Coherent Structures in Large-Eddy Simulations of a Nonprecipitating Stratocumulus-Topped Boundary Layer

Paolo Davini Laboratoire de Météorologie Dynamique/IPSL, École Normale Supérieure, PSL Research University, CNRS, Paris, France, and Istituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle Ricerche, Torino, Italy

Search for other papers by Paolo Davini in
Current site
Google Scholar
PubMed
Close
,
Fabio D’Andrea Laboratoire de Météorologie Dynamique/IPSL, École Normale Supérieure, PSL Research University, CNRS, Paris, France

Search for other papers by Fabio D’Andrea in
Current site
Google Scholar
PubMed
Close
,
Seung-Bu Park Department of Earth and Environmental Engineering, Earth Institute, Columbia University, New York, New York

Search for other papers by Seung-Bu Park in
Current site
Google Scholar
PubMed
Close
, and
Pierre Gentine Department of Earth and Environmental Engineering, Earth Institute, Columbia University, New York, New York

Search for other papers by Pierre Gentine in
Current site
Google Scholar
PubMed
Close
Open access

Abstract

The properties of coherent convective structures are analyzed in a nonprecipitating marine nocturnal stratocumulus-topped boundary layer (STBL) with a series of high-resolution large-eddy simulations (LESs). A new classification method based on octant analysis—using vertical velocity and two passive scalars—is introduced to systematically define convective structures in both the cloudy and the cloud-free regions. It is therefore possible to detect and track updrafts, downdrafts, and their turbulent shells (both ascending and subsiding), together with the entraining air from the inversion layer or the free troposphere. The geometrical and thermodynamical characteristics (e.g., areal fraction, temperature, liquid and total water mixing ratio, buoyancy) of those structures are then accurately described, and particular attention is given to their respective contributions to the turbulent transport of mass, heat, and moisture. It is shown that updrafts, downdrafts, and entrainment are equally important to describe the STBL dynamics. Conversely, it is found that shells, although they partially contribute to the mass transport, have a negligible contribution to the turbulent fluxes of heat and moisture.

Denotes content that is immediately available upon publication as open access.

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

Corresponding author: P. Davini, pdavini@lmd.ens.fr

Abstract

The properties of coherent convective structures are analyzed in a nonprecipitating marine nocturnal stratocumulus-topped boundary layer (STBL) with a series of high-resolution large-eddy simulations (LESs). A new classification method based on octant analysis—using vertical velocity and two passive scalars—is introduced to systematically define convective structures in both the cloudy and the cloud-free regions. It is therefore possible to detect and track updrafts, downdrafts, and their turbulent shells (both ascending and subsiding), together with the entraining air from the inversion layer or the free troposphere. The geometrical and thermodynamical characteristics (e.g., areal fraction, temperature, liquid and total water mixing ratio, buoyancy) of those structures are then accurately described, and particular attention is given to their respective contributions to the turbulent transport of mass, heat, and moisture. It is shown that updrafts, downdrafts, and entrainment are equally important to describe the STBL dynamics. Conversely, it is found that shells, although they partially contribute to the mass transport, have a negligible contribution to the turbulent fluxes of heat and moisture.

Denotes content that is immediately available upon publication as open access.

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

Corresponding author: P. Davini, pdavini@lmd.ens.fr

1. Introduction

Stratocumulus clouds are ubiquitous over the globe—with year-round coverage of around 20%—and have a significant impact on Earth’s radiation budget (Hartmann et al. 1992; Eastman and Warren 2014). They are typically found on the cold eastern edges of the oceans in areas of large-scale subsidence characterized by strong lower-tropospheric stability (Klein and Hartmann 1993; Wood and Bretherton 2006).

The fractional coverage of those low-level clouds is controlled by the dynamics of the stratocumulus-topped boundary layer (STBL), which consists of a well-mixed layer of several hundreds of meters lying below a cloud deck, which is in turn capped by a thin stable inversion layer and by the free troposphere (e.g., Albrecht et al. 1995; Wood 2012). The dynamics of the STBL are challenging for two main reasons: 1) the concurrence of several thermodynamical and turbulent processes in action and 2) the relative thin region over which the most of these processes are occurring.

Indeed, a typical marine STBL shows an abrupt change of the moisture and temperature profiles within an inversion layer of a few tens of meters (Wood 2012). In this thin layer, convective instability, driven by cloud-top radiative cooling, controls the turbulent motions of the underlying mixed layer (Lilly 1968; Randall 1980; Deardorff 1981; Stevens et al. 1999). The entrainment of warm and dry tropospheric air into the STBL, favoring evaporative cooling and mixing, further contributes to local changes in buoyancy (Deardorff 1976; Stevens 2002; Mellado 2017).

Many observational studies have investigated the STBL properties, often focusing on entrainment (e.g., Wood and Field 2000; Gerber et al. 2005; Ghate et al. 2015). Large-eddy simulations (LESs) and direct numerical simulations (DNSs), even though with some caveats, have provided a detailed insight into the STBL dynamics (Moeng et al. 1996; Bechtold et al. 1996; Stevens et al. 2003b, 2005b; Mellado et al. 2009; Yamaguchi and Randall 2012; de Lozar and Mellado 2015).

The STBL is in the climate community spotlight also because stratocumulus clouds represent a large source of uncertainty in global climate model simulations (Bony and Dufresne 2005; Nam et al. 2012). Realistic simulation of low-level clouds is still problematic, in large part because these are the results of the interplay of the representation of many physical processes (Bechtold and Siebesma 1998; Dal Gesso et al. 2015; Bellon and Geoffroy 2016). This considerably affects both the simulation of present-day climate and the spread of future climate projections (Brient and Bony 2013; Tsushima et al. 2016).

Therefore, in order to provide a reasonable representation of the stratocumulus clouds dynamics, an accurate parameterization of the STBL is mandatory. And in order to do so, a comprehensive knowledge of the turbulent fluxes within the STBL must be achieved.

Several works have investigated the organized turbulent transport in the STBL with a special focus on downdrafts, updrafts, and entrainment (e.g., Schumann and Moeng 1991; Moeng and Schumann 1991; Krueger 1993; Kollias and Albrecht 2000). However, the definition of these convective structures is typically unclear or case specific and usually is based on vertical velocity sign or magnitude (Nicholls 1989; Schumann and Moeng 1991; Moeng et al. 1992). The analysis of the entrainment, even when it is done through a passive scalar emitted in the free troposphere (e.g., Kurowski et al. 2009; Pedersen et al. 2016), is usually not included in a comprehensive and complete analysis of the turbulent transport within the STBL. To this day, no systematic study has investigated updrafts, downdrafts, and entrainment in a unique framework and has evaluated each contribution to the overall turbulent transport of the STBL. Nonetheless, the identification of convective structures—including the role of entrainment—is of key importance to improve our knowledge and to correctly parameterize stratocumulus clouds.

Recently, Park et al. (2016) provided a new estimate of the turbulent transport for shallow convection, making use of the concept of coherent structures (Robinson 1991; Haller 2015). The definition of coherent structures can widely vary in literature: usually they are detected as the dominant larger-scale structures within the turbulent flow that are characterized by similar thermodynamical and/or scalar properties over a defined spatial or temporal extent (e.g., Khanna and Brasseur 1998; Haller and Yuan 2000; Farge et al. 2001; Schoppa and Hussain 2002; Kim and Park 2003; Li and Bou-Zeid 2011).

Our work aims to provide a more detailed look at the thermodynamical and transport properties of the STBL using high-resolution LES and coherent structures analysis. This is made in a standard nonprecipitating nocturnal marine stratocumulus case, the Second Dynamics and Chemistry of the Marine Stratocumulus field study Research Flight 01 (DYCOMS II RF01; Stevens et al. 2003a, 2005b). We here use the approach developed for shallow convection by Park et al. (2016), which identified coherent structures using two passive scalars and the vertical velocity field. However, considering the marked dynamical differences between shallow convection and the STBL, we adapt their method in order to focus on the most known convective structures of the STBL, that is, updrafts, downdrafts, and entrainment. Finally, we provide a measure of the turbulent fluxes of each coherent structure identified, evaluating their contribution to the STBL dynamics.

2. Data and methods

a. The model and the case study

We use the University of California, Los Angeles, large-eddy simulation model (UCLA-LES; Stevens et al. 1999, 2005b; Stevens and Seifert 2008). UCLA-LES solves the prognostic equations for the velocity field, liquid water potential temperature , and total water mixing ratio . They are integrated with a third-order Runge–Kutta scheme on a three-dimensional Arakawa C grid. A fourth-order centered difference scheme is used for momentum advection and monotone upwind scheme is used for scalar advection. The subgrid fluxes of heat, moisture, and momentum are parameterized using the Smagorinsky model, which includes a stability correction. Both the eddy Prandtl number and eddy Schmidt number are 1/3, and the Smagorinsky coefficient is set to 0.2. Full details on the code can be found in the official UCLA-LES documentation (Stevens 2010).

The DYCOMS II RF01 case study is selected (Stevens et al. 2003a, 2005b): it represents a nonprecipitating nocturnal maritime boundary layer where the stratocumulus cloud layer is persistent throughout the night. The inversion layer is found at about 850 m, and it is characterized by a temperature jump of about 10 K.

In our simulations, UCLA-LES is set to consider only the reversible conversion between liquid water and water vapor, neglecting any form of precipitation. Surface fluxes are set according to the DYCOMS II RF01 protocol, with latent heat set to 115 W m−2 and sensible heat set to 15 W m−2, while the simplified radiative scheme from Stevens et al. (2005b) is used, providing a radiative cooling dependent on the liquid water mixing ratio of about 60 W m−2. Finally, geostrophic winds are set to Ug = 6.5 m s−1 and Vg = −5 m s−1.

b. Large-eddy simulations

LESs of STBL dynamics require extremely high horizontal and vertical resolution—on the order of 1 m isotropically (Matheou et al. 2016). The spatial dimension of the cloud holes (where entrainment occurs) can be as small as 5 m (Gerber et al. 2005), and the sharp inversion layer needs to be properly resolved in order to avoid an overestimation of the entrainment (Stevens et al. 2005b).

We tested the UCLA-LES model in eight different configurations, changing both the horizontal and vertical grids and the domain size. The simulation length was fixed to 4 h as in the DYCOMS II RF01 LES intercomparison project (Stevens et al. 2005b).

In general, good agreement was found with observations, even at the lowest resolution (80 m × 80 m × 5 m). However, improvements were observed following grid refinement, while the results are not sensitive to the domain size (see the appendix for details). Therefore the configuration used for the rest of the work is the one with the highest horizontal and vertical resolution. It has a domain size of about 5 km × 5 km, using a 10-m horizontal grid (512 × 512 grid points) with a nonregular vertical grid. The vertical grid spacing is refined (about 1 m) close to the surface, then it becomes coarser in the heart of the STBL (up to about 16 m), and it gets again finer at the top of the mixed layer, with a fixed value of 1 m throughout the inversion layer. The grid spacing increases again above the inversion (up to about 55 m) to reach the final height of about 1500 m with 258 levels. In the upper 10 layers (about 400 m) a sponge layer is present to avoid propagation of gravity waves. The grid spacing as a function of height, compared to the 5-m vertical grid used in other sensitivity simulations, is reported in Fig. 1a.

Fig. 1.
Fig. 1.

(a) UCLA-LES vertical grid spacing for the nonregular grid (circles) and the 5-m regular one (diamonds). For the nonregular grid, the resolution in the inversion layer is 1 m. Vertical profiles of (b) and (c) at the first time step (dotted) and averaged during the last hour of simulation (solid). Red lines indicate defined as the maximum gradient of .

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

c. The scalars: and

To detect the coherent structures in the STBL, two passive scalars are added to the simulations. A first scalar is introduced with a linearly increasing profile with height: it is relaxed to its original local values at each time step with a characteristic relaxation time . The second scalar behaves similarly, but it is relaxed with a characteristic time to a step function profile, which is 0 below the STBL height and 1 above it (i.e., in the free troposphere). The STBL height is defined at each time step by the model at the height of maximum gradient (Gentine et al. 2015).

The equations governing the relaxation of the two scalars can be written as
e1
e2

The vertical profiles of the scalars at the beginning of the simulation and after 4 h are reported in Figs. 1b and 1c.

The scalars are used for the definition of the coherent structures, which is described in section 3. While is used to follow the movement of the air masses within the STBL, is used to identify and track the entrainment of air from the free troposphere. It is thus important to correctly define the relaxation time scales of the two scalars.

This has been done with a series of specific sensitivity experiments, aimed at investigating the coherent structures sensitivity to the relaxation times and . We found that for values of larger than the STBL eddy turnover time is well mixed in the STBL, and that the coherent structures definition we adopted is almost insensitive to further changes, especially in terms of contribution to turbulent fluxes. We therefore fixed an optimal value of τFT = 3200 s. The corresponding sensitivity experiments and discussion are presented in the appendix.

Conversely, the definition of is more subtle. Also in this case the sensitivity of the heat and moisture fluxes to the relaxation time is minor. Considering that describes the fast motion of the air masses within the STBL itself (as opposed to , which is used to detect the slower process associated with free-tropospheric entrainment), we fixed a shorter time scale at half of the eddy turnover time so that τBL = 400 s. A more detailed discussion on this is presented in the appendix.

3. Identification of the coherent structures

To identify the coherent structures, we make use of the octant analysis (Volino and Simon 1994; Park et al. 2016). Here a spatial field is divided into eight clusters—named octants—according to the signs of the anomalies of three variables. Octant analysis is an extension of better-known quadrant analysis (Raupach 1981; Sullivan et al. 1998), where only two fields are used. To present our methodology in a detailed way, we start introducing the quadrant analysis, which is based on the anomalies of w and . After that, we will present the octant analysis including the anomalies of .

a. Quadrant analysis

Figure 2 shows a snapshot at the last time step of the simulation (after 4 h) of the horizontal cross section at 0.9zi (785 m), that is, already within the cloud layer. The detailed cloud structure can be seen in Fig. 2a, with circular clouds with a radius of about 1 km, surrounded by thin cracks where the liquid water content drops abruptly. The cloud pattern resembles the observed patchy stratocumulus cloud deck obtained from satellite imagery (Stevens et al. 2005a; Wood 2012).

Fig. 2.
Fig. 2.

Horizontal cross sections at 0.9zi for (a) liquid water mixing ratio, (b) vertical velocity, (c) quadrants, (d) , (e) , and (f) octants at t = 4 h.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

Strong updrafts are evident in the core of the clouds, with maximum velocity of about 3 m s−1 (Fig. 2b). The presence of updrafts is confirmed by the low values of in this region (Fig. 2d), indicating that such air is coming from lower levels. Conversely, negative vertical speeds are generally placed along the cloud edges, where the high concentrations of suggest that entrainment is occurring (Fig. 2e). However, the vertical velocity field exhibits important spatial fluctuations, and descending air masses can also be found in the core of the updrafts (Fig. 2b).

Entrainment is even more evident when looking at the vertical cross section at x = −5 m in Fig. 3. Here the cloud hole at about y = −1400 m is associated with a plume of injected into the STBL (Fig. 3c) with negative vertical velocities, pointing to the presence of an entrainment event. More generally, the cross section is dominated by a large updraft at about y = 500 m (Fig. 3d) that is carrying low concentrations of up to the inversion (Fig. 3b).

Fig. 3.
Fig. 3.

Vertical cross sections at X = −5 m for (a) liquid water mixing ratio, (b) , (c) , (d) vertical velocity, (e) quadrants, and (f) octants at t = 4 h.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

As mentioned above, in order to identify the coherent structures, we start introducing the quadrant analysis. We make use of the signs of the anomalies from the horizontal slab average of and w to identify rising and descending air masses: (i.e., , where angle brackets identify the horizontal average) identifies the direction of movement of the air parcels and (i.e., ) their provenance.

The result of the quadrant analysis is shown in Fig. 2c and Fig. 3e. Air coming from below and with positive velocities defines Octant 4 (marked in red): this clearly identifies the updrafts. Similarly, Octant 7 (marked in dark blue) is descending air coming from above , identifying the downdrafts.

In addition to these two “main” convective structures, air with the same signs of updrafts and downdrafts is often found to have opposite but weaker vertical velocity. In analogy to what was done in shallow convection by Park et al. (2016), we defined them as the subsiding and ascending shells (marked in light blue and orange). However, as it will be discussed in detail below, they are characterized by a different dynamics and therefore we name them “turbulent shells” instead of “returning shells.”

b. Octant analysis

Even if the quadrant analysis provides significant information on the convective structures, we would like to distinguish between the part of the downdraft air that is merely the recirculation within the STBL and the part that is coming from the free troposphere. To fully define the coherent structures, we thus generalize to octant analysis using a third variable (SFT). However, the quadrant-to-octant step is performed in an atypical way, avoiding the slab-average anomalies of . Indeed, using the slab averaging also for would lead to a misleading definition of the octants: at lower levels, small positive anomalies compared to smaller slab-average values would be indeed spuriously considered as entrainment.

We therefore introduce a threshold so that : for values of a particle is considered as originating from the free troposphere while for the particle is considered to be part of the mixed-layer air. This threshold can be thought as an analogy of what is done in mixing ratio analysis (which uses a minimum mixing ratio to distinguish between tropospheric and boundary layer air) or with Lagrangian particle tracking (which uses a maximum tracking duration to define the entrained air).

Combining the signs of together with the four quadrants introduced above eight different clusters are obtained, as shown in Table 1, in Figs. 2f and 3f. At the lower levels of the STBL, since the concentration of is low and so , the previously defined quadrants (i.e., updraft, downdraft, ascending, and subsiding shells) are the same as before. However, in the proximity of the inversion, air with concentrations of the higher than —representing parcels of free-tropospheric provenance—is found.

Table 1.

The eight octants defined according to the different anomalies of vertical velocity, scalar, and tracer.

Table 1.

The new clusters are found in the cloud region only: Octant 5 (light green) represents air coming from aloft that is descending (positive and negative velocity)—therefore with the same properties of downdraft air—but originated from the free troposphere : this can be defined as the entrainment, and it is evident along the cloud holes in Fig. 2f or in the entraining plumes themselves, as seen in Fig. 3f [similar to what shown by Yamaguchi and Randall (2012)]. We next define Octant 1 (pink), which has the same and properties as Octant 5 but opposite vertical velocity. Finally, the last two octants—Octant 2 (yellow) and Octant 6 (dark green)—represent the updrafts approaching the free troposphere (with ), and mixing there, acquiring positive anomalies of . These are mainly found along cloud edges within the inversion layer but are relatively infrequent.

Since we are not aiming to define coherent structures in the free troposphere, octants are defined only up to values of smaller or equal to 0.99.

c. The threshold definition

An important question is obviously how to objectively determine the above-defined threshold . A first guess is the mixed-layer mean: however, this definition would be affected by the discretization of the vertical grid and by the large values of in the proximity of . We thus introduce a threshold that is based on a thermodynamical difference between tropospheric and mixed-layer air. Indeed, the free-tropospheric air is warm and dry: when this air gets entrained into the mixed layer, it contributes negatively to the upward turbulent heat flux. On the other hand, boundary layer descending air is colder than its environment and, thus, contributes to the upward turbulent heat flux with a positive term. The threshold is thus defined as the value of that maximizes the difference between the vertically integrated turbulent heat flux of the octant representing the entrainment (, , and ) and the one representing the STBL-originated downdrafts (, , and ). We are trying to maximize the separation between the descending air that is warm (coming from the inversion layer or the free troposphere) from the descending air that is cold (coming from the PBL).

Formally, we define a cost function to be maximized as follows:
e3
where is the model lid and the subscripts identify the conditional mean over the entrainment and downdraft octants. The cost function is unimodal, as can be seen in Fig. 4a. A Brent convergence algorithm (Brent 1973) is used to search for the maximum: the procedure of optimization is illustrated in Figs. 4b and 4c. The threshold is the value of thr that maximizes the cost function and consequentially the separation between the heat fluxes contribution of the entrainment and downdraft octants. The threshold is allowed to vary in time to follow the fluctuations in the concentration of in the STBL and to take into account the noncontinuity of the entrainment events, providing a robust and adaptive measure of the entrainment. We note that what is called “entrainment” should be not be intended as strictly dry (i.e., without liquid water) free-tropospheric air. Indeed, as the air mixes in the inversion and down into the mixed layer, it becomes moister and colder and thus “disappears” quickly. Our definition of entrainment is thus tracking warm air coming from the free troposphere as long as its contribution to the downward heat transport is positive.
Fig. 4.
Fig. 4.

(a) Cost function values as a function of threshold thr evaluated for the last time step of the simulation (at t = 4 h). Vertical profiles of turbulent heat flux for the (b) entrainment octant and (c) downdraft octant for a set of values of thr. The colors of the profiles correspond to the values of the points shown in (a). The dashed lines show the heat flux profiles for the maximum value of the cost function, that is, for thr0.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

4. Properties of the coherent structures

a. Main features

A more integrated analysis of the octants is provided by investigating the horizontal cross section at six different levels in Fig. 5. Here we see that Octants 4 (red) and 7 (dark blue) (i.e., updrafts and downdrafts) dominate the lower and middle parts of the STBL. Around those main structures, Octants 3 (orange) and 8 (light blue) represent what are defined as turbulent shells (ascending and subsiding, respectively). These octants track air with the same and values but with opposite vertical velocity, suggesting the presence of turbulent mixing and velocity fluctuations—generally placed at the edge of the updraft–downdraft cores as can be seen in Figs. 3 and 5.

Fig. 5.
Fig. 5.

Horizontal cross sections for octants at 0.05, 0.2, 0.5, 0.8, 0.95, and 1 × zi at t = 4 h.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

As mentioned above, in the upper layers entrainment (Octant 5; light green) becomes more frequent. Entrained air gets mixed by turbulence and “transformed” into standard downdraft air (Octant 7; dark blue) as it descends into the STBL: as the air descends it levels out its anomalies (and its thermodynamical properties) with the rest of the mixed layer. Indeed, at levels below a half of the STBL height, only very few grid points are detected as entrainment (Octant 5, light green). Such mixing is evident also in Fig. 3, where cloud holes are associated with entrained air and farther down in the STBL downdrafts (Octant 7, dark blue) progressively replace entrainment (Octant 5; light green).

At higher levels the less-frequent Octant 1 (pink) is identified: this has the same and values as the entrainment octant but opposite velocity. It is mainly found in the inversion layer in areas of low vertical velocity, highlighting near-laminar flow and gravity waves. Indeed, the presence of gravity waves is clearly visible in the proximity of , where waves generated by the overshooting updrafts propagate in the inversion layer (Figs. 5e,f).

Finally, the last two octants—Octant 2 (yellow) and Octant 6 (dark green)—are infrequent too, and they are mainly found in the inversion layer along the edges and the top of the updrafts (Fig. 5f). They detect regions where updraft air, mostly originating from lower levels mixes with free-tropospheric air (thus showing a high concentration of ). This appears when updrafts overshoot and mixing occurs along the cloud edges: in other words, they identify the place where cloud-top detrainment is occurring.

Further insight onto the properties of the different octants is provided by the scatterplot of Fig. 6. This shows that the thermodynamical properties of the air masses are strongly associated with the and anomalies—on which octants are based—providing further physical evidence of the robustness of the octant definition. Entrainment characterizes the tail of dry and warm air in the upper portion of the STBL (light green), which then mixes and becomes downdraft air as it descends (dark blue), becoming colder but remaining relatively dry. Conversely, updrafts are warm and moist (red): at lower levels they are characterized by large temperature and moisture anomalies (Figs. 6a,b) that lose intensity going to upper STBL levels. It is also interesting to see that the mixing octants (Octants 2 and 6; yellow and dark green, respectively; Figs. 6e,f) are characterized by warmer but drier air than updrafts and in the same time by moister but colder air than entrainment. It is thus very likely that they represent the transition between the overshooting updrafts and the creation of entrainment, that is, when cloud-top detrainment prepares the ground for entrainment of inversion-layer air through successive mixing events (Gerber et al. 2013).

Fig. 6.
Fig. 6.

Scatterplot for total water mixing ratio and liquid potential temperature at 0.05, 0.2, 0.5, 0.8, 0.95, and 1 × zi at t = 4 h. Colors identify the different octants. Each point represents a grid point on the plane; diamonds represent the average value for each octant at the given level. Vertical and horizontal lines identify the slab-average value. Please note the different axes for 0.95zi and 1zi.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

Figure 6 shows also that not only the and anomalies but also the thermodynamical properties (i.e., and ) of the turbulent shells are similar to the ones from the their corresponding “main” octants. Indeed, the diamonds marking the average of the Octants 7 and 3 (downdraft and ascending shells, respectively) and of Octants 4 and 8 (updraft and subsiding shells, respectively) at each level are superimposed in large part of the STBL. This further suggests that turbulent shells—both ascending and subsiding shell—are part of the turbulent motion in the STBL and they cannot be compared directly with the ones identified in shallow convection (Heus and Jonker 2008).

b. Vertical profiles

At first glance, the presence of downdrafts and updrafts may suggest the presence of a symmetric circulation. In a broader sense, the STBL shares many common features with Rayleigh–Bénard convection: a warm surface below and a radiatively cooled surface above. However, as we will show, the free-tropospheric entrainment (due to the absence of a rigid lid at the top) and the phase change of water generate important differences in the vertical profiles of updrafts and downdrafts.

As previously noted in Fig. 2, updrafts and downdrafts dominate the octants’ frequency, as can be seen by the vertical profiles in Fig. 7a. Combined, they cover about 75% of the area in the most of the STBL. As expected, approaching the inversion layer, the entrainment octant replaces the downdraft octant, reaching values of about the 30% at . In the vicinity of no downdraft octant is found: the maximum height at which we identify downdrafts is about 10–20 m below the maximum level reached by updrafts, showing that the octants correctly identify the presence of overshoots.

Fig. 7.
Fig. 7.

Vertical profiles averaged over the last hour of simulation according to each octant for (a) octant frequency, (b) total water mixing ratio , (c) liquid potential temperature , (d) liquid water mixing ratio l, (e) vertical velocity w, (f) buoyancy, (g) turbulent kinetic energy, and (h) radiative cooling. Solid green line is zi; dashed black line is the mean vertical profile. Only values when the octant frequency is higher than 2% are plotted.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

Turbulent shells occupy most of the remaining horizontal area, with a higher frequency of occurrence at the surface and at the top, where the presence of the physical and thermodynamical lids favor the occurrence of a returning circulation: this can also be seen in the turbulent kinetic energy [TKE = (1/2)〈u2 + υ2 + w2〉] profile of Fig. 7g, in which subsiding shells are characterized by large values of TKE. Turbulent shells show weaker vertical speeds (about 1/3 of their “main” octants; Fig. 7e).

The vertical profiles of liquid and total water mixing ratio and liquid water potential temperature, shown in Figs. 7b–d, confirm the results of Fig. 6. The updraft octant is composed of warm and moist air, downdrafts are cold and relatively dry, and entrainment is dry and warm. It is however interesting to point out that the liquid water content of the entrainment octant is not null, suggesting that our definition of entrainment is not bound to the presence of pure dry air, but it is tracking the motion of parcels of free-tropospheric air that can then mix within the cloud layer generating cloudy air. As a consequence, entrained air exhibits weak but nonzero radiative cooling (Fig. 7h).

As expected, buoyancy (Fig. 7f) is strongly affected by phase change of water: updrafts are characterized by positively buoyant air with large positive vertical velocity and they increase their buoyancy into the cloud layer owing to condensation of water vapor. Interestingly the properties of the downdrafts are almost symmetrical in vertical to the updrafts, with a large negative vertical velocity and a minimum in buoyancy magnitude in the middle of the STBL. They experience also a further decrease in buoyancy approaching the surface, owing to the local increase of environmental temperature—caused by surface fluxes.

Finally, it is interesting to note that Octants 2 and 6, which represent the turbulent mixing in the inversion layer, have large values of TKE (and strong negative speed for Octant 6). Moreover, when compared to entrainment (Octants 1 and 5), they also show stronger radiative cooling, colder temperatures, and considerably negative values of buoyancy. This further suggests that a large part of the mixing in the inversion layer is identified by Octants 2 and 6.

As the air descends into the column, it loses memory of its origin (i.e., decays) and it becomes “standard” entrainment (Octants 5 and 1). At lower levels entrainment becomes less-frequent mixing with the downdraft octant. However, since Octant 5 is composed of negatively buoyant air with a large negative vertical speed, the buoyancy anomalies further accelerate the vertical velocity of the downdrafts. As a result both updrafts and downdrafts reach maximum speed values in the core of the STBL, but the largest vertical velocities are attained by the entrainment octant (Fig. 7e).

c. Shape

The shape of the updrafts and downdrafts structures (Fig. 8) can be studied introducing the definition of “circularity” of each structure: this is the ratio c = 4π[Area/(Perimeter)2]. A circle will have c = 1 while an asymptotic stretched ellipse (i.e., a line) will have c = 0.

Fig. 8.
Fig. 8.

Circularity profiles for updraft (Octant 4; red), downdraft (Octant 7; solid blue), and full downdraft (Octant 7 + Octant 5; dotted blue), averaged during the last hour of simulation. Solid green line is zi.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

We identify individual updrafts and downdrafts (Octant 4 and Octant 7, respectively) at each level and then compute their circularity. To take into account the fact that entrainment and downdraft octants belong to the same vertical movement, circularity for the two octants jointly is also estimated (Octants 5 and 7). The average profile of circularity is then computed as the area-weighted circularity at each level. Only structures having a horizontal area at least of 0.05 km2 are considered.

Figure 8 shows that updrafts evolve from a stretched-streaks structure to round thermals as they move up into the STBL (as seen also in Fig. 5). Downdrafts behave similarly: they are relatively stretched aloft in the inversion layer and become more circular close to the surface. This is typical of convective instability that organizes in thermal sheets (Stull 1988). However, wind shear tends to increase this feature, especially close to the surface, as commonly seen in a boundary layer with shear (Park et al. 2016). When averaged throughout the STBL, downdrafts are more elongated than updrafts (c = 0.034 for entrainment/downdrafts and c = 0.047 for updrafts), which means that for the same area the perimeter of updraft structures is smaller by 20%.

d. Turbulent fluxes

We finally investigate the relative contribution of each coherent structure to the turbulent transport of mass, heat, and moisture (Fig. 9). Indeed, this is essential for the development of a physically based parameterization of any boundary layer in weather and climate models (e.g., Rio et al. 2010).

Fig. 9.
Fig. 9.

Vertical profiles averaged over the last hour of simulation according to each octant for (a) turbulent mass flux , (b) turbulent heat flux , and (c) turbulent total water flux . Solid green line is zi; dashed black line is the mean vertical profile. Only values when the octant frequency is higher than 2% are plotted.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

The largest transport of mass is carried out by updrafts and downdrafts, while within the inversion layer the entrainment octant makes for a large part of transport. When considered together, entrainment and downdraft octants roughly match—with opposite signs—the vertically integrated mass transport of updrafts (about 205 kg m−1 s−1). Turbulent shells (both subsiding and ascending) have a smaller but not negligible contribution to mass transport that integrated over the whole STBL is about 10%–15% of the share of the updraft octant.

In terms of heat transport, as expected by the thr0 definition, downdraft and entrainment have opposite contributions to the turbulent heat flux. The global contribution of the three main octants (updraft, downdraft, and entrainment) to the overall turbulent heat flux profile is complex (Fig. 9b). Updrafts dominate the positive transport in the lower part of the column and downdrafts in the higher levels. However, the negative heat flux carried by entrainment is the largest term of the decomposition, especially in the cloud region. It is interesting to note that the updrafts contribution to the heat flux transport becomes negative approaching the inversion owing to higher environmental temperature aloft. In this region—owing to the large gradient of temperature—also secondary octants acquire nonnegligible relevance.

Moisture transport is simpler, with positive values associated with the mixing of both surface moisture and dry entrained air into the STBL. In our experiment, the contribution of the updrafts is about twice that of downdrafts/entrainment with the exception of the upper part of the STBL, where entrainment becomes more important. Integrated over the whole column, about the 65% of the moisture transport is associated with updraft octant, and only 20% (25%) to downdraft (entrainment) octant.

The turbulent shells found in the STBL deserve a final comment: they are clearly different from those observable in shallow or deep convection (Heus and Jonker 2008; Glenn and Krueger 2014). As shown in Fig. 9, their contribution to the mass flux is minor, whereas their overall heat and moisture fluxes are negligible. This confirms the idea that they represent mostly turbulent mixing occurring along the edges of updrafts and downdrafts. While in the shallow convection we have sparse convection surrounded by large portions of stagnant and slightly subsiding air, in the STBL turbulence is present everywhere, and it is characterized by updrafts and downdrafts strongly interacting with each other; it is not unusual to see updrafts and downdrafts that collide in the middle of the STBL.

5. Conclusions

In the present work we have investigated the properties of coherent structures in a nonprecipitating nocturnal marine stratocumulus-topped boundary layer. This has been done with a series of high-resolution large-eddy simulations performed with UCLA-LES in the DYCOMS II RF01 case study. We have introduced a novel method of analysis based on two different passive scalars and the octant analysis, allowing a qualitative and quantitative analysis of coherent structures in the STBL.

The method is weakly sensitive to the relaxation times of and and is adaptive to the concentration of in the boundary layer, detecting not only updrafts and downdrafts (and their shells) but also the entrainment of free-tropospheric air. The latter is identified using an optimization procedure aimed at finding the maximum separation between the negative turbulent heat flux associated with the entrainment and the positive one associated with downdrafts.

The resulting octant analysis identifies eight different coherent structures that are described in terms of their thermodynamical properties: this is summarized by the schematic diagram presented in Fig. 10. The surface fluxes generate elongated and irregular “streak” updrafts (Octant 4; red), made by warm, moist, and positively buoyant parcels that acquire strong positive vertical velocity and rapidly reach the lifting condensation level, acquiring further buoyancy and here generating the stratocumulus cloud deck. As the updrafts ascend, they partially mix with the rest of STBL (along their turbulent subsiding shells marked in light blue). When updrafts reach the inversion layer they begin to lose buoyancy, penetrating irregularly in the free troposphere and forming cloud domes rather than a flat surface. The overshooting air turbulently mixes with free-tropospheric air along the cloud edges (dark green and yellow in Fig. 10) and generates gravity waves that propagate in the nonturbulent free troposphere.

Fig. 10.
Fig. 10.

Schematic diagram representing the STBL turbulent coherent structures identified with the octant analysis. Each color is representative of each octant.

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

At this stage, the inversion deflects horizontally the air masses—creating a horizontal pressure gradient that favors entrainment—and, following continuity, forces them back to the STBL. In the meantime the cooling, both evaporative and radiative, acts on the newly mixed air reducing its buoyancy. Such a mixed air mass starts its descent as entrainment, penetrating along cloud edges and elongated cloud holes (Octant 5; light green), whereas air that has never left the STBL is identified as a standard downdraft (Octant 7; dark blue). Turbulent shells develop on both sides of these coherent structures (pink and orange, respectively). As it descends, entrainment air is further mixed with STBL air and finally disappears, transferring its negative speed and buoyancy to downdrafts. Finally, this air hits the surface and, by continuity, gives rise to new updrafts.

Given the high degree of turbulence and the complex three-dimensional nature of the flow, the real circulation is obviously more complex that the one depicted above: the schematic in Fig. 10 provides a distinction among the different coherent structures.

The method presented here allows a detailed quantification of the turbulent fluxes associated to each structure and can be easily exported to all the STBL configurations. In this direction, we ran a simulation with 20 m × 20 m × 5 m resolution and 256 × 256 grid points of the DYCOMS II RF02 case study following the setup of Ackerman et al. (2009). Even though the results are considerably different owing to the different setups between the two experiments—RF02 includes drizzle, a weaker inversion, a larger liquid water path, etc.—the combination of relaxed scalars, threshold detection, and octants is robust. It correctly identifies updrafts, downdrafts, entrainment, and their turbulent shells (not shown).

Most importantly, this work highlights a nonnegligible contribution of the downdrafts to mass, heat, and moisture fluxes in the STBL, pointing to a potential weakness of most of the present-day parameterizations of stratocumulus convection in weather and climate models. Most of them include an eddy diffusion scheme (e.g., Lock et al. 2000) usually in combination with a nonlocal mass flux component to characterize an ensemble of updrafts [known as eddy diffusivity–mass flux approach (EDMF); Köhler et al. 2011]. Those EDMF schemes—although they can represent some STBLs in a satisfactory way (Sušelj et al. 2013)—are based on the assumption that only a small areal fraction is covered by updrafts. Also, they do not include an explicit representation of boundary layer entrainment. From a physical point of view these assumptions are not valid, and they lead to unrealistic response to changes in radiative and surface forcing (Dal Gesso et al. 2015). As shown by Fig. 9, the contribution to turbulent transport of downdrafts is important and should be accounted for in order to reproduce correctly the STBL dynamics. Such contribution becomes striking when looking at turbulent heat flux and its subdivision into a STBL component and an entrained one.

The relative contributions of the coherent structures—especially to heat and moisture turbulent fluxes—are likely dependent on external forcing (as the strength of the radiative cooling or the intensity of the surface fluxes) and on the mean state (as temperature and moisture profiles). As can be easily imagined, two different structures can be affected in different ways by different forcing.

This analysis paves the way to further studies: quantifying the different contributions of each coherent structure as a function of several external forcing, such as the radiative cooling, the wind shear, the vertical stability, and the surface fluxes. With the adoption of coherent structures analysis, we can provide a robust way to estimate the turbulent fluxes sensitivity in order to inform parameterization development—as has been done in shallow convection (Rio et al. 2010)—accounting for each different contribution, especially the ones from updrafts, downdrafts, and entrainment.

Acknowledgments

PD and FD gratefully acknowledge the funding from the European Union’s Horizon 2020 research and innovation program COGNAC under the European Union Marie Skłodowska-Curie Grant Agreement 654942. PG acknowledges the funding from the National Science Foundation EAR 1552304 CAREER grant and NASA New Investigator Program (NIP) NNX14AI36G.

APPENDIX

Sensitivity Experiments

a. Resolution and domain size

STBL simulation by LES is known to be affected by domain size and horizontal and vertical resolution (e.g., Pedersen et al. 2016). We explored this sensitivity with eight DYCOMS II RF01 simulations (4 h long) for which 3D data have been stored only for the last hour of simulation (sampling it every 120 s).

Keeping constant the domain size, 80-, 40-, 20-, and 10-m horizontal grids have been tested. For the two finer grids, both a 5-m regular vertical grid and a nonuniform vertical grid (with 1-m resolution in the inversion layer) have been used. The vertical grid spacing of the two grids is shown in Fig. 1a. All the above-mentioned simulations have a fixed domain size of 2.56 km × 2.56 km: in order to test possible sensitivity to the domain size, an 80 m × 80 m × 5 m simulation over a ~41 km × 41 km domain was also run.

A comparison with the observations (Stevens et al. 2005b) shows that even at lower resolution the model performances are fairly good (Fig. A1). Vertical profiles of total water mixing ratio and liquid potential temperature are in agreement with original data. Finer grids, both vertical and horizontal, increase the liquid water content and the radiative cooling at the top of the cloud. As a consequence, at higher resolution the STBL is colder than for a coarser grid. Finer grids also reduce the UCLA-LES bias in vertical velocity variance and skewness. Finally, by comparing the two experiments at 80 m × 80 m × 5 m grid, one with small and the other with large domain, it is possible to see that domain size is not affecting the statistics of any variable, similarly to what was reported by Pedersen et al. (2016).

Fig. A1.
Fig. A1.

Vertical profiles averaged over the last hour of simulation for the sensitivity experiments to horizontal and vertical resolution. The different grids used are marked in the legends. Dashed lines show the cloud base and the cloud top. Dots show the observational values retrieved from Stevens et al. (2005b).

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

Figure A2 shows the time evolution of entrainment velocity (evaluated as ) and cloud limits, suggesting that these are weakly influenced by the resolution. As commonly seen in many DYCOMS II RF01 LESs, an overestimation of the cloud-bottom height is found. Higher resolution improves the cloud cover, which is close to one at the highest resolution (10 m × 10 m × 1A, where A marks the fact that the vertical grid spacing is irregular—up to 1 m in the inversion layer—as described in Section 2b) as observed. A deeper liquid water path and higher values of turbulent kinetic energy and maximum vertical speed are also found. Even if a completely stationarity cannot be achieved owing to presence of constant surface fluxes and entrainment, the temporal variability in the last hour of simulation is minor. The simulation at 80 m × 80 m × 5 m on the larger domain is a smoothed averaged of the one on the small domain, suggesting again a negligible sensitivity to domain size.

Fig. A2.
Fig. A2.

Time evolution of (a) liquid water path, (b) turbulent kinetic energy, (c) cloud fraction, (d) maximum vertical velocity, (e) entrainment velocity (1-min running mean), and (f) cloud-base, cloud-top, and inversion heights for the sensitivity experiments to horizontal and vertical resolution. Dots show the observational values retrieved from Stevens et al. (2005b).

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

Finally, it is interesting to study the sensitivity of the octant analysis to the different resolutions. This is done in Figs. A3a and A3b, where the variability of the octant frequencies and of the turbulent fluxes is shown (averaged over the STBL and over the last hour of simulation). These quantities are weakly sensitive to resolution: in general, finer grids lead to a larger mass transport by updrafts and downdrafts. Conversely, they reduce the amount of entrainment octant. Finally, while moisture fluxes seems to be insensitive to the different grids, Fig A3b shows that turbulent heat fluxes are sensitive to it. The most evident variability is seen for the updrafts, which moves from a negative averaged transport for coarse resolution to a positive one: this is actually driven by the change in the environmental temperature in the STBL. Low-resolution simulations are warmer (owing to the reduced radiative cooling) and therefore the heat flux contribution by updrafts tends to be negative.

Fig. A3.
Fig. A3.

Scatterplots representing the sensitivity to resolution and domain size of the STBL vertically averaged (a) octant frequencies vs octant turbulent mass flux and (b) octant turbulent heat flux vs octant turbulent total water flux . The outer colors represent the octant on which the conditional mean is evaluated; the inner color identifies the sensitivity experiment. (c),(d) As in (a) and (b), respectively, but for sensitivity to the relaxation time . (e),(f) As in (a) and (b), respectively, but for sensitivity to the relaxation time .

Citation: Journal of the Atmospheric Sciences 74, 12; 10.1175/JAS-D-17-0050.1

More generally, the lack of convergence of several variables suggests that LES of the DYCOMS II RF01 case would likely benefit of a further increase of resolution, especially in its horizontal component. A uniform grid with less of 2.5-m resolution seems to be necessary to achieve convergence for the main thermodynamical fields (Matheou et al. 2016). However, a 1-m grid—without changing the number of grid points—would lead to an undesirably small domain of 512 m × 512 m (a factor of 100 in terms of area). Given that in the present work we aim to characterize the statistical properties of the coherent structures within STBL, such a reduced-size domain would likely invalidate our analysis (Sullivan and Patton 2011). The 10 m × 10 m simulation represents thus the most reasonable trade-off between domain size and grid refinement, keeping constant the available computing time.

b. relaxation time

The sensitivity analysis of the octant frequencies to the relaxation time has been carried out with a series of 4-h simulations with a smaller domain (128 m × 128 m) and fixed horizontal and vertical resolution (20 m × 20 m × 5 m).

Considering the generalized convective vertical velocity in the stratocumulus case following Deardorff (1980),
ea1
which in our experiments is w ~ 1.05–1.1 m s−1, and considering a STBL height around 870 m, we can estimate an eddy turnover time of about τ = 800–850 s. This value—13–14 min—is similar to what is generally reported in literature for stratocumulus convection (Stevens et al. 2005b).

Seven different relaxation times have tested, ranging from τ/2 to 8τ (i.e., from 400 to 6400 s). Analogous to what was done for resolution, Figs. A3c and A3d show the sensitivity of the different octants-averaged STBL-averaged quantities. As expected, a fast relaxation time reduces the frequency and the mass flux for the entrainment octant, whereas a slow relaxation time increases it. However, for large enough decay times , becomes well mixed in the STBL, and a convergence for the values of downdraft and entrainment frequencies is achieved. Conversely, the other octants are not sensitive to . Heat and moisture fluxes—for entrainment and downdraft octants—are slightly more affected; however, they show the same convergence for large enough values of . Indeed, setting a large relaxation time is physically reasonable considering the long time needed by the parcels to move from the near-laminar free troposphere (where vertical speed are almost null) to the turbulent STBL (e.g., Yamaguchi and Randall 2012).

Therefore = 3200 s is set, but any values higher than this will have negligible effect on our results.

c. relaxation time

With a similar setup as the one for the relaxation time, we run sensitivity experiments to investigate the dependence of our results to the relaxation time, ranging from τ/8 to 4τ (i.e., from 100 to 3200 s). Figures A3e and A3f show that for the strongest sensitivity is associated with the turbulent shells (i.e., ascending and subsiding shells) and their main coherent structures (i.e., downdrafts and updrafts, respectively). The sensitivity to is only apparently relevant: indeed, air moving from one octant to another is characterized by small vertical velocities and small temperature and water mixing ratio anomalies. Consequently, the change in the vertical turbulent fluxes of heat and moisture is almost negligible (Fig. A3f).

It is important to highlight that, owing to the turbulent nature of the flow within the STBL, it is actually impossible to uniquely define when a particle stops to be part of the ascending (subsiding) shell and starts to be part of the updraft (downdraft) structures. This is because mixing and thermodynamical process operates continuously and most thresholds are arbitrary. An extremely short implies that all the particles belong to the downdraft and updraft octants. As long as we increase the relaxation time, more particles are transferred from the updraft octant to the ascending shell one and from the downdraft octant to subsiding shell one. In the limit of infinite , the profile of will be well mixed in the STBL: it will be impossible to distinguish between shells and their corresponding main octants, making useless.

It is thus important to assess the properties of the eddies within the STBL to correctly set . We know that for the largest eddies in the DYCOMS II RF01 STBL we can evaluate , , and . Considering the eddy kinetic energy scaling and assuming eddies as isotropic, an eddy of will have a typical . Therefore, eddy turnover time ranges from about 400 s for the smallest eddies up to about 800 s for the largest ones.

From a physical point of view, this implies that a very short —smaller than 400 s—would imply that all particles originate close to the height of interest; that is, the transport would appear artificially diffusive. On the other hand, a too-large —larger than 800 s—would “kill” the contribution from small-scale eddies, which will be completely mixed, thus neglecting their contribution to the overall transport. Therefore, even if an apparent convergence is seen for large relaxation times, it is mandatory to assume an intermediate value.

Therefore, τBL = 400 s is chosen as it is representative of the time scale of the smallest eddies resolved by the model. Indeed, such a value is a good trade-off between the capacity of describing the small-scale turbulence in our LES and the possibility of identifying the larger-scale nonlocal transport in the STBL.

REFERENCES

  • Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 10831110, https://doi.org/10.1175/2008MWR2582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Albrecht, B. A., M. P. Jensen, and W. J. Syrett, 1995: Marine boundary layer structure and fractional cloudiness. J. Geophys. Res., 100, 14 20914 222, https://doi.org/10.1029/95JD00827.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bechtold, P., and P. Siebesma, 1998: Organization and representation of boundary layer clouds. J. Atmos. Sci., 55, 888895, https://doi.org/10.1175/1520-0469(1998)055<0888:OAROBL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bechtold, P., S. K. Krueger, W. Lewellen, E. Van Meijgaard, C. Moeng, D. Randall, A. Van Ulden, and S. Wang, 1996: Modeling a stratocumulus-topped PBL: Intercomparison among different one-dimensional codes and with large eddy simulation. Bull. Amer. Meteor. Soc., 77, 20332042, https://doi.org/10.1175/1520-0477(1996)077<2033:MASTPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bellon, G., and O. Geoffroy, 2016: How finely do we need to represent the stratocumulus radiative effect? Quart. J. Roy. Meteor. Soc., 142, 23472358, https://doi.org/10.1002/qj.2828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bony, S., and J.-L. Dufresne, 2005: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett., 32, L20806, https://doi.org/10.1029/2005GL023851.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brent, R. P., 1973: Algorithms for Minimization without Derivatives. Courier Corporation, 195 pp.

  • Brient, F., and S. Bony, 2013: Interpretation of the positive low-cloud feedback predicted by a climate model under global warming. Climate Dyn., 40, 24152431, https://doi.org/10.1007/s00382-011-1279-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dal Gesso, S., J. Van Der Dussen, A. Siebesma, S. De Roode, I. Boutle, Y. Kamae, R. Roehrig, and J. Vial, 2015: A single-column model intercomparison on the stratocumulus representation in present-day and future climate. J. Adv. Model. Earth Syst., 7, 617647, https://doi.org/10.1002/2014MS000377.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1976: On the entrainment rate of a stratocumulus-topped mixed layer. Quart. J. Roy. Meteor. Soc., 102, 563582, https://doi.org/10.1002/qj.49710243306.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495527, https://doi.org/10.1007/BF00119502.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1981: On the distribution of mean radiative cooling at the top of a stratocumulus-capped mixed layer. Quart. J. Roy. Meteor. Soc., 107, 191202, https://doi.org/10.1002/qj.49710745112.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • de Lozar, A., and J. P. Mellado, 2015: Evaporative cooling amplification of the entrainment velocity in radiatively driven stratocumulus. Geophys. Res. Lett., 42, 72237229, https://doi.org/10.1002/2015GL065529.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eastman, R., and S. G. Warren, 2014: Diurnal cycles of cumulus, cumulonimbus, stratus, stratocumulus, and fog from surface observations over land and ocean. J. Climate, 27, 23862404, https://doi.org/10.1175/JCLI-D-13-00352.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Farge, M., G. Pellegrino, and K. Schneider, 2001: Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett., 87, 054501, https://dx.doi.org/10.1103/PhysRevLett.87.054501.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., G. Bellon, and C. C. van Heerwaarden, 2015: A closer look at boundary layer inversion in large-eddy simulations and bulk models: Buoyancy-driven case. J. Atmos. Sci., 72, 728749, https://doi.org/10.1175/JAS-D-13-0377.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerber, H., G. Frick, S. Malinowski, J. Brenguier, and F. Burnet, 2005: Holes and entrainment in stratocumulus. J. Atmos. Sci., 62, 443459, https://doi.org/10.1175/JAS-3399.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerber, H., G. Frick, S. Malinowski, H. Jonsson, D. Khelif, and S. K. Krueger, 2013: Entrainment rates and microphysics in POST stratocumulus. J. Geophys. Res. Atmos., 118, 12 09412 109, https://doi.org/10.1002/jgrd.50878.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghate, V. P., M. A. Miller, B. A. Albrecht, and C. W. Fairall, 2015: Thermodynamic and radiative structure of stratocumulus-topped boundary layers. J. Atmos. Sci., 72, 430451, https://doi.org/10.1175/JAS-D-13-0313.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glenn, I. B., and S. K. Krueger, 2014: Downdrafts in the near cloud environment of deep convective updrafts. J. Adv. Model. Earth Syst., 6, 18, https://doi.org/10.1002/2013MS000261.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., 2015: Lagrangian coherent structures. Annu. Rev. Fluid Mech., 47, 137162, https://doi.org/10.1146/annurev-fluid-010313-141322.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., and G. Yuan, 2000: Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 147, 352370, https://doi.org/10.1016/S0167-2789(00)00142-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hartmann, D. L., M. E. Ockert-Bell, and M. L. Michelsen, 1992: The effect of cloud type on Earth’s energy balance: Global analysis. J. Climate, 5, 12811304, https://doi.org/10.1175/1520-0442(1992)005<1281:TEOCTO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heus, T., and H. J. Jonker, 2008: Subsiding shells around shallow cumulus clouds. J. Atmos. Sci., 65, 10031018, https://doi.org/10.1175/2007JAS2322.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Khanna, S., and J. G. Brasseur, 1998: Three-dimensional buoyancy- and shear-induced local structure of the atmospheric boundary layer. J. Atmos. Sci., 55, 710743, https://doi.org/10.1175/1520-0469(1998)055<0710:TDBASI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kim, S.-W., and S.-U. Park, 2003: Coherent structures near the surface in a strongly sheared convective boundary layer generated by large-eddy simulation. Bound.-Layer Meteor., 106, 3560, https://doi.org/10.1023/A:1020811015189.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klein, S. A., and D. L. Hartmann, 1993: The seasonal cycle of low stratiform clouds. J. Climate, 6, 15871606, https://doi.org/10.1175/1520-0442(1993)006<1587:TSCOLS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Köhler, M., M. Ahlgrimm, and A. Beljaars, 2011: Unified treatment of dry convective and stratocumulus-topped boundary layers in the ECMWF model. Quart. J. Roy. Meteor. Soc., 137, 4357, https://doi.org/10.1002/qj.713.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kollias, P., and B. Albrecht, 2000: The turbulence structure in a continental stratocumulus cloud from millimeter-wavelength radar observations. J. Atmos. Sci., 57, 24172434, https://doi.org/10.1175/1520-0469(2000)057<2417:TTSIAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Krueger, S. K., 1993: Linear eddy modeling of entrainment and mixing in stratus clouds. J. Atmos. Sci., 50, 30783090, https://doi.org/10.1175/1520-0469(1993)050<3078:LEMOEA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurowski, M. J., S. P. Malinowski, and W. W. Grabowski, 2009: A numerical investigation of entrainment and transport within a stratocumulus-topped boundary layer. Quart. J. Roy. Meteor. Soc., 135, 7792, https://doi.org/10.1002/qj.354.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Li, D., and E. Bou-Zeid, 2011: Coherent structures and the dissimilarity of turbulent transport of momentum and scalars in the unstable atmospheric surface layer. Bound.-Layer Meteor., 140, 243262, https://doi.org/10.1007/s10546-011-9613-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lilly, D. K., 1968: Models of cloud-topped mixed layers under a strong inversion. Quart. J. Roy. Meteor. Soc., 94, 292309, https://doi.org/10.1002/qj.49709440106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lock, A., A. Brown, M. Bush, G. Martin, and R. Smith, 2000: A new boundary layer mixing scheme. Part I: Scheme description and single-column model tests. Mon. Wea. Rev., 128, 31873199, https://doi.org/10.1175/1520-0493(2000)128<3187:ANBLMS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Matheou, G., D. Chung, and J. Teixeira, 2016: On the synergy between numerics and subgrid scale modeling in LES of stratified flows: Grid convergence of a stratocumulus-topped boundary layer. Proc. VIII Int. Symp. on Stratified Flows, San Diego, CA, IAHR, 8 pp., http://escholarship.org/uc/item/3rx2n92v.

  • Mellado, J. P., 2017: Cloud-top entrainment in stratocumulus clouds. Annu. Rev. Fluid Mech., 49, 145169, https://doi.org/10.1146/annurev-fluid-010816-060231.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mellado, J. P., B. Stevens, H. Schmidt, and N. Peters, 2009: Buoyancy reversal in cloud-top mixing layers. Quart. J. Roy. Meteor. Soc., 135, 963978, https://doi.org/10.1002/qj.417.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moeng, C.-H., and U. Schumann, 1991: Composite structure of plumes in stratus-topped boundary layers. J. Atmos. Sci., 48, 22802291, https://doi.org/10.1175/1520-0469(1991)048<2280:CSOPIS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moeng, C.-H., S. Shen, and D. A. Randall, 1992: Physical processes within the nocturnal stratus-topped boundary layer. J. Atmos. Sci., 49, 23842401, https://doi.org/10.1175/1520-0469(1992)049<2384:PPWTNS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moeng, C.-H., and Coauthors, 1996: Simulation of a stratocumulus-topped planetary boundary layer: Intercomparison among different numerical codes. Bull. Amer. Meteor. Soc., 77, 261278, https://doi.org/10.1175/1520-0477(1996)077<0261:SOASTP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nam, C., S. Bony, J.-L. Dufresne, and H. Chepfer, 2012: The “too few, too bright” tropical low-cloud problem in CMIP5 models. Geophys. Res. Lett., 39, L21801, https://doi.org/10.1029/2012GL053421.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nicholls, S., 1989: The structure of radiatively driven convection in stratocumulus. Quart. J. Roy. Meteor. Soc., 115, 487511, https://doi.org/10.1002/qj.49711548704.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Park, S.-B., P. Gentine, K. Schneider, and M. Farge, 2016: Coherent structures in the boundary and cloud layers: Role of updrafts, subsiding shells, and environmental subsidence. J. Atmos. Sci., 73, 17891814, https://doi.org/10.1175/JAS-D-15-0240.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pedersen, J. G., S. P. Malinowski, and W. W. Grabowski, 2016: Resolution and domain-size sensitivity in implicit large-eddy simulation of the stratocumulus-topped boundary layer. J. Adv. Model. Earth Syst., 8, 885903, https://doi.org/10.1002/2015MS000572.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Randall, D. A., 1980: Conditional instability of the first kind upside-down. J. Atmos. Sci., 37, 125130, https://doi.org/10.1175/1520-0469(1980)037<0125:CIOTFK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Raupach, M., 1981: Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech., 108, 363382, https://doi.org/10.1017/S0022112081002164.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rio, C., F. Hourdin, F. Couvreux, and A. Jam, 2010: Resolved versus parametrized boundary-layer plumes. Part II: Continuous formulations of mixing rates for mass-flux schemes. Bound.-Layer Meteor., 135, 469483, https://doi.org/10.1007/s10546-010-9478-z.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Robinson, S. K., 1991: Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech., 23, 601639, https://doi.org/10.1146/annurev.fl.23.010191.003125.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schoppa, W., and F. Hussain, 2002: Coherent structure generation in near-wall turbulence. J. Fluid Mech., 453, 57108, https://doi.org/10.1017/S002211200100667X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schumann, U., and C.-H. Moeng, 1991: Plume fluxes in clear and cloudy convective boundary layers. J. Atmos. Sci., 48, 17461757, https://doi.org/10.1175/1520-0469(1991)048<1746:PFICAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers. Quart. J. Roy. Meteor. Soc., 128, 26632690, https://doi.org/10.1256/qj.01.202.

  • Stevens, B., 2010: Introduction to UCLA-LES, version 3.2.1. MPI Doc., 20 pp., https://www.mpimet.mpg.de/fileadmin/atmosphaere/herz/les_doc.pdf.

  • Stevens, B., and A. Seifert, 2008: On the sensitivity of simulations of shallow cumulus convection to their microphysical representation. J. Meteor. Soc. Japan, 86A, 143162, https://doi.org/10.2151/jmsj.86A.143.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., C.-H. Moeng, and P. P. Sullivan, 1999: Large-eddy simulations of radiatively driven convection: Sensitivities to the representation of small scales. J. Atmos. Sci., 56, 39633984, https://doi.org/10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., and Coauthors, 2003a: Dynamics and chemistry of marine stratocumulus—DYCOMS-II. Bull. Amer. Meteor. Soc., 84, 579593, https://doi.org/10.1175/BAMS-84-5-579.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., and Coauthors, 2003b: On entrainment rates in nocturnal marine stratocumulus. Quart. J. Roy. Meteor. Soc., 129, 34693493, https://doi.org/10.1256/qj.02.202.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., G. Vali, K. Comstock, R. Wood, M. C. van Zanten, P. H. Austin, C. S. Bretherton, and D. H. Lenschow, 2005a: Pockets of open cells and drizzle in marine stratocumulus. Bull. Amer. Meteor. Soc., 86, 5157, https://doi.org/10.1175/BAMS-86-1-51.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stevens, B., and Coauthors, 2005b: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus. Mon. Wea. Rev., 133, 14431462, https://doi.org/10.1175/MWR2930.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 670 pp.

    • Crossref
    • Export Citation
  • Sullivan, P. P., and E. G. Patton, 2011: The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulation. J. Atmos. Sci., 68, 23952415, https://doi.org/10.1175/JAS-D-10-05010.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sullivan, P. P., C.-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci., 55, 30423064, https://doi.org/10.1175/1520-0469(1998)055<3042:SOTEZC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sušelj, K., J. Teixeira, and D. Chung, 2013: A unified model for moist convective boundary layers based on a stochastic eddy-diffusivity/mass-flux parameterization. J. Atmos. Sci., 70, 19291953, https://doi.org/10.1175/JAS-D-12-0106.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tsushima, Y., and Coauthors, 2016: Robustness, uncertainties, and emergent constraints in the radiative responses of stratocumulus cloud regimes to future warming. Climate Dyn., 46, 30253039, https://doi.org/10.1007/s00382-015-2750-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Volino, R. J., and T. W. Simon, 1994: An application of octant analysis to turbulent and transitional flow data. J. Turbomach., 116, 752758, https://doi.org/10.1115/1.2929469.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wood, R., 2012: Stratocumulus clouds. Mon. Wea. Rev., 140, 23732423, https://doi.org/10.1175/MWR-D-11-00121.1.

  • Wood, R., and P. R. Field, 2000: Relationships between total water, condensed water, and cloud fraction in stratiform clouds examined using aircraft data. J. Atmos. Sci., 57, 18881905, https://doi.org/10.1175/1520-0469(2000)057<1888:RBTWCW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wood, R., and C. S. Bretherton, 2006: On the relationship between stratiform low cloud cover and lower-tropospheric stability. J. Climate, 19, 64256432, https://doi.org/10.1175/JCLI3988.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yamaguchi, T., and D. A. Randall, 2012: Cooling of entrained parcels in a large-eddy simulation. J. Atmos. Sci., 69, 11181136, https://doi.org/10.1175/JAS-D-11-080.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 10831110, https://doi.org/10.1175/2008MWR2582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Albrecht, B. A., M. P. Jensen, and W. J. Syrett, 1995: Marine boundary layer structure and fractional cloudiness. J. Geophys. Res., 100, 14 20914 222, https://doi.org/10.1029/95JD00827.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bechtold, P., and P. Siebesma, 1998: Organization and representation of boundary layer clouds. J. Atmos. Sci., 55, 888895, https://doi.org/10.1175/1520-0469(1998)055<0888:OAROBL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bechtold, P., S. K. Krueger, W. Lewellen, E. Van Meijgaard, C. Moeng, D. Randall, A. Van Ulden, and S. Wang, 1996: Modeling a stratocumulus-topped PBL: Intercomparison among different one-dimensional codes and with large eddy simulation. Bull. Amer. Meteor. Soc., 77, 20332042, https://doi.org/10.1175/1520-0477(1996)077<2033:MASTPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bellon, G., and O. Geoffroy, 2016: How finely do we need to represent the stratocumulus radiative effect? Quart. J. Roy. Meteor. Soc., 142, 23472358, https://doi.org/10.1002/qj.2828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bony, S., and J.-L. Dufresne, 2005: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett., 32, L20806, https://doi.org/10.1029/2005GL023851.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brent, R. P., 1973: Algorithms for Minimization without Derivatives. Courier Corporation, 195 pp.

  • Brient, F., and S. Bony, 2013: Interpretation of the positive low-cloud feedback predicted by a climate model under global warming. Climate Dyn., 40, 24152431, https://doi.org/10.1007/s00382-011-1279-7.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dal Gesso, S., J. Van Der Dussen, A. Siebesma, S. De Roode, I. Boutle, Y. Kamae, R. Roehrig, and J. Vial, 2015: A single-column model intercomparison on the stratocumulus representation in present-day and future climate. J. Adv. Model. Earth Syst., 7, 617647, https://doi.org/10.1002/2014MS000377.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1976: On the entrainment rate of a stratocumulus-topped mixed layer. Quart. J. Roy. Meteor. Soc., 102, 563582, https://doi.org/10.1002/qj.49710243306.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495527, https://doi.org/10.1007/BF00119502.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1981: On the distribution of mean radiative cooling at the top of a stratocumulus-capped mixed layer. Quart. J. Roy. Meteor. Soc., 107, 191202, https://doi.org/10.1002/qj.49710745112.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • de Lozar, A., and J. P. Mellado, 2015: Evaporative cooling amplification of the entrainment velocity in radiatively driven stratocumulus. Geophys. Res. Lett., 42, 72237229, https://doi.org/10.1002/2015GL065529.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eastman, R., and S. G. Warren, 2014: Diurnal cycles of cumulus, cumulonimbus, stratus, stratocumulus, and fog from surface observations over land and ocean. J. Climate, 27, 23862404, https://doi.org/10.1175/JCLI-D-13-00352.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Farge, M., G. Pellegrino, and K. Schneider, 2001: Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett., 87, 054501, https://dx.doi.org/10.1103/PhysRevLett.87.054501.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gentine, P., G. Bellon, and C. C. van Heerwaarden, 2015: A closer look at boundary layer inversion in large-eddy simulations and bulk models: Buoyancy-driven case. J. Atmos. Sci., 72, 728749, https://doi.org/10.1175/JAS-D-13-0377.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerber, H., G. Frick, S. Malinowski, J. Brenguier, and F. Burnet, 2005: Holes and entrainment in stratocumulus. J. Atmos. Sci., 62, 443459, https://doi.org/10.1175/JAS-3399.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerber, H., G. Frick, S. Malinowski, H. Jonsson, D. Khelif, and S. K. Krueger, 2013: Entrainment rates and microphysics in POST stratocumulus. J. Geophys. Res. Atmos., 118, 12 09412 109, https://doi.org/10.1002/jgrd.50878.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ghate, V. P., M. A. Miller, B. A. Albrecht, and C. W. Fairall, 2015: Thermodynamic and radiative structure of stratocumulus-topped boundary layers. J. Atmos. Sci., 72, 430451, https://doi.org/10.1175/JAS-D-13-0313.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Glenn, I. B., and S. K. Krueger, 2014: Downdrafts in the near cloud environment of deep convective updrafts. J. Adv. Model. Earth Syst., 6, 18, https://doi.org/10.1002/2013MS000261.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., 2015: Lagrangian coherent structures. Annu. Rev. Fluid Mech., 47, 137162, https://doi.org/10.1146/annurev-fluid-010313-141322.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haller, G., and G. Yuan, 2000: Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 147, 352370, https://doi.org/10.1016/S0167-2789(00)00142-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hartmann, D. L., M. E. Ockert-Bell, and M. L. Michelsen, 1992: The effect of cloud type on Earth’s energy balance: Global analysis. J. Climate, 5, 12811304, https://doi.org/10.1175/1520-0442(1992)005<1281:TEOCTO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heus,