Posted by Jeff Id on October 25, 2010
M10 linked again here, starts with the following equation.
dQ = CvdT + PdV + L dλ — Eq 1, from first law of Thermo.
From this equation many on and offline comments were sent to me which claimed heat release was a far greater concern, gavin placed one here although his latent heat number is not correct, in fact it’s kind of all over the place. We’ve learned that several climate models assume exactly this fact though so I tried to improve on Gavin’s calculation below.
One Kg of saturated water at 30 C at constant pressure. What happens to the temperature when all the water is magically condensed in the atmosphere at constant pressure (isobaric).
Total vapor amount from this table = 0.027Kg which corresponds to 4 percent of 1 atmosphere vapor pressure.
Heat of vaporization/condensation for H20 at 30C is 2,429,000 J/kg. Converting 0.027 Kg of condensation releases 65,000 Joules of energy.
Q =k*P*(V2 – V1)/(k – 1)
Where k is the ratio Cp/Cv for air = 1000/713 = 1.4
V2-V1 = Q(k-1)/(kP) = 65000J *(1.4-1)/(1.4*101000) = 0.184 m^3
Saturated air from the table linked above has a density of 1.11 Kg/m^3 or 0.896 m^3/kg.
So 1.1kg/(.896+.184) = 1.01kg/m^3 density. The net volume change is 20% by this equation directly but in order to do it we lost 4 percent of the vapor pressure. So in the end we got about 16 percent expansion. With a 5 x differential between the effect of mass loss from condensation to the energy released.
This means that the energy content of the water L dλ is the biggest factor in Eq 1.
dQ = CvdT + PdV + L dλ
Basically this shows that condensation does cause gas expansion, it also causes an additional volume loss which is not usually incorporated in models but is a smaller factor.
Switching topics, in Eq 37 of M10, the formula was derived completely without any latent heat, and when observed values were plugged in, the equation predicted reasonable values for some of the biggest storms on earth – again while apparently completely ignoring latent heat of condensation. Considering that the latent heat is so much larger, this should be a surprising result.
So far with respect to models we’ve received comments from several experts who agree that the condensation volume loss is a minor factor, so how is it that M10 can predict such good values from basic equations.
Now my question is, is this equation really independent from latent heat when observed values for h gamma and w/u are plugged in. The fact that γ itself is inserted at 0.03 doesn’t convince me that we still aren’t just seeing mixed effects that happen to give reasonable values for dp/dx which is the reason the post is titled independence. Would observed hv and w/u already contain the effects of latent heat of condensation?
The relevent paragraph form section 4.1 is here:
The argument Anastassia made by email, was that the thermal heat release was so inefficient that it can be neglected. Climate and weather models exist that use only the latent heat effect. Maybe someone can provide more insight but somebody is wrong, models or M10.