Shallow cumulus convection: A validation of large-eddy simulation against aircraft and Landsat observations

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1 Q. J. R. Meteorol. Soc. (3), 9, pp doi:.56/qj..93 Shallow cumulus convection: A validation of large-eddy simulation against aircraft and Landsat observations By R. A. J. NEGGERS, P. G. DUYNKERKE and S. M. A. RODTS Royal Netherlands Meteorological Institute, De Bilt, the Netherlands Institute for Marine and Atmospheric Research, Utrecht, the Netherlands (Received 5 March ; revised November ) SUMMARY Large-eddy simulation (LES) results of shallow cumulus convection are directly evaluated against in-cloud aircraft measurements, as made during the Small Cumulus Microphysics Study (SCMS). To this purpose an LES case is rst constructed, based on available observations (during SCMS). Then the simulations are directly compared with the in-cloud measurements by using conditionally sampled elds. An advantage of the SCMS dataset is the combination of a range of different surface measurements, in-cloud measurements by an aircraft at many levels in the cloud layer, and the availability of high-resolution Landsat images. The results show that given the correct initial and boundary conditions the LES concept is capable of realistically predicting the bulk thermodynamic properties of temperature, moisture and liquid-water content of the cumulus-cloud ensemble as observed in SCMS. Furthermore, the vertical component of the in-cloud turbulent kinetic energy and the cloud size distribution in LES were in agreement with the observations. These results support the credibility of cloud statistics as produced by LES in general, and encourage its use as a tool for testing hypotheses and developing parametrizations of shallow cumulus cloud processes. Several hypotheses which make use of conditionally sampled elds were tested on the SCMS data. The magnitudes and the decrease with height of the bulk entrainment rate following from the SCMS data con rm the typical values of the order of magnitude of 3 m as reported in several other recent LES studies. An alternative formulation of the lateral entrainment rate as a function of the liquid-water content and the mean lapse rate agrees well with the original form based on the conserved variables. Applying an often-used simpli ed equation for the cloud vertical velocity to the aircraft measurements results in a reasonably closed budget. KEYWORDS: Aircraft measurements LES SCMS. INTRODUCTION Shallow cumulus cloud elds cover large areas in the trade-wind regions (Riehl et al. 95). These cloudy boundary layers are considered to play an important role in the ocean atmosphere exchange of sensible and latent heat, in the radiative energy budget at the earth s surface and at the top of the atmosphere, and in the vertical transport of momentum. In order to estimate the large-scale budgets of heat and moisture for this type of convection, several eld experiments have been organized in the past (e.g. Augstein et al. 973; Holland and Rasmusson 973; Yanai et al. 973; Nitta and Esbensen 974). More recently, sensitivity studies on shallow cumulus convection in general-circulation models (GCM) have shown that it has a signi cant impact on the resolved model climate (Tiedtke et al. 988; Nordeng 994; Slingo et al. 994; Gregory 997). However, operational cumulus convection schemes in GCMs still have problems in correctly representing important aspects of shallow-cumulus-topped boundary layers (Stevens et al. ; Siebesma et al. 3). Parametrizations of shallow convection in GCMs are to represent the impact on the resolved ow of whole cloud populations in a single GCM grid box. A common approach in many single-column models (SCMs) is to make a decomposition between cloudy and non-cloudy air, and to predict the vertical pro les of the average properties of these two fractional areas (Asai and Kasahara 967; Tiedtke 989). This method is also known as the top-hat approach, and applying such a decomposition to data is known as conditional sampling. It is important to realize that the vertical pro les of Corresponding author, present af liation: Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, CA neggers@atmos.ucla.edu c Royal Meteorological Society, 3. 67

2 67 R. A. J. NEGGERS et al. these conditional averages are controlled heavily by the changing cloud population with height (as large clouds reach greater heights than small clouds). Therefore, if reliable bulk cloud averages are to be obtained from aircraft measurements, it is necessary that the aircraft visits many clouds of all sizes and does so at many heights, preferably in a relatively short period of time. In the last decades, large-eddy simulation (LES) has become an important new tool in boundary-layer research. It provides cloud data at resolutions which are almost impossible to realize in eld experiments measuring real clouds. LES has already been used widely to study the turbulent structure of clear and cloudy boundary layers. A series of intercomparison studies as part of the Global Energy and Water Cycle Experiment (GEWEX) Cloud System Studies (GCSS) show that most current LES models agree on the basic structure of shallow cumulus cloud layers (Stevens et al. ; Brown et al. ; Siebesma et al. 3). Due to the typically high resolutions in space and time LES is suitable for studying conditionally sampled properties of shallow cumulus cloud elds (e.g. Schumann and Moeng 99; Siebesma and Cuijpers 995; Wang and Stevens ; De Roode and Bretherton 3). However, despite these encouraging results, some important LES results on cumulus clouds still remain unsupported by observations, mainly due to the scarcity of suitable in-cloud measurements. For example, after the many intercomparisons mentioned above, there has still not been a qualitative comparison of the in-cloud thermodynamics and turbulence as produced by LES with direct measurements inside natural shallow cumulus clouds. Another LES result yet unsupported by measurements is the typical decrease of the cloud fraction with height in simulated cumulus cloud elds. But perhaps the most important issue in the parametrization of cumulus convection is the interaction or mixing between shallow cumulus clouds and the dry air of their surrounding environment. To describe the changing conditionally sampled pro les with height, a bulk lateral entrainment rate has to be used which is an average over the whole ensemble of clouds (Simpson and Wiggert 969; Gregory ). The little observational evidence for lateral entrainment rates presented to this date indicates that it is of the order of magnitude km (Raga et al. 99; Barnes et al. 996). However, these values are associated with single clouds, and no bulk lateral entrainment rates derived from observations have yet been published. These issues emphasize that thorough and critical evaluation of LES results against measurements inside real clouds remains necessary for a better understanding of the strong and weak points of LES. This paper describes such an attempt, using direct incloud measurements of turbulence and thermodynamics by aircraft made during the Small Cumulus and Microphysics Study (SCMS) in Florida in 995. An LES case is constructed based on data from surface instruments and radiosondes of a certain day during this experiment on which a diurnal cycle over land was observed. It features a steadily growing clear convective boundary layer in the morning, resulting in a well developed cumulus cloud layer later on that day. On the same day, the National Center for Atmospheric Research (NCAR) C-3 aircraft made measurements in many clouds of all sizes at multiple levels. From both the LES and the aircraft data, conditionally sampled averages of rst and second statistical moments of thermodynamic and turbulent properties are calculated. Using these results, bulk entrainment rates are calculated and some widely used parametric formulations are evaluated. Finally, several cloud size distributions are calculated in LES. These are compared with distributions as derived from high-resolution Landsat images of the SCMS area. The instrumentation and observations of SCMS used in this study are described in section. The LES case based on these measurements is presented in section 3.

3 A VALIDATION OF LES AGAINST OBSERVATIONS 673 9Ê N 8Ê N 7Ê N 6Ê N 5Ê N km 5 83Ê W 8Ê W 8Ê W 8Ê W Figure. A map of Florida. The Small Cumulus Microphysics Study (SCMS) campaign was situated near Cocoa Beach, Cape Canaveral. The ground stations PAM and PAM3 are indicated by the black dots. The radiosondes were released in the close vicinity of the PAM3 station. The area of ight RF and the area covered by the Landsat 5 image are indicated by the rectangles. The methods of conditional sampling as used on the measurements and on the LES data are discussed in section 4. The parametric formulations which are evaluated in this study are brie y described in section 5. The results are presented in section 6, and the nal discussion and conclusions can be found in section 7.. INSTRUMENTATION AND OBSERVATIONS IN SCMS The Small Cumulus Microphysics Study took place from 7 July until 3 August 995 in Florida, near Cocoa Beach, just north of Cape Canaveral (see Fig. ). On 5 August a clear convective boundary layer developed over land in the early morning. It deepened in time, and during the course of the morning a shallow cumulus cloud layer developed. The clouds were categorized as shallow non-precipitating cumulus with a cloud fraction of %. This particular golden day was part of a period in which persistently every day a shallow-cumulus-topped boundary layer developed. Observations of the geometrical and microphysical structure of the cumulus clouds in this period in SCMS have been reported by Knight and Miller (998), French et al. (999) and Laird et al. (). The large-scale conditions did not change signi cantly during this period, nor were they very large compared with the local forcing by the surface uxes. These conditions make the 5 August a suitable day on which to base an

4 674 R. A. J. NEGGERS et al. a) Surface heat flux [W/m ] 6 4 H LE F net G PAM b) Surface heat flux [W/m ] 6 4 H LE F net G PAM UTC time [hr] Figure. Measured time series of the of the latent (LE) and sensible (H) surface heat uxes at ground stations (a) PAM and (b) PAM3 on 5 August 995. For completeness the net incoming radiation (F net ) and the soil heat ux (G) are also plotted. LES case. This section describes the instruments and measurements of SCMS which are relevant to the set-up of the LES case and to the comparison between the model and observations. Two Portable Automated Mesonet (PAM) meteorological stations were employed in the SCMS campaign. For a detailed description we refer to Horst and Oncley (995) and Militzer et al. (995). The stations named PAM and PAM3 were situated approximately 5 km westward from Cape Canaveral (see Fig. ). The near-surface uxes of momentum and virtual temperature are measured by eddy correlation, using a three-component sonic anemometer. The water-vapour ux is calculated from the directly measured virtual heat ux by means of a virtual Bowen ratio, measured as the ratio of the virtual-heat ux to the water-vapour ux. The water-vapour ux can then be used to extract the sensible-heat ux from the measured virtual-heat ux. Figure shows that clearly an inbalance exists in the PAM3 measurements between the incoming net radiation plus the soil heat ux on one side and the latentplus sensible-heat ux on the other. What causes this gap is unknown. Temperature and humidity are measured at a height of m with a Vaisala 5Y Humitter that includes a platinum-resistance thermometer and a solid-state capacitance sensor for relative humidity. The corresponding time series are plotted in Fig. 3. Near the PAM3 station radiosondes were released at intervals of approximately 3 hours, giving the vertical pro les of the temperature, speci c humidity, wind direction and wind speed (see Fig. 4). This range of measurements is used to align the LES case with reality, as will be described in section 3.

5 A VALIDATION OF LES AGAINST OBSERVATIONS 675 a) Temperature [ƒc] PAM PAM3 b) Total specific humidity [g/kg] PAM PAM UTC time [hr] Figure 3. Measured time series of (a) the temperature (T ) and (b) the total speci c humidity (q T ) at m height at ground stations PAM and PAM3. a) b) 3.8 UTC 5.8 UTC 8. UTC.7 UTC RF 8. UTC 3.8 UTC 5.8 UTC 8. UTC.7 UTC RF 8. UTC Potential temperature [K] 5 5 Total specific humidity [g/kg] Figure 4. Radiosoundings of (a) the potential temperature and (b) the total speci c humidity near station PAM3. The idealized pro les based on the vertical ascent of ight RF at 8 UTC are also shown for comparison.

6 676 R. A. J. NEGGERS et al. The C-3 aircraft operated by NCAR carried instruments measuring turbulence, thermodynamics and microphysics. A detailed description of the instrumentation on this aircraft and the statistical quality of the resulting cloud averages is given by Rodts et al. (submitted to J. Atmos. Sci.). The temperature was measured by a platinum-resistance thermometer, model Rosemount EAL. Measurements of temperature inside clouds can be inaccurate as a result of wetting or icing of the sensor (e.g. Lenschow and Pennell 974; Heyms eld 979). Therefore, the measured temperatures during SCMS ight RF as used in this paper were corrected by Rodts et al. (submitted to J. Atmos. Sci.) for the impact of wetting, using a factor which was derived by comparing the Rosemount device with the on-board dew-point thermometer (a thermoelectric hygrometer). Using such a device for calibrating thermometers or correcting temperatures has been done earlier by Raga et al. (99). The liquid-water content (q l ) of the clouds is obtained from a Particle Volume Monitor (PVM). The details of this method and its application in SCMS have been published by Gerber et al. (), and will therefore not be described in great detail here. Flight RF on 5 August started at 8 UTC and lasted until UTC. Immediately after take-off, the aircraft made a vertical sounding up to 4 km, giving the vertical pro les of the temperature and speci c humidity in the ight area. This was followed by a descent to lower altitudes where the clouds were located. Three consecutive hours of measurements at multiple levels in the cloud layer then followed. The area of the ight-path is shown in Fig.. These measurements are used to calculate conditionally averaged pro les in the cloud layer, as will be described in section 4. On August 995 high-resolution images were taken of the cloud elds over Florida by the Landsat 5 satellite. These images can be used to calculate so-called cloud size densities, or the probability density function for the cloud population as a function of cloud size. Cloud size densities have been calculated in the past of many observed cloud populations, using a variety of methods (e.g. Plank 969; Wielicki and Welch 986; Cahalan and Joseph 989; Benner and Curry 998). The availability of high-resolution Landsat images of the SCMS area as well as an LES case directly based on (almost) simultaneous observations enable a direct and straightforward comparison between the simulated and observed cloud size densities (e.g. Neggers et al. 3). Only the images of August were available to us, which is 5 days later than the day selected for simulation. Nevertheless, the large-scale conditions did not change signi cantly during the period of 5 August, and the diurnal development of the shallow cumulus clouds was observed to be roughly the same every day. The method of calculation of the cloud size densities is described in the appendix. A more thorough analysis of the cloud size densities which can be obtained from this Landsat image is presented by Rodts et al. (submitted to J. Atmos. Sci.). The image was taken at 453 UTC. The size of the area captured by the image is 68.5 km 68.5 km, with a horizontal grid spacing of 3 m. The image captured 84 individual cumulus clouds. 3. SET-UP OF THE LES CASE The aim is to construct a case for LES for which the development in time stays as close as possible to the range of different kinds of measurements made during the day. Once that is achieved, the resulting cloud properties can be studied and compared in detail with the available in-cloud observations. Therefore, let us rst consider the initial pro les of the two basic thermodynamic variables, the potential temperature (µ) and the total speci c humidity (q T ). The vertical soundings by the radiosondes and the

7 A VALIDATION OF LES AGAINST OBSERVATIONS 677 aircraft are used for this purpose. Figure 4 gives an overview of all radiosonde pro les. Also plotted are the idealized pro les derived from the aircraft sounding at 8 UTC. It is clear that the latter sounding is about K cooler and g kg more moist than the radiosounding of the inland station PAM3 at the corresponding time, over the whole depth of the boundary layer. This must be due to the close proximity of relatively cool and moist sea air. The second signi cant difference between the aircraft and the radiosonde pro le at 8 UTC is that the inversion height in the ight area is about 5 m higher. The nal remarkable feature in the radiosoundings is the decrease of the inversion height between 8 and 5 UTC. In the successive radio soundings, the inversion clearly rises in time. Fortunately, the lapse rates in the conditionally unstable layer and the inversion in the two different soundings agree very well (see Fig. 4). One problem, however, is the observed initially sinking inversion, because it can never be explained by a developing convective boundary layer which is driven by increasing surface heat uxes in time. In that case the inversion height would only increase. The sinking inversion in this particular period could have been caused by large-scale subsidence, or perhaps by some residual layer which was still present in the morning. In any case, the LES model cannot be expected to resolve this sinking inversion in the early hours, and accordingly it is neglected. The initial inversion height is obtained by estimating the growth of the inversion height between and 8 UTC from the radio soundings and subtracting it from the inversion height in the aircraft sounding at 8 UTC. The resulting initial pro les are displayed in Fig. 5(a). Apart from the initial thermodynamic state, realistic boundary conditions have to be provided to the LES model during the numerical simulation. The measurements of the surface energy balance are used to extract the surface latent- and sensible-heat uxes (see Fig. 5(b)). The reason for the surface energy inbalance at PAM3 is unknown and, therefore, the LES surface uxes are based on the more balanced PAM data only. After several test runs we assumed a sinusoid shape for the surface uxes, with the maximum at 8 UTC and corresponding values of and 3 W m for the sensible- and latent-heat uxes respectively. Another boundary condition in the model is the roughness length at the surface z D :35 m, which is a typical value for at land surfaces. The radiative and large-scale forcings which may act on the area have to be accounted for in the simulation. Unfortunately, a network of radiosonde measurements over the area was not available, and as a consequence no variational analysis could be applied to estimate the large-scale advective temperature and moisture forcings. The offset in temperature and moisture between the soundings of the inland radiosondes and the aircraft at the coast only suggests that there was a large-scale cooling and moistening tendency, probably due to the proximity of the ocean. Measurements of radiative heating tendencies were also not available to us. This lack of observational data makes it very dif cult to make a realistic estimate of the forcing tendencies. As the main interest of this paper lies in evaluating characteristic LES results used in entrainment models and budget studies, a small offset between the mean temperature or moisture pro le in LES and reality is acceptable. However, it is important to realize that in order to compare the LES cloud eld with the in-cloud observations, it is essential that the simulated cloud base and cloud top are located at the correct heights. Fortunately, measurements of the time series of the surface temperature and speci c humidity at the PAM stations are available, which can be used to align the development of the thermodynamic state of the boundary layer in LES during the day (see Fig. 3). These gures clearly show that the mixed layer warms up

8 678 R. A. J. NEGGERS et al. a) b) Total specific humidity [g/kg] Initial. UTC RF 8. UTC 4 H LES LE LES H PAM LE PAM q q t Surface heat flux [W/m ] Potential temperature [K] Figure 5. (a) The initial pro les for large-eddy simulation (LES) at UTC of the potential temperature µ and the total speci c humidity q T of the Small Cumulus Microphysics Study (SCMS) case. The idealized pro les based on the vertical ascent of ight RF at 8 UTC are also shown for comparison. (b) The surface latent (LE) and sensible (H) heat uxes as measured at ground station PAM, as well as the corresponding values imposed on LES as a boundary condition. in the morning. There is less consensus between the two PAM stations about the speci c humidity. To summarize, the LES case is designed to reproduce the heights of cloud base, cloud top and the inversion as observed by the aircraft and the radiosondes, using the large-scale tendencies as a tool for calibration and the measured surface time series as a constraint. This resulted in a net temperature forcing of 3 K day. This tendency is set to decrease with height towards zero just above the inversion. The moisture forcing is set to zero. Finally, based on the radiosonde data the initial mean horizontal wind in the simulation is set to ( 4; 4) m s in the zonal and meridional directions. The geostrophic wind is also set to these values, as detailed information of this forcing was unavailable. 4. CONDITIONAL SAMPLING Once the LES case is constructed, a method has to be chosen to compare the LES results with the available observations. To this purpose we use the conditional sampling technique. In this method, horizontal averages are calculated over a certain area de ned by some criterion, which can be the presence of liquid water, a positive vertical velocity, a positive buoyancy, or any combination. In other words, a decomposition is made in which the horizontal slice is split up into two areas. Conditionally sampled elds are

9 A VALIDATION OF LES AGAINST OBSERVATIONS 679 widely used in budget studies using LES (e.g. Schumann and Moeng 99; Siebesma and Cuijpers 995; Wang and Stevens ; De Roode and Bretherton 3) and in parametrizations of convection in GCMs (e.g. Asai and Kasahara 967; Tiedtke 989) which make use of the top-hat approximation. In LES, the conditionally sampled elds at a certain height and at a certain time are calculated using Á c D X c ij Á ij ; () N D where Á is the variable to be sampled, i and j are the horizontal coordinates, and N D is the total number of cloudy points in the domain. c ij D if the point is cloudy and if it is non-cloudy. In addition, next to the cloud criterion the more stringent criterion known as the cloud core (Siebesma and Cuijpers 995) is used, de ned as the cloudy air which is also positively buoyant. The conditional averages resulting from both criteria will be evaluated. Using ight legs to obtain conditionally sampled elds requires a different technique, because the aircraft cannot measure everywhere in the domain at the same time. Therefore, it is assumed that an ergodic equivalent can be used, in this case a time average, Á c D X c t Á t : () N T Here t is the time during the ight and N T is the total number of measurements during the ight which meet the criterion. Hereafter, an overbar will denote a domain average, the superscript c will denote a cloud average, and the superscript e will denote an average over the dry environment. These two de nitions essentially give the same value of Á c if (i) the cloud ensemble is in steady state, and (ii) the aircraft ies enough straight legs through the whole domain to catch a realistic size distribution of sampled clouds. Typically, a cloud ensemble consists of many small clouds and fewer large clouds (Plank and 969; Wielicki and Welch 986; Cahalan and Joseph 989; Benner and Curry 998). The contribution to Á c as a function of cloud size is a balance between cloud number and individual contribution per cloud, and it is therefore important to sample a realistic distribution. Flight RF ew in straight legs of about km length, after which it turned around to stay in the same area. The km length of each straight leg is comparable to the domain size of LES and, therefore, it is reasonable to assume that the sampled cloud size distribution is of comparable quality. One of the objectives of the SCMS campaign was to study the microphysics of the larger cumulus clouds, in particular the onset of precipitation in warm cumulus. For that reason the C-3 closely collaborated with a ground-based CP dual-wavelength radar (Knight and Miller 998) in order to guide the aircraft after each turn towards a certain large specimen (Laird et al. ). Each preferred large cloud was traced in time by the radar in order to sample it as often as possible during its lifetime. Accordingly, this may lead to a slight overestimation of the cloud fraction. On fractional properties like the mass ux this does have a signi cant impact. It is a lesser problem for the cloud averages derived using () as they are independent of the cloud fraction, but nevertheless a possible bias in the conditionally sampled statistics cannot be dismissed completely. ij 5. PARAMETRIZATIONS Once the pro les of the conditionally sampled elds are known, hypotheses and parametrizations which are based on such decompositions can also be evaluated. t

10 68 R. A. J. NEGGERS et al. One good example is the bulk entrainment rate (² c ) of the cloud ensemble, which is normally calculated by using the simpli ed lateral mixing equation ² c D.Á (Betts 975; Anthes 977; Tiedtke 989; Raga et al. 99). This entrainment rate can be interpreted as the inverse of the depth in which the excess over the environment.á c Á/ has decreased by a factor e. Á c can be the liquid-water potential temperature µ l c or the total speci c humidity qt c, which are both conserved variables for moist adiabatic ascent. Any change of Á c is, therefore, caused by either a diabatic process like mixing or a statistical process like a changing cloud size distribution with height. The LES results on the Barbados Oceanographic and Meteorological Experiment (BOMEX) of Siebesma and Cuijpers (995) gave typical values of ² D :5 to 3 3 m, decreasing with height in the cloud layer. The process of cloud mixing has also been formulated in terms of other variables. A de nition which is often used is the ratio between the measured q l at a certain height in a cloud and the moist adiabatic value ql ad of an undiluted parcel that has risen from cloud base to that height (e.g. Warner 955; Raga et al. 99). Measurements have shown that typically this ratio strongly decreases with height in a shallow cumulus cloud layer towards values around.3.4 near cloud top, which points to signi cant mixing. How does de nition (3) relate to this ratio? One possible approach is to write µ l c as a function of µ c and q c l, ql c D c p5 L.µ c µ l c /; (4) where c p is the speci c heat at constant pressure, L is the speci c latent heat of the phase change between water vapour and liquid water, and 5 is the Exner function. At this point we introduce a hypothesis based on observations and LES results, which show that in shallow cumulus the cloud-average potential temperature at a certain height (µ c ) is typically almost equal to that of the environment, (3) µ c ¼ µ: (5) This hypothesis will be evaluated later using the SCMS aircraft observations (see Fig. (a)). Next, the typical very small cloud cover of shallow cumulus allows for the well known assumption µ l ¼ µ. Substituting µ c ¼ µ l in (4) and then using (3) with Á µ l gives q c l D ² c c p 5 c : (6) Substituting (4) for µ c l and again using (5) in the lapse rate nally gives ² c D q c l c p ln.qc ln.5 /: (7) The advantage of assumption (5) is that ² c is now only dependent on the pressure, the mean temperature lapse rate and the liquid-water content. The rst two properties can be obtained from data measured by a radiosonde, while a ground radar can remotely measure cloud liquid-water content. This enables the calculation of the bulk entrainment rate without direct in-cloud measurements by aircraft. On the other hand, assumption (5) needs to be highly accurate for (7) to give representative values for ² c. The individual

11 A VALIDATION OF LES AGAINST OBSERVATIONS 68 a) b) 3 3 Initial. UTC 3.3 UTC 5.3 UTC 7.3 UTC 9.3 UTC.3 UTC 3.3 UTC Potential temperature [K] Initial. UTC 3.3 UTC 5.3 UTC 7.3 UTC 9.3 UTC.3 UTC 3.3 UTC 5 5 Total specific humidity [g/kg] Figure 6. The development in time of the domain-averaged pro les of (a) the potential temperature and (b) the total speci c humidity, as produced by large-eddy simulation. terms of (7) and the validity of assumption (5) will be evaluated with the aircraft data and LES results. Knowledge of the entrainment rate is useful in other parametrizations. The clouds are responsible for the bulk of the vertical transport in the cloud layer (Siebesma and Cuijpers 995). Accordingly, SCMs in GCMs are often equipped with a simpli ed equation for the vertical velocity of / D ² c.w c / C B c ; (8) (Simpson and Wiggert 969; Siebesma et al. 3). The term on the left-hand side is the advection term, and the rst term on the right is the mixing term enhanced by a factor to account for the impact of pressure perturbations. It is assumed here that the domain-average vertical velocity w can be neglected. B c stands for the buoyancy of the clouds, which is reduced by a factor to account for loss of potential (gravitational) energy to sub-plume turbulence. Simpson and Wiggert (969) suggested D and D =3, while the operational European Centre for Medium-Range Weather Forecasts (ECMWF) model uses D =3. Siebesma et al. (3) applied (8) as a rising-plume model for the BOMEX case and compared it with LES results. Here (8) will be applied to the RF data, using D and D =, the latter being an intermediate value between the two mentioned above. 6. RESULTS The LES model used in this study is described in detail in Cuijpers and Duynkerke (993). The LES simulation was performed on a domain of 6.4 km 6.4 km 5 km.

12 68 R. A. J. NEGGERS et al. a) 3 clouds level of minimum buoyancy flux b) Projected cloud fraction [%] projected cloud fraction liquid water path UTC time [hr] Liquid water path [g/m ] Figure 7. (a) Structure of the simulated cloud layer. The dotted lines mark the period for which in-cloud measurements by the RF ight are available. (b) The time series of the projected cloud fraction (left axis) and the liquid-water path (right axis) as produced by large-eddy simulation. The corresponding grid spacing was 5 m 5 m 4 m. A centred-difference integration scheme was used. In the time integration a time step of second was used to prevent any possible numerical instability. A damping layer is implemented in the model above 37 m to supress arti cial gravity-wave re ection at the top. An all-ornothing condensation scheme is used, meaning that any grid box is entirely saturated or entirely unsaturated. Hourly averaged vertical pro les were calculated of the basic thermodynamic variables. Every 3 s several cloud properties were evaluated. A period of hours was simulated, covering the daytime cycle from 7 to 9 local time (which corresponds to UTC to UTC). (a) Daytime development We commence with a short description of the initial development of the convective boundary layer in LES until 8 UTC after which in-cloud observations are available. Figure 6 shows the pro les of µ and q T in this period. The potential temperature gradually increases with time in the mixed layer. It initially moistens, followed by a long period of drying. This is consistent with the surface measurements as shown in Fig. 3. The temperature and moisture in the conditionally unstable layer do not change much in time, which is also apparent in the radiosoundings.

13 A VALIDATION OF LES AGAINST OBSERVATIONS 683 a) b) 3 3 RF environment LES environment RF cloud LES cloud LES core Liquid water potential temperature [K] RF environment LES environment RF cloud LES cloud LES core 5 5 Total specific humidity [g/kg] Figure 8. Pro les of (a) the liquid-water potential temperature and (b) the total speci c humidity. Both the mean and cloudy averages are plotted. Observations are plotted as unconnected data points, and large-eddy simulation (LES) results are plotted as connected lines. The cloud-core average value in LES is also plotted for comparison, de ned as the average over all cloudy points which are also buoyant. Figure 7(a) illustrates that the rst clouds appear in LES at about 33 UTC. At rst the cloud layer is very shallow, but within two hours it deepens to about 5 m. From 5 UTC the growth of the cloud top is controlled by the rise of the capping inversion. The development of the cloud fraction and the integrated liquid-water path is shown in Fig. 7(b). The cloud fraction peaks at 4% at 5 UTC, at the same time as the clouds rst reach the inversion. From then on there is a steady decrease of the cloud fraction with time. These basic results are qualitatively similar to the results of the intercomparison study between many LES models of GCSS working group on a quite similar diurnal cycle, as observed over the Southern Great Plains site of the Atmospheric Radiation Measurement (ARM) programme (Brown et al. ). (b) Thermodynamic state From this point onwards we focus on the period between 8 and UTC during which the in-cloud measurements were taken by ight RF. In order to get reliable statistics, all measurements are averaged over this 3-hour period. Figure 8 shows the pro les in this period of the conserved thermodynamic variables in LES and the observations. The conditionally sampled pro les are only shown at those heights where the cloud (or cloud core) fraction was higher than %. Figure 8(a) illustrates that there is a remarkable agreement between LES and the observations concerning the liquid potential temperature (µ l ) in both the environment and the clouds. The same is true for the total speci c humidity (q T ), although the observations show a slightly drier layer near cloud base (see Fig. 8(b)). As the exact magnitude of the conserved thermodynamic variables is highly tunable in LES, it is perhaps more interesting to look

14 684 R. A. J. NEGGERS et al. a) b) RF cloud LES cloud LES core RF cloud LES cloud LES core c c) q l - ql [K] d) 4 6 c q t - qt [g/kg].5.5 RF Cloud LES cloud LES core RF cloud LES cloud LES core q l c / z [K/m] -.5 q t c / z [g / kg m] Figure 9. Pro les of the excess of the cloudy averages of µ l c and qt c over the dry environment (panels (a) and (b), respectively) and the gradients of the cloudy averages (panels (c) and (d) respectively), for the RF ight observations and large-eddy simulation (LES).

15 A VALIDATION OF LES AGAINST OBSERVATIONS 685 a) b).5.5 RF cloud LES cloud LES core RF cloud LES cloud LES core moist adiabat Liquid water content [g/kg] 5 5 Cloud fraction [%] Figure. (a) Pro les of the liquid-water content ql c. RF ight observations are plotted as unconnected data points, and large-eddy simulation (LES) results are plotted as connected lines. Both the mean and cloudy averages are plotted. The cloud-core average in LES is included for comparison. The dash dotted line is the moist adiabatic pro le starting at cloud base. (b) The pro le of the cloud fraction. at the excess values and the vertical gradients of the conditionally sampled variables. Figures 9(a) and (b) show that in general the increasing excess values with height of the cloudy averages in LES are supported by the observations. Near cloud base the speci c- humidity excess of the clouds is somewhat under-predicted by LES. The simulated lapse rates are also in good agreement with the observations, see Figs. 9(c) and (d). The liquid-water content of the clouds (ql c ) is shown in Fig. (a), which illustrates that the LES pro le of ql c lies very close to the measured values. Apparently the allor-nothing condensation scheme in LES is capable of predicting cloud liquid-water contents which increase with height by the same order of magnitude as is measured by these instruments in natural clouds. The ratio of the liquid-water content to its moist adiabatic value is roughly.3.4 in the top of the cloud layer, and comparable to the values found by Warner (955) and Raga et al. (99). A current issue in the LES community is the question of whether the decreasing cloud fraction with height in shallow cumulus cloud layers as typically observed in LES is realistic. The pro le of the cloud fraction is shown in Fig. (b). The six heights for which data points exist show considerable scatter, but the cloud fraction certainly does not decrease with height. However, note that the time spent inside clouds is increased by the decision to adjust the ight path towards certain large cumulus clouds. Accordingly, the aircraft measurements give a cloud fraction which is not necessarily equal to the actual cloud fraction of the population.

16 686 R. A. J. NEGGERS et al. a) b).5 RF e (q l ) RF e (q t ) LES e (q l ) LES e (q t ) RF e (q l ) LES e (q l ) LES e (q t ) Cloud entrainment rate [m - ] Figure. Bulk entrainment rates (²) of the cloud eld, using several different methods of calculation. These are applied to both the RF ight observations and the large-eddy simulation (LES) results. Panel (a) shows (3) with ² as a function of q T and µ l, panel (b) shows (7) based on ql c In panel (b) (3) is shown again for reference. (c) Lateral mixing The fact that LES produces realistic conditionally sampled elds in this case encourages the use of LES as a tool to evaluate hypotheses or parametrizations which are based on conditional averages. A good example of such a parametrization is the bulk entrainment rate ² c which can be calculated from these averages using (3). Figure (a) illustrates that the order of magnitude is comparable to that found in observational and LES studies (Raga et al. 99; Siebesma and Cuijpers 995). In fact, if µ c l is used in (3), the observed ² c even decreases with height and lies close to the LES entrainment rate. The decrease with height of ² c is mainly a result of the increase with height of the excess.µ l c µ l /, while the lapse l c =@z is fairly constant with height. In the lower half of the cloud layer, the calculation of ² c using qt c gives much lower results for the observations compared with LES. This is caused by both a smaller lapse rate near cloud base and a larger excess of the observed qt c. Figure (a) illustrates that the measured cloud-average liquid-water content is considerably smaller than the moist adiabat, pointing to signi cant cloud mixing. Equation (7) relates the bulk mixing rate ² c de ned in (3) to the liquid-water content ql c, using assumption (5). Figure (b) shows that (7) corresponds well with (3) in both LES and the observations except at cloud base and the inversion. This is caused by assumption (5) (see Fig. (a)), which applies remarkably well in the bulk of the cloud layer but no longer at its boundaries. For completeness, Fig. (b) shows the individual terms of (7). It is clear that the pressure term in (7) can be neglected, and that the measurements support the LES budgets for the two remaining terms.

17 A VALIDATION OF LES AGAINST OBSERVATIONS 687 a) b) RF q RF q c LES q LES q c Potential temperatures [K].5.5 LES lapse rate term RF " " c LES q l gradient term RF " " LES pressure term Entrainment rate budgets [m - ] Figure. (a) Pro les of the mean and cloudy averaged potential temperature. (b) Budgets of Eq. (7), based on both the large-eddy simulation (LES) results and the RF ight data. The lapse-rate term corresponds to the rst term on the right-hand side in Eq. (7), the ql c gradient term to the second term, and the pressure term to the third. (d) Vertical transport The conditionally sampled vertical velocity w c is plotted in Fig. 3(a). Despite the scatter in the observations we can say that the magnitude of the in-cloud vertical velocity in LES is comparable to the observations. The mass ux plotted in Fig. 3(b) is the product of the cloud fraction and cloud-average vertical velocity. In contrast to LES, the measured mass ux clearly increases with height. This is remarkable, as LES and the measurements agree very well on other conditionally sampled dynamical properties which are independent of the cloud fraction, like the pro les of vertical velocity and the bulk entrainment rate. It is obvious that the increasing mass ux is mainly caused by the measured increasing cloud fraction with height (see Fig. (b)). As noted earlier, the cloud fraction as obtained from the aircraft measurements might be an overestimation of the actual cloud fraction, due to the adjustment of the ight path towards large cumulus clouds. As a consequence, no conclusion can be made about the decrease or increase of the mass ux with height based on these data. The buoyancy within the clouds is an important source for the production of turbulent kinetic energy (TKE) in the cloud layer. Figure 4(a) shows that LES is in good agreement with the observations on magnitude and shape of the vertical pro le of the virtual potential-temperature (µ v ) excess of the clouds. The clouds are only marginally buoyant, with a µ v excess of only a few tenths of a kelvin. The negative buoyancy at cloud base and cloud top is present in both the observations and LES, indicating the statically stable layers which envelope the conditionally unstable cloud layer. The LES results show that the cloud core is signi cantly more buoyant than the clouds and also

18 688 R. A. J. NEGGERS et al. a) b).5.5 RF cloud LES cloud LES core RF cloud LES cloud Vertical velocity [m/s].5..5 Mass flux [m/s] Figure 3. Pro les of (a) the cloud-average vertical velocity and (b) the vertical mass ux by the clouds. RF ight observations are plotted as unconnected data points, and large-eddy simulation (LES) results are plotted as connected lines. The cloud-core average in LES is also included. has a much higher vertical velocity, re ecting the fact that the production of TKE in the clouds is associated with the relatively high buoyancy of the cloud-core elements. Next, some vertical-velocity budgets are calculated from the RF data. Figure 4(b) illustrates that the cloud-average vertical acceleration is almost negligible. The simpli ed budget equation (8) reproduces this feature reasonably well in the region where the clouds are marginally buoyant. Apparently the factors and which are included in the equation to account for sub-plume turbulence and pressure perturbations, and of which the particular values were originally tuned for the cloud core, also result in a balanced budget for the cloud-average vertical velocity. (e) Cloud turbulence After evaluating the cloud-average vertical velocity it is interesting to take a closer look at the TKE in the cloud layer. The vertical component of the TKE of the domain is the vertical-velocity variance ¾w, de ned by ¾ w D.w w/ X.w i w/ ; (9) N where w is the horizontally domain-averaged vertical velocity and N is the number of sampled points. By using (9) we choose not to evaluate the sub-cloud-scale turbulence (i.e. the turbulence around the cloud mean, w c ). The reason for this is that the i

19 A VALIDATION OF LES AGAINST OBSERVATIONS 689 a) b).5 RF cloud LES cloud LES core moist adiabat.5 z (wc ) B c - e (w c ) - e (w c ).5 + Bc.5 - c q v - qv [K] w c budgets [m s - ] Figure 4. (a) Pro les of the virtual potential-temperature excess of the clouds. RF ight observations are plotted as unconnected data points, and large-eddy simulation (LES) results are plotted as connected lines. (b) Pro les of the simpli ed budget (8) for the vertical velocity of the cloud, based on RF ight data. The shaded areas represent the range between the entrainment rates following from the use of µ l c and qt c in (3). See text for further information. main interest of this paper lies in evaluating well known bulk parametrizations and assumptions, and that LES cannot yet be expected to reliably resolve motions on small sub-cloud scales. Comparing measurements of the vertical-velocity variance around w with that produced by LES is a critical test for the capacity of LES to resolve the organized cloudy convective motions. Very few observations of conditional averages of this vertical-velocity variance have yet been reported, while it is an important variable in parametrizations relying on TKE to model the intensity of vertical transport by an ensemble of shallow cumulus clouds. The results for the domain and environmental average variance are shown in Fig. 5(a). The LES results include the subgrid part of the TKE, for which the Royal Netherlands Meteorological Institute (KNMI) LES model has a prognostic equation (Schumann 975; Cuijpers and Duynkerke 993). The average over the whole domain (¾w m ) in LES is much smaller than the observed value. This is probably due to the high cloud fraction resulting from the ight legs, which might be an overestimation of the actual cloud fraction as explained earlier. However, ¾w m in LES is even somewhat smaller than the observed average over the dry environment (¾w e ). This means that in any case the turbulent activity of the environment is slightly under-predicted in LES. The pro le of ¾w m in SCMS can be compared with ight A on 4 June 99 of the Atlantic Stratocumulus Transition Experiment (ASTEX, see Albrecht et al. 995), during which cumulus clouds were sampled and ¾w m was measured (De Roode and Duynkerke 997).

20 69 R. A. J. NEGGERS et al. a) b) 3 3 RF mean RF environment LES mean RF cloud LES cloud Height [m] s w [m s ] c (s w ) [m s - ] Figure 5. Pro les of the vertical-velocity variance ¾w. Panel (a) shows the domain and dry environmental averages, and panel (b) shows the cloud averages. RF ight observations are plotted as unconnected data points, and large-eddy simulation (LES) results are plotted as a solid line. The SCMS data are consistent with the ASTEX data in that ¾w m slightly increases with height in the cloud layer, with a maximum near cloud top. The cloud-average variance is plotted in Fig. 5(b), illustrating that LES reproduces the observed pro le of the cloud-average vertical-velocity variance.¾w c / remarkably well. The observed ¾w c is much larger than ¾ w e, which re ects that the generation of TKE in a cumulus layer mainly takes place inside the clouds, and that the environment is fairly laminar and statically stable. Clearly the linear increase of.¾w c / with height above cloud base in LES is supported by the aircraft data. The good agreement between ¾w c in LES and the aircraft measurements gives con dence in the capacity of LES to realistically resolve cloud turbulence, although it remains important to investigate the resolution dependence of these second statistical moments in LES (e.g. Brown 999). A recent example of a parametrization based on LES results on turbulent variances is the similarity theory for shallow cumulus developed by Grant and Brown (999). Use is made of the typical linear increase of the in-cloud vertical-velocity variance with height in the cloud layer in LES, a feature now supported by the in-cloud measurements of SCMS presented here. ( f ) Cloud size densities Finally, the geometrical properties of the cloud population are evaluated by using cloud size densities (see Fig. 6(a)). The densities are normalized by the total number of clouds N. The bin sizes are equal to the horizontal grid spacings of the LES and Landsat elds, being 5 m and 3 m respectively. The general outcome of Neggers et al. (3) was that the densities in LES are well described by a power law, scaling

21 A VALIDATION OF LES AGAINST OBSERVATIONS 69 a) N* / N.. LandSat LES b = -.7. b).5 Cloud size [m].4 LandSat LES a p [ % / m ] Cloud size [km] Figure 6. (a) Log log scale plot of the cloud size density normalized by the total number of clouds N. The solid line represents the t.l/ l :7 by Neggers et al. (3), which is based on large-eddy simulation (LES) results on a range of different cumulus cases. (b) The cloud size decomposition of the vertically projected (or shaded ) cloud fraction p. The area covered by the histogram is equal to the total shaded cloud fraction. up to a certain size (the scale break). Accordingly, we evaluate the cloud size densities here by comparing power-law exponents and scale-break sizes. It is clear that both the densities of LES and Landsat are well described by a power law at the relatively small cloud sizes, with an exponent of :7. The area of scaling is about one decade wide, up to a scale break at about 8 m which is reproduced by LES. At sizes larger than the scale break, the densities decay rapidly with cloud size. This decay is not as strong in Landsat as in LES: signi cantly larger clouds occur in the Landsat images. The limited domain size of 6:4 km in LES may prevent the growth of clouds at these largest sizes.

22 69 R. A. J. NEGGERS et al. The corresponding cloud size decomposition of the vertically projected (or shaded ) cloud fraction is plotted in Fig. 6(b). Very clearly an intermediate dominating size exists at the same size as the scale break in the cloud number density,, as was shown by Neggers et al. (3). The largest sizes in Landsat do not occur in the LES domain. At sizes smaller than the scale-break size, the LES clouds contribute relatively too much to the shaded cloud fraction compared with Landsat. However, the total shaded cloud fraction, equal to the area covered by the histogram, does not differ that much in Landsat and LES, being 9.7% and about %, respectively. This means that more small clouds occur in LES. Apart from a possible physical reason, there might be other explanations for the differences between LES and Landsat. Firstly, the cloud elds are not exactly the same. The LES case is based on measurements on 5 August, while the Landsat image is made on August. Secondly, note that the Landsat image was taken at 453 UTC. Figure 7(b) shows that this time corresponds to the very beginning of the daytime cycle of clouds in LES. To get qualitative distributions in LES, we had to sample over a 3-hour period before a number of sampled clouds was reached which was comparable to the number captured by the Landsat image. In this period the cloud size distribution may change signi cantly. We realize that these differences may frustrate a direct comparison. However, from that point of view the similarities we found between LES and Landsat only get more meaningful: the power-law exponents agree remarkably well in LES and Landsat, and in both cases a scale break is present (implying a dominating size in the cloud-fraction decomposition). These results, therefore, emphasize the universality of the functional form which describes the cloud size densities and decompositions in shallow cumulus cloud populations (e.g. Neggers et al. 3). This conclusion could be illustrated by dividing the cloud fraction decompositions by the total number of clouds N. As p and are closely related by (A.3) (see the appendix), it is evident from Fig. 6(a) that the size decompositions of LES and Landsat would collapse at the sizes below the scale break. 7. DISCUSSION AND CONCLUSIONS We choose to use conditionally sampled elds to evaluate the LES results against in-cloud measurements. This means that in fact we are studying the bulk properties of the cloud ensemble. Apart from giving good statistical averages, this also enables the analysis of some well known parametrizations which are based on such ensemble-average properties. The SCMS dataset was chosen for this study because of the combination of a range of different surface measurements, in-cloud measurements by aircraft at many levels in the cloud layer, and the availability of high-resolution Landsat images. The results show that LES accurately predicts the thermodynamic and turbulent state of the shallow cumulus cloud layer. More speci cally, the rst statistical moments of many thermodynamic variables as well as the vertical-velocity variance of the cloud ensemble agree well with the aircraft observations. The evaluation of the cloud size distributions of LES against Landsat supports the results on size statistics of Neggers et al. (3). While Siebesma and Jonker () showed that LES reproduced the typical morphology of individual cumulus clouds, these results show that the same is true for cumulus-cloud populations. Unfortunately, the aircraft data cannot provide reliable vertical pro les of the cloud fraction and mass ux, due to the choice to adjust the ight path towards certain large cumulus clouds during the ight. The question of whether the cloud fraction and mass ux in shallow cumulus decreases or increases with height could not be answered.