#### 3.1. Validation set-up

To investigate the analytical expressions for and δ in more detail and to assess the validity, we use LES of the Dutch Atmospheric LES model (DALES; Cuijpers and Duynkerke, 1993) for three shallow convection cases. Two of the cases are more or less steady state shallow convection cases over tropical oceans designed from the field campaigns BOMEX (Barbados Oceanographic and Meteorological Experiment; Siebesma and Cuijpers, 1995) and RICO (Rain in Tropical Cumulus over the Ocean; Rauber *et al.*, 2007). For RICO we use the 24 h composite run. (More information about this case and the experimental set-up of the composite run can be found online at www.knmi.nl/samenw/rico.) The main differences between these two cases concern the cloud depth ( m for BOMEX and m for RICO) and the mass flux profiles (more variable in RICO). The third case is based on an idealization of observations made at the Southern Great Plains ARM (Atmospheric Radiation Measurement Program) site on 21 June 1997 (Brown *et al.*, 2002). The ARM case describes the development of daytime shallow cumulus convection over land. After approximately 5 h of simulation, at 1130 LT (local time), clouds start to develop at the top of an initially clear convective boundary layer. From this moment on, the cloud layer grows to a maximum depth of 1500 m at 1630 LT, after which it starts to decrease. Finally, at the end of the day at 1930 LT, all clouds collapse. For the ARM case we solely present results for the cloudy period. Because the ARM case is non-steady, it is pre-eminently suited as a thorough test of our expressions.

For all cases, precipitation is turned off in the LES model (only RICO observations show some light rain) and cloud base level is defined as the height where the mass flux is at its maximum (de Rooy and Siebesma, 2008). For BOMEX and RICO the first hour is excluded for spin-up reasons.

All presented LES results are hourly averaged and based on the cloud core sampling, i.e. all LES gridpoints that contain liquid water and are positively buoyant () are considered to be part of the cloudy updraught. Note that recently a sampling method based on passive tracers has been developed (Couvreux *et al.*, 2010) giving comparably good estimates of the total turbulent transport of moist conserved variables in the cloud layer. In principle such a sampling method could be used as an alternative for the core sampling to evaluate the analytical expressions.

As mentioned by Siebesma *et al.* (2003), the plume model breaks down near the inversion because a simple bulk approach with a single positive entrainment rate is not able to represent the behaviour of the core fields. It is therefore justifiable to exclude the top 15% of the cloudy layer (where the cloud layer is defined as the layer where *a*_{c} > 0). Also negative and/or δ_{qt} values indicate that the bulk approach breaks down and these situations are therefore excluded from our evaluation. However, when an expression or parametrization of or δ results in a negative value, it is cut off to zero, as would be done in practice. Note that this cut-off (instead of maintaining the negative value) has no significant impact on the results.

#### 3.2. Results

Before we show the results of the analytical expressions against LES, the sensitivity of Eqs. (27) and (28) for α is investigated. This is done by varying α and showing in Figure 3, for all three investigated cases together, the overall performance of Eqs. (27) and (28) in terms of the RMSE defined as

- (29)

where and *i* is an index over all presented (*N* = 1009) results. Figure 3 reveals that the optimal α for and δ_{w} (0.62) coincides and this value is also quite close to the values found by the aforementioned LES experiments (0.6, Voogd, 2009; 0.67, Simpson and Wiggert, 1969). Hereafter all presented results are based on α = 0.62.

The generally good correspondence of the expressions with the reference lateral mixing coefficients, including the correct height dependence, is confirmed by Figure 5. For clarity reasons, only some selected hours are plotted. Figure 5(a) reveals that the relatively large overestimations in the entrainment at 1330 LT (Figure 4(a)) occur near cloud base. This is also the case for 1230 and 1430 LT (not plotted). However, the aforementioned overestimations of the entrainment at 1930 LT show a much different height dependence with a maximum error at 1800 m, in the middle of the cloud layer (Figure 5(c)). Noteworthy in the detrainment profile plots (Figures 5(b, d)) is the large variation in time. (Note the different *x*-axis scale for and δ.) As explained by de Rooy and Siebesma (2008), the large δ values as observed during the first cloudy hours of the ARM case (Figures 5(b) and 4(b)) are for an important part caused by the relative shallowness of the cloud layers. In a bulk sense (averaged over the cloud layer depth), shallower layers inevitably lead to larger (1/*M*)∂*M*/∂*z* and (1/*a*_{c})∂*a*_{c}/∂*z* terms (). Under the usual divergent conditions, these terms only affect the detrainment ((22) and (28)), explaining the large δ values observed during the first cloudy hours of ARM. Besides the depth of the cloud layer, the shape of the mass flux profile, and therewith δ, is also influenced by environmental conditions (e.g. Derbyshire *et al.*, 2004) as well as properties of the updraught itself (de Rooy and Siebesma, 2008). The above-mentioned arguments support the approach of de Rooy and Siebesma (2008) to describe the mass flux profile with a fixed function for but a flexible parametrization for δ to account for the variations in the shape from hour to hour and case to case (e.g. a strong or zero decrease of the mass flux in the lowest half of the cloud layer). An interesting variation on this approach is given by Neggers *et al.* (2009) describing changes in the cloud fraction profile based on thermodynamical arguments.

The overall picture of Figures 6(a) and (c) is that both terms in (27) for , i.e. and (1/*w*_{c})∂*w*_{c}/∂*z*, are of the same order of magnitude, with the buoyancy term being somewhat larger. However, for δ_{w} (Figures 6(b) and (d)) the situation is different. Because the sum of the buoyancy and the vertical velocity terms in (28) for δ_{w}, i.e.

result mostly in small negative values for a large part of the cloud layer, the strongly fluctuating (1/*a*_{c})∂*a*_{c}/∂*z* term clearly dominates the height and time variation in δ. As a result, δ_{w} can be reasonably well approximated by − (1/*a*_{c})∂*a*_{c}/∂*z* with generally underestimations near cloud base and overestimations near cloud top. Also a direct comparison between δ_{qt} and − (1/*a*_{c})∂*a*_{c}/∂*z* for all three cases together gives reasonable results (Figure 7(a)).

Table I. Root Mean Square Error of different expressions for and δ against and δ_{qt} respectively for the ARM, BOMEX and RICO case together (*N* = 1009). Results for and δ_{w} are based on α = 0.62 and for on γ = 0.31. | RMSE |
---|

| 3.45 × 10^{−4} |

| 4.87 × 10^{−4} |

| 1.01 × 10^{−3} |

δ_{w} | 3.85 × 10^{−4} |

Yet another approximation can be made by simply ignoring the (1/*w*_{c})∂*w*_{c}/∂*z* term (Figure 6(a) and (c)) in (27), leading to the following expression, as proposed by Gregory (2001)

- (31)

where γ represents a tuning constant. Gregory (2001) used γ = 1/12 and found a 50% underestimation of his expression against LES for BOMEX. The sensitivity of (31) for γ is shown in Figure 3 which suggests a much higher optimal value, namely γ = 0.31. To demonstate the potential of (31), we show results with the latter optimal value. Figure 7(c) for all three cases reveals reasonable results for (31) but less good than with the full analytical expression 27 (compare Figure 7(c) with Figures 4(a) and 8(a) or see Table I or Figure 3). Especially for BOMEX and RICO, the (1/*w*_{c})∂*w*_{c}/∂*z* term (not shown) has relatively large values near cloud base and cloud top. Consequently, in Figure 7(c) reveals inevitably a bend in the scatterplot with overestimations near cloud base and top and underestimations around the middle of the cloud layer. Another indication that the influence of (1/*w*_{c})∂*w*_{c}/∂*z* cannot be ignored in an expression for comes from examining the profiles in Figure 5(c) in detail. For example, looking just above 1500 m for 1730 LT, atypically increases slightly with height. This increase is caused by the increase of − (1/*w*_{c})∂*w*_{c}/∂*z* at the corresponding height (Figure 6(c)). Although such increases in are generally at the approximately correct heights, they seem to be somewhat stronger than in , especially for hour 1930 LT. As well as Gregory, Mironov (2009) and Rio *et al.* (2010) also mentioned the term in an expression for the lateral mixing coefficients.

As a validity check we also present results for the steady state cases BOMEX and RICO (Figure 8). Again the correspondence between the usual LES diagnosed mixing coefficients and and δ_{w} is good. While the analytical expressions overestimated the entrainment for ARM, they seem to underestimate the detrainment for BOMEX and RICO somewhat. The RMSE for the analytical expressions 27 and 28 for all cases together are presented in Table I.

In comparison with δ, the variations from hour to hour and case to case in are small (Figure 5) and describing with some fixed (non-dimensionalized) function from cloud base to cloud layer top seems more feasible than a fixed function for δ (also de Rooy and Siebesma, 2008). But despite the relative small variation in profiles, the results clearly show overall smaller entrainment rates for the ARM case than for the BOMEX and RICO case (compare Figure 4(a) with Figure 8). This is caused by the smaller term, and more specifically the larger vertical velocity in the ARM case. For example, at cloud base *w*_{c} ≈ 1.5 m s^{−1} for ARM whereas *w*_{c} ≈ 0.7 m s^{−1} for BOMEX and RICO. The higher velocities during the ARM case can be related to the more vigorous convection in the subcloud layer with strong surface heating above land. If we are able to make an adequate estimate of the vertical velocity of the updraught at cloud base in a NWP or climate model, this velocity can be used to refine the often-applied parametrization where is a fixed function of height (e.g. Siebesma *et al.*, 2003), i.e. parametrize the starting value of at cloud base with a function of the vertical velocity of the updraught and assume e.g. a *z*^{−1} lapse rate for the rest of the cloud layer. This is an example of how insight given by the analytical expressions can be used for parametrization developments. Note that the ARM case has a much deeper subcloud layer than the other two cases, enabling not only larger accelerations of the updraught thermals but also the development of larger thermals that will normally have smaller values. A separation between these positively correlated updraught properties, high updraught vertical velocity and large sizes of the thermals, cannot be made here.