a. Background

The initial value problem consisting of a uniformly stratified fluid evolving under the action of a constant surface buoyancy flux is perhaps the simplest representation of the dry convective boundary layer. It shall serve as the basis for our study of the effects of cumulus convection and thus merits a brief review. Other than parameters describing the properties of the working fluid, the only parameters in this problem are the stratification, which is measured by the Brunt–Väisälä frequency, N ², the surface buoyancy flux denoted by B₀, and the time, t, since initialization. It follows dimensionally that the growth law (1) describes the evolution of a mixed layer growing at the same rate as the convective layer, so long as the evolution of the layer is independent of the Prandtl and Reynolds number and (tN)^1/3. The latter is a nondimensional measure of the ratio of a convective time scale to a gravity wave time scale within the stratified layer and might be expected to measure the importance of energy radiation by waves in the free troposphere. In this limit, the thickness of the inversion layer; that is, that layer whose stratification is greater than the background fluid and is often associated with the presence of penetrative convection, can be shown to scale with the depth of the convecting layer. And so the growth of boundary layer as a whole (irrespective of whether or not is well mixed) can be parameterized by a model consisting of only a single length scale, for example, a simple, or zeroth-order, mixed layer. For such a model it also follows that the depth of the mixed layer, h_m, is given for all time by

View Expanded

The scale height h_* is, by assumption, the only length scale in the problem. It is determined by the external parameters and is often called the encroachment depth (e.g., Carson and Smith 1974; Driedonks 1982; Fedorovich 1998) as it describes the growth rate of the layer in the absence of penetrative convection and when an inversion layer is not readily evident. For this case k = 0 equivalently α = 1. The salient aspect of (2) is that it predicts a scaling regime in which h ∼ t^1/2.

Figure 1 shows both the initial profile and the simulated boundary layer structure from a large-eddy simulation for this initial value problem. Although not shown, it turns out that the growth law (2) is well satisfied by this simulation. Here the height, h_m, of the mixed layer used to scale the growth of the actual convective layer is determined by matching the enthalpy of the two layers within the warmed layer, subject to the overall constraint of enthalpy conservation. Different choices for the mixed layer height, such as the height of the minimum buoyancy flux, h_B, or the height of the maximum buoyancy gradient h_θ, would lead to slightly different values of k.¹

b. Prototype problem

To understand how cumulus convection affects the evolution of the developing thermal boundary layer we extend the canonical problem of the dry convective boundary layer to include clouds. To do so, we propose the modified initial value problem illustrated by Fig. 2. This figure shows a growing and moistening thermal boundary layer at two times, before (dashed lines) and after (dotted lines) the development of clouds. The initial conditions are shown by the solid gray lines. In analogy to the dry convective boundary layer, the modified problem consists of the growth of a thermodynamic boundary layer into a layer whose density is uniformly stratified over a saturated water surface. The surface temperature is varied to maintain a constant surface buoyancy flux, B₀, which is the only source of driving of the flow. Given this basic framework the only unconstrained aspect of the problem is the specification of the initial humidity profile. Ideally, one would like a profile that allows the cloud layer to develop over a reasonable range of heights, and permits its development thereafter to proceed in some statistically invariant manner. The profile chosen with these ideas in mind is the following:

View Expanded

It corresponds to a relative humidity profile that decreases exponentially with height. Equation (3) also ensures that the initial equivalent potential temperature, θ_e ≈ θ exp[q_υL/(c_pT)], decreases with height. Such a negative gradient in θ_e is both a realistic representation of the lower tropical troposphere, as well as a necessary condition for the growth of a conditionally unstable cloud layer.

Our extension of the canonical problem of the growth of the dry convective boundary layer to allow for cumulus convection introduces three additional parameters (not including microphysical, radiative, or diffusive parameters). These being the two parameters, λ and q₀, introduced explicitly by (3) as well as the moisture scale height

View Expanded

which depends on Θ. The other constants have their usual meanings and are defined in Table 1.

Our motivation for fixing B₀ is twofold: it maintains continuity with the standard representation of the dry convective layer, which simplifies the dynamics of the subcloud layer; and it allows the surface fluxes to be represented by a single parameter. This means that the surface Bowen ratio β ≡ c_pQ₀/(LR₀), where Q ≡ w′θ′ and R ≡ w′q′ is free to vary. We relate the surface fluxes, Q₀ and R₀, to the near surface thermodynamic state as

View Expanded

Both θ₀ and q₀ are determined by the surface temperature and pressure, assuming saturation to determine q₀; θ₀₊ and q₀₊ correspond to the thermodynamic state just above the surface; and V is a constant set to 0.01 m s⁻¹.

Some insight into the behavior of the moist system can be obtained by first studying how a mixed layer would evolve if it deepened following (2). In this case, the mixed layer profiles, θ_m and q_m, must evolve in a way that satisfy their respective conservation laws, namely,

View Expanded

View Expanded

Given q_m, θ_m, and the surface pressure p₀ we can calculate the lifting condensation level, η(t). So long as h_m < η, one would expect the mixed layer representation of the flow evolution to be qualitatively correct. Thereafter the solutions can be expected to increasingly depart from the actual evolution of the layer, as the assumption of a mixed layer evolution implicit in (6)–(7) and the growth law (2) fails to account for the effects of clouds.

The evolution of h_m and η, as predicted by the mixed layer model is shown in the left panel of Fig. 3. Because of the temporal evolution of Q₀ and R₀ the integrals are evaluated numerically with α = 1.3, Γ = 6 × 10⁻³ K m⁻¹, and B₀ = 7 × 10⁻⁴ m² s⁻³. Here we note that η equilibrates relatively quickly and h_m grows steadily in time, following the t^1/2 rule as prescribed by (2). Eventually h_m exceeds η, and clouds are predicted to develop with bases near 600 m after about 8 h. These key features, cloud-base height and cloud onset time are shown as a function of B₀ and λ in the right panel of the same figure. Cloud onset time is most strongly determined by λ through its regulation of η, while the height at which h_m = η depends on both λ and B. We note that realistic cloud-base heights (500–700 m being a good tropical rule of thumb) are evident over much of the parameter space.

Because the prediction of the mixed layer model, given by (6)–(7) with (2), does not incorporate cloud processes, it provides a reference for measuring how the development of the cloud layer in the large-eddy simulations (which do incorporate cloud processes) affects the evolution of the boundary layer as a whole, and how this varies as a function of the major parameters of the system. Already from Fig. 2, in which the profiles of θ and q are taken from the large-eddy simulations to be presented subsequently, one can deduce that clouds must have some effect. Whereas (2) would have predicted the boundary layer depth to have doubled in the period between 24 000 and 96 000 s, its actual depth has almost tripled. Using large-eddy simulation as a proxy for the evolution of actual clouds, our goal in the following sections is to better understand this underlying tendency of the clouds to accelerate the growth of the boundary layer.

a. Large-eddy simulation model

The code used to perform the large-eddy simulations is presented in some detail in the appendix of Stevens et al. (2005). Here we briefly summarize by noting that it solves the anelastic equations2 of Ogura and Phillips (1962) over three spatial dimensions. Centered differences in space and time are used to represent momentum advection, and monotone-centered-slope-limited upwinding is used for the advection of scalars. The Smagorinsky subgrid model is employed to represent subgrid fluxes of both scalars and momentum. The prognostic model variables are the three components of the velocity vector, the liquid-water potential temperature, θ_l = θ exp[−q_lL/(c_pT)] introduced by Betts (1973), and the total water mixing ratio, q. Liquid water, q_l, is diagnosed assuming equilibrium conditions and no rain is allowed to develop. As we shall discuss later, this latter assumption becomes increasingly untenable as the cloud layer deepens.

The model is initialized with a horizontally homogeneous mean state and no mean wind. To specify an initial state of constant N ², θ_i(z) is calculated based on q_i(z). To break the slab symmetry of the initial conditions random fluctuations are added to the potential temperature field near the surface. The model is then stepped forward in time, with the sea surface temperature held homogeneous over the domain, but varying in time so as to maintain B₀ at the desired mean value given (5), with θ₀₊ and q₀₊ given by their values at the first model level above the surface. Eleven simulations are analyzed, most on a grid whose horizontal spacing is 75 m and whose vertical grid is 5 m near the surface and stretched by a constant factor of 2.5% per grid interval. This stretching results in a grid spacing of about 30 m near z = 1200 m and 120 m near the model top at 4750 m. The standard horizontal domain consists of 96 points in the horizontal and 131 points in the vertical. Simulations on grids twice as large, or twice as fine, suggest that the results we center our analysis around are not especially sensitive to modest changes in the calculation grid, although the larger-domain simulations do yield more reliable statistics at late times in situations when the cloud layer is relatively deep and the finer resolution simulations are more robust at very early times; that is, when h < 200 m.

b. Time evolution

In Fig. 4 we show the time evolution of distinguished layers (Fig. 4a) and cloud fractions (Fig. 4b) from a simulation of the prototype problem with the SST chosen so that c_pρ₀Θg⁻¹B₀ = 25 W m⁻², λ = 1500 m, Γ = 6 K km⁻¹, and q₀ = 0.8q_s,0. Figure 5 shows the evolution of the sea surface temperature (SST) and surface fluxes from this same simulation.

Two regimes are clearly evident: a cloud-free regime at early times (t < t₁ = 8.4 h), and a regime with approximately constant cloud fraction at later times (t > t₂ = 13 h). The times t₁ and t₂ are defined when the cloud first exceeds 1% of its value averaged over the final 2 h of the simulation, while t₂ corresponds to the time at which the cloud fraction first exceeds 90% of that value. The time of cloud onset, t₁ = 8.4 h is predicted relatively well by the mixed layer equations (which predicted h_m = η at 8.3 h). Subsequent to t₂, η tracks h_B (the height of the minimum buoyancy flux) indicating that the latter effectively measures the depth of the subcloud layer at late times. For comparison, we note that the profiles in Fig. 2 are taken from the base simulation at t = 8 and t = 30 h, respectively.

At early times, that is, t < t₁, the boundary layer grows with the square root of time. This is true irrespective of how one measures the boundary layer depth, with h_* < h_B < h_m < h_θ each existing in fixed ratio with respect to one another, especially toward the latter part of the early period when the boundary layer depth is much larger than the grid mesh over which it is represented, that is, h ≫ Δx. By any measure of the layer’s depth, h > h_*, hence penetrative convection is contributing to the growth of the layer. The signature of penetrative convection is also evident in the development of a thin layer, centered near h_B, in which θ < θ_i. This implies the development of stronger temperature and moisture gradients than were initially present, and is often referred to as an inversion layer because the stratification is sometimes sufficient for the temperature to actually be increasing at h_θ. Because such a layer emerges spontaneously, treating it as a controlling parameter that should be specified externally, as is common practice (e.g., Nieuwstadt et al. 1991; Sullivan et al. 1998), only adds unnecessary complication.

The surface buoyancy flux is calculated as

View Expanded

For fixed β (Bowen ratio) and B₀, the SST (equivalently θ₀) should increase as θ₀₊ (equivalently θ_m), that is, also proportional to t^1/2. In Fig. 5 we see that β is decreasing with time, indicating that the SST must increase slightly less rapidly than t^1/2. The actual time rate of change of the SST and the surface fluxes is predicted by the bulk model to within a constant offset (0.15 K, −9.5 W m⁻², and 0.5 W m⁻² to the SSTs, latent and sensible heat fluxes, respectively), which arises as a result of a surface layer in the simulations. The emergence of such a layer (the precise characteristics of which likely are sensitive to the grid) results in a slightly larger SST and a larger surface Bowen ratio than predicted by (6)–(7) with θ_m and q_m substituted for θ₀₊ and q₀₊ in the calculation of the surface fluxes.

At late time, that is, t > t₂, the boundary layer enters a qualitatively different scaling regime, wherein we witness the emergence of a cloud layer whose top, h, grows linearly with time, t. This is evident irrespective of how one measures the depth of the layer. At late time the cloud fraction is stationary at about 10% and the SSTs and surface fluxes increasingly depart from what would have been expected had the solutions remained cloud free. In this late time regime, the SST has to increase more markedly to maintain the same buoyancy flux, and the surface Bowen ratio tends to decrease more strongly. The latter is not unlike the evolution of the Bowen ratio along trajectories of low-level winds in the trades (e.g., Riehl et al. 1951) and suggests that our specification of a constant surface buoyancy flux is not unduly artificial. For those readers who are concerned that the rate of increase in the SST in Fig. 5 is unrealistic we note that (i) the actual rate of SST increase, and hence surface buoyancy flux, ends up being immaterial to the arguments we are making; (ii) because we study the problem in the absence of radiative cooling in the atmosphere, for which 2 K day⁻¹ is a typical value, it is necessary to increase the surface temperature more rapidly to maintain the same rate of driving of the flow; and (iii) the rate of SST increase implied by the prescribed surface buoyancy flux is varied later in this study thereby substantiating our first point.

The cloud layer at late times (Fig. 2) consists of a deepening layer that has been both cooled and humidified relative to the initial sounding. Cloud fraction peaks at the base of this layer and decreases through the depth of the cooled zone. The capping inversion is now much more pronounced than it was previously and caps a cooled layer that is no longer a well-mixed extension of the lower boundary layer, but follows a conditionally unstable lapse rate, Γ_c ≃ 0.0022 K m⁻¹. The maximum of cloud fraction near the base of the cloud layer, and the secondary maximum in the gradient of q near 900 m are signatures of a cloud-base transition layer. All of these features are quite realistic, and in accord with the growing body of literature on trade-cumulus convection (see Siebesma 1998; Stevens 2005, for recent reviews). These results also suggest that properties such as the trade inversion, the cloud-base transition layer, and the level of conditional instability in the cloud layer, should be viewed as emergent properties of the simulation, rather than parameters, or boundary conditions, to be specified externally. The merit of this distinction becomes evident when deciding what parameters to use to describe the asymptotic behavior of the layer.

On the basis of this analysis alone, one can begin to glimpse how shallow moist convection changes the world. Trade winds admitting only dry convection would be shallower, and surface heat fluxes would be less, leading to less cooling of the surface ocean, and less charging (moistening) of the atmosphere in anticipation of deep convection (see also Tiedtke 1989; McCaa and Bretherton 2004; Zhu and Bretherton 2004, for similar conclusions). In the next section we will try to put these changes in the theoretical framework sketched out in the introduction.

a. The energetics of ballistic growth

We begin by considering the budget of buoyancy as measured by the density potential temperature θ_ρ, which following Emanuel (1994) we write as

View Expanded

This differs from the definition of density potential temperature used by Betts and Bartlo (1991), corresponding instead to what they (and others) call the virtual potential temperature for cloudy air. We adopt the Emanuel terminology because the literature is ambiguous as to whether liquid water effects should be included in the virtual temperature, and because θ_ρ makes an explicit reference to the density, differences in which underlies all convection.

The profile of θ_ρ is shown at two different times in Fig. 6. The evolution of the layer is marked by θ_ρ increasing with time in the subcloud and cloud layers, and decreasing with time at the top of the cloud layer: thus defining an inversion layer, which like for the dry convective boundary layer, cools as it deepens. Quantitatively the rate of increase in θ_ρ is remarkably constant through the cloud and subcloud layers (Fig. 6, middle panel). Moreover, as we shall soon show, the rate of warming is commensurate with what one would expect had the subcloud layer been evolving without the subsequent development of clouds.

From (9) and the definition of θ_l, fluctuations in θ_ρ can be linearly related to fluctuations in θ_l, q, and q_l as follows:

View Expanded

where a₁, a₂, and a₃ are thermodynamic parameters, with a₂ given by (8). If we neglect fluctuations among thermodynamic perturbations, then a₁ and a₃ can also be expressed as effective constants:

View Expanded

where Θ_ρ, Θ_l, q_l, and T are taken to be some fixed value representative of the cloud layer. In cumulus layers q_l ≈ 1 × 10⁻⁵ kg kg⁻¹, so for Θ_l ≈ Θ_ρ, a₁ ≃ 1 and a₃ ≃ L/(c_pT) − R_υ/R_d ≃ 7. At this level of approximation perturbations in θ_ρ are indistinguishable from perturbations in θ_υ,l the liquid-water virtual potential temperature (e.g., Emanuel 1994).

Equation (10) provides a basis for studying the budget of buoyancy. By correlating fluctuations in θ_ρ with fluctuations in w it follows that

View Expanded

where Q_l and R_l denote the vertical kinematic flux of θ_l and q_l, respectively. This defines the buoyancy flux B ≡ (g/Θ)Q_ρ, which measures the exchange between turbulence kinetic and mean-flow potential energy.

To see how the divergence of the buoyancy flux relates to the temporal evolution of buoyancy we recognize that in horizontally homogeneous layers, for which there exist no sources of θ_l or q (i.e., in the absence of radiation and precipitation), then

View Expanded

where C is the net rate of condensation. Associating fluctuations in (10) with changes in time, it follows from above that

View Expanded

This equation emphasizes that in saturated flows, buoyancy is not conserved—condensation acts as a source term in the buoyancy budget. However, in the case when

View Expanded

which in the limit corresponds to q_l being stationary (the case of unsaturated layers being the trivial case), then

View Expanded

which is conserved. Hence, in the stationary limit of q_l, θ_ρ evolves according to

View Expanded

It turns out that this flux, Q̃_ρ, proves central to understanding the energetics of the cloud-layer growth. From (14) and (17) we note that

View Expanded

which defines Q̃_ρ, to be proportional to that part of the buoyancy flux not associated with perturbations in q_l. This equation emphasizes the fact that in unsaturated layers Q̃_ρ, = Q_ρ = (Θ/g)B. At a level of approximation commensurate with our determination of the constants a₁ and a₃, Q_ρ is equivalent to the flux of liquid-water virtual static energy (e.g., Bretherton and Wyant 1997; Bretherton and Park 2007, manuscript submitted to J. Atmos. Sci.), while Q̃_ρ is what Lewellen and Lewellen (2002) call the “dry” virtual potential temperature flux.

To the extent that the evolution of the cloud layer is well described by (17), which as we show below turns out to be the case, then quasi steadiness³ implies

View Expanded

In other words, Q̃_ρ must be linear. This condition is reasonably well satisfied below the level z_a for the simulation analyzed in Fig. 6. The profile of Q̃_ρ in this figure makes two further points: (i) its slope is more or less the same in the cloud and subcloud layer, as we would expect from the fact that the cloud and subcloud layer are warming (as measured by θ′_ρ) at the same rate; and (ii) in the subcloud layer Q̃_ρ is indistinguishable from the shape it would assume in the absence of an overlying cloud layer. This latter point is the foundation of almost all models of cumulus convection (e.g., Betts 1973). To the extent that clouds merely act to equilibrate the density of the cloud layer to that of the subcloud layer, which itself evolves according to (1), then

View Expanded

which follows from simple linear extrapolation of Q̃_ρ given its surface value Q̃_ρ,0, and its value, kQ̃_ρ,0, at the top of the subcloud layer.

Equation (20) helps explain why the cumulus-topped layer grows more efficiently than the cloud-free convective boundary layer. For both, the divergence of Q̃_ρ at z = h drives the growth of the layer. However in cloud-free convective layers Q̃_ρ ∝ B, and thus is energetically constrained to be some fraction of the surface buoyancy flux B₀. This leads to constraints of the form (1). For cloudy layers this energetic constraint is relaxed, for example, Eq. (18), and increasingly negative Q̃_ρ can be supported by the transport of liquid water in the clouds, thus allowing Q̃_ρ at h to scale with h, thereby supporting a different (ballistic) scaling regime for the evolution of the layer depth.

These relationships are illustrated in Fig. 7, where we plot the three terms in (18) in buoyancy units. Throughout the cloud layer, B is nearly constant and significantly smaller than ga₃R_l. In the inversion (between z_b and z_a) B is nearly zero, and so the divergence of g/ΘQ̃_ρ is supported almost entirely by the convergence of the liquid water flux at these levels. This helps quantify the discussion in the introduction, showing that, at least in this prototype problem, the growth of the layer is best interpreted as resulting from the enthalpy of vaporization driving an increase in mean flow potential energy. Unlike in dry layers, the turbulence kinetic energy plays at most an indirect role, in that it supports the circulations that deposit the liquid water in the inversion, and the subsequent mixing that leads to its evaporation.

b. Detailed budget

To better quantify these ideas, in this section we examine the budget of the inversion layer in more detail, thus addressing the question as to what role the inversion layer thickness, and the character of the buoyancy jump at cloud top, play in the evolution of the layer.

In the limit where q_l is stationary following the layer, the weak form of the conservation law (14) is given by the integral of (17) between z_a and z_b. Assuming to a first approximation that z_a and z_b evolve at the same rate, so that dh/dt = dz_a/dt = dz_b/dt yields

View Expanded

This equation is the basis for linking flux laws of the form (1) to growth laws of the form (2), but for layers of finite thickness (cf. Sullivan et al. 1998). An analysis of our baseline large-eddy simulation (LES) shows that (21) is satisfied to within 1%, indicating the validity of the assumptions used in its derivation. Assuming that Q̃_ρ(z_b) is negligible compared to Q̃_ρ(z_a) and that both terms on the lhs of (21) are proportional to one another; that is,

View Expanded

allows us to express (21) as

View Expanded

In the dry convective boundary layer, the assumption of a single length scale implies that Δθ_ρ ∝ Γh and Q̃_ρ(z_a) ≈ constant, in which case by (23) hdh ∝ dt or h ∝ t^1/2 so long as (22) prevails. In contrast, if Q̃_ρ(z_a) ∝ h, as we found to be the case for the cumulus-topped boundary layer, then (23) implies that h ∝ t so long as Δθ_ρ continues to scale with h.

To further develop these ideas we explore a similarity hypothesis analogous to that used to scale the dry convective boundary layer. Specifically, we hypothesize that an equivalent (bulk) cloud-topped layer of some depth h, consisting of a well-mixed subcloud layer, a cloud layer whose lapse rate Γ_c is constant, and an infinitesimally thin inversion layer⁴ deepening under the action of an equivalent θ_ρ flux, Q̃_ρ ∝ (h − η), can scale the growth of the actual cloud-topped boundary layer as represented by our simulations. An example of such a layer, constructed from the 33–36-h mean profiles of θ_ρ is given by the dotted line in Fig. 6. For such a layer

View Expanded

By conservation of enthalpy

View Expanded

which given Γ_c defines θ̂_ρ. Substituting (20) and (24) into (23) leads to an expression for the growth rate of the cumulus layer,

View Expanded

where κ, the nondimensional deepening rate, is a constant that carries the proportionality in (23). An aspect of (26) worth drawing attention to is that the moisture scale height λ in the initial profiles does not appear explicitly in the growth rate. It is implicit in that it helps determine η and may determine the extent to which our assumption (15) remains valid. This suggests that, to the extent (26) adequately describes the growth of the cloud layer, once η is set the amount of moisture in the free troposphere has little effect on the rate of growth of the cloud layer (insofar as precipitation remains unimportant), even if it may end up being important to the ensuing structure of that layer. Numerical integration of (26), with η and k fixed, thetascirc_ρ given by (25) with Γ_c = 2.2 K km⁻¹, and h = η as an initial conditions, yields values of h that increase linearly at late times (after a few hours). This result is commensurate with Fig. 4 and suggests that (26) is at least qualitatively correct.

As a quantitative test, we evaluate whether (26) can scale the actual growth rate from a range of simulations chosen to sample the parameter space of the prototype problem, details of which are summarized in Table 2. In Fig. 8 we plot (26) versus the actual deepening rate dh_θ/dt as determined by the best-fit linear slope to h_θ for the period t > t₂, which corresponds to the period of stationary cloud fraction in Fig. 4. In evaluating (26) we take k = −0.3 from our simulations for the dry convective boundary layer. Figure 8 shows that the actual growth rate of the cloud layer is well predicted (scaled) by (26) given a nondimensional growth rate of κ = 0.8. The dispersion in k among simulations is only about 10% over a nearly fourfold change in dh_θ/dt. Also evident in this figure is that much of the scatter can be associated with the use of a single value for Γ_c. The lapse rates in the cloud layer vary slightly, but systematically, among the simulations (see Table 2), and unaccounted for variations in Γ_c account for a significant amount of the scatter in the predicted versus the actual growth rates. The variation of κ for the same case, but simulated using different grids and evaluated at different times (marked by open squares), is similar to the variation of κ among simulations with different parameter sets. This suggests that the simulations probably do not sufficiently sample the flow to provide more quantitative information about systematic variations in quantities like the lapse rate or the inversion layer thickness and their effect on the growth of the layer.

In summary, the growth of the cloud layer in this prototype problem is well described by assuming that (i) changes in the cloud water field at a given level contribute negligibly to the evolution of θ_ρ; (ii) the increase in θ_ρ in the cloud layer tracks changes in the subcloud layer; and (iii) the energetics of the subcloud layer are, to leading order, indistinguishable from those of a dry convective boundary layer. The last assumption requires the cloud-base buoyancy flux to be a fixed fraction of its value at the surface, and the first two require the equivalent θ_ρ flux at the top of the cloud layer to scale with the depth of the cloud layer following (20).

a. Thermodynamic constraints

The first question can be posed formally as follows:

View Expanded

In words: why does the density potential temperature in the cloud layer so closely track its value in the subcloud layer? Perhaps to preserve the stability of the cloud layer with respect to unsaturated perturbations from the subcloud layer? Maintaining the cloud layer at neutral buoyancy with respect to saturated perturbations from the subcloud layer requires that θ_s (the saturated value of θ_e) in the cloud-layer track θ_e in the subcloud layer; that is,

View Expanded

Figure 9 shows that this constraint is also satisfied by the simulation. Satisfying both the unsaturated (27) and saturated (28) constraints, and assuming that the cloud and subcloud layer are moistening at the same rate, effectively partitions the moisture and heat flux divergences such that

View Expanded

which upon integrating allows one to differentiate between the heat and moisture fluxes themselves.

b. Relation to previous work

With a little scrutiny the thread of the ideas presented above is apparent in the fabric of much previous work, beginning with that of Betts (1973). While Betts framed his study around the enthalpy budget of an existing cloud layer, his ideas encapsulate much of the behavior of our prototype problem of a developing convective layer under the action of a fixed surface buoyancy flux. Of the many similarities, his invocation of the assumption that the liquid water is stationary, and the way in which he formulates the problem as an extension of the development of the cloud-free convective boundary layer are especially important. His results also anticipate the linear growth regime, which we highlight in the present work. In particular, the unnumbered equation following his Eq. (62) implies that the cloud layer will grow linearly with time in the case when the inversion layer thickness scales with the depth of the cloud layer as a whole. Although Betts only addresses the stability and energetics of the system as a way to address the question of the inversion layer thickness, some of the similarities in the two studies follow from the fact that the effective buoyancy flux around which we center our analysis so closely tracks the flux of θ_l around which Betts centers his analysis. This follows largely from (17) and (29), which together imply that

View Expanded

In other words, to a first approximation the divergence of the enthalpy (as measured by c_pθ_l) flux is a good approximation to the divergence of the density flux. These similarities necessarily carry over to the finding of Betts (1973, 1975, that the liquid water flux, R_l, is a dominant contributor to the flux of θ_l, and our finding that the R_l term (e.g., Fig. 7) supports the large downward flux of buoyancy in the cloud layer. The similarity between Q̃_ρ and Q_l also explains why the scaling of Moeng (2000), which was also based on an analysis of the budget of θ_l, so effectively scaled the growth of the stratocumulus layers she studied.

In their study of different shallow cloud regimes, Lewellen and Lewellen (2002) do focus on the energetics of deepening cloud layers, and in so doing suggest a constraint [their Eq. (10)] essentially identical to our (20) but phrased in terms of what they call D, the buoyancy flux for dry layers. They show its appropriateness for shallow cumulus layers, particularly in so far as they rise into an upper layer of stratocumulus. From this perspective we are simply arguing for the more general validity of their result. In so doing we provide an argument for its interpretation and energetic support (the liquid water flux) especially in layers where the cloud layer is far from uniform (i.e., in situations that violate their assumption 2).

c. Implications for ongoing work

While the linear growth law of the cloud layer will only be realized in the limit of a constant surface buoyancy flux and uniform outer layer stratification, our understanding of why this regime is manifest should also help us understand the behavior of cumulus-topped layers more generally. In particular, because θ_ρ evolves under the divergence of the flux Q̃_ρ, which is closely related to Q_l, both of which must be linear⁵ for the quasi-steady evolution of the layer, it is not surprising that arguments based on the enthalpy budget are so successful in describing the boundary layer evolution. More importantly, this suggests that simple models that treat the evolution of the enthalpy budget consistently should be able to well represent the rate of deepening of cloud-topped boundary layers. That said, because the parameter complexity of simple parameterizations (in terms of dimensional constants) often exceeds that of the underlying flow, it is less clear that parameterizations forced under the conditions of this simple problem recover the simple scaling of (26). As such the prototype problem we propose here might prove to be a useful benchmark for parameterization schemes.

A better understanding of the mechanism by which cumulus-topped layers deepen is also useful when thinking about processes that may arrest the ballistic growth of the trade wind layer. These include large-scale subsidence, precipitation, and unfavorable gradients in θ_e. As the cloud layer deepens, precipitation can be expected to become increasingly efficient at depleting the liquid-water flux, and thus arresting the growth of the layer. Preliminary simulations of our prototype case with a simple microphysical scheme (to be discussed further in a subsequent paper) indeed show that as the precipitation rate increases the cloud layer ceases to deepen. These simulations also show that Γ_c,, which is approximately fixed in the nonprecipitating regimes, increases markedly as precipitation increases. Because the depth of the cloud layer necessary to produce a precipitation flux large enough to balance the upward flux of liquid water depends on the ambient concentration of condensation nuclei this suggests yet another mechanism whereby the chemical state of the atmosphere (as measured by the atmospheric aerosol) may affect the structure of larger-scale circulations, and hence climate. The ability of precipitation to quench the growth of the cloud layer may also help explain simulations of stratocumulus whereby increasing precipitation, or sedimentation, fluxes appear to reduce the entrainment rate (e.g., Stevens et al. 1998; Ackerman et al. 2004; Bretherton et al. 2007).

Last, the mechanisms we discuss here suggest a new twist on how we usually view the role of large-scale θ_e gradients (cf. Squires 1958; Randall 1980; Deardorff 1980). For negative θ_e gradients, the ballistic (water injection) growth mechanism provides an efficient means for deepening the boundary layer. In unfavorable θ_e gradients the injection of water into the inversion layer can lessen, but not obliterate, unfavorable density gradients. In the latter case, the rate of working against the outer-layer stratification, by a negative buoyancy flux, is essential to sustain boundary layer growth. Because doing work on the outer-layer stratification proves to be an inefficient way to grow a boundary layer, such situations would appear to be more favorable for moistening the cloud layer and the development of a layer of stratiform clouds at its top.