Mineral Dust and its Microphysical Interactions with Clouds - Mineral Dust: A Key Player in the Earth System - page 295

where N 1 and N 2 are the number of liquid droplets at the beginning and end of
the time step, t , respectively. J het is the heterogeneous nucleation rate coefficient
and s is the surface area of nucleant per droplet. Traditionally, J het has been
described with classical nucleation theory where a contact angle parameter is used
as an independent measure of how efficiently a material will nucleate ice. If this
equation holds, then we would expect a first-order decay of droplets at constant
temperature and also values of J het from experiments with different cooling rates
to be the same. The only known IN sample which is consistent with this model is
kaolinite (KGa-1b) from the clay mineral society (Herbert et al. 2014 ; Murray et al.
2011 ). Unfortunately, the assumption of particle uniformity does not hold for other
atmospherically relevant aerosol types; instead, their ice-nucleating ability exhibits
diversity with some particles having a greater propensity to nucleate ice than others
(Herbert et al. 2014 ; Broadley et al. 2012 ; Wheeler and Bertram 2012 ; Niedermeier
et al. 2011b ; Lüönd et al. 2010 ; Barahona 2012 ).
In order to account for the diverse nature of ice-nucleating particles, a number
of authors introduced models in which a distribution of ice-nucleating abilities was
incorporated into a stochastic model (Broadley et al. 2012 ; Niedermeier et al. 2011b ;
Wright and Petters 2013 ; Marcolli et al. 2007 ; Stoyanova et al. 1994 ; Barahona
2012 ). In these models, the effects of many different ice-nucleating particles are
summed; however, these models are complex. Barahona ( 2012 ) reinforces the view
that IN tend to be diverse, commenting that inferring the aerosol heterogeneous
nucleation properties from measurements of onset conditions may have significant
error.
An alternative approach to dealing with the diverse nature of atmospheric ice
nuclei is to make the assumption that the time dependence of nucleation is of
secondary importance compared to the particle-to-particle variability of ice nuclei.
It has been shown that individual droplets freeze over a wide range of temperatures,
whereas the variability in freezing temperature on repeated freezing cycles is much
smaller (Vali 2008 ; Wright et al. 2013 ). This shows that in many samples it is the
nature of the IN in the droplets which primarily determines freezing temperature and
the stochastic effects are second order, at least under some conditions. Observations
such as these have given rise to the singular approximation in which the time
dependence of nucleation is neglected. In the singular description, nucleation occurs
instantaneously on an active site once a characteristic temperature is reached. The
fraction of droplets frozen as a function of temperature and saturation ( S ) can be
expressed as (Connolly et al. 2009 ; Murray et al. 2012 ;Demott 1995 )

n ice .T;S/
N tot

f ice .T;S/ D D 1 exp . n s .T;S/s/ (12.5)

where n ice is the number of frozen droplets, N tot is the total number of frozen and
unfrozen droplets, and n s is the active site density. n s is the cumulative number of
nucleation sites per surface area that become active on decreasing T and increasing
S . When IN are immersed in pure water, S is defined by water saturation; hence, n s
is then reported as a function of T only.