Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part I: Transport

Transcription

1 1 Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part I: Transport Ming Zhao and Philip H. Austin Atmospheric Science Programme, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada Short title:

2 2 Abstract. This paper is the first in a two-part series in which the life cycles of numerically simulated shallow cumulus clouds are systematically examined. The life cycle data for six clouds with a range of cloud-top heights are isolated from an equilibrium trade cumulus field generated by a large eddy simulation (LES) with a uniform resolution of 2 m. A passive subcloud tracer is used to partition the cloud life-cycle transport into saturated and unsaturated components; the tracer shows that on average cloud convection occurs in a region with time-integrated volume roughly 2-3 times that of the liquid water-containing volume. All six clouds exhibit qualitatively similar vertical mass flux profiles, with net downward mass transport at upper levels and net upward mass flux at lower levels. This downward mass flux comes primarily from the unsaturated cloud-mixed convective region during the dissipation stage, and is evaporatively driven. Unsaturated negatively buoyant cloud mixtures dominate the buoyancy and mass fluxes in the upper portion of all clouds while saturated positively buoyant cloud mixtures dominate the fluxes at lower levels. Small and large clouds have distinct vertical profiles of heating/cooling and drying/moistening, with small clouds cooling and moistening throughout their depth, while larger clouds cool and moisten at upper levels and heat and dry at lower levels. The simulation results are compared to the predictions of conceptual models commonly used in shallow cumulus parameterizations.

3 3 1. Introduction The importance of shallow cumulus convection for the redistribution of atmospheric heat and moisture is well established. Numerous approaches have been used to represent the mixing effect of these unresolved boundary layer clouds in large-scale models, including eddy diffusivity approximations, moist convective adjustment schemes and mass flux parameterizations (e.g. Tiedtke et al. 1988; Betts 1986; Arakawa and Schubert 1974). Mass flux parameterizations are the most sophisticated of these approaches; they are built around specific conceptual models that constrain the way in which clouds can interact with the environment. Examples of such conceptual models include the entraining plume model (e.g Stommel 1947; Arakawa and Schubert 1974), episodic mixing and buoyancy sorting models (EMBS) (e.g. Raymond and Blyth 1986; Kain and Fritsch 199; Emanuel 1991; Emanuel and Zivkovic-Rothman 1999) and the transient cumulus model of von Salzen and McFarlane (22). All of these mass flux parameterizations must account either implicitly or explicitly for the size distribution of the unresolved clouds. For example Arakawa and Schubert (1974) represent the cloud population by a spectrum of entraining plumes with different entrainment rates or, equivalently, cloud-top heights. Implicit representations of the cloud size distribution model the net effect of the cloud population as a single entraining/detraining plume with variable entrainment and detrainment rates (Kain and Fritsch 199; Siebesma and Holtslag 1996). The free parameters for all of these schemes can be adjusted to reproduce the average vertical mass flux profile as inferred from measurements of the large scale forcing in equilibrium cumulus convection. An obvious drawback to tuning the parameterized mass flux profile to observations is that it limits the conditions under which the parameterization can be used. If the size spectrum and the conceptual cloud model can be considered independently, it becomes possible to introduce new observational constraints on the parameterization. As Grinnell et al. (1996) point out, the conceptual models listed above make distinct predictions for both the vertical mass transport within individual clouds, and the manner in which this transport varies with height due to cloud entrainment and detrainment. An entraining plume model predicts increasing mass transport with

4 4 height due to lateral entrainment, with all detrainment taking place at cloud top. In an EMBS model, both entrainment and detrainment take place at all heights within the cloud layer, so that mass transport may actually decrease with height above cloud base. Transient cumulus models introduce the cumulus life cycle into the bulk representation of the entraining and detraining plume so that the lifetime-averaged cloud fraction and mass flux may also decrease with height. The EMBS models also differ in their treatment of the detrainment of mixed cloud air into the environment. For example, the EMBS scheme of Emanuel (1991) (E91 below) detrains cloud mixtures into their unsaturated neutral buoyancy levels (UNBL), defined as the neutral buoyant level (NBL) obtained by reversible adiabatic evaporation of the detraining parcel s liquid water content. This detrainment criterion essentially defines the cloud boundary to encompass the evaporating mixed region, so that the evaporation directly cools the cloud air (Emanuel 1994). Transmission of this cooling to the environment is realized through environmental compensating upward motion induced by cloud downdrafts. In contrast, the EMBS model of Raymond and Blyth (1986), in common with entraining plume models, directly detrains liquid water into the environment. A similar approach is used by Kain and Fritsch (199) who detrain neutral and negatively buoyant saturated mixtures directly into the environment. Both the profiles of vertical mass flux and the cloud heating and moistening rates are sensitive to this detrainment criterion (see, e.g. Zhao and Austin 23). Direct observations of the vertical profile of vertical mass flux within cumulus clouds can help distinguish between these competing conceptual models. Doppler radar, used in concert with in-situ aircraft observations, provides one tool for examining the vertical mass flux profile over the life-cycle of individual clouds. Grinnell et al. (1996) combined two ground-base Doppler radars and an instrumented aircraft to compute the vertical mass flux in trade wind cumulus clouds during the Hawaiian Rainband Project of 199. They found that, averaged over the cloud s lifetime, the vertical profile of net vertical mass flux was negative in the upper portion of the cloud layer and positive at lower levels. The cloud life-cycle was characterized by a growth phase in which the mass flux was upward at all levels, a mature stage in which there was downward mass flux at

5 upper levels, and a dissipation stage during which the mass flux was everywhere downward. A complementary approach is provided by large-eddy simulations (LES) and cloud-resolving models which have been increasingly used to study cumulus convective transport (e.g. Siebesma and Cuijpers 199; Brown 22; Stevens et al. 21). Siebesma and Cuijpers (199) for example, found that the vertical profile of cloud cover for the BOMEX trade cumulus boundary layer was maximum near cloud-base and decreased monotonically to zero with height, indicating numerous small clouds. This feature of shallow convection is common to 11 different LES codes (Siebesma et al. 23). The LES-determined cloud size distribution has also been studied by Neggers et al. (23), who found consistency between the modeled cloud field and satellite observations showing a power-law size distribution dominated by small clouds. The effects of this cloud size distribution are also seen in observational studies by Nitta (197), who used a spectrum of entraining plumes to show that small clouds dominate the mass flux profile in the lower part of the cloud layer. Siebesma and Holtslag (1996) showed that large detrainment rates were required in order to represent the transport due to this cloud population by a single plume. The examination of the individual clouds embedded in a simulated cloud ensemble has until recently received little attention. There have been numerous simulations of single cumulus clouds developing in a conditionally unstable atmosphere (e.g. Klaassen and Clark 198; Grabowski and Clark 1993; Bretherton and Smolarkiewicz 1989) However, the simulated structure and time evolution of these clouds depend on the trigger that releases the stored energy. The most common initialization technique involves specifying a positively buoyant impulse in the subcloud layer; the evolution of clouds initialized in this manner may be sensitive to the size and shape of the initial impulse. This is in general inappropriate in low-buoyancy environments such as shallow convection (Carpenter et al. 1998). In principal, this ad-hoc specification of the cloud environment and initialization can be overcome by studying individual clouds embedded in LES runs that are independent of initial conditions. There is reason to think that individual shallow clouds may be realistically simulated in such models. For example, Siebesma and Jonker () show that the fractal dimension of

6 6 the simulated individual cloud boundaries at 1 m resolution are in excellent agreement with observations. As mentioned above, Neggers et al. (23) retrieved cloud size distributions from several LES datasets at m resolution and found them to be consistent with satellite observations. These models appear to have reached the stage where they can resolve both the cloud-ensemble statistics and some of the individual cloud dynamics. In this paper, we use an LES model run to examine the life cycle of the vertical cloud mass flux, with an emphasis on studying the effects of unsaturated convection within the cloud-mixed region using passive tracers. We use a set of six clouds to examine the role of cloud size in the cloud ensemble transport. Our objective is to test some of the physical assumptions underlying conceptual models of cumulus clouds used in cumulus parameterizations. In a companion paper (Part II) we focus on cloud mixing dynamics for these same six clouds. In Section 2 we briefly describe the LES model, the simulation setup and the approach used to isolate and identify individual clouds and cloud-mixed convective regions. In Section 3 we present the simulated individual clouds and their life cycles with emphasis on the vertical transport of mass, the role of buoyancy in convective mass flux and thermodynamic transport based on conditional sampling of cloud mixtures. In Section 4 we test the sensitivity of these results to the choice of threshold used to identify the unsaturated convective mixed region. We further discuss the model results and their implications for conceptual models of cumulus mixing in Section. 2. Approach a. The LES model and case description We use the Colorado State University Large Eddy Simulation/Cloud Resolving Model; a detailed description of this model is given in Khairoutdinov and Randall (23). The equations of motion are written using the anelastic approximation. The prognostic thermodynamic variables are the liquid water static energy, total non-precipitating water and total precipitating water (precipitation is switched off for this shallow convection case study). The subgrid-scale model

7 7 employs a 1.-order closure based on the prognostic subgrid-scale turbulent kinetic energy (Stull 1988), while the advection of momentum is computed with second-order finite differences in flux form with kinetic energy conservation. The equations of motion are integrated using the third-order Adams-Bashforth scheme with a variable time step. All prognostic scalar variables are advected using the fully three-dimensional positive definite and monotonic scheme of Smolarkiewicz and Grabowski (199). We use a three-dimensional domain with 26x26x128 grid points, 2 m uniform resolution in both the horizontal and vertical, periodic horizontal boundary conditions and a time step of 1. s. We choose the undisturbed trade wind boundary layer from Phase III of the Barbados Oceanographic and Meteorological Experiment (BOMEX). A detailed description can be found in Siebesma et al. (23) and siebesma/gcss/bomex.html. The convection in the boundary layer is driven by specified surface fluxes with latent and sensible heat fluxes of 1 and 8 Wm 2 respectively. Figure 1 panels a,b,c show the initial profiles for the BOMEX simulation: a well-mixed sub-cloud layer between approximately - m, a conditionally unstable cloud layer between - 1 m, and an overlying inversion layer. Figure 1 panels d,e,f show the prescribed large scale forcing which includes large scale subsidence, clear-sky radiative cooling and advective drying in the lower subcloud layer. Although idealized, the BOMEX case setup insures a consistent set of large-scale forcings which establish an equilibrium for a shallow cumulus cloud boundary layer in the LES. The model-produced equilibrium vertical profiles of temperature and moisture closely match the observed BOMEX atmosphere (Nitta and Esbensen 1974; Siebesma and Cuijpers 199). The first three hours of the simulation are considered to be spin-up, in which the system reaches its steady-state equilibrium. We run the LES for 6 hours; all the selected individual clouds are sampled between hours 3 to 6. b. Isolating individual clouds Since the BOMEX environment has strong westward ambient wind (Figure 1c) the simulated individual clouds move rapidly from east to west. Therefore the distance that an individual cloud

8 8 travels during its life cycle is much larger than its horizontal size. To isolate individual clouds over their life-cycle and obtain time-dependent -dimensional cloud data we must sample the cloud in a reference frame that moves to the west at approximately the mean horizontal velocity. We choose a translation velocity U = 7. ms 1, which is close to the horizontally and vertically-averaged mean velocity within the cloud layer. Because the simulated cloud field data is stored every t = 3 s, this choice of U insures that as each succeeding 3 second snapshot is shifted by a translation distance d = U t, it is moved an integral number of gridcells d/ x. The resulting translated time series maintains the exact relative position between the different snapshots in gridcell coordinates. Consistent with the model, we use periodic horizontal boundary conditions when we apply the translation. The time-sampled cloud field moves much more slowly in the translated reference frame, allowing us to use fixed boxes to contain individual clouds throughout their life-cycle. Note that the Galilean invariance of the Navier-Stokes equations ensures that the behavior of the fluid is unchanged in this new inertial reference frame. Even with the above technique, isolating individual clouds throughout their life-cycle is not always easy because of the overlap of the evolving clouds. This is particularly true for larger clouds, due to their greater volumes, longer lifetimes, and the tendency for new clouds to develop nearby. During the isolation process we therefore always try to select a box which includes a single cloud with as much surrounding environment as possible while excluding other clouds. In this study we examine 6 clouds with cloud-top heights (defined as the maximum height reached by cloud liquid water) ranging from m to m. We refer to them below (ordered smallest to largest) by the labels A, B, C, D, E, F. c. Distinguishing the convective region from the environment To help with the isolation of individual clouds, and more importantly to provide an objective way to identify the cloud-mixed convective region (which may be unsaturated), we use a passive numerical tracer ζ initialized to g kg 1 above cloud-base and 1 g kg 1 below cloud-base. We

9 9 first run the LES, obtain a particular realization of the time dependent cloud fields and select interesting clouds. Since we now know the start and end times and the cloud-base height of each cloud, we rerun the LES with the tracer initialized at a single time step just before individual cloud emerges from the subcloud layer. Only advection is applied to the subcloud layer tracer; the tracer variable aids in tracking the destination of subcloud air based on the resolved velocity. In this simulation, cloud convection is the only process which can bring subcloud air to appreciably higher levels. Dry thermals may also overshoot the cloud-base height for a short distance and bring some tracer to levels slightly above cloud-base. Although the overshooting dry thermals always sink back after entraining some environmental air near the cloud-base level, they do contaminate the time-dependent cloud data near cloud-base. In addition, implicit diffusion due to the advection scheme may also transport tracer across the sharp tracer interface near the cloud-base level. Therefore data sampled near the cloud-base level must be interpreted with caution. Given the isolated individual clouds A-F, we can further identify their associated cloud-mixed regions based on a simple criterion ζ > ζ, where ζ is a threshold value of subcloud layer tracer mixing ratio. Figure 2 shows an example of the joint frequency distribution (JFD) of subcloud tracer mixing ratio ζ and liquid water mixing ratio q c for all liquid water-containing gridcells during the lifetime of cloud E (a similar pattern holds for the other clouds). As the figure shows, for ζ >. g kg 1 the identified mixed region begins to exclude a significant number of liquid water containing gridcells, which are generally located at the upshear side of the clouds during their ascending stage (not shown). Therefore we choose ζ =. g kg 1 for all 6 selected clouds; this threshold eliminates less than 1% of the liquid water volume from the sampled cloud. This choice of tracer level is conservative in the sense that it selects the smallest mixed region which does not exclude a significant number of liquid water containing gridcells. This criterion, however, does not guarantee that the chosen region is still convective, since some mixed volume may simply contain detrained subcloud layer air which has reached equilibrium with the stratified environment. Figure 3 shows the JFD of w and θ v within the

10 1 unsaturated cloud mixed region (ζ > ζ and q c = ) for each cloud over its life-cycle. Here θ v = θ v θ v, θ v is the gridcell value and represents a horizontal model domain average (which is approximately equal to the unmixed cloud environment average). The figure shows that the distribution mode is centered around w = and θ v =, values that correspond to the quiescent environment. To eliminate this detrained mixed-region from the sampled mixed region and obtain the convective mixed region, we remove those gridcells that satisfy the criterion: (ζ > ζ and q c = and θ v θ v, and w w ), where θ v, and w are adjustable positive threshold values. Choosing large values for θ v, and w selects a smaller and more active unsaturated mixed region; arbitrarily large θ v, and w will eliminate all the unsaturated mixed region and reduce the sampled data to saturated cloud. In principal, the choice of θ v, and w may depend on the unmixed environment s θ v and w variability. The shaded region on each panel of Figure 3 shows the set of gridcell ( θ v, w) values that characterize at least 99.6% of the gridcells in the unmixed environment (i.e., those gridcells with q c = and ζ <.1 g kg 1 ). For the smaller clouds (panels 3a,b,c), these environment gridcells lie approximately within the range θ v <.2 K, w <. ms 1 ). For the larger clouds (panels 3d,e,f) the environment spans a much larger range of ( θ v, w) (note the different scales between left and right panels). The larger environmental variability shown in panels 3d,e,f is limited to the inversion levels; beneath the inversion the environmental ( θ v, w) fluctuations have roughly the same range in all clouds. Although large clouds, which penetrate the inversion, generate internal gravity waves which dominate the environmental ( θ v, w) fluctuations at upper levels, these waves make a negligible contribution to the net vertical mass transport. Below we use, for simplicity, the thresholds θ v, =.2 K and w =. ms 1 suggested by the environmental variability of the small clouds A-C, for all clouds at all levels. In Section 4 we will test the sensitivity to these thresholds by choosing several different values of θ v, [, 1.2] K and w [, 1.2] ms 1 for the large clouds. Note that our (ζ, w, θ v, q c ) criteria have to be simultaneously satisfied in order to identify a gridcell which will not become

11 11 convective. For example, a neutrally buoyant region with zero vertical velocity may still be potentially convective if it has liquid water, since further mixing with the subsaturated environment may generate negative buoyancy and therefore downward velocity. To summarize, we distinguish different sampled regions: 1) the mixed region (all grid-cells with ζ > ζ ); 2) the detrained mixed-region (DMR) (ζ > ζ and q c = and θ v θ v, and w w ); 3) the convective-mixed region (CMR) (ζ > ζ and not DMR); 4) the unsaturated convective-mixed region (CMR and q c = ); ) the liquid water (or saturated) cloud (CMR and q c > ). 3. Results a. Life-cycle overview Figure 4 shows the top and base height of each cloud over its lifetime, together with the maximum updraft and downdraft velocity. The cloud-top and base height at each time step are determined by the highest and lowest levels which contain liquid water. Small clouds A, B and C have a maximum height around - m and do not reach the inversion layer (1 m). Clouds D, E, F are large clouds which penetrate into the inversion. In contrast to their very different cloud-top heights, all clouds have essentially the same cloud-base heights which do not vary with time prior to the dissipation stage. The uniform cloud-base height is due to the homogeneous heat flux specified over the ocean surface in this simulation. Individual cloud lifetimes end when all liquid water is evaporated (possibly before all the unsaturated cloud mixtures reach their neutral buoyancy levels (NBL)). Lifetime animations 1 of these clouds show that small clouds A, B, and C contain a single ascending updraft which decays after reaching its maximum height, while large clouds D, E and F tend to contain 2 to 3 pulse-like updrafts. Each succeeding pulse ascends and decays similarly, but reaches a maximum height lower than its predecessor. Due to these multiple pulses, large clouds D, E and F have longer lifetimes, as 1 Animations available from

12 12 shown in Figure 4a. It should also be noted that large clouds in the simulated cloud field often organize in a group of cells and evolve through a more complex life-cycle, which may persist for up to one hour before complete dissipation. Compared with these long-lived clouds, our selected large clouds D, E, and F might be interpreted as single, clearly isolated, large convective cells or elements with vertical penetration as high as the inversion layer. Panels 4b, 4c show the time evolution of the maximum updraft and downdraft velocities within individual clouds. The maximum upward velocity of each cloud always occurs before arrival at the maximum height, while the strongest downdraft occurs slightly after arrival at the maximum height. This pattern is associated with the penetration of the ascending cloud-tops above their NBLs, where the cloud-tops decelerate as they continue to rise. The collapse of cloud-top and the associated mixing and evaporation triggers the strongest downdrafts. In a companion paper (Zhao and Austin, Part II) we discuss in detail the intermittent behavior of the simulated clouds and the cloud mixing dynamics. Figure, panels a,b show the time evolution of the volumes of the saturated and unsaturated convective regions for each cloud. During the clouds ascending stage the unsaturated convective region is smaller than the saturated region for all six clouds. By the time the clouds reach their maximum height, the unsaturated convective regions have volumes that are roughly half (small clouds) or about 2/3 (large clouds) of their saturated volumes. After the clouds reach their maximum height, the liquid water cloud volumes begin to shrink due to continuous evaporation, while the unsaturated convective regions continue to increase due to turbulent mixing. A short time before the clouds liquid water is completely evaporated, the unsaturated convective regions reach their maximum volume and begin to rapidly decrease. This decrease is due to the exclusion of mixtures which have reached equilibrium with the environment based on the ( θ v,, w ) thresholds. When the evaporation of the liquid water content is nearly complete, the unsaturated convective regions are still roughly as large as the maximum liquid-water containing cloud. This indicates that the lifetime of individual clouds as defined by the existence of liquid water tends to be slightly shorter than the convective period associated with the turbulent mixing of individual

13 13 clouds. Integration over individual cloud lifetimes reveals that the volume of the total convective mixed region (saturated plus unsaturated) is about 2-3 times the size of the saturated region (see the ratios at the top of Panels a and b). Translating these volumes to equivalent cylindrical radii implies that the horizontal radius of the total convective mixed region is roughly 1. times larger than the radius of the visible cloud. Panels c,d show the time evolution of the vertical mass flux integrated over the two regions of Panels a,b. For each cloud the mass flux is generally positive in the saturated volume and negative in the unsaturated region with flux in the liquid water cloud volume dominating the earlier phase while the flux in the unsaturated volume dominates the later phase of convection. Panels e,f show that there is a transition in the total vertical mass flux from positive to negative at approximately one-half of the cloud lifetime and a resulting sinusoidal time variation for the volume integrated vertical mass flux. Integrated over the total cloud life-cycle, this produces a negative net vertical mass flux for small clouds and a positive net vertical mass flux for large clouds. b. Vertical profiles of the vertical mass flux Figures 6, panels a,b show the vertical profiles of cloud lifetime-averaged vertical mass flux for the 6 clouds. All clouds have net downward mass flux in their upper levels. The downward mass flux for small clouds extends deeper into the cloud layer while for inversion-penetrating clouds D, E and F the net downward mass flux is limited to approximately the upper 1/3 of the cloud depth, producing net downward mass flux within the inversion. The cloud-top downward mass flux of the simulated clouds is consistent with the aircraft and radar observations of Hawaiian trade cumulus clouds reported by Raga et al. (199) and Grinnell et al. (1996). It is also consistent with the result of Zhao and Austin (23) (hereafter ZA3), who diagnosed the cloud ensemble vertical mass flux using an EMBS model based on the BOMEX equilibrium soundings and large-scale forcings. The vertical profiles of convective mass flux, partitioned into saturated and unsaturated components, are given in panels 6c,d; they show that the downward mass flux comes

14 14 primarily from the unsaturated cloud-mixed region while the saturated component produces, on average, upward mass flux throughout the cloud depth (although a significant number of saturated downdrafts do exist near cloud-top). In Section we discuss the implications of this vertical profile of vertical mass flux for the shallow cumulus parameterization problem. A life-cycle view of the vertical mass flux profiles is shown in Figure 7. The total (saturated plus unsaturated) vertical mass fluxes are partitioned into periods which are nearly equally spaced over the six cloud lifetimes. As the figure shows, during the developing stage all clouds produce net upward mass flux. When the clouds reach their mature stage (i.e. cloud-tops reach their maximum heights) downward mass flux begins to increase. Downward mass flux occurs first near cloud-top for the large clouds (D,E,F) while for small clouds downward mass flux appears first at lower levels, before the cloud-tops reach their maximum heights. During the dissipation stage all clouds have net downward mass flux throughout their depth. Thus, at every level, the time history of the vertical mass flux resembles the sinusoids of panels e and f; near cloud base the upward flux during the growth phase is larger than the downward flux at the dissipation stage, while near cloud-top this pattern is reversed. In general, this life cycle of the vertical profile of the cloud vertical mass flux through the growth and dissipation stages is again consistent with the radar observations of Grinnell et al. (1996), who found negative net mass flux in the upper portion of the cloud layer at the dissipation stage and positive mass flux at lower levels during the growth phase for small non-precipitating Hawaiian clouds. c. The role of buoyancy in vertical mass transport The buoyancy-sorting hypothesis suggests that all air coming from above an observation level should have negative or neutral buoyancy at the observation level, while air arriving from below should have positive or neutral buoyancy. In another words, cloud mixtures tends to move following their buoyancy and produce a positive buoyancy flux. To test the buoyancy-sorting hypothesis with our simulated clouds, we next evaluate the contribution of buoyancy-direct and buoyancy-indirect mass fluxes to the convective mass transport. Here we define buoyancy-direct

15 1 motion as vertical motion that is positively correlated with the buoyancy; buoyancy-indirect or counter-buoyancy transport is characterized by a negative velocity-buoyancy correlation. Figure 8a shows the JFD of (q t, θ v ) over all gridcells at a level within the inversion (1612. m above the surface) over the life-cycle of cloud E. Note the strong correlation between mixture buoyancy and the total water mixing ratio q t which, as a conserved variable, is usually well correlated with the mixture s cloud-environment mixing fraction. The square symbol marks the (q t, θ v ) of cloud-base air lifted adiabatically to this level, while the circle represents the (q t, θ v ) of the mean environment. The well-defined V shape indicates the strong nonlinear dependence of the mixture buoyancy on mixing fraction caused by phase change. The quadrants separate these convective cloud-environment mixtures into 4 categories: saturated positively buoyant (SP), saturated negatively buoyant (SN), unsaturated negatively buoyant (UN) and unsaturated positively buoyant (UP). Panel 8b shows a contour plot of the bin-averaged vertical velocity using the gridcells of Panel 8a. There is both buoyancy-direct and buoyancy-indirect motion, with unsaturated mixtures dominating the downward motion while saturated mixtures dominate the upward motion. Figure 9 shows the partitioned vertical profile of the vertical mass flux for the 4 different categories for each of the clouds. The saturated, positively buoyant mixtures (SP) tend to dominate the upward mass flux, with maxima peaking near cloud-base, while unsaturated negatively buoyant mixtures (UN) tend to dominate the downward mass flux, with maxima peaking near cloud-top. Both are buoyancy direct. However, significant counter-buoyancy transport also occurs. In particular, Figure 9 shows that saturated negatively buoyant mixtures (SN) on average transport mass upward and unsaturated positively buoyant mixture (UP) on average transport mass downward. The JFD of θ v and w within the saturated cloudy region for each cloud is shown in Figure 1. While most of the saturated positively buoyant mixtures are positively correlated with vertical velocity, this is not true for saturated negatively buoyant mixtures. In fact, the majority of the saturated negatively buoyant mixtures in Figure 1 have positive vertical velocity. This mixture component can be produced by a mixing process in which momentum exchange allows

16 16 a mixture to maintain positive velocity while evaporation produces buoyancy reversal. These upward-moving, negatively buoyant mixtures may also be caused simply by initially positively buoyant mixtures carried past their NBL by their inertia. For both cases, these mixtures might not return to their initial NBL if mixing continues along their counter-buoyant trajectories. In particular, when significant mixing causes saturated parcels to evaporate all their liquid water, they will descend as unsaturated mixtures. We find a relatively small number of saturated downdrafts but a large number of these unsaturated downdrafts, indicating that further mixing and phase change is a dominant feature of counter-buoyancy transport in the simulated clouds. In contrast, unsaturated negatively buoyant mixtures do not experience this phase change, and the unsaturated positively buoyant mixtures are generated simply by initially unsaturated negatively buoyant mixtures overshooting their NBL from above. Although further mixing may allow these mixtures to do some net thermodynamic transport, the impact should generally be small since, as will be shown in Section 3d, these mixtures usually have thermodynamic properties very close to their environment as they reach their NBL. Figure 11 summarizes our discussion of the role of buoyancy in vertical convective mass transport. It shows the buoyancy flux partitioned into positively buoyant updrafts (PU), negatively buoyant downdrafts (ND), positively buoyant downdrafts (PD) and negatively buoyant updrafts (NU). The sum of PU and ND gives all the buoyancy direct (BD) transport while the sum of PD and NU gives all the buoyancy indirect (BI) transport. In general, buoyancy-direct transport dominates the overall convective mass flux and produces positive buoyancy flux throughout the individual cloud depths, supporting the buoyancy-sorting hypothesis. However, significant counter-buoyancy transport does exist, particularly from saturated negatively buoyant and unsaturated positively buoyant mixtures at higher cloud levels for large clouds (c.f. panels 11d, e and f). This counter-buoyancy transport is primarily associated with mixture inertia, and indicates that a buoyancy-sorting model which transports all cloud-environment mixtures based solely on their buoyancy may potentially over-estimate the buoyancy flux. This is in contrast to the tendency of entraining plume models to under-estimate the buoyancy flux, as shown by Siebesma

17 17 and Cuijpers (199). d. The nature of the unsaturated downdrafts As shown in Figure 9 the unsaturated negatively buoyant mixtures dominate the downward mass flux. More information about the thermodynamic properties of these downdrafts is given in Figure 12, which is a mixing diagram using (θ l, q t ) coordinates following Taylor and Baker (1991). Data in the scatter plot is taken from all gridcells at the m inversion level over the lifetime of cloud E (c.f. Figure 8). These mixtures are distributed along a mixing line between the thermodynamic coordinates of cloud-base environmental air (symbol: x) and those for air at (or slightly above) the current observation level (symbol: +). This linear mixing line between cloud-base air and air at or slightly above the observation level is a feature of numerous in-situ observations in cumulus clouds. Also shown on the diagram are the average environmental sounding (solid line), θ v isopleths at m (dash-dot), and the saturation line at m (dashed). As the figure shows, nearly % of the mixture range at this level is unsaturated; these mixtures have negative buoyancy as large as their saturated counterparts. These unsaturated mixtures are often neglected in analyses of in-situ aircraft measurements and numerical simulations. The continuous character of this mixing distribution, however, underscores the fact that unsaturated mixtures are an integral part of cloud mixing throughout the cloud life cycle. Figure 12 also shows some unsaturated mixtures which are positively buoyant at m. As discussed previously, these mixtures are primarily due to unsaturated mixtures generated at higher levels which have overshot their NBLs from above. Figure 13 and 14 show partitioned lifetime-averaged vertical profiles of the θ l and q t difference between each category of mixtures and the environment. The partitioning of the 4 categories is as in Figure 8. As noted above and in Section 3c, the unsaturated positively buoyant mixtures (UP) for all clouds do have θ l and q t nearly indistinguishable from their environment, supporting the idea that they are primarily due to unsaturated negatively buoyant mixtures overshooting their NBLs. Figure 13 and 14 also show that the unsaturated negatively buoyant

18 18 mixtures (UN), which are primarily responsible for the downward mass flux, are on average systematically cooler and moister than the environment. This again indicates that they must be associated with cloud mixing and evaporation, since downdrafts that were instead purely mechanically forced by saturated updrafts would be drier and warmer than the surrounding environment. e. Thermodynamic fluxes and tendencies Figures 1 and 16 show the fluxes of two conserved variables θ l, q t partitioned into the 4 mixture categories as in Figure 9. The θ l and q t fluxes produced by negatively buoyant saturated and unsaturated mixtures always have opposite sign and tend to cancel throughout the cloud depth for all clouds, while the θ l and q t flux contributions due to positively buoyant unsaturated mixtures are small throughout the cloud depth. Thus saturated positively-buoyant mixtures best represent the overall conserved variable transport in these clouds. However, this is not true near the cloud-top for large clouds, where all saturated air becomes negatively buoyant when reaching the maximum cloud-top height. Siebesma and Cuijpers (199) used an LES simulation of BOMEX convection to examine how well the turbulent flux of a conserved variable χ can be represented by a top-hat model of the form: w χ = M ρ (χ c χ e ) (1) where M is the vertical mass flux, χ e is the cloud environment mean and χ c the cloud mean with cloud determined by one of three different criteria based on gridcell liquid water content q c, vertical velocity w, and buoyancy B. They label the three criteria as the cloud decomposition (q c > ), the updraft decomposition (w > and q c > ) and the cloud core decomposition (w > and q c > and B > ). Siebesma and Cuijpers found that (1) gave the best approximation to the LES flux when χ c was determined by the cloud core decomposition. The underlying

19 19 reason for this can be seen in Figures 1 and 16: although the saturated negatively buoyant fluxes are non-negligible, they are approximately canceled through most of the cloud layer by the unsaturated negatively buoyant mixtures, leaving the saturated positively buoyant flux as most representative of the net θ l, q t transport. While the saturated positively buoyant mixtures may provide a good approximation to the net θ l and q t fluxes, they fail to adequately represent the buoyancy flux. Figure 17 shows that the buoyancy flux in the upper 1/3 of all clouds is dominated by unsaturated negatively buoyant mixtures, while the contribution from saturated negatively buoyant mixtures is very small. Therefore, including the unsaturated convective region is crucial for the representation of both the vertical mass flux and the buoyancy flux. Figure 18 shows the total lifetime averaged θ l and q t fluxes for the 6 clouds. The θ l fluxes for small clouds tend to monotonically increase with height while q t fluxes tend to monotonically decrease with height. Except near cloud base, this vertical distribution of the small cloud θ l and q t fluxes indicates cooling and moistening of the environment throughout their cloud depth. In contrast, the large clouds tend to have minimum θ l and maximum q t fluxes at their mid-levels, indicating cooling and moistening at upper cloud and inversion layers and warming and drying at lower cloud layers. The corresponding tendency profiles due to the transport of each cloud are shown in Figure 19. The panels show the magnitude of the heating and moistening rates found from the flux profiles of Figure 18. As that figure showed, small clouds cool and moisten throughout nearly their entire depth. The average values for these tendencies are approximately.2 K day 1 and.1 g kg 1 day 1 for the three smallest clouds. The large clouds have heating rates within the cloud layer that are roughly 1 times larger than those for the small clouds, and drying rates about 3-4 times larger (although the net heating and drying rate of cloud D is close to zero within the cloud layer). Within the inversion, all of the large clouds produce strong cooling and moistening with a peak near the inversion base. These simulation results are consistent with the results of Esbensen (1978), who combined a laterally entraining plume model to represent clouds

20 2 that penetrate into the inversion with a bulk model of the shallower cloud circulation below the inversion base. He found that the deep, inversion penetrating clouds are primarily responsible for the warming and drying of the lower cloud layer, while the shallower clouds are primarily responsible for moistening below the inversion base. 4. Sensitivity As discussed in Section 2c, while the threshold values of θ v, =.2 K and w =. ms 1 may be appropriate for small clouds, it is unlikely that they will eliminate all of the detrained mixed region for the large clouds within the inversion. This is because, for these clouds, the environmental variations of θ v and w are larger than these two threshold values. However, underestimating the detrained mixed region should not significantly influence the net vertical transport of mass, since detrained air should not contribute much to the net vertical mass transport (although it does dramatically influence the calculated volume of the unsaturated convective-mixed region for the large clouds). In this section, we examine the impact of the threshold values of θ v, and w on both the time evolution of the unsaturated convective region and its associated vertical mass transport. Figure 2 shows the time evolution of the unsaturated convective-mixed region and the associated vertical mass flux for the three large clouds (D,E,F) with 4 different ( θ v,, w ) thresholds in the range [-1.2 K, -1.2 m s 1 ]. When both θ v, and w are chosen to be zero (case 1), all of the cloud mixed region is assumed to be convective. This produces a large increase in the volume of the unsaturated convective region but only a slight increase in the net downward vertical mass flux, compared to threshold pair 2. This indicates that the detrained mixed region does not contribute significantly to the net vertical mass flux. As the threshold pair is increased beyond θ v, >.4 K and w =.4 m s 1 (cases 3-4), the net vertical mass flux begins to decrease. This decrease is more significant within the lower cloud layers since more mixtures are eliminated by this criterion (note that the mixtures variability decreases as the height decreases). Within the inversion (above 1 m) the mass flux due to unsaturated downdrafts is insensitive

21 21 to these 4 choices of threshold value, although the time-integrated volume of unsaturated air is reduced to only 2% of the saturated volume for the highest threshold pair (case 4) for clouds D and E and 4% for F. This indicates that most of the unsaturated downward mass flux within the inversion comes from an unsaturated convective-mixed region that is only 2-4% of size of the saturated cloud, supporting the idea that unsaturated downdrafts are confined within a small volume and are truly part of the convective cloud. When the two threshold values are chosen to be arbitrarily large, clearly we expect that all of the unsaturated mixed region will be eliminated, and the convective region is then reduced to saturated cloud air. Our choice of threshold (approximately case 2 in Figure 2) eliminates the large majority of detrained mixtures while including nearly all of the convective transport.. Discussion As we noted in Section 1, the vertical mass flux profile of a trade-cumulus boundary layer consists of the superposed contribution of many individual cumulus clouds. We have used a high-resolution LES to examine in detail the life-cycle of six simulated clouds taken from a cloud field in equilibrium with its large scale forcing. With the help of a passive tracer, we have partitioned the cloud life-cycle transport into saturated and unsaturated components, and have shown that on average cloud convection occurs in a region with time-integrated volume roughly 2-3 times that of the liquid water-containing volume. All six clouds exhibit qualitatively similar vertical mass flux profiles, with net downward motion at upper levels and net upward motion at lower levels. The results indicate that this downward mass flux comes primarily from the unsaturated convective-mixed region during the dissipation stage. Vertical profiles of the partitioned cloud-environment θ l and q t difference show that the unsaturated negatively buoyant mixtures are consistently cooler and moister than the environment, indicating that the downdrafts are driven by evaporative cooling. This unsaturated negatively buoyant convective mixed region provides the dominant contribution to both the buoyancy and mass fluxes in the upper portion of the cloud layer, while the saturated positively buoyant cloud mixtures dominate the fluxes at

22 22 lower levels. However, small and large clouds have distinct vertical profiles of heating/cooling and drying/moistening, with small clouds cooling and moistening throughout their depth, while larger clouds cool and moisten at upper levels and heat and dry at lower levels. a. Comparison with entraining plume and EMBS models These results can be used to evaluate the two conceptual models that have been used as building blocks for cumulus parameterizations. Most existing parameterizations are built around the idea of an entraining plume or parcel (e.g Arakawa and Schubert 1974). An entraining plume model represents an individual convective element or cloud using an ascending subcloud air parcel that continuously entrains environmental air with a specified entrainment rate. The entrained environmental air is homogenized instantaneously and the plume is finally detrained into its NBL. In contrast, an EMBS model (e.g. Emanuel 1991) assumes that an element of subcloud air ascends adiabatically to a particular level and undergoes dilution that generates a spectrum of mixtures. These mixtures are then evaporated and vertically displaced to their individual NBLs where they are detrained into the environment. Both conceptual models are highly simplified pictures of the transport associated with the individual convective elements of real cumulus clouds. Figure 21 shows the very different mass flux profiles produced by the entraining plume and EMBS approaches. For each panel we use the equilibrium BOMEX environmental soundings. The individual lines of Figure 21a show the mass flux profiles for 1 different choices of entrainment rate ranging from.1 to.3 m 1, normalized by the mass flux at cloud base. Panel 21b shows the corresponding mass flux profiles for an EMBS model in which a subcloud air parcel with unit mass flux is lifted to a specific level. A uniform distribution of mixtures is generated. These mixtures follow their adiabats, evaporating any liquid water; they are subsequently detrained at their unsaturated neutral buoyancy level. As the panels show, the entraining plume model produces an exponentially increasing vertical mass flux for an individual convective element or cloud while the vertical mass flux produced by an EMBS-modeled convective element decreases with height, becoming negative at upper levels. The two models also differ in the way

23 23 in which the massflux profile changes with cloud height or size. For the entraining plume model the vertical mass flux at a given level depends on both the entrainment rate and the level s height above the cloud-base, with smaller clouds producing more mass flux at a fixed level due to their larger entrainment rate. In contrast, for the EMBS example the contribution of an individual convective element to the mass flux at a particular level increases with increasing cloud size. Comparison of these profiles with Figure 6 reveals that neither the vertical mass flux profiles of the liquid water cloud region nor those of the convective mixed region are consistent with the entraining plume prediction, although the mass flux of the largest cloud F does increase slightly with height in the middle of cloud layer. Large clouds show no tendency to have smaller normalized vertical mass flux at lower levels, which again contradicts the entraining plume model prediction. In contrast to the qualitative failure of the entraining plume model s prediction of the vertical mass flux of a single cloud or convective element, comparison of Figure 6 and Figure 21b shows that an EMBS model qualitatively captures the principal features of the mass transport of individual clouds/convective elements. In particular, the model-produced net vertical mass flux is downward near cloud-top and upward near cloud-base. The prediction of net negative vertically-integrated vertical mass flux for small clouds, and net positive vertically-integrated vertical mass flux for larger clouds, is also consistent with the LES simulation. Note however that a monotonically decreasing vertical mass flux is not a necessary feature of an EMBS model for precipitating clouds, since mixtures may have an unsaturated NBL above their mixing level after the fallout of some precipitation. Figure 22 shows an example of the θ l and q t tendencies produced by the spectrum of entraining plumes and EMBS elements in Figure 21, assuming a vertical resolution of 1 m and a unit subcloud convective element with mass flux equal to 1 gm 2 s 1. The magnitude of both heating/cooling and drying/moistening are significantly larger in this entraining plume element than in the EMBS element. In this sense, the transport produced by an entraining plume model of a single convective element is much more efficient than that of an EMBS model. Dramatic differences exist not only in the way that the ascending parcels entrain environmental air (which