Diagnosis of systematic differences between multiple parametrizations of warm rain microphysics using a kinematic framework



This paper introduces the Kinematic Driver Model (KiD) and its application. KiD is a 1D modelling framework developed at the UK Met Office to facilitate a consistent intercomparison of the microphysics parametrizations employed in a number of cloud-resolving models and numerical weather prediction models. The development of the KiD model was born out of GCSS intercomparison results, which show a wide range of microphysical behaviour and a need to explain the behaviour of different microphysics schemes without the complication of dynamic or radiative feedbacks. This paper describes the KiD model and one of the standard test cases for warm microphysics. Results from a comparison of warm microphysics schemes are presented. The schemes tested produce very different surface precipitation rates and onset timings, as well as different accumulated precipitation. In general, single-moment bulk schemes produce the most rapid onset of precipitation and the widest range in accumulated surface precipitation. Sensitivity tests within the KiD show that including a diagnostic intercept parameter for the rain size distribution reduced variation between the single-moment schemes. However, the early onset of precipitation is a persistent feature of single-moment schemes. Double-moment bulk schemes exhibit improved timing for the precipitation onset relative to an explicit bin model and comparison of accumulated precipitation shows much better agreement. However, double-moment schemes tend to produce relatively large peak precipitation rates, particularly with the deeper cloud. Further sensitivity tests within the KiD demonstrate that using a three-moment scheme, with variable shape parameter, significantly improves the simulation peak precipitation rate relative to an explicit bin model. The tests and results presented in this paper are described to act as a reference for other users to compare with their microphysics schemes and aid microphysics parametrization development. Copyright © 2012 British Crown copyright, the Met Office. Published by John Wiley & Sons Ltd.

1. Introduction

The Global Energy and Water Exchange (GEWEX) Cloud System Study (GCSS) project was set up to facilitate the development and testing of improved cloud parametrizations for climate and numerical weather prediction (NWP) models (Randall et al., 2003). GCSS aims to achieve this by using observations and cloud (system)-resolving models (CRMs), which act as the best numerical representation of the cloud system, to improve understanding of the physical processes that occur within specific cloud systems and thus inform and guide parametrization development.

Recent GCSS CRM intercomparisons, particularly those focusing on boundary layer and polar clouds, have highlighted a large spread in important simulated parameters such as surface precipitation rate, liquid water content and ice water content (in the polar case). For example, the intercomparison based on DYCOMS-II demonstrated a substantial range in inter-model surface precipitation rates for the drizzling, stratocumulus-topped boundary layer case (see Ackerman et al., 2009). Ackerman et al. (2009) argue that this range results from differences in model dynamics. Although evidence is presented for this claim, the fact that the microphysics and dynamics are coupled in these simulations means that it is difficult to say to what extent the spread can be attributed to the different numerical methods for dynamics alone or interactions between dynamics and microphysics.

The recent Rain In Cumulus over the Ocean (RICO) CRM intercomparison demonstrates that for an idealized trade wind cumulus case the general evolution and structure of the cloud field are quite similar amongst participating models (vanZanten et al., 2011). However, as with DYCOMS-II, a wide range of surface precipitation rates are produced by the different models. The RICO intercomparison shows that the timing of precipitation development and the amount that develops vary greatly between the different models. In contrast to the DYCOMS-II intercomparison, the range in surface precipitation is attributed to differences in the microphysics schemes. For example, there is a strong dependence of precipitation on the representation of cloud droplet number concentration, as well as the actual number concentration of cloud droplets.

In general, GCSS CRM intercomparisons have demonstrated that dynamical and microphysical differences between models cause much of the divergence between simulation results from the same case. When considering the role of microphysics, the different intercomparison models use a wide range of schemes with varying complexity and underlying assumptions. These schemes can be separated into two broad groups, namely bulk and size-bin resolved schemes. In bulk schemes all microphysical processes are described in terms of integral parameters, such as mass-mixing ratio or number concentration. The simplest bulk schemes are single-moment (1M) schemes in which a microphysical partition, e.g. cloud water, rainwater or ice, is represented by a single prognostic equation. In the more sophisticated double-moment (2M) bulk microphysics representation a microphysics partition is described by both mass and number concentration, with both moments being predicted and advected. Both the 1M and 2M schemes employ assumptions and parameters to describe cloud drop size distribution, particle fall speeds and conversions between microphysical partitions, e.g. the autoconversion of cloud to rain. Different schemes can, however, have a different range of values over which the scheme starts to produce precipitation and this can play an important role in the timing and amount of precipitation (Stevens and Seifert, 2008).

In contrast to bulk schemes, size-bin resolved schemes make far fewer assumptions as they explicitly solve the equations thought to govern the evolution of the drop size distribution. In such schemes, hydrometeors are described using drop size (mass or number) distribution functions that are discretized into several tens of bins. Bin schemes can be single moment or double moment. In general, bin schemes are considered to be capable of producing a more realistic representation of cloud microphysics, that does not need to be tuned for specific clouds. As a result, bin schemes have been used to act as the standard to which simpler bulk schemes are tuned (e.g. Khairoutdinov and Kogan, 2000). Having said that, there is spread in the results produced by different bin models. For example, single-moment bin models based on number or mass, that assume a fixed cloud drop size in a bin, can accelerate precipitation when compared to analytic solutions and two-moment schemes (Tzivion et al., 1987). Further, the largest spread in precipitation rates from the RICO intercomparison results from comparison of the three bin models (vanZanten et al., 2011).

1.1. Holism versus reductionism

A ‘holistic’ approach to investigating the impact of a variety of microphysics schemes, as well as the importance relative to other model components in a full 3-D CRM simulation, is challenging; it is difficult to isolate microphysics effects in the presence of radiative and dynamic feedbacks. Furthermore, 3-D CRMs are computationally expensive, so executing a large number of sensitivity tests is demanding. In an effort to overcome some of these issues and aid development and testing of microphysics schemes some researchers have taken a ‘reductionist’ approach and made use of kinematic frameworks (e.g. Clark, 1974; Petch et al., 1997; Morrison and Grabowski, 2007; Seifert, 2008; Seifert and Stevens, 2010). In a kinematic framework the flow is prescribed, allowing advective transport and particle sedimentation while avoiding the complexity caused by feedbacks between dynamics and microphysics. Although this method does not include an accurate representation of cloud dynamics, research employing such a method has been shown to provide very useful insight. Clark (1974) used both a 1D and a 2D kinematic framework to test an aerosol cloud parametrization. These tests highlighted the importance of vertical resolution when determining cloud droplet number concentration, which has since been shown to be important in 3D CRM simulations (e.g. Hill et al., 2009). More recently, a kinematic framework has been exploited to compare bin and bulk warm microphysics (Morrison and Grabowski, 2007; Seifert and Stevens, 2010), develop and test new bulk ice microphysics parametrizations (Morrison and Grabowski, 2008) and new bulk subgrid cloud-mixing schemes (Morrison and Grabowski, 2008) and investigate the controls on precipitation efficiency in shallow cumulus (Seifert and Stevens, 2010). As the kinematic framework is generally run as a 1D or 2D model, multiple sensitivity simulations are much more feasible than in a 3-D CRM.

Holistic development of a General Circulation Model (GCM), which requires continual adjustment and improvement, is equally challenging. For the most part, advances are made through small increments to individual components (exceptions to this would be step changes in model behaviour through the introduction of previously missing processes, assimilation of new observations or changes to resolution through availability of additional computational resources).

The process of holistic development seeks to make changes to one or many aspects of the model and then compare the full model to relevant observations (e.g. Brooks et al., 2011). This comparison usually takes place on a set of metrics which are chosen to represent the key quantities of interest. As with a CRM, this holistic approach is ultimately necessary in GCM development, since only by putting all the components together can we establish that any isolated improvements translate to the benefit of the full model when combined with its other elements.

However, the holistic approach has a number of drawbacks:

  • The nature of large and complex models makes them susceptible to the issue of compensating errors; errors in one component are offset (and thus difficult to detect) by opposing errors in another component.

  • The varied temporal and spatial behaviour (e.g. of GCMs) means that examination of ‘broad-brush’ metrics will not differentiate between the many regimes within a varied climate.

  • Metrics cannot always capture enough detail, e.g. total surface precipitation may be correct, but vertical distribution (and redistribution) of water and energy may not be.

  • Where errors are observed, it is difficult to decouple the processes to identify the sources of the error.

  • Large, complex models are computationally expensive, narrowing the range of exploration of the parameter space of poorly constrained variables.

In combination with the holistic approach, a reductionist approach is able to help overcome some of these problems and allows more rapid and targeted development. GCSS intercomparisons have typically taken a reductionist approach, but combined them with observational data and tried to ensure that any conclusions can translate back to the GCM.

For example, the recent TWP-ICE intercomparison encapsulates a range of models of varying complexity; CRMs for assessment of detailed cloud structure, SCMs for investigation of the behaviour of GCM model physics, as well as regional and global model runs. In some of the reduced components of the intercomparison, i.e. the SCMs and CRMs, multiple simulations were performed using different microphysics parametrization or ‘ensembles’ of forcing data (Varble et al., 2010; Fridlind et al., 2012; Davies, personal communication).

Similarly, the recent SHEBA intercomparison (Morrison et al., 2011) of Arctic mixed-phase stratocumulus recognizes the lack of ability of microphysics parametrizations in predicting ice number concentration and so artificially reduces the problem by fixing this number across participating models. Reducing the system in this way allows a more consistent evaluation of the components of the physics and better advances our understanding of the behaviour of mixed-phase clouds than was achieved in earlier studies (Klein et al., 2009).

These reduced systems have served very well in advancing our understanding of the physics of clouds and the associated parametrizations. However, even in these reduced systems there are still multiple interactions which can obfuscate the behaviour of individual elements of the physics. In particular, the microphysics–dynamics(–radiation) interaction can be very complicated. The recent RICO intercomparison demonstrated orders-of-magnitude difference in surface precipitation rates for drizzling shallow cumulus and this could not be directly attributed to different choices in the complexity of the microphysics (e.g. 1M, 2M or bin), nor could it be attributed to the dynamics (e.g. grid resolution, numerical diffusion) or a combination of the two.

In this paper, we introduce a 1D kinematic framework, the Kinematic Driver (KiD) model, which has been developed at the UK Met Office. The KiD framework hopes to offer another element to the reductionist approach by constraining the dynamics of the problem and isolating the microphysics. In this work we show how such a framework can be used and in particular shed some light on why microphysics in the Met Office LEM RICO simulations produces much greater precipitation fluxes than some of the other schemes. Although the use of kinematic frameworks is not new, the KiD model has been developed to be a community tool that is freely available and as such can be widely used.

2. The KiD model

The KiD model has been specifically designed to act as a basic dynamic driver, so that it is simple to add different microphysics schemes and perform consistent testing of those schemes using a common advective component. It is not designed with an accurate representation of atmospheric dynamics in mind, but rather as a consistent and flexible framework for forcing microphysics schemes in a way that is consistent with observed cloud dynamical fields or 3D-model-generated dynamical fields.

The default advection scheme in the KiD is the Total Variance Diminishing (TVD) scheme of Leonard et al. (1993), known as ULTIMATE. The ULTIMATE scheme is a monotonicity preserving scheme in one-dimensional flows, which is one of the main advection schemes available in the Met Office Large Eddy Model. Since the KiD model is only designed for a 1D column, there will be no mass conservation for a divergent vertical velocity. In order to mitigate this problem, a divergence term can be added such that a horizontally homogeneous condition is assumed for scalars and a linear gradient of the horizontal velocity is assumed and determined from mass continuity (in two dimensions).

The model grid is defined in terms of ‘full’ model levels and ‘half’ levels. The prognostic variables are held on the full model levels. The vertical velocity and density are held on both levels such that the grid can be used as a Lorenz- or Charney–Phillips type (see Arakawa and Konor, 1996), thus permitting the addition and use of other advection schemes.

The prognostic variables are potential temperature, water vapour mixing ratio, hydrometeors and aerosols. The hydrometeors by default are cloud, rain, ice, snow and graupel, but these are each represented in a flexible derived type and as such can represent any number of species or habits. Each instance of this derived type, e.g. cloud, rain, ice, snow or graupel, is described by the number of moments and number of bins. For example, if the number of bins is one, then the scheme will invoke a bulk microphysical representation. If the number of moments is one, then the scheme will be 1M or 2M if it is two and so on. Aerosol can be similarly represented.

Further details about interfacing a new microphysics scheme, and running and controlling the KiD input and output, are documented by Shipway and Hill (2011). The code and documentation are available from http://appconv.metoffice.com/microphysics

2.1. Microphysics schemes

Motivation for this comparison arises from the desire to better understand the relatively high precipitation rates that were produced by the Met Office LEM in the GCSS RICO intercomparison (vanZanten et al., 2011). Although the other microphysics schemes used in that study are not currently available, many microphysics schemes have been included in the KiD model and a mini-intercomparison of a subset of these schemes is presented.

The focus of the intercomparison presented is warm microphysical processes, i.e. schemes and cases presented include no ice/cold processes. The most detailed microphysics scheme included in this intercomparison is the Tel Aviv University (TAU) size-bin resolved microphysics (Tzivion et al., 1987). The TAU scheme requires a single prognostic variable for aerosol, in which aerosols are described with a log-normal aerosol distribution with a mean radius of 0.1 µm and a standard deviation of 1.5 µm. In the current work, the TAU is essentially the same as that described in Hill et al. (2009), except that 34 bins are used for cloud drop mass and number distribution and there is no subgrid mixing assumption. The TAU scheme is assumed to be the best microphysical estimate and as such provides a baseline to which all other schemes are compared.

It is worth noting that this version of the TAU scheme shares a common base with that used in the RAMS model in the RICO intercomparison, but uses a different collection kernel. In contrast to the LEM, the RAMS model produced very low precipitation rates in the RICO intercomparison.

The other schemes included in this study are six bulk microphysics schemes: three 1M and three 2M schemes, i.e. 1M cloud and 2M rain. Table 1 defines each of the schemes and the constants that are assumed for the size spectra relations and the fall speeds. In all schemes the particle size distribution for a hydrometeor is defined as a gamma size distribution, which has the general form

equation image(1)

where N0 is the distribution intercept parameter, μ is the shape parameter and λ is the slope parameter. In LEM2.4sm and UM7_3, N0 is assumed to be a simple function of λ:

equation image(2)
Table 1. Description of the bulk microphysics schemes tested. μ is the shape parameter for the scheme and N0 is the intercept parameter, used to define the cloud drop size distribution. Swann (1998) is the modified Kessler scheme; KK2000 is from Khairoutdinov and Kogan (2000).
DescriptionMass–diameter and size spectra relationsAutoconversionFall speed constants
  • a

    The Thompson 1M, Thompson 2M and Morrison 2M are as made available with Weather Research & Forecasting Model (WRF) version 3.1, with some minor bug corrections added.

Single-moment (1M)
LEM 1MNa = 1.1e15, Nb = 0.0Swann (1998)ar = 362.0, br = 0.65
(LEM2.4sm)μ = 2.5 fr = 0.0, gr = 0.5
UM operational 1MNa = 26.6, Nb = 1.57Tripoli and Cottonar = 386.8, br = 0.67
(UM7_3)μ = 0.0(1980)fr = 0.0, gr = 0.4
Thompson 1MaN2 = 2e6 ,N1 = 9e9Berry and Reinhardtar = 4854, br = 1
(Thompson07)μ = 0.0(1974) (modified)fr = 195.0, gr = 0.5
Double-moment (2M)
LEM 2Mμ = 2.5Swann (1998)ar = 362.0, br = 0.65
(LEM2.4dm)  fr = 0.0, gr = 0.5
Thompson 2Maμ = 0.0Berry and Reinhardtar = 4854, br = 1
(Thompson09) (1974) (modified)fr = 195.0, gr = 0.5
Morrison 2Maμ = 0.0KK2000ar = 841, br = 0.8
(Morrison)  fr = 0.0, gr = 0.54

In UM7_3 standard settings, because Nb is not equal to 0.0, N0 is dependent on λ. In the LEM2.4sm, because Nb is 0.0, N0 is constant and equal to Na.

Thompson07 also assumes a gamma size distribution of the form given in Eq. (1); however, the intercept parameter varies depending on rain mass (qr), as follows (see Thompson et al., 2008):

equation image(3)

where qr0 =0.0001 kg kg−1. This adaptable N0 is used to mimic drizzle drops as well as larger raindrops, so that there is a smoother transition from non-settling cloud drops to typical raindrop sizes (Thompson et al., 2008). In LEM2.4sm or UM7_3 the transition from cloud drops to raindrop sizes is assumed to be instantaneous.

In the 2M schemes the intercept parameter (N0) is a function of the total number of raindrops m−3 (Nr):

equation image(4)

Thompson09 and Morrison set μ = 0, N0 = Nrλ, while μ = 2.5 in the LEM2.4dm. However, μ can be modified in Thompson09 and LEM2.4dm, while it is fixed to zero in Morrison and so cannot be easily modified.

All the bulk microphysics schemes use a saturation adjustment to calculate condensation and evaporation rates. The TAU scheme has a prognostic supersaturation treatment as described in Stevens et al. (1996).

In the TAU scheme the growth of cloud drops to raindrops is initiated by coalescence of cloud drops, which results in larger drops and more efficient collision–coalescence, and thus more growth. The bulk microphysics schemes tested do not explicitly include these processes so it is necessary to transfer water from the cloud partition to the rain partition to initiate the warm rain processes. This transfer of mass is performed by the autoconversion process. Table 1 defines the autoconversion method used in each scheme. As demonstrated and discussed later, this process is important in determining the onset and rate of precipitation.

Table 1 also presents fall speed constants, which are used to calculate the terminal velocity for the distribution (V(D)) using the following:

equation image(5)

Table 1 shows that there are obvious differences in the set-up of each of the 1M schemes and 2M schemes. LEM2.4sm uses a non-zero μ and a fixed N0. These settings are designed for tropical conditions (Swann, 1998). In contrast, Thompson07 chooses 0.0 for μ and has a variable N0, which was designed to simulate drizzle, as well as heavier precipitation. Such a difference is also apparent in the comparison of the 2M schemes. All schemes employ different autoconversion parametrizations which have been developed for a wide range of cloud types. For example, Swann (1998) uses a Kessler autoconversion method that has been modified to be more representative of tropical deep convection. In contrast, KK2000 is derived from curve fitting to large eddy simulations of marine stratocumulus. Thompson09 uses a more general Berry and Reinhardt (1974) scheme with an assumed gamma distribution for the droplet spectra (Berry and Reinhardt, 1974, droplet spectra originally followed a generalized Golovin distribution). The schemes presented in Table 1 also have very different fall-speed constants.

3. KiD set-up and tests

The aim of the intercomparison presented in this work is to demonstrate how the KiD model can be used to understand the importance of the differences between microphysics schemes (Table 1) when simulating warm shallow clouds. The KiD model as described in Shipway and Hill (2011) includes a number of test cases aimed at exploring the conditions seen in a variety of atmospheric conditions. These range from warm rain conditions, as will be used here, to mixed-phase and ice-phase scenarios. In this section we briefly describe one of the standard KiD warm test cases, which, with a small modification, is the basis of the two test cases used in the current work.

3.1. Test cases

The test case known as warm 1 provides a simple updraught, which is constant in height, to advect vapour and hydrometeors. The strength of the updraft follows a half-period of a sinusoid peaking at 2 m s−1, after which there is no further forcing of the microphysical and thermodynamic fields. The temperature field is kept fixed throughout the simulation so as to minimize feedback from the different microphysical schemes (in a full 3D simulation the temperature within an updraft core would be further influenced by entrainment mixing and so reduce the temperature variations that may be seen from vertical advection alone). Thus the updraft velocity is given as

equation image(6)

where t1 =600 s and w1 =2 m s−1.

The initial profiles of temperature and moisture in warm 1 are set to be similar to the initial profiles used in the GCSS RICO composite intercomparison (Table 2). In an effort to keep the set-up as simple as possible, no divergent term is applied in this test case.

Table 2. The initial water vapour mixing ratio (qv) and potential temperature profiles (θ) for warm 1 and W2/W3.
Height (m)warm 1W2 and W3
 qvθ qvθ

The simplicity of the set-up of warm 1 makes it a very useful benchmark. However, the cloud that results from a simulation of warm 1 tends to produce a peak liquid water content (LWC) low down in the cloud, which is not usually observed in warm cumulus clouds and is not well suited to direct comparison with the 3D RICO intercomparison. To improve the realism of the cloud simulated with KiD, in the current work, warm 1 is modified to have a weaker gradient of moisture in the lower layers (Table 2). When w1 = 2 m s−1 this modification causes LWC to increase with height and maximum liquid water path to reduce. This modified simulation is referred to as W2 throughout the rest of this paper. In addition to W2, simulations are performed using the same profiles as W2 but with w1 = 3 m s−1. This simulation is referred to as W3 and, as will be shown in the next section, the effect of increasing w1 is to deepen the cloud and increase the liquid water path (LWP) relative to W2. W3 is simulated to investigate whether the behaviour of microphysics schemes identified in W2 is consistent with the behaviour in a deeper, more vigorous cloud.

3.2. Standard simulation set-up

For the initial comparison, W2 and W3 were simulated using the microphysics schemes described in Table 1. In all these simulations the time-step was set to 1.0 s and the vertical resolution was 25 m. The initial number concentration in the bulk schemes was assumed to be 50 cm−3. In the TAU scheme, the CCN concentration was set at 50 cm−3, which was maintained as a constant background CCN concentration throughout; i.e. CCN was not removed (returned) from the distribution following activation (complete evaporation of a droplet). This method follows Stevens et al. (1996) and was chosen as it was felt to be the closest approximation to the bulk saturation adjustment. Initially, W2 and W3 were simulated with no precipitation processes, so that only condensation and evaporation are considered. These simulations were then repeated with precipitation processes switched on; i.e. sedimentation, autoconversion, accretion and rain evaporation were included in the bulk schemes, while sedimentation and collision–coalescence were included in the TAU scheme.

4. Results

As noted, current intercomparisons of bin microphysics schemes show significant spread in results. However, it is speculated that this spread may well result from differences in model dynamics rather than the microphysical representation. In principle, a bin representation should provide a more robust solution with less reliance on assumptions about the population distribution. Although we must caution over-interpretation of the realism of the bin model results, we use the TAU as a benchmark against which the bulk models are compared.

Figure 1 shows time–height plots of LWC from W2 and W3 simulated with the TAU scheme with no precipitation processes switched on. In W2, liquid water begins to condense at around 2 min, while in W3 condensation starts a little earlier. Due to the timing of the prescribed updraught, in both cases the maximum liquid water content is reached after 10 min, once the updraft has ceased. W3 produces a deeper cloud with a maximum height of 2700 m, compared with 2000 m in the W2 set-up. W3 also produces a larger maximum LWC and hence a greater LWP. After 10 min in both cases the liquid water content achieved is maintained for the rest of the simulation as there are no other processes or dynamics to deplete or increase the water.

Figure 1.

Liquid water content from (a) W2 and (b) W3 simulated with the TAU with no precipitation processes included.

As indicated in section 3.1, the initial thermodynamic profiles have been modified from the original KiD set-up. This allows a slightly more realistic in-cloud LWC profile. Although the uniformity of the profiles represents a barrier to achieving full realism using this approach and the two cases considered here will not necessarily be representative of all convective updraughts seen in nature, the aim of intercomparing and assessing some of the characteristics of the microphysics schemes remains viable.

When precipitation processes are included, liquid water begins to convert to rainwater after about 5 min in W2 (Figure 2(c)) and a little earlier in W3 (Figure 2(d)). Unlike the bulk schemes, within the TAU scheme there is no separate definition for cloud and rain particles. To facilitate comparison with the bulk schemes, a cut-off radius r0 is defined such that drops larger than this radius are considered to be rain and droplets smaller are considered as cloud. In the initial results presented, r0 = 25 µm. This 25 µm boundary resides in the spectral minimum that is often observed (e.g. Wood, 2000) and which is exploited in bulk schemes to differentiate those larger drops (rain) with appreciable fall velocity and the smaller droplets (cloud) which fall very slowly. We further examine sensitivity to this choice of diagnostic cut-off in section 4.1.

Figure 2.

Liquid water content from (a) W2 and (b) W3 and rain water content from (c) W2 and (d) W3 simulated with the TAU with precipitation processes included.

In W2, once rainwater develops through collision–coalescence, it begins to sediment, reaching the surface at around 17 min (Figure 2(c)). The larger LWC associated with W3 results in a more efficient collision–coalescence process and a more rapid development of rainwater. This leads to an earlier onset of precipitation, with rainwater reaching the surface at around 14 min (Figure 2(d)). In both cases cloud water that does not convert to rain remains suspended in the column (Figures 2(a) and (b)). Due to the lower LWC, and hence lower collision efficiencies in W2, cloud water remains in the cloud for longer than that in the W3 case. Thus the larger LWC produced in W3, relative to the W2 case, leads to more rapid development of rainwater, earlier onset of surface precipitation and higher precipitation rates.

When precipitation processes are not simulated, all schemes, irrespective of microphysical complexity, produce a very similar simulation of LWP (Figure not shown). Including precipitation processes, however, leads to significant differences in the LWP simulated by each scheme (Figures 3(a) and (b) and 4(a) and (b)). These differences are caused by variations in the conversion of cloud water to rainwater (Figures 3(c) and (d) and 4(c) and (d)), as well as the timing and amount of precipitation produced by each scheme (Figures 3(e) and (f) and 4(e) and (f)).

Figure 3.

(a, b) Cloud water path (kg m−2), (c, d) rainwater path (kg m−2) and (e, f) surface precipitation rate (mm h−1) from W2 when precipitation processes are included. Parts (a) (c) and (e) show the 1M schemes compared to the TAU, where the solid line represents the TAU scheme, the dotted line shows LEM2.4sm, the dashed line shows Thompson07 and the dot-dashed line shows UM7_3. Parts (b), (d) and (f) show the 2M schemes compared to TAU, where the solid line represents the TAU scheme, the dotted line shows LEM2.4dm, the dashed line shows Thompson09 and the dot-dashed line shows Morrison.

Figure 4.

As Figure 3 but for W3.

Figures 3(c) and (d) show that when simulating W2 all bulk schemes except Morrison start to convert cloud to rainwater earlier than the TAU scheme. The early production of rainwater is associated with an earlier onset of surface precipitation, with 1M schemes precipitating earlier than 2M schemes, which is consistent with Morrison and Grabowski (2007). As stated in section 2.1, rainwater initially forms in bulk schemes by autoconversion (Figures 5(a) and (b)). Once rainwater is formed, rainwater mass can increase by rain collecting cloud droplets, a process known as accretion (Figures 5(c) and (d)), and to a lesser extent further autoconversion. The early production of rainwater demonstrated in Figures 3(c) and (d) show that the autoconversion method in all schemes except Morrison is creating rainwater too rapidly relative to the TAU scheme (Figures 5(a) and (b)).

Figures 3(c) and (e) show that simulations of the W2 case with LEM2.4sm and UM7_3 produce a very similar timing for the development of rainwater and onset of surface precipitation, although UM7_3 produces a higher surface precipitation rate. With both schemes, however, the onset of surface precipitation is several minutes earlier than that simulated with the TAU scheme. Simulation of W2 with Thompson07 is characterized by a later onset of surface precipitation, relative to the other 1M schemes, but the onset is still 3 min earlier than the TAU scheme. In contrast to the other 1M schemes, maximum precipitation rate produced by Thompson07 is over half that produced by the TAU scheme. In contrast, while LEM2.4sm and UM7_3 produce a rainwater path (RWP) that is lower than, or similar to that produced by either Thompson07 or TAU, LEM2.4sm and UM7_3 produce a larger maximum surface precipitation rate (Figures 3(c) and (e)). Thus LEM2.4sm and UM7_3 appear to precipitate more readily than either Thompson07 or the TAU.

Figures 3(d) and (f) show that increasing microphysical complexity from 1M to 2M microphysics tends to improve the timing for the onset of surface precipitation but does not necessarily improve the simulation of rainwater or surface precipitation rate relative to the TAU scheme. Thompson09 and LEM2.4dm produce a peak surface precipitation rate that is between 50% and 75% greater than that produced by the TAU scheme. Focusing on the onset of surface precipitation and the maximum rate, the 2M results show the largest difference compared to the TAU. For example, comparing LEM2.4sm and dm to the TAU, although LEM2.4dm exhibits an improvement in the timing of surface precipitation onset relative to the TAU, the maximum precipitation rate increases relative to both the LEM2.4sm and TAU. Likewise, comparing Thompson07 and Thompson09 to the TAU scheme shows that Thompson09 tends to produce an earlier onset and a larger precipitation maximum than both Thompson07 and TAU. Figures 3(d) and (f) do show, however, that the Morrison scheme produces RWP timing and a maximum precipitation rate that compare well with the TAU scheme.

Figure 4 shows simulations of W3 performed with the different schemes. As shown for W2, all bulk schemes except Morrison generate rainwater earlier than the TAU and this is associated with an earlier onset of surface precipitation. Using two moments does not improve the simulation of precipitation rate, relative to the 1M schemes or TAU, but as with the W2 case using two moments does tend to improve the timing for precipitation onset at the surface relative to the TAU scheme. Interestingly, while the W2 case simulated with the Morrison scheme appears to compare well with the TAU simulation of W2, with W3 such an agreement is no longer a feature. Instead, simulations of W3 with all 2M schemes produce a maximum precipitation rate that is around three times greater than that produced by the TAU scheme. This result indicates that changes in microphysical complexity may not improve details of the results from a scheme for all conditions.

4.1. Rainwater budget

To examine the behaviour of the bulk schemes and how they relate to the bin representation, it is useful to consider the production and loss terms through the budget for rain. Figures 5 and 6 show time series of column-integrated process rates, while Table 3 summarizes these by presenting the maximum precipitation rate, the accumulated surface precipitation (Iprecip), as well as the time-integrated budget terms, i.e. autoconversion (Iaut), accretion (Iacw) and rain condensation (Icond) and evaporation (Ievp).

Figure 5.

Time series of vertically integrated rain process rates (kg m−2) from W2 simulations: (a, b) autoconversion, (c, d) accretion, (e, f) condensation and (g, h) evaporation. Parts (a) (c), (e) and (g) for 1M schemes and parts (b), (d), (f) and (h) for 2M schemes.

Figure 6.

As Figure 5 but for W3.

Table 3. Maximum surface precipitation rate (mm h−1), accumulated surface precipitation (mm) (Iprecip) and budget terms for Iprecip, i.e. accumulated autoconversion (Iaut), accretion (Iacw), rate of condensation on to raindrops (Icond), rain evaporation rate (Ievp) and change in rainwater path (ΔRWP) for the 1 h simulation from W2 and W3. The last column indicates the amount of rainwater left in the column at the end of the simulation.
Microphysics schemePeak precip. (mm h−1)Iprecip (mm)Iaut (kg m−2)Iacw (kg m−2)Icond (kg m−2)Ievp (kg m−2)ΔRWP (kg m−2)
TAU [40 µm]4.9515.710.832.370.
TAU [32 µm]4.9515.710.832.370.120.301.202.560.0030.050.400.380.100.16
TAU [25 µm]4.9515.710.832.370.160.301.
TAU [20 µm]4.9515.710.832.370.080.090.841.130.431.760.400.380.120.19

The budget terms are calculated by integrating the process rates over the column at each time step, which gives the contribution of the process to the rainwater path. This contribution is then integrated over time for the first hour of the simulation to give the values presented in Table 3.

The column rainwater budget is given by

equation image(7)

where ΔRWP is the change in the rainwater path over the time interval of integration.

In order to make a useful comparison between the TAU scheme tendencies and those of the bulk schemes, we must derive bulk rain production and loss terms from the bin tendencies. Each of the conversion terms is dependent upon the critical radius, r0, chosen to differentiate between cloud and rain such that bins with lower bounds greater than or equal to this value are considered to be rain. Throughout the rest of the paper, we take r0 = 25 µm; however, it is interesting to examine the behaviour of these derived tendencies using a range of possible values.

The autoconversion term is then taken as the total increment to the bin whose lower bound corresponds to a radius r0, due to collisions between droplets in bins smaller than r0. The accretion term is taken as the total increment to all bins larger than r0 due to collisions with bins smaller than r0. Evaporation and sedimentation tendencies are likewise calculated using a consistent value of r0. Since all bins can grow through condensation, we must also consider the possibility that there is a source of rain through condensation (hence the inclusion of Icond in Eq. (7)).

This method of determining autoconversion and accretion rates is consistent with that used by Wood (2005), who looked at solutions based on stochastic collection equations with a continuous spectrum, and that used by Khairoutdinov and Kogan (2000), who also derived these quantities from an explicit bin representation. We note that Wood (2005) used a value for r0 of 20 µm, while Khairoutdinov and Kogan (2000) use 25 µm (this is stated for their accretion calculation, but is not explicitly stated for autoconversion). This work also presents results from a 32 and 40 µm cut-off for estimation of autoconversion to investigate sensitivity to this cut-off. Table 3 and Figures 5 and 6 provide diagnostics calculated using this range of values of r0.

While there is little sensitivity to changes in the value of r0 when calculating surface precipitation (plots not shown), there is a sensitivity in some of the derived process rates. The partitioning between cloud and rain is primarily imposed to differentiate between two populations with different sedimentation characteristics. However, in general, and specifically for the bulk schemes considered here, the two populations also treat condensation processes differently, with condensate only forming on the smaller droplets and no mass added to the rain category. As can be seen from Figures 5 and 6, condensation represents a significant source of mass categorized as rain if r0 = 20 µm. This is still the case but to a lesser extent if r0 = 25 µm. Given this information, we would suggest that a threshold of at least r0 = 25 µm should be specified if using derived process rates from a bin model for comparison or development of bulk processes (32 µm may be even better). The alternative would be to allow for bulk rain schemes to adopt a condensation term, but since partitioning condensation between cloud and rain depends upon the details of the drop distributions and that sedimentation velocity of the bins that are seen to grow is small, this would add an unnecessary complication.

The accretion rate shows very little sensitivity to r0 other than when using the 20 µm threshold, while the evaporation of rain shows no significant sensitivity. This is because it is only the larger drops (r > 40 µm) which have significant fall speed and so fall below the cloud to the subsaturated air. As one might expect, for a lower threshold there is an earlier onset of autoconversion. For the 20 µm threshold, however, the rate is always fairly low in comparison to values obtained using the higher thresholds. It is interesting to note that the Morrison scheme (which uses the autoconversion and accretion rates of Khairoutdinov and Kogan) produces autoconversion rates that are close to those diagnosed using r0 = 20 µm, but the accretion rate is nearer that for r0 = 25 µm for the stronger updraft case in Figure 6.

Table 3 provides a useful summary of the production mechanisms characteristic to each scheme and demonstrates that the contribution from each process varies significantly between schemes. Given the discussion above, comparison of the bulk schemes with the TAU model continues using values derived with r0 = 25 µm.

For the 1M schemes, comparison with the TAU scheme shows that for both cases the LEM2.4sm and UM7_3 produce more Iprecip and Iaut but less Iacw and Ievp than TAU. Further, the relative contribution of autoconversion and accretion to rain production in LEM2.4sm and UM7_3 also differ from that in the TAU. For example, in TAU simulations of W2 (W3) accretion accounts for 92% (91%) of the rain production but in LEM2.4sm accretion only accounts for 43% (60%) of rain production.

Rain mass is lost from the system through evaporation and precipitation. Table 3 shows that rain evaporation (Ievp) from the LEM2.4sm and UM7_3 is less than half the Ievp from the TAU scheme. The relatively low Ievp coupled with the relatively large Iprecip indicate that both 1M schemes are sedimenting too rapidly and evaporating too little relative to the TAU, which leads to a larger surface precipitation. This is further evidence that the LEM2.4sm and UM7_3 precipitate too readily.

Simulations of both cases with Thompson07 produce less Iprecip than TAU simulations. Rain production, i.e. Iaut + Iacw, is similar to that from TAU, with accretion in Thompson07 accounting for 83% (82%) of this production, which is more in line with TAU results than the other SM schemes. Thompson07, however, simulates at least double the rain evaporation rates of TAU and a similar ΔRWP. Thus, in contrast to the other 1M schemes, the low Iprecip results from a large rain evaporation term.

Table 3 shows that the 2M schemes exhibit more consistent behaviour when compared to one another. In general, budget terms also tend to exhibit better agreement with those from the TAU scheme than the 1M schemes do. For example, for both W2 and W3 the Iprecip from all 2M schemes is closer in value with TAU than any from the 1M schemes. Furthermore, the relative contribution of accretion and autoconversion to rain production is comparable to the TAU scheme. For example, LEM2.4dm simulation of W2 (W3) accretion accounts for 86% (92%) of rain production, while in Thompson09 simulations accretion accounts for 83% (84%) of rain production. The LEM2.4dm tends to evaporate more, while the Morrison evaporates less than the TAU scheme. However, the Ievp from the 2M schemes show better agreement with the Ievp from the TAU than the 1M schemes do.

5. Sensitivity tests

The W2 and W3 cases (section 4) show that while the LWP simulated by six bulk schemes and one bin scheme compare very well when only condensation and evaporation processes are included, the addition of precipitation processes leads to significant differences in the cloud evolution simulated by the schemes. Such differences result from differing microphysical assumptions and the level of complexity within the different schemes. Section 4 shows that the 1M schemes produce the largest spread of total surface precipitation, rain production (Iaut + Iacw) and rain evaporation. In contrast, the 2M schemes produce integrated totals which are in better agreement with each other and the TAU scheme; however, the 2M schemes produce significantly larger peak precipitation rates. In this section the KiD model is employed to investigate the role of different standard settings in the 1M and 2M schemes and how they influence their behaviour relative to other schemes.

5.1. Single-moment bulk schemes

In general, the maximum precipitation rate produced in simulations with the 1M schemes agrees reasonably well with the TAU simulations. However, the onset of surface precipitation from the 1M simulations is much earlier than either the TAU or the 2M schemes. Furthermore, the 1M schemes produce the largest range of Iprecip, rain production (Iaut + Iacw) and Ievp. Table 3 shows that while the 1M schemes produce similar rainwater production to the TAU, LEM2.4sm and UM7_3 produce more Iprecip and less Ievp than the TAU, while Thompson07 produces significantly less Iprecip and more Ievp than the TAU (and the other 1M schemes).

A potential source of the large difference in Ievp and Iprecip between LEM2.4sm, UM7_3 and Thompson07 is the different intercept and shape parameters assumed by the schemes. As described in section 2.1, Thompson07 uses a diagnostic intercept parameter (N0), whereas LEM2.4sm and UM7_3 employ simpler fixed intercept parameters. Also, the LEM2.4sm has a shape parameter and intercept parameter that is tuned for tropical deep convection and hence may not be appropriate for the present test cases. For both W2 and W3, the N0 diagnosed from Thompson07 is large, which is representative of small, drizzle-size drops (Figure not shown). Such drops will fall slowly and hence will have a prolonged residence time in the column, leading to greater evaporation rates and less surface precipitation accumulation. In contrast, in the UM7_3 and, in particular the LEM2.4sm, the intercept and shape parameter are representative of rain-size drops, which will fall faster, have a lower residence time in the column and thus less evaporation and a higher surface precipitation rate.

To understand the influence of these differences in intercept and shape parameter settings, W2 and W3 were rerun with the 1M schemes, with all schemes adopting the diagnostic N0 from Thompson07. In addition, for LEM2.4sm we set μ = 0, so that its settings are consistent with Thompson07.

Table 4 shows the rainwater budget terms, as described in section 4. Use of a diagnostic N0 in the UM7_3 and LEM2.4sm results in an increase in Ievp and a decrease in Iprecip, so that the 1M schemes are in much better agreement with each other than the comparison shown in section 4. As with Thompson07, the N0 diagnosed by the UM7_3 and LEM2.4sm is large and this results in a reduction in drop fall velocity, which prolongs drop residence time in the column, leading to enhanced evaporation and reduced surface precipitation. It is interesting to note that changing N0 reduces the contribution of autoconversion to the rain production term in both the UM7_3 and LEM2.4sm. This results from changes in drop size distribution associated with increased N0, which leads to enhanced collection of cloud water, which is shown by the increased accretion rate and hence less cloud water is available for autoconversion.

Table 4. Accumulated autoconversion (Iaut), accretion (Iacw) surface precipitation (Iprecip) and rain evaporation rate (Ievp) for the standard 1M simulations of W2 and W3, with a diagnostic N0.
Microphysics schemeIprecip (kg m−2)Iaut (kg m−2)Iacw (kg m−2)Ievp (kg m−2)

Thus employing a diagnostic N0 results in a more consistent behaviour amongst 1M schemes as well as better agreement between the 1M schemes and the TAU scheme. It is noted, however, that the onset of surface precipitation from all 1M schemes is still very early relative to the TAU scheme. Extra tests (not shown) in which fall speed–diameter relationships are varied can slightly change the timing of precipitation onset relative to the TAU, but the 1M schemes still precipitate much earlier than any of the other schemes tested.

The tests with 1M schemes and KiD model show that, while certain parameters can be tuned, e.g. the N0 representation, to improve agreement amongst schemes and behaviour of schemes relative to the TAU, it is difficult to improve all aspects of a 1M microphysics simulation. For the cases presented, a persistent feature of simulations with the 1M schemes is a tendency to produce surface precipitation too early. This is the case for both the standard and modified parameter settings in a scheme and hence seems to be an underlying issue with 1M schemes. This result is consistent with 2-D kinematic simulations presented in Morrison and Grabowski (2007).

5.2. Double-moment bulk schemes

Comparison of the 1M and 2M schemes show that the budgets and the relative contribution of the terms from the 2M schemes are in better agreement with that produced by the TAU scheme. Furthermore, in general, the onset of surface precipitation from 2M schemes is also closer to that produced by the TAU scheme. These results suggest that increasing microphysical complexity improves overall results relative to the TAU. However, the 2M simulations of both cases, but in particular W3, produce a larger peak precipitation rate than that generated by both the 1M and TAU scheme. In general, the 2M schemes produce less rainwater but similar Iprecip in W2 and more Iprecip in W3. As Ievp is comparable with that produced by the TAU scheme, the larger peak precipitation rates that result from less rain production within the 2M schemes result from larger sedimentation rates.

A known limitation of the 2M schemes is that the shape parameter is fixed in space and time and hence it is invariant during sedimentation. In reality, big drops fall fast whereas small drops persist in the column. This size sorting leads to narrowing of the distribution in mass space, while broadening it in physical space (Stevens and Seifert, 2008). While 1M schemes are unable to simulate this size sorting, 2M schemes begin to. The extent of this size sorting is, however, dependent on the shape parameter with a fixed μ of 0, producing excessive size sorting (Stevens and Seifert, 2008; Milbrandt and McTaggart-Cowan, 2010). Simulations of W2 and W3 exhibit the result of excessive size sorting through the large maximum precipitation rate that occurs for a short period of time. The effect is more obvious in W3 because the cloud is deeper, with higher LWC, which will rain more readily, leading to enhanced size sorting. Milbrandt and Yau (2005), Stevens and Seifert (2008) and Milbrandt and McTaggart-Cowan (2010) have demonstrated that permitting μ to vary in space and time by employing a triple-moment bulk scheme or 2M scheme with a diagnostic μ improves the simulation of sedimentation relative to analytic solutions. It is worth noting that, while comparison of the 2M schemes and TAU scheme shows obvious differences in surface precipitation and timing, such differences are not apparent in the comparison presented in Seifert and Stevens (2010). The reason for the apparent inconsistency is that certain parameters such as the shape parameter were tuned in the bulk scheme used in Seifert and Stevens (2010) in order to agree with the bin scheme.

So far in this work no effort has been made to tune the 2M bulk schemes, as the aim has been to understand and demonstrate the differences between schemes. However, motivated by the results of others, W2 and W3 are simulated with a new flexible triple-moment bulk scheme that has been developed at the Met Office. We will refer to this scheme simply as 3M. See the Appendix for a brief overview of this scheme.

Figure 7 shows a comparison of RWP and surface precipitation rates from the TAU and the triple-moment 3M scheme. In addition, Figure 7 shows results from the Morrison scheme and the LEM2.4dm with KK2000 autoconversion and accretion and the Uplinger fall speed relationship (Uplinger, 1981), which is most similar to the processes used in the 3M scheme. Modifying the LEM2.4dm in this way results in the onset of precipitation at the surface being delayed, in both cases, so that the timing is closer to the TAU scheme. Furthermore, the timing of the maximum precipitation is also closer to that of the TAU. The modified LEM2.4dm is, however, still producing a relatively large maximum precipitation rate of almost triple that produced by the TAU scheme. This suggests that simply modifying the processes within the double-moment schemes will not prevent the large maximum precipitation rates simulated in W3.

Figure 7.

Rainwater path (kg m−2) and the surface precipitation rate (mm h−1) from (a, c) W2 and (b, d) W3. All plots show the triple-moment 3M scheme, Morrison and LEM2.4dm, with KK2000 autoconversion, accretion and Uplinger fall speed relationship, compared to the TAU scheme.

Using the 3M triple-moment scheme, which also implements the KK2000 autoconversion and accretion and the Uplinger fall speed relationship, results in surface precipitation and RWP that compare very well with those produced by the TAU scheme. This is the case for both W2 and W3. This demonstrates that including the third moment and hence relaxing the assumption of an invariant shape parameter leads to a better simulation of the maximum precipitation rate and the onset of this maximum. This is consistent with the sedimentation-only tests presented in Milbrandt and Yau (2005), Stevens and Seifert (2008) and Milbrandt and McTaggart-Cowan (2010).

The above experiments are interesting because the early intense precipitation from LEM2.4dm KiD simulations is a feature of 3D RICO simulations. The KiD simulations of W2 and W3 presented here demonstrate a similar tendency to the RICO simulations. Using the KiD model it has been possible to identify potential causes of this early precipitation and develop a scheme that corrects this issue. Such an analysis was made possible due to the relatively small computational cost of the KiD and its isolation of the microphysical components from any dynamic influence or feedback. However, to confirm these results it is necessary to perform 3D simulations with the suggested settings. This is beyond the scope of this paper, but will be the main topic of a follow-up paper.

6. Summary

This paper has introduced the Kinematic Driver model (KiD), which is a kinematic modelling framework developed at the UK Met Office as a generic ‘plug and play’ dynamical driver for cloud microphysics schemes. The principal idea behind this development is to facilitate the consistent intercomparison of microphysics parametrizations employed in a number of cloud-resolving models and numerical weather prediction models. The need for such a development was born out of GCSS intercomparison results, which show a wide range of microphysical behaviour, and a need to explain the behaviour of different microphysics schemes without the complication of dynamic or radiative feedbacks.

In this demonstration of the KiD model, we seek to understand the different behaviour of the LEM as seen in the RICO intercomparison. To this end we carry out a mini-intercomparison of warm-cloud microphysics schemes of differing complexity. The schemes included consist of three single-moment schemes, three schemes with single -moment cloud and double-moment rain and one size-resolved scheme, which is considered as the best estimate (although it is acknowledged that bin schemes proved to give the biggest spread in the full 3D RICO comparison). Each scheme is used to simulate two test cases: one which represents a shallow, weakly precipitating cumulus and the other a slightly deeper, more vigorous cumulus with heavier precipitation. The profiles for both cases are based on the initial specifications for the RICO GCSS intercomparison. The results from these simulations have been used to demonstrate how the KiD model can be used as a tool to understand differing behaviour of microphysics schemes.

The KiD simulations presented demonstrate that single-moment schemes produce the earliest onset of surface precipitation, which is consistent with Morrison and Grabowski (2007). The standard settings in single moment schemes result in the largest spread in the accumulated surface precipitation, precipitation budget terms and partitioning of the budget terms. It is demonstrated within the KiD frameworks that much of this variation stems from the representation of the raindrop size distribution intercept parameter. Including a diagnostic intercept parameter in all single-moment schemes, such as that described in Thompson et al. (2008), reduces this spread and causes the accumulated surface precipitation to be in better agreement with the ‘best estimate’, i.e. the size-resolved scheme.

In general, simulations with the more complex double-moment rain schemes (including the LEM scheme used in the RICO intercomparison) exhibit better inter-scheme agreement than the 1M schemes and better agreement with the size-resolved scheme. This is demonstrated for timing of the onset of precipitation, the accumulated surface precipitation, and partitioning of the total precipitation budget. A feature of the 2M schemes, however, is that all the schemes tested produce larger precipitation peaks than any of the other schemes. This problem becomes considerably worse with the deeper cloud. While modifying autoconversion, accretion methods and sedimentation fall speed relationships can improve the precipitation onset timing, such changes do not improve the maximum precipitation rate. It is argued that the large maximum precipitation rate results from excessive size sorting, caused by having an invariant shape parameter, which leads to unrealistically large sedimentation rates. Within the KiD framework it is demonstrated that using a triple-moment bulk microphysics scheme leads to considerable improvement in the comparison of the bulk schemes with the TAU scheme.

Thus, as well as introducing the KiD model, this paper has demonstrated that the KiD model is a useful tool for comparing and understanding the behaviour of different microphysics schemes. Furthermore, it is has been shown that the KiD model is a useful development tool that can be used to efficiently tune existing schemes and/or develop new schemes. The high precipitation flux seen in the LEM double-moment scheme is consistent with the results in the RICO intercomparison, while the larger rain evaporation rate may also be consistent with development of deeper, stronger cold pool formation and the start of a feedback onto the dynamics. It is noted that, while the results presented offer a significant insight into the behaviour of different microphysics schemes, relating such results to GCSS CRM intercomparisons requires testing within a 3D intercomparison simulation. Such testing is beyond the context of this paper, but will be the focus of future work.

Finally, while this paper has focused on using the KiD model to undertake a warm microphysics scheme intercomparison, the KiD model can and has been used to understand different systems. For example, Field et al. (2012) simulated orographic wave clouds to investigate ice processes and Dearden et al. (2010) used the KiD to investigate aerosol–cloud interactions. Onishi and Takahashi(personal communication) have also used the warm rain and deep convective cases to benchmark a new microphysics scheme. This demonstrates the flexibility of the KiD model for both development and research.


Triple-moment implementation

Section 5.2 demonstrates the benefit of having a third prognostic moment to describe the evolution of the assumed rain spectrum. Milbrandt and Yau (2005) first proposed a closure for such a scheme which forms the basis in 3M. We give a brief overview of the construction of the triple-moment scheme used here.

Given the definition of the particle size distribution of the form

equation image(8)

(cf. Eq. (1)), we define the pth moment of the distribution as

equation image(9)

where we allow p to be any real non-negative number. It is apparent from (9) that Nr is simply given by the 0th moment. Furthermore, given a mass–diameter relationship of the form

equation image(10)

we readily see that the bulk mass mixing ratio of the rain distribution is related to the dth moment by

equation image(11)

where ρ is the density of air. For spherical particles (as is assumed here for rain), d will be 3 and so mass relates to the 3rd moment.

Manipulation of Eq. (9) reveals that, given p1≠p2≠p3, we can combine three independent moments such that the resulting expression is independent of Nr and λ. That is,

equation image(12)

provided that we choose k1 = p2p3, k2 = p3p1 and k3 = p1p2.

Similarly, given p1≠p2, an expression which is independent of Nr, but depends upon both λ and μ, can be formulated as

equation image(13)

provided l1 = −l2 = 1/(p1p2).

Thus, given any three moments (which may represent or be deduced from our prognostic variables), we are able to reconstruct the distribution by first inverting the relation Eq. (12) and then Eq. (13). The value of Nr and thence any other moment of the distribution follows using Eq. (9). Furthermore, it is worth noting for consideration in solving Eq. (12) numerically that the function G1(μ) is monotonic in μ and increasing(decreasing) if the product k1k2k3 is positive (negative).

In the implementation used in this paper, we choose p1 = 0, p2 = 3, p3 = 6. Increments to moments p1 and p2 can then be directly calculated from any of the process rates employed in existing 2M bulk parametrizations. In this instance we choose the representation of KK2000 for accretion and autoconversion; evaporation follows diffusional growth concepts of Byers (1965) and self-collection follows Beheng (1994). These are all similar to the processes used in the Morrison 2M scheme.

To calculate increments to p3, we simply derive them through Eq. (12) from p1, p2 and their related increments, given certain assumptions about the way μ evolves (Milbrandt and Yau, 2005): if there is no pre-existing rain, then we assume the distribution is initialized with μ = μ0 = 2.5. Thence, from Eq. (12):

equation image(14)

For warm rain, this is only the case with the autoconversion term. For other source/sink terms, we assume that μ varies very slowly, i.e. equation image, so that ΔM(p3) can be found from equating Eq. (12) before and after the distribution evolves, i.e.

equation image(15)

Sedimentation represents the strongest mechanism by which the shape of the distribution evolves. The sedimentation calculation can then be carried out on each moment separately –it is this that then sees significant evolution of μ and tempers the overactive size sorting.


The authors would like to thank Jonathan Wilkinson for his help in extracting and installing UM_7_3 in the KiD model. We would like to thank Jon Petch, Paul Field and Steve Derbyshire for their helpful comments on the manuscript. Thanks also to Greg Thompson and Hugh Morrison for giving us permission to make their microphysics codes available in the KiD model. We would further like to thank the two anonymous reviewers, whose comments were very helpful in improving this paper.