326 MONTHLY WEATHER REVIEW VOLUME 120Parameterization of Convective Precipitation in Mesoscale Numerical Models: A Critical Review JOHN MOLINARI AND MICHAEL DUDEKDepartment of Atmospheric Science, State University of New York at Albany, Albany, New York(Manuscript received 30 November 1990, in final form 24 June 1991) Current approaches for incor0orating cumulus convection into mesoscale numerical models are divided intothree groups. The traditional approach utilizes cumulus parameterization at convecfively unstable points andexplicit (nonl~xameterized) condensation at convectively stable points. The fully explicit approach uses explicitmethods regardless of stability. The hybrid approach parameterizes convective scale updral~ and downdrafts,but "detrains" a fraction of parameterized cloud and precipitation particles to the grid scale. This allows thepath and phase changes of such particles to be explicifiy predicted over subsequent time steps. The traditional approach provides the only alternative for numerical models with grid spacing too large toresolve mesoseale structure. As grid spacing falls below 50 kin, the traditional approach becomes increasinglylikely to violate fundamental scale-separation requirements of parameterizafion, particularly if mesoscale organization of convection is parameterized as well. The fully explicit approach has no such limits, but it hasrepeatedly failed in mesoscale models in the presence of large convective instability. Although it is preferableunder certain specialized circumstances, the fully explicit approach cannot provide a general solution for modelswith grid spacing above 5-10 km. The hybrid approach most cleanly separates convective-scale motions from the slow growth, fallout, andphase changes of detrained hydrometeors that produces mesoscale organization of convection. It is argued thatthis charaaerisfic removes the need to parameterize the mesoscale and thus reduces the scale-separation pwblcmsthat may arise when the traditional approach is used. The hybrid approach provides in principle the preferredsolution for mesoscale models, though such promise has yet to be fully realized. In the absence of large rotation, the fundamental assumptions of cumulus parameterization begin to breakdown once grid spacing falls below 20-25 kin. For models with such resolution, the time scale oftbe convectionbeing parameterizcd approaches the characteristic time scale of the grid, and parameterized and unparameterizedconvective clouds often exist simultaneously in a grid column. Under such ambiguous circumstances, successfulsimulations have been produced only because parameterized convection rapidly gives way in the model to itsgrid-scale counterpart. It is essential to understand the interactions between implicit and explicit clouds thatproduce this transition, and whether they represent physical processes in nature, before cumulus parameterizationcan be widely used in such high-resolution models. In a broader sense, more detailed analysis of why convectiveparameterizations succeed and fail is needed.1. Introduction The pioneering work of Riehl and Malkus (1958)showed that in convectively unstable regions, verticaltransports of mass and moist static energy were notaccomplished by the synoptic-scale circulation but byindividual cumulonimbus clouds. The concept of cumulus parameterization in numerical models was required to incorporate these otherwise unrcsolvablesubgrid-scale transports. Subsequent development hasbeen active, as indicated by the large number of reviewand overview papers in recent years (Ooyama 1982;Frank 1983; Arakawa and Chen 1987; Tiedtke 1988;Cotton and Anthes 1989). Without exception, these Corresponding author address.' Dr. John Molinari, Department ofAtmospheric Sciences, State University of New York a~ Albany, Albany, NY 12222,authors agree that cumulus parameterization is required in large-scale numerical models (grid spacinggreater than 50-100 kin) at convectively unstable gridpoints. As computer power has continued to advance, veryhigh-resolution mesoscale numerical models have beendeveloped. Cotton and Anthes (1989) suggest that theconceptual basis for cumulus parameterization becomes "muddy" and "not well posed" when the modelgrid spacing falls below about 50 km. Along these lines,some researchers have omitted cumulus parameXerizafion in high-resolution models and instead directlysimulated cumulus convection on the grid (e.g., Yamasaki 1977; Rosenthal 1978). Conversely, otherthors have designed cumulus parametefizafions expressly for grid spacings well below $0 km (Fritsch andChappell 1980; Frank and Cohen 1987). The choiceof what and whether to parameterize in mesosealemodels has been further complicated by the recognitionc 1992 American Meteorological SocietyFEBRUARY 1992 MOLINARI AND DUDEK 327that convection in nature often develops mesoscale organization. This resolvable mesoscale structure develops from initially unresolvable cumulonimbus cloudsand thus provides a major challenge to mesoscalemodelers. As a result of the above difficulties, greateruncertainty exists in cumulus parameterization thanin any other aspect of mesoscale numerical weatherprediction. In a thorough discussion of the conceptual basis forcumulus parameterization, Arakawa and Chen(1987)noted that in principle the solution to the problemsabove lies in using a grid spacing on the order of 100m. This does not remove the necessity of parameterizing microphysics and turbulence, nor does it addressthe source of initial data on such scales. It does, however, allow the direct simulation of clouds and thuseliminates the need for cumulus parameterization.Unfortunately, computer power will not currently allow use of 100-m resolution in three-dimensional mesoscale models. Even if such resolution were possiblewithin the decade, interpretation of literally billions ofmodel output points would be extraordinarily challenging due to the stochastic nature of the scales beingsimulated (Ooyama 1982). The need for simplermodels that represent larger scales of motion will notdisappear. As a result, the question of how to parameterize cumulus convection will likely remain for anindefinite period. In the current paper, the goal is to provide a framework for choosing the form for cumulus parameterization in mesoscale models. The discussion will beconfined primarily to three-dimensional models, because limited-area two-dimensional models are (orsoon will be) able to use cloud-model resolution foridealized studies. In this paper, the term mesoscalemodel will refer to any hydrostatic model with gridspacing between 10 and 50 km, which could includeregional or even global models of such resolution. Cumulus parameterization in current-generation generalcirculation models (grid spacing > 80-100 km), forwhich the problem is better defined, will not be addressed. Parameterization in theoretical models suchas wave CISK (Lindzen 1974; Raymond 1986) will beomitted, as will nonhydrostatic cloud models, for whichthe parameterization problem exists more with regardto cloud microphysics and turbulence. Only thermodynamic aspects of the problem will be considered because convective momentum transports remain toopoorly understood to parameterize. Parameterizationof slantwise convection will be dealt with only in termsof how it may impact upright cumulus parameterization. Emphasis will be given to (i) conceptual groupingof current approaches in mesoscale models; (ii) fundamental and practical limitations of each; (iii) problems presented by mesoscale organization of convection and by slantwise convection; (iv) recommendations for the optimum approach as a function of modelgrid spacing; and (v) suggestions for directions of futurework. Definitions and terminology. Cumulus parameterization requires the creation of subgrid-scale implicitclouds, which vertically transport heat, water vapor,and other quantities, generally in the absence of gridscale saturation. Closure assumptions are required todefine the relationship between these implicit cloudsand the grid-scale variables. For mesoscale models, theform of closure may have to difl~r from that for largescale models (Fritsch and Chappell 1980; Frank 1983).Details of the closure issue will not be discussed in thecurrent paper. A conceptual classification of closureassumptions has been provided by Arakawa and Chen(1987). The unparameterized or explicit condensationmethod allows clouds to exist only when grid-scale saturation occurs. To the extent possible in a hydrostaticmodel, the cloud is directly simulated on the grid. Thisdefinition of the explicit method holds, regardless ofthe level of microphysics, from the simplest form whereno cloud stage is present and condensed water is assumed to fall immediately as rain, to a complex formin which prediction equations exist for cloud water andice, rainwater, and various forms of frozen particles.In all of these explicit formulations, no subgfid-scaleclouds are accounted for. A single characteristic unambiguously distinguishes the two methods: in implicitmethods, the properties of the cloud(s) differ fromthose of the grid; in explicit methods, cloud and gridare synonymous. In large-scale numerical weather prediction models,both implicit and explicit methods have traditionallybeen used, following the historical division of precipitation in nature into convective and stratiform, respectively (see, for example, Houghton 1968). Thus,cumulus parameterization was used at convectivelyunstable grid points, and explicit condensation at nonconvective (i.e., convectively stable) grid points(Krishnamurti and Moxim 1971 ). In practice, nonconvective condensation represents simply the removal of supersaturation (see, for example, Kanamitsu1975; also described by Molinari and Corsetti 1985,appendix B). In mesoscale models, this broad consensus on theform of incorporating cumulus convection has not beenmaintained. In Table 1, a conceptual division of currentapproaches is proposed. The traditional approach usesimplicit or explicit methods depending upon localconvective stability, as described earlier.~ The fully explicit approach uses only the explicit formulation, re ~ In large-scale models, nonconvective precipitation may also beparameterized, typically in terms of areal coverage (Sundqvist 1978 ).Such variations will not be considered in this discussion ofmesoscalemodels.MONTHLY WEATHER REVIEW VOLUME 120 TABLE 1. Description of approaches to cumulus parameterization in mesoscale models. Convectively unstable Convectively stable Approach points pointsTraditional Implicit ExplicitFully explicit Explicit ExplicitHybrid Hybrid Explicitgardless of convective stability. A third category, whichwill be labeled hybrid, is proposed in this paper. Thehybrid approach (see, for instance, Frank and Cohen1987) uses cumulus parameterization to provide a vertical distribution of cloud and precipitation particlesin convectively unstable regions. A fraction of theseparticles is "detrained" into the cloud environment,then is predicted explicitly over subsequent time stepsusing nonconvective forecast equations and advectionby grid-scale motions. In convectively unstable regions,the hybrid approach is thus partly implicit and partlyexplicit. The hybrid approach should be distinguished frompure cumulus parameterization found in the traditionalapproach. For example, Fritsch and Chappell (1980)and Emanuel ( 1991 ) compute condensate in their entraining updrafts and downdrafts and incorporate theinfluence of evaporation of convectively generatedcondensate. These and other similar procedures do notclassify as hybrid, however, because microphysical influences must be incorporated all at once. Water content is implicit and not carried to subsequent time steps,and it does not communicate with its grid-scale counterpart. In practice, the definition of hybrid used here requires grid-scale, nonconvective prediction equationsfor cloud and precipitation particles. In addition, theseequations must contain convective source terms inwhich implicit particles are transferred to the grid scale.Thus, the presence of cumulus parameterization plusnonconvective microphysical equations alone is insufficient. For example, Zhang and Gao (1989) use theFritsch-Chappell (1980) cumulus parameterization atunstable grid points, plus rather detailed microphysicalforecast equations at stable grid points or upon supersaturation. This qualifies as a traditional approach because the cumulus parameterization does not supplyexplicit precipitation particles to the grid scale. It willbe argued in section 4 that the difference between thetraditional and hybrid approaches is fundamental. For large-scale models, only the traditional approachhas been utilized (e.g., Krishnamurti et al. 1983; Albrecht 1983; Betts 1986; Tiedtke 1989 ). In mesoscalemodels, however, each of the three approaches has beentried, with a bewildering variety of results. This reviewwill attempt to establish a basis for the choice betweenapproaches.2. The traditional approacha. Arguments in favor of cumulus parameterization The traditional approach remains the dominantchoice in numerical weather prediction models, including (to the authors' knowledge) 100% of operational models. The traditional approach requires cumulus parameterization at convectively unstable gridpoints when conditions are met for its initiation. Thisimplicit inclusion of cumulus clouds has been justifiedfor the following reasons. (i) The preferred scale for convective instabili~ty(initially) is less than the grid spacing of any hydrostaticmodel (e.g., Lilly 1960). If saturation were allowed inthe presence of convective instability, unstable growahwould be aliased onto the smallest resolvable., scales ofthe model. Cumulus parameterization attempts toavoid this problem by allowing convective instabilityto be removed without grid-scale saturation in unstablelayers. (ii) Significant convective precipitation (and thusa net heat source in the column) occurs in nature whenthe grid-scale area is unsaturated. A model that allowedcondensation only upon grid-scale saturation wouldfail to reproduce this heat source. (iii) Large vertical fluxes of heat, moisture, andother quantities by cumulus convection occur on scalesunresolvable by hydrostatic model grids. A model thatdid not implicitly include such subgrid-scale sourceterms could not accurately predict grid-scale evolutionwhen convection was active. The sum of net condensation heating (or moistening) in the column [(ii) above] and convective eddyflux convergence [ (iii) above ] can in principle be measured from observations using residuals from large~scalebudgets. These residuals are known as the apparentheat source Q~ and the apparent moisture sink 02(Yanai et al. 1973). Molinari and Dudek (1986) notedthat if the vertical profiles of Q~ and Q2 differ, eddyfluxes are nonzero and cumulus parameterization isrequired. If Q~ and Q2 are nearly identical, heating isdetermined by the local condensation rate and parameterization becomes unnecessary. On the synoptic scale in convectively unstable regions, precipitation occurs without mean saturationand vertical subgrid-scale fluxes are large, so that Q~ isnonzero and differs from Q2 (Riehl and Malkus 1958;Yanai et al. 1973). Observations thus support the needfor cumulus parameterization in models with gridspacing on the order of 100 km or larger. In practice,the omission of cumulus parameterization in suchmodels may produce gross underestimates of observedconvectiv$ precipitation (Molinari and Corsetti 1985). In contrast to the synoptic scale, data on the scaleof mesoscale model grid points are insufficient to accurately determine the budget residuals that make upFEBRUARY 1992 MOLINARI AND DUDEK 329Q~ and Q2. Even high-resolution Severe EnvironmentalStorm and Mesoscale Experiment (SESAME) data hadat best an 80-km mean station separation and a 3-htime resolution. Kuo and Anthes (1984) found thatspatial and temporal averaging were required to produce a coherent signal, and the resultant Ql and Q2were valid only for large-scale models. Current datasets thus cannot prove the need for theparameterization of cumulus convection in mesoscalemodels. Instead, numerical models themselves mustbe used empirically to make such determinations. Theresults of such numerical experiments will be discussedin the following sections.b. Arguments against cumulus parameterization in mesoscale models Regardless of what observations might show, conceptual problems arise in cumulus parameterization asgrid spacing decreases. Arakawa and Chen( 1987 ) discussed the need for a distinct scale separation and'agap in the energy spectrum between the scale beingparameterized and that being explicitly resolved by thegrid. They noted the lack of a spectral gap in naturebetween the cloud scale and mesoscale. In practice,this means that in a mesoscale model, explicit cloudsmay form at a grid point while essentially similarclouds are simultaneously being parameterized. Calculation techniques are such that modelers do not allow"double counting": energy and moisture are conservedin such situations because grid-scale condensation iscomputed at the end of the time step after all otherprocesses have acted (e.g., Kanamitsu 1975). Nevertheless, the presence of the same physical process inparameterized and unparameterized forms at the samegrid point seems ambiguous at best. This difficulty canbe avoided in practice only by a cumulus parameterization that always maintains subsaturation in convectively unstable layers. As the grid spacing becomessmaller, however, grid-scale saturation is more likelyto be observed, and an approach that prevented itsoccurring would not simulate nature. This issue willbe addressed further in section 4. As part of the spectral gap assumption in cumulusparameterization, the subgrid-scale eddies are assumedto have a time scale much smaller than grid-resolvabledisturbances (Ooyama 1971). This allows the integrated influence of the life cycle of convection to beinserted during a given time step in the model. In reality, the deep convection may take 1 h (or more, ifmesoscale organization is included; see section 2c) toproduce this cumulative effect. In mesoscale models,the time scale of grid-scale motions may be comparableto the characteristic life cycle of the convection beingparameterized. Related to the time-scale issue is the fractional coyerage of convection over a grid area, which is formallyassumed to be much less than unity in some cumulusparameterizations (e.g., Arakawa and Schubert 1974).Although this assumption can be extended if the fractional coverage is assumed to represent only areas ofactive updrafts (Ogura and Kao 1987), it must breakdown when the grid spacing becomes small enough tobe filled by active cloud in nature. Ooyama (1982) addressed the time-scale problemin terms of local rotational constraints. Ooyama notedthat under strong rotation, the local deformation radiuscan shrink enough to produce a long-lasting, inertiallystable disturbance. In such cases, divergent circulationsare controlled by the slowly varying primary circulation. By this reasoning, the time-scale separation requirement is indeed met, even in mesoscale models,under sufficiently strong rotation. Ooyama noted it wasthis characteristic that allowed the success of cumulusparametedzation in numerical simulation of maturehurricanes. He warned against uncritically using thesame approaches to understand hurricane formation,for which rotational constraints are weaker and theconstraints imposed by closure may be too restrictive. Arakawa and Chert (1987) note that two closureconditions are required because four unknowns existin the cumulus parameterization problem (Q~, Q2,OT/Ot, and Oq/Ot), but only two equations. As notedby Ooyama (1982), these closures constrain, sometimes strongly, the allowed interactions between thegrid scale and the convection. For mesoscale models,the lack of observations makes it impossible to verifyany current closure. As a result, this fundamental aspectof mesoscale cumulus parameterizations remains onan ad hoc basis. The ambiguities arising from questionable scaleseparation assumptions and closure conditions maycause problems in practice as well. Frank (1983) notedthat when rotational constraints are weak, local divergent circulations may become dominant in mesoscalemodels. Grid-scale forcing may initiate parameterizedconvection, but once heat is released at the model gridpoint, the local circulations under weak forcing maygrow rapidly in a manner determined by the details ofthe cumulus parameterization. The implicit assumption that the grid scale deterministically controls theconvection (Arakawa and Chen 1987) appears to beviolated. Most cumulus parameterizations have freeparameters whose actual value in nature is unknown;under the above circumstances the solution may become overly sensitive to such parameters (Rosenthal1979). The arguments against cumulus parameterization may be summarized as follows: (i) at some small grid spacing, which is less than 50 km and greater than 5 km, the requirements for scale separation in time and space between convection and grid-scale flows may not be met except locally under strong rotation; and (ii)330 MONTHLY WEATHER REVIEW VOLUME 120for any model with less than 50-km grid spacing, keyaspects of the parameterization remain ad hoc due tothe lack of observations on such small scales. In thefollowing section, it will be shown that the presence ofmesoscale organization of convection makes theseproblems worse.c. Complications of mesoscale organization The role of cloud-scale downdrafts in removingconvective instability by cooling the planetary boundary layer has long been accepted. Over the last twodecades, however, the presence of unsaturated downdrafts on scales much larger than individual cumulusclouds has been recognized (Zipser 1969, 1977; Houze1982; Lcary and Rappaport 1987). These downdraftsare part of a mesoscale organization of upper-tropospheric updrafts and lower-tropospheric downdrafts,which typically evolves as follows: (i) Intense individualcumulonimbus cells break out over an area in the presence of large convective instability and favorable,though often not intense, dynamical foming (e.g.,Maddox 1983); (ii) These cells detrain large numbersof frozen hydrometeors, mostly in the upper troposphere (Houze 1989); (iii) These frozen particles areadvected by upper-tropospheric flow and slowly fallout; as they fall they may grow by vapor deposition ifan anvil has already developed (Rutledge and Houze1987; Houze 1989; Dudhia 1989); (iv) When theseparticles reach the freezing level they melt and laterevaporate or reach the ground as widespread stratiformrain; (v) As a result of (iii) and (iv), fusion heatingoccurs above the freezing level and cooling by meltingoccurs immediately below the freezing level, as well asevaporative cooling occurring in the lower troposphere;(vi) Associated with these heating and cooling patterns,widespread upward motion occurs in the anvil anddownward motion occurs in the region of stratiformrain below. The downdraft region typically remainsunsaturated and relatively warm, and a strong inversionoften develops above a cool, moist boundary layer(Betts 1973). As a result, the column may stabilizeand limit active convection to the leading edge of thesystem (Leary and Rappaport 1987). The mesoscalecirculations may extend 100 km or more from theconvective sources. This evolution may take 6 h ormore, although in a propagating system it may be experienced more rapidly at a given location (Leary andRappaport 1987). A significant fraction of total rainfallwith the convective system occurs in the stratiform region (Cheng and Houze 1979; Gamache and Houze1982). Recently, Gallus and Johnson ( 1991 ) provideda well-documented example of mesoscale organizationin a severe squall line in the PRE-STORM experiment. As a result of these processes, the vertical distributionof heating for the mature mesoscale convective systemdiffers significantly from that for isolated cumulonimbus (Houze 1982; Johnson 1984). Hartman et al.(1984) found that a "mature cluster" heating, profile,which included mesoscale influences, reproducedtropical divergent circulations in a linear steady-sta~emodel more accurately than a heating profile: characteristic of isolated cumulonimbus clouds. The authorssuggested that the mesoscale heating profile representsthe dominant mode of diabatic heating in the tropics.Taken together, the studies reviewed in this sectionindicate that mesoscale organization of convectioncannot be overlooked. Mesoscale organization is also present in the pressureand wind fields, often with rather complex verticalstructure. It appears, however, that this structure develops primarily as a result of the diabatic sources andsinks associated with the processes described above(Zhang and Gao 1989; Chen and Cotton 1988). As aresult, this paper will not deal directly with surface mesohighs and mesolows, midlevel vortices, and othermanifestations of mesoscale organization of convection. Rather, it will be assumed that if the heat sourcesassociated with convective processes are well simulated,the wind and pressure structure will follow in themodel. Further detail concerning the mesoscale structure of convective systems is provided by Houze(1989), Cotton and Anthes (1989), and Emanuel andSanders (1983). The development of mesoscale organization in precipitating convection complicates the already difficultcumulus parameterization problem. In large-scalemodels, for which the entire mesoscale circulation remains subgrid scale, Arakawa and Chen(1987) provideconvincing evidence that mesoscale effects are parameterizable in principle. Emanuel (1991 ) has proposeda specific parameterization of unsaturated mesoscaledowndrafts for large-scale models. The mesoscalemodeler, however, faces additional difficulties. Initiallythe convection is subgrid scale and must be parameterized. As the convective cluster matures, mesoscaledowndrafts and associated stratiform rain cover an increasing portion of the grid area. Eventually, as thecolumn stabilizes, convective updrafts may stop, leaving only the mesoscale circulation (Johnson and gMete1982); or the convectively active region may propagateto an adjacent grid point, leaving a mesoscale circulation in its wake (Leary and Rappaport 1987). In either case, the upper-tropospheric updraft and lowertropospheric unsaturated downdraft become (at theoriginal point) convectively stable, grid-scale phenomena, unaccompanied by the convective scale, and areno longer parameterized. The mesoscale modeler mustsimulate (i) the vertical distributions of subgrid-scaleheat, moisture, and hydrometeor som:ces due tosubgrid-scale convection; (ii) their time variation overthe life cycle of the system, as mesoscale downtdraftsincrease in areal coverage; and (iii) the explicit updraftFEBRUARY 1992 MOLINARI AND DUDEK 331downdrafi couplet, saturated aloft and unsaturated below, which is left behind. Microphysical processes play a major role in the development of mesoscale organization. In a traditionalapproach with no parameterized source of hydrometeors on the grid scale, no precipitation particles areavailable to be advected horizontally and verticallyduring subsequent time steps. It is possible empiricallyto simulate the evolution of vertical motion and stability with a traditional approach, so that the cumulusparameterization produces an appropriate updraftdowndraft couplet on the grid scale in the correct timeperiod (Molinari and Corsetti 1985). Once this occursin a model with traditional cumulus parameterization,however, the explicit formulation must then spin upcloud water and ice, as well as precipitation, from zero.This produces a spurious gap in the model precipitationwhile such processes occur. Alternatively, stratiformrain may never develop to the intensity observed(Molinari and Corsetti 1985). It thus appears that mosttraditional approaches, because they do not supply thegrid scale with hydrometeors, cannot reproduce a realistic transition between convective and stratiformrain. An apparent exception to the views in this paragraph, the work by Zhang and Gao (1989), will bediscussed in section 4b. A more fundamental problem with parameterizationof mesoscale processes arises as well. The life cycle ofconvective systems described above may take hours todevelop (Houze 1982). The time-scale separation requirement is thus more severe in mesoscale modelswhen mesoscale convective organization is included inthe parameterization because the life cycle of thesubgrid-scale eddies is clearly not less than that of gridscale motions. Molinari and Corsetti ( 1985 ) mitigatedthis problem by delaying the onset of parameterizedmesoscale downdrafts for 2 h, then linearly increasingtheir influence to its full value in the parameterizationover an additional 2 h. The cost of this procedure isthat a single parameter, whose true value is unknown,must be used to represent all the processes that enterinto the delay in nature. The fully explicit approach provides an alternativeto the difficulties above. Such complex processes asgeneration, advection, and interaction ofhydrometeorscan be addressed using direct prediction equations,without the need for arbitrary parameters relating thegrid scale and the cloud. The following section exploresthis alternative to cumulus parameterization.3. The fully explicit alternativea. Benefits Yamasaki (1977) and Rosenthal (1978)were thefirst to successfully adopt into mesoscale models thefully explicit alternative, that is, use of the explicit approach in both convectively unstable and stable regions.The approach has several immediate benefits over thetraditional approach. No scale-separation or fractionalarea assumptions are required, nor are closure assumptions, because no implicit clouds exist. As notedby Rosenthal (1978), the fully explicit approach allowsa broad spectrum of interactions between the convective scale (to the extent it is resolved) and larger scales.In Rosenthal's hurricane simulations, squall linespropagated across the incipient storm with little or nointeraction, but eventually the convection coupled withthe grid-scale vortex and the hurricane developed. Inprinciple, the fully explicit approach may simulate closure assumptions associated with any form of cumulusparameterization, without restricting the interaction toany one. For instance, the Kuo (1974) closure relatingconvective intensity to moisture convergence, whichprobably holds for strongly forced, high relative humidity cases, could be simulated, while simultaneouslythe area-averaged fields might follow the ArakawaSchubert (1974) quasi-equilibrium assumption. Similarly, regions or scales where moisture convergence orquasi-equilibrium assumptions were not met could alsobe simulated. In addition,, convective momentumfluxes, which are not well enough understood to parameterize in most situations, are directly simulatedby the fully explicit approach in a manner consistentwith fluxes of other quantities. If computer limitations were not a factor, the fullyexplicit approach would be the logical choice. As notedearlier, a grid spacing on the order of 100 m wouldallow direct simulation of individual clouds in a nonhydrostatic model and move the parameterizationquestion down in scale to turbulence and microphysics.Although the latter is anything but well understood,the 100-m mesh would make cumulus parameterization as it is practiced today obsolete. Because computerpower does not allow such a solution, the resolutionrequirements must be relaxed. It must be asked whetherthe fully explicit approach retains its benefits at somecurrently achievable resolution sufficiently well to makeit superior in practice to cumulus parameterization.This question is addressed below for hydrostatic modelswith 10-50-km grid spacing.b. Explicit method successes 1 ) TROPICAL CYCLONES The fully explicit approach in mesoscale models hashad its greatest success in the modeling of hurricanes(Yamasaki 1977; Rosenthal 1978; Jones 1986; Rotunno and Emanuel 1987; Baik et al. 1990). Yamasaki(1977), like Willoughby et al. (1984), used cloudmodel resolution in a nonhydrostatic model. Such workis not relevant to the current discussion of models with10-50-km grid spacing. In each of the remaining fully332 MONTHLY WEATHER REVIEW VOLUME 120explicit hurricane simulations, grid spacing (and thuscloud diameter) of 10-20 km imposed obvious limitations, but the explicit approach made it possible tointerpret mature hurricane structure while avoidingarbitrary assumptions of cumulus parameterizations(Rosenthal 1979). Nevertheless, the fully explicit approach has been tested in only a narrow way in hurricane simulation. The formation process has not beensimulated in three dimensions [the initial vortex ofJones (1986) was already tropical storm strength], andno real-data simulations have been attempted. A realdata case makes greater demands on the fully explicitapproach because cloud water and rainwater are poorlyobserved at best, and their initial values in disturbedregions may greatly influence the simulation. In theresults noted above, the models were integrated to aquasi-steady state, and the final results did not dependon initial microphysical quantities. In contrast to thefully explicit approach, the traditional approach hasbeen used both for hurricane formation studies in threedimensions (Kurihara and Tuleya 1981 ) and for realdata prediction of hurricanes (e.g., Krishnamurti 1989;Heckley et al. 1987). The fully explicit approach remains inadequately tested in hurricane simulation. 2) MIDLATITUDE CYCLOGENESIS A second area in which the fully explicit approachhas had some success is in the prediction of explosivecyclogenesis in midlafitudes. Anthes et al. (1983) madereal-data simulations of an Atlantic storm on 45- and22.5-km grid spacing using the simplest form of explicitcondensation, whereby all condensed water fell immediately as rain. They had mixed success; the predicted intensity was superior to a parallel integrationwith a Kuo (1974) cumulus parameterizafion, but thetrack was far worse. Liou et at. (1987) also used a simple form of fully explicit heating on an 80-kin gridspacing for a rapidly intensifying cyclone over thesoutheastern United States. Compared to a parallel integration with a Kuo form of parameterization, theexplicit approach better forecasted storm intensity withno degradation of the track forecast. In the parameterized integration, precipitation started too soon andcovered too wide an area. Even in the parameterizedintegration, however, most of the total latent heat release occurred in nonparameterized heating, indicatingthat upright convective instability was relatively rare. Kuo and Reed (1988), in a case of explosive cyclogenesis over the midlatitude Pacific Ocean, used detailed microphysical equations following Hsie andAnthes (1984), on both 80- and 40-kin grid spacings.Although they did not capture all of the deepening,their fully explicit simulations were quite successful inpredicting track and intensity. The parameterized simulation (which was a traditional, nonhybrid approach)eventually achieved the appropriate intensity, but theintensification itself was associated with latent heatingfrom nonparameterized heating. Kuo and Reed (1988)attributed a delay in development in the parameterizedcase to high levels of maximum heating versus the fullyexplicit case. Kuo and Low-Nam (1990) tested nine cases of explosive oceanic cyclogenesis in order to avoid thetential case-study dependence of previous work. Averaged over the nine cases, the Arakawa-.Schubert( 1974) parameterized integrations and the fully exphcitintegrations did almost equally well. The major reasonfor this was that the Arakawa-Schubert scheme conriders only boundary-layer-based convection, and thusrarely turned on in these frontal cyclone cases. Instead,extensive nonparameterized precipitation occurredahead of the warm front in both the Arakawa-Schubertand the fully explicit experiments, and storm track andintensity were similar. By contrast, a traditional approach based on Kuo (1974) produced significantamounts of high-based convection in the region of thewarm front. The resulting lapse rate along the frontalsurface remained unstable, whereas slantwise neutralitywould be expected in midlatitude oceanic cyclones(Emanuel 1988). Overa.~, the fully explicit integrahonswere the equal or better of their parameterized counterparts. It must be noted, however, that little verifyingdata exist in explosively growing midlatitude oceanicstorms, and traditional approaches have also successfully reproduced what little was known from observations (Liou et al. 1990; Chang et al. 1989; Monobianco 1989; Sanders 1987; Nuss and Anthes 1987;Uccellini et al. 1987). More detailed verification isneeded before a consensus Can be reached on the optimum approach for condensation heating in midlatitude cyclogenesis. 3) OTHER PHENOMENA Bougeault and Geleyn (1989) compared a fully explicit approach on a 10-km grid with two Kuo-basedcumulus parameterizations for a Florida sea-breezecase. The fully explicit approach produced far betterrainfall forecasts than the others in this single case. The explicit approach has also been used successfullyin frontal simulations (Ross and Orlanski 1978, 1982;Hsie et al. 1984) with grid spacings ran~Sng from 20to 61.5 km. These works are difficult to incorporateinto the current framework for a variety of reasons.Ross and Orlanski effectively parameterized cumulusconvection within the explicit approach by enhancingvertical diffusion in regions of(i) convective instability(1978 paper) or (ii) large, presumably diabaticallyforced, vertical motion ( 1982 paper). This explicit approach with parameterized convective diffusion doesnot represent a fully explicit approach as described inthis paper. Hsie et al. (1984) used an initial relativehumidity of 98% at all grid points. This unrealisticallyFEBRU^RY 1992 MOLINARI AND DUDEK 333large value artificially avoided the problem of initiationof convection in the explicit approach, which will beaddressed below.c. Explicit method failures With the exception of the Bougeault and Geleyn(1989) results, the above successful applications of thefully explicit approach have the following in common:(i) strong grid-scale forcing and (ii) lack of intenseconvective instability. When either or both of theseconditions are absent, fully explicit simulations havebeen considerably worse. Dudek ( 1988 ) used a 40-kmgrid spacing in a modified version of the NCAR MM4mesoscale model (Anthes and Warner 1978) to simulate two cases of formation of mesoscale convectivecomplexes. These provided an extreme test for explicitcondensation because the observed MCCs did not formin nature until more than 12 h into the simulation. Inboth cases the onset of precipitation was unrealisticallydelayed, areal coverage of rainfall was strongly underestimated, localized overprediction of rainfall occurred,and the location of rainfall maxima was in much greatererror than the parameterized control integration. Figure I shows the evolution of temperature andmoisture in a fully explicit integration at a point thatexperienced overprediction of rainfall. The grid columnshown was initially unstable and experiencing upwardmotion due to large-scale forcing. Conditions were suchthat cumulus parameterization would have initiatedhad it been present. In the fully explicit approach,however, relative humidity simply kept rising until saturation was reached, initially near 700 mb in this case.The existence of saturation in a conditionally and convectively unstable sounding produced an unrealisticevolution: strong localized heating in the vertical generated superadiabatic lapse rates. The explicit bulkRichardson number-dependent diffusion in the modeladjusted these to nearly dry adiabatic (Fig. lb). Eventually a deep dry-adiabatic but saturated layer formedas the heating layer deepened (Fig. lc). This layer rapidly overturned, producing intense precipitation on thegrid scale and ultimately a near-neutral lapse rate (Fig.ld; note that the neutral lapse rate includes waterloading effects and, thus, differs from the traditionaldefinition). Figure 1 displays all the weaknesses of thefully explicit approach: (i) Rainfall was unrealistically delayed because precipitation could not occur until the grid saturated, andthe grid-scale updraft was not vigorous enough to produce saturation in a timely manner. (ii) Once the grid saturated in the convectively unstable layer, the instability could not be removed byeddies as in nature because updrafts and downdraftscould not occur simultaneously. Instead, only the gridscale ~ acted and advected the high low-level 0e upward,producing a tropospheric-deep layer of convective instability, similar to that shown by Molinari and Dudek(1986). This represents the reverse of what was foundin nature by Riehl and Malkus (1958) and Betts(1974), in which the area-averaged midtropospheric0e minimum remained in place during convection. (iii) This absolutely unstable column overturned onthe scale of the grid, stabilizing only by entraining airfrom adjacent grid points, while producing extremeoverprediction of rainfall in the process. A structureremarkably similar to Fig. I c was produced by P. Pauley (personal communication 1988) using a fully explicit approach in a midlatitude explosive cyclogenesiscase, as simulated by the nested-grid model at the National Meteorological Center. The behavior described above has several similaritiesto that shown by Molinari and Dudek (1986; also anMCC case), but occurred in a model with a much moresophisticated multilevel boundary layer and more detailed microphysics, including the inhibiting effects ofwater loading and evaporation of rain. The evidencesuggests that the fully explicit approach was entirelyunsuitable on a 40-km grid spacing for these cases withmodest grid-scale forcing and large instability. Zhang et al. (1988) found similar problems withfully explicit approaches in mesoscale models. Usinga 25-kin grid spacing, they noted that a fully explicitapproach produced too much rain and too intense asurface mesolow in an MCC case. They attributed theoverprediction to the unrealistically low level of maximum heating that occurred with the explicit formulation, which produced excessive low-level spinup. Inaddition, a squall line observed in nature was not simulated because the low-level transient upward motionthat initiated it failed to produce saturation on the gridscale. Doubling the resolution (to 12.5 km) producedthe same errors and also introduced a spurious secondmesolow that was apparently orographically induced.These results support the weaknesses of the fully explicitapproach described above: the unrealistically low levelof maximum heating relates to the lack of parameterized vertical eddy fluxes, and the failure to simulatethe squall line arises from the spurious delay in thestart of convection. Zhang et al. (1988) argued thatcumulus parameterization was required even for a 10km grid spacing and that optimum results also requireddetailed microphysical forecast equations for nonconvective precipitation. Sardie and Warner (1985) and Nordeng (1987)found spurious overprediction of polar lows with thefully explicit method on grid spacings of 80 and 50kin, respectively. They attributed the problem to excessive heat release at low levels, which again pointsto the need for parameterized vertical eddy fluxes. Inboth cases, a traditional approach with cumulus parameterization produced a superior simulation.334 MONTHLY WEATHER REVIEW VOLUME 120 100 20O400 s00600 ~700~oo ~- /~oo ~ / ~ /.//~"~//~';.,'.:,"t--.J~~ooo F" "',,/ / '~-,/ ,,'"' -~o -~o -~o o ~o 20 3olOO2003004005006007008009OO1000-30 ,I ~- --'/"~'".- ~-k ~,, '..' .~~i,.,~~/'-,/' .~ x.,~ ",~.,'~-.,"~ ",~ /'%/,~<,,. ,~ ,., ,'_. ,,. ~,.,,~~..:~:I/~' /~z' -/'~k,,.,, ~, .' '~~~'.,,:,!i..,:~~ / %f / ,'~>~//~,<~',~'~/..~ / ~ .~"?.~ .~,~::'"'~ -20 -10 0 10 20 30lOO200'3OO40050060070O8009001000-30J '%' \%, \~' "'"~'x \~\ / /. N. \, / ',, y' ....~, " / ~ I" ,1' - ,,( ,, , ~. / / - / / '~ -,' ,,, ,.~ ~ '~~ ~~:;~( .~/ /-~, ,/7-/:~/ ~ /~,.,.,:~ ....,,,-~ / ~ / ,."~::,t / '~'~ / ~ ,."~ ,... ~.';-'./~,~.,/~/ ~/ / ~/ //.;t~./~/,1,? ~-20 -10 0 10 20 30 I00200300 "..,, ., ', ,,. ,.~ ,,~..." .!'.,,/...,oo~lO00 -~0 -20 -10 0 lO 20 30FlG. 1. Evolution of the sounding at a point of excessive precipitation in the fully explicit integration of Dudek (1988) at (a) hour 3; (b) hour 6; (c) hour 9; and (d) hour 12. Tripoli (1986) and Tripoli et al. (1986) used a 14km grid spacing and attempted to reproduce an orographically induced mesoscale convective system thatwas simulated earlier with a high-resolution (1-km)nonhydrostatic model. The authors found that (i) convection initiated 2 h later than the control and (ii)excessive precipitation and a spuriously intense mesoscale convective system (MCS) occurred once theexplicit condensation turned on. Similar difficultieswere encountered by Kalb (1987) using a 70-km gridspacing.d. Summary of results for the fully explicit approach As noted above, the fully explicit approach is mostlikely to succeed when grid-scale forcing is large andinstability is relatively small. Large forcing ensures thatvertical motions will be sufficient to minimize the delays in precipitation onset caused by the need for gridscale saturation and spinup of cloud and rainwater.Modest convective instability suggests the vertical eddyfluxes are sufficiently small to minimize errors in thevertical distribution of explicit heating. Conversely,when grid-scale forcing is weak and/or instability islarge, the onset of saturation is spuriously delayed.Once saturation occurs, energy is released essentiallyas conditional instability of the first kind over an unrealistically large area, producing localized excessiveprecipitation. In addition to the problems described above, thefully explicit approach has not successfully simulatedFEBRUARY 1992 MOLINARI AND DUDEK 335mesoscale organization of convection in a model with10-kin or greater grid spacing. In principle, explicitforecast equations for microphysical particles shouldallow the key processes of lateral transport and subsequent phase changes of hydrometeors to be modeleddirectly. The results reported in section 3c showed,however, that the explicit method does not accuratelysimulate the onset of intense convection, except possibly under strong local forcing. This early period ofintense convection provides much of the source of laterstratiform rain in mesoscale downdrafts. The fully explicit approach has thus proven to be no panacea, evenon grid spacings as fine as I0 or 15 km, where, as notedearlier, cumulus parameterization has difficulties aswell. The obstacles that the traditional approach and thefully explicit approach face in the important problemof mesoscale organization have given rise to a thirdalternative: the hybrid approach. This alternative ispresented in the following section, along with theFritsch-Chappell scheme, which alone among traditional approaches has simulated mesoscale organization of convection.4. Alternative methodsa. Hybrid approaches The hybrid approach is a cumulus parameterizationin that implicit clouds are defined on the basis of, butdifferent from, grid-scale fields, so that closure conditions must be specified. Unlike traditional cumulusparameterizations, however, once convective sourcesof condensate are determined, a fraction of this (implicit) condensate is added to the grid-scale cloud andrainwater equations. In effect, the hybrid approachparameterizes convective-scale evaporation, condensation, and vertical eddy fluxes, but does not parameterize heat and moisture sources that arise from themore slowly evolving process of detrained precipitation-sized particles falling between (or downwind of)convective clouds. This latter process, with its associated phase changes, develops independently of thecumulus parameterization over subsequent time steps. Kreitzberg and Perkey (1976) originated the idea ofdetraining a fraction ofparameterized rainwater to thegrid scale. Their grid-scale microphysical equations(Kreitzberg and Perkey 1977) did not, however, contain source terms from the convective parameterization, so the communication between implicit and explicit processes was not as complete as in later approaches. Kreitzberg and Perkey (1976) also were thefirst to propose using a Lagrangian cloud model in amesoscale model cumulus parameterization. This step,which allows realistic inclusion of microphysics, is acritical component of the most successful of currentapproaches, including the Frank-Cohen (1987) andFritsch-Chappell (1980) methods discussed in thispaper. Hammarstrand (1987) and Sundqvist et al. (1989)defined and explicitly predicted cloud water generatedwithin a Kuo (1974) cumulus parametefization. Thesedo not represent hybrid approaches as described here,because little communication occurred between theconvective and stratiform cloud water. Hammarstrandassumed that each grid point could have only convective or stratiform cloud water, but not both. Sundqvistet al. convert implicit cloud water to explicit only whenT ~< -20-C. Neither approach provides for the conversion of implicit rainwater to the grid scale. Brown(1979) developed a more complete approach with formal prediction equations for cloud water and rainwater.Brown included no ice phase and detrained only cloudwater, not precipitation; the latter occurred in nonconvective equations only when cloud water exceededa critical value. Because only cloud water was detrained,evaporation was excessive in detraining layers, andBrown's vertical distribution of heating had too muchcooling aloft. Yamasaki (1987) developed a hybrid approach inwhich source terms from parameterized clouds wereincluded in explicit equations for both cloud water andrainwater. The source term took the form of a detrainment in the cloud water equation. In the rainwaterequation, implicit rain was converted to explicit rainat a constant rate. Yamasaki applied the scheme in atropical cyclone model. His solution produced moreextensive and realistic mesoscale rainband structurethan had been achieved with earlier traditional andfully explicit formulations. Yamasaki did not, however,address the process by which mesoscale organizationdeveloped in the model. The most complete hybrid approach was presentedby Frank and Cohen (1987) and Cohen and Frank(1987). They included convective source terms in gridscale prediction equations for cloud water and ice,rainwater, and one form of frozen precipitation. Aswith Kreitzberg and Perkey (1976), Frank and Cohen(1987) specified the fraction of convective precipitationthat fell within clouds (thus, in parameterized cumulusdowndrafts) and that fell between clouds (i.e., that hadbeen detrained from implicit clouds); the latter wastreated explicitly in subsequent time steps. Using a 25km grid spacing, Cohen and Frank (1987) were ableto simulate many aspects of mesoscale organization inGARP Atlantic Tropical Experiment (GATE) convective systems. When upper-tropospheric detrainmentof hydrometeors was suppressed, they found that cloudlifetimes were shorter, grid-scale (stratiform) precipitation did not occur, and mesoscale structure did notdevelop. Dudhia (1989) used the Frank-Cohen approachwith more sophisticated ice microphysics and a 10-kmgrid spacing to simulate the mesoscale structure of awinter monsoon cluster. In his simulations, parameterized convective updrafts smoothly evolved to grid336 MONTHLY WEATHER REVIEW VOLUME I20scale mesoscale downdrafis below and updrafts aloftas the system propagated, so that at a grid point thetransition from convective to stratiform precipitationoccurred without discontinuity. Dudhia (1989) showedthat the mesoscale updraft aloft was driven by fusionheating associated with vapor deposition onto ice andsnow in the anvil. Removing ice-phase physics virtuallyeliminated stratiform precipitation. The benefits of the hybrid approach lie in its potentialfor realistic simulation of mesoscale organization. Itallows convective updrafts and downdrafts (which areparameterized) to coexist with stratiform rain (whichis explicit), as is often observed (Churchill and Houze1984). The hybrid approach accomplishes this by directly incorporating the processes that occur in nature:detrainment of hydrometeors and subsequent phasechanges of such hydrometeors. A traditional approachdoes not include these subgrid-scale sources of particles. The hybrid approach also has one major advantageover the fully explicit formulation. As noted earlier,the explicit approach does poorly at initiating convection at the right time and place, especially under weakforcing, even for a I0- or 15-km grid spacing. Onceinitiation occurs in the explicit approach, spurious instability often follows, as was shown in section 3c. Inthe hybrid approach, the initiation of convection at agrid point is determined by the cumulus parameterization. Although this too is imperfect, it is likely tobe superior to explicit approaches, especially for gridspacings of 30-50 km (Molinari and Dudek 1986). The hybrid approach thus has clear advantages overboth the traditional and fully explicit approaches. Unfortunately, these advantages are accompanied by bothconceptual and practical limitations. It shares with purecumulus parameterization the difficulty of choosing aclosure, which cannot be verified in mesoscale modelswith current datasets. For very fine grids, in whichcharacteristic time scales approach those of the convective scale, the hybrid approach may suffer the samescale separation problems as the traditional approach.For example, Dudhia (1989) noted on a 10-km gridspacing that (i) results were sensitive to parameters ofthe convective parameterization; and (ii) if convectiveinstability passed some threshhold, grid-scale instabilitydeveloped that resembled that occurring with fully explicit approaches, except parameterized heating replaced explicit condensation. An additional limitation of the hybrid approacharises because detrained particles are carded by gridscale motions. In nature, such particles fall betweenclouds and thus experience an environment that differsfrom the grid-scale average. This difference may belarge; Soong and Ogura (1980), in nonhydrostaticcloud-model integrations, found that the average between-cloud w was close to an order of magnitude lessthan its area-averaged counterpart. In hybrid approaches in mesoscale models, the resultant waterloading, evaporation, and other processes may befluenced adversely. The importance of this limitationvaries with model resolution: for 40-50-km g2rid spa~>ing, terminal velocity is usually much greater than either grid-area averaged or between-cloud vertical velocity, so that particle evolution may not differ dramatically in the two cases. For a 10-kin grid spacing,however, the grid-scale updraft may approach instrength a convective updraft in nature, and convectiveprecipitation intended to fall between clouds may beunrealistically suspended. The resultant distortion ofparticle interactions may be significant, especially forfrozen particles with small terminal velocities. The one remaining problem of the hybrid approachis the lack of knowledge of actual precipitation detrainment profiles for deep convection. The fractionof frozen or liquid particles that detrains, and its verticaldistribution, depend upon buoyancy, conversion ofcloud matter to precipitation in the updraft, turbulence,and many other poorly understood processes. The uncertainty is mirrored by values used in previous work:Kreitzberg and Perkey (1976) specified precipitationdetrainment as decreasing upward; Frank and Cohen(1987) as increasing upward; and Dudhia (1989) asfixed with height. In addition, it is conceivable thatwith strong forcing but marginal convective instability,precipitation detrainment in nature may nearly vanishbecause a mesoscale saturated updraft develops ratherthan intense localized cumulonimbus clouds. The results of Frank and McBride (1989) give indirect evidence for this. They found during some stages of atropical system near Australia that Ql and Q2 werenearly equal and representative of saturated updrafts,with no evidence ofmesoscale unsaturated downdrafis.In current hybrid approaches, the lack of developmentof mesoscale updraft-downdraft couplets cannot besimulated because precipitation detrainment is fixed.The detrainment problem is solvable in principle,however, and recent buoyancy sorting detrainmentmodels (Raymond and Blyth 1986; Kain and Fritsch1990) may contribute to an appropriate formulation.Emanuel (1991) has applied the Raymond-Blythscheme in a traditional cumulus parameterization designed for general circulation models. In summary, the hybrid approach shares with cumulus parameterization the difficulties of defining aclosure condition on the mesoscale and has potentialproblems with vertical advection of convectively generated particles. Both problems become significant asgrid spacing reaches 10-15 km. The current rather arbitrary specification of precipitation detrainment profiles should not present a major obstacle. Offsetting the weaknesses of hybrid approaches arethe desirable separation of fast and slow processes intoparameterized and explicit forms and the simulationof mesoscale organization and evolution superior toalmost all traditional approaches. Because mesoscaleFEBRUARY 1992 MOLINARI AND DUDEK 337organization occurs so frequently and so strongly affectssome fundamental inputs to the grid scale such as thevertical heating distribution, these characteristics of thehybrid approach offer significant advantages.b. Fritsch-Chappell scheme Zhang and Gao (1989) produced a remarkably successful simulation ofmesoscale organization in a PRESTORM convective system, including the verticalstructure of flow through the line, midlevel vorticitygeneration, and surface mesohighs and lows. They didso using a traditional approach made up of the FritschChappell (1980) cumulus parameterization plus formalmicrophysical forecast equations for nonconvectiveprecipitation, with a grid spacing of 25 km. It was argued in the previous section that this method wouldnot succeed because a cumulus parameterization inwhich no hydrometeors were detrained to the grid scalecould simulate neither mesoscale organization nor thetransition from convective to stratiform precipitation. Zhang and Fritsch (1987; 1988), Zhang and Gao(1989), and Zhang (personal communication 1990)have described the processes by which mesoscale organization was simulated. The cumulus parameterization produced saturation in the column by (i) detraining moisture and (ii) driving a grid-scale circulationthat by itself produces saturation. Figures from Zhangand Fritsch (1988) and Zhang and Gao (1989) showthat saturation developed in a column in which 0eslowly decreased upward, producing an absolutely unstable column. As a result, these clouds behaved asgrid-scale cumulus clouds; Zhang and Fritsch ( 1987 ),for instance, show a maximum instantaneous grid-scalecondensation heating rate of nearly 800-C day-~. Inlater papers (Zhang et al. 1988; Zhang and Gao 1989 ),these explicit convective clouds were kept under controlby the presence of evaporation and water loading inthe explicit microphysical equations. Because the gridscale clouds developed while cumulus parameterizationwas still active, they filled the column with hydrometeors and helped produce a smooth transition fromparameterized to grid-scale condensation. Two difficulties occur with this method. The firstrelates to details of the detrainment process. TheFritsch-Chappell scheme evaporates all liquid waterabove the equilibrium level of the parameterized cloud("anvil evaporation"). If the detrainment level is saturated, however, this liquid water is evaporated at thenext lower unsaturated level, rather than being allowedto remain as liquid at its detraining level (this is anessential difference between a traditional approach anda hybrid approach). This has the effect of saturatingthe column at progressively lower levels. In additionto anvil evaporation, the Fritsch-Chappell scheme detrains vapor, but not cloud water or ice. The differencecan be significant: vapor detrained into a saturated environment immediately becomes liquid, but only afterheat is released; no such heat release occurs with particledetrainment. Similarly, particles detrained into an unsaturated environment become vapor, but only aftercooling occurs. Thus, the vertical distribution of heatingmay differ significantly when vapor and not cloud particles are detrained. As a result of the above evaporationand detrainment characteristics, the Fritsch-Chappellcumulus parameterization appeared to simulate realistic mesoscale organization of convection by a somewhat different process than occurs in nature. The remaining difficulty of the Zhang and Gao(1989) simulation relates to the ambiguity that ariseswhen implicit and explicit clouds of the same type coexist, as was noted in section 2b. The hybrid approachalso allows parameterized precipitation to coexist withexplicit grid-scale precipitation. This coexistence,however, is fundamentally different from that in Zhangand Gao (1989), because in the hybrid approach, parameterized clouds are convective (i.e., have small timeand space scales), while the explicit clouds have muchlonger time scales. The hybrid method appears to morecleanly separate implicit and explicit clouds, and in away that reflects processes in nature. In the Zhang andGao simulation, parameterized clouds and explicitclouds both behave as convective, and their interrelationships are difficult to untangle. It should be emphasized that the Fritsch-Chappellapproach has been the focus of this section preciselybecause it has been so successful. No cumulus parameterization has done as well in real-data cases. It is clearthat many characteristics of the approach are extremelywell suited to simulating severe convective events. Thekey property of the parameterization is that the intensity of parameterized convection is proportional to theamount of convective available potential energy(CAPE), a point emphasized by Fritsch and Chappell(1980) in their original work. This allows enormousamounts of moisture to be transported vertically (anddetrained) when instability is large, a characteristic notpresent in, for instance, the Kuo (1974) and ArakawaSchubert (1974) schemes. The successful simulationsof Zhang and Fritsch ( 1987 ), Zhang et al. ( 1988 ), andZhang and Gao (1989) depend upon the use of theFritsch-Chappell approach. It is proposed, however,that certain aspects of the detrainment parameterization may be producing the right result for the wrongreason. If(i) cloud water and ice were detrained to thegrid-scale cloud-water equation, (ii) cloud-to-precipitation conversion was defined in the parameterizedupdrafts, and (iii) the vertical profile of precipitationdetrainment to the grid scale was defined, the schemewould become a hybrid approach. This procedurewould retain the beneficial characteristics of theFritsch-Chappell approach in mesoscale models, whileformally including precipitation detrainment, which isimportant in nature.338 MONTHLY WEATHER REVIEW VOLUME 120 Any cumulus parameterization in which microphysical quantities are tracked, including those of Arakawa and Schubert ( 1974; see Lord 1979) and Emanuel(1991), can in principle be converted to hybrid. TheKuo (1974) approach would require a one-dimensionalcloud model along the lines proposed by Anthes(1977). No clear-cut way exists to convert to hybridthe approaches of Betts (1986) or moist convectiveadjustment (Kurihara 1973). Finally, it should be noted that as grid spacing decreases to 20 km or less, the simultaneous occurrenceof both subgrid-scale and grid-scale convective cloudscan occur with the hybrid approach as well (Chen1990). This problem may relate primarily to violationof scale separation requirements, as discussed in section2b. This issue will be addressed further in section 6.5. Slantwise convection The existence of upright convective instability hasbeen assumed throughout this paper, while instabilityon a slanted path (Bennetts and Hoskins 1979) hasbeen ignored. Slantwise (symmetric) instability carriesits own mesoscale organization, typically in the formof bands. It is not the intent in the current paper toreview slantwise convection, which could fill an additional manuscript. Rather, the issue is how the possible presence of slantwise instability impacts the choiceof a parameterization for upright convection. Emanuel (1983) has noted that in the presence ofboth types, upright instability dominates. This impliesthat slantwise convection can be separated in numericalmodels from upright convection, in that slantwise convection would be considered only for upright stablecases. As is true of upright convection, further choicesmust be based on empirical evidence from numericalprediction models because observations are limited andthe theory of symmetric instability is not developedfor primitive equation dynamics. Knight and Hobbs ( 1988 ), using a fully explicit approach, simulated banded structure in the presence ofslantwise convective instability on a 10-km grid. Reducing the grid spacing did not significantly change thesolution, suggesting that 10 km is sufficient resolution.Symmetrically unstable bands also formed at 40-kmresolution, but were wider and fewer in number thanin the high resolution integrations. The fully explicitapproach failed to simulate banded structure at 80-kmresolution. The results suggest that for mesoscale models the fully explicit approach is capable of simulatingslantwise convective instability if resolution is sufficient. Symmetric neutrality has been observed in midlatitude explosive cyclogenesis cases (Emanuel 1988),suggesting that symmetric instability had previouslyoccurred. Kuo and Low-Nam (1990) simulated sucha neutral state with a fully explicit approach in thevicinity of warm fronts using a grid spacing of 40 kin.Kuo and Low-Nam also simulated such neutralitying a traditional approach with the Arakawa-Schubertcumulus parameterization, but explicit condensationalone was responsible for the neutral regiom In thesame case, a Kuo (1974) cumulus parameterizationallowed slantwise convective instability to remain,which is probably less realistic. Nordeng (1987) simulated two polar lows using a50-km grid spacing. Nordeng included an upright parameterization, a slantwise parameterization (using aclosure following Kuo 1974), and explicit condensationfor fully stable regions. The slantwise parameterizationproduced a sounding closer to slantwise neutral andshifted maximum vertical motion toward the frontalregion compared to when only the upright parameterization was included. Nevertheless, the slantwise convection parameterization had a much smaller influenceon the simulations than did upright parameterization. The limited results reported above make it difficultto make definitive conclusions. Three options for simulating slantwise convection in mesoscale models arise:(i) parameterize slantwise convection separately fromboth upright convection and explicit grid-scale condensation; (it) use a fully explicit approach with noparameterization of either upright or slamwise convection; and (iii) use a traditional or hybrid approachfor parameterization of upright convection, leavingslantwise convection to occur explicitly on the grid. For the first option, a closure condition must be provided to relate the grid scale and subgrid scale. Becausedata on the mesoscale are lacking, and the nonlinearevolution of localized regions of symmetric instabilityremains uncertain (Thorpe and Rotunno 1989), theappropriate closure condition is not known. Definitionof slantwise neutrality, which would be required in aparameterization, must assume some sort of dynamicbalance. No such balance can be defined in a primitiveequation model and the resulting "neutral" soundingwill in fact be stable in the presence of cyclonic vorticity(Nordeng 1987). Option two, the fully explicit approach, bypassesboth of these problems if resolution is sufficient. It allows localized unstable regions to evolve on the gridwith no constraints from poorly understood closureconditions. By explicitly resolving the shmtwise updrafts, a "primitive equation neutrality" can be generated that includes all physical and dynamical processes. The fully explicit approach cannot be 'used,however, if upright instability exists in other parts ofthe forecast region, as noted in section 3c. As a result,the fully explicit approach is appropriate in mesoscalemodels only when slantwise, but not upright, instabilityis present on the model grid, a rather narrow conditionunlikely to be met in most real-data applications. Itthus appears that traditional or hybrid approaches thatparameterize upright convection but allow slantwiseFEBRU^RY 1992 MOLINARI AND DUDEK 339convection to evolve on the grid scale (option three)will prove to be most generally applicable. Care mustbe taken, however, to ensure that the upright cumulusparameterization does not adversely influence development of explicit slantwise convection, as noted byKuo and Low-Nam (1990). On the basis of these brief arguments, it will be assumed in this paper that slantwise convection in mesoscale models can be incorporated as part of the upright stable formulation in a traditional or hybrid approach and requires no special parameterization.Further discussion of parameterization of slantwiseconvection is provided by -manuel (1983), Seltzer etal. (1985), Thorpe (1986), and Nordeng (1987).6. Discussiona. Choice of a cumulus parameterization The simulation of mesoscale organization of convection presents a major challenge for mesoscale numerical weather prediction models. A significant fraction of the water in the mesoscale circulations originatesin subgrid-scale convective updrafts, but the mesoscaleupdrafts and downdrafts themselves are not subgridscale in either space or time and must be resolved explicitly. The hybrid approach parameterizes only theconvective scale while allowing mesoscale structuresdriven by detrained hydrometeors to develop slowlyand separately through the grid-scale equations. As aresult, although implicit and explicit clouds are simultaneously present, they fundamentally differ incharacter and no ambiguity is present. Arakawa andChen(1987) argued that at least some part ofmesoscaleorganization has to be parameterized due to the lackof a spectral gap between cloud and mesoscale. It isproposed that the hybrid approach, by separating outthe forcing mechanism for the mesoscale component,removes the need to parameterize the mesoscale anyfurther. This property may allow the hybrid approachto be used for smaller grid spacings than the traditionalapproach without encountering severe scale-separationproblems. These conceptual benefits of the hybrid approach have not been clearly stated by its originators. The essence of the hybrid scheme can be seen asfollows. Assume it is possible to define a "perfect" mesoscale cumulus parameterization that, if it persists longenough, produces grid-scale saturation and moist neutrality simultaneously, as well as appropriate lowertropospheric subsidence and upper-tropospheric updrafts. The model could then in principle smoothlymake the transition to explicit grid-scale microphysics.This scheme would fail, however, if the cumulus parameterization were not supplying the grid with hydrometeors, because the grid-scale equations wouldthen have to spin up cloud and precipitation particles,and a spurious gap in precipitation would occur. Thehybrid approach avoids this problem by separating themechanisms that produce mesoscale organization. Thebest of the current cumulus parameterizations, that ofFritsch and Chappell (1980), simulates mesoscale organization somewhat differently by producing gridscale saturation and subsequently generating particlesin explicit updrafts while the cumulus parameterizationis still active. It is argued that their approach would beimproved if cloud and especially precipitation detrainment to the grid scale were added. Either way, gridscale forecast equations for microphysical quantities,including frozen and liquid precipitation, are requiredon the grid scale, a point emphasized by Zhang et al.(1988) and Zhang and Gao (1989). Figure 2 presents recommended solutions as a function of grid spacing. The optimum solution dependsmore upon local rotational constraints than upon gridspacing, as noted by Ooyama (1982). In most applications of mesoscale models, however, several dynamical regimes are likely to be present, so Fig. 2 representsthe most general solution for each grid spacing. For large-scale models (grid spacing > 50-60 kin),the traditional approach provides the only solution.The hybrid approach cannot drive realistic mesoscalecirculations in such models because the mesoscalecannot be resolved. For models with grid spacing lessthan 2-3 km, the fully explicit approach is clearly superior to parameterized approaches, even though a 1km grid spacing can fully simulate only the largest ofconvective clouds (Lilly 1990). It is between these two extremes of resolution thatthe choices become more complex. For the hybrid approach to be effective, grid spacing must be smallenough to resolve mesoscale organization, but not sosmall that scale separation problems arise. It is suggested that the hybrid approach is the preferred choicefor grid spacings from 20 or 25 to 50 km. For grid spacings from about 3 to 20-25 km, it remains uncertain whether a general solution exists.When grid-scale forcing is large and convective instability is small or moderate, the fully explicit approachGRID SPACING (KM)0.1 1 10 20 30 40 50 60 70FULLY .... ? .... HYBRID .... TRADITIONALEXPLICIT FIG. 2. Proposed form for cumulus parameterization in regionalmesoscale models as a function of grid spacing. The scale is logarithmicbelow 10 km and linear above. The question mark indicates the lackof an obvious solution, and the dots represent transition regions between the choices. It is assumed the model covers a wide enougharea that the approach must simulate convective effects over a rangeof thermodynamic and inertial stability regimes.340 MONTHLY WEATHER REVIEW VOLUME 120may suffice (see, for instance, Rosenthal 1978). In suchcases the grid-scale forcing quickly produces saturation,and the vertical distribution of heating by the explicitapproach differs only slightly from that in nature. Theexplicit approach otherwise fails at these intermediategrid spacings because resolution is insufficient to realistically model cloud initiation and subgrid-scaletransports. Cumulus parameterization also has problems for gridspacings between 3 and 25 km, even with the hybridapproach. The possibility that convective clouds willgrow directly on the grid scale increases with increasingresolution. As this occurs, it becomes difficult to identify separate physical processes with the grid-scale andsubgrid-scale clouds. Under such circumstances, successful simulations are likely only if parameterizedclouds quickly become secondary to grid-scale convective clouds. The process by which parameterizedconvection produces the transition to saturation mustbe understood and compared to that occurring in nature. Cumulus parameterization remains insufficientlytested on such fine grids.b. Recommendations for further work The cumulus parameterization problem in mesoscale models can be addressed using four types of studies: observational, semiprognostic, cloud ensemblemodeling, and mesoscale modeling. Observational studies provide the primary means forimproving individual cumulus parameterizations.Arakawa and Chen(1987) described the need forstudies of individual convective elements, the largerscale influences of such elements, and the interactionsbetween convection and the larger scale. A criticalquestion for the topics discussed in this paper is todetermine under what circumstances convective updrafts evolve into deep mesoscale saturated updrafts,versus when they will evolve to the classic mesoscalestructure of Houze (1982), with most of the stratiformrain falling in lower tropospheric downdrafts. Logicwould suggest that under strong forcing and marginalinstability, saturated mesoscale updrafts would form;detrainment would be small, thus limiting mesoscaledowndrafts. For large instability and modest forcing,convective updrafts would detrain huge amounts ofwater and mesoscale downdrafts would dominate,pushing the convection to a leading edge where unstable air remains. Observational studies can establish theanswers to such questions, but space and time resolution of the observations most likely will have to begreater than in many previous data collection effortssuch as SESAME. Semiprognostic studies (Lord 1979; Krishnamurtiet al. 1980; Grell et al. 1991 ) provide a means of evaluating from real data the instantaneous heating andmoistening profiles and rainfall rates produced by cumulus parameterizations. Such tests are limited, becausethey do not measure time evolution. This particularlyholds for the hybrid approach, in which several hoursof integration are required for evaluation. New~rtheless,semiprognostic tests provide a necessary step in tlaeevaluation of a cumulus parameterization. Cumulus ensemble models, which resolve individt~alclouds but cover a wide region, provide another majortool for improving individual cumulus parameterizations. Given the lack of observations on small scales,such models provide the greatest potential fi~r testingclosures and other aspects of cumulus parameteri:mtion. Most current efforts fix the large-scale forcing anddetermine the convective response (e.g., Soong andOgura 1980; Tap et al. 1987; Kreuger 1988). Advancing computer power will eventually allow the large scaleas well as the cloud scale to be predicted, so that twoway interaction between the convection and largerscales can be simulated. At the very least, interactivecumulus ensemble models will tell whether relativelysimple assumptions, such as those regarding precipitation detrainment profiles, can be relevant for a widerange of stability and grid-scale forcing. In general, thevalidity of both closure assumptions and parametervalues can be tested using suitable averaging of cumulusensemble model output. Encouraging progress in thisdirection has been made by Xu (1991). Because cumulus parameterization contains manyempirical aspects, its ultimate value must be measuredby actual performance in numerical models. The interpretation of such behavior can be extraordinarilycomplex for several reasons. First, cumulus momentumtransports may often be as important as heat andmoisture transports (Schubert et al. 1980), but an appropriate parameterization of such transports in upright convection has proven elusive. Second, otherphysical processes interact with parameterized convection in models in a highly complex nonlinear l?ashion, making intercomparison studies tricky to interpret.As an example, Molinari and Corsetti (1985) foundthat inclusion of cumulus and mesoscale downdraftsin a cumulus parameterization dramatically reducedrain volume to a value much closer to that obse.,~ed.Using the same cumulus parameterization in a differentcase study, but with the NCAR MM4 model (Anthesand Warner 1978), Dudek (1988) found only slightreductions in rain volume when downdrafts were included. The differences appeared to relate (Dudek1988) to the more sophisticated multilevel planetaryboundary layer parameterization in the latter model.The higher PBL resolution made the excitation of convective updrafts adjacent to cold downdrafts morelikely than in the bulk aerodynamic low-resolutionboundary layer used earlier. The earlier compz~risonstudy apparently gave misleading results about cuxnulusparameterization because the boundary-layer resolution was inadequate. Interactions of radiative processes~EBRUP~RY 1992 MOLINARI AND DUDEK 341and orography with parameterized cumulus cloudsfurther complicate interpretation of model output. Caution is required in the analysis of model outputin the vicinity of strong convection, because energyoften enters the model grid at the smallest resolvablescales with both parameterized and fully explicit approaches. These limiting scales are where finite-difference truncation produces the greatest inaccuracies(Haltiner and Williams 1980). Simultaneously, explicitdiffusion strongly damps such scales, producing an energy cycle (convective source, diffusive sink) that isunlikely to resemble that observed in nature. Because of the factors listed above, it becomes essential to establish the robustness of results in cumulusparameterization. This can be accomplished by doingmultiple case studies using one or more cumulus parameterizations. Two examples are work by Kuo andLow-Nam (1990) and Anthes et al. (1989). Kuo andLow-Nam chose nine case studies of explosive cyclogenesis and compared two traditional cumulus parameterizations with a fully explicit approach (discussedin section 3b). Anthes et al. (1989) also compared several cumulus parameterization approaches for a largenumber of case studies and 50-100-km grid spacingand determined statistically significant differences invarious quantities between each set of experiments.Because verifications are not required (rather, differences from control integrations are examined), similarstudies for the high-resolution models discussed in thispaper could establish the overall sensitivity of the resultsto various physical processes, including cumulus parameterization.c. Final comments The evidence presented earlier from numerical modeling studies strongly suggests that cumulus pa rameterization must be included in mesoscale models. In such models, it is not possible to claim to have un derstood the interaction of the convective scale and mesoscale, because cumulus parameterization fixes that interaction a priori through closure conditions. In the words of Ooyama (1982), modelers must not "play the game with loaded dice." Even a perfect forecast on the mesoscale does not mean the process by which in dividual clouds produced a mesoscale disturbance has been understood. Interpretation is further complicated by the development within models of explicit convec tive clouds that coexist with parameterized clouds and only crudely simulate real clouds. Such difficulties oc cur even with the hybrid approach once grid spacing drops to 20 km or below. It is conceivable that for study of convectively driven disturbances, mesoscale modelers should either increase grid spacing to avoid grid-scale cumulus clouds, or decrease grid spacing to 1-2 km to avoid the need for cumulus parameteriza tion. Regardless, it is argued that caution must be ex ercised in interpreting model output when cumulusparameterization is used in high-resolution models.Additional detailed studies of both successes and failures of cumulus parameterizations are needed. Keyser and Uccellini (1987) have emphasized thepotential value of mesoscale numerical models, whichprovide a uniform, high-resolution, dynamically consistent dataset, for understanding natural phenomenathat otherwise cannot easily be observed. A mesoscalemodel provides a powerful tool, but the more powerfulthe tool, the more care is required to interpret its results.Such care is particularly warranted when cumulus parameterization is involved. Acknowledgments. The authors are greatly indebtedto Dr. Lloyd Shapiro of the Hurricane Research Division (HRD) of NOAA for his thorough critical reviewof the manuscript. One of us (JM) has benefited fromconversations over several years with Dr. K. V. Ooyama of HRD/NOAA. We thank Dr. R. A. Anthes,Director of UCAR, and Dr. Y. H. Kuo of NCAR, forproviding the NCAR MM4 model and related documentation. Numerical simulations were carded out atthe National Center for Atmospheric Research, whichis supported by the National Science Foundation. Thiswork was supported by National Science FoundationGrant ATM8902487. REFERENCESAlbrecht, B. A., 1983: A cumulus parameterization for climate studies of the tropical atmosphere. Part I: Model formulation and sen sitivity tests. J. Atmos. $ci., 40, 2166-2182.Anthes, R. A., 1977: A cumulus parameterization scheme utilizing a one-dimensional cloud model. Mon. Wea. Rev., 105, 270 286. , and T. T. Warner, 1978: Development of hydrodynamic models suitable for air pollution and other mesometeorological studies. Mon, Wea. Rev., 106, 1045-1078. , Y.-H. Kuo, and J. R. Gyakum, 1983: Numerical simulations of a case of explosive marine cyclogenesis. Mon, Wea. Rev., 111, 1174-1188.--, , E.-Y. Hsie, S. Low-Nam, and T. W. Bettge, 1989: Es timation of skill and uncertainty in regional numerical models. Quart. J. Roy. Meteor. Soc., 115, 763-806.Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. At mos. Sci., 31, 674-701.--., and J.-M. Chert, 1987: Closure assumptions in the cumulus parameterization problem. Short and Medium Range Numerical Weather Prediction. T. Matsuno, Ed., Meteor. Soc. Japan, Uni versal Academy Press, 107-131,Balk, J.-J., M. DeMaria, and S. Raman, 1990: Tropical cyclone sim ulations with the Betls convective adjustment scheme. Part II: Sensitivity experiments. Mon. Wea. Rev., 118, 529-541.Bennetts, D. A., and B. J. Hoskins, 1979: Conditional symmetric instability--a possible explanation for frontal rainbands. Quart. J, Roy. Meteor. Soc., 105, 945-962.Betts, A. K., 1973: A composite mesoscale cumulonimbus budget. J. Atmos. Sci., 30, 597-610. , 1974: Thermodynamic classification of tropical convective soundings. Mon. Wea. Rev., 102, 760-764. ,1986: A new convective adjustment scheme. 1: Observational and theoretical basis. Quart. J. Roy. Meteor Soc., 112, 677 692.342 MONTHLY WEATHER REVIEW VOLUME 120Bougeault, P., and J. F. Geleyn, 1989: Some problems of closure assumptions and scale dependency in the parameterization of moist deep convection for numerical weather prediction. Meteor. Atmos. Phys., 40, 123-135.Brown, J. M., 1979: Mesoscale unsaturated downdrafts driven by rainfall evaporation: a numerical study. J. Atmos. Sci., 36, 313 338.Chang, S., K. Brehme, R. Madala, and K. Sashegyi, 1989: A numerical study of the East Coast snowstorm of 10-12 February 1983. Mon. Wea. Rev., 117, 1768-1778.Chen, S., 1990: A numerical study of the genesis of extratropical convective mesovortices. Ph.D. thesis, Department of Meteo rology, Pennsylvania State University, 178 pp.Chert, S., and W. R. Cotton, 1988: The sensitivity of a simulated extratropical mesoscale convective system to longwave radiation and ice-phase microphysics. J. Atmos. Sci., 45, 3897-3910.Cheng, C. P., and R. A. Houze, Jr., 1979: The distribution of con vective and mesoscale precipitation in GATE radar echo patterns. Mon. Wea. Rev., 107, 1370-1381.Churchill, D. D., and R. A. Houze, Jr., 1984: Development and structure of winter monsoon cloud clusters on 10 December 1978. J. Atmos. Sci., 41, 933-960.Cohen, C., and W. M. Frank, 1987: Simulation of tropical convective systems. Part II: Simulations of moving cloud lines. J. Atmos. Sci., 44, 3800-3820.Cotton, W. R., and R. A. Anthes, 1989: Storm and Cloud Dynamics. Academic Press, 883 pp.Dudek, M., 1988: Numerical study of the formation of mesoscale convective complexes. Ph.D. dissertation, Department of At mospheric Science, State University of New York at Albany, 266 pp.Dudhia, J., 1989: Numerical study of convection observed during the winter monsoon experiment using a mesoscale, two-dimen sional model. J. Atmos. Sci., 46, 3077-3107.Emanuel, K. A., 1983: On assessing local conditional symmetric in stability from atmospheric soundings. Mort. Wea. Rev., 111, 2016-2033.---, 1988: Observational evidence of slantwise convective adjust ment. Mon. Wea. Rev., 116, 1805-1816. ,1991: A scheme for representing cumulus convection in large scale models. J. dtmos. Sci., 48, 2313-2335. , and F. Sanders, 1983: Mesoscale meteorology. Rev. Geoph. Space Phys., 21, 1027-1042.Frank, W. M., 1983: The cumulus parameterization problem. Mon. Wea. Rev., 111, 1859-1871. , and C. Cohen, 1987: Simulation of tropical convective systems. Part I: A cumulus parameterization. J. Atmos. Sci, 44, 3787 3799. , and J. L. McBride, 1989: The vertical distribution of heating in AMEX and GATE cloud clusters. J. Atmos. Sci., 46, 3464 3478.Fritsch, J. M., and C. F. Chappell, 1980: Numerical prediction of convectively driven mesoscale pressure systems. Part I: Con vective parameterization. J. Atmos. Sci., 37, 1722-1733.Gallus, W. A., and R. H. Johnson, 1991: Heat and moisture budgets of an intense midlatitude squall line. J. Atmos. Sci., 48, 122 146.Gamache, J. F., and R. A. Houze, Jr., 1982: Mesoscale air motions associated with a tropical squall line. Mon. Wea. Rev., ll0, 118 135.Grell, G., Y. H. Kuo, and R. Pasch, 1991: Semiprognostic tests of cumulus parameterization schemes in the middle latitudes. Mon. Wea. Rev., 119, 5-31.Haltiner, G. J., and R. T. Williams, 1980: Numerical Prediction and Dynamic Meteorology. John Wiley and Sons, 477 pp.Hammarstrand, U., 1987: Prediction of cloudiness using a scheme for consistent treatment of stratiform and convective conden sation and cloudiness in a limited area model. Short andMedium Range Numerical Weather Prediction. T. Matsuno, Ed., Meteor. Soc. Japan, Universal Academy Press, 187-198.Hartman, D. L., H. H. Hcndon, and R. A. Houze, Jr., 1984: Some implications of the mesoscale circulations in tropical cloud clus ters for large-scale dynamics and climate. J. Atmos. Sci., ,11,Heckley, W. A., M. J. Miller, and A. K. Betts, 1987: An example of hurricane tracking and forecasting with global analysis-fore casting system. Bull. Amer. Meteor. Soc., 68, 226-229.Houghton, H. G., 1968: On precipitation mechanisms and their ar tificial modification. J. Appl. Meteor.. 7, 851-859.Houze, R. A., 1982: Cloud clusters and large-scale vertical motions in the tropics. J. Meteor. Soc. Japan, 60, 396-410.---, 1989: Observed structure of mesoscale convective systems and implications for large-scale heating. Quart. J. Roy. 3leteor. Soc., 115, 425-462.Hsie, E.-A., and R. A. Anthes, 1984: Simulations of frontogenesis in a moist atmosphere using alternative parameterizations of con densation and precipitation. J. Atmos. Sci., 41, 2701-2716, , and D. Keyser, 1984: Numerical simulation of fronto genesis in a moist atmosphere. J. Atmos. Sci., 41, 2581-2594.Johnson, R. H., 1984: Partitioning tropical heat and moisture budgets into cumulus and mesoscale components: Implications for cu mulus parameterization. Mon. Wea. Rev., 112, 1590-1601.---, and D. C. Kriete, 1982: Thermodynamic and circulation char acteristics of winter monsoon tropical mesoscale convection. Mon. Wea. Rev., ll0, 1898-1911.Jones, R. W., 1986: Mature structure and motion of a model tropical cyclone with latent heating by the resolvable scales. Mort. IVea. Rev., 114, 973-990.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.Kalb, M. W, 1987: The role of convective parameterization in the simulation of a Gulf Coast precipitation system. Mort. Wea. Rev., 115, 214-234.Kanamitsu, M., 1975: On numerical prediction over a global tropical belt. Ph.D. Thesis, Department of Meteorology, Florida :State University, 281 pp.Keyser, D., and L. W. Uccellini, 1987: Regional models: Emerging research tools for synoptic meteorologists. Bull. Amer. Meteor. Soc., 68, 306-320.Knight, D. J., and P. V. Hobbs, 1988: The mesoscale and microscale structure and organization of clouds and precipitation in mid latitude cyclones. Part XV: A numerical modeling study of frontogenesis and cold-frontal rainbands. J. Atmos. Sci., 45, 915 930.Kreitzberg, C., and D. Perkey, 1976: Release of potential instability. Part I: A sequential plume model within a hydrostatic primitive equation model. J. Atmos. Sci., 33, 456-475. , and --, 1977: Release of potential instability. Part II: The mechanism of convective/mesoscale interaction. J. Atmos. Sci., 34, 1569-1595.Kreuger, S. K, 1988: Numerical simulation of tropical cumulus clouds and their interaction with the subcloud layer. J. Atmos. Sci., 45, 2221-2250.Krishnamurti, T. N., 1989: Prediction of the life cycle ofa superty phoon with a high-resolution global model. Bull. Amer. Meteor. Soc., 70, t218-1230.--., and W. J. Moxim, 1971: On parameterization of convective and nonconvective latent heat release. J. Appl. Meteor., 10, 3--., Y. Ramanathan, H. L. Pan, R. J. Pasch, and J. Molinari, 1980: Cumulus parameterization and rainfall rates I. Mon. Wea. Rev., 108, 465-472. , S. Low-Nam, and R, Pasch, 1983: Cumulus parameterization and rainfall rates II. Mon. Wea. Rev., 111, 815-828.Kuo, H. L., 1974: Further studies of the parameterization of the influence of cumulus convection on large scale flow. J. Atmos. ScL, 31, 1232-1240.Kuo, Y.-H., and R. A. Anthes, 1984: Accuracy of heat and moisture budgets using SESAME-79 field data as revealed by observingFEBRU^RY 1992 MOL1NARI AND DUDEK 343 system simulation experiments. Mort. Wea. Rev., 112, 1465 1481.--, and R. J. Reed, 1988: Numerical simulation of an explosively deepening cyclone in the eastern Pacific. Mort. Wea. Rev., 116, 2081-2105. , and S. Low-Nam, 1990: Prediction of nine explosive cyclones over the western Atlantic Ocean with a regional model. Mon. Wea. Rev., 118, 3-25.Kurihara, Y., 1973: A scheme of moist convective adjustment. Mon. Wea. Rev., 101, 547-553, , and R. E. Tuleya, 1981: A numerical simulation study on the genesis of a tropical storm. Mon. Wea. Rev., 109, 1629-1653.Leary, C. A., and E. N. Rappaport, 1987: The life cycle and internal structure of a mesoscale convective complex. Mon. Wea. Rev., 115, 1503-1527.Lilly, D. K., 1960: On the theory of disturbances in a conditionally unstable atmosphere. Mon. 14/ca. Rev., 88, 1-17. , 1990: Numerical prediction of thunderstorms--has its time come? Quart. J. Roy. Meteor. Soc., 116, 779-798.Lindzen, R. S., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31, 156-179.Liou, C.-S., R. L. Elsberry, and R. M. Hodur, 1987: Impact ofpa rameterized cumulus convection on meso-alpha scale cyclone prediction. Short and Medium Range Numerical Weather Pre diction. T. Matsuno, Ed., Meteor. Soc. Japan, Universal Acad emy Press, 595-604. , C. H. Wash, S. M. Heikkinen, and R. L. Elsberry, 1990: merical studies ofcyclogenesis events during the second intensive observation period (IOP-2) of GALE. Mort. Wea. Rev., 118, 218-233.Lord, S. J., 1979: Verifications of cumulus parameterizations using GATE data. Proc., Impact of GATE on Large-Scale Numer. Modeling of the Atmosphere and Ocean, National Academy of Sciences, Woods Hole, MA, 182-191.Maddox, R. A., 1983: Large-scale meteorological conditions asso ciated with midlatitude, mesoscale convective complexes. Mon. Wea. Rev., 111, 1475-1493.Molinari, J., and T. Corsetti, 1985: Incorporation of cloud-scale and mesoscale downdrafts into a cumulus parameterization: Results of one- and throe-dimensional integrations. Mon. Wea. Rev., 113, 485-501.--, and M. Dudek, 1986: Implicit versus explicit convective heating in numerical weather prediction models. Mort. Wea. Rev., 114, 1822-1831.Monobianco, J., 1989: Explosive East Coast cyclogenesis: Numerical experimentation of model-based diagnostics. Mon. Wea. Rev., 117, 2384-2405.Nordeng, T. E., 1987: The effect of vertical and slantwise convection on the simulation of polar lows. Tellus, 39A, 354-375.Nuss, W. A., and R. A. Anthes, 1987: A numerical investigation of low-level processes in rapid cyclogenesis. Mon. Wea. Rev., 115, 2728-2743.Ogura, Y., and C. Y. J. Kao, 1987: Numerical simulation of a tropical mesoscale convective system using the Arakawa-sehubert cu mulus parameterization. J. Atmos. Sci., 44, 2459-2476.Ooyama, K. V., 1971: A theory on parameterization of cumulus convection. J. Meteor. Soc. Japan, 49, 744-756. ,1982: Conceptual evolution of the theory and modeling of thetropical cyclone. J. Meteor. Soc. Japan, 60, 369-379.Raymond, D. J., 1986: Prescribed heating of a stratified atmosphere as a model for moist convection. J. Atmos. Sci., 43, 1101-1111.--, and A. M. Blyth, 1986: A stochastic mixing model for non precipitating cumulus clouds. J. Atmos. Sci., 43, 2708-2718.Riehl, H., and J. S. Malkus, 1958: On the heat balance of the equa torial trough zone. Geophysica, 6, 503-538.Rosenthal, S. L., 1978: Numerical simulation of tropical cyclone development with latent heat release by the resolvable scales. I: Model description and preliminary results. J. Atmos. Sci., 35, 258-271.--, 1979: The sensitivity of simulated hurricane development to cumulus parameterization details. Mon. Wea. Rev., 107, 193 197. , and , 1982: The circulation associated with a cold front. Part II: Moist case. J. Atmos. Sci., 35, 445-465.Rotunno, R., and K. A. Emanuel, 1987: An air-sea interaction theory for tropical cyclones. Part II: Evolutionary study using a non hydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542-561.Rutledge, S. A., and R. A. Houze, 1987: A diagnostic model of the trailing stratiform region ofa midlatitude squall line. J. Atmos. Sci., 44, 2640-2656.Sanders, F., 1987: Skill of NMC operational dynamic models in pre diction of explosive cyclogenesis. Wea. and Forecasting, 2, 322 336.Sardie, J. M., and T. T. Warner, 1985: A numerical study of the development mechanisms of polar lows. Tellus, 37A, 460-477.Schubert, W. H., J. J. Hack, P. L. Silva Dias, and S. R. Fulton, 1980: Geostrophic adjustment in an axisymmetric vortex. J. Atmos. Sci., 37, 1464-1484.Seltzer, M. A., R. E. Pasarelli, and K. A. Emanuel, 1985: The possible role of symmetric instability in the formation of precipitation bands. J. Atmos. Sci., 42, 2207-2219.Soong, S.-T., and Y. Ogura, 1980: Response of tradewind cumuli to large-scale processes. J. Atmos. Sci., 37, 2035-2050.Sundqvist, H., 1978: A parameterization scheme for non-convective condensation including prediction of cloud and water content. Quart. J. Roy. Meteor. Soc., 104, 677-690. , E. Berge, and J. E. Kristjansson, 1989: Condensation and cloud parameterization with a mesoscale numerical weather prediction model. Mon. Wea. Rev., 117, 1641-1657.Tao, W.-K., J. Simpson, and S.-T. Soong, 1987: Statistical properties of a cloud ensemble: A numerical study. J. Atmos. Sci., 44, 3175-3187.Thorpe, A. J., 1986: Parameterization of mesoscale processes. Proc.,Phys. Parameterization for Numerical Models of the Atmosphere,ECMWF, Shinfield Park, Reading, U.K., 205-233.--., and R. Rotunno, 1989: Nonlinear aspects of symmetric insta bility. J. Atmos. Sci., 46, 1285-1299.Tiedtke, M., 1988: Parameterization of cumulus convection in large scale models. Physically Based Modelling and Simulation of Climate and Climate Change. M. Schlesinger, Ed., Reidel, 375 431. ,1989: A comprehensive mass flux scheme for cumulus param eterization in large-scale models. Mon. Wea. Rev., 117, 1779 1800.Tripoli, G, J., 1986: A numerical investigation of an orogenic me soscale convective system. Atmos. Pap. No. 401,209 pp. [Avail able from Department of Atmospheric Sciences, Atmospheric Sciences Building, Colorado State Univ., Fort Collins, CO, 80523.] , C. Tremback, and W. R. Cotton, 1986: A comparison of a numerically simulated mesoscale system with explicit and pa rameterized convection. 23rd Conf Cloud Physics and Radar Meteorology, Amer. Meteor. Soc. Snowmass, CO, J131-J134.Uccellini, L. W., R. A. Petersen, K. F. Brill, P. J. Kocin, and J. J. Tucillo, 1987: Synergistic interactions between an upper level jet streak and diabatic processes that influence the development of a low-level jet and secondary coastal cyclone. Mort. Wea. Rev.. 115, 2227-2261.Willoughby, H. E., H.-L. Jin, S. J. Lord, and J. M. Piotrowicz, 1984: Hurricane structure and evolution as simulated by an axisym metric, nonhydrostatic numerical model. J. Atmos. Sci., 41, 1169-1186.Xu, K. M., 1991: The coupling of cumulus convection with large scale processes. Ph.D. dissertation, Department of Atmospheric Sciences, University of California at Los Angeles, 250 pp.Yamasaki, M., 1977: A preliminary experiment of the tropical cyclonewithout parameterizing the effects of cumulus convection. J.Meteor. $oc. Japan, Series II, 55, 11-31.344 MONTHLY WEATHER REVIEW VOLOM- 120 , 1987: Parameterization of cumulus convection in a tropical cyclone model. Short and Medium Range Numerical Weather Prediction. T. Matsuno, Ed., Meteor. Soc. Japan, Universal Academy Press, 665-678.Yanai, M., S. Esbensen, and J. Chu, 1973: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets, d. Atmos. Sci., 30, 611-627.Zhang, D.-L., and J. M. Fritsch, 1987: Numerical simulation of the meso-g-scale structure and evolution of the 1977 3ohnstown flood. Part II: Inertially stable warm-core vortex and the me soscale convective complex../. Atmos. Sci, 44, 2593-2612.--, and ,1988: Numerical sensitivity experiments of varying model physics on the structure, evolution, and dynamics of two mesoscale convective systems. J. Atmos. Sci., 45, 261-293.--, and K. Gao, 1989: Numerical simulation of an intense squall line during 10-11 June 1985 Pre-Storm. Part II: Rear inflow, surface pressure perturbations, and strafiform precipilafion. Mon. Wea. Rev., 117, 2067-2094.--, E.-Y. Hsie, and M. W. Moncrieff, 1988: A compm~son of ex plicit and implicit predictions of convective and stratiform pre cipitating weather systems with a mesoq3-scale numerical model. Quart. J. Roy. Meteor. Soc., 114, 31-60.Zipser, E. J., 1969: The role of organized unsaturated convective downdrafts in the structure and rapid decay of an equatorial disturbance. J. Appl. Meteor., 8, 799-814.--, 1977: Mesoscale and convective scale downdmfV.; as distinct components of squall-line structure. Mort. Wea. Rev., 105, 1568 1589.