## Sensitivity studies of developing convection in a cloud-resolving model

### By J. C. PETCH ^{∗}

*Atmospheric Processes and Parametrizations, Met Office, UK* (Received 22 April 2005; revised 4 July 2005)

### S ^{UMMARY}

### Cloud-resolving models (CRMs) remain an important tool for providing detailed process information about convection. In this short paper I focus on the development of deep convection and consider what can be considered a minimum expense benchmark simulation for comparison with a numerical weather-prediction model. To decide this a range of sensitivity studies are presented to aspects of the experimental set-up which strongly impact the computational expense. Many of the sensitivities shown in these CRM experiments are quite different to those seen in previous papers which have tended to focus more on deep active convection. Here it is shown that for the case-study presented a minimum expense benchmark simulation must be a 3D simulation. A 200 m horizontal grid length and a domain of 25 km are also required to capture the most important processes.

### K EYWORDS : Dimensionality Diurnal Cycle Resolution

### 1. I NTRODUCTION

## Development of convection is important in numerical weather-prediction (NWP) and was recently the focus of the Global Energy and Water Cycle Experiment (GEWEX) *Cloud System Study (GCSS) working group 4 case-study (Grabowski et al. 2006)* *and also considered in Guichard et al. (2004) both of which used single-column* model (SCM) and Cloud-resolving model (CRM) results to focus on aspects related *to the development of convection. Using the case-study described in Grabowski et al.*

## (2006), hereafter G06, the Met Office CRM is being used to provide information for use in the development of its high-resolution NWP model (Forbes 2005, personal communication). This involves a direct comparison between the two models with the CRM using the same grid length as the NWP model as well as the CRM run to provide a ‘benchmark simulation’. In this case a ‘benchmark simulation’ is the best possible simulation in terms of representing the important processes given current typical computer constraints. The comparison between the CRM and NWP model will be described in a later paper (Forbes 2005, personal communication). This paper presents a series of sensitivity studies to determine what can be considered a benchmark simulation while using the minimum possible amounts of computer expense, hereafter referred to as a minimum benchmark run.

## In the development of precipitating convection over land, the boundary-layer trans- port of moisture and growth of shallow cumulus clouds play a key role and should *be predominantly resolved in any benchmark simulation. Petch et al. (2002) used 3D* simulations of shallow cumulus clouds and 2D simulations of deep convection to show that generally a horizontal grid length of between 100 and 250 m (depending on the depth of the sub-cloud layer) was required to resolve boundary-layer clouds and there- fore realistically reproduce the development of convection. However, the recent results presented in G06 suggested that there may be notable differences between 2D and 3D simulations for the development of deep convection. For this reason there are still strong computational constraints if we wish to resolve the important processes in 3D and have a large enough domain to represent deep convection. Therefore, in terms of a bench- mark simulations we must consider: if 3D is needed, what grid length is needed to resolve the important transports and how big a domain is needed. While investigating the

### ∗ Corresponding address: Met Office, FitzRoy Road, Exeter, EX1 3PB, UK. e-mail jon.petch@metoffice.gov.uk

### Crown copyright, 2006. c

### 345

## impact of grid length, I also consider the role of subgrid mixing in the model, and when considering 2D–3D differences, I also consider sensitivity to shear.

## Many of the most recent studies of deep convection over land have used 2D CRMs *with a horizontal domain of 250 or 500 km (Xu et al. 2002; Guichard et al. 2004; G06)* with a very small number of 3D models run at coarser resolution and a smaller domain.

*In Xu et al. (2002) differences between 2D and 3D versions of the same model were* linked to domain size issues and differences in 2D–3D dynamics, but unlike G06, the differences were small compared to inter-model differences. Previous papers which have investigated 2D–3D differences in CRM simulations also used resolutions of 1 km or coarser and were of active deep convection strongly forced from aloft by large-scale *cooling. The general consensus from these papers (e.g. Grabowski et al. 1998; Donner* *et al. 1999; Tompkins 2000; Petch and Gray 2001) was that the use of a 3D model often* has a relatively small impact on the mean thermodynamics of the simulation other than statistical issues (far fewer points are representing the convection in 2D, i.e. Xu and Randall 1996). The discussion in the papers which did note some differences between *2D and 3D (e.g. Donner et al. 1999; Tompkins 2000; Petch and Gray 2001) was not* relevant to the development stage of convection considered in this paper.

## G06 presented a useful way of modelling the development phase given typical com- putational constraints and the potential need for high resolution, three dimensions and a domain suitable for deep convection. This involved running a small high-resolution 3D simulation for the early stages of development and then mapping this on to lower- resolution larger grids as the clouds deepen. In G06 this step up in domain size was made three times, so during the final 2 hours of the simulation the horizontal grid length was 400 m and with a domain size of 50 or 76 km in both horizontal directions. This bench- mark was an important component of understanding what is required to represent the development of deep convection and the work presented in this paper compliments this.

## In particular, I consider the sensitivity to all the constraints imposed on the simula- tions (domain size, resolution and 2D–3D differences) showing if suitable choices were made. I also address the potential need for higher resolution later in the simulation when boundary-layer transport may still be important as well as entrainment associated with the smaller eddies in the free troposphere. Section 2 briefly describes the model and experimental framework, section 3 discusses sensitivity to resolution, section 4 describes sensitivity to domain size and section 5 considers the differences between 2D and 3D. Section 6 describes the final benchmark run and section 7 discusses implications of the sensitivities for different uses of a CRM.

### 2. M ODEL DESCRIPTION AND EXPERIMENTAL DESIGN

## Both the model and the main aspects of the experimental framework have been

## described in other papers and therefore only the most relevant information will be given

## here. The model used is based on the Met Office Large Eddy Model described in Shutts

## and Gray (1994). The treatment of moisture and cloud microphysics is described in

## Swann (1998), Brown and Heymsfield (2001) and Petch and Gray (2001). Aspects of

## resolution, horizontal domain size and the use of 2D and 3D experiments are the focus

## of this paper and discussed in later sections. A vertical domain of 20 km is used with

## the number of vertical levels chosen to give a vertical grid length no larger than the

## horizontal grid length in the free troposphere and stretched to give shorter grid lengths

## in the lowest 2 km. The top 5 km of the grid include a damping layer to absorb gravity

## waves but convection is capped well below 15 km by the tropopause.

## The subgrid turbulence scheme is likely to be important when considering the grid lengths required to reproduce the basic development of the clouds (e.g. Cheng and Xu 2006). The subgrid turbulence scheme in the Met Office model is based on the Smagorinsky–Lilly model and is essentially a three-dimensional version of a first-order mixing-length closure. The subgrid stress tensor *τ* *ij* and the subgrid scalar flux *Y* *j* of a scalar *X are parametrized through*

*τ* _{ij} *= −ρl* ^{2} *S(u* _{i} *x* _{j} *+ u* _{j} *x* _{i} *)F* _{m} *,* *Y* *j* *= −ρl* ^{2} *S(Xx* *j* *)F* *h* *,*

_{ij}

_{i}

_{j}

_{j}

_{i}

_{m}

## where

*S* ^{2} = ^{1} _{2} *(u* _{i} *x* _{j} *+ u* _{j} *x* _{i} *)* ^{2} *.*

_{i}

_{j}

_{j}

_{i}

## Here *(u* 1 *, u* 2 *, u* 3 *) is the velocity and ρ is the density and l is the neutral subgrid* length-scale. This has a constant value of *l* 0 , except very close to the surface where it *becomes proportional to distance from the surface as described in Brown et al. (1994).*

## Stability dependence is introduced through *F* *m* and *F* *h* which are functions of the local Richardson number, which is calculated taking account of the effects of moist processes as described in MacVean and Mason (1990). The effects of *F* *m* and *F* *h* are to give a sharp fall-off of subgrid mixing efficiency in stable conditions, with no mixing at all when the Richardson number exceeds a critical value of 0 *.25. There is no explicit diffusion apart* from that given by the subgrid model, although the third-order advection scheme will also lead to some dissipation of scalar variance.

## The experiments used here are based on those presented in G06 but for simplicity the prescribed temperature tendencies to represent radiative forcing are not included (sensitivity to this term has been tested and shown to be negligible). The forcing is therefore purely the latent- and sensible-heat fluxes at the surface with the winds relaxed back to the initial values on a two-hour time-scale. The simulations are initialized close to sunrise which is 0730 LST ( UTC −4 h) and all time-series plots show hours after start of run. The surface fluxes and the initial temperature and moisture profiles used for the experiments are shown in Fig. 1 and discussed in more detail in G06. In this and all time series shown in this paper a temporal average of 12 minutes is used. To show how the run evolves over the 6 hours, Fig. 2 shows time–height contour plots of total cloud water content and upward mass flux from a 3D simulation with a 100 m grid length. It can be seen that cloud forms after about 2 hours at a height of around 1 km, this then deepens steadily to 6 km after 6 hours. Other runs will be shown to deepen more quickly than this one but this does give a general picture of how the convection develops during the 6 hour period.

## A wide range of simulations have been carried out to understand the sensitivities

## and a large number of diagnostics have been tested. In total over 200 different runs

## with the model have been carried out, over 60 of which are 3D. For simplicity the

## plots shown in this paper use a subset of these runs and a small number of the

## most representative diagnostics. In each case total hydrometeor content is shown for

## consistency and then one further diagnostic representative of many others often used in

## GCSS WG4 intercomparison projects. Table 1 summarizes the parameter space which

## has been covered, but details of which options are tried in specific runs are described

## when the results are discussed. To address issues of predictability raised in Petch (2004),

## 2D and 3D runs with a domain size less than 80 km are plotted as the mean of several

## ensemble members; predictability issues are discussed in section 4.

### 0 1 2 3 4 5 6 Time (hours after start of run)

### 0 100 200 300 400 500 600

Surface heat flux (W m-2)

### Latent heat flux Sensible heat flux

### 0 2 4 6 8 10 12 14 16 18

### Water vapour (g kg

^{-1}

### )

### 0 2 4 6 8 10 12

### Height (km)

### Water vapour

### 290 300 310 320 330 340 350

### Temperature (K) 0

### 2 4 6 8 10 12

### Height (km)

### Temperature

### (a) (b)

### Figure 1. (a) Time series of surface heat fluxes, and (b) initial conditions of potential temperature and moisture.

### (a) (b)

### Figure 2. Time–height contour plots from a 3D simulation with a 100 m horizontal grid length. (a) Domain average values of total hydrometeor content (g kg ^{−1} ), and (b) upward mass flux (g m ^{−2} s ^{−1} ) defined as the

### vertical flux of all upward moving buoyant points with a cloud water content greater than 0.01 g kg ^{−1} .

### TABLE 1. S UMMARY OF THE CRM RUNS USED FOR SENSITIVITY EXPERIMENTS INVOLVING CHANGES IN RESOLUTION , DOMAIN SIZE , SUBGRID MIXING , SHEAR AND THE INTRODUCTION OF

### A THIRD DIMENSION

### Parameter Min (2D run) Max (2D run) Min (3D run) Max (3D run)

### Horizontal grid length 50 m 1 km 100 m 1 km

### Mixing length-scale 230 m 0.1 m 230 m 0.1 m

### Horizontal domain size 10 km 2000 km 10 km 100 km

### Imposed wind shear 0 14 m s ^{−1} over 7 km Standard winds Standard winds

### 3. S ENSITIVITY TO RESOLUTION AND SUBGRID MIXING

## In this section I consider the sensitivity to resolution and subgrid mixing in both *two- and three-dimensional runs. Petch et al. (2002) showed that the required horizontal* grid length was dependent on the depth of the sub-cloud layer. Figure 2 shows an initial *sub-cloud layer depth of a little less than 1 km, so Petch et al. (2002) would suggest a* grid length of approximately 200 m will be required.

## Figure 3 shows time series of total hydrometeor content and maximum cloud-top

## height in the domain from 2D and 3D runs for a range of horizontal grid lengths.

### 0 1 2 3 4 5 6 Time after start of run (hours)

### 0 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 50m 100 m 200 m 250 m 500 m 1 km

### (a)

### 0 1 2 3 4 5 6

### Time after start of run (hours) 0

### 2000 4000 6000 8000 10000 12000 14000

CLoud top height (m)

### 50m 100 m 200 m 250 m 500 m 1 km

### (b)

### 0 1 2 3 4 5 6

### Time after start of run (hours) 0

### 0.2 0.4 0.6 0.8 1

Total hydrometeor content (kg m-2)

### 100 m 200 m 250 m 500 m 1 km

### (c)

### 0 1 2 3 4 5 6

### Time after start of run (hours) 0

### 2000 4000 6000 8000 10000 12000 14000

Cloud top height (m)

### 100 m 200 m 250 m 500 m 1 km

### (d)

### Figure 3. Time series for a range of horizontal grid lengths with standard subgrid mixing. (a) Total hydrometeor content from 2D runs, (b) maximum cloud-top height from 2D runs, (c) total hydrometeor content from 3D runs, and (d) maximum cloud-top height from 3D runs. Note that no 50 m grid-length runs were carried out in 3D.

## In each case the vertical grid length was at least as fine as the horizontal and no coarser than 250 m in the lower-resolution runs. The subgrid mixing-length-scale was set to be a factor of 0.23 of the horizontal grid length. The domain was the largest carried out for any given resolution (sensitivity to domain will be discussed later). It can be seen that in 2D there is reasonable agreement in the time of development of cloud in the 50, 100 and 200 m grid lengths with significant delays when 250 m, 500 m or 1 km grid lengths are used. The lower-resolution runs tend to develop more rapidly once the cloud has formed. In terms of the cloud-top height there are some quite notable differences between the 200 and 100 m run but it still shows a closer development to 100 than 250 m. It is worth noting that maximum cloud-top height is not a very robust diagnostic as it only requires one point in the domain to have a small amount of cloud water present (0.01 g kg ^{−1} ) for that level to be defined as cloudy.

## The 3D runs also show similar sensitivity to the 2D with 250 m and longer grid

## lengths showing significant delays in the initial development of convection and a more

## rapid growth when it does begin. Due to impacts of domain size restrictions discussed

## in the following section, no 50 m runs were carried out in 3D but again there is good

## agreement between the 100 and 200 m grid lengths in maximum cloud-top height and in

## the total hydrometeor content. It is very noticeable that there is a big difference between

## the 250 and 200 m grid-length runs suggesting there is a threshold between these grid

## lengths where the simulation is better resolved. While there are notable differences

### 0 1 2 3 4 5 6 Time after start of run (hours)

### 0 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 100 m 250 m 500 m 1 km

### (a)

### 0 1 2 3 4 5 6

### Time after start of run (hours) 0

### 1 2 3 4 5 6

### Cloud centre of mass (km)

### 100 m 250 m 500 m 1 km

### (b)

### Figure 4. Time series for a range of horizontal grid lengths with no subgrid mixing in 3D simulations. (a) Total hydrometeor content and (b) cloud centre of mass.

## between the 2D and 3D simulations (which are discussed later) the general sensitivity to resolution looks very similar for 2D and 3D runs.

## A further sensitivity considered here is the impact of the parametrized subgrid mixing when the resolution is changed. Mason and Brown (1999) considered in some detail the sensitivity of simulations of convection to the horizontal mixing length used in the subgrid model. Here, I simply carry out a range of resolution runs with no parametrized subgrid mixing; the only subgrid mixing comes from the advection scheme and this is sufficient for numerical stability. As with the resolution tests using standard mixing, sensitivity to resolution is very similar for 2D and 3D runs, so here I focus on results from 3D cases. Figure 4 shows time series of total hydrometeor content and cloud centre of mass (defined using the same definition as G06) for a range of horizontal grid lengths. It can be seen that there is much less sensitivity to resolution in these mean fields when compared to the results seen with subgrid transport included. However, there is still evidence of delays and too rapid development of the 500 m and 1 km runs. To better understand these differences it is useful to look at the structures of clouds in the different runs.

## Figures 5 and 6 show clouds in a sub-region of the total domain region from four 2D runs at two times to show the impact of grid length on cloud sizes. The times chosen are just after clouds have formed (Fig. 5) and after they have deepened above 6 km (Fig. 6) and the runs chosen are the 50 m, 100 m and 1 km runs with and without parametrized subgrid mixing. The subregion has been chosen as typical of the whole domain, but focusing in shows the cloud structure and size in a little more detail. It can be seen that several clouds form in the 50 and 100 m runs with a typical width and height of a little less than 1 km, whereas the 1 km run without mixing produces clouds with a width of around 3–5 km at a similar time. The large delay in the 1 km run with standard mixing can be seen to be due to a lack of variability developing in the boundary layer to the extent that mostly a large deck of cloud forms with little structure.

## When deep convection does get going then the two 1 km runs have clouds with a similar

## size (widths of ∼6 km) with the 50 and 100 m runs showing a little more structure and

## slightly smaller cloud sizes (widths of ∼4 km). There is a suggestion of convergence in

## the 100 and 50 m runs in cloud structure, but clear evidence that the 1 km run aliases

## this on to larger scales. This is true of 500 m and to some extent 250 m too, but to keep

## this paper short these are not shown here. In later sections, when lower-resolution runs

### (a) (b)

### (c) (d)

### Figure 5. Shaded areas represent regions with a total hydrometeor mixing ratio greater than 0.05 g kg ^{−1} at a time shortly after the first clouds have formed. Each run is focused on a 40 km region which is typical of the full domain. Results from 2D runs using a grid length of (a) 50 m, (b) 100 m, (c) 1 km and no subgrid mixing, and

### (d) 1 km and standard subgrid mixing.

### (a) (b)

### (c) (d)

### Figure 6. Shaded areas represent regions in the domain with a total hydrometeor mixing ratio greater than 0.05 g kg ^{−1} at a time shortly after clouds have penetrated above 6 km. Each run is focused on a 40 km region which is typical of the full domain. Results from 2D runs using a grid length of (a) 50 m, (b) 100 m, (c) 1 km and

### no subgrid mixing, and (d) 1 km and standard subgrid mixing.

## are shown, these are with no subgrid mixing because they give a better representation of the development of convection than those with the subgrid mixing.

### 4. SENSITIVITY TO DOMAIN SIZE

## If there is a need for 3D runs then computational constrains quickly put limits on the domain size. In this section I consider the impact of the domain size in 2D and 3D simulations. For 2D runs a grid length of 100 m was used as high-resolution runs have been shown to better resolve key processes and 2D does not impose strong computational constraints. For the 3D runs a grid length of 500 m was used with low mixing. This produced a general development like the 100 m run and allows larger domains to be tested, but would perhaps not be suitable for a final benchmark run due to the unrealistic nature of the shallow clouds described in the previous section. A further issue of using the small domain sizes is that of predictability as discussed in Petch (2004). To address this, the smaller 3D domain sizes (less than 100 km) and all the 2D domain tests are produced from an ensemble of runs created using different white noise in the initial conditions as described in Petch (2004).

## Figure 7 shows total hydrometeor content from 2D and 3D simulations using a

## range of domain sizes. Also shown in Fig. 7 is the spread in the ensemble of runs used

## to create the mean values of the 3D 10 and 50 km domains. It can be seen that a domain

## size of 50 km or less in 2D is suppressing the development of convection but domain

### 0 1 2 3 4 5 6 Time (hours)

### 0 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 10 km 25 km 50 km 100 km 200 km

### (a)

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 5 km 10 km 25 km 50 km 100 km

### (b)

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### Min Value Max Value Mean Value (9 runs)

### (c)

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### Min Value Max Value Mean Value (9 runs)

### (d)

### Figure 7. Time series for a range of domain sizes for 2D and 3D runs with domain sizes less than 100 km using an ensemble mean. Total hydrometeor content from (a) 2D runs using 100 m grid length and standard mixing, (b) 3D runs with 500 m grid length and no mixing, (c) the spread in ensemble members used to make up the mean 3D 10 km domain size, and (d) the spread in ensemble members used to make up the mean 3D 50 km domain size. The spread of ensembles is shown using the mean (thick solid line), a standard deviation either side of the

### mean (shaded area) and the maximum and minimum values from the ensemble members (dashed line).

## sizes of 100 km and above tend to show the same development. In 3D, a 25 km square domain gives very similar results to 50 and 100 km square domains but 5 and 10 km domains suppress the convection. The plots showing the range of total hydrometeor contents from the ensembles in 3D suggests that with a 10 km domain (Fig. 7(c)) there is significant spread in the results, whereas with a 50 km square domain (Fig. 7(d)) this is greatly reduced. While not included here for brevity, ensemble members using a 25 km square domain produced a relatively small amount of spread and the 5 km square domain had notably larger spread than the 10 km domain. For the 2D simulations used, there is notable spread with 100 km domain sizes but with 200 km this is reduced to more like that seen for the 50 km square domain in 3D.

### 5. S ENSITIVITY TO 2 D –3 D ASPECTS

## Even from previous plots it was clear there was notable differences between 2D

## and 3D with and without subgrid mixing and independent of grid size chosen. In this

## section I focus in more detail on this difference. Figure 8(a) shows total hydrometeor

## content from 2D and 3D runs using a 100 m grid length where both orientations are

## used in 2D. It is very clear that, while the orientation can make a small difference

## (due to the shear imposed on the simulation), the 3D runs are very different to this

### 0 1 2 3 4 5 6 Time (hours)

### 0 0.2 0.4 0.6 0.8 1

Total hydrometeor content (kg m-2)

### 2D E-W domain 2D N-S domain 3D 25x25km 3D 50x2km

### (a)

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

Total hydrometoer content (kg m-2)

### Control - no shear 0.7 1.4 2.8 4.3 8.6

### (b)

### Figure 8. Total hydrometeor content from a range of runs using 100 m grid length. (a) 2D runs using both a north–south and east–west orientation (two runs for the east–west are included to show typical spread of this) as well as a 3D run, and (b) 2D 100 m grid-length runs using a range of prescribed shear from the surface to 7 km.

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

Total hydrometeor content (kg m-2)

### 100 m 2D 100 m 3D 500 m 2D 500 m 3D

### (a)

### 0 1 2 3 4 5 6

### Time (hours) 0

### 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 2D 3D 0.8 m 3D 1.2 m 3D 1.6 km 3D

### (b)

### Figure 9. Total hydrometeor content comparison for 2D and 3D simulations: (a) 2D and 3D runs using 100 m grid length and 500 m grid length with no subgrid mixing, and (b) 2D and square 3D run with 3D runs using a narrow domain in one direction. The square 3D run had a horizontal domain size of 50 × 50 km. The narrow runs

### all had a long domain of 100 km with the narrow size varying from 0.8 to 1.6 km.

## even when one of the dimensions is only 2 km (almost quasi-2D in terms of the larger circulations allowed). Figure 8(b) shows the impact of forcing a range of shears on the 2D simulations. Basically shear can be seen to suppress the development of the convection in 2D runs but this effect is significantly smaller than differences between 2D and 3D simulations for reasonable values of shear. Very large values or shear in 2D are needed to match the standard 3D results (the shear of 4.3 m s ^{−1} km ^{−1} uses a wind speed increasing from 0 at the surface to 40 m s ^{−1} at 7 km), whereas the shear in the standard runs was approximately 1.5 m s ^{−1} km ^{−1} in the east–west direction and 1 m s ^{−1} km ^{−1} in the north–south direction.

## Figure 9(a) shows the difference between 2D and 3D with both a 100 m and a 500 m

## grid length (the 500 m using no subgrid mixing). It is clear that the differences between

## 2D and 3D runs exist at both grid lengths and are larger than the differences due to the

## use of the different grid lengths. Figure 9(b) shows the impact of a long but very narrow

## domain size in 3D simulations. It shows that, for a narrow domain size in one direction,

## the development begins like a 3D simulation but after a given time it tends towards a 2D

## simulation. The time of the drift towards 2D behaviour is later as the domain width is

### 0 1 2 3 4 5 6 Time after start of run (hours)

### 0 0.2 0.4 0.6 0.8 1

### Total hydrometeor content (kg m

-2### )

### 25 km; 200m 50 km; 200m 80 km; 200m 25 km; 100m

### (a)

### 0 1 2 3 4 5 6

### Time after start of run (hours) 0

### 1000 2000 3000 4000 5000 6000

Cloud centre of mass (m)

### 25 km; 200m 50 km; 200m 80 km; 200m 25 km; 100m

### (b)

### Figure 10. Time series comparing a suggested minimum benchmark run (3D; 200 m grid length 25 km domain) with larger-domain and higher-resolution runs. (a) Total hydrometeor content and (b) cloud centre of mass.

## increased. Once the width is above 1.6 km the results look like the full size 3D runs for all 6 hours. This behaviour is likely to be due to the fact that the size of the convective cells grow in time and become wrapped across the domain, with this happening sooner with a very narrow width. Further discussions about the use of a narrow domain are included in the following section.

### 6. A MINIMUM BENCHMARK SIMULATION

## Based on the sensitivities shown in the previous sections it is possible to make a suggestion of a minimum benchmark simulation for this particular case. This can only be general guidance as the choices are strongly dependant on the focus of the study.

## However, given the large differences between runs shown here, I would suggest that a minimum run is a 3D simulation with a grid length of 200 m and a domain size of at least 25 km. This should not require ensembles and be a reasonable match to a larger domain higher-resolution run. In this section I test this hypothesis.

## Figure 10 shows a comparison of total hydrometeor content and cloud centre of mass from the suggested minimum benchmark run (3D; 25 km domain; 200 m grid length) against runs with larger domains and a run with the same domain size but higher resolution. It is clear that, when compared to the big differences seen throughout this paper, this minimum benchmark run is a good match to significantly more expensive runs. One exception may be the cloud centre of mass (Fig. 10(b) at the early stages before 2.5 hours) but, as can be seen in Fig. 10(a), this is at a time when there is negligible amounts of cloud water so is unlikely to be of any significance.

## Here I focus on cloud structures and compare the minimum benchmark run to a 100 m grid length run also with a 25 km domain. Figure 11 shows the clouds at 1.3 and 6 km after 3 and 6 hours, respectively, from the 100 and 200 m grid-length runs. It can be seen that the 200 m run has very similar cloud sizes and spacing to the 100 m run during both the shallow and the deeper phases. The shallow clouds in both cases have a diameter of 2 km or less and the deep clouds have a diameter of up to 5 km. This suggests that the 200 m grid length in a 3D simulation is not too coarse to represent the clouds in this development case.

## One further issue to consider is the use of a non-square domain such as that shown

## in Fig. 9(b). This gave a reasonable realization of mean fields if the width was greater

## than 1.6 km. However, there are two main reasons why care should be taken with the use

### (a) (b)

### (c) (d)

### Figure 11. Shaded areas represent regions with a total hydrometeor mixing ratio greater than 0.05 g kg ^{−1} from a horizontal slice of the domain. Each plot is focused on a 20 km square region which is typical of the full domain.

### Results from (a) 200 m grid-length run after 3 hours at a height of 1.3 km, (b) 100 m grid-length run after 3 hours at a height of 1.3 km, (c) 200 m grid-length run after 6 hours at a height of 6 km, and (d) 100 m grid-length run

### after 6 hours at a height of 6 km.

## of a rectangular domain for this case. The first is that the 25 km domain was only just

## big enough not to need ensembles of runs by the end of the simulation. As discussed

## in Petch (2004), this tends to be related to the number of clouds in the domain, and

## reducing the domain size in one direction is likely to significantly reduce the number

## of clouds thus introducing a need for ensembles. Secondly, the clouds sizes towards the

## end of the simulation in the square domain are of the order of 5 km, which will wrap

## across the narrow domain. These two issues can be seen in Fig. 12 which shows the

## same as Fig. 11 but for a 50 km by 2 km domain. It is clear that after 3 hours both runs

## give a reasonable realization of clouds, but after 6 hours there are only three main clouds

## in the domain which Petch (2004) has shown would need an ensemble of runs to give a

### (a) (b)

### Figure 12. Shaded areas represent regions with a total hydrometeor mixing ratio greater than 0.05 g kg ^{−1} from a horizontal slice from a run with 200 m grid length and 50 km × 2 km domain. (a) Clouds after 3 hours at a height

### of 1.3 km, and (b) clouds after 6 hours at a height of 6 km.

## good statistical representation of this situation. Also clear is that the clouds are wrapped around the domain in the narrow direction after 6 hours which is influencing the cloud geometry.

### 7. C ONCLUSIONS

## There are still computational constraints on the use of a third dimension and domain size for CRM simulations of deep convection, particularly when considering the development of deep convection in the diurnal cycle. In the work presented here the Met Office CRM is run in 2D and 3D, using a range of horizontal grid lengths with and without parametrized subgrid mixing and using a range of domain sizes. The aim is to understand the relative sensitivities to all these aspects as well as to design a minimum benchmark simulation for comparisons such as with a high-resolution NWP model (Forbes 2005, personal communication). In this case a minimum benchmark is a run which is the best choice of resolution, domain and whether to use 2D or 3D but uses the minimum computational expense. Through showing the sensitivities to each aspect it is hoped that as computational power is increased it will be clear where these additional resources should be applied. For example, it may be apparent that microphysics is the key focus of a study so we may wish to keep to the minimum requirements of domain and resolution so we can include far more complex microphysics.

## A wide range of results were analysed from many runs of the model. A small subset of basic but representative diagnostics were shown with total hydrometeor content in the domain used throughout all comparisons for consistency. Results for this specific experiment, which focused only on the cloud produced during the early stages of deep convection, showed that:

## (i) the use of 3D had the largest impact on the development of convection and this is clearly important;

## (ii) the horizontal grid length needed to be 200 m or less to resolve clouds and produce sensible cloud sizes;

## (iii) if no parametrized subgrid mixing is used then grid lengths longer than 200 m can be used to reproduce the general development of the convection but cloud sizes differ from the truly resolved simulations and this may be important for some studies;

## (iv) a domain size of 100 km is needed if the run is 2D, and 25 km square if the run is 3D otherwise there is evidence of suppression of convection;

## (v) ensembles of 2D runs less than 100 km and 3D runs less than 25 km square showed significant spread before the end of the 6 hour period;

## (vi) a rectangular domain was considered but a similar area to a 25 km square would be needed to avoid the requirement of ensembles;

## (vii) a minimum benchmark run for this case is a 3D run using a grid length of

## 200 m and a square domain size of 25 km; this did not produce a very large spread

## in ensemble members and compared favourably to higher-resolution and larger-domain simulations, which were significantly more expensive.

## It is important to note that these findings are for a specific case and included only basic analysis of domain properties and cloud sizes. The initial conditions, vertical wind profiles and surface forcing will all have an impact on the sensitivities seen in this work.

## It should also be stressed that some sensitivities shown, such as dependence on grid length, will be affected by the subgrid scheme and the numerics of the advection scheme used, so are specific to the Met Office CRM. For example, it is clear that the less well- resolved simulations are a closer match to the benchmark run when no subgrid mixing is included. However, for a model with a less diffusive advection scheme this may well not be the case.

## Although these findings are case specific, the findings were robust across a range of diagnostics and the sensitivities shown do give a starting point for understanding what set-up will be needed in a different case. The findings have confirmed that the general decision of G06 to use 3D and increase the domain size was well founded and the choices were good with perhaps the final domain of 70 km square a little larger than required. However, it has also demonstrated that there may be an option to have a simpler framework for the benchmark where a basic 3D simulation is run throughout.

### A CKNOWLEDGEMENT

## I would like to acknowledge Andy Brown, Steve Derbyshire, Roy Kershaw and Richard Forbes for help during this work. I would also like to thank Wojtek Grabowski for his work on this GCSS working group 4 case-study and two anonymous reviewers who provided useful comments to improve this work.

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