Guest Post by Willis Eschenbach
In my previous posts, yclept “Greenhouse Efficiency” and “The Multiplier”, I described a metric I’d developed to look at how successful the very poorly named “greenhouse effect” was at warming the surface. The metric was the upwelling surface longwave radiation (in watts per square meter, W/m2) divided by the solar power actually absorbed by the system (solar minus albedo reflections).
And of course, since radiation emitted by an object can be used to determine the temperature, this metric also measures how efficiently the incoming sunshine is converted to surface temperature.
Here’s how that metric has changed over time, as discussed in my two previous posts.
Figure 1. Greenhouse multiplier. The multiplier is calculated as upwelling longwave surface radiation divided by incoming solar radiation (after albedo reflections). A multiplier of 2 would mean that the surface would be radiating two W/m2 of energy for each one W/m2 of solar energy actually entering the system. This shows that the greenhouse has increased the incoming solar radiation by about two-thirds, as measured at the surface.
I got to thinking about that, and after a while I realized that that doesn’t tell the whole story. I realized that the answer was distorted because I hadn’t included advection.
Advection is the horizontal transport of heat. Generally, it’s in the form of moving ocean and atmosphere. It generally flows, as you might expect, from warm to cold—from the equator to the poles. Here’s a map showing the average transport, both export (red) and import (blue) of power.
Figure 2. Advection (horizontal transport) of power from the tropics to the poles.
The issue with not including advection comes up especially in the polar regions. There, a large amount of the power input is from advection, and little is from sunlight. So it looks as thought the power in the sunlight is highly multiplied, but it’s not—the extra power is from advected atmosphere and ocean.
Since I wanted a measure of the total watts out divided by watts in, the true multiplier, I had to include the advected energy. Since the net energy advected is about zero, I didn’t expect it would change the overall average multiplier by much. But I did expect it to be more accurate on a gridcell-by-gridcell basis since it was no longer missing the advected energy.
And indeed, this was the result. Figure 3 shows the earlier calculations as in Figure 1 (blue), plus the calculation including the advected energy (red).
Figure 3. A comparison of the two metrics, which either include (red) or omit (blue) the advection.
Now, there are some interesting things about this figure.
First, as I’d hoped, regarding the standard deviation (SD) of the detrended results, it is smaller when we include the advected power. This means they cluster more tightly around the trend line. The SD of the original method (blue) is 0.0040, and of the method including advection (red), it’s 0.0026.
Including the advection also corrects the problems at the poles which import copious power, and the problems at the tropics, which export the same.
Figure 4. Average multiplier, both with (red) and without (blue) the advection. Average CERES data, Mar 2000 to Feb 2021
I love the surprises of science. The surprise in this one for me was that once we’ve included the effects of advection, the multiplier is pretty much equal from the North Pole down to the north tip of Antarctica.
Next, a small digression. Ramanathan pointed out that we can measure the poorly-named “greenhouse effect” directly. It is the amount of upwelling longwave power absorbed by the clouds, aerosols, and greenhouse gases in the atmosphere. Note that the power absorbed ends up back at the earth’s surface.
The size of the “greenhouse effect” is measured as the upwelling longwave at the surface minus the upwelling longwave at the top of the atmosphere (TOA). The difference between the two is the “greenhouse effect”, in watts per square meter.
Here’s the most surprising oddity. It turns out that when we include advection in our surface power changes, the new multiplier is exactly equal to one plus the greenhouse effect (measured as above) divided by available solar energy. Math in the footnotes.
And this lets us understand what is happening in Figure 2. The blue trend is the change in surface upwelling per unit of incoming energy. This is measured above in W/m2, but it can be converted using the Stefan-Boltzmann equation to the surface temperature. That multiplier has been decreasing.
The red trend is the trend of the change in total surface power, not just the radiation but the advection as well, per unit of incoming energy. That multiplier has been increasing as we’d expect given increasing levels of CO2.
And that is very interesting. It shows that overall, increasing greenhouse gases increase the amount of downwelling radiation per unit of incoming solar power. And in fact, they are increasing at the rate expected from the increasing concentration of CO2. But that’s not what is expected overall.
Figure 6. Changes in the efficiency of the “greenhouse effect”, as measured by the multiplier.
The first reason that the increase is less than expected is that there are other greenhouse gases besides CO2 (methane, N20, chlorofluorocarbons). So with those other gases, the increase in greenhouse efficiency as measured by the multiplier should have been more than it actually has been.
There is also the purported positive cloud feedback and the water vapor feedback. Like the effect of other greenhouse gases, these should also have increased the multiplier.
So it appears that there are unknown countervailing forces preventing a larger increase in greenhouse efficiency despite increases in a variety of greenhouse gases. However, the net of all of these is a slight increase in greenhouse efficiency.
But curiously, this is counterbalanced by a reduction (per unit of incoming solar power) in the amount of increase in surface temperatures, along with a corresponding increase in the advection.
And the net result of the two is that in Figure 2 the multiplier shown by the blue line (how efficiently the system multiplies the incoming energy into surface temperature) is trending down, despite the increasing efficiency of the “greenhouse effect” due to increasing GHGs as shown by the red line.
“Simple physics”?
I don’t think so.
w.
MATH: Here are the equations showing that
surface upwelling + advection / available solar
is equal to
one plus the greenhouse effect (as defined by Ramanathan) / available solar.
Where:
- SOLAR = TOA incoming solar power
- SWtoa = Upwelling (reflected) shortwave measured at the top of the atmosphere
- LWtoa = Upwelling (emitted) longwave measured at the top of the atmosphere
- LWsurfup = Upwelling longwave at the surface
Advection is measured as the amount of solar power (shortwave) entering the gridcell (SOLAR) minus the amount of radiated power leaving the gridcell to space (SWtoa + LWtoa). The difference must be advected, except for a very small fraction that raises or lowers the surface temperature and can be neglected at this level of analysis.
My Usual Note: When you comment PLEASE quote the exact words you’re discussing. I can defend my own ideas. I can’t defend someone else’s random claim about what they think I said.
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The Diabatic Model fully matches the Vacuum Planet Equation only when I average Flux and not Temperature.
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Philip the S-B law when solved in the reverse direction gives you the upper limit on temperature, not the actual temperature.
The reason is that VPE assumes a 1/4 flat disk with zero variance. I have demonstrated the math numerous times.
Maximum temp works for GHG theory because it is sufficient to show max temp -18+33 would not be possible without ghg.
Your model does not have zero variance. So what temperature are you trying to match? A maximum or gridded average?
Another approach would be to solve min/max/avg in your model then try and narrow the limits.
Ferd,
You have lost me here. Please provide a link to show this work you are referring to.
Philip
Search WUWT for the holder inequality.
What it shows is that avg(T) < avg (T^4).
So, we know GISS and others have built their temperaturs as avg(T) by sampling surface temps and get 15C as the answer.
And we know that using avg(T^4)
climate scientists get an average temp of 15C.
And we know from the holder inequality that these results cannot be equal, that avg(T) < avg (T^4), so at least one and possibly both calculations of 15C are wrong.
For exanple
avg (2,2),=2
Avg (1,3)=2
Avg(2^4,2^4)=16
Avg(1^4,3^4)=41
The holder inequality shows that S-B Law is not reversible unless you know the variance.
Therefore your hot cold model is fundamentally different than a disk of uniform temperarure, because you have non zero variance..
Ferd,
This is where I lose track of your thoughts.
Above you also state this
The VPE is NOT modelling a flat disk!
Please tell me that you do recognise that the VPE is modelling the illuminated surface of a sphere with surface area 4Pir^2 and that the power intercepted from the solar beam has a cross-sectional area of Pir^2 so the power dilution (intercept to surface distribution) will be 1/4.
The does not make the relationship into a flat disk.
Ferd,
Please let me be blunt. In our work we are presenting a re-design of the standard concept by changing the power dilution factor from 1/4 – total instantaneous surface illumination of the whole globe, to 1/2 – accurate representation of the real single hemisphere illumination.
Consequently, we must follow all of the rest of the standard logic. I cannot match your understanding of Hölder’s inequality and so need to stop here.
Thank you for raising the issue of lapse rate I have corrected 8 K/km MELR to 6.5 K/km MELR in the tables where necessary and added this comment to our text.
Ferd,
A good night’s sleep and I wake with a new thought. You say:
I am going to assume that by variance you mean variation and so I will pick up from there. As I stated above the purpose of our work is to match the logic flow of the prior authors’ work. We have demonstrated that their work creates a single surface flux value of 390 W/m^2 and we have achieved this value with our DAET model by averaging surface flux, thereby matching them.
Once you move into the realm of averaging surface temperature (which is where I think that you are) then the issue of power flux variation over the surface of the lit hemisphere from zenith to terminator becomes paramount and we move into the complex realm spherical geometry.
Please note that when dealing with the averaging of surface temperature we approach this issue from the domain of meteorology and not that of radiative physics. Here is our technique of how we achieve this:
Return to Earth: A New Mathematical Model of the Earth’s Climate
So why does climate science with its flat surface toy model ignore spherical geometry? Because the planetary disk can NEVER intercept more power from the irradiance of the solar beam than the cross-sectional planar surface equivalent area of its disk shadow silhouette.
Ferd,
A final comment on flat surface models and spherical geometry.
Note that in all of this climate modelling everyone uses the Bond Albedo.
The Bond Albedo is a spherical filter that incorporates all of the geometric effects of the illumination of the lit hemisphere by the incident solar beam into its computation. Consequently the need to apply Hölder’s inequality is avoided.
This comment by Joseph Postma, astrophysicist, must have been really disturbing for Willis.
Get rid of the sphere, it is the shell itself which is its own power source. The shell itself produces the power inside its own material. There’s no sphere inside the hollow cavity of the shell. The power production is 240W, the surface area of the shell is 1m^2, and the shell is negligible thickness. What’s the temperature of the shell? What is its surface flux?
leitmotif September 22, 2022 3:48 pm
“Disturbing”? Don’t make me laugh.
Sorry to disappoint you, leitmotif, but the comment by Joseph Postma, half-ass-trophysicist, was a joke. All it did was prove he didn’t understand the example.
w.
“The difference must be advected, except for a very small fraction that raises or lowers the surface temperature and can be neglected at this level of analysis.”
Is that really true for portions of the earth’s surface that have ice that melts and refreezes each year?
Using the CERES data, there is a calculation of the total amount of SW radiation absorbed by the surface in the Arctic here:
https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2021GL095813
Meanwhile melting a mass of ice 1 meter thick takes about 333MJm-2. So quite a chunk of the absorbed radiation goes to melting ice in these regions – being returned when the ice refreezes, of course. The concentration of both melting and refreezing into relatively short periods should give a noticeable effect.
I am going to assume that by variance you mean variation
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No. I means statistical variance. The power of the difference about the mean.
Calculate average temperature using simple averages and s-b for 3 cases of a 2 hemisphere earth with temps:
15C, 15C
10C, 20C
0C , 30C
These all have the same avg but different variance. Does this average also hold true using S-B?
If you average radiation does S-B give you 15C in all 3 cases? The holder inequality says it won’t.
If not how do you resolve gridded avg temps like GISS and astronomer’s temp of earth?
I don’t think you will ever get an answer. There should be a variance provided along with all of the climate science results but there never is. What does the Global Average Temp mean if you don’t have a variance to go along with it?
Isn’t this an empirical confirmation of Javier’s hypothesis, that meridional transport is having a primar climatic influence?