Parameterization of Ice Nucleation

The above nucleation rate is for a single particle, and is actually the growth kernel

ܭ_௞ in Eq. (2.7) of the nucleation process. To get the bulk nucleation rate for a

population (i.e., one mode) of IN, the group nucleation rate for ice nuclei in the k^th

moment can be expressed as:

ܫ_௞ ൌ ׬ ܬ_௛௘௧ή ݎ^௞ή ݊ሺݎሻ݀ݎ, (2.37)

where I_kis the bulk nucleation rate of the k^thmoment, and ܬ_௛௘௧ is the single-particle

nucleation rate as given in Eqs. (2.27) or (2.31). Due to the complicate form of ܬ_௛௘௧,

analytical solution of Eq. (2.37) cannot be derived. So, the parameterization of the

bulk nucleation rate was attempted using the SNAP scheme. In the parameterization

processes, the numerical solutions of Eq. (2.33) were calculated by separating particle

spectrum into 100 bins and systematically varying the dependent variables within

possible ranges of values (see Table 2.4). Since the parameterization results of

immersion freezing and deposition nucleation are in the same mathematical form, only

the results of immersion freezing is given below as an example.

Applying the MSA method to Eq. (2.37), the nucleation rate can be obtained quite

straightforwardly:

ܫ_௞ ൎ ܬሚ_௛௘௧ߤ^௞׬ ݊ሺݎሻ݀ݎ ൌ ܬሚ௛௘௧ߤ^௞ܯ_଴ (2.38)

where ܬሚ_௛௘௧ is calculated by using the modal size P (which is a constant) to replace r in

Eq. (2.31). Figure 2.8 (blue circles) compares the parameterization results of MSA and

the numerical solutions of immersion freezing rate, showing that the MSA method

significantly underestimated the immersion rate especially at lower nucleation rates, and

the error increased with higher σ.

When applying the SNAP-KT method, the complex growth kernel needs to be

transformed into functional forms of ݎ^௔ or ‡š’ሺܾŽ^ଶݎሻ, which is not easy to do for the

ice nucleation rate because of the highly nonlinear form of f in Eq. (2.33). Since the

parameter f in Eq. (2.32) are in square root and exponential forms, there are two

transformations needed using SNAP-KT. Chen et al. [2008] provided a conversion

formula for f which can be used to transform the exponential term:

Ž݂ ൎ  ܽ_ଵ൅ ܽ_ଶŽሺͳ െ ݉ሻ ൅ ܽ_ଷŽ ^௥

௥_೒ (2.39)

where ܽ_ଵ = -0.06841440, ܽ_ଶ ൌ1.983733 and ܽ_ଷ ൌ -0.05734672. The

transformation of ඥ݂ can be done by letting (Chen et al., 2013):

ඥ݂ ൎ ܽ_ସݎ^ೌయమ (2.40)

where ܽ_ସ ؠ ඥ‡š’ሺܽ_ଵ൅ ܽ_ଶŽሺͳ െ ݉ሻ െ ܽ_ଷŽ ݎ_௚ሻ is independent of r, and

Ž݂ ൎ  Ž ൬ܾ_ଵ൅ ܾ_ଶŽ ^௥

௥_೒൰ ൅ ܾ_ଷ൅ ܾ_ସŽሺͳ െ ݉ሻ ൅ ܾ_ହሺͳ െ ݉ሻ (2.41)

where b₁= 4.51029, b₂= -0.11301, b₃= -1.60130, b₄= 2.00589, and b₅= -0.458392.

Applying this formula to simplify f in the exponential term in Eq. (2.33) one can get:

‡š’ሺܤ݂ሻ ൎ ‡š’ሺܿ_ଵሻ ݎ^௖మ (2.42)

where ܤ ؠ ^ο௚೒

௞_ಳ், ܿ_ଵ ൌ ܤሺܾ_ଵെ ܾ_ଶŽݎ_௚ሻ‡š’ሺܾ_ଷ^ᇱሻ, ܿ_ଶ ൌ ܤܾ_ଶ‡š’ሺܾ_ଷ^ᇱሻ, and ܾ_ଷ^ᇱ ൌ ܾ_ଷ൅

ܾ_ସŽሺͳ െ ݉ሻ ൅ ܾ_ହሺͳ െ ݉ሻ are all independent of r. The above constant coefficients

ܽ and ܾ varies with the type of nuclei, and the values shown are specifically for Asian

dust. The fitting for eq.(2.40) and eq.(2.42) are applicable for contact angel T^{from 1ͼ}

to 110ͼ. Combining the above equations, the group nucleation rate derived using the

SNAP-KT method becomes:

Figure 2.8 shows that the SNAP-KT parameterization (purple crosses) is closer to the

numerical solution than the results of MSA, but still slightly overestimated it at lower

immersion freezing rates.

The essence of SNAP-IT is to find a correction term to modify the result of MSA,

i.e., Eq. (2.38). The main assumption of the MSA method is the use of μ to replace r

in the growth kernel, which is therefore the main source of error. Note that the MSA

method is in effect assuming V=0 (i.e., monodispersed size distribution). So its error

must be associated with σ, and one can also expect that the error grows with increasing V. Another complication stems from the highly non-linear term f which contains the

factor q (i.e. rN/rg) that is dependent on r. Therefore, the correction term ݃_ଵ was

derived as a function ofV and q. The fitting results shown in Figure 2.9 reveal that the

SNAP-IT can provide fairly good fitting using the derived formula for ݃_ଵ:

݃_ଵൌ ‡š’ሾܽ_ଵή ߪ^ଶ൅ ܽ_ଶή ‡š’ሺെݍതሻሿ (2.44)

where ܽ_ଵ and ܽ_ଶ are constant coefficient (listed in Appendix 2.A) which changes with

the nucleation modes, and the fitting surface is shown in Figure 2.9a. .

However, it is found that the parameterization derived with the SNAP-IT method is

only slightly better than the SNAP-KT method, and significant error still exist at low

immersion freezing rates. A further improvement was obtained by using the

SNAP-OS method, which is similar to SNAP-IT but the correction term ݃_ଶ is used to

generate an optimal size P’ that can be used to substitute P in the MSA method. The

fitting formula for ݃_ଶis obtained as:

݃_ଶ ൌ ‡š’ ቀܽ_ଵή ɐ^ଶ൅^௔మ

௤ത^మቁ, (2.45)

where the constant coefficient are listed in appendix A, and the fitting surface is shown

in Figure 2.9b. Figure 2.8 shows that SNAP-OS method (green triangles) provides

highly accurate parameterization even at very low immersion freezing rates. The

fitting using SNAP-IT and SNAP-OS for the immersion freezing mode are shown in

Figure 2.9.

Table 2.5 and Table 2.6 show the coefficient of determination (R²) and mean

relative errors of the above parameterization schemes for immersion freezing rate.

One can see that the results of SNAPs are better than the results of MSA, and SNAP-OS

had the highest determination coefficient. At the same time, the mean errors in SNAPs

were much lower than that of MSA, and SNAP-OS has the lowest mean errors.

However, the SNAP methods required more CPU time than MSA (Table 2.7), ranging

from the 74% for SNAP-KT and 19% for SNAP-OS. All of above results shows

SNAP-OS is the best way to deal with the immersion freezing rate, so it was also

directly applied in the parameterization of deposition nucleation rate for this study.

Details of the parameterization for the deposition nucleation mode are ignored here, but

the fitting surface is shown in Figure 2.10.