Most of what you have written is correct. There are multiple ways to answer this question.

A doubling of CO2 was calculated by Myhre to reduce LWR radiative cooling to space by 3.7 W/m2. The calculation assumes nothing else changes and is sometimes referred to as an instantaneous doubling. Under this hypothetical situation, the law of conservation of energy demands that temperature rise (somewhere below the tope of the atmosphere or TOA). That rise will continue until the planet emits 3.7 W/m2 and no radiative imbalance at the TOA exists.

To determine how much warming is needed to eliminate this imbalance, we need to know how much more radiation the planet emits for each degC its temperature rises. This is called the climate feedback parameter (W/m2/K).

As you noted, one can postulate that the Earth emits 239 W/m2 because it behaves like a blackbody at 255:

W = -oT^4

dW/dT = -4oT^3 = -3.76 W/m2/K

According to this model, the 3.7 W/m2 radiative imbalance from 2XCO2 will be negated by 0.98 K of warming. (By convention, the negative sign represents heat lost by the planet, though this convention is sometimes ignored.)

As you noted, others (presumably including this post) postulate a graybody model for the Earth, where emission of 239 W/m2 is the result of a surface temperature of 288 K and an emissivity of 0.615 (or slight variants of these values).

W = -eoT^4

dW/dT = -4oeT^3 = -3.33 W/m2

According to this model, the 3.7 W/m2 radiative imbalance from 2XCO2 will be negated by 1.11 K of warming.

A few postulate a model where the Earth behaves like at blackbody at 288, but this model doesn’t explain why the Earth emits 239 W/m2. Most people would call this “wrong”, but technically one is free to postulate any model one wants.

Climate models divide the planet up into roughly a million grid cells with realistic temperatures and calculate how much more radiation they absorb and emit. From their output, climate scientists have determined that a warming planet emits 3.21 W/m2/K (+/-0.05 W/m2/K) more radiation to space – assuming nothing else changes. According to this model, the 3.7 W/m2 radiative imbalance from 2XCO2 will be negated by 1.15 K of warming. This is what most people call the no-feedbacks climate sensitivity.

If you want to be more sophisticated, the following model I devised may be useful. Others explain this differently. The planet’s radiative imbalance (I) is:

I = (S/4)*(1-a) – eoTs^4

where a is albedo, S is incoming solar radiation, Ts is surface temperature and e is the planet’s effective emissivity given an average Ts of 288 K.

dI/dTs = -4eoTs^3 – (oTs^4)*(de/dTs) – (S/4)*(da/dTs)

dI/dTs is the climate feedback parameter. Here emissivity hasn’t been treated as a constant. The first term is called Planck feedback and is -3.3 W/m2. The third term is the change in reflection of SWR by clouds and the surface, called cloud SWR and ice albedo feedbacks. The sum of the first and second terms is called LWR feedback. The second term includes: a) the effect on emissivity of rising humidity (water vapor is a GHG reducing emissivity) with temperature, b) the change in emissivity due rising humidity causing more warming higher in the atmosphere than at the surface, and c) the change in emissivity of clouds due to a change in their altitude/temperature or composition. These are called water vapor, lapse rate and cloud LWR feedbacks. (And while we are being more sophisticated, the composition of atmosphere predicted by climate model changes with warming and those changes result in a doubling of CO2 producing an effective forcing (reduction of radiative cooling to space) from 2.4 to 4.4 W/m2.)

No one knows the correct values for de/dTs, da/dTs, or feedbacks, but they can be abstracted from the output of climate models. Most climate models predict dI/dTs is around -1 W/m2/K. Current forcing is about 2.5 W/m2 and ocean heat uptake is about 0.7 W/m2, so current warming of roughly 1 degK is sending about 1.8 W/m2 more radiation to space (emission of LWR and reflection of SWR). That’s -1.8 W/m2/K; almost half as much warming at steady state as predicted by climate models. This approach is called an energy balance model.

Assuming I haven’t made any mistakes, all of these answers begin with different models/assumptions and are as “correct” or “wrong” as the assumptions on which they are based. Confusion arises because different people use different assumptions without making them clear.

]]>I would like to discuss the following section:

Tnew4/Told4 = (239 + 3.7)/239

where Tnew = the temperature we want to determine, Told = 15°C or 288K

We get Tnew = 289.1K or a 1.1°C increase.

Why do you take the value of 239? The current temperature is not because of the solar radiation of 239 W/m2, its due to the solar radiation of 239 W/m2 + the current radiation forcing of 157 W/m2 which brings it to 396 W/m2. You are going to raise the total energy warming the planet from this value of 396 W/m2 to 399.7 W/m2.

So we get

Tnew4/Told4 = (396 + 3.7)/396

where Tnew = the temperature we want to determine, Told = 15°C or 288K

We get Tnew = 288.67K or a 0.67°C increase.

What you are doing is calculating the ratio by which an earth without an atmosphere would increase given an extra 3.7 W/m^2. Then you multiply that ratio with the temperature of the earth with an atmosphere to get to the end value.

Off course it requires way less energy to heat non greenhouse gas -18 degree Earth than it does to heat pre industrial revolution +15 degree greenhouse earth.

Put differently the 239 comes from how much the sun shines on the top of the atmosphere and how much the earth radiates into space from the top of the atmosphere. This value is in equilibrium as you explained in another post. Obviously when we get more greenhouse gasses this value will not change! It will be the 396 W/m^2 that will change. So this should show in your calculations.

I also looked at calculations I came across at other places that try to derive the climate sensitivity parameter from the Boltzmann equation.

I came across this: landa = Ts / 4*sigma*Te^4

Here: http://web.ma.utexas.edu/mp_arc/c/11/11-16.pdf

She used Ts = 288 and Te = 255, again I would argue that 255 (temperature of a blackbody radiation 239 W/m^2) has no place in these calculations and it should also equate to 288. (289 is closer if I take your value of 396 btw)

Then that equation simplifies to landa = 1 / 4*sigma*T^3

Similar to the one I found here:

clivebest.com/blog/?p=4923

Where he uses an emissivity factor to make the earth a greybody instead of a blackbody.

Now you wrote on another page that the earth could be seen as close to a blackbody. So if we take the simplified formula and T = 289 then that solves for landa to 0.1827. Significantly lower than the 0.3 commonly quoted.

This could only come close to 0.3 (but not quite, 0.285) if we use an emissivity of 0.64 which wikipedia claims of the earth taking cloud cover into account.

But would this not already be effectively a feedback loop of the water vapor?

And even when not half of the atmosphere by mass (pressure, same thing if you think about it) lies under 5000 meter altitude. So half of the radiative forcing should take place here under the clouds (assuming that the average cloud lies at 5000 meter) and thus the emissivity that matters for that half at least is that of the earth’s surface which is quite close to 1, lets say 0.95. So the total emissivity should be somewhere in between 0.64 and 0.95 then.

However clouds are vapor and does not absorb the same wavelengths as CO2 albeit there’s some overlap as you showed in part 1. Thus assigning a lower emissivity to the earth due to the clouds seems to be a mistake from the getgo. The source of the radiation is the Earth and giving it a lower emissivity (which would increase the effect of CO2) because it already has gasses surrounding it that traps radiation that the CO2 cant trap a second time seems illogical to say the least.

Thus the question remains. Where does the 1-1.2 degree no-feedback calculation comes from? Because I get different figures using two different, albeit it similar methods.

Thank you for your attention.

I hope you can shine a good light on this.

Regards,

Jan

To a first approximation, the deep ocean is simply a large heat sink. ECS is the reciprocal of the climate feedback parameter: dOLR/dTs + dOSR/dTs, meaning that steady state climate change depends on forcing and the climate feedback parameter. Chaotic fluctuations in the currents linking the surface and the deep ocean may be the most important source of internal/unforced climate variability, but (on paper) they don’t change long-term steady state temperature. For energy balance models:

TCR = F_2x*(dT/dF) ECS = F_2x*(dT/(dF-dQ))

TCR/ECS = 1 – dQ/dF

where dQ is ocean heat uptake. So, the only thing a larger ocean heat uptake does is increase the amount of time it takes to approach steady-state and not steady-state itself.

Figure 2B in the paper we were discussing shows the differences in ocean heat uptake between various IPCC models, expressed as the effective depth of the mixed layer that contains all of the heat ocean absorbs over time (assuming a uniform heat accumulation in that mixed layer). That value ranges +/- 25%.

https://www.pnas.org/content/111/47/16700

We can track transport of heat by physical mixing through two proxies: CFCs and C-14 from atmospheric testing of atomic bombs, but I haven’ found any good comparisons between observations and hindcasts.

]]>Thoughts to the SW warming: (The energy budget in reality). As 93% of this warming goes into oceans, the behavour of oceans will be the critical point. Most of the warming goes out again in the night. Another part of the energy has a seasonal outlet. Some great cycles have an outlet of some years, as Pacific and Atlantic currents. And still others will have decadal and even millennial cycles, as ocean overturning and water sinking. So how can this energy transport be simulated in models? My guess is that informed guessing comes closer than mathematics.

And how do the CO2 level influence this energy transport? ]]>

The canonical view isn’t wrong; it is just poorly expressed – because the consensus has a very CO2-centric view. Consider ANY forcing, an imbalance at the TOA that is imposed by a permanent change (not the temporary changes associated with internal variability in chaotic systems). If forcing is positive, the retained energy is going to warm the planet until the balance at the TOA is restored by an increase in emission of LWR OR reflection of SWR. The planet’s radiative imbalance (I) is given by the equation below where OLR is negative:

I = ASR + OLR

dI/dTs = dASR/dTs + dOLR/dTs

Taking the derivative gives us the climate feedback parameter dI/dTs (W/m2/K), which is what we need to know to figure out how much warming (deltaT) will correct the imbalance caused by a forcing F:

F + (dI/dTs)*deltaT = 0

Equilibrium warming depends only on the climate feedback parameter, not how it is partitioned between its LWR and SWR components. The “canonical view” misses the importance, generality and physics of the climate feedback parameter by discussing its reciprocal K/(W/m2) and converting W/m2 to doublings of CO2 to get ECS.

dI/dTs must be negative (or you have a runaway GHE) and the biggest negative term is Planck feedback (-3.3 W/m2/K for a graybody model). If your forcing is from the sun, you certainly expect OLR to increase, despite the fact that the forcing is in the SWR channel.

The forcing caused by rising GHGs is no different, except that we have no way to physically measure that forcing. Instead, we fall back on radiative transfer calculations, which require us to specify the temperature and composition of the atmosphere through which radiation is traveling. Thus we do calculations imagining an abrupt doubling of CO2. However, long before CO2 can double, the planet has begun to warm, changing both OLR and ASR.

If dASR/dTs were zero, what would we measure at the TOA? For an instantaneous doubling of CO2, we would expect OLR to drop 3.6 W/m2 and gradually return to normal as the planet warms.

Now lets suppose dASR/dTs and dOLR/dTs are both equal magnitude and have the same (negative) sign. Balance at the TOA will be restored by a 1.8 W/m2 increase in OLR and a 1.8 W/m2 increase in reflected SWR (technically both values are negative, since they are heat lost by the planet). In this case only half of the drop in OLR (1.8 W/m2) will be recovered by steady state warming; the rest will come from decreasing ASR (1.8 W/m2).

Now let’s suppose that dASR/dTs has the opposite sign as dOLR/dTs and half the magnitude. That would mean that for every 2 W/m2 more OLR lost to space by warming, 1 additional W/m2 of ASR would be coming in. To get a NET 3.6 W/m2 out, we need 7.2 W/m2 of increased OLR (technically a negative number) and 3.6 W/m2 more solar radiation will be getting past the clouds. Halfway through this warming process, OLR will have increased 3.6 W/m2 – restoring “balance” in the LWR channel – and ASR will have increased 1.8 W/m2 – making it appear as it ASR is now doing all of the warming. However, this isn’t right, because the planet IS WARMER at this halfway point and would be emitting 1.8 W/m2 more LWR simply because of its higher temperature, and leaving half of the forcing from rising GHGs in place.

You cited a press release by MIT. The paper can be found at the link below. Figure 3 shows us how mutually inconsistent climate models are in partitioning the climate feedback parameter into its dOLR/dTs and dASR/dTs components.

https://www.pnas.org/content/111/47/16700

Figure 3A in the paper shows how much varies AOGCMs disagree with each other about dASR/dTs and dOLR/dTs. The surprise in the figure is the square symbol and dashed circle claiming to be observational estimates for dASR/dTs and dOLR/dTs. These observational estimates are calculated in the Supplementary Material. Those values are derived from CERES data for 2000-13, using the same kind of linear regressions as Dessler (2018) except that OLR and SWR are analyzed separately. dOLR/dTs is -2.0 +/- 0.4 W/m2/K (one STD, r^2 = 0.28) and dOSR/dTs = +0.8 +/- 0.4 W/m2 (r^2 = 5%). Of course, the data from seasonal warming indicates to me that monthly SWR isn’t a simply function of Ts and that some components are lagged, explaining why the r^2 for SWR is so low and invalidating the slope as a measure of SWR feedback.

]]>http://news.mit.edu/2014/global-warming-increased-solar-radiation-1110

““So there are two types of radiation important to climate, and one of them gets affected by CO2, but it’s the other one that’s directly driving global warming — that’s the surprising thing,” says Armour, who is a postdoc in MIT’s Department of Earth, Atmospheric and Planetary Sciences.”

“The study sorts out another tricky climate-modeling issue — namely, the substantial disagreement between different models in when shortwave radiation takes over the heavy lifting in global warming.”

“The finding was a curiosity, conflicting with the basic understanding of global warming,” says lead author Aaron Donohoe, a former MIT postdoc who is now a research associate at the University of Washington’s Applied Physics Laboratory. “It made us think that there must be something really weird going in the models in the years after CO2 was added. We wanted to resolve the paradox that climate models show warming via enhanced shortwave radiation, not decreased longwave radiation.”

“Donohoe, along with MIT postdoc Kyle Armour and others at Washington, spent many a late night throwing out guesses as to why climate models generate this illogical finding before realizing that it makes perfect sense — but for reasons no one had clarified and laid down in the literature.”

“I think the default assumption would be to see the outgoing longwave radiation decrease as greenhouse gases rise, but that’s probably not going to happen,” Donohoe says. “We would actually see the absorption of shortwave radiation increase. Will we actually ever see the longwave trapping effects of CO2 in future observations? I think the answer is probably no.”

Why was this so surprising? There had been some sceptical voices to the canonical wiev. Why couldn`t they just say that the scientific consensus on greenhuose warming was a big misunderstanding, and that the canonical view was just plain wrong?

]]>Even if you use something like LES to simulate the large scales and make use of the fact that backscatter is very small and energy flows in the mean to the smallest resolved scales, you will have almost no chance in getting the subgrid information you would like to have with big grid cells.

People tried to use stochastic models to generate some structure mimicking some kind of isotropic turbulence. Nevertheless, trying to find accurate prediction on the subgrid scale is impossible, in my opinion, esp. for higher moments. I say so doing those simulations for years and developing models myself.

Most one- or two-equation RANS models are a lot worse, relying on crude eddy viscosity concepts with often only one external length scale. Those models, when applied to wings etc., are perfectly able to get integral quantities reliably (e.g. drag or lift, used by the big car and air-plane companies), but fail already in prediction detachment or reattachment, reliably. As you said, Reynolds stress models have become really good but due to the higher cost aren’t used in industry at all – I know of no German manufacturer having used any for production stuff. Nevertheless I consider those important, as they are able to deal with anisiotropy.

The problem is not, that those RANS models lack physics. It is that the physics involved is often crude or simplified, lacking important mechanisms and therefore not being able to predict strongly anisotropic turbulent flows requiring a multitude of length and time scales.

LES is a much better way to go and my experience is that you get excellent results, but only if you can afford the computational expense and resolution required to resolve the high shear/gradient regions and have a mesh of the order of the scales you want to make predictions about.

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