# Twomey effect

The Twomey effect describes how additional cloud condensation nuclei (CCN), possibly from anthropogenic pollution, may increase the amount of solar radiation reflected by clouds. This is an indirect effect (or radiative forcing) by such particles, as distinguished from direct effects (forcing) due to enhanced scattering or absorbing radiation by such particles not in clouds.

Cloud droplets normally form on aerosol particles that serve as CCN. Increasing the number concentration of CCN can lead to formation of more cloud droplets, which, in turn, have smaller size. The increase in number concentration increases the optical depth of the cloud, which results in increase in the cloud albedo making clouds appear whiter. Satellite imagery often shows trails of cloud or of enhanced brightness of cloud behind ocean-going ships due to this effect. The decrease in global mean absorption of solar radiation due to increases in CCN concentrations exerts a cooling influence on climate; the global average magnitude of this effect over the industrial era is estimated as between -0.3 and -1.8 Wm−2.[1]

## Derivation

Assume a uniform cloud that extends infinitely in the horizontal plane, also assume that the particle size distribution peaks near an average value of ${\displaystyle {\bar {r}}}$.

The formula for the optical depth of a cloud:

${\displaystyle \tau =2\pi h{\bar {r}}^{2}N}$

Where ${\displaystyle \tau }$ is the optical depth, ${\displaystyle h}$ is cloud thickness, ${\displaystyle {\bar {r}}}$ is the average particle size, and ${\displaystyle N}$ is the total particle density.

The formula for the liquid water content of a cloud is:

${\displaystyle LWC={\tfrac {4}{3}}\pi {\bar {r}}^{3}\rho _{L}hN}$

Where ${\displaystyle \rho _{L}}$is the density of air.

Taking our assumptions into account we can combine the two to derive this expression:

${\displaystyle \tau ={\tfrac {3}{2}}{\tfrac {LWC}{\rho _{L}{\bar {r}}}}}$

If we assume liquid water content (${\displaystyle LWC}$) is equal for the cloud before and after altering the particle density we obtain:

${\displaystyle {\bar {r_{2}}}={\bar {r_{1}}}\left({\frac {N_{1}}{N_{2}}}\right)^{\frac {1}{3}}}$

Now we assume total particle density ${\displaystyle N}$ is increased by a factor of 2 and we can solve for how ${\displaystyle {\bar {r_{1}}}}$ changes when ${\displaystyle N}$ is doubled.

${\displaystyle {\bar {r_{2}}}}$= ${\displaystyle 0.79{\bar {r_{1}}}={\bar {r_{1}}}\left({\frac {N_{1}}{2N_{1}}}\right)^{\frac {1}{3}}}$

We can now take our equation that relates ${\displaystyle \tau }$ to ${\displaystyle LWC}$ to solve for the change in optical depth when the particle size is reduced.

${\displaystyle \tau _{2}={\frac {\tau _{1}}{0.79}}=1.26\,\tau _{1}}$

In more general terms, the Twomey effect states that for a fixed liquid water content ${\displaystyle LWC}$ and cloud depth, the optical thickness can be represented by:

${\displaystyle \tau \varpropto N^{\tfrac {1}{3}}}$

This brings us to the conclusion that increasing the total particle density also increases the optical depth, illustrating the Twomey Effect mathematically.