as a relaxation term toward an ambient aerosol concentration with a time scale . The LESs suggest the existence of three distinct cloud regimes corresponding to different aerosol relaxation times. Specifically, for a small , the simulations are characterized by a large aerosol concentration, weak precipitation, relatively thick cloud depth, and closed cells. For a moderate , the simulated clouds exhibit open cellular patterns, in accordance with low aerosol concentration and moderate precipitation that oscillates in time. For a large , the aerosol may be depleted by intense drizzling and accordingly the cloud disappears. The critical aerosol relaxation times that separate these regimes vary with the ambient aerosol number concentration and cloud depth. Solutions from the low-order dynamical model with parameters relevant to the LESs are in general consistent with the LES results and provide further insight into the interplay among clouds, aerosol, and precipitation.

Corresponding author address: Qingfang Jiang, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93940-5502. E-mail: jiang@nrlmry.navy.mil

Abstract

The impact of aerosol replenishment in a stratocumulus-topped boundary layer (STBL) on cloud morphology and albedo is examined using a large-eddy simulation (LES) model in conjunction with a prey–predator-type dynamical model following Koren and Feingold. In both the LES and the prey–predator models, the aerosol replenishment is represented as a relaxation term toward an ambient aerosol concentration with a time scale . The LESs suggest the existence of three distinct cloud regimes corresponding to different aerosol relaxation times. Specifically, for a small , the simulations are characterized by a large aerosol concentration, weak precipitation, relatively thick cloud depth, and closed cells. For a moderate , the simulated clouds exhibit open cellular patterns, in accordance with low aerosol concentration and moderate precipitation that oscillates in time. For a large , the aerosol may be depleted by intense drizzling and accordingly the cloud disappears. The critical aerosol relaxation times that separate these regimes vary with the ambient aerosol number concentration and cloud depth. Solutions from the low-order dynamical model with parameters relevant to the LESs are in general consistent with the LES results and provide further insight into the interplay among clouds, aerosol, and precipitation.

Corresponding author address: Qingfang Jiang, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93940-5502. E-mail: jiang@nrlmry.navy.mil
Keywords: Boundary layer; Drizzle; Cloud droplets; Cumulus clouds; Large eddy simulations; Nonlinear models

1. Introduction

Because of its significant role in global energy balance (e.g., Hartmann and Short 1980), subtropical marine stratocumulus (MSc) has been the focus of several major field campaigns over the past two decades [e.g., Monterey Area Ship Track (MAST; Durkee et al. 2000); Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS II; Stevens et al. 2003); Eastern Pacific Investigation of Climate Processes in the Coupled Ocean–Atmosphere System (EPIC; Bretherton et al. 2004); the second Marine Stratus/Stratocumulus Experiment (MASE-II; Lu et al. 2009); and Eastern Pacific Emitted Aerosol Cloud Experiment (E-PEACE; Russell et al. 2013)]. The Variability of American Monsoon Systems (VAMOS) Ocean–Cloud–Atmosphere–Land Study (VOCALS; Wood et al. 2011b) over the southeastern Pacific (SEP) represented another major international effort to improve understanding and modeling of the coupled ocean–atmosphere–land system over subtropical oceans. Besides field observations, modern large-eddy simulation (LES) models play an instrumental role in advancing our understanding of the formation and variability of MSc (e.g., Savic-Jovcic and Stevens 2008; Xue et al. 2008; Wang and Feingold 2009; Mechem et al. 2012). Regardless of these efforts, some fundamental dynamical and physical processes related to MSc formation and variability are still not well understood and consequently are difficult to properly parameterize in global and climate models (e.g., Ma et al. 1996; Kiehl and Gent 2004).

Besides synoptic-scale forcing, the variability of MSc is also influenced by dynamical interactions among clouds, aerosol, and precipitation, which are further complicated by processes related to surface fluxes, turbulence transport, entrainment, and radiation. The role that precipitation plays in modulating cloud albedo and morphology has been emphasized by several recent studies. Particularly, precipitation is believed to be crucial for the transition from closed to open cells (Stevens et al. 2005; Wood et al. 2009). In the meantime, the intensity of MSc precipitation is found to be sensitive to the aerosol concentration; polluted clouds with higher cloud droplet concentration tend to produce weaker precipitation (Wood 2012). On the other hand, strong precipitation may deplete cloud condensation nuclei (CCN) and consequently MSc becomes unsustainable in the absence of CCN replenishment (Ackerman et al. 1993). In a VOCALS case study, Wang et al. (2011) found that an hourly replenishment rate of accumulation mode particles of 1 cm−3 uniformly over the depth of the stratocumulus-topped boundary layer (STBL) was sufficient to maintain open cell circulations. In the real world, the loss of CCN due to precipitation washout can be compensated by entrainment from the aerosol-enriched free troposphere and injection of sea salt and reactive gas from the sea surface [e.g., dimethyl sulfide (DMS); Ayers and Gras 1991; Clarke e al. 2006]. In addition, horizontal advection may play a role in aerosol resupply over near-shore regions where large horizontal aerosol gradient exists. For example, the VOCALS Regional Experiment (VOCALS-REx) documented a marked offshore aerosol gradient from high concentration along the central Chilean coastline to cleaner conditions over the open ocean (Allen et al. 2011), and the observed cloud properties are in general correlated with the aerosol gradient (Bretherton et al. 2010).

Previous LES studies demonstrated that the MSc morphology, cloud albedo, and precipitation are sensitive to the CCN number concentration. In LES models, the aerosol effect is typically represented in one of four different ways, depending on the complexity of the cloud microphysics. They are 1) using a fixed cloud droplet number concentration (e.g., Savic-Jovcic and Stevens 2008; Berner et al. 2011), 2) using a fixed CCN number concentration with cloud droplet activation driven by supersaturation (e.g., Xue et al. 2008; Jiang and Wang 2012), 3) calculating the aerosol budget without source functions (therefore, the CCN number concentration decreases as a result of collection and precipitation processes; e.g., Wang and Feingold 2009), and 4) calculating a full aerosol budget with source functions (e.g., Kazil et al. 2011). In the current study, we adopt a different approach. While the CCN budget is computed as in the approaches 3 and 4 above, the aerosol replenishment is represented as a relaxation term that is inversely proportional to a constant relaxation time. The objective is to understand the sensitivity of the MSc and STBL to the aerosol repopulation. In addition to LESs, we also examine a prey–predator-type aerosol–cloud–precipitation model following Koren and Feingold (2011, hereafter KF11).

The remainder of this paper is organized as follows. The large-eddy simulation model and model configuration are described in section 2. The LES results are presented in section 3. Section 4 includes results from the KF11 model, and section 5 contains concluding remarks.

2. LES model and numerical configuration

The large-eddy simulation version of the Coupled Ocean–Atmospheric Mesoscale Prediction System (COAMPS-LES1) is used for this study. A detailed description of the COAMPS-LES model can be found in Golaz et al. (2005) and Wang et al. (2012). For this study, a monotonic advection scheme following Blossey and Durran (2008) and a two-moment microphysical scheme based on Feingold et al. (1998) are employed. For simplicity, the shortwave radiation is not taken into account.

The computational domain contains 351 × 351 grid points in the horizontal with a grid spacing of 200 m and periodic conditions applied along lateral boundaries. The horizontal grid spacing used in this study is comparable to other similar studies [e.g., Δx = 300 m in Wang et al. (2010); Δx = 125 m in Berner et al. (2011)] and is fine enough to resolve the MSc cellular structure. There are 100 vertical model levels with a uniform grid spacing of 30 m. The model top is located at 3 km, where the potential temperature and water vapor gradients are fixed for all simulations.

The model is initialized using the wind, potential temperature and moisture profiles (Fig. 1) that are constructed from observations obtained during VOCALS-REx research flight 6 (RF-6) on 28 October 2008 (Wood et al. 2011a). RF-6 observed pockets of open cells embedded in a stratocumulus deck over the southeastern Pacific. A detailed description of the cloud characteristics and large-scale conditions documented by the VOCALS-REx RF-6 can be found in Wood et al. (2011a). These profiles have been used in an LES study of gravity wave effects on MSc by Jiang and Wang (2012). The mean winds are northwesterly above the STBL and are in geostrophic balance. The STBL is approximately 1.3 km deep and capped by a sharp inversion of 12.5 K, above which the water vapor mixing ratio decreases sharply.

Fig. 1.
Fig. 1.

(a) Horizontal wind components, (b) potential temperature, and (c) water vapor mixing ratio profiles derived from VOCALS RF-6 measurements.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

A subsidence term is included with a constant divergence of 2 × 10−6 s−1 from the sea surface to the bottom of the inversion, representing synoptic-scale forcing. This value is slightly larger than those used in Berner et al. (2011, 1.33 × 10−6 s−1) and Wang et al. (2010, 1.67 × 10−6 s−1) and results in a better balance with the growth of the STBL depth because of entrainment in our simulations.

The CCN concentration budget equation can be symbolically written as
e1
where the left-hand side denotes the Lagrangian increase rate of CCN number concentration (comprising dry, cloud, and drizzle particles) Nd; the first right-hand-side (rhs) term represents an exponential relaxation toward the given ambient concentration N0 with an e-folding relaxation time ; and the second rhs term denotes the aerosol washout rate by precipitation through collection–coalescence and sedimentation processes following Feingold et al. (1998). As in many previous LES studies, here we use terms “aerosol” and “CCN” interchangeably. The integration time is 14 h for all the simulations, and the first 2 h are considered the spinup period during which no aerosol replenishment is activated.

3. LES results

To understand the sensitivity of the simulated MSc to the aerosol relaxation time, two sets of simulations have been conducted with N0 = 100 and 200 cm−3 and a range of aerosol relaxation times (i.e., 0.5–24 h). We focus on the results from five simulations corresponding to N0 = 200 cm−3 and = 0.5, 1, 3, 6, and 9 h (Figs. 2–4), respectively.

Fig. 2.
Fig. 2.

Plan views of pseudoalbedo at (top to bottom) t = 5, 8, 11, and 14 h for five simulations with N0 = 200 cm−3 and (left to right) = 0.5, 1, 3, 6, and 9 h.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

Fig. 3.
Fig. 3.

Time series of domain-averaged variables from the five simulations (see legend) in Fig. 2 are shown, namely, (a) the proportional change of the STBL depth, (b) cloud fraction, (c) CWP, (d) surface rain rate, (e) the maximum vertical velocity, and (f) the cloud droplet number concentration.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

Fig. 4.
Fig. 4.

Time–height sections of horizontally averaged variables derived from the same five simulations in Fig. 2 with (left to right) increasing aerosol relaxation time as shown in Figs. 2 and 3. The variables include (top) cloud water qc (g kg−1, color) and liquid potential temperature θl (contours, min: 289 K, and interval: 1 K); (second row) rainwater flux , where wT denotes raindrop terminal velocity (mm day−1, color) and (contours, min: 0.01 m2 s−2, interval: 0.1 m2 s−2); (third row) buoyancy flux (W m−2, color) and cloud droplet number Nc (contours, start: 10 cm−3, interval: 50 cm−3); and (bottom) (m3 s−3, color) and water vapor mixing ratio qυ (contours, start: 7 g kg−1, interval: 0.5 g kg−1).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

Shown in Fig. 2 are the pseudoalbedo patterns at 5, 8, 11, and 14 h. The pseudoalbedo is calculated from the modeled cloud and rainwater in each column following Feingold et al. (1997). It is evident that the cloud patterns for = 0.5 and 1 h are similar, both of which are characterized by closed cellular patterns. The cloud morphologies for these short-relaxation-time simulations resemble that from the corresponding simulation with a constant aerosol number concentration, Nd = 200 cm−3, in Jiang and Wang (2012, their Fig. 11). Self-organized cellular patterns are well developed at 5 h, with the cloud coverage nearly 100% throughout these simulations. In contrast, the cloud morphologies are dramatically different in the other three simulations with = 3 h or longer. At t = 5 h, the albedo fields for the simulations with are characterized by scattered highly reflective clusters surrounded by less-reflective (i.e., dark or dark gray) areas—patterns more typical for open cells (three rightmost columns in Fig. 2). Between 8 and 14 h, the less-reflective areas expand substantially, separated by thin highly reflective filaments, and by the end of these simulations, the cloud morphologies are evidently open cellular. Quantitatively, the cells in the = 6-h simulation are more open than in the = 3-h simulation at the corresponding hours. However, the differences in the albedo and cloud morphologies between = 6 and 9 h are rather subtle. In summary, Fig. 2 suggests the existence of two distinct cloud regimes, namely, a fast-replenishment closed cell (FRCC) regime corresponding to < 3 h and a slow-replenishment open cell (SROC) regime corresponding to > 3 h.

The dramatic difference between the simulations with and is also evident in Fig. 3, which shows the evolution of some domain-average variables. The simulated boundary layer depths tend to grow with time for all the simulations, implying that the cloud-top entrainment effect overpowers the enforced large-scale divergence. The growth of the STBL depth for the two simulations with a smaller is noticeably faster than the other three, implying stronger boundary layer (BL)–top entrainment. The domain-averaged cloud fraction (CF) is near unity and the cloud water path (CWP) remains relatively large for these two simulations, consistent with the faster growth of the BL depths. The average cloud droplet number stays above 100 cm−3 throughout these two simulations, and the rain rate is virtually zero.

On the contrary, the CF and CWP exhibit a substantial decrease in the SROC simulations. Consequently, the precipitation is much stronger than that from simulations in the FRCC regime, though the rain rate tends to oscillate in time and lags behind the decrease of CWP. The significantly more intense precipitation in these simulations is consistent with the marked reduction in the number concentration of the cloud droplets (<50 cm−3). In addition, the maximum vertical velocity in the SROC simulations are considerably larger than in the FRCC simulations, implying much enhanced updrafts likely driven by the evaporative cooling and cold pooling in the lower STBL associated with more intense precipitation (Feingold et al. 2010; Jiang and Wang 2012). While the CF and CWP in the simulation with = 6 h are noticeably smaller than that with = 3 h, the difference between the two simulations for = 6 h and = 9 h (or 12, 18, and 24 h; not shown) is rather small, suggesting that the simulations tend to converge toward the nonreplenishment limit when is sufficiently large (for this set, when > 6 h). Finally, oscillations in the rain rate and maximum w are evident for the SROC simulations, which are in qualitative agreement with previous studies (e.g., Feingold et al. 2010). It is noteworthy that the oscillation of the CWP is less pronounced, likely due to contributions from multiple processes such as cloud physics, precipitation, evaporation, surface fluxes, turbulent transport of moisture, cloud-top entrainment, radiative cooling, and large-scale forcing.

The time–height sections shown in Fig. 4 reinforce the existence of two distinct regimes and provide further insights into the complex dynamical and physical interactions between different elements of the cloud–aerosol–turbulence–precipitation system and their dependence on . For simulations in the FRCC regime (i.e., the left two columns), the rainwater flux is small and the cloud water shows little decrease during the course of the simulations. A strong positive buoyancy flux is evident in the cloud layer, which helps maintain strong turbulence in the STBL. The third-moment of vertical velocity measures the asymmetry of turbulence and convective circulations. The time–height section of is characterized by negative values in the upper STBL and weak positive values in the lower STBL, suggesting closed-cellular circulations dominated by strong and narrow downdrafts amid weak broad updrafts. For the SROC regime simulations, the rainwater flux (cloud water) is markedly larger (less) than the FRCC simulations. The turbulence and mesoscale circulations in the STBL are modified considerably by the evaporation of rainwater below the cloud level and reduction in the longwave radiative cooling associated with less cloud water. Specifically, the turbulence (see contours in the second row) in the upper STBL is substantially weakened, implying the occurrence of decoupling (e.g., Stevens et al. 1998). Compared to the FRCC simulations, the positive in-cloud buoyancy flux becomes smaller, while the buoyancy flux in the lowest approximately 300 m becomes positive on account of evaporative cooling and cold pooling above the surface. In the upper STBL, is characterized by large positive values indicative of intense and more localized updrafts in accordance with the development of open cells. The decoupling in the SROC simulations also reduces the vertical turbulence transport of moisture and accounts for the increased vertical gradient of the water vapor mixing ratio qυ.

Similar sensitivity of the cloud morphology and STBL characteristics on the aerosol relaxation time is found in the other group of simulations of N0 = 100 cm−3 with two exceptions. First, the cloud morphology for the simulation with bears close resemblance to that for N0 = 200 cm−3 and , with comparable rain rates and maximum updrafts, implying that the critical aerosol relaxation time that separates the FRCC and SROC regimes is shorter for N0 = 100 cm−3. In fact, as shown in Jiang and Wang (2012, their Fig. 11), the cloud morphology is open cellular in the simulation with a fixed CCN concentration, N0 = 50 cm−3, implying that the FRCC regime virtually disappears for N0 = 50 cm−3 or smaller for the VOCALS RF-6 example examined in this study. Second, substantially more precipitation (i.e., ~2 mm day−1) occurs in the simulation with N0 = 100 cm−3 and , which depletes the CCN and causes the STBL to collapse around t = 4 h (Fig. 5). Consequently, turbulence, radiative cooling at the inversion top, and mesoscale circulations (e.g., ) exhibit a sudden weakening after 4 h. This simulation suggests the existence of the third regime, characterized by strong precipitation, aerosol depletion, demise of clouds, and collapse of the STBL (AD regime; Wang and Feingold 2009).

Fig. 5.
Fig. 5.

Time–height sections of the same set of horizontally averaged variables shown in Fig. 4, but derived from the simulation with N0 = 100 cm−3 and = 24 h.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

4. Predator–prey model solutions

KF11 demonstrated that a simple low-order predator–prey-type model can provide consistent interpretation of the aerosol–cloud–precipitation interaction processes reported in previous LES studies. The objective of this section is to gain further insights into the dependence of the cloud morphology and precipitation on aerosol replenishment time scale over a larger parameter space, through exploring solutions of the computationally low-cost KF11 model. The KF11 model is composited of the following two ordinary differential equations:
e2
e3
where is the precipitation rate. This system illustrates the interdependence of three variables: the cloud depth H (prey, related to cloud liquid water path); rain (predator); and cloud droplet concentration Nd, a proxy for the aerosol (or CCN). The first terms on the right-hand side of the two equations represent an exponential relaxation toward the ambient cloud depth H0 and aerosol number concentration N0 associated with large-scale forcing. The relaxation time constants ( and ) are presumably determined by water vapor and aerosol sources, turbulent transport, boundary layer top entrainment, etc. The second terms correspond to the decrease of H and Nd due to precipitation and are formulated based on previous observational, analytical, and LES studies as elaborated in KF11. The second term in (3) can be thought as a parameterized form of the volume average of the washout term in (1).

The subscript T denotes a time lag of T, accounting for the delay of precipitation due to rain-forming processes, which is approximated as a constant in KF11 and this study. There are three empirical constants in (2) and (3)—namely, c1, c2, and α—the values, units, and origins of which can be found in KF11. KF11 emphasized the dependence of solutions on H0 and N0 with and nearly fixed (i.e., 60–84 min). For this study, we reexamine system (2) and (3) with emphasis on the dependence of its steady-state and time-dependent solutions on .

a. Steady-state solutions

Letting and in (2) and (3) and eliminating Nd, after some manipulations, we obtain a steady-state nondimensional quartic equation in terms of the nondimensional cloud depth,
e4
where , , and . The corresponding cloud droplet number concentration and rain rate are given by
e5
e6
respectively. Here is the reference rain rate. As variables H, Nd, and R are nonnegative, a physical root of (4) should satisfy . Equation (4) has been solved numerically using Newton’s method for a wide range of parameters (i.e., H0, N0, , and ). Over the parameter space explored, (4) has either a single root or no physical root. For the examples shown in Fig. 6, the sensitivity of the system to the aerosol relaxation time is evident. With a small (i.e., fast aerosol replenishment), the nondimensional cloud depth, aerosol number concentration, and rain rate are close to unity. The rain rate increases nearly exponentially with increasing aerosol relaxation time, and the cloud depth and aerosol number concentration decrease accordingly. For the two examples corresponding to a thicker ambient cloud depth, the decrease (increase) of the cloud depth and aerosol number concentration (rain rate) is more rapid than the other two with a thinner cloud depth. Physically, a thicker cloud depth leads to more intense precipitation, which depletes the aerosol faster. It is noteworthy that, for the two examples corresponding to H0 = 500 m and N0 = 250 and 500 cm−3, steady-state solutions only exist approximately for < 3 and 5 h, respectively. For a larger , the aerosol washout by precipitation is faster than the replenishment, and accordingly the aerosol is depleted. Given the same cloud depth, the decrease of the nondimensional cloud depth and aerosol number with is noticeably faster with a smaller N0, which induces stronger precipitation.
Fig. 6.
Fig. 6.

The steady-state (a) normalized cloud depth, (b) aerosol number concentration, and (c) rain rate obtained by numerically solving (4)–(6) are shown as a function of the aerosol relaxation time for four example solutions. The four sets of curves correspond to H0 and N0 pairs of 500 m and 500 cm−3 (black), 500 m and 250 cm−3 (red), 250 m and 500 cm−3 (green), and 250 m and 250 cm−3 (blue).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

In summary, over the parameters examined, there is either a single or no physical root to (4). Corresponding to the increase of the aerosol relaxation time, a physical solution to (4) falls into one of the three regimes—namely, fast replenishment (i.e., small ), weak precipitation (accordingly, the normalized cloud depth and aerosol concentration are near unity); slow replenishment (i.e., moderate ), moderate precipitation (accordingly, the normalized cloud depth and aerosol concentration are substantially less than unity in Fig. 6); and no-steady-state solution (i.e., relatively large ) regimes. A few aspects of these solution regimes are worth mentioning. First, the transition between the first two regimes is rather gradual, as opposed to the transition between the second and third regimes, which is clearly defined by (Fig. 6). Second, the regime boundaries vary with the ambient cloud depth and CCN concentration. Furthermore, there is an apparent parallel between the three solution regimes derived from the KF11 model and the three regimes identified from the LESs in the previous section.

b. Time-dependent solutions

For a given set of parameters, a time-dependent solution can be obtained by numerically integrating (2) and (3). Figure 7a shows the evolution of H, Nd, and R for H0 = 200 m, N0 = 200 cm−3, T = 0.25 h, = 1 h, and a range of values. These solutions exhibit significant sensitivity to the aerosol replenishment time, qualitatively resembling that of the LESs. For = 0.5 and 1 h, both the cloud depth and the aerosol concentration quickly reach equilibrium after the initial decrease. The state of equilibrium is characterized by weak precipitation and a balance between a relatively small scavenging rate and fast replenishment of aerosol. Both the aerosol number concentration and cloud depth remain high in the equilibrium state, resembling the LESs in the FRCC regime. For , the evolution of all three variables is characterized by damped oscillations around an equilibrium state. The oscillation of the cloud depth lags behind that of the aerosol, a characteristic of predator–prey solutions. Compared to the solutions in the fast replenishment and weak precipitation regime, these equilibrium states correspond to much more intense precipitation, reduced aerosol concentration, and thinner cloud depth, which are characteristics of the LES results in the SROC regime. The equilibrium cloud depth and aerosol concentrations (precipitation) decrease (increases) slowly with increasing for and tend to converge for larger .

Fig. 7.
Fig. 7.

Two sets of example solutions to system (2) and (3) are shown. The cloud depth, cloud droplet number concentration, and rain rate between 1 and 9 h are plotted for (a) H0 = 200 m, N0 = 200 cm−3, = 1 h, and = 0.5, 1, 3, 6, 9 and 100 h; and (b) a set of solutions showing sensitivity to N0, H0, and τH, including the control [same as τH = 1 h in (a)], τH = 3 h, N0 = 500 and 50 cm−3, and H0 = 500 and 150 m (other parameters are as in the control).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

Additional solutions to system (2) and (3) have been obtained over a broad parameter space and some results are shown in Fig. 7b. This system appears to be relatively insensitive to and T and more sensitive to H0 and N0. In general, a larger N0 induces weaker precipitation and accordingly less CCN concentration reduction, and a larger H0 does the opposite. It is interesting that for H0 = 500 m, the cloud depth and aerosol exhibit (undamped) steady oscillations with minima equal to zero, and no steady state can be reached. Physically, intense precipitation depletes aerosol and causes the STBL to collapse. However, the rapid recovery of clouds and aerosol concentration in the KH11 solution after the aerosol depletion is not observed in the LES (Fig. 5). It is also noteworthy that the critical condition for having a steady-state solution varies with the aerosol relaxation time (Fig. 6). A shorter aerosol relaxation time tends to weaken precipitation, prevent the depletion of aerosol, and therefore promote damped-oscillation or steady-state solutions.

c. Linear analysis

To further understand the time-dependent solutions, especially the damped oscillations, it is instructive to linearize system (2) and (3) around an equilibrium state (i.e., He, Ne). For simplicity, we ignore T, let H = He + H′ and Nd = Ne + N′, and linearize (2) and (3), assuming and . To the first order of approximation, we obtain
e7
e8
Combining (7) and (8) yields a second-order ordinary differential equation in terms of . Using scaling and , we obtain a typical damped-oscillation equation in terms of the nondimensional cloud depth perturbation :
e9
where the coefficients and . Letting in (9), we obtain the characteristic equation
e10
e11
Some interesting property of the linearized KF11 system can be inferred from these equations. According to (11), there are three possible solutions (e.g., Wylie 1995), namely,

  1. if then the system is overdamped, characterized by an exponential decay toward its equilibrium state without oscillation;

  2. if , then the system is critically damped without oscillation; and

  3. if then the system is underdamped, characterized by damped oscillations with a frequency , and an e-folding damping time is given by .

In addition, as the real part of beta [Re(β)] is always negative, any physical equilibrium state (i.e., 0 < He < H0 and 0 < Ne < N0) of system (2) and (3) is linearly stable [i.e., state (He, Ne) is an attractor]. For a few example equilibrium states, the dependence of the solution characteristics on the aerosol relaxation time is shown in Fig. 8. According to Fig. 8, for a small , the system is underdamped, characterized by damped oscillation. However, the e-folding damping time is positively correlated with the aerosol relaxation time. Accordingly, if is very short (i.e., <1 h in our examples), no oscillation is observed because of fast damping. For some cases, with a large , the system is overdamped and accordingly, oscillations are absent as well. In addition, because in (9), there is no local bifurcation for any physical equilibrium state.
Fig. 8.
Fig. 8.

(top) Damping time and (bottom) oscillation frequency as a function of the aerosol relaxation time derived from the linearized solution (11). Four examples are shown, corresponding to the equilibrium states (H0, N0) = (500, 500), (500, 250), (250, 500), and (250, 250) (m, cm−3), respectively.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

5. Concluding remarks

In this study, the dependence of the marine stratocumulus morphology and STBL structure on the aerosol relaxation times is investigated through diagnosing LESs and examining a low-order prey–predator-type model. The aerosol replenishment rate is represented by an exponential relaxation term toward a given ambient concentration N0 with an e-folding relaxation time scale in both the LES model and the prey–predator model. Regardless of the substantial difference in the level of complexity between the two models, both models suggest the significance of the aerosol replenishment to precipitation, development of mesoscale circulations in STBL, and cloud morphology.

The results are summarized in a τN–N0 regime diagram (Fig. 9). Over the range of the ambient CCN concentration and the aerosol relaxation time examined, three distinct cloud regimes are identified, namely, fast-replenishment closed cell (FRCC), slow-replenishment open cell (SROC), and aerosol-depletion (AD) regimes. The FRCC regime is associated with fast aerosol replenishment, which helps to maintain a relatively high CCN concentration and results in weak precipitation. Accordingly, the cloud morphologies are predominantly closed cellular with a large cloud fraction and albedo, and mesoscale circulations are weak in the STBL. The SROC regime is characterized by slow aerosol replenishment (i.e., a relatively large ), moderate precipitation that oscillates in time, open-cellular MSc, and strong mesoscale circulations. As the aerosol replenishment is slow, the CCN number concentration decreases because of the rain scavenging effect. The reduced CCN concentration in turn enhances the precipitation. Such a positive feedback mechanism pushes the system into the intense precipitation regime until the aerosol replenishment, which increases with the decrease of the aerosol number concentration, becomes faster than the aerosol loss to scavenging. Afterward, the system swings back toward weaker precipitation. As a result, in the SROC regime, the precipitation intensity oscillates in time around a new equilibrium state with much reduced CCN concentration, and the cloud morphology is predominantly open cellular. A third regime exists when the ambient atmosphere is clean (i.e., low ambient aerosol number concentration) and/or the aerosol replenishment is slow relative to the washout by precipitation. Accordingly, strong precipitation occurs, which leads to the depletion of CCN, the demise of the MSc, and the collapse of the STBL (Fig. 5).

Fig. 9.
Fig. 9.

Schematic regime diagram created based on the LES and KF11 results. Two boundary curves (AB and CD) separate the τN–N0 diagram into three distinct regimes (i.e., FRCC, SROC, and AD).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0128.1

A few more aspects of the regime diagram are worth noting. First, the regime boundaries in Fig. 9 are not universal. They vary with other parameters, such as the large-scale conditions specified in the LES model and the ambient cloud depth in the KF11 model. According to the KF11 model, a thicker ambient cloud tends to enhance precipitation, and consequently the two boundary curves should shift upward toward higher N0. For a thinner ambient cloud, the two curves should move downward toward lower N0, associated with generally weaker precipitation. Second, while a FRCC solution can transit into a SROC solution by increasing from the left to the right of the CD curve, the changes in terms of precipitation intensity and cloud morphology are rather gradual, as opposed to a catastrophic change across the AB curve. Third, for a relatively small N0 (i.e., N0 < 50 cm−3 in this study), the precipitation is moderate or intense and accordingly the clouds are open cellular, regardless of the aerosol relaxation time. On the other hand, if the air is heavily polluted (e.g., N0 ~ 1000 cm−3), there is virtually no precipitation and accordingly the clouds are closed cellular, independent of . Furthermore, it is worth noting that for the LES examples, the cloud morphology seems to be most sensitive to the aerosol relaxation time on the order of 1–3 h, which is comparable to the typical period of open-cell oscillations documented in Feingold et al. (2010). Finally, to put the current work into perspective, in the regime diagram (Fig. 9), and , corresponding to approaches 2 (i.e., using a fixed CCN number concentration) and 3 (calculating the aerosol budget without source functions) discussed in the introduction, respectively.

Acknowledgments

This research was supported by the Office of Naval Research (ONR) Program Element (PE) 0601153 N. The authors want to thank Drs. H. Wang and G. Feingold for helpful discussions about the two-moment cloud microphysics and the predator–prey model. The primary sponsor of VOCALS is the U.S. National Science Foundation. Computational resources were supported by a grant of HPC time from the Department of Defense Major Shared Resource Centers.

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Allen, G., and Coauthors, 2011: South east Pacific atmospheric composition and variability sampled along 20°S during VOCALS-REx. Atmos. Chem. Phys., 11, 5237–5262.

    • Search Google Scholar
    • Export Citation

  • Ayers, G. P., and J. L. Gras, 1991: Seasonal relationship between cloud condensation nuclei and aerosol methanesulphonate in marine air. Nature, 353, 834–835.

    • Search Google Scholar
    • Export Citation

  • Berner, A. H., C. S. Bretherton, and R. Wood, 2011: Large-eddy simulation of mesoscale dynamics and entrainment around a pocket of open cells observed in VOCALS RF06. Atmos. Chem. Phys., 11, 10 525–10 540.

    • Search Google Scholar
    • Export Citation

  • Blossey, P. N., and D. R. Durran, 2008: Selective monotonicity preservation in scalar advection. J. Comput. Phys., 227, 5160–5183.

  • Bretherton, C. S., and Coauthors, 2004: The EPIC 2001 stratocumulus study. Bull. Amer. Meteor. Soc., 85, 967–977.

  • Bretherton, C. S., R. Wood, R. C. George, D. Leon, G. Allen, and X. Zheng, 2010: Southeast Pacific stratocumulus clouds, precipitation and boundary layer structure sampled along 20°S during VOCALS-REx. Atmos. Chem. Phys., 10, 10 639–10 654.

    • Search Google Scholar
    • Export Citation

  • Clarke, A. D., S. R. Owens, and J. Zhou, 2006: An ultrafine sea-salt flux from breaking waves: implications for cloud condensation nuclei in the remote marine atmosphere. J. Geophys. Res., 111, D06202, doi:10.1029/2005JD006565.

    • Search Google Scholar
    • Export Citation

  • Durkee, P. A., K. J. Noone, and R. T. Bluth, 2000: The Monterey Area Ship Track Experiment. J. Atmos. Sci., 57, 2523–2541.

  • Feingold, G., R. Boers, B. Stevens, and W. R. Cotton, 1997: A modeling study of the effect of drizzle on cloud optical depth and susceptibility. J. Geophys. Res., 102 (D12), 13 527–13 534.

    • Search Google Scholar
    • Export Citation

  • Feingold, G., R. L. Walko, B. Stevens, and W. R. Cotton, 1998: Simulations of marine stratocumulus using a new microphysical parameterization scheme. Atmos. Res., 47–48, 505–528.

    • Search Google Scholar
    • Export Citation

  • Feingold, G., I. Koren, H. Wang, H. Xue, and W. Brewer, 2010: Precipitation-generated oscillations in open cellular cloud fields. Nature, 466, 849–852.

    • Search Google Scholar
    • Export Citation

  • Golaz, J.-C., S. Wang, J. D. Doyle, and J. M. Schmidt, 2005: COAMPS-LES: Model evaluation and analysis of second- and third-momentum vertical velocity budgets. Bound.-Layer Meteor., 116, 487–517.

    • Search Google Scholar
    • Export Citation

  • Hartmann, D. L., and D. A. Short, 1980: On the use of earth radiation budget statistics for studies of clouds and climate. J. Atmos. Sci., 37, 1233–1250.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., and S. Wang, 2012: Impact of gravity waves on stratocumulus variability. J. Atmos. Sci., 69, 3633–3651.

  • Kazil, J., H. Wang, G. Feingold, A. D. Clarke, J. R. Snider, and A. R. Bandy, 2011: Modeling chemical and aerosol processes in the transition from closed to open cells during VOCALS-REx. Atmos. Chem. Phys., 11, 7491–7514.

    • Search Google Scholar
    • Export Citation

  • Kiehl, J. T., and P. R. Gent, 2004: The Community Climate System Model, version 2. J. Climate, 17, 3666–3682.

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    • Search Google Scholar
    • Export Citation

  • Lu, M.-L., A. Sorooshian, H. H. Jonsson, G. Feingold, R. C. Flagan, and J. H. Seinfeld, 2009: Marine stratocumulus aerosol-cloud relationships in the MASE-II experiment: Precipitation susceptibility in eastern Pacific stratocumulus. J. Geophys. Res., 114, D24203, doi:10.1029/2009jd012774.

    • Search Google Scholar
    • Export Citation

  • Ma, C.-C., C. R. Mechoso, A. W. Robertson, and A. Arakawa, 1996: Peruvian stratus clouds and the tropical Pacific circulation: A coupled ocean–atmosphere GCM study. J. Climate, 9, 1635–1645.

    • Search Google Scholar
    • Export Citation

  • Mechem, D. B., S. E. Yuter, and S. P. de Szoeke, 2012: Thermodynamic and aerosol controls in southeast Pacific stratocumulus. J. Atmos. Sci., 69, 1250–1266.

    • Search Google Scholar
    • Export Citation

  • Russell, L. M., A. Sorooshian, J. H. Seinfeld, B. A. Albrecht, and A. Nenes, 2013: Eastern Pacific Emitted Aerosol Cloud Experiment. Bull. Amer. Meteor. Soc., 94, 709–729.

    • Search Google Scholar
    • Export Citation

  • Savic-Jovcic, V., and B. Stevens, 2008: The structure and mesoscale organization of precipitating stratocumulus. J. Atmos. Sci., 65, 1587–1605.

    • Search Google Scholar
    • Export Citation

  • Stevens, B., W. R. Cotton, G. Feingold, and C. Moeng, 1998: Large-eddy simulations of strongly precipitating, shallow, stratocumulus-topped boundary layers. J. Atmos. Sci., 55, 3616–3638.

    • Search Google Scholar
    • Export Citation

  • Stevens, B., and Coauthors, 2003: Dynamics and Chemistry of Marine Stratocumulus—DYCOMS-II. Bull. Amer. Meteor. Soc., 84, 579–593.

  • Stevens, B., G. Vali, K. Comstock, R. Wood, M. C. Van Zanten, P. H. Austin, C. S. Bretherton, and D. H. Lenschow, 2005: Pockets of open cells (POCs) and drizzle in marine stratocumulus. Bull. Amer. Meteor. Soc., 86, 51–57.

    • Search Google Scholar
    • Export Citation

  • Wang, H., and G. Feingold, 2009: Modeling mesoscale cellular structures and drizzle in marine stratocumulus. Part I: Impact of drizzle on the formation and evolution of open cells. J. Atmos. Sci., 66, 3237–3255.

    • Search Google Scholar
    • Export Citation

  • Wang, H., G. Feingold, R. Wood, and J. Kazil, 2010: Modelling microphysical and meteorological controls on precipitation and cloud cellular structure in southeast Pacific stratocumulus. Atmos. Chem. Phys., 10, 6347–6362.

    • Search Google Scholar
    • Export Citation

  • Wang, H., Rasch P. J., and G. Feingold, 2011: Manipulating marine stratocumulus cloud amount and albedo: A process-modeling study of aerosol-cloud-precipitation interactions in response to injection of cloud condensation nuclei. Atmos. Chem. Phys., 11, 4237–4249, doi:10.5194/acp-11-4237-2011.

    • Search Google Scholar
    • Export Citation

  • Wang, S., X. Zheng, and Q. Jiang, 2012: Strongly sheared stratocumulus convection: An observationally based large-eddy simulation study. Atmos. Chem. Phys., 12, 5223–5235, doi:10.5194/acp-12-5223-2012.

    • Search Google Scholar
    • Export Citation

  • Wood, R., 2012: Stratocumulus clouds. Mon. Wea. Rev., 140, 2373–2423.

  • Wood, R., M. Köhler, R. Bennartz, and C. O’Dell, 2009: The diurnal cycle of surface divergence over the global oceans. Quart. J. Roy. Meteor. Soc., 135, 1484–1493.

    • Search Google Scholar
    • Export Citation

  • Wood, R., C. S. Bretherton, D. Leon, A. D. Clarke, P. Zuidema, G. Allen, and H. Coe, 2011a: An aircraft case study of the spatial transition from closed to open mesoscale cellular convection over the southeast Pacific. Atmos. Chem. Phys., 11, 2341–2370.

    • Search Google Scholar
    • Export Citation

  • Wood, R., and Coauthors, 2011b: The VAMOS Ocean-Cloud-Atmosphere-Land Study Regional Experiment (VOCALS-REx): Goals, platforms, and field operations. Atmos. Chem. Phys., 11, 627–654.

    • Search Google Scholar
    • Export Citation

  • Wylie, C. R., 1995: Advanced Engineering Mathematics. McGraw-Hill, 1362 pp.

  • Xue, H., G. Feingold, and B. Stevens, 2008: Aerosol effects on clouds, precipitation, and the organization of shallow cumulus convection. J. Atmos. Sci., 65, 392–406.

    • Search Google Scholar
    • Export Citation
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Journal of the Atmospheric Sciences

  • Ackerman, A. S., O. B. Toon, and P. V. Hobbs, 1993: Dissipation of marine stratiform clouds and collapse of the marine boundary layer due to the depletion of cloud condensation nuclei by clouds. Science, 262, 226–229.

    • Search Google Scholar
    • Export Citation

  • Allen, G., and Coauthors, 2011: South east Pacific atmospheric composition and variability sampled along 20°S during VOCALS-REx. Atmos. Chem. Phys., 11, 5237–5262.

    • Search Google Scholar
    • Export Citation

  • Ayers, G. P., and J. L. Gras, 1991: Seasonal relationship between cloud condensation nuclei and aerosol methanesulphonate in marine air. Nature, 353, 834–835.

    • Search Google Scholar
    • Export Citation

  • Berner, A. H., C. S. Bretherton, and R. Wood, 2011: Large-eddy simulation of mesoscale dynamics and entrainment around a pocket of open cells observed in VOCALS RF06. Atmos. Chem. Phys., 11, 10 525–10 540.

    • Search Google Scholar
    • Export Citation

  • Blossey, P. N., and D. R. Durran, 2008: Selective monotonicity preservation in scalar advection. J. Comput. Phys., 227, 5160–5183.

  • Bretherton, C. S., and Coauthors, 2004: The EPIC 2001 stratocumulus study. Bull. Amer. Meteor. Soc., 85, 967–977.

  • Bretherton, C. S., R. Wood, R. C. George, D. Leon, G. Allen, and X. Zheng, 2010: Southeast Pacific stratocumulus clouds, precipitation and boundary layer structure sampled along 20°S during VOCALS-REx. Atmos. Chem. Phys., 10, 10 639–10 654.

    • Search Google Scholar
    • Export Citation

  • Clarke, A. D., S. R. Owens, and J. Zhou, 2006: An ultrafine sea-salt flux from breaking waves: implications for cloud condensation nuclei in the remote marine atmosphere. J. Geophys. Res., 111, D06202, doi:10.1029/2005JD006565.

    • Search Google Scholar
    • Export Citation

  • Durkee, P. A., K. J. Noone, and R. T. Bluth, 2000: The Monterey Area Ship Track Experiment. J. Atmos. Sci., 57, 2523–2541.

  • Feingold, G., R. Boers, B. Stevens, and W. R. Cotton, 1997: A modeling study of the effect of drizzle on cloud optical depth and susceptibility. J. Geophys. Res., 102 (D12), 13 527–13 534.

    • Search Google Scholar
    • Export Citation

  • Feingold, G., R. L. Walko, B. Stevens, and W. R. Cotton, 1998: Simulations of marine stratocumulus using a new microphysical parameterization scheme. Atmos. Res., 47–48, 505–528.

    • Search Google Scholar
    • Export Citation

  • Feingold, G., I. Koren, H. Wang, H. Xue, and W. Brewer, 2010: Precipitation-generated oscillations in open cellular cloud fields. Nature, 466, 849–852.

    • Search Google Scholar
    • Export Citation

  • Golaz, J.-C., S. Wang, J. D. Doyle, and J. M. Schmidt, 2005: COAMPS-LES: Model evaluation and analysis of second- and third-momentum vertical velocity budgets. Bound.-Layer Meteor., 116, 487–517.

    • Search Google Scholar
    • Export Citation

  • Hartmann, D. L., and D. A. Short, 1980: On the use of earth radiation budget statistics for studies of clouds and climate. J. Atmos. Sci., 37, 1233–1250.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., and S. Wang, 2012: Impact of gravity waves on stratocumulus variability. J. Atmos. Sci., 69, 3633–3651.

  • Kazil, J., H. Wang, G. Feingold, A. D. Clarke, J. R. Snider, and A. R. Bandy, 2011: Modeling chemical and aerosol processes in the transition from closed to open cells during VOCALS-REx. Atmos. Chem. Phys., 11, 7491–7514.

    • Search Google Scholar
    • Export Citation

  • Kiehl, J. T., and P. R. Gent, 2004: The Community Climate System Model, version 2. J. Climate, 17, 3666–3682.

  • Koren, I., and G. Feingold, 2011: Aerosol-cloud-precipitation system as a predator-prey problem. Proc. Natl. Acad. Sci. USA, 108, 12 227–12 232.

    • Search Google Scholar
    • Export Citation

  • Lu, M.-L., A. Sorooshian, H. H. Jonsson, G. Feingold, R. C. Flagan, and J. H. Seinfeld, 2009: Marine stratocumulus aerosol-cloud relationships in the MASE-II experiment: Precipitation susceptibility in eastern Pacific stratocumulus. J. Geophys. Res., 114, D24203, doi:10.1029/2009jd012774.

    • Search Google Scholar
    • Export Citation

  • Ma, C.-C., C. R. Mechoso, A. W. Robertson, and A. Arakawa, 1996: Peruvian stratus clouds and the tropical Pacific circulation: A coupled ocean–atmosphere GCM study. J. Climate, 9, 1635–1645.

    • Search Google Scholar
    • Export Citation

  • Mechem, D. B., S. E. Yuter, and S. P. de Szoeke, 2012: Thermodynamic and aerosol controls in southeast Pacific stratocumulus. J. Atmos. Sci., 69, 1250–1266.

    • Search Google Scholar
    • Export Citation

  • Russell, L. M., A. Sorooshian, J. H. Seinfeld, B. A. Albrecht, and A. Nenes, 2013: Eastern Pacific Emitted Aerosol Cloud Experiment. Bull. Amer. Meteor. Soc., 94, 709–729.

    • Search Google Scholar
    • Export Citation

  • Savic-Jovcic, V., and B. Stevens, 2008: The structure and mesoscale organization of precipitating stratocumulus. J. Atmos. Sci., 65, 1587–1605.

    • Search Google Scholar
    • Export Citation

  • Stevens, B., W. R. Cotton, G. Feingold, and C. Moeng, 1998: Large-eddy simulations of strongly precipitating, shallow, stratocumulus-topped boundary layers. J. Atmos. Sci., 55, 3616–3638.

    • Search Google Scholar
    • Export Citation

  • Stevens, B., and Coauthors, 2003: Dynamics and Chemistry of Marine Stratocumulus—DYCOMS-II. Bull. Amer. Meteor. Soc., 84, 579–593.

  • Stevens, B., G. Vali, K. Comstock, R. Wood, M. C. Van Zanten, P. H. Austin, C. S. Bretherton, and D. H. Lenschow, 2005: Pockets of open cells (POCs) and drizzle in marine stratocumulus. Bull. Amer. Meteor. Soc., 86, 51–57.

    • Search Google Scholar
    • Export Citation

  • Wang, H., and G. Feingold, 2009: Modeling mesoscale cellular structures and drizzle in marine stratocumulus. Part I: Impact of drizzle on the formation and evolution of open cells. J. Atmos. Sci., 66, 3237–3255.

    • Search Google Scholar
    • Export Citation

  • Wang, H., G. Feingold, R. Wood, and J. Kazil, 2010: Modelling microphysical and meteorological controls on precipitation and cloud cellular structure in southeast Pacific stratocumulus. Atmos. Chem. Phys., 10, 6347–6362.

    • Search Google Scholar
    • Export Citation

  • Wang, H., Rasch P. J., and G. Feingold, 2011: Manipulating marine stratocumulus cloud amount and albedo: A process-modeling study of aerosol-cloud-precipitation interactions in response to injection of cloud condensation nuclei. Atmos. Chem. Phys., 11, 4237–4249, doi:10.5194/acp-11-4237-2011.

    • Search Google Scholar
    • Export Citation

  • Wang, S., X. Zheng, and Q. Jiang, 2012: Strongly sheared stratocumulus convection: An observationally based large-eddy simulation study. Atmos. Chem. Phys., 12, 5223–5235, doi:10.5194/acp-12-5223-2012.

    • Search Google Scholar
    • Export Citation

  • Wood, R., 2012: Stratocumulus clouds. Mon. Wea. Rev., 140, 2373–2423.

  • Wood, R., M. Köhler, R. Bennartz, and C. O’Dell, 2009: The diurnal cycle of surface divergence over the global oceans. Quart. J. Roy. Meteor. Soc., 135, 1484–1493.

    • Search Google Scholar
    • Export Citation

  • Wood, R., C. S. Bretherton, D. Leon, A. D. Clarke, P. Zuidema, G. Allen, and H. Coe, 2011a: An aircraft case study of the spatial transition from closed to open mesoscale cellular convection over the southeast Pacific. Atmos. Chem. Phys., 11, 2341–2370.

    • Search Google Scholar
    • Export Citation

  • Wood, R., and Coauthors, 2011b: The VAMOS Ocean-Cloud-Atmosphere-Land Study Regional Experiment (VOCALS-REx): Goals, platforms, and field operations. Atmos. Chem. Phys., 11, 627–654.

    • Search Google Scholar
    • Export Citation

  • Wylie, C. R., 1995: Advanced Engineering Mathematics. McGraw-Hill, 1362 pp.

  • Xue, H., G. Feingold, and B. Stevens, 2008: Aerosol effects on clouds, precipitation, and the organization of shallow cumulus convection. J. Atmos. Sci., 65, 392–406.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) Horizontal wind components, (b) potential temperature, and (c) water vapor mixing ratio profiles derived from VOCALS RF-6 measurements.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Plan views of pseudoalbedo at (top to bottom) t = 5, 8, 11, and 14 h for five simulations with N0 = 200 cm−3 and (left to right) = 0.5, 1, 3, 6, and 9 h.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Time series of domain-averaged variables from the five simulations (see legend) in Fig. 2 are shown, namely, (a) the proportional change of the STBL depth, (b) cloud fraction, (c) CWP, (d) surface rain rate, (e) the maximum vertical velocity, and (f) the cloud droplet number concentration.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Time–height sections of horizontally averaged variables derived from the same five simulations in Fig. 2 with (left to right) increasing aerosol relaxation time as shown in Figs. 2 and 3. The variables include (top) cloud water qc (g kg−1, color) and liquid potential temperature θl (contours, min: 289 K, and interval: 1 K); (second row) rainwater flux , where wT denotes raindrop terminal velocity (mm day−1, color) and (contours, min: 0.01 m2 s−2, interval: 0.1 m2 s−2); (third row) buoyancy flux (W m−2, color) and cloud droplet number Nc (contours, start: 10 cm−3, interval: 50 cm−3); and (bottom) (m3 s−3, color) and water vapor mixing ratio qυ (contours, start: 7 g kg−1, interval: 0.5 g kg−1).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Time–height sections of the same set of horizontally averaged variables shown in Fig. 4, but derived from the simulation with N0 = 100 cm−3 and = 24 h.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    The steady-state (a) normalized cloud depth, (b) aerosol number concentration, and (c) rain rate obtained by numerically solving (4)–(6) are shown as a function of the aerosol relaxation time for four example solutions. The four sets of curves correspond to H0 and N0 pairs of 500 m and 500 cm−3 (black), 500 m and 250 cm−3 (red), 250 m and 500 cm−3 (green), and 250 m and 250 cm−3 (blue).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Two sets of example solutions to system (2) and (3) are shown. The cloud depth, cloud droplet number concentration, and rain rate between 1 and 9 h are plotted for (a) H0 = 200 m, N0 = 200 cm−3, = 1 h, and = 0.5, 1, 3, 6, 9 and 100 h; and (b) a set of solutions showing sensitivity to N0, H0, and τH, including the control [same as τH = 1 h in (a)], τH = 3 h, N0 = 500 and 50 cm−3, and H0 = 500 and 150 m (other parameters are as in the control).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (top) Damping time and (bottom) oscillation frequency as a function of the aerosol relaxation time derived from the linearized solution (11). Four examples are shown, corresponding to the equilibrium states (H0, N0) = (500, 500), (500, 250), (250, 500), and (250, 250) (m, cm−3), respectively.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Schematic regime diagram created based on the LES and KF11 results. Two boundary curves (AB and CD) separate the τN–N0 diagram into three distinct regimes (i.e., FRCC, SROC, and AD).

Fig. 1.

(a) Horizontal wind components, (b) potential temperature, and (c) water vapor mixing ratio profiles derived from VOCALS RF-6 measurements.
Fig. 2.

Plan views of pseudoalbedo at (top to bottom) t = 5, 8, 11, and 14 h for five simulations with N0 = 200 cm−3 and (left to right) = 0.5, 1, 3, 6, and 9 h.
Fig. 3.

Time series of domain-averaged variables from the five simulations (see legend) in Fig. 2 are shown, namely, (a) the proportional change of the STBL depth, (b) cloud fraction, (c) CWP, (d) surface rain rate, (e) the maximum vertical velocity, and (f) the cloud droplet number concentration.
Fig. 4.

Time–height sections of horizontally averaged variables derived from the same five simulations in Fig. 2 with (left to right) increasing aerosol relaxation time as shown in Figs. 2 and 3. The variables include (top) cloud water qc (g kg−1, color) and liquid potential temperature θl (contours, min: 289 K, and interval: 1 K); (second row) rainwater flux , where wT denotes raindrop terminal velocity (mm day−1, color) and (contours, min: 0.01 m2 s−2, interval: 0.1 m2 s−2); (third row) buoyancy flux (W m−2, color) and cloud droplet number Nc (contours, start: 10 cm−3, interval: 50 cm−3); and (bottom) (m3 s−3, color) and water vapor mixing ratio qυ (contours, start: 7 g kg−1, interval: 0.5 g kg−1).
Fig. 5.

Time–height sections of the same set of horizontally averaged variables shown in Fig. 4, but derived from the simulation with N0 = 100 cm−3 and = 24 h.
Fig. 6.

The steady-state (a) normalized cloud depth, (b) aerosol number concentration, and (c) rain rate obtained by numerically solving (4)–(6) are shown as a function of the aerosol relaxation time for four example solutions. The four sets of curves correspond to H0 and N0 pairs of 500 m and 500 cm−3 (black), 500 m and 250 cm−3 (red), 250 m and 500 cm−3 (green), and 250 m and 250 cm−3 (blue).
Fig. 7.

Two sets of example solutions to system (2) and (3) are shown. The cloud depth, cloud droplet number concentration, and rain rate between 1 and 9 h are plotted for (a) H0 = 200 m, N0 = 200 cm−3, = 1 h, and = 0.5, 1, 3, 6, 9 and 100 h; and (b) a set of solutions showing sensitivity to N0, H0, and τH, including the control [same as τH = 1 h in (a)], τH = 3 h, N0 = 500 and 50 cm−3, and H0 = 500 and 150 m (other parameters are as in the control).
Fig. 8.

(top) Damping time and (bottom) oscillation frequency as a function of the aerosol relaxation time derived from the linearized solution (11). Four examples are shown, corresponding to the equilibrium states (H0, N0) = (500, 500), (500, 250), (250, 500), and (250, 250) (m, cm−3), respectively.
Fig. 9.

Schematic regime diagram created based on the LES and KF11 results. Two boundary curves (AB and CD) separate the τN–N0 diagram into three distinct regimes (i.e., FRCC, SROC, and AD).
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