## 1. Introduction

Observations and numerical simulations suggest the following conceptual picture of entrainment and mixing in cumulus clouds: Eddies not much smaller than the cloud itself engulf discrete blobs of environmental air. Turbulent deformation stretches and compresses the entrained blobs into successively thinner sheets until the Kolmogorov scale is reached and molecular diffusion becomes significant and changes the local properties of the air. This view is consistent with the mixing process envisaged by Broadwell and Breidenthal (1982) and Baker et al. (1984). Inertial range scaling indicates that the time for complete homogenization of an entrained blob initially of size *s* is of order (*s*^{2}/*ϵ*)^{1/3} where *ϵ* is the dissipation rate (Baker et al. 1984). Typical values of *s* and *ϵ* in a cumulus cloud are 100 m and 10^{−2} m^{2} s^{−3}. For these values, the mixing timescale is 100 s, which is not insignificant compared to the lifetime of a cumulus cloud thermal.

Three-dimensional numerical simulations of small cumulus clouds suggest that entrainment is associated with large structures not much smaller than the size of the cloud (Grabowski and Clark 1993). Paluch and Baumgardner (1989) conclude from their finescale measurements in nonprecipitating continental cumulus turrets that mixing involves both bulk entrainment, which produces coarse mixtures of cloud and clear air on the scale of meters or tens of meters, and finescale (molecular) mixing, which changes the local microphysical properties. More recent observations (e.g., Brenguier 1993) indicate that cloud structures with very sharp interfaces exist on the smallest observable (centimeter) scales. An analysis of droplet counts also suggests that variability exists on all scales down to about 1 cm (Baker 1992). These observations are consistent with the mixing process described above.

The range of scales that must be included in a model to fully represent entrainment and mixing is very large for atmospheric flows. The largest turbulent eddy in a cumulus cloud may be a kilometer across, while the smallest are roughly the size of the Kolmogorov scale, which is about a millimeter. Explictly simulating the entire range of motions responsible for entrainment and mixing in a cumulus cloud with a three-dimensional numerical model is very demanding computationally and is clearly not feasible now or in the foreseeable future.

For some purposes, such as for studying the dynamics of supercell thunderstorms, it is not necessary to explicitly simulate the full range of scales because the dynamics are little affected by processes on scales smaller than a few hundred meters. However, in small cumulus clouds, the finescale internal structure may significantly affect the cloud dynamics. The buoyancy of a volume of fluid containing entrained air and cloud air depends on the degree of molecular mixing or homogenization that has occurred (e.g., Grabowski 1993; Krueger 1993). In such a volume, most cloud droplets do not evaporate until complete mixing has taken place. As a result, there is a significant time interval between entrainment and the subsequent evaporative cooling and decrease in buoyancy. Both Grabowski (1993) and Krueger (1993) concluded that by the time mixing and evaporative cooling have occurred in a cloud, the entrained air is likely to be far away from the entrainment zone (at the cloud top or edge), and direct feedback to the entrainment process is unlikely. A model that does not represent the finescale internal structure cannot properly simulate the mixing process. Such models will generally mix too rapidly and may lead to erroneous conclusions regarding under what conditions evaporative cooling associated with mixing of entrained air may directly feed back to the entrainment process and lead to cloud-top entrainment instability.

It is now widely accepted that entrainment and subsequent mixing influences the growth of cloud droplets in cumulus clouds (Blyth 1993). The resulting finescale structure allows different droplets in the same region of the cloud to experience different microphysical environments. Blyth notes however that there has been a considerable debate about the mixing process itself, and he concludes that “it is unclear at this time what the real effects of entrainment on cloud droplets are.”

In summary, the dynamical effects of entrainment in large precipitating and small nonprecipitating cumuli can be adequately simulated by low and high resolution multidimensional models, respectively. However, relating the evolution of the droplet size spectrum to the entrainment and mixing process remains a challenge due to the large range of scales involved.

In this paper we describe an extension of the model used by Krueger (1993) to study entrainment and mixing in the stratus-topped boundary layer that can be used to study entrainment and mixing in cumulus clouds. The new model, called the “explicit mixing parcel model” (EMPM), depicts the finescale internal structure of a rising thermal in a cumulus cloud using a 1D domain. The internal structure evolves in the EMPM as a consequence of a sequence of discrete entrainment events and an explicit representation of turbulent mixing based on Kerstein’s (1988) linear eddy model. In this version of the EMPM, subgrid-scale (eddy) diffusion is found to be adequate for representing the effects of the smallest turbulent eddies. In addition, a simple parameterization is used to determine the local condensation or evaporation rates. If a droplet growth model is incorporated and the grid size is reduced so that the Kolmogorov scale is resolved, the EMPM can predict the local microphysical environments of individual cloud droplets (Su et al. 1996).

To place the EMPM in context, the next section reviews modeling approaches used to study entrainment and mixing in cumulus clouds. The primary purpose of this paper is to describe the EMPM (sections 3 and 4) and to test its entrainment parameterization by comparing the model’s predictions of the in-cloud variability of Hawaiian cumulus cloud main turrets to the aircraft measurements made by Raga et al. (1990) (section 5). Additional predictions of the model regarding the dependence of the internal cloud structure on the entrained blob size are also provided in section 5. A summary and conclusions constitute section 6.

## 2. Modeling entrainment and mixing in cumulus clouds

Three types of models have been used to study the effects of entrainment and mixing on cumulus dynamics and microphysics. These are high resolution multidimensional models (“large-eddy” or “cloud” models), entraining parcel models, and one-dimensional finescale mixing models (“small-eddy” models). In cloud models, the large-eddy structure is simulated, which in turn determines the entrainment and mixing processes. In parcel models, there is no internal structure, the entrainment rate is specified, and the entrained air is mixed instantaneously. In small-eddy models, the finescale structure is simulated, the entrainment process is parameterized, and the entrained air mixes at a finite rate.

Table 1 lists the features of several models of each type that have been used to study the effects of entrainment and mixing on cumulus dynamics and microphysics.

In a model, we consider entrainment and turbulent deformation (i.e., turbulent mixing) to be “explicit” when it has a visible impact on the model’s representation of the internal structure of the cumulus cloud (or cloud parcel). This is clearly the case when entrainment or turbulent deformation is carried out by the 2D or 3D flow field predicted by the model. Entrainment and turbulent deformation may also be “explicit” in a model that does not predict the flow field. In such a model, entrainment typically involves discrete entrainment events, while turbulent deformation is determined by a specified strain rate (or eddy size) distribution. In a model that does not represent the internal structure, entrainment and turbulent deformation are both “implicit.” In this case, entrainment is usually parameterized as a continuous process, while turbulent deformation is “implicit” in the model’s formulation.

Molecular mixing is “explicit” only in those models that resolve the Kolmogorov scale (actually, the Batchelor scale). In all other models that resolve spatial structure, molecular diffusion is represented “implicitly.” In these models, the effects of subgrid-scale turbulent deformation are parameterized by replacing the molecular diffusivities with much larger turbulent or eddy diffusivities.

The three cloud models listed in Table 1 (K91, BG93, and GC93) explicitly represent the cloud-scale motions, including the (large) entraining eddies. Such multidimensional models generally cannot afford to include detailed microphysical parameterizations, although some have. The grid size in K91 is 250 m, which is the smallest grid size used to date in a 3D cloud model that explicitly predicts the evolution of the droplet size spectrum. In a 2D cloud model with 25-m resolution (BG93), the droplet size spectrum evolution is predicted in a parameterized way. A 3D cloud model with a 3-m grid size (GC93) has been used to study the detailed structure and circulation of a rising thermal. This “large-eddy simulation” (LES) approach appears to be adequate for studying the dynamical effects of entrainment on cumulus clouds.

In a LES of a cumulus cloud, turbulent deformation of entrained air is resolved only partially. Turbulent deformation by subgrid-scale eddies and homogenization by molecular diffusion must still be parameterized. With a grid size of 3 m (the smallest grid size used to date), the smallest resolvable scale is 6 m. A typical mixing timescale for a blob of size *s* = 6 m is 15 s. Recall that for a blob size of 100 m, the corresponding mixing timescale is 100 s. Thus, a LES with a 3-m grid size will explicitly simulate 85% of the turbulent deformation that occurs after entrainment of a 100-m size blob. In comparison, in a model with a grid size of 250 m, entrainment of a 100-m blob is a subgrid-scale process; none of the turbulent mixing process is explicitly simulated.

Extensive use has been made of entraining parcel models to study the effects of entrainment and mixing in cumulus clouds (e.g., Pruppacher and Klett 1997). In these models, a parcel represents a thermal in a growing cumulus tower that can entrain environmental air as it ascends. The conventional assumption made in this approach is that the parcel is homogeneous at all times, which implies that entrained air is instantly mixed into the parcel. This assumption has made detailed microphysical calculations, including droplet growth calculations, feasible. However, by assuming a homogeneous (instantaneous) mixing process, no information on the internal structure of clouds can be obtained.

Hill and Choularton (1986) developed a parcel model (HC86) in which entrainment occurs discontinuously and nonuniformly across the cloud. The initial diameter of the cloud was 2000 m, while the diameter of the entrained blob was taken as 200 m. The parcel height interval between entrainment events ranged from 300 m to 1300 m. The mixing rate between the blob and the cloud was set either at a constant value or at a value dependent on the relative velocities of the cloud and the blob. The material mixed into the blob was instantly mixed through its volume.

Jensen and Baker (1989) and Grabowski (1993) recognized that turbulent deformation and molecular diffusion could both be explicitly represented in a 1D small-eddy model. Such a model could also predict droplet growth according to the local microphysical environment. In the models developed by Jensen and Baker (1989) and Grabowski (1993), turbulent deformation is represented by a specified strain rate that continuously increases the scalar gradients. The strain rate represents the effect of an eddy of a specific size.

In Jensen and Baker’s model (JB89), turbulent deformation is modeled as a compressive strain that continuously shrinks the domain. The strain represents the effect of an eddy of the domain size. Grabowski’s model (G93) includes a constant, uniform strain rate. In neither model are the effects of smaller and larger eddies included. Jensen and Baker applied their model to study the evolution of droplet spectra during isobaric mixing, given an initial distribution of cloudy and clear air. Grabowski used his model to describe the early stages of the isobaric mixing process between a thin cloudy sheet with initial width of 10 cm and cloud-free air.

Kerstein (1988) developed a mixing model, the linear eddy model, that is similar in basic concept to Jensen and Baker’s but incorporates the effects of eddies of all sizes between the Kolmogorov scale and the integral scale in a manner consistent with inertial-range scaling laws. Applications of this model show that it is capable of accurately describing many features of turbulent mixing (e.g., Kerstein 1991). Krueger (1993) used the linear eddy model (K93) with subgrid-scale (eddy) diffusion to simulate the mixing of air entrained into the downdraft of a large convective eddy in a stratus-topped boundary layer.

In this paper we describe an extension of Krueger’s (1993) model that can be used to study entrainment and mixing in cumulus clouds. The new model, called the explicit mixing parcel model (EMPM), represents the finescale internal structure of a rising thermal in a cumulus cloud using a 1D domain. The internal structure evolves in the EMPM as a consequence of a sequence of discrete entrainment events and an explicit representation of turbulent mixing based on Kerstein’s linear eddy model. (In this version of the EMPM, as in K93, eddy diffusion is found to be adequate for representing the effects of the smallest turbulent eddies.) The EMPM conceptually links the conventional parcel model, with no internal structure, and multidimensional cloud models, with cloud-scale internal structure. Because it is a 1D model, the EMPM can resolve a much larger range of scales than can multidimensional cloud models, although entrainment must be parameterized and the turbulence properties specified. Unlike conventional parcel models, the EMPM predicts the internal variability of the parcel that results from multiple entrainment events. Unlike other small-eddy models, the EMPM uses a well-founded finite-rate mixing model (the linear eddy model) that can include the effects of the full range of turbulent eddy sizes.

## 3. The explicit mixing parcel model (EMPM)

The explicit mixing parcel model (EMPM) is similar to the model used by Krueger (1993) to study the mixing of air entrained into a cloud-topped boundary layer during a single entrainment event. The EMPM differs primarily by allowing multiple entrainment events.

The EMPM includes two principal processes: entrainment of environmental air into the parcel and finite-rate mixing within the parcel. An additional feature of the model is that these processes are represented in a 1D spatial domain. By reducing the description of the internal structure to one dimension, high spatial resolution can be achieved. This high resolution is needed to accurately represent the finescale structure of the scalar fields and the in-cloud processes—such as condensation, evaporation, and chemical reactions—that are determined by the local scalar fields.

The EMPM predicts the evolution of one or more scalar fields within the parcel. The fields predicted within the EMPM’s 1D domain represent those of the parcel and are assumed to be *statistically* homogeneous, although a high degree of instantaneous spatial variation typically exists as a result of entrainment and mixing. The assumption of statistical homogeniety within a cloud parcel is supported by observations, which show no systematic trend of the liquid water content from the edge toward the middle of the cloud (e.g., Austin et al. 1985). Statistical homogeniety also makes it appropriate to use the periodic boundary condition for the fields.

In principle, any number of scalar fields *ϕ*_{i} (*x*) (where *x* is the single spatial coordinate and *i* indicates the field) can be computed by the EMPM. In the present work, two scalar quantities are predicted by the EMPM: the total water mixing ratio, *q*_{w} = *q* + *l,* where *q* is the water vapor mixing ratio and *l* is the liquid water mixing ratio, and the liquid water static energy, *s*_{l} = *s* − *Ll,* where *s* = *c*_{p}*T* + *gz* is the dry static energy, *L* is the latent heat per unit mass of water vapor, *c*_{p} is the specific heat of moist air at constant pressure, *T* is the temperature, *g* is the acceleration of gravity, and *z* is the height. These two scalars are approximately conserved during moist adiabiatic processes. Knowing *q*_{w} and *s*_{l}, we can determine *T, q,* and *l* if we assume that the liquid water and vapor are in equilibrium (see the appendix). This assumption implies that everywhere in the parcel there is either exact saturation, in which case liquid water may be present, or subsaturation, with no liquid water, and is only approximately true when droplets are evaporating.

The evolution of the predicted scalars as the parcel rises is governed by the entrainment and mixing processes within the parcel. The parcel’s rise rate, entrainment rate, and turbulence characteristics are all specified inputs to the model, as are the parcel’s initial scalar fields and the environmental scalar profiles. The manner in which the entrainment and mixing processes are carried out by the EMPM are described next.

### a. Entrainment in the EMPM

The primary source of internal variability in individual cumulus cloud towers is entrained blobs or wisps of environmental air that take a significant length of time to mix. In the conventional parcel model, the assumption of internal uniformity implies that entrained air mixes instantaneously. This is equivalent to assuming that the entrained blob size is infinitesimal. To allow finite-rate mixing, entrainment in the EMPM is represented by a series of discrete *entrainment events.* Each event involves inserting a segment of length *d* of environmental air from the current parcel level into a randomly selected location of the computational domain. We refer to this new segment of air as the *entrained blob.*

Typically, the parcel size will increase rapidly with height due to entrainment. For computational convenience, we maintain a constant domain size by removing a randomly selected segment of the computational domain of size *d* immediately before inserting each entrained blob. For simplicity, we simply *replace* the selected segment with one of environmental air. Because the parcel is assumed to be statistically homogeneous, the removal of randomly selected segments does not affect the parcel’s statistical properties.

*z*)

_{avg}, is determined by the fractional rate of entrainment per unit height,

*λ,*the entrained blob size

*d,*and the domain size

*D.*The fractional rate of entrainment is the fractional rate of change of the parcel mass,

*m,*with height:

*z*between two successive entrainment events is randomly selected from a Poisson distribution with mean (Δ

*z*)

_{avg}.

In cumulus clouds, entrainment of environmental air occurs as a result of cloud-scale vortical motions. The size of the entrained blobs is expected to have a distribution around some characteristic size determined by the cloud and turbulence dynamics. In a 3D LES of a cumulus cloud, the rate of entrainment and the sizes of the entrained blobs are determined by the resolved flow field. In the EMPM, both the entrainment rate and the characteristic entrained blob size are specified. For the case studied, the fractional rate of entrainment per unit height, *λ,* was determined from the observed in-cloud profiles of a conserved scalar. However, the characteristic entrained blob size, *d,* was not measured. For this reason, we carried out a parametric study by varying the blob size over a reasonable range. (See section 4b for further discussion of these two parameters.)

### b. Finite-rate mixing: The linear eddy model

As mentioned already, the internal variability of cumulus clouds is a result of entrainment and mixing. With only entrainment acting, variability (as measured by scalar variance) would continually increase. However, the turbulent eddies that entrain air also mix it. As described in the introduction, turbulent mixing of air entrained into a cumulus cloud occurs in two stages. First, advection by the turbulent eddies deforms (stretches) the entrained air. This increases the area of contact (the interfacial area) between the entrained air and the cloud air and increases the scalar gradients in the interfacial zone. During the second stage, the interfacial area becomes very large as the thicknesses of the sheets and filaments of entrained air approach the Kolmogorov scale (typically a few millimeters in a cumulus cloud) and molecular diffusion rapidly smooths out the large scalar gradients.

Kerstein (1988) recognized that turbulent deformation and molecular diffusion could both be explicitly represented in a 1D model and developed the linear eddy model, which provides an approximate, yet physically based description of turbulent mixing. Namely, molecular diffusion is accounted for explicitly by numerically integrating the diffusion equation, while turbulent deformation is represented by random rearrangement events.

Since 1988, the linear eddy model has been used to simulate mixing in a variety of flows including homogeneous, statistically steady flows (Kerstein 1991; McMurtry et al. 1993) and spatially developing flows (Kerstein 1988, 1989, 1990, 1992). These applications served both to test aspects of the model and to provide mechanistic interpretations of measured properties in a unifying conceptual framework. The applications to spatially developing flows collectively demonstrate that the diverse phenomenology observed in such flows may be viewed as various manifestations of a simple underlying kinematic picture. In such applications, configuration-specific aspects are reflected in the initial and boundary conditions of the computations and in the model analogs of quantities such as Re (Reynolds number), Sc (Schmidt number), and Da (Damköhler number), but the underlying kinematic picture is the same in all cases. The unification of these diverse phenomena achieved by the linear eddy model is remarkable.

*ϕ*

_{i}is implemented in the linear eddy model by numerically integrating the diffusion equation,

*D*

_{M}is the molecular diffusivity.

It is the implementation of turbulent deformation that is the key feature of the linear eddy model, and what distinguishes it from JB89’s and G93’s mixing models. Turbulent deformation is represented by a sequence of discrete events that punctuate the continuous diffusion process. Each event represents the effect of an eddy of size *l* and does so by instantaneously rearranging the scalar field in a randomly selected segment of length *l* using the “triplet map” (Kerstein 1991). This map, illustrated in Fig. 1, causes an increase of the scalar gradient within the segment, analogous to the effect of compressive strain (deformation) in turbulent flow.

*f*(

*l*). Rearrangement events occur at a rate Λ per unit length. Kerstein (1991) determined these parameters by recognizing that the rearrangement events induce a random walk of a fluid element, by interpretating the corresponding diffusivity as the turbulent diffusivity

*D*

_{T}, and by incorporating the Kolmogorov (inertial range) scaling relations. Furthermore, the segment sizes are limited to the range

*η*≤

*l*≤

*L,*where

*η*and

*L*are the model analogs of the Kolmogorov and integral length scales of the turbulent flow. For a high-Re turbulent flow, Kerstein (1991) found that

*f*(

*l*) and Λ can be expressed as

*D*

_{M}(the molecular diffusivity that appears in the scalar diffusion equation),

*D*

_{T},

*η,*and

*L.*

The size of the largest turbulent eddy, *L,* is specified from observations. We can use *L* and measurements of the turbulent kinetic energy dissipation rate, *ϵ,* to specify the turbulent diffusivity, *D*_{T}. Krueger (1993) showed that *D*_{T} ≈ 0.1*L*^{4/3}*ϵ*^{1/3}. Note that in this formulation, there is no feedback from the mixing process to *D*_{T}.

For many applications, it is not necessary to resolve the Kolmogorov scale of the turbulent flow. At a sufficiently large Reynolds number, Re ≡ (*L*/*η*)^{4/3}, the linear eddy model results become statistically independent of Re (see section 4b). When the physical Kolmogorov scale is not resolved, the molecular diffusivity *D*_{M} is simply replaced by the turbulent diffusivity due to eddies smaller than *η.* This is *D*_{T}(*η*/*L*)^{4/3} (Tennekes and Lumley 1972).

An example taken from Krueger (1993) will illustrate how the linear eddy model works. Consider the mixing of two distinct air masses, one of which consists of entrained air and occupies 5% of the linear eddy model’s 1D domain. Krueger considered cases for which the entrained air initially consists of 1, 3, or 9 blobs. The initial blob size is thus inversely proportional to the number of blobs. The blobs were assigned a value of one, while the remainder of the domain was assigned a value of zero. The initial fields are shown in Fig. 3 of Krueger (1993). The linear eddy model was used to simulate the mixing process for each case. The results for each case are based on an ensemble of 100 realizations.

As mixing proceeds, the scalar variance decreases in each case. In Fig. 2a, the scalar variance for each case is plotted versus the elapsed time *t* scaled by *τ*_{L}, a large-eddy timescale proportional to (*L*^{2}/*ϵ*)^{1/3}. It is evident that the variance decay rate depends on the blob size, with a greater rate for a smaller blob size. Since inertial range scalings are built into the linear eddy model, the dependence of the mixing timescale, *τ*_{d} ≡ (*d*^{2}/*ϵ*)^{1/3}, on the blob size, *d,* should be reflected in the variance decay rates. When the scalar variance for each case is plotted versus *t*/*τ*_{d}, (as shown in Fig. 2b), the curves collapse fairly well for *t*/*τ*_{d} < 5 and are very close for *t*/*τ*_{d} ≤ 1. Note that the scaling arguments that lead to the definition of the mixing timescale *τ*_{d} are not applicable for *t*/*τ*_{d} ≫ 1 (Mell et al. 1991). For further discussion of linear eddy simulations of mixing in homogeneous turbulence, see McMurtry et al. (1993).

The example just described demonstrates the relationship between the entrained blob size and the subsequent scalar variance evolution. In cumulus clouds, variance is produced by multiple entrainment events. The in-cloud variance level is thus determined by the relative rates of variance production by entrainment and variance decay by mixing.

### c. EMPM implementation

By combining the linear eddy model described in section 3b with the entrainment parameterization described in section 3a, the EMPM is able to represent the effects of entrainment, turbulent deformation, and molecular diffusion on the internal structure of the parcel.

The evolution of a parcel as it ascends from cloud base is calculated using the EMPM as shown schematically in Fig. 3. The EMPM’s 1D scalar fields are initially uniform and set equal to the observed horizontally averaged cloud base values. As the parcel rises above cloud base at a specified rate based on observations, entrainment events occur at irregular intervals. The entrained blobs are mixed by the linear eddy model’s rearrangement events—which increase the scalar gradients—and by eddy diffusion.

Many realizations (independent calculations) of parcel evolution are made with the EMPM for each set of parcel parameters in order to provide a precise statistical representation of the entrainment and mixing processes, which are both modeled as stochastic processes in the EMPM. Each realization differs from the others in the ensemble in its sequence of entrainment intervals and its set of rearrangement events. Each simulation described in the next section consisted of an ensemble of 100 realizations.

## 4. Simulations

We used the EMPM to simulate entrainment and mixing in Hawaiian cumulus cloud “main turrets” observed by Raga et al. (1990, hereafter denoted RJB). RJB identified “main turrets” from aircraft measurements as cloudy segments that showed active growth as characterized by peak liquid water content and vertical velocity, and vertical coherence between aircraft flight levels. These are also parts of the clouds where precipitation has not yet developed. RJB found that below the inversion the main turrets consisted almost entirely of updrafts. This differs from the up- and downdraft structure often found in small continental cumuli. RJB’s observations suggest that the main turrets are the result of individual rising thermals and that buoyancy sorting (i.e., selective detrainment) of sub-turret-scale parcels (due to negative buoyancy generated by mixing of entrained air and subsequent evaporative cooling) is not significant below the inversion. RJB also found that the average thermodynamic characteristics of the main turrets below the inversion appear to be well-described by a conventional parcel model with constant *λ.* Based on these results, it appears appropriate to use the EMPM to predict the bulk properties of the main turrets observed by RJB.

### a. Cloud base conditions and environmental profiles

To predict the evolution of the properties of a parcel starting from cloud base at *p* = *p*_{b}, the EMPM needs the observed in-cloud horizontal mean values at cloud base, *q*_{w}*p*_{b}) and *s*_{l}*p*_{b}), to initialize the EMPM’s 1D scalar fields (which are assumed to be spatially uniform) and the observed environmental profiles, *q*_{we}(*p*)*s*_{le}(*p*)

RJB composited aircraft penetrations of 17 cumulus cloud turrets to obtain profiles of in-cloud ensemble means and variances. The variances are consequently due to both cloud-to-cloud variations and in-cloud variations. We introduced cloud-to-cloud variations into the ensemble of realizations by varying *q*_{w}*p*_{b}) and *s*_{l}*p*_{b}). We assumed that the total water mixing ratio variations were distributed normally with the observed ensemble mean and variance values. The corresponding cloud base values of *s*_{l} were obtained from those of *q*_{w} by assuming saturated conditions with no liquid water.

Table 2 lists the cloud base conditions used in the simulations: the ensemble mean cloud base pressure, *p*_{b}, and the ensemble mean cloud base values of *q*_{w} and *s*_{l}, 〈*q*_{w}(*p*_{b})〉 and 〈*s*_{l}(*p*_{b})〉.

The composite environmental profiles, *q*_{we}(*p*)*s*_{le}(*p*)*T*_{e}(*p*), *q*_{e}(*p*), and *z*(*p*) during 15 aircraft soundings. The observed cloud-to-cloud variations may have also been partially due to variations in the environmental profiles. However, we have not found reports of the variance of these profiles.

The parcel’s pressure level *p*∗ ≡ *p*_{b} − *p* at any time is *p*∗(*t*) ≡ ^{t}_{0}*ω dt,* where *ω* ≡ *dp*∗/*dt* = *w dp*∗/*dz, w* is the parcel’s ascent rate, and *dp*∗/*dz* ≈ 10 Pa m^{−1}.

### b. Entrainment and mixing parameters

The fractional rate of entrainment *λ,* dissipation rate *ϵ,* and the parcel ascent speed *w* are based on observations of Hawaiian cumulus clouds reported by RJB. The remaining parameters, *d, D, L, η,* and the grid interval Δ*x,* were chosen based on the considerations discussed below. Table 2 lists the entrainment and mixing parameter values used.

The largest eddy size *L* is not a crucial parameter as long as it is significantly larger than the entrained blob size *d.* RJB reported that the active cloud turrets were 2–3 km wide. This sets an upper bound on *L* since it cannot be larger than the observed cloud widths. Cumulus clouds consist of many thermals with internal circulations, so *L* may be significantly less than the cloud width. Therefore we set *L* = 500 m.

The EMPM results become statistically independent of the model Kolmogorov scale, *η,* for Re = (*L*/*η*)^{4/3} ≥ 100, or *η* ≤ 0.03*L.* This has also been noticed in other studies that used the linear eddy model (Krueger 1993; McMurtry et al. 1993). Although a Reynolds number of 100 is a relatively low value for such statistical independence, it is not surprising since the linear eddy model is based on high Reynolds number scaling laws. Therefore, regardless of the value of the model Reynolds number, the EMPM simulations should exhibit high Reynolds number effects. However, when small-scale physical processes are specifically of interest (e.g., chemical reactions or droplet growth), the above resolution criterion will not necessarily be valid. When such processes are explicitly represented, greater resolution is required. For example, Su et al. (1996) resolved the physical Kolmogorov scale in their study of the effects of entrainment and mixing on cloud droplet growth.

For *L* = 500 m, *η* ≤ 15 m is required to satisfy the Reynolds number independence criterion. Our value of *η,* 5 m, satisfies this criterion. The grid size Δ*x* is chosen so that *η* is the size of the smallest segment that undergoes rearrangement events. When using the triplet map, six grid cells are the minimum that can participate in a rearrangement event. Therefore, Δ*x* = *η*/6 ≈ 0.8 m.

The specification of the characteristic entrained blob size *d* is more difficult since the distribution of entrained blob sizes in cumulus clouds is not known. An indirect way to determine *d* is to vary it in the EMPM with the remaining parameters set at their observed values. The occurrence of a good match between predicted and measured quantities that are sensitive to *d* would suggest that the chosen value of *d* is realistic. To follow this approach, a reasonable range of values for *d* must first be identified. We can do this by relating *d* to the domain size, *D.*

In the EMPM, the ratio *σ* ≡ *d*/*D* is the fractional volume of the domain occupied by entraining air during an entrainment event. We can interpret *σ* as the fractional area of a thermal’s surface occupied by entraining air if we assume that an entrainment event in a (spherical) thermal of radius *R* occurs by incorporating a volume of air of fractional area *σ* and depth *R*/3. Then *σ* = *w*_{e}/*w*∗, where *w*_{e} is the average entrainment velocity and *w*∗ is the characteristic velocity of the entraining eddies. We assume that *w*∗ ∼ *w* and that *w*_{e} = *αw.* Then *σ* ∼ *α.* Laboratory experiments suggest that *α* ∼ 0.2 for a spherical parcel (Cotton 1975). However, conventional entraining parcel model studies indicate that values of *α* about half as large are needed to accurately predict cumulus cloud-top heights (Cotton 1975). Therefore, we set *σ* = 0.1.

The domain size *D* is closely related to the observed cloud width. A range of 500–2000 m for *D* is plausible based on the clouds observed by RJB. The corresponding range for *d* = *σD* is then 50–200 m.

“Signatures” of entrainment events appear to exist in cumulus clouds. RJB identified relatively homogeneous regions as such signatures. They found that the average width of such regions, which they called the “entrainment length,” was a few hundred meters for the Hawaiian trade cumuli they studied. Such regions would be larger than the associated entrained blob size. The observed entrainment lengths are therefore in accord with our estimated range of entrained blob sizes.

### c. Comparison to observations

We performed three simulations of rising thermals in the main turrets of Hawaiian cumulus clouds with the EMPM to compare to the observations reported by RJB. These simulations differed only in their values of *d* (50, 100, 200 m) and *D* (500, 1000, 2000 m). For reference, we also performed one simulation in which the entrained air was instantaneously mixed throughout the parcel so that the parcel was homogeneous (and therefore without internal structure) at all times.

RJB defined “in-cloud” to include only those 1-sec (85 m) segments that contained no 0.02-sec (1.7 m) segments without cloud droplets. They then assumed that the in-cloud segments were saturated and derived the air temperature from the water vapor mixing ratio measured by the Lyman-*α* hygrometer. To compare our simulations to RJB’s measurements, we used an essentially identical sampling method. We included as “in-cloud” only those 100-gridpoint (83 m) segments that did not include any two-gridpoint (1.7 m) segments with zero liquid water.

In addition to this conditional sampling method, described above, we also used a complete sampling method that included as “in-cloud” all EMPM grid points. Results based on the two sampling methods will generally differ because clear-air segments of varying sizes occur in the EMPM simulations (see Fig. 11). We include results from both sampling methods because it is not evident which sampling method is more appropriate for making comparisons between the model results and observations. For example, conditional sampling underestimates to some degree the extent of “in-cloud” segments because the EMPM assumes that droplets immediately evaporate in subsaturated regions, while complete sampling obviously overestimates the extent of “in-cloud” segments.

Our comparison to RJB’s aircraft measurements illustrates a noteworthy feature of the EMPM. Because of the EMPM’s high spatial resolution, the EMPM’s results are analogous to those obtained from aircraft penetrations and can be analyzed in the same manner.

## 5. Results

Figure 4 shows the total water mixing ratio, *q*_{w}(*x*), at *p* = 900 mb and *p* = 850 mb from the EMPM for one realization. Three types of regions are evident: 1) fairly uniform with high values of *q*_{w}, 2) fairly uniform consisting of recently entrained air, with low values of *q*_{w}, and 3) highly variable with intermediate values of *q*_{w}. At 900 mb, the fairly uniform regions with high values consist of unmixed cloud base air, while at 850 mb they may correspond to the homogeneous, significantly diluted parcels identified by RJB and hypothesized to be the end result of entrainment and mixing in a limited volume.

### a. Conserved scalars

Figure 5 contains profiles of the in-cloud ensemble means of the liquid water static energy, 〈*s*_{l}〉, and the total water mixing ratio, 〈*q*_{w}〉, while Fig. 6 contains profiles of the corresponding in-cloud standard deviations, *s*^{′}_{l}

In Figs. 5 and 6, the heavy solid lines are the composite profiles obtained by RJB from aircraft measurements. The “instant mixing” profiles are obtained from the EMPM when the entrained blobs are immediately mixed throughout the parcel. For reference, the adiabatic (no entrainment) parcel profiles and the environment profiles are also included in the mean profile plots.

The mean profiles of the conserved quantities *s*_{l} and *q*_{w} obtained from the EMPM using complete sampling should depend only on the fractional rate of entrainment and not on the entrained blob size (or other aspects of how turbulent mixing is represented). The overlapping gray lines in Fig. 5 confirm this expectation. However, the corresponding mean profiles obtained from the EMPM using conditional sampling do depend on the entrained blob size because the spatial distribution of liquid water (upon which the conditional sampling method is based) is determined by the turbulent mixing process (see section 5d).

By comparing mean profiles from an instant mixing entraining parcel model with the measured (conditionally sampled) profiles, RJB estimated the entrainment rate. This approach ignores the parcel model profiles’ dependence on the entrained blob size. However, the dependence appears to be within the range of measurement uncertainty.

The EMPM standard deviation profiles in Fig. 6 exhibit a significant dependence on the entrained blob size, and also on the sampling method. Only the conditionally sampled *q*^{′}_{w}

### b. Liquid water mixing ratio and buoyancy

In the previous section we showed that finite-rate mixing is necessary to reproduce the in-cloud variability of the conserved quantities *s*_{l} and *q*_{w} observed in Hawaiian cumulus clouds by RJB. However, finite-rate mixing is not necessary to match the observed mean profiles of *s*_{l} and *q*_{w}. Are these conclusions valid for nonconserved quantities such as the liquid water mixing ratio, *l,* and the buoyancy?

*B*

*T*

_{υ}

*T*

_{υe}

*B*as the buoyancy. The appendix describes how

*l*and

*B*are obtained from

*s*

_{l}and

*q*

_{w}.

Figure 7 presents the profiles of the in-cloud ensemble means of the liquid water mixing ratio, 〈*l*〉, normalized by the adiabatic liquid water mixing ratio obtained using the ensemble mean cloud base conditions, *l*_{a}, and the buoyancy, 〈*B*〉. Figure 8 shows the in-cloud standard deviations of the liquid water mixing ratio, *l*′, and the buoyancy, *B*′. The figures include EMPM in-cloud profiles for entrained blob sizes of 50, 100, and 200 m obtained using both conditional sampling and complete sampling. These figures also include the observed and instant mixing profiles, plus the adiabatic profile for 〈*B*〉.

We noted above that the mean profiles of the conserved quantities *s*_{l} and *q*_{w} obtained from the EMPM using complete sampling do not depend on how turbulent mixing is represented. However, Fig. 7 illustrates that the profiles of 〈*l*〉/*l*_{a} and 〈*B*〉 obtained from the EMPM using complete sampling do depend on how turbulent mixing is represented because 〈*l*〉/*l*_{a} and 〈*B*〉 depend on the degree of mixing.

Figure 7 shows that the mean profiles obtained from the EMPM for the three entrained blob sizes using conditional sampling and complete sampling differ in two aspects. The conditionally sampled mean profiles have larger values of 〈*l*〉/*l*_{a} and 〈*B*〉 than their corresponding completely sampled mean profiles (as one would expect from the difference in the sampling methods) and exhibit a greater dependence on the entrained blob size. However, no set of mean profiles for a particular blob size matches the observations best.

Figure 7 illustrates that instant mixing always produces the smallest values of 〈*l*〉/*l*_{a} and 〈*B*〉. This figure also shows that 1) all of the profiles of 〈*l*〉/*l*_{a} obtained from the EMPM (for the three entrained blob sizes and for both sampling methods) more closely match the observed profile than does the instant mixing profile and 2) that all of the values of the vertical average of 〈*B*〉 obtained from the EMPM more closely match the average observed value than does the average of the instant mixing profile. In addition, the nonbuoyancy levels predicted by the EMPM are all closer to the observed level than is the instant mixing parcel model’s nonbuoyancy level.

Figure 7 also shows that the profiles of 〈*l*〉/*l*_{a} predicted by the EMPM agree well with those observed except near cloud base (where measurement uncertainties are magnified) and in the inversion layer (near *p* = 800 mb). It is likely that in the observed cumulus clouds, only the most buoyant (least diluted) parcels penetrated into the trade inversion. The EMPM with an entrained blob size of 100 m predicts that local buoyancy values range from −1 to +1.5 K at the inversion base (830 mb). Instead of carrying the entire parcel mass upward into the inversion as the EMPM does, RJB’s profile of 〈*B*〉 suggests that cumulus clouds detrain the less buoyant air as they rise into the inversion.

Figure 8 shows that the sampling method has little impact on the profiles of *l*′ and *B*′ obtained from the EMPM for the three entrained blob sizes. The figure also shows, just as for the mean profiles, that all of the profiles of *l*′ obtained from the EMPM more closely match the observed profile than does the instant mixing profile and that all of the values of the vertical average of *B*′ obtained from the EMPM more closely match the observed value than does the average of the instant mixing profile.

In summary, the profiles of 〈*l*〉/*l*_{a}, 〈*B*〉, *l*′, and *B*′ presented in Figs. 7 and 8 clearly show that finite-rate mixing of entrained blobs of 50–200-m size is required to match the observations. However, it is difficult to select results for a particular entrained blob size as the “best fit” due to measurement uncertainties.

### c. Unmixed cloud base air

An interesting quantity obtained from the EMPM that can be compared with measurements when they become available is the ensemble-average fraction of unmixed cloud base air (Fig. 9). The fraction of unmixed cloud base air is defined as the fraction of the total number of grid points for which *q*_{w}(*p*_{b}) − *q*_{w} ≤ 0.02*q*_{w}(*p*_{b}). Based on this definition, Fig. 9 shows that unmixed cloud base air is rarely found in these simulations above *p* = 880 mb for *d* = 50 m and above *p* = 840 mb for *d* = 100 m. For *d* = 200 m, about 1% of cloud-base air reaches *p* = 810 mb without being mixed. The curve labeled “entrainment only” shows that most of the decrease of the fraction of unmixed cloud-base air is due to entrainment; the remainder is due to mixing.

RJB reported that there were parcels of undiluted cloud-base air at all levels up to and above the inversion base at 840 mb in almost all of the clouds sampled. This suggests, based on the EMPM results, that some of the entrained blobs were 100 m or larger in extent in these clouds.

### d. The characteristic entrained blob size

Here we propose two methods for determining the characteristic entrained blob size from aircraft measurements.

An easily measurable quantity clearly related to the characteristic entrained blob size is the in-cloud fraction of clear air. “Clear air” exists in the EMPM whenever a 0.83-m segment (one grid cell) contains no liquid water. Figure 10 shows profiles of the ensemble mean clear-air fraction obtained from the EMPM for the three entrained blob sizes. The clear-air fraction ranges from about 0.10 to 0.25 and significantly increases as the blob size increases. This reflects the greater time required to mix larger blobs. By comparing EMPM results to observations of the clear-air fraction, one may estimate the characteristic entrained blob size.

A more direct measurement of the characteristic entrained blob size can be made with the pdf of the clear-air segment size. Figure 11 shows the pdfs (averaged over all heights and normalized by the entrained blob size) obtained from the EMPM for the three entrained blob sizes. There is a peak in each pdf at the entrained blob size. But more importantly, it is evident that the pdfs are similar when the clear-air segment size is scaled by the entrained blob size. This allows the characteristic entrained blob size to be determined directly from the pdf.

## 6. Summary and conclusions

The model used by Krueger (1993) to study entrainment and mixing of thermodynamic properties in the stratus-topped boundary layer has been extended to represent these processes in cumulus clouds. The new model, called the “explicit mixing parcel model,” depicts the finescale internal structure of a rising thermal in a cumulus cloud using a 1D domain. The internal structure evolves in the EMPM as a consequence of a sequence of discrete entrainment events and an explicit representation of turbulent mixing based on Kerstein’s (1988) linear eddy model. In this version of the EMPM, subgrid-scale (eddy) diffusion is found to be adequate for representing the effects of the smallest turbulent eddies. In addition, a simple parameterization is used to determine the local condensation or evaporation rates.

By reducing the description of the internal structure of a parcel to one dimension, high spatial resolution can be achieved. The scalar fields predicted within the EMPM’s 1D domain represent those of the parcel and are assumed to be *statistically* homogeneous, although a large amount of instantaneous spatial variation typically exists as a result of entrainment and mixing.

In the EMPM, each entrainment event involves inserting a segment of environmental air of length *d* from the current parcel level into a randomly selected location of the 1D computational domain. We refer to this new segment of air as the *entrained blob.* The average frequency of the entrainment events depends on the fractional rate of entrainment *λ,* the parcel ascent rate *w,* and the entrained blob size *d,* all of which are specified parameters.

In the EMPM, turbulent mixing is represented by the linear eddy model, which separately treats turbulent deformation that results in scale contraction and molecular diffusion that ultimately mixes fluid properties. It is the implementation of turbulent deformation that is the key feature of the linear eddy model. Turbulent deformation is represented by a sequence of discrete rearrangement events. Each event represents the effect of an eddy of size *l* and does so by instantaneously rearranging the scalar field in a randomly selected segment of length *l* using the “triplet map.” This map causes an increase of the scalar gradient within the segment, analogous to the effect of compressive strain (deformation) in turbulent flow.

The distribution of segment sizes and the frequency of rearrangement events are determined by the largest eddy size *L,* the smallest resolved eddy size *η,* and the turbulent diffusivity *D*_{T}. The turbulent diffusivity can be determined from *L* and the turbulent kinetic energy dissipation rate *ϵ.*

For given cloud base and environmental conditions, the in-cloud properties predicted by the EMPM depend primarily on *w, λ, d,* and *ϵ.* The largest eddy size *L* is not a crucial parameter as long as it is significantly larger than the entrained blob size *d.* For *bulk* in-cloud properties (but not for droplet spectra evolution), the smallest resolved eddy size is likewise not a key parameter because of Reynolds number similarity. In other words, for predicting bulk properties, eddy diffusion is adequate for the representing the effects of the smallest turbulent eddies (those less than 5 m in size in the case considered).

Only *λ* changes the ensemble means of conserved quantities such as the liquid water static energy and the total water mixing ratio. For a given *λ,* the quantities *w, d,* and *ϵ* determine the degree of mixing or internal structure and, therefore, influence the ensemble means of unconserved quantities such as the liquid water mixing ratio and the buoyancy, and the variances of all quantities. The parameters *w, d,* and *ϵ* determine the mixing height scale, *w*(*d*^{2}/*ϵ*)^{1/3}. For larger values of this scale, a parcel will be relatively less mixed at a given height and thus have more internal variability. Of the three mixing parameters, two are reasonably well measured quantities (*w* and *ϵ*), while the remaining parameter (*d*) is not.

A reasonable range of values for *d* was identified by relating *d* to the EMPM’s domain size, *D,* through similarity arguments and parcel model predictions of cumulus cloud-top heights reported by Cotton (1975). We concluded that *d*/*D* ∼ 0.1.

To evaluate the EMPM’s entrainment parameterization, the model was used to predict the bulk properties of Hawaiian cumulus cloud main turrets observed by RJB. They found that the average thermodynamic characteristics of these turrets below the inversion appear to be well-described by a conventional parcel model with constant *λ.* RJB’s measurements also indicate that below the inversion the turrets were not subject to significant sub-turret-scale buoyancy sorting (i.e., selective detrainment). All of the quantities required by the EMPM except for the entrained blob size were obtained from the observations. Based on the range of cloud widths observed by RJB, entrained blob sizes ranging from 50 m to 200 m were used. Profiles of in-cloud means and variances of thermodynamic properties calculated by the EMPM for entrained blob sizes of 50 m, 100 m, and 200 m and by a parcel model with instantaneous mixing of entrained air (equivalent to the EMPM with an infinitesimal entrained blob size) were compared to those observed. The observed mean profiles of two conserved scalars are reproduced by both mixing representations, but the observed mean liquid water mixing ratio and buoyancy profiles, all of the observed variance profiles, and the observed nonbuoyancy level are better reproduced by the EMPM. For entrained blob sizes of 100 m and 200 m, undiluted cloud base air reaches the inversion base in the EMPM, as was observed. These results indicate that the EMPM’s entrainment parameterization is adequate for these cloud turrets and that the characteristic entrained blob size is about 100 m. Additional results suggest that the characteristic entrained blob size may be estimated from aircraft measurements of the clear-air segment size distribution. The model results also demonstrate that the finescale structure represented by the EMPM’s 1D domain can be directly compared to high-frequency aircraft measurements.

Overall, the in-cloud structure simulated by the EMPM resembles that observed in the Hawaiian cumulus cloud main turrets. This suggests that the entrainment and mixing process as represented in the EMPM capture the essential features of the processes that occur in these cumulus cloud turrets. However, the EMPM is based on several simplifying assumptions whose impact should be evaluated. We have assumed that the parcel is statistically homogeneous, entrainment events occur instantaneously, the entrained blob size is constant, there is no preferential detrainment of negatively buoyant air, and the parcel ascent rate is constant. Except for the first two, these assumptions can easily be relaxed.

We view the EMPM as both a conceptual and a practical link between the conventional parcel model, with no internal structure, and multidimensional cloud models, with cloud-scale internal structure. The 1D EMPM can represent finescale internal structure revealed by aircraft measurements, which multidimensional cloud models cannot resolve. Furthermore, if the EMPM’s grid size is reduced so that the Kolmogorov scale is resolved and a droplet growth model is incorporated, the EMPM can predict the local microphysical environments of individual cloud droplets and is therefore capable of simulating the evolution of droplet spectra in cumulus clouds (Su et al. 1996).

In the EMPM, the cloud-scale structure is parameterized (via the entrainment rate, entrained blob size, and ascent rate) in the simplest way possible. We consider the simplicity of the entrainment parameterization to be an appealing aspect of the model. This simplicity allowed us to specify all model parameters except for the entrained blob size from observations. In future applications of the linear eddy model, the cloud-scale structure could be specified in much greater detail, either along a parcel trajectory obtained from a 3D LES (e.g., Stevens et al. 1996) or by coupling a LES to a subgrid-scale model based on the linear eddy model.

## Acknowledgments

Graciela Raga kindly provided the composite observed in-cloud and environmental profiles. The authors thank Alan Kerstein and Philip Austin for helpful discussions, and Ilga Paluch, Wojciech Grabowski, and an anonymous reviewer for their constructive comments. This research was supported by the Office of Naval Research under Grant N00014-91-J-1175 and by the National Science Foundation under Grant CTS 9258445. A grant of computer time from the Utah Supercomputer Institute, which is funded by the State of Utah and the IBM Corporation, is gratefully acknowledged.

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## APPENDIX

### Parcel Temperature

*T, q,*and

*l*from

*s*

_{l}and

*q*

_{w}we first assume that the air is saturated. Then,

*q*

*q*

*T, p*

_{e}

*q*∗(

*T, p*

_{e}) is the saturation mixing ratio. The subscript

*e*denotes the environmental value(s) at the same level. (The environmental values are presumed to be known.) We assume that the in-cloud pressure is the same as the environmental pressure

*p*

_{e}. Following Arakawa and Schubert (1974), we approximate

*q*by

*q*

^{∗}

_{e}

*q*∗(

*T*

_{e},

*p*

_{e}). Then

*c*

_{p}≈

*c*

_{pd}(1 + 0.85

*q*

_{e}) (Emanuel 1994),

*c*

_{pd}is the specific heat of dry air at constant pressure,

*h*=

*s*

_{l}+

*Lq*

_{w}, and

*h*

^{∗}

_{e}

*c*

_{p}

*T*

_{e}+

*L*

*q*

^{∗}

_{e}

*gz*is the saturation value of the moist static energy of the environment. If

*q*

_{w}−

*q*≥ 0, the air is saturated as assumed, so

*T*and

*q*are given by the values obtained above, and

*l*=

*q*

_{w}−

*q.*The error in

*T*due to linearizing

*q*∗(

*T, p*

_{e}) is generally less than 0.1 K for |

*T*−

*T*

_{e}| ≤ 2 K, which is usually the case for small cumulus clouds.

*q*

_{w}−

*q*< 0, the saturation assumption is invalid. In this case,

*l*= 0,

*q*=

*q*

_{w}, and

*s*=

*s*

_{l}. We then obtain

*T*from

*T*

_{υ}

*T*

*T*

_{e}

*δq*

*l*

*δ*= 0.608. The excess virtual temperature is

*T*

_{υ}

*T*

_{υe}

*T*

*T*

_{e}

*T*

_{e}

*δ*

*q*

*q*

_{e}

*l*

Table 1. Key features of several models used for simulating entrainment and mixing in cumulus clouds. In the last three columns E = explicit and I = implicit.

Table 2. Parameter values used for simulating entrainment and mixing in Hawaiian cumulus clouds.