Twomey effect

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 water.

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.

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