a. Present formulation of the ECMWF model

At present, the existence of convection is tested by an undiluted ascent of a parcel that is initialized at the lowest model level in the surface layer, with the temperature and humidity at that level. If this undiluted parcel finds a cloud base—that is, the lowest level where it becomes supersaturated—with a parcel buoyancy larger than −0.5 K everywhere between the surface and the cloud-base level, then convection is initiated. This choice is simply related to the assumption that a rising parcel has enough kinetic energy to penetrate through subcloud levels despite a negative buoyancy down to −0.5 K. Subsequently, an initial guess of cloud-top height is made by extending the undilute ascent of the parcel, including condensation effects, until it finds its zero buoyancy level. The level where this occurs is used as the first guess for cloud-top height. The distinction between deep and shallow convection is based on the cloud depth, that is, the difference between this initial guess of cloud-top height and cloud-base height. If the cloud depth exceeds 200 hPa, deep convection is assumed, otherwise shallow convection.

The same undiluted ascent is also used to obtain cloud-base values for temperature and humidity. Since no vertical momentum equation is used in the subcloud layer, the in-cloud vertical velocity at cloud base is arbitrarily set to 1 m s⁻¹. Finally, the cloud-base mass flux M_b is determined. In the case of deep convection, M_b is dictated from the assumption that the convection reduces the convective available potential energy (CAPE) toward zero over a specified timescale τ (Fritsch and Chappell 1980; Gregory et al. 2000). This relaxation time τ is typically of the order of 1 h, but its precise value is made dependent on the horizontal model resolution. In the case of shallow convection, this CAPE closure is not used. Instead, a boundary layer equilibrium closure is used: the cloud-base mass flux M_b is chosen so as to sustain a neutral budget for the moist static energy h in the subcloud layer:

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The right-hand side of (1) contains the sources and sinks of h due to large-scale advection, subsidence, radiative cooling or heating, and turbulent fluxes of the boundary layer scheme. Usually, the dominant contribution originates from the surface flux, in which case this closure reduces to

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b. A new parcel model

A new model for strong updrafts in the convective boundary layer has recently been developed for use in the ECMWF parameterization package (Siebesma and Teixeira 2000). Its motivation grew out of the increasing need to have a more coherent parameterization of turbulent transport in the clear and cloudy boundary layers. To this purpose, one single formulation for strong updrafts that can be used in both the clear and cloudy boundary layers has been formulated (Fig. 1). The ultimate objective is to parameterize the transport associated with these strong updrafts (in both clear and cloudy conditions) with one single mass-flux scheme, whereas the transport due to the remaining smaller eddies can be parameterized by a more local diffusive approach. However, in this study we use this new updraft model only as a more physical update for the presently used undiluted parcel ascent described in section 2a.

A proper definition of strong updrafts that is valid in both the cloud layer and in the subcloud layer is not without problems. The widely used cloud core definition (Siebesma and Cuijpers 1995)—that is, updrafts that are positively buoyant and within clouds—obviously can not be used in the subcloud layer and the dry boundary layer. Therefore we define strong updrafts as those areas that contain the highest positive vertical velocities. More precisely, in the large eddy simulation (LES) analysis that has been used to design and evaluate this parameterization (van Ulden and Siebesma 1997; Siebesma and Teixeira 2000), strong updrafts are simply defined as a fraction a_u of all the grid points on a horizontal slab that contain the highest vertical velocities. By choosing a value of a_u ≃ 0.03, it has been checked that this definition coincides with the cloud core definition near cloud base so that a continuous formulation between the cloud layer and subcloud layer is guaranteed.

The model that has been proposed for the strong updrafts is a simple entraining plume model (Siebesma and Teixeira 2000). It is most conveniently defined in terms of moist conserved variables, such as the moist static energy h = c_pT + gz + Lq_υ and the total specific humidity q_t = q_υ + q_l (Betts 1973). It is worthwhile noting that there are limitations to the description of convection using the entraining plume assumption. Several authors (e.g., Raymond and Blyth 1986) have shown that observations of cumulus convection can be matched more closely when applying a multiple parcel ascent. However, since the main part of the convection scheme used here is based on an entraining plume model, it is appropriate to base our extension to the decision-making model on the same assumption.

A generic equation for the evolution with height of a conserved updraft property ϕ_u can be written as

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where ε is the fractional entrainment rate. Assuming no subgrid variability within the updraft, a simple additional calculation of the saturation specific humidity q_s(p, T) allows a determination of specific humidity q_υ, cloud liquid water/ice content q_l, and temperature T. A simple “microphysics” parameterization is added to the updraft equations by removing any condensate in excess of 0.5 g kg⁻¹. The water-loading effect of the remaining condensate is taken into account in the parcel buoyancy calculations. The effects of freezing are taken into account through a gradual transition from water to ice in the temperature range 273.16–250.16 K.

The fractional entrainment rate has been the subject of many recent studies (Siebesma and Cuijpers 1995; Grant and Brown 1999; Gregory 2001; Neggers et al. 2002). LES studies of the convective boundary layer indicate that the fractional entrainment rate behaves in the lower part of the boundary layer as (van Ulden and Siebesma 1997; Siebesma and Teixeira 2000; de Roode et al. 2000)

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For shallow cumulus convection, typical values of ε ≃ 1 ∼ 3 × 10⁻³ m⁻¹ are reported, both from LES studies (Siebesma and Cuijpers 1995) and observational studies (Raga et al. 1990). For deep convection, cloud-resolving models (CRMs) suggest values of ε of the order of 10⁻⁴ m⁻¹. Here (4) is used as a parameterization in the simple updraft model (3); it has the correct behavior in the lower part of the subcloud layer, and it interpolates smoothly to typical values found for shallow and deep convection. Moreover, simple scaling arguments in the cloud layer also support (4) (Siebesma 1998).

The updraft model is initialized by taking the mean value at the lowest model level z₁ and adding an excess that scales with the surface flux (Troen and Mahrt 1986):

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where w′ϕ^′_s is the surface flux. The value of the prefactor b ≃ 1 is based on LES results (Siebesma and Teixeira 2000). For the standard deviation σ_w near the surface, we use an empirical expression based on a combination of atmospheric data, tank measurements, and LES data (Holtslag and Moeng 1991):

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where w∗ is the usual convective velocity scale, and u∗ is the friction velocity.

In addition, an equation for the vertical velocity (Simpson and Wiggert 1969) is employed:

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with a buoyancy term B = g(θ_υ,u − θ_υ)/θ_υ as an extra source term. The coefficients c₁ and c₂ take into account the pressure perturbations and subplume turbulence terms. The same values, c₁ = 2 and c₂ = 1/3, that are currently already used in the full convection parameterization are adopted for the first-guess ascent. These values are varied to study the sensitivity of this choice.

Finally, the definition of the discrete cloud-base level is also altered. In the original formulation this was the lowest model level where the rising parcel is supersaturated. In the new formulation the exact cloud-base pressure is calculated. The discrete model level closest to this pressure is then defined to be the discrete cloud-base level. This way a systematic overestimation of cloud-base height is removed.

The new formulation of the first-guess parcel ascent has a number of advantages over the previous one. First, the overly simple dependence on buoyancy together with an artificial threshold in virtual temperature excess of −0.5 K, as used in the previous trigger function, becomes obsolete. Such a simple trigger function assumes that a parcel always has the same amount of kinetic energy that is just enough to overcome a potential barrier near cloud base of −0.5 K. In the present situation the amount of kinetic energy gained by the rising parcel in the subcloud layer is explicitly calculated by the vertical velocity equation (7). If cloud base is found while the vertical velocity is still positive, cumulus convection is initiated. The new preliminary cloud-top height is now simply defined as the height at which the updraft vertical velocity vanishes. This way the overshoot of convective clouds into an inversion is taken into account. Second, the present form of the entrainment rate of the updraft model has the advantage that the precise choice of the initialization height z₁ becomes less relevant. More precisely, if ϕ has a logarithmic profile, as is common in the surface layer, one can show that for ε ∼ 1/z, the excess ϕ_u − ϕ becomes height independent. As a result, the only height dependence is in the excess formulation (5).

a. Motivation and experiment setup

Before introducing a new parameterization into the full GCM and assessing its impact on either forecast performance or model climate, it is desirable to understand its workings in a more simplified setup. The most commonly used of such setups is the so-called single-column model (SCM) (e.g., Randall et al. 1996). A single column of the GCM is extracted and, by prescribing the large-scale conditions normally predicted in the full GCM, the response of the physical parameterizations to that forcing can be studied easily in detail. Although this approach is computationally cheap and in widespread use, one of its major drawbacks is the scarcity of suitable case prescriptions currently available.

Here a different approach is followed. The direct impact of the new algorithm described in section 2 is examined by performing a single time step integration of the full GCM rather than relying on SCM simulations. That way the parameterizations that are the subjects of the comparison are exposed to a large number of identical atmospheric conditions, and their response can be studied without allowing for feedback. This provides a first indication of how the introduction of any new parameterization will directly affect the model. This method was successfully applied before by Jakob and Klein (1999, 2000) when investigating the impact of a parameterization of cloud fraction effects on microphysics.

It is obvious that this method is not sufficient for understanding what impact a changed parameterization has on the full GCM. However, it will provide crucial information on the detailed working of the scheme, which will assist in the interpretation of the results of longer model integrations in which complex feedback processes will have occurred.

In order to facilitate the comparison, the ECMWF model was integrated using various versions of the first-guess parcel treatment for one time step at a horizontal resolution of T_L95, with 60 model levels in the vertical. A description of the different model versions can be found in Table 1. The initial conditions for all experiments were arbitrarily chosen to be those of 1 May 1987 and are taken from ECMWF's reanalysis (Gibson et al. 1997) archive. The results are analyzed on the model's native reduced linear grid (Hortal 1999) to avoid any influence of interpolation between grid points.

b. Results

As described in section 2, the decision-making algorithms under investigation provide the following information to the convection scheme: the existence of surface-driven convection, an initial estimate of convective cloud depth on which the decision on the type of convection is based, and the updraft properties at cloud base. In this section these quantities will be analyzed in the single–time step experiments for the various model versions in Table 1.

The most basic decision that the decision-making algorithms need to provide is that of the existence of surface-driven convection at a given model grid point. Furthermore, an initial estimate of the expected cloud depth is used in both algorithms to determine the prevalent type of convection, that is, deep or shallow. Figure 2 shows the relative frequency of occurrence of surface-driven convection separated into the deep and shallow types allowed for by the parameterization. The results are taken from the tropical and subtropical belt only (30°N–30°S, approximately 6000 model grid points). This area will be loosely referred to as the Tropics hereafter.

The parameterization using the new algorithm produces significantly less grid points with convection than the current one. The total fraction of points with convection is reduced from 0.74 in CTRL to 0.5 in NCUB. The ratio of deep to shallow convective points in CTRL is 0.32; it reduces to 0.27 in NCUB, indicating a relatively stronger influence of the new algorithm on deep convection. The various sensitivity experiments lead to variations in both the number of deep and shallow points. However, all of the experiments yield significantly less points with shallow convection. Further investigation reveals that the largest drop in convection occurs because the vertical velocity equation at many points yields negative w_u well before cloud base is reached. The current algorithm allows points with negative T_υ-excess up to −0.5 K to still be classified as convective, while in the new algorithm negative excess will quickly lead to vanishing w_u, and hence the updraft will terminate.

There is a large sensitivity to the choice of entrainment coefficient. The smaller the entrainment, the larger the number of convective points as expected. None of the entrainment coefficients, however, yields as much convection as the current algorithm, not even the undiluted ascent. This indicates the importance of other parts of the NCUB algorithm. Although for deep convection low entrainment leads to similar or even larger occurrences compared to CTRL, the occurrence of shallow convection remains significantly lower throughout. As outlined above, this is mainly because of the use of the vertical velocity instead of parcel buoyancy for the decision making. Perhaps surprisingly, the results are not very sensitive to the choice of coefficients in the vertical velocity equation, indicating some robustness in the use of this parameter for the decision making.

Changing the algorithm for choosing the location of the discrete cloud-base level also has a nonnegligible effect on the number of convective points. It is remarkable that this effect is larger than that of the choice of vertical velocity equation coefficients. Further investigation reveals that the change in convection occurrence, as seen in Fig. 2, is caused by two separate effects. First and foremost, a significant number of points (351 out of 2380) that have shallow convection in NCUB have no convection at all in CLEV. This can be understood as a result of poor vertical discretization. Clouds that are single-layer clouds in NCUB—that is, those for which the vertical velocity drops to zero immediately above cloud base—will be assigned as “nonconvective” in CLEV, since the cloud-base level here coincides with the first layer where the vertical velocity is zero. The drop in occurrence of shallow convection is partly compensated for by a shift from deep to shallow convection (127 out of 649 points).

Now that the occurrence of convection has been determined, some of the updraft properties at cloud base will be compared. First, the frequency distribution of updraft vertical velocity is shown in Fig. 3. Note that CTRL is absent in these figure panels since its first-guess algorithm does not evaluate the vertical velocity. The distributions of all experiments peak well away from zero w_u. This indicates that the points that were deemed nonconvective have been filtered out well before reaching cloud base. NCUB provides a distribution with a peak around +0.6 m s⁻¹. The sensitivity to the choice of coefficients is as expected. Larger c₂ and smaller c₁ lead to higher values of w_u. Since the entrainment rate enters the vertical velocity equation, it is not surprising to see a large sensitivity to the choice of ε. The choice of the discrete cloud-base level to be always the top level of the first level with condensation leads to a shift toward higher w_u. This can be understood by realizing that (i) as shown above, this choice leads to less convective points where the weaker updrafts have been filtered out and (ii) the lower part of the convective updrafts is usually marked by acceleration.

Because of its strong relation to parcel buoyancy, another crucial cloud-base parameter to investigate is the virtual temperature excess. This excess has a direct influence on CAPE and is strongly related to the moist static energy excess, both of which are important ingredients of the closure assumptions used in the parameterization. Figure 4 shows the frequency distribution of this variable for the Tropics. There is a large difference between the CTRL and the NCUB models. While CTRL shows a broad distribution of T_υ excess peaking around 0.3 K, NCUB exhibits a much narrower distribution, with its peak just above zero. Both models have a number of cases with negative values, with an explicit cutoff at −0.5 K in CTRL. At first thought, one would ascribe the differences in the distributions to the fact that NCUB is a strongly entraining plume, while the updraft in CTRL is undiluted. However, the sensitivity to entrainment rate shown in Fig. 4 suggests that although lower entrainment leads to higher excess values on average, the shape of the distribution does not change significantly. Changes to the vertical velocity equation show an even smaller sensitivity, and the results are therefore omitted from the figure. Interestingly, it is the CLEV model that leads to a similar distribution shape as found in the CTRL model, indicating that the choice of the discrete level for cloud base plays a significant role in determining the shape of distribution.

This can be understood by considering the evolution of the virtual potential temperature, θ_υ, during a parcel ascent together with an environmental profile. In the subcloud layer the virtual temperature of the parcel, θ_υ,u, is conserved and higher than that of the environment. In a typical convective environment, the subcloud layer is well-mixed, and hence the value of θ_υ is independent of height (θ_υ(z) = const). As soon as cloud base is reached condensation occurs, and from that point θ_υ,u follows a moist adiabatic profile, while the environment profile typically lies between that of a moist and a dry adiabat. This implies that above cloud base θ_υ,_u is increasing faster than θ_υ, with the consequence that the farther above true cloud base the discrete cloud base is chosen, the larger the parcel excess in θ_υ and hence T_υ. By always choosing the model level above true cloud base as the discrete cloud-base level, as done in CTRL and CLEV, the parcel excess is necessarily larger than in all other experiments, as is evident in Fig. 4.

The final cloud-base property considered here is the cloud-base mass flux. Figure 5 shows the frequency distribution of this quantity. Note that in this figure the frequency of occurrence has been normalized with the total number of convective points. Hence, only relative changes at points that are convective can be deduced. It is evident that the average cloud-base mass flux in NCUB is significantly larger than in CTRL. Another obvious feature is a number of spikes at selected mass-flux values that become dramatically large with NCUB but are already present in CTRL. This leads to two questions. Why are the mass fluxes larger, and what leads to the observed peaks in the frequency distribution?

The first question can be answered by considering the closure for shallow convection [see Eq. (1)]. This closure is a typical example for an equilibrium closure, where it is assumed that the convective flux of moist static energy from the subcloud into the cloud layer is such that the mean subcloud moist static energy is conserved. It is obvious from (1) that the cloud-base mass flux directly depends on the parcel excess in moist static energy at cloud base. It can be inferred from Fig. 4 that this excess is significantly smaller in NCUB than in CTRL, leading to larger values of cloud-base mass flux.

The maxima in the frequency distribution at certain mass-flux values have been identified as a numerical artifact. The experiments performed here use the nominal time step for the T95L60 resolution, which is 1 h. With this relatively large time step and the high vertical resolution of the ECMWF model near the surface, the (explicit) solution of the mass-flux equations in the convection scheme can only remain stable if the local Courant–Friedrichs–Lewy (CFL) criterion is fulfilled. This criterion implies a maximum mass flux in a given layer that is defined as

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where Δp is the depth of the model layer, and Δt is the model time step. Assuming a surface pressure of 1013.25 hPa, the values for maximum mass flux for ECMWF model layers at typical cloud-base heights (200–600 m) are 0.027, 0.034, 0.042, 0.05, and 0.058 kg m⁻² s⁻¹, respectively, well in-line with the location of the maxima in Fig. 5. This result clearly illustrates the increasing need for an implicit solver for the mass-flux scheme, as the CFL criterion is violated more often if models increase the time step and refine the vertical resolution.

The decision on the occurrence of convection in both algorithms is entirely based on subcloud layer properties and the fulfilment of some criteria at cloud base (a virtual temperature excess of more than −0.5 K in the current scheme or positive updraft vertical velocity in the new algorithm). In its current formulation, the next important decision that needs to be made in the ECMWF convection parameterization is the location of cloud top. This is so because the decision on convection type (shallow versus deep), which influences the choice of closure and entrainment rate, depends directly on cloud depth. In the CTRL model, cloud top is determined using an entirely thermodynamic criterion proposed by Arakawa and Schubert (1974), while NCUB uses a dynamical criterion based on finding at what height the updraft vertical velocity reverses sign (for details see section 2).

Figure 6 shows the cloud depth as determined by the CTRL and NCUB first-guess algorithms as well as the sensitivity in this case to the entrainment rate only (the sensitivity to w and CLEV is small). It is obvious that the two algorithms used in CTRL and NCUB give hugely different results. In CTRL many points are initially diagnosed to have convection deeper than 8 km, while in NCUB most of the points have cloud depths less than 2000 m. The sensitivity to entrainment rate is large, with higher (lower) entrainment rates yielding shallower (deeper) clouds. It is noteworthy that even the undilute version of NCUB diagnoses more shallow clouds than the original algorithm.

Given this result it is a legitimate question how the CTRL model provides the large number of shallow convective points seen in Fig. 2. The answer lies in the iterative nature of the scheme. After the first-guess calculation, which is the subject of this study, two iterations of a much more complex updraft calculation follow. It is obvious from Fig. 6 that the first-guess calculation performed in CTRL is very inconsistent with the later calculations. This is further emphasized in Fig. 7, which compares the first guess to the final cloud depth for both CTRL and NCUB. It is evident that, although far from perfect, the NCUB algorithm provides a first guess much more in-line with the final result of the convection parameterization.