VI. INSTABILITY Expanding Eq. 1 to first order in δT : 1 ν 0 T 0 dδT dt ≈ [ df$_{b}$ ( T ) dT − 4 T 0 (1 − f$_{b}$ ( T 0 )) ] δT, (4) ν 0 ≡ σ$_{SB}$T 3 0 C$_{p}$ . (5) where δT grows or decays with an exponentiation rate ν : 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 : ν ν 0 = T 0 df$_{b}$ ( T ) dT − 4 (1 − f$_{b}$ ( T 0 )) . (6) p$_{v}$ ( T ) ∝ T 3 / $^{2}$exp ( − Δ H/k$_{B}$T ) , (7) 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) f$_{b}$ ( T ) ≈ f$_{b}$ ( T 0 ) ( τ 0 ( T ) τ 0 ( T 0 ) ) 1 / 2 ≈ f$_{b}$ ( T 0 ) ( p$_{v}$ ( T ) p$_{v}$ ( T 0 ) ) 1 / 2 . (8) df$_{b}$ ( T ) dT ∣ T 0 = 1 T 0 f$_{b}$ ( T 0 ) ( Δ H 2 k$_{B}$T 0 + 3 4 ) (9) ν ν 0 = f$_{b}$ ( T 0 ) [ Δ H 2 k$_{B}$T 0 + 3 4 − 4 (1 − f$_{b}$ ( T 0 )) ] , (10) Then and 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, ν ν 0 = 13 . 2 f$_{b}$ ( T 0 ) − 4 . (11) 5