content_image/370905.jpg content_image/370910.jpg content_image/370906.jpg content_image/370911.jpg content_image/370913.jpg content_image/370914.jpg content_image/370915.jpg content_image/370920.jpg
## VI. INSTABILITY 

Expanding Eq. 1 to first order in δT : 

where 

δT grows or decays with an exponentiation rate ν : 

<img src='content_image/370905.jpg'>

<img src='content_image/370910.jpg'>

<img src='content_image/370906.jpg'>

To estimate the derivative in Eq. 6 we use an approximate equation 10 for the saturation water vapor pressure over a surface of temperature T : 

<img src='content_image/370911.jpg'>

where Δ H = 2260 J/g is the enthalpy of evaporation of water and k$_{B}$ is Boltzmann’s constant. For a Lorentzian line profile with τ 0 ≫ 1, the width and f$_{b}$ at a specified optical depth O (1) 1 2 1 2 

<img src='content_image/370913.jpg'>

Then 

and 

<img src='content_image/370914.jpg'>

<img src='content_image/370915.jpg'>

where we have made the tacit assumptions 1 − f$_{b}$ ( T 0 ) > τ − 1 / 2 0 , which amount to assuming that there is enough unblocked spectrum for the line to broaden according to Eq. 8. The appropriate value of τ 0 in this condition is an average over all lines that are optically thick at their cores. This is likely to be dominated by the very numerous lines for which τ 0 exceeds unity by a factor of a few, but not by orders of magnitude, so these results may be valid for f$_{b}$ ≲ 0 . 7. The model breaks down as f$_{b}$ → 1 because then there is little unblocked spectrum into which the lines can broaden. 

Evaluating at a sea surface T 0 = 290 $^{◦}$K, 

<img src='content_image/370920.jpg'>

