**1. Introduction**

72 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Zdunkowski W. & Bott A., (2004), *Thermodynamics of the Atmosphere: A Course in Theoretical* 

Precipitation efficiency is one of important meteorological parameters and has been widely used in operational precipitation forecasts (e.g., Doswell et al., 1996). Precipitation efficiency has been defined as the ratio of precipitation rate to the sum of all precipitation sources from water vapor budget (e.g., Auer and Marwitz, 1968; Heymsfield and Schotz, 1985; Chong and Hauser, 1989; Dowell et al., 1996; Ferrier et al., 1996; Li et al., 2002; Sui et al., 2005) after Braham (1952) calculated precipitation efficiency with the inflow of water vapor into the storm through cloud base as the rainfall source more than half century ago. Sui et al. (2007) found that the estimate of precipitation efficiency with water vapor process data can be more than 100% or negative because some rainfall sources are excluded or some rainfall sinks are included. They defined precipitation efficiency through the inclusion of all rainfall sources and the exclusion of all rainfall sinks from surface rainfall budget derived by Gao et al. (2005), which fixed precipitation efficiency to the normal range of 0-100%.

In additional to water vapor processes, thermal processes also play important roles in the development of rainfall since precipitation is determined by environmental thermodynamic conditions via cloud microphysical processes. The water vapor convergence and heat divergence and its forced vapor condensation and depositions in the precipitation systems could be major sources for precipitation while these water vapor and cloud processes could give some feedback to the environment. Gao et al. (2005) derived a water vapor related surface rainfall budget through the combination of cloud budget with water vapor budget. Gao and Li (2010) derived a thermally related surface rainfall budget through the combination of cloud budget with heat budget. In this chapter, precipitation efficiency is defined from the thermally related surface rainfall budget (*PEH*) and is calculated using the data from the two-dimensional (2D) cloud-resolving model simulations of a pre-summer torrential rainfall event over southern China in June 2008 (Wang et al., 2010; Shen et al., 2011a, 2011b) and is compared with the precipitation efficiency defined from water vapor related surface rainfall budget (Sui et al., 2007) to study the efficiency in thermodynamic aspect of the pre-summer heavy rainfall system.

The impacts of ice clouds on the development of convective systems have been intensively studied through the analysis of cloud-resolving model simulations (e.g., Yoshizaki, 1986;

Thermodynamic Aspects of Precipitation Efficiency 75

and graupel, respectively; overbar denotes a model domain mean; prime is a perturbation from model domain mean; and superscript o is an imposed observed value. The comparison between (1) and (2) shows that the net condensation term (*Sqv*) links water vapor, heat, and

> () () () () () () () () () () ( ) ()

(2c)

*DEP SDEP GDEP MLTS MLTG SACW o SFW o GACW o IACR o GACR o SACR o GFR o RACS o SMLT o GMLT o IHOM oo IMLT o*

*P TT P TT P TT P TT P TT P TT P TT P TT P TT P TT P TT P TT*

Following Gao et al. (2005) and Sui and Li (2005), the cloud budget (1c) and water vapor

*P wq r Tr r z* 

*P wq s Ts s z* 

*P wq g Tg g z* 

[ ][ ] *l l q q u w*

[ ] *<sup>v</sup> WVT <sup>q</sup> <sup>Q</sup> <sup>t</sup>*

*x z*

*Q PPP P WVOUT CND DEP SDEP GDEP* [ ][ ][ ][ ] (5f)

*Q PP P WVIN REVP MLTG MLTS* [ ][ ][ ] (5g)

[ ]*<sup>l</sup>*

*CM <sup>q</sup> <sup>Q</sup> <sup>t</sup>*

' ' ( ) [ ][ ][ ] *o o v vv WVF q q uq Qu w*

*x zx*

*P Q S CM* = *QWVS* = *Q Q WVOUT WVIN* (3)

*QQQ WVT WVF WVE* = *QWVS* (4)

*P PPP Srs <sup>g</sup>* (5a)




(5e)

(5h)

' ''

 [ ][ ][ ] *o o v vv q qq uww x zz* (5i)

*Q E WVE s* (5j)

'

budget (1a) are mass integrated and their budgets can be, respectively, written as

*PP P P P P*

 

cloud budgets.

where

18

( ),

*IDW oo o*

*P T TT*

Nicholls, 1987; Fovell and Ogura, 1988; Tao and Simpson, 1989; McCumber et al., 1991; Tao et al., 1991; Liu et al., 1997; Grabowski et al., 1999; Wu et al., 1999; Li et al., 1999; Grabowski and Moncrieff, 2001; Wu, 2002; Grabowski, 2003; Gao et al., 2006; Ping et al., 2007). Wang et al. (2010) studied microphysical and radiative effects of ice clouds on a pre-summer heavy rainfall event over southern China during 3-8 June 2008 through the analysis of sensitivity experiments and found that microphysical and radiative effects of ice clouds play equally important roles in the pre-summer heavy rainfall event. The total exclusion of ice microphysics decreased model domain mean surface rain rate primarily through the weakened convective rainfall caused by the exclusion of radiative effects of ice clouds in the onset phase and through the weakened stratiform rainfall caused by the exclusion of ice microphysical effects in the development and mature phases, whereas it increased the mean rain rate through the enhanced convective rainfall caused by the exclusion of ice microphysical effects in the decay phase. Thus, effects of ice clouds on precipitation efficiencies are examined through the analysis of the pre-summer heavy rainfall event in this chapter. Precipitation efficiency is defined in section 2. Pre-summer heavy rainfall event, model, and sensitivity experiments are described in section 3. The control experiment is discussed in section 4. Radiative and microphysical effects of ice clouds on precipitation efficiency and associated rainfall processes are respectively examined in sections 5 and 6. The conclusions are given in section 7.

## **2. Definitions of precipitation efficiency**

The budgets for specific humidity (*qv*), temperature (*T*), and cloud hydrometeor mixing ratio (*ql*) in the 2D cloud resolving model used in this study can be written as

$$\frac{\partial q\_{\upsilon}}{\partial t} = -\frac{\partial (\boldsymbol{u}^{\boldsymbol{\prime}} \boldsymbol{q}\_{\upsilon}^{\boldsymbol{\prime}})}{\partial \mathbf{x}} - \overline{\boldsymbol{u}}^{\boldsymbol{o}} \frac{\partial q\_{\upsilon}^{\boldsymbol{\prime}}}{\partial \mathbf{x}} - \overline{\boldsymbol{w}}^{\boldsymbol{o}} \frac{\partial q\_{\upsilon}^{\boldsymbol{\prime}}}{\partial \boldsymbol{z}} - \boldsymbol{w}^{\boldsymbol{o}} \frac{\partial \overline{q}\_{\upsilon}}{\partial \boldsymbol{z}} - \frac{1}{\rho} \frac{\partial}{\partial \boldsymbol{z}} \overline{\rho} \boldsymbol{w} \boldsymbol{q}\_{\upsilon}^{\boldsymbol{\prime}} - \boldsymbol{S}\_{q\upsilon} - \overline{\boldsymbol{u}}^{\boldsymbol{o}} \frac{\partial \overline{q}\_{\upsilon}}{\partial \boldsymbol{x}} - \overline{\boldsymbol{w}}^{\boldsymbol{o}} \frac{\partial \overline{q}\_{\upsilon}}{\partial \boldsymbol{z}} \tag{1a}$$

$$\frac{\partial T}{\partial t} = -\frac{\partial}{\partial \mathbf{x}} (\overline{u}^o + \stackrel{\cdot}{u}) T^\uparrow - \pi \overline{u}^o \frac{\partial}{\partial \mathbf{x}} - \pi \overline{w}^o \frac{\partial}{\partial z} (\overline{\theta} + \stackrel{\cdot}{\theta}) - \pi w \frac{\partial \overline{\theta}}{\partial z} - \frac{\pi}{\overline{\rho}} \frac{\partial}{\partial z} (\overline{\rho} w^\prime \theta^\prime) + \frac{Q\_{\text{cn}}}{c\_p} + \frac{Q\_R}{c\_p} \tag{1b}$$

$$\frac{\partial \overline{q}\_{l}}{\partial t} = -\frac{\partial (uq\_{l})}{\partial x} - \frac{1}{\rho} \frac{\partial}{\partial z} \overline{\rho} wq\_{l} + \frac{1}{\rho} \frac{\partial}{\partial z} \overline{\rho} (w\_{\Gamma r} q\_{r} + w\_{\Gamma s} q\_{s} + w\_{\Gamma \xi} q\_{\xi}) + S\_{qv} \tag{1c}$$

where

$$S\_{qv} = P\_{\rm CND} + P\_{\rm DEP} + P\_{\rm SDEP} + P\_{\rm GDEP} - P\_{\rm REVP} - P\_{\rm MLTS} - P\_{\rm MLTG} \tag{2a}$$

$$Q\_{cn} = L\_v S\_{qv} + L\_f P\_{18} \tag{2b}$$

is potential temperature; *u* and *w* are zonal and vertical components of wind, respectively; is air density that is a function of height; cp is the specific heat of dry air at constant pressure; *Lv*, *Ls*, and *Lf* are latent heat of vaporization, sublimation, and fusion at *To*=0oC, respectively, *Ls*=*Lv*+*Lf*,; *Too*=-35 oC; and cloud microphysical processes in (2) can be found in Gao and Li (2008). *QR* is the radiative heating rate due to the convergence of net flux of solar and IR radiative fluxes. wTr, wTs, and wTg in (1c) are terminal velocities for raindrops, snow, and graupel, respectively; overbar denotes a model domain mean; prime is a perturbation from model domain mean; and superscript o is an imposed observed value. The comparison between (1) and (2) shows that the net condensation term (*Sqv*) links water vapor, heat, and cloud budgets.

$$\begin{aligned} P\_{18} &= P\_{DEP} + P\_{SDEP} + P\_{GDEP} - P\_{MLTS} - P\_{MLTG} \\ &+ P\_{SACV}(T < T\_o) + P\_{SFW}(T < T\_o) + P\_{GACV}(T < T\_o) \\ &+ P\_{LACR}(T < T\_o) + P\_{GACR}(T < T\_o) + P\_{SACR}(T < T\_o) \\ &+ P\_{GFR}(T < T\_o) - P\_{RACS}(T > T\_o) - P\_{SMLT}(T > T\_o) \\ &- P\_{GMLT}(T > T\_o) + P\_{HOOM}(T < T\_{oo}) - P\_{MLT}(T > T\_o) \\ &+ P\_{IDW}(T\_{oo} < T < T\_o) \end{aligned} \tag{2c}$$

Following Gao et al. (2005) and Sui and Li (2005), the cloud budget (1c) and water vapor budget (1a) are mass integrated and their budgets can be, respectively, written as

$$P\_S - Q\_{CM} = Q\_{WWS} = Q\_{WVOLIT} + Q\_{WVDC} \tag{3}$$

$$Q\_{\rm WVT} + Q\_{\rm WVF} + Q\_{\rm WVE} = Q\_{\rm WVS} \tag{4}$$

where

74 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Nicholls, 1987; Fovell and Ogura, 1988; Tao and Simpson, 1989; McCumber et al., 1991; Tao et al., 1991; Liu et al., 1997; Grabowski et al., 1999; Wu et al., 1999; Li et al., 1999; Grabowski and Moncrieff, 2001; Wu, 2002; Grabowski, 2003; Gao et al., 2006; Ping et al., 2007). Wang et al. (2010) studied microphysical and radiative effects of ice clouds on a pre-summer heavy rainfall event over southern China during 3-8 June 2008 through the analysis of sensitivity experiments and found that microphysical and radiative effects of ice clouds play equally important roles in the pre-summer heavy rainfall event. The total exclusion of ice microphysics decreased model domain mean surface rain rate primarily through the weakened convective rainfall caused by the exclusion of radiative effects of ice clouds in the onset phase and through the weakened stratiform rainfall caused by the exclusion of ice microphysical effects in the development and mature phases, whereas it increased the mean rain rate through the enhanced convective rainfall caused by the exclusion of ice microphysical effects in the decay phase. Thus, effects of ice clouds on precipitation efficiencies are examined through the analysis of the pre-summer heavy rainfall event in this chapter. Precipitation efficiency is defined in section 2. Pre-summer heavy rainfall event, model, and sensitivity experiments are described in section 3. The control experiment is discussed in section 4. Radiative and microphysical effects of ice clouds on precipitation efficiency and associated rainfall processes are respectively examined in sections 5 and 6.

The budgets for specific humidity (*qv*), temperature (*T*), and cloud hydrometeor mixing ratio

*l Tr r Ts s Tgg qv*

 *Q LS LP cn v qv f* 18 (2b)

 

*o o v v qv q q Su w*

*x z* (1a)

> *<sup>Q</sup> <sup>Q</sup> <sup>w</sup> z cc*

(1b)

*p p*

 

(1c)

*SP P P P P P P qv CND DEP SDEP GDEP REVP MLTS MLTG* (2a)

'

 

( ) 1 1 ( ) *l l*

 

*q uq wq w q w q w q S txz z*

is potential temperature; *u* and *w* are zonal and vertical components of wind, respectively; is air density that is a function of height; cp is the specific heat of dry air at constant pressure; *Lv*, *Ls*, and *Lf* are latent heat of vaporization, sublimation, and fusion at *To*=0oC, respectively, *Ls*=*Lv*+*Lf*,; *Too*=-35 oC; and cloud microphysical processes in (2) can be found in Gao and Li (2008). *QR* is the radiative heating rate due to the convergence of net flux of solar and IR radiative fluxes. wTr, wTs, and wTg in (1c) are terminal velocities for raindrops, snow,

 

( ' ') *cn <sup>R</sup>*

(*ql*) in the 2D cloud resolving model used in this study can be written as

 ' ' ( ) *o o* <sup>1</sup> *v v v vv <sup>v</sup> q uq q q q uww w q t x x z zz*

' ' ' ' ( ) ( ) *<sup>o</sup> <sup>T</sup> o oo u uT u w <sup>w</sup> t x xz z* 

 

'' ' '

 

The conclusions are given in section 7.

where

**2. Definitions of precipitation efficiency** 

$$P\_S = P\_r + P\_s + P\_g \tag{5a}$$

$$P\_r = \rho w\_{Tr} q\_r \mid\_{z=0} \tag{5b}$$

$$P\_s = \overline{\rho w} \upsilon\_{Ts} q\_s \mid\_{z=0} \tag{5c}$$

$$P\_{\mathcal{S}} = \overline{\rho w}\_{T\mathcal{S}} q\_{\mathcal{S}}\big|\_{z=0\text{ }\prime} \tag{5d}$$

$$Q\_{\rm CM} = -\frac{\partial [q\_l]}{\partial t} \quad \text{[} \ln \frac{\partial \, q\_l}{\partial \mathbf{x}} \text{]} - [w \frac{\partial q\_l}{\partial z} \text{]} \tag{5e}$$

$$\left[\left[\left[Q\_{\rm INVOUT}\right]\right]\right] = \left[\left[P\_{\rm CND}\right]\right] + \left[\left[P\_{\rm DEP}\right]\right] + \left[\left[P\_{\rm GDEP}\right]\right] \tag{56}$$

$$Q\_{\rm MVIN} = -[P\_{\rm REVP}] - [P\_{\rm MLTG}] - [P\_{\rm MLTS}] \tag{5g}$$

$$Q\_{\rm VWT} = -\frac{\partial \left[ q\_{\rm v} \right]}{\partial t} \tag{5h}$$

$$\mathbf{Q}\_{\rm NVF} = -[\overline{u}^{o}\frac{\partial \overline{q}\_{\nu}}{\partial \mathbf{x}}] - [\overline{w}^{o}\frac{\partial \overline{q}\_{\nu}}{\partial z}] - [\frac{\partial (\overset{\cdot}{u}\overset{\cdot}{q}\_{\nu})}{\partial \mathbf{x}}] - [\overline{u}^{o}\frac{\partial \overline{q}\_{\nu}}{\partial \mathbf{x}}] - [\overline{w}^{o}\frac{\partial \overline{q}\_{\nu}}{\partial z}] - [\mathbf{w}^{\cdot}\frac{\partial \overline{q}\_{\nu}}{\partial z}] \tag{5i}$$

$$Q\_{\rm WVE} = E\_s \tag{5j}$$

Thermodynamic Aspects of Precipitation Efficiency 77

decrease of local hydrometeor concentration/hydrometeor convergence (*QCM* >0) or increase of local hydrometeor concentration/hydrometeor divergence (*QCM* <0). *PSWV* =

4

*<sup>P</sup> PEWV*

5

efficiency (*PEWV*) is exactly same to *LSPE2* defined by Sui et al. (2007).

**3. Pre-summer rainfall case, model, and experiments** 

*<sup>P</sup> PEH*

1

*i*

 

1

*i*

where *Qi*=(*QWVT, QWVF, QWVE, QCM*); *Si*=(*SHT, SHF, SHS, SLHLF, SRAD*); *H* is the Heaviside function, *H(F)*=1 when *F*>0, and *H(F)*=0 when *F* 0. Large-scale heat precipitation efficiency (*PEH*) is first introduced in this study, whereas large-scale water vapor precipitation

The pre-summer rainy season is the major rainy season over southern China, in which the rainfall starts in early April and reaches its peak in June (Ding, 1994). Although the rainfall is a major water resource in annual water budget, the torrential rainfall could occur during the pre-summer rainfall season and can lead to tremendous property damage and fatalities. In 1998, for instance, the torrential rainfall resulted in over 30 billion USD in damage and over 100 fatalities. Thus, many observational analyses and numerical modeling have been contributed to understanding of physical processes responsible for the development of presummer torrential rainfall (e.g., Krishnamurti et al., 1976; Tao and Ding, 1981; Wang and Li, 1982; Ding and Murakami, 1994; Simmonds et al., 1999). Recently, Wang et al. (2010) and Shen et al. (2011a, 2011b) conducted a series of sensitivity experiments of the pre-summer torrential rainfall occurred in the early June 2008 using 2D cloud-resolving model and studied effects of vertical wind shear, radiation, and ice clouds on the development of torrential rainfall. They found that these effects on torrential rainfall are stronger during the decay phase than during the onset and mature phases. During the decay phase of convection on 7 June 2008, the increase in model domain mean surface rain rate resulting from the exclusion of vertical wind shear is associated with the slowdown in the decrease of perturbation kinetic energy due to the exclusion of barotropic conversion from mean kinetic energy to perturbation kinetic energy. The increase in domain-mean rain rate resulting from the exclusion of cloud radiative effects is related to the enhancement of condensation and associated latent heat release as a result of strengthened radiative cooling. The increase in the mean surface rain rate is mainly associated with the increase in convective rainfall, which is in turn related to the local atmospheric change from moistening to drying. The increase in mean rain rate caused by the exclusion of ice clouds results from the increases in the mean net condensation and mean latent heat release caused by the strengthened mean radiative cooling associated with the removal of radiative effects of ice clouds. The increase

( ) *S*

*HQ Q*

() ( ) *S*

*HS S HQ Q*

*i i CM CM*

*i i*

(9a)

(9b)

From (8), precipitation efficiencies can be respectively defined as

*PSH* = *PS*.

Here, *PS* is precipitation rate, and in the tropics, *Ps*=0 and *Pg*=0, *PS=Pr*; *Es* is surface evaporation; [()] () *<sup>t</sup> b z z dz* , *zt* and *zb* are the heights of the top and bottom of the model atmosphere, respectively.

The heat budget (1b) is mass integrated and can be written as

$$\rm S\_{HT} + S\_{HF} + S\_{HS} + S\_{LHLF} + S\_{RAD} = Q\_{WVS} \tag{6}$$

where

$$S\_{HT} = \frac{c\_p}{L\_v} \frac{\partial \lbrack T \rbrack}{\partial t} \tag{7a}$$

$$S\_{HF} = \frac{c\_p}{L\_v} \left[ \frac{\partial}{\partial \mathbf{x}} (\overline{u}^o + \mathbf{u}^\cdot) \mathbf{T}^\cdot + \pi \overline{u}^o \frac{\partial}{\partial \mathbf{x}}^o + \pi \overline{w}^o \frac{\partial}{\partial z} (\overline{\theta} + \theta^\cdot) + \pi w^\cdot \frac{\partial \overline{\theta}}{\partial z} \right] \tag{7b}$$

$$S\_{HS} = -\frac{c\_p}{L\_v} H\_s \tag{7c}$$

$$S\_{LHLF} = -\frac{L\_f}{L\_v} < P\_{18} > \tag{7d}$$

$$S\_{RAD} = -\frac{1}{L\_v} < Q\_R > \tag{7e}$$

*Hs* is surface sensible heat flux.

The equations (3), (4), and (6) indicate that the surface rain rate (*PS*) is associated with favorable environmental water vapor and thermal conditions through cloud microphysical processes (*QWVOUT*+*QWVIN*). Following Gao and Li (2010), the cloud budget (3) and water vapor budget (4) are combined by eliminating *QWVOUT*+*QWVIN* to derive water vapor related surface rainfall equation (*PSWV*),

$$\mathbf{P\_{SW}} = \mathbf{Q\_{WVT}} \pm \mathbf{Q\_{WW}} \pm \mathbf{Q\_{WVE}} \pm \mathbf{Q\_{CM}} \tag{8a}$$

In (8a), the surface rain rate (*PSWV*) is associated with local atmospheric drying (*QWVT* >0)/moistening (*QWVT* <0), water vapor convergence (*QWVF* >0)/divergence (*QWVF* <0), surface evaporation (*QWVE*), and decrease of local hydrometeor concentration/hydrometeor convergence (*QCM* >0) or increase of local hydrometeor concentration/hydrometeor divergence (*QCM* <0). Similarly, the cloud budget (3) and heat budget (6) are combined by eliminating *QWVOUT*+*QWVIN* to derive thermally related surface rainfall equation (*PSH*),

$$\mathbf{P\_{SH}} = \mathbf{S\_{HT}} + \mathbf{S\_{HF}} + \mathbf{S\_{HS}} + \mathbf{S\_{LHEF}} + \mathbf{S\_{RAD}} + \mathbf{Q\_{CM}} \tag{8b}$$

In (8b), the surface rain rate (*PSH*) is related to local atmospheric warming (*SHT* >0)/cooling (*SHT* <0), heat divergence (*SHF* >0)/convergence (*SHF* <0), surface sensible heat (*SHS*), latent heat due to ice-related processes (*SLHLF*), radiative cooling (*SRAD* >0)/heating (*SRAD* <0), and

Here, *PS* is precipitation rate, and in the tropics, *Ps*=0 and *Pg*=0, *PS=Pr*; *Es* is surface

*SHT* <sup>=</sup> *<sup>p</sup>* [ ] *v c T L t* 

*S u uT u w w*

*SHS* <sup>=</sup> *<sup>p</sup>*

*SLHLF* = <sup>18</sup> *f v L*

*SRAD* <sup>=</sup> <sup>1</sup>

*L*

*v <sup>Q</sup> <sup>L</sup>*

The equations (3), (4), and (6) indicate that the surface rain rate (*PS*) is associated with favorable environmental water vapor and thermal conditions through cloud microphysical processes (*QWVOUT*+*QWVIN*). Following Gao and Li (2010), the cloud budget (3) and water vapor budget (4) are combined by eliminating *QWVOUT*+*QWVIN* to derive water vapor related

PSWV=QWVT+QWVF+QWVE+QCM (8a)

PSH=SHT+SHF+SHS+SLHLF+SRAD+QCM. (8b)

In (8b), the surface rain rate (*PSH*) is related to local atmospheric warming (*SHT* >0)/cooling (*SHT* <0), heat divergence (*SHF* >0)/convergence (*SHF* <0), surface sensible heat (*SHS*), latent heat due to ice-related processes (*SLHLF*), radiative cooling (*SRAD* >0)/heating (*SRAD* <0), and

In (8a), the surface rain rate (*PSWV*) is associated with local atmospheric drying (*QWVT* >0)/moistening (*QWVT* <0), water vapor convergence (*QWVF* >0)/divergence (*QWVF* <0), surface evaporation (*QWVE*), and decrease of local hydrometeor concentration/hydrometeor convergence (*QCM* >0) or increase of local hydrometeor concentration/hydrometeor divergence (*QCM* <0). Similarly, the cloud budget (3) and heat budget (6) are combined by

eliminating *QWVOUT*+*QWVIN* to derive thermally related surface rainfall equation (*PSH*),

*<sup>o</sup> <sup>p</sup> o oo*

' ' ' ' [( ) () ]

 

*s v c H L*

*P*

*R*

*L x x z z* 

*dz* , *zt* and *zb* are the heights of the top and bottom of the model

*SHT* + *SHF* + *SHS* + *SLHLF* + *SRAD* =*QWVS* (6)

 

(7b)

(7a)

 

(7c)

(7d)

(7e)

evaporation; [()] () *<sup>t</sup>*

atmosphere, respectively.

where

*b z z* 

*HF*

*Hs* is surface sensible heat flux.

surface rainfall equation (*PSWV*),

*v c*

The heat budget (1b) is mass integrated and can be written as

decrease of local hydrometeor concentration/hydrometeor convergence (*QCM* >0) or increase of local hydrometeor concentration/hydrometeor divergence (*QCM* <0). *PSWV* = *PSH* = *PS*.

From (8), precipitation efficiencies can be respectively defined as

$$PEWV = \frac{P\_S}{\sum\_{i=1}^{4} H(Q\_i)Q\_i} \tag{9a}$$

$$PEH = \frac{P\_S}{\sum\_{i=1}^{5} H(S\_i)S\_i + H(Q\_{CM})Q\_{CM}}\tag{9b}$$

where *Qi*=(*QWVT, QWVF, QWVE, QCM*); *Si*=(*SHT, SHF, SHS, SLHLF, SRAD*); *H* is the Heaviside function, *H(F)*=1 when *F*>0, and *H(F)*=0 when *F* 0. Large-scale heat precipitation efficiency (*PEH*) is first introduced in this study, whereas large-scale water vapor precipitation efficiency (*PEWV*) is exactly same to *LSPE2* defined by Sui et al. (2007).

### **3. Pre-summer rainfall case, model, and experiments**

The pre-summer rainy season is the major rainy season over southern China, in which the rainfall starts in early April and reaches its peak in June (Ding, 1994). Although the rainfall is a major water resource in annual water budget, the torrential rainfall could occur during the pre-summer rainfall season and can lead to tremendous property damage and fatalities. In 1998, for instance, the torrential rainfall resulted in over 30 billion USD in damage and over 100 fatalities. Thus, many observational analyses and numerical modeling have been contributed to understanding of physical processes responsible for the development of presummer torrential rainfall (e.g., Krishnamurti et al., 1976; Tao and Ding, 1981; Wang and Li, 1982; Ding and Murakami, 1994; Simmonds et al., 1999). Recently, Wang et al. (2010) and Shen et al. (2011a, 2011b) conducted a series of sensitivity experiments of the pre-summer torrential rainfall occurred in the early June 2008 using 2D cloud-resolving model and studied effects of vertical wind shear, radiation, and ice clouds on the development of torrential rainfall. They found that these effects on torrential rainfall are stronger during the decay phase than during the onset and mature phases. During the decay phase of convection on 7 June 2008, the increase in model domain mean surface rain rate resulting from the exclusion of vertical wind shear is associated with the slowdown in the decrease of perturbation kinetic energy due to the exclusion of barotropic conversion from mean kinetic energy to perturbation kinetic energy. The increase in domain-mean rain rate resulting from the exclusion of cloud radiative effects is related to the enhancement of condensation and associated latent heat release as a result of strengthened radiative cooling. The increase in the mean surface rain rate is mainly associated with the increase in convective rainfall, which is in turn related to the local atmospheric change from moistening to drying. The increase in mean rain rate caused by the exclusion of ice clouds results from the increases in the mean net condensation and mean latent heat release caused by the strengthened mean radiative cooling associated with the removal of radiative effects of ice clouds. The increase

Thermodynamic Aspects of Precipitation Efficiency 79

Fig. 1. Temporal and vertical distribution of (a) vertical velocity (cm s-1) and (b) zonal wind (m s-1) from 0200 LST 3 June – 0200 LST 8 June 2008. The data are averaged in a rectangular box of 108-116oE, 21-22oN from NCEP/GDAS data. Ascending motion in (a) and westerly

In the control experiment (C), the model is integrated with the initial vertical profiles of temperature and specific humidity from NCEP/GDAS at 0200 LST 3 June 2008. The model is integrated with the initial conditions and constant large-scale forcing at 0200 LST 3 June for 6 hours during the model spin-up period and the 6-hour model data are not used for analysis. The comparison in surface rain rate between the simulation and rain gauge observation averaged from 17 stations over southern Guangdong and Guangxi reveals a fair agreement with a gradual increase from 3-6 June and a rapid decrease from 6-7 June (Fig. 2). Their RMS difference (0.97 mm h-1) is significantly smaller than the standard derivations of simulated (1.22 mm h-1) and observed (1.26 mm h-1) rain rates. The differences in surface rain rate between the simulation and observation can reach 2 mm h-1, as seen in the previous studies (e.g., Li et al., 1999; Xu et al., 2007; Wang et al., 2009). The differences may partially be from the comparison of small hourly local sampling of rain gauge observations over 35% of model domain over land and no rain gauge observations over 65% of model domain over ocean with large model domain averages of model data in the control experiment with imposed 6-hourly large-scale forcing. The convection may be affected by land-ocean contrast and orography; these effects are included in the large-

wind in (b) are shaded.

scale forcing imposed in the model.

in mean rain rate caused by the removal of radiative effects of water clouds corresponds to the increase in the mean net condensation.

The pre-summer torrential rainfall event studied by Wang et al. (2010) and Shen et al. (2011a, 2011b) will be revisited to examine the thermodynamic aspects of precipitation efficiency and effects of ice clouds on precipitation efficiency. The cloud-resolving model (Soong and Ogura, 1980; Soong and Tao, 1980; Tao and Simpson, 1993) used in modeling the pre-summer torrential rainfall event in Wang et al. (2010) is the 2D version of the model (Sui et al., 1994, 1998) that was modified by Li et al. (1999). The model is forced by imposed large-scale vertical velocity and zonal wind and horizontal temperature and water vapor advections, which produces reasonable simulation through the adjustment of the mean thermodynamic stability distribution by vertical advection (Li et al., 1999). The modifications by Li et al. (1999) include: (1) the radius of ice crystal is increased from m (Hsie et al., 1980) to 100m (Krueger et al., 1995) in the calculation of growth of snow by the deposition and riming of cloud water, which yields a significant increase in cloud ice; (2) the mass of a natural ice nucleus is replaced by an average mass of an ice nucleus in the calculation of the growth of ice clouds due to the position of cloud water; (3) the specified cloud single scattering albedo and asymmetry factor are replaced by those varied with cloud and environmental thermodynamic conditions. Detailed descriptions of the model can be found in Gao and Li (2008). Briefly, the model includes prognostic equations for potential temperature and specific humidity, prognostic equations for hydrometeor mixing ratios of cloud water, raindrops, cloud ice, snow, and graupel, and perturbation equations for zonal wind and vertical velocity. The model uses the cloud microphysical parameterization schemes (Lin et al., 1983; Rutledge and Hobbs, 1983, 1984; Tao et al., 1989; Krueger et al., 1995) and solar and thermal infrared radiation parameterization schemes (Chou et al., 1991, 1998; Chou and Suarez, 1994). The model uses cyclic lateral boundaries, and a horizontal domain of 768 km with 33 vertical levels, and its horizontal and temporal resolutions are 1.5 km and 12 s, respectively.

The data from Global Data Assimilation System (GDAS) developed by the National Centers for Environmental Prediction (NCEP), National Oceanic and Atmospheric Administration (NOAA), USA are used to calculate the forcing data for the model over a longitudinally oriented rectangular area of 108-116oE, 21-22oN over coastal areas along southern Guangdong and Guangxi Provinces and the surrounding northern South China Sea. The horizontal and temporal resolutions for NCEP/GDAS products are 1ox1o and 6 hourly, respectively. The model is imposed by large-scale vertical velocity, zonal wind (Fig. 1), and horizontal temperature and water vapor advections (not shown) averaged over 108-116oE, 21-22oN. The model is integrated from 0200 Local Standard Time (LST) 3 June to 0200 LST 8 June 2008 during the pre-summer heavy rainfall. The surface temperature and specific humidity from NCEP/GDAS averaged over the model domain are uniformly imposed on each model grid to calculate surface sensible heat flux and evaporation flux. The 6-hourly zonally-uniform large-scale forcing data are linearly interpolated into 12-s data, which are uniformly imposed zonally over model domain at each time step. The imposed large-scale vertical velocity shows the gradual increase of upward motions from 3 June to 6 June. The maximum upward motion of 18 cm s-1 occurred around 9 km in the late morning of 6 June. The upward motions decreased dramatically on 7 June. The lower-tropospheric westerly winds of 4 - 12 m s-1 were maintained during the rainfall event.

in mean rain rate caused by the removal of radiative effects of water clouds corresponds to

The pre-summer torrential rainfall event studied by Wang et al. (2010) and Shen et al. (2011a, 2011b) will be revisited to examine the thermodynamic aspects of precipitation efficiency and effects of ice clouds on precipitation efficiency. The cloud-resolving model (Soong and Ogura, 1980; Soong and Tao, 1980; Tao and Simpson, 1993) used in modeling the pre-summer torrential rainfall event in Wang et al. (2010) is the 2D version of the model (Sui et al., 1994, 1998) that was modified by Li et al. (1999). The model is forced by imposed large-scale vertical velocity and zonal wind and horizontal temperature and water vapor advections, which produces reasonable simulation through the adjustment of the mean thermodynamic stability distribution by vertical advection (Li et al., 1999). The modifications by Li et al. (1999) include: (1) the radius of ice crystal is increased from m (Hsie et al., 1980) to 100m (Krueger et al., 1995) in the calculation of growth of snow by the deposition and riming of cloud water, which yields a significant increase in cloud ice; (2) the mass of a natural ice nucleus is replaced by an average mass of an ice nucleus in the calculation of the growth of ice clouds due to the position of cloud water; (3) the specified cloud single scattering albedo and asymmetry factor are replaced by those varied with cloud and environmental thermodynamic conditions. Detailed descriptions of the model can be found in Gao and Li (2008). Briefly, the model includes prognostic equations for potential temperature and specific humidity, prognostic equations for hydrometeor mixing ratios of cloud water, raindrops, cloud ice, snow, and graupel, and perturbation equations for zonal wind and vertical velocity. The model uses the cloud microphysical parameterization schemes (Lin et al., 1983; Rutledge and Hobbs, 1983, 1984; Tao et al., 1989; Krueger et al., 1995) and solar and thermal infrared radiation parameterization schemes (Chou et al., 1991, 1998; Chou and Suarez, 1994). The model uses cyclic lateral boundaries, and a horizontal domain of 768 km with 33 vertical levels, and its horizontal and temporal resolutions are 1.5

The data from Global Data Assimilation System (GDAS) developed by the National Centers for Environmental Prediction (NCEP), National Oceanic and Atmospheric Administration (NOAA), USA are used to calculate the forcing data for the model over a longitudinally oriented rectangular area of 108-116oE, 21-22oN over coastal areas along southern Guangdong and Guangxi Provinces and the surrounding northern South China Sea. The horizontal and temporal resolutions for NCEP/GDAS products are 1ox1o and 6 hourly, respectively. The model is imposed by large-scale vertical velocity, zonal wind (Fig. 1), and horizontal temperature and water vapor advections (not shown) averaged over 108-116oE, 21-22oN. The model is integrated from 0200 Local Standard Time (LST) 3 June to 0200 LST 8 June 2008 during the pre-summer heavy rainfall. The surface temperature and specific humidity from NCEP/GDAS averaged over the model domain are uniformly imposed on each model grid to calculate surface sensible heat flux and evaporation flux. The 6-hourly zonally-uniform large-scale forcing data are linearly interpolated into 12-s data, which are uniformly imposed zonally over model domain at each time step. The imposed large-scale vertical velocity shows the gradual increase of upward motions from 3 June to 6 June. The maximum upward motion of 18 cm s-1 occurred around 9 km in the late morning of 6 June. The upward motions decreased dramatically on 7 June. The lower-tropospheric westerly

winds of 4 - 12 m s-1 were maintained during the rainfall event.

the increase in the mean net condensation.

km and 12 s, respectively.

Fig. 1. Temporal and vertical distribution of (a) vertical velocity (cm s-1) and (b) zonal wind (m s-1) from 0200 LST 3 June – 0200 LST 8 June 2008. The data are averaged in a rectangular box of 108-116oE, 21-22oN from NCEP/GDAS data. Ascending motion in (a) and westerly wind in (b) are shaded.

In the control experiment (C), the model is integrated with the initial vertical profiles of temperature and specific humidity from NCEP/GDAS at 0200 LST 3 June 2008. The model is integrated with the initial conditions and constant large-scale forcing at 0200 LST 3 June for 6 hours during the model spin-up period and the 6-hour model data are not used for analysis. The comparison in surface rain rate between the simulation and rain gauge observation averaged from 17 stations over southern Guangdong and Guangxi reveals a fair agreement with a gradual increase from 3-6 June and a rapid decrease from 6-7 June (Fig. 2). Their RMS difference (0.97 mm h-1) is significantly smaller than the standard derivations of simulated (1.22 mm h-1) and observed (1.26 mm h-1) rain rates. The differences in surface rain rate between the simulation and observation can reach 2 mm h-1, as seen in the previous studies (e.g., Li et al., 1999; Xu et al., 2007; Wang et al., 2009). The differences may partially be from the comparison of small hourly local sampling of rain gauge observations over 35% of model domain over land and no rain gauge observations over 65% of model domain over ocean with large model domain averages of model data in the control experiment with imposed 6-hourly large-scale forcing. The convection may be affected by land-ocean contrast and orography; these effects are included in the largescale forcing imposed in the model.

Thermodynamic Aspects of Precipitation Efficiency 81

To investigate effects of ice clouds on precipitation efficiency, two sensitivity experiments are examined in this study. Experiment CNIR is identical to C except that the mixing ratios of ice hydrometeor are set to zero in the calculation of radiation. Experiment CNIR is compared with C to study radiative effects of ice clouds on rainfall responses to the largescale forcing. Experiment CNIM is identical to C except in CNIM ice clouds (the ice hydrometeor mixing ratio and associated microphysical processes) are excluded. The comparison between CNIM and CNIR reveals impacts of the removal of microphysical efficient of ice clouds on rainfall responses to the large-scale forcing in the absence of radiative effects of ice clouds. The hourly model simulation data are used in the following

Model domain mean surface rain rate starts on 3 June 2008 with the magnitude of about 1 mm h-1 (Fig. 3), which corresponds to the weak upward motions with a maximum of 2 cm s-1 at 6-8 km (Fig. 1a). The rain rate increases to 2 mm h-1 as the upward motions increase up to over 6 cm s-1 on 4 June. When the upward motions weaken in the evening of 4 June and a weak downward motion occurs near the surface, the mean rainfall vanishes. As upward motions pick their strengths on 5 June, the mean rain rate intensifies (over 2 mm h-1). The mean rainfall reaches its peak on 6 June (over 4 mm h-1) as the upward motions have a maximum of over 20 cm s-1. The upward motions rapidly weaken on 7 June, which leads to the significant reduction in the mean rainfall. Thus, four days (4, 5, 6, and 7 June) are defined as the onset, development, mature, and decay phases of the rainfall event, respectively. During 3-6 June, the mean rainfall is mainly associated with the mean water vapor convergence (*QWVF*>0) in water vapor related surface rainfall budget and the mean heat divergence (*SHF*>0) in thermally related surface rainfall budget. Local atmospheric drying (*QWVT*>0) and moistening (*QWVT*<0) occur while the mean local atmospheric cooling (*SHT*<0) prevails. The mean hydrometeor loss/convergence (*QCM*) has small hourly fluctuations. The mean radiative heating during the daytime and mean radiative cooling during the nighttime have the much smaller magnitudes than the mean heat divergence and the mean local heat change do in thermally related surface rainfall budget. On 7 June, the mean water vapor related surface rainfall budget shows that the rainfall is associated with local atmospheric drying while water vapor divergence prevails. The mean thermally related surface rainfall budget reveals that the rainfall is related to heat divergence while the

The calculation of precipitation efficiency using model domain mean model simulation data shows that *PEWV* generally is higher than *PWH* (Fig. 4a) because the rainfall source from the mean water vapor convergence in water vapor related surface rainfall budget is weaker than the rainfall source from the mean heat divergence in thermally related surface rainfall budget (Fig. 3). This suggests that the precipitation system is more efficient in the consumption of rainfall source from water vapor than in the consumption of the rainfall source from heat. The root-mean-squared (RMS) difference between *PEWV* and *PEH* is 24.4%. Both *PEWV* and *PEH* generally increase as surface rain rate increases (Fig. 5a). This indicates that the precipitation system generally is more efficient for high surface rain rates than for low surface rain rates. The ranges of *PEWV* and *PEH* are smaller when surface rain rate is higher than 3 mm h-1 (70-100%) than when surface rain

discussions of this study.

**4. The control experiment: C** 

heat divergence cools local atmosphere.

rate is lower than 3 mm h-1 (0-100%).

Fig. 2. Time series of surface rain rates (mm h-1) simulated in the control experiment (solid) and from rain gauge observation (dash).

Fig. 3. Time series of model domain means of (a) *PSWV* (dark solid), *QWVT* (light solid), *QWVF* (short dash), *QWVE* (dot), *QCM* (dot dash), and (b) *PSH* (dark solid), *SHT* (light solid), *SHF* (short dash), *SHS* (dot), *SLHLF* (long short dash), *SRAD* (long dash), and *QCM* (dot dash) in C. Unit is mm h-1.

Fig. 2. Time series of surface rain rates (mm h-1) simulated in the control experiment (solid)

Fig. 3. Time series of model domain means of (a) *PSWV* (dark solid), *QWVT* (light solid), *QWVF* (short dash), *QWVE* (dot), *QCM* (dot dash), and (b) *PSH* (dark solid), *SHT* (light solid), *SHF* (short dash), *SHS* (dot), *SLHLF* (long short dash), *SRAD* (long dash), and *QCM* (dot dash) in C. Unit is

and from rain gauge observation (dash).

mm h-1.

To investigate effects of ice clouds on precipitation efficiency, two sensitivity experiments are examined in this study. Experiment CNIR is identical to C except that the mixing ratios of ice hydrometeor are set to zero in the calculation of radiation. Experiment CNIR is compared with C to study radiative effects of ice clouds on rainfall responses to the largescale forcing. Experiment CNIM is identical to C except in CNIM ice clouds (the ice hydrometeor mixing ratio and associated microphysical processes) are excluded. The comparison between CNIM and CNIR reveals impacts of the removal of microphysical efficient of ice clouds on rainfall responses to the large-scale forcing in the absence of radiative effects of ice clouds. The hourly model simulation data are used in the following discussions of this study.
