The Common.jl
module includes
a parameterization for difference in water activity between a H2SO4
solution droplet and ice. This can be used in immersion and homogeneous
freezing parameterizations of nucleation rate coefficient, J
.
The parameterization is based on Baumgartner2022, Koop2000,
and Luo1995.
and the water activity of ice at the same temperature, a_{w,ice}(T)
. When the
droplet is in equilibrium with its surroundings, a_w
is equivalent to relative
humidity. Otherwise, a parameterization can be found in the Common.jl
file and
comes from Koop2002,
where p_{sol}
is saturated vapor pressure of water above solution, p_{sat}
is saturated vapor pressure above pure liquid water, and p_{i,sat}
is saturated
vapor pressure above ice. p_{sol}
is determined in mbar using a parameterization
for supercooled, binary H_2SO_4/H_2O
solution from Luo1995 which is only
valid for 185K < T < 235K
:
where x
is the weight fraction of sulphuric acid in the droplets
(i.e. if droplets are 10% sulphuric acid by mass, x = 0.1
), w_h = 1.4408x
,
and temperature is in Kelvins. Note that x
should vary as the size of the droplet
changes.
!!! note
There is a need to find a parameterization for p_{sol}
at temperatures warmer than 235K for mixed phase clouds.
For now, the equation used to find water activity of a droplet at equilibrium at temperatures warmer than 235K is taken from Baumgartner2022 equation 4:
S_i
is water vapor saturation raito with respect to ice and defined by
where p_v
is the ambient partial pressure of water vapor.
!!! note
Only use the above parameterization for S_i in the application
that it is not available. i.e. for a parcel model, use the S_i
calculated in the parcel model.
p_{sat}
is currently defined using the Thermodynamics.jl
library.
An alternative is using the parameterization for p_{sol}
and setting
x = 0
. We choose to use the Thermodynamics.jl
to keep saturated vapor
pressure of pure water consistent throughout CloudMicrophysic.jl
.
Similarly, p_{i,sat}
is defined through the Thermodynamics.jl
library. It is
possible to calculate it using chemical potential as done in Koop2000, however,
we will use the current parameterization for consistency since they give similar results.
To verify that our parameterizations for p_{i,sat}
and p_{sat}
from
Thermodynamics.jl
are valid at cold temperatures, we can compare to various
other vapor pressure parameterizations as found in Baumgartner2022.
Here, CM
refers to CloudMicrophysics.jl
and all dashed curves are different
implementations of p_{i,sat}
and p_{sat}
(p_{sat}
labelled as
p_{liq}
to emphasize ice vs liquid phase of the pure water).
include("plots/Baumgartner2022_fig5.jl")
To verify that our parameterizations for water activty using Thermodynamics.jl
is ok, we plot critical water activity (water activtiy at which freezing occurs)
against the other variations found in Baumgartner2022. Though not
matching exactly, our parameterization is relatively close to the other parameterizations.
Shown in red is the water activity over ice using our parameterization. With these two lines
plotted (critical water activity of the droplet and ice water activity), we create a phase
diagram. Under the red line is liquid, above the critical water activity is ice, and between
the two curves is supercooled liquid.
Another plot to test if our parameterization is reasonable is plotting against other parameterizations
of water activity (as opposed to critical water activity) as a function of temperature. Plotted in
green are various ways to compute water activity over ice. using p(0,T)
refers to how the denominator,
a_w
, is calculated. By default, this is parameterized assuming a pure liquid droplet with
Thermodynamics.jl
. using p(0,T)
implies that the parameterization of vapor pressure of a solution
droplet is used but setting concentration of H2SO4 to zero. using \mu
refers to the parameterization
used in Koop2000 where water activity is dependent on chemical potential.
include("plots/T_vs_wateractivity.jl")
Taking the difference between any pair of blue and green lines will give a
\Delta a_w(T)
.
Since all the blue lines are similar and all the green lines are similar, we can
assume that our parameterization of pure liquid and ice water activities are reasonable.