Willis added a tropopause break to separate the troposphere from the upper atmosphere. Excellent move! The values on his drawing are true to the more recent K&T cartoon. In analysis I did I used the older version.

In the post I did before I found that a ratio of temperature change from the atmosphere should equal the ratio of the energy flux change in the atmosphere. The flux at the surface 392Wm-2, is supposed to be the energy radiated by a black body at a temperature of 288K degrees. This flux would assume the application of emissivity of the planet and is based on the average temperature. The calculated temperature at the Top of the Atmosphere is 255K degrees for a radiant flux of 237Wm-2. Note that this value is slightly larger than the 235 on their old cartoon and slight lower than the general estimate of 240Wm-2. The difference between 392 and 237 is 155Wm-2 just as it was in the older cartoon. The actually average surface temperature is fairly accurately estimated at 288K and the flux at the TOA is fairly accurately measurable along with the temperature of 255 degrees K. the largest uncertainty appears to be the actual flux rate at the surface. Measuring that flux rate is extremely complicated and the accuracy of the measuring instruments is roughly 2% of the range. For a pure black body, the uncertainty in the emissivity is very small. For a gray body, the emissivity is a bit more debatable, but generally thought to be fairly accurate. The issue is that 288-255=33 has to have a consistent ratio with range of flux change in the atmosphere. So 33/288-30 = 1 and 155/(392-237) = 1 should have a direct relationship by the equation dF/dT=4F/T or a change in T=4F. The estimated sensitivity of the Earth surface to a doubling of CO2 is related to the change in flux by 3.7 1T=3.7F in a change. 3.7 is close to 4, 3.7/4=0.925, which is the estimated emissivity of the atmosphere. Pretty simple. However, in my previous post, the emissivity of the atmosphere appears to be 0.85 and not 0.925. 0.85/0.925=0.92 or 8% less than estimated. That difference could lead to a +/-8% error where contrasting errors could be 16%. Complimentary errors would negate.

Or vice versa depending on how you look at it.

In the flux ratio, since its change is 4 times that of temperature, that error would be multiplied by a factor of 4. So it would only take a relatively small error measuring temperature to result in a fairly large error in calculated flux. Since flux is not easily measured, it makes the use of calculating flux very common. A small error in flux measured would result in a smaller error of temperature calculated. So it would be easy for me or anyone else to have big issues if temperature measurement is wrong and the effective emissivity is off by 8 percent. The best estimate for effective emissivity (e)is 0.85, and 3.7Wm=-2 is estimate to cause a 1.2K increase in temperature, we can arraign the dT=3.7/(0.825*4)=3.7/3.3=1.12K and see that 1.12/1.2=93% of the estimated warming. Effective emissivity is however the variable to be changed causing the increase change in flux. If the Earth were a perfect black body,e=1, so the change in flux from the surface would be 155Wm-2. If the Earth were a regular back body, e=0.926, then the flux change from the surface would be would be 143.5Wm-2, so the surface flux would be 237+143.5=380Wm-2 Well, the Earth's surface is closer to a black body than the atmosphere. A one degree increase in the surface temperature would produce an increased flux of .926*4=3.7 Wm-2. The Earth though is mainly water and the emissivity of water is 0.99, so a one degree increase in the temperature of the oceans would cause a 4Wm-2 increase in flux. Conversely, one degree increase in surface air temperature would only cause a 3.3/4=0.825 degree increase in the ocean temperature, That increase in flux would produce 3.7/3.3=1.12K increase in temperature.

The ocean air interface is the greenhouse effect, radiant heat leaving the ocean surface at 100Wm-2 is restricted to 82 Wm-2 when it hits the atmosphere, conductive heat from water flows at 10,000Wm-2 and is restricted to 100Wm-2, latent heat from the ocean is restricted by the relative humidity of the air. All three modes of heat transport are bottle necked at the surface. Chop created by winds increase the cooling of the oceans, but not the absorption of energy. The Earth's real thermostat.

So looking at the K&T drawing for the surface, Fc=24, Fl=76 and Fr=392-24-76= 292 Where Fc is the thermals heat loss, Fl is the latent heat loss and Fr is the radiant heat loss. Fr though is balanced by down welling radiation created by the radiative bottleneck change from emissivity nearly 1 to 0.825. At the surface we have 292*.825=239Wm-2 leaving, 292-239=53Wm-2, which is the value that should be used for radiative absorption by the atmosphere, not the 18 implied by the 339 in and 321out for the troposphere shown of the drawing. The total of all the downwelling created by Fc, Fl and Fr is 392-239=153Wm-2. Since I undoubtedly had some rounding errors, 155Wm-2 is close enough for government work. So dF/dT=4(cFc+lFl+eFr)/T is perfectly valid at the surface, as long as the coefficients,c and l are small in comparison to e.

**Update: I fixed a small error and missed one calculation. 239-155=84 which would be Frnet or the net radiative flux through the atmosphere from the surface proper. 51 of that should be absorbed in the troposphere and 33 radiated to space. Since water as a liquid and solid in the atmosphere can radiate in the atmospheric window, that should be shown at the top of the troposphere**

There appears to be an issue with the K&T drawing right at the surface. DWR should be between 206 and 239 not 321.

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