## 1. Introduction

In most global forecast models and general circulation models (GCMs), the cloud cover and boundary layer structure are not well represented in regions of persistent marine stratocumulus, particularly in the subtropics. Over the past 15 yr, field experiments and fine-scale numerical models have given us a reasonable understanding of stratocumulus-capped boundary layers (SCBLs). They suggest that a successful parameterization of the SCBL must (a) faithfully represent the tight coupling between cloud formation, convective turbulence, microphysical, radiative and surface fluxes, and (b) handle the internal stratification or “decoupling” of an SCBL that can be driven by some combination of high surface latent heat fluxes, daytime absorption of insolation within clouds, and drizzle. Criterion a allows for the formation and maintenance of a SCBL, while criterion b allows for the formation of cumulus under the main stratocumulus layer, which is often an important intermediary in the thinning and breakup of subtropical stratocumulus.

Almost no GCM parameterization used today satisfies both these criteria. Many GCMs assume unsaturated thermodynamics in their turbulence schemes. In many SCBLs, however, the dominant source of turbulent kinetic energy is buoyancy fluxes within the cloud layer. SCBLs in such models must be maintained unphysically by a mixture of dry convection and shallow cumulus convection. Bougeault (1985), Bechtold et al. (1995), and others have shown that 1.5 and higher order turbulence closure models (TCMs) can represent the dynamics and internal stratification of SCBLs quite well if they are formulated using moist thermodynamics and if the vertical grid resolution is 100 m or less. At coarser vertical resolution, this method breaks down (Bechtold et al. 1996) due to underresolution of the inversion and the cloud-layer structure.

In sections 2 and 3 of this paper, we describe a general purpose boundary layer parameterization that can simulate the boundary layer structure and cloud properties in SCBLs even at GCM vertical resolution. A 1.5-order TCM based on thermodynamic variables conserved for moist reversible adiabatic processes is used for the internal structure of the boundary layer. In neutral to unstable conditions, an entrainment parameterization, rather than the TCM, is used to specify the vertical fluxes of heat, moisture, and momentum at the boundary layer top. An approach similar in spirit (insofar as it combined an internally varying boundary layer with an entrainment parameterization) was proposed many years ago by Randall (1976) in his pioneering doctoral thesis and was used for five years in the University of California, Los Angeles (UCLA) GCM. At the time, his state-of-the-art GCM had only nine levels for the entire atmosphere, which is perhaps too coarse for this type of approach. This approach was superceded in the UCLA GCM by a parameterization in which the planetary boundary layer (PBL) was taken to be a mixed layer whose top evolves according to an entrainment closure, and forms the bottom sigma level of the rest of the GCM (Suarez et al. 1983; Randall et al. 1985). This approach can elegantly represent entraining cloud-topped mixed layers, but does not attempt to resolve the internal structure of the PBL, and therefore cannot easily handle boundary layer decoupling or stable boundary layers. A PBL parameterization of this type, but with four layers internal to the PBL, is currently under development by the UCLA GCM group (A. Arakawa 2000, personal communication). Beljaars and Viterbo (1998) combined a nonlocal mixing scheme for dry convective boundary layers with an entrainment closure with good results; this approach is now implemented in the European Center for Medium-Range Weather Forecasts forecast model. Lock et al. (2000) have recently implemented a scheme coupling an entrainment parameterization to a nonlocal parameterization of moist PBL convection in the 40-layer U.K. Meteor. Office GCM, with encouraging results. Our approach is the first to couple a TCM to an entrainment parameterization.

Regardless of the PBL mixing and entrainment parameterizations used, important implementation issues arise that have not been adequately treated in the litterature. The first is how to represent the entrainment interface given a discrete vertical grid. In section 4, three implementations of the entrainment interface ere compared in a single-column modeling framework at both coarse and fine vertical resolution. In the “prognostic inversion” approach, the PBL depth is a prognostic variable, in the “reconstructed inversion” approach, the PBL height and the capping inversion strength are reconstructed from thermodynamic values at the grid points, and in the “restricted inversion” approach, the PBL height is restricted to lie on a flux level of the host model grid. The first two approaches permit a stratocumulus top and base to lie between grid levels and evolve continuously with time, but are more complicated to implement in a three-dimensional model. A continuously varying cloud thickness is desirable for the simulation of stratocumulus clouds, so that cloud microphysics and cloud feedbacks on turbulence and radiation are captured despite the coarseness of the grid.

The second implementation issue is how to ensure that radiation, vertical advection, and (if present) cumulus convection are handled by the host GCM consistently with the inversion representation in the PBL parameterization. Regarding radiation, Stevens et al. (1999) show how large eddy simulations (LES) of entrainment rate into radiatively driven boundary layers can be affected by underresolution of the radiative cooling zone (see their appendix C). Lenderink and Holtslag (2000) showed how poor performance of a classical TCM at low resolution, in which the inversion top gets unrealistically locked to a fixed grid level, can arise due to feedbacks between vertical advection of a poorly resolved inversion and turbulent mixing. In section 5, we propose a simple method to treat vertical advection consistently with the presence of an inversion, which we use with the profile reconstruction and restricted inversion approach.

To determine some parameters of the scheme, demonstrate its general performance at both high and low vertical resolution, and to compare the three possible inversion implementations, we present simulations of a dry convective boundary layer (section 6) and a stratocumulus to trade-cumulus transition (section 7). In these simulations, we have coupled the TCM to our own“host single-column model” consisting of cloud, radiation, surface fluxes, and vertical advection parameterizations. For simplicity, precipitation and shallow convection have not been included.

## 2. Turbulence closure model formulation

*u*and

*υ*of the horizontal wind velocity, and two thermodynamic variables that are approximately conserved for moist reversible adiabatic processes. These are the linearized versions of the liquid water potential temperature

*θ*

_{l}(Betts 1973) and the total water mixing ratio

*q*

_{t},Here and elsewhere, conventional notation is used where possible. The Exner function Π =

*p*

_{ref}/

*p*)

^{Rd/Cp}

*F*

_{r}, and the precipitation flux

*F*

_{P}.

*D*

_{h}denote a time-derivative operator following the horizontal motion. We vertically discretize using a staggered grid with indices increasing with height. The basic prognostic variables are carried at the integer-indexed levels

*j,*and their fluxes are carried at the intermediate levels

*j*+

^{1}/

_{2}. The vertical difference operator

*δ*

^{j}

*X*is defined as

*X*

^{j+1/2}−

*X*

^{j−1/2}. The pressure thicknesses of the thermodynamic layers are

*δ*

^{j}

*p*and the distances between grid points are

*δ*

^{j+1/2}

*p.*We define the total energy fluxand the total water fluxOverbars indicate ensemble averaged quantities and primes indicate deviations from the ensemble average over the grid cell. Bars over prognostic variables are dropped for clarity. With this notation, the vertically discretized budget equations for the prognostic variables arewhere

*U*

_{g}and

*V*

_{g}are the horizontal components of the geostrophic wind and

*ω*

### a. Turbulent fluxes

*θ*

_{l},

*q*

_{t}, and momentum within the PBL are computed using an eddy diffusivity

*K*

_{h}for the conserved thermodynamic variables and an eddy viscosity

*K*

_{m}for the horizontal velocity components, both derived from the 1.5-order turbulence closure:Following Mellor and Yamada (1982, hereafter MY82),

*K*

_{h}and

*K*

_{m}are related to the turbulent kinetic energy (TKE)

*e*

^{j+1/2}, a master turbulent master length scale (

*l*

^{j+1/2}), and stability functions

*S*

^{j+1/2}

_{h}

*S*

^{j+1/2}

_{m}

*K*

_{e}

*η*

_{Te}

*K*

_{m}

*K*

_{m}is the geometrically averaged eddy viscosity between adjacent flux levels (Stevens et al. 1999; Patankar 1980) and

*η*

_{Te}

### b. Turbulent length scale

*l*

*kz*

*kz*

*λ*

*k*the von Kármán constant, set to 0.4. To achieve the observed scaling of TKE in a deepening dry-convective PBL, we have chosen the asymptotic length scale

*λ*proportional to the PBL depth

*z*

_{i}:

*λ*

*η*

_{l}

*z*

_{i}

*η*

_{l}

*e*

^{1/2}. This prevents parcels from buoyantly accelerating to unrealistic vertical velocities as they traverse such layers. In stably stratified layers, the parcel deceleration is calculated from its buoyancy following Bougeault and Lacarrère (1989). Parcels are assumed to reach a height at which their vertical velocity goes to zero. The distances reached by the upward and downward parcels before reaching zero vertical velocity—or the PBL top/base—are, respectively,

*l*

_{u}and

*l*

_{d}. The parcel length scale is finally

*l*

_{p}

*η*

_{l}

*l*

_{u}

*l*

_{d}

*l*

*kz*

*kz*

*l*

_{p}

*l*

_{p}=

*η*

_{l}

*z*

_{i}and (15) is equivalent to (12). However, if internal stratification of the PBL is strong enough,

*l*

_{p}can locally scale with the depth of each convective layer with the proportionality factor

*η*

_{l}. Throughout this paper, the model performance is evaluated using the Blackadar length scale [Eq. (12)]. In section 7b, we investigate the sensitivity of simulated decoupling to the use of the parcel length scale.

## 3. PBL entrainment

Convective boundary layers are usually topped by a stratified entrainment interface, in which large-eddy motions drive smaller entraining eddies that incorporate free-tropospheric air into the PBL. In this section we describe our parameterization for entrainment. In section 4 we will discuss how to implement this or another entrainment parameterization into the TCM.

### a. Specification of entrainment parameterization

*w*

_{e}. This involves the specification of an eddy length scale

*L*and velocity scale

*U*:

*w*

_{e}

*AU*

*AU*

^{3}

*L*

_{i}

*b*

*A*is a nondimensional parameter called the entrainment efficiency, Δ

_{i}

*b*=

*g*Δ

_{i}

*θ*

_{υ}/

*θ*

_{υ,0}is the buoyancy jump across the inversion, and Ri =

*L*Δ

_{i}

*b*/

*U*

^{2}is a bulk Richardson number. In our parameterization, we follow (A13), neglecting the generally small contribution of liquid water to the inversion stability and approximating Δ

_{i}

*θ*

_{υ}≈ Δ

_{i}

*θ*

_{υl}, where

*θ*

_{υl}=

*θ*

_{l}(1 + 0.608

*q*

_{t}) is the liquid water virtual potential temperature, an adiabatically conserved variable for reversible processes that reduces to

*θ*

_{υ}in unsaturated air.

### b. Integral closure

*L*=

*z*

_{i}and

*U*equal to the convective velocity scale accounting for all buoyancy production terms within the PBL:Because

*U*and

*L*are scales based on the entire PBL, we call this an integral closure. Nicholls and Turton (1986) proposed, based on a fit to observations and laboratory experiments of dry CBLs and SCBLs,Here Δ

_{m}

*b*is the linearized average buoyancy of all possible mixtures between purely cloudy air and purely free-tropospheric air, relative to the cloudy air. The term

*a*

_{2I}

*E*determines the evaporative enhancement of entrainment. In appendix B, we relate

*E*to cloud-top liquid water and Δ

_{i}

*θ*

_{υl}, a necessary step for implementation of this parameterization. Whether this entrainment parameterization approach is appropriate, and whether this parameterization for

*A*represents the evaporative cooling feedback appropriately is a matter of debate; other approaches could be used in its place (e.g., Lock and McVean 1999; Moeng et al. 1999; van Zanten et al. 1999). Here, we focus on a variant of this approach (a local entrainment closure) that can naturally be generalized to decoupled PBLs or sheared PBLs.

### c. Local entrainment closure

*i*’) are used in the Turner–Deardorff closure. For a 1.5-order TCM, we can choose

*L*=

*l*

^{i}and

*U*=

*e*

^{i}

*l*

^{i}is characteristic of the size of the eddies generating entrainment. Hence, with the Blackadar length scale model,

*l*

^{i}

*kz*

_{i}

*kz*

_{i}

*λ*

*l*

^{i}

_{u}

*l*

^{i}

_{p}

*η*

_{l}

*l*

^{i}

_{d}

*l*

_{i}=

*kz*

_{i}/(1 +

*kz*

_{i}/

*l*

^{i}

_{p}

*A*is chosen to have the same form as (18),

*A*

*A*

_{L}

*a*

_{1L}

*a*

_{2L}

*E*

*a*

_{1L}= 0.16. Due to the different TKE profiles for stratocumulus-topped PBLs compared to dry convective PBLs, the optimal match of the local entrainment parameterization with the integral closure is achieved with

*a*

_{2L}= 15, rather than 60, as discussed in section 7a.

## 4. Implementation of entrainment parameterization

*p*

_{i}(

*t*), across which the thermodynamic variables have jumps Δ

_{i}

*θ*

_{l}(

*t*) and Δ

_{i}

*q*

_{t}(

*t*) that must be determined at each time step. If

*M*+

^{1}/

_{2}indexes the highest flux level at or below the inversion, profiles in conserved variables are assumed to be well mixed from

*p*

_{M−1/2}to

*p*

_{i}. The TCM is used to compute the fluxes at and below level

*M*−

^{1}/

_{2}, and the entrainment parameterization is used to compute the fluxes at

*p*

_{i}. Fluxes at

*M*+

^{1}/

_{2}are computed in different ways depending on the implementation approach chosen (see later). The turbulent fluxes of any conservative variable

*X*at

*p*

_{i}are (Lilly 1968)We now discuss each of the approaches in turn.

### a. Approach 1: Prognostic inversion

*p*

_{i}moves continuously with time according to(the cumulus mass flux across the inversion would also appear on the right-hand side if present). If the inversion lies between thermodynamic levels

*p*

_{M}and

*p*

_{M+1}, flux level

*M*+

^{1}/

_{2}is set to

*p*

_{i}. Other flux levels are at their nominal fixed pressures. If

*p*

_{i}rises above the next thermodynamic level

*p*

_{M+1},

*p*

_{M+3/2}is set to

*p*

_{i}, and

*p*

_{M+1/2}becomes a flux level internal to the PBL where the TCM is used to compute turbulent fluxes. Advantages of this approach are (i) that cloud liquid water can be diagnosed fairly precisely from the profiles in conserved variables and saturation mixing ratio below the inversion height, (ii) that inversion jumps in conservative variables are physically realistic, and (iii) that cloud properties vary quasi-continuously as the inversion moves up and down. This enables proper accounting for entrainment drying/warming of the PBL, and accurate computation of the cloud–radiation interactions. The main disadvantage of this approach is that it is impractical to implement in a three-dimensional host numerical model, in which other processes such as horizontal advection are computed on a fixed grid.

### b. Approach 2: Reconstructed inversion

The concept of reconstructing the inversion properties (height and jumps) from grid points profiles was proposed to us by B. Stevens. It attempts to retain the advantages of the prognostic inversion but using a fixed grid. It is useful to first discuss how a sharp inversion that may continuously rise or sink is represented within a fixed grid (Stevens et al. 1999; appendix C). The ×’s in Fig. 1b show the mean values of a particularly significant conserved variable, *θ*_{υl}, over each fixed grid layer, corresponding to the *θ*_{υl} profile shown in Fig. 1a. The value of *θ*^{M+1}_{υl}*ambiguous layer* *M* + 1 (Stevens et al. 1999) that includes the inversion is a weighted average of below-inversion and above-inversion profiles of *θ*_{υl}. Other conserved scalars *X* will have qualitatively similar behavior (Fig. 1b). If the inversion rises through the ambiguous layer, *X*^{M+1} will evolve toward its below-inversion value, as more of the ambiguous layer is made of PBL air. The turbulent fluxes of *X* are zero by definition at flux level *M* + ^{3}/_{2} atop the ambiguous layer, and are diagnosed at flux level *M* + ^{1}/_{2} by calculating entrainment fluxes at *p*_{i} and assuming linear variation of total energy, water, and momentum fluxes between *p*_{i} and *p*_{M−1/2}. The flux convergence in *M* + 1 is how entrainment contributes to the tendency of *X*^{M+1}.

The reconstructed inversion algorithm attempts to reconstruct the profiles of the variables in the ambiguous layer from their gridpoint value, and to deduce *w*_{e} and the entrainment fluxes at the inversion. It is based on writing conservation equations for conserved variables in the ambiguous layer using upward and downward extrapolation (see the detailed presentation in appendix C). Once these profiles are known, saturation water vapor pressure and cloud liquid water and cloud fraction (appendix D2) can also be precisely estimated and passed to the radiation scheme.

### c. Approach 3: Restricted inversion

_{i}

*X*

*X*

^{M+1}

*X*

^{M}

*M*+ 1 is a combination of PBL and free-tropospheric air, this jump may be much smaller than the true inversion jump. Since our entrainment closure correspondingly overestimates

*w*

_{e}[it varies as (Δ

_{i}

*θ*

_{υl})

^{−1}], entrainment fluxes at

*p*

_{M+1/2}obtained from (22) remain meaningful (see section 7a). The inversion cannot explicitly rise due to entrainment, but the turbulent fluxes act to mix layers

*M*and

*M*+ 1 with an “equivalent diffusivity”

*K*=

*w*

_{e}

*δ*

^{M+1/2}

*z.*Acting unopposed, this would systematically bring the properties of the ambiguous layer toward those of the PBL, reducing the stratification at flux level

*M*+

^{1}/

_{2}until ultimately the diagnosed PBL top jumps up to

*M*+

^{3}/

_{2}. Note that when both the ambiguous layer and the boundary layer top are saturated, the evaporative cooling efficiency is set to zero.

## 5. Vertical advection

*X*often have large gradients, so the advection scheme in a host GCM may produce large discretization errors. In a conventional

*E*−

*l*model for example, Lenderink and Holtslag (2000) have shown that this leads to unrealistic entrainment rates, especially when mean subsidence is strong. For our model, vertical advection is also a concern for the reconsructed and restricted inversion implementations. In these implementations, there is an ambiguous layer that is interpreted as a combination of below-inversion and above-inversion air. This leads to a natural way to handle vertical advection across the ambiguous layer. We writewhich is discretized asDifferent schemes (e.g., centered, upwinded, etc.) correspond to different choices for discretizing

*ωX*at flux levels. However, within the ambiguous layer, we have described how to reconstruct a profile of

*X,*including a discontinuity at the inversion. This profile dictates the values of

*X*

^{M+1/2}and

*X*

^{M+3/2}at the flux levels bounding the ambiguous layer. At the top of the ambiguous layer

*M*+ 1, the air comes from above the PBL and

*X*

^{M+3/2}is computed through downward extrapolation (see appendix C). At

*M*+

^{1}/

_{2}, as the air comes from inside the PBL, we set

*X*

^{M+1/2}=

*X*

^{M}, unless the inversion lies exactly at

*p*

_{M+1/2}, in which case we set

*X*

^{M+1/2}=

*X*

^{M+1}. This discretization of

*ωX*at

*p*

_{M+1/2}and

*p*

_{M+3/2}is used both with the reconstructed and restricted inversion approaches. Thus, subsidence strongly affects the ambiguous layer (corresponding to a lowering inversion) but does not create spurious gradients inside the PBL. At flux levels not adjoining the ambiguous layer, the host model advection scheme is used to calculate the advective fluxes. For our simulations, we use the scheme presented in appendix D, section a. In section 7c(2), we compare the performance of this scheme with an upwind advection scheme used at all flux levels, including those adjoining the ambiguous layer.

## 6. Simulations of a dry CBL

The dry convective boundary layer (CBL) heated from below has been extensively studied both observationally, using water tanks, and with LES, with consistent results. In this section, we show how simulations of a dry CBL with our parameterization determine the convective length scale parameter *η*_{l} [see Eq.(15)], the turbulent transport parameter *η*_{Te}*a*_{1L} for the local entrainment closure. In addition, we use it as an initial test of the skill of the three implementations of entrainment at varying grid resolutions. We have also performed successful simulations of a diurnal cycle. We will not discuss these simulations, since in a stable boundary layer the entrainment closure is not used and the model reduces to a standard 1.5 TCM, whose performances and sensitivity to model formulation can be found in the literature (see e.g., Brasseur et al. 1998).

We initialize the model with no moisture (*q*_{t} = 0) or TKE, and a vertical profile of *θ*_{l} (=*θ*) consisting of a 500-m-deep mixed layer capped by a 4-K inversion, with uniform static stability of 3.4 K km^{−1} above. A steady surface heat flux of 100 W m^{−2} is imposed throughout the 6-h simulation.

### a. Specification of *η*_{l} and *η*_{Te}

*η*

_{Te}

Since the TKE dissipation rate is related to the turbulent length scale, this can be used to determine *η*_{l}. The vertical distribution of TKE is primarily controlled by the efficiency of TKE transport, that is, *η*_{Te}*η*_{l} and *η*_{Te}*θ*_{l} simulated by our model with *η*_{l} = 0.085, *η*_{Te}*e*/*w*^{2}_{∗}*λ* were used in the Blackadar formulation, this would not be the case.)

As shown by Moeng and Wyngaard (1989), and earlier discussed by Zeman and Lumley (1976), buoyancy effects and gradients in vertical velocity variance, rather than down-gradient diffusion of *e,* are the main contributors to the budget of *w*′*e*′*K*_{e} = *η*_{Te}*K*_{m} with *η*_{Te}*K*_{e}, the vertical profile of TKE transport simulated by the model compares well with Moeng and Sullivan’s (1994) formulation (also shown in Fig. 3b).

### b. Local entrainment closure

*η*

_{l}

*z*

_{i}. Hence the dry entrainment efficiency for local entrainment closure,

*a*

_{1L}, can be estimated from (19) and (16):

*a*

_{1L}

*a*

_{1I}

*η*

_{l}

*e*

^{i}

*w*

^{2}

_{∗}

^{−3/2}

*a*

_{1I}= 0.2,

*η*

_{1}= 0.085, and

*e*/

*w*

^{2}

_{∗}

*a*

_{1L}= 0.16. A simulation using the local entrainment closure with this value for

*a*

_{1L}, inversion reconstruction, and 15-hPa vertical resolution is almost indistinguishable from the simulation with the integral closure discussed above.

### c. Sensitivity to entrainment implementation and vertical resolution

Figure 4 shows the inversion height for the three implementations of the entrainment closure, and 15-mb vertical resolution typical of many regional models. Figure 5 shows corresponding results with a coarser 31-level vertical grid, representative of current GCMs. All three approaches deepen the CBL at a similar rate following surface warming at both fine and coarse resolution. The scaled model profiles of TKE and buoyancy production look almost identical to those in Fig. 3, even with 31-level vertical resolution and a flux-level-restricted inversion. In contrast, a recent model intercomparison by Ayotte et al. (1996) found that most currently used GCM parameterizations were not properly simulating the buoyancy flux profile in the CBL.

## 7. Marine stratocumulus advecting over rising SST

The main basis for evaluating different versions of the moist PBL parameterization presented in this paper is an idealized simulation of an SCBL advecting over rising SST. We test the model’s ability to sustain stratocumulus cloud of realistic thickness at GCM vertical resolution, and to simulate the decoupling of the cloud layer from the subcloud layer that is observed to follow both daytime cloud solar absorption (Nicholls 1984; Nicholls and Leighton 1986; Hignett 1991) and systematic SST increase (Bretherton et al. 1995; Krueger et al. 1995; Bretherton and Wyant 1997). This is a necessary precursor to formation of shallow cumulus within the SCBL, which are thought to lead to the ultimate breakup of the capping stratocumulus layer (Wyant et al. 1997).

Following Bretherton and Wyant (1997), each simulation is initialized with a shallow, well-mixed SCBL over a relatively cool (286 K) SST, topped by a strong (13 K) inversion. The SST is increased at 1.5 K day^{−1}, while the free-tropospheric thermodynamic sounding is held fixed. This mimics the changing boundary conditions encountered by a subtropical SCBL advected equatorward by the trade winds off the west coast of a continent. Large-scale subsidence is specified assuming a constant divergence of 3 × 10^{−6} s^{−1}. Simulations start at midnight local time and last for 5 days. Table 1 gives other relevant boundary conditions, while Table 2 summarizes the model configurations used in the different simulations. The model time step is 8 min.

### a. Simulations using 15-mb vertical resolution

#### 1) Integral versus local entrainment closure

Figure 6a shows the time evolution of cloud base, cloud top, and surface lifting condensation level (LCL) during simulation IP. During days 1–3, the simulation exhibits a realistic diurnal cycle of cloud thickness, 300 m in early morning, thinning to 150–200 m by noon, comparable to observations reported during the First International Satellite Cloud Climatology Program Regional Experiment (FIRE) (Hignett 1991). In this figure, the internal stratification of the SCBL can be assessed from the difference between the surface LCL and cloud base. During the first three nights, these are nearly coincident, implying the SCBL is well mixed. During the first three mornings, surface LCL drops slightly while cloud base rises, indicating a slightly less well-mixed SCBL. Later in the simulation, the now deeper SCBL remains stratified or decoupled throughout the diurnal cycle, and is composed of a surface-driven convective layer, a conditionally unstable layer with intermittent turbulence, and a radiatively driven upper convective layer, as found in both observations and eddy-resolving models (Albrecht et al. 1995; Bretherton and Pincus 1995; Krueger et al. 1995). In section 7b, we will examine the feedbacks responsible for SCBL stratification in our parameterization.

Figure 6b shows results obtained with the local entrainment closure (simulation LP). For this simulation, we chose a value *a*_{2L} = 15 such that when the boundary layer remains well mixed (the first 3–4 days), LP matches the integral closure results of IP. Here, *a*_{2L} is smaller than *a*_{2I} because the concentration of buoyancy production at cloud top makes for a larger *e*/*w*^{2}_{∗}

#### 2) Comparison of inversion implementations at 15-mb resolution

Figure 7 compares the PBL evolution with the three different inversion implementations at 15-mb vertical resolution. As in the dry convection case, simulations with prognostic and reconstructed inversions give very similar PBL evolution, though the latter exaggerates the diurnal fluctuations of inversion height. For the reconstructed and restricted inversion, we use the special treatment of vertical advection discussed in section 5. Use of upwind advection prevents the reconstructed inversion from deepening comparably to the prognostic inversion. The restricted inversion simulation also gives very satisfactory results at this resolution with our vertical advection scheme.

#### 3) Advection–radiation–entrainment interplay with the restricted inversion

*θ*

_{l}and

*q*

_{t}jumps across the inversion, Fig. 8 shows that entrainment fluxes in simulation LR do have a similar evolution as in the reconstructed inversion simulation. This reflects an important property of our entrainment closure that allows it to work successfully with a restricted inversion. With the local entrainment closure (19), the entrainment flux of any conserved scalar

*X*can be writtenAs the ambiguous layer is cooled and moistened by entrainment, its

*X*is brought toward its value at the top of the SCBL, but so is its

*θ*

_{υl}(which is another scalar conserved for adiabatic processes). Hence the ratio Δ

_{i}

*X*/Δ

_{i}

*θ*

_{υl}remains nearly constant. The TCM maintains a fairly steady TKE during virtual deepening. If the evaporative enhancement factor

*E*is evaluated using jumps between the PBL top and the level above the ambiguous layer, it is also insensitive to virtual deepening. Hence, the entrainment fluxes are insensitive to the virtual deepening process, except for a few hours preceeding the jump of the inversion to the next flux level. During this period of time, the ambiguous layer is significantly saturated. Radiation then adds up with entrainment to cool the ambiguous layer, and therefore, does not contribute to TKE generation inside the boundary layer. As the evaporative cooling efficiency is also set to zero if the ambiguous layer is saturated, entrainment fluxes significantly decrease shortly before the inversion jumps to the next flux level.

### b. Mechanisms of SCBL response to external forcings

In this section, we discuss the response of the simulated SCBL to the diurnal cycle of insolation and to SST rise, and present how decoupling occurs in the TCM. The primary forcings are radiative flux divergence and surface fluxes, shown in Fig. 9a for simulation LD (other simulations are similar). The strong diurnal cycle in the net radiative flux divergence over the SCBL is due to cloud shortwave absorption. Surface sensible heat fluxes are small, while latent heat fluxes rise threefold over the simulation. Figures 9b and 9c show some measures of the SCBL response to these forcings. The convective velocity scale *w*∗ (computed from the vertically integrated buoyancy flux) responds to the strong diurnal cycle in net cloud radiative cooling, but does not have a large trend over the simulation. The entrainment rate *w*_{e} has a corresponding diurnal cycle and rises steadily as the inversion weakens from 13 to 7 K with the warming of the boundary layer. These variations in *w*_{e} contribute to keep entrainment fluxes of *θ*_{l} in close balance with the net SCBL radiative flux divergence (Bretherton and Wyant 1997). The minimum in buoyancy flux across the boundary layer (

#### 1) Diurnal cycle in the early phase

During the early phase of the simulation, nighttime TKE profiles simulated by the model (Fig. 10a) are comparable to those in a typical shallow nocturnal marine SCBL observed during FIRE (Hignett 1991; Duynkerke and Hignett 1993). The simulated daytime TKE profile is somewhat larger and more vertically uniform than observed (Fig. 10a). Although environmental conditions were quite similar between FIRE and our simulation, the observed daytime cloud thinning was more pronounced than in our model. This was partly due to a strong diurnal cycle in the mean inversion height (Duynkerke and Hignett 1993), suggesting a diurnal cycle in the mean subsidence. We believe this may account for the model simulation differences, because TKE is driven mainly by buoyancy fluxes within the cloud layer, so a thinner cloud cannot support as much TKE. The proximity of the observation site with the coast may also have an impact on the observations reported by Hignett (1991) (B. Stevens 2000, personal communication).

#### 2) Processes controlling decoupling in the TCM

Observations (Hignett 1991; Rogers et al. 1995; de Roode and Dyunkerke 1997) and LES (Krueger et al. 1995; Bretherton and Wyant 1997; Stevens 2000) indicate that a distinct minimum in TKE and a negative buoyancy flux at the top of the surface-driven mixed layer are characteristic of the decoupled SCBL. Although the TCM realistically stratifies the SCBL during daytime and the decoupled phase, TKE does not develop a pronounced subcloud minimum (Fig. 10b). Instead, it is maintained there against buoyancy destruction and dissipation by turbulent transport (Fig. 11a). So, how is eddy diffusivity *K*_{h} suppressed at this level to produce stratification? Recall that *K*_{h} is a product of *e*^{1/2}, *l,* and the stability functions *S*_{h} [which are functions of normalized stratification *N*^{2}*l*^{2}/*e*; see Galperin et al. (1988)]. Although *e*^{1/2} and *l* remain sizeable in the subcloud layer even with SCBL stratification, the *S*_{h} is rather sensitive to stratification. In neutral stratification, *S*_{h} is less than 10% as large as in strongly unstable stratification, and in weakly stable stratification seen in the decoupled phase, *S*_{h} drops a further factor of 10. Variations in *S*_{h} clearly dominate the variations in *K*_{h} in the decoupled regime (Figs. 11b–d).

#### 3) Sensitivity of decoupling to the master turbulent length scale

In the decoupled regime, a single Blackadar length scaling with the SCBL depth is no longer appropriate. Hence, we also performed a deepening/warming simulation with the Bougeault parcel-displacement-based master length scale [see Eq. (2.2)], which is sensitive to internal stratification inside the SCBL. Figure 12 shows that in the two convective layers the parcel length scales with the convective layer depth, while it drops in the stably stratified layer below the cloud base. It is smaller than the Blackadar length scale throughout the decoupled PBL. The profile of *K*_{h} has a shape similar to the one simulated with the Blackadar length scale, except in the upper convective layer where it is significantly smaller. Consequently, the upper part of the PBL is more stratified with the parcel length scale. As turbulent dissipation is proportional to *l*^{−1}, but buoyancy fluxes are little affected by the length scale, TKE is reduced throughout the boundary layer. This brings TKE profiles simulated by the model in the decoupled phase (Fig. 13) in better overall agreement with the profiles presented by Rogers et al. (1995) and de Roode and Duynkerke (1997). In particular the observed TKE minimum below cloud base is now well captured by the model. During these observations, characteristic of decoupled boundary layers during nighttime and daytime, a substantial fraction of the TKE was localized within cumuli rising into a stratocumulus [broken in flight 5 of de Roode and Duynkerke (1997)]. The simulations presented here do not include an explicit parameterization of shallow cumuli. The TKE scheme imperfectly represents their effects by intermittent pulses of cloud and TKE rising from the surface LCL, but it would be more realistic to include a separate shallow cumulus parameterization. We will present results from such an approach in a forthcoming paper.

Figure 14 compares the evolution of the PBL over seven days of the deepening/warming simulation using the two length scale models. Bulk features of the PBL are remarkably insensitive to the length scale model, even though TKE and *K*_{h} explicitly depend on the length scale. One reason is that the entrainment closure (19) relates the entrainment rate to *e*^{3/2}/*l,* that is, turbulent kinetic energy dissipation. Averaged over the upper convective layer, dissipation balances buoyancy production, which is mainly radiatively driven and not sensitive to the turbulence scheme. Hence, the entrainment rate is insensitive to the turbulent length scale. Similarly, the stable layer can be thought of as a valve that regulates the cloud thickness so that entrainment warming (which increases for a thicker cloud as the evaporative enhancement factor *E* rises) stays in balance with radiative cooling. As long as some mechanism (in our parameterization the stability functions) permits this valve to form, the thermodynamic profiles are insensitive to the details of the parameterization. However, the parcel length scale enables more realistic simulation of TKE in decoupled boundary layers.

### c. Low-resolution deepening/warming simulations

Low-resolution (31 level) results (Fig. 15) with the prognostic inversion approach are very similar to those presented in Fig. 7 for the high-resolution case. With the two other approaches, decreasing the resolution introduces a bias on the liquid water path (LWP) (Table 2). With the reconstructed inversion, the cloud thickness and cloud depth are too large at lower resolution, and the PBL depth is underestimated by 100 m or so. Differences in LWP are close to 30% on average over the whole simulation. This increase in cloud depth may be due to the mixed layer assumption made to reconstruct thermodynamic profiles between *p*_{M−1/2} and *p*_{i}. At coarse resolution, it is common that the cloud base (where negative buoyancy fluxes and decoupling typically initiate) is above *p*_{M−1/2}. This may impede the development of a stable layer below cloud base, delay decoupling, enhance vertical mixing, and increase cloud depth.

With a restricted inversion, the model is still able to sustain the stratocumulus at the 31-level resolution and the PBL evolution is qualitatively similar, but the cloud is systematically too thin. The location of the Sc base is well captured, but the PBL top is underestimated, so liquid water path is only half as large on average as in simulation IP (Table 2). This leads to reduced longwave cooling across the SCBL in the coupled regime, delayed deepening, and abrupt decoupling when the cloud top jumps to the next flux level.

#### 1) Sensitivity to the *a*_{2L} parameter

We tried to compensate for the LWP bias of the restricted inversion approach by retuning the evaporative enhancement feedback parameter *a*_{2L}. Smaller *a*_{2L} means that for a given TKE, higher cloud-top liquid water content (a thicker cloud) is needed to generate a given entrainment rate. Thus one might expect *a*_{2L} to regulate equilibrium stratocumulus cloud thickness. We show in Table 3 results obtained with *a*_{2L} = 10 and *a*_{2L} = 5. As expected, this parameter has a strong impact on the liquid water content of the cloud, which increases by a factor 1.5 with *a*_{2L} = 5. However, both the PBL top and cloud base are seriously underestimated with *a*_{2L} = 5 because the entrainment rate is too small. Consequently, the surface layer is too cold and too moist (Table 3). As the PBL remains too shallow, it also remains well mixed even over warmer SST, and cannot support a transition toward cumulus convection. Hence, retuning of *a*_{2L} is not advisable.

#### 2) Sensitivity to vertical advection

In the ambiguous layer, vertical advection is of comparable importance to entrainment. Inversion reconstruction suggested a natural treatment of vertical advection at the flux levels bounding the ambiguous layer (section 5). However, it is obviously simpler to just use the host model vertical advection scheme throughout. To test how this would affect our results, we performed simulations similar to LD_{31} and LR_{31} but with upwind advection. For the reconstructed inversion, this fails completely. With a restricted inversion, use of upwind advection lowers the simulated liquid water path (Table 3) by an average of 40% since the PBL warms, dries, and deepens faster than with our advection scheme. This bias is of the same sense and magnitude as the bias introduced by use of a restricted inversion versus reconstructed inversion. Other advection schemes tend to perform worse; hence use of our advection scheme around the ambiguous layer is important in obtaining reasonable results, especially with coarse vertical resolution. When the subsidence is larger than the simulated entrainment rate, the model does not tend to lock the PBL top on a flux level, as is the case with the *E* − *l* model (Lenderink and Holtslag 2000), either with our advection scheme or upwind advection. This again illustrates the utility of treating the inversion using an explicit entrainment parameterization.

## 8. Conclusions

We present a new PBL parameterization for large-scale models that combines 1.5-order turbulent closure within the PBL with an explicit entrainment parameterization at the top of a convective PBL. The parameterization is designed to provide efficient and accurate simulations of cloud-topped boundary layers given limited vertical resolution. The convective PBL is assumed to be topped by an infinitely thin inversion where we apply an entrainment closure. We use a Turner–Deardorff entrainment parameterization that relates entrainment rate to boundary layer TKE, an eddy length scale, and inversion strength. Following Nicholls and Turton (1986), the entrainment rate into cloud-topped boundary layers is considerably enhanced by mixing-induced evaporative cooling. The formulation of the TCM is classical [it is based on the Mellor and Yamada (1982) formulation and uses moist conserved thermodynamic variables] except for the following important feature: in the case of an SCBL, the cloud-top LW radiative flux divergence is all applied at the inversion, rather than being distributed through a grid layer. This helps ensure a reasonable profile of buoyancy production even at coarse vertical resolution. The vertical transport of TKE has also been artificially increased in order to obtain proper matching of the TKE profile with LES of various convective PBLs.

We compared three different implementations of the entrainment closure in a single-column model. The prognostic inversion implementation provides good simulations of cloud-topped and dry convective boundary layers at all resolutions tested. Like some GCMs, it uses an inversion-following coordinate level. Despite its good performance in our simulations, the prognostic inversion has potential difficulties for GCM use. First, it is not compatible with the fixed vertical grid levels used in most GCMs. Second, in some situations the PBL top may discontinuously change in time or space, requiring very careful treatment of a prognostic inversion. Examples include decay of the daytime dry convective boundary layer over land or PBL decoupling into two distinct turbulent layers.

The two other approaches, because they are diagnostic in nature, make the PBL scheme more widely applicable to a whole range of situations. The grid levels are fixed and at the PBL top there is an ambiguous layer whose layer-mean properties are intermediate between the PBL top and the free troposphere. We can interpret the ambiguous layer as a partial layer of PBL air overlaid by free-tropospheric air. This structure is explicitly diagnosed by the reconstructed inversion implementation. This provides simulations nearly as good as the prognostic inversion. Its implementation in a large-scale model requires that vertical advection and the cloud scheme account for the presence of PBL air in the ambiguous layer. The restricted inversion approach, in which the internal structure of the ambiguous layer is ignored, is simplest to implement in a large-scale model. It produced good simulations of penetrative dry-convective boundary layers at both vertical resolutions tested (15 mb and a coarser 31-level variable mesh). For stratocumulus, it also produces very good results at l5-mb resolution. However, as it systematically truncates the PBL top below its actual value, the restricted inversion produces stratocumulus clouds with too low a liquid water path at lower vertical resolution. This bias might be correctable by applying the “profile reconstruction” algorithm solely for estimates of liquid water path, which would then be passed to the radiation scheme.

With a restricted or reconstructed inversion, the model performance is strongly dependent on the vertical advection scheme. We propose a correction to the advective fluxes at the flux levels bounding the ambiguous layer inspired by profile reconstruction. Our approach for vertical advection is essential for obtaining good results with the reconstructed inversion; for the restricted inversion, it largely improves the results compared to an upwind scheme (giving 50% higher liquid water path).

In theory, our PBL scheme is compatible with a variety of entrainment parameterizations, and in fact there is considerable debate about the most appropriate parameterization for entrainment into an SCBL. However, the success of the scheme with a restricted inversion does depend on three important features of our entrainment parameterization. First, the entrainment rate is inversely proportional to the jump in a conserved variable (*θ*_{υl}), so the entrainment fluxes are not sensitive to the fraction of boundary layer air in the ambiguous layer above the PBL, and do not vary spuriously as the ambiguous layer evolves. Second, the entrainment rate increases rapidly with cloud-top liquid water content due to the evaporative cooling feedback. In practice, the strength of this feedback helps regulates the cloud thickness even when the cloud is barely vertically resolved. Finally, our entrainment closure makes the PBL model quite insensitive to the particular choice of the convective length scale in the turbulent closure model.

It is our great pleasure to thank Bjorn Stevens for many fruitful discussions on various aspects of this work. We are also grateful to him, David Randall, and Stephan de Roode for making useful suggestions on previous versions of the manuscript. The research was supported by NASA Grant NAG5-6101.

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# APPENDIX A

## TKE and Turbulent Fluxes

### TKE equation

*e*on flux levels is discretized as follows:The subscripts

*n*or

*n*+ 1 indicate the time step where the terms are computed, while

*δt*is the time increment

*t*

_{n+1}−

*t*

_{n}. For stability, a backward Euler scheme is used. It is modified to allow a linear tridiagonal implicit solution for the

*e*

^{j+1/2}by evaluating at the old time level

*n*those coefficients that multiply each term,

*e*

_{n+1}. Resolved-scale TKE advection is neglected.

*P*

_{b}and shear production

*P*

_{s}are computed aswhere

*N*

^{2}is the moist buoyancy frequency (

*A*12) and

*K*

_{h}is the eddy-diffusivity (10). Dissipation

*D*

_{e}is modeledIn (A4),

*l*

_{ϵ}is a length scale for dissipation taken proportional to the turbulent length scale (

*l*

_{ϵ}=

*B*

_{1}

*l*

_{e}) with

*B*

_{1}= 5.87 (equivalent to MY82 after appropriate translation of notation). Note that for stability, the formulation of the dissipation term is partly implicit. Turbulent transport

*T*

_{e}is parameterized based on downgradient diffusion,whereAs justified in section 6, we use an enhanced eddy viscosity for TKE,with

*η*

_{Te}(fixed to 5) a nondimensional coefficient tuned against a variety of convective BL simulations.

*F*

^{M+1}

_{n+1}

*e*must be specified at the surface pressure level

*p*

_{1/2}=

*p*

_{s}. Here, TKE is diagnosed according to surface similarity theory and production–dissipation balance (MY82):

*e*

_{s}

*B*

^{2/3}

_{1}

*u*

^{2}

_{∗}

*u*∗ is the surface friction velocity.

When this scheme was initially implemented, all diffusivities were calculated from TKE, thermodynamic variables, and velocities at time level *n.* In strongly convective boundary layers at high vertical resolution, this approach produced 2*δt* oscillations. These oscillations are due to the very strong dependence of the stability functions (and hence the diffusivities) on the static stability in this regime, which leads to stiffness of the system of prognostic equations. We now use a predictor–corrector approach for calculating the diffusivities that cure this problem at the expense of additional computations. This approach makes the time stepping closer to a fully implicit backward Euler approach: it is more stable and removes the 2*δt* oscillations.

### Computation of *N*^{2}

*N*

^{2}consistent with moist thermodynamics. Fluctuations in

*θ*

_{υ}can be written aswhere primes denote departures from horizontal average. The dimensionless thermodynamic coefficients

*c*

_{t}and

*c*

_{q}are quite different in locally saturated air than in unsaturated air; in both cases they are weak functions of

*p*and

*T*specified following Cuijpers and Duynkerke (1993). In a layer partly filled with cloud, we use a weighted average of saturated and unsaturated values of

*c*

_{t}and

*c*

_{q}:where

*υ*

_{s}is the effective saturation fraction. There is some degree of arbitrariness in specifying

*υ*

_{s}. A common approach, used by Ballard et al. (1991), Bechtold et al. (1995), and others, is to specify

*υ*

_{s}to be the cloud fraction at the level of interest; however, this introduces a strong dependence of the PBL model on the host model cloud microphysical scheme. Instead, we suggest that within the context of MY82, (

*N*

^{2})

^{j+1/2}at a given flux level

*j*+

^{1}/

_{2}is best thought of as a “bulk” buoyancy frequency across the layer from

*p*

_{j}to

*p*

_{j+1}. We distinguish two cases. If level

*j*+ 1 is saturated, we let

*υ*

^{j+1/2}

_{s}

*p*

_{j}to

*p*

_{j+1}that is saturated in the horizontal mean (computed using our finite-volume interpretation of the thermodynamic profiles discussed in section a of appendix D). If level

*j*+ 1 is unsaturated in the mean, we let

*υ*

^{j+1/2}

_{s}

*υ*

^{j+1/2}

_{s}

*j*+

^{1}/

_{2}as follows:A corollary to the approximation

*υ*

^{M+1/2}

_{s}

*θ*

_{υl}≡

*θ*

_{l}(1 +

*r*

_{m}

*q*

_{t}) is the liquid water virtual potential temperature and

*r*

_{m}= 0.608. Note that

*θ*

_{υl}is equal to

*θ*

_{υ}in unsaturated air and is conserved in adiabatic displacements of a fluid parcel. This approximation has the attractions of numerical efficiency, and of ready transferability to the restricted inversion entrainment implementation (discussed in section 4c). Because jumps at the top of Sc are large, we have found that this simplification does not alter the model performance.

### Buoyancy flux at the inversion

*M*+ 1 just above the PBL is unsaturated, all the longwave flux divergence across the uppermost thermodynamic layer containing PBL air is located exactly at the inversion. With the reconstructed inversion, this LW flux divergence is

*δ*

^{M+1}

*F*

_{lw}and we can write the buoyancy flux just below the inversion aswhere

*w*′

*θ*

^{′}

_{υ}

^{i}is the entrainment flux of

*θ*

_{υ}. With the restricted or prognostic inversion, the index

*M*replaces

*M*+ 1 in the above expression. This enables the TCM to reasonably capture the forcing of convection by cloud-top longwave cooling even at low resolution.

# APPENDIX B

## Evaporative Enhancement Formulation

*E*

_{m}

*b*

_{i}

*b*

*χ*

_{s}

_{i}

*b*

_{s}

_{i}

*b*

*χ*

_{s}is the mixing fraction of above-PBL air in an exactly saturated mixture of cloudy and above-PBL air, Δ

_{i}

*b*

_{s}= (

*g*/

*θ*

_{0})(

*β*Δ

_{i}

*θ*

_{e}−

*θ*

_{0}Δ

_{i}

*q*

_{t}), and the thermodynamic parameter

*β*is defined as in Randall (1980). We define

*Q*as −

*q*

_{l}/Δ

_{i}

*q*

_{t}, where

*q*

_{l}is the cloud-top liquid water content. We also define the inversion stability parameter

*R*≡

*C*

_{p}ΠΔ

*θ*

_{e}/(

*L*Δ

*q*

_{t}). We write Δ

_{i}

*b*

_{s},

*χ*

_{s}, and Δ

_{i}

*b*in terms of

*Q*and

*R*:where

*k*

_{d}= 1 − Π

*ϵr*

_{m}and

*ϵ*=

*C*

_{p}

*θ*

_{0}/

*L.*Therefore, we have

*θ*

_{e}in favor of jumps in

*θ*

_{υl}. Since

*R*can be written asWe introduce the parameters

*r*= −

*L*Δ

*q*

_{t}/(

*C*

_{p}ΠΔ

*θ*

_{υl}) and

*λ*=

*Lq*

_{l}/(

*C*

_{p}Δ

*θ*

_{υl}Π). Then

*R*and

*Q*can be written in terms of

*r*and

*λ*:Substituting (B8) into (B3) and (B5) gives.Substituting these expressions into (B1), the evaporative enhancement term can be written

*E*

*λF*

*r, λ, p, T*

*F*(

*r, λ, p, T*) around 0.8.

# APPENDIX C

## Profile Reconstruction

The base of the ambiguous layer (flux level *M* + ^{1}/_{2}) is identified as the first flux level above the surface that is saturated, has a Richardson number Ri^{M+1/2} > Ri_{c} = 0.3, and is below a minimum PBL top pressure set to 700 mb. If these criteria fail to identify the PBL top (in that case the PBL is not cloud topped), *M* + ^{1}/_{2} is found as the lowest level where Ri^{M+1/2} > Ri_{c}. Here, Ri_{c} is chosen to significantly exceed the maximum gradient Richardson number of 0.19 for a boundary layer in production–dissipation balance permitted by the Galperin et al. (1988) stability functions.

The algorithm described below calculates a unique inversion pressure satisfying the following four conditions once the thermodynamic layer *M* is found.

- The inversion lies somewhere in the ambiguous layer, between
*p*_{M+1/2}and*p*_{M+3/2}. - Below the inversion,
*θ*_{υl}= . In the part of the ambiguous layer above the inversion,*θ*^{M}_{υl}*θ*_{υl}is extrapolated downward from its overlying profile. - The vertical average of the reconstructed profile of
*θ*_{υl}over the ambiguous layer is .*θ*^{M+1}_{υl} - The inversion jump Δ
_{i}*θ*_{υl}> Δ_{0}, a threshold value we take as 0.1 K.

*θ*

_{υl}profile. We define

*s*

^{M+3/2}

_{θυl}

*δ*

^{M+3/2}

*θ*

_{υl}/

*δ*

^{M+3/2}

*p*as the slope of

*θ*

_{υl}between

*p*

_{M+1}and

*p*

_{M+2}and similarly define

*s*

^{M+5/2}

_{θυl}

*s*

^{M+3/2}

_{θυl}

*s*

^{M+5/2}

_{θυl}

*θ*

_{υl}downward from

*p*

_{M+2}to

*p*

_{M+3/2}, we use

*s*

^{+}= max(

*s*

^{M+5/2}

_{θυl}

*s*

^{M+3/2}

_{θυl}

*θ*

^{M+3/2}

_{υl}

*θ*

^{M+2}

_{υl}

*s*

^{+}

*p*

_{M+3/2}

*p*

_{M+2}

*θ*

_{υl}down into the ambiguous layer:We must choose

*p*

_{i}to produce the correct vertical average of

*θ*

_{υl}:Let

*μ*= (

*p*

_{M+3/2}−

*p*

_{i})/

*δ*

^{M+1}

*p*be the above-inversion mass fraction of the ambiguous layer

*M*+ 1. Substituting (C2) into (C3), we obtain the quadratic equationThe physically meaningful solution with 0 <

*μ*⩽ 1 can be shown to always be the smaller root of (C4). It is possible for the reconstructed inversion to lie on the flux level

*M*+

^{1}/

_{2}, that is,

*μ*= 1.

*μ*is found, Δ

_{i}

*θ*

_{υl}can be computed as

_{i}

*θ*

_{υl}

*θ*

^{M+3/2}

_{υl}

*s*

^{+}

*μδ*

^{M+1}

*p*

*θ*

^{M}

_{υl}

_{0}, we consider the reconstruction of

*θ*

_{υl}by (C2) to be successful and compute the jumps in other variables as explained later. If not, we require that Δ

_{i}

*θ*

_{υl}= Δ

_{0}. If we knew the inversion height (i.e.,

*μ*), the above-inversion slope of

*θ*

_{υl}would have to beThere is a unique

*μ*for which the resulting reconstructed

*θ*

_{υl}profile has the correct vertical average (C3). The equation for

*μ*has the form (C4), but with

*s*

^{+}found from (C6). It simplifies to a linear equation with the solutionIt can again be shown that 0 <

*μ*< 1.

*p*

_{i}is found, the profiles of the conservative variables

*θ*

_{l},

*q*

_{t},

*u,*and

*υ*in the layer

*M*+ 1 are found as follows. Let

*X*be any one of these variables. We again assume that

*X*=

*X*

^{M}between

*p*

_{M−1/2}and

*p*

_{i}. Since

*p*

_{i}is known, we can deduce the mean value

*X*

^{+}of

*X*over the above-inversion fraction of layer

*M*+ 1 by equating the layer mean of

*X*to

*X*

^{M+1}:With

*X*

^{M+3/2}being known from

*X*

^{M+2}and

*s*

^{M+3/2}

_{X}

*X.*

*X*has a slope equal to the gradient

*p*

_{i}is not a flux level, we must now use the entrainment fluxes to specify the turbulent energy and water fluxes at the highest flux level

*M*+

^{1}/

_{2}within the PBL. Between this flux level and the inversion height, the conservative variables have the same values of

*X*

^{M}as in the layer below. Hence the total energy, water, and momentum fluxes vary linearly between

*p*

_{M−1/2}and

*p*

_{i}, and they can be linearly interpolated to

*p*

_{M+1/2}. At

*p*

_{i}, the total energy and water fluxes areThe fluxes at

*p*

_{M−1/2}are computed using turbulence closure and a radiation scheme. Hence,withIf the boundary layer top is saturated, it is assumed that all the radiative flux divergence in the thermodynamic layer containing the inversion occurs within the cloud. Hence

*F*

^{i}

_{r}

*F*

^{M+3/2}

_{r}

# APPENDIX D

## Single-Column Model Formulation

In this section, we summarize our single-column model within which we implemented our PBL scheme.

### Finite-volume interpretation

The model equations are all formulated in terms of fluxes across the boundaries of thermodynamic layers, so it is appropriate to regard the gridpoint values as averages over their thermodynamic layer, a “finite volume” interpretation of the equations.

*j,*we approximate a prognostic variable

*X*(=

*θ*

_{l},

*q*

_{t},

*u,*or

*υ*) asWhile any specification of the slope

*s*

_{j}will guarantee that the layer average of

*X*will be

*X*

_{j}, we adopt the following specification, which is popular in finite-volume-based advection schemes (Carpenter et al. 1990) because it is second-order accurate and does not create maxima and minima in the profile that are not present in the gridpoint averages. We let

*s*

_{j+1/2}

*δ*

^{j+1/2}

*X.*

*s*

_{j−1/2}and

*s*

_{j+1/2}have opposite signs, we take

*s*

_{j}= 0. If they have the same sign, we take

*s*

_{j}to be the one of these slopes that has the smaller absolute value. Note that the resulting profile of

*X*will have discontinuities at the flux levels. We also assume that

*s*

_{M}= 0, that is, that the layer below the inversion is a mixed layer, a simplification that is quite accurate in practice for a convective PBL in which vertical turbulent mixing minimizes vertical gradients of

*X.*This simplification is useful mainly for a reconstructed inversion; for consistency of presentation we have made it for the other inversion implementations as well.

The treatment of the inversion layer depends on the inversion implementation. With a prognostic inversion, no special treatment is needed. With the reconstructed inversion, inversion reconstruction automatically gives us the desired profiles. If a restricted inversion is used, we assume that the ambiguous layer gridpoint value of any variable uniformly represents the entire ambiguous layer (Fig. 1).

This interpretation of the profiles is used in the single-column model when specification of variables at the flux levels is necessary. For vertical advection, we use (26). At flux levels not adjoining the ambiguous layer, we use the values of *X*^{j+1/2} from the finite-volume profile for the upwind thermodynamic layer scheme.

### Statistical cloud model

*q*

_{t}and

*θ*

_{l}betwen grid points using our finite-volume reconstruction. Since this profile is well mixed between the highest grid point

*M*within the boundary layer and the inversion pressure

*p*

_{i}, the PBL will be cloud topped if

*q*

_{sat}

*T*

_{i}

*p*

_{i}

*q*

^{M}

_{t}

*p*

_{i}is given by

*q*

^{M}

_{t}

*q*

_{sat}(

*T*

_{i},

*p*

_{i}). We calculate the saturation mixing ratio

*q*

_{s}just above and below each flux level and assume (in approximate accordance with our finite volume model) that saturation mixing ratio varies linearly between these points, and (if inversion reconstruction is used) between the highest PBL flux level and the inversion. From the profiles of

*q*

_{t}and

*q*

_{s}, we can deduce the stratocumulus cloud base

*p*

_{cb}and the vertically averaged liquid water content within each thermodynamic layer. The cloud base may lie above the highest grid point within the PBL; that is, the PBL may be diagnosed to be cloud topped even if all of the gridpoint values are subsaturated.

To compute cloud–radiation interaction, cloud fraction is estimated using a statistical cloud scheme and assuming a Gaussian probability density function for conserved variables (Bougeault 1981). The saturation deficit variance is estimated as the sum of the variance implied by turbulent moments and a fixed value (taken to be 0.02 times the saturation mixing ratio) linked to fluctuations of conserved variables at the mesoscale.

### Radiation and surface fluxes

*ϵ*

*e*

^{−1.66κLWP}

*μ*m.

Surface fluxes are also computed as in the NCAR CCM3 GCM (Kiehl et al. 1998), using surface similarity theory.

Large-scale and boundary conditions used in the deepening/warming experiment

Summary of the various deepening/warming experiments. The liquid water path is the mean over a 5-day simulation

Summary of the sensitivity experiments performed at low resolution with the restricted inversion. The boundary layer top (*z*_{i}) and the LWP are 5-day averaged. Here, Δ_{s}*θ*_{l} is the surface value minus the value of *θ*_{l} at the first grid level and is computed at the end of the 5-day simulation