Twomey's seminal 1959 paper provided lower and upper bound
approximations to the estimation of peak supersaturation within an
updraft and thus provides the first closed expression for the number
of nucleated cloud droplets. The form of this approximation is
simple, but provides a surprisingly good estimate and has subsequently
been employed in more sophisticated treatments of nucleation
parametrization. In the current paper, we revisit the lower bound
approximation of Twomey and make a small adjustment that can be used
to obtain a more accurate calculation of peak supersaturation under
all potential aerosol loadings and thermodynamic conditions. In order
to make full use of this improved approximation, the underlying
integro-differential equation for supersaturation evolution and the
condition for calculating peak supersaturation are examined. A simple
rearrangement of the algebra allows for an expression to be written
down that can then be solved with a single lookup table with only one
independent variable for an underlying lognormal aerosol
population. While multimodal aerosol with

Atmospheric aerosols are fundamental to the formation of clouds. They
provide the substrate onto which liquid droplets can form, overcoming
the energy barriers associated with clustering of water molecules

The approach generally used to ascertain how many of the total
population of the underlying interstitial aerosol will grow into
droplets is based upon solving a system for the time variation of
supersaturation. Supersaturation is key, since a given aerosol
particle will activate to form a droplet at a particular critical
supersaturation,

Here, the cooling term is represented by

Equation (

Equations (

Twomey obtained this bound using a simple geometric argument that
considers the areas of triangles bounded above by a line whose slope
is given by

The following sections revisit this lower bound approximation, develop
a slight improvement and subsequently employ this revised
approximation in a new parametrization scheme for inclusion in cloud
resolving models and General Circulation Models. Section

The peak supersaturation,

Lower bound approximation to integral under the supersaturation curve. The red hatched region represents the lower bound approximation of Twomey, the black hatched region represents the revised approximation.

Retaining Twomey's approximation in Eq. (

Twomey used a differential activity spectrum of the form

This represents a superposition of

Here

The resulting expression when using this information in
Eq. (

The first thing to note about this expression is that all the thermodynamic and dynamic information, i.e. temperature, pressure and vertical velocity, is held on the left hand side of this equation. Thus for fixed aerosol characteristics, the right hand side need only be calculated once to be used in a variety of thermodynamic states. However, if aerosol characteristics also vary in space and time (as is usually the case in a GCM) we cannot exploit this feature of the equation without precalculating all possible aerosol states.

Nevertheless, the computationally challenging part of evaluating the
right hand side comes from the integral terms. In this form, in
addition to

For single mode aerosol, the right hand side of
Eq. (

For a multimodal distribution, Eq. (

The method used in Sect.

Figure

However, if we maintain this trapezoidal representation, then it can
be clearly seen that taking the gradient of the upper edge to be equal
to the mean gradient of the supersaturation curve,
i.e.

A feature of the evolution of the supersaturation gradient is that we
know that for small values of time,

It is noted that this functional form, i.e. with argument

To investigate the behaviour of

We choose to fit a curve through these data of the form

Having facilitated a suitable expression for

Our revised approximation for the inner integral of

Scaled

In order to understand the impact of this revised approximation, the
relation (

Evolution of supersaturation using Whitby marine aerosol loading.
Solid line shows numerical integration of the full equation set in Eq. (

Section

Evaluation of peak supersaturation for the parametrization with the
original (Eq.

The recent paper of

However, as a sample demonstration, plots are presented in
Figs.

As Fig.

The first thing to note from these plots is the discrepancy between
the full numerical implementation and the results obtained from the
numerical integration of Eq. (

Despite these differences, it is apparent from the data in
Fig.

A more comprehensive analysis of the performance of this and other
parametrizations has been conducted by

A new method for parametrizing peak supersaturation,

This method in itself represents a much more computationally efficient method for determining aerosol activation and links through to the underlying aerosol physicochemistry. However, another benefit is that it is not constrained to use the fixed form of the lower bound approximation of Twomey (1959), which is frequently used to make analytic integration feasible, and so a more accurate approximation is derived. This newly derived approximation perhaps lacks the elegance of Twomey's original estimate, but is shown to faithfully reproduce the evolution of supersaturation and the calculation of the peak supersaturation across a range of scenarios.

The underlying equation set on which the parametrization is built
makes a number of physical assumptions, which are common to many
well used activation parametrizations

A complete derivation of Eq. (

The expression for

List of symbols.

These approximations remove the dependence of the right hand side
of Eq. (

To complete the derivation of Eq. (

By substituting Eq. (

The choice to approximate

However, Pinsky et al. (2013) use a scale analysis to demonstrate that the
time at which peak supersaturation is achieved,

The result of Pinsky et al. (2013) further provides a basis for testing the
validity of the parametrization of the gradient of

Solution space for the parameters

Thus the mean

The works published in this journal are distributed under the Creative
Commons Attribution 3.0 License. This license does not affect the Crown
copyright work, which is reusable under the Open Government Licence (OGL).
The Creative Commons Attribution 3.0 License and the OGL are interoperable
and do not conflict with, reduce or limit each other.^{©}Crown copyright
2014

The author would like to thank Daniel Partridge for providing useful
feedback on the performance of the parametrization and Steve Ghan for
providing the data for comparison in Figs.