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Figure 7. Profiles of horizontally averaged solar heating rates and IR heating rates for four different ice crystal types. The warm stable case is plotted in left two panels, and the cold stable case is shown in right two panels (adopted from Liu et al.., 2003b).

Figure 8. Ice crystal capacitance as a function of ice crystal mass and habit (adopted from Liu et al., 2003b).

Figure 8. Ice crystal capacitance as a function of ice crystal mass and habit (adopted from Liu et al., 2003b).

observational data on not only the size but also the shape distributions of ice crystals in them.

The domain-averaged ice water content and number concentration are shown in Fig. 9 for both the warm and cold stable cases. The effect of ice crystal habit is evident here, too. During the growing stage, rosettes tend to consume more water vapor and thus suppress the homogeneous nucleation process (the homogeneous freezing of haze solution droplets, as parametrized in Part I of the article). Consequently, the ice number concentration for rosettes is less than that for the other ice habits. However, the

Figure 9. Domain averaged ice water content and number concentration for warm stable and cold stable case (adopted from Liu et al., 2003b).

resulting ice water content still exceeds those for the other ice habits because their large growth rates produce correspondingly larger crystals. This is confirmed by the profiles of mean size for rosettes in Fig. 6, where the warm case at 10 min already shows a mean size in excess of 100 ¡m for rosettes, while the other ice types still average less than 70 ¡m.

### 3.4. Summary and conclusions

In summary, it has been found that the cloud development is very sensitive to ice crystal habit. This is mainly because the different ice crystal habits have distinctly different capacitances and optical properties, and all these factors contribute differently to ice crystal growth rates and hence to cloud development. The most important factor is the capacitance — among ice particles of a given mass, rosettes have the greatest capacitance and hence the largest growth rate, with successively smaller capacitances for columns, hexagonal plates and spherical particles. Bullet rosettes generally grow 2-3 times faster than other crystal types. Ice crystal habit influences the homogeneous nucleation process. Bullet rosettes consume large amounts of the water vapor available in the cloud layer, due to their largest growth rate and thus severely suppress the initial nucleation process. As a result, cirrus clouds consisting of bullet rosettes have significantly fewer ice crystals nucleated than those with other ice crystal types. The IR heating rate is also more sensitive to ice crystal habit than the solar heating rate is. The role of aggregation in the development of cirrus has also been examined; it tends to reduce the optical depth of the cloud and is thus likely to reduce the radiative destabilization of the cloud layer.

### 4. Capacitance of Ice Crystals

The results in the previous section indicate that the growth rate of ice crystals has great impacts on the behavior of cloud radiative and dynamical properties. Since in most cloud scale models, as in our cirrus model, the ice growth rate is calculated using the electrostatic analogy, the capacitance of the ice crystal is the central quantity in this analogy. Clearly, it is necessary to obtain the values of capacitances of ice crystals considered in the cloud scale model if their growth rates are to be determined accurately.

The capacitance can be determined either experimentally or theoretically. Experimentally, one can construct metallic models of ice crystals; charge them with electricity and then measure the capacitance of the model directly. Some earlier measurements were done this way (e.g. McDonald, 1963; Podzimek, 1966). Aside from the fact that such techniques are usually quite involved, it is difficult to make models that cover a wide variety of shapes and sizes of ice crystals in clouds. In the following, we report on the theoretical calculation results of ice capacitances performed recently by Chiruta and Wang (2003, 2005).

Chiruta and Wang (2003) gave an outline of this technique. To calculate the capacitance of an ice crystal (assumed to be an electrical conductor), we need to determine the electric potential distribution around an ice crystal. Obviously the potential has to satisfy the Laplace equation (no space charge needs to be considered). This is a differential equation, and boundary conditions are needed to uniquely determine the solutions. The surface potential is assumed to be known (and is usually taken as 1 in the nondimensional formulation), whereas the potential at infinity is 0. Since numerical techniques are used in most cases (because of the complicated crystal shape), the "infinity" simply means the outer boundary of the computational domain. In this way, the potential distribution around an ice crystal is determined. Integrating the potentials over a certain equipo-tential surface will give the total charge inside this surface (which, in reality, resides on the crystal surface). Knowing the total charge and the potential difference, we can determine the capacitance.

The central difficulty in this technique is how to specify the inner boundary surface which coincides with the crystal surface. For simpler shapes such as hexagonal columns and plates, it is feasible to map the boundary surface points directly onto the computational grids, as was done by Chiruta and Wang (2005). For more complicated shapes such as rosettes, it is convenient to use mathematical formulas to generate the surface points when such formulas are available. This was done by Chiruta and Wang (2003), who used the mathematical formula described by Wang (1999). In both papers, Chiruta and Wang (2003, 2005), the numerical technique used to solve the Laplace equation was the finite element technique.

In this way, Chiruta and Wang computed the capacitances of seven rosette ice crystals and nine hexagonal columns (both solid and r = ff(j - cos4 9 - sinJ <p f ,■ = <,$- cosJ 6 )" (l - sin'! ,5cp J r = cosJ 9 )" (l -sinJ 2cp J

r = ff(j - cos4 9 - sinJ <p f ,■ = <,$- cosJ 6 )" (l - sin'! ,5cp J r = cosJ 9 )" (l -sinJ 2cp J

Figure 11. The nine simulated columnar ice crystals considered in Chiruta and Wang (2005). The hollow columns (left) and solid columns (right) have the same external dimensions. The small hexagonal disk (upper right) is a disk whose thickness is the distance between the tips of the two opposing cavities. Its capacitance is also calculated to serve as a reference (adopted from Chiruta and Wang, 2005).

hollow). Figures 10 and 11 show the seven bullet rosettes and nine solid and hollow columns that they treated in their papers. Geometrical properties of these ice crystals have been given in the above-cited papers.

Figures 12 and 13 show the calculated capacitances of the rosettes and columns, respectively. Figure 12 demonstrates that the capacitance of rosettes cannot be approximated by spheres, oblate or prolate spheroids. Rather, it is a nonlinear function of the number of lobes. The more lobes a rosette has, the closer is its capacitance to that of a sphere of the same radius. This seems to be logical, as one can imagine that a rosette with an infinite number of lobes would be effectively the same as a sphere. For rosettes with few lobes, the capacitances are smaller. This implies that their growth rates are smaller correspondingly, and may result in significant impact on the radiative properties of the cloud over an extended period.

Figure 13 demonstrates that the capacitance of a hollow hexagonal column is the same as that of a solid column of the same dimension and aspect ratio. This implies that their mass growth rates are the same. But since the hollow column has a smaller mass than the solid column, this implies that the hollow column will grow faster linearly (assuming that the main direction of growth is along the c axis, which is likely the case). The different linear dimensions of ice crystals would also result in different magnitudes of radiative interaction and hence different radiative properties of the cloud.

Naturally, the remarks made in the previous two paragraphs represent only the effect of the ice growth rate alone. To really assess the overall impacts, one needs to incorporate these results into a cloud scale model so that the interactions between the dynamics, radiation and cloud microphysics can be considered together. Our group is working on this research currently.

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