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  • View in gallery

    Development of a convective boundary layer under the action of a constant surface enthalpy flux ρ0cpQ0. Shown are the initial profile θi (gray solid line); the profile calculated using LES at t = 48 000 s (dashed line); and equivalent mixed layer profile, θm (dark solid line). Also shown are the four heights, h*, hB, hm, and hθ, described in the text.

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    Development of a moist thermal layer under the action of a constant surface buoyancy flux B0 ≃ 7 × 10−4 m2 s−3. Shown are the initial profiles of θi, qi (solid); and the profiles of 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, q0,+ = 0.8qs,0, and Γ = 6 × 10−3 K m−1.

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    (left) Time series of hm (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 B0 and λ for k = 1.3.

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    (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 hθ for t < 8.4 h and t > 13 h, respectively. The early time fit is proportional to t1/2 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 0.2hB. (b) Time series of cloud fraction, defined as the percent of columns with condensate.

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    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 hm given by (2).

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    Density potential temperature, θρ 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 θρ between hours 36 and 33; and (right) average of effective θρ flux during 33–36 h.

  • View in gallery

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

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    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 W is calculated using the actual cloud lapse from each simulation rather than a fixed constant are shown by gray points.

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    (left) Profiles of θe and θs from the base simulation as presented in Fig. 6, and (right) the change in time of θe and θs.

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On the Growth of Layers of Nonprecipitating Cumulus Convection

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  • 1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
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Abstract

A prototype problem of a nonprecipitating convective layer growing into a layer of uniform stratification and exponentially decreasing humidity is introduced to study the mechanism by which the cumulus-topped boundary layer grows. The problem naturally admits the surface buoyancy flux, outer layer stratification, and moisture scale as governing parameters. Large-eddy simulations show that many of the well-known properties of the cumulus-topped boundary layer (including a well-mixed subcloud layer, a cloud-base transition layer, a conditionally unstable cloud layer, and an inversion layer) emerge naturally in the simulations. The simulations also quantify the differences between nonprecipitating moist convection and its dry counterpart. Whereas dry penetrative convective layers grow proportionally to the square root of time (diffusively) the cumulus layers grow proportionally to time (ballistically). The associated downward transport of warm, dry air results in a significant decrease in the surface Bowen ratio. The linear-in-time growth of the cloud layer is shown to result from the transport and subsequent evaporation of liquid water into the inversion layer. This process acts as a sink of buoyancy, which acts to imbue the free troposphere with the properties of the cloud layer. A simple model, based on this mechanism, and formulated in terms of an effective dry buoyancy flux (which is constrained by the subcloud layer’s similarity to a dry convective layer), is shown to provide good predictions of the growth of the layer across a wide range of governing parameters.

Corresponding author address: Bjorn Stevens, Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, 405 Hilgard Ave., Box 951565, Los Angeles, CA 90095-1565. Email: bstevens@atmos.ucla.edu

Abstract

A prototype problem of a nonprecipitating convective layer growing into a layer of uniform stratification and exponentially decreasing humidity is introduced to study the mechanism by which the cumulus-topped boundary layer grows. The problem naturally admits the surface buoyancy flux, outer layer stratification, and moisture scale as governing parameters. Large-eddy simulations show that many of the well-known properties of the cumulus-topped boundary layer (including a well-mixed subcloud layer, a cloud-base transition layer, a conditionally unstable cloud layer, and an inversion layer) emerge naturally in the simulations. The simulations also quantify the differences between nonprecipitating moist convection and its dry counterpart. Whereas dry penetrative convective layers grow proportionally to the square root of time (diffusively) the cumulus layers grow proportionally to time (ballistically). The associated downward transport of warm, dry air results in a significant decrease in the surface Bowen ratio. The linear-in-time growth of the cloud layer is shown to result from the transport and subsequent evaporation of liquid water into the inversion layer. This process acts as a sink of buoyancy, which acts to imbue the free troposphere with the properties of the cloud layer. A simple model, based on this mechanism, and formulated in terms of an effective dry buoyancy flux (which is constrained by the subcloud layer’s similarity to a dry convective layer), is shown to provide good predictions of the growth of the layer across a wide range of governing parameters.

Corresponding author address: Bjorn Stevens, Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, 405 Hilgard Ave., Box 951565, Los Angeles, CA 90095-1565. Email: bstevens@atmos.ucla.edu

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.

So while it has long been appreciated, at least by some, that the maintenance of the trades depends on the transport of liquid water, this view is worth contrasting with ideas used to explain the growth, or maintenance, of layers maintained by other types of shallow convective. Dry convective boundary layers are generally understood to deepen as a result of turbulent convective eddies working against the stratification of the capping layer. Simply put, the growth rate dh/dt is expressed as the ratio between the entrainment buoyancy flux, Bh, and the jump in buoyancy, Δb, across the capping layer. Here Bh measures the rate at which turbulence energy is converted to potential energy, and Δb 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, B0) of the layer, that is,
i1520-0469-64-8-2916-e1
The entrainment flux ratio, 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

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, N2, 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, hm, is given for all time by
i1520-0469-64-8-2916-e2
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 ht1/2.

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, hm, 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, hB, or the height of the maximum buoyancy gradient hθ, 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:
i1520-0469-64-8-2916-e3
It corresponds to a relative humidity profile that decreases exponentially with height. Equation (3) also ensures that the initial equivalent potential temperature, θeθ exp[qυL/(cpT)], decreases with height. Such a negative gradient in θe 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
i1520-0469-64-8-2916-e4
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 βcpQ0/(LR0), where Qwθ and Rwq is free to vary. We relate the surface fluxes, Q0 and R0, to the near surface thermodynamic state as
i1520-0469-64-8-2916-e5
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, θm and qm, must evolve in a way that satisfy their respective conservation laws, namely,
i1520-0469-64-8-2916-e6
i1520-0469-64-8-2916-e7
Given qm, θm, and the surface pressure p0 we can calculate the lifting condensation level, η(t). So long as hm < 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 hm 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 hm grows steadily in time, following the t1/2 rule as prescribed by (2). Eventually hm 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 hm = 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.

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, θl = θ exp[−qlL/(cpT)] introduced by Betts (1973), and the total water mixing ratio, q. Liquid water, ql, 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 N2, θi(z) is calculated based on qi(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 cpρ0Θg−1B0 = 25 W m−2, λ = 1500 m, Γ = 6 K km−1, and q0 = 0.8qs,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 hm = η at 8.3 h). Subsequent to t2, η tracks hB (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* < hB < hm < hθ 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 hB, in which θ < θi. 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 hθ. 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
i1520-0469-64-8-2916-e8
For fixed β (Bowen ratio) and B0, the SST (equivalently θ0) should increase as θ0+ (equivalently θm), that is, also proportional to t1/2. In Fig. 5 we see that β is decreasing with time, indicating that the SST must increase slightly less rapidly than t1/2. 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 θm and qm 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, Γc ≃ 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 t1/2) 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

We begin by considering the budget of buoyancy as measured by the density potential temperature θρ, which following Emanuel (1994) we write as
i1520-0469-64-8-2916-e9
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 θρ makes an explicit reference to the density, differences in which underlies all convection.

The profile of θρ is shown at two different times in Fig. 6. The evolution of the layer is marked by θρ 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 θρ 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 θl, fluctuations in θρ can be linearly related to fluctuations in θl, q, and ql as follows:
i1520-0469-64-8-2916-e10
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:
i1520-0469-64-8-2916-e11
where Θρ, Θl, ql, and T are taken to be some fixed value representative of the cloud layer. In cumulus layers ql ≈ 1 × 10−5 kg kg−1, so for Θl ≈ Θρ, a1 ≃ 1 and a3L/(cpT) − Rυ/Rd ≃ 7. At this level of approximation perturbations in θρ are indistinguishable from perturbations in θυ,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 θρ with fluctuations in w it follows that
i1520-0469-64-8-2916-e12
where Ql and Rl denote the vertical kinematic flux of θl and ql, respectively. This defines the buoyancy flux B ≡ (g/Θ)Qρ, 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 θl or q (i.e., in the absence of radiation and precipitation), then
i1520-0469-64-8-2916-e13
where C is the net rate of condensation. Associating fluctuations in (10) with changes in time, it follows from above that
i1520-0469-64-8-2916-e14
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
i1520-0469-64-8-2916-e15
which in the limit corresponds to ql being stationary (the case of unsaturated layers being the trivial case), then
i1520-0469-64-8-2916-e16
which is conserved. Hence, in the stationary limit of ql, θρ evolves according to
i1520-0469-64-8-2916-e17
It turns out that this flux, ρ, proves central to understanding the energetics of the cloud-layer growth. From (14) and (17) we note that
i1520-0469-64-8-2916-e18
which defines ρ, to be proportional to that part of the buoyancy flux not associated with perturbations in ql. This equation emphasizes the fact that in unsaturated layers ρ, = Qρ = (Θ/g)B. At a level of approximation commensurate with our determination of the constants a1 and a3, Qρ 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 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
i1520-0469-64-8-2916-e19
In other words, ρ must be linear. This condition is reasonably well satisfied below the level za for the simulation analyzed in Fig. 6. The profile of 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 θρ) 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
i1520-0469-64-8-2916-e20
which follows from simple linear extrapolation of ρ given its surface value ρ,0, and its value, kρ,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 ρB, 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 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, B is nearly constant and significantly smaller than ga3Rl. In the inversion (between zb and za) B is nearly zero, and so the divergence of gρ 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 ql is stationary following the layer, the weak form of the conservation law (14) is given by the integral of (17) between za and zb. Assuming to a first approximation that za and zb evolve at the same rate, so that dh/dt = dza/dt = dzb/dt yields
i1520-0469-64-8-2916-e21
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 ρ(zb) is negligible compared to ρ(za) and that both terms on the lhs of (21) are proportional to one another; that is,
i1520-0469-64-8-2916-e22
allows us to express (21) as
i1520-0469-64-8-2916-e23
In the dry convective boundary layer, the assumption of a single length scale implies that Δθρ ∝ Γh and ρ(za) ≈ constant, in which case by (23) hdhdt or ht1/2 so long as (22) prevails. In contrast, if ρ(za) ∝ h, as we found to be the case for the cumulus-topped boundary layer, then (23) implies that ht so long as Δθρ 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 Γc is constant, and an infinitesimally thin inversion layer4 deepening under the action of an equivalent θρ flux, ρ ∝ (hη), 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 θρ is given by the dotted line in Fig. 6. For such a layer
i1520-0469-64-8-2916-e24
By conservation of enthalpy
i1520-0469-64-8-2916-e25
which given Γc defines θ̂ρ. Substituting (20) and (24) into (23) leads to an expression for the growth rate of the cumulus layer,
i1520-0469-64-8-2916-e26
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, thetascircρ given by (25) with Γc = 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 dhθ/dt as determined by the best-fit linear slope to hθ 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 dhθ/dt. Also evident in this figure is that much of the scatter can be associated with the use of a single value for Γc. The lapse rates in the cloud layer vary slightly, but systematically, among the simulations (see Table 2), and unaccounted for variations in Γc 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 θρ; (ii) the increase in θρ 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 θρ flux at the top of the cloud layer to scale with the depth of the cloud layer following (20).

5. Discussion

Based on the above, some issues emerge that warrant further discussion. One is why the cloud-layer value of θρ 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

The first question can be posed formally as follows:
i1520-0469-64-8-2916-e27
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 θs (the saturated value of θe) in the cloud-layer track θe in the subcloud layer; that is,
i1520-0469-64-8-2916-e28
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
i1520-0469-64-8-2916-e29
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 θl around which Betts centers his analysis. This follows largely from (17) and (29), which together imply that
i1520-0469-64-8-2916-e30
In other words, to a first approximation the divergence of the enthalpy (as measured by cpθl) 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, Rl, is a dominant contributor to the flux of θl, and our finding that the Rl term (e.g., Fig. 7) supports the large downward flux of buoyancy in the cloud layer. The similarity between ρ and Ql also explains why the scaling of Moeng (2000), which was also based on an analysis of the budget of θl, 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 θρ evolves under the divergence of the flux ρ, which is closely related to Ql, 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 θe. 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 Γc,, 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 θe gradients (cf. Squires 1958; Randall 1980; Deardorff 1980). For negative θe gradients, the ballistic (water injection) growth mechanism provides an efficient means for deepening the boundary layer. In unfavorable θe 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.

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 θρ 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.

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Fig. 1.
Fig. 1.

Development of a convective boundary layer under the action of a constant surface enthalpy flux ρ0cpQ0. Shown are the initial profile θi (gray solid line); the profile calculated using LES at t = 48 000 s (dashed line); and equivalent mixed layer profile, θm (dark solid line). Also shown are the four heights, h*, hB, hm, and hθ, described in the text.

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

Fig. 2.
Fig. 2.

Development of a moist thermal layer under the action of a constant surface buoyancy flux B0 ≃ 7 × 10−4 m2 s−3. Shown are the initial profiles of θi, qi (solid); and the profiles of 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, q0,+ = 0.8qs,0, and Γ = 6 × 10−3 K m−1.

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

Fig. 3.
Fig. 3.

(left) Time series of hm (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 B0 and λ for k = 1.3.

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

Fig. 4.
Fig. 4.

(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 hθ for t < 8.4 h and t > 13 h, respectively. The early time fit is proportional to t1/2 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 0.2hB. (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

Fig. 5.
Fig. 5.

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 hm given by (2).

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

Fig. 6.
Fig. 6.

Density potential temperature, θρ 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 θρ between hours 36 and 33; and (right) average of effective θρ flux during 33–36 h.

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

Fig. 7.
Fig. 7.

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

Fig. 8.
Fig. 8.

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 W is calculated using the actual cloud lapse from each simulation rather than a fixed constant are shown by gray points.

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

Fig. 9.
Fig. 9.

(left) Profiles of θe and θs from the base simulation as presented in Fig. 6, and (right) the change in time of θe and θs.

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

Table 1.

List of symbols.

Table 1.
Table 2.

Summary of simulations.

Table 2.

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.

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