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 is

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

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 number density of cloud droplets.

The formula for the liquid water content of a cloud is

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

where ${\displaystyle \rho _{L}}$  is the density of water.

Taking our assumptions into account we can combine the previous two equations to yield

${\displaystyle \tau ={\frac {3}{2}}{\frac {h\,LWC}{\rho _{L}{\bar {r}}}}}$ 

To derive the effect of changing ${\displaystyle N}$  while keeping ${\displaystyle h}$ , ${\displaystyle \rho _{L}}$  and ${\displaystyle LWC}$  constant, from the last equation we can write

${\displaystyle \tau \propto {\frac {1}{\bar {r}}}}$ 

and from the equation for ${\displaystyle LWC}$  we can write

${\displaystyle {\bar {r}}^{3}\propto {\frac {1}{N}}}$ 

therefore

${\displaystyle \tau \propto N^{1/3}}$ 

This illustrates the Twomey Effect mathematically, that is, for a constant liquid water content, ${\displaystyle LWC}$ , increasing the number density of cloud droplets, ${\displaystyle N}$ , increases the optical depth of the cloud.

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