The Macroscopic Entrainment Processes of Simulated Cumulus Ensemble. Part II: Testing the Entraining-Plume Model

Chichung Lin Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, California

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Akio Arakawa Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, California

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Abstract

According to Part I of this paper, it seems that ignoring the contribution from descendent cloud air in a cloud model for cumulus parameterization (CMCP), such as the spectral cumulus ensemble model in the Arakawa–Schubert parameterization, is an acceptable simplification for tropical deep convection. Since each subensemble in the spectral cumulus ensemble model is formally analogous to an entraining plume, the latter is examined using the simulated data from a cloud-resolving model (CRM). The authors first follow the analysis procedure of Warner. With the data from a nonprecipitating experiment, the authors show that the entraining-plume model cannot simultaneously predict the mean liquid water profile and cloud top height of the clouds simulated by the CRM. However, the mean properties of active elements of clouds, which are characterized by strong updrafts, can be described by an entraining plume of similar top height.

With the data from a precipitating experiment, the authors examine the spectral cumulus ensemble model using the Paluch diagram. It is found that the spectral cumulus ensemble model appears adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different entrainment characteristics. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum. To further investigate whether the fractional rate of entrainment is an appropriate parameter for characterizing cloud types in the spectral cumulus ensemble model, the authors stratify the simulated saturated updrafts (subcloud elements) into different types according to their eventual heights and calculate the cloud mass flux and mean moist static energy for each type. Entrainment characteristics are then inferred through the cloud mass flux and in-cloud moist static energy. It is found that different types of subcloud elements have distinguishable thermodynamic properties and entrainment characteristics. However, for each cloud type, the fractional rate of entrainment is not a constant in height but tends to be larger at lower levels and near cloud top. In addition, the in-cloud moist static energy at cloud base considerably deviates from the mean in the subcloud layer, indicating that the effects due to inhomogeneity of the planetary boundary layer should be taken into account in a CMCP as well.

Corresponding author address: Dr. Chichung Lin, Dept. of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, CA 90095.

Email: cclin@fuji.atmos.ucla.edu

Abstract

According to Part I of this paper, it seems that ignoring the contribution from descendent cloud air in a cloud model for cumulus parameterization (CMCP), such as the spectral cumulus ensemble model in the Arakawa–Schubert parameterization, is an acceptable simplification for tropical deep convection. Since each subensemble in the spectral cumulus ensemble model is formally analogous to an entraining plume, the latter is examined using the simulated data from a cloud-resolving model (CRM). The authors first follow the analysis procedure of Warner. With the data from a nonprecipitating experiment, the authors show that the entraining-plume model cannot simultaneously predict the mean liquid water profile and cloud top height of the clouds simulated by the CRM. However, the mean properties of active elements of clouds, which are characterized by strong updrafts, can be described by an entraining plume of similar top height.

With the data from a precipitating experiment, the authors examine the spectral cumulus ensemble model using the Paluch diagram. It is found that the spectral cumulus ensemble model appears adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different entrainment characteristics. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum. To further investigate whether the fractional rate of entrainment is an appropriate parameter for characterizing cloud types in the spectral cumulus ensemble model, the authors stratify the simulated saturated updrafts (subcloud elements) into different types according to their eventual heights and calculate the cloud mass flux and mean moist static energy for each type. Entrainment characteristics are then inferred through the cloud mass flux and in-cloud moist static energy. It is found that different types of subcloud elements have distinguishable thermodynamic properties and entrainment characteristics. However, for each cloud type, the fractional rate of entrainment is not a constant in height but tends to be larger at lower levels and near cloud top. In addition, the in-cloud moist static energy at cloud base considerably deviates from the mean in the subcloud layer, indicating that the effects due to inhomogeneity of the planetary boundary layer should be taken into account in a CMCP as well.

Corresponding author address: Dr. Chichung Lin, Dept. of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, CA 90095.

Email: cclin@fuji.atmos.ucla.edu

1. Introduction

Cumulus parameterization is a closure problem in which the degree of freedom is larger than the number of equations. A cloud model of cumulus parameterization (CMCP), which describes the statistical thermodynamic properties of a cumulus ensemble under given large-scale conditions, provides important constraints on moist convective processes and thus can be considered as a part of the closures necessary for parameterizing cumulus convection. Among the important physical processes to be included in a CMCP is cumulus entrainment, which can significantly dilute cloud properties and increase cloud mass flux. This process has been formulated in many existing cumulus parameterization schemes (e.g., Arakawa and Schubert 1974; Kreitzberg and Perkey 1976; Fritsch and Chappell 1980; Kain and Fritsch 1990; Emanuel 1991). However, a consensus on how to formulate the process in a CMCP has not been reached yet, even though many efforts have been made to understand cumulus entrainment (Reuter 1986; Blyth 1993).

Lin and Arakawa (1997, hereafter Part I) address the problem of formulating entrainment effects in a CMCP with a unique approach. Since it is difficult to obtain synchronous observations covering a population of clouds, they use a cloud-resolving model (CRM), which covers a large domain and resolves individual cumulus clouds, to infer sources of entrainment into cumulus ensembles. They argue that, although the simulations are two-dimensional and turbulent flows are parameterized, much can still be learned through the simulated data in view of the CRM’s success in simulating cumulus feedback on large-scale thermodynamic fields, an objective that cumulus parameterization is intended to achieve.

With the Paluch (1979) diagram, they show that the linear patterns on the diagram in observational studies can be reproduced with the simulated data and can be interpreted, as suggested by Taylor and Baker (1991), in ways other than two-point mixing. It is also concluded that, as far as tropical deep convection is concerned, the sources of a cumulus cloud at one level are from locations of various heights, indicating a continuous series of entrainment events occurring throughout the cloud depth. However, they do not observe a cloud air parcel descending more than several hundred meters. Therefore, they suggest that ignoring the contribution from descendent cloud air in a CMCP, such as the spectral cumulus ensemble model in the Arakawa–Schubert (1974) parameterizations, is an acceptable simplification.

Motivated by the results of Part I, we proceed to examine the entraining-plume model, which describes the properties of each subensemble in the spectral cumulus ensemble model, using the simulated data from a cloud-resolving model that has been described in Part I. The simulated dataset used in this study is summarized in the next section. In section 3, we analyze the simulated data following the approach used by Warner (1970), who compared the observed liquid water content and cloud top height to those predicted by an entraining-plume model. Since Warner’s conclusion was based on observations of nonprecipitating small clouds, we use the data of a nonprecipitating experiment to parallel Warner’s study.

In sections 4 and 5, the validity of the spectral cumulus ensemble model is examined using the data from a precipitating experiment. With the Paluch diagram, we compare the cloud properties predicted by the spectral cumulus ensemble model to those simulated by the CRM in section 4. Based on the results from section 4, it seems that the spectral cumulus ensemble model is adequate if different types of clouds in the spectrum are interpreted as cloud air parcels with different fractional rates of entrainment. With this interpretation, in section 5, we further investigate whether a fractional rate of entrainment is an appropriate parameter for characterizing a cloud type in the spectral cumulus ensemble model. The simulated saturated updrafts are first stratified into different types according to their eventual heights. The entrainment characteristics of each type are then inferred based on the cloud mass flux and in-cloud moist static energy calculated from the stratified saturated updrafts. Finally, conclusions are given in section 6.

2. The numerical experiments

The numerical experiments were performed using the CRM developed by Krueger (1988; see also Xu and Krueger 1991). The experiments have been described in Part I. Briefly, the first experiment, A10, simulated tropical oceanic cumulus ensemble with given large-scale advective and radiative cooling effects. The width of the domain is 256 km, and the horizontal resolution is 500 m. The depth of the model domain is 19 km with a stretched vertical coordinate. The vertical resolution is 100 m near the surface and 1 km near the model top. The other experiment, A12, is identical to experiment A10, except that the precipitating processes are absent. It is expected that the dynamics of simulated clouds would be different without precipitating processes (see Part I). The motivation of experiment A12, however, is to parallel the situations of nonprecipitating cumuli that Warner’s (1970) analysis was based upon and to see whether we can reproduce Warner’s results using the simulated data.

3. Analysis following Warner’s (1970) approach

Warner (1970) compared the observations made over many nonprecipitating small cumuli observed along the eastern Australian coast to the predictions by an entraining-plume model. The entraining-plume model is described by equations for conservation of heat, moisture, and momentum. It is assumed that a fractional rate of entrainment, μ, is related to a radius, r, by
i1520-0469-54-8-1044-e3-1
where α is an empirical coefficient set to 0.1. To use the model to predict plume properties at each level, initial conditions at the plume base need to be specified in addition to cloud environment sounding. With the plume top defined as the level where the vertical velocity becomes or tends to become negative, Warner concludes that the predicted cloud top height and liquid water content cannot simultaneously match those from observations.

Following Warner’s analysis procedure, we compare the ratio of liquid water content to its adiabatic value for clouds simulated by the CRM, (W/Wa)CRM, to that predicted by the entraining-plume model, (W/Wa)plume. The clouds1 are chosen from experiment A12 to parallel nonprecipitating cumuli in Warner’s analysis. (The reason that we do not choose the small clouds from experiment A10 is that nearly all these clouds eventually precipitate after their tops reach levels higher than 4 km at certain stages of their life cycles.) Since the clouds in experiment A12 have different top heights, the average of (W/Wa)CRM should be limited to clouds whose top heights are comparable. We obtain the (W/Wa)CRM profile using the clouds whose ultimate tops are between 4 and 5 km between 0200 MT (model time) and 0300 MT (the ultimate top of a cloud means the highest level the cloud ever reaches in its entire life cycle). The profile is truncated below the 0.5-km level and above the 3.5-km level to ensure that there is sufficient data at each level to estimate (W/Wa)CRM.2

The (W/Wa)plume profiles are calculated with the radius of the plume fixed in height. The cloud environment sounding is assumed approximately equal to the means over the entire domain between 0200 MT and 0300 MT. The plume base, following Warner (1970), is set to be 150 m below the lower end of the (W/Wa)CRM profile. The temperature and vertical velocity at the base of the plume are taken from the cloud environment sounding at that level. Saturation is assumed at the plume base.

The (W/Wa)CRM profile and the (W/Wa)plume profiles, obtained in such a way as explained above, are shown in Fig. 1. We see that, as concluded by Warner (1970), the plume model cannot simultaneously predict (W/Wa)CRM and cloud top height. If sufficient entrainment is introduced to obtain (W/Wa)CRM, the plume cannot reach the 4–5-km height interval. Conversely, if an appropriate amount of entrainment is introduced to obtain a plume reaching the 4–5-km height interval, (W/Wa)plume would be significantly larger than (W/Wa)CRM. We also notice that the (W/Wa)plume profiles are very sensitive to the initial conditions of the plume. For example, when the plume base is set to the 0.6-km level [the level 100 m above the lower end of the (W/Wa)CRM profile in Fig. 1] and the initial conditions are taken from the means averaged over only the cloudy areas of the CRM domain (instead of the entire domain) at that level, the (W/Wa)plume profiles (as shown in Fig. 2a) are very different from those in Fig. 1. Nevertheless, Warner’s conclusion is still valid even though different initial conditions are used to predict the (W/Wa)plume profiles. The entraining-plume model with a fixed radius, therefore, is incapable of describing the mean properties of clouds with similar top heights.

On the other hand, we find that the entraining-plume model can be used to describe the mean properties of active elements of the clouds. Instead of using all the data from the clouds whose eventual tops are between 4 and 5 km, we now obtain the (W/Wa)CRM profile using a subset consisting of elements whose vertical velocities are greater than 1.5 m s−1. The (W/Wa)plume profiles are then obtained with the plume base at the 0.75-km level3 and initial conditions given by the means over the cloudy area of the CRM domain at that level. It is found that the (W/Wa)CRM profile is very close to the (W/Wa)plume profile with a radius of 250 m, which predicts a cloud top of 4.7 km (Fig. 2b).

It should be noted that the criterion for active cloud elements is empirical; using 1.5 m s−1 as the criterion yields the best agreement between the (W/Wa)CRM profile and a (W/Wa)plume profile. Furthermore, this agreement does not imply similarity between an active cloud element and an entraining-plume in view of the fact that the plume radii in Fig. 2b are all but one smaller than the grid size used in the CRM. Nevertheless, it turns out that the mean properties of active elements for clouds whose top is within a certain range can be formally described by an entraining-plume of similar top height.

4. Testing the spectral cumulus ensemble model using the Paluch diagram

In this section the data from experiment A10 are used to test two versions of the spectral cumulus ensemble model in the Arakawa–Schubert parameterization using the Paluch diagram. One is the original version of the spectral cumulus ensemble model (Arakawa and Schubert 1974), in which each cloud type is characterized by a fixed fractional rate of entrainment, λ. The other is the version used in the relaxed Arakawa–Schubert scheme (Moorthi and Suarez 1992), in which each cloud type is characterized by a fixed entrainment rate, μ. While the cloud mass flux profiles for the former (hereafter referred to as the λ-version model) are exponential in height, those for the latter (hereafter referred to as the μ-version model) are linear in height. This implies that the clouds in the μ-version model would experience more dilution at lower levels than their comparable cloud types (that is, the same cloud top heights) in the λ-version model. Therefore, it is expected that some types of small clouds that exist in the λ-version model may disappear in the μ-version model.

It should be pointed out that comparing the properties predicted by the spectral cumulus ensemble model to those of visual clouds is not straightforward. In the spectral cumulus ensemble model, clouds are classified into different types according to their cloud top heights. This method of classification, however, is difficult to apply to visual clouds because different elements of a cloud often reach different heights. In some cases, we cannot even distinguish one cloud from another when clouds congest together. We thus interpret various types of clouds in the spectral cumulus ensemble model as various types of subcloud elements. The thermodynamic properties of subcloud elements simulated by the CRM are then compared with those predicted by the spectral cumulus ensemble model. In the predictions, the conversion coefficient from suspended condensate to precipitate is set to be 0.0015 m−1, the effects due to ice–liquid phase change are parameterized following Lord (1978) and Lord et al. (1984) (see the appendix), and the base conditions of each cloud type are given by the mean properties averaged over the entire CRM domain at the 0.25-km level to parallel the spectral cumulus ensemble model, in which the mean properties of the mixed layer are used.

Figure 3 shows the Paluch diagram with in-cloud data taken from different levels of the entire model domain during the time period from 0650 MT to 0840 MT in experiment A10. The curve characterizing one cloud type of the λ-version model is represented by a dashed line, and the fractional rate of entrainment associated with each cloud type is labeled at the bottom of the dashed line. The calculations of model-predicted clouds are terminated when the zero buoyancy condition is satisfied. We see that the in-cloud data at each level spread over the (h, Q) space, corresponding to the coexistence of various cloud types. Moreover, the data taken from a higher observation level tend to correspond to a narrower spectrum that shifts to smaller fractional rates of entertainment.

Figure 4 is the same as Fig. 3 except that the μ-version model is used. We see that the curves that represent various cloud types in the μ-version model are similar to those in the λ-version model, especially for the cloud types with smaller entrainment rates. This is not surprising because the exponential profiles (in the λ-version model) can be well approximated by linear profiles (in the μ-version model) when the exponents are small. However, we find that the curves predicted by the μ-version model cannot cover all the subcloud elements from the 2-km and 4-km levels, no matter what range of entrainment is used. The λ-version model does a better job than the μ-version in this regard.

As demonstrated by Figs. 3 and 4, the distributions of in-cloud data in the Paluch diagram do resemble a spectrum of clouds as described by both the λ-version and μ-version models. It appears that the spectral cumulus ensemble model is adequate if different types of clouds in the spectrum are interpreted as subcloud elements with different fractional rate of entrainment. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum.

5. Inferring entrainment characteristics from cloud mass flux and in-cloud moist static energy profiles

Motivated by the results of section 4, we interpret a type of cloud in the spectral cumulus ensemble model as a group of subcloud elements that reach the same height interval. One advantage of this interpretation is that each type of cloud is, by definition, composed of elements that have the same top heights, thus bypassing the possibility that detrainment may occur at any levels below cloud top. In the following, we use the data from experiment A10 to investigate whether the fractional rate of entrainment is indeed an appropriate parameter for characterizing a cloud type in the spectral cumulus ensemble model. Our approach is to infer entrainment characteristics of the stratified saturated updrafts based on their cloud mass flux and in-cloud moist static energy profiles.

a. The procedure of stratifying saturated updrafts

Saturated updrafts are selected for analysis following the procedure as described below. The simulated data are checked throughout the entire CRM domain with a space interval of 0.5 km (in x direction) × 0.25 km (in z direction) and with a time interval of 2 min. If the air parcel at a data point satisfies the cloudiness criterion, which has been described in Part I, and has vertical velocity greater than 0.25 m s−1, we calculate the forward trajectory of the air parcel to determine its eventual height. The eventual height of an air parcel is defined as the level where the air parcel’s vertical velocity is smaller than 0.05 m s−1. If the vertical excursion of the air parcel exceeds 0.5 km, it is considered a saturated updraft. The selected saturated updrafts are then stratified into different types according to their eventual heights. Hereafter we refer to a family of subcloud elements whose eventual heights are between N and N + 1 km as a cloud type with its top at the N-km level. By deep (shallow) cloud type we mean N is large (small) in a relative sense. In this paper, cloud types whose top heights are below the 6-km level are referred to as “shallow,” and cloud types whose top heights are above the 10-km level are referred to as “deep.” Those types whose top heights are in between are referred to as “middle.”

Figure 5 shows one example of stratified saturated updrafts in a two-parameter space: the abscissa is moist static energy and the ordinate is height. Each cross in the two-parameter space represents the moist static energy and the height of a saturated updraft at the time it was recorded. Each panel of Fig. 5 shows one type of saturated updraft, where each cross eventually arrived at the height interval as indicated in the left upper corner of the panel. We see that deeper cloud types tend to have higher moist static energy overall. However, this tendency is not obvious between cloud base and the 3-km level, where moist static energy decreases rapidly for all four cloud types.

In the rest of this section, we present vertical profiles of normalized cloud mass flux and in-cloud moist static energy for the types that have sufficiently large data sizes (at least 50 data points at cloud-base level). For each cloud type, the cloud mass flux profile is obtained by summing up the products of vertical velocity by density at each level over all grid points. The profile is then normalized by cloud base mass flux obtained in the same way. The in-cloud moist static energy profile is obtained simply by taking average over the elements of that type at each level over all grid points. It should be noted that data points associated with convection-induced gravity waves can be falsely selected as saturated updrafts through the procedure we described above and contaminate the calculated profiles. In the cases we studied, gravity waves typically have an amplitude of less than 0.5 km. Thus, in this paper, we show the profiles only up to the (N − 0.5)-km level for a cloud type whose top heights are at the N-km level.

b. The vertical profiles of normalized cloud mass flux and in-cloud moist static energy in experiment A10

The normalized cloud mass flux profiles for the time periods 9–28 h is shown in Fig. 6. First of all, we see that deeper cloud types generally have smaller normalized cloud mass flux at each level. This coincides with the concept that deeper clouds, which usually have larger cores, take longer to dilute. Perhaps the most important feature of Fig. 6 is that the entrainment rates are not constant in height for middle and deep cloud types, whose mass flux increased most significantly at lower levels and near cloud top. As for shallow cloud types, their mass flux profiles seem to be approximately exponential.

Part of the feature identified above can be viewed from the standpoint of in-cloud moist static energy profiles (h-profiles). Along with domain-averaged h- and h*-profiles (h* is saturation moist static energy), the in-cloud h-profiles are shown in Fig. 7. We see that moist static energy of all cloud types (deep or shallow) decreases considerably in a layer immediately above cloud base, and the types appear so similar in this layer that we can hardly distinguish one type from the other. Such a layer is hereafter referred to as a “transition layer.” Since the variation of in-cloud moist static energy below the freezing level is mainly caused by entrainment, all cloud types must have comparably large entrainment rate in the transition layer. Above this layer, h-profiles begin to branch out and different cloud types become distinguishable. However, the moist static energy of deep cloud types does not decrease significantly near cloud top where large entrainment rates are indicated by normalized cloud mass flux profiles. This is because the differences of moist static energy between clouds and their environment become smaller at higher levels, and at the same time, in-cloud moist static energy is increased by latent heat release from freezing approximately above the 4.5-km level. For some deep cloud types, moist static energy even increases at middle and upper levels.

We also notice that, for all cloud types, in-cloud moist static energy is very close to domain-averaged saturation moist static energy at the cloud base level. It suggests that in-cloud buoyancy at cloud base is approximately zero. On the other hand, in-cloud moist static energy is expected to be approximately equal to the domain-averaged saturation moist static energy at cloud top if zero buoyancy is assumed at that level. It cannot be verified in Fig. 7 because the top 1-km parts of the profiles are truncated.

c. Discussion

According to the results presented above, when saturated updrafts are stratified into different types based on their eventual heights, different types do have distinguishable thermodynamic properties and entrainment characteristics, indicating that a spectral representation of clouds using one parameter (cloud top height) is meaningful. It is, therefore, instructive to compare the simulated in-cloud moist static energy profiles (Fig. 7) with the profiles calculated using the λ-version spectral cumulus ensemble model (Fig. 8). With the base conditions of the spectral cumulus ensemble model given by the mean properties averaged over the CRM domain at the 0.25-km level, we find that, above the 1.5-km level, the in-cloud moist static energy profiles predicted by the spectral cumulus ensemble model apparently resemble those simulated by the CRM except near cloud top of middle clouds.

It should be pointed out, however, that the agreement above the 1.5-km level between the results from the spectral cumulus ensemble model and those from the CRM simulation is actually due to the offset of two assumptions in the spectral cumulus ensemble model. First of all, the base conditions in the spectral cumulus ensemble model are assumed to be the mean properties of the subcloud layer (SCL), while the thermodynamic properties of cloud roots considerably deviate from the means at their comparable levels in the CRM simulation. From Fig. 7, we see that the moist static energy of the mean sounding at the 0.25-km level is 339.1 × 103 m2 s−2. However, the moist static energy of cloud roots obviously exceeds 341.0 × 103 m2 s−2, because the moist static energy of the simulated clouds is already 341.0 × 103 m2 s−2 at the 0.5-km level. Second, for each cloud type in the spectral cumulus ensemble model, the fractional rate of entrainment is assumed to be constant in height, while all cloud types have comparably large fractional rates of entrainment at lower levels in the CRM simulation. Due to offset of these two effects, the in-cloud moist static energy profiles predicted by the spectral cumulus ensemble model apparently match those from the CRM simulation at higher levels.

Nevertheless, we cannot ignore the difference between the cloud mass fluxes inferred from the CRM simulation and those predicted by the spectral cumulus ensemble model. For example, in Fig. 6, which is the result from the CRM simulation, the normalized cloud mass flux for the cloud-element type whose top is located between 9 and 10 km is 27.5 at the 8.5-km level. On the other hand, in Fig. 8, which is the result predicted by the spectral cumulus ensemble model, the in-cloud moist static energy profile whose top is located between 9 and 10 km corresponds to a fractional rate of entrainment of 12% km−1, which yields a normalized cloud mass flux of 2.6 at the 8.5-km level. The much smaller cloud mass flux predicted by the spectral cumulus ensemble model than that from the CRM simulation is at least partially due to the fact that the spectral cumulus ensemble model neglects larger entrainment rate at lower levels.

In view of this, further improvement of a CMCP may rely on a formulation of entrainment profiles that can account for the low-level dilution of clouds. Another issue to be addressed is how to formulate the effects of horizontal inhomogeneity of the SCL properties on cumulus clouds. A better understanding of the SCL processes, including the interaction between the downdraft air and the SCL air, may provide guidance for determining base conditions from large-scale variables.

6. Conclusions

Motivated by the results of Part I, which indicates that ignoring the contribution from descendent cloud air in a CMCP is an acceptable simplification, we proceed to examine the entraining-plume model using the simulated data from a CRM. With the data from a nonprecipitating experiment, A12, we tested the entraining-plume model following a procedure outlined by Warner (1970). It is found that the entraining-plume model cannot simultaneously predict the mean liquid water profile and cloud top height simulated by the CRM. However, the mean properties of active elements of clouds, which are characterized by strong updrafts, can be formally described by an entraining-plume of a similar height.

With the Paluch diagram, we further tested the spectral cumulus ensemble model in the Arakawa–Schubert parameterization using the data from a precipitating experiment, A10. Our results show that the distributions of in-cloud data in the Paluch diagram do resemble a spectrum of clouds as described by the spectral cumulus ensemble model, which appears adequate if different types of cloud in the spectrum are interpreted as subcloud elements with different fractional rate of entrainment. The resolved internal structure of clouds can thus be viewed as a manifestation of a cloud spectrum.

To investigate whether the fractional rate of entrainment is indeed an appropriate parameter for characterizing subcloud elements, we stratified saturated updrafts (subcloud elements) in experiment A10 into different types according to their eventual heights, and infer entrainment characteristics for each type based on the calculated cloud mass flux and mean moist static energy. Our results show that different types of subcloud elements have distinguishable thermodynamic properties and entrainment characteristics, indicating that a spectral representation of clouds using one parameter (cloud top height) is meaningful. However, for each cloud type, fractional rate of entrainment is not constant in height but tends to be larger at lower levels and near cloud top.

The simulated in-cloud moist static energy profiles from experiment A10 are also compared to those predicted by the spectral cumulus ensemble model. It is found that, above the 1.5-km level, the in-cloud moist static energy profiles predicted by the spectral cumulus ensemble model apparently resemble those simulated in experiment A10 except near cloud top of middle clouds. It should be pointed out, however, that the agreement above the 1.5-km level between the results of the spectral cumulus ensemble model and those from experiment A10 is actually due to the offset of two effects. First, the base conditions in the spectral cumulus ensemble model are assumed to be the mean properties of the subcloud layer, while the thermodynamic properties of cloud roots considerably deviate from the means at their comparable levels. Second, for each cloud type in the spectral cumulus ensemble model, fractional rate of entrainment is assumed to be constant in height, while all cloud types have comparably large fractional rates of entrainment at lower levels. Further improvement of a CMCP may rely on a formulation of entrainment profiles that can account for the low-level dilution of clouds together with a formulation of the cloud-root properties.

Acknowledgments

The authors would like to thank Professor Steven Krueger and Dr. Kuan-Man Xu for their help in using the cloud ensemble model. Thanks are also extended to Professor Michio Yanai and Ms. Christine Holten for their valuable comments. We are especially grateful to Professors Steven Krueger and Nilton Rennó, Dr. Kuan-Man Xu, and two anonymous reviewers for their thorough reviews, which have led to significant improvement of the original manuscript. This material is based on work supported jointly by NSF Grant ATM-9224863, NASA Grants NAG 5-789 and 5-2591, WESTGEC Grant 91-026, and DOE CHAMMP Grant DE-FG03-91ER61214. The computations were performed at the Department of Atmospheric Sciences at UCLA and the NCAR Scientific Computing Division. NCAR is sponsored by the NSF.

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APPENDIX

Including the Freezing Effects in the Arakawa–Schubert Spectral Cumulus Ensemble Model

Moist static energy can increase significantly above the melting level due to latent heat release from ice–liquid phase change. To consider this effect in the spectral cumulus ensemble model in the Arakawa–Schubert parameterization and its modified version, some assumptions need to be made. Following Lord et al. (1984), we assume that all supersaturated water vapor is condensed and partitioned between cloud water and cloud ice as a linear function of temperature. Furthermore, following Lord (1978), the ice production from supercooled water per unit height, I (unit: m−1), is assumed to be a linearly increasing function of the cloudy air temperature Tc and the amount of liquid water W as described below:
i1520-0469-54-8-1044-ea-1
where TF ≡ −40°C, TCR ≡ −0°C, and a0 is a length scale to be specified. We use a value of 100 m for a0 in our calculations.

Fig. 1.
Fig. 1.

Predicted profiles of W/Wa (the ratio of liquid water content to its adiabatic value) for plumes of different initial radii (dashed lines). The number associated with each dashed line shows the initial radius (unit: m). The thick solid line represents the mean W/Wa profile of the clouds simulated by the CRM.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 2.
Fig. 2.

(a) Same as Fig. 1 except that the plume-base level is at the 0.6-km level and the base conditions are given by the air properties averaged over the cloudy areas of the CRM domain (instead of the entire domain) at that level. (b) Same as (a) except that plume-base level is at the 0.75-km level and the thick solid line represents the mean values of W/Wa for in-cloud data whose vertical velocity is larger than 1.5 m s−1.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 3.
Fig. 3.

Paluch diagram with data points from different levels of the entire model domain during the time period from 0650 MT to 0840 MT. The dashed lines represent the predicted (h, Q) lines by the λ-version spectral cumulus ensemble model, with the associated fractional rates of entrainment (unit: % km−1) indicated at the bottoms of the lines. The solid line is the mean environmental sounding, with error bars showing the standard deviations of the sounding. Different symbols are used to represent in-cloud data from different levels (see the explanation in the upper-left corner of the panel). The same levels of the predicted (h, Q) lines are connected by dotted lines and labeled with their corresponding heights.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 4.
Fig. 4.

Same as Fig. 3 except that the dashed lines represent different cloud types predicted by the μ-version spectral cumulus ensemble model. The numbers at the bottoms of dashed lines indicate their associated entrainment rates (unit: % km−1).

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 5.
Fig. 5.

A scatter diagram of saturated updrafts whose eventual heights are between (a) 3 and 4 km, (b) 6 and 7 km, (c) 9 and 10 km, and (d) 12 and 13 km for experiment A10. The abscissa is moist static energy (unit: 103 m2 s−2), and the ordinate is height (unit: km).

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 6.
Fig. 6.

The profiles of normalized cloud mass flux for different cloud types with their tops at the 3-km, 4-km, . . . and 13-km level, respectively. Each cloud type is represented by a solid line with an arrowhead, whose level is 0.5 km below its corresponding cloud top. These profiles are based on all the subcloud elements sampled from the entire CRM domain during the period from 0400 MT to 2800 MT in experiment A10.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 7.
Fig. 7.

The profiles of in-cloud moist static energy (unit: 103 m2 s−2) for different cloud types with their tops at the 3-km, 4-km, . . . and 13-km level, respectively. Each cloud type is represented by a solid line with an arrowhead, whose level is 0.5 km below its corresponding cloud top. These profiles are based on all the subcloud elements sampled from the entire CRM domain during the period from 0400 MT to 2800 MT in experiment A10. The domain and time averages of moist static energy (solid lines) and saturation moist static energy (dashed lines) are also shown with error bars indicating their standard deviations.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

Fig. 8.
Fig. 8.

The vertical profiles of in-cloud moist static energy for different cloud types predicted by the λ-version spectral cumulus ensemble model of the Arakawa–Schubert cumulus parameterization. The corresponding fractional rate of entrainment for each cloud type is indicated at the top of the profile (unit: % km−1). The domain and time averages of moist static energy (solid lines) and saturation moist static energy (dashed lines) are also shown with error bars indicating their standard deviations.

Citation: Journal of the Atmospheric Sciences 54, 8; 10.1175/1520-0469(1997)054<1044:TMEPOS>2.0.CO;2

1

The criterion for cloudy area is the same as that used in Part I of this paper.

2

In total, 12 clouds whose ultimate tops are between 4 and 5 km are sampled during this period but only 3 of them are in their mature stages (reaching their ultimate tops) at the time they are sampled. Therefore, we have less data near the ultimate top. In order to estimate (W/Wa)CRM using at least 10 data points at each level, we have to truncate the profile above the 3.5-km level and below the 0.5-km level. While the 3.5-km level is considerably lower than the ultimate top, the 0.5-km level is approximately the same level as the observed cloud base.

3

Notice that the (W/Wa)CRM profile is now truncated below the 0.75-km level to ensure sufficient data at each level.

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  • Fig. 1.

    Predicted profiles of W/Wa (the ratio of liquid water content to its adiabatic value) for plumes of different initial radii (dashed lines). The number associated with each dashed line shows the initial radius (unit: m). The thick solid line represents the mean W/Wa profile of the clouds simulated by the CRM.

  • Fig. 2.

    (a) Same as Fig. 1 except that the plume-base level is at the 0.6-km level and the base conditions are given by the air properties averaged over the cloudy areas of the CRM domain (instead of the entire domain) at that level. (b) Same as (a) except that plume-base level is at the 0.75-km level and the thick solid line represents the mean values of W/Wa for in-cloud data whose vertical velocity is larger than 1.5 m s−1.

  • Fig. 3.

    Paluch diagram with data points from different levels of the entire model domain during the time period from 0650 MT to 0840 MT. The dashed lines represent the predicted (h, Q) lines by the λ-version spectral cumulus ensemble model, with the associated fractional rates of entrainment (unit: % km−1) indicated at the bottoms of the lines. The solid line is the mean environmental sounding, with error bars showing the standard deviations of the sounding. Different symbols are used to represent in-cloud data from different levels (see the explanation in the upper-left corner of the panel). The same levels of the predicted (h, Q) lines are connected by dotted lines and labeled with their corresponding heights.

  • Fig. 4.

    Same as Fig. 3 except that the dashed lines represent different cloud types predicted by the μ-version spectral cumulus ensemble model. The numbers at the bottoms of dashed lines indicate their associated entrainment rates (unit: % km−1).

  • Fig. 5.

    A scatter diagram of saturated updrafts whose eventual heights are between (a) 3 and 4 km, (b) 6 and 7 km, (c) 9 and 10 km, and (d) 12 and 13 km for experiment A10. The abscissa is moist static energy (unit: 103 m2 s−2), and the ordinate is height (unit: km).

  • Fig. 6.

    The profiles of normalized cloud mass flux for different cloud types with their tops at the 3-km, 4-km, . . . and 13-km level, respectively. Each cloud type is represented by a solid line with an arrowhead, whose level is 0.5 km below its corresponding cloud top. These profiles are based on all the subcloud elements sampled from the entire CRM domain during the period from 0400 MT to 2800 MT in experiment A10.

  • Fig. 7.

    The profiles of in-cloud moist static energy (unit: 103 m2 s−2) for different cloud types with their tops at the 3-km, 4-km, . . . and 13-km level, respectively. Each cloud type is represented by a solid line with an arrowhead, whose level is 0.5 km below its corresponding cloud top. These profiles are based on all the subcloud elements sampled from the entire CRM domain during the period from 0400 MT to 2800 MT in experiment A10. The domain and time averages of moist static energy (solid lines) and saturation moist static energy (dashed lines) are also shown with error bars indicating their standard deviations.

  • Fig. 8.

    The vertical profiles of in-cloud moist static energy for different cloud types predicted by the λ-version spectral cumulus ensemble model of the Arakawa–Schubert cumulus parameterization. The corresponding fractional rate of entrainment for each cloud type is indicated at the top of the profile (unit: % km−1). The domain and time averages of moist static energy (solid lines) and saturation moist static energy (dashed lines) are also shown with error bars indicating their standard deviations.

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