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Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part II: Mixing Dynamics Zhao, Ming; Austin, Philip H. May 31, 2005

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Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part II: Mixing Dynamics MING ZHAO* AND PHILIP H. AUSTIN Atmospheric Science Programme, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 2 June 2003, in final form 1 September 2004) ABSTRACT This paper is the second in a two-part series in which life cycles of six numerically simulated shallow cumulus clouds are systematically examined. The six clouds, selected from a single realization of a large- eddy simulation, grow as a series of pulses/thermals detached from the subcloud layer. All six clouds exhibit a coherent vortical circulation and a low buoyancy, low velocity trailing wake. The ascending cloud top (ACT), which contains this vortical circulation, is associated with a dynamic perturbation pressure field with high pressure located at the ascending frontal cap and low pressure below and on the downshear side of the maximum updrafts. Examination of the thermodynamic and kinematic structure, together with passive tracer experiments, suggests that this vortical circulation is primarily responsible for mixing between cloud and environment. As the cloud ACTs rise through the sheared environment, the low pressure, vortical circulation, and mixing are all strongly enhanced on the downshear side and weakened on the upshear side. Collapse of the ACT also occurs on the downshear side, with subsequent thermals ascending on the upshear side of their predecessors. The coherent core structure is maintained throughout the ACT ascent; mixing begins to gradually dilute the ACT core only in the upper half of the cloud’s depth. The characteristic kinematic and dynamic structure of these simulated ACTs, together with their mixing behavior, corresponds closely to that of shedding thermals. These shallow simulated clouds, however, reach a maximum height of only about four ACT diameters so that ACT mixing differs from predictions of self-similar laboratory thermals. 1. Introduction This paper continues the investigation of the life cycle of numerically simulated shallow cumulus clouds begun in Zhao and Austin (2005, hereafter Part I). In Part I we describe the large-eddy simulation (LES) case setup, the isolation of individual cloud life cycle data from the simulated cloud field, and the use of a tracer to identify the cloud-mixed convective region. We ex- amine in detail the mass and thermodynamic transport associated with the life cycle of individual clouds with special emphasis on the unsaturated part of cumulus convection, the impact of buoyancy on the cloud verti- cal convective transport, and the role of cloud size in the cloud ensemble transport. Part II continues this in- vestigation using the same set of six clouds. Our focus in Part II is on understanding the details of the simulated cumulus mixing dynamics, particularly the mixing asso- ciated with the active ascending cloud elements. Our objective is to produce a qualitative and quantitative description of the simulated cloud mixing life cycle that can be compared against the conceptual models that form the basis of parameterizations of shallow cumulus cloud mixing and transport (e.g., Arakawa and Schu- bert 1974; Emanuel 1991; Zhao 2003). Detailed aircraft measurement of the structure of or- dinary cumulus clouds date back to the work of Stom- mel (1947), Malkus (1954), Warner (1955), and Warner and Squires (1958). These observations have estab- lished that the properties of such clouds cannot be ex- plained solely by the adiabatic ascent of air from cloud base. The observations indicated that the horizontally averaged liquid water content and buoyancy of cumulus clouds are far smaller than their respective adiabatic values. It is natural to assume that most of the depar- ture from adiabatic is due to mixing through the cloud sides. The observations showed, however, that there is little systematic variation in cloud properties across a cloud and that dry holes occur frequently in the bases of clouds, in contrast to laboratory plumes (e.g., Warner * Current affiliation: NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey. Corresponding author address: Dr. Ming Zhao, NOAA/ Geophysical Fluid Dynamics Laboratory, Princeton University, Forrestal Campus, Box 308, U.S. Route 1, Princeton, NJ 08542. E-mail: ming.zhao@noaa.gov MAY 2005 Z H A O A N D A U S T I N 1291 JAS3415 1955). Later observations showed that undilute sub- cloud air (USCA) occurs in all levels within cumulus clouds (e.g., Heymsfield et al. 1978; Jensen et al. 1985; Austin et al. 1985). These measurements raised doubts about the validity of treating clouds as entraining plumes. For example, Warner (1970) pointed out that it was impossible for such models to simultaneously re- produce realistic values of liquid water content and cloud-top height (the cloud-top liquid water paradox); if the entrainment rate is adjusted to predict the observed height, then the predicted liquid water con- tent is much too high. On the other hand, Squires (1958) proposed that the turbulent mixing of dry environmental air near the tops of clouds can result in substantial negative buoyancy; this may cause strong penetrative downdrafts. Emanuel (1981) showed that the penetrative downdrafts may be potentially as strong or stronger than convective up- drafts. Squires argued that vertical penetrative down- drafts rather than horizontal mixing was the principal mechanism responsible for cumulus cloud dilution. Squire’s hypothesis received some support from the ob- servations of Paluch (1979), who used two conserved variables to deduce the environmental origin of air sampled by glider inside Colorado cumuli. The interiors of these nonprecipitating cumuli were shown to be ex- tremely inhomogeneous, with some samples corre- sponding to nearly undilute ascent cloud-base air and others composed of various mixtures of subcloud air with environmental air from near cloud top. These re- sults were corroborated by a number of other studies (e.g., Boatman and Auer 1983; Jensen et al. 1985). However, it is difficult to imagine how cloud updrafts could be constrained to entrain only upon reaching their ultimate tops. Using the analysis technique of Paluch (1979), Blyth et al. (1988) examined more than 80 Montana cumuli and found that the source of entrained air was close to, or slightly above, the observation height of the aircraft at many different levels in the clouds. This result sug- gests that the entrainment process in growing clouds is modified cloud-top entrainment, where cloud top means the ascending cloud top (ACT) of the growing cloud. These and other considerations motivated Blyth et al. to propose a conceptual cloud model based on a shedding thermal. In this model, entrainment occurs near the ACT, and mixed parcels subsequently descend around the edge of the advancing thermal core into a trailing wake region driven by a toroidal circulation. While supporting the idea of ascending cloud-top en- trainment, Jonas (1990) presented a different picture in which the environmental air from near cloud top is transported in a thin, subsiding layer to lower levels and is then entrained laterally into the cloud. The cloud tracer experiments of Stith (1992) give some support to this latter picture. Although these in situ studies have provided insight into cumulus mixing dynamics, there is still no general agreement on the detailed mechanism by which air enters and is mixed into a cumulus cloud (Blyth 1993). Laboratory studies of buoyant plumes and thermals, together with similarity assumptions, have provided an- other tool to understand cumulus mixing. Conceptual cumulus models based on bubbles/thermals were pro- posed by Scorer and Ludlam (1953) and Scorer and Ronne (1956). These and other laboratory experiments (e.g., Scorer 1957; Woodward 1959) showed that ther- mals mix environmental fluid into their tops as well as through their wakes during their ascent. Furthermore, Scorer and Ronne found that parts of thermals were sometimes shed into the wake in a stratified ambient fluid. This led them to propose a shedding thermal model of the sort later adopted by Blyth et al. (1988). More recently, Johari (1992) found that entrainment occurs as ambient fluid moves in thin sheets around the perimeter of the thermals and enters through the rear of the thermal. This is in contrast to the classical picture of mixing taking place along the advancing front of the thermal (Scorer 1958) and (as mentioned above) is sup- ported by the in situ observations of Jonas (1990) and Stith (1992). Johari also noted the presence of unmixed ambient fluid deep inside the thermal, captured not by small-scale motions along the thermal’s leading edge but by the larger-scale motion set up by the advancing vortical circulation. He concluded that it is the large- scale vortical circulation associated with the global mo- tion of the thermal that dominates entrainment and mixing. Laboratory experiments have illustrated the detailed flow structure of individual thermals in an idealized environment. Cumulus clouds, however, differ from laboratory thermals in several respects. Specifically, they are characterized by 1) phase change, which can produce either positive or negative buoyancy through condensation or mixing and evaporation; 2) environ- mental wind and wind shear; 3) subcloud buoyancy fluctuations that are continuously generated from a tur- bulent boundary layer; 4) ambient vertically varying stratification; and 5) Reynolds numbers that are far larger than those that can be produced in current tank experiments. High-resolution numerical simulation is a third tool that has been increasingly used to study both individual cumulus dynamics and cumulus ensemble statistics. Nu- merous cloud simulations attempt to model the evolu- tion and structure of one or a few convective clouds. Typically, these models assume an atmosphere that possesses a given quantity of convective available po- tential energy, which is released over a limited period by the clouds. Klaassen and Clark (1985) used a speci- fied surface heat flux and a 2D model with interactive grid nesting to examine in detail the entrainment mechanism of a small cumulus cloud. They found that small-scale horizontal density gradients were respon- sible for the onset of cloud–environment interface in- stabilities, while evaporative cooling played only a mi- 1292 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 nor role in mixing. Grabowski and Clark (1993) ex- tended these simulations to 3D and obtained results similar to those of Klaassen and Clark, indicating that dynamical effects rather than evaporation are more sig- nificant in cumulus entrainment. Carpenter et al. (1998a) pointed out difficulties with a simple specification of the surface heat flux, arguing that fully turbulent motion in the boundary layer is required to produce realistic cloud entrainment. They first arbitrarily generate turbulent motion on all resolv- able scales and then reduced the heat flux to more re- alistic values. Carpenter et al. used this initialization and a nested mesh with an inner 50-m resolution to study the entrainment and detrainment in New Mexi- can cumulus congestus. The authors conducted parcel trajectory and conserved variable analyses of the mod- eled clouds and found that cumulus entrainment and detrainment is primarily associated with the ascending cloud top and its induced toroidal circulation. Their results lend some support to the shedding thermal model previously described in Blyth et al. (1988). In contrast to these individual cloud simulations, an LES is continually forced by specified large-scale con- ditions, and many convective clouds are simulated over a sufficient period so that they collectively come into equilibrium with the forcing. Unless multiple equilibria are possible, an LES can be expected to reach a statis- tical steady state that is independent of initial condi- tions (Emanuel 1994). Therefore, an LES provides an alternative approach to generate cumulus clouds with fully developed boundary layer turbulence in a more realistic large-scale environment, albeit with some sac- rifice of model resolution. We can select individual clouds from an equilibrium convective field and apply passive numerical tracers to systematically diagnose the isolated cloud mixing behavior. In this paper we make a first attempt to examine in detail the mixing dynamics of convective elements generated in these LES fields, using the six clouds described in Part I. This analysis complements the cloud in situ observations, laboratory tank experiments, and previous numerical simulations that have motivated a variety of conceptual models of cumulus mixing. In Part I, section 2, we describe the LES, the Barba- dos Oceanic and Meteorological Experiment (BOMEX) initial conditions and forcings, and the pro- cedures used to distinguish convective clouds from the environment. To review briefly: the simulation was per- formed with the Colorado State University System for Atmospheric Modeling (Khairoutdinov and Randall 2003) on a model domain of 6.4 km  6.4 km  3.2 km. We isolated six clouds (labeled A–F) with cloud-top heights ranging from 1 to 2 km and used a numerical tracer to track the evolution of both the saturated cloud and the unsaturated cloud-mixed region. In section 2a of this paper, we present the pulsating behavior of the simulated cumulus growth. The cloud growth rate and thermal turnover time are quantified in section 2b. In section 2c, we present the cloud kine- matic structure, cloud-top vortical circulation, and an elevated tracer experiment for cumulus entrainment. In section 2d, we link the cloud-top vortical circulation to the baroclinic torque and present the perturbation pres- sure field. The dilution rate and the mixture distribu- tion in the ascending cloud top are quantified in section 2e. The continued mixing between low-buoyancy, low- vertical-velocity cloud mixtures and the environment is briefly described in section 2f. In section 3 we discuss a picture of cumulus mixing that is consistent with the simulation results. 2. Results a. Pulsating cloud growth Animations of clouds A–F (Zhao and Austin 2004) show that an individual cloud consists of one or more coherent updraft pulses/thermals that grow and decay over 10 to 15 min periods. Small clouds (A, B, C) have a single ascending pulse that dissipates without reach- ing the inversion. Large clouds (D, E, F) consist of a series of pulses, each of which detaches from a subcloud layer thermal, strengthens in the middle of the cloud layer, and collapses after reaching a maximum height within or below the inversion. The first pulse of these large clouds is the strongest, reaching its maximum height within the inversion, while the subsequent pulses are weaker and attain maximum heights lower than their predecessors. There is also a tendency for an elon- gated pulse to break into two elements, with the lower element strengthening when the upper one collapses after reaching its maximum height. In this case, the lower pulse always develops on the upshear side while the upper pulse collapses into the downshear side. The pulsating character of cloud growth is most distinct be- ginning at about 400 m above the cloud base (1 km above the surface). All of the pulse elements tend to become dynamically detached1 from their subcloud layer parent thermals some distance above cloud base. Figure 1 shows an example of this pulsating ascent as viewed at a fixed cloud level for cloud E. When the updraft first arrives at the 1237.5-m level at 500 s, it brings a large proportion of relatively undilute sub- cloud air, producing the largest anomalies of liquid wa- ter potential temperature l, total water qt, virtual po- tential temperature  , and vertical velocity w for both the cloud mean value (dashed line) and the most undi- lute cloud grid cell (i.e., the grid cell extreme value: solid line). Between 500 and 800 s these values quickly 1 Here, by dynamically detached, we mean the pulse elements will not be significantly influenced by the cloud air below, since these elements ascend faster and tend to distance themselves from the cloud material below. MAY 2005 Z H A O A N D A U S T I N 1293 decrease (l increases) as more dilute cloud mixtures with low vertical velocities are left in the trailing wake of the advancing top. Further mixing with environmen- tal air tends to continually dilute these cloud mixtures until the next ascending thermal penetrates this level and brings a new anomaly at approximately 1000 s. Cloud E dissipates following the second pulse. Figure 2 shows this pulsating structure at all cloud levels as time–height contours of l, qt,  , and w of the most undilute cloud parcel. The two growth pulses, which are spaced approximately 500 s apart, are present at all cloud levels above 1 km. The second pulse is weaker and moves upward less rapidly than the initial thermal. The structure of the second pulse is harder to discern in the mean fields (not shown) because it as- cends into the remnants of the previous thermal. Due to this difficulty in separating a pulse from the remnants of previous thermals, we limit the following analysis to the first thermal for each cloud. This initial thermal is the simulated counterpart to the ascending turret of a de- veloping cloud, and will be referred to below as the ascending cloud top (ACT). The sharp contrast be- tween an ACT and its environment makes it easy to isolate, particularly with the use of a subcloud layer tracer as described in section 2c of Part I. Animation of the entire simulated cloud field shows that pulsating growth is a characteristic feature of all simulated clouds. This is consistent with numerous ob- servational studies describing the growth of cumulus clouds either as individual bubbles or as a collection of multiple bubbles (e.g., Ludlam and Scorer 1953; Malkus and Scorer 1955; Saunders 1961; Blyth et al. 1988; Blyth and Latham 1993; Barnes et al. 1996; French et al. 1999). Pulsating cloud growth is also reported in other high-resolution numerical simulations of isolated cumu- lus clouds (e.g., Carpenter et al. 1998a). These single cloud simulations, however, typically discard the early phase of the convective clouds in which the ACT is most easily isolated. This is necessary due to the unrealistic boundary layer turbulence and ambient environment caused by the model’s initial lack of motion on all resolvable scales (Carpenter et al. 1998a). In contrast, for the LES-ensemble cloud simu- lation presented here, individual clouds develop from a fully turbulent boundary layer that is in equilibrium with the large-scale environment. With the isolation technique described in Part I, we can obtain uncontami- nated 5-dimensional simulation data for the ACT of each selected cloud. These pseudoclouds provide an excellent opportunity to investigate the kinematics and dynamics of the ACT and its mixing behavior. In the following sections we begin with a description of the bulk properties of the ACT and then examine in detail the kinematic structure and dynamics of the ACT for clouds A–F. b. Growth rate and thermal turnover time scale The cloud-top height zT(t) (the maximum height that contains liquid water at time t) was shown in Part I (Fig. 5a) to increase at an approximately exponential rate during the growth phase of individual clouds. To quan- tify the difference between clouds A–F, Fig. 3a repro- duces Fig. 5a of Part I in semilogarithmic coordinates. There is a near-linear dependence of log(zT) on time t for all clouds, motivating the definition of the ACT growth rate  as   dlogzT dt , 1 where  is determined by a linear fit to d(log zT)/dt from Fig. 3a during the growth phase. A vertical length scale of an ACT may be chosen as the maximum height zT,max achieved by the ACT. Figure 3b shows the varia- tion of growth rate  versus zT,max. This growth rate tends to vary near-linearly with zT,max, with the ACTs of larger clouds having higher growth rates. The solid line in Fig. 3b is a linear least squares fit of  to zT,max for the six clouds. Since the growth rate  is nearly constant during an individual ACT ascent (Fig. 3a) the thermal turnover time t* can be computed by integrat- ing (1) t*   zT,0 zT,max 1  d logzT  1  logzT,maxzT,0 , 2 where zT,0  zB represents the cloud-top height when it first emerges from the subcloud layer. Figure 3c shows the ACT thermal turnover time for the six clouds; with FIG. 1. An example of the pulsating character of cloud E. (a) Dashed line: time evolution of 	l, the l difference between liq- uid water cloud horizontal mean and environment at height 1237.5 m. Solid line: the l difference between the most undilute cloud grid cell and environment. (b) As in (a) but for qt. (c) As in (a) but for . (d) As in (a) but for vertical velocity w. 1294 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 the exception of cloud C, all clouds have approximately the same thermal turnover time of 600 s (solid line). Our 600-s t* represents the average time required for air in a turret (ACT) to travel from cloud base to maxi- mum cloud top. Neggers et al. (2002) used instanta- neous snapshots of an LES shallow cumulus cloud field to determine a similar time scale for a given idealized parcel ascending through a cloud’s depth. They aver- aged the vertical velocity along the three-dimensional path of the strongest updraft in a particular cloud. They then used the ratio of the cloud depth to this averaged maximum vertical velocity to estimate c, the time re- quired for the air in the strongest updraft to rise from cloud base to cloud top. Using a cloud population with a range of thicknesses, they found c  300 s to be approximately independent of cloud depth (as do we: when we follow the maximum updraft for our six clouds, we also find an approximate time constant of c  350 s that is roughly independent of cloud depth). The larger value of t* determined from Fig. 3 indicates that air contained within the coherent structure of the ACT does not simply rise along the instantaneous maximum vertical velocity in the cloud field. Rather it tends to be vertically constrained and circulates within the ACT, rising on average with about half the speed of the major updrafts. In the next section we show this coherent circulation more explicitly. c. Kinematic structure of the ACT Individual ascending thermals have a spatial scale smaller than the cloud envelope. Observational studies of the kinematic structure of these ascending thermals is difficult, due primarily to their highly transient nature (Blyth et al. 1988). They are, however, the building blocks of individual clouds and hence the cloud en- semble and, therefore, play a fundamental role in cu- mulus convective transport and cloud environment mixing. Below we examine in detail the kinematic structure of the first ascending thermal, that is, the ACT of the simulated clouds. We use cloud E as an example to illustrate characteristic features present in all six selected clouds. Figure 4 shows vertical cross sections (x–z plane) of the resolved wind vectors from cloud E at 7.5, 8.5, 9.5, and 10.5 min after emergence from the subcloud layer. The ambient mean wind u0  7.5 m s 1 and the ver- tical velocity of the ascending cloud–environment inter- face, wT(t) (where the subscript T indicates the current cloud-top height) have been subtracted from the cor- responding velocity field. The wT(t) at each time step is found using the growth rate E  0.0017 s 1 from Fig. 3b and the current cloud-top height zT via (1); that is, wT(t)  dzT(t)/dt  EzT(t). A distinct vortical circula- tion exists within the top part of the growing cloud FIG. 2. Time–height variation of the differences (	l, 	qt, 	, 	w) for the most undilute grid cell for cloud E: (a) l (K) difference between the most undilute grid cell and environment. (b) As in (a) but for qt (g kg 1). (c) As in (a) but for  (K). (d) As in (a) but for vertical velocity w (m s 1). MAY 2005 Z H A O A N D A U S T I N 1295 [approximately 14 model levels (350 m) below zT], which is seen in 3-dimensional visualizations to consist of ascent along the central (not strictly axisymmetric) vertical axis and descent around the periphery. Below this coherent internal circulation, the cloud- top relative vertical velocity (denoted as CT-relative below) is generally near zero or negative (i.e., Figs. 4a–d at 800, 900, 1000, and 1200 m). Air parcels located below these levels generally cannot approach the mov- ing cloud top given this resolved flow. Motivated by this distinct kinematic structure, we define the ACT as the upper part of the growing cloud; it encompasses the coherent vortical circulation. We will refer to the region below the ACT, which is more dilute and has low ver- tical velocity, as the trailing wake of the advancing ACT. The cloud–environment mixtures in the trailing wake will also be called passive cloud mixtures below to distinguish them from the active cloud mixtures within the ACT. In addition, we will refer to the shallow, fron- tal cap of the ACT, which extends about 4 to 6 model levels below the cloud–environment interface and has strong horizontal divergence, as the ascending frontal cap (AFC; see box, Fig. 4a). Note that the AFC repre- sents the top edge of the ascending liquid cloud rather than the updrafts. Therefore, it has a mean vertical ve- locity close to the ascending speed of the cloud– environment interface. As Fig. 4 shows, there is a strong updraft core in the center of the ACT with maximum vertical velocity ap- proximately twice that of the AFC mean vertical veloc- ity (or equivalently, wT). The core updraft velocity is gradually reduced to near zero (CT- relative) within the AFC, which results in strong horizontal divergence. Near the edge of the ACT there is a distinct overturn- ing circulation with downward CT-relative vertical ve- locity roughly equal to that of the central updrafts. The overall internal circulation is weak when the ACT just emerges from subcloud layer (Fig. 4a), is gradually strengthened in the middle of the cloud layer (Figs. 4b,c,d), and tends to be rapidly weakened after pen- etration of the inversion layer. Before we discuss how the vortical circulation influ- ences the cloud entrainment and mixing, we show in Fig. 5 the vortical circulation in y–z cross sections cut along the x  550 m lines of Fig. 4. Clearly, the coherent ACT vortical circulation is three-dimensional although it is not axisymmetric; the downshear side circulation is more complete while the upshear side circulation tends to be weakened (note that the ambient wind shear in the north–south direction diminishes above 1 km). We will discuss this asymmetrical ACT vortical circulation and the ambient shear later in this section and in section 2d. To examine how the vortical circulation influences ACT–environment mixing, we release a tracer  with a uniform mixing ratio of   1 g kg1 in an elevated layer between 1100 and 1200 m prior to the ACT pen- etration of this layer (Fig. 4a). Figure 4b shows that, when the ACT reaches the tracer layer, it displaces ambient fluid upward while ambient  is incorporated into a thin layer by direct contact with the ACT surface. The entrained air does not, however, continually erode FIG. 3. (a) Time evolution of the ascending cloud-top height zT for clouds A–F. (b) The growth rate  vs maximum cloud-top height zT,max for each cloud. (c) The thermal turnover time t* vs maximum cloud-top height zT,max for each cloud. Solid line: av- erage t* neglecting cloud C. 1296 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 into the center of the ACT but is, instead, advected downward along the edge to the rear of the ACT (e.g., Figs. 4b and 4c, at 1100 to 1200 m) leaving a nearly -free AFC after the penetration (Figs. 4c and 4d). En- vironmental air entrained through the ACT top, after being circulated to the ACT rear, is further mixed into the ACT center through the vortical circulation (Fig. 4d, downshear side between 1200 and 1400 m). The strong AFC divergence prevents any deep erosion of the ACT core from the ACT top, while the circulation along the ACT edge tends to quickly remove the more dilute cloud mixtures from the AFC, leaving the AFC as the most active and buoyant region throughout the ascent. In general, these simulation results support the schematic shedding thermal model proposed by Blyth and coauthors (e.g., Blyth et al. 1988; Blyth 1993). Environmental air is also directly engulfed into the ACT from the ACT rear through an organized vortical circulation on a large scale (comparable to ACT size). Figure 6 shows x–y cross sections of the horizontal ve- locity field at two different vertical levels and three different times. Figures 6a and 6b show the horizontal cross sections at a level 1137.5 m. Wind vectors are overlaid with the shaded contour of liquid water mixing ratio qc. At 8.5 min this 1137.5-m level is 75 m below the current cloud-top height [zT (8.5 min)  1212.5 m]. The flow field reflects the AFC mass divergence; no clear resolvable-scale entraining eddies appear. The mixing FIG. 4. Vertical cross section (x–z slices cut following the y  650 m lines in Fig. 5) of the internal flow pattern within the ACT of cloud E at 7.5, 8.5, 9.5, and 10.5 min after emerging from the subcloud layer. Solid line: qc 
 0.01 g kg 1 contour. Arrows: u–w wind vectors for the convective mixed region defined in section 2c of Part I (wind vectors outside this region are masked). The ambient mean wind u0  7.5 m s 1 has been subtracted from horizontal velocity u while the cloud-top vertical velocity wT has been subtracted from the vertical velocity w for each panel. The vertical shear of the horizontal mean wind is from left (west) to right (east) as shown in Fig. 1c of Part I. The arrows at the top left corner of each panel show the vertical velocity scale (3.8 m s1). Shading: the mixing ratio of a tracer , which is initially uniform (  1 g kg1) between 1100 and 1200 m at the 7.5-min time step. Dashed lines of constant x indicate the x coordinates of the y–z slices of Fig. 5. Dashed lines of constant z indicate the z coordinates of the x–y slices of Fig. 6. MAY 2005 Z H A O A N D A U S T I N 1297 along the edge of cloud and environment is primarily through the parameterized subgrid-scale eddy diffu- sion. At 9.5 min this same level now samples the ACT rear [zT (9.5 min)  1362.5 m]; Fig. 6b shows the clear indication of resolvable-scale entraining eddies. The circulation along these horizontal slices indicates that the vorticity vector is not strictly along the hori- zontal plane (Figs. 4 and 5), as predicted for an ideal- ized vortex ring or a spherical vortex, but also has a vertical component (Fig. 6). These entraining eddies have scales comparable to the ACT size itself and they directly engulf the environmental air and move it deep into the ACT center. Figures 6c,d show horizontal slices through another vertical level (1287.5 m) at 9.5 and 10.5 min. We observe a pattern similar to that shown in Figs. 6a,b. Notice that the engulfment of environmental air from the ACT rear is located on the downshear side [east (right) of the cloud] in this sheared environment. Overall, there is a net convergence at the ACT rear, in contrast to the net divergence at the ACT top (or AFC). In contrast to the ACT upper-interface mixing, which is primarily driven by subgrid-scale eddies, the ACT rear entrainment is primarily through resolvable large- scale entraining eddies. In comparison, these large- scale eddies associated with the vortical circulation must dominate the entrainment and mixing of the simu- lated clouds throughout their ascent. Furthermore, the mixed parcels first entrained from either the ACT top or sides are carried by the vortical circulation to the ACT rear where some of the mixtures are further wrapped into the vortex center, while others may stay behind depending on their vertical velocity and buoy- ancy. Therefore, the coherent vortical circulation strongly modulates the ACT’s entrainment and mixing of environmental air. The effect of ambient wind shear on the ACT’s vor- tical circulation and mixing can also be seen in Figs. 4 and 5. The vertical shear of the horizontal mean wind is from west (left) to east (right) in Fig. 4 and from south (left) to north (right) in Fig. 5. Comparing the four FIG. 5. As in Fig. 4 except these are y–z cross sections cut following the x  550 m lines in Fig. 4. Arrows: –w wind vectors for the convective mixed region defined in section 2c of Part I. The ambient mean wind 0 is from right (north) to left (south) with the shear direction point to north; 0 is small and is not subtracted from . The arrows at the top left corner of each panel show the scale for vertical velocity 4.1 m s1. Dashed lines of constant y indicate the y coordinate of the x–z slices of Fig. 4. Dashed lines of constant z indicate the z coordinates of the x–y slices of Fig. 6. 1298 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 panels in Fig. 4 and also the four panels in Fig. 5 shows that, when the ACT rises through the sheared environ- ment, the vortical circulation is continuously tilted to- ward the downshear side with the downshear-side cir- culation enhanced while the upshear side is weakened. For this reason, the overall ACT mixing and subse- quent mixing-induced downdrafts are enhanced at the downshear side of the cloud [note the unsaturated downdrafts (arrows outside the qc 
 0.01 g kg 1 con- tour) on the downshear side of the cloud in Figs. 4c,d and Figs. 5a–d]. The ACT eventually collapses to the downshear side after penetrating the inversion, while newly developed thermals ascend through the upshear side of the cloud (not shown). This ambient shear effect can be understood by ex- amining the ambient vorticity field produced by the vertical shear of the horizontal mean wind. For ex- ample, in Fig. 4 the ambient vorticity vector points northward (into the page) and, hence, has the same sign as the downshear side vorticity but with sign opposite that of the upshear side vorticity. The vorticity at both sides is horizontal and parallel to the ambient vorticity. Thus, the ambient vorticity enhances the downshear- side circulation while weakening the upshear-side cir- culation. Note, however, that the ACT vortical circula- tion is three-dimensional. When the horizontal vorticity is parallel to the ambient shear vector, the effect of the ambient wind shear is to tilt the horizontal vorticity toward the vertical, which can be seen from the tilting/ twisting term of the vorticity equation (e.g., Holton 1992). Therefore, the ambient shear is ultimately respon- sible for the generation of the horizontal entraining ed- dies shown in Figs. 6b,d. In section 2d, we will further explore some dynamics of the vortical circulation. Figure 7 shows the time–height variation of the hori- zontal mass divergence of the liquid water cloud region (left panels) and convective mixed region (right panels) for the large clouds D, E, and F. For all three clouds, FIG. 6. Horizontal cross section cut following the horizontal lines shown in Figs. 4 and 5. Solid line: qc 
 0.01 g kg 1 contour. Arrows: u– wind vectors. The ambient mean wind u0  7.5 m s 1 has been subtracted from the horizontal velocity u. The arrows at the top left corner of each panel show the vertical velocity scale 2.3 m s1. Shading: the liquid water mixing ratio qc (g kg 1). Dashed lines of constant y indicate the y coordinate of the x–z slices of Fig. 4. Dashed lines of constant x indicate the x coordinate of the y–z slices of Fig. 5. MAY 2005 Z H A O A N D A U S T I N 1299 the divergence zone associated with the shallow AFC is evident throughout the ACT ascent between roughly 200 and 600 s; it overlays a relatively thick convergence zone that ascends with the cloud top. After penetrating the inversion at 600 s, the strength of the convergence and divergence pattern is weakened. When the ACTs reach their maximum height at 700 to 800 s they col- lapse and accelerate downward; this, together with the subsequent arrival of ascending thermals, produces a strong divergence zone near the inversion base. When comparing the left and right panels of Fig. 7 we see large areas of strong divergence (dark-shaded region at approximately 1300 to 1600 m after 800 s) on the right panels that are absent from the left panels. This indi- FIG. 7. Time–height variation of the air–mass horizontal divergence {i.e., contours of Dt,k (kg m 1 s1)  A  [(w)/z]dA for time step t, vertical level k} integrated over different cloud areas A: (a) A calculated for the liquid water cloud region of cloud D. (b) As in (a) but for convective mixed region. (c) As in (a) but for cloud E. (d) As in (b) but for cloud E. (e) As in (a) but for cloud F. (f) As in (b) but for cloud F. 1300 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 cates that the persistent divergence zone occurs primar- ily in the unsaturated convective mixed region follow- ing the collapse of the ACTs. As we will show in section 2e, the strong divergence is associated with the detrain- ment of cloud–environment mixtures at the layer where the cloud buoyancy decreases with height. This prefer- ential detrainment is consistent with both the buoy- ancy-sorting detrainment hypothesis of Raymond and Blyth (1986) and the numerical and analytical results of Bretherton and Smolarkiewicz (1989) and Taylor and Baker (1991), which indicate that cloud detrainment occurs at levels with decreasing cloud buoyancy. While the simulated ACTs present a complex three- dimensional structure of the vortical circulation, the general pattern in which ascent is along the central ver- tical axis and descent is around the periphery is consis- tent with the laboratory results of ascending thermals, in situ observations of cumulus clouds, and the concep- tual shedding thermal model proposed by Blyth et al. (1988) (Scorer 1957; Johari 1992; Stith 1992; Carpenter et al. 1998b). The impact of ambient wind shear on the simulated clouds is furthermore consistent with obser- vational evidence showing that downdrafts and mixing occur preferentially on the downshear side of a cloud, while updrafts tend to persist on a cloud’s upshear side (e.g., Warner 1977). d. Dynamical structure of the ACT To obtain more insight into the internal circulation and mixing behavior of the ACT, we need to further examine its dynamics. In this section, we first examine the buoyancy field in the simulated ACT and explain the simulated vortical circulation using the concept of baroclinic torque. We thenfurther examine the pertur- bation pressure field and partition it into dynamic and buoyancy components following Wilhelmson and Ogura (1972). 1) BUOYANCY DISTRIBUTION AND BAROCLINIC TORQUE Figure 8 shows the buoyancy contours of the cross section of Fig. 4c. We see a mushroom-shaped buoyant core (e.g., 0.03 m s2 contour) that consists of the AFC centered at 1300 m and a trailing stem located within the ACT between 1100 and 1250 m. There is a strong horizontal buoyancy gradient, particularly at the down- shear side (note the negatively buoyant area at the downshear side of cloud edge). During the ascent of the ACT, more buoyant parcels at the ACT center tend to move rapidly upward while the less buoyant parcels at the ACT edge tend to move more slowly. This horizon- tal buoyancy difference provides a net torque that acts to rotate the cloud parcels clockwise on the right-hand side and anticlockwise on the left-hand side of the buoyancy maximum in Fig. 8. The baroclinic torque and its role in generating the horizontal vorticity can be seen from a two-dimensional (x–z plane) vorticity equation derived under the Bouss- inesq approximation, which is valid for this shallow FIG. 8. The cross section of Fig. 4c but with contours of the buoyancy B (m s2). The convective mixed region is shaded. Thick dotted line: qc 
 0.01 g kg 1 contour. Arrows as in Fig. 4c. MAY 2005 Z H A O A N D A U S T I N 1301 convection case (although the actual LES equations are anelastic): d dt   t  u x  w z  ux  wz   Bx , 3 where u and w are the wind components along the x and z directions respectively, B is buoyancy, and   (u/z  w/x) is the vorticity component perpendicu- lar to the x–z plane, with positive vorticity pointing into the page (Fig. 8). The vorticity production terms on the rhs of (3) are due respectivelyto the divergence and the horizontal buoyancy gradient. If the system initially contains no vorticity (  0), the only vorticity produc- tion term comes from the horizontal buoyancy-gradient term. From Fig. 8, we see that the buoyancy-gradient term should produce positive  on the right-hand side of the buoyancy maximum and negative  on the left-hand side of the buoyancy maximum, consistent with the simulation results. This buoyancy-gradient term can be formally related to the solenoidal term in the general vorticity equation (e.g., Holton 1992). The solenoidal term involves the density difference along a constant pressure surface and is generally called the baroclinic torque (e.g., Klaassen and Clark 1985). Here, this den- sity difference is primarily due to the cloud condensa- tional heating. The strong horizontal buoyancy gradient in the simulated clouds indicates that the baroclinic torque is primarily responsible for the generation of the vortical circulation of the simulated ACTs. While the baroclinic torque may be the primary source for the initial production of the vorticity for the simulated ACTs, this initially horizontal vorticity, as discussed in section 2c, must also be redistributed into vertical vor- ticity in the sheared environment (when the horizontal vorticity component is parallel to the shear vector). This will result in large ACT-scale horizontal eddies which might further break into smaller-scale eddies. This is a probable explanation for the complex three- dimensional structure of the entraining eddies shown in Figs. 4–6. 2) PERTURBATION PRESSURE Figures 9a–c show the same vertical cross section as in Fig. 8 but with contours of perturbation pressure p, the vertical component of the perturbation pressure gradient force (PPGF) (PPGF  ( 1p/z), and the sum of PPGF and B. The subgrid-scale momentum flux term is, in general, a small term compared with either PPGF or B and will be neglected below. Figure 9a shows a high pressure region centered on the AFC at 1380 m with a low pressure region below and on the downshear (right) side of the vertical velocity maxi- mum at 1200 m. Notice the tendency for the perturba- tion pressure contours to parallel the wind vectors around the vortical circulation. The ACT, accelerating into the quiescent environ- ment, produces a perturbation pressure field that tends to slow down its ascent (see Fig. 9b). Figure 9c shows that near the shallow AFC region (1300 to 1400 m) the PPGF is larger in magnitude than the buoyancy force and produces a net downward force, indicating that the FIG. 9. (a) As in Fig. 8, but with contour of perturbation pres- sure field p (Pa). Arrows as in Fig. 4c. (b) As in (a) but with contours of PPGF (m s 2). (c) As in (a) but with contours of PPGF  B (m s 2). 1302 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 vertical momentum flux into the AFC must be positive in order to maintain its ascent. This net downward force experienced by individual parcels works effectively to cap the fast moving parcels and is associated with their overturning motion within the ACT. Due to this net downward force, the most buoyant parcels decelerate and spread outward as they approach the AFC. Mixing with surrounding air along the leading edge of the ACT tends to dilute cloud parcels exposed to the environ- ment, while evaporation rapidly reduces their buoy- ancy. When these parcels circulate downward along the ACT edge to the ACT rear (i.e., between 1050 and 1200 m in Fig. 9c) the combined effect of the downward circulation, dilution, and evaporation produces large values of the CT-relative downward vertical velocity. By continually shedding these cloud–environment mixtures, the AFC region remains the most buoyant part of the ACT throughout its ascent. The CT-relative downward transport of mixed air and the enhanced en- trainment due to the vortical circulation at the rear of the thermal tends to push the more buoyant core into a comparatively narrow shaft below the AFC. This tends to intensify the horizontal buoyancy gradient and pro- duces stronger vortical circulation. This is consistent with the fact that the vortical circulation is strengthened throughout its ascent in the cloud layer (below the in- version). Below we further explore the perturbation pressure by partitioning it into dynamical and buoyancy compo- nents following Wilhelmson and Ogura (1972). The di- agnostic equation for the perturbation pressure under the anelastic approximation may be written as (Wil- helmson and Ogura 1972): 2p xi 2   xi uiujxj  ijxj I  B xi i3 II , 4 where p is the nonhydrostatic perturbation pressure,   (z) is the base state mean density that varies only with altitude. Here, ui (i  1, 2, 3) are the resolved wind components along the x, y, and z directions respec- tively; ij is the subgrid-scale momentum ui flux in the j direction. The equation is written in summation nota- tion and repeated indices are summed. Following Wil- helmson and Ogura (1972), the first two terms on the rhs are labeled the dynamic contribution (I) and the third term is labeled the buoyancy contribution (II) to the p Laplacian. Since (4) is linear in p, the total per- turbation pressure p can be decomposed into the dy- namic perturbation pressure pd and the buoyancy per- turbation pressure pb. These two terms can be found by solving 2pd xi 2  I 5a 2pb xi 2  II. 5b Figures 10a,b show the partitioned dynamic PPGF [PPGFd  ( 1pd/z] and buoyancy PPGF [PPGFb  (1)pb/z] which sum to the total PPGF shown in Fig. 9b. Both PPGFs are negative and act to decel- erate the air parcels within the AFC. However PPGFd is nearly one order of magnitude larger than PPGFb within the AFC. Thus, pd is expected to largely deter- mine the distribution of the total perturbation pressure p within the ACT. A contour plot of pd (not shown) indicates that it does indeed have a very similar pattern to that of Fig. 9a. In section 2c we have described the impact of the ambient shear on the ACT’s vortical circulation and ACT entrainment. The impact of this ambient wind shear on the dynamic perturbation pressure field can also be seen in Fig. 9a; it enhances the low pressure region on the downshear side of the ACT while dimin- FIG. 10. (a) As in Fig. 9b but with PPGF replaced by PPGFd (m s 2). (b) As in (a) but with PPGFd replaced by PPGFb (m s 2). MAY 2005 Z H A O A N D A U S T I N 1303 ishing or weakening the low pressure region on the upshear side. e. Dilution of the ACT In section 2c we defined the ACT as the upper part of the growing cloud containing the vortical circulation. To quantify the dilution rate of the ACT we sample it by choosing all the saturated grid cells located in the 14 model levels below zT (	h  350 m below the current zT) for large clouds and the 7 model levels below zT (	h  175 m below the current zT) for small clouds. These vertical limits generally include the cloud-top vortical circulation for each of the six selected clouds. The ACT mean properties are computed by averaging all selected grid cells. We also define the most undilute core within the ACT (called the ACT core below) as the average of the 2  2  2 grid cells that have the largest 	qt  qt,c  qt, where qt,c denotes the total water of a cloudy grid cell and qt is the horizontal model domain aver- aged qt. Similarly, the most dilute cloudy parcels within the ACT are defined as the average of the 2  2  2 grid cells that have the smallest 	qt. These two statistics are used as a measure of ACT inhomogeneity. Since an ACT ascends with time, the time histories of an ACT’s mean properties may be directly converted to vertical profiles of those properties based on the time- dependent height. Thus, the vertical profiles of the ACT properties presented below show the time evolu- tion of the ACT in a Lagrangian frame of reference. Figure 11 shows the vertical profiles (time histories) of buoyancy (given as the virtual potential temperature difference 	  ,c  ) for the ACT mean (solid), the ACT core (dashed), and the most dilute air within ACT (dot–dash). The vertical height at a given time is taken as the average geometric center of each category of grid cells. The vertical profile of cloud-mean buoy- ancy, which is calculated by averaging over the entire life cycle for all liquid water containing grid cells at each level, is also plotted (dotted) for comparison. First, we see that the interior of the ACT is nonhomo- geneous for all six clouds; the ACT core remains nearly undilute during much of its ascent (see also Fig. 12), thus producing large positive buoyancy values. In con- trast, the most dilute cloudy grid cells are negatively buoyant throughout the ACT ascent for all clouds. Al- though not as buoyant as the ACT core, the ACT mean is systematically more buoyant than the cloud mean (dotted). The buoyancy excess of the ACT mean in comparison with the cloud mean is significant, consid- ering that the cloud mean is only marginally buoyant. For large clouds, the cloud-mean buoyancy becomes negative at roughly 150 m below the inversion base (Figs. 11d–f). In the literature, the cloud-top height and cloud ver- tical velocity are usually determined by the cloud-mean buoyancy assuming a steady state. When integrating ww/z  B(z), with B(z) determined from the cloud- mean buoyancy, we find that the resulting velocity pro- file (not shown) always underestimates the cloud-top height. This is expected since cloud growth is associated with the ACT properties rather than the cloud-mean properties, which may be significantly biased to low values by the passive cloud mixtures detrained from the active ascending ACT. This result is consistent with the liquid water/cloud-top paradox first noted by Warner (1970). Figures 11d–f show, in contrast, that for the large clouds the ACT mean is significantly more buoy- ant and maintains its positive buoyancy up to the base of the inversion, while the most buoyant ACT cores maintain their positive buoyancy up to the middle of the inversion. These clouds ascend as long as their ACT core is positively buoyant; therefore the dilution rate of the ACT core is particularly relevant to the determina- tion of the cloud-top height. Also shown in Figs. 11d–f is the fact that large clouds overshoot their ACT core NBL by roughly 200 m. Figures 12a,b show the vertical profiles (time histo- ries) of l and qt for the ACT core for each cloud. For comparison, we also plot on Figs. 12a,b the vertical profiles of l and qt calculated based on an entraining plume model. Figures 12a,b show the dilution rate for the large clouds (D, E, F) is less than that for the smaller clouds (A, B, C) for both conserved variables. For the largest cloud F, the ACT core is diluted at a rate approximately equal to an entrainment rate 0.0003 m1 (dotted line); this is the entrainment rate used in the cumulus parameterization scheme of Tiedtke (1989). Although, as Siebesma and Cuijpers (1995) point out, this entrainment rate is one order of magnitude smaller than the dilution rate for the cloud-mean properties; it better predicts the dilution rate of the ACT core and thus the cloud-top height. However, as Figs. 12a,b show, the shapes of the vertical profiles of l and qt for the ACT core still differ qualitatively from those pre- dicted by an entraining plume, because the ACT cores experience no dilution until they reach the upper half of their zT,max. Figures 12c,d further show that, with the exception of cloud A, when the profiles are normalized by the individual cloud depth they collapse into a single profile, showing that the ACT cores stay essentially unmixed in the lower half of their depth and are gradu- ally diluted in the upper half of their depth. While individual clouds may achieve very different cloud-top heights, they have nearly the same cloud-root thermodynamic properties. This can be seen from Figs. 12a,b, where the cloud-base l and qt for all clouds are nearly the same [although large clouds tend to be slightly moister (0.1–0.3 g kg1 in Fig. 12b) than small clouds]. Given this, the dilution rates (and, therefore, cloud-top heights) are most probably caused by differ- ences in ACT size. Indeed, as Fig. 13 shows, the ACTs of large clouds (D, E, F) are nearly twice as big as those of the small clouds (A, B, C) when measured in terms of the ACT volume-equivalent spherical radius. How- ever, it is likely that this size dependency will be dimin- ished when an ACT exceeds a threshold size large 1304 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 enough to allow some air within the ACT to reach or overshoot its NBL without dilution. In this case, the cloud-top height will be solely determined by the ACT core thermodynamic properties and the environmental sounding. Another important feature shown in Fig. 13 is that the ACT size tends to decrease with height for small clouds (A, B, C) but stays relatively constant be- low the inversion and decreases above the inversion for large clouds. In no case does the ACT show a signifi- cant increase of size with height. This indicates that the ACT must detrain cloud environment mixtures as well as entraining environmental air throughout its ascent. The buoyancy profiles of Fig. 11 indicate the inho- mogeneous nature of the ACT interior. Figure 14 fur- ther shows the distributions of l and qt in the ACT of cloud E at several different heights (or times). Consis- tent with Fig. 11, the ACT is not uniformly diluted during its ascent. This would lead to a narrow distribu- tion with mode properties that continuously shift to- ward environmental values with increasing height. In- FIG. 11. (a)–(f) Vertical profiles/time histories of the buoyancy of the ACT mean (solid line), ACT core (dashed line), and the most dilute air (dotted–dash line) within ACTs for clouds A–F, respectively. The vertical height is the height of the geometric center of each region at 30-s time intervals. Dotted line: the vertical profile of the liquid water cloud lifetime mean buoyancy for each cloud (note the different scales between left and right panels). MAY 2005 Z H A O A N D A U S T I N 1305 stead, the (l, qt) distributions maintain modes corre- sponding to a relatively undilute core with a tail continuously extending to environmental values, al- though the modes also show slight dilution above 1200 m. This mixture distribution and the mixing behavior of the ACT core in which there is little dilution until the ACT has traveled approximately two diameters, is con- sistent with the shedding thermal model of Blyth (1993) rather than a continuously entraining homogeneous plume model. f. Passive mixing The previous sections have focused on the kinematics and dynamics of individual ascending thermals, which may be viewed as fundamental elements or agents of cumulus convection. The ascending thermals are inher- ently nonsteady and produce the pulsating growth char- acteristic of the simulated clouds. The mixing behavior of ascending thermals controls the maximum cloud-top height and dominates the cloud upward vertical mass transport. However, as noted above, during the ascent of an individual thermal a volume of more dilute cloudy air is left behind at the trailing wake. These mixtures may be generated either through the detrainment of ACT mix- tures or simply by enhanced mixing in the wake of the ACT. They are characterized by small or negative buoyancy and low vertical velocity. To distinguish them from the active ascending element, we will refer to them as passive cloud mixtures. In contrast to the active ACT mixing, no distinctive coherent structures are ob- served in these passive cloud regions (compare, for ex- ample the velocity structure in Fig. 4 at 1100 m at 8.5 and 10.5 min). The continued mixing of this passive cloud results in more negative buoyancy and a net downward motion. As shown in Part I, much of the descent motion is realized in the unsaturated cloud mixed convective region in the simulated clouds. Figure 15 shows an example of a cloud mixing life cycle at a fixed layer (1187.5–1312.5 m) that brackets the 1237.5-m vertical level shown in Fig. 1. Three joint distributions are plotted as functions of vertical velocity w and buoyancy 	. Figure 15a shows contours of the joint frequency distribution (JFD) computed from the liquid-water-containing grid cells within the FIG. 12. Vertical profiles of l and qt for the ACT core for clouds A–F. (a) Liquid water potential temperature l. Dotted line shows an entraining plume prediction with entrainment rate e  0.0003 m 1 for comparison. (b) As in (a) but for the total water qt. (c) As in (a) but normalized by the cloud-top heights for each cloud. (d) As in (c) but for qt. 1306 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 1187.5–1312.5-m layer during the life cycle of cloud E. Figures 15b,c show distributions of q̂t, the bin-averaged total water, and t̂, the bin-averaged cloud mixture ages defined as the time since emergence of the cloud from the subcloud layer. The Fig. 15c contours show a steady increase in mixture age moving from the positively buoyant updrafts in the upper right of the figure to the negatively buoyant downdrafts in the lower left. Com- paring this trend with Fig. 15b shows that the increase in mixture age is associated with a decrease in qt, which indicates the dilution of cloud mixtures with time. Spe- cifically, at about 600 s (upper right corner of Fig. 15c), the mixtures have qt  16.2 g kg 1 (Fig. 15b) with 	 values greater than 1 K. Comparison of the three panels shows that the ACT arrives at this level as relatively undilute subcloud layer air with high values of total water (q̂t 
 16 g kg 1) and buoyancy (	 
 1 K), with q̂t dropping below 13.4 g kg 1 by the end of the cloud life cycle. Notice, however, the inhomogeneity of the ACT vertical velocity, which ranges between 1 and 6.5 m s1 for bins along the 600-s contour. Examining (w, 	) bins along a line of increasing t̂ shows that, as time passes, there is a larger volume of more dilute cloudy air with low total water, buoyancy, and vertical velocity. Specifically, as t̂ increases from 550 to 1200 s in Fig. 15c, the number of grid cells increases steadily (Fig. 15a), while the total water, buoyancy, and vertical velocity of those grid cells decreases (Fig. 15b). This progression is due to the continuous dilution of passive cloudy mix- tures with environmental air; these mixtures become negatively buoyant and move downward as they age. 3. Discussion We have simulated an ensemble of shallow cumulus clouds using an LES with an initial sounding and large- scale forcings consistent with the BOMEX field experi- ment. From this cloud ensemble we selected six indi- vidual clouds with cloud-top heights ranging from 1 to 2 km and, with the help of a subcloud tracer, observed their evolution from initial growth to final dissipation. In Part I we focused on the vertical mass flux profile in both the saturated and unsaturated convective mixed region. In this paper we have presented the kinematic and dynamic details of the saturated regions of the six clouds. All six clouds share these common features: 1) A growth phase that is characterized by the ascent of a single pulse (clouds A, B, C) or a series of pulses (clouds D, E, F) detached from the subcloud layer ther- mal. 2) A cloud-cap region that contains the largest values of horizontal divergence, vertical downward per- turbation pressure gradient force (PPGF), and buoy- ancy. We refer to this as the ascending frontal cap (AFC); it is approximately 75 to 150 m thick in the simulated clouds. 3) A larger region extending through the cloud top that encompasses the coherent vortical internal (CT-relative) circulation. For the quantitative analyses we approximate this region by the upper 175 m (small clouds) and 350 m (large clouds). This as- cending cloud top (ACT) is marked by a dynamic per- turbation pressure pattern that is consistent with the vortical internal circulation. 4) A trailing wake charac- terized by passive cloud–environment mixtures with low values of vertical velocity and buoyancy. 5) A nearly constant thermal turnover time of approximately 600 s. 6) An ACT interior that has an inhomogeneous distribution of both thermodynamic properties and ver- tical momentum, with a relatively undilute core struc- ture. 7) An ACT core that experiences no dilution dur- FIG. 13. Vertical profile of the spherical volume-equivalent radius of the ACT for clouds A–F. FIG. 14. Histograms of (a) l and (b) qt for all ACT grid cells for cloud E at four different heights (or equivalently, times). The legend heights show the ACT heights zT(t) at the selected times (390, 480, 570, and 660 s). MAY 2005 Z H A O A N D A U S T I N 1307 ing the initial half of its ascent and gradual dilution during the later half of its ascent. The elevated tracer experiment (Figs. 4–6) indicates that environmental air is entrained into the ACT both through its top and sides. Either route, however, is strongly modulated by the ACT’s vortical circulation. The coherent ACT vortical circulation may be directly related to the horizontal buoyancy gradient within the ACT, which decreases from the ACT center to ACT edge, although the precise spatial distribution is far from symmetric (see Fig. 8). This horizontal buoyancy gradient produces a net torque, which tends to rotate the flow during its ascent. The result is an internal cir- culation where the ascent is along the central vertical axis while the descent is around its periphery. The vor- tical circulation drives the large-scale entraining eddies and wraps environmental air from the ACT rear into the rotating center where smaller-scale eddies produce further mixing. The large entraining eddies, which have scales comparable to the ACT size, dominate the ACT entrainment, mixing, and dilution. These eddies display a complex three-dimensional structure with a clear preference for enhanced entrainment and mixing at the downshear side of the clouds. For all simulated clouds, the ACT flow structure [calculated after removal of the cloud-top mean velocity (wT(t)] shows remarkable consistency with the dynamic perturbation pressure field (Fig. 9a). The strong downward PPGF within the AFC serves effectively as a lid for individual air parcels contained in the ACT and associates with the overturn- ing motion, while the low pressure generated near the ACT edge serves as centripetal force for the vortical circulation (Yau 1979). These simulation results, in combination with the cloud animations, paint the following picture of shallow cumulus mixing dynamics. The subcloud layer air tends to ascend in discrete elements that detach from their subcloud layer parent thermals. In contrast to the tra- ditional parcel models (adiabatic or entraining plume), these ascending thermals both entrain and detrain on their way toward their maximum height. As a result of this continuous entrainment and detrainment, the as- cending thermals are inhomogeneous both in their ther- modynamic and dynamic properties. Note that, while the dilution of an individual ascending thermal may be continuous, the occurrence of each thermal in a cumu- lus cloud is typically episodic. For example, as shown in Fig. 1, the arrival of ascending thermals at a particular level is intermittent in both time and space. The dy- namic perturbation pressure appears as the organizer of this group of inhomogeneous parcels. Specifically, it slows down fast-moving parcels and accelerates slow- moving parcels. The whole entity ascends at a speed approximately one-half that of the major updraft in the ACT center. The coherent vortical circulation strongly shapes the way that an ACT mixes with its environ- ment. The coherence of the ACT mixing makes it quali- tatively different from the dilution of passive cloud at the trailing wake of the ACT. The vortical circulation of the simulated ACT is con- sistent with the laboratory results of ascending thermals across a range of stratifications (e.g., Scorer and Ronne FIG. 15. An illustration of the mixing life cycle within a fixed cloud layer (between 1187.5 and 1312.5 m for cloud E). (a) Contours of the joint frequency distribution (unit: number of grid cells) of 	 and w for the liquid water cloud region. (b) As in (a) but for the bin- averaged total water q̂t (g kg 1) for grid cells in each (	, w) bin. (c) As in (a) but for the bin-averaged cloud age t̂ (unit: seconds) since emergence of the cloud from the subcloud layer. 1308 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62 1956; Scorer 1957; Sanchez et al. 1989; Johari 1992). Our simulation shows, as do the recent laboratory re- sults of Sanchez et al. and Johari, that the thermals that characterize shallow moist convection are not self- similar. Sanchez et al. (1989) found that the transition from an accelerating phase (near field) to a self-similar phase (far field) occurs when a thermal has traveled a distance of about six times its diameter. The maximum height reached by these simulated ACTs is typically four ACT diameters. This is possibly the reason that the hollow-cored thermals suggested by Scorer (1957) are not observed. Instead, the simulated ACT main- tains a mushroomlike core with a shrinking stem con- taining the most undilute subcloud air. These simulation results also strongly support the schematic shedding thermal model proposed by Blyth and coauthors (Blyth et al. 1988; Blyth 1993; Carpenter et al. 1998b). Blyth et al. (1988) proposed this ascending cloud-top mixing mechanism primarily to explain air- craft in situ observations showing that the conserved thermodynamic properties of cumulus mixtures tend to fall along a straight line between undilute subcloud air and environmental air near or above the observational level. Our numerical examination of the cloud kine- matic structure, perturbation pressure field, and tracer transport provide more direct evidence of such mixing and its dominant role in the dynamics of these simu- lated clouds. The fact that none of the simulated ACTs undergo significant size increases during ascent indi- cates that the simulated ACTs must detrain as well as entrain. The distribution of ACT thermodynamic prop- erties and the dilution history of the ACT core are also consistent with a mixing picture characterized by gradual core erosion and a continuous shedding of the ACT–environment mixtures during ascent, although the simulations also indicate that some mixed parcels may be carried upward for some distance before being shed in the ACT wake as either saturated or unsaturat- ed mixtures. The simulation results have indicated the important role of the dynamic perturbation pressure in the ACT’s vertical momentum budget. A detailed understanding of ACT mixing is particularly relevant to questions about the ultimate height of an ascending thermal. We are currently using a Lagrangian analysis to examine the vertical momentum budget of the ACT and AFC. These results will be used to evaluate the impact of the vertical perturbation pressure gradient force and en- trainment on the vertical momentum prediction, and therefore on the determination of cloud-top height. Acknowledgments. We thank Marat Khairoutdinov for making his model available and for assistance with the case setup. The simulations were primarily per- formed using the facilities of the Geophysical Disaster Computational Fluid Dynamics Centre at the Univer- sity of British Columbia; we thank Henryk Mod- zelewski for Beowulf cluster computing support. The manuscript was significantly improved by the com- ments of Steven Esbensen, Mark Holzer, Chin-Hoh Moeng, and two anonymous reviewers. This work was supported through funding of the Modeling of Clouds and Climate Proposal by the Canadian Foundation for Climate and Atmospheric Sciences, the Meteorological Service of Canada, and the Natural Sciences and Engi- neering Research Council. The first author gratefully acknowledges support from a University of British Co- lumbia University Graduate Fellowship. REFERENCES Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701. Austin, P. H., M. B. Baker, A. M. Blyth, and J. B. Jensen, 1985: Small-scale variability in warm continental cumulus clouds. J. Atmos. Sci., 42, 1123–1138. Barnes, G. M., J. Fankhauser, and W. D. Browning, 1996: Evolu- tion of the vertical mass flux and diagnosed net lateral mixing in isolated convective clouds. Mon. Wea. Rev., 124, 2764– 2784. Blyth, A. M., 1993: Entrainment in cumulus clouds. J. Appl. Me- teor., 32, 626–641. ——, and J. 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