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Influence of the cloud shell on tracer budget measurements of LES cloud entrainment. Dawe, Jordan T.; Austin, Philip H. 2011

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The Influence of the Cloud Shell on Tracer Budget Measurements of LESCloud EntrainmentJORDAN T. DAWE AND PHILIP H. AUSTINDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada(Manuscript received 30 August 2010, in final form 20 July 2011)ABSTRACTDirect measurements of rates of entrainment into and detrainment from cumulus cloud cores obtainedfrom LESmodel cloud fields produce values twice as large as those produced from tracer budget calculations.This difference can be explained by two effects: the presence of a shell of air around the cloud cores that ismoister than the mean environment and air at the edge of the cloud core that is drier than the mean core, andthe tendency for the mean tracer values of the entrained fluid to be greater than the mean tracer values of thecloud shell. Preferential entrainment of shell air that is moving upward faster than the mean shell createsstrong vertical momentum fluxes into the cumulus cloud core, thereby making the assumption that cumuluscloud cores entrain fluid with zero vertical momentum incorrect. Variability in the properties of the moistcloud shell has strong impacts on entrainment values inferred from tracer budget calculations. These resultsindicate that the dynamics of the cloud shell should be included in parameterization of cumulus clouds used ingeneral circulation models.1. IntroductionThe rate at which air is entrained into and detrainedfrom cumulus clouds affects cloud properties, cloud-topheight, and vertical transports of heat andmoisture. Propersimulation of cumulus subgrid-scale fluxes in general cir-culation models (GCMs) depends on the accurate param-eterization of entrainment of environmental propertiesinto the clouds and detrainment of cloud properties intothe environment (Bechtold et al. 2008; de Rooy andSiebesma 2010).Entrainment and detrainment may be defined math-ematically asE521A?n^(u2ui),0rn^  (u 2 ui) dl, (1a)D51A?n^(u2ui).0rn^  (u 2 ui) dl, (1b)where E and D are the entrainment and detrainmentrates (kg m23 s21), r is the density of air (kg m23 s21),u is the velocity of the air (m s21), ui is the velocity of thecloud surface (m s21), A is the area of the cloud (m2), n^is a unit vector directed out the cloud surface, and thepath integral is taken around the cloud surface at a con-stant vertical level (Siebesma 1998). Entrainment anddetrainment are thus caused by differences between themotion of the cloud surface and the motion of the air.This includes not just mixing processes, but also adia-batic processes such as condensation of water vapor atcloud base.Many parameterizations use the cloud core asthe region over which to consider entrainment and de-trainment, defined as regions having condensed liquidwater, positive buoyancy, and upward vertical velocity. Inthis case, the motion of the cloud core surface is simplysubstituted for themotion of the cloud surface inEqs. (1).Entrainment and detrainment rates impact GCM pa-rameterizations in several ways. First, profiles of cloudvertical mass flux are usually calculated from parame-terized entrainment values using the continuity equationfor a simple entraining plume to represent an ensembleof cumulus clouds:r?a?t1?Mcore?z5 E 2 D. (2)Here a is the fractional cloud core area and Mcore isvertical cloud core mass flux (kg m22 s21). The levelCorresponding author address: Jordan T. Dawe, Department ofEarth and Ocean Sciences, University of British Columbia, 6339Stores Road, Vancouver, BC V6T 1Z4, Canada.E-mail: jdawe@eos.ubc.caDECEMBER 2011 DAWE AND AUST IN 2909DOI: 10.1175/2011JAS3658.1 2011 American Meteorological Societywhere the mass flux profile goes to zero then defines thelocation of the cloud ensemble top. Thismass flux profileis combined with the entrainment rate of environmentalair into the cloud and the detrainment rate of cloud airinto the environment to generate vertical profiles ofcloud water vapor, condensate, and temperature, andthese profiles are then used to calculate the moisteningof the environment by detrainment of cloud fluid (Tiedtke1989; Kain and Fritsch 1990). Precipitation rates arealso generated from the mass flux and tracer profilesproduced from the entrainment and detrainment pro-files. This wide variety of effects make entrainmentrate one of the strongest controls on the climate sen-sitivity of GCMs (Stainforth et al. 2005; Rougier et al.2009).Large-eddy simulation (LES) is the primary tool usedto study cloud entrainment. LES mass entrainment anddetrainment rates are typically obtained using budgetsof conserved tracer variables to infer the amount of fluidexchange between the cloud ensemble and the sur-rounding air. Siebesma and Cuijpers (1995) derive thefollowing equations for entrainment and detrainment ofmass from the ensemble of cloud core plumes:Ef(fcore 2 fenv) 5 2Mcore?fcore?z2?raw9f9core?z2 ra?fcore?t1 ar?f?t forcing(3a)andDf(fcore 2 fenv) 5 2Mcore?fenv?z1?r(1 2 a)w9f9env?z1 r(1 2 a)?fenv?t2 r(1 2 a)?f?t forcing, (3b)wheref (units denoted by [f]) represents any conservedtracer, such as the total specific humidity qt (kilograms ofwater per kilogram of moist air) or the liquid-water moiststatic energy h (J kg21);w is vertical velocity (m s21); thesub- and superscripts ??env?? and ??core?? denote horizon-tally averaged values conditionally sampled in the cloudenvironment and core, respectively; ??forcing?? refers totracer sources and sinks, such as radiation or large-scalesubsidence, not included in the other terms; primed valuesrepresent anomalies relative to the horizontal mean;overbars represent horizontal averaging; and Ef(z) andDf(z) are the total mass entrainment rate into and de-trainment rate from the cloud core inferred from thetracer budget (kg s21 m23). Under this budget formula-tion, Ef(z) and Df(z) consist of all horizontal mass ex-changes between clouds and their environment, includingadvective processes, diffusive processes, and thermody-namic phase changes. We shall refer to values calculatedby this method as ??tracer budget?? entrainment anddetrainment. For convenience, the various tracer andentrainment/detrainment rate subscripts used below aresummarized in the appendix.Alternatively, entrainment and detrainment of mass canbe calculated directly from the LES velocity and tracerfields. Romps (2010) recently presented a technique tomeasure local (grid scale) mass entrainment rate e(x, y, z)and detrainment rate d(x, y, z). His Eq. (2) ise 2 d 5??t(Ar) 1 $  (ruA). (4)Here A is the ??activity?? of the fluid, where A is 1at cloud core points and 0 otherwise. The values ofe 2 d are averaged over the time that a grid cell ex-periences mass fluxes between an active and an inactivepoint; positive e2 d values are considered to be purelye and negative values, d. Summing these point mea-surements horizontally gives Ed(z) and Dd(z), the totalmass entrained into and detrained from the cloud corefield (kg s21 m23), where the subscript d indicates thatthese quantities were calculated directly from themodel velocity and tracer fields. We shall refer to en-trainment and detrainment values calculated by thismethod as ??direct?? entrainment and detrainment.Romps found that such direct calculation of the en-trainment and detrainment mass fluxes produced valuesroughly twice as large as tracer budget calculations.Rompsattributed this difference to the tracer budget calcula-tion assumption that fluid exchanged between cloudsand environment has the mean properties of the cloudensemble or environment at that level, respectively. Stud-ies of the dense, descending shell of moist air that formsaround trade wind cumulus clouds (Jonas 1990; Rodtset al. 2003; Heus and Jonker 2008; Jonker et al. 2008;Heus et al. 2009; Wang and Geerts 2010) suggest thatthe cloud shell properties are quite different than thecore or environment properties, bolstering Romps? hy-pothesis. Since fluid exchanges between clouds and en-vironment must pass through this shell, it is likely that itplays an important role in entrainment and detrainmentdynamics.Below we examine the sources of the discrepancy inentrainment and detrainment values calculated via tracerbudgets and directly using Eq. (4). We show that the dis-crepancy is explained by two effects: the presence of theshell of moist air around the cloud cores and drier air at2910 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68the edge of the cloud core, and preferential entrainmentof shell air with higher average humidity and upward ve-locity than the mean shell properties, which enhancestracer fluxes between the clouds and the environment.We derive a relation to transform the direct mass entrain-ment and detrainment rates into tracer budget valuessuitable for use in one-dimensional simple entrainingplume cloud parameterizations, and then use thesetransformed fluxes to evaluate the impact of the shellon tracer budget entrainment and detrainment rates ofspecific humidity and vertical velocity. Finally, we ex-amine the dynamics that drives the preferential entrain-ment of air with higher than average specific humidityand vertical velocity.2. Model descriptionAll LES calculations in this paper were made using theSystem forAtmosphericModeling (SAM;Khairoutdinovand Randall 2003). Two model runs were performed,configured as standard Global Energy and Water CycleExperiment (GEWEX) Cloud System Studies (GCSS;Randall et al. 2003) experiments: a Barbados Ocean-ographic and Meteorological Experiment (BOMEX;Siebesma et al. 2003) run and an Atmospheric Radia-tion Measurement Study (ARM; Brown et al. 2002) run.The BOMEX run was performed on a 6.4 km 3 6.4 kmhorizontal3 3.2 km vertical domain with 25-m grid sizein all directions for 6 h, and the first 3 h of simulation werediscarded. The ARM run was performed on a 7.68 km 37.68 km 3 4.5 km domain with 30-m grid size. Precip-itation was disabled in both runs.We have implemented the direct entrainment calcu-lation scheme of Romps (2010) in SAM, allowing us tocalculate the mass of air entrained into and detrainedfrom cloud core directly from model r, u, and A. Romps[2010, his Eq. (4)] also presents a method for calculatinglocal entrainment and detrainment rates for any modelvariable in the same framework as ourEq. (4) but neglectsforcing terms. These terms are significant for quantitiessuch as vertical momentum, so we modify Romps?equation to include their effects:ef 2 df 5??t(fAr) 1 $  (fruA) 2 rASf, (5)where Sf is any nonadvective and nondiffusive source orsink term for f, such as precipitation for qt or pressuregradient for w (units of [f] s21). Diffusion is excludedfrom Sf as it is part of the entrainment/detrainmentprocess and so is included in the ef 2 df term of theequation. The inclusion of these source/sink terms al-lows us to expand the definition of f to include non-conserved fluid properties.As with Eq. (4), the local (ef)(x, y, z) and (df)(x, y, z)must be horizontally summed to give the total entrain-ment into or detrainment out of the cloud ensemblefor any fluid property, but since f can be negative forproperties such as vertical velocity, it is possible for en-trainment to reduce and for detrainment to increase thevarious properties of the cloud core. To accommodatethis effect, if the average value of f is positive over thetime that a grid cell experiences mass fluxes between anactive and an inactive grid cell, then positive ef 2df values are considered to be purely ef and negativevalues, df. However, if the average off is negative, thenpositive ef 2 df values are considered to be purely(df)(x, y, z) and negative values, (ef)(x, y, z).The obvious way to calculate the average value of fis to perform a flux-weighted calculation, so that f 5(ef2 df)/(e2 d).However, doing so for positive definitequantities, such as qt, sometimes results in negative f.To see the reason for this, consider a situation in whiche2 d integrated over the period that a grid cell is active isfound to be slightly bigger than zero for a grid cell.The Romps algorithm would assign e to a small valueand d to be zero, but this is only one of many equallyvalid choices; as long as e ? d, the net flux measuredby the algorithm would be satisfied. At the same time,(eq)d2 (dq)d f[(ef)d2 (df)d] usingqt forfg is found to benegative, presumably due to e and d having similar mag-nitudes while the detraining air has higher humidity thanthe entraining air. In this case, [(eq)d 2 (dq)d]/(e 2 d)will be negative, even though qt is always positive. To avoidthis problem, we calculate f as a simple time average forthe purpose of determining if ef 2 df is assigned to efor to df. Horizontal summation of (ef) and (df) thengives (Ef)d(z) and (Df)d(z), the total entrainment anddetrainment of a property for the cloud ensemble (unitsof [f] kg s21 m23) calculated directly from the modelvelocity and property fields.3. Relationship between direct and tracerbudget entrainmentRomps (2010) established that the direct estimate ofmass entrainment and detrainment yields values roughlytwice the size of those calculated via conserved tracerbudgets. Furthermore, examination of the ratios of themass entrainment and detrainment calculated via a totalspecific water budget (Eq, Dq) to the directly calculatedvalues (Ed, Dd) over the diurnal cycle of an ARM LESreveals significant changes over the course of the day(Fig. 1). Thus, the direct and tracer budget measure-ments of E and D are not only significantly different butalso have differing dynamics, which may need to beaccounted for in large-scale parameterizations of cloudDECEMBER 2011 DAWE AND AUST IN 2911entrainment and detrainment. In this section we exam-ine the sources of disagreement between direct andtracer budget estimates of mass entrainment into anddetrainment from the cloud core.a. E and D cloud shell correctionRomps attributed the differences between (Ef, Df)and (Ed, Dd) to the assumption made by Siebesma andCuijpers (1995) that fluid being entrainedor detrained hasthe properties of the mean environment or cloud core,respectively. If we examine the horizontal mean specifichumidity of the fluid at the ??cloud core edge?? (cloud coremodel grid cells that are nearest-neighbor adjacent tononcore cells), which presumably is the fluid being de-trained, we see that it is indeed drier than the mean core(Fig. 2a). Similarly, the fluid available for entrainment inthe ??cloud core shell?? (noncore model grid cells that arenearest-neighbor adjacent to core cells) is moister thanthe mean environment.Budget equations that explicitly distinguish betweenthe mean cloud core and environment properties andthe properties of the entraining and detraining fluid al-low us to transform (Ed, Dd) values into correspondingtracer budget values, and back again. We start ourderivation with the observation that both the tracerbudget and direct values of E and D are consistentwith the continuity equation [Eqs. (2) and (4)]. Thisimplies thatEf 2 Df 5 Ed 2 Dd. (6)Similarly, the entrainment and detrainment rates of fluidproperties must be consistent with the total propertybudget, giving usEffenv 2 Dffcore 5 EdfE 2 DdfD, (7)where fE and fD represent the f of the fluid being en-trained or detrained, respectively. Combining theseequations and solving for Ef and Df in turn results inEf,T 5 Ed 2 EdfE 2 fenvfcore 2 fenv1Ddfcore 2 fDfcore 2 fenv ,(8a)Df,T 5 Dd 2 EdfE 2 fenvfcore 2 fenv1Ddfcore 2 fDfcore 2 fenv .(8b)Here the subscript T indicates that these Ef and Dftracer budget values have been calculated by trans-formation of direct values, not via Eqs. (3). Ideally thesetransformed values should be identical to the tracerbudget values, but numerical errors may result in dif-ferences between the two calculationmethods and so weuse the subscript T to make clear the method used tocalculate each entrainment and detrainment value. Thebracketed terms represent the bias introduced by as-suming that entrained/detrained air has the properties ofthe mean environment and core. Thus, to convert from(Ed, Dd) to (Ef,T, Df,T), both Ed and Dd must be re-duced by EdA 1 DdB, where A 5 (fE 2 fenv)/(fcore 2fenv) and B 5 (fcore 2 fD)/(fcore 2 fenv).Note that rearrangement ofA givesfE5Afcore1 (12A)fenv, meaning that A can be thought of as the fractionof mean core air in a mixture of mean core and meanenvironment air needed to produce the properties ofthe entrained fluid. Similarly,fD5Bfenv1 (12B)fcoreand B can be thought of as the fraction of mean envi-ronment air in a mixture of mean core and mean envi-ronment air needed to produce the properties of thedetrained fluid.FIG. 1. Ratio of the total specific water tracer budget (a) en-trainment and (b) detrainment values to the directly calculatedvalues over the duration of the ARM model run.2912 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68Alternatively, we can solve for Ed and Dd, arriving atEd,T 5 Ef 1 EffE 2 fenvfD 2 fE1 Dffcore 2 fDfD 2 fE .(9a)Dd,T 5 Df 1 EffE 2 fenvfD 2 fE1 Dffcore 2 fDfD 2 fE .(9b)Again, the T subscript indicates these Ed and Dd valueshave been calculated by transformation of Ef and Df,not via Eq. (4). In this case, to convert from (Ef, Df) to(Ed,T,Dd,T), bothEf andDf must be increased byEfa1Dfb, where a5 (fE2 fenv)/(fD2 fE) and b5 (fcore2fD)/(fD 2 fE).We now have relationships allowing us to transformthe unbiased Ed andDd values into biased tracer budgetEf and Df values, which are better suited for simpleentraining plume parameterization of cloud fields. Com-parison of Eq and Dq (Ef and Df inferred using totalspecific moisture qt as the tracer) with Ed and Dd showsthat the direct entrainment and detrainment magnitudesare significantly larger than the tracer budget values(Figs. 2b,c, gray and dotted lines). Using Eq. (8) to cal-culate Eq,T and Dq,T with qE 5 qedge, the horizontalmean humidity in the cloud edge, and qD 5 qshell, thehorizontal mean humidity in the cloud shell, results invalues quite close to the tracer budget values above themiddle of the cloud layer. The transformation also du-plicates the negative detrainment values near cloud basethat are typically produced by tracer calculations.b. Preferential entrainment of moist ascending airRelative to the tracer budget values, theEq,T andDq,Tvalues calculated using qE5 qedge and qD5 qshell are stilltoo large near cloud base.We can explain this differenceas being the result of the mean fluid property values ofthe entrained and detrained air being different than themean values of the shell and edge air, respectively. Usingthe mean shell and edge values of properties to trans-form the direct entrainment and detrainment assumes thatany fluid parcel in the shell or edge is equally likely to beentrained or detrained. In reality,mixing relatively dry airinto the cloud core is more likely to cause evaporation,which will drive detrainment, while mixing relativelymoist air into the cloud core is more likely to producea saturated fluid mixture, resulting in entrainment. Thissuggests that the moistest shell parcels are more likely toundergo entrainment than the average shell parcel, andthe driest edge parcels are more likely to detrain than theaverage edge parcel.We can directly calculate the effective fluid propertyvalues at which entrainment occurs by taking the totalfluid property entrainment (Ef)d [calculated viaEq. (5)] and dividing it by the total mass entrainmentEd so that fentrain 5 (Ef)d/Ed. Similarly, the effectiveproperty values of the detraining air can be found fromFIG. 2. Result of transforming direct entrainment values into equivalent tracer budget values using mean cloud core shell and edgeproperties from the BOMEX simulation. (a) Mean profiles of the total specific humidity in the cloud core (thick black line), cloud coreedge (thin black line), cloud core shell (thin gray line), and cloud core environment (thick gray line). These qt values are used to transformdirectly calculated values of (b) entrainment and (c) detrainment (gray line) into equivalent tracer budget values (black line). The tracerbudget entrainment and detrainment are shown for comparison (dotted lines).DECEMBER 2011 DAWE AND AUST IN 2913fdetrain 5 (Df)d/Dd. Examination of these values fromthe BOMEX simulation using qt for f shows that qentrainis moister than qshell (Fig. 3a), indicating that entrainmentoccurs preferentially at the moistest parts of the shell.Conversely, there is little difference between qdetrain andqedge, indicating that the detrained parcels tend to havethe mean moisture of the cloud core edge. Using qentrainand qdetrain to transform Ed and Dd reduces the largeentrainment and detrainment values near cloud basefound using the mean shell and edge properties (solidblack line, Figs. 3b,c), bringing Eq,T and Dq,T into agree-ment with Eq and Dq.4. Eq, Eh, and Ew differencesEquation (8a) or (8b) implies that the tracer budgetmethod will measure different entrainment and detrain-ment values for fluid properties with differing values ofA 5 (fE 2 fenv)/(fcore 2 fenv) and B 5 (fcore 2 fD)/(fcore 2 fenv), respectively. Supporting this, Romps(2010, his Fig. 2) showed that changing the sourcesand sinks applied to an artificial numerical tracer re-sulted in large changes in measured tracer budget en-trainment rates. With this in mind we compare Ef,T andDf,T values produced by h and w with those producedusing qt.Liquid water moist static energy shows a similar relativedistribution of core, edge, shell, environment, entrained,and detrained properties when compared to qt, indicatinga tight coupling between these variables in the clouddynamics. Because these properties are so tightly cou-pled,Eh,T andDh,T values are nearly identical to theEq,Tand Dq,T (not shown).Vertical velocity shows very different relative profilescompared to qt or h (cf. Figs. 4a and 3a). There is a muchwider spread in the w values, with the shell having nearlyzero vertical velocity and the edge being halfway betweenthe core and the environment. The value of wdetrain isslightly larger than the value ofw in the cloud core edge,while wentrain is much larger than w in the shell, be-coming roughly the same value as wedge. Since wentrainand wdetrain are both larger than wshell and wedge, thisimplies that rapidly rising air is both preferentially en-trained and detrained over slowly rising air. These ef-fective entrainment and detrainment w values produceEw,T and Dw,T values (solid black line, Figs. 4b, 4c) thatare quite different than the entrainment and detrainmentproduced by qt and h (dotted line, Figs. 4b,c); Ew,T isnegative near cloud base while Eq,T is positive, and bothEw,T and Dw,T are half the magnitude of Dq,T overmuch of the cloud layer.Finally, we examine the temporal variability of A 5(fE2 fenv)/(fcore2 fenv) andB5 (fcore2 fD)/(fcore2fenv) from the transformation Eqs. (8a) and (8b) in theARMmodel run. SinceA represents the fraction of coreair in a mixture of core and environmental air that hasthe properties of fE, and B represents the fraction ofenvironmental air in amixture of core and environmentalFIG. 3. Result of transforming direct entrainment values into equivalent tracer budget values using effective entrainment and de-trainment properties from the BOMEX simulation. (a) Mean profiles of qentrain (black line), and qdetrain (dotted line), overlaid on themean total specific water values of the core, edge, shell, and environment. These qt values are used to transform directly calculated valuesof (b) entrainment and (c) detrainment (gray line) into equivalent tracer budget values (black line). The Siebesma tracer budget en-trainment and detrainment are shown for comparison (dotted lines).2914 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68air that has the properties of fD, A and B provideinformation about the reentrainment of previouslydetrained core fluid and detrainment of previouslyentrained environmental fluid occurring at differentmodel heights. However, A and B cannot be consid-ered exact mixing fractions, as the source heights ofentrained or detrained air mixtures may be differentthan the height at which they entrain or detrain, andnot all mixtures of core and environment propertiesare equally likely to undergo entrainment.The values ofA andB calculated for qt both show strongchanges over the ARM diurnal cycle (Figs. 5a,b). Nearcloud base, A is nearly 1 while B is nearly 0, implyingthat both the entrained and detrained air have theproperties of the cloud core air. This is due to the mainentrainment process at cloud base being condensationof rising thermals instead of mixing; buoyant updraftssimply condense without modification of their proper-ties. Similarly, the air that detrains from the clouds atcloud base is almost undiluted becausemost entrainmentat this level comes from the well-mixed subcloud layerbelow and so has properties nearly identical to the cloudcore. At cloud top, A is also nearly 1 while B is nearly0, implying that both the entrained and detrained airhave the properties of the cloud core air. This is due tocloud core detrainment in the inversion being driven bythe cloud becoming negatively buoyant as it enters thesteep inversion temperature gradient (Wu et al. 2009).This process is adiabatic, and so the air detraining fromthe core is undiluted by the environment. Conversely,much of the air surrounding the remaining cloud core thatis available for entrainment was previously detrainedfrom the corewithoutmixing, and so has the properties ofthe core air.As the clouds detrain into the inversion over the courseof the day, they cause the inversion to rise. This meansthat points in themidcloud layer are less influenced by theadiabatic entrainment and detrainment processes oc-curring at cloud base and cloud top, and so the effects ofmixing become more prominent. By the end of the day,A within the cloud layer has values near 0.6, suggestingthat a significant amount of reentrainment of air previouslydetrained from the core still occurs.Also,Bhas values near0.2, indicating that air detraining from the core is rela-tively undiluted by environmental air; this makes sense,since relatively little dilution by environmental air is re-quired to cause the core air to become neutrally buoyantand detrain.When calculated for w, A reaches a value around 0.4and B goes to 0.6 near the middle of the cloud layer bythe end of the day. The differences between these valuesand the values ofA and B calculated for qt are the resultof buoyancy and pressure gradient forces on themixtures,and the stronger tendency for upward-moving shell par-cels to be entrained relative to the tendency to entrainmoister shell parcels. In other words, a relatively dry,rapidly ascending shell parcel is more likely to be en-trained than a relatively moist, slowly ascending shellparcel. This is especially apparent near cloud base wherethe values ofA are larger than 1, due to themean entrainedFIG. 4. Result of transforming direct entrainment values from the BOMEX simulation into equivalent w budget values. (a) Meanprofiles of the effectivew values being entrained (black line), and detrained (dotted line), overlaid on the meanw values of the core, edge,shell, and environment. The entrained and detrained w values are used to transform directly calculated values of (b) entrainment and (c)detrainment (gray line) into equivalent tracer budget values (black line). The entrainment and detrainment values transformed using qtare shown for comparison (dotted lines).DECEMBER 2011 DAWE AND AUST IN 2915parcels having a larger upward velocity than the meancore parcels.Performing these calculations with fixed values of(fcore 2 fenv), to remove changes due to movement ofthe mean environment and core profiles, shows similarresults. Changes in the properties of the entraining anddetraining fluid due to the dynamics of mixing and en-trainment in the shell clearly are active in determiningthe rates at which properties entrain and detrain.5. Causes of preferential entrainment of moistascending airThe reason that shell air that is moister and ascendingfaster than the mean shell is more likely to be entrainedcan be seen by comparing instantaneous snapshots of themodel values of local mass entrainment e, moisture en-trainment eqt, and vertical velocity entrainment ew. SincetheRomps (2010) method of calculating e and d requirestaking time averages, it is unsuitable for calculating in-stantaneous entrainment fields. Instead, we use an alter-native method we have devised that substitutes spatialinterpolation for time averaging (DaweandAustin 2011).This alternative method results in slightly smaller valuesof e and d than those produced by Romps? method, butthe two calculations show good agreement in variability.The eqt and ew fields are calculated simply bymultiplyingthe value of e by the values of qt and w, respectively.Comparing the e, eqt, and ew fields shows that e and eqthave a very similar spatial pattern, but ew is concentratedin regions where strong updrafts enter the cloud core(Fig. 6). The reason for this can been seen by examiningthe buoyancy, condensed liquid water, and vertical ve-locity fields that define the cloud core. Of these threefields, buoyancy is the strongest constraint determining ifair is part of the core. However, regions exist far abovecloud base where air has become negatively buoyant butmaintains upward velocity and condensed liquid water.As this air continues to rise, more condensation occurs,which heats the updraft, makes it positively buoyant,and thus entrains it into the core. In this way, entrain-ment is positively correlated with both qt and w. Thisprocess occurs fairly often in our model cloud field, asevidenced both by our manual examination of theFIG. 5. Variation in (a) the fraction of core air in a mixture of core and environmental air needed to produce themean humidity entrained by the clouds, (b) the fraction of environmental air in a mixture of core and environmentalair needed to produce the mean humidity detrained by the clouds, (c) the fraction of core air in a mixture of core andenvironmental air needed to produce the mean vertical velocity entrained by the clouds, and (d) the fraction ofenvironmental air in a mixture of core and environmental air needed to produce the mean vertical velocity detrainedby the clouds, over the duration of the ARM model run.2916 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68output fields and by the size of the difference betweenwshell and wentrain in the mean profiles.While the tendency for rapidly ascending shell air tobe entrained more often than the slower parts of theshell was found for cloud core entrainment, we wouldlike to emphasize that this process is not an artifact ofthe cloud core sampling; similar results appear when weperform entrainment calculations for simple cloudy re-gions (areas of condensed liquid water). In this case,vertical advection of air can drive condensation, con-verting environment air into cloud air and thus drivingentrainment of air into the cloud.6. DiscussionConsidering all these results, we now turn to the mostimportant question of all: which entrainment value is theright one? The unsatisfying answer is that it depends onthe purpose for which the entrainment is to be used.Consider a cumulus cloud plume model based ona simplified form of the continuity equation that assumesthe cloud fraction is constant,?Mcore?z5 E 2 D, (10)a cloud budget equation for a conserved variable f thatassumes mean vertical advection is balanced by en-trainment of mean environmental properties,Mcore?fcore?z5 E(fenv 2 fcore), (11)and a simple detrainment forcing equation,r?fenv?t5 D(fcore 2 fenv). (12)The term fenv is input to the parameterization from theGCM. If we assume we have a perfect parameterizationof Mcore and fcore at cloud base with which to constructmean core mass flux and tracer profiles, we wish the EandD values to produce a profile of ?fenv/?t to force theGCM that agrees with LES results for a similar meanenvironmental profile. The E and D we desire then areFIG. 6. Instantaneous vertical cross section of (a) directly calculated cloud coremass entrainment, (b) humidity entrainment, (c) verticalvelocity entrainment, (d) buoyancy, (e) condensed liquid water, and (f) vertical velocity of a single BOMEX simulation cloud, illustratingthe tendency to entrain shell air that is rising faster than the mean shell. Black lines indicate the edge of the cloud core in each figure.DECEMBER 2011 DAWE AND AUST IN 2917closer to Ef and Df than to Ed and Dd but neverthelessare not identical to Ef and Df because of neglect of thetime tendency and Reynolds flux budget terms in Eqs.(11) and (12). This also implies that we should havedifferent Ef and Df values for properties with differentdistribution patterns around the clouds.Using values near Ed and Dd instead would requiremodifying Eq. (11) toMcore?fcore?z5 E(fcore 2 fE) 2 D(fcore 2 fD) (13)and Eq. (12) tor?fenv?t5 D(fD 2 fenv) 2 E(fE 2 fenv). (14)Now, instead of calculating different E and D values foreach tracer we wish to model, we must instead calculatefE and fD values for each property that is entrained ordetrained. While it is possible this would produce a bet-ter parameterization, it seems simpler to fold the effectsof fE and fD into the E and D values and keep theequations in their less complex forms. On the otherhand, the true values of the mass entrainment and de-trainment are important for comparison of LES resultswith field studies using Doppler radar to estimate en-trainment and detrainment velocities, or possibly for cal-culations of aerosol reactions whose chemical propertiesare dependent on the concentration of liquid water inthe air (Hoppel et al. 1994). They are also vital for di-agnosing mass exchanges of individual clouds in an LESensemble, for which a simple ??environment?? and ??cloudcore?? mean tracer budget may be difficult to define.The large positive values of wentrain are clearly in-consistent with the often-made assumption that fluidentrained into the cloud has negligible vertical mo-mentum (Siebesma et al. 2003). This is reflected in thesmaller values of the transformed Ew,T and Dw,T shownin Fig. 4. The negative values of Ew, Eq, and Dw pro-duced near cloud base emphasize the artificial nature ofthe tracer budget entrainment and detrainment. Rather,Ef and Df should be interpreted simply as mathemati-cal quantities that satisfy both the continuity Eq. (2) andthe tracer budget of the cloud core under the assumptionthat the core entrains mean environment fluid and de-trains mean cloud core fluid.As both BOMEX and ARM model simulationsinvolved nonprecipitating shallow cumulus, we haveignored the effects of precipitation. Precipitation isgenerally not considered part of the turbulent mixingprocesses associated with entrainment and detrainmentin parameterization, instead being represented by a sinkterm in the liquid water budget (Tiedtke 1989; Kain andFritsch 1990). Nevertheless, incorporating precipitationinto the tracer budget and direct entrainment calcula-tions would be relatively simple. The precipitation fluxdivergence would be a new sink/source forcing term inSiebesma?s Eq. (3), andwould be part of the forcing termrASf in Romps? Eq. (5), resulting in precipitation fluxdivergence not being counted as part of the detrainment.Specifying the advection terms would be somewhattrickier since, depending on the complexity of the mi-crophysics scheme, moisture might be advected as a sin-gle qt field or advected as separate hydrometeor classes.However, this would simply mean adding extra advec-tion terms for each hydrometeor class. Once these ef-fects were properly incorporated into the calculations,the transformations between (Ed, Dd) and (Eq, Dq)would be unchanged.7. ConclusionsWe have explained the differences between values ofcloud core entrainment and detrainment of mass cal-culated via tracer budget and direct calculations as theresult of the presence of a shell of moist air around thecloud cores and drier air at the edge of the cloud core,and the tendency for themean tracer value of the entrainedfluid to be greater than themean tracer value of the cloudshell. Relaxing the assumption made by Siebesma andCuijpers (1995) that entrained and detrained fluid hasthe properties of the environment and core, respectively,allows us to transform direct entrainment/detrainmentvalues into corresponding tracer budget values suitable foruse in simple entraining plume parameterizations ofcumulus convection.The moistest, fastest-rising regions of the shell areentrained more often than drier, slower-rising regions.This tendency results from thermodynamic phase changesin rising air parcels. Condensation in upward-moving,saturated, but negatively buoyant air parcels causes la-tent heating, increasing the air?s buoyancy and causing itto entrain into the core. This effect suggests that thedynamics of the moist cloud shell have a role in medi-ating fluxes between the clouds and the environment.Transforming directly calculated values of entrainmentand detrainment into tracer budget values results indifferent entrainment and detrainment values for totalspecificwater qt and liquid-watermoist static energy h thanfor vertical velocity w. Furthermore, the tracer budgetvalues of entrainment and detrainment compatible withqt, h, and w exhibit different patterns of variability overthe course of a diurnal cycle in a GCSS ARM LES. Thissuggests that cumulus cloud parameterizations based onbulk models should use different E and D values for qtand h than forw, and it raises the possibility that different2918 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68entrainment and detrainment rates apply to other cloudproperties, such as aerosol concentrations.Direct entrainment and detrainment calculations arepowerful tools that should be used to help improve ourunderstanding of the dynamics ofmass exchanges betweenclouds and their environment, with an eye to folding theseeffects into the simplest parameterizations possible. Doingso has the potential to improve GCM parameterization ofthe magnitude and variability of mass and tracer ex-changes between clouds and their environment.Acknowledgments. Support for this workwas providedby the Canadian Foundation for Climate and Atmo-spheric Science through theCloudAerosol Feedback andClimate network. We thank Marat Khairoutdinov formaking SAM available to the cloud modeling com-munity. We would also like to thank David Romps andtwo anonymous reviewers whose comments significantlyimproved the quality of this paper. All figures were gen-erated using the matplotlib library in the Python pro-gramming language.APPENDIXTable of NotationTable A1 shows the variables, along with their defi-nitions and units, used in this study.REFERENCESBechtold, P., M. Ko?hler, T. Jung, F. Doblas-Reyes, M. Leutbecher,M. J. Rodwell, F. Vitart, and G. Balsamo, 2008: Advances insimulating atmospheric variability with the ECMWF model:From synoptic to decadal time-scales. Quart. J. Roy. Meteor.Soc., 134, 1337?1351, doi:10.1002/qj.289.Brown, A. R., and Coauthors, 2002: Large-eddy simulation of thediurnal cycle of shallow cumulus convection over land. Quart.J. Roy. Meteor. Soc., 128, 1075?1093.Dawe, J. T., and P. H. Austin, 2011: Interpolation of LES cloudsurfaces for use in direct calculations of entrainment and de-trainment. Mon. Wea. Rev., 139, 444?456.de Rooy, W. C., and A. P. Siebesma, 2010: Analytical expressionsfor entrainment and detrainment in cumulus convection.Quart.J. Roy. Meteor. Soc., 136, 1216?1227, doi:10.1002/qj.640.Heus, T., and H. J. J. Jonker, 2008: Subsiding shells around shallowcumulus clouds. J. Atmos. Sci., 65, 1003?1018.??, C. F. J. Pols, H. J. J. Jonker, H. E. A. Van den Akker, andD. H. Lenschow, 2009: Observational validation of the com-pensating mass flux through the shell around cumulus clouds.Quart. J. Roy. Meteor. Soc., 135, 101?112.Hoppel, W. A., G. M. Frick, J. Fitzgerald, and R. E. Larson, 1994:Marine boundary-layer measurements of new particle for-mation and the effects nonprecipitating clouds have on aerosolsize distribution. J. Geophys. Res., 99, 14 443?14 459.Jonas, P. R., 1990: Observations of cumulus cloud entrainment.Atmos. Res., 25, 105?127.Jonker, H. J. J., T. Heus, and P. P. Sullivan, 2008: A refined view ofvertical mass transport by cumulus convection. Geophys. Res.Lett., 35, L07810, doi:10.1029/2007GL032606.Kain, J. S., and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective pa-rameterization. J. Atmos. Sci., 47, 2784?2802.TABLE A1. List of symbols.Symbol Unit Definition First occurrenceE(z), D(z) kg m23 s21 Cloud core mass entrainment, detrainment rate Eq. (2)Ef(z), Df(z) kg m23 s21 Mass entrainment, detrainment rate calculatedusing the Siebesma tracer budgetEqs. (3a), (3b)e(x, y, z), d(x, y, z) kg m23 s21 Local mass entrainment, detrainment rate calculateddirectly from model velocity and tracer fieldsEq. (4)Ed(z), Dd(z) kg m23 s21 Cloud core mass, detrainment rate calculated from thehorizontal sum of e(x, y, z) or d(x, y, z)Section 1(ef)(x, y, z), (df)(x, y, z) [f] kg m23 s21 Local cloud core f entrainment, detrainment rate calculateddirectly from model velocity and tracer fieldsEq. (5)(Ef)d(z), (Df)d(z) [f] kg m23 s21 Cloud core f entrainment, detrainment rate calculated from thehorizontal sum of (ef)(x, y, z) or (df)(x, y, z)Section 2Ef,T(z), Df,T(z) kg m23 s21 Mass entrainment, detrainment rate calculated by transformingdirect values into budget valuesEqs. (8a), (8b)Ed,T(z), Dd,T(z) kg m23 s21 Cloud core mass entrainment, detrainment rate calculated bytransforming tracer budget values into direct valuesSection 1f(x, y, z) [f] Any fluid property, such as qt (kg kg21), h (J kg21), or w (m s21) Section 1fcore(z) [f] Mean cloud core f Section 3afedge(z) [f] Mean cloud edge f Section 3afshell(z) [f] Mean cloud shell f Section 3afenv(z) [f] Mean environment f Section 3afentrain(z) [f] Effective value of f being entrained calculated from (Ef)d/Ed Section 3bfdetrain(z) [f] Effective value of f being detrained calculated from (Df)d/Dd Section 3bfE(z) [f] Placeholder for the value of f assumed to be entraining Eq. (8a)fD(z) [f] Placeholder for the value of f assumed to be detraining Eq. (8b)DECEMBER 2011 DAWE AND AUST IN 2919Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolvingmodeling of the ARM summer 1997 IOP: Model formulation,results, uncertainties, and sensitivities. J.Atmos. Sci., 60, 607?625.Randall, D., and Coauthors, 2003: Confronting models with data:The GEWEXCloud Systems Study.Bull. Amer. Meteor. Soc.,84, 455?469.Rodts, S. M. A., P. G. Duynkerke, and H. J. J. Jonker, 2003: Sizedistributions and dynamical properties of shallow cumulusclouds from aircraft observations and satellite data. J. Atmos.Sci., 60, 1895?1912.Romps, D. M., 2010: A direct measure of entrainment. J. Atmos.Sci., 67, 1908?1927.Rougier, J., D. M. H. Sexton, J. M. Murphy, and D. Stainforth,2009: Analyzing the climate sensitivity of the HadSM3 climatemodel using ensembles from different but related experi-ments. J. Climate, 22, 3540?3557.Siebesma, A. P., 1998: Shallow cumulus convection. BuoyantConvection in Geophysical Flows, E. J. Plate, Ed., KluwerAcademic, 441?486.??, and J. W. M. Cuijpers, 1995: Evaluation of parametric as-sumptions for shallow cumulus convection. J. Atmos. Sci., 52,650?666.??, and Coauthors, 2003: A large-eddy simulation intercomparisonstudy of shallow cumulus convection. J. Atmos. Sci., 60, 1201?1219.Stainforth, D. A., and Coauthors, 2005: Uncertainty in predictionsof the climate response to rising levels of greenhouse gases.Nature, 433, 403?406.Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulusparameterization in large-scale models. Mon. Wea. Rev., 117,1779?1800.Wang, Y., and B. Geerts, 2010: Humidity variations across the edgeof trade wind cumuli: Observations and dynamical implica-tions. Atmos. Res., 97, 144?156, doi:10.1016/j.atmosres.2010.03.017.Wu, C.-M., B. Stevens, and A. Arakawa, 2009: What controls thetransition from shallow to deep convection? J. Atmos. Sci., 66,1793?1806.2920 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 68


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