# Twomey effect

The Twomey effect describes how cloud condensation nuclei (CCN), possibly from anthropogenic pollution, may increase the amount of solar radiation reflected by clouds. This is an indirect effect.

Aerosol particles can act as CCN's creating more droplets which have a smaller size distribution. The reduction in size distribution increases the Optical depth of the cloud. This increases the cloud albedo as clouds appear whiter and larger, leading to a cooling of between -0.3 and -1.8 Wm−2.[1] For example, on satellite imagery we observe trails of white clouds from ships crossing the oceans due to this effect.

## 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.