Just a detail on this page: the link to Kiehl/Tremberth inbetween has gone out of life 😦

Interested readers might find the text under

http://journals.ametsoc.org/doi/abs/10.1175/1520-0477%281997%29078%3C0197%3AEAGMEB%3E2.0.CO%3B2

]]>Steve,

The difference in surface area is very small.

Typically, for reasons that can get quite technical (see note), the values at the tropopause are used. This is around 15km above the surface.

But if instead we used the values at the top of the stratosphere this would be only 50km.

The radius of the earth is 6370km.

So the area at the surface, using a sphere (which is a bit of an approximation) is 4.pi.6370^{2} = 510M km^{2}.

And the area at the tropopause is 4.pi.6385^{2} = 512M km^{2}.

Another way to look at it, the ratio is (6385/6370)^{2} = 1.005 or 0.5% difference.

If we considered the top of the stratosphere, the ratio is (6420/6370)^{2} = 1.016 or 1.6% difference.

The graphical representations of the atmosphere above the earth are for education, they are never (?) drawn to scale.

Technical note: this is probably best explained in Wonderland, Radiative Forcing and the Rate of Inflation.

]]>But that is not correct. The surface area of the sphere at TOA is much larger than the surface area of the sphere formed by Earth’s surface. So – a given amount of radiation is radiating outward from a given sphere, and penetrating a larger sphere. The radiation per unit of surface area (square meter) will always go down at the surface of the larger sphere,even if there is no absorption.

The surface area of earth is roughly 4 pi (6,360 km)^2

I do not know what radius was used for the TOA surface.

But it should be easy to determine the amount of emission decrease per square meter caused by simple diffusion of radiation, subtract that from the earth’s surface value, and then determine absorption up to the TOA by subtracting the TOA emission value from the previous result.

Or – have I missed something?

]]>A temperature profile of -15 K/km is defined as a lapse rate of 15 K/km. So by convention a temperature profile of -15 K/km has a lapse rate *greater than *a temperature profile of -3 K/km. Any lapse rate greater than the adiabatic lapse rate is unstable because a packet of air when raised to a higher altitude will be warmer and less dense than the air around it at the new altitude. That means buoyancy will keep the packet moving upwards, In other words, convection will ensue. The adiabatic lapse rate is inversely proportional to the heat capacity at constant pressure of the air, Cp. Increased humidity increases Cp and lowers the adiabatic lapse rate, i.e. makes the adiabatic temperature profile decrease less rapidly with altitude.

Pat: It may also help to know that lapse rate is defined as -dT/dz. So lapse rates are usually positive numbers that tell us how much the temperature DROPS with altitude. When SOD talks about a “temperature profile” of -15K/km, he is also talking about a lapse rate of 15K/km.

Life gets more confusing when looking at plots of temperature (x-axis) vs altitude, which is on the y-axis by tradition (even though altitude is the independent variable). Slope on such plots has units of km/K – the reciprocal of the units for lapse rate. Regions with a gentle slope downward from left to right are those with the greatest lapse rate (or most negative temperature profile). It seems a little strange to me that flat or nearly flat regions are convectively unstable (as are regions with positive slope), while the stable regions are those with a steep negative slope.

In the gray atmosphere (Figure 2.9 above), the atmosphere unfortunately doesn’t have enough GHG to produce a regions with a lapse rate larger (or flatter) than 6.5 K/km that would be convectively unstable. In our atmosphere, the concentration of water vapor increases dramatically at lower altitudes, so the curve for pure radiative equilibrium flattens dramatically and doesn’t intersect the x-axis until well above 300 degK.

]]>Pat,

I’m usually unsure (and probably inconsistent article to article) about which words to use to compare two lapse rates due to the negative value implied. So apologies for any confusion caused.

What you have written is correct.

Just to confirm:

If the temperature profile due to radiation alone was -15K/km then convection would take over – because the atmosphere would be unstable, and if the temperature profile due to radiation alone was -3K/km then convection would not take place – because the atmosphere would be stable.

If the “adiabatic lapse rate” is a lower (absolute) value than the environmental lapse rate then the atmosphere is stable.

I’ve written at length about the lapse rate in:

Density, Stability and Motion in Fluids – some basics about instability

Potential Temperature – explaining “potential temperature” and why the “potential temperature” increases with altitude

Temperature Profile in the Atmosphere – The Lapse Rate – lots more about the temperature profile in the atmosphere

]]>I have just come across your site and i have to say how really useful it is. There is so much well explained. I am puzzled by your statement

“if the energy transfer from radiation at any point in their vertical profile resulted in a temperature gradient less than that from convection then use the known temperature profile at that point. And if it was greater than the temperature gradient from convection then we don’t have to think about convection in this “slice” of the atmosphere.”

Should this not be the other way round. At higher altitudes the lapse rate is lower in the “gray model” and is it not here that we dont have to think about convection because radiation dominates? Or have I mistaken some reasoning? ]]>