1. Melting of ice particles:
When ice particles fall below 0 ˚C they begin to melt, but the process takes some time since heat transfer needs to occur (heat from ambient environment has to supply the latent heat to effect the phase change)
We can apply the same simplified model as before: hailstone settling through population of cloud droplets, accreting them, but at temperature > 0 ˚C
Put in rate of dry collection here

2. Setting the rate of energy conduction through air equal to the rate of energy lost through evaporation, solve for the ambient temperature that yields a surface temperature = T0 (0 ˚C):
An ice particle in cloud (RH~100%) may begin melting near 0 ˚C, but outside the cloud, melting may not begin until ambient temperatures are a few degrees above 0 ˚C, because at lower RH there is a driving force for evaporation, which cools the drop, and thus inhibits melting.

3. Points:
Melting distance increases with density (more mass for given radius)
Melting time increases with particle size
Melting distance is shorter in water saturated environments
Evaporation not allowed, thus all absorbed heat can go to phase change
In subsaturated environment, some goes into latent heat of evaporation

4. Back to Supercooling and Freezing
Derivation from HW
Koop’s plot – looking at it more closely (melting, freezing)
Using Koop’s parameterization, relate fraction frozen to nucleation rate (spreadsheet)

5. Derivation: Method 1 p is constant for the supercooled drop
latent heat is a function of T only

6. Derivation: Method 2 (Pruppacher & Klett)
Starting point

7. Approach is to find mL-mS by differencing: (mL-mV) - (mS-mV)
So write equation 6-5 for each of these transitions
We integrate from T= K to the final supercooled temperature T
At K, ea,i = ea
Also at K, Ls=Lm,o+Le,0 (latent heats of sublimation, evaporation, and melting)
Although latent heat varies with T, assume here an average, constant value
So difference equation becomes:
Pure water

8. integrate

9. Homogeneous nucleation of ice from solution (Koop et al.)
(freezing of ice from bulk solutions; no nucleation barrier)
273.15
235
(-38 C)
The nucleation rate is a function of temperature, but smaller drops require a higher nucleation rate to achieve the same nucleation efficiency as larger drops – so this “onset” curve really depends on solution drop size too
pure water
(freezing of ice from solution drops; nucleation barrier)
(solution composition)
(relative humidity)

10. How can we calculate the lines on this plot?
What equilibrium conditions are represented by this line?
In a bulk solution, for a chosen temperature T, this is where saturation must exist for the ice phase: the vapor pressure in the ambient air equals the vapor pressure over ice
We can write this because all the phases have to be in equilibrium
Express activity relative to saturation of the liquid (here, supercooled water)
Water activity = vapor pressure of ice at T / vapor pressure of supercooled water at T
.. … … … ..
….. ….. ….
The calculation of this line (melting, Tm) is found in the Excel workbook, “Murphy&Koop_water_vapor_pressure_calcs.xls”
See sheet, “vp calcs”, which contains Murphy and Koop’s polynomials for vapor pressures over supercooled water and over ice. Column V forms this ratio, shown in the plot, “Koop plot”

11. How can we calculate the lines on this plot?
How did Koop et al. get this line?
From looking at data for onset of ice formation in many solutions, Koop et al. deduced that this line was simply the bulk liquid-ice phase line shifted along the x-axis, by Daw = 0.305
The drops need to dilute more than the bulk liquid before ice formation can begin
The calculation of this line is found in the Excel workbook, “Murphy&Koop_water_vapor_pressure_calcs.xls”
See sheet, “vp calcs”. Column W applies the Koop Daw, and shown in the plot, “Koop plot”
Notice that aw > 1 for temperatures warmer than 235 K  no ice formation!
Note also that we could find freezing temperature Tf, from Tf = Tm(aw-Daw)
For the plot on the right, we are plotting this same homogeneous nucleation line, except as RH_i instead of RH_w. See columns AA and AD.

12. Nucleation rates and fraction frozen
In experiments with the CFDC, we control T and we also generally use monodisperse drops as test particles. Therefore we can compute the fraction frozen in a time t using
However, we are measuring the fraction frozen during the residence time in the CFDC (~10 sec), so generally we’d like to invert this equation and obtain the nucleation rate, Ji(T)
For “onset” conditions, we typically look for a small fraction frozen, 1% of the test particles or even smaller.
The calculation is outlined in the Excel workbook, “Koop_calcs_for_ATS620”
At the top, the equations presented by Koop et al for computing J are shown
At the left, the sheet scans through choices of water activity, using the kappa parameterization to find the amount of water associated with the drop as well as the RH in equilibrium with it. We must choose a T and a dry particle diameter for the calculation.
At the right, the nucleation rate is computed and scaled by the drop volume.
The frozen fraction is computed as above.
Try changing residence time, particle hygroscopicity and dry particle size to see the effects on F.
Plot nucleation rate as a function of ambient RH, in log-log space and lin-lin space.