## 1. Introduction

What sets the depth of layers of shallow cumulus convection? The rate at which such layers deepen determines the rate at which dry free-tropospheric air is mixed to the surface, and hence the surface fluxes. The deepening rate also helps determine the time of onset of precipitation, which is closely tied to the depth of the cloud layer (Byers and Hall 1955). The impact of cumulus convection on surface fluxes has been shown to significantly affect larger-scale circulations and the skill of medium-range weather forecasts (Tiedtke 1989). The modulating effect of cloud-layer depth on precipitation may be important to the diurnal cycle of convection over land (Khairoutdinov and Randall 2006) as well as aerosol effects on clouds (Albrecht 1989). In addition, layers of fair-weather cumuli, such as those found in the trades, are increasingly being recognized as a key point of departure among models of the climate system, with much of the variance in model-based estimates of climate sensitivity being attributed to differences in the modeled response of layers of shallow convection (Bony and Dufresne 2005; Wyant et al. 2006).

The question of the growth of layers of shallow, nonprecipitating, cumuli has long been phrased in terms of the maintenance of the trade wind layer. In addressing this very question, Riehl et al. (1951) write:

It is well known that the bases of the cumuli have a nearly uniform height, but that the tops are very irregular. Some are found within the cloud layer, many near the inversion base, and some within the inversion layer as active clouds penetrate the base. As shown by visual observation and many photographs, the tops of these clouds break off and evaporate quickly. In this way moisture is introduced into the lower portions of the inversion layer, and the air there situated gradually takes on the characteristics of the cloud layer.

Their findings, as embodied by this quote, helped establish the view that the cumulus-induced transport (and subsequent evaporation) of liquid water into the trade wind inversion layer is the principal mechanism counteracting the drying and warming that accompanies the prevailing subsidence. Riehl et al.’s point of view, based on budget analyses of the trade winds of the northeast Pacific, received further support from similar analyses of data collected as part of the Atlantic Tradewind Experiment (ATEX; Augstein et al. 1973) and Barbados Oceanographic and Meteorological Experiments (BOMEX; Nitta and Esbensen 1974).

These ideas were put in a theoretical context by Betts (1973), whose model of the cumulus layer was based on the budgets of enthalpy in two sublayers: an upper, stably stratified, inversion layer, identified with divergence in the enthalpy flux, net evaporation, and cooling; and a lower, conditionally unstable, cloud layer, identified with a convergent enthalpy flux, net condensation, and warming. Betts showed how this enthalpy flux is largely carried by the transport of liquid water, which is condensed at one level (the cloud layer), and evaporated at another (within the inversion), thus cementing the conceptual foundations of the earlier budget studies.

*dh*/

*dt*is expressed as the ratio between the entrainment buoyancy flux,

*, and the jump in buoyancy, Δ*

_{h}*b*, across the capping layer. Here

*measures the rate at which turbulence energy is converted to potential energy, and Δ*

_{h}*b*measures the stability of the interface. (Throughout we use subscript

*h*and 0 to denote inversion and surface values of quantities that vary with height.) For idealized situations this leads to the familiar hypothesis wherein the rate of working against the overlying stratification (the buoyancy flux at the top of the layer) is set to a fixed fraction,

*k*, of the surface driving (as measured by the surface buoyancy flux,

_{0}) of the layer, that is,

*k*, is taken to be a universal constant (Ball 1960; Betts 1973; Tennekes 1973; Deardorff 1974) that observations, experiments, or simulations attempt to deduce, with

*k*= −0.2 being a common reference.

This buoyancy point of view, which emphasizes exchanges between turbulence kinetic and mean flow potential energy, also colors our view of the growth of stratocumulus-topped boundary layers. Most parameterizations strive to represent the growth of the layer by constraining the rate of working at cloud top to the net rate of buoyancy production, due to either radiative cooling, or surface fluxes, within the layer (Turton and Nicholls 1987; Stevens 2002). Such a way of thinking about the energetics of a growing boundary layer is also evident, at times, in discussions of layers maintained by cumulus convection. For instance, Wyant et al. (1997) emphasize this analogy and define an entrainment mass flux based on the relative energetics of cumulus updrafts and the stability of the inversion. This results in an entrainment law similar to (1).

In an attempt to reconcile the thermodynamic point of view of the earlier studies, with the energetic point of view manifest in discussions of the growth of layers maintained by other forms of shallow convection, we revisit the question of the growth of the trade wind layer. Our analysis is centered around large-eddy simulations of an idealized, or prototype, trade wind cumulus layer. Although most studies of trade wind convection are based on abstractions of observed cases (e.g., Sommeria 1976; Nicholls and LeMone 1980; Siebesma and Cuijpers 1995) we focus on an idealized case in an effort to limit the number of parameters on the one hand, and help extract essential phenomena on the other. Such an approach, both common and illustrative in studies of deep precipitating convection (Held et al. 1993; Emanuel and Bister 1996; Tompkins and Craig 1998; Pauluis and Held 2002), is beginning to be used more often in studies of shallow convection (Grant and Brown 1999; Grant 2001). The remainder of this paper is organized as follows: The idealization we explore is described in section 2, and large-eddy simulations thereof are presented in section 3. Section 4 interprets the energetics of the simulations. In section 5 we discuss some of the issues raised by this analysis, including its relation to previous work, and implications for ongoing studies. Section 6 concludes with a brief summary of our major findings.

## 2. Setup

### a. Background

*N*

^{2}, the surface buoyancy flux denoted by

_{0}, 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*, is given for all time by

_{m}*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*, would lead to slightly different values of

_{θ}*k*.

^{1}

### b. Prototype problem

_{0}, 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:

*θ*≈

_{e}*θ*exp[

*q*/(

_{υ}L*c*)], decreases with height. Such a negative gradient in

_{p}T*θ*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.

_{e}*λ*and

*q*

_{0}, introduced explicitly by (3) as well as the moisture scale height

_{0}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*

_{p}_{0}/(

*L*

_{0}), where

*w*′

*θ*′

*w*′

*q*′

_{0}and

_{0}, to the near surface thermodynamic state as

*θ*

_{0}and

*q*

_{0}are determined by the surface temperature and pressure, assuming saturation to determine

*q*

_{0};

*θ*

_{0+}and

*q*

_{0+}correspond to the thermodynamic state just above the surface; and

*V*is a constant set to 0.01 m s

^{−1}.

*θ*and

_{m}*q*, must evolve in a way that satisfy their respective conservation laws, namely,

_{m}*q*,

_{m}*θ*and the surface pressure

_{m},*p*

_{0}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

_{0}and

_{0}the integrals are evaluated numerically with

*α*= 1.3, Γ = 6 × 10

^{−3}K m

^{−1}, and

_{0}= 7 × 10

^{−4}m

^{2}s

^{−3}. Here we note that

*η*equilibrates relatively quickly and

*h*grows steadily in time, following the

_{m}*t*

^{1/2}rule as prescribed by (2). Eventually

*h*exceeds

_{m}*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

_{0}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

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.

## 3. Large-eddy simulations

### 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*/(

_{l}L*c*)] introduced by Betts (1973), and the total water mixing ratio,

_{p}T*q*. Liquid water,

*q*, 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.

_{l}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

_{0}at the desired mean value given (5), with

*θ*

_{0+}and

*q*

_{0+}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 _{p}ρ*

_{0}Θ

*g*

^{−1}

_{0}= 25 W m

^{−2},

*λ*= 1500 m, Γ = 6 K km

^{−1}, and

*q*

_{0}= 0.8

*q*

_{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* < *t*_{1} = 8.4 h), and a regime with approximately constant cloud fraction at later times (*t* > *t*_{2} = 13 h). The times *t*_{1} and *t*_{2} are defined when the cloud first exceeds 1% of its value averaged over the final 2 h of the simulation, while *t*_{2} corresponds to the time at which the cloud fraction first exceeds 90% of that value. The time of cloud onset, *t*_{1} = 8.4 h is predicted relatively well by the mixed layer equations (which predicted *h _{m}* =

*η*at 8.3 h). Subsequent to

*t*

_{2},

*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* < *t*_{1}, 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*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,

_{θ}*h*≫ Δ

*x*. 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*. 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

_{i}*h*. 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.

_{θ}*β*(Bowen ratio) and

_{0}, the SST (equivalently

*θ*

_{0}) should increase as

*θ*

_{0+}(equivalently

*θ*), that is, also proportional to

_{m}*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*and

_{m}*q*substituted for

_{m}*θ*

_{0+}and

*q*

_{0+}in the calculation of the surface fluxes.

At late time, that is, *t* > *t*_{2}, 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.

## 4. The energetics of the growing cloud layer

Although not shown, the flow visualization suggests that evaporative cooling at the top of the cloud layer may play an important role in the anomalous (*t* versus *t*^{1/2}) growth of the cloud layer. These images show the now familiar tendency (e.g., Reuter and Yau 1987; Zhao and Austin 2005a, b) of the cumuli to become enveloped, or crowned, by a downdraft sheath as they penetrate and evaporate into the inversion layer. To emphasize the differences between the scaling regimes at early (cloud free) and late times, we borrow from the terminology of stochastic processes and refer to the former as a diffusive, and the latter as a ballistic, growth regime. In this section we focus on the question as to the extent to which the ballistic growth regime is analogous to the growth of dry convective layers, whose anomalous (nonencroachment) growth is supported by the conversion of turbulence kinetic energy into mean-flow potential energy.

### a. The energetics of ballistic growth

*θ*, which following Emanuel (1994) we write as

_{ρ}*θ*makes an explicit reference to the density, differences in which underlies all convection.

_{ρ}The profile of *θ _{ρ}* is shown at two different times in Fig. 6. The evolution of the layer is marked by

*θ*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

_{ρ}*θ*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.

_{ρ}*θ*, fluctuations in

_{l}*θ*can be linearly related to fluctuations in

_{ρ}*θ*,

_{l}*q*, and

*q*as follows:

_{l}*a*

_{1},

*a*

_{2}, and

*a*

_{3}are thermodynamic parameters, with

*a*

_{2}given by (8). If we neglect fluctuations among thermodynamic perturbations, then

*a*

_{1}and

*a*

_{3}can also be expressed as effective constants:

*, Θ*

_{ρ}*,*

_{l}*q*and

_{l},*T*are taken to be some fixed value representative of the cloud layer. In cumulus layers

*q*≈ 1 × 10

_{l}^{−5}kg kg

^{−1}, so for Θ

*≈ Θ*

_{l}*,*

_{ρ}*a*

_{1}≃ 1 and

*a*

_{3}≃

*L*/(

*c*) −

_{p}T*R*/

_{υ}*R*≃ 7. At this level of approximation perturbations in

_{d}*θ*are indistinguishable from perturbations in

_{ρ}*θ*

_{υ}_{,}

*the liquid-water virtual potential temperature (e.g., Emanuel 1994).*

_{l}*θ*with fluctuations in

_{ρ}*w*it follows that

*and*

_{l}*denote the vertical kinematic flux of*

_{l}*θ*and

_{l}*q*respectively. This defines the buoyancy flux

_{l},*g*/Θ)

*, which measures the exchange between turbulence kinetic and mean-flow potential energy.*

_{ρ}*θ*or

_{l}*q*(i.e., in the absence of radiation and precipitation), then

*q*being stationary (the case of unsaturated layers being the trivial case), then

_{l}*q*,

_{l}*θ*evolves according to

_{ρ}_{ρ}, proves central to understanding the energetics of the cloud-layer growth. From (14) and (17) we note that

_{ρ}, to be proportional to that part of the buoyancy flux not associated with perturbations in

*q*. This equation emphasizes the fact that in unsaturated layers

_{l}_{ρ,}=

_{ρ}= (Θ/

*g*)

*a*

_{1}and

*a*

_{3},

*is equivalent to the flux of liquid-water virtual static energy (e.g., Bretherton and Wyant 1997; Bretherton and Park 2007, manuscript submitted to*

_{ρ}*J. Atmos. Sci.*), while

_{ρ}is what Lewellen and Lewellen (2002) call the “dry” virtual potential temperature flux.

^{3}implies

_{ρ}must be linear. This condition is reasonably well satisfied below the level

*z*for the simulation analyzed in Fig. 6. The profile of

_{a}_{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

_{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

_{ρ}given its surface value

_{ρ,0}, and its value,

*k*

_{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 _{ρ} at *z* = *h* drives the growth of the layer. However in cloud-free convective layers _{ρ} ∝ _{0}. This leads to constraints of the form (1). For cloudy layers this energetic constraint is relaxed, for example, Eq. (18), and increasingly negative _{ρ} can be supported by the transport of liquid water in the clouds, thus allowing _{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, *ga*_{3}* _{l}*. In the inversion (between

*z*and

_{b}*z*)

_{a}*g*/Θ

_{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.

*q*is stationary following the layer, the weak form of the conservation law (14) is given by the integral of (17) between

_{l}*z*and

_{a}*z*. Assuming to a first approximation that

_{b}*z*and

_{a}*z*evolve at the same rate, so that

_{b}*dh/dt*=

*dz*/

_{a}*dt*=

*dz*/

_{b}*dt*yields

_{ρ}(

*z*

_{b}) is negligible compared to

_{ρ}(

*z*) and that both terms on the lhs of (21) are proportional to one another; that is,

_{a}*θ*∝ Γ

_{ρ}*h*and

_{ρ}(

*z*) ≈ constant, in which case by (23)

_{a}*hdh*∝

*dt*or

*h*∝

*t*

^{1/2}so long as (22) prevails. In contrast, if

_{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 Δ

*θ*continues to scale with

_{ρ}*h*.

*h*, consisting of a well-mixed subcloud layer, a cloud layer whose lapse rate Γ

*is constant, and an infinitesimally thin inversion layer*

_{c}^{4}deepening under the action of an equivalent

*θ*flux,

_{ρ}_{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*is given by the dotted line in Fig. 6. For such a layer

_{ρ}*defines*

_{c}*θ̂*. Substituting (20) and (24) into (23) leads to an expression for the growth rate of the cumulus layer,

_{ρ}*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*given by (25) with Γ

_{ρ}*= 2.2 K km*

_{c}^{−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*for the period

_{θ}*t > t*

_{2}, 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

*. 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*

_{c}*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*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

_{ρ}*θ*flux at the top of the cloud layer to scale with the depth of the cloud layer following (20).

_{ρ}## 5. Discussion

Based on the above, some issues emerge that warrant further discussion. One is why the cloud-layer value of *θ _{ρ}* so closely tracks the value in the subcloud layer. Another is how our ideas relate to the significant body of earlier literature that addresses similar topics, and the last is what implications our findings have for ongoing research.

### a. Thermodynamic constraints

*θ*(the saturated value of

_{s}*θ*) in the cloud-layer track

_{e}*θ*in the subcloud layer; that is,

_{e}### b. Relation to previous work

*θ*around which Betts centers his analysis. This follows largely from (17) and (29), which together imply that

_{l}*c*) 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,

_{p}θ_{l}*, is a dominant contributor to the flux of*

_{l}*θ*, and our finding that the

_{l}*term (e.g., Fig. 7) supports the large downward flux of buoyancy in the cloud layer. The similarity between*

_{l}_{ρ}and

*also explains why the scaling of Moeng (2000), which was also based on an analysis of the budget of*

_{l}*θ*, so effectively scaled the growth of the stratocumulus layers she studied.

_{l}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

_{ρ}, which is closely related to

*, both of which must be linear*

_{l}^{5}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

*, 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).*

_{c,}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*gradients, the ballistic (water injection) growth mechanism provides an efficient means for deepening the boundary layer. In unfavorable

_{e}*θ*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.

_{e}## 6. Summary

We have constructed an idealized representation of a shallow cumulus layer to explore how the development of cumulus clouds affects the growth of the PBL. We pose the problem as one of a thermal boundary layer growing into a uniformly stratified layer whose specific humidity decays exponentially. The growth is maintained by the action of a constant surface buoyancy flux, which is specified by assuming a fixed wind speed and a saturated lower boundary whose temperature is allowed to evolve in time.

In the period before clouds develop, the problem is identical to that of a dry convective boundary layer, for which the layer deepens self-similarly with the square root of time. When clouds develop they quickly equilibrate to maintain a constant cloud fraction near 10%, after which the boundary layer deepens at a rate that increases linearly with time (ballistically). The more rapid deepening of the boundary layer enhances the downward mixing of heat and dry air, which by the surface specification implies enhanced moisture fluxes, and diminished surface sensible heat fluxes. Many characteristics of shallow cumulus layers are apparent in this prototype problem. These include a well-mixed subcloud layer, a transition layer (most evident in moisture gradients), a conditionally unstable cloud layer, and an inversion layer into which clouds penetrate and detrain.

The tendency of the cloud layer to grow linearly in time is explained in terms of a mechanism that is fundamentally different than that used to support the growth of dry convective boundary layers. In the cloud-topped layers, the convective clouds penetrate into a conditionally unstable capping layer and detrain (inject) their liquid water there. The subsequent evaporation of which imbues the capping layer with the properties of the cloud layer. This injection mechanism is best measured by what we call the equivalent flux of density potential temperature. Its value at the top of the cloud layer is constrained by the subcloud layer energetics (which are isomorphic to that of a dry convective boundary layer) and the tendency for the cloud layer values of *θ _{ρ}* to change commensurately with the values in the subcloud layer. The development of a simple bulk model based on these concepts is shown to provide a satisfactory prediction of the growth rate for a suite of large-eddy simulations configured to sample a wide span of the problem’s parameter space.

While we have made progress in understanding a highly idealized problem, one could argue that many of our idealizations are untenable: even shallow clouds can readily develop precipitation, radiative cooling cannot be ignored, nor can the vertical shear in the horizontal wind. And while the trade wind layer tends to advect over steadily warming water, with progressively decreasing Bowen ratios, they probably do not do so at a rate that maintains a constant surface buoyancy flux. As such one might rightly inquire as to the relevance of our arguments. Notwithstanding the simplifications made in framing the problem, to the extent the simulations we describe behave like a real cumulus-topped layer would in a similar situation, we believe that the present study makes a number of fundamental contributions. Foremost, it provides a framework for thinking about more realistic problems, in so doing it clarifies the link between the evolution of the density and enthalpy fields, the latter being the basis of many past studies of trade–cumulus layers. Finally, it defines a useful test problem, and a rich set of constraints that can be used to evaluate and improve more general models (or parameterizations) of shallow convection.

## Acknowledgments

The exploration of this problem was motivated by Kerry Emanuel, whose pounding of the projection screen in want of a theory of moist convection during a lecture at the Institute on Pure and Applied Mathematics 2002 summer school motivated me to develop this simple framework. The ideas developed here also benefited greatly from subsequent conversations and comments by Alan Betts, Christopher Bretherton, Kerry Emanuel, Chin-Hoh Moeng, Pier Siebesma, and João Teixeira. Louise Nuijens and Verica Savic-Jovcic are also thanked for their comments on an early version of this manuscript, which improved the presentation. Computational resources were made available by NCAR. Financial support has been provided by NSF Grants ATM-00342625, ATM-9985413, and NASA Grant NGT5-30499.

## REFERENCES

Ackerman, A. S., M. P. Kirkpatrick, D. E. Stevens, and O. B. Toon, 2004: The impact of humidity above stratiform clouds on indirect aerosol climate forcing.

,*Nature***432****,**1014–1017.Albrecht, B. A., 1989: Aerosols, cloud microphysics and fractional cloudiness.

,*Science***245****,**1227–1230.Augstein, E., H. Riehl, F. Ostapoff, and V. Wagner, 1973: Mass and energy transports in an undisturbed Atlantic trade wind flow.

,*Mon. Wea. Rev.***101****,**101–111.Ball, F. K., 1960: Control of inversion height by surface heating.

,*Quart. J. Roy. Meteor. Soc.***86****,**483–494.Betts, A. K., 1973: Non-precipitating cumulus convection and its parameterization.

,*Quart. J. Roy. Meteor. Soc.***99****,**178–196.Betts, A. K., 1975: Parametric interpretation of trade wind cumulus budget studies.

,*J. Atmos. Sci.***32****,**1934–1945.Betts, A. K., and J. Bartlo, 1991: The density temperature and the dry and wet virtual adiabats.

,*Mon. Wea. Rev.***119****,**169–175.Bony, S., and J-L. Dufresne, 2005: Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models.

,*Geophys. Res. Lett.***32****.**L20806, doi:10.1029/2005GL023851.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.Bretherton, C. S., P. N. Blossey, and J. Uchida, 2007: Cloud droplet sedimentation, entrainment efficiency, and subtropical stratocumulus albedo.

,*Geophys. Res. Lett.***34****.**L03813, doi:10.1029/2006GL027648.Byers, H. R., and R. K. Hall, 1955: A census of cumulus-cloud height versus precipitation in the vicinity of Puerto Rico during the winter and spring of 1953–1954.

,*J. Meteor.***12****,**176–178.Carson, D. J., and F. B. Smith, 1974: Thermodynamic model for the development of a convectively unstable boundary layer.

*Advances in Geophysics,*Vol. 18A, Academic Press, 111–124.Deardorff, J. W., 1974: Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer.

,*Bound.-Layer Meteor.***7****,**81–106.Deardorff, J. W., 1980: Cloud top entrainment instability.

,*J. Atmos. Sci.***37****,**131–147.Deardorff, J. W., E. G. Willis, and D. K. Lilly, 1974: Comments on non-precipitating cumulus convection and its parameterization.

,*Quart. J. Roy. Meteor. Soc.***100****,**122–123.Driedonks, A. G. M., 1982: Models and observations of the growth of the atmospheric boundary layer.

,*Bound.-Layer Meteor.***23****,**283–306.Emanuel, K. A., 1994:

*Atmospheric Convection*. Oxford University Press, 580 pp.Emanuel, K. A., and M. A. Bister, 1996: Moist convective velocity and buoyancy scales.

,*J. Atmos. Sci.***53****,**1111–1143.Fedorovich, E., 1998: Bulk models of the atmospheric convective boundary layer.

*Buoyant Convection in Geophysical Flows,*E. J. Plate et al., Eds., Vol. 513 of C, Kluwer Academic, 265–290.Grant, A. L. M., 2001: Cloud based fluxes in the cumulus-capped boundary layer.

,*Quart. J. Roy. Meteor. Soc.***127****,**407–422.Grant, A. L. M., and A. R. Brown, 1999: A similarity hypothesis for shallow-cumulus transports.

,*Quart. J. Roy. Meteor. Soc.***125****,**1913–1936.Held, I. M., R. S. Hemler, and V. Ramaswamy, 1993: Radiative–convective equilibrium with explicit two-dimensional moist convection.

,*J. Atmos. Sci.***50****,**3909–3927.Khairoutdinov, M., and D. Randall, 2006: High-resolution simulation of shallow-to-deep convection transition over land.

,*J. Atmos. Sci.***63****,**3421–3436.Lewellen, D. C., and W. S. Lewellen, 2002: Entrainment and decoupling relations for cloudy boundary layers.

,*J. Atmos. Sci.***59****,**2966–2986.McCaa, J. R., and C. S. Bretherton, 2004: A new parameterization for shallow cumulus convection and its application to marine subtropical cloud-topped boundary layers. Part II: Regional simulations of marine boundary layer clouds.

,*Mon. Wea. Rev.***132****,**883–896.Moeng, C-H., 2000: Entrainment rate, cloud fraction, and liquid water path of PBL stratocumulus clouds.

,*J. Atmos. Sci.***57****,**3627–3643.Nicholls, S., and M. A. LeMone, 1980: The fair weather boundary layer in GATE: The relationship of subcloud fluxes and structure to the distribution and enhancement of cumulus clouds.

,*J. Atmos. Sci.***37****,**2051–2067.Nieuwstadt, F. T. M., P. J. Mason, C-H. Moeng, and U. Schumann, 1991: Large-eddy simulation of the convective boundary layer: A comparison of four computer codes.

*Turbulent Shear Flows 8: Selected Papers from the Eighth International Symposium on Turbulent Shear Flows,*F. Durst et al., Eds., Springer-Verlag, 353–367.Nitta, T., and S. Esbensen, 1974: Heat and moisture budget analyses using BOMEX data.

,*Mon. Wea. Rev.***102****,**17–28.Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere.

,*J. Atmos. Sci.***19****,**173–179.Pauluis, O., and I. M. Held, 2002: Entropy budget of an atmosphere in radiative–convective equilibrium. Part I: Maximum work and frictional dissipation.

,*J. Atmos. Sci.***59****,**125–139.Randall, D. A., 1980: Conditional instability of the first kind upside-down.

,*J. Atmos. Sci.***37****,**125–130.Reuter, G. W., and M. K. Yau, 1987: Mixing mechanisms in cumulus congestus clouds. Part II: Numerical simulations.

,*J. Atmos. Sci.***44****,**798–827.Riehl, H., T. C. Yeh, J. S. Malkus, and N. E. la Seur, 1951: The north–east trade of the Pacific Ocean.

,*Quart. J. Roy. Meteor. Soc.***77****,**598–626.Siebesma, A. P., 1998: Shallow convection.

*Buoyant Convection in Geophysical Flows,*E. J. Plate et al., Eds., Vol. 513 of C, Kluwer Academic, 441–486.Siebesma, A. P., and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection.

,*J. Atmos. Sci.***52****,**650–666.Sommeria, G., 1976: Three-dimensional simulation of turbulent processes in an undisturbed trade wind boundary layer.

,*J. Atmos. Sci.***33****,**216–241.Squires, P., 1958: Penetrative downdraughts in cumuli.

,*Tellus***10****,**381–385.Stevens, B., 2002: Entrainment in stratocumulus mixed layers.

,*Quart. J. Roy. Meteor. Soc.***128****,**2663–2690.Stevens, B., 2005: Atmospheric moist convection.

,*Annu. Rev. Earth Planet. Sci.***33****,**605–643.Stevens, B., W. R. Cotton, G. Feingold, and C-H. Moeng, 1998: Large-eddy simulations of strongly precipitating, shallow, stratocumulus-topped boundary layers.

,*J. Atmos. Sci.***55****,**3616–3638.Stevens, B., G. Vali, K. Comstock, R. Wood, M. C. van Zanten, P. H. Austin, C. S. Bretherton, and D. H. Lenschow, 2005: Pockets of open cells and drizzle in marine stratocumulus.

,*Bull. Amer. Meteor. Soc.***86****,**51–57.Sullivan, P. P., C-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer.

,*J. Atmos. Sci.***55****,**3042–3064.Tennekes, H., 1973: A model for the dynamics of the inversion above a convective boundary layer.

,*J. Atmos. Sci.***30****,**558–567.Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models.

,*Mon. Wea. Rev.***117****,**1779–1800.Tompkins, A. M., and G. C. Craig, 1998: Radiative convective equilibrium in a three-dimensional cloud-ensemble model.

,*Quart. J. Roy. Meteor. Soc.***124****,**2073–2097.Turton, J. D., and S. Nicholls, 1987: A study of the diurnal variation of stratocumulus using a multiple mixed layer model.

,*Quart. J. Roy. Meteor. Soc.***113****,**969–1009.Wyant, M. C., C. S. Bretherton, H. A. Rand, and D. E. Stevens, 1997: Numerical simulations and a conceptual model of the stratocumulus to trade cumulus transition.

,*J. Atmos. Sci.***54****,**168–192.Wyant, M. C., C. S. Bretherton, J. T. Backmeister, J. T. Kiehl, I. M. Held, S. A. Klein, and B. J. Soden, 2006: A comparison of low-latitude cloud properties and their response to climate change in three AGCMs sorted into regimes using mid-tropospheric vertical velocity.

,*Climate Dyn.***27****,**261–279.Zhao, M., and P. H. Austin, 2005a: Life cycle of numerically simulated shallow cumulus clouds. Part I: Transport.

,*J. Atmos. Sci.***62****,**1269–1290.Zhao, M., and P. H. Austin, 2005b: Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics.

,*J. Atmos. Sci.***62****,**1291–1310.Zhu, P., and C. S. Bretherton, 2004: A simulation study of shallow moist convection and its impact on the atmospheric boundary layer.

,*Mon. Wea. Rev.***132****,**2391–2409.

Development of a moist thermal layer under the action of a constant surface buoyancy flux _{0} ≃ 7 × 10^{−4} m^{2} s^{−3}. Shown are the initial profiles of *θ _{i}*,

*q*(solid); and the profiles of

_{i}*θ*and

*q*calculated using LES at

*t*= 24 000 s (dashed) and

*t*= 96 000 s (dotted). At the latter time, the cloud fraction is also plotted on the left. Other parameters for this problem include

*λ*= 1500 m,

*q*

_{0,+}= 0.8

*q*

_{s}_{,0}, and Γ = 6 × 10

^{−3}K m

^{−1}.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Development of a moist thermal layer under the action of a constant surface buoyancy flux _{0} ≃ 7 × 10^{−4} m^{2} s^{−3}. Shown are the initial profiles of *θ _{i}*,

*q*(solid); and the profiles of

_{i}*θ*and

*q*calculated using LES at

*t*= 24 000 s (dashed) and

*t*= 96 000 s (dotted). At the latter time, the cloud fraction is also plotted on the left. Other parameters for this problem include

*λ*= 1500 m,

*q*

_{0,+}= 0.8

*q*

_{s}_{,0}, and Γ = 6 × 10

^{−3}K m

^{−1}.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Development of a moist thermal layer under the action of a constant surface buoyancy flux _{0} ≃ 7 × 10^{−4} m^{2} s^{−3}. Shown are the initial profiles of *θ _{i}*,

*q*(solid); and the profiles of

_{i}*θ*and

*q*calculated using LES at

*t*= 24 000 s (dashed) and

*t*= 96 000 s (dotted). At the latter time, the cloud fraction is also plotted on the left. Other parameters for this problem include

*λ*= 1500 m,

*q*

_{0,+}= 0.8

*q*

_{s}_{,0}, and Γ = 6 × 10

^{−3}K m

^{−1}.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Time series of *h _{m}* (solid) and

*η*(dashed) for a mixed layer growing into a layer with the same initial conditions and surface buoyancy flux as shown in Fig. 2. (right) Cloud-base height,

*η*, (solid contours) and cloud onset time in h (dashed) as a function of

_{0}and

*λ*for

*k*= 1.3.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Time series of *h _{m}* (solid) and

*η*(dashed) for a mixed layer growing into a layer with the same initial conditions and surface buoyancy flux as shown in Fig. 2. (right) Cloud-base height,

*η*, (solid contours) and cloud onset time in h (dashed) as a function of

_{0}and

*λ*for

*k*= 1.3.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Time series of *h _{m}* (solid) and

*η*(dashed) for a mixed layer growing into a layer with the same initial conditions and surface buoyancy flux as shown in Fig. 2. (right) Cloud-base height,

*η*, (solid contours) and cloud onset time in h (dashed) as a function of

_{0}and

*λ*for

*k*= 1.3.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(a) Time series of distinguished heights for the base simulation described in Fig. 2. A gray dotted line (which is effectively overlaid by other lines) shows the fit to *h _{θ}* for

*t*< 8.4 h and

*t*> 13 h, respectively. The early time fit is proportional to

*t*

^{1/2}and the late time fit is proportional to

*t*. Also shown by the dashed line is the lifting condensation level

*η*based on the average value of

*θ*and

*q*over a layer from 0 to 0.2

*h*

_{B}. (b) Time series of cloud fraction, defined as the percent of columns with condensate.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(a) Time series of distinguished heights for the base simulation described in Fig. 2. A gray dotted line (which is effectively overlaid by other lines) shows the fit to *h _{θ}* for

*t*< 8.4 h and

*t*> 13 h, respectively. The early time fit is proportional to

*t*

^{1/2}and the late time fit is proportional to

*t*. Also shown by the dashed line is the lifting condensation level

*η*based on the average value of

*θ*and

*q*over a layer from 0 to 0.2

*h*

_{B}. (b) Time series of cloud fraction, defined as the percent of columns with condensate.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(a) Time series of distinguished heights for the base simulation described in Fig. 2. A gray dotted line (which is effectively overlaid by other lines) shows the fit to *h _{θ}* for

*t*< 8.4 h and

*t*> 13 h, respectively. The early time fit is proportional to

*t*

^{1/2}and the late time fit is proportional to

*t*. Also shown by the dashed line is the lifting condensation level

*η*based on the average value of

*θ*and

*q*over a layer from 0 to 0.2

*h*

_{B}. (b) Time series of cloud fraction, defined as the percent of columns with condensate.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Time series of (a) sea surface temperature, (b) surface latent enthalpy flux, and (c) surface dry enthalpy flux. Dotted lines are estimates based on bulk model (6)–(7) adjusted to account for the lack of a surface layer in the bulk model, and with *h _{m}* given by (2).

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Time series of (a) sea surface temperature, (b) surface latent enthalpy flux, and (c) surface dry enthalpy flux. Dotted lines are estimates based on bulk model (6)–(7) adjusted to account for the lack of a surface layer in the bulk model, and with *h _{m}* given by (2).

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Time series of (a) sea surface temperature, (b) surface latent enthalpy flux, and (c) surface dry enthalpy flux. Dotted lines are estimates based on bulk model (6)–(7) adjusted to account for the lack of a surface layer in the bulk model, and with *h _{m}* given by (2).

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Density potential temperature, *θ _{ρ}* budget: (left) Profiles at 33 h (solid) and 36 h (dashed), and an idealized bulk profile (dotted) based on the average between 33 and 36 h; (middle) difference in

*θ*between hours 36 and 33; and (right) average of effective

_{ρ}*θ*flux during 33–36 h.

_{ρ}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Density potential temperature, *θ _{ρ}* budget: (left) Profiles at 33 h (solid) and 36 h (dashed), and an idealized bulk profile (dotted) based on the average between 33 and 36 h; (middle) difference in

*θ*between hours 36 and 33; and (right) average of effective

_{ρ}*θ*flux during 33–36 h.

_{ρ}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Density potential temperature, *θ _{ρ}* budget: (left) Profiles at 33 h (solid) and 36 h (dashed), and an idealized bulk profile (dotted) based on the average between 33 and 36 h; (middle) difference in

*θ*between hours 36 and 33; and (right) average of effective

_{ρ}*θ*flux during 33–36 h.

_{ρ}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Fluxes of liquid water (dotted), buoyancy (dashed), and equivalent buoyancy (solid) in buoyancy units for base simulation.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Fluxes of liquid water (dotted), buoyancy (dashed), and equivalent buoyancy (solid) in buoyancy units for base simulation.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Fluxes of liquid water (dotted), buoyancy (dashed), and equivalent buoyancy (solid) in buoyancy units for base simulation.

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Predicted vs actual growth rate for simulations in Table 2. The three simulations solved using different numerical formulations (grids) are shown by the open squares. Simulations in which

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Predicted vs actual growth rate for simulations in Table 2. The three simulations solved using different numerical formulations (grids) are shown by the open squares. Simulations in which

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

Predicted vs actual growth rate for simulations in Table 2. The three simulations solved using different numerical formulations (grids) are shown by the open squares. Simulations in which

Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Profiles of *θ _{e}* and

*θ*from the base simulation as presented in Fig. 6, and (right) the change in time of

_{s}*θ*and

_{e}*θ*.

_{s}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Profiles of *θ _{e}* and

*θ*from the base simulation as presented in Fig. 6, and (right) the change in time of

_{s}*θ*and

_{e}*θ*.

_{s}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

(left) Profiles of *θ _{e}* and

*θ*from the base simulation as presented in Fig. 6, and (right) the change in time of

_{s}*θ*and

_{e}*θ*.

_{s}Citation: Journal of the Atmospheric Sciences 64, 8; 10.1175/JAS3983.1

List of symbols.

Summary of simulations.

^{1}

The fact that *k* depends on how the layer’s depth is defined underscores the point that it does not measure the ratio of the actual minimum buoyancy flux to the surface buoyancy flux, but rather that value required by a mixed-layer model to match the growth of the actual layer (cf. Deardorff et al. 1974).

^{2}

The analysis of our simulations is performed in a way that accounts for the fact that the background density varies with height. For ease of exposition however the results have been presented in the Boussinesq limit. Throughout we have checked our analysis to ensure that it is not qualitatively sensitive to the height variations of the background density in the simulations.

^{3}

Recall that in the context of turbulent boundary layers this term refers to the condition that the shape of the profile of a particular quantity does not change in time, that is, stationary gradients.

^{4}

This fundamentally amounts to assuming that the inversion layer does not introduce a new length scale.

^{5}

If other sources such as radiation are active, then the linearity constraint of quasi steadiness applies to the sum of the radiative and turbulent fluxes.