4.5 Prognostic Condensate and Precipitation Parameterization

4.5.1 Introductory comments

The parameterization of non-convective cloud processes in CAM 3.0 is described in Rasch and Kristjánsson [144] and Zhang et al. [200]. The original formulation is introduced in Rasch and Kristjánsson [144]. Revisions to the parameterization to deal more realistically with the treatment of the condensation and evaporation under forcing by large scale processes and changing cloud fraction are described in Zhang et al. [200]. The equations used in the formulation are discussed here. The papers contain a more thorough description of the formulation and a discussion of the impact on the model simulation.

The formulation for cloud condensate combines a representation for condensation and evaporation with a bulk microphysical parameterization closer to that used in cloud resolving models. The parameterization replaces the diagnosed liquid water path of CCM3 with evolution equations for two additional predicted variables: liquid and ice phase condensate. At one point during each time step, these are combined into a total condensate and partitioned according to temperature (as described in section 4.5.3), but elsewhere function as independent quantities. They are affected by both resolved (e.g. advective) and unresolved (e.g. convective, turbulent) processes. Condensate can evaporate back into the environment or be converted to a precipitating form depending upon its in-cloud value and the forcing by other atmospheric processes. The precipitate may be a mixture of rain and snow, and is treated in diagnostic form, i.e. its time derivative has been neglected.

The parameterization calculates the condensation rate more consistently with the change in fractional cloudiness and in-cloud condensate than the previous CCM3 formulation. Changes in water vapor and heat in a grid volume are treated consistently with changes to cloud fraction and in-cloud condensate. Condensate can form prior to the onset of grid-box saturation and can require a significant length of time to convert (via the cloud microphysics) to a precipitable form. Thus a substantially wider range of variation in condensate amount than in the CCM3 is possible.

The new parameterization adds significantly to the flexibility in the model and to the range of scientific problems that can be studied. This type of scheme is needed for quantitative treatment of scavenging of atmospheric trace constituents and cloud aqueous and surface chemistry. The addition of a more realistic condensate parameterization closely links the radiative properties of the clouds and their formation and dissipation. These processes must be treated for many problems of interest today (e.g. anthropogenic aerosol-climate interactions).

The parameterization has two components: 1) a macroscale component that describes the exchange of water substance between the condensate and the vapor phase and the associated temperature change arising from that phase change Zhang et al. [200]; and 2) a bulk microphysical component that controls the conversion from condensate to precipitate [144]. These components are discussed in the following two sections.

4.5.2 Description of the macroscale component

As in Sundqvist [169] and Rasch and Kristjánsson [144], the controlling equations for the water vapor mixing ratio, temperature, and total cloud condensate are written as

$$\frac{{\partial q}}{{\partial t}} = A_q - Q_{} + E_r$$ (4.109)

$$\frac{{\partial T}}{{\partial t}} = A_T + \frac{L}{{c_p }}(Q - E_r )$$ (4.110)

$$\frac{{\partial l}}{{\partial t}} = A_l + Q - R_l ,$$ (4.111)

where $A_q$, $A_T$, and $A_l$ are tendencies of water vapor, temperature, and cloud water from processes other than large-scale condensation and evaporation of cloud and rain water. $A_q$, $A_T$ and $A_l$ include advective, expansive, radiative, turbulent, and convective tendencies. The convective tendencies include evaporation of convective cloud and convective precipitation. For simplicity, all these processes are collectively called advective tendencies. They are assumed to be uniform across the whole model grid cell, although this assumption can be relaxed as discussed in Zhang et al. [200]. $Q$ is the grid-averaged net stratiform condensation of cloud meteors (condensation minus evaporation). $E_r$ is the grid-averaged evaporative rate of rain and snow. $R_l$ is the conversion rate of cloud water to rain and snow. This section is devoted to the determination of the term $Q$ in equations (4.109)-(4.111).

The controlling equation of relative humidity $U$, when written on a pressure surface, can be derived from (4.109) and (4.110) as

$$\frac{{\partial U}}{{\partial t}} = \alpha \frac{{\partial q}}{{\partial t}} - \beta \frac{{\partial T}}{{\partial t}}$$ (4.112)

$$= \alpha A_q - \beta A_T - \gamma (Q - E_r )$$ (4.113)
where

$$\alpha = \frac{1}{{q_s }},$$ (4.114)

$$\beta = \frac{q}{{q_s^2 }}\frac{{\partial q_s }}{{\partial T}},$$ (4.115)

$$\gamma = \alpha + \frac{L}{{c_p }}\beta .$$ (4.116)

Note that $\alpha$, $\beta$, and $\gamma$ are all positive. They can be viewed as the efficiencies of moisture advection, cold advection, and net evaporation in changing the relative humidity $U$. Changing $U$ can alter the fractional cloud cover. As in Sundqvist [169] and Rasch and Kristjánsson [144], ice saturation is not separately considered here; rather, it is approximated by a weighted average $q_s (T)$ of the saturation mixing ratios over ice and water. The dependence of $q_s$ on pressure is not made explicit since pressure enters into the calculation only as a parameter.

Equations (4.109)-(4.113) are applicable on both the grid scale and sub-grid scale as long as $Q$, $E_r$, and $R_l$ are appropriately defined. In the following, a hat denotes variables in the cloudy portion of a grid box to distinguish them from variables of the whole grid box, and $\cfrac$ denotes the fractional cloud coverage. For the portion of the grid box that is cloudy before and after the calculation of fractional condensation (i.e., the cloudy area that does not experience clear-cloudy conversion), equation (4.113) becomes

$$\alpha \hat A_q - \hat \beta \hat A_T - \hat \gamma \hat Q = 0.$$

This follows from the assumption that $E_r = 0$ and $U = 1$ in the saturated cloud interior. Thus the condensation rate in this portion of the grid box is

$$\hat Q = \frac{{\alpha \hat A_q - \hat \beta \hat A_T }}{{\hat \gamma }}$$ (4.117)

and the in-cloud condensate equation becomes

$$\frac{{\partial \hat l}}{{\partial t}} = \hat A_l + \frac{{\alpha \hat A_q - \hat \beta \hat A_T }}{{\hat \gamma }} - \hat R_l .$$ (4.118)

Since the total cloud water can be written as $l = \cfrac\hat l$, it follows that

$$\frac{{\partial l}}{{\partial t}} = \cfrac\frac{{\partial \hat l}}{{\partial t}} + \hat l^* \frac{{\partial \cfrac }}{{\partial t}}$$ (4.119)

The symbol $\hat l^*$ denotes the mean cloud condensate of the newly formed or dissipated clouds within a time step. The first term on the right hand side of the above equation represents the evolution of cloud water within existing clouds, and the second term represents the change in cloud water associated with expansion and contraction of cloud boundaries. Theoretically, newly formed or dissipated clouds should have zero cloud water content, except for detrained cloud from cumulus. However, because of the finite time step in the integration of the cloud water equation, the second term may be nonzero. Rasch and Kristjánsson [144] set $\hat l^* = \hat l$, and the same closure is used in CAM 3.0. Inserting (4.118) and the relations $R_l = \cfrac\hat R_l$ as well as $A_T = \hat A_T$, $A_q = \hat A_q$, and $A_l = \hat A_l$ into (4.111) yields:

$$\hat l^* \frac{{\partial \cfrac }}{{\partial t}} = (1 - \cfrac )A_l + Q - \cfrac (\frac{{\alpha A_q - \hat \beta A_T }}{{\hat \gamma }})$$ (4.120)

This equation states that the condensation rate is linked with fractional cloudiness change as required by the total water budget. Equation (4.120) is not integrated in the present formulation. Instead, it is used to calculate the condensation rate as follows.

The fractional cloud cover and grid-scale relative humidity are related by

$$\cfrac = \cfrac (U,b)$$ (4.121)

where $b$ denotes a generic variable describing vertical stability, local Richardson number, cumulus mass flux, etc. The term $b$ varies with space and time. This equation is assumed to be valid when the relative humidity $U$ is larger than a threshold value $U_{00}$, which is the minimum grid-scale relative humidity at which clouds are present.

Taking partial derivatives of the equation (4.121) with respect to time gives

$$\frac{{\partial \cfrac }}{{\partial t}} = \frac{{\partial \cfrac ... ...t}} + \frac{{\partial \cfrac }}{{\partial b}}\frac{{\partial b}}{{\partial t}}$$

With the definitions

$$F_a = \frac{{\partial \cfrac }}{{\partial U}}$$ (4.122)
and

$$F_b = [(\frac{{\partial \cfrac }}{{\partial b}})/(\frac{{\partial \cfrac }}{{\partial U}})]\frac{{\partial b}}{{\partial t}} ,$$ (4.123)

the time derivative of cloud amount becomes

$$F_a^{ - 1} \frac{{\partial \cfrac }}{{\partial t}} = \frac{{\partial U}}{{\partial t}} + F_b$$ (4.124)

It is assumed that $F_a$ and $F_b$ can be calculated without the knowledge of the condensation rate. Substituting the relative humidity equation (4.113) into equation (4.124) yields

$$F_a^{ - 1} \frac{{\partial \cfrac }}{{\partial t}} = \alpha A_q - \beta A_T - \gamma (Q - E_r) + F_b$$ (4.125)

Eliminating ${\partial \cfrac }/{\partial t}$ between (4.120) and (4.125) gives

$$Q = c_q A_q - c_T A_T - c_l A_l + c_r E_r + \sigma \hat l^* F_b$$ (4.126)
with

$$c_q = \frac{\alpha }{{\hat \gamma }}\cfrac + \left(1 - \frac{\gamma }{{\hat \gamma }}\cfrac\right)\sigma \alpha \hat l^*$$ (4.127)

$$c_T = \frac{{\hat \beta }}{{\hat \gamma }}\cfrac + \left(1 - \frac{\g... ...}{{\hat \gamma }}\frac{{\hat \beta }}{\beta }\cfrac\right)\sigma \beta \hat l^*$$ (4.128)

$$c_l = \left(1 - \cfrac\right)\sigma F_a^{ - 1}$$ (4.129)

$$c_r = \sigma \gamma \hat l^*$$ (4.130)
where

$$\sigma = \frac{1}{{F_a^{ - 1} + \gamma \hat l^* }} .$$ (4.131)

All coefficient variables are positive, and all are non-dimensional except for $C_T$ and $\beta$ which have units of 1/K. Equation (4.126) is valid when $U \ge U_{00}$. The terms in the equation have the following physical interpretation. Moist advection (positive $A_q$) and cold advection (negative $A_T$) produce condensation. Evaporation of rain/snow water (positive $E_r$) also produces cloud condensation because it changes the mean relative humidity, thus increasing cloud amount and cloud water. Import of cloud water (positive $A_l$) leads to evaporation. The reason is that it increases cloud fraction, thus requiring a higher clear-sky relative humidity which has to be generated by evaporation. The increase of cloud fraction from a non-water source through $F_b$, however, requires condensation.

To evaluate $F_a$, the cloud routine is called twice each time step with relative humidity perturbed by one percent (indicated by a $*$ superscript) while holding all other variables in the model fixed. Thus,

$$F_a \approx \frac{{\Delta \cfrac }}{{\Delta U}} = \frac{{\cfrac ^* - \cfrac }}{{U^* - U}}.$$

In this implementation, all $b$ variables are assumed fixed in the stratiform condensation calculation, and therefore $F_b = 0$. Since a top-hat distribution is adopted for the cloud water distribution, $\hat l_{}^* = \hat l$.

The effects of convection on cloud cover are introduced through the convective tendencies. Detrainment of cloud water from the Zhang and McFarlane [199] convection scheme is used as input in the calculation of $A_l$, $A_T$ and $A_q$. In the original version of the Zhang and McFarlane [199] parameterization, the detrained cloud water from convection was assumed to evaporate.

The calculation is carried out by categorizing each model grid into one of four cases:

• If $U = 1$, $Q$ is calculated from (4.117);
• if $1 > U \ge U_{00}$, $Q$ calculated from (4.126);
• if $U < U_{00}$ but $l > 0$, $Q = - l$; and
• if $U < U_{00}$ and $l = 0$, $Q = 0$.

The use of the threshold relative humidity follows from equation (4.121).

4.5.3 Description of the microscale component

The condensation process has been determined by forcing terms and closure assumptions described in the previous subsection rather than an approach in which a supersaturation is calculated and CCN can nucleate and grow. Therefore the whole microphysical calculation reduces to modeling the process of conversion of cloud condensate to precipitation. The microscale component of the parameterization determines the evaporation $E_R$ and conversion of condensate to precipitate $R_l$.

The formulation follows closely the bulk microphysical formulations used in smaller scale cloud resolving models rather than those of Sundqvist [169]. A method based upon cloud resolving models makes an explicit connection between the formation of precipitate and individual physical quantities like droplet or crystal number, shape of size distribution of precipitate, etc. It also separates the various processes contributing to precipitation more strongly, and makes diagnosis more straightforward. Because these quantities must represent an ensemble of cloud types in any given region (or grid volume) the new formulation still involves gross approximations, but it is much easier to control the parameterizations and understand their individual impact when the processes are isolated from each other.

As in Sundqvist [169], the parameterization is expressed in terms of a single predicted variable representing total suspended condensate. Within the parameterization, however, there are four types of condensate expressed as mixing ratios: a liquid and ice phase for suspended condensate with minimal fall speed ($q_l$ and $q_i$) and a liquid and ice phase for falling condensate, i.e. precipitation ($q_r$ and $q_s$). Currently, only the suspended condensates ($q_l$ and $q_i$) are integrated in time; the other quantities are diagnosed as described below.

Before beginning the microphysical calculation, the total condensate is decomposed into liquid and ice phases assuming the fraction of ice is

$$f_i = \frac{T - T_{max}}{T_{min}-T_{max}}, \quad T_{min} \leq T \leq T_{max}$$ (4.132)

with $f_i(T<T_{min}) =1$ and $f_i(T>T_{max}) =0$. $T$ is the grid volume temperature. The bounds are adjustable constants with current settings $T_{min}=-40^\circ$ C and $T_{max}=-10^\circ$ C. Observations and more detailed microphysical models show a broad range of ratios of liquid to ice in clouds, and it is difficult to be certain of an appropriate range for this parameter.

Liquid and ice mass mixing ratios ($\ell$ and $I$) are independently advected, diffused, and transported by convection. The detrained liquid from the ZM convection is all added to the cloud liquid, since the ZM scheme does not have an ice phase. After the convection and sedimentation (see below), the liquid and ice are recalculated from the total cloud condensate

$$\ell_{n\prime} = (\ell_n + I_n)(1-f_i)$$ (4.133)

$$I_{n\prime} = (\ell_n + I_n)f_i .$$ (4.134)

The heating due to the change in cloud ice is

$$Q^k = L_f \frac{I_n - I_{n\prime}}{\delta t}.$$ (4.135)

The stratiform cloud condensate tendency is computed and partitioned according to $f_i$. The excess heating due to cloud ice production instead of cloud liquid production is included with the evaporation and freezing of precipitation below.

The in-cloud liquid water mixing ratio is

$$\hat q_l = (1-f_i) q_c / \cfrac$$ (4.136)

and the in-cloud ice water mixing ratio is assumed to be

$$\hat q_i = (f_i)q_c / \cfrac .$$ (4.137)

The grid volume mean quantities have been converted to in-cloud quantities by dividing the mean mixing ratios by the cloud fraction.

The evaporation of precipitation is computed for each source of precipitation using the same expressions, following Sundqvist [169]. The precipitate falling from above can be a mixture of snow and rain. The flux of total precipitation $F^{k+}$ on each interface is

$$F^{k+} = F^{k-} + \frac{\delta^k p}{g} (P^k - E^k)$$ (4.138)

where $P^k$ and $E^k$ are precipitation production and evaporation, respectively. $P^k$ is determined by the convection or stratiform microphysics routines and

$$E^k = k_e (1 - c^k) \left(1-\min(1,\frac{q^k}{q_*^k})\right) (F^{k-})^{1/2}$$ (4.139)

where $k_e$ is an adjustable constant and $c^k$ is the fractional cloud area. The $(1 - c^k)$ factor represents a random overlap assumption; precipitation falling into the existing cloud in a layer does not evaporate. For stratiform precipitation, $k_e=1\times 10^{-5}$, while for convective precipitation, $k_e$ is considered to be an adjustable parameter and is specified according to the table in appendix C.

Two bounds are applied to $E^k$:

1. $E^k \leq \frac{q_*^k - q^k}{\delta t}$, to prevent supersaturation;
2. $E^k \leq F^{k-}\frac{g}{\delta^k p}$, to prevent $F^{k+}<0$. Note that precipitation is not permitted to evaporate in the layer in which it forms;

Exactly the same procedure is applied to snow,

$$F_s^{k+} = F_s^{k-} + \frac{\delta^k p}{g} (P_s^k - E_s^k - M^k)$$ (4.140)

where $P_s^k = f_{s} P^k$ is the snow production, $f_s(T)$ is the snow production fraction, $M^k$ is the melting rate and

$$E_s^k = E^k F_s^{k-}/F^{k-}$$ (4.141)

so snow evaporates in proportion to the fraction of snow in the precipitation flux on the upper interface.

The snow production fraction is simple function of temperature

$$f_s = \frac{T - T_{s,max}}{T_{s,min}-T_{s,max}}, \quad T_{min} \leq T \leq T_{max}$$ (4.142)

with $f_s(T<T_{s,min}) =1$ and $f_s(T>T_{s,max}) =0$. $T$ is the grid volume temperature. The bounds are adjustable constants with current settings $T_{min}=-5^\circ$ C and $T_{max}=0^\circ$ C.

Falling precipitation is not permitted to freeze. Snow is produced only by the assumed snow fraction $f_s$ in the production term. Snow does not melt unless it it falls into a layer with $T^k>0$ C, in which case $M^k = F_s^k \frac{g}{\delta^k p}$ so that all the snow melts.

The net heating rate due to freezing, melting and evaporation of precipitation is

$$Q^k = -L_v E^k + L_f(P_i^k - E_s^k - M^k)$$ (4.143)

This is the method by which the heating due to $L_f$ is included for all condensation processes. For convective precipitation, $P_i^k \equiv P_s^k$, while for stratiform precipitation, $P_i = f_i C^k$ where $C^k$ is the net condensation rate in the cloud. Both the cloud ice fraction and the snow production fractions are determined by $f_i$, with $P_s^k$ coming from the cloud ice. For stratiform precipitation, the above equations are iterated once to allow the first estimate of the heating to change $T$ and consequently $q_*$ (but not $f_i$) for the 2nd iteration.

Cloud liquid and ice particles are allowed to sediment using independent settling velocities, similar to the form described by Lawrence and Crutzen [102]. The liquid and ice settling fluxes are computed at interfaces, from velocities and concentrations at midpoints, using a SPITFIRE solver [145]. The resulting flux at each interface is constrained to be smaller than the mass of liquid or ice in the layer above. This constraint does not allow for particles falling into the layer from above.

Sedimenting particles evaporate if they fall into the cloud free portion of a layer. No bound is applied to prevent supersaturation of the layer. This will be accounted for in the subsequent cloud condensate tendency calculation. Maximum overlap is assumed for stratiform clouds, so particles only evaporate if the cloud fraction is larger in the layer above. The overlapped fraction is

$$f_o = \min\left(\frac{f_c^k}{f_c^{k-1}},1\right)$$ (4.144)

The ice velocity $v_i$ is a function only of the effective radius $R_e$ (see Section 4.8.4 for more information and a plot), which itself is a function only of $T$. For $R_e < 40\times 10^{-6}$ m, the Stokes terminal velocity equation for a falling sphere is used

$$v_i = \frac{2}{9} \frac{\rho_w g R_e^2}{\eta}$$ (4.145)

where $\eta=1.7\times 10^{-5} \rm kg m/s$ is the viscosity of air and the density of air has been neglected compared to the density of water.

For $R_e > 40\times 10^{-6}$ m, the Stokes formula is no longer valid and we use a linear dependence of $v_i$ on $r = 10^{-6}\times R_e$

$$v_i(r) = v_i(40) + (r-40) \frac{v_{400} - v_i(40)}{400 - 40}$$ (4.146)

where $v_{400} = 1.0$ m/s is the assumed velocity of a 400 micron sphere, close to the value suggested by Locatelli and Hobbs [114].

The liquid particle velocity depends only on whether the cloud is over land or ocean, as is true of the liquid effective radius. The net liquid velocity $v_l$ is

$$v_l = v_l^{land} f^{land} + v_l^{ocean} f^{ocean}$$ (4.147)

where $f^{land}$ and $f^{ocean}$ are the land and ocean fractional areas of the cell, respectively. The ocean fraction may contain sea ice. The velocities are $v_l^{land} = 1.5$ and $v_l^{ocean} = 2.8$ cm/s.

It is assumed that there are five processes that convert condensate to precipitate:

$\bullet$ The conversion of liquid water to rain (PWAUT) follows a formulation originally suggested by Chen and Cotton [31]:

$$PWAUT = C_{l,aut} {\hat q_l}^2 \rho_a / \rho_w ( \hat q_l \rho_a / \rho_w N)^{1/3} H(r_{3l}-r_{3lc}).$$ (4.148)

Here $\rho_a$ and $\rho_w$ are the local densities of air and water respectively, and $N$ is the assumed number density of cloud droplets. $C_{l,aut} = 0.55 \pi^{1/3}k (3/4)^{4/3} (1.1)^4$, and $k = 1.18 \times 10^6$ cm$^{-1}$ sec$^{-1}$ is the Stokes constant.

$N$ is set to $400/cm^3$ over land near the surface, $150 /cm^3$ over ocean, and $75 /cm^3$ over sea ice. The number density also varies with distance from land by a factor equal to the distance to the nearest land point divided by 1000 km and multiplied by the cosine of latitude. The provides a sharper transition from land properties to ocean properties near the poles.

The terms $r_{3l}$ and $r_{3lc}$ are the mean volume radii of the droplets and a critical value below which no auto-conversion is allowed to take place, respectively. $H$ is the Heaviside function with the definition $H(x) = (0,1)$ for $x (<,\ge) 0$. The volume radius $r_{3l} = [(3 \rho_a q_l)/(4 \pi N\rho_w)]^{1/3}$. The standard value for the critical mean volume radius at which conversion begins is $15\mu$m. Baker [9] has shown that this parameterization results in collection rates that far exceed those calculated in more realistic stochastic collection models. This is because the parameterization is based upon a collection efficiency corresponding to a cloud droplet distribution that has already been substantially modified by precipitation. Austin et al. [7] suggest that a much smaller choice is appropriate prior to precipitation onset. Therefore the parameterization is adjusted by making $C_{l,aut} \rightarrow 0.1C_{l,aut}$ when the precipitation flux leaving the grid box is below 0.5 mm/day.

$\bullet$ The collection of cloud water by rain from above (PRACW) follows Tripoli and Cotton [173]

$$PRACW = C_{racw}\rho^{3/2} \hat q_l q_r$$ (4.149)

where $C_{racw} = 0.884 (g/(\rho_w \; 2.7 \times 10^{-4}))^{1/2} s^{-1}$ is derived by assuming a Marshall-Palmer distribution of rainwater falling through a uniformly distributed cloud water field, and $q_r$ is determined iteratively.

$\bullet$ The auto-conversion of ice to snow (PSAUT) is similar in form to that originally proposed by Kessler [85] for liquid processes and Lin et al. [111] for ice. However, it includes a temperature dependence similar to that proposed in Sundqvist [169]

$$PSAUT = C_{i,aut} H(\hat q_i-q_{ic}).$$ (4.150)

The rate of conversion of ice ($C_{i,aut}$) to snow is set to $10^{-3} s^{-1}$ when the ice mixing ratio exceeds a critical threshold $q_{ic}$. The threshold is set to $q_{ic,warm}$ at $T = 0^{\circ}C$ and $q_{ic,cold}$ at $T=-20^{\circ}C$. Values for $q_{ic,warm}$ and $q_{ic,cold}$ are given in Appendix C. The threshold varies linearly in temperature between these two limits.

$\bullet$ The collection of ice by snow (PSACI) follows Lin et al. [111], although it has been rewritten in the form:

$$PSACI = C_{sac}e_i \hat q_i.$$ (4.151)

where $e_i$ ($= 1$) is an ice collection efficiency. The coefficient of collection is

$$C_{sac} = c_7 \rho^{c_8}_a \tilde P^{c_5}$$ (4.152)

Here, $c_5$, $c_7$ and $c_8$ are constants arising from the assumed shape of the snow distribution.

The coefficients of the equation (4.152) arise from some algebraic manipulation of the expressions appearing in Lin et al. [111]. They in turn depend upon the specification for parameters describing an exponential size distribution for graupel-like snow. The parameter values used in Lin et al. [111] are adopted in the CAM 3.0 implementation. The parameters are a slope parameter $d = 0.25$; an empirical parameter $c = 152.93$ controlling the fall speed of graupel-like snow; and the assumed integrated number density of snow $N_s = 3. \times 10^{-2}$. The constants appearing in equation (4.152) can be expressed as

$$c_1 = \pi N_s c \Gamma(3+d) / 4$$ (4.153)

$$c_2 = 6 (\pi \rho_s N_s)^{d+4} / \bigl[c \Gamma(4+d) \rho_0^{0.5}\bigr]$$ (4.154)

$$c_5 = (3+d)/(4+d)$$ (4.155)

$$c_6 = (3+d)/4$$ (4.156)

$$c_7 = c_1 \rho_0^{0.5} c_2^{c_5}/(\rho_s N_s)^{c_6}$$ (4.157)
and

$$c_8 = -0.5/(4+d) .$$ (4.158)

Here $\Gamma$ is the Gamma function, $\rho_s = 0.1$ is the density of snow, and $\rho_0 = 1.275 \times 10^{-3}$ is a reference air density at the surface. All constants have been expressed in CGS units. The constants follow from integrating the geometric collection of a uniform distribution of suspended cloud liquid or ice over the size distribution of snow.

The collection of liquid by snow (PSACW) also follows Lin et al. [111]:

$$PSACW = C_{sac}e_w \hat q_l.$$ (4.159)

where $e_w$ is the water collection efficiency. Lohmann and Roeckner [115] note that the work by Levkov et al. [106] suggests that the riming process is too efficient using the standard values. There the collection efficiency is reduced by an order of magnitude to $e_w = 0.1$.