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Temperature changes $\Delta T$ are related to radiative forcing $\Delta F$ linearly (to a first approximation):

$\Delta T = \lambda \Delta F$

I was thinking how to estimate the climate sensitive parameter $\lambda $. If the Earth had no atmosphere and we treat it as a black-body (reflecting 30% of the light) we get a temperature of 255K. With an atmosphere (and the greenhouse effect), we get a temperature of 288K, which is 33K higher. The atmosphere absorbs 23% of solar radiation. The solar irradiance is 342 W/m$^{2}$, which means 78.66 W/m$^{2}$ is absorbed by the atmosphere. This implies:

$\lambda = 33 / 78.66 = 0.42 \frac{K}{W/m^{2}}$

The total increase in radiative forcing (RF) since the industrial revolution is 2.72 W/m$^{2}$ and would give a temperature increase of 1.14K, which is very consistent with current observations. CO2 at 420 ppm accounts for 2.17 W/m$^{2}$ of the RF, while at 560 ppm for 3.71 W/m$^{2}$, which would give an additional 0.65K of warming, or roughly 1.8K overall since the industrial revolution. The Wikipedia article https://en.wikipedia.org/wiki/Radiative_forcing gives a value of $\lambda$ of 0.8, almost double the above estimate. What's wrong with the above reasoning? I know there are many complexities and would like to know some of them. However, it just strikes me that the above calculation should hold and be at least a first estimate of $\lambda$.

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First, the biggest problem with your calculation is that the radiative forcing is defined as the difference in radiated energy flow up and down at the top of the atmosphere, and you are using the shortwave incoming energy absorbed by the atmosphere - an entirely different quantity.

The downward flow of energy at the top of the atmosphere is unaffected by atmospheric absorption, which all occurs below that altitude after it has been measured. And you don't mention the upward flow from the atmosphere emitting longwave radiation to space at all.

There are several other problems; some minor. The absorption of incoming shortwave radiation doesn't have anything to do with the greenhouse effect, which is to do with the altitude of outgoing emission. The effects are often non-linear (e.g. the Stefan-Boltzman law says black body radiation power emitted is proportional to temperature to the fourth power) so taking two points on a curve and calculating the gradient gives different answers depending on the points selected. The Bond albedo of the Earth you have used includes the effect of reflection from clouds, which wouldn't exist if there was no atmosphere. If it's reflecting light it's called a grey body, not a black body.

However, some of the problems are not from your analysis, but arise from this definition of climate sensitivity, which is technically inconsistent. It is defined as the ratio of the energy imbalance at the top of the atmosphere to the change in equillibrium surface temperature. First, if there is an energy imbalance, there can be no equilibrium, and the temperature will keep on rising or falling indefinitely. Second, there are effects that change the surface temperature that have nothing to do with the top of atmosphere energy imbalance, and don't change it - giving an infinite climate sensitivity. Third, the energy imbalances reported would give rise to very fast rates of change of temperature that are not observed. For comparison, 1 W/m$^2$ is 31 MJ/m$^2$ per year, or 3.1 GJ/m$^2$ per century. A cubic metre of water takes around 4.2 MJ to heat it up 1 C. There are about $10^4$ kg of air above each square metre, taking about 10 MJ to heat by 1 C. A sustained heat imbalance of several Watts per square metre over most of a century would result in massive and very obvious warming.

It appears that what is being calculated is the result of a step change in input on the initial energy imbalance before equilibrium is restored. If you increase incoming radiation by 1 Wm$^{-2}$, the temperature rises until the outgoing radiation also increases by 1 Wm$^{-2}$ and equilibrium is restored. The amount of heat absorbed depends on how long it takes to do so. But in most cases this is a thought-experiment. The input effect is varied continuously, and the climate responds continuously.

This means that the meaning of the numbers depends on the particular details of the calculation used. What is considered part of the effect and what is part of the climate's response? What effects are included or excluded? On what timescale? I suspect it is a useful rule-of-thumb number for climate scientists comparing the sensitivities of different climate models, but it is very specific to the way they calculate it, and difficult to interpret or use in other contexts as an objective physical quantity.

As a side note, I would observe that the 'typical' value of 0.8 probably comes from the climate models, almost all of which have been running hotter than actual observations of the climate, most by a factor of 2 or more. At what point they should be considered scientifically falsified is of course a controversial subject, but there's clearly at the very least a broad band of uncertainty, so I wouldn't worry about getting an answer a little bit different.

And finally, I'd note that the mechanism of the greenhouse effect in a convective atmosphere does not change the balance of radiation at the top of the atmosphere. The effect is the result of the moist adiabatic lapse rate - which is the rate of temperature change with altitude due to compression/expansion of air as it falls and rises convectively, 6.5 K/km on Earth - and the average altitude of emission of longwave radiation to space, which for our IR-opaque atmosphere full of greenhouse gases is about 5 km above the surface. The air at 5 km altitude equilibrates at 255 K, the surface is $6.5\times 5=32.5$ K warmer due to compression of the air that descends. More greenhouse gases raises the altitude of emission to space, increasing the compressive heating from there down to the surface, but it doesn't change the energy balance at the top of the atmosphere. (Plus, if the greenhouse gas is water vapour, it reduces the moist adiabatic lapse rate, which reduces surface heating. It's complicated.) The IR-visible emitting surface still equilibrates at 255 K - it's just higher up. To get an energy imbalance, you have to do something like increase the altitude of emission to space but leave all the temperatures the same - a hypothetical situation which doesn't happen. Convection erases such imbalances in a matter of minutes to hours, while greenhouse gas changes have been over periods of decades. And there's no reason why the resulting calculated surface temperature sensitivity should be the same as for any other contributor to climate change.

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    $\begingroup$ Thanks for this. There is a basic question: If only the top of the atmosphere counts for radiative forcing, why it is used for greenhouse gases which are dispersed throughout the whole atmosphere? Either way, I think you got the idea for the calculation I was trying to do: use two different states of the planet (with and without the atmosphere) to get the climate sensitivity. Is there some simple modification to make the calculation more consistent or realistic? Or are there too many unknowns? $\endgroup$
    – Sebby
    Jan 9, 2023 at 0:51
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    $\begingroup$ As I said, I suspect the reason it's used is that it is an easy way to compare disparate effect and models on the same scale. Personally, I think the climate is far too complicated, with too many unknowns and the observational data too short and too poor quality to be making reliable estimates in the current state of the science. Simply put: we don't know. I keep an eye on researchers like Nic Lewis (e.g. link.springer.com/article/10.1007/s00382-022-06468-x ) who have a better idea of the state of the art in this area. But to my eye it's unjustified assumptions all the way down. $\endgroup$ Jan 9, 2023 at 1:43
  • $\begingroup$ By the way, your method of picking two states of the planet to get an estimate of sensitivity reminded me strongly of Idso's 'natural experiments' paper ( int-res.com/articles/cr/10/c010p069.pdf ). Frankly, I'd not put any credence in the results he gets, either. But it's interesting for what it says about how wide a range of numbers you can get by dividing observed temperature changes by heat inputs, and the potential dangers of that approach. I thought you might find it interesting. $\endgroup$ Jan 9, 2023 at 2:18
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    $\begingroup$ Thanks again. This is very useful and what I was looking for. $\endgroup$
    – Sebby
    Jan 9, 2023 at 9:56

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