#### 4.1. Effects of Ice PSD on Bulk Process Rates

All models in this intercomparison use the same mass-size, capacitance-size, and fall speed-size relationships for ice particles. In bin microphysics schemes (DHARMA-bin and SAM-bin), these relationships control the rates for ice processes, such as depositional growth and sedimentation, in each size bin and therefore directly govern the evolution of ice PSDs. For bulk microphysics schemes, however, the shape of the ice PSD has to be specified in order to obtain the integrated process rate from individual particle properties. For example, in five models (DHARMA-2M, METO, SAM-2M, UCLALES, WRFLES, and WRFLES-PSU), ice PSD are assumed to follow a gamma distribution in the form

- (2)

where *D* is the crystal maximum dimension, *ν* and *λ* are two parameters of the distribution, , *N*_{i} is the total ice number concentration, and Γ is the gamma function. (Note that here maximum dimension is the same as the diameter of the prescribed ice spheres, which have a density less than one tenth that of bulk ice.) In the default configuration of the scheme used in this intercomparison, the shape parameter *ν* is set to zero and the distribution is reduced to an exponential ice PSD [*Morrison et al*., 2005]

- (3)

where *λ* and *N*_{0} are the slope and intercept parameters, respectively. Ice number concentration, *N*_{i}, and ice water content, *q*_{i}, related to the 0th and 3rd moments of the PSD, respectively, can be expressed as

- (4)

- (5)

The two parameters of the PSD in equation (3) are therefore uniquely defined by *N*_{i} and *q*_{i} predicted by a two-moment microphysics scheme as

- (6)

Ice PSDs in two other models (COSMO and UCLALES-SB) are based on a modified gamma distribution (MGD) expressed in terms of ice particle mass, *m*, rather than size [*Seifert and Beheng*, 2006]

- (7)

Noteworthy for the purpose of this intercomparison and, more generally, for any comparisons of different microphysical schemes is that the parameters of the MGD change when the distribution type changes (e.g., size versus mass) [*Petty and Huang*, 2011]. Thus, an exponential distribution in terms of *D* (*ν* = 0 in equation (2)) becomes a MGD in terms of *m* with *µ* = 1/3 and *ν*_{m} = −2/3 [*Seifert and Beheng*, 2006]. Similarly, the default values in COSMO and UCLALES-SB (*ν*_{m} = 0 and *µ* = 1/3 in equation (7)) result in a gamma distribution with *ν* = 2 in terms of *D* (equation (2)) [*Petty and Huang*, 2011]. In the following discussion, the distributions are considered in terms of ice particle diameter (*D*), unless stated otherwise.

The shape of the distribution is important because it affects the process rates for sedimentation, depositional growth, and sublimation of ice particles. Because these processes depend on different moments of the PSD, it is instructive to examine relationships among the moments of distributions with different shape parameters. For a gamma distribution in the form of equation (2), the moment *p* is given by

- (8)

The ratio of any moment to the same moment of the exponential distribution (i.e., *ν* = 0) is

- (9)

Figure 9 illustrates how this ratio changes as a function of *p* and *ν*. Because both exponential and gamma distributions are constructed to represent the same *N*_{i} and *q*_{i}, the ratio is unity for *p* = 0 and *p* = 3. By definition, the ratio is also unity for *ν* = 0. For any positive *ν* and *p* between 0 and 3, the ratio is larger than unity, while for *p* larger than 3 the ratio is smaller than unity (Figure 3a). Thus, moments three and lower are larger for the gamma distribution than for the exponential, while the opposite is true for higher moments (Figure 9b). The ratio levels off for *ν* larger than 6 or so (Figure 9c).

With respect to the simulations analyzed here, the role of the PSDs stems from dependency of the depositional growth and sublimation rates and sedimentation effects on different moments. The rate of water vapor deposition on ice crystals, being proportional to the first moment of the PSD, increases as the shape parameter *ν* increases and ice spectrum becomes narrower for given *N*_{i} and *q*_{i}. Indeed, the maximum deposition rate in two models using an effective *ν* = 2 in calculation of the deposition rate (COSMO and UCLALES-SB) is larger than in models that assume an exponential distribution (DHARMA-2M, SAM-2M, UCLALES, and WRFLES) for comparable values of IWP and *N*_{i} (Figure 10). Notably, the two simulations using bin microphysics schemes have a maximum deposition growth rate larger than 2 × 10^{−5} g m^{−3} s^{−1}, the upper–bound threshold for rapid glaciation of mixed-phase clouds found in *Morrison et al*. [2011], but nevetheless maintain quasi steady state mixed-phase clouds, indicating that other processes also play significant roles and suggesting that the threshold may depend on some microphysical assumptions.

Another important process affected by the shape of the distribution is sedimentation. While the fall speed *V*_{i} of an individual ice crystal is prescribed as a function of size ( , where *a*_{v} is defined in the Appendix Simulation Setup), it is the integral over the size spectrum that is important for the water budget. Furthermore, for any but monodisperse size spectrum, sedimentation affects distributions of mass and number mixing ratios differently. The mass-weighted fall speed (*V*_{mi}), which controls sedimentation of *q*_{i}, is

- (10)

Since *V*_{mi} is proportional to moment 3.5 of the PSD, it is smaller for a gamma distribution with *ν* = 2 than for an exponential distribution with the same *q*_{i} by about 15% (Figure 9c). Both slower growth rates and faster mass-weighted fall speeds contribute to smaller IWPs predicted by models using exponential ice PSD for ice1 (cf. DHARMA-2M, SAM-2M, UCLALES, and WRFLES in Figure 3c.). For higher ice number concentrations, as in ice4, these models also predict smaller IWP (Figure 3e) than other ensemble members. Models with narrower ice PSDs, such as COSMO and UCLALES-SB with an effective *ν* = 2 in sedimentation calculations, predict stronger precipitation in both ice1 and ice4 runs (Figure 6). In ice4, however, the difference in IWP among different bulk schemes diminishes with time (Figure 3e) as precipitation exceeds the resupply of moisture to the cloud layer from below. For this set of runs, the only two models that result in a significantly higher IWP are models with bin microphysics (DHARMA-bin and SAM-bin).

The number-weighted fall speed for ice particles, which controls sedimentation of *N*_{i}, is

- (11)

Being controlled by the 0.5th moment of the PSD, the effect of *ν* on *V*_{ni} is stronger and opposite in sign to the effect on *V*_{mi} (Figure 9c): *V*_{ni} for a gamma distribution with *ν* = 2, for example, exceeds *V*_{ni} for an exponential distribution by ∼30%.

The ratio *V*_{mi}/*V*_{ni}, which indicates the efficiency of size sorting by sedimentation, can be expressed as a function of spectral shape parameter

- (12)

#### 4.2. Sensitivity of Simulations to Ice Size Distribution

Given the large sensitivity of bulk process rates to the shape of ice PSD, it is important to understand a realistic range of the shape parameter. One way to obtain this information is through the analysis of simulations conducted using bin microphysics schemes. These schemes do not make assumptions on the shape of the ice size spectrum and explicitly predict the PSDs, notwithstanding uncertainties in the accuracy of these predictions due to assumptions about particle properties, neglect of aggregation in the presented simulations, numerical representations, and other issues. Figure 12 illustrates the variability of the shape parameter in gamma distributions approximating ice size spectra from the DHARMA-bin ice4 simulation, in which *ν* is computed from the relative dispersion of *D* using the first three moments of the PSD. The domain-average value of *ν* (computed from the domain averages of each of the first three moments) is around 3, but the parameter ranges from 0 to 15 with a pronounced height dependency (Figures 12c and 13). Near the surface, *ν* is small and the ice distributions are nearly exponential. The spectra become narrower with altitude and, in the depositional growth region (between 400 and 800 m levels), *ν* can be 10 or higher with the horizontal mean value of 6 at z = 400 m (Figure 12c). There is no clear correlation between horizontal variability of *ν* and *q*_{c} or *q*_{i} (Figure 12).

To confirm the role of ice PSD in the evolution of the cloud, sensitivity simulations with altered ice treatments are conducted. Table 3 lists DHARMA and SAM-2M sensitivity experiments, results from which for the ice4 configuration are presented in Figures 14 and 15. Using a gamma ice PSD (*ν* = 3) instead of exponential (*ν* = 0) in DHARMA-2M simulations leads to a boost in depositional growth rate of ice, a close match of the resulting LWP evolution to that in DHARMA-bin simulations considered as a reference (Figure 14a) and a substantial increase in IWP relative to the default bulk configuration (Figure 14c). Very good agreement in the net longwave radiative flux at the surface, which is determined by LWP in the radiation parameterization, is also achieved with this setup (not shown). The remaining underprediction of IWP in these runs relative to the bin microphysics is presumably because the depositional growth rate is underestimated in the 400–600 m layer, where the spectra in the DHARMA-bin ice4 simulations are narrower (*ν* ≈ 6, Figure 13b) than those in modified DHARMA-2M (*ν* = 3). Analogous experiments with SAM-2M produce similar results. When a narrower gamma ice PSD with *ν* = 3 is used instead of the default exponential distribution (*ν* = 0), the total condensed water path changes little (Figure 14f), but its partitioning between liquid and ice phases is altered drastically, bringing both LWP and IWP in much closer agreement with SAM-bin (Figures 14b and 14d). An agreement in precipitation rate is similarly improved (Figures 15a and 15b).

Table 3. Sensitivity Experiments for the Ice PSD EffectsCase | Ice PSD Treatment | Model(s) |
---|

*ν* = 0 | Exponential ice PSD, same as default | DHARMA-2M, SAM-2M |

*ν* = 3 | Gamma ice PSD (equation (2)) with *ν*=3 | DHARMA-2M, SAM-2M |

*ν* = 0, *V*_{ni} = *V*_{mi} | Same as *ν* = 0, but no size sorting | SAM-2M |

*ν* = 3, *V*_{ni} = *V*_{mi} | Same as *ν* = 3, but no size sorting | SAM-2M |

*ν* = 3, *V*_{ni} = *V*_{mi}, *V*_{mi}(*ν* = 0) | Same as *ν* = 3, but no size sorting and *V*_{mi} is computed for exponential PSD | SAM-2M |

*ν* = 3, *V*_{ni} = *V*_{mi}, *Deps*(*ν* = 0) | Same as *ν* = 3, but no size sorting and depositional growth is computed for exponential PSD | SAM-2M |

When SAM-2M runs with *ν* = 0 and *ν* = 3 are repeated with the size-sorting effect turned off by setting *V*_{ni}=*V*_{mi}, only small changes are seen in LWP, IWP, and precipitation. Although the size-sorting effect in these simulations is small regardless of *ν*, it is only the case because of the constraint on ice concentration imposed in the current setup. When size sorting is turned off, a 70% larger column integrated ice nucleation rate is required to maintain the prescribed *N*_{i} in the mixed-phase cloud for *ν* = 0 (cf. cases with *ν* = 0 and *ν* = 0, *V*_{ni} = *V*_{mi} in Figure 15a). The effect becomes smaller for larger *ν* because the difference between *V*_{ni} and *V*_{mi} is reduced for narrower spectra (Figure 11). Consequently, only a 30% increase in the nucleation rate is needed to offset the size sorting for *ν* = 3 (cf. cases with *ν* = 3 and *ν* = 3, *V*_{ni} = *V*_{mi} in Figure 15a). If *N*_{i} were to vary and a relatively constant ice nucleation rate were to be prescribed (or predicted), size sorting would have a significant impact on IWP, particularly in models assuming broad (exponential) ice PSDs.

Relative roles of ice PSD effects on LWP and IWP via changes in two processes, namely the crystal fall speed and depositional growth rate, are quantified as follows: Simulations are repeated with a gamma PSD and no size sorting (i.e., *ν* = 3, *V*_{ni} = *V*_{mi}) for one process rate while assuming an exponential PSD in computing the other process rate. The two simulations, Deps(*ν* = 0) and *V*_{mi}(*ν* = 0), produce very similar IWPs, which are approximately halfway between those for SAM-bin and SAM-2M. This suggests that accounting for ice PSD is equally important for both processes in order to predict the correct IWP. The LWP evolution, however, is clearly dominated by the size distribution effect on the depositional growth rate as seen in the tight grouping of curves in Figure 14a depending on whether *ν* = 0 or *ν* = 3 is used in computing the ice crystal growth rate.

To further test the effect of PSD assumptions on ice crystal growth, the parameters of *f*_{m} (equation (7)) in COSMO are varied so that the integrated depositional growth rate of ice is reduced to match that in DHARMA-2M and SAM-2M. With this modification, the mean LWP increases, particularly for ice4 (not shown), and COSMO shifts closely toward the cluster of DHARMA-2M, SAM-2M, UCLALES, and WRFLES points in LWP-IWP phase space seen in Figure 7. In another set of runs using COSMO's default scheme, which accounts for ventilation due to the crystals' sedimentation velocity, the depositional growth rate of ice is increased relative to that from intercomparison simulations and liquid water cloud is completely desiccated before the end of the ice4 simulation. In ice1 simulations with ventilation effects, the mean LWP decreases and IWP increases to the point where they are comparable to those from the ice4 run with reduced depositional growth. Thus, the combined effect of a narrow size ice PSD and ventilation is comparable to quadrupling the ice particle concentration with a broader PSD and no ventilation.

#### 4.3. Sublimation Effects on Ice Mass and Number Concentration

According to the microphysics setup, the ice number concentration is essentially fixed within the mixed-phase cloud, i.e., when liquid is also present. In the absence of liquid water, however, ice concentration is allowed to evolve, leading to a significant spread in *N*_{i} among the models below the liquid cloud base (Figure 16). One of the main reasons for this is the unconstrained effect of sublimation on *N*_{i}. Depositional growth increases *q*_{i} but does not affect *N*_{i}. This is not necessarily true for sublimation, which can reduce both *N*_{i} and *q*_{i}, and models use different approaches to account for sublimation-induced reduction in *N*_{i}. Five models (DHARMA-2M, SAM-2M, UCLALES, WRFLES, and WRFLES-PSU) assume that sublimation reduces *N*_{i} by the same fraction as it reduces *q*_{i}, i.e., (d*N*_{i}/*N*_{i})_{sub}=(d*q*_{i}/*q*_{i})_{sub}, or equivalently that the mean ice size is preserved during sublimation. This assumption leads to nearly constant mean ice size below 400 m in these models (Figure 17). In COSMO, UCLALES-SB, and METO, *N*_{i} does not change when ice sublimates until the mean ice particle mass falls below a prescribed minimum. That minimum mass is then used to compute updated *N*_{i} from *q*_{i} after sublimation. When *N*_{i} remains nearly constant as *q*_{i} declines throughout the sublimation zone (COSMO, UCLALES-SB, and METO, Figure 18), ice crystals become smaller and fall slower, leading to further decrease in size due to longer exposure to subsaturated conditions. Consequently, these models have the smallest ice particles near the surface (Figure 17). Regional Atmospheric Modeling System (RAMS) employs lookup tables developed from parcel model simulations with bin ice microphysics to obtain (d*N*_{i}/*N*_{i})_{sub} from (d*q*_{i}/*q*_{i})_{sub} depending on environmental conditions, parameters of the gamma PSD, and ice crystal habit [*Harrington et al*., 1995].

The effects of these different specifications can be gleaned from comparisons with simulations using bin microphysics schemes, which compute the change in *N*_{i} due to sublimation explicitly without invoking additional assumptions and therefore provide a more physically based treatment of the effect of sublimation on the parameters of the ice PSD. The bin models show a reduction in both *N*_{i} and *q*_{i} when precipitating ice particles approach the surface (Figures 16 and 18, DHARMA-bin and SAM-bin), leading to mean crystal size in the sublimation region below 400 m (Figure 17) that is not constant but varies less than in models assuming constant *N*_{i} during sublimation.

Note that sedimentation in a sublimation region can lead to an increase in *N*_{i}. When sublimation reduces ice particle size and, therefore, *V*_{ni}, sedimentation leads to convergence of *N*_{i} in that layer, which, if not compensated by a sufficient reduction of *N*_{i} due to sublimation, will result in a *N*_{i} increase, as seen in COSMO, METO, and RAMS profiles in Figure 16.