Abstract

A scheme is described that provides an integrated description of turbulent transport in free convective boundary layers with shallow cumulus. The scheme uses a mass-flux formulation, as is commonly found in cumulus schemes, and a 1.5-order closure, involving turbulent kinetic energy and eddy diffusivity. Both components are active in both the subcloud and cloud layers. The scheme is called “mass flux–diffusion.” In the subcloud layer, the mass-flux component provides nonlocal transport. The scheme combines elements from schemes that are conceptually similar but differ in detail. An entraining plume model is used to find the mass flux. The mass flux is continuous through the cloud base. The lateral fractional entrainment rate is constant with height, while the detrainment-rate profile reduces the mass flux smoothly to zero at the cloud top. The eddy diffusivity comes from a turbulent kinetic energy–length scale formulation. The scheme has been implemented in a simple one-dimensional (single column) model. Results of simulations of a standard case that has been used for other model intercomparisons [Atmospheric Radiation Measurement (ARM), 21 June 1997] are shown and indicate that the scheme provides good results. The model also simulates the profile of a conserved scalar; this capability is applied to a case from the 1999 Southern Oxidants Study Nashville (Tennessee) experiment, where it produces good simulations of vertical profiles of carbon monoxide in a cloud-topped boundary layer.

Introduction

Cumulus clouds commonly occur atop convective boundary layers over land. If the cloud fraction is not too large and no precipitation is occurring, the clouds are called “shallow” or “fair weather” cumulus, even if the vertical extent of the cloud is 1 km or more. These cloud layers are generally not treated in numerical models or, if they are, are treated separately from the subcloud boundary layer. However, these cloud layers can have significant effects on the vertical transport of energy, water, and chemical constituents (pollutants). Models without shallow cumulus, or with inadequate representations of vertical transport resulting from cumulus, will produce incorrect profiles of chemical constituents, whether those constituents are emitted at the surface or transported aloft.

Brevity precludes going into detail of the extensive history and taxonomy of turbulence schemes for the dry boundary layer here. The reader may refer to the standard textbooks—for example, Stull (1988) and Garratt (1992). Dry boundary layer schemes fall into two basic categories: local and nonlocal. Local schemes are based on an analogy to molecular diffusion. Nonlocal schemes are based on the observation that most of the energy in the convective boundary layers is carried by structures (“large eddies”) whose size is comparable to the depth of the layer and therefore is much larger than the model grid spacing. Local schemes may be modified by various means to allow for the observed countergradient transport of buoyancy in the upper half (roughly) of the layer. In the absence of such modifications, local schemes produce simulated profiles with unrealistic gradients (Lock et al. 2000).

Schemes for shallow convection are reviewed by Siebesma (1998). One common approach involves the definition of an updraft carrying a certain amount of mass and having specific properties (temperature, water vapor content, etc.) and therefore is called the mass-flux approach. The updraft mass flux and properties are modified by lateral entrainment and detrainment (not to be confused with vertical entrainment at the top of the dry boundary layer). The entrainment- and detrainment-rate profiles are the critical elements of mass-flux schemes.

Siebesma and Teixeira (2000) introduced the idea of an integrated scheme for turbulent vertical transport in cloud-topped convective boundary layers, which they called an “advection–diffusion” scheme. Variations of this idea have been presented by Soares et al. (2004), where it is called “EDMF.” The scheme presented here owes much to their work but differs at some points. The basic concept is to combine eddy diffusion (“K theory”), a time-honored dry boundary layer scheme, with mass flux, the most common cumulus convection scheme, in a single formulation for the whole cloud-topped boundary layer. I call my scheme “mass-flux–diffusion,” or “M-K” for short.

Extending the mass-flux approach downward into the subcloud layer provides a nonlocal transport component in that layer, eliminating the need for countergradient correction terms. The mass-flux term alone, however, is not adequate as the only representation of turbulence in the subcloud layer. Petersen et al. (1998) show that the mass-flux component is about two-thirds of the total flux in such a layer. The remainder can be thought of as the results of smaller-scale (“subgrid”) transport and is therefore appropriately represented by a local scheme, such as eddy diffusion. Here, the eddy diffusivity is derived from turbulent kinetic energy (TKE) and a length scale. The length scale profile has a specified shape.

The need for a local turbulence component in the cloud layer is less obvious. When the cloud fraction is very small, turbulence outside the clouds should be negligible. A potential barrier exists between the subcloud layer and the cloud layer, and the profile of turbulence intensity has two maxima (Siebesma 1998). Only the most energetic updrafts can overcome the potential barrier and form clouds, and all of the transport through the cloud base is by these strongest updrafts. At larger cloud fractions, including stratocumulus, a variety of phenomena can occur and a variety of turbulence and stability profiles may be present. Local and nonlocal transport may take place in varying proportions depending on the strength of the potential barrier between the two layers. The formulation that we report here uses TKE to define the amount of small-scale turbulence that is present, and uses a specified length scale to convert that TKE to eddy diffusivity. TKE is also used by Soares et al. (2004).

If one uses separate boundary layer (subcloud) and cumulus schemes, the closure assumption at cloud base is a difficult question (Siebesma and Holtslag 1996). In my M-K scheme, I simply assume that the mass flux is continuous across the cloud base, as do Siebesma and Teixeira (2000) and Jakob and Siebesma (2003). This is equivalent to assuming that the strongest updrafts are those that form clouds, which is precisely the basic idea of this scheme.

The limitations of the M-K scheme presented here should be stated up front. It is applicable only to surface-driven free convection and only to shallow cumulus at small cloud fractions (not stratocumulus). However, it should be straightforward to extend the formulation to other conditions.

The motivation to work on this problem comes from its importance for atmospheric chemistry and regional air quality applications. My formulation directly addresses the vertical transport of constituents by cloud and their detrainment into the partly cloudy layer. In this paper, I demonstrate the use of the M-K scheme to produce profiles of a conserved scalar. The profiles are compared with measurements of carbon monoxide (CO) that were measured by an aircraft during the 1999 Southern Oxidants Study (SOS) Nashville (Tennessee) experiment on a day when shallow cumulus were an important factor.

Several related approaches have appeared in the literature. Wang and Albrecht (1990) used a mass-flux parameterization in the dry boundary layer. Lappen and Randall (2001a, b,c) developed a parameterization for both the dry and moist convective PBL. Their approach combines mass-flux and higher-order closure and uses assumed probability distributions. They also provide a review of previous similar work. Assumed probability density functions also underlie the work of Golaz et al. (2002). De Roode et al. (2000) argue that mass-flux and (more common) Reynolds-averaged formulations are, to some degree, equivalent. Cheinet and Teixeira (2003) give an alternative formulation using separate eddy diffusivities for the clear and cloudy parts of the column.

Given this wealth of literature, the reader may ask why yet another parameterization is needed. While none of the elements of the scheme is original, they are combined in a different way. Most significant is that the scheme is applied to a different kind of problem—that of chemical or tracer transport. The Nashville case has interesting features that are not present in other well-documented cases.

Formulation of the M-K scheme

The M-K scheme is formulated in the conserved variables θl (liquid water potential temperature) and qt (total water mixing ratio). Two passive scalars, S1 and S2, are also carried. The basic mass-flux and diffusion equation (Siebesma and Teixeira 2000) is

 
formula

where ψ is θl, qt, S1, or S2. The mass flux M is in kinematic units (m s−1). Throughout, subscript u designates the updraft, and unsubscripted variables designate the environment. This equation implies the assumption that the updraft fraction is small enough that the downdraft characteristics are negligibly different from the environmental average, or, equivalently, that the overall average is not affected by the updraft. The updraft represents the largest (cloud forming) scales. Given the flux, the time evolution of any quantity is

 
formula

Subcloud layer

The plume (thermal) model in the subcloud layer (Siebesma and Teixeira 2000) is

 
formula

where ɛ is the fractional (lateral) entrainment rate (m−1).

The updraft properties (other than velocity) are initialized at the surface by adding to the environment value an excess that scales with the surface flux and w* as

 
formula

where ce is a constant taken to be 10. This differs from the initialization used by Siebesma and Teixeira [2000; their Eq. (5)] by using the convective velocity scale w* rather than the vertical velocity variance in the denominator. The vertical velocity variance is a strong function of height (and thus of model resolution) near the surface, while w* is well defined and independent of resolution. The constant ce is larger than the comparable coefficient of Siebesma and Teixeira (2000), because w* is larger than the vertical velocity variance near the surface.

The fractional entrainment rate is constant with height,

 
formula

and the fractional detrainment rate in the absence of cloud is

 
formula

These rates are the most sensitive part of the formulation, defining essentially all aspects of the model results. They were chosen to be broadly consistent with large-eddy simulation (LES) results for the cloud layer (Siebesma 1998; Brown et al. 2002) because similar values are used in that layer (below). However, I could not simply use the exact profiles found in those papers in this simplified model. Instead, I sought profiles that allowed the model to operate in a stable manner and produce plausible mass-flux profiles. The constant fractional entrainment rate is exactly that chosen by Soares et al. (2004) [their Eq. (9)] and is in the middle of the range proposed by Siebesma [1998; his Eq. (51)].

The dry or sub-cloud-layer height hd is taken to be the height where the (dry) updraft velocity wu first goes to zero. The updraft velocity is

 
formula

where a = 1/3 and b = 2 are coefficients whose possible values are discussed in Siebesma et al. (2003). The lower boundary condition is w0 = Cww*, where w* is the convective velocity scale and Cw is a constant (=0.5). In this calculation, latent heat release is not taken into account, because the purpose of the calculation is to determine the height that is reached by a dry (noncondensing) thermal. The updraft penetrates the capping stable layer to an extent that is determined by the stability and the convective velocity scale. This allows for realistic top entrainment. It should be noted that some similar entraining plume schemes (e.g., Jakob and Siebesma 2003) use a vertical velocity that is computed with condensation.

Turbulent kinetic energy is governed by a prognostic equation:

 
formula

where Cd = 0.16 (Duynkerke and Driedonks 1987) is a proportionality constant and lɛ is a length scale for TKE dissipation (=2.5l, see below). The first term on the right-hand side is the buoyancy production, the second term is the turbulent transport and pressure correlation parameterized with eddy diffusivity, and the third term is parameterized dissipation. This equation is strictly valid only when there is no mean wind or when shear production and advection are negligible. The same formulation is used by Teixeira and Cheinet (2004). Here, latent heat release is taken into account in the computation of the buoyancy flux. Calculation of the buoyancy flux in partly cloudy conditions is nontrivial. In this case, the formulation of Lewellen and Lewellen (2004) is used. The assumption is made that the environment is unsaturated and the updraft is either entirely unsaturated (in the subcloud layer or in the absence of cloud) or entirely saturated (in the cloud layer). Then, the buoyancy flux can be calculated from Eq. (2) (environment) and Eq. (3) (updraft) of Lewellen and Lewellen (2004). Strictly speaking, this limits the applicability of the M-K scheme to small cloud fractions. However, because the TKE in the cloud layer is only used to determine the amount of small-scale mixing (eddy diffusivity) in that layer, it may be an acceptable approximation up to moderate cloud fractions.

Given TKE, the eddy diffusivity K is then derived as

 
formula

where K has units of meters squared per second, Ck = 0.4 is a constant, and l is a length scale defined below. The canonical value of Ck is somewhat larger (0.55; Duynkerke and Driedonks 1987), but this is reasonable because in the present M-K scheme some of the mixing is carried out by the mass flux. With the larger value, the model still works reasonably well but shows signs of too much diffusion. A small minimum diffusion, tapered to zero at the top of the domain, is added for numerical stability.

The critical part of a TKE–l closure is the selection of a length scale. Here, the formulation for the subcloud layer is

 
formula

This is a simple expression of the concepts that the eddy size is constrained by the surface (the first term) and by the inversion (the second term). It is a modification of a standard formula [e.g., Garratt 1992, his Eqs. (8)–(54)] where the second term is usually a constant (“asymptotic”) length. The basic idea of making the length scale proportional to the distance a parcel can travel against the stability is the result of Bougeault and Lacarrere (1989).

The mass flux is initialized as M = CMw*, where CM = 0.04, and evolves according to

 
formula

Cumulus layer

Cloud is diagnosed when the lifting condensation level (LCL) of the updraft air is below the dry thermal top hd. The cloud-top height hct is taken as the “limit of convection” (Stull 1988) for the updraft starting at the LCL. This is found by computing the parcel temperature with the moist-adiabatic lapse rate and comparing the virtual potential temperature θυ of the parcel with the mean (environment) θυ. In other words, the cloud top is calculated assuming an undiluted parcel. If the limit of convection is less than or equal to hd within the model resolution, the cloud is “forced,” that is, driven by the momentum of the thermal rather than by latent heat release; otherwise, the cloud is “active.”

The formulation of the cumulus layer also includes both mass-flux and diffusion elements, like Soares et al. (2004) but unlike the original formulation of Siebesma and Teixeira (2000) who did not use any diffusion in the cloud layer. The cloud model is the same as the plume (thermal) model (3) in the subcloud layer. Vertical velocity is not used in the cumulus layer. The length scale l for determining the eddy diffusivity K is computed as for the subcloud layer (10), but the length scale above the middle of the subcloud layer is forced to remain >100 m; in practice, this means that the length scale in the cloud layer is fixed at 100 m. The fixed minimum of 100 m is rather arbitrary, but it reflects that the eddy diffusion in the cloud layer is intended to provide the small-scale mixing, while the mass flux handles the larger scales.

The mass flux at cloud base is carried through unchanged from the sub-cloud-layer model and continues to evolve within the cumulus layer according to (11), with the same fractional entrainment rate ɛ but a different fractional detrainment rate

 
formula

The form of the fractional detrainment rate is chosen to produce a smooth transition from a balance between entrainment and detrainment at cloud base to dominance of detrainment near cloud top. This results in a smooth decrease in mass flux with increasing height in the cloud layer—a physically realistic result for most shallow cumulus situations.

Model framework, and initial and boundary conditions

The M-K scheme was tested in a simple 1D model framework, coded as a Matlab script, with an explicit second-order solution of the discretized equations. The time step was 0.8 s and the vertical grid spacing was 25 m, with the top of the model at 4 km. This is a very crude numerical method; a more sophisticated solution method would allow the time step to be increased to a value that is more practical for use in a larger modeling system. An initial profile of each of the conserved variables was specified. At the lower boundary, surface fluxes of each variable were specified at each time step. There is no surface model as such. The rates of change of all variables were forced to zero at the top level.

Evaluation—ARM case

The M-K model was evaluated using the “Atmospheric Radiation Measurement (ARM) case,” which is an idealization of the situation of 21 June 1997 at the southern Great Plains site of the ARM Program. This case was the basis of extensive intercomparisons of LES models (Brown et al. 2002) and of single-column models (Lenderink et al. 2004), and it has also been used to evaluate other models. The various LES models that are intercompared produced generally consistent results. The single-column models were compared with the LES results from the Royal Netherlands Meteorological Institute (KNMI) LES, which is also used here (data available online at http://www.knmi.nl/samenw/eurocs/ARM). The observations showed fair-weather cumulus developing in the morning and dissipating in late afternoon atop a continental convective boundary layer. The intercomparisons used a slightly modified version of the measured sounding to account for advection during the day. For this work, the initial sounding was further modified by removing the stable layer at the lowest level, because this model does not handle stable cases. The surface fluxes were modified to eliminate the negative fluxes at the beginning of the run for the same reason.

Figures 1 and 2 show the M-K model results and compare them with the KNMI LES and the observations. Cloud is first produced after 1500 UTC, which is slightly later than the observations and LES results. The M-K model produces forced cumulus (as a result of overshooting thermals, not latent heat release) until approximately 1700 UTC, after which time the clouds are active. The M-K model is warmer than the LES in the mixed layer, as a result of the modifications of the initial sounding and surface flux described above. Because the moisture in the mixed layer remains the same, the M-K model cloud base is higher than that of the LES result, near 1 km AGL between 1500 and 1700 UTC, and is closer to that of the observations. Cloud base remains generally consistent with the observations throughout the run. The M-K model mixes a little more vigorously than the LES below 2 km and a little less vigorously above 2 km, indicated by a more pronounced change of slope of the θl and qt profiles at that level. As a result, cloud top in the M-K model is generally lower than in the LES results (not shown, see Brown et al. 2002). For example, the M-K model cloud top at 2030 UTC is approximately 2100 m when the LESs show 2500–2600 m. The modeled cloud top rises monotonically rather than declining in the late afternoon as in the LES. The cloud does not fully dissipate near the end of the run, but it does become intermittent.

Fig. 1.

The M-K model initial profiles and results for the ARM case. (a) Profiles of θl at time 0 (1130 UTC, dashed) and 2030 UTC (solid). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top, all at 2030 UTC. (b) Profiles of qt at 1130 (dashed) and 2030 (solid) UTC. (c) The θl flux (solid), mass-flux contribution to θl flux (dotted), and θυ flux (dashed) at 2030 UTC. (d) Time series of hd (solid), cloud base (dashed), and cloud top (upper broken line). Vertical dotted line marks 2030 UTC, the time of the profiles in (a)–(c).

Fig. 1.

The M-K model initial profiles and results for the ARM case. (a) Profiles of θl at time 0 (1130 UTC, dashed) and 2030 UTC (solid). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top, all at 2030 UTC. (b) Profiles of qt at 1130 (dashed) and 2030 (solid) UTC. (c) The θl flux (solid), mass-flux contribution to θl flux (dotted), and θυ flux (dashed) at 2030 UTC. (d) Time series of hd (solid), cloud base (dashed), and cloud top (upper broken line). Vertical dotted line marks 2030 UTC, the time of the profiles in (a)–(c).

Fig. 2.

Comparison of M-K model with KNMI LES and observations for the ARM case. (a) Profiles of θl at 2030 UTC: M-K model (solid), LES (dashed), and observations (dotted). (b) Profiles of qt at 2030 UTC, same line types as in (a). (c) Time series of hd from M-K model (solid) and height of minimum potential temperature flux from LES (dashed). (d) Time series of M-K model cloud base (solid), LES cloud base (dashed), and observed cloud bases from two sites (plus and circle symbols). In (c) and (d), vertical dotted line marks 2030 UTC, the time of the profiles in (a) and (b).

Fig. 2.

Comparison of M-K model with KNMI LES and observations for the ARM case. (a) Profiles of θl at 2030 UTC: M-K model (solid), LES (dashed), and observations (dotted). (b) Profiles of qt at 2030 UTC, same line types as in (a). (c) Time series of hd from M-K model (solid) and height of minimum potential temperature flux from LES (dashed). (d) Time series of M-K model cloud base (solid), LES cloud base (dashed), and observed cloud bases from two sites (plus and circle symbols). In (c) and (d), vertical dotted line marks 2030 UTC, the time of the profiles in (a) and (b).

Flux profiles from the M-K model (Fig. 1) indicate that the mass-flux term makes its greatest contributions to the θl flux near the surface and in the middle of the cloud layer, contributing about one-half of the total flux in both places. The buoyancy flux profile is similar to the LES result (Brown et al. 2002, their Fig. 7) in the subcloud layer and near the cloud base but has considerably less positive flux in the cloud layer. The ratio of the minimum buoyancy flux (below cloud base) to the surface flux is −0.2, which is a commonly accepted value for dry boundary layers with no shear; however, see Angevine (1999) for a discussion of this ratio and its limitations in nonideal conditions.

Figure 3 shows the profiles of the lateral fractional entrainment and detrainment rates, mass flux, diffusion coefficient, updraft–environment differences, and updraft liquid water. The updraft mass flux in the M-K model is quite similar in magnitude and shape to the core mass flux of LES (Brown et al. 2002, their Fig. 8d), making an allowance for the somewhat different cloud top that is mentioned above. A direct comparison of TKE in the M-K model and the KNMI LES is difficult, because the M-K model does not have horizontal wind components, which dominate the LES TKE below about 300 m. The M-K model TKE is similar in shape and about 3 times the magnitude of the LES ww′ in the subcloud layer but is considerably smaller in the cloud layer. The M-K scheme produces a roughly constant diffusion coefficient K in the lower and middle parts of the cloud layer, which is roughly consistent with the profile found by Cheinet and Teixeira (2003), Stevens et al. (2001), and Siebesma et al. (2003).

Fig. 3.

Internal model variables for the ARM case at 2030 UTC. (a) Lateral fractional entrainment rate ɛ (solid profile) and fractional detrainment rate δ (dashed). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top. (b) Mass flux (×10, m s−1, solid), diffusion coefficient (/100, m2 s−1, dashed), and TKE (m2 s−2, dotted). (c) Difference between updraft and environment θl. (d) Difference between updraft and environment qt (solid) and liquid water ql (dashed).

Fig. 3.

Internal model variables for the ARM case at 2030 UTC. (a) Lateral fractional entrainment rate ɛ (solid profile) and fractional detrainment rate δ (dashed). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top. (b) Mass flux (×10, m s−1, solid), diffusion coefficient (/100, m2 s−1, dashed), and TKE (m2 s−2, dotted). (c) Difference between updraft and environment θl. (d) Difference between updraft and environment qt (solid) and liquid water ql (dashed).

Nashville case

On 14 July 1999, the National Oceanic and Atmospheric Administration (NOAA) WP-3 research aircraft made a flight in the vicinity of Nashville as part of the Southern Oxidants Study 1999 Nashville summer intensive. The flight was designed to characterize the urban pollutant plume. A set of stacked flight legs traversed the urban plume across the wind over Nashville, and a second set was flown approximately 30 km downwind. The legs over Nashville were flown at approximately 1400 LST, and the downwind legs were at approximately 1500 LST. Instruments aboard the aircraft measured a variety of chemical species. The boundary layer was topped by a shallow cumulus layer of moderate cloud fraction, and the chemical profiles showed definite signs of vertical transport in the cloud layer. This case was simulated using the 1D model incorporating the M-K scheme. Because it is relatively passive, CO was chosen as the species to compare with the modeled tracer.

No soundings were available in the morning, so initial profiles were devised to yield correct 1200 LST soundings. The model run started at 0900 LST. Figure 4 shows the initial profiles and model results at 1300 LST for a run that was intended to simulate the conditions over Nashville (downtown run), as well as the time evolution of cloud base, cloud top, and dry thermal top. Figure 5 compares the model profiles with measured profiles from a sonde that was launched at Old Hickory, northeast of Nashville, and from the aircraft west of Nashville. There are some differences between the two soundings reflecting regional variability, but the basic features are similar.

Fig. 4.

The M-K model initial profiles and results for Nashville 14 Jul 1999 downtown run. (a) Profiles of θl at time 0 (0900 LST, dashed) and 1300 LST (solid). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top, all at 1300 LST. (b) Profiles of qt at 0900 (dashed) and 1300 LST (solid). (c) The θl flux (solid), mass-flux contribution to θl flux (dotted), and θυ flux (dashed) at 1300 LST. (d) Time series of hd (solid), cloud base (dashed), and cloud top (upper broken line). Vertical dotted line marks 1300 LST, the time of the profiles in (a)–(c).

Fig. 4.

The M-K model initial profiles and results for Nashville 14 Jul 1999 downtown run. (a) Profiles of θl at time 0 (0900 LST, dashed) and 1300 LST (solid). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top, all at 1300 LST. (b) Profiles of qt at 0900 (dashed) and 1300 LST (solid). (c) The θl flux (solid), mass-flux contribution to θl flux (dotted), and θυ flux (dashed) at 1300 LST. (d) Time series of hd (solid), cloud base (dashed), and cloud top (upper broken line). Vertical dotted line marks 1300 LST, the time of the profiles in (a)–(c).

Fig. 5.

Comparison of M-K model results with measurements for Nashville downtown run. (a) Profiles of θl at 1200 LST. Solid line is M-K model result, dashed line is from sonde launched at 1200 LST, and plus signs are from aircraft sounding at approximately 1230 LST. (b) Profiles of qt at 1200 LST. Line types as in (a). (c) Time series of hd from M-K model (solid) and BL top from wind profiler (dashed). (d) Time series of M-K model cloud base (dashed) and cloud top (solid line) compared with ceilometer-measured cloud base (circles) and cloud top from profiler reflectivity (x symbols). Vertical dotted lines in (c) and (d) mark 1200 LST, the time of the profiles in (a) and (b).

Fig. 5.

Comparison of M-K model results with measurements for Nashville downtown run. (a) Profiles of θl at 1200 LST. Solid line is M-K model result, dashed line is from sonde launched at 1200 LST, and plus signs are from aircraft sounding at approximately 1230 LST. (b) Profiles of qt at 1200 LST. Line types as in (a). (c) Time series of hd from M-K model (solid) and BL top from wind profiler (dashed). (d) Time series of M-K model cloud base (dashed) and cloud top (solid line) compared with ceilometer-measured cloud base (circles) and cloud top from profiler reflectivity (x symbols). Vertical dotted lines in (c) and (d) mark 1200 LST, the time of the profiles in (a) and (b).

Figure 5 also shows cloud-base and cloud-top heights measured at the Cornelia Fort site, just downwind of downtown Nashville. Cloud base was measured by a laser ceilometer. Cloud top was diagnosed from the reflectivity patterns of a 915-MHz radar wind profiler. The measurements at this site are discussed in more detail by Angevine et al. (2003). The model cloud base agrees very well with the measurements once the model produces cloud, at approximately 1100 LST. Model cloud top is lower than the (rather uncertain) measurements. Profiles of the key internal variables in the model are shown in Fig. 6. This case has a lower LCL, a greater distance between LCL and dry thermal top, and a much deeper cloud layer with more liquid water than the ARM case (Fig. 3). The mass flux below cloud is slightly larger. TKE is slightly larger below the cloud and is substantially larger in the lower cloud layer, resulting (by design) in a substantially larger eddy diffusivity in the cloud layer. As in the ARM case, here the M-K scheme produces a roughly constant diffusion coefficient K in the lower and middle parts of the cloud layer.

Fig. 6.

Internal model variables for Nashville downtown run at 1300 LST. (a) Lateral fractional entrainment rate ɛ (solid profile) and fractional detrainment rate δ (dashed). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top. (b) Mass flux (×10, m s−1, solid), diffusion coefficient (/100, m2 s−1, dashed), and TKE (m2 s−2, dotted). (c) Difference between updraft and environment θl. (d) Difference between updraft and environment qt (solid) and liquid water ql (dashed).

Fig. 6.

Internal model variables for Nashville downtown run at 1300 LST. (a) Lateral fractional entrainment rate ɛ (solid profile) and fractional detrainment rate δ (dashed). Dotted line is cloud base (LCL), solid horizontal line is dry thermal top hd, and dot–dash line is cloud top. (b) Mass flux (×10, m s−1, solid), diffusion coefficient (/100, m2 s−1, dashed), and TKE (m2 s−2, dotted). (c) Difference between updraft and environment θl. (d) Difference between updraft and environment qt (solid) and liquid water ql (dashed).

To compare the aircraft CO measurements with a 1D model, it was assumed that the background wind was uniform and, therefore, that time and distance along the wind were interchangeable with suitable scaling. An initial CO profile was chosen based on the values that are measured at the ends of the flight legs, outside of the urban plume. A temporal pattern of CO flux from the surface was set up so that an appropriate amount of CO was introduced into the air that would be sampled, at the time it passed over the city. Figure 7 shows the kinematic heat, moisture, and CO fluxes that are used in the simulation. Three profiles were extracted from the aircraft flight patterns, with one each representing background, downtown, and downwind conditions. The background profile represents CO at the edges of the urban plume. The downtown profile corresponds to values that are measured in the urban plume during crosswind transits centered over the Cornelia Fort site at several levels. The downwind profile represents similar transits that are approximately 2-h transit time downwind. Three simulations were run to produce corresponding model output. Relative to the background simulation, the downtown and downwind simulations had more heat and CO flux. The CO flux in the downwind case was shifted 2 h earlier in time to simulate the earlier passage over the city of the air sampled downwind.

Fig. 7.

Surface fluxes specified to the M-K model for the Nashville case. (a) Kinematic heat flux for downtown and downwind runs (solid) and background run (dashed); kinematic moisture flux for all cases (dotted). (b) Kinematic CO flux for downtown run (solid) and background run (dashed).

Fig. 7.

Surface fluxes specified to the M-K model for the Nashville case. (a) Kinematic heat flux for downtown and downwind runs (solid) and background run (dashed); kinematic moisture flux for all cases (dotted). (b) Kinematic CO flux for downtown run (solid) and background run (dashed).

Simulated and measured CO profiles are shown in Fig. 8. The simulations resemble the measurements closely. Measured and modeled CO over downtown Nashville is enhanced relative to the background throughout a deep vertical layer. Farther downwind, CO concentrations in the lowest measured levels are about the same as in the downtown profile, but more CO has been introduced (resulting from passage over more of the city) and mixed higher into the atmosphere (resulting from the longer mixing time). There are indications that the model may still have too much potential barrier at cloud base, resulting in a larger change in slope of the CO profile there than in the measurements.

Fig. 8.

The CO profiles from aircraft measurements (symbols) and M-K model simulations (lines) for Nashville case: background (+ symbols and solid line), downtown (o symbols and dashed line), and downwind (x symbols and dotted line) are shown.

Fig. 8.

The CO profiles from aircraft measurements (symbols) and M-K model simulations (lines) for Nashville case: background (+ symbols and solid line), downtown (o symbols and dashed line), and downwind (x symbols and dotted line) are shown.

Sensitivity tests

In this section, the sensitivity of the M-K model results to the free parameters and features of the model is discussed. First, the impact of the mass-flux term and of cloud transport is examined. Then, the effects of changing lateral fractional entrainment and detrainment rates are explored. Last, relatively insensitive features are mentioned.

The mass-flux term is included in the model scheme primarily to integrate the sub-cloud- and cloud-layer formulations. However, the mass-flux term also provides nonlocal transport in the subcloud layer. This has a small effect by reducing the potential temperature and humidity gradients in the mixed layer and allowing the mixed-layer top to rise more quickly. Figure 9 shows profiles of θl and qt in the Nashville case for the full model scheme, with no mass flux, and with no cloud.

Fig. 9.

Profiles of (a) θl and (b) qt for Nashville downtown run testing sensitivity to model transport components.Solid lines are normal downtown run, dashed line is run without mass flux, and dotted line is run with no cloud.

Fig. 9.

Profiles of (a) θl and (b) qt for Nashville downtown run testing sensitivity to model transport components.Solid lines are normal downtown run, dashed line is run without mass flux, and dotted line is run with no cloud.

Vertical transport by clouds is critical to the model performance in the Nashville case. To confirm this, the model was run with no cloud by forcing the lifting condensation level to 4 km AGL at all times. The resulting profiles are shown in Figs. 9 and 10. The model forms a classical well-mixed dry boundary layer with a sharp inversion. Carbon monoxide from the surface is not transported upward above the inversion, and becomes much too concentrated in the lower levels.

Fig. 10.

As Fig. 8, but showing results with no cloud.

Fig. 10.

As Fig. 8, but showing results with no cloud.

The most sensitive parts of the M-K scheme are the lateral fractional entrainment and detrainment rates. Profile shapes and magnitudes for these are still an active subject of research (see discussion below). The lateral fractional entrainment-rate profile affects the temperature and water vapor difference between updraft and environment and therefore the amount that thermals overshoot their level of neutral buoyancy, which governs the vertical entrainment rate at the boundary layer top. It also affects the mass flux at cloud base and therefore the cloud-layer properties. The lateral fractional detrainment-rate profile affects where the updraft heat and moisture are deposited. The chosen profile produces an approximately linear decrease in mass flux with height in the cloud layer, so that the updraft heat and moisture are distributed approximately uniformly through the layer. The entrainment and detrainment profiles must stay within a relatively narrow range, or the model will switch back and forth between cloudy and noncloudy regimes as heat and moisture build up within the subcloud layer or are too rapidly transported out of that layer. This intermittency was observed in several of the single-column models included in the intercomparison study of Lenderink et al. (2004).

Results from the M-K scheme are not sensitive to reasonable changes in other parameters. Table 1 shows a few of the many possible permutations of parameter values. For the table, each parameter was changed by plus and minus 10%. Four measures of the model result for the ARM case were chosen—the dry thermal top, cloud top, and mass flux at cloud base, all at 2030 UTC, and the time of first cloud onset. Changing the fractional entrainment rate by 10% results in approximately a 4% change in hd, no change in hct, a 25% change in the mass flux at cloud base, and a few minutes change in the cloud onset. For the constant part of the fractional detrainment rate, a 10% change results in a 3% change in hd, 2%–3% change in hct, a 25% change in mass flux, and almost no change in the cloud onset. None of the other constants has more than a very small effect, except for the mass-flux initialization constant CM, which changes the cloud-base mass flux by almost 10% (nearly a linear change) but has little effect on the other measures. Larger changes in some parameters would probably have nonlinear effects if they pushed the model out of its nominal operating envelope.

Table 1.

Sensitivity of the model run for ARM case to ±10% changes in several parameters. For each parameter, the nominal values are presented in the middle of the three rows. For the detrainment rate, the change was applied to the constant part only. The final parameter, lmin, is the minimum length scale in the cloud layer for determination of K [Eq. (9)]. Note that the middle row of each set has the same resulting values in the last four columns; it is included for ease of reading the table and to make clear the nominal value of each of the independent variables.

Sensitivity of the model run for ARM case to ±10% changes in several parameters. For each parameter, the nominal values are presented in the middle of the three rows. For the detrainment rate, the change was applied to the constant part only. The final parameter, lmin, is the minimum length scale in the cloud layer for determination of K [Eq. (9)]. Note that the middle row of each set has the same resulting values in the last four columns; it is included for ease of reading the table and to make clear the nominal value of each of the independent variables.
Sensitivity of the model run for ARM case to ±10% changes in several parameters. For each parameter, the nominal values are presented in the middle of the three rows. For the detrainment rate, the change was applied to the constant part only. The final parameter, lmin, is the minimum length scale in the cloud layer for determination of K [Eq. (9)]. Note that the middle row of each set has the same resulting values in the last four columns; it is included for ease of reading the table and to make clear the nominal value of each of the independent variables.

Discussion and conclusions

A mass-flux–diffusion (M-K) scheme to represent cloud-topped convective boundary layers has been presented. The scheme is conceptually simple and tied to physical phenomena. The scheme, when implemented in a column model, produces results that compare well with observations and with LES model results. The scheme also simulates profiles of long-lived chemical species (CO) under conditions that could not be simulated correctly without taking the cloud-layer vertical transport into account.

The scheme presented here was inspired by and is conceptually similar to that of Siebesma and Teixeira (2000), which has also been followed up by Soares et al. (2004). It differs from each of these, however, in several aspects, and they differ from each other. One of these aspects is the cloud-base mass-flux closure. The M-K scheme carries the subcloud mass flux through cloud base unchanged, as do Siebesma and Teixeira (2000). This is the simplest possible closure. What it means, in concept, is that the subcloud mass flux represents only those thermals that form cloud. These presumably cover less of the area at any given height than the larger fraction of thermals represented by the subcloud mass flux in the other related schemes, but the difference does not seem to be important. The most important functions of the subcloud mass flux are to produce correct gradients and reasonable top entrainment (when cloud is not present). The total amount of mass flux does not seem to be critical to the fulfillment of these functions. Soares et al. (2004) change the mass flux at cloud base according to a diagnosed cloud fraction. Grant (2001) used LES of marine cumulus situations to conclude that the mass flux can be parameterized as a constant fraction of the sub-cloud-layer convective velocity scale w*. Neggers et al. (2004) compared three cloud-base mass-flux closure methods for the ARM case and concluded that a closure based on w* performed well, although a more sophisticated version, also including the cloud-core fraction, was a little better. The M-K scheme that is presented here uses w* to initialize the entraining plume at the surface, and because the fractional entrainment and detrainment rates are almost the same, the mass flux remains constant up to cloud base. The result is that the cloud-base mass flux is proportional to w*. The proportionality constant in the M-K scheme [Eq. (11)] is CM = 0.04, which is a little larger than the value of 0.03 that is recommended by Grant (2001).

The mass-flux profile also differs among the family of similar schemes. The M-K scheme mass flux is roughly constant with height in the subcloud layer and decreases monotonically in the cloud layer. The vertical velocity profile produces a variable updraft area fraction that is about 3% in the middle of the subcloud layer. Siebesma and Teixeira (2000) specify a mass-flux profile that is proportional to the vertical velocity variance, therefore having a peak near the middle of the subcloud layer. Jakob and Siebesma (2003), in their entraining plume model, specify a fixed updraft area fraction (3%) and use a prognostic vertical velocity, which would result in a similarly peaked mass-flux profile. Soares et al. (2004) specify a larger fixed updraft area (10%), which will result in a smaller mass flux. Whether the mass flux is constant or peaked in the middle of the subcloud layer is apparently not critical; the important thing is to have a reasonable mass flux in the upper part of the layer, and at cloud base. One physical argument for constant mass flux is that the updraft represents the cores of the most energetic thermals, which transit the subcloud layer with minimal change. A mass flux varying with height may be more appropriate to represent the entire distribution of (positive) vertical velocities, but this would in principle require reformulating the scheme in terms of updraft and downdraft rather than updraft and environment.

Another difference is the specific choice of lateral fractional entrainment- and detrainment-rate profiles, which are the key sensitive parameters in the scheme. In most similar schemes, the fractional entrainment rate depends inversely on height near the surface. That dependence was not found to be necessary in this scheme, and so it was removed for simplicity. Even so, the mass-flux initialization was not found to sensitively affect the results. Unlike some similar schemes, when cloud is not present, the thermal also detrains in such a way that the mass flux is roughly constant with height in the lower and middle boundary layer and decreases near the boundary layer top. When cloud is present, the fractional entrainment- and detrainment-rate profiles are such that the mass flux remains roughly constant below the cloud base and decreases roughly linearly to the cloud top. These behaviors are both physically appealing and consistent with LES results (Brown et al. 2002). This would, of course, not be correct for situations in which anvils or stratus were produced atop the shallow cumulus layer.

The use of eddy diffusivity in the cloud layer is another difference between this scheme and Siebesma and Teixeira (2000), although it is used by Soares et al. (2004). Here, it was introduced so that the scheme could reproduce the SOS Nashville case, in which the cloud-base potential barrier was smaller than in the ARM case. The diffusivity in both layers is based on prognostic TKE and a length scale. This formulation was chosen in order to make the scheme easily extensible to conditions other than free convection. In the ARM case, the model has less TKE in the cloud layer than LES, probably because it has less positive buoyancy flux. Further tuning might improve this, but too much mixing can lead to intermittency (Lenderink et al. 2004). The length scale in the subcloud layer is a simple function of distance from the boundaries (surface and dry thermal top). Bougeault and Lacarrere (1989) present a more elaborate form of the length scale that is based on the distance an eddy can travel against the stability in either direction. In the cloud layer, the length scale is fixed at 100 m. More complex functions could be used, such as that proposed by Lenderink and Holtslag (2004), but the simple fixed scale seems to work well. The combination of the fixed length scale and the prognosed TKE profile produces a roughly constant eddy diffusivity in the lower and middle parts of the cloud layer, which is roughly consistent with the profile found by Cheinet and Teixeira (2003), Stevens et al. (2001), and Siebesma et al. (2003). Teixeira and Cheinet (2004) and Cheinet and Teixeira (2003) present an alternative formulation for K that replaces the length scale with a time scale; the implications of that alternative would be worth exploring.

The current M-K scheme diagnoses cloud top by moist-adiabatic ascent of an undiluted parcel from cloud base (LCL). This is simple and stable. Other schemes (Jakob and Siebesma 2003; Soares et al. 2004) instead use updraft vertical velocity to diagnose the cloud top. Because the lateral fractional entrainment and detrainment rates are such that the mass flux decreases to small values near the cloud top, any difference in cloud-top diagnosis should result in only small differences in thermodynamic or scalar profiles. In fact, within the model, the cloud top only serves to define the profile of detrainment. However, exploration of the alternative choice for cloud-top diagnosis would be worthwhile.

Previous work on chemical transport by clouds was reported by Lin et al. (1994). They incorporated a parameterization of subgrid convective cloud transport into a 3D regional chemistry model and compared the results with aircraft observations. The primary emphasis was on the deep convective cloud. Because of the relatively coarse vertical grid spacing in their model, they used a very simple representation of shallow cloud. However, their Fig. 7 shows some impact on the chemical profiles from the shallow cloud vertical transport.

The current version of the M-K model works for free convection only. The model will be extended to handle stable and transitional conditions by including equations for wind. Making smooth transitions rather than abruptly switching between parameterizations is important to handling the stable–convective and convective–stable transitions. Basing the eddy diffusivity on TKE should facilitate handling conditions other than free convection.

Conceptually, the M-K model involves only a single updraft and therefore a single cloud. The properties of the updraft and cloud therefore represent some aggregate of all of the real updrafts and clouds in the grid box. The ensemble approach of Neggers et al. (2000) would improve the representation at the cost of increased complexity.

Acknowledgments

The author is most grateful for the generosity of Pier Siebesma, Roel Neggers, and Joao Teixeira, who spent considerable time introducing him to an unfamiliar specialty. This work was begun while the author was a visitor at the Royal Netherlands Meteorological Institute (KNMI), the hospitality of which is gratefully acknowledged. Michael Trainer of the NOAA Aeronomy Laboratory provided and analyzed the aircraft data. The aircraft CO measurements were made by John Holloway and David Parrish. The sounding data were provided by Mohammed Ayoub of the University of Alabama at Huntsville, in cooperation with the NOAA Climate Monitoring and Diagnostics Laboratory. Geert Lenderink provided the LES results, which were produced by Roel Neggers. Andrew Brown provided the ARM observations. The careful reviews of three anonymous reviewers contributed substantially to the improvement of the manuscript.

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Footnotes

Corresponding author address: Dr. Wayne M. Angevine, NOAA Aeronomy Lab R/AL3, 325 Broadway, Boulder, CO 80305. wayne.m.angevine@noaa.gov