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A Conceptual Model for Entrainment in Cumulus Clouds Article in Journal of the Atmospheric Sciences · July 2005 DOI: 10.1175/JAS3499.1
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A Conceptual Model for Entrainment in Cumulus Clouds AMIT AGRAWAL Department of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai, India (Manuscript received 28 November 2003, in final form 14 January 2005) ABSTRACT Cumulus clouds are generally modeled as a plume, and the model works well up until the cloud base is encountered, beyond which the continuously entraining model does not appear appropriate. Although it has been known for a long time that cumulus clouds have very little lateral entrainment, the reason for it is not evident. An expression for the mean streamwise velocity profile as a Gaussian multiplied by a fourth-order polynomial factor is hereby proposed such that the mass and momentum fluxes are decoupled. This model suggests that cumulus clouds differ from other shear flows in that the former need not interact with the ambient for some downstream distance. The proposed model replicates a number of characteristics of cumulus clouds like a conserved mass flux with varying momentum flux, and may therefore be employed to describe them, in a time-averaged sense, beyond the cloud base.
1. Introduction Free-shear flows (e.g., jets and plumes) are known to entrain ambient air, first through engulfment by large eddies, then with a subsequent increase in surface area by straining, and finally by mixing at the smallest scales (Baker et al. 1984). The added mass leads to an increase of the characteristic length scale with the downstream coordinate. Following Morton et al. (1956), the mean entrainment velocity can be related to some characteristic velocity in the flow, and the entrainment coefficient remains constant with downstream distance for a simple jet and plume. However, for flows where selfsimilarity has not been achieved it has been suggested that entrainment should not be related to the mean velocity. For example, Morton (1968) suggested that entrainment should instead be related to the Reynolds stress, while Telford (1975) has suggested that entrainment should be related to the local level of turbulence. However, the model of Morton et al. (1956) has been found to be most widely applicable [see, e.g., its application to geophysical flows in Turner (1986)]. It is well known that lateral entrainment is almost nonexistent in cumulus clouds, suggesting that the mass
Corresponding author address: Dr. Amit Agrawal, Dept. of Mechanical Engineering, Indian Institute of Technology, Powai, Mumbai 400076, India. E-mail:
[email protected]
© 2005 American Meteorological Society
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flux should be approximately constant. Early attempts by Morton (1957), Squires and Turner (1962), and others to describe entrainment in cumulus clouds using a laterally entraining model, where air at a certain level comes solely from that level or below, failed to make realistic predictions in a cumulus cloud. Warner (1970) shows that the standard lateral entrainment model for plumes cannot simultaneously predict the height of ascent and air–water ratio in cumulus clouds. If the air– water ratio is matched, then the ascent height is underpredicted, while matching the ascent height of the cloud results in too little dilution. Heymsfield et al. (1978) found undiluted cloud base air at all levels within cumulus clouds. Paluch (1979) shows that the properties of air within cumulus clouds are attributable primarily to the mixing of air from below the cloud base with environmental air from near the cloud top, again attesting to the lack of lateral mixing. See Emanuel (1994) for an excellent review of entrainment in cumulus clouds. It is interesting to note that apart from cumulus clouds some other geophysical systems (e.g., magma) may exhibit reduced entrainment as well (Bergantz and Breidenthal 2001). It has therefore been well realized that continuous entrainment of the type found in free shear flows cannot occur in cumulus clouds. At best, lateral entrainment will be intermittent in nature. Telford (1975) proposed that the air rises from the cloud base, mixes with the environment, and then seeks a new level of neutral
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buoyancy. Raymond and Blyth (1986) refined this model and introduced the buoyancy-sorting hypothesis. The buoyancy-sorting hypothesis suggests that all air coming from above the observation level should have negative or neutral buoyancy at the observation level, while air arriving from below should have positive or neutral buoyancy. The lateral mixing model of free shear flows has also been substituted by a vertical mixing model whereby mixing at the cloud top is primarily responsible for the dilution of a cumulus cloud. Mixing at the cloud top results in negatively buoyant parcels of air that descend downward. Emanuel (1981) through a self-similarity assumption based model showed that the dry air engulfed near the top of the cumulus cloud could descend a considerable distance through the cloud by entraining and evaporating cloud water. Breidenthal (2003) and Govindrajan (2002) proposed a vortex-based approach to explain the phenomena of reduced entrainment in free shear flows under some specific conditions like off-source heating. However, none of the above approaches have satisfactorily addressed the issue of cumulus cloud entrainment. Although the mechanisms responsible for reduction of lateral entrainment in cumulus clouds are not immediately clear, a simple and realistic model is nonetheless needed to describe cumulus clouds beyond the cloud base to predict, for example, the air–water ratio and height of ascent. A time-averaged model for entrainment in cumulus clouds is hereby proposed. In the model, the mass flux can remain conserved for some downstream distance, while the momentum flux can vary. This is in contrast with the conventional wisdom that the mass and momentum fluxes are coupled (see, e.g., Turner 1986), and the mass flux should vary in free shear flows while the momentum flux may or may not vary.
2. Model for cumulus cloud The rising updraft of air is somewhat akin to a plume with continuous lateral entrainment before the cloud base is encountered (Fig. 1). Mixing of ambient and cloud air may still occur just beyond the cloud base. In the author’s experience with an off-source volumetrically heated jet, which was studied in an effort to understand entrainment in cumulus clouds (Agrawal et al. 2004b; Agrawal and Prasad 2004), mixing at the beginning of the heat injection zone is actually much higher for the heated case as compared to its unheated counterpart. Therefore, at least some entrainment should occur in cumulus clouds for a small downstream extent beyond the cloud base. This mixing of dry ambient air
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FIG. 1. Schematic of the ascending cloud and the various processes.
with moist cloud air may result in reevaporation of some water droplets near the cloud edge, producing regions of negative buoyancy there (Grabowski 1993). This negative buoyancy will create downdrafts near the edges of the cloud. A change in the mean velocity profile of the rising plume from a Gaussian one to one with downdrafts [described by Eq. (1) below] can therefore be expected. The above argument also implies that most of the ambient air is mixed near the edges and is eventually carried by the downdrafts. To address the simultaneous presence of both updrafts and downdrafts, a streamwise velocity profile for a cumulus cloud as a Gaussian multiplied by a fourthorder polynomial factor is considered here. The proposed expression is obtained by a suitable modification of the velocity profile for volumetrically heated jets described in Agrawal et al. (2004b):1
1 The velocity profile U/Uc ⫽ (1 ⫹ B12) exp(⫺2) was obtained empirically by Agrawal et al. (2004b) for a volumetrically heated jet to describe the flattened Gaussian streamwise velocity profile. This profile correctly predicted the cross-stream variation of the radial velocity and temperature. It was however realized that the expression is not suitable for cumulus clouds because it predicts a larger than normal mass flux with heating, which was subsequently corroborated by the experiments of Agrawal and Prasad (2004) and numerical simulations of Agrawal et al. (2004a). However, the model becomes appropriate for cumulus clouds upon addition of a fourth-order term.
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FIG. 2. Proposed velocity profiles at three successive downstream locations in a cumulus cloud. Equation (3) represents the lowermost location.
Ucloud ⫽ 共B1 ⫹ B22 ⫺ B3 4兲 exp共⫺2兲, Uc
共1兲
where Ucloud is the mean streamwise velocity, Uc is the mean centerline velocity, and ⫽ r/b (where r is the radial or cross-stream coordinate, and b is the width or the characteristic length). Here, B1 ⬎ 0 while B2 and B3 can take either positive or negative values. Because of the interest in the shape of the velocity profiles, only the radial variation of streamwise velocity will be explored here (and not the axial variation of Uc). Note that setting B3 ⫽ 0 reduces the polynomial factor to second order and recovers the volumetrically heated jet profile. Further setting B2 ⫽ 0 recovers a normal (Gaussian) jet profile. As will soon be evident, this fourth-order polynomial model has very different properties from both Gaussian and second-order polynomial models, and can represent the time-averaged velocity profile of a cumulus cloud. The mass flux, cloud is obtained by integrating Eq. (1) as
cloud b2Uc
⫽ B1 ⫹ B2 ⫺ 2B3.
共2兲
Let us consider velocity profiles represented by Eq. (1) at three successive downstream locations in a cumulus cloud. The coefficients in Eq. (1) for the three profiles have been selected such that the normalized mass flux [in Eq. (2)] remains constant at 1.3 units:
U1 ⫽ 共1.9 ⫹ 0.62 ⫺ 0.6 4兲 exp共⫺2兲, Uc
共3兲
U2 ⫽ 共2 ⫹ 0.12 ⫺ 0.4 4兲 exp共⫺2兲, Uc
共4兲
U3 ⫽ 共2.1 ⫺ 0.42 ⫺ 0.2 4兲 exp共⫺2兲. Uc
共5兲
Figure 2 reveals that all three velocity profiles exhibit both positive and negative velocities. On comparing Fig. 2 with the actual profiles of cumulus clouds (e.g., Warner 1977) one can immediately associate positive velocities with updrafts and negative velocities with downdrafts that exists in cumulus clouds. It can be easily verified that the momentum flux is different for the three profiles. An increase in momentum flux can occur, for example, due to buoyancy addition by the release of latent heat upon condensation (Bhat and Narasimha 1996). It is also apparent from Fig. 2 that the intensity of the downdraft decreases as one moves progressively downstream. In fact, a separate examination of the upward and the downward moving mass fluxes indicates that both of these reduce from Eq. (3) through Eq. (5). This implies that the downward mass flux is absorbed into the upward flux in order to maintain a constant net mass flux. In other words, our model suggests a redistribution of mass from downdrafts to updrafts between successive downstream locations resulting in a constant
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mass flux within the cumulus cloud. The system described by Eq. (1) differs from other shear flows in that the former is stable in the sense that it need not interact with the ambient, and the net mass flux within this system is conserved.
3. Discussion Let us now compare the properties of Eq. (1) with field measurements of cumulus clouds. Warner (1977) found that the velocity obtained by averaging the vertical velocities across the width of the cloud increases with altitude above the cloud base for the initial 60% of the cloud height, in general agreement with our model. Further, the relative magnitudes of the updrafts and downdrafts represented by Eqs. (3)–(5) are comparable to Warner’s measurements. Thus the model can represent cumulus clouds by tuning B1, B2, and B3 such that the relative magnitudes of up- and downdrafts matches between observation and model. Carpenter et al. (1998) used numerical simulations of cumulus congestus clouds to separately compute the downward and upward mass fluxes and found that both of these quantities decrease with the downstream coordinate, while their difference is conserved. As explained above, our model reproduces this feature. One would expect most of the mixing between the up- and downdrafts to occur in the interfacial region. The field observations of Paluch (1979) do indicate maximum mixing in this region. Further, most of the ambient air moves with the downdrafts, and therefore the fluid close to the centerline may remain almost undiluted. This observation is again confirmed by the field measurements of Heymsfield et al. (1978) and Paluch (1979). Extrapolating Eqs. (3)–(5) to further downstream locations suggests that velocities in the downdraft region will become weaker with the downstream coordinate to the point that they vanish altogether. Further redistribution of mass within the cloud is therefore not possible beyond this downstream distance, and the system will become unstable with respect to conserving mass within the body of the cloud. At this point, lateral entrainment of dry ambient air will recur, resulting in evaporation of water droplets near the edges and the reestablishment of downdrafts. However, due to the complex nature of turbulent flows, on an instantaneous basis one might encounter lateral entrainment (or a lack thereof) intermittently in both space and time. This picture of intermittently entraining cumulus clouds is consistent with the buoyancy-sorting hypothesis of Telford (1975) and Raymond and Blyth (1986), the computations of Carpenter et al. (1998), and the field observations of Taylor
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and Baker (1991). The buoyancy-sorting hypothesis states that, as the air within the ascending cloud mixes with the ambient air at a particular level, the mixed air will detach from the undiluted core and attain equilibrium at that level. Emanuel (1994) shows that the buoyancy-sorting hypothesis can lead to intermittent entrainment in agreement with our model. Finally, four aspects about the model should be noted. First, although regions of downdrafts are assumed to occur near the edges of the cloud, this assumption can be relaxed. The model assumes redistribution of mass between the up- and downdraft regions, and will therefore work for downdrafts distributed anywhere across the entire cloud body. Second, any instantaneous realization of a cloud will be different from the picture suggested here, which only considers a timeaveraged velocity profile. Nevertheless the author believes that the model can adequately describe a cumulus cloud in a time-averaged sense. Third, it is possible that Eq. (1) proposed here is not unique and other mathematical forms could be found that incorporate the basic properties of clouds outlined above. Fourth, the proposed model needs experimental or numerical verfication. This can possibly be achieved in a laboratory by using the approach of Turner (1966) and Johari (1992), which successfully creates downdrafts.
4. Conclusions The primary contribution of this paper is to describe a conceptual system where the mass and momentum fluxes are decoupled from each other, that is, one can change independently of the other. A specific example of a constant mass flux with a varying momentum flux has been provided. Such decoupled systems differ substantially from conventional free shear flows and are important because they occur in nature, notably cumulus clouds. The model is characterized by regions of positive and negative velocities denoting the updrafts and downdrafts encountered in a cumulus cloud. The basic premise of the model is that redistribution of mass between updrafts and downdrafts conserves the overall mass flux within the cloud. The underlying principle of the model conforms with the buoyancy-sorting hypothesis. The system need not interact with its surroundings for some downstream length, that is, as long as the downdrafts are present. Beyond this point, at which the downdrafts more or less vanish, the cloud will entrain laterally and the mass flux will increase. However, the lateral entrainment will recreate the downdraft regions, and the system may revert to one that does not interact with its surroundings; that is, the existence of downdrafts are critical for conserving the mass flux within the cumulus cloud.
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The model is shown to predict a number of observed characteristics of cumulus clouds. It is an advance over existing models because it explains how cumulus clouds experience little overall lateral entrainment and why their interaction with the ambient can cease for some downstream distance. Further, the model clarifies the role of updrafts and downdrafts, which appears crucial in cumulus clouds. This simple streamwise velocity profile expression can therefore be employed by modelers to describe an ascending cumulus cloud beyond its cloud base. In conjugation with streamwise variation of the mean centerline velocity, the model can be used to estimate length scale of downdrafts and height of cloud among other quantities of interest. Acknowledgments. I am grateful to Profs. A. K. Prasad and H. Wang of the University of Delaware, for useful discussions on the problem. REFERENCES Agrawal, A., and A. K. Prasad, 2004: Evolution of a turbulent jet subjected to volumetric heating. J. Fluid Mech., 511, 95–123. ——, B. J. Boersma, and A. K. Prasad, 2004a: Direct numerical simulation of a turbulent axisymmetric jet with buoyancy induced acceleration. Flow Turbul. Combust., 73, 277–305. ——, K. R. Sreenivas, and A. K. Prasad, 2004b: Velocity and temperature measurements in an axisymmetric jet with cloudlike off-source heating. Int. J. Heat Mass Transfer, 47, 1433– 1444. Baker, M. B., R. E. Breidenthal, T. W. Choularton, and J. Latham, 1984: The effects of turbulent mixing in clouds. J. Atmos. Sci., 41, 299–304. Bergantz, G. W., and R. E. Breidenthal, 2001: Non-stationary entrainment and tunneling eruptions: A dynamic template for eruption processes and magma mixing. Geophys. Res. Lett., 28, 3075–3078. Bhat, G. S., and R. Narasimha, 1996: A volumetrically heated jet: Large-eddy structure and entrainment characteristics. J. Fluid Mech., 325, 303–330. Breidenthal, R. E., 2003: The vortex as a clock. Proc. Symp. on Advances in Fluid Mechanics, Bangalore, India, Jawaharlal Nehru Centre for Advanced Scientific Research, 246–254.
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Carpenter, R. L., Jr., K. K. Droegemeier, and A. M. Blyth, 1998: Entrainment and detrainment in numerically simulated cumulus congestus clouds. Part II: Cloud budgets. J. Atmos. Sci., 55, 3433–3439. Emanuel, K. A., 1981: A similarity theory for unsaturated downdrafts within clouds. J. Atmos. Sci., 38, 1541–1557. ——, 1994: Atmospheric Convection. Oxford University Press, 580 pp. Govindrajan, R., 2002: Universal behaviour of entrainment due to coherent structures in turbulent shear flow. Phys. Rev. Lett., 88, 134 503–134 506. Grabowski, W. W., 1993: Cumulus entrainment, fine-scale mixing, and buoyancy reversal. Quart. J. Roy. Meteor. Soc., 119, 935– 956. Heymsfield, A. J., P. N. Johnson, and J. E. Dye, 1978: Observations of moist adiabatic ascent in northeast Colorado cumulus congestus clouds. J. Atmos. Sci., 35, 1689–1703. Johari, H., 1992: Mixing in thermals with and without buoyancy reversal. J. Atmos. Sci., 49, 1412–1426. Morton, B. R., 1957: Buoyant plumes in a moist atmosphere. J. Fluid Mech., 2, 127–144. ——, 1968: Turbulence structures in cumulus models. Proc. Int. Conf. on Cloud Physics, Toronto, ON, Canada, Int. Association of Meteorology and Atmospheric Sciences. ——, G. I. Taylor, and J. S. Turner, 1956: Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. London, 234, 1–23. Paluch, I. R., 1979: The entrainment mechanism in Colorado cumuli. J. Atmos. Sci., 36, 2467–2478. Raymond, D. J., and A. M. Blyth, 1986: A stochastic model for nonprecipitating cumulus clouds. J. Atmos. Sci., 43, 2708– 2718. Squires, P., and J. S. Turner, 1962: An entraining model for cumulonimbus updrafts. Tellus, 14, 422–434. Taylor, G. R., and M. B. Baker, 1991: Entrainment and detrainment in cumulus clouds. J. Atmos. Sci., 48, 112–121. Telford, J. W., 1975: Turbulence entrainment, and mixing in cloud dynamics. Pure Appl. Geophys., 113, 1067–1084. Turner, J. S., 1966: Jets and plumes with negative and reversing buoyancy. J. Fluid Mech., 26, 779–792. ——, 1986: Turbulent entrainment: The development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech., 173, 431–471. Warner, J., 1970: On steady-state one-dimensional models of cumulus convections. J. Atmos. Sci., 27, 1035–1040. ——, 1977: Time variation of updraft and water content in small cumulus clouds. J. Atmos. Sci., 34, 1306–1312.
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