1. Introduction
Boundary layer clouds have radiative properties that influence the earth’s energy balance. Also, they exchange heat, momentum, and moisture between the boundary layer and the free troposphere by means of turbulent transport. For these reasons, an accurate representation of cloud-topped boundary layers is important for general circulation models. Therefore, it is necessary to develop parameterizations for these types of boundary layers, which are for computational reasons, relatively simple but, on the other hand, sufficiently accurate. There are a variety of cloud parameterization schemes that range from the very simple to rather complex. For instance, to prognose the evolution of a well-mixed stratocumulus-topped boundary layer, the mixed-layer model is the simplest one to use. In such a model, the vertical fluxes of temperature and moisture are rather simple functions of the surface fluxes, entrainment velocity, and thermodynamic jumps across the inversion, cloud-base height, radiation, and precipitation (Nicholls 1984; Wang 1993; Bretherton and Wyant 1997). However, when the stratocumulus deck is broken and inhomogeneous, the assumption that the boundary layer is well mixed is no longer valid, and the mixed-layer scheme therefore cannot be used. For the same reason cumulus cloud fields, which develop in conditionally unstable layers, cannot be represented by such a simple mixed-layer model, either.
Unless the height-dependent coefficient Kψ is prescribed, it must be predicted and is dependent on the flow variables. For instance, the turbulent-mixing coefficient can be parameterized as a function of a velocity scale and a length scale. In a one-and-a-half-order closure model, a prognostic equation for the turbulent kinetic energy (TKE) equation is included (Duynkerke and Driedonks 1987) and the square root of the TKE is used as a typical velocity scale. Because in a convective boundary layer (CBL) the flux can flow counter to the vertical mean gradient, a correction term can be included (Holtslag and Moeng 1991) that is referred to as nonlocal closure.
In many general circulation models different parameterization schemes are in use, as is illustrated in Fig. 1. Typically, the large-scale thermodynamic tendencies due to cumulus convection are computed from a mass-flux scheme, whereas for dry convection and stratocumulus, a K-diffusion parameterization is applied. This can lead to model inconsistencies when stratocumulus is gradually replaced by cumulus, a type of boundary layer that is frequently observed in the subtropics (Bretherton et al. 1995; Martin et al. 1995; de Roode and Duynkerke 1996; de Roode and Duynkerke 1997; Wang and Lenschow 1995). To overcome the problem of convection-dependent schemes, Randall et al. (1992) formulated a “second-order bulk boundary layer” model as a compromise between a higher-order closure model and a mass-flux model. They developed a theoretical framework for a single mass-flux scheme that could deal with the simulation of all types of convective atmospheric boundary layers, ranging from the convective boundary layer to a cumulus-topped boundary layer.
In this paper, a prognostic equation for the variance using the mass-flux approach will be derived by mathematically manipulating the updraft and downdraft mass-flux equations for conserved variables (Arakawa and Schubert 1974; Tiedtke 1989; Siebesma and Cuijpers 1995). The resulting variance equation is nearly identical to the one presented by Randall et al. (1992). However, they obtained the variance equation by a direct substitution of relationships between Reynolds-averaged higher-order moments and mass-flux variables in the Reynolds-averaged variance equation. The primary difference between the variance equation of Randall et al. (1992) and the one presented in this paper lies in a slightly different formulation of the dissipation term. Here it will be shown that the parameterization of the dissipation of variance is a simple, yet unique, function of the lateral entrainment and detrainment rates. In this process, the lateral mixing rates represent typical inverse dissipative timescales. As a physical explanation, lateral mixing tends to decrease the difference between the updraft and downdraft properties. Eventually, we will discuss analogies in the closure problems encountered in mass-flux and Reynolds-averaged equations. Some illustrative examples are presented from LES results and aircraft observations made in a cumulus field.
2. Basic equations
a. Reynolds decomposition
b. Mass-flux decomposition
3. Lateral entrainment and detrainment
a. Comparison of Reynolds-averaged and mass-flux variance equations
It should be stressed that the mass-flux variance dissipation term does not directly arise from the molecular diffusivity term in (2.2), but rather is a result of the conditionally sampled horizontal flux divergence term Lex in (2.7) and its parameterization by (2.8). The analogies between the mass-flux and Reynolds-averaged equations can be explained by the similar way they are derived. To obtain the prognostic Reynolds-averaged variance equation, the prognostic equation for ψ′ is multiplied by ψ′, whereas in deriving (2.12), the prognostic equation (A.8) for (ψu − ψd) is multiplied by (ψu − ψd). In the Reynolds-averaged prognostic variance equation the dissipation term acts to decrease the variance, while the lateral entrainment–detrainment term in (2.12) tends to decrease the square of the difference between the updrafts and downdrafts, which is also the variance in the mass-flux representation.
A simulation of the CBL (Table 1) was performed with the Institute for Marine and Atmospheric Utrecht/Royal Meteorological Institute of the Netherlands (IMAU/KNMI) LES model (Cuijpers and Duynkerke 1993) using a central difference scheme. To test the similarities between the mass-flux and Reynolds-averaged variance equations the LES results were used to compute the variance budgets for the potential temperature. For the Reynolds-averaged variance budget we only considered the contributions due to the resolved motions because the subgrid fluxes are very small with an exception for the lowest model levels. Conditional sampling was performed by applying the updraft–downdraft decomposition. The lateral entrainment and detrainment rates were calculated following the method described in Siebesma and Cuijpers (1995). Basically, the net lateral exchange is determined as a residual from the budget equations, and we are using (2.8)–(2.9) to compute the lateral entrainment and detrainment rate. In the boundary layer the subsidence velocity is typically one order of magnitude smaller than the convective mass flux, and therefore, for simplicity, we did not prescribe any large-scale advection. As is clear from Fig. 2, the production term, which is given by the product of the vertical flux and the mean gradient, becomes negative in the bulk of the boundary layer. This means that the flux and the gradient have the same sign and therefore the downgradient formulation (1.1) cannot give a correct flux. Since the dissipation term is negative by definition, variance in the bulk of the boundary layer needs to be produced by the vertical transport term. Indeed, in these regions the transport terms in (2.4) and (2.12) both produce variance. Generally, both variance budgets exhibit the same features; a maximum dissipation and production near the surface and the top of the boundary layer, and production by the transport terms in the bulk. Because the triple moment (
b. Expressions for the fractional entrainment and detrainment coefficients
1) Cumulus convection (σ → 0)
2) Convection with zero vertical velocity skewness (σ = 0.5)
3) Surface layer scaling
According to a parameterization for the updraft fraction in a CBL proposed by Young (1988a), the updraft fraction varies between σ = 0.5 (z = 0) and σ = 0.48 at z = 0.1zi, with zi the boundary layer height. Therefore it is reasonable to assume that the second term is much smaller than the first term on the rhs of (3.9). To reduce (3.9) further we will neglect, for simplicity, updraft fraction variations with height by using the approximation σ = 0.5. By (2.12) this means that we effectively neglect the effect of vertical transport of variance.
c. Examples of lateral entrainment and detrainment
As another illustration, in-cloud and environmental values of the total water content and liquid water potential temperature in a cumulus cloud field were conditionally sampled. The observations were made from the National Center for Atmospheric Research (NCAR) C-130 aircraft during the Small Cumulus Microphysics Study near Florida (Gerber 1999). During flight RF12, which took place from 1300 to 1615 LT (local time) on 5 August, the observed cumuli had a base at 400 m and cloud top at 3000 m. The aircraft observations were made between cloud base and about 2000 m. The estimated cloud fraction was about 15%. We selected clouds that had a horizontal extent larger than 500 m. From the results shown in Fig. 4 we calculated the total water content gradient and the difference between the in-cloud value and the environmental value. After substituting these numbers into (3.7) we find ε ≈ 1.5 × 10−3 m−1. The same approach for the liquid water potential temperature leads to a similar result, namely, ε ≈ 1.3 × 10−3 m−1. This result agrees well with the observations of Raga et al. (1990). The horizontally averaged dissipation εqt was calculated from the Fourier spectra of the total water content in and outside the cloud. The fractional detrainment δ needed in the mass-flux variance dissipation was calculated from the vertical mass-flux gradient and the continuity equation (2.9), and (3.7). The results from legs at approximately the same height were averaged and are shown in Fig. 5. The dissipation obtained with the two different methods shows that the dissipation in the cumulus field tends to increase with height. According to the mass-flux expression of the variance, this is due to a larger difference with increasing height between the cloud and environmental value of the total water content.
4. Flux parameterizations
a. Convection with zero vertical velocity skewness (σ = 0.5)
b. Cumulus convection (σ → 0)
c. Flux expressions for arbitrary σ: Flux correction terms
The numerator in the two terms on the rhs of (4.14) is a function of dynamical parameters only and has values that typically lie in the range between 1 and 1.5, with minima near the bottom and top of the boundary layer and a maximum at about 0.6z/zi. Thus, the numerator effectively tends to reduce the eddy mixing coefficient and the multiplication factor of the flux gradient.
5. Summary and discussion
In this paper we have derived a prognostic variance equation in the mass-flux approach for an arbitrary conserved variable (Arakawa and Schubert 1974; Siebesma and Cuijpers 1995; Tiedtke 1989) and compared it with the variance equation in the Reynolds-averaged approach. Mass-flux equations include a parameterization for the net lateral exchange of mass per unit of time between the updrafts and downdrafts in terms of lateral entrainment (Es) and detrainment (Ds) rates. The major findings of this research are as follows.
The prognostic variance equation in the Reynolds-averaged and the mass-flux approach do have striking similarities; they both contain a gradient-production, a transport, and a dissipative term. In the mass-flux approach and for a top-hat distribution, the prognostic variance equation derived in this paper is nearly identical to the one presented by Randall et al. (1992).
The sum of the lateral entrainment and detrainment rate is equivalent to the inverse of the characteristic timescale applied in the Reynolds-averaged closure of the molecular dissipation (André et al. 1978; Canuto et al. 1994; Mellor and Yamada 1982).
In the lower half of the convective boundary layer the fractional entrainment coefficient ε follows a power law z−α, with α a number close to 1.
For a skewed vertical velocity distribution, σ ≠ 0.5, the mass-flux equations bear a solution that accounts for countergradient flux transport, similar to the formulation presented by Wyngaard and Weil (1991).
In the mass-flux approach, a variance destruction term appears in the variance equation that arises from the lateral entrainment and detrainment terms. Except for this term, the prognostic mass-flux variance equation derived in this paper resembles the one presented by Randall et al. (1992). The analogies between the mass-flux and Reynolds-averaged equations can be explained by the similarities in their derivation. To obtain the Reynolds-averaged variance equation, the prognostic equation for ψ′ is multiplied by ψ′, whereas in deriving the mass-flux variance equation, Eq. (A.8) for (ψu − ψd) is multiplied by (ψu − ψd). In the Reynolds-averaged prognostic variance equation the dissipation term acts to decrease the variance, while the lateral entrainment/detrainment term in (2.12) tends to decrease (the square of) the difference between the updraft and downdraft properties, which is also the variance in the mass-flux representation.
Several solutions for the vertical turbulent flux of a scalar have been presented. In the updraft–downdraft decomposition, σ is closely related to the vertical velocity skewness and is an important parameter for flux parameterizations. As is summarized in Table 2, the vertical flux of a scalar is downgradient when the updraft fraction equals σ = 0.5, which is equivalent to a vertical velocity skewness Swm = 0. When σ → 0, which is the limit for convection with a large vertical velocity skewness, the relevant gradient is given by the in-cloud gradient. Also we conclude that it is necessary to include the effect of lateral detrainment in the compensating subsidence formulation (Arakawa and Schubert 1974; Randall et al. 1992) since it represents the effect of lateral mixing on the mean in-cloud vertical gradient. For an arbitrary σ the general solution for the flux contains a (nonlocal) correction term that is similar to that presented by Wyngaard and Weil (1991), who derived the same correction term from a Taylor expansion. These results imply that the updraft fraction needs to be parameterized, which is similar to parameterizing the third-order moment of the vertical velocity in a Reynolds-averaged closure, as is clear in Eq. (4.11). However, the general mass-flux solution gives an additional term for the nonlocal part of the vertical flux, which acts to modify the eddy diffusivity coefficient.
Typically, in the mass-flux equations the diffusivity parameter Kψ depends on the mass flux and the fractional entrainment and detrainment coefficients. Computing the mass flux as a (vertical) velocity scale is in fact similar to introducing a TKE or
Acknowledgments
The first author is very grateful to Dr. David Randall, who enabled him to visit Colorado State University to work on mass-flux equations. The authors would like to thank Dr. Hans Cuijpers for his assistance with the LES model. The investigations were supported by the Netherlands Geosciences Foundation (GOA) with financial aid (Grant 750.295.03A) from the Netherlands Organization for Scientific Research (NWO). This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from NWO. The aircraft data collected by means of the C-130 of NCAR during SCMS were kindly supplied by Dr. C. A. Knight. The authors acknowledge NCAR and its sponsor, the National Science Foundation, for the use of the observational data. The helpful suggestions from two anonymous reviewers are also very much appreciated.
REFERENCES
André, J. C., G. De Moor, P. Lacarrère, G. Therry, and R. du Vachat, 1978: Modeling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci.,35, 1861–1883.
Arakawa, A., 1969: Parameterization of cumulus convection. Proc. WMO/IUGG Symp. Numerical Weather Prediction, Tokyo, Japan, Japan Meteorological Agency, IV, 8, 1–6.
——, and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment: Part I. J. Atmos. Sci.,31, 674–701.
Bretherton, C. S., and M. C. Wyant, 1997: Moisture transport, lower-tropospheric stability, and decoupling of cloud-topped boundary layers. J. Atmos. Sci.,54, 148–167.
——, P. Austin, and S. T. Siems, 1995: Cloudiness and marine boundary layer dynamics in the ASTEX Lagrangian Experiments. Part II: Cloudiness, drizzle, surface fluxes, and entrainment. J. Atmos. Sci.,52, 2724–2735.
Canuto, V. M., F. Minotti, C. Ronchi, and R. M. Ypma, 1994: Second-order closure PBL model with new third-order moments: Comparison with LES data. J. Atmos. Sci.,51, 1605–1618.
Coulman, C. E., 1978: Boundary-layer evolution and nocturnal inversion dispersal, Part II. Bound.-Layer Meteor.,14, 493–513.
Cuijpers, J. W. M., and P. G. Duynkerke, 1993: Large eddy simulation of trade-wind cumulus clouds. J. Atmos. Sci.,50, 3894–3908.
De Laat, A. T. J., and P. G. Duynkerke, 1998: Analysis of ASTEX-stratocumulus observational data using a mass-flux approach. Bound.-Layer Meteor.,86, 63–87.
de Roode, S. R., and P. G. Duynkerke, 1996: Dynamics of cumulus rising into stratocumulus as observed during the first “Lagrangian” experiment of ASTEX. Quart. J. Roy. Meteor. Soc.,122, 1597–1623.
——, and ——, 1997: Observed Lagrangian transition of stratocumulus into cumulus during ASTEX: Mean state and turbulence structure. J. Atmos. Sci.,54, 2157–2173.
Duynkerke, P. G., and A. G. M. Driedonks, 1987: A model for the turbulent structure of the stratocumulus-topped atmospheric boundary layer. J. Atmos. Sci.,44, 43–64.
Garratt, J. R., 1994: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.
Gerber, H., 1999: Comments on “A comparison of optical measurements of liquid water content and drop size distribution in adiabatic regions of Florida cumuli.” Atmos. Res.,50, 3–19.
Greenhut, G. K., and S. J. S. Khalsa, 1982: Updraft and downdraft events in the atmospheric boundary layer over the equatorial Pacific Ocean. J. Atmos. Sci.,39, 1803–1818.
Holtslag, A. A. M., and C.-H. Moeng, 1991: Eddy diffusivity and countergradient transport in the convective atmospheric boundary layer. J. Atmos. Sci.,48, 1690–1698.
Krueger, S. K., G. T. McLean, and Q. Fu, 1995: Numerical simulation of the stratus-to-cumulus transition in the subtropical marine boundary layer. Part II: Boundary-layer circulation. J. Atmos. Sci.,52, 2851–2868.
Lamb, R. G., 1978: A numerical simulation of dispersion from an elevated point source in the convective planetary boundary layer. Atmos. Environ.,12, 1297–1304.
Lenschow, D. H., and P. L. Stephens, 1980: The role of thermals in the convective boundary layer. Bound.-Layer Meteor.,19, 509–532.
Manton, M. J., 1977: On the structure of convection. Bound.-Layer Meteor.,12, 491–503.
Martin, G. M., D. W. Johnson, D. P. Rogers, P. R. Jonas, P. Minnis, and D. A. Hegg, 1995: Observations of the interaction between cumulus clouds and warm stratocumulus clouds in the marine boundary layer during ASTEX. J. Atmos. Sci.,52, 2902–2922.
Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys.,20, 851–875.
Nicholls, S., 1984: The dynamics of stratocumulus: Aircraft observations and comparisons with a mixed layer model. Quart. J. Roy. Meteor. Soc.,110, 783–820.
——, 1989: The structure of radiatively driven convection in stratocumulus. Quart. J. Roy. Meteor. Soc.,115, 487–511.
Petersen, A. C., C. Beets, H. van Dop, P. G Duynkerke, and A. P. Siebesma, 1999: Mass-flux characteristics of reactive scalars in the convective boundary layer. J. Atmos. Sci.,56, 37–56.
Raga, G. B., J. B. Jensen, and M. B. Baker, 1990: Characteristics of cumulus band clouds off the coast of Hawaii. J. Atmos. Sci.,47, 338–355.
Randall, D. A., Q. Shao, and C.-H. Moeng, 1992: A second-order bulk boundary-layer model. J. Atmos. Sci.,49, 1903–1923.
Schumann, U., and C.-H. Moeng, 1991a: Plume fluxes in clear and cloudy convective boundary layers. J. Atmos. Sci.,48, 1746–1757.
——, and ——, 1991b: Plume budgets in clear and cloudy convective boundary layers. J. Atmos. Sci.,48, 1758–1770.
Siebesma, A. P., 1996: On the mass flux approach for atmospheric convection. Proc. ECMWF Seminar: New Insights and Approaches to Convective Parametrization, Shinfield Park, Reading, United Kingdom, ECMWF, 25–51.
——, 1998: Shallow cumulus convection. Buoyant Convection in Geophysical Flows, E. J. Plate et al., Eds., Kluwer Academic Publishers, 41–82.
——, and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. J. Atmos. Sci.,52, 650–666.
——, and A. A. M. Holtslag, 1996: Model impacts of entrainment and detrainment rates in shallow cumulus convection. J. Atmos. Sci.,53, 2354–2364.
Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, 666 pp.
Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev.,117, 1779–1800.
Wang, Q., and D. H. Lenschow, 1995: An observational study of the role of penetrating cumulus in a marine stratocumulus-topped boundary layer. J. Atmos. Sci.,52, 2778–2787.
Wang, S., 1993: Modeling marine boundary-layer clouds with a two-layer model: A one-dimensional simulation. J. Atmos. Sci.,50, 4001–4021.
——, and B. A. Albrecht, 1990: A mean-gradient model of the dry convective boundary layer. J. Atmos. Sci.,47, 126–138.
Wyngaard, J. C., and J. C. Weil, 1991: Transport asymmetry in skewed turbulence. Phys. Fluids A,3, 155–162.
——, and C.-H. Moeng, 1992: Parameterizing turbulent diffusion through the joint probability density. Bound.-Layer Meteor.,60, 1–13.
Young, G. S., 1988a: Turbulence structure of the convective boundary layer. Part II: Phoenix 78 aircraft observations of thermals and their environment. J. Atmos. Sci.,45, 727–735.
——, 1988b: Turbulence structure of the convective boundary layer. Part III: The vertical velocity budgets of thermals and their environment. J. Atmos. Sci.,45, 2039–2049.
APPENDIX A
Derivation of the Prognostic Variance Equation in Mass-Flux Variables
APPENDIX B
Surface Layer Similarity Relationship
Schematic of various types of atmospheric boundary layers and its representation in general circulation models.
Citation: Journal of the Atmospheric Sciences 57, 10; 10.1175/1520-0469(2000)057<1585:ABMFAR>2.0.CO;2
Variance budget for (a) the Reynolds-averaged equations according to Eq. (2.4) and (b) the mass-flux equations according to Eq. (2.12). Line styles as indicated in the legend.
Citation: Journal of the Atmospheric Sciences 57, 10; 10.1175/1520-0469(2000)057<1585:ABMFAR>2.0.CO;2
Fractional lateral entrainment and detrainment coefficients for a clear convective boundary layer. Line styles as indicated in the legend.
Citation: Journal of the Atmospheric Sciences 57, 10; 10.1175/1520-0469(2000)057<1585:ABMFAR>2.0.CO;2
Conditionally sampled total water content in cumulus clouds (qt,c) and environment (qt,e). Also indicated are vertical gradients calculated with a linear fit.
Citation: Journal of the Atmospheric Sciences 57, 10; 10.1175/1520-0469(2000)057<1585:ABMFAR>2.0.CO;2
Horizontal mean dissipation calculated from inertial subrange theory and according to the mass-flux variance equation (3.2) in a cumulus cloud field.
Citation: Journal of the Atmospheric Sciences 57, 10; 10.1175/1520-0469(2000)057<1585:ABMFAR>2.0.CO;2
Characteristics of the large eddy simulation of a clear convective boundary layer. The boundary layer height h is defined as the level where the buoyancy flux has a minimum value.
Summary of vertical flux solutions derived from the mass-flux variance Eq. (2.12).
Summary of mass-flux variables and the analogous Reynolds-averaged closure variables.