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Episodic mixing and buoyancy-sorting representations of shallow cumulus convection Zhao, Ming 2003

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Episodic Mixing and Buoyancy-Sorting Representations of Shallow Cumulus Convection by Ming Zhao M.Sc, Nanjing Institute of Meteorology, 1993 B.Sc, Nanjing Institute of Meteorology, 1990 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF Doc to r of Ph i losophy in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Earth and Ocean Sciences) We accept this thesis as conforming to the required standard The University of British Columbia July 2003 © Ming Zhao, 2003 UBC Rare Books and Special Collections - Thesis Authorisation Form http://www.library.ubc.ca/spcoll/thesauth.html In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g ain s h a l l not be allowed without my w r i t t e n permission. r The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada lof 1 09/15/2003 07:20 PM A b s t r a c t This dissertation combines a diagnostic study, a numerical simulation and a theory of cumulus mixing to investigate shallow cumulus convection and its representation in large-scale models. The objective is to facilitate the development of a cumulus parameterization scheme and also to improve our understanding of the mixing dynamics, transport and life-cycle of shallow cumulus clouds. Applying.an episodic mixing and buoyancy sorting (EMBS) model to shallow clouds requires specification of the undilute eroding/mixing rate (UER), a cloud detrainment criterion and the cloud-environment mixing distribution. A diagnostic equation is for-mulated for the U E R based on a convective equilibrium constraint. A particular UER profile is retrieved by applying the diagnostic equation to an observed trade cumulus boundary layer. Based on this framework, a study of the U E R retrieval sensitivity to variations in the mixing distribution and the detrainment criterion is carried out. The retrieved U E R decreases exponentially with height above cloud-base, suggesting a strong modulation by the cloud size distribution. The EMBS-diagnosed vertical mass flux is downward within the inversion layer and upward within the cloud layer, indicating the important role of cloud evaporation in cloud detrainment. A large eddy simulation (LES) is conducted to provide a further evaluation of the EMBS model and the diagnostic results. Numerical tracers are used to identify the cloud-mixed region. Six clouds with a range of heights are isolated from the LES field. The results reveal that the lifetime-averaged vertical mass-flux profile decreases with height and becomes negative within the upper cloud layers. The simulated clouds produce cool-ing/moistening in their upper portion of the penetration depth and warming/drying in the lower portion of their penetration depth. The success of the EMBS approach is found to be primarily due to its correct representation of the growing and dissipating phase of a cloud life-cycle. Unsaturated downdrafts are evaporatively driven and dominate the overall mass and buoyancy fluxes at upper cloud layers for all simulated clouds. The explicit simulation also strengthens a conclusion made in the diagnostic study that an equilibrium cloud size distribution can be seen as the product of the large scale forcing and dynamics of individual convective clouds. ii Contents A b s t r a c t i i C o n t e n t s i i i L i s t o f F i g u r e s v i i i A c k n o w l e d g e m e n t s x i i 1 I n t r o d u c t i o n 1 1.1 The importance of shallow cumulus convection 1 1.2 Shallow cumulus representation and conceptual models of cumulus mixing 3 1.3 Observational studies of cumulus mixing 5 1.4 Laboratory investigation of cumulus mixing 8 1.5 Numerical simulation of shallow cumulus clouds 10 1.6 Research objectives and methodology 13 2 A n O v e r v i e w o f E M B S M o d e l s 17 2.1 Introduction 17 2.2 Mass flux parameterization 19 2.3 E M B S model notation 20 2.4 Detrainment criterion 24 2.5 Mixing distributions 29 iii 2.6 Undilute eroding rate 31 2.7 Summary 32 3 A D i a g n o s t i c S t u d y o f t he E M B S M o d e l 33 3.1 Introduction 33 3.2 Diagnostic equations for the UER 34 3.3 B O M E X convective equilibrium 38 3.4 UER retrieval ".. 39 3.5 Linking the UER to cloud size distribution 42 3.5.1 Comparing retrieved UER to cloud size distribution 42 3.5.2 Single cloud and cloud ensemble effects 44 3.6 Sensitivity tests 47 3.7 Discussion 53 3.8 Summary 56 4 L i f e C y c l e o f N u m e r i c a l l y S i m u l a t e d S h a l l o w C u m u l u s C l o u d s . P a r t I: T r a n s p o r t 58 4.1 Introduction 58 4.2 Approach 60 4.2.1 The LES model and case description 60 4.2.2 Isolating individual clouds 63 4.2.3 Distinguishing the convective region from the environment . . . . 64 4.3 Results 68 4.3.1 Life-cycle overview 68 4.3.2 Vertical profiles of the life-time averaged vertical mass flux . . . . 73 4.3.3 The role of buoyancy in vertical mass transport 76 4.3.4 The nature of the unsaturated downdrafts 81 iv 4.3.5 Thermodynamic fluxes and tendencies 84 4.4 Sensitivity 90 4.5 Discussion 92 4.5.1 Comparison with entraining plume and EMBS models 93 4.5.2 Unsaturated convection 97 4.5.3 Role of cloud-size distribution in cloud-ensemble transport . . . . 99 5 Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part I I : Mix ing Dynamics 101 5.1 Introduction 101 5.2 Pulsating cloud growth 102 5.3 Growth rate and thermal turn-over timescale 106 5.4 Kinematic structure of the A C T 108 5.5 Dynamical structure of the A C T 118 5.5.1 Buoyancy distribution and baroclinic torque 118 5.5.2 Perturbation pressure 121 5.6 Dilution of the A C T 126 5.7 Passive mixing 132 5.8 Discussion 135 6 Cloud-top determination 140 6.1 Introduction 140 6.2 Approaches using NBL of a plume 141 6.3 Lagrangian vertical momentum budget 143 6.4 Discussion 151 7 Summary 153 7.1 Diagnostic study 153 v 7.2 Conceptual model for the transport by individual convective elements . . 155 7.3 Conceptual model for the mixing dynamics of convective elements . . . . 157 7.4 Unsaturated convection, downdrafts and cloud evaporation 159 7.5 Buoyancy-sorting hypothesis 161 7.6 A conceptual model for shallow cumulus ensemble transport 163 7.7 Suggestions for future work 165 Appendix A List of Symbols and Acronyms 169 A . l Symbols 169 A.2 Acronyms 172 Appendix B Multi-mixing of positively buoyant mixtures 175 Appendix C Consistency and stability of the UER inversion 177 C.1 Consistency of solutions 177 C. 2 Stability of solutions 179 Appendix D BOMEX case description and LES ensemble statistics 182 D. 1 B O M E X field experiment and B O M E X GCSS model-intercomparison . . 182 D. 2 A comparison of ensemble statistics with other models 185 D.2.1 Time evolution and steady state 186 D.2.2 Fluxes and variances 186 D. 2.3 Cloud fractions and vertical mass fluxes 189 Appendix E A description of the LES model 193 E. 1 Model equations and dynamical framework 193 E.2 Subgrid-scale model/parameterization 194 E.3 Boundary conditions 195 E. 3.1 Upper boundary 195 vi E.3.2 Lower boundary . . .• 196 E.3.3 Lateral boundaries 197 E.4 Numerical methods 197 A p p e n d i x F E n t r a i n i n g p l u m e m o d e l c a l c u l a t i o n 198 A p p e n d i x G D y n a m i c p e r t u r b a t i o n p res su re 200 v i i List of Figures 2.1 Vertical profiles of mixtures generated by the episodic mixing hypothesis 21 2.2 6i, qt and 6V of mixtures from Figure 2.1 following buoyancy sorting . . . 25 2.3 An illustration of the treatment of positively buoyant mixtures 27 2.4 Mixing distributions for a selection of EMBS models 29 2.5 UER for B O M E X cumuli following assumptions in RB86 and EZ99 . . . 31 3.1 An illustration of the episodic mixing and buoyancy sorting model . . . . 37 3.2 B O M E X equilibrium state 38 3.3 Diagnosed fluxes and tendencies from the EMBS model 40 3.4 A comparison of the diagnosed UER with assumptions and cloud-size dis-tribution 43 3.5 Single cloud-induced net tendencies of Q\ and qt 45 3.6 Decomposed tendencies for the 1400-m-deep cloud of Figure 3.5 46 3.7 Sensitivity of the retrieved UER based on mixing distributions PI — P4 . 48 3.8 Tendencies computed for the EMBS model for a single convective element 50 3.9 As in Figure 3.8, but using the multi-mixing algorithm 50 3.10 As in Figure 3.8, but using the NBL detrainment criterion 52 3.11 Net tendencies using different detrainment criteria and the retrieved UER 53 4.1 Initial sounding and large scale forcing profiles for the LES B O M E X case 61 viii 4.2 Joint frequency distribution of subcloud layer tracer mixing ratio and liq-uid water mixing ratio 66 4.3 JFD of A9V and w within the unsaturated cloud-mixed region 67 4.4 Life-cycle of cloud-top, cloud-base height, the maximum upward and down-ward vertical velocity for clouds A - F 69 4.5 Time evolution of the volumes, mass flux of the liquid water cloud and unsaturated convective-mixed regions 70 4.6 Vertical profiles of the lifetime-averaged vertical mass flux for the convective-mixed region 73 4.7 Time-height variation of the vertical mass flux integrated over the hori-zontal convective mixed region A 75 4.8 JFD of A0V and qt for all the convective-mixed air at a single level over the life-cycle of cloud E 77 4.9 Vertical profiles of the partitioned vertical mass flux 78 4.10 JFD of A9V and w within the liquid water cloud region for clouds A - F . . 79 4.11 Vertical profiles of the partitioned buoyancy flux 80 4.12 Convective cloud mixtures plotted on a (9i, qt) conserved variable diagram 82 4.13 Vertical profiles of the 9\ difference between each of the 4 mixture categories and the environment for clouds A - F 83 4.14 As in Figure 4.13, but for qt 84 4.15 Vertical profiles of the partitioned 9i flux 85 4.16 As in Figure 4.15, but for qt flux • • • 85 4.17 As in Figure 4.15, but for 9V flux 86 4.18 Vertical profiles of 9\ and qt flux averaged over individual cloud lifetimes for clouds A - F 87 ix 4.19 Vertical profiles of environmental 9\ and qt tendencies due to the convective transport of clouds A - F over their lifetimes 89 4.20 Sensitivity of the volume of the unsaturated convective-mixed region and the corresponding vertical mass flux to the choice of (A9Vto, WQ) threshold 91 4.21 Normalized vertical profiles of vertical mass flux produced by a spectrum of convective elements based on the equilibrium B O M E X environment . . 94 4.22 Vertical profiles of 9i and qt tendencies produced by the spectrum of con-vective elements of Figure 4.21 96 5.1 An example of the pulsating character of cloud E 103 5.2 Time-height variation of A9i, Aqt, A9V, Aw for the most undilute gridcell 104 5.3 Time evolution of A C T height, the growth rate and thermal turn-over time 107 5.4 Vertical x-z cross sections of the internal flow pattern within the A C T . . 109 5.5 As in Figure 5.4, except for y-z cross sections I l l 5.6 Horizontal x-y cross sections cut following the lines in Figures 5.4 and 5.5 113 5.7 Time-height variation of the cloud air-mass horizontal divergence . . . . 116 5.8 The cross-section of Figure 5.4c but with contours of buoyancy B . . . . 119 5.9 Contour of perturbation pressure p, PPGFV, and PPGFV + B 122 5.10 As in Figure 5.9b but for PPGFd and PPGFb 124 5.11 Vertical A9V profiles for the A C T mean, A C T core, and the most dilute air 127 5.12 Vertical profiles of 9L and qt for the A C T core for clouds A - F 129 5.13 Vertical profiles of the A C T volume-equivalent radius for clouds A - F . . . 131 5.14 Histograms of 9L and qt for all A C T gridcells for cloud E at 4 heights/times 132 5.15 An illustration of the mixing life-cycle within a fixed cloud layer 133 6.1 Cloud-top estimations based on the NBLs 141 6.2 As in Figure 6.1 but for cloud B 142 6.3 Vertical profiles of vertical acceleration on the RHS of (6.12) 148 x 6.4 As in Figure 6.3, but for the ACTs of clouds D, E and F 150 7.1 A conceptual model for trade-wind cumulus ensemble transport 164 C.1 Consistency of the solutions between UT and Uq 178 C. 2 Stability of the solutions oiUT 181 D. 1 B O M E X ship array, 1969 183 D.2 Time series of the total cloud cover, liquid water path and T K E 187 D.3 Mean vertical profiles averaged over the last 3 hours of #/, qt, u, v, and qc 188 D.4 Turbulent flux profiles averaged over the last 3 h for 8i, qt, 0V, and qc . . 190 D.5 Vertical profiles of momentum flux, T K E and velocity variance 191 D.6 Vertical profiles of cloud and cloud core fraction, cloud and cloud core vertical mass flux 192 xi A c k n o w l e d g e m e n t s I would like to make special thanks to my supervisor Prof. Phil Austin for his abundant support, encouragement and advice during the course of this work. The knowledge, advice and support I received from Phil ranged not only over cloud and atmospheric sciences but also included computer technology and English. He introduced me to various software utilities and computer languages that proved to be very helpful in this research and probably throughout my future career. For all that Phil has done for me I am sincerely grateful. I am also sincerely thankful for the constructive suggestions and comments from my Ph.D. committee members: Prof. Mark Holzer, Prof. Norm McFarlane, Prof. Douw Steyn and Prof. Roland Stuff I am grateful to Prof. Kerry Emanuel and Dr. Pier Siebesma for making their model and research results available through the Internet. I owe special thanks to Dr. Marat Khairoutdinov for providing his C S U - C R M / L E S model and assisting in the B O M E X case setup. I thank Prof. Min Xue for making the A R P S model available. Although the simulation results using A R P S are not presented in this dissertation they significantly improved my knowledge of numerical modelling. I would also like to thank Charles Leung for his computer assistance. The large eddy simulations were primarily performed using the facilities of the Geo-physical Disaster Computational Fluid Dynamics Centre (GDCFDC) at the University of British Columbia; I thank Henryk Modzelewski for Beowulf cluster computing support. I would also like to thank Prof. Matt Choptuik for kindly allowing me to use the Physics department's Linux cluster. I gratefully acknowledge the 3 year support from the University Graduate Fellowship and a one-time T .K . Lee scholarship from the University of British Columbia. 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 Engineering Research Council. I thank my friends Mike Barton, Stephan de Wekker, Charles Leung, Johnson Zhong, Kuansheng Geng for the good times. Finally, I wish to thank my wife Lihua, for her patience, support and encouragement during this course of this work. M I N G Z H A O The University of British Columbia July 2003 xii C h a p t e r 1 I n t r o d u c t i o n 1 . 1 The importance of shallow cumulus convection Over large areas in the tropics, subtropics, and mid-latitudes during summertime, the atmosphere is dotted with shallow, usually non-precipitating, cumulus clouds. These clouds typically cover only a small fraction of the sky, yet have a substantial impact on the vertical distributions of heat and moisture in the lower troposphere. By venting air from the subcloud mixed layer toward the free troposphere, shallow cumulus convection strongly influences boundary layer depth, cloud cover, and winds. In the subtropics, a sizable fraction of stratocumuli are also underlain and sustained by shallow cumulus convection. Unlike stratocumuli, shallow cumulus clouds have much less direct effect on radiative transfer due to its small cloud cover (e.g., 15%). However, through regulation of the lower atmospheric temperature and water vapor profiles, they significantly impact the earth's radiation budget and regional and global circulations, making .them important components of the earth weather and climate system (e.g., Riehl et al., 1951; Riehl, 1954; Tiedtke et a l , 1988; Betts and Ridgway, 1989; Betts, 1997; Gregory, 1997). "Trade cumuli", which are ubiquitously observed over the subtropical oceans in trade-wind regimes, are a particular example of shallow cumulus clouds. These clouds transport 1 heat and moisture upward and greatly enhance surface evaporation. The moisture col-lected in the trade wind areas is transported downstream to the intertropical convergence zone (ITCZ), where it is finally released as latent heat in deep convective tropical distur-bances. Since this latent heat released in the ITCZ is the engine of the Hadley (and also Walker) circulation, the surface evaporation upstream in the undisturbed trade wind areas can be regarded as the main fuel supply for this circulation. In this sense, the existence of trade cumuli intensifies the large-scale circulation (Siebesma and Cuijpers, 1995). Indeed, by comparing a 50-day integration of the E C M W F (European Centre for Medium-Range Weather Forecasts) general circulation model (GCM) including a trade cumulus parameterization with an integration without such a parameterization, Tiedtke et al. (1988) have demonstrated that the parameterization increases surface latent heat flux, especially above the subtropical oceans, by as much as 50 W m ~ 2 to more realistic values. Precipitation in the ITCZ is enhanced by up to 10 mm day - 1 . The hydrological cycle is intensified and hence also the Hadley circulation, resulting in stronger and more realistic subtropical anti-cyclones especially over the Atlantic Ocean. In the subtropics, the vertical mixing of heat and moisture by trade cumuli is important to counteract the drying and warming effects of the large-scale subsidence induced by the Hadley circulation. As a result, a quasi-steady thermodynamic state of the cloud and inversion layer in the undisturbed trade wind regions can be maintained. Over a longer time scale, this quasi-equilibrium state (the boundary layer depth and vertical distribution of temperature and humidity) feeds back on the sea surface temperature (SST) through its impact on the radiation and surface fluxes. While the importance of shallow cumulus convection for the weather and climate has been well established, the approaches to representing shallow cumulus convection in large-scale and general circulation models are much more controversial. 2 1.2 Shallow cumulus representation and conceptual models of cumulus mixing Current atmospheric GCMs are typically designed to resolve processes with horizontal scales of a few hundred kilometers, vertical scales of tens to hundreds of meters and temporal scales.of hours and longer (von Salzen and McFarlane, 2002). Thus, individual cumulus clouds and even organized convective systems are only subgrid-scale phenomena, and their collective effect on the large-scale (resolved) flow has to be represented by resolvable-scale variables. The principal underlying assumption in representations of chaotic subgrid-scale processes is that the statistical properties of such processes can be deduced from a knowledge of the resolved variables (Emanuel, 1994). The formulation of the collective effects of subgrid-scale cumulus clouds in terms of the prognostic grid-scale variables may include clouds either explicitly or implicitly. For instance, the schemes generally classified as convective adjustment, such as Kuo (1965), Manabe et al. (1965), Betts (1986), and Betts and Miller (1986) simulate the effects of moist convection by adjusting the lapse rates of temperature and moisture to specified profiles within the local grid column. In these parameterization schemes the physical properties of the clouds are entirely implicit. Adjustment schemes are relatively simple and easy to apply, but the prediction of temperature and water vapor profiles is severely limited by the imposed profiles. Moreover, there is little physical basis for the universality of the water vapor profiles used (Emanuel, 1991; Emanuel and" Zivkovic-Rothman, 1999). Another family of schemes focuses on the parameterization of cumulus convection using the mass-flux approach. This involves coupling a simple cloud model to an algo-rithm for determining the mass flux pumped upward through cloud-base. Historically, the cloud model has been either a spectrum of entraining plumes (Arakawa and Schubert, 1974), a spectrum of subcloud-scale drafts (Emanuel, 1991), or a single bulk entraining-3 detraining plume (e.g., Tiedtke, 1989; Gregory and Rowntree, 1990; Kain and Fritsch, 1990; von Salzen and McFarlane, 2002). Each cloud model is built on a different set of parametric assumptions regarding mixing, entrainment, and detrainment between the cloud and environment. This ultimately depends on an understanding of cumulus mixing and the conceptual model of cumulus mixing one has in mind. One important complication for a cumulus parameterization scheme is that it is typ-ically designed to represent an ensemble of cumulus clouds which, as suggested by ob-servations, consists of a size or height distributed population of clouds or convective elements. For a bulk entraining-detraining plume model, this cloud-size distribution is accounted for implicitly and the performance of such schemes can only be compared with the net effect of the cloud ensemble. The bulk schemes are difficult to evaluate based on individual cloud observations. For spectral convective element models (e.g., Arakawa and Schubert, 1974; Emanuel, 1991), the modelled convective elements are intended to rep-resent individual clouds or convective drafts and, therefore, may in principle be directly compared with observed single clouds or convective elements. As a building block of the spectrum of convective elements, two quite different con-ceptual models of cumulus mixing have been proposed: the entraining plume (EP) model (e.g., Arakawa and Schubert, 1974) and the episodic mixing and buoyancy sorting (EMBS) model (e.g., Raymond and Blyth, 1986; Emanuel, 1991). An entraining plume model represents an individual convective element or cloud using an ascending subcloud air parcel that continuously entrains environmental air at a specified entrainment rate. The entrained environmental air is homogenized instantaneously and the plume is finally detrained into its neutral buoyancy level (NBL). In contrast, an EMBS model 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 vertically dis-placed to their individual NBLs where they are detrained into the environment. Both 4 conceptual models are highly simplified pictures of the transport associated with the in-dividual convective elements of real cumulus clouds. Conceptual models have been very difficult to test observationally, especially for shallow convection with its small length and time scales (Grinnell et al., 1996). Because of this, cumulus parameterizations are typically tested by examining their impact on forecasts or climatology, or occasionally by comparison with a few regional datasets. It seems clear that accurate representation of the statistical properties of convection must rely on a correspondingly accurate representation of the properties of the con-vective elements themselves (Emanuel, 1994). To understand and accurately quantify the cumulus effect on large-scale flow, one must recognize the penetrative nature of the cumulus convection, i.e., how eddies entrain air into, and detrain air from, a cumulus cloud (Bretherton, 1997). Indeed, it has been argued that an effective representation of cumulus convection must be built solidly on the physics and microphysics of cloud processes as deduced from observations, numerical cloud models, and theory (Emanuel, 1991). This dissertation is based on these arguments and is an attempt to obtain a better understanding and representation of shallow cumulus convection. 1.3 Observational studies of cumulus mixing Detailed aircraft measurement of the structure of ordinary cumulus clouds date back to the work of Stommel (1947), Malkus (1954), Warner (1955), Warner and Squires (1958). These observations have established that the properties of such clouds cannot be explained 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 departure from adiabatic is due to mixing through the cloud sides. The observations showed, however, that there is little systematic variation in cloud properties 5 across a cloud and that dry "holes" occur frequently in the bases of clouds, in contrast to laboratory plumes (e.g., Warner, 1955). Later observations showed that undilute subcloud air (USCA) occurs in all levels within cumulus clouds (e.g., Heymsfield et a l , 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 reproduce 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 content is much too high. On the other hand, Squires (1958) proposedthat the turbulent mixing of dry envi-ronmental 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 updrafts. Squires argued that vertical penetrative downdrafts rather than horizontal mixing was the prin-cipal mechanism responsible for cumulus cloud dilution. Squire's hypothesis received some support from the observations 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 non-precipitating cumuli were shown to be extremely inhomogeneous, with some samples corresponding to nearly undilute ascent cloud-base air and others composed of various mixtures of subcloud air with environmental air from near cloud-top. These results 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 6 result suggested 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 A C T , and mixed parcels subsequently descend around the edge ofthe advancing thermal core into a trailing wake region driven by a toroidal circulation. While supporting the idea of ascending cloud-top entrainment, 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). The hypothesis of two-sample mixing between undilute subcloud air and the environmental air near each observation level is primarily inferred from the observed character of the mixing-line structure for the conserved ther-modynamic variables of the cloud mixture. However, two-sample mixing is a sufficient, but not necessary, condition for the mixing-line structure. In fact, Taylor and Baker (1991) illustrate that a "buoyancy-sorting" mechanism may also explain the existence of such a nearly linear mixing-line structure. The buoyancy-sorting idea was first introduced by Telford (1975) and later refined by Raymond and Blyth (1986). The buoyancy-sorting hypothesis assumes that individual cloud mixtures tend to seek their individual NBL be-fore being detrained into the environment, i.e., positively buoyant cloud parcels tend to move upward while negatively buoyant parcels tend to move downward. With a care-fully designed conserved-variable diagram (Paluch diagram), Taylor and Baker (1991) were able to show that their observed cloud mixtures have a strong tendency to move following their individual buoyancy, as predicted by the buoyancy-sorting hypothesis. 7 To summarize, the following are the well-observed characteristics of cumulus clouds: 1) Dilution: cumulus clouds are highly dilute; the bulk of a cumulus cloud is composed mostly of entrained air. 2) Undilute core: nearly undilute subcloud air is observed throughout all levels of cumulus clouds, and cumulus cloud tops often approach or reach the NBL defined by undilute ascending subcloud air. 3) Downdrafts: downdrafts can be as strong as updrafts in non-precipitating shallow cumulus clouds. 4) Inhomogeneity: cumulus clouds are highly inhomogeneous with individual cloud mixtures undergoing vertical displacement via buoyancy-sorting. 5) Mixing line: mixtures sampled in a cumu-lus cloud tend to fall along a straight line connecting environmental air near the cloud base with air above or near the observational levels when plotted on a conserved variable diagram. While these observed cumulus properties provide important constraints on cumulus mixing, few cumulus observational studies of the detailed flow structure of the entraining eddies have been reported. The detailed mixing mechanisms are largely inferred rather than directly observed. The difficulties of making direct observation of the time-evolving 3 D flow structure and accurate measurements of entrainment into and detrainment from cumulus clouds have slowed progress in modelling these processes. Next, I discuss other tools for understanding cumulus mixing, which may complement these observational studies. 1.4 Laboratory investigation of cumulus mixing Laboratory studies of buoyant plumes and thermals, together with similarity assump-tions, are additional tools to understand the cumulus mixing. These experiments have provided insight into the conversion of buoyancy into motion in simple flows. In contrast to aircraft in-situ observations of real clouds the laboratory tank experiments often pro-vide detailed information on the flow structure and its evolution with time. Combined 8 with similarity theory, these experiments laid the foundation for early cumulus models. Morton et al. (1956) performed laboratory experiments on buoyant plumes and ther-mals in stratified environments. These experiments confirmed that a plume (thermal) does entrain laterally, quantified the rate of lateral entrainment and showed the plume overshooting its NBL, then spreading out around its NBL. Morton et al. assumed that plumes and thermals maintain geometric similarity and entrain air at a rate proportional to their vertical velocity as they rise. These experiments appeared to be consistent with the entraining plume model. Conceptual cumulus models based on bubbles/thermals were proposed by Scorer and Ludlam (1953) and Scorer and Ronne (1956). These and other laboratory experiments (e.g., Scorer, 1957; Woodward, 1959) showed that thermals 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 recent laboratory experiments have helped extend these earlier views of ther-mals. Sanchez et al. (1989) found that thermals starting from rest entrain ambient fluid at a much smaller rate compared with self-similar thermals. The transition from near-field (accelerating phase) to far-field (similarity phase) behavior occurs when the thermal has traveled a distance of about six diameters. Sanchez et al. proposed inefficient mixing during the near-field phase as an explanation for the presence of undilute cloud-base air at high cloud levels. 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 differs from the classic picture of mixing taking place along the advancing front of the thermal (Scorer, 1958) but is supported by the in-situ observations of Jonas (1990) and Stith (1992). Johari's finding indicates that a thermal's entrainment is pri-9 marily due to the larger-scale motion set up by the advancing vortical circulation rather than small-scale motions along the thermal's leading edge. He concluded that clouds may share both near- and far-field properties of laboratory thermals. Despite many insights provided by these laboratory experiments, cumulus clouds differ significantly from laboratory plumes or thermals. Specifically, they are characterized by: 1) phase change, which can produce either positive or negative buoyancy through condensation or mixing and evaporation; 2) environmental wind and wind-shear; 3) sub-cloud buoyancy fluctuations that are continuously generated from a turbulent boundary layer; 4) ambient vertically varying stratification; 5) Reynolds numbers that are far larger than those that can be produced in current tank experiments. Although several attempts have been made to include some of these factors (e.g., ambient stratification or evaporation) the similarity between the mixing mechanism of the laboratory plumes or thermals and that of the real cumulus clouds is generally not well established. 1.5 Numerical simulation of shallow cumulus clouds High resolution numerical simulation is the third tool that has been increasingly used to study both individual cumulus dynamics and cumulus ensemble statistics. These "large eddy simulations" (LES) explicitly calculate the largest eddies which are responsible for most of the turbulent transport of heat, moisture, and momentum while parameterizing the net effect of small eddies. There are several distinct strengths to the LES approach. For example, most of the laboratory limitations may be eliminated in a LES, while a LES can also provide detailed 3D time-evolving turbulent fields, which may be used to examine the time evolution of coherent structures and their contributions to turbulent transport. Moreover certain quantities, such as the perturbation pressure field, are available from a LES but not readily obtained from field observation or laboratory experiments. LES also permit controlled numerical experiments to isolate individual physical processes and 10 the use of passive synthetic tracers to systematically diagnose the cloud mixing behav-ior. These strengths make LES a very useful tool for investigating the cumulus mixing problem. In the case of cumulus convection, two quite different categories of LES may be distinguished (Emanuel, 1994). In the first, the atmosphere is assumed to possess a given quantity of convective available potential energy which is released over a limited period by one or a few convective clouds (called a single cloud simulation below). The single cloud simulation attempts to model the evolution and structure of the cloud. The formation of cloud depends on the perturbation or "trigger" that releases the stored energy. The most common initialization technique involves specifying a positive buoyant impulse/bubble in the subcloud layer. However, as noted by several modelers (e.g., Klaassen and Clark, 1985; Carpenter et a l , 1998a), the evolution of clouds initialized using this "bubble approach" may be sensitive to the shape, size and magnitude of the initial perturbation. Furthermore, as pointed out by Carpenter et al. (1998a) such an approach is not appropriate for clouds occurring in a low-buoyancy environment such as shallow cumulus convection. To generate more realistic clouds, Klaassen and Clark (1985) instead directly specified the surface heat flux with a Gaussian pattern. Using a 2D slab-symmetric model with interactive grid nesting and a maximum resolution of 5 m, Klaassen and Clark examined in detail the entrainment mechanism of a simulated small cumulus cloud. They found that small-scale horizontal density gradients were responsible for the onset of cloud-environment interface instabilities, while evaporative cooling played only a minor role in mixing. These cloud-top interface instabilities were further examined by Grabowski and Clark (1991) with improved numerics. Grabowski and Clark (1993) extended these simu-lation to 3D and obtained results similar to those of Klaassen and Clark, indicating that dynamical effects rather than evaporation are more significant in cumulus entrainment. 11 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. To resolve this deficiency, they applied a large surface heat flux with Gaussian functions at four points for the first simulation hour to generate motion on all resolvable scales throughout all vertical levels. They then reduced the heat flux to more realistic values and limited it to a single Gaussian function centered in the middle of the model domain. Carpenter et al. used this initialization and a nested mesh with an inner. uniform resolution of 50 m to study the entrainment and detrain-ment in simulated New Mexican cumulus congestus. These authors conducted a detailed parcel-trajectory and conserved-variable analysis of the modeled 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 of Blyth et al. (1988) discussed in Section 1.3. In the other category of cumulus simulation, the atmosphere is continually forced by specified large-scale conditions, and an ensemble of clouds are simulated over a sufficient period so that the clouds collectively come into equilibrium with the forcing (called an ensemble cloud simulation below). Unless multiple equilibria are possible, the ensemble cloud simulation can be expected to reach a statistical steady-state that is independent of initial conditions (Emanuel, 1994). Historically, the ensemble cloud simulation has been primarily used to quantify cumulus ensemble transport. In contrast, little attention has been given to the examination of individual clouds or convective elements embedded in a simulated cumulus field. However, this type of LES provides a natural way to generate individual cumulus clouds with fully developed boundary layer turbulence in a more realistic ambient environment, albeit with some sacrifice of model resolution compared with single cloud simulations. There is reason to think that individual shallow clouds may be realistically simulated in such models. For example, Siebesma and Jonker 12 (2000) show that the fractal dimension of the simulated individual cloud boundaries at resolutions as large as 100 m is in excellent agreement with observations. Neggers et al. (2003) retrieved cloud-size distributions from several LES datasets at 50 m resolution and found them to be consistent with satellite observations. These results indicate that with current computational power, LES have reached the stage where they can resolve both the cloud-ensemble statistics and some of the individual cloud dynamics. In this dissertation I will conduct a high resolution LES of shallow cumulus field and systematically investigate the transport and mixing dynamics associated with individual cumulus clouds of different sizes. 1.6 Research objectives and methodology The observed properties of cumulus clouds weigh strongly against the entraining plume model and tend to favor the EMBS model. However, many aspects of the EMBS model are not entirely plausible and have not been adequately and quantitatively tested (Bretherton, 1997). For instance, one assumption is that after a mixing event, each mix-ture either ascends or descends to its NBL without interacting with other mixtures. Since most mixtures are likely to form within large eddies that are part of the cloud, it seems unlikely that these mixtures simply move independently under their own buoyancy. Also, cumulus clouds have a life cycle and it is not clear how this life cycle may impact the buoyancy-sorting process. In particular, it is unclear whether negatively buoyant mix-tures are immediately rejected by a cumulus updraft or detrained only after the collapse of the entire updraft. The EMBS model has several free parameters; as will be discussed in Chapter 2, there is considerable uncertainty about how to choose these parameters, particularly when the model is used to represent an ensemble of clouds. In view of the preceding, this dissertation contains a systematic evaluation of the para-metric assumptions of the EMBS models using a relatively simple case, namely shallow 13 non-precipitating cumulus clouds over an ocean surface. My ultimate goal is to achieve a better understanding of cumulus mixing, the cumulus cloud life cycle and a better rep-resentation of shallow cumulus convection in large-scale models. I begin in Chapters 2 and 3 with a diagnostic study. Chapter 2 gives an overview of the EMBS model, model notations and current closure assumptions. Three uncertainties are discussed using the undisturbed trade-wind boundary layer from Phase III of the Barbados Oceanographic and Meteorological Experiment (BOMEX). In Chapter 3, a diagnostic approach is for-mulated to retrieve a major free parameter: the undilute eroding/mixing rate, given the observed convective equilibrium of the B O M E X case. A multi-mixing treatment of pos-itively buoyant mixtures is also introduced in this diagnosis. In this chapter, I further test the EMBS model sensitivity to different closure assumptions based on this diagnostic framework. I conclude Chapter 3 with a discussion of the parametric implications of the diagnostic results. Since the EMBS model provides only an approximation of the transport of real cu-mulus clouds, the diagnostic results in Chapter 3 are highly dependent on the validity of the EMBS model itself. To further investigate the several potentially important physi-cal aspects of convection revealed by the diagnostic study, and also to provide a direct evaluation to the EMBS hypotheses, I use a LES to perform a detailed analysis of six simulated shallow clouds and their life cycles in Chapters 4 and 5. Chapter 4 focuses on the mass and thermodynamic transport with special emphasis on the unsaturated part of cumulus convection, the role of buoyancy, and the role of cloud-size distribution in cumulus-ensemble transport. Chapter 5 focuses on the detailed picture of cumulus mix-ing dynamics as revealed in the simulated clouds. Chapter 6 uses the simulated cloud data to evaluate methods of cloud-top determination, with a special emphasis on the Lagrangian budget of the vertical momentum of the ascending cloud top. In summary, the questions to be addressed in this dissertation cover the following 14 EMBS constraints: How good are current assumptions of the free parameters in an EMBS model? How can we retrieve a major free parameter: the undilute eroding/mixing rate based on an observed convective equilibrium and large-scale forcing? How can the retrieved free parameter be physically understood? What are the implications for large-scale parameterization? Conceptual model of transport: Which conceptual model is more realistic in the representation of the transport associated with an individual shallow convective clouds/elements: an'entraining plume model or an EMBS model? Conceptual model of mixing dynamics: How can we use an LES to systematically diagnose cumulus mixing properties and improve our understanding of cumulus mixing? What is the coherent structure of cumulus mixing? What is the role of the buoyancy gradient and its associated vortical circulation in cumulus mixing? What is the conceptual picture of cumulus mixing dynamics revealed by the LES? Unsaturated convection, downdrafts and cloud evaporation: What is the role of the unsaturated component of cumulus convection? What is the sensitivity of an EMBS model to the treatment of detrainment? Buoyancy-sorting hypothesis: How well do numerically simulated cumuli conform to the buoyancy-sorting hypothesis? How does the cumulus life cycle affect the vertical transport of mass and buoyancy? Conceptual model of cumulus ensemble transport: What is the role of the cloud-size distribution in cumulus ensemble transport? How can we understand the cloud-size distribution typically observed in a cumulus field? What is a useful conceptual picture of cumulus ensemble transport? 15 The work presented in Chapters 2 and 3 also appears as Zhao and Austin (2003a). Chapters 4 and Chapter 5 have been submitted as Zhao and Austin (2003b,c). 16 Chapter 2 An Overview of EMBS Models 2.1 Introduction As discussed in Chapter 1, over the last three decades observations and models of cumulus clouds have raised doubts about the applicability of the entraining plume concept. Obser-vational studies in individual clouds from a wide range of field experiments suggest that the distribution of cloud properties at any given height is typically non-homogeneous, and that cloudy air often has a composition consistent with two-sample mixing between undilute sub-cloud air (USCA) and the environmental air from near or above the mixing level (e.g., Blyth et al., 1988; Paluch, 1979). Recently, a family of cloud models charac-terized by episodic mixing of USCA and environmental air has been designed to account for the inhomogeneity of cloud entrainment and mixing (e.g., Raymond and Blyth, 1986; Emanuel, 1991; Hu, 1997). A key feature of episodic cumulus mixing is that individual samples of subcloud air may travel long distances with relatively little mixing, only to mix vigorously with neighboring samples. This kind of advective-like and horizontally highly-inhomogeneous turbulent mixing is fundamentally nonlocal and has been called transilient turbulent mix-ing (Stull, 1984, 1993). The transilient nature of cumulus mixing can be clearly seen from 17 both aircraft measurements and high-resolution numerical simulations for which USCA parcels are typically present throughout the cloud layer (e.g., Paluch, 1979; Jensen et al., 1985; Raga et a l , 1990; Carpenter et a l , 1998c). However, many of the details of tran-silient mixing occurring in cumulus clouds are still a mystery. Emanuel (1994) has spec-ulated that the differences between cumulus mixing and more homogeneous laboratory plumes or thermals may be related to the phase changes of water substance that occur in cumulus clouds. It may also simply be due to the extremely high Reynolds number of atmospheric convection, which have not been achieved in either laboratory or numerical models (Raymond and Blyth, 1986). From a cumulus modelling and parameterization point of view, a simple hypothesis is clearly needed. Current EMBS models idealize a cloud as a core updraft of subcloud air, part of which mixes with environmental air at each cloud level. The mixing is envisioned to occur in eddies that create a spectrum of mixtures ranging from pure subcloud air to pure ambient air. Each mixture is then assumed to rise or fall to its NBL. The more dilute mixtures (mixtures containing more environmental air) are negatively buoyant and sink, while less dilute mixtures continue to rise without further mixing. The modelled vertical motion following an episodic mixing event is based on the buoyancy sorting idea which, as discussed in Section 1.3, recog-nizes that individual samples of cloud-environment mixture tend to move toward their NBL (e.g., Telford, 1975; Taylor and Baker, 1991). Although highly idealized, EMBS models account for in-cloud inhomogeneity, resolve the cloud-top liquid water paradox (see Section 1.3) and also accommodate features such as downdrafts that can not be represented explicitly in entraining plume models. However, EMBS models require the specification of three pieces of information for which there are few observational constraints: the mixing rate at which USCA is eroded into the environment (which I will refer.to as the "undilute eroding rate" or UER), the probability distribution of the mixing fraction (called the "mixing distribution" below) 18 following a mixing event, and an algorithm (called the "detrainment criterion" below) to both account for subsequent mixing between the mixed parcel and its environment, and to redistribute cloud-environment mixtures through the column following the detrainment of cloud air. In this chapter, I introduce the mass flux parameterization, the EMBS model notation and discuss the range of closure assumptions adopted in current EMBS models based on the B O M E X sounding. In Chapter 3, I formulate a diagnostic approach to retrieve one of the major free parameters and test the model's sensitivity to different assumptions and discuss the parametric assumptions based on the diagnostic results. 2.2 Mass flux parameterization For an atmosphere containing non-precipitating cumulus clouds with small cloud-fraction, the conservation equations for atmospheric entropy and total water, averaged over a region large enough to contain many clouds but small compared to the large-scale system of interest, can be written as (e.g., Arakawa and Schubert, 1974; Emanuel, 1994) ^ + V „ . V e i + J A + Q R _ + (2.1a) ot oz p oz p i n dqt , r dqt Mt dqt D. . ,n 1, . ot oz p oz p I II where Q\ is liquid water potential temperature and qt is total water mixing ratio; both are conserved variables during moist adiabatic motion. V # is the horizontal velocity vector, w is the vertical velocity, p is the base state mean density which varies only with height and QR represents the 9i tendency due to radiation (see Appendix A for a complete list of symbols and units). The RHS terms of (2.1) are tendencies due to cumulus subgrid scale transport. Note that I have neglected all unresolved processes which are not related to moist convection, and precipitation is also neglected throughout the thesis for this 19 shallow cumulus convection. The derivation of the subgrid scale cumulus transport terms on the right of (2.1) has assumed that the cloud fraction is small so that the average over the cloud free envi-ronment is approximately equal to the average of the whole area. Term I represents the effect of cloud-induced environmental compensating vertical motion, with Mf denot-ing the average vertical cloud mass flux. This mass flux can take on either positive or negative values for typical shallow cumulus layers, either heating and drying or cooling and moistening the environment. Term II represents lateral detrainment of the entropy and moisture difference between cloudy air (fyc, qt>c) and the environment; it is generally a moistening and cooling effect. Terms' i" and II are coupled by cumulus entrainment and mixing, one of the most important processes in non-precipitating cumulus dynamics and thermodynamics. To close (2.1), we need to further parameterize Mf, the detrained cloud properties 0/>c, qt}C and detrainment rate of cloud air-mass D. These quantities depend on the individual cloud model underlying terms I and II. 2.3 EMBS model notation In this section I introduce the procedure for calculating terms I and II of (2.1) on a vertically discretized grid for an EMBS model. Given a vertical level i between cloud-base (i = ICB) and cloud-top (i = ICT), I denote the upward flux of USCA at interface level i + 1/2 as Fl+l/2, and write the change of F over the depth of level i as Ul = pi-i/2 _ pi+i/2^ w h e r e jji i g t h e e r o d e d U S C A mass flux at discretized level i with units of kg m~ 2 s _ 1 . The UER at level i, which is defined as the vertical gradient of the undilute mass flux, is given by El = —(dF/dz)1 ~ Ul/Az, where Az is the layer thickness. El has units of k g m ~ 3 s - 1 and is resolution independent. Given these definitions, Fb, the undilute cloud-base mass flux, is Fb = J2i=JcB Ul-At each level i, N mixtures are generated between USCA with values of entropy and 20 total water (#",<?") and environmental air wi th Assuming that mixing is strictly linear, the conserved thermodynamic variables for individual mixture j are given by: el'j = o-j6i + (l-aj)0? (2.2a) ql'j = **4+(l-<ritet (2.2b) where a* is the fraction of environment air i n the jth mixture in the range 0 < o~j < 1. Figure 2.1: a, b, c): Vertical (8i, qt, 6V) profiles of mixtures generated by the episodic mixing hypothesis. The solid lines are the B O M E X sounding. Black circles represent ascending U S C A . A t each level, a portion mixes with environmental air and generate a spectrum of mixtures (dots). Red dots are positively buoyant, saturated mixtures; green dots are negatively buoyant, saturated mixtures and blue dots are negatively buoyant, unsaturated mixtures. Numbered labels are described in the text. Dashed line in c) is the moist adiabat. d) The mixture spectrum generated at the cloud top level (2000 m). a is the fraction of environmental air in mixtures. Symbols shown in d) are described in the text. 21 Figure 2.1 gives an example of (9\'3, q\'3) mixtures for a representative trade cumulus sounding of 9i, qt and 9V (BOMEX phase III, see Section 3.3 for additional detail). The sounding has the typical layer structure for shallow cumulus convection: a well-mixed sub-cloud layer between approximately 0 - 500 m (not shown), a conditionally unstable cloud layer between 500 - 1500 m, and an overlying inversion layer (1500 - 2000 m). The cloud-base height (600 m) and cloud-top height (2000 m) are determined respectively by the lifting condensation level (LCL) and the NBL of the-USCA at the top of the surface layer at 120 m (not shown). The vertical distance between levels in Figure 2.1 is 40 meters. The dots at each level represent N = 20 mixtures generated using (2.2), ranging linearly from purely USCA (0% environment air) to highly dilute (95% environment) air. To provide a specific focus for discussion, four numbered boxes in Figure 2.1 show the coordinates for three negatively buoyant mixtures with liquid water contents ranging from unsaturated (mixture 1) to saturated with small liquid water content (0.22 g k g - 1 , mixture 2) to saturated with large liquid water content (1.87gkg _ 1, mixture 3). Mixture 4 at the 1000 m level is positively buoyant and saturated. Before I discuss the 9V values of Figure 2.1c, I consider Figure 2.Id, which shows 9\3 and 9lv'J for a single mixing spectrum generated at cloud top. The panel shows that while the mixtures' conservative variables 9\3 and q\'3 (q\,J not shown) are strictly linear during the mixing, the mixtures' buoyancy variable varies non-linearly with the mixing fraction (cr-7). The panel also shows that all mixtures at the cloud top level are negatively buoyant due to the evaporation of liquid water during the mixing process. Since the environmental air is warm and dry, only a small proportion of environmental air is needed for an USCA-environment mixture to achieve significant negative buoyancy (e.g. mixture 3). The blue and green dots in Figure 2.1c are negatively buoyant mixtures generated by mixing between positively buoyant USCA and the environment throughout the cloud 22 levels. Two general features can be seen from the panel. The first is that only a very small fraction of saturated mixtures are negatively buoyant below the inversion, while a much larger fraction are negatively buoyant above the inversion. This is consistent with the finding in LES studies that few saturated downdrafts are present below the inversion, but a significant number are observed above the inversion (de Roode and Bretherton, 2003). Secondly, about half of the cloud-environment mixtures are unsaturated and neg-atively buoyant, if we assume a uniform mixing distribution. The fraction of unsaturated mixtures should be increased beyond this, given that, in non-precipitating cumuli, all saturated mixtures have to be further mixed and eventually evaporated. Since unsatu-rated mixtures are invisible and have thermodynamic properties close to those of their environment, they can be easily overlooked in either observations or analysis of numerical simulations. Compared with the saturated part of the cumulus cloud little quantification has been done on this part of cumulus convection, which may also play an important role in vertical mass transport. This will be extensively discussed in Chapter 4. The points of Figure 2.1 span all possible cloud environment mixtures with 0 < aJ < 1. Left undetermined is the probability distribution of mixing fraction P(o-j) for the mixture spectrum generated at each model level. To illustrate my use of the mixing distribution in an EMBS model, consider U\ the eroded USCA mass flux at level i. The model first divides Ul into undilute sub-parcels according to an assumed mixing distribution. Each sub-parcel is labeled with index j , and will mix with environment air, with environment fraction given by oK The resulting eroded USCA mass flux for sub-parcel j is therefore: where P(aj) is the normalized distribution of the eroded USCA mass flux in the spectrum of mixtures. The total mass flux of the mixture j is: (2.3) u (2.4) m' 1 - oi 23 Given the thermodynamic properties and the individual mixture's m1'3 , we can cal-culate its detrainment level and its associated mass flux contribution to levels between the mixing level i and its detrainment level. In the next section I discuss the choice of detrainment level, while in Section 2.5 I review four choices of P(crJ). 2.4 Detrainment criterion Given 9\3, q\J and a pressure level, we can calculate all other thermodynamic properties of a mixture, as illustrated in Figure 2.1c for 9V. To account for the evaporation effect to a mixture's buoyancy we require an additional variable, the liquid water virtual potential temperature 9\3. This is the 9V obtained by reversible adiabatic evaporation of a mixture liquid water, and is nearly conserved for adiabatic motion in shallow clouds (Emanuel, 1994). In the buoyancy sorting stage of an EMBS model, 9V or 9Lv is compared against the environment to determine the detrainment level of each mixture. Raymond and Blyth (1986) (RB86 below) for example, assume that mixed parcels exit the cloud at their NBL, where the parcel and environment have equal 9V. A problem with this detrainment, as pointed out by Taylor and Baker (1991), is that cloudy air detrained at its NBL will become negatively buoyant upon mixing with its sub-saturated environment. Emanuel (1991) (E91 below) addresses this by detraining mixtures at their unsaturated neutral buoyancy level (UNBL), defined as the level where the parcel and environment have equal 9iv. This is equivalent to defining a cloud-environment boundary such that all evaporation of liquid water occurs within cloud. Such a cloud boundary is larger than the actual visible cloud and includes a near-cloud region which is free of liquid water but part of the cloud-induced convective circulation. Figure 2.2 shows the very different consequences of these two detrainment assumptions for shallow cumulus clouds. Panels a-c, and d-f of Figure 2.2 show the final location of each of the mixtures in Figure 2.1, given.the detrainment assumptions of RB86 and 24 Figure 2.2: ft, qt and ft, of mixtures from Figure 2.1 following buoyancy sorting. The left panels (a-c) represent the N B L (RB86) detrainment assumption while the right panels (d-f) represent U N B L (E91) detrainment. Dot colors and labels are as in Figure 2.1. 25 E91 respectively. As required by buoyancy sorting, all mixtures in panels 2.2c and 2.2f collapse (with small deviations due to finite vertical resolution) to the environmental 9V. Labelled mixtures 1-4 correspond to those of Figure 2.1, transported to their new equilibrium levels. A comparison of panels 2.2c and 2.2f reveals that the detrainment criteria of both RB86 and E91 require unsaturated mixture 1 to descend dry adiabatically to roughly 1800 m and detrain. Similarly mixture 2 with little liquid water descends first moist adiabatically for a short distance (parallel to the moist adiabat shown in Figure 2.1c), then follows a dry adiabat (vertically) to detrain at roughly 1500 m under both detrainment assumptions. In contrast mixture 3, with its larger liquid water content, is detrained at a NBL of approximately 1600 m under the detrainment criterion of RB86, where its 9V matches that of the environment. Under the UNBL detrainment assumption of E91, the mixture evaporates its liquid water and descends further to « 1200 m, where the environment and mixture 9\v values are equal. Finally, positively buoyant mixture 4 is transported well into the inversion under the detrainment assumption of RB86, but evaporates its liquid water and descends to 700 m using the UNBL criterion of E91. Under the NBL detrainment criterion, mixtures 3 and 4, which are comparatively undilute following initial mixing, preserve large amounts of liquid water when reaching their inversion-level NBL. Compared to the drier mixtures 1 and 2, mixtures 3 and 4 have much larger 9\ and qt differences relative to the environment. Figures 2a and 2b show this divergence of 9i and qt values above the inversion. However, liquid water-containing mixtures such as 3 and 4 presumably do not remain neutrally buoyant when further mixing and evaporation occurs. Evaporative cooling may drive them downward; the UNBL detrainment criterion accounts for this effect by detraining these mixtures at lower levels (Figure 2.2d-f). For mixture 3, UNBL detrainment occurs 400 m below its initial NBL. This further descent seeks the level where cloudy air ultimately detrains and is therefore physically more plausible; I adopt the UNBL detrainment criterion for 26 negatively buoyant mixtures in the diagnostic study of Chapter 3. As Figure 2.2 shows, the two detrainment criteria differ sharply in their treatment of positively buoyant mixtures. Given the UNBL detrainment criterion and an unsaturated environment for which 8\v — 8V increases monotonically with height, all non-precipitating cloud mixtures at level i satisfy 8\3 < 8\v and will descend. In particular, positively buoyant mixtures with their larger liquid water contents often detrain well below the mixing level due to strong evaporative cooling. These same positively buoyant mixtures, given the NBL detrainment criterion, produce updrafts rather than downdrafts, and account for significant transport into the inversion layer. Figure 2.3: An illustration of the treatment of positively buoyant mixtures under NBL, UNBL and multi-mixing. Solid line: the environmental 8V sounding. Open circles: the ascent of USCA; filled circles: a mixture spectrum generated at 1400 m. Solid arrows: the moist adiabatic ascent to mixtures' NBL (RB86); dashed arrow: the adiabatic descent to mixtures' UNBL (E91); thick arrows: the multi-mixing treatment of positively buoyant mixtures. Figure 2.3, which is similar to Figure 2.1c but retains only a single mixture spec-trum at 1400 m, shows schematically the difference between the treatment of positively buoyant mixtures under NBL, UNBL and multi-mixing detrainment described below and in Section 3.2. The initially generated mixing spectrum (at the 1400 m level) is sepa-27 rated into mixtures that are neutrally or negatively buoyant (black circles to the left of the environmental 9V sounding) or positively buoyant (with 9^3 > 9V). Notice that the relatively undilute positively buoyant mixtures either ascend moist-adiabatically to near cloud top and detrain under NBL detrainment or descend adiabatically to near cloud-base and detrain under UNBL detrainment (the UNBL levels are not necessarily the intersection level of the mixtures moist adiabat and the environmental sounding). Given fixed amount of USCA, the convective upward mass flux and its associated 9\ and qt transport are significantly reduced by detrainment at the UNBL, compared with NBL detrainment. In real cumulus clouds, positively buoyant cloudy mixtures must undergo further mix-ing. These mixtures should be diluted and detrained somewhere between the NBL and the UNBL of the initial mixture, as illustrated in Figure 2.3. I introduce a simple form of multi-mixing in Section 3.2 and Appendix B, in which positively buoyant mixtures are mixed on their way toward their NBL. For simplicity, after each mixing event the positively buoyant mixtures are grouped and averaged into a single updraft parcel, con-tinue ascending to the next higher level and pursue further mixing until further mixing generates only neutrally and/or negatively buoyant mixtures. During this process only neutrally and negatively buoyant mixtures are detrained. The detrainment levels are chosen to be the mixtures' UNBL, i.e. this detrainment criterion is UNBL detrainment limited to neutrally and negatively buoyant mixtures. The addition of multi-mixing ac-counts for the further mixing of the active cloudy parcels. As a result, the total amount of entrained environmental air is increased, and model-produced mixtures are not solely mixtures between USCA and the environment. I use this modified detrainment criterion in the diagnosis of the UER in Chapter 3. 28 2.5 Mixing distributions There is little observational or laboratory evidence to constrain the mixing distribution P(aj) required by (2.3). In Figure 2.4 I present four distributions that have been dis-cussed in the context of EMBS parameterizations. Shown in each panel are distributions of eroded USCA (u), environmental air (e), and their sum (u + e) as a function of environ-mental fraction a. The curves are normalized so that the total u in the spectrum is unity. Distributions PI and P2 make two different assumptions about uniform mixing. E91 (PI) P1 1.2 E D ) - 0 . 8 3 u+e (total - 1 - u (undiluted) e (environment) sum(u) =1.00 sum(u+e) =3.14 0.2 0.4 0.6 0.8 1 Fraction of environment air (a) P 3 1.2 0.8 - * - u+e (total) —•— u (undiluted) — — e (environment) sum(u) =1.00 sum(u+e) =2.00 0.2 0.4 0.6 0.8 1 Fraction of environment air (0) 0.2 0.4 0.6 0.8 Fraction of environment air (a) P 4 1.2 E u> " 0 . 8 x i 0.6 CO ^ 0 . 4 N E 0.2 - * - u+e (total) —1— u (undiluted) — — e (environment) sum(u) =1.00 sum(u+e) =2.00 0.2 0.4 0.6 0.8 1 Fraction of environment air (a) Figure 2.4: Mixing distributions as a function of environmental air fraction a for a selection of EMBS models. PI is from Emanuel (1991) (eq. 7); P2 is from Raymond and Blyth (1986); P3 is from Kain and Fritsch (1990); P4 is determined following Cohen (2000). 29 assumes a fixed amount of USCA in each mixture, with more environmental air entrained into mixtures with larger a3. RB86 (P2) assume a uniform probability of formation of mixing fractions, so that the resulting total mass flux (environment plus USCA) of each mixture is constant, and parcels with a higher environmental mass fraction a3 contain less sub-cloud air. Kain and Fritsch (1990) assume that cumulus turbulence tends to mix USCA and en-vironment air in equal portions, and apply a Gaussian distribution centered at a — 50% with standard deviation given as 0.2 (hereafter called P3). Recently, Cohen (2000) sug-gested an anti-Gaussian distribution of cloud mixtures based on his numerical simulation of cumulonimbus clouds. Distribution P4 is an analytic function which approximates the shape of Cohen's mixture distribution. The four normalized distributions of USCA (given by the thick lines in Figure'2.4) are: Pl(a3) = \ . (2.5a) (1 - a3) P2(a3) = ^ a ) (2.5b) P3(a3) = (1 - a3)G(a3) (2.5c) PA{a3) = ) (2.5d) £ ( 1 / V ) 3 = 1 where G(a3) is a Gaussian distribution with mean=0.5 and standard deviation=0.2. Note from Figure 2.4 that the total amount of entrained environmental air is larger in distribution PI than in distributions P2-P4. In Chapter 3, I use N = 81 with a3 bins equally spaced within the range 0.1 < a3 < 0.9. The EMBS model sensitivity to each of these distributions will also be discussed in Chapter 3. 30 2.6 Undilute eroding rate A major unresolved problem of the EMBS models is the determination of the vertical profile of the UER (e.g Emanuel and Zivkovic-Rothman, 1999, below EZ99). The dif-ficulty is magnified by the fact that a cumulus parameterization tries to represent an ensemble of cumulus clouds in which typically more than one type of cloud coexists (e.g., Plank, 1969). In principle, the vertical profile of UER may also be influenced by the a) 1.5 Figure 2.5: Undilute eroding rate E normalized by EQ for B O M E X cumuli (see Figure 2.1) following RB86 (dashed line) and EZ99 (solid line). cloud type/size/height distribution in the cumulus field. In RB86, the U E R is taken to be constant with height, an assumption which distributes the undilute cloud-base mass flux equally into each of the model levels. However, the objective of RB86 is to simulate a single cloud. EZ99 motivated by Bretherton and Smolarkiewicz (1989)'s modelling work, suggest a formulation based on the general idea that entrainment and detrainment rates are functions of the vertical gradients of buoyancy in clouds. Specifically, E\ normalized 31 by the cloud layer averaged UER E0, is: \ EL EQ H ~Az ( | A B | + AAz){ (2.6) ICT \j=ICB {\AB\+AAz)j J where H is the cloud layer depth, EQ = (l/H) ^ i = I C B ElAz, A is a constant mixing parameter (0.006 m - 1 ) , B is the buoyancy of the ascending USCA and AB is the change in undilute buoyancy over an height interval Az. Note that (H EQ) is the undilute cloud-base mass flux, Fb, and (AzE1) is the eroded USCA mass flux U\ Figure 2.5 shows E/E0 for the B O M E X soundings based on both RB86 and EZ99. It is not clear whether or not these assumptions are adequate in the representation of shallow cumulus clouds. However, it seems that the determination of a cloud ensem-ble's UER should also takes into account the cloud size distribution in a cumulus field. In Chapter 3, I will diagnose the UER based on the EMBS models and an observed convective equilibrium, i.e. the B O M E X case. 2.7 Summary Episodic mixing and buoyancy-sorting (EMBS) models have been proposed as a phys-ically more realistic alternative to entraining plume models of cumulus convection. In this chapter I outline the EMBS approach for a representative trade-wind boundary layer (BOMEX) cumuli and introduce the EMBS model notation and the three free pa-rameters, i.e. the undilute eroding rate (UER), detrainment criterion and probability distribution of mixing fraction. Current assumptions for these free parameters have been summarized and illustrated for the B O M E X sounding. Considerable uncertainties exist in the specification of these parameters. In particular the different assumptions of de-trainment criterion and their potential impact on the EMBS model have been discussed. A multi-mixing treatment of positively buoyant mixtures has also been briefly described. 32 C h a p t e r 3 A D i a g n o s t i c S t u d y o f t h e E M B S M o d e l 3.1 Introduction As discussed in Chapter 2, given P(o^) and a detrainment criterion, the final specifi-cation required by an- EMBS model is the UER as a function of vertical level. From a diagnostic point of view, given P(aj) and a detrainment criterion it is possible to derive a vertical profile of UER based on an observed equilibrium state and large-scale forcings for shallow cumulus convection. In this chapter, I adapt an EMBS model to study con-vection in a field of shallow non-precipitating cumulus clouds. I first diagnose the UER and compare it with current assumptions. Based on this framework I then examine the transport predicted by the EMBS model for common choices for the mixing distribution and the detrainment criterion. In Section 3.2 I derive a diagnostic expression for the UER, given an algorithm for detraining cloud-environment mixtures and an atmosphere in equilibrium with large scale forcings for heat and moisture. In Section 3.3 I briefly describe the B O M E X case and data. I present the diagnostic results in Section 3.4. In Section "3,5 I offer an interpretation for the form of the resulting U E R in terms of the 33 individual transport contributions of clouds with a range of sizes. In Section 3.6, I test the sensitivity of the retrieved UER to different assumptions of mixing distribution, and the impact of the detrainment criterion on the EMBS model. Finally, in Section 3.7 I conclude the chapter with a discussion of the implications of the diagnostic results for cumulus parameterization in large-scale model. 3.2 Diagnostic equations for the U E R In this section I develop a discretized version of Eq. (2.1) which, for non-precipitating clouds, is linear in the UER and can therefore be solved given an equilibrium atmosphere. The discretization includes a treatment of simplified multi-mixing for positively buoyant mixtures as outlined is Section 2.4. At any level i between cloud-base and cloud-top (ICB <i< ICT), the total eroded USCA mass flux is,represented by Ul. The mass flux of an individual mixture j involved in mixing with part of the eroded Ul is determined by Eq. (2.4), which I rewrite as: 1 - a* Ul = XJUl (3.1) If mhj is neutrally or negatively buoyant with respect to its environment (#*J < 9lv), I detrain it by finding its detrainment level k (k < i) based on 9\3 = 9* (i.e. the UNBL). Knowing the detrainment level k, I can find the mass flux contribution of the mixture m M to any model level n f 0 n > i -rtfj = -XjUi k <n<i (3.2) 0 n < k 0 n = k = i MFX 34 The detrainment contributions of m M to the environmental ft, qt at any level n are , '9]'3 - 6,)rriJ = (9)'3 - eftX'U* n = k DTLT™ = { (3.3) 0 n^k , (qiJ - q?)rriJ = U'3 - qftX'U* n = k DQT1'1'3 = { (3.4) 0 n ^ k I sum over the set {j} of all mixtures that are negatively or neutrally buoyant to obtain the total mass flux and detrainment contributions at level n caused by the USCA mixing event at level i. MFXn'1'3 = MF^U1 (3.5a) {J} DTLn'%'° = DT'W (3.5b) {j} DQT'*'3 = DQn'lUl (3.5c) {j} MFn'\ DTn'\ DQn'1 represent respectively the [/^-normalized total mass flux and de-trained ft and qt tendencies at level n due to the mixing of USCA at level i. Both the mixing-induced mass flux and the detrained tendencies are linear functions of U\ The treatment of the set {/} of all mixtures that are positively buoyant (#*J > 9ZV), is described in the Appendix B. To summarize, the average ft and qt of set {j1} is calculated. The mixtures are grouped into a single parcel and carried to the next higher level. New mixtures are generated; positively buoyant mixtures continue to merge, ascend and mix while other mixtures are detrained. This process continues until new mixtures are all neutrally or negatively buoyant mixtures and subsequently are detrained. Multi-mixing completes the mixing and detrainment of a single convective element Ul which is eroded at level i. Without multi-mixing, the detrainment relationships of (3.5) can be reduced to those of either RB86 or E91, given appropriate choices for the detrainment level k. This algorithm is carried out for all levels i (ICB < i < ICT) and produces three two-dimensional arrays MFn'\ DTn'\ DQn'\ Thus, assuming an equilibrium atmosphere, 35 a discretized version of (2.1) may be formulated as ICT ICT z £ ] E u % + E M F n ' l u l D T n ' l u l = (3-6a) PAzS U n + 1 ^WB J P / X Z i=ICB / A x n [" ICT ICT 1 7CT h £ E ^ + E M p n ' l u l + M I E ^ n ' l f / l = (3-6b) V ^ 2 / [_i=n+l l = / C B J P L A Z i=ICB where <f>j> and $nq represent the sum of all large-scale forcings of environmental 6i and qt respectively. Inserting (3.5) into (3.6), we find two sets of linear equations for U. Given the large-scale forcings (<% and 0") and mixing distribution P{crj), (3.6a) and (3.6b) may be solved either independently or as a coupled system. Having obtained the U profile the UER profile is simply E = U/Az. I show in Appendix C that, for B O M E X forcings, independent solutions of (3.6a) and (3.6b) give consistent solutions of the U E R profile. In the diagnosis of Section 3.4 I simply retrieve the UER by (3.6a), followed by smoothing using a 3-point running mean. The smoothed UER is then substituted into (3.6a) and (3.6b) to verify that the residual tendencies are small compared to the uncertainties of the data. I discuss the details of the inversion technique and the consistency and stability of the solutions in Appendix C. Figure 3.1 summarizes episodic mixing and buoyancy sorting as calculated using the framework described above and in Appendix B. The left panel of Figure 3.1 illustrates the ascent of USCA from cloud-base to cloud-top (the cloud-base height and cloud-top height are determined respectively by the L C L and the NBL of the USCA at the upper surface-layer). The small cutout in the left panel shows a single "convective element" of USCA which is eroded at vertical level i. The right panel of Figure 3.1 shows initial and subsequent mixing events between this eroded USCA element Ul and the environment. I distinguish between the remain-ing positively buoyant USCA and positively buoyant mixtures, which experience more efficient dilution and therefore can only ascend a short distance from their mixing level depending on their average buoyancy. I specify the same mixing distribution P(cr) for 36 both the initial and subsequent mixing events, and test the sensitivity of the model to the mixing distribution in Section 3.6 below. The positively buoyant mixtures are al-ways homogenized and undergo further ascent and mixing until all new mixtures are neutrally or negatively buoyant and are detrained at their NBL (unsaturated mixtures) or UNBL (saturated mixtures). Applying the UNBL detrainment criterion enforces the complete evaporation of saturated mixtures before their final detrainment. The multi-mixing treatment of positively buoyant mixtures accounts for the further mixing and dilution of positively buoyant cloudy air, permits the existence of upward moving mix-tures and eliminates the deep penetrative downdrafts otherwise caused by detraining positively buoyant mixtures at their UNBL. Episodic Mixing Buoyancy Sorting Detrainment jc loudbase (ICB) undilute subcloud air IIIUIU-II multi mixing multi-mixing mixing distribution P(aJ) O NBL O U N B L B positively buoyant saturated mixtures H neutrally and negatively buoyant saturated mixtures • neutrally and negatively buoyant unsaturated mixtures Figure 3.1: An illustration of the episodic mixing and buoyancy sorting model. Left panel: the episodic mixing process between USCA (black) and environment (white). The small cutout on the left panel illustrate the mixing with the environment of a single convective element Ul at level i. Right panel: the buoyancy sorting process, showing initial and secondary mixing given mixing distribution P{o~j), with detrainment of neutrally and negatively buoyant mixtures at their NBL (unsaturated) or UNBL (saturated) mixtures. The three dots in the upper right hand corner indicate further mixing events, which continue until there are no positively buoyant mixtures. 37 3.3 B O M E X convective equilibrium I use vertical sounding and forcing profiles taken from the G E W E X Cloud Systems Study B O M E X inter-comparison, which provides a prescribed large-scale subsidence rate and rates for radiative cooling and subcloud layer advective drying (Siebesma et a l , 2003). The forcing data is consistent with B O M E X Phase III (22 to 26 June, 1969) which pro-vides a 4 day period during which the non-precipitating trade cumulus boundary layer was in steady-state, with the large-scale advective and radiative forcings in balance with the cloud response. Although idealized, the B O M E X case study insures a consistent set of large-scale forcings with which an equilibrium state for a shallow cumulus cloud boundary layer is established in large eddy simulations (LES). The model-produced equi-i QL_I 1 . . . , 01 ' ' — 1 — -2 0 2 4 , 6 8 -8 -6 -4 -2 0 2 de/dt (Kday - 1) dq/dt (gkg"T) Figure 3.2: B O M E X equilibrium state, a) liquid water potential temperature #/. b) total water mixing ratio qt. c) solid line: 9i tendency due to subsidence: —w^; dashed line: radiative cooling: —QR; d) solid line: qt tendency due to subsidence: —w^, dashed line: large scale advective drying: — V# • \7qt-38 librium vertical profiles of temperature and moisture closely match the observed B O M E X atmosphere (Holland and Rasmusson, 1973; Nitta and Esbensen, 1974b; Siebesma and Cuijpers, 1995; Siebesma et al., 2003). Figure 3.2a,b show the horizontally averaged vertical profiles of the B O M E X equi-librium LES output for ft and qt (Siebesma et a l , 2003). The figure shows the typical layer structure for shallow cumulus convection: a well-mixed sub-cloud layer between approximately 0 - 500 m, a conditionally unstable cloud layer between 500 - 1500 m, and an overlying inversion layer (1500 - 2000 m). Figure 3.2c,d shows the large-scale forcings of ft and qt calculated based on the specified large-scale subsidence, radiative cooling, and sub-cloud layer advection. Given the observed equilibrium state, we expect that the cloud response will balance the net large-scale forcing throughout the cloud layer and the inversion layer. As indicated in Figure 3.2c,d the cloud ensemble must cool and moisten the inversion layer and upper cloud layer and slightly heat and moisten the lower cloud layer. 3.4 U E R retrieval In this section I apply the UER diagnostic equation of (3.6) to the B O M E X convective equilibrium shown in Figure 3.2 . For this retrieval, I follow E91 and assume mixing distribution PI, so that at each model level Ul is equally divided and then mixed with different proportions of environment air. Figure 3.3 shows the diagnosed results. Figure 3.3a presents the U E R retrieved by solving (3.6a) for the forcings of Figure 3.2c. Within the cloud layer, the UER decreases exponentially with increasing height. Near the base of the inversion, it increases be-fore decreasing almost linearly throughout the inversion layer. As shown in Figure 3.3c (dotted line), with the diagnosed UER the EMBS model produces a cumulus-induced ft tendency that balances the large-scale ft tendency. As a consistency check the retrieved 39 UER is put back .into ,the EMBS model; Figure 3.3d (dotted line) shows the resulting net qt tendency. Exact balance for the qt tendency is not expected, given uncertainties in the specified sounding and forcing data; however, the net qt tendency in Figure 3.3d is within the range of the residual LES tendencies (personal communication, P. Siebesma http://www.knmi.nl/~siebesma/gcss/tend.3d.html). In Appendix C.1 I present ad-ditional results showing the consistency of the 9i, qt retrievals. d0 |/dt(Kday"1) dq/dt (gkg~1day~1) Figure 3.3: Diagnosed fluxes and tendencies from the EMBS model, a) U E R retrieved by solving (3.6a). b) solid line: upward mass flux due to ascending USCA; dashed line: downward mass flux due to descent of initially negatively buoyant mixtures; dotted line: net mass flux produced by initially positively buoyant mixtures through multi-mixing; dot-dashed line: the total net mass flux produced by the EMBS model; crosses: the cloud mass flux obtained from an LES case study, c) solid line: total large-scale forcing of 8\\ dot-dashed line: induced 9i tendency due to cloud net mass flux; dashed line: cloud lateral detrained tendency; dotted line: the total net tendency, d) as in c) but for qt. Figure 3.3b shows the mass flux calculated using the diagnosed UER, partitioned 40 into undilute ascending updrafts (solid line), downdrafts induced by initially negatively buoyant mixtures (dashed-line) and downdrafts induced by initially positively buoyant mixtures through multi-mixing (dotted line). The cloud net mass flux is given by the dot-dashed line, while the line marked by crosses shows the net cloud mass flux from the liquid water-containing grid cells of the B O M E X LES (Siebesma et al., 2003). The net mass flux from the LES is significantly larger than the EMBS-diagnosed profile, which is downward within the inversion layer and upward below the inversion. For the EMBS model, the vertical integral of the 9V tendency induced by the net cloud mass flux is close to zero, consistent with the fact that non-precipitating clouds only transport 9V and do not act as net sources or sinks of 9V. This disparity between the LES and EMBS-produced mass flux is due primarily to different definitions of the cumulus cloud boundary. Recall that the UNBL detrainment criterion, which defines the cloud boundary to include a surrounding region of unsaturated air, incorporates cloud evaporation and the associated unsaturated convective downdrafts into the net cloud mass flux. In the LES, in contrast, clouds are defined as saturated cloudy parcels which have liquid water. Indeed, in Figure 3.3c, we see little lateral detrained 9i tendency (dashed line) while the detrained water vapor tendency is still significant in Figure 3.3d (dashed line), as parcels reach their UNBL with a small 9\ deficit but with larger excess qt. (See Figure 2.2d and 2e, where detrained qt differences are larger than 9i differences). An additional feature of the partitioned EMBS mass flux apparent in Figure 3.3b is the counterintuitive fact that, in this EMBS model, positively buoyant mixtures produce a net downward mass flux (dotted line). This downward motion is due to the evaporation of liquid water in these mixtures via two processes, as illustrated in Figure 2.3. One is explicit mixing with environmental air and the generation of new mixture spectra; the other is the moist adiabatic descent of saturated, negatively buoyant mixtures prior to their detrainment. Both occur before the final detrainment of cloud-processed air to the 41 environment at the equilibrium level (UNBL). As a result, all liquid-water evaporation is realized within clouds and the absorbed latent heat cools the cloud air itself, rather than directly cooling the environment. This evaporative cooling drives cloud air parcels down-ward. Thus, the cloud evaporative cooling of the environment is represented through the net downward mass flux, rather than through the direct local detrainment of liquid water. In reality, much of this evaporation occurs at the dissipation stage and is characterized by a significant number of unsaturated downdrafts. Given this life-cycle perspective, the EMBS models (with UNBL detrainment) can be understood as a simple representa-tion of the two stages of a cumulus life cycle: growth, incorporated through the ascent of USCA, and dissipation, represented by the mixing, evaporation, and detrainment of both the USCA and the initially positively buoyant mixtures. 3.5 Linking the UER to cloud size distribution 3.5.1 Comparing retrieved U E R to cloud size distribution Figure 3.4 compare the retrieved vertical profile of U E R with their parametric assump-tions in RB86 and EZ99. The EZ99 parameterization produces increased erosion of USCA due to the sudden decrease of cloud buoyancy in the vicinity of the inversion base (1400 m in Figure 3.4a). The EMBS-retrieved UER displays a corresponding E increase in Figure 3.4b. In contrast, in the cloud layer the retrieved U E R decreases exponentially from cloud-base (600 m) to just below the inversion base. Given that the equilibrium soundings of Figure 3.2a,b are the cumulative product of a size-distributed cloud population, we compare the retrieved U E R in Figure 3.4b with a shallow cumulus cloud size distribution calculated from LES results by Neggers et al. (2003). This distribution has a power-law form that is typical of several studies of shallow cumulus size distributions (e.g., Cahalan and Joseph, 1989; Kuo et a l , 1993; Benner and 42 Figure 3.4: a) Undilute eroding rate E normalized by EQ (see Eq. 2.6) for B O M E X cumuli following RB86 (dashed line) and EZ99 (solid line), b) A comparison of the diagnosed UER (solid line) and a cloud size/height distribution from shallow cumuli (dashed line). Curry, 1998). This size/height distribution for clouds with diameters less than « 1 km is given by N(h) = a(h - hb)b (3.7) where, b = —1.7, hb is cloud-base height (600 m for B O M E X ) , h is cloud-top height, h—hb is cloud depth or cloud size, Here, a is a free scaling parameter and I have transformed (3.7) from cloud horizontal size to cloud height assuming an aspect ratio of 1 (see, e.g., Betts, 1973). Figure 3.4b compares the retrieved UER and the normalized cloud height/size dis-tribution for small to medium size clouds (thicknesses < 1 km). Both profiles decrease sharply in the first 200 m above cloud-base and then decrease steadily with increasing cloud height. In Section 3.5.2, I examine the d6i/dt and dqt/dt profiles determined by the EMBS model applied to single clouds of varying sizes, and show that the retrieved UER profile from an ensemble of clouds in Figure 3.4b is qualitatively consistent with the superimposed effect of individual clouds with a size distribution determined by (3.7). 43 3.5.2 Single cloud and cloud ensemble effects Given the fact that a shallow cumulus boundary layer consists of a broad range of cloud sizes, I next examine the total convective effect [i.e., the sum of terms I and II in Eq. (2.1)], for four clouds with respective thicknesses of 1400 m, 900 m, 400 m and 200 m. For each of these clouds, I calculate the tendencies d9i/dt and dqt/dt using the EMBS model described in Section 3.2, with constant UER and the mixing distribution PI. The assumption of a constant UER for individual clouds is supported by the 2-dimensional numerical simulation of Bretherton and Smolarkiewicz (1989), which indicates that the entrainment of environmental air is relatively uniform, while detrainment can occur at preferred levels. A constant UER has also been used by Raymond and Blyth (1986) to successfully reproduce the observed cloud height, mass flux and cloud detrainment in individual non-precipitating cumulus clouds. My objective is to represent in a very simple way the size dependent transport of individual clouds. In Section 4.3.5 I will repeat this analysis with clouds sampled from an LES. I take the cloud-base mass flux to be a linear function of cloud thickness Ah (Raymond and Blyth, 1986): Fb = XAh (3.8) where x is chosen to be 1.25 x 10~5 kgm~ 3 s _ 1 . This x value produces net cooling and moistening tendencies at the inversion that are roughly equal to the forcing at that level for the 1400-m-thick cloud. Scaling Fb by the cloud thickness is consistent with the idea that deeper clouds need larger quantities of USCA to reach higher into the upper environment. Figure 3.5 shows the convective tendencies for the four individual clouds. We see that the two thickest clouds produce roughly symmetric cooling and moistening near their cloud-top and warming and drying near their cloud-base. For the 400-m-thick cloud, moistening and cooling tends to dominate most of the cloud depth except near its 44 2000 1800 1600 ^,1400 £ 1 2 0 0 1000 800 600 a) ' ' ' ' ' * • \ \ \ \ f \ \ \ V i ^ v \ 1 \ V •. V d9/dt (Kday~ 0 1 1 5 dq/dt (gkg~1day ) 10 Figure 3.5: Single cloud-induced net tendencies of ft and qt (sum of terms / and II of (2.1) calculated from an EMBS). a) ft tendencies due to individual clouds of different size; dotted line: large cloud (thickness A/i=1400 m); dot-dash line: medium cloud (A/i=900 m); dashed line: small cloud (Ah = 400 m); solid line: smallest cloud (A/i=200 m). b) as in a) but for qt tendencies. base. The smallest cloud cools and moistens the environment throughout its depth. Figure 3.6 shows the decomposed tendencies as in Figure 3.3c,d, but for the single 1400-m-thick cloud. We see most of the cooling and moistening near the cloud-top is due to the cloud downdraft-induced environmental compensating upward motion, while most of the warming and drying near the cloud-base is due to the cloud updraft-induced envi-ronmental compensating subsidence. As in Figure 3.3c, the cloud detrained temperature tendency is close to zero. The cloud detrained water tendency is significant throughout most of the cloud layer, with a maximum near the base of the inversion layer. The cloud-top cooling and moistening is not concentrated in a single layer at cloud-top, as would be expected with an entraining plume model. Instead it extends through roughly half the cloud depth, gradually decreasing to zero. In contrast with the two thickest clouds, the vertical tendency profiles for the two smallest clouds in Figure 3.5 are dominated by moistening and cooling. This is due to the 45 de/dt (Kday 1) dq/dt (gkg~1day~1) Figure 3.6: Decomposed tendencies for the 1400 m deep cloud of Figure 3.5. a) dotted-dashed line: induced 9\ tendency due to cloud net mass flux; dashed line: cloud lateral detrained tendency; solid line: the total net tendency, b) as in a) but for qt. fact that many cloud-top mixtures descend to their UNBL detrainment level at cloud-base for these cases. This produces a downward net mass flux throughout the smallest cloud, with corresponding induced upward motion in the environment. This result is consistent with observations from the Small Cumulus Microphysics Study (SCMS) that show that while large clouds transport net mass upward, small clouds (depth < 350 m) have average downward transport, while accounting for 50% of the cloud cover (Duynkerke, 1998). Since the USCA (#/, qt) is slightly cooler and moister than the cloud-base environment, the cooling and moistening is distributed approximately uniformly within the very shallow cloud depth. In general, the convective tendencies produced by the EMBS model for individual clouds of different thicknesses are consistent with the results of Esbensen (1978), who used a laterally entraining plume model to represent clouds that penetrate into the inversion, combined with a bulk model of shallower cloud circulations below the inversion base. He found that the deep clouds are primarily responsible for the warming and drying of 46 the lower cloud layer, while the shallower clouds are primarily responsible for moistening below the inversion base. Comparing Figure 3.5 with the total forcing shown in Figure 3.2, it is apparent that the total forcing requires a cloud response that cools and moistens the inversion layer and upper part of the cloud layer and slightly warms and moistens the lower part of the cloud layer. No single cloud modeled by the simple E M B S assumptions of Figure 3.5 can accomplish both these effects. I discuss the contribution of a size-distributed cloud population to the ensemble cloud response in Section 3.7 below. 3.6 Sensitivity tests In Section 3.4 I retrieved the U E R for an equilibrium boundary layer given a particular mixing distribution P I and the multi-mixing detrainment criterion. In Figure 3.7 I com-pare the U E R retrieved using (3.6a) for each of the mixing distributions of Eq. (2.5). The retrieved UERs shown in Figure 3.7a are nearly identical across mixing distributions P1-P4. Similarly, when each U E R is inserted into (3.6b) and the net dqt/dt is calculated, the tendency agreement is essentially independent of the choice of mixing distribution. Thus, provided the E M B S model detrains only neutrally and negatively buoyant mix-tures, with positively buoyant mixtures pursuing further mixing, the details of the initial mixing distribution have a very small impact on the convective transport produced by the E M B S parameterization. To illustrate why the proposed multi-mixing treatment of positively buoyant mixtures reduces the E M B S model's sensitivity to mixing distribution, I examine a single convec-tive element U% which is brought to a particular level i and subsequently mixed with the environment. As indicated in the LHS of (3.6), Ul influences environmental thermody-namic tendencies in three ways. First, it heats and dries all the levels n below i through compensating environmental subsidence caused by its adiabatic ascent. After mixing 47 E (kgrrfV 1) X10'4 dq/dt (gkg^day-1) Figure 3.7: a) The retrieved UER based on mixing distributions PI — PA using (3.6a). b) The residual qt tendencies calculated from (3.6b), given the U E R of Figure 3.7a. at level i, mixtures descend to their UNBLs and penetrate specific distances below i, exiting the cloud given the UNBL detrainment criterion. These penetrative downdrafts cool and moisten their penetrated levels through compensating environmental upward motion. Finally, detrained mixtures also moisten and slightly cool their detrainment lev-els. The sum of these three tendencies determines the total net tendency at any level n caused by the ascent, mixing and detrainment of a particular convective element Ul and its mixtures. Figure 3.8 illustrates the net tendencies and net mass flux induced by a unit USCA (Ul = 1 gm^s" 1 ) at a particular level i (1400 m). I have assumed single mixing and the UNBL detrainment criterion for this calculation. First, we see that this par-ticular level's mixing of USCA does not influence any higher levels. This is because, as discussed in Section 2.4, the UNBL detrainment criterion combined with the single mixing assumption requires that all mixtures detrain below their mixing level for non-precipitating clouds. Immediately below the mixing level (1400 m) we see a region (about 400 m deep) experiencing net cooling and moistening. This is because most mixtures 48 generated at the mixing level descend and are detrained at levels below 1100 m, and therefore give a net downward motion. This can be seen in the vertical profile of the net mass flux (Figure 3.8c) which decreases with height above 600 m and becomes negative at 1100 m, rather than maintaining a constant flux of 1 g m ~ 2 s _ 1 . Between 1000-1100 m, Figures 3.8a,b show that the net tendencies change sign from cooling and moistening to heating and drying. This is because the mixtures' downdrafts are not as large as the updrafts of adiabatic ascending USCA. Therefore, the net tendencies are dominated by warming and drying induced by adiabatic ascent. Mixture detrainment also contributes to moistening below the mixing level (not shown). The three panels in Figure 3.8 show the tendency and mass flux profiles produced using different mixing distributions. PI gives a comparatively concentrated cooling and moistening immediately below the mixing level. This is because PI entrains more en-vironment air; most PI mixtures are highly dilute and generate a relatively large but shallow downward mass flux near the mixing level (Figure 3.8c). The P2 and P4 dis-tributions have relatively larger proportions of nearly undilute mixtures, which tend to sink long distances and detrain near cloud-base (under UNBL detrainment). These mix-tures tend to reduce both upper level cooling and moistening and lower level heating and drying. P3 has an enhanced proportion of denser mixtures. P3 mixtures tend to cause strong downdrafts concentrated at upper to middle levels (1100-1400 m). Due to their limited liquid water content, they cannot sink as deeply as the positively buoyant mixtures, given UNBL detrainment. Therefore, P4 gives maximum net upward mass flux below 1000 rn. Figure 3.9 repeats the calculation shown in Figure 3.8, with single-mixing replaced by the multi-mixing algorithm described in Appendix B. In contrast to Figure 3.8, Figures 3.9a,b show that the vertical extent and the magnitude of the cooling and moist-ening is now roughly equal for all mixing distributions, and the transition to heating and 49 Figure 3.8: a) (9/ tendencies computed for the EMBS model using the LHS of (3.6) for a single convective element U% = 1 g m s" -i (\— 35; corresponding to height z=1400 m) given single mixing with distributions P1-P4 and the UNBL detrainment criterion, b) As in a) but for the qt tendencies, c) Total mass flux due to the ascent, mixing and detrainment of the Ul element. 2000 1800 1600 H 1400 'o 1200 1000 800 600 a) 2000 2000 -1 -0 .5 C d0/dt (Kday - 1 ) 0.5 -1 0 J ,2 dq/dt (gkg- T day- 1 ) c) - 2 a 1 2 M F (grri s - 1 ) Figure 3.9: As in Figure 3.8, but using the multi-mixing algorithm described in Section 3.2 and Appendix B. 50 drying occurs near 1150-1200 m for P1-P4. Note that the influence of the mixing level now extends above 1400 m, because positively buoyant mixtures are now permitted to as-cend. However, the grouping and homogenizing of these mixtures after each mixing event rapidly dilutes the mixtures and limits their region of influence to roughly 150 m above the 1400 m mixing level. The constant upward mass flux (equal to the adiabatic USCA mass flux 1 gm~ 2 s _ 1 ) below 1100 m shown in Figure 3.9c indicates that no mixtures descend below 1100 m for all mixing distributions. Comparison between Figure 3.9c and Figure 3.8c indicates that the reduction of upward mass flux below 1100 m in Figure 3.8c is mainly due to the detrainment of positively buoyant'mixtures at. their UNBL. From Figure 3.9, we see that multi-mixing acts to remove the signature of the orig-inal mixing distribution by requiring positively buoyant mixtures to undergo additional mixing and detrain at levels higher than their initial UNBL. In contrast to single-mixing, multi-mixing tends to entrain larger quantities of environmental air and produce modi-fied mixing distributions for which dilute mixtures are most probable. Thus multi-mixing modifies distributions P2-P4 so that these distributions take on the characteristics of PI, with its large fraction of environmental air and highly dilute mixtures. I find that the EMBS model is much more sensitive to the choice of detrainment criterion than to the specification of the mixing distribution. Figure 3.10 repeats the convective element calculation shown in Figure 3.8 and Figure 3.9, substituting NBL detrainment. Under the NBL detrainment criterion, the mixing event at the 1400 m significantly cools and moistens the middle inversion layer (1600-1800 m). This is be-cause the positively buoyant mixtures ascend to their N B L well within the inversion and produce large detrainment tendencies for both ft and qt. Using Figures 3.8, 3.9 and 3.10 we can compare the net tendencies caused by the ascent, mixing and detrainment of the single convective element U\ We see that, given fixed U\ NBL detrainment enhances convective transport, UNBL detrainment reduces convective transport and transport 51 2 0 0 0 r 1800 1600 t(m) 1400 CT) 'CD 1 2 0 0 -X 1 0 0 0 -8 0 0 -6 0 0 -2 0 0 0 2 0 0 0 r - 2 - 1 0 , 1 de /dt(Kday _ 1) c) dq /d t ( g k g - ' d a y - 1 ) 0 1 2 M F (grrfV 1 ) Figure 3.10: As in Figure 3.8, but using the NBL detrainment criterion. given multi-mixing detrainment lies between these two extremes, with more convective cooling and moistening near the mixing level (see Figure 2.3). Figure 3.11 shows the impact of the detrainment criterion on the entropy and moisture tendencies for all cloud levels with the retrieved E profile in Figure 3.4b and mixing distribution PI. As in the single-level example of Figure 3.10, UNBL detrainment gives the smallest convective transport, multi-mixing gives slightly larger convective transport while NBL detrainment gives the largest convective transport and produces extremely strong cooling and moistening near the top of the inversion. These large cooling and moistening tendencies are due to the fact that nearly all positively buoyant mixtures below the inversion are able to ascend well into the inversion, given the NBL detrainment and the B O M E X sounding as shown in Figure 2.2a-c. These positively buoyant mixtures contain large amounts of liquid water and produce extremely large and concentrated cooling and moistening near the top of the inversion. Due to these anomalously large tendencies we find no positive-definite vertical profile of the U E R that satisfies both (3.6) 52 and the assumption of direct detrainment at the NBL for the B O M E X case. This leads 600 -60 me to conclude that, at least for the B O M E X case, the UNBL detrainment criterion is more realistic than NBL detrainment for an EMBS model. 2000 1800 1600 ,^1400 SI g> £1200 1000 800 a) / (i" V \ i V / ' • /' / ' i i i - i i i - i 1 -40 -20 de/dt (Kday" 20 600 -20 1 4 0 1 dq/dt (gkg-1day-1) 80 Figure 3.11: Net 9i tendencies (panel a) and qt tendencies (panel b) given three different detrainment criteria and the retrieved UER profile from Figure 3.4b. Solid line: multi-mixing; Dashed line: detrain all mixtures (single mixing) at their U N B L (following E91); Dot-dashed line: detrain all mixtures (single mixing) at their NBL (following RB86). 3.7 Discussion I have applied an EMBS model to an equilibrium sounding of a trade-wind cumulus atmo-sphere. My objective is to use the equilibrium constraint and known large-scale forcing to better understand the detailed behavior of this type of model, particularly the impact of undetermined free parameters on the ability of the model to represent the convective transport of shallow clouds. This approach is similar to that of Nitta (1975), who used the spectral entraining plume modelof Arakawa and Schubert (1974)'to diagnostically determine the contribution of clouds in a particular size range (or equivalently, with a particular entrainment rate) to the total cloud mass flux. A similar approach can be 53 applied to a bulk entraining and detraining plume model, which is often used in shallow cumulus parameterizations. Bulk entraining and detraining plume models require specification of the vertical profile of the entrainment and detrainment rate. While the entrainment rate might be constrained by laboratory results, the detrainment parameterization is currently ad hoc, and detrained parcels do not exit the cloud at their NBL, but detrain arbitrarily through-out the layer (Siebesma and Holtslag, 1996). The corresponding unknowns for an EMBS model of non-precipitating clouds are the UER, the detrainment criterion and the mixing distribution. For a specific choice of mixing distribution and detrainment criterion, I have retrieved a unique vertical profile of the UER which satisfies an equilibrium constraint. I have investigated the sensitivity of this retrieval to variations in the mixing distribution and the detrainment criterion, and find that, given the requirement that mixtures de-train at their NBL, there is no positive-definite vertical U E R profile consistent with the observed equilibrium state and the measured large-scale forcings. As discussed in Sec-tion 3.6, this is due to the fact that, for the B O M E X environment, all positively buoyant mixtures ascend into the inversion, where the requirement that they detrain at the NBL produces large updrafts and associated large detrainment of total water and entropy. Given the alternative assumption of E91, i.e. single mixing with all mixtures de-trained at their UNBL and mixing distribution PI, it is possible to retrieve a physically plausible profile of the UER. I have introduced a simple form of multi-mixing to treat pos-itively buoyant mixtures, and with this addition, the EMBS model is nearly insensitive to the choice of mixing distribution. The characteristic UER retrieved via this diagnos-tic approach is maximum at cloud-base, decreasing rapidly through the cloud layer and increasing slightly near the base of the inversion. This vertical structure is consistent with the cloud-base maximum in the entrainment rate diagnosed by Nitta (1975), who argued that the maximum was the product of the large number of small clouds present 54 in the B O M E X boundary layer. As Figure 3.5 shows, an EMBS model applied to the B O M E X environment predicts that, over a range of cloud sizes, individual cumulus clouds cool and moisten the environ-ment at cloud-top and warm and dry the environment at cloud-base. The smallest clouds, in contrast, cool and moisten throughout their depth, as discussed in Section 3.5.2. Given the cumulative effect of the clouds, as represented by the large-scale forcing data (Fig-ure 3.2c,d), a size-distributed population is required, so that the cloud-top moistening effect of more numerous smaller clouds can counteract the cloud-base-drying effect of less frequent larger clouds, with their larger undilute vertical mass flux and correspondingly larger convective transport-induced tendencies. This counterbalancing transport requires a size distribution for which the cloud num-ber density increases monotonically with decreasing cloud size. An endpoint is reached for the smallest clouds in the ensemble, which only cool and moisten throughout their depth. These small clouds are also called forced clouds; they are little more than tracers of subcloud thermals, and do not vent subcloud air into upper levels or contribute to the cloud-scale circulation (Stull, 1985). Nevertheless, these forced clouds play the important role of moistening and cooling throughout the shallow cloud-base layer, balancing heating and drying due to larger, deeper and more active clouds. The accurate representation of these cloud-base processes is crucial to the prediction of cloud-base thermodynamic properties, the cloud-base mass flux, and therefore the cloud-layer subcloud-layer inter-action. The EMBS model provides an example of the unified representation of condensation and evaporation through induced convective drafts rather than through direct local mix-ing and detrainment of liquid water. However, condensation and evaporation do show important differences in non-precipitating cumuli. In particular, condensation may not necessarily be involved with mixing while evaporation is almost certainly associated with 55 mixing. In this sense, the E M B S model can be interpreted as a simple representation of the cloud life cycle, with the growth phase/condensation represented by the upward transport of USCA, and the dissipation/evaporation phase represented by mixing and detrainment. The E M B S model's symmetric treatment of condensation and evapora-tion through convective drafts is physically appealing, given that the two processes both rapidly influence the temperature of the far field environment but are confined to small turbulent regions. In the E M B S model, these downdrafts produce net downward mass flux within the inversion layer; observations indicate that they are a dominant feature of the dissipation stage of shallow cumulus clouds. As I have shown here, the E M B S model requires no more free parameters than a bulk entraining and detraining plume model, given the U N B L detrainment criterion and the demonstrated lack of sensitivity to the choice of the mixing distribution. I have also suggested that constraining the U E R requires consideration of the cloud size distribution. 3.8 Summary Episodic mixing and buoyancy-sorting (EMBS) models have been proposed as a physi-cally more realistic alternative to entraining plume models of cumulus convection. Ap-plying these models to shallow non-precipitating clouds requires assumptions about 1) the rate at which undilute sub-cloud air is eroded into the environment; 2) an algorithm to calculate the eventual detrainment level of cloud-environment mixtures; and 3) the probability distribution of mixing fraction. A diagnostic approach is used to examine the sensitivity of an E M B S model to these three closure assumptions, given equilibrium con-vection with known large-scale forcings taken from phase III of B O M E X . The undilute eroding rate (UER) is retrieved and found to decrease exponentially with height above cloud-base, suggesting a strong modulation by the cloud size distribution. The E M B S 56 model is also used to calculate convective transport by individual clouds of varying thick-ness. No single cloud from this ensemble can balance the large-scale B O M E X forcing; the observed equilibrium requires a population of clouds with a cloud size distribution that is maximum for small clouds and decreases monotonically with cloud size. The EMBS model depends sensitively on the assumptions governing the detrainment of positively buoyant mixtures. In particular, given the requirement that positively buoy-ant mixtures detrain at their neutral buoyancy level, there is no positive-definite undilute eroding rate that is consistent with the B O M E X forcing. The model is less sensitive to the assumed distribution of cloud-environment mixtures, given a multiple-mixing treat-ment of positively buoyant parcels and detrainment at the unsaturated neutral buoyancy level (UNBL). 57 Chapter 4 Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part I: Transport 4.1 Introduction The ID diagnostic study in Chapter 3 has provided some insight into shallow cumulus convection and the EMBS representation of convective transport. However, the diag-nostic results may rely heavily on the validity of the EMBS conceptual model itself. In addition, a ID model cannot offer further information about the detailed dynamics of cumulus mixing. In order to establish the validity of the findings presented in Chapter 3, the results should in principle be compared with real observational data. Unfortunately, such data are lacking. As discussed in Chapter 1, an alternative approach is to use high-resolution three-dimensional numerical models (i.e., LES) to simulate these clouds and make such a comparison. However, it must be kept in mind that these.numerically sim-ulated clouds are not real clouds and therefore they can supplant neither observations nor experiments (Stevens and Lenschow, 2001). Nevertheless, these numerically sim-58 ulated "pseudo-clouds" [following the terminology of "pseudo-fluid" and "pseudo-fluid simulation" introduced by Stevens and Lenschow (2001)] provide synthetic, yet detailed, descriptions of phenomena that can then be used to help shape ideas regarding the be-havior of real clouds. The results must be finally validated against the observations. Indeed, it has been stressed that numerical models are most successful when used in conjunction with observations and theory. When used properly they may constitute a powerful means of advancing understanding (e.g., Emanuel, 1994). LES have demonstrated that they have ability to reproduce many of the characteris-tics of the B O M E X boundary layer (e.g., Siebesma and Cuijpers, 1995; Siebesma et al., 2003). Direct comparison between LES (numerical simulation) and the EMBS (theoret-ical) results presented in Chapter 3 would be particularly useful. This requires that the LES be sampled using a definition of the cloud boundary that corresponds to that used in the EMBS model, where the convective region includes the sub-saturated near-cloud environment with its unsaturated downdrafts. An additional sampling requirement for the EMBS evaluation is that the effect of individual clouds be tracked through both growth and dissipation stages in the LES. The introduction of tracers would be par-ticularly helpful in determining the cloud convective envelope and isolating individual cloud life cycles from simulated time-evolving cloud fields. As discussed in Chapter 1, numerical simulations have particular advantages due to the fact that they can be con-trolled, repeated and throughly examined. This motivates the use of an LES with passive numerical tracers. In this chapter I conduct LES including tracers to quantify cumulus mixing and its associated convective transport, particularly for the evaluation of EMBS models. The LES cloud ensemble statistics for the B O M E X case have been thoroughly examined and reported in Siebesma and Cuijpers (1995), a recent paper (Siebesma et al., 2003) describing the B O M E X inter-comparison of the G E W E X Cloud Systems Study (GCSS), 59 and the website http://www.knmi.nl/~siebesma/gcss/bomex.html. I first compared my LES cloud ensemble transports with these GCSS intercomparison results to gain confidence in the simulation (see Appendix D for these results). Following this, I used the LES to address the following: • Cloud evaporation and the role of the unsaturated part of cumulus convection in cloud transport. • Cloud life cycles and their impact on the convective mass flux. • The validity, of the buoyancy-sorting hypothesis. • The role of cloud-size distribution in cloud ensemble transport In Section 4.2 I 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 4.3 I present the simulated individual clouds and their life cycles with emphasis on the vertical transport of mass, heat, and moisture based on conditional sampling of cloud mixtures. In Section 4.4 I test the sensitivity of these results to the choice of threshold used to identify the unsaturated convective mixed region. I further discuss the model results and their implications for conceptual models of cumulus mixing in Section 4.5. 4.2 Approach 4.2.1 T h e L E S mode l and case descr ipt ion The Colorado State University Large Eddy Simulation/Cloud Resolving Model (CSULES) is used in this study; a detailed description of this model is given in Appendix E and in Khairoutdinov and Randall (2003). The equations of motion are written using the 60 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 employs a 1.5-order closure based on the prognostic subgrid-scale turbulent kinetic energy (e.g., 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 defi-nite and monotonic scheme of Smolarkiewicz and Grabowski (1990). Periodic boundary -1 -0.5 0 0.5 -4 -2 0 2 - 2 - 1 0 1 w(cms"1) dO/dt (Kday ) dqt/dt (gkg"1day_1) Figure 4.1: Initial sounding and large scale forcing profiles for the LES B O M E X case (see text for details), a) liquid water potential temperature ft. b) total water mixing ratio qt. c) horizontal wind (only have east-west component u while north-south component v is initialized as zero), d) large-scale subsidence, e) clear air radiative cooling, f) large-scale advective drying. conditions are applied for both east-west and north-south lateral boundaries; the top 61 boundary is treated as free-slip with a sponge-layer to damp the spurious reflection of upward propagating gravity waves, while the bottom boundary uses prescribed sensible and latent heat flux as will be presented below. I use a three-dimensional domain with 256 x 256 x 128 grid points with uniform grid-spacing Ax = Ay = Az = 25 m in both the horizontal and vertical, with a model time step of 1.5 s. I choose the undisturbed trade wind boundary layer from Phase III of the B O M E X case which has been described in Section 3.3. A more detailed description of the LES initialization can be found in Siebesma and Cuijpers (1995), Siebesma et al. (2003), and http://www.knmi.nl/~siebesma/gcss/bomex.html. I have also included a short de-scription of the B O M E X case and a comparison between my simulated ensemble statistics and 11 other model outputs in Appendix D. Convection in the boundary layer is driven by specified surface fluxes with latent and sensible heat fluxes of 153.4 and 9.46 W m " 2 (or 8 x 10~3 K m s - 1 and 5.2 x 10~5 k g k g _ 1 m s _ 1 ) , respectively. Figure 4.1 panels a,b,c show the initial profiles for the GCSS B O M E X inter-comparison simulation (available from http://www.knmi.nl/~siebesma/gcss/bomex.html). There is a well-mixed sub-cloud layer between approximately 0 - 500 m, a conditionally unstable cloud layer between 500 - 1500 m, and an overlying inversion layer. The wind profiles are initialized as easterly with north-south component v set to zero. Figure 4.1 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, these sounding and forcing profiles are close to those presented in Holland and Rasmusson (1973) and Nitta and Esbensen (1974b) for the B O M E X square during phase III. Siebesma and Cuijpers (1995) give a comparison between these initialization conditions and the B O M E X ob-servations [e.g., Fig 1 in Siebesma and Cuijpers (1995)]. The first three hours of the simulation are considered to be spin-up, during which the system reaches its steady-state equilibrium. The LES was run for a total of 6 hours; all the selected individual clouds 62 are sampled between hours 3 to 6. 4.2.2 Isolat ing ind iv idua l clouds Since the B O M E X environment has strong easterly ambient wind (Figure 4.1c) the sim-ulated individual clouds move rapidly from east to west. Therefore the distance that an individual cloud travels during its life cycle is much larger than its horizontal size. To isolate individual clouds over their life-cycle and obtain time-dependent 5-dimensional (1 time + 3 space + 1 variable) cloud data we must sample the cloud in a reference frame that moves to the west at approximately the mean horizontal velocity. The coordinate in the moving frame x n e w is x - x0(t) = x - U0t (4.1) where x is the coordinate in the fixed frame and x0(t) = Uot is the coordinate of the origin of the moving frame in the fixed frame. I choose a translation velocity UQ — —7.5 ms - 1 , which is close to match the horizontally and vertically-averaged mean velocity within the cloud layer. Because the simulated cloud-field data is stored every At0 = 30 s, this choice of UQ insures that as each succeeding 30-second snapshot is shifted by a displacement Ax0 = UoAto, it is moved an integral number of gridcells, i.e., Ax0 U0At0 f A 0 , —— = — = integer (4.2) A x Ax The resulting translated time series maintains the exact relative position between the different snapshots in gridcell coordinates. Consistent with the LES, periodic boundary conditions are used when I apply the translation. The time-sampled cloud field moves much more slowly in the translated reference frame, allowing me 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 (e.g., Pope, 2000). 63 Even with the above technique, isolating individual clouds throughout their life-cycle is not always easy because of 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 I therefore always try to select a box which includes a single cloud with as much cloud-free surrounding environment as possible. In this study I examine 6 clouds with cloud-top heights (defined as the maximum height reached by cloud liquid water) ranging from 1000 m to 2000 m. They are independent of each other in both time and space and will be referred below (ordered smallest to largest) by the labels A, B, C, D, E, F. 4.2.3 Distinguishing the convective region from the environ-ment 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), the mass-mixing ratio of a passive numerical tracer, £, is initialized to 0 above cloud-base and 1 g k g - 1 below cloud-base. I first run the LES, obtain a particular realization of the time dependent cloud fields and select individual clouds. Since I now know the start and end times and the cloud-base height of each cloud, I rerun the LES with the tracer initialized at a single time step just before an individual cloud emerges from the subcloud layer. Only advection 1 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 1 A simulation with both advection and subgrid-scale diffusion applied to the subgrid-scale tracer produces little difference. This is partly due to the fact that the current advection scheme (Smolarkiewicz and Grabowski, 1990) can include an effective subgrid scale model when no explicit turbulence is applied, as has been demonstrated in Margolin et al. (1999). 64 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 numerical 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 , the cloud mixed region associated with each cloud will be defined as those gridcells above cloud-base level which initially contain no subcloud layer tracer C but now contain C > Co due to the cloud upward transport and mixing of subcloud air with its upper environment. Here Co is a threshold value of subcloud layer tracer mixing ratio. Ideally, the Co should be chosen so that cloud mixed region defined by C > Co includes all the liquid water cloud region. Figure 4.2 shows an example of the joint frequency distribution (JFD) of subcloud tracer mixing ratio C and liquid water mixing ratio qc 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 Co > 0.05 g k g - 1 the cloud-mixed region defined by C > Co begins to exclude a certain number of liquid water containing gridcells, located at the upshear side of the clouds during the ascending stage (not shown). Therefore, I choose Co = 0.05 g k g - 1 for all 6 selected clouds; with this threshold the sampled cloud mixed region include nearly all (> 99%) of liquid water cloud. This choice of tracer level is conservative in the sense that it selects the smallest mixed region which includes nearly all 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 4.3 shows the JFD of w and A6V within the unsaturated cloud mixed region (C > Co and qc = 0) for each cloud over 65 Figure 4.2: Joint frequency distribution of subcloud layer tracer mixing ratio C and liquid water mixing ratio qc for all liquid water-containing gridcells during the life-cycle of cloud E. The contoured unit is number of gridcells. The dashed line indicates the Co = 0.05 g kg" 1 threshold. its life-cycle. Here A9V = 9V — (9V), 9V 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 on w = 0 and A9V = 0, values that correspond to the quiescent environment. To eliminate this detrained mixed-region from the sampled mixed region and obtain the convective mixed region, I remove those gridcells that satisfy the criterion: (( > Co and qc = 0 and \A9V\ < A9vfi and IH < ^o), where A9Vj0 and w0 are adjustable positive threshold values. Choosing large values for A9v>o and wo selects a smaller and more active unsaturated mixed region; arbitrarily large A9Vt0 and w0 will eliminate all the unsaturated mixed region and reduce the sampled data to saturated cloud. In principal, the choice of A9Vt0 and wo may depend on the unmixed environment's A9V and w variability. The shaded region on each panel of Figure 4.3 shows the set of gridcell (A9V, w) values that charac-terize at least 99.6% of the gridcells in the unmixed environment (i.e., those gridcells with qc — 0 and C < 0.01 gkg - 1 ) . For the smaller clouds (Figure 4.3a,b,c), these environment gridcells lie approximately within the range \A9V\ < 0.2 K, \w\ < 0.5 ms - 1 ) . For the 66 Figure 4.3: Contours: joint frequency distribution of A9V and w within the unsaturated cloud-mixed region (defined as qc = 0 g k g - 1 and £ > 0.05 g kg - 1 ) for clouds A - F (panels a-f respectively). Shaded region: the range of A9V and w values that encompass more than 99.6% of the unmixed gridcells, defined as qc = O g k g - 1 and £ < 0.01 g k g - 1 . Percentages for shaded regions are A: 99.64%, B: 99.86%, C: 99.84%, D: 99.94%, E: 99.95%, F: 99.96%). 67 larger clouds (Figure 4.3d,e,f) the environment spans a much larger range of (A8v,w) (note the different scales between left and right panels). This larger environmental vari-ation is limited to the inversion levels; beneath the inversion the environmental (A8V, w) fluctuations have roughly the same range in all clouds. Although large clouds, which penetrate the inversion, generate internal gravity waves which dominate the environmen-tal ( A f t , w) fluctuations at upper levels, these waves make a negligible contribution to the net vertical mass transport. Below I use, for simplicity, the thresholds A f t j 0 = 0.2 K and wo = 0.5 m s - 1 suggested by the small clouds A - C , for all clouds at all levels. In Section 4.4 I will test the sensitivity to these thresholds by choosing several different values of A f t j 0 € [0,1.2] K and WQ 6 [0,1.2] m s - 1 for the large clouds. Note that the (£, w, A8V, qc) criteria have to be simultaneously satisfied in order to identify a gridcell which will not become 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, 5 different sampled regions are distinguished: 1) the mixed region (all grid-cells with C > Co); 2) the detrained mixed-region (DMR) (C > Co and qc = 0 and | A f t | < A f t > 0 and \w\ < w0); 3) the convective-mixed region (CMR) (C > Co and not DMR); 4) the unsaturated convective-mixed region (CMR and qc = 0); 5) the liquid water (or saturated) cloud (CMR and qc > 0). 4.3 Results 4.3.1 Life-cycle overview Figure 4.4 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 68 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 1000 - 1400 m and do not reach the inversion layer (1500 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 1500 500 1000 Time (seconds) 500 1000 Time (seconds) 1500 Figure 4.4: Life-cycles of clouds A-F . a) cloud-top heights (subscript T) , b) cloud-base heights (subscript B). c) maximum upward velocities, d) maximum downward velocities. evaporated (possibly before all the unsaturated cloud mixtures reach their NBL). Lifetime 6 9 x 10 A: 1.2. B:1.4, CM x 10 D:1.8, E:1.6, F:1. 1.5 E. a) E O > 0.5 - - Ds \ b) - - Du • - • Es • - • E n Fs Fu I + \ ' \ j t \ \ j / V \ / / / / . ' * ' * ' * *» * X / 7 ^ N \ / / / ' A A , \ / ' 1 » X \ / / / ' > s V \ '•''If \ \ >,' I ' ' J •* j— 400 600 800 Time (seconds) 1200 500 1000 Time (seconds) 1500 1500 400 600 800 Time (seconds) 1500 Time (seconds) Figure 4.5: a) Time evolution of the volumes of the liquid water cloud and unsaturated convective-mixed regions for small clouds A, B and C. The numbers at the top of panels a and b show the ratio of the unsaturated/saturated time-integrated volume for each cloud. The legend code denotes the cloud (A-F) and region type (saturated/unsaturated or (s/u)) (see Section 4.2.3 for full selection criteria), b) As in a) but for large clouds D, E and F (note the different scale), c) Time evolution of the volume-integrated vertical mass flux for the saturated and unsaturated convective-mixed region for small clouds A-C. d) As in c) but for large clouds D-F. e) Time evolution of the volume-integrated vertical mass flux for the total convective-mixed region (saturated plus unsaturated) for small clouds A - C . f) As in e) but for large clouds D-F. 70 animations 2 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 shown in Figure 4.4. 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, the selected large clouds D, E, and F might be interpreted as single, clearly isolated, large convective cells or elements which penetrate as high as the inversion layer. Figure 4.4c, 4.4d show the time evolution of the maximum updraft and downdraft velocities within individual clouds. The maximum upward velocity of each cloud al-ways 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 pene-tration 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 evapora-tion triggers the strongest downdrafts. In Chapter 5 I discuss in detail the intermittent behavior of the simulated clouds and the cloud mixing dynamics. Figure 4.5, 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' 2Animations available from h t tp : / /www.eos .ubc .ca / research /c louds 71 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 ( A f t ) 0 , wo) 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 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 Figure 4.5a and Figure 4.5b). Translating these volumes to equivalent cylindrical radii implies that the horizontal radius of the total convective mixed region is roughly 1.5 times larger than the radius of the visible cloud. Figure 4.5c,d show the time evolution of the vertical mass flux integrated over the two regions of Figure 4.5a,b. For each cloud the mass flux is generally positive (upward) in the saturated volume and negative (downward) in the unsaturated volume with flux in the saturated volume dominating the earlier phase, while the flux in the unsaturated volume dominates the later phase of convection. Figures 4.5e,f show that there is a transition in the total volume-integrated vertical mass flux from positive to negative beginning at approximately one-half of the cloud lifetime with a roughly oscillatory time variation. Integrated over the cloud life-cycle, this produces a negative net vertical mass flux for small clouds and a positive net vertical mass flux for large clouds. 72 4.3.2 Vertical profiles of the life-time averaged vertical mass flux Figures 4.6, 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. 1600 1400 E~ 1200 ^ 1000 800 600 -1 a) - - A . _ . B C 0 1 2 Mass flux (kgs~1) X 1 0 4 2000 1800 1600 -§-1400 CD o 1200 X 1000 800 600 - 5 0 5 Mass flux (kgs~1) x 10 1600 1400 E J 2 0 0 £ 1000 800 600 - 2 C) - - As • - • Bs Cs - - Au • - • Bu s / A Cu / y \ \ y \ \ t\ ^ \ \ s « V n \ Z v • \ A „ . ^"~«%. ' ~- • -1 0 Mass flux (kgs~ x 10 2000 1800 1600 -§1400 St CO o 1200 I * 1 ) \ » / 's 1 ( 1000 *\ A 800 600 Mass flux (kgs 1) x 10 4 Figure 4.6: a) Vertical profiles of the lifetime-averaged vertical mass flux for the convective-mixed region of small clouds A, B and C. b) As in a) but for large clouds D, E and F. c) As in a) but partitioned into contributions from saturated and unsatu-rated regions, with legend codes as in Figure 4.5. d) As in c) but for large clouds D, E , and F. 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 73 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 cumuli reported by Raga et al. (1990), Grinnell et al. (1996) and with the observations of mountain-induced cumulus congestus clouds reported by Raymond and Wilkening (1982, 1985). It is also consistent with the result in Chapter 3, where I diagnosed the cloud ensemble vertical mass flux using an EMBS model based on the B O M E X equilibrium soundings and large-scale forcings. The vertical profiles of convective mass flux, partitioned into saturated and unsaturated components, are given in Figure 4.6c,d; they show that the downward mass flux comes 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 4.5 I discuss the implications of this vertical profile of vertical mass flux for the shallow cumulus parameterization problem. A life-cycle view of the vertical distribution of vertical mass flux is shown in Figure 4.7. The vertical mass fluxes integrated over the horizontal convective mixed area at different vertical levels and time-steps are contoured and shaded in the time-height space for clouds A-F . 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) the downward mass flux begins to increase. Net 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. The intensity of the downward mass fluxes are as strong as the upward mass fluxes. Thus, at every level, the time history of the vertical mass flux resembles the time variation of Figure 4.5e-f; near cloud-base the upward flux during the growth phase is larger than the 74 1400 1300 1200 -.1100 E a) 1000 900 800 700 600 200 400 600 800 1000 1200 Time (s) 200 400 600 800 1000 1200 Time (s) 1400 1300 1200 ,1100 11000 c) 900 800 700 600 200 400 600 800 1000 1200 Time (s) 1500 1500 500 1000 Time (s) 1500 Figure 4.7: Time-height variation of the vertical mass flux [i.e., contours of (JApwdA)tik (unit: 1 0 0 0 kgs - 1 ) for timestep t, vertical level k] integrated over the horizontal convec-tive mixed region A for clouds A - F (panels a-f respectively). 75 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. 4.3.3 T h e role of buoyancy in ver t ica l mass t ranspor t The buoyancy-sorting hypothesis suggests that cloud mixtures coming from above an ob-servation level should have negative or neutral buoyancy at the observation level, while mixtures arriving from below should have positive or neutral buoyancy. In other words, cloud mixtures tend to move following their buoyancy and produce a positive buoy-ancy flux. To test the buoyancy-sorting hypothesis with these simulated clouds, I next evaluate the contribution of "buoyancy-direct" and "buoyancy-indirect" mass fluxes to the convective mass transport. Here I define buoyancy-direct 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 4.8a shows the JFD of (qt, A9V) over all gridcells at a level within the inversion (1612.5 m above the surface) over the life-cycle of cloud E. Note the strong correlation between mixture buoyancy and the total water mixing ratio qt which, as a conserved variable, is usually well correlated with the mixture's cloud-environment mixing fraction. The square symbol marks the (qt, A9V) of cloud-base air lifted adiabatically to this level, while the circle represents the (qt, A9V) of the mean environment. The well-defined V shape indicates the strong nonlinear dependence of the mixture buoyancy on mixing frac-tion caused by phase change (see Figure 2.Id). The quadrants separate these convective cloud-environment mixtures into 4 categories: saturated positively buoyant (SP), satu-76 10 12 qt (gkg"n) 10 12 qt (gkg^  Figure 4.8: a) Joint frequency distribution (unit: number of gridcells) of A8V and qt for all the convective-mixed air at a single level (1612.5 m) over the life-cycle of cloud E. The quadrants separate saturated (right) from unsaturated (left), positively buoyant (up) from negatively buoyant (down) mixtures. The square symbol represents cloud-base air lifted adiabatically to this level, while the circle represents the mean environment at this level, b) Bins as in a) but contours are the bin-averaged vertical velocity w ( m s - 1 ) . Dashed contours: downward w, solid contours: upward w rated negatively buoyant (SN), unsaturated negatively buoyant (UN) and unsaturated positively buoyant (UP). Figure 4.8b shows a contour plot of the bin-averaged vertical velocity w using the same gridcells and times of Figure 4.8a. There is both buoyancy-direct and buoyancy-indirect motion, with unsaturated mixtures dominating the downward motion, while saturated mixtures dominate the upward motion. Figure 4.9 shows the partitioned vertical profile of the vertical mass flux for the 4 different categories for each of the clouds. The sat-urated, 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 4.9 shows that saturated negatively buoyant mixtures (SN) on average transport mass upward and unsaturated positively buoyant mixture (UP) on average 77 transport mass downward. 1600 1400 §-1200 £ S1000 x 800 600 1600 1400 §-1200 £ •5 1000 X 800 -1 600 -1 1600 1400 §-1200 £ s iooo X 800 600 -1 -0.5 0 Mass flux (kgs 0.5 1 - h X10 4 b) -0.5 0 0.5 1 Mass flux (kgs - 1) X 1 0 4 c) -0.5 0 0.5 1 2000 E, 1500 £ D) x 1000 2000 E 1500 go X 1000 - 5 2000 £ 1500 go X 1000 Mass flux (kgs d) Mass flux (kgs ) X 1 0 4 Mass flux (kgs x 10 Figure 4.9: Vertical profiles of the vertical mass flux partitioned into contributions from 4 mixture categories of Figure 4.8. Panels a-f correspond to clouds A - F . The JFD of A8V and w within the saturated cloudy region for each cloud during its lifetime is shown in Figure 4.10. While most of the saturated positively buoyant mixtures have positive vertical velocity, this is not true for saturated negatively buoyant mixtures. In fact, the majority of the saturated negatively buoyant mixtures in Figure 4.10 have positive vertical velocity. This mixture component can be produced by a mixing process in which momentum exchange allows a mixture to maintain positive velocity while evap-oration produces buoyancy reversal. These upward-moving, negatively buoyant mixtures 78 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. There is a relatively small number of saturated downdrafts, but a large number of the unsaturated downdrafts in the simulated clouds indicating that further mixing and phase change is a dominant feature of counter-buoyancy transport in the simulated clouds. -2 0 A 9 v (K) 10 ~ 5 i CO E -5 -4 e) i i T~{ i i 0 1 A 6 V ( K ) -2 0 A 6 V ( K ) Figure 4.10: Joint frequency distribution (unit: number of,gridcells) of A6V and w within the liquid water cloud region for clouds A - F (panels a-f respectively). 79 In contrast, unsaturated negatively buoyant mixtures do not experience this phase change, and the unsaturated positively buoyant mixtures are generated primarily by initially unsaturated negatively buoyant mixtures overshooting their NBL from above. Although further mixing may allow these mixtures to do some net heat and moisture transport, the impact should generally be small because, as will be shown in Section 4.3.4, these mixtures usually have thermodynamic properties very close to their environment as they reach their NBL. 1600 1400 §-1200 £ S 1000 X 800 600 0 1 2 3 4 Buoyancy flux (W) ^ Q6 b) - 1 0 1 2 3 4 Buoyancy flux (W) 1 Q 6 0 1 2 3 4 Buoyancy flux (W) X ^ Q6 2000 E 1500 X 1000 2000 E 1500 co X 1000 2000 E 1500 d) 0 2 4 Buoyancy flux (W) X -| Q7 0 2 4 Buoyancy flux (W) X -| Q7 CO '<*> x 1000 0 5 Buoyancy flux (W) 1 Q ? Figure 4.11: Vertical profiles of the buoyancy flux partitioned into contributions from 4 mixture categories: positively buoyant updrafts (PU), negatively buoyant downdrafts (ND), positively buoyant downdrafts (PD) and negatively buoyant updrafts (NU) mix-tures. The two thick lines are buoyancy-direct (BD=PU+ND) and buoyancy-indirect (BI=PD+NU). Panels a-f are for clouds A - F respectively. 80 Figure 4.11 summarizes my 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 (Bl) trans-port. 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 buoy-ant mixtures at higher cloud levels for large clouds (cf. Figure 4.11d-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 and Cuijpers (1995). 4.3.4 T h e nature of the unsaturated downdrafts As shown in Figure 4.9 the unsaturated negatively buoyant mixtures dominate the down-ward mass flux. More information about the thermodynamic properties of the cloud-environment mixtures is given in Figure 4.12, which is a mixing diagram using (9i, qt) coordinates following Taylor and Baker (1991). Values in the scatter plot are taken from all gridcells at the 1612.5 m inversion level over the lifetime of cloud E (cf. Figure 4.8). These mixtures are distributed along a mixing line between the thermodynamic coor-dinates of cloud-base environmental air (symbol: x) and those for air at (or 100-200 m 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 81 observations in cumulus clouds. Also shown on the diagram are the average environmental sounding (solid line), ft isopleths at 1612.5 m (dash-dot), and the saturation line at 1612.5 m (dashed). As the figure shows, nearly 50% of the mixture range at this level is unsaturated; these mixtures have negative buoyancy of the same magnitude 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, 18r 16k 14 'o)12 10 ' — 1 1 — '-•XT) 62.5m ' "•-.•..», **• • 1 i . , X 587.5m _ * ' *• • , _ \ \ I SP _ \ \ \ **••( X \ V X \ V \ \ \ ( \ ' "1 S1^.5m^jkJ' "••v." " -\ \ \ \ \ \ \ \ ^ ' * ^ ^ « v * \ \ ^ - M 4 1 2 S J t i ^ f c t . | M 1 **" \ \ ^ ^ w L \ V 1 \ X \ \ \ \ - ""'\ \ \ 1S12.5m*B^'» \ V \ \ -v \ \ \ \ |^jy \ \ x y \ \ \ V \ N y \ \ \ \ 1 6 v 1 2 5 m " i, UP \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \ \ \ \ \ \ \ \ X l 7 1 2 . ^ m \ \ \ \ \ \ \ t \ \ . \ \ " \ \ \ \ \ \ \ \ * ^ j1812.5np -\ \ \ \ \ \ 1 * • "» 1 • \ \ \ 298 300 302 304 306 308 6, (K) Figure 4.12: Convective cloud mixtures plotted on a (ft, qt) conserved variable diagram. Points: all C M R gridcells at the 1612.5 m level during the lifetime of cloud E, shaded to denote the four mixture categories, which are labeled as in Figure 4.8. Solid line: the vertical profile of environmental (ft, qt), with the values at 1612.5 m indicated by (+) and values at other height by • . The circle indicates the surface layer mean properties, while the (x) indicates the cloud-base mean properties. Thin dash-dot lines: ft isopleths for air at this level. Thick dash-dotted line: the zero buoyancy isopleth (ft = 305.6 K) at this level. Dashed line: the saturation curve. underscores the fact that unsaturated mixtures are an integral part of cumulus mixing throughout the cloud life cycle. Figure 4.12 also shows some unsaturated mixtures which are positively buoyant at 1612.5 m. As discussed previously, these mixtures are primarily 82 due to unsaturated mixtures generated at higher levels which have overshot their NBLs from above. 2000 E 1500 • 1000 2000 E 1500 • 1000 2000 E 1500 Figure 4.13: Vertical profiles of the 9\ difference between each of the 4 mixture cate-gories and the environment for clouds A - F (panel a-f respectively). Legend labels as in Figure 4.9. Figures 4.13 and 4.14 show partitioned lifetime-averaged vertical profiles of the 9i and qt difference between each mixture category and the environment. The partitioning of the 4 categories is as in Figure 4.8. As noted above and in Section 4.3.3, the unsaturated positively buoyant mixtures (UP) for all clouds do have 9i and qt nearly indistinguishable from their environment, supporting the idea that they are primarily due to unsaturated negatively buoyant mixtures overshooting their NBLs. Figures 4.13 and 4.14 also show that the unsaturated negatively buoyant 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 83 -1 0 1 , 2 3 0 2 4 6 8 Aq,(gkg"') A q ( ( g k g ) Figure 4.14: As in Figure 4.13, but for qt. updrafts would be drier and warmer than the surrounding environment. 4.3.5 Thermodynamic fluxes and tendencies Figures 4.15 and 4.16 show the fluxes of two conserved variables 9i, qt partitioned into the 4 mixture categories as in Figure 4.9. The 9i and qt 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 9[ and qt flux contributions due to positively buoyant unsaturated mixtures are small throughout the cloud depth. Thus it is the saturated positively-buoyant mixtures that 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. 84 1600 1400 |-1200 ?1000 800 600 1600 1400 i-1200 hooo 800 600 1600 1400 5 1000 I 800 600 SP SN UN UP ' flux (W) x10 b) - 5 0 e flux (W) x 10 o) 0 i flux (W) 2000 ' 1 £ 1 5 0 0 i V \ 1 : • 1 1000 I -5 0 9 flux (W) 2000 , E 1500 • 1000 e 1500 • 1000 x10 e) x 10 Figure 4.15: Vertical profiles of Q\ flux partitioned into contributions from the 4 mixture categories. Legend labels as in Figure 4.9. x 10 Figure 4.16: As in Figure 4.15, but for qt flux. 85 Siebesma and Cuijpers (1995) used an LES simulation of B O M E X convection to examine how well the turbulent flux of a conserved variable x £ {®u Qt} can be represented by a top-hat model of the form: W X = —{Xc-Xe) P (4.3) where M is the cloud vertical mass flux, Xe is the cloud environment mean and Xc the cloud mean with cloud determined by one of three different criteria based on gridcell liquid water content qc, vertical velocity w, and buoyancy B. They label the three criteria 2000 r 1600 1400 § - 1 2 0 0 £ •5 1000 X 800 600 1600 1400 § -1200 £ •5 1000 X 800 600 1600 1400 § -1200 £ •5 1000 x 800 a) -1 0 1 2 3 4 Buoyancy flux (W) x ^ 6 b) 0 1 2 3 4 Buoyancy flux (W) x 1 Q 6 c) 600 -1 0 1 2 3 4 Buoyancy flux (W) x I Q 6 E 1500 X 1000 2000 E 1500 £ X 1000 d) 0 1 2 3 4 Buoyancy flux (W) x -| Q 7 e) 0 1 2 3 4 Buoyancy flux (W) 1 Q 7 0 1 2 3 4 Buoyancy flux (W) x Figure 4.17: As in Figure 4.15, but for 9V flux. as the cloud decomposition (qc > 0), the updraft decomposition (w > 0 and qc > 0) and 86 the cloud-core decomposition (w > 0 and qc > 0 and B > 0). Siebesma and Cuijpers found that (4.3) gave the best approximation to the LES flux when Xc was determined by the cloud-core decomposition. The underlying reason for this can be seen in Figures 4.15 and 4.16: 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 ft, qt transport. 1600 1400 £.1200 21 £ 1000 800 600 a) -15 -10 - 5 e. flux (W) 1600 1400 I! 1200 £ 1000 800 600 b) x 10 2 4 q ( flux (W) x 10 ^ flux (W) x 10 2 4 q ( flux (W) x 10 Figure 4.18: Vertical profiles of ft flux (left panels) and qt flux (right panels) averaged over individual cloud lifetimes for clouds A-F . Upper panels show small clouds A, B and C, lower panels show large clouds D, E and F. While the saturated positively buoyant mixtures may provide a good approximation to the net ft and qt fluxes, they fail to adequately represent the buoyancy flux. Figure 4.17 shows that the buoyancy flux in the upper 1/3 of all clouds is dominated by unsaturated 87 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 4.18 shows the total lifetime-averaged 8\ and qt fluxes for the 6 clouds. The 61 fluxes for small clouds tend to monotonically increase with height while qt fluxes tend to monotonically decrease with height. Except near cloud-base, this vertical distribution of the small cloud 81 and qt fluxes indicates cooling and moistening of the environment throughout their cloud depth. In contrast, the large clouds tend to have minimum 81 and maximum qt fluxes at their mid-levels, indicating cooling and moistening at upper part of cloud layer and inversion layer and warming and drying at lower cloud layers. Note however, that individual non-precipitating clouds only transport &i and qt; the vertical integration of the convergence of 8\ and qt flux must be zero. This indicates that 81 and qt fluxes must vanish somewhere below cloud-base. Since individual clouds are only the visible part of subcloud layer thermals that are rooted into the surface layer (e.g., LeMone and Pennell, 1976) this suggests that these small clouds essentially have transport behavior similar to that of the large clouds, i.e., warming and drying their lower penetration depth and cooling and moistening their upper penetration depth. Here the penetration depth includes both the cloud layer and subcloud layer; for the small clouds the lower penetration depth is limited to the subcloud layer. The corresponding tendency profiles due to the transport of each cloud are shown in Figure 4.19. The panels show the magnitude of the heating/cooling and drying/moistening rates found from the flux profiles of Figure 4.18 assuming individual cloud fluxes extend the model horizontal domain area,(6.4 x 6.4km2). As that figure showed, small clouds cool and moisten throughout nearly their entire depth except near cloud-base. The av-erage values for these tendencies are approximately 0.02 K d a y - 1 and 0.07 g k g - 1 day - 1 for the three smallest clouds. The large clouds have heating rates within the cloud 88 layer that are roughly 10 times larger than those for the small clouds, and drying rates about 5 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 1600 1400 £1200 £ 1000 800 600 dO/dt (Kday"') -0.2 b) dq/dt (gkg- 1day- 1) 0.2 -0.5 0 de/dt (Kday~n) 0.5 0 0.5 ,1 .1.5 dq/dt (gkg- 1day- 1) Figure 4.19: Vertical profiles of environmental Q\ and qt tendencies due to the convective transport of clouds A - F over their lifetimes. Upper panels show small clouds A-C; lower panels show large clouds D-F. strong cooling and moistening with a peak cooling and moistening rate of approximately 0.5 — I K day - 1 , 1 — 2 g k g - 1 day - 1 near the inversion base. Comparing these values with the equilibrium large-scale forcing shown in Figure 3.2c,d indicates that approximately 4 - 8 large inversion penetration clouds are needed at any given time in the model domain. To counteract the cloud-base warming and drying caused by the large clouds, approxi-mately 40 - 80 small clouds are needed. These rough estimates are consistent with the 89 snapshots of the simulated cloud fields (not shown). In general, the differing role of small and large clouds in the cloud-layer heat and mois-ture budgets revealed from the LES are consistent with the results of Esbensen (1978), who combined a laterally entraining plume model to represent clouds 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. 4.4 Sensitivity As discussed in Section 4.2.3, while the threshold values of A8Vto = 0.2 K and w0 — 0.5 m s - 1 for determining the detrained mixed-region 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 A8V 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 trans-port (although it does dramatically influence the calculated volume of the unsaturated convective-mixed region for the large clouds). In this section, I examine the impact of the threshold values of A8Vto and wo on both the time evolution of the unsaturated convective region and its associated vertical mass transport. Figure 4.20 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 (A0„ i O , w0) thresholds in the range [0-1.2 K, 0-1.2 m s - 1 ]. When both A8V,0 and wo are chosen to be zero (case ul), 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 90 x 1 0 ? (u1):4.1, (u2):1.7, (u3):0.6, (u4):0.2 0 x 10 500 1000 Time (seconds) (u1):4.1, (u2):1.6, (U3):0.6, (u4):0.2 1500 2000 1800 1600 £1400 x: 1200 1000 800 600 Ds Du1 . - - Du2 ' - • Du3 Du4 b) >^  • [j I -II 1 \ '• 1" \ 11 -~ - • y, --2 0 2 Mass flux (kgs - 1) -2 0 2 Mass flux (kgs - 1) 500 1000 Time (seconds) -2 0 2 Mass flux (kgs - 1) x 10 Es Eu1 _ _ C O d) ' - ' Eu3 • • " Eul / ' 1 \ » 1 \l : t jf\ - -7 1 \ i : S ^ i x 10 Fs Fu1 - - Fu2 Fu3 Fu4 f) /> \ \ 1 » \ •:-\\ . . Y . » . . i » i i J \ r x 10 Figure 4.20: Sensitivity of the volume of the unsaturated convective-mixed region and the corresponding vertical profile of the vertical mass flux to the choice of (A0V^, WQ) thresh-old, with the saturated region shown for comparison. The 3 letter legend codes denote the cloud (D-F), saturated/unsaturated region (s/u) and one of 4 threshold choices (1-4). The numbers at the top of panels a, c, and e give the ratio of the unsaturated/saturated volume for each threshold choice. The threshold pairs are: ul) (0 K, 0 m s - 1 ) , u2) (0.4 K, 0.4 ms" 1), u3) (0.8 K, 0.8 ms" 1), u4) (1.2 K, 1.2 ms" 1). a) Volume ofthe un-saturated convective-mixed region for cloud D. b) As in a) but for the corresponding vertical profile of the vertical mass flux, 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. 91 slight increase in the net downward vertical mass flux, compared to threshold pair u2. This indicates that the detrained mixed region does not contribute significantly to the net vertical mass flux. As the threshold pair is increased beyond A9Vt0 > 0.4 K and WQ = 0 .4ms - 1 (cases u3-u4), 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 with height). Within the inversion (above 1500 m) the mass flux due to unsaturated downdrafts is insensitive to these 4 choices of threshold value, although the time-integrated volume of unsaturated air is reduced to only 20% of the saturated volume for the highest threshold pair (case u4) for clouds D and E and 40% 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 20-40% of size of the saturated cloud, supporting the idea that unsaturated downdrafts are confined within a small volume and are truly part of the cloud convection. 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. The choice of threshold in Section 4.2.3 (approximately case u2 in Figure 4.20) eliminates the large majority of detrained mixtures while including nearly all of the convective transport. 4.5 Discussion The vertical mass-flux profile of a trade-wind cumulus boundary layer results from the contributions of many individual cumulus clouds. I 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. W i t h the help of a passive tracer, I have par-titioned the cloud life-cycle transport into saturated and unsaturated components, and have shown that on average cloud convection occurs in a region with a time-integrated 92 volume roughly 2-3 times that of the liquid water-containing volume. All six clouds ex-hibit 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 dissi-pation stage (Figures 4.6 and 4.7). Vertical profiles of the partitioned cloud-environment differences A6i and Aqt show that the unsaturated negatively buoyant mixtures are con-sistently cooler and moister than the environment, indicating that the downdrafts are driven by evaporative cooling (Figures 4.13 and 4.14). This unsaturated negatively buoy-ant 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 lower levels. However, small and large clouds have distinct vertical profiles of heating/cooling and drying/moistening rate to the environmental atmosphere, with small clouds cooling and moistening throughout their depth, while larger clouds cool and moisten at upper cloud depth and heat and dry at lower cloud depth. When the cloud vertical depth is extended to include the subcloud layer, small and large clouds show similar transport profiles; both cool and moisten their upper penetration environment and warm and dry their lower penetration environment. 4.5.1 Compar i son w i t h entraining p lume and E M B S models These LES 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; Neggers et al., 2002). An entraining plume model represents an individual convec-tive 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 at its NBL (see Ap-93 pendix F for details about this calculation). 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 (see Chapter 2 and 3 for details about this calculation). Both conceptual models are highly simplified pictures of the transport associated with the individual convective elements of real cumulus clouds. Normalized mass flux Normalized mass flux Figure 4.21: Normalized vertical profiles of vertical mass flux produced by a spectrum of convective elements ascending through the simulated equilibrium B O M E X environment. The three thick solid lines have roughly the same vertical height as small, middle, and large B O M E X clouds (or convective elements), a) Element profiles calculated using an entraining plume model; each entraining plume stops at its N B L with respect to the environment, b) As in a) but with each element calculated using an EMBS model. Figure 4.21 shows the very different mass-flux profiles produced by the entraining plume and EMBS approaches. For each panel I use the equilibrium B O M E X environ-mental soundings. The individual lines of Figure 4.21a show the mass flux profiles for 15 different choices of entrainment rate ranging from 0.0001 to 0.003 m _ 1 , normalized by the mass flux at cloud base. Figure 4.21b shows the corresponding mass flux profiles for an EMBS model in which a subcloud air parcel with unit-mass flux (1 gm~ 2 s - 1 ) is lifted to 94 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 el-ement 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 in which the mass flux profile changes with the height/size of cloud or convective element. 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 4.6 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 4.6 and Figure 4.21b shows that an EMBS model qualitatively captures the principal features of the mass transport of these simulated individual clouds/convective elements. In particular, the model-produced net vertical mass flux is downward near cloud-top and upward near cloud-base. The pre-diction 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 95 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. 2000 1800 1600 £1400 co £ 1200 1000 800 600 -2. de/dt (Kday~K r — c ) . i — • — 0 5 , , 1 0 dq/dt (gkg-1day-1) 15 600 -0.5 0 0.5. . 1 dq/dt (gkg"1day-1) Figure 4.22: Vertical profiles of di and qt tendencies produced by the spectrum of convec-tive elements of Figure 4.21. The cloud-base mass flux for each element is 1 gm~ 2 s _ 1 . a) and c): entraining plume model, b) and d): EMBS model. Figure 4.22 shows an example of the 6i and qt tendencies produced by the spectrum of entraining plumes and EMBS elements in Figure 4.21, assuming a vertical resolution of 100 m and a unit subcloud convective element with mass flux equal to 1 g m _ 2 s _ 1 . The magnitude of both heating/cooling and drying/moistening is significantly larger in this entraining plume element than in the EMBS element. In this sense, the transport 96 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 produces different heating and drying profiles) but also in the way that they detrain mixed parcels. An entraining plume element detrains all mixed parcels at a single level at cloud-top and produces very large, concentrated cooling and moistening. In contrast, the EMBS model spreads this cooling over the upper-half of its ascent through downward transport of evaporated mixtures. Comparison with Figure 4.19 shows that the EMBS model prediction is again more consistent with the LES results. 4.5.2 Unsaturated convection The simulation indicates that the decay of individual clouds of convective elements is always associated with a significant amount of downward mass transport, which begins with the collapse of the ascending turret and is further enhanced by mixing and evapora-tion. As shown in Figure 4.6, many of these downdrafts are unsaturated; they dominate the overall mass transport near the individual cloud-tops. This result is consistent with the EMBS representation and is also consistent with the diagnostic results in Chapter 3. In Chapter 3, I used a B O M E X convective equilibrium sounding and the large-scale forc-ings to diagnose the vertical profile of the cloud-ensemble mass flux. I found that this diagnosed cloud mass flux is significantly smaller than the saturated cloud mass flux ob-tained from the LES in Siebesma et al. (2003). In particular, the EMBS model-diagnosed cloud vertical mass flux is downward within the inversion layer and upward within the cloud layer, while the LES saturated cloud mass flux is upward throughout both the cloud and inversion layers. I attribute this difference to the different definitions of cloud boundary in the two models. In the EMBS model, the modeled clouds include not only the liquid water cloud but also the unsaturated convective region, which can play an 97 important role in the vertical mass transport. In contrast the cloud mass flux obtained in the LES of Siebesma et al. (2003) is calculated only for the liquid water containing gridcells. In this chapter I introduce a subcloud-layer tracer into the LES to explicitly track the convective mixed region associated with individual clouds. Figure 4.6 shows that by including the unsaturated convective region in the vertical mass flux calculation, all of the inversion penetrating clouds do indeed produce net downward mass flux within the inversion. For non-precipitating cumulus clouds all condensed water must be evaporated over a very short life-cycle consisting of a growth and a dissipation stage (Figure 4.7). The EMBS model attempts to represent this growing phase through adiabatic lifting of sub-cloud air parcels, and the dissipation phase through mixing, evaporation and the descent of mixtures to their unsaturated NBLs. As shown in the partitioned mass flux (Fig-ures 4.9), for these six clouds the growth phase is dominated by transport associated with positively buoyant saturated mixtures while the dominant transport in the dissi-pation phase is via unsaturated negatively buoyant mixtures; in both phases of the life cycle mixture transport tends to be buoyancy-direct. However, a significant amount of buoyancy-indirect transport can also be seen in Fig-ures 4.9 and 4.11. In particular, saturated negatively-buoyant mixtures on average trans-port air upward while unsaturated positively buoyant mixtures on average transport air downward. This buoyancy-indirect transport is primarily associated with cloud-mixture inertia and momentum mixing and may be important for large clouds at higher cloud layers. The significant amount of buoyancy-indirect transport indicates that an EMBS model, which transports every mixture based solely on its buoyancy, could potentially overestimate the buoyancy flux. This result also supports a recent parameterization ap-proach by Bretherton et al. (2003) who modified the buoyancy-sorting model of Kain and Fritsch (1990) by explicitly including a fraction of negatively buoyant air into their 98 parameterized bulk entraining and detraining updraft. An important fact associated with this counter-buoyancy transport is that mixing continues after the individual mixtures penetrate their NBLs, so that a significant number of saturated mixtures become unsaturated prior to, or just after, the transition to negative velocity. Under the usual definition of cloud, these transformed unsaturated mixtures are typically excluded from the calculation of the cloud mass flux and are treated as environmental air. The fact that the existing literature reports relatively few downdrafts in numerically simulated clouds is consistent with this mixture exclusion (e.g., Siebesma and Cuijpers, 1995; de Roode and Bretherton, 2003). Finite resolution and the bulk representation (all or nothing) of grid-scale saturation in numerical models tends to overestimate the rate of cloud evaporation, so that some of these seemingly unsaturated mixtures may actually be saturated in real clouds. However, the dominant role of the unsaturated mixtures in this simulation strongly suggests that unsaturated downdrafts should also be an important feature of real clouds. 4.5.3 Role of cloud-size distribution in cloud-ensemble trans-port The cloud-ensemble mass flux is modulated by both the cloud-size distribution and the transport associated with individual cloud types (heights); a cumulus parameterization scheme needs to account for both (Arakawa and Schubert, 1974). From a diagnostic point of view, knowing one helps to deduce the other, given the observed net effect of the cloud ensemble on the large-scale flow (i.e., the heating/cooling and drying/moistening rates). As Figure 4.19 shows, the simulated large clouds cool and moisten near cloud-top while heating and drying near cloud-base. In contrast, small clouds tend to cool and moisten throughout most of their depth except near cloud-base level. This result is consistent with the diagnostic results of Esbensen (1978). This result also confirms 99 the diagnostic results in Chapter 3, where I use an EMBS model with a simple constant eroding rate to predict the vertical profile of convective tendencies of individual clouds with different size/height. As discussed there, the size-differentiated vertical profile of heating/cooling and drying/moistening rate can be used to understand the observed cloud-size distribution and the resulting cloud-ensemble transport. The simple EMBS diagnostic results of Chapter 3 and the more realistic explicit simulations reported in this chapter paint the following conceptual picture of equilib-rium convection with a size-distributed cloud ensemble. During the adjustment toward equilibrium, smaller clouds precondition the environment by continuously cooling and moistening their upper environment. In this way, future ascending subcloud air is sub-ject to less evaporation and becomes more buoyant and therefore is able to reach higher levels. However, taller clouds, once developed, tend to heat and dry their lower envi-ronment and therefore counteract the effect of small clouds. As shown in Figure 4.19, large clouds have heating and drying rates considerably larger than small clouds and therefore, would suppress convection in a population with similar numbers of large and small clouds. If convection is to be sustained, more numerous small clouds must coun-terbalance the heating and drying effect of large clouds within the cloud layer. When the cloud ensemble reaches equilibrium the heat and water vapor circulation can be de-scribed as follows: the smallest "forced clouds" (Stull, 1985) feed only from subcloud layer thermals and are suppressed by all larger clouds, while clouds of intermediate size are supported by both subcloud layer thermals and the smallest clouds. In this way, the largest clouds feed from the subcloud layer not only directly from associated penetrative ascending thermals, but also indirectly from all smaller clouds. An entire population of clouds must work in concert to transport and redistribute heat and water vapor out of the subcloud layer and through the cloud layer. Thus, the cloud-size distribution can be seen as the result of the equilibrium large-scale forcing and the individual cloud dynamics. 100 Chapter 5 Life Cycle of Numerically Simulated Shallow Cumulus Clouds. Part II: M i x i n g Dynamics 5.1 Introduction This chapter continues the investigation of the life cycle of numerically simulated shallow cumulus clouds. In Chapter 4 I have described the LES model and case setup, the isolation of the simulated life-cycle data for six individual clouds from the LES cloud field and the use of a subcloud layer tracer to identify the cloud-mixed convective region. In this chapter, I use the same set of six clouds to examine the detailed mixing dynamics in these simulated clouds, in particular the mixing associated with the active ascending cloud elements. The 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 shallow cumulus parameterizations. Historically, most of the detailed numerical studies of cumulus mixing mechanisms focus on the single cloud simulation. However, as I have described in Chapter 1, this 101 type of simulation suffers from unrealistic representations of boundary-layer turbulence and the ambient environment. In contrast, the ensemble cloud simulation conducted in Chapter 4 generates clouds in a more natural way. The literature contains few detailed investigations of the individual convective elements embedded in the ensemble cloud field. My analysis in this chapter complements the in-situ cloud observations, laboratory tank experiments, and previous numerical simulations which, as discussed in Chapter 1, have motivated a variety of conceptual models of cumulus mixing. In Section 5.2 I present the pulsating behavior of the simulated cumulus growth. The cloud growth rate and thermal turn-over time are quantified in Section 5.3. In Section 5.4 I present the cloud kinematic structure, cloud-top vortical circulation, and an elevated tracer experiment for cumulus entrainment. Section 5.5 links the cloud-top vortical circulation to the baroclinic torque and presents the perturbation pressure field and its physical interpretation. The dilution rate and the mixture distribution in the ascending cloud-top is quantified in Section 5.6. The continued mixing between low-buoyancy, low-vertical-velocity cloud mixtures and the environment is briefly described in Section 5.7. In Section 5.8 I present a picture of cumulus mixing that is consistent with the simulation results. 5.2 Pulsating cloud growth Animations1 of clouds A - F show that an individual cloud consists of one or more coherent updraft pulses/thermals which grow and decay over 10 to 15 minute periods. Small clouds (A, B, C) have a single ascending pulse which dissipates without reaching 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 Animat ions available from h t tp : / /www.eos .ubc .ca / research /c louds 102 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 elongated 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 clearest, beginning at about 400 m above the cloud-base (as 1 km above the surface). All of the pulse elements tend to become dynamically detached 2 from their subcloud layer parent thermals some distance above cloud-base. 0 -0.5 g -1 cr>— < -1.5 -2 -2.5 i 1.5 1 0.5 0 -0.5 a) ''/\ / < "0 500 1000 1500 Time (s) A f\ \ \ | * \ \ /v\ A < 1 b) \ \j >v 6 _ 4 I C / ) ^2 500 1000 1500 Time (s) A <i) - A / \ "0 500 1000 1500 Time (s) 500 1000 1500 Time (s) Figure 5.1: An example of the pulsating character of cloud E. a) Dashed line: time evolu-tion of A#;, the 6i difference between liquid-water cloud horizontal mean and environment at height 1237.5 m. Solid line: the 8i difference between the most undilute cloud gridcell and environment, b) As in panel a) but for qt. c) As in panel a) but for 8V. d) As in panel a) but for vertical velocity w. 2Here, by dynamically detached, I 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. 103 Figure 5.1 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 seconds it brings a large proportion of relatively undilute subcloud air, producing the largest anomalies of liquid water potential temperature ft, total water qt, virtual potential temperature ft and vertical velocity w for both the cloud mean value (dashed-line) and the most undilute cloud gridcell (i.e. the grid-cell extreme value) (solid-line). Between 500 and 800 seconds these values quickly decrease (ft increases) as more dilute cloud mixtures with low vertical velocities are left in the trailing wake of the advancing top. Further 2000 -^1500 x 1000 500 500 1000 1500 Time (s) 2000 -£-1500 500 500 1000 1500 Time (s) 2000 2000 500 500 1000 Time (s) 1500 500 500 1000 Time (s) 1500 Figure 5.2: Time-height variation of the differences (Aft, Aqt, Aft , Aw) for the most undilute gridcell for Cloud E. a) ft (K) difference between the most undilute gridcell and environment, b) As in a) but for qt (gkg - 1 ) . c) As in a) but for ft ( K). d) As in a) but for vertical velocity w ( m s - 1 ) . mixing with environmental air tends to continually dilute these cloud mixtures until the 104 next ascending thermal penetrates this level and brings a new anomaly at approximately 1000 seconds. Cloud E dissipates following the second pulse. Figure 5.2 shows this pulsating structure at all cloud levels as time-height contours of 6[, qt, 6V and w of the most undilute cloud parcel. The two growth pulses, which are spaced approximately 500 seconds 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 ascends into the remnants of the previous thermal. Due to this difficulty in separating a pulse from the remnants of previous thermals, I limit the following analysis to the first thermal for each cloud. This initial thermal is the simulated counterpart to the ascending turret of a developing cloud, and will be referred to below as the ascending cloud-top (ACT). The sharp contrast between an A C T and its environment makes it easy to isolate, particularly with the use of a subcloud layer tracer as described in Section 4.2.3. Animation of the entire simulated cloud field shows that pulsating growth is a char-acteristic feature of all simulated clouds. This is also consistent with numerous obser-vational studies describing the growth of cumulus clouds either as individual bubbles or as a collection of multiple bubbles (e.g., Ludlam and Scorer, 1953; 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 cumulus clouds generated from a conditionally unstable environment with specified surface heating (e.g., Carpenter et al., 1998a). These single cloud simulations, however, typically discard the early phase of the convective clouds, in which the A C T is most easily isolated. This is necessary due to the unrealistic boundary layer turbulence and ambient environment caused by the models' initial lack of motion on all resolvable scales (Carpenter et al., 1998a). In contrast, for the LES-ensemble cloud simulation presented here, individual clouds develop from a 105 fully turbulent boundary layer that is in equilibrium with the large-scale environment. With the isolation technique described in Section 4.2.2, I can obtain uncontaminated 5-dimensional simulation data for the A C T of each selected cloud. These pseudo-clouds provide an excellent opportunity to investigate the kinematics and dynamics of the A C T and its mixing behavior. In the following sections I begin with a description of the bulk properties of the A C T and then examine in detail the kinematic structure and dynamics of the A C T for clouds A-F . 5.3 Growth rate and thermal turn-over timescale The cloud-top height zr(t) (the maximum height which contains liquid water at time t) was shown in Figure 4.4a to increase at an approximately exponential rate during the growth phase of individual clouds. To quantify the difference between clouds A - F , Figure 5.3a reproduces Figure 4.4a in semi-logarithmic coordinates. There is a near-linear dependence of log(zx) cm time t for all clouds, motivating the definition of the A C T growth rate a as a=dJ^A (5.1) dt where a is determined by a linear fit to d(\ogzr)/dt from Figure 5.3a during the growth phase. A vertical length scale of an A C T may be chosen as the maximum height ZT,max achieved by the A C T . Figure 5.3b shows the variation of growth rate a 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 Figure 5.3b is a linear least-squares fit of a to ZT,max for the six clouds. Since the growth rate a is nearly constant during an individual ACTs' ascent (Figure 5.3a) the thermal turn-over time £* can be computed by integrating (5.1): *. = r,m" - dlog W = - l o g ( ^ = ) (5.2) JzTi0 a a ZT,0 106 where ZT,O = ZB represents the cloud-top height when it first emerges from subcloud layer. Figure 5.3c shows the A C T thermal turn-over time for the six clouds; with the exception of cloud C all clouds have approximately the same thermal turn-over time of 600 s (solid line). 1000 1200 1400 1600 1800 2000 2200 r,maxv ' Figure 5.3: a) Time evolution of the ascending cloud-top height ZT for clouds A-F . b) The growth rate a versus maximum cloud-top height zr,max for each cloud, c) The thermal turn-over time £* versus maximum cloud-top height zx,max for each cloud. Here the thermal turnover time represents the average time required for air in a turret (ACT) to travel from cloud-base to maximum cloud-top. Neggers et al. (2002) used instantaneous snapshots of LES shallow cumulus cloud field to determine a similar timescale for a given idealized parcel ascending through a cloud's depth. They averaged 107 the vertical velocity along the 3-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 r c , the time required 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 T c = 300 s to be approximately independent of cloud depth. When I adopt the approach of Neggers et al. for my six clouds I also find an approximate time constant of 350 s that is, like £*, roughly independent of cloud depth. The larger value of £* determined from Figure 5.3 indicates that air contained within the coherent structure of the A C T 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 A C T , rising on average with about half the speed of the major updrafts. In the next section I show this coherent circulation more explicitly. 5.4 Kinematic structure of the A C T In Section 5.2 I showed that the numerically simulated cumuli tend to grow in a series of intermittent ascending pulses/thermals that collapse after reaching their maximum height. These thermals have a spatial scale smaller than the individual 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 ensemble, and therefore, play a fundamental role in cumulus convective transport and cloud environment mixing. Below I examine in detail the kinematic structure of the first ascending thermal, i.e., the A C T of the simulated clouds. I use cloud E as an example to illustrate characteristic features present in all 6 selected clouds. Figure 5.4 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 minutes after emergence from the subcloud layer. 108 Figure 5.4: Vertical cross section (x-z slices cut following the y=650 m lines in Figure 5.5) of the internal flow pattern within the A C T of cloud E at 7.5, 8.5, 9.5, 10.5 minutes after emerging from the subcloud layer. Solid line: qc > 0.01 g k g - 1 contour. Arrows: u-wwind vectors for the convective mixed region defined in Section 4.2.3. (wind vectors outside this region are masked) The ambient mean wind UQ = -7.5 ms" 1 has been subtracted from horizontal velocity u while the cloud-top vertical velocity wj 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 Figure 4.1c. The arrows at the top-left corner of each panel show the vertical velocity scale (3.8 m s - 1 ) . Shading: the mixing ratio of a tracer r\ which is initially uniform (rj = 1 gkg - 1 ) between 1100-1200 m at the 7.5 min timestep. Dashed lines of constant x indicate the x coordinates of the y-z slices of Figure 5.5. Dashed lines of constant z indicate the z coordinates of the x-y slices of Figure 5.6. 109 The ambient mean wind UQ = —7.5 m s - 1 and the vertical velocity of the ascending cloud-environment interface, %(t) (where the subscript T indicates the current cloud-top height) have been subtracted from the corresponding velocity field. The u>T(t) at each time step is found using the growth rate ag = 0.0017 s - 1 from Figure 5.3b and the current cloud-top height ZT via (5.1), i.e., writ) = dzr(t)/dt = aEZr(t). A distinct vortical circulation exists within the top part of the growing cloud [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 (de-noted as CT-relative below) is generally near zero or negative (i.e., Figure 5.4a at 800 m, 5.4b at 900m, 5.4c at 1000 m and 5.4d at 1200 m). Air parcels located below these levels generally cannot approach the moving cloud-top given this resolved flow. Motivated by this distinct kinematic structure, I define the A C T as the upper part of the growing cloud; it encompasses the coherent vortical circulation. I will refer to the region below the A C T , which is more dilute and has low vertical velocity, as the trailing wake of the advancing A C T . 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 A C T . In addition, I will refer to the shallow, frontal cap of the A C T , which extends about 4-6 model levels below the cloud-environment interface and has strong horizontal divergence, as the ascending frontal cap (AFC). As Figure 5.4 shows, there is a strong updraft core in the center of the A C T with maximum vertical velocity approximately twice that of the A F C mean vertical velocity (or equivalently, WT)- The core updraft velocity is gradually reduced to near zero (CT-relative) within the A F C which results in strong horizontal divergence. Near the edge of the A C T there is a distinct overturning circulation with downward CT-relative vertical 110 velocity roughly equal to that of the central updrafts. The overall internal circulation is weak when the A C T just emerges from subcloud layer (Figure 5.4a), is gradually strengthened in the middle of the cloud layer (Figure 5.4b,c,d) and tends to be rapidly weakened after penetration of the inversion layer. C l o u d E: 7 .5 m in (w =1.9 m s " 1 ) C l o u d E: 8 .5 m in (w T =2.1 m s " 1 ) y (m) y (m) Figure 5.5: As in Figure 5.4 except these are y-z cross sections cut following the x=550 m lines in Figure 5.4. Arrows: v-w wind vectors for the convective mixed region defined in Section 4.2.3. The ambient mean wind v0 is from right (north) to left (south) with the shear direction point to north (see Figure D.3d). VQ is small and are not subtracted from v. The arrows at the top-left corner of each panel show the scale for vertical velocity 4.1 m s - 1 . Dashed lines of constant y indicate the y coordinate of the x-z slices of Figure 5.4. Dashed lines of constant z indicate the z coordinates of the x-y slices of Figure 5.6. Before I discuss how the vortical circulation influences the cloud entrainment and mixing, I show in Figure 5.5 the vortical circulation in y-z cross sections cut along the x = 111 550 m lines of Figure 5.4. Clearly, the coherent A C T vortical circulation is 3 dimensional although it is not axisymmetric; the downshear side circulation is more complete while the upshear side circulation tends to be weakened (note the ambient wind shear in north-south direction diminishes above 1 km see Figure D.3c). I will discuss this asymmetrical A C T vortical circulation and the ambient shear later in this section and in Section 5.5. To examine how the vortical circulation influences ACT-environment mixing, I release a tracer q with a uniform mixing ratio of rj — 1 g k g - 1 in an elevated layer between 1100-1200 m prior to the A C T penetration of this layer (Figure 5.4a). Figure 5.4b shows that when the A C T reaches the tracer layer it displaces ambient fluid upward while ambient q is incorporated into a thin layer by direct contact with the A C T surface. The entrained air does not, however, continually erode into the center of the A C T , but is instead advected downward along the edge to the rear of the A C T (e.g., Figure 5.4b and 5.4c, at 1100-1200 m) leaving a nearly ?7-free A F C after the penetration (Figure 5.4c and 5.4d). Environmental air entrained through the A C T top, after being circulated to the A C T rear, is further mixed into the A C T center through the vortical circulation (Figure 5.4d, downshear side between 1200-1400 m). The strong A F C divergence prevents any deep erosion of the A C T core from the A C T top, while the circulation along the A C T edge tends to quickly remove the more dilute cloud mixtures from the A F C , leaving the A F C 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 co-authors (e.g., Blyth et al., 1988; Blyth, 1993). Environmental air is also directly engulfed into the A C T from the A C T rear through an organized vortical circulation on a iarge scale (comparable to A C T size). Figure 5.6 shows x-y cross sections of horizontal velocity field at two different vertical levels and three different times. Figure 5.6a,b 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. 112 At 8.5 min this 1137.5 m level is 75 m below the current cloud-top height [2x(£)=1212.5 m]. The flow field reflects the A F C mass divergence; no clear resolvable-scale entraining eddies appear. The mixing along the edge of cloud and environment is primarily through the parameterized subgrid-scale eddy diffusion. At 9.5 min this same level now samples the A C T rear [2>r(£)=1362.5 m], Figure 5.6b shows the clear indication of resolvable-scale entraining eddies. The circulation along these horizontal slices indicates that the vorticity E: h=1137.5 m (8.5 min, 2^=1212.5 m) 300 400 500 600 700 800 x(m) E: h=1287.5 m (9.5 min, ^=1362.5 m) 300 400 500 600 700 800 x(m) E: h=1137.5 m (9.5 min, ^=1362.5 m) 300 400 500 600 700 800 x(m) E: h=1287.5 m (10.5 min, 2^=1537.5 m) 300 400 500 600 700 800 x(m) Figure 5.6: Horizontal cross section cut following the horizontal lines shown in Figures 5.4 and 5.5. Solid line: qc > 0.01 g k g - 1 contour. Arrows: u-v wind vectors. The ambient mean wind UQ — -7.5 ms" 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 s - 1 . Shading: the liquid water mixing ratio qc ( gkg - 1 ) . Dashed lines of constant y indicate the y coordinate of the x-z slices of Figure 5.4. Dashed lines of constant x indicate the x coordinate of the y-z slices of Figure 5.5. 113 vector is not strictly along the horizontal plane (Figure 5.4, Figure 5.5) as predicted for an idealized vortex ring or a spherical vortex, but also has a vertical component (Figure 5.6). These entraining eddies have scales comparable to the A C T size itself and they directly engulf the environmental air and move it deep into the A C T center. Figure 5.6c,d show horizontal slices through another vertical level (1287.5 m) at 9.5 min and 10.5 min. We observe a pattern similar to that shown in Figure 5.6a,b. Notice that the engulfment of environmental air from the A C T rear is located on downshear sides [east (right) of the cloud] in this sheared environment. Overall, there is a net convergence at the A C T rear, in contrast to the net divergence at the A C T top (or AFC) . In contrast to the A C T upper-interface mixing, which is primarily driven by subgrid-scale eddies, the ACT-rear entrainment is primarily through resolvable large-scale en-training eddies. In comparison these large-scale eddies associated with the vortical cir-culation must dominate the entrainment and mixing of the simulated clouds throughout their ascent. Furthermore, the mixed parcels first entrained from either the A C T top or sides are carried by the vortical circulation to the A C T rear, where some of the mixtures are further wrapped into the vortex center while others may stay behind, depending on their vertical velocity and buoyancy. Therefore, the coherent vortical circulation highly modulates the ACT's entrainment and mixing of environmental air. The effect of ambient wind shear on the ACT's vortical circulation and mixing can also be seen in Figure 5.4 and 5.5. As shown in Figure D.3c,d, the vertical shear of the horizontal mean wind is from west (left) to east (right) in Figure 5.4 and from south (left) to north (right) in Figure 5.5. Comparing the four panels in Figure 5.4 and also the four panels in Figure 5.5 shows that when the A C T rises through the sheared environment the vortical circulation is continuously tilted towards the downshear side with the downshear-side circulation enhanced while the upshear side is weakened (note the vertical variation of shear in Figure D.3c,d; the south-north shear diminishes above 1 km). For this reason, 114 the overall A C T mixing and subsequent mixing-induced downdrafts are enhanced at the downshear side of the cloud [note the unsaturated downdrafts (arrows outside the qc > 0.01 g k g - 1 contour) on the downshear side of the cloud in Figure 5.4c,d and Figure 5.5 a,b,c,d). The A C T 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 examining the ambient vorticity field produced by the vertical shear of the horizontal mean wind. For example, in Figure 5.4 the ambient vorticity vector points northward (into the page) and hence has the same sign as the downshear side vorticity but the opposite sign 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 circulation. Note, however that the A C T vortical circulation is 3 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 vertical, which can be seen from the tilting/twisting term of the vorticity equation (e.g., p. 104, Holton, 1992). Therefore, the ambient shear is ultimately responsible for the generation of the horizontal entraining eddies shown in Figure 5.6 b,d. In Section 5.5.2, I will further explore some dynamics of the vortical circulation. Figure 5.7 shows the time-height variation of the horizontal 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, the divergence zone associated with the shallow A F C is evident throughout the A C T ascent between roughly 200-600 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 ~ 700-800 s they 115 2000 1800 1600 If ruoo o x 1200 1000 a) # % 600 ^ 0 ^ 2.0°, *. o <m -4oo * <<? 0 ,op a _AO0 0 i >* 600 800 1000 1200 Time (s) 400 600 800 1000 1200 Time (s) 1000 600 600 400 200 E 0 ght -200 Hei -400 -600 -800 -1000 1000 800 600 400 200 (m) 0 Z a -200 Hei -400 -600 -600 -1000 1000 BOO 6CC 400 200 E 0 !c a> -200 Hei -400 -600 -800 -1000 2000 1800 1600 1400 1200 1000 8 0 0 ^ 200 2000 1800 1600 1200 1000 * O 3oo 8 h, ,400 2.00 0 400 600 800 Time (s) 1000 1200 d) *^ t(fc 0 -200 1000 /JsL f % fi <fi > -200 600 800 Time (s) 1000 1200 200 400 600 800 1000 1200 Time (s) 200 400 600 800 Time (s) 1000 1200 1000 800 600 400 200 0 -200 -400 -600 -800 -1000 1000 800 600 400 200 0 -200 -400 -600 -800 -1000 1000 800 600 400 200 0 -200 -400 -600 -800 -1000 Figure 5.7: Time-height variation of the air-mass horizontal divergence [i.e., contours of k g m - 1 s _ 1 ) = JA —[d(pw)/dz]dA for timestep 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. 116 collapse and accelerate downward, this together with the subsequent arrival of ascending thermals, produces a strong divergence zone near the inversion base. Comparing the left and right panels of Figure 5.7 indicates that this persistent divergence zone occurs primarily in the unsaturated convective-mixed region following collapse of the ACTs. The strong divergence is associated with the detrainment of cloud-environment mixtures at the layer where the cloud buoyancy decreases with height, as can be seen below in Figure 5.11d,e,f. This preferential detrainment is consistent with both the buoyancy-sorting detrainment hypothesis of Raymond and Blyth (1986) and the numerical and analytical result of Bretherton and Smolarkiewicz (1989) and Taylor and Baker (1991) which indicate that cloud detrainment occurs at levels with decreasing cloud buoyancy. This divergence-convergence pattern, which is associated with the vortical internal circulation of an ascending thermal, is most clearly visible for the first thermal, i.e., the A C T . However, a similar, although much weaker, divergence-convergence pattern can also been seen around the top of the second updraft at about 1000 seconds and slightly below the inversion base (e.g., in Figure 5.7a,c). The weakness of this pattern is due both to the reduced buoyancy of the second thermal and to the masking effects of the remnants of the first thermal. While the simulated ACTs present a complex 3 dimensional structure of the vortical circulation, the general pattern, in which ascent is along the central vertical axis and descent is around the periphery, is consistent with the laboratory results of ascending thermals, in-situ observations of cumulus clouds and the conceptual shedding thermal model proposed by Blyth et al. (1988) (Scorer, 1957; Johari, 1992; Stith, 1992; Carpenter et al., 1998c). The impact of ambient wind shear on the simulated clouds is furthermore consistent with observational evidence showing that downdrafts and mixing occur pref-erentially on the downshear side of a cloud, while updrafts tend to persist on a cloud's upshear side, (e.g., Warner, 1977). 117 5.5 Dynamical structure of the A C T To obtain more insight into the internal circulation and mixing behavior of the A C T , we need to further examine its dynamics. In this section, I first examine the buoyancy field in the simulated A C T and explain the simulated vortical circulation using the concept of baroclinic torque. I then further examine the perturbation pressure field and partition it into dynamic and buoyancy component following Wilhelmson and Ogura (1972) and present a physical interpretation to the dynamic perturbation pressure. 5.5.1 Buoyancy distribution and baroclinic torque I begin with the LES model's momentum equation in tensor form (Khairoutdinov and Randall, 2003): duj = duj | ldpUjUj = 1 dp | ^ ldrl3 | / duA _ dt dt p dxj p dxi 1 p dxj \ dt J l s The terms on the right side are respectively the nonhydrostatic perturbation pressure gradient force (PPGF), the buoyancy force, the convergence of momentum flux due to unresolved subgrid scale eddies and the model-specified large-scale forcings. No large-scale forcing is applied to vertical momentum in this LES. Here all terms and variables represent the model gridcell mean, p = p(z) is the base state mean density which varies only with altitude, (i=l, 2, 3) are the resolved wind components along the x, y and z directions respectively, p is the nonhydrostatic perturbation pressure and buoyancy is defined as B = —g{p'/p) ~ <~[T P^ + 0.608(^ — (qv)) — qc], which includes the effect of liquid water (qc) loading (Emanuel, 1994). The vapor mixing ratio is denoted by qv, T is temperature and (•) represents the horizontal domain average, is the subgrid scale momentum pUi flux in the j direction (see Appendix E). The subscript l.s. denotes the large-scale forcing, 5^ is the Kronecker delta and t is time. Figure 5.8 shows the buoyancy contours of the cross-section of Figure 5.4c. We see 118 a mushroom-shaped buoyant core (e.g., 0.03 m s - 2 contour), which consists of the A F C centered at 1300 m and a trailing stem located within the A C T between 1100-1250m. There is a strong horizontal buoyancy gradient, particularly at the downshear side (Note 1 4 5 0 r 1 4 0 0 -1 3 5 0 -1 3 0 0 : 1 2 5 0 -> § 1 2 0 0 -N 1 1 5 0 -1 1 0 0 -1 0 5 0 -1 0 0 0 -9 5 0 -2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 x(m) Figure 5.8: The cross-section of Figure 5.4c but with contours of the buoyancy B ( m s - 2 ) . The convective mixed region is shaded. Thick dotted line: qc > 0.01 g k g - 1 contour. Arrows are the same as in Figure 5.4c. the negatively buoyant area at the downshear side of cloud edge). During the ascent of the A C T , more buoyant parcels at the A C T center tend to move rapidly upward while the less buoyant parcels at the A C T edge tend to move slower. This buoyancy difference provides a net torque to rotate the cloud parcels clockwise at the right side and anti-clockwise at the left side of the buoyancy maximum in Figure 5.8. Below I explain the A C T vortical circulation with the baroclinic torque using the 2-dimensional momentum equation under the Bousinesq approximation. This is valid for 119 this shallow convection case (although the actual LES equations are anelastic). du . du ^ du I dp (5 4a) dt dx dz po dx dw . dw dw. 1 dp „ ... dt dx dz po dz For simplicity, the subgrid-scale diffusion, and large-scale forcing terms are neglected and the density p is replaced by a constant density po (note under the Bousinesq approxi-mation, the density variation is neglected except when it is coupled with the buoyancy term). Now I differentiate the u equation with respect to z and the w equation with respect to x: d .du. d , du du. 1 d .dp. . . d t ^ = - a ^ + ^ - ^ f e ) . ( 5 ' 5 a ) d.dw. d , dw ^ dw. 1 d .dp.^dB (55b) dt dx dx dx dz p0 dx dz dx Subtracting (5.5b) from (5.5a) yields the vorticity equation du du duj duo. .du dw. dB . . dt = m+iudx- + wd;) = u{Tx + ^ ] - d x - ( 5 - 6 ) where, u — (du/dz — dw/dx) is the vorticity component perpendicular to the x-z plane, with positive vorticity pointing into the page (Figure 5.8). The vorticity production terms on the RHS of (5.6) are due respectively to the divergence and the horizontal buoyancy gradient. If the system initially contains no vorticity (to — 0) the only vorticity production term comes from the horizontal buoyancy-gradient term. From Figure 5.8, we see that the buoyancy-gradient term should produce positive UJ on the right side of the buoyancy maximum and negative UJ on the left 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., p. 104, 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 density difference is primarily due to the cloud condensational 120 heating. The strong horizontal buoyancy gradient in the simulated clouds means 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 primarily source for the initial production of the vorticity for the simulated ACTs, this initially horizontal vorticity, as discussed in Section 5.4, must also be redistributed into vertical vorticity in the sheared environment (when horizontal vorticity component is parallel to the shear vector). This will result in large ACT-scale horizontal eddies which may further break into smaller scale eddies. This is a probable explanation for the complex 3 dimensional structure of the entraining eddies shown in Figures 5.4-5.6. 5.5.2 Perturbation pressure Figure 5.9 (panels a-c) shows the same vertical cross section as in Figure 5.8 but with contours of perturbation pressure p, the vertical component of the P P G F (PPGFV), and the sum of PPGFV and B. The subgrid-scale momentum flux term is in general a small term compared with either PPGFV or B and will be neglected below. Figure 5.9a shows a high pressure region centered on the A F C at 1380 m, with a low pressure region below and on the downshear (right) side of the vertical velocity maximum at 1200 m. Notice the tendency for the perturbation pressure contour to parallel to the wind vector around the vortical circulation. I will come back to this point later. The A C T , accelerating into the quiescent environment, produces a perturbation pres-sure field that tends to slow down its ascent. Figure 5.9c shows that near the shallow A F C region (1300-1400 m) the PPGFV is larger in magnitude than the buoyancy force and produces a net downward force. This net downward force experienced by individual parcels works effectively to cap the fast moving parcels, and drives their overturning mo-tion within the A C T . Due to this net downward force, the most buoyant parcels spread 121 outward as they approach the A F C . Mixing with surrounding air along the leading edge of the A C T tends to dilute cloud parcels exposed to the environment, while evaporation rapidly reduces their buoyancy. When these parcels circulate downward along the A C T edge to the A C T rear (i.e., between 1050-1200 m in Figure 5.9c) the combined effect of the downward circulation, dilution, and evaporation produces large values of the C T -relative downward vertical velocity. By continually shedding these cloud-environment 950 200 300 400 500 x ( m ) 600 700 800 800 Figure 5.9: a) As in Figure 5.8, but with contour of perturbation pressure field p (Pa). Arrows are the same as in Figure 5.4c. b) As in a) but with contours of PPGFV ( m s - 2 ) . c) As in a) but with contours of PPGFV + B ( m s - 2 ) . mixtures, the A F C region remains the most buoyant part of the A C T throughout its ascent. The CT-relative downward transport of mixed air and the enhanced entrainment 122 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 A F C . This tends to intensify the hor-izontal buoyancy gradient and produces stronger vortical circulation. This is consistent with the fact that the vortical circulation is strengthened throughout its ascent in the cloud layer (below the inversion). The net downward force within the A F C shown in Figure 5.9c indicates that the vertical momentum flux into the A F C must be positive in order to maintain its ascent. In Chapter 6 I will examine the vertical momentum budget of the A F C / A C T . In this section, I further explore the physical interpretation of perturbation pressure by partitioning it into dynamical and buoyancy components based on the diagnostic pressure equation under the anelastic approximation. Following (5.3) I differentiate the momentum equations for u, v, w with respect to x, y, z and sum them together; the resulting equation is then combined with the anelastic continuity equation leading to the diagnostic equation for the perturbation pressure: d2P = d dpujUj dnj dpB ^ dxj dxi dxj dxj Q~ l 3 The equation is written in summation notation and repeated indices are summed. Fol-lowing Wilhelmson and Ogura (1972), the first two terms on the RHS are labelled the dynamic contribution (I) and the third term is labelled the buoyancy contribution (II) to the p Laplacian. Since (5.7) is linear in p, the total perturbation pressure p can be decomposed into the dynamic perturbation pressure pd and the buoyancy perturbation pressure pb. These two terms can be found by solving: d2pd 2 = / (5.8a) 3 X ; = / / . (5.8b) d2pb dx 123 800 Figure 5.10: a) As in Figure 5.9b but with PPGFV replaced by PPGFd (ms" 2). b) As in a) but with PPGFd replaced by PPGFb (ms" 2). Figure 5.10a and 5.10b show the partitioned dynamic P P G F [PPGFd = (-1/p)dpd/dz] and buoyancy P P G F [PPGFb = (-1/p)dpb/dz] which sum to the total PPGFV = (—l/p)dp/dz shown in Figure 5.9b. Both PPGFs are negative and act to decelerate the air parcels within the A F C . However PPGFd is nearly one order of magnitude larger than PPGFb within the A F C . Thus pd is expected to largely determine the distribution of the total perturbation pressure p within the A C T . A contour plot of pd (not shown) indicates that it does indeed have a very similar pattern to that of Figure 5.9a. Yau (1979) presents a physical interpretation of the dynamic perturbation pressure using an analytically tractable steady-state cloud model. In the following I will briefly discuss the relevancy of Yau's finding to these simulation results; a detailed derivation of (5.9) and (5.10) is given in Appendix G. Yau recasts (5.7) using the Boussinesq approximation and shows the dependence of pd on the kinetic energy and the rotational component of the flow. This yields the Bernoulli equation along a stream or vortex line, - U • U + — = constant 2 Po (5.9) Here U is the velocity in vector form. Eq. (5.9) explains pd as a consequence of the Bernoulli effect, namely the conversion between the kinetic and potential energy of the 124 flow. Yau also obtains a constraint on pd normal to a curved streamline: po on Rs Here n is the unit vector normal to the streamline, s is the distance along the streamline, U is the air velocity in the direction of s and Rs the radius of curvature. Eq. (5.10) suggests that the dynamic perturbation pressure can also arise from the requirement that the gradient of Pd normal to the streamline serves as the centripetal force for the curved trajectory of the air parcel. While Yau's derivation is based on the steady state assumption I find these interpre-tations hold for my simulation results, even though the simulated clouds are not steady. For example, Figure 5.9a shows that the high-pressure region within the A F C is associ-ated with low kinetic energy while the lower pressure region below and to the right of the vertical velocity maximum is associated with the curved vortical circulation within the A C T . Furthermore, a detailed examination of Figure 5.9a also indicates that the wind vectors (streamlines) tend to cross the pd contours blowing outward from the rotating center. This indicates the PPGFd does not supply enough centripetal force to balance the centrifugal forces (pseudo force) arising from the rotating wind (5.10). This net mass outflow will further decrease the low pressure in the rotation center. Since the ascending A C T is unsteady, we expect the pressure/mass adjustment to lag changes in the wind field. The mass divergence and the corresponding drop in pd indicate that the vorticity is intensifying, consistent with the fact that the A C T vortical circulation is strengthened during its ascent (before reaching inversion). In general, the tendency of Pd contours be oriented parallel to the wind vectors is consistent with the fact that the perturbation pressure does not generate vorticity, as indicated in (5.6). Figure 5.9a also shows that the dynamic pressure gradient acts to decelerate cloud parcels near the A C T top (or AFC) and accelerate them near the A C T rear with the A F C deceleration forces apparently much larger than the forces driving the 125 ACT-rear acceleration. In Chapter 6 I will examine the averaged P P G F and its impact to vertical momentum budget of the A F C / A C T . In Section 5.4 I have described the impact of the ambient shear on the ACT's vortical circulation and A C T entrainment. The impact of this ambient wind shear on the dynamic perturbation pressure field can also be seen in Figure 5.9a; it enhances the low pressure region on the downshear side of the A C T while diminishing or weakening the low pressure region on the upshear side. 5.6 Dilution of the A C T In Section 5.4 I defined the A C T as the upper part of the growing cloud containing the vortical circulation. To quantify the dilution rate of the A C T I sample it by choosing all the saturated gridcells located in the 14 model levels below ZT (Ah = 350 m below the current zj) for large clouds and the 7 model levels below ZT (Ah = 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 A C T mean properties are computed by averaging all selected gridcells. I also define the most undilute core within the A C T (called the A C T core below) as the average of the 2x2x2 gridcells which have the largest Aqt = qt,c — (Qt), where qttC denotes the total water of a cloudy gridcell and (qt) is horizontal model domain averaged qt. Similarly, the most dilute cloudy parcels within the A C T are defined as the average of the 2x2x2 gridcells which have the smallest Aqt. These two statistics are used as a measure of A C T inhomogeneity. Since an A C T 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 A C T properties presented below show the time evolution of the A C T in a Lagrange frame of reference. Figure 5.11 shows the vertical profiles (time histories) of buoyancy (given as the 126 1400 1200 £ 1000 O X 800 600 -0.5 1400 1200 1000 800 600 a) ( 1 * A • Vf"^ t 2000 E 1500 '(D x 1000 0 0.5 A O (K) V v ' b) 1 \ \ > V - • . • 1 \ f ; y . * < . . ' \ —. I / • \ / II *• 0 0.5 A O (K) 2000 1500 1000 2000 1500 1000 d) \ 1 ; N \ \ / / % / / / -2 0 A 9 (K) r e) \ \ \ \ : \ / / / -2 0 2 A O (K) -0.5 0 0.5 A O (K) V v ' Figure 5.11: Panels a-f show the vertical profiles/time histories of the buoyancy of the A C T mean (solid line), A C T core (dashed line) and the most dilute air (dotted-dash line) within ACTs for clouds A - F (panels a-f) respectively. The vertical height is the height of the geometric center of each region at 30 second 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). 127 virtual potential temperature difference A8V = 8VtC-(9v)) for the A C T mean (solid), the A C T core (dashed) and the most dilute air within A C T (dot-dash). The vertical height at a given time is taken as the average geometric center of each category of gridcells. The vertical profile of cloud-mean buoyancy which is calculated by averaging over the entire life-cycle for all liquid water containing gridcells at each level is also plotted (dotted) for comparison. First, we see that the interior of the A C T is non-homogeneous for all six clouds (note the large buoyancy difference between the A C T core and the most dilute cloudy gridcells within the ACT) . As shown in Figure 5.11 and also in Figure 5.12 below, the A C T core remains nearly, undilute during much of its ascent, thus producing large positive buoyancy values. In contrast, the most dilute cloudy gridcells are negatively buoyant throughout the A C T ascent for all clouds. Although not as buoyant as the A C T core, the A C T mean is systematically more buoyant than the cloud mean (dotted) throughout the cloud depth for all clouds. The buoyancy excess of the A C T mean in comparison with the cloud mean is significant, considering 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 (Figure 5.lld-f). In the literature, the cloud-top height and cloud vertical velocity are usually de-termined by the cloud mean buoyancy assuming a steady state. When integrating wdw/dz = B(z) [ w(z) = (2 f* B(z)dz)1/2] with B(z) determined from the cloud-mean buoyancy, I find that the resulting velocity profile (not shown) always under-estimates the cloud-top height. This is expected since cloud growth is associated with the A C T 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 A C T . This result is consistent with the liquid-water/cloud-top paradox first noted by Warner (1970), who found that it is impossible using a steady homogeneous plume model to simulta-neously predict both the observed cloud-mean liquid water content and the observed 128 cloud-top height. Figure 5.11d,e,f show, in contrast, that for the large clouds the A C T mean is significantly more buoyant and maintains its positive buoyancy up to the base of the inversion, while the most buoyant A C T cores maintain their positive buoyancy up to the middle of the inversion. These clouds ascend as long as their A C T core is positively buoyant; therefore the dilution rate of the A C T core is particularly relevant to the determination of the cloud-top height. Also shown in Figure 5.11d-f is the fact that large clouds overshoot their A C T core NBL by roughly 200 m. In Chapter 6 I will further discuss about the cloud-top determination. e,(K) q^gkg" 1) Figure 5.12: Vertical profiles of 9t and qt for the A C T core for clouds (A-F). a) Liquid water potential temperature 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-129 Figure 5.12a,b show the vertical profiles (time histories) of ft and qt for the A C T core for each cloud. For comparison, I also plot on Figure 5.12a,b the vertical profiles of ft and qt calculated based on an entraining plume model with the form (see Appendix F): f j = e(x e-Xe) (5.11) where, x ^ {ft; Qt} and Xe ~ (x) is environmental values while Xc is for cloud; e is the fractional entrainment rate. Figure 5.12a,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 A C T core is diluted at a rate approximately equal to an entrainment rate e = 0.0003 m _ 1 (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 A C T core and thus the cloud-top height. However, as Figure 5.12a,b show, that the shapes of the vertical profiles of ft and qt for the A C T core still differ qualitatively from those predicted by an entraining plume, because the A C T cores experience no dilution until they reach the upper half of their z^max- Figure 5.12c,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 A C T cores stay essentially unmixed in the lower half of their depth and are gradually 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 Figure 5.12a,b, where the cloud-base ft and qt for all clouds are nearly the same (although large clouds tend to be slightly moister (0.1-0.3 g k g - 1 in Figure 5.12b) than small clouds). Given this, the dilution rates (and, therefore, cloud-top heights) are most probably caused by differences in A C T size. Indeed, Figure 5.13 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 130 2000 r 1800H 1600H - e - A — B - B - C - - D . . . . E F \ E 1400 \ 600 60 80 100 120 140 ACT radius (m) 160 180 200 Figure 5.13: Vertical profile of the spherical volume-equivalent radius of the A C T for clouds A-F . A C T volume-equivalent spherical radius. However, it is likely that' this size dependency will be diminished when an A C T exceeds a threshold size large enough to allow some air within the A C T to reach or overshoot its NBL without dilution. In this case the cloud-top height will be solely determined by the A C T core thermodynamic properties and the environmental sounding. Another important feature shown in Figure 5.13 is that the A C T size tends to decrease with height for small clouds (A,B,C) but stays relatively constant below the inversion and decreases above the inversion for large clouds. In any case, the A C T does not show significant increase of size with height. This indicates that the A C T must detrain cloud-environment mixtures as well as entraining environmental air throughout its ascent. The buoyancy profiles of Figure 5.11 indicate the inhomogeneous nature of A C T interior. Figure 5.14 further shows the distributions of 9i and qt in the A C T of cloud E at several different heights (or times). Consistent with Figure 5.11, the A C T is not uniformly diluted during its ascent. This would lead to a narrow distribution with mode 131 properties that continuously shift toward environmental values with increasing height. Instead the (ft, qt) distributions maintain modes corresponding to a relatively undilute core with a tail continuously extending to environmental values, although the modes also show slight dilution above 1200 m. This mixture distribution and the mixing behavior of Figure 5.14: Histograms of ft (panel a) and qt (panel b) for all A C T gridcells for cloud E at four different heights (or equivalently, times). The legend heights show the A C T heights zT(t) at the selected times (390s, 480s, 570s, 660s). the A C T core, in which there is little dilution until the A C T has traveled « 2 diameters, is consistent with the shedding thermal model of Blyth (1993) rather than a continuously entraining homogeneous plume model. 5.7 Passive mixing The previous sections have focussed on the kinematics and dynamics of individual as-cending thermals, which may be viewed as fundamental elements or agents of cumulus 132 convection. The ascending thermals are inherently non-steady and produce the pulsating growth characteristic 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 before that during the ascent of an individual thermal a 2 Figure 5.15: An illustration of the mixing life-cycle within a fixed cloud layer (between 1187.5-1312.5 m for cloud E). a) Contours of the joint frequency distribution (unit: num-ber of gridcells) of A9V and w for the liquid water cloud region, b) As in a) but for the bin-averaged total water qt (gkg - 1 ) for gridcells in each (A8V , w) bin. c) As in a) but for the bin-averaged cloud age i (unit: seconds) since emergence of the cloud from the subcloud layer). volume of more dilute cloudy air is left behind at the trailing wake. These mixtures may be generated either through the detrainment of A C T mixtures or simply by enhanced mixing in the wake of the A C T . They are characterized by small or negative buoyancy 133 and low vertical velocity. To distinguish them from the active ascending element, I have referred them as the passive cloud mixtures. In contrast to the active A C T mixing, no distinctive coherent structures are observed in these passive cloud regions (compare, for example the velocity structure in Figure 5.4 at 1100 m at 8.5 minutes and 10.5 minutes). The continued mixing of these passive cloud results in more negative buoyancy and a net downward motion. As presented in Chapter 4, much of the descent motion is realized in the unsaturated cloud mixed convective region in the simulated clouds. Figure 5.15 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 Figure 5.1. Three joint distributions are plotted as functions of vertical velocity w and buoyancy A9V. Fig-ure 5.15a shows contours of the joint frequency distribution (JFD) computed from the liquid water-containing gridcells within the 1187.5-1312.5 m layer during the life-cycle of cloud E. Figure 5.15b and 5.15c show distributions of qt, the bin-averaged total water and t, the bin-averaged cloud mixture ages defined as time since emergence of the cloud from the subcloud layer. Comparison of the three panels shows that the A C T arrives at this level as relatively undilute subcloud layer air with high values of total water (qt > 16.2 g kg - 1 ) and buoyancy (A9V > 1 K). Notice, however, the inhomogeneity of w, which ranges from 1 - 6.5 ms" 1 at this time. Examining (w, A6V) bins along a line of increasing t 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 t increases from 550 to 1200 seconds in Figure 5.15c, the number of gridcells increases steadily (Figure 5.15a), while the total water, buoyancy and vertical velocity of those gridcells decreases (Fig-ure 5.15b). This progression is due to the continuous dilution of passive cloudy mixtures with environmental air; these mixtures become negatively buoyant and move downward as they age. 134 5.8 Discussion I have simulated an ensemble of shallow cumulus clouds using a three-dimensional high-resolution LES with an initial sounding and large-scale forcings consistent with the B O M E X field experiment (see Appendix D and http: //www.knmi .nl/~siebesma/gcss/ bomex.html). From this cloud ensemble six individual clouds with cloud-top heights ranging from 1-2 km are selected and observed from initial growth to final dissipation. All six clouds share 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 sub cloud-layer thermal. 2) A cloud-cap region that contains the largest values of horizontal divergence, vertical perturbation pressure gradient force (PPGF) and buoy-ancy. I refer to this as the ascending frontal cap (AFC); it is approximately 75-150 m thick in the simulated clouds. 3) A larger region extending through the cloud-top which encompasses the coherent vortical internal (CT-relative) circulation. For the quantitative analyses I approximate this region by the upper ~ 175 m (small clouds) and ~ 350 m (large clouds). This ascending cloud top (ACT) is marked by a dynamic perturbation pressure pattern that is consistent with the vortical internal circulation. 4) A trailing wake characterized by passive cloud-environment mixtures with low values of vertical velocity and buoyancy. 5) A nearly constant thermal turn-over time of-approximately 600 seconds. 6) An A C T interior that has an inhomogeneous distribution of both ther-modynamic properties and vertical momentum, with a relatively undilute core structure. 7) An A C T core that experiences no dilution during the initial half of its ascent and gradual dilution during the later half of its ascent. The elevated tracer experiment (Figures 5.4-5.6) indicates that environmental air is entrained into the A C T both through its top and sides. Either route, however, is strongly modulated by the ACT's vortical circulation. Tracer entrained through the A C T top is transported from the A F C region downward through the A C T edge and is 135 finally incorporated into the turret from the A C T rear. This mixture trajectory allows the A F C to maintain its buoyancy maximum throughout its ascent. Lateral entrainment also occurs, as the A C T vortical circulation directly engulfs environmental air into the strong convergence zone at its rear. These entraining eddies have scales comparable to the A C T size and dominate the A C T entrainment, mixing and dilution. These large-scale entraining eddies display a complex 3 dimensional structure. The coherent A C T vortical circulation may be directly related to the horizontal buoy-ancy gradient within the A C T , which decreases from A C T center to A C T edge, although the precise spatial distribution is far from symmetric (see Figure 5.8). This horizontal buoyancy gradient produces a net torque which tends to rotate the flow during its ascent. The net result is an internal circulation where the ascent is along the central vertical axis while the descent is around its periphery. The vortical circulation drives the large-scale entraining eddies and wraps environmental air from A C T rear into the rotating center where smaller-scale eddies produce further mixing. This process tends to shrink the most-buoyant core in the A C T rear to a narrow stem below the A F C and results in a mushroom-like core structure which further strengthens the horizontal buoyancy gradient and intensifies the vortical circulation. The dynamic perturbation pressure is directly associated with the A C T ascent and its internal circulation. For all the simulated clouds, the A C T flow structure [calculated after removal of the cloud-top mean velocity (iux(i))] shows remarkable consistency with the dynamic perturbation pressure field (as shown in Figure 5.9a). The strong downward P P G F within the A F C serves effectively as a lid for individual air parcels contained in the A C T and drives the overturning motion, while the low pressure generated near A C T edge serves as centripetal force for the vortical circulation. These simulation results, in combination with the cloud animations, paint the follow-ing picture of shallow cumulus mixing dynamics. The subcloud layer air tends to ascend 136 in discrete elements that detach from their subcloud layer parent thermals. In contrast to the traditional parcel models (adiabatic or entraining plume), these ascending ther-mals both entrain and detrain on their way toward their maximum height. As a result of this continuous entrainment and detrainment, the ascending thermals are inhomogeneous both in their thermodynamic and dynamic properties. Note that while the dilution of an individual ascending thermal may be continuous, the occurrence of each thermal in a cumulus cloud is typically episodic. For example, as shown in Figure 5.1, the arrival of ascending thermals at a particular level is intermittent in both time and in space. The dynamic perturbation pressure is 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 A C T center. The coherent vortical circulation strongly shapes the way that an A C T mixes with its environment. The coherence of the A C T mixing makes it qualitatively different from the dilution of passive cloud at the trailing wake of the A C T . The vortical circulation of the simulated A C T is consistent with the laboratory re-sults of ascending thermals across a range of stratifications (e.g., Scorer and Ronne, 1956; Scorer, 1957; Sanchez et al., 1989; Johari, 1992). My simulation shows, as do the recent laboratory results of Sanchez et al. and Johari, that the thermals that characterize shal-low 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 4 A C T diameters. This is possibly the rea-son that the hollow-cored thermals suggested by Scorer (1957) are not observed. Instead the simulated A C T maintain a mushroom-like core with a shrinking stem containing the most undilute sub-cloud air. . These simulation results also strongly support the schematic shedding thermal model 137 proposed by Blyth and co-authors (Blyth et a l , 1988; Blyth, 1993; Carpenter et al., 1998c). Blyth et al. proposed this ascending cloud-top mixing mechanism primarily to explain aircraft in-situ observations showing that the conserved thermodynamic proper-ties of cumulus mixtures tend to fall along a straight line between undilute subcloud air and environmental air near or above the observational level. My numerical examination of the cloud kinematic 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 indicates that the simulated ACTs must detrain as well as entrain. The distribution of A C T thermodynamic properties and the dilution history of the A C T 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 sim-ulations also indicate that some mixed parcels may be carried upward for some distance before being shed in the A C T wake as either saturated or unsaturated mixtures. The effect of ambient wind shear on the vortical circulation, dynamic perturbation pressure field, and the A C T mixing is very clear in these simulations. As an A C T rises through the sheared environment, the vortical circulation, associated low pressure, and mixing are strongly enhanced on the downshear side and are weakened on the upshear side. The ambient wind shear also tilts the parallel (to the ambient shear vector) com-ponent of horizontal vorticity into vertical and generates horizontal eddies at the down-shear side. As a result, I observe a clear preference for more dilute cloud-environmental mixtures and downdrafts on the downshear side of the clouds, consistent with in-situ observations (e.g., Warner, 1977). For the same reason, all the ACTs eventually collapse to the downshear side while subsequent thermals ascend into the upshear side of their predecessors. The simulation results have indicated the important role of the dynamic perturbation 138 pressure in the ACT's vertical momentum budget. A detailed understanding of A C T mixing is particularly relevant to questions about the ultimate height of an ascending thermal. In Chapter 6, I focus on cloud-top determination and present a Lagrangian budget analysis of the vertical momentum of the A F C and A C T . These results can be used to evaluate the impact of the vertical perturbation pressure gradient force and entrainment on the vertical momentum prediction, and therefore on the determination of cloud-top height. 139 C h a p t e r 6 C l o u d - t o p d e t e r m i n a t i o n 6.1 Introduction In Chapter 4 and 5 I have shown that while the simulated clouds have fairly uniform cloud-base heights there is significant variation in their cloud-top heights. The uniform cloud-base height, coincides well with the lifting condensation level (LCL) defined by surface layer'air. An obvious question is: what controls the cloud-top height of individual clouds? An individual cloud-top height determines the maximum depth over which this cloud may influence its environment. In cumulus parameterization schemes for large scale models the maximum height reached by a cloud ensemble is determined by the largest or highest clouds. This makes an accurate estimate of the highest cloud-top height particularly important. Many cumulus models estimate a cloud-top height based on the NBL determined either by an adiabatic ascending parcel/plume or a parcel/plume with a specified en-trainment rate (e.g., Arakawa and Schubert, 1974; Emanuel, 1991). Some other schemes make use of a simplified vertical momentum equation (e.g., Gregory, 2001; von Salzen and McFarlane, 2002). In this chapter, I evaluate these approaches using the simulated clouds. In particular, the detailed simulation results provide the potential to estimate 140 the terms of the vertical momentum budget, such as the perturbation pressure and en-trainment, which are very difficult to observe in real clouds. 6.2 Approaches using NBL of a plume Figure 6.1 extends the entraining plume comparison of Figure 5.12, showing the dilution properties of the largest cloud F together with plume-model estimations (see Eq. 5.11), given several entrainment rates . For this calculation the cloud-root properties are chosen F ev(K) q g^kg"1) Figure 6.1: Cloud-top estimations based on the NBLs of adiabatic or entraining plumes for cloud F. The lines show profiles for the environment, A C T mean, A C T core, and three entraining plume estimations with entrainment rates of 0.002 m _ 1 , 0.0008 m - 1 and 0 (adiabatic). (a) liquid water potential temperature (b) total water mixing ratio qt. (c) virtual potential temperature (including liquid water loading) 9V. (d) liquid water mixing ratio qc. 141 as the cloud mean at cloud-base level, which is nearly identical to the surface-layer mean. The panels show vertical profiles of ft, qt, ft and cloud liquid water content qc for the environment and for the cloud mean, A C T mean, and A C T core from the simulated clouds, where these averages are calculated as in Section 5.6. Also shown on each panel are thermodynamic profiles for a plume with three different entrainment rates e. All four panels show that an entrainment rate of 0.002 m _ 1 successfully captures the cloud mean ft, qt, ft, and qc below the inversion, in agreement with the LES results of Siebesma and Cuijpers (1995). This entrainment rate fails, however, above the inversion (note from Figure 6.1c,d that the plume becomes unsaturated at about 1700 m while the simulated cloud-top height reaches 2000 m). Figure 6.2: As in Figure 6.1 but for cloud B and entraining plume estimations with an entrainment rate of 0.002 m _ 1 and 0 (adiabatic). 142 The four panels also show that an entrainment rate of about 0.0008 m _ 1 makes reasonable estimations for the vertical profiles of the A C T mean properties. The cor-responding NBL of the A C T mean is about 400 m below the cloud-top height, but it should be remembered that the A C T itself has a vertical extent of 350 m. Although the A C T core does experience some dilution in the upper half of the cloud and has an NBL at 1750 m, overshooting carries it beyond this point to the undilute NBL (e = 0) at 2000 m. In contrast, the NBL defined by undilute ascending subcloud air (USCA) appears to correctly estimate the cloud-top height for Cloud F. While the NBL defined by USCA may provide a reasonable estimate of cloud-top height for the largest cloud, it clearly fails as a predictor for small clouds A, B, and C. Figure 6.2 shows an example of the dilution properties for small cloud B compared with both an entraining plume with e=0.002 m _ 1 and adiabat ascent (e = 0). As all panels show, there is a rapid transition from adiabatic (no mixing) to diabatic (mixing) ascent for the A C T core at roughly half the cloud depth. Beginning at 975 m the A C T core suddenly experiences rapid dilution and ascent stops at 1300 m, well below the undilute NBL. This mixing behavior is consistent with a picture of the A C T core that is continually eroded until at some height even the last part of undilute air is exposed and subsequently diluted. Similar behavior is seen for cloud F at about 1500 m; The difference in these transition heights is consistent with the cloud sizes shown in Figure 5.13: namely the larger A C T size protects the A C T core of cloud F from erosion until it has reached the inversion, while for cloud B core erosion begins in the middle cloud layer. 6.3 Lagrangian vertical momentum budget While many cumulus models simply use the NBL to determine the cloud-top height, there is also a long history of direct use of the vertical momentum equation. This approach was initially employed to model the ascending turret (e.g., A C T / A F C ) of an individual 143 developing cloud in a Lagrangian frame (e.g., Levine, 1959; Simpson and Wiggert, 1969). More recently it has also been used to model the bulk vertical momentum of a steady-state cloud ensemble (Bretherton et al., 2003; Gregory, 2001; Siebesma et al., 2003). Below I will stick to its original meaning, namely modelling the vertical momentum of an A C T / A F C (e.g., Simpson and Wiggert, 1969) given the fact that individual shallow cumulus clouds are never in a steady state and the lifetime averaged vertical profile of vertical momentum are not useful in cloud-top determination. The ID vertical momentum equation used by these authors contains terms for the buoyancy, vertical P P G F (or form drag) and entrainment with the general form: ^ = i ^ = * ™ . T - « * (CD where the subscript T denotes the A F C / A C T , WT and BT are vertical velocity and buoyancy respectively, and PPGFVT represents the net effect of the perturbation pressure or form drag. The last term represents the momentum exchange between the A F C / A C T with its surrounding environment, for which it has been assumed the plume behaves as an entraining plume with a constant fractional entrainment rate per unit height e. The major concern with using this equation is the uncertainty in estimates of PPGFVIT and the validity of using an entraining plume to treat both the momentum exchange and the buoyancy terms. In practice, e is often determined based on laboratory entraining plume experiments (e.g., Simpson and Wiggert, 1969): 1 dM 0.2 , . M dz R K ' where M is the mass flux of the A F C / A C T and R is the A F C / A C T radius at cloud-base level. The pressure perturbation PPGFV<T is parameterized as a linear combination of BT and — ewT. Thus, (6.1) becomes: dwr 1 dwr „ , 9 , „ x IT = 2 ^ 7 = - 6 o t - ( 6 ' 3 ) 144 where, a and b are two free coefficient for example, Simpson and Wiggert (1969) chose o = 2/3 and 6 = 2. Nevertheless, there remains uncertainty as to the appropriate formulation (Bretherton et a l , 2003). In the remainder of this section I will use the simulated cloud fields to estimate the vertical momentum budget for an ascending A F C / A C T . I first derive an averaged verti-cal momentum equation which may be applied to the A F C / A C T . The simulated cloud data are then used to evaluate the individual buoyancy, vertical P P G F and momentum exchange terms. I begin with the budget equation for the vertical momentum of any model gridcell: dw = 1 dpujw | B I dp 1 dr3j ^ dt p dxj pdz p dxj ' where all symbols are defined as in Eq. (5.3). A spatially homogeneous filtering operation on an arbitrary variable £ may be defined using kernel convolution (Pope, 2000, p. 561): f ( x ) = / G ( r , * ) £ ( x - r , t ) d r , (6.5) Jv where the integration is over the entire model domain, x and r are gridcell coordinates and the homogeneous filter function (or kernel) G(r, t) is defined for a discrete domain as I reV(t) G ( T , t ) = { m (6.6) [ O rtV(t), where V(t) is an arbitrarily shaped 3D volume which may change with time and represent an ascending A F C / A C T , N(t) is the total number of gridcells in V(t) and the form of G(r , t) insures normalization. Applying (6.5) to filter the vertical momentum budget equation (6.4): -v • —rr-V —K v dw _ - = ^ _ l ^ P 1 dpujw 1 dr3j ^ ^ dt pdz p dxj p dxj 145 this yields the filtered budget equation: -v v dujV f w(x r t)9G^' ^ dr = BV l d p ' 1 d p U j W ' 1 ^ (6 8) dt JD dt pdz p dxj p dxj x Note that since G is independent of x but a function of time t, the operation of filtering and differentiating commute with respect to x but do not commute with respect to t; this results in one more term (X) which represents the effect of volume V(t) variation with time. Note also that after the discretized filtering operation the domain has the same number of gridcells, with each gridcell containing the mean value for the volume V(t) centered at that cell. Adding a mean advection term to both sides of the equation yields dwv | vdwv | x = g V l d P V ldpujwv | wydwv ldr3jv dt 3 dxj pdz p dxj 3 dxj p dxj A mean substantial derivative based on the filtered velocity u]v can be further defined as ft'^i' (610) where d /d t represents the rate of change following a point moving with the mean (filtered) velocity v~v (Pope, 2000, p. 582). A Lagrangian form of the spatially filtered vertical momentum equation can then be constructed as dVwv = g V ldpV ldpujWV —ydwv 1 dr3jV x I pdz p dxj 3 dxj p dxj S 7 V The first and the second term on the right are respectively the volume-averaged buoyancy and vertical PPGF. The third and fourth terms on the right together represent the wv tendency contribution produced by the convergence of sub-volume scale (LES model-resolved) fluxes of vertical momentum through the boundary of the volume V(t). The fifth term represents the wv tendency contribution due to the convergence of subgrid 146 scale (LES model-parameterized) fluxes of vertical momentum through the boundary of the volume. The last term is due to the variation of V(t) (and therefore G) with time. For a fixed volume V(t) = V0, X=0. The four terms grouped together as SfV will be referred to below as the vertical momentum flux convergence term or simply the vertical momentum flux term. I further decompose the perturbation pressure p into the buoyancy component pb and the dynamic component pd- This yields -v_v -T-^—V ± ^ - = B V - \ ^ (6-12) d t P d z p d z PPGFV Eq. (6.12) indicates that if we follow a moving volume V(t) with the volume mean velocity UjV we can predict the change of the volume-mean vertical velocity wv with time through its averaged buoyancy B , vertical P P G F (PPGFV = PPGFb + PPGFd ), and the net vertical momentum flux SfV through its boundary. Notice the similarity between (6.12) and (6.1). A particular advantage of (6.12) over (6.1) is that (6.12) makes no assumptions regarding the mixing behavior of the ascending thermal and, in principle, the momentum flux term SfV can be calculated from the LES output. However, an accurate estimate of SfV, particularly the term X, requires high frequency output. Since the output frequency for my individual clouds is determined by Eq. (4.2), the model grid-spacing, and the model timestep, output at intervals less than 30 seconds is currently not available. Below, I will take a simple approach and treat the momentum flux term SfV as a residual (although the above provides a theoretical derivation for a Lagrangian vertical momentum equation which may be useful for future evaluation based on LES output). I choose V(t) as either an ascending A F C (150 m or 6 model levels below zT) or A C T (350 m or 14 model levels below ZT) of a developing cloud. Then the LHS term of (6.12) can be estimated using the time difference of the AFC/ACT-averaged vertical velocity 147 -V following its ascending trajectory while the B and PPGFV can be directed calculated based on each individual snapshot, with SfV estimated as the residual. Given a time series of identified A F C / A C T s , I can construct the vertical profiles of each term in (6.12) for an ascending A F C / A C T based on its height at succeeding time-steps. Therefore, the 2000 2000 r 0.05 Figure 6.3: Vertical profiles of vertical acceleration on the RHS of (6.12) for the AFCs of -v cloud D, E , and F (panel a, b, and c respectively), solid: B , dot-dashed: Sf , dashed: PPGFbV, dotted: PPGF/ , "+" line: total dvwv/d"ton the LHS of (6.12) vertical profiles of the vertical momentum budget below are essentially the Lagrangian budget following the A F C / A C T trajectories. My objective is to provide information about the averaged impact of P P G F and momentum flux term in the vertical momentum budget of an A F C or A C T . -v v 148 Figure 6.3 shows the vertical profiles of B*, PPGFb , PPGF/ , Sf* and their sum d wv/d t for the A F C of large clouds D, E , and F. We see that throughout the A F C as-v • —v -pz—v —v cent PPGFb opposes B (notice PPGFb becomes positive when B becomes negative above the inversion), maintaining a magnitude of ~ 15-20% of BV for these simulated clouds. This is in agreement with Yau (1979), although PPGFbV has a magnitude about one half that estimated by Yau. Yau obtained steady-state solutions for the PPGFs of slab and axisymmetric clouds with specified forcing functions of buoyancy and vertical velocity. He found that PPGFb acted against B with a magnitude of 30-50% of B, which he specified as a vertically varying sinusoid. However, his calculated PPGFs depend on assumed model geometry: the PPGFs using a slab-symmetric geometry are twice as large as those found using an axisymmetric cloud. v Consistent with the results discussed in Section 5.5.2, PPGFd is approximately 5-10 v times larger than PPGFb and dominates the total P P G F within the A F C . In compari-—v son, the buoyancy B is smaller than the total P P G F within the A F C throughout most of the ascent, indicating individual parcels moving into the A F C on average decelerate. —v The vertical momentum flux Sf is as large as buoyancy and produces upward accel-eration for the A F C . Above the inversion buoyancy decreases to negative values while momentum fluxes becomes the primarily source for upward acceleration of A F C . Com-pared with these large individual terms, the net A F C acceleration dfwvjdYt is rather small, with magnitude approximately 1/3 (clouds D, E) to 1/2 (cloud F) of BV, i.e., only 30 - 50% of the AFC's mean buoyancy is used for upward acceleration while the rest (in combination with SfV) counteracts the PPGF. v —v An important fact about the PPGFd and Sj is that they have opposite sign and tend to largely cancel, leaving a relatively small negative residual (which is still larger v v —v than PPGFb ) throughout the ascent. Also, both PPGFd, and Sf depend sensitively on the specified depth of the averaging volume V. When V is enlarged beyond the A F C 149 -V -V to include more A C T levels, both the PPGFd and Sf terms decrease sharply, although their sum remains relatively unchanged. Figure 6.4 shows the same vertical profiles as 2000 1800 1600 § 1 4 0 0 'CD I 1200 1000 800 2000 1800 1600 • § 1 4 0 0 !E 5? 'cu I 1200 1000 800 -0.02 dw/dt (ms"') a) . . . s, _ _ P P Q F b P P G F a ••*• sum 2000 1800 1600 § 1 4 0 0 x: O l 'CD I 1200 1000 800 :±. . . « . : - ; * : c) *< • **. ••••' ' . ' \ . ... 1, _ _ P P G F b . . . . P P G F a • +• sum • ' l : '•.-' I '• * .«. i • i > * 1 : 1 i : 1 + >.R \ / ir V / >'•: •' • i i - * * < b) ' 1 " " - ^ — T ''••iff " • : i r ; •i i --. •-' ; ^ / : . " : l l » ' ^ / * f V i 1 -- l • i *.( + / + / - 8 S , P P G F l i i ' i . . . . P P G F d • sum -0.02 0 dw/dt (ms - 2) 0.02 -0.02 0 dw/dt (ms" 0.02 Figure 6.4: As in Figure 6.3, but for the ACTs of clouds D, E and F. in Figure 6.3 but with an averaging volume V(t) that includes the whole A C T , with a vertical depth of 350 m. The magnitudes of both the averaged vertical momentum flux SfV and PPGFd are significantly reduced, in particular, the momentum flux SfV now v changes sign and becomes slightly negative. PPGFd is also now much smaller than the —V V V buoyancy B . As indicated in Figure 5.9, this large variation of PPGFd and Sf within the A C T is associated with the A C T internal circulation. Internally, the distribution of PPGFd tends to organize the population of inhomogeneous cloud parcels so that they 150 ascend together (decelerating fast moving parcels while accelerating the slow moving parcels). Externally, the averaged PPGF/ works to slow down the overall A C T ascent. 6.4 Discussion Accurate estimation of individual cloud-top heights remains a challenging task due to the fact it requires a detailed understanding of the mixing behavior of the penetrative ascending convective element. In this chapter I have applied several commonly used methods to estimate cloud-top height and compared them with the simulated clouds. The NBL defined by the USCA appears to reasonably estimate the cloud-top height for the largest cloud in the simulated cloud field but fails as a predictor for small clouds. Individual cloud-top height appears to depend on both its subcloud layer thermodynamic property and its size. As shown in Figure 6.2, this size-dependence arises through the protection of the undilute cores from erosion. Small clouds are subject to complete core erosion well before their undilute NBL and therefore cannot ascend further while large clouds can carry their undilute cores to higher cloud levels (inversion base) before being completely eroded. Ascending cloud-tops that overshoots the actual N B L also play a role in determining the maximum ascending height. To evaluate the approach to cloud-top.determination based on the vertical momentum equation, I have conducted a Lagrangian budget analysis for the averaged vertical mo-mentum of the A F C / A C T . This analysis shows that PPGFb always opposes buoyancy —v with a magnitude proportional to B . Therefore, it could potentially be parameterized by a product of BV and a negative coefficient (~ 0.15 - 0.2) as suggested by the simula-tion results. Within the interior of the A C T (i.e., the AFC) , there is also a clear coupling v —v v between PPGFd and Sf , with PPGFd opposing the momentum flux. However, when the averaged size approaches the size of the whole A C T this coupling tends to dimin-v ish and both PPGFd and Sf become negative. While the simulation results support, 151 in general, the practice of parameterizing PPGF*V in terms of B" and Sf they also highlight the difficulty in estimating cloud-top height based on the vertical momentum equation. For the A F C , the total acceleration is a small term compared with individual large terms, such as BV, PPGF^ and SfV. Thus, small (in percentage) errors in estimat-v —v ing PPGFd and Sf would lead to large errors to the total acceleration. For the A C T , the buoyancy BV becomes a distinct large term but the large A C T size (therefore coarse resolution) appears to degrade its capability in representing the cloud-top interface. 152 C h a p t e r 7 S u m m a r y In this dissertation I have combined a diagnostic approach, an explicit numerical sim-ulation and a theory of cumulus mixing to study shallow cumulus convection and its representation in large-scale models. The objective of this research is not only to fa-cilitate the development of a cumulus parameterization scheme but also to improve our understanding of cumulus mixing dynamics, transport and the cloud life cycle. As dis-cussed in Chapter 1, I believe that an accurate representation of the statistical properties of convection must rely on a correspondingly accurate representation of the properties of the convective elements themselves. To accurately quantify the effect of cumulus clouds on the large-scale flow, one must recognize the penetrative nature of cumulus convec-tion. Thus an effective parameterization of cumulus convection must be built solidly on the physics and microphysics of cloud processes as deduced from observations, numerical cloud models, and theory (Emanuel, 1991, 1994; Bretherton, 1997). 7.1 Diagnostic study This dissertation begins with a diagnostic study of the episodic mixing and buoyancy-sorting (EMBS) representation of shallow cumulus convection. Chapter 2 provides an overview of the EMBS approach and current uncertainties. In particular, applying the 153 EMBS approach to shallow non-precipitating clouds requires assumptions about the rate at which undilute sub-cloud air is eroded into the environment (the undilute eroding rate or UER), an algorithm to calculate the eventual detrainment level of cloud-environment mixtures (detrainment criterion), and the probability distribution of mixing fraction (mixing distribution). Each of the free parameters/assumptions points to one aspect of the uncertainty in our understanding of the cumulus mixing process. To provide an observational constraint on these free parameters, in Chapter 3 I for-mulate a diagnostic equation for the undilute eroding rate (UER) given a known mixing distribution, detrainment criterion and a shallow cumulus convective equilibrium. Ap-plying the diagnostic equation to the B O M E X case I have retrieved a particular vertical profile of the U E R which satisfies the B O M E X equilibrium constraint. Based on this diagnostic framework I have also investigated the sensitivity of the retrieval to varia-tions in the mixing distribution and the detrainment criterion, and find that, given the requirement that mixtures detrain at their neutral buoyancy level (NBL), there is no positive-definite vertical UER profile consistent with the B O M E X equilibrium state and the large-scale forcings. As I noted in Chapter 3, the model is less sensitive to the assumed mixing distribution, given a multiple-mixing treatment of positively buoyant mixtures and detrainment at the unsaturated neutral buoyancy level (UNBL). The retrieved U E R decreases exponentially with height above cloud-base, suggesting a strong modulation by the cloud size distribution. When the EMBS model is used to calculate convective transport by individual clouds of varying thickness I find that no single cloud from this ensemble can balance the large-scale B O M E X forcing; the observed equilibrium requires a population of clouds with a cloud size distribution that is maximum for small clouds and decreases monotonically with cloud size. The EMBS model-diagnosed cloud ensemble vertical mass flux is downward within the inversion layer and upward within the cloud layer, in contrast to that obtained from previous 154 LES (Siebesma and Cuijpers, 1995; Siebesma et al., 2003) which show upward mass flux throughout both the cloud and inversion layers with a magnitude significantly larger than the EMBS diagnosis. I attribute this difference to the different definitions of the cloud boundary in the two models. In the EMBS model, the modeled clouds include not only the liquid water cloud but also the unsaturated convective region while, in these LES, cloud is defined as the liquid water containing gridcells. These diagnostic results revealed two physical aspects of shallow cumulus convection that are particularly important and require further investigation. Specifically, they are the role of the cloud size distribution in cloud ensemble transport and the role of cloud evaporation in cloud-mixture detrainment. The EMBS model is sensitive to assumptions about each of these. To further clarify these uncertainties and moreover to provide a more direct evaluation of the EMBS concept and its ability to represent shallow cumulus convection, in Chapters 4, 5 and 6 I carry out a large eddy simulation (LES) with a corresponding numerical analysis of simulated individual cloud life cycles. I also use passive numerical tracers to identify the cloud-mixed convective region and systematically diagnose cumulus mixing properties. These investigations together with the diagnostic study in Chapters 2 and 3 are intended to answer the questions raised in the end of Chapter 1. The following summarizes the major results. 7.2 Conceptual model for the transport by individ-ual convective elements One of the most important goals of this dissertation is to evaluate and refine the con-ceptual models of shallow cumulus convective transport associated with the lifetime of individual convective clouds/elements. Conceptual models such as the entraining plume model and the EMBS model have been proposed and used as building blocks for ensemble 155 cumulus parameterizations. Simulated life cycle data for individual clouds isolated from the LES fields reveal the following characteristics of the modelled transport: • The lifetime averaged vertical profile of vertical mass flux decreases with height and becomes negative in the upper cloud layers. The vertical mass flux of small clouds tends to decrease more rapidly above cloud-base than that of large clouds. • The vertical profile of convective tendencies due to transport by individual convec-tive clouds/elements produces cooling and moistening in upper penetration depths and warming and drying in lower penetration depths. I define the penetration depth to include the unsaturated subcloud layer, since the cloud root often orig-inates from surface layer: it extends to the highest level reached by cloud liquid water. • Individual convective clouds/elements have a distinct life cycle consisting of ascent, mixing and descent, and final detrainment. As a result, the vertical mass flux at any cloud level varies in a roughly oscillatory (although asymmetrical) way with time during the cloud life cycle. These simulation results decisively rule out the notion that transport due to these simu-lated individual shallow convective clouds/elements can be represented by an entraining plume model. The conflicts between an entraining plume model and the simulation results are due to the facts: • An entraining plume model produces an exponentially increasing vertical mass flux with height. • The vertical mass flux of small clouds increases more rapidly from cloud-base than for large clouds, due to the corresponding larger entrainment rate. 156 • An entraining plume model produces concentrated cooling and moistening at a single cloud-top level and warming and drying throughout the cloud's depth. In contrast, an EMBS model captures the principal features of the mass transport of individual convective elements and the qualitative dependence of vertical mass flux on cloud/element size. Furthermore, the prediction of convective tendencies based on an EMBS model is also broadly consistent with the simulation results. In view of the preceding I conclude that the EMBS model is a more realistic conceptual model for the transport associated with individual convective clouds/elements. An entraining plume model fails in the representation of the transport of individual convective clouds/elements for two reasons: 1) It does not recognize that individual convective clouds/elements detrain as well as entrain. 2) It oversimplifies the cloud evaporation and detrainment, which is an inherent part of the life cycle of individual convective clouds/elements. These two failures cause the entraining plume's vertical redistribution of heat and moisture to be highly skewed to the highest level, and vertical convective transport is therefore overestimated. As will be seen in the following sections, although the ascent of individual convective clouds/elements is not strictly adiabatic (as assumed by an EMBS model), the EMBS model appears to capture the principal features of the individual convective transport through a correct representation of cloud evaporation, detrainment and the cloud life cycle. 7.3 Conceptual model for the mixing dynamics of convective elements The simulation results clearly show the existence of subcloud scale convective elements, which are the primary agent of cumulus convection. The occurrence of each convective element in a cumulus cloud is typically episodic and produces the pulsating character 157 of cumulus growth. These "large convective eddies" are likely to generate large scale variability (inhomogeneity) for a large cloud, which simultaneously contains many of these eddies, with each being in a different stage of its life cycle. For the simulated B O M E X cumuli, these coherent convective elements are not much-smaller than the individual cloud envelope. The detailed analysis of the ascending cloud-top of each isolated cloud reveals the following character of mixing behavior and dynamics for an ascending convective element. • Ascending convective elements both entrain and detrain on their way toward their maximum height, which is roughly 4 times the element diameter. Ascending ele-ments have a constant thermal turn-over time of roughly 600 seconds. • Mixtures within the ascending convective elements are typically inhomogeneous with a relatively undilute core. Mixing begins to dilute the element core only in the upper half of the cloud depth. Mean buoyancies of an ascending element are significantly larger than cloud mean buoyancies at any given level. • There exists a distinct vortical internal circulation with ascent along the central vertical axis and descent around the periphery. The vortical circulation is marked by a distinct dynamic perturbation pressure pattern with high pressure centered at the ascending frontal cap and a low pressure below and on the downshear side vortex center. The coherent convective element ascends at approximately half the speed of the element's central updraft. • The coherent vortical circulation is driven by baroclinic torque caused by the hori-zontal buoyancy gradient. The initially horizontal vorticity is also tilted to vertical in a sheared environment and generate horizontal eddies. The large scale entrain-ing eddies associated with the vortical circulation directly engulf environmental air from the rear of the ascending element and are responsible for most of the 158 entrainment. • As the ascending elements 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 thermals occurs on the downshear side, with subsequent thermals ascending on the upshear side of their predecessors. • All ascending elements tend to evolve an asymmetric, mushroom-like core with a shrinking stem containing the relatively undilute air at later stages of the life cycle. The characteristic kinematic and dynamic structure of the simulated ascending cloud-tops, together with their core dilution behavior, lend support to the schematic shedding thermal model proposed by Blyth and co-authors (Blyth et al., 1988; Blyth, 1993). Al-though these simulation results favor a mixing picture characterized by gradual core erosion and a continuous shedding of the cloud-environment mixtures during ascent, the fact that some dilute mixtures may be carried upward for a certain distance before being shed into the wake has important implications for a further refinement of current EMBS models. This is also consistent with a multi-mixing treatment of positively buoyant mixtures. 7.4 Unsaturated convection, downdrafts and cloud evaporation Although cumulus convection is most prominently associated with the striking appear-ance of liquid water clouds, unsaturated air (invisible to the eye) may also participate in convective transport in the non-precipitating cumulus clouds. The simulation results reveal the following character of unsaturated convection. • Unsaturated downdrafts tend to develop either near the .edge (downshear side) of 159 an ascending element or after a convective element passes a particular level (for small clouds). • Large numbers of unsaturated downdrafts do not occur until the collapse of an entire convective element at its dissipation stage, indicating that unsaturated downdrafts are an inherent part of the life cycle of individual convective clouds/elements. • On average, the unsaturated convection occurs in a region roughly 1-2 times the size of the liquid water-containing cloud. Unsaturated convection dominates the convective downdrafts of the simulated clouds, although significant numbers of saturated downdrafts are also observed near cloud-top during the dissipation stage. • Unsaturated downdrafts are evaporatively driven and dominate the overall lifetime-averaged mass and buoyancy flux at upper cloud layer for all the simulated clouds. The primary role of the convective downdrafts is to redistribute the evaporative cooling and moistening which tend to concentrate near individual cloud-tops. Although these downdrafts have a vertical extent of only a few hundreds meters (e.g., 500 m for cloud F), this is a significant fraction of the individual cloud depth (e.g., 1400 m for cloud F). When cooling and moistening are continuously generated near cloud-top, the downdrafts serve to remove this cooled air and spread it downward into a deeper layer. Therefore such transport, just like the convective updrafts induced by cloud condensational heating, is fundamentally nonlocal and, in principal, should not be represented as local mixing or direct detrainment. The fact that, for shallow non-precipitating cumuli, all condensed water must be re-evaporated over a very short life-cycle emphasizes the need for a sym-metrical representation of both condensational heating and evaporative cooling through their induced convective drafts. Since finite resolution and the bulk representation of grid-scale saturation in the LES tends to overestimate the rate of cloud evaporation, it is not clear how many of these modelled unsaturated mixtures may actually be saturated 160 in real clouds. Nevertheless, the dominant role of unsaturated downdrafts in these simu-lated clouds underscores the importance of including the unsaturated part of convection in shallow cumulus parameterizations. 7.5 Buoyancy-sorting hypothesis The buoyancy-sorting hypothesis suggests that cloud mixtures move following the in-dividual mixtures' buoyancy, namely mixtures coming from above an observation level should have negative or neutral buoyancy at this level, while mixtures arriving from be-low should have positive or neutral buoyancy. I have used a conserved variable diagram following Taylor and Baker (1991) as a tool to test this hypothesis. Unfortunately, for the B O M E X case, the environmental sounding itself is already fairly linear when plotted on this diagram (Figure 4.12), making it difficult to use for this particular case. An alternative approach is to examine the conditionally sampled buoyancy and mass fluxes, since the buoyancy-sorting hypothesis requires that all cloud mixtures produce positive buoyancy flux. The simulation results show: • Buoyancy-direct transport dominates the overall buoyancy flux throughout the depth of individual clouds. In particular, saturated, positively buoyant mixtures (SP) dominate the upward mass flux while unsaturated negatively buoyant mix-tures (UN) dominate the downward mass flux. This broadly supports the buoyancy-sorting hypothesis. • A significant amount of counter-buoyancy transport does exist, especially at higher cloud levels for large clouds. In particular, saturated negatively-buoyant mixtures on average transport air upward while unsaturated positively buoyant mixtures on average transport air downward. 161 As discussed in Section 7.3, individual cloud mixtures are generated within the large eddies of convective elements. These eddies have a coherent internal vortical circulation. Negatively buoyant mixtures formed in these large eddies are not immediately rejected from the coherent updrafts following their own buoyancy. Instead they are often carried upward for a short distance before being shed into the wake of ascending thermals as either saturated or unsaturated- mixtures. Above the inversion, many upward-moving, negatively buoyant mixtures are produced due to the mixtures' inertia, which carries them beyond their NBL. As discussed in Section 7.4 and shown in Figure 4.7, a large fraction of the downward transport of negatively buoyant cloud mixtures does not occur until the collapse of an entire convective element at its dissipation stage. Therefore, the dominant role of buoyancy-direct transport in cloud convective mass flux appears to be primarily associated with the life cycle of individual convective clouds/elements, although the individual mixtures' buoyancy does play a role in partitioning or sorting which mixtures are shed behind and which are kept within an ascending element. Indeed, the simulation results show that the growth-phase mass transport is domi-nated by saturated positively buoyant mixtures with a positive buoyancy flux peaking at lower levels, while the decay phase is dominated by unsaturated negatively buoyant mixtures with a positive buoyancy flux peaking at upper levels. Both are buoyancy-direct. Although an EMBS model captures the principal buoyancy-direct portion of the life cycle transport, the significant amount of counter-buoyancy transport 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 and Cuijpers (1995). 162 7.6 A conceptual model for shallow cumulus ensem-ble transport As revealed from both observations, the I D diagnostic study and the 3D explicit nu-merical simulation, shallow cumulus convection consists of a population of convective clouds with different heights/sizes. In fact, even a single cloud often consists of a series of subcloud-scale convective elements, each of which may have a different penetration height. Cumulus ensemble transport is therefore a product of the size spectrum of con-vective clouds/elements and their individual transport. Achieving a better understanding of an individual convective cloud/element helps in deducing the dynamics of the cloud ensemble transport and therefore the cloud size distribution. Figure 7.1 gives a schematic illustration of how the cloud ensemble responds to the large-scale forcing in a typical trade-wind cumulus boundary layer such as the B O M E X case. Convective elements originating from the surface layer may penetrate to a range of vertical depths from just above cloud-base to the top of the inversion. As discussed in Section 7.2, individual convective elements have a rather particular vertical transport profile, i.e., cooling and moistening near their upper penetration depth and warming and drying near their lower depth. This character of the individual transport, combined with the large scale forcing and the observed convective equilibrium reveals the following features of the dynamics of cloud ensemble transport: • At a given level in the cloud layer, larger and smaller clouds produce competing effects on the environmental atmosphere, i.e., smaller clouds cool and moisten the environment while larger clouds warm and dry the environment. • Larger clouds have larger convective tendencies, this requires that smaller clouds exist in larger numbers in order for the cloud ensemble transport to balance the large-scale forcing. Therefore, the characteristic form of the observed cloud-size 163 distribution, in which cloud number density increases monotonically with decreas-ing cloud size, can be qualitatively understood as a result of a given large-scale forcing and the transport of individual convective clouds/elements. dt dt Figure 7.1: A conceptual model for trade-wind cumulus ensemble transport. Red denotes the large-scale forcing: surface sensible heat flux (SHF) and latent heat flux ( L H F ) , clear air radiative cooling, and large-scale subsidence. Green denotes the typical 3 layer structure of trade-wind boundary layer sounding of ft and qt. Blue denotes the ft and qt tendencies due to cumulus transport; thin solid lines represent individual transport due to large, middle, and small clouds with thick dashed lines representing the total ensemble transport required to balance the large-scale forcings. Note that large and small clouds induced tendencies are not in quantitative proportion. • In a convective equilibrium state, the smallest clouds are sustained only from the subcloud layer and suppressed by al l larger clouds, while larger clouds are sustained not only directly by subcloud layer penetrating thermals but also indirectly by smaller clouds. A n entire population of clouds must work in concert to transport and redistribute heat and water vapor out of the subcloud layer and through the 164 cloud and inversion layer. 7.7 Suggestions for future work The diagnostic study, combined with the explicit numerical simulation, suggests several avenues to improve shallow cumulus parameterizations. • For an EMBS model, the parameterization of the U E R should incorporate the cloud-size distribution. This might be attempted either explicitly by specifying a base state vertical distribution of UER or implicitly by applying a quasi-equilibrium constraint and a cloud work function for each cloud type, following the spirit of Arakawa and Schubert (1974). • For an EMBS' model, the detrainment algorithm should take into account the evap-orative effect of a mixture's liquid water content. Although the U N B L detrainment criterion provides a simple way to do so, the numerical simulation suggests that it oversimplifies the evaporation process associated with most positively buoyant and some negatively buoyant saturated mixtures. In contrast, a multi-mixing treatment is a more appropriate idealization of the LES results. I have proposed an example of such a multi-mixing algorithm in the diagnostic study. With this example, the EMBS model becomes less sensitive to the mixing distribution. However, there may be other simple ways to incorporate such an effect into an EMBS parameterization. • The large mass flux from unsaturated downdrafts associated with cloud evaporation suggest a symmetric representation of both condensational heating and evaporative cooling through their induced convective drafts. This result has broad implications for any mass flux representation of shallow cumulus convection. This needs to be emphasized, given the fact that most current schemes still try to simplify the cloud's final evaporation through the direct detrainment of liquid water into the 165 environment. Such direct detrainment can lead to unrealistic representations of cumulus transport near cloud-top. • The numerical simulation results suggest the important role of the perturbation pressure in the cloud vertical momentum budget. The Lagrangian analysis of A F C / A C T vertical momentum suggests that it may be possible to couple the buoy-ancy P P G F parameterization to the A F C / A C T buoyancy and relate the dynamic P P G F to the A F C / A C T momentum flux convergence term. • The 5 dimensional simulated cloud data isolated over an individual cloud life cycle are useful for the evaluation and calibration of other shallow cumulus parameteriza-tion schemes. An example is the'shallow transient cumulus model recently proposed in von Salzen and McFarlane (2002). This model also identifies the importance of the transient shallow cumulus life cycle effect and accounts for it through a final detrainment term. It should be noted that a shallow cumulus parameterization requires not only a cloud mixing model which determines the vertical variation of vertical mass flux and cloud de-trainment but also a closure algorithm which determines the cloud-base mass flux. This algorithm in turn depends sensitively on subcloud layer processes. Historically, the cloud-base mass flux is often determined by imposing a moisture balance for the subcloud layer such that the moisture content is maintained in the presence of large-scale transport, turbulent transports and convective transport (e.g., Tiedtke, 1989). More recently, Ray-mond (1995) proposed a theory of subcloud-layer quasi-equilibrium, which states that convective mass fluxes will adjust so that air within the subcloud layer remains neutrally buoyant when displaced upward to just above the top of the subcloud layer. A formula-tion based on this hypothesis has been implemented in the EMBS model of Emanuel and Zivkovic-Rothman (1999). The success of the LES opens many areas of investigations on topics such as this. Examples of investigations of subcloud layer processes include: 166 • Cloud root properties and cloud root coherent structure. • Properties of the transition layer near cloud base, and cloud-layer subcloud layer interaction. • Cloud-base mass flux and its control mechanism The numerical simulation, tracer technique and the cloud life cycle analysis presented in this dissertation demonstrate the potential of using numerical simulations to provide insight into cumulus mixing dynamics and associated vertical transport. These numerical results need to be compared with real cumulus observations. Direct observations of the wind structure around an ascending element are relatively rare, primarily due to their highly transient nature. As Blyth et al. (1988) point out, most aircraft observations start too late in the cloud life cycle to capture the A C T vortical circulation. Aircraft in-situ observations also suffer from the inability to define the 3D flow structure. Due to these difficulties, details of cumulus entrainment have been inferred rather than directly measured (Blyth et a l , 1988; Blyth and Latham, 1993). However, new remote sensing techniques such as millimeter cloud radar have demonstrated the potential to measure the features of the flow structure of these ascending clouds (Kollias et a l , 2001). In principal, techniques presented here, such as the analysis of the cloud-top relative velocity field, may be used to retrieve the kinematic structure of the ACT's internal circulation from radar data. Direct aircraft penetrations and tracer studies may be used in conjunction with these remote sensing measurements to provide more details about A C T mixing. It is also possible to measure the unsaturated cloud-mixed environment using techniques such as Bragg scattering (Grinnell et al., 1996; Kollias et al., 2001). Given the importance of the transient life cycle to shallow cumulus dynamics, I suggest intensive measurements of single clouds throughout their life cycle. This requires early detection of developing clouds. With this, the measurement of an A C T growth rate and thermal turn-over time can be easily obtained. 167 In this dissertation, precipitation is not considered for these shallow cumulus clouds. When convective clouds become deep enough, precipitation starts to develop. Radar observations have shown that some trade cumuli indeed precipitate. How does the pre-cipitation impact the shallow cumulus dynamics, transport and life cycle? Furthermore, how does the precipitation impact the spatial organization and regeneration of the cumu-lus cloud field? These questions are related to physical processes at a range of scales from the microphysical scale (microns) to the ensemble cloud field scale (tens of kilometers or more); representing and understanding these processes will be more challenging than for the non-precipitating case considered there. Recently the RICO (Rain in Cumulus over the Ocean) experiment ( h t t p : / / r i c o . a t m o s . u i u c . e d u / i n d e x . h t m ) has been proposed to provide a comprehensive investigation of these scientific issues using aircraft in-situ observation, remote sensing measurements, and numerical simulation. This experiment will provide an excellent opportunity to directly compare numerical simulations with cloud observations. Techniques developed in this dissertation may prove useful in this new experiment. Both ID and 3D numerical simulation will continue to serve as im-portant tools to obtain insight into the corresponding physical processes involved with precipitating cumulus clouds. 168 A p p e n d i x A L i s t o f S y m b o l s a n d A c r o n y m s .1 Symbols Symbol Unit Description t s Time P k g m- 3 Air density 9 _ 0 ms Acceleration due to gravity V Pa Pressure w ms" 1 Vertical velocity Oi K Liquid water potential temperature 9V K Virtual potential temperature (including liquid water loadin ®iv K Liquid water virtual potential temperature Qt kg kg" 1 Total water mixing ratio Qv kg k g - 1 Water vapor mixing ratio Qc kg k g - 1 Liquid water mixing ratio VH m s - 1 Large-scale horizontal wind vector QR K s - 1 Radiation-induced temperature tendency M, kg m - 2 s _ 1 Averaged cloud vertical mass flux D kg m - 3 s - 1 Detrainment rate of cloud air mass 169 Symbol unit Description El kg m - 3 s _ 1 Undilute eroding rate at level i Ul kg m - 2 s _ 1 Eroded USCA mass flux at level i Fb k g m ~ 2 s - 1 Undilute cloud base mass flux pi+1/2 k g m - 2 s _ 1 Upward flux of USCA at an interface level i + 1/2 M F n , i — Total mixture-induced mass flux at level n due to the eroding of USCA at level i, normalized by Ul DTn,i K Total mixture-induced detrainment of 9i at level n due to the eroding of USCA at level-i, normalized by Ul DQn,i kg k g - 1 Total mixture-induced detrainment of qt at level n due to the eroding of USCA at level i, normalized by Ul a3 — Fraction of environment air in the jth mixture P(aJ) — Normalized distribution of the eroded USCA in the generated spectrum of mixtures Pl(aj) — Mixing distribution based on Emanuel (1991) P2(aj) — Mixing, distribution based on Raymond and Blyth (1986) P3(aj) — Mixing distribution based on Kain and Fritsch (1990) PA(a3) — Mixing distribution based on Cohen (2000) u1'3 k g m _ 2 s _ 1 Eroded USCA mass flux component in mixture j generated at level i mh3 kg m~ 2 s - 1 Total mass flux of mixture j generated at level i q\ kg k g - 1 Environmental total water mixing ratio at leveli 9\ K Environmental liquid water potential temperature at level i 9lv K Environmental virtual potential temperature at level i kg k g - 1 Total water mixing ratio of USCA 170 Symbol unit Description Of K Liquid water potential temperature of USCA 9];° K Virtual potential temperature (including liquid water loading) of mixture j generated at level i 9\'3 K Liquid water potential temperature of mixture j generated at level i 9\'3 K Liquid water virtual potential temperature of mixture j gener-ated at level i qt,J k g k g - 1 Total water mixing ratio of mixture j generated at level i ICB — level of cloud-base ICT - level of cloud-top £ g k g - 1 Subcloud layer tracer mixing ratio rj g k g - 1 Elevated layer trace mixing ratio Co g k g - 1 C threshold value for cloud-mixed region A9V$ K A9V threshold value for convective-mixed region WQ m s - 1 w threshold value for convective-mixed region UQ m s - 1 Constant velocity for the moving reference used to sample indi-vidual clouds zr(t) m Ascending cloud-top height which is function of time ZT,max m Maximum height of an ascending cloud-top ZT,O m An A C T height when it first emerges from subcloud layer ZB m Cloud-base height a s _ 1 Cloud growth rate defined as d(logzr)/dt t* s Ascending cloud-top thermal turn-over time e m _ 1 Fractional entrainment rate 171 Symbol unit Description wT(t) ms 1 Vertical velocity of cloud-top environment interface, i.e Ui m s _ 1 Resolved wind components in each direction, i=l,2,3 Pb Pa Buoyancy perturbation pressure Pd Pa Dynamic perturbation pressure B ms Buoyancy PPGFb m s - 2 Vertical buoyancy perturbation pressure gradient force PPGFd m s - 2 Vertical dynamic perturbation pressure gradient force PPGFV _ 0 ms Total vertical perturbation pressure gradient force (•> — Horizontal model domain average -V — Volume mean A0t K Oi - (Oi), Oi difference A0V K 0V - (0V), 0V difference Aqt kg kg" 1 qt - (qt), qt difference Aw m s - 1 w - (w), w difference A.2 Acronyms A C T Ascending cloud-top p. 110 A F C Ascending frontal cap p. 110 BD Buoyancy direct p. 80 Bl Buoyancy indirect p. 80 B O M E X Barbados Oceanographic and Meteorological Experiment p. 14 C M R Convective-mixed region p. 68 172 CSULES Colorado State University Large Eddy Simulation Model p. 60 C T Cloud-top p. 80 DMR Detrained mixed-region p. 68 EMBS Episodic mixing and buoyancy sorting p. 4 E91 Emanuel (1991) p. 24 EZ99 Emanuel and Zivkovic-Rothman (1999) p. 31 GCSS G E W E X Cloud Systems Study p. 59 JFD Joint frequency distribution p. 65 L C L Lifting condensation level p. 22 LES Large eddy simulation p. 10 NBL Neutral buoyancy level p. 4 ND Negatively buoyant downdrafts p. 80 NU Negatively buoyant updrafts p. 80 PD Positively buoyant downdrafts p. 80 P P G F Perturbation pressure gradient force p. 118 PU Positively buoyant updrafts p. 80 RB86 Raymond and Blyth (1986) p. 24 SCMS Small cumulus microphysics study p. 46 SN Saturated negatively buoyant p. 77-78 173 SP Saturated positively buoyant p. 77-78 UER Undilute eroding rate p. 18 UN Unsaturated negatively buoyant p. 77-78 UNBL Unsaturated neutral buoyancy level UP Unsaturated positively buoyant p. 77-78 USCA Undilute sub-cloud air p. 17 174 A p p e n d i x B M u l t i - m i x i n g o f p o s i t i v e l y b u o y a n t m i x t u r e s To account for further mixing of positively buoyant mixtures I group them and calculate their total mass flux: m {r} ° and the mass-averaged 9i and qt\ " " E ^ M - " E , , ) S ) < B ' 2 a ) = O^T^ f _ 5 B L | f (B.2b) Note that [7l cancels from the expressions for Of and qf. This makes the mass flux and the detrainment contributions due to further mixing depend linearly on the initially eroded U\ We let the homogenized parcel S ascend to the next higher level i + 1 (the ascend-ing S-induced upward mass flux is added to MFn'1), pursue further mixing with the environment at level i + 1 and generate a new spectrum of mixtures. P(a3) P(a3) rhl>3 = - ^ S = ^—^BlUl = Cl'3Ul (B.3) I — Q-J 1 — <j3 175 The spectrum of newly generated mixtures rhh3 is treated in the same way as in the initial mixing, i.e. neutrally and negatively buoyant mixtures are detrained and posi-tively buoyant mixtures are grouped and continue their ascent and further mixing. The mixtures' buoyancy is evaluated at the current mixing level and their contribution to the mass flux is calculated based on the current mixing level and their detrainment level. The detrained mixtures' induced mass flux and detrainment tendencies are added to MFn'\ DTn'1 and DQn'1 through (3.5). The multi-mixing events (B.1-B.3) continue until further mixing between the grouped positively buoyant mixtures and their environment gener-ate only neutrally and negatively buoyant mixtures or cloud top is reached. In practice the multi-mixing process always stops well before cloud top due to the continuous ho-mogenization and dilution of the positively buoyant mixtures. The multi-mixing process completes the Ul mixing initiated at level i. 176 A p p e n d i x C C o n s i s t e n c y a n d s t a b i l i t y o f t h e U E R i n v e r s i o n C.1 Consistency of solutions I rewrite (3.6) in matrix form: ATU = <f>T (C. la ) AgU = <f>g (C . lb ) where U, (fir and 4>q are respectively vectors of eroded USCA mass flux and the total large-scale forcings of 61 and qt. AT and AQ are 61 and qt coefficient matrices generated using (3.6) with a specified mixing distribution and the UNBL detrainment criterion with the multi-mixing treatment of positively buoyant mixtures. AT and AQ have dimensions of M x M, where M — 35 is the number of vertical cloud levels; the model's vertical resolution is 40 m. Since (C.la) and (C.lb) provide two constraints on one quantity U I may solve (C.la) and (C.lb) independently. However due to uncertainties in the specified soundings and large-scale forcings of 6i and qt we don't expect a single U to satisfy both equations exactly. Below I compare these two U solutions as a check on the consistency of the 177 retrieval, using mixing distribution PI (solutions using other mass distributions yield similar results). Since both AT and Aq are well conditioned I may invert both (C.la) and (C.lb) directly1. 2000 "E1500 CD 1000 2000 If 1500 CD I 1000 U(kgrrf2s-1) x 1 0 " 2000 "E 1500 sz CD X 1000 -5 2000 1=1500 g> 'CD X 1000 0 5 U (kgm-2s-r) x 10 10 -3 -10 -5 Residual de/dt (Kday-1) .5 10 Residual dq(/dt (gkg day ) Figure C.1: a) Exact solutions of (C.la) (UT, solid line) and (C.lb) (Uq, dashed line) calculated by direct inversion, b) Smoothed solutions (UT,s, Uq,s) of a) after applying a 3-point running average, c) Residual tendencies from (C.la) given the smoothed solutions UT,S a n d UgtS, i.e. ATUT,S — 4>T (solid line) and ArUq:S — 4>T (dashed line), d) Residual tendencies from (C.lb) given the smoothed solutions: i.e. AqUqtS — 4>q (solid line) and AqUx^s — <Aj (dashed line). Figure C. la shows a comparison between U retrieved by the direct inversion (i.e. mul-tiplication by A^jq) of (C.la) and (C.lb). The comparison shows good agreement be-tween the two retrievals, although both profiles exhibit small scale variability, a feature Condition numbers are Ar i=153.21, AT2=153.03, AT3=8567.73, AT4=U5.72, 4,1=83.47, Ag2=164.00, A93=750.06, Ag4=162.60, given multi-mixing and the UNBL detrainment criterion, with Axn/qn denoting the condition number for mixing distribution Pn 178 also seen below when I introduce noise into AT or fa- Figure C.lb shows the solutions of Figure C. la after applying a 3-point running average. When the smoothed solutions UT,S and UQTS are inserted into the corresponding equations (C.la) and (C.lb) the residual tendencies are very small (see solid lines in Figure C.lc,d), with \\ATUT,S — </>T||/||</>T|| = 0.04 and ||A g£/g ] S - 0 g | | / | |0 g | | = 0.06 (where || • || is the vector norm). When I cross-substitute the smoothed solution UT,S into (C.lb), and UQTS into (C.la) equation, the residual tendencies are | |A 9 C/r ) S — c6 g||/||0 g|| = 0.18 and \\ATUQJS — 0 T | | / | | 0 T | | = 0.23 (see dashed lines in Figure C.lc,d). I find similar residuals when I try to reproduce the Oi or qt budgets using a bulk entraining and detraining plume model with diagnosed parameters from one ofthe budget equations. This suggests that the slightly larger residual tendencies found with cross-substitution are due to the imperfect coupling of 0i and qt in the large-scale forcing and environmental sounding data, i.e. when any cloud model exactly reproduces the budget equation (2.1a) it will produce a residual tendency in (2.1b) (and vice versa). Nevertheless, such differences are within the uncertainties of the LES residual tendencies, (personal communication, P. Siebesma h t t p : //www.knmi . n l / ~ s i e b e s m a / g c s s / t e n d 3 d . html) In addition to direct inversion I have also used linear least-squares to solve (C.l) subject to a smoothness constraint. This approach yields solutions nearly identical to the direct retrievals of Figure C.lb. C.2 Stability of solutions As mentioned above, the Ax/q coefficient matrices for all mixing distributions are char-acterized by small condition numbers. I therefore expect U solutions to be insensitive to the introduction of noise through round-off errors. In the following, I further explore the sensitivity of the retrieved U to noise which may exist in the soundings or large-179 scale forcings. In this calculation each term <f>T of the forcing vector <f>T is replaced by a perturbation of the form: where rj is a uniformly distributed random number in the range (-1,1) and a is an am-plitude in the range [0,0.15]. Figure C.2a shows U solutions obtained for different noise levels applied to the forcing for 61 equation (C.la). In Figure C.2b, the noisy U solutions are smoothed with a 3-point running mean as in Section 3.3. As Figure C.2a, b show, although the exact solutions vary with the introduction of noise, the three smoothed solutions are nearly identical. In Figure C.2c I insert the smoothed solutions into (C.la) and plot the residual tendency, ATUT,S — 4>T\ the residual tendencies are very small at all levels. The largest of the normalized deviations (defined by \\ATUT,S — </>T||/||0TH), is 0.07. Repeating the random perturbation procedure for the coefficient matrix AT, other mixing distributions P2-P4, and the qt equation (C.lb) yields similar results. (C.2) 180 2000 CD 1500 1000 a) I none - - 5 % ' - • 1 0 % • " • 1 5 % . 2000 'co X 1500 1000 - 5 2000 £ 1500 2z go 'cu X 1000 0 5 U T (kgirfV1) x 10 10 -3 b) I none - - 5 % 1 0 % • • " 1 5 % . U° T s (kgm -V) x 1 Q -10 -3 Residual 0 tendency (Kday 1) Figure C.2: Sensitivity of the solution of (C.la) to noise in the forcing term 4>T with relative perturbation magnitudes in the range [0, 0.15] (see text), a) Exact solutions given by direct inversion of (C.la). b) As in a) but smoothed by a 3-point running mean, c) The residual tendencies produced by substituting the smoothed solutions into (C.la). 181 A p p e n d i x D B O M E X c a s e d e s c r i p t i o n a n d L E S e n s e m b l e s t a t i s t i c s D.1 B O M E X field experiment and B O M E X GCSS model-intercomparison The effect of cumulus convection, on its large-scale (e.g., G C M grid-scale) environment may be deduced by measuring the large-scale environment instead of directly measuring the cumulus clouds themselves. This technique uses data from arrays of atmospheric soundings to calculate large-scale tendencies and the advection of heat and water vapor, which are then used to infer transports of heat and water by the subgrid-scale cumulus convection. This technique was pioneered by Yanai et al. (1973) and further developed by Ogura and Cho (1973), Nitta and Esbensen (1974b), Nitta (1975), Esbensen (1978) and others. A well-known case is the Barbados Oceanographic and Meteorological Experiment (BOMEX). Figure D.1 shows the fixed-ship array during B O M E X observation periods I, II, and III, May 1 - July 1, 1969. The season was chosen to provide a wide range of convective activity without well-developed storms (Holland and Rasmusson, 1973). Data 182 were obtained from rawinsondes launched every 1.5 hour from the four corner ships of the ship array, which covers a 500 x 500 km 2 square centered at 15° N, 56°30' W. The primary objective of the Air-Sea Interaction Program, or "Core Experiment" of B O M E X was to determine the rate of transfer of water vapor, heat, and momentum from the tropical ocean to the atmospheric. '20-111-Figure D. l : B O M E X fixed-ship array during Phases I, II, and III. May 1 - July 1, 1969. (From Nitta and Esbensen (1974b)) During phase III of B O M E X there is a 4-day period (June 22-26, 1969) which is marked by undisturbed trade-wind weather with the atmospheric boundary layer in a well-defined steady state. The atmospheric mass, heat, and water budgets during this period have been extensively investigated and reported in Holland and Rasmusson (1973), Nitta and Esbensen (1974b), and Nitta (1975). The objective of this Appendix is not to supply the detailed observational results but rather to summarize some of the major observations, while referring the details to these references. Holland and Rasmusson (1973) deduced surface sensible and latent heat fluxes of approximately 14 W m - 2 and 169 W m ~ 2 respectively. Nitta and Esbensen (1974b) presented the vertical profile of the large-scale subsidence rate with a maximum value « 0.006 m s - 1 peaking near the base of 183 the trade inversion (~ 1500m) and decreasing approximately linearly to zero. Nitta and Esbensen (1974b) also presented a near 2 K d a y - 1 radiative cooling below the inversion base, decreasing to a small value at the top of the inversion. Above the trade inversion, as shown in Holland and Rasmusson (1973), the radiative cooling approximately balanced warming due to the large-scale subsidence. The only significant diagnosed large scale advection term is a low level drying of about 1 g kg day - 1 (Holland and Rasmusson, 1973). Based on these B O M E X observational results, Siebesma and Cuijpers (1995) designed simple yet realistic large-scale forcings to drive their LES code. Their model-produced equilibrium state is in agreement with the B O M E X observations. Note that the pro-cesses above the trade inversion are not of interest for the LES study and therefore are simplified in the specification of the LES forcing. Siebesma and Cuijpers (1995) have demonstrated that the LES has the ability to reproduce many of the characteristics of the B O M E X boundary layer. Recently, the G E W E X Cloud System Studies (GCSS) Boundary Layer Cloud Working Group has undertaken a model intercomparison study for this undisturbed phase III of the B O M E X case (Siebesma et al., 2003). As described in Siebesma et al. (2003), there are four reasons for this choice: 1) It is a trade wind cu-mulus case with vertical profiles that are typical for a large part of the trade wind region. 2) There are no mesoscale complications. 3) There are no transitions between cumulus and stratocumulus. 4) There is no precipitation observed near the surface. Therefore, the undisturbed phase III of B O M E X is a simple but realistic case. For the detailed specification of the LES initialization (i.e., Figure 4.1) see Siebesma et al. (2003) and http://www.knmi.nl/~siebesma/gcss/bomex.html. 184 D.2 A comparison of ensemble statistics with other models Eleven independently developed model codes from different countries and organizations have participated in the GCSS B O M E X model intercomparison. These model codes cover a wide range of subgrid parameterization schemes and numerical representations (Siebesma et al., 2003). This intercomparison study provides an excellent opportunity to first evalu-ate my LES results by comparing the ensemble statistics with the intercomparison results (which are available from http://www.knmi.nl/~siebesma/gcss/bomex.html). Below I briefly present this comparison primarily to demonstrate that my simulation results (denoted as CSULES below) match other models in terms of ensemble statistics. For a detailed and complete description of these results, see Siebesma et al. (2003). The model abbreviations and references are listed in Table D . l . Table D. l : Model abbreviations and references Name Description N C A R National Center for Atmospheric Research (Moeng, 1984) MPI Max-Planck-Institut fur Meteorologie (Chlond, 1992) KNMI Royal Netherlands Institute of meteorology (Cuijpers and Duynkerke, 1993) U O K University of Oklahoma (Khairoutdinov and Kogan, 1999) W V U West Virginia University (Lewellen et al., 1996) U K M O 'United Kingdom Meteorological Office (Shutts and Gray, 1994) RAMS Regional Atmospheric Modeling System (Pielke et al., 1992) U C L A University of California Los Angeles (Stevens et al., 1996) UW University of Washington (Wyant and Bretherton, 1997) INM Instituto Nacional de Meteorologia (Cuxart et al., 2000) E U L A G The Eulerian/semi-Lagrangian model (Smolarkiewicz and Margolin, 1997) CSULES Colorado State University (Khairoutdinov and Randall, 2003) 185 D.2.1 Time evolution and steady state Figure D.2 shows the time evolution of the total cloud cover, the vertically integrated liquid water path and the vertically integrated T K E . These time series show a spin-up period during the first two hours of the simulations followed by a gradual approach to steady state. During the last 3 hours all models produce a steady state value for the cloud cover, although the vertically integrated T K E still shows an increasing trend and tends to level off only over longer integration times. Given these time series, I follow Siebesma et al. (2003) and calculate the LES ensemble statistics using the last three hours of my simulation. Figure D.3 shows the vertical profiles of the horizontal mean values of 61, qt, u, v, and qc. All models produce an equilibrium state characterized by a well-mixed subcloud layer, a conditionally unstable layer and an inversion layer, in agreement with the observed steady state. The most direct observational verification of the steady state is provided by the mean vertical profiles of 9\ and qt (Siebesma and Cuijpers, 1995; Siebesma et al., 2003). D.2.2 Fluxes and variances Figure D.4 shows the vertical profiles of turbulent fluxes for 0/, qt, 9V and qc.. All models produce decreasing Oi and qt fluxes from the surface to the upper cloud layer. From the upper part of the cloud layer the 9\ flux starts to increase with height while the qt flux decreases more sharply with height. The 9V flux decreases from the surface reaching a negative value near the cloud base. It then starts to increase and attains a maximum value near the middle of cloud layer before gradually decreasing to zero at the top of the inversion. Liquid water qc fluxes increase within the lower cloud layer and decrease at the upper cloud and inversion layer, with a maximum near the upper cloud layer. Figure D.5a'shows the vertical profile of horizontal velocity u flux. It decreases 186 700 j-j % <*» I 500 ui * 400 300-200 -100 NCAR MPI KNMI UOK >\ WVU • i UKMO i RAMS UCLA • UW • EULAG I* CSULES 100 150 200 Tim© {min; 300 350 Figure D.2: T ime series of a) the total cloud cover b) the vertically integrated l iquid water path and c) the vertically integrated T K E . The total cloud cover is defined as the fraction of vertical columns that contain cloud water. 187 2200 2000 1800 1600 1400 £ 1 2 0 0 g) ® 1000 800 600 400 200 a) -NCAR MPI M KNMI UOK WVU UKMO - - RAMS - I - - UCLA — uw - - INM EULAG CSULES 300 302 306 308 310 2200 2000 1800 1600 1400 Ej200 — •1000 800 600 400 200 0 - 1 1— L V 1 I 1 I NCAR MPI KNMI UOK WVU UKMO - - RAMS - - UCLA m -— UW — INM EULAG i I\ CSULES \ \ 8 10 12 q, (9*8) 14 16 18 --! 1 ' J *\S. \ ' » / NCAR ' \ \ \ \ MPI KNMI UOK WVU ! 1 L_ UKMO - - RAMS — UCLA — UW - - INM EULAG CSULES 0.005 0.01 q c (p/kg) Figure D.3: Mean vertical profiles averaged over the last 3 hours of a) l iquid water poten-t ial temperature 6i, b) total water mixing ratio qt, c) the horizontal velocity components u and v, and d) l iquid water mixing ratio qc. 188 monotonically with height to zero in the middle of the cloud layer. The v flux is nonzero due to Coriolis effects, but it is a small term and is not shown here. Figure D.5b,c,d show the resolved turbulent kinetic energy (TKE) and its horizontal and vertical components. The T K E decreases monotonically with height is most uncertain in the inversion. A double-peaked structure for vertical velocity variance is common to all model results. D.2.3 C l o u d fractions and ver t ica l mass fluxes Figure D.6a,b show the vertical profiles of cloud fraction and cloud core fraction. Here cloud is defined as all liquid-water-containing gridcells while the cloud core is defined as positively buoyant liquid-water-containing gridcells. We see that the model results agree remarkably well with a maximum around cloud base and a monotonically decreasing cloud or cloud core cover with height. The shape of these profiles is due to the fact that, in the simulated cloud fields, individual clouds have roughly the same cloud-base height but different cloud-top heights. Figure D.6c,d show the cloud vertical mass flux and cloud core vertical mass flux. All models give a systematic decrease of the mass flux with height. Again these model results agree very well. Given the fact that these model codes cover a wide range of subgrid parameterization schemes and numerical representations, the agreement between these results is encour-aging. This indicates that the ensemble statistics that are explicitly calculated by LES are not sensitive to errors in the parameterization of subgrid scale effects. This provides a sufficient justification for the LES approach in investigating the B O M E X trade-wind boundary layer (see the justification conjecture C2 in Stevens and Lenschow, 2001). 189 Figure D.4: Vert ical profiles of turbulent fluxes averaged over the last 3 h of a) l iquid water potential temperature ft, b) total water mixing ratio qt, c) vir tual potential temperature ft, and d) l iquid water mixing ratio qc. 190 Figure D.5: Vertical profiles of a) momentum u flux b) turbulent kinetic energy c) vertical velocity variance and d) horizontal velocity variance averaged over the last 3 h. 191 Figure D.6: Vertical profiles of a) cloud fraction b) cloud core fraction c) cloud vertical mass flux d) cloud core vertical mass flux. 192 A p p e n d i x E A d e s c r i p t i o n o f t h e L E S m o d e l This appendix contains a description of only those parts of the CSU L E S / C R M , which are relevant to the B O M E X simulation. For a complete description of this model see Khairoutdinov and Randall (2003). E.1 Model equations and dynamical framework The dynamics equations are based on the anelastic approximation. The momentum, mass and scalar conservation equations are written in tensor notation as dui 1 3 , . 1 dp . _, , , TT \ f®ui ~dt = - P d x - { p u ^ + r*} --Pdx~ + 5 « B + - u » ] + {-dJ dxj d h l ld<PU^FhaH(d4) +fdhl dt pdx^— \ d t j r a d \dt dqt Id (dqt -{pUjqt + Fqtj) + (E.1) dt q t J / \dt where U{ are the resolved wind components, p is the base state mean density which varies only with altitude, p is nonhydrostatic perturbation pressure, / is Coriolis parameter, Ugj is the prescribed geostrophic wind. The subscript rad denotes the radiation-induced hi tendency, the subscript Is. denotes the prescribed large-scale forcing tendency, 5i3 is 193 the Kronecker delta and t is time. The subgrid-scale stress tensor is denoted by r^, F^j and Fqtj are subgrid-scale scalar fluxes, and hi, qt and B are liquid water static energy, total water mixing ratio and buoyancy, defined as hi = cpT + gz - Lvqc qt = qc + qv (E.2) B = -gP- « g + 0.608( 9 w - (gv)) - qc^j where cp is specific heat at constant pressure, Lv is the latent heat of evaporation, g the gravitational acceleration, qc the liquid water mixing ratio, qv the water vapor mixing ratio and (•) represents the horizontal domain average. Given a vertical pressure level or height, qc and T can be diagnosed using an iterative method from the conserved prog-nostic thermodynamic variables hi and qt. Gridcells are assumed to be either completely unsaturated or completely saturated, which is also called the "all-or-nothing" subgrid-scale condensation scheme. Super-saturation with respect to water vapor is prohibited. The model equations are approximated and solved in the discretized coordinates based on a finite-difference representation. E.2 Subgrid-scale model /parameterization The subgrid-scale flux terms r -^, Fhlj and Fqtj in (E.1) have to be parameterized. They are parameterized similar to Deardorff (1980) (Note that in this section, the overbar denotes the LES grid-scale mean while the prime denotes the LES subgrid scale perturbation). rl3 = puiUj = -pKM ^— + — FHA = ph>lU> = -PKH-~ . (E.3) dq~t Fqtj = Pliu'j = ~pKH 194 where KM and KH are parameterized as KM = CklxyJl, KH = ^ - , Pr= 1 A S = ( A x . A r A z f Pr (1 + f i ) min(As, 0.76^) TV > 0 As /y < o (E-4) Nl = m l ( h , + 0 - 6 1 T c p q t ) outside cloud Ns=i^i- (hll{rLvt7PJ^) inside cloud where l\ is a subgrid-scale mixing length which is required not to exceed the grid scale As; Pr = KM/KH is the turbulent Prandtl number; N is the buoyancy frequency outside (Nu) or inside clouds (Ns). And e = \u'2 — \{u'2 + v'2 + w'2) is the turbulent kinetic energy, whose prognostic equation is written as ^ = A ^ + 5 l 3 B ^ . - M ^ - 1 - ^ - 1 - d ^ - e (E.5) dt p dxj 1 1 3 dxj p dxj p dxj where the following parameterizations apply B'w' = =6'vw> = —KHN 0„ 'du'A* e3/2 t = v\ dxj) l3 (E.6) ^ e ^ 1 /„./ 1 /„.>2 , „,/2 , „„/2\ E.3 Boundary conditions E.3.1 Upper boundary Free-slip (stress-free) rigid where z = i7 is the model upper boundary. 195 Preserving vertical gradients of thermodynamic variables (hi, qt) (SO =(jl) =constant (E.8) Sponge-layer damping In order to minimize the spurious reflection of upward propagating gravity waves, the model use a sponge layer for damping perturbations of $ £ (tt, v, w, hi, qt). <I> = <l>-(<!>-<!>„) / ? A ' , (E.9) where RD(Z) is the vertical profile of the damping time scale and has the form H-z RD(Z) = RD,MM ( J H~Z° (E.IO) where zry = 2400 m is the minimum height to start the Rayleigh damping; RD,™,™ is the minimum damping time-scale applied at the model upper boundary z — H; RD,max is the maximum damping time-scale applied at the base of damping layer and the sponge layer damps the perturbations from the initial base state <E>o £ (tin, VQ, WQ, hio, qto)-E.3.2 Lower boundary-Rigid wall with specified surface fluxes Advective fluxes are zero and subgrid-scale heat fluxes are specified as Fhl3 = ph'tw' = 9.46 Wm"' (E.H) LvFqti = Lvpq'tw' = 153.4 W m " 2 The subgrid-scale momentum flux (wind stress) u* 2 is calculated based on similarity theory (Businger, 1973) with the specified buoyancy flux and ocean surface roughness 196 z0=0.0001 m. L z k = 0.4 (E.12) 4.8f - 2 l n ^ - - l n { - ^ + 2tan~1 7T 2 ' x=(l- 19.3f )^4 E.3.3 La te ra l boundaries Periodic boundary condition are used for both east-west and south-north boundaries. E.4 Numerical methods The model uses the staggered Arakawa C-type grid, which defines the velocity compo-nents at the sides and pressure and all other scalar quantities at the center of an individual grid cell. The equations of motion are integrated using the third-order Adams-Bashforth scheme with a variable time step. This Adams-Bashforth scheme uses previous solution values at time levels n — 2, n — 1, and n to obtain the solution at time level n + 1 by assuming a polynomial interpolant through previously computed solutions. A 2nd-order centered difference scheme is used for advection of momentum. The advection scheme for all scalars is the forward-in-time multidimensional positive definite advection transport algorithm (MPDATA) with non-oscillatory option. This scheme is second-order-accurate, nonlinear and monotonic. The details of the scheme are described in Smolarkiewicz (1983, 1984); Smolarkiewicz and Clark (1986); Smolarkiewicz and Grabowski (1990). In order to diagnose the perturbation pressure, a corresponding Poisson equation is solved using fast Fourier transforms in both horizontal directions and a tridiagonal matrix solver in the vertical direction: The algorithm is rather efficient and takes only 6% of the C P U time needed for the dynamical framework calculations (Khairoutdinov and Randall, 2003). 197 A p p e n d i x F E n t r a i n i n g p l u m e m o d e l c a l c u l a t i o n An entraining plume model represents an individual convective element or cloud using an ascending subcloud air parcel that continuously entrains environmental air with a speci-fied entrainment rate. The entrained environmental air is homogenized instantaneously and the plume is finally detrained at its NBL. The following describes the calculation of the mass flux and thermodynamics properties associated with an entraining plume. The critical assumption of the calculation is a constant fractional entrainment rate e due to the plume's lateral entrainment, which is defined as IdM 0.2 e = « — (F.l) M dz R K ' where M is the plume's vertical mass flux. The relationship of e ~ 0.2/R is based on laboratory experiment with R representing the plume radius at the cloud-base level. Given a specified e, M can be directly integrated as M{z) = M(zb)exp[{z - zb)e] (F.2) where zb is the cloud-base height. M(zb) is-the cloud-base vertical mass flux. Given e, the conservation equation for the plume's thermodynamic properties Xc (x £ {0i,Qt}) is written as • • , ^ = EXe = eMXe (F.3) 198 where Xe represents environment x values and E = eM is the entrained environmental air mass flux per unit height. Plugging (F.l) into (F.3) yields the vertical dilution equation for Xc Given a specified e, an environmental sounding and the cloud properties at cloud-base level, we can calculate 6i,qt at each model level. Knowing 9i,qt and the level's pressure/height we can calculate with an iterative method the plume's buoyancy with respect to its environment, and therefore determine if the plume will continue to rise or detrain. The resulting vertical profile for the plume vertical mass and detrainment fluxes can be then put into Eq. (2.1) to calculate the induced environmental tendency for 81, qt. (FA) 199 A p p e n d i x G D y n a m i c p e r t u r b a t i o n p r e s s u r e Yau (1979) presented a physical interpretation of the dynamic perturbation pressure pd under a steady state assumption. In this appendix, I follow his argument and provide the derivation for Eqs. (5.9) and (5.10) . For the shallow convection case we may neglect the density variation with height p(z) = p0 and simplify Eq. (5.7) using the Bousinesq approximation. In the analytic model of Yau (1979) no subgrid-scale momentum fluxes were considered and viscous forces were also neglected, therefore the diagnostic equation for the dynamics perturbation pressure Pd can be written in a vector form as: - V 2 p d = - V • ( U • V U ) (G.l) Po where U is the velocity vector. By means of vector identity, the right of (G.l) can be further expanded into two parts: —S72pd = - V 2 ( - U • U ) + fi • fi - U • ( V x fi) (G.2) Po v 2 / * v ' Ta where fi = V x U is the vorticity vector. (G.2) indicates that pd represents the response to the kinetic energy (Term la) and to the rotational part of the flow (Term lb). To obtain additional insight Yau assumed a steady state and neglected the buoyancy term in the momentum equation Eq. 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