A More Accurate Multiplier

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 energy.

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.

Figure 5. Figure 2, shown again

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.


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.


  • 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.

via Watts Up With That?


September 17, 2022 at 12:35PM

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