Precipitation efficiency is an important physical parameter in convective systems and has been applied to determine the rainfall intensity in operational precipitation forecasts (e.g. Doswell et al., 1996). Since Braham (1952) calculated precipitation efficiency with the inflow of water vapour into the storm through cloud base as the rainfall source more than half century ago, precipitation efficiency has been defined as the ratio of the precipitation rate to the sum of all precipitation sources. This definition of large-scale precipitation efficiency (LSPE) has been modified and widely applied in modelling studies and operational forecasts (e.g. Auer and Marwitz, 1968; Heymsfield and Schotz, 1985; Chong and Hauser, 1989; Doswell et al., 1996; Ferrier et al., 1996; Li et al. 2002; Tao et al., 2004; Sui et al., 2005). Due to the fact that prognostic cloud microphysical parametrization schemes are used in cloud-resolving modelling of convective processes, precipitation efficiency is also defined through cloud microphysical budgets as cloud microphysics precipitation efficiency (CMPE; e.g. Weisman and Klemp, 1982; Lipps and Hemler, 1986; Ferrier et al., 1996; Li et al., 2002; Sui et al., 2005). While estimates of CMPE and LSPE can be more than 100% and LSPE estimates can be negative, they are altered to fall within the normal range of 0–100% through the inclusion of all rainfall sources and the exclusion of all rainfall sinks from the surface rainfall budget for LSPE (Gao et al., 2005), and the cloud microphysical budget for CMPE (Sui et al., 2007).
While the precipitation efficiencies in the previous studies have been defined in the surface rainfall budget derived from water vapour and cloud budgets and in the cloud microphysical budget derived from the microphysical budgets of five cloud species (cloud water, rain, cloud ice, snow, and graupel; e.g. Li et al., 2002; Sui et al., 2007), we argue that the precipitation efficiency can be defined only in the budget where precipitation rate is a diagnostic term. An example of such a primitive budget is the rain microphysical budget in the Tropics. Thus, the rain microphysical budget is used to define rain microphysics precipitation efficiency (RMPE) and its estimate from grid-scale simulation data serves as the ‘true’ precipitation efficiency in this study. LSPE and CMPE may deviate from RMPE because only rainfall sources are used to estimate precipitation efficiency. Do CMPE and LSPE deviate from RMPE? What causes the differences? Can water vapour process data be used to estimate precipitation efficiency? These questions will be discussed by analyzing a 21-day two-dimensional (2D) cloud-resolving model simulation that is forced by the large-scale forcing derived from the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). In the next section, the cloud model, forcing, and experiment are described. The results are presented in section 3. The summary is given in section 4.
2. Model and experiment
The cloud-resolving model used in this study is the 2D version of the Goddard Cumulus Ensemble Model, which was originally developed by Soong and Ogura (1980), Soong and Tao (1980), and Tao and Simpson (1993) and was modified by Li et al. (1999). The model has prognostic equations of potential temperature, specific humidity, mixing ratios of cloud water, raindrop, cloud ice, snow, and graupel, and perturbation momentum. The model also includes the cloud microphysical parametrization schemes (Lin et al., 1983; Rutledge and Hobbs, 1983, 1984; Tao etal., 1989; Krueger et al., 1995) and interactive solar and thermal infrared radiation parametrization schemes (Chou et al., 1991, 1998; Chou and Suarez, 1994). The model uses cyclic lateral boundaries, a horizontal domain of 768 km, a horizontal grid resolution of 1.5 km, 33 vertical levels, and a time step of 12 s. Detailed model descriptions can be found in Gao and Li (2008).
The model is forced by zonally uniform vertical velocity, zonal wind, and thermal and moisture advection based on 6-hourly TOGA COARE observations within the Intensive Flux Array (IFA) region (Zhang, personal communication, 1999). The calculations are based on a constrained, variational method applied to column-integrated budgets of mass, heat, moisture, and momentum as proposed by Zhang and Lin (1997). Hourly sea surface temperature (SST) at the Improved Meteorological (IMET) surface mooring buoy (1.75°S, 156°E) (Weller and Anderson, 1996) is also imposed in the model. The model is integrated from 0400 LST on 18 December 1992 to 1000 LST on 9 January 1993 (a total of 486 h). Figure 1 shows the time evolution of the vertical distribution of the large-scale vertical velocity and zonal wind and the time series of the SST, which are imposed in the model during the integrations. The 21-day simulation data have been applied to the analysis of precipitation processes including the roles of surface evaporation (Cui and Li, 2006), ice microphysics (Gao et al., 2006), precipitation efficiency (Li et al., 2002; Sui et al., 2005, 2007), and diurnal variation (Gao et al., 2009). Hourly simulation data are used in this study.
3.1. RMPE: ‘true’ precipitation efficiency
The mass-integrated rain microphysical budget in the Tropics is used to define RMPE, which can be written as
Here, PS is surface rain rate; is air density, which is height dependent only; wTr is terminal velocity for rain, qr is the mixing ratio of rain; z is vertical coordinate; u and w are the zonal and vertical components of wind, respectively; RPI denotes the rainfall source/sink terms from rain microphysical processes, which are defined in Table I, and T0 = 0°C. , where zt and zb are the heights of the top and bottom of the model atmosphere respectively.
Table I. List of microphysical processes and their parametrization schemes.
KFLC: Krueger et al. (1995). LFO: Lin et al. (1983).
RH83, RH84: Rutledge and Hobbs (1983,1984). TSM: Tao et al. (1989).
Evaporation of liquid from graupel surface
Evaporation of melting snow
Evaporation of raindrops
Melting of cloud ice
Condensation of supersaturated vapour
Melting of graupel
Melting of snow
Accretion of cloud ice
Collection of cloud water
Accretion of snow
Autoconversion of cloud water
Deposition of cloud water
Accretion of rain
Homogeneous freezing of cloud water
Deposition of supersaturated vapour
Conversion of cloud ice
Collection of cloud ice
Accretion of cloud water
Deposition of cloud water
Deposition from cloud ice
Accretion of raindrops
Deposition of vapour
Collection of cloud ice
Accretion of raindrops
Accretion of snow
Accretion of cloud water
Riming of snow
Deposition of vapour
Freezing of raindrops
Thus, RMPE is defined as
where RSRB (= RSR + H(QRM)QRM) is the rainfall source from rain microphysical budget;
is the rainfall source from rain microphysical processes, and
H is the Heaviside function,
RMPE is calculated using hourly data and accumulating rainfall sources (RSRB) from each model grid over the model domain, which serves as the ‘true’ precipitation efficiency.
3.2. CMPE versus RMPE
Sui et al. (2007) used the cloud microphysical budget to define precipitation efficiency (CMPE). The cloud microphysical budget can be expressed by
Here, ql = qc + qr + qi + qS + qg, where qc,qr,qi,qS,qg are the mixing ratios of cloud water, raindrops, cloud ice, snow, and graupel, respectively; PI denotes rainfall source/sink terms from cloud microphysical processes, which are defined in Table I. Thus, CMPE is defined as
where is the rainfall source from cloud microphysical processes.
Rainfall sources are used to calculate precipitation efficiency, whereas rainfall sinks are excluded, which can yield the difference between RMPE and CMPE. This is demonstrated in Figure 2 where RMPE is larger than CMPE. RMPE and CMPE are calculated by accumulating rainfall sources from each model grid over the model domain in Figure 2.
Since the cloud microphysical budget is derived by combining mass-integrated microphysical budgets of cloud water, rain, cloud ice, snow, and graupel, the difference between RMPE and CMPE can be contributed to by microphysical budgets of cloud water, cloud ice, snow, and graupel. The budgets can be written as
CWPI, CIPI, SPI, and GPI denote rainfall source/sink terms from microphysical processes of cloud water, cloud ice, snow, and graupel, respectively. The microphysical processes in (6e)–(6h) are defined in Table I, and T00 = −35°C.
Since the rainfall sources are obtained by taking positive values for (5a)–(5d), we may get
where , , , are the rainfall sources from cloud water, from cloud ice, from snow and from graupel microphysical processes, respectively.
Equations (7a)–(7d) show the possible contributions of microphysical budgets of cloud water, cloud ice, snow, and graupel to the difference between RMPE and CMPE. This can be demonstrated in Figure 3, which shows RSC versus RSR,RSCW, RSCI, RSS, and RSG, respectively, and in Figure 4, which shows H(QCM)QCM versus H(QRM)QRM, H(QCWM)QCWM, H(QCIM)QCIM, H(QSM)QSM, and H(QGM)QGM, respectively. The graupel and cloud water microphysical budgets contribute more to the difference in rainfall sources between RMPE and CMPE than the cloud ice and snow microphysical budgets do, while the cloud microphysical budget (3) is primarily attributable to the rain microphysical budget (1).
3.3. LSPE versus RMPE
While cloud information is usually unavailable from conventional data, water vapour processes can be estimated with available conventional data. Sui etal. (2007) showed that large-scale precipitation efficiency (LSPE) is defined as
where is the rainfall source from water vapour and cloud budgets, QI = (QWV T,QWV F,QWV E,QCM), where QWV T is the local vapour change, QWV F is vapour convergence, and QWV E is the surface evaporation rate. LSPE (8) can be derived from the surface rainfall budget (Gao et al., 2005; Cui and Li, 2006), which combines the mass-integrated cloud microphysical budget (3) with the mass-integrated water vapour budget, which can be expressed as
The comparison between RMPE (2) and LSPE (8) indicates that LSPE=RMPE only when RSRB = RSWVCB. This is not the case, as indicated in Figure 5. The rainfall source from the rain microphysical budget (RSRB = RSR + H(QRM)QRM) is generally smaller than the rainfall source from the water vapour and cloud microphysical budget when the water vapour and cloud microphysical budgets are averaged over areas smaller than 192 km (Figures 5(a)–(d)), whereas it is generally larger than RSWVCB when the water vapour and cloud microphysical budget is averaged over areas larger than 384 km (Figures 5(e, f)). As a result, LSPE is significantly different from RMPE (Figure 6). The root-mean-squared (RMS) differences between RMPE and LSPE are 20.7 to 37.5% (Table II), which are significantly larger than the RMS difference between RMPE and CMPE (15.3%) and the standard deviation of RMPE (18.0%).
Table II. RMS differences between RMPE and estimates of LSPE using grid-scale (1.5 km) and model-domain mean (768 km) data and data averaged over the areas of 12 km, 96 km, 192 km, and 386 km.
RMS difference (%)
Many previous studies showed the effects of vertical wind shear on the development of convective systems and associated rainfall (e.g. Pastushkov, 1975; Corbosiero and Molinari, 2002; Wang et al., 2009; Shen et al., 2011). Vertical wind shear and its standard deviation, σ, is calculated using the vertical zonal-wind difference between 11 km and 3.7 km (maximum westerly wind) and categorized wind-shear data into three types: strong shear (wind shear larger than 2σ), moderate shear (wind shear between σ and 2σ), and weak shear (wind shear less than σ). The RMS differences between LSPE and RMPE are 22.6% for strong shear, 23.3% for moderate shear, and 20.5% for weak shear when LSPE is calculated using large-scale data averaged over the area of 384 km. The standard deviations of RMPE are 18.2% for strong shear, 17.2% for moderate shear, and 18.1% for weak shear. The RMS differences between RMPE and LSPE are larger than the standard deviations of RMPE. The estimate of precipitation efficiency with water vapor process data may not capture the variation of the true precipitation efficiency. Therefore, water vapour process data cannot be used to estimate precipitation efficiency.
In this study, precipitation efficiency (RMPE) is first defined through a rain microphysical budget where precipitation rate is a diagnostic term and is considered to be the ‘true’ precipitation efficiency when it is calculated by accumulating rainfall source from each model grid over the model domain. RMPE is then compared with cloud microphysics precipitation efficiency (CMPE) defined through a cloud microphysical budget and large-scale precipitation efficiency (LSPE) through a water vapour budget. The precipitation efficiencies are calculated using hourly data from a 21-day 2D cloud-resolving model simulation with imposed large-scale vertical velocity, zonal wind and horizontal advection obtained from TOGA COARE data. The calculations with accumulations of rainfall sources from each model grid over the entire model domain show that CMPE is generally smaller than RMPE. The difference between RMPE and CMPE is primarily from the graupel and cloud water microphysical budgets. The comparison between RMPE and LSPE shows that their RMS differences are larger than the standard deviation of RMPE. This suggests that water vapour process data may not be used to estimate precipitation efficiency. Since this study only uses 2D simulation data with idealized cyclic lateral boundaries, 3D model simulations are needed to investigate temporal and spatial dependence of precipitation efficiency through analyzing relations between RMPE, CMPE, and LSPE and to evaluate the calculations of precipitation efficiency with large-scale water vapour process data.
The authors thank Prof. M. Zhang (State University of New York at Stony Brook) for his TOGA COARE forcing data, and two anonymous reviewers for their constructive comments. This work is supported by the National Key Basic Research and Development Project of China no. 2009CB421505, and the National Natural Sciences Foundation of China under grant nos. 40930950 and 41075043.