(1-a)*S/4 = eoT^4

GHG’s reduce the rate of LWR cooling to space, so rising GHGs represent a reduction in emissivity. A 3.7 W/m2 reduction in OLR from 2XCO2 reduces e to 0.0601. Climate scientists call that a forcing. If S (solar irradiation) changes, that is a solar forcing, such as the Maunder Minimum. The amount of sulfate particles reflecting incoming SWR changes albedo (a) after a volcanic eruption, that is a volcanic forcing. For any of these “forcings”, we can calculate the change in equilibrium temperature. For a reduction in emissivity to 0.601, that change is 1.15 K and is called the no-feedbacks climate sensitivity.

Climate scientists convert changes in e and a (both dimensionless ratios) into changes in W/m2 by multiplying by oT^4 and S/4 respectively (which obscures the underlying physics). So, whether e, a or S changes, they can be expressed as a change in forcing F.

(1-a)*S/4 = eoT^4 + F

It turns out that the emissivity and albedo of our planetary spacecraft vary with temperature:

a = a_0 + da/dT e = e_0 + de/dT

Emissivity changes with temperature because: a) water vapor (a GHG) rises with temperature, b) more humidity causes larger changes in the temperature of the upper atmosphere than the surface (where we measure temperature change), and c) because the height/temperature of cloud tops may change differently from surface temperature. Climate scientists call these derivatives water vapor, lapse rate and LWR cloud feedbacks.

Albedo changes with temperature because: a) surface snow and ice cover change with temperature and b) cloud cover and reflectivity may change with temperature. Climate scientists call these derivatives ice-albedo and SWR cloud feedback.

Hopefully, there is now no excuse for confusing forcing (a simple change in a, e or S expressed in W/m2) and feedback (a temperature-DEPENDENT change in a or e expressed in W/m2/K).

If we know a, S, e, de/dT, da/dT and F, we can calculate the equilibrium change in T. When F is caused by a doubling of CO2, we call the change in T ECS.

If we want to be a little more sophisticated:

d(eoT^4)/dT = (de/dT)*oT^4 + 4oeT^3

where 4oeT^3 = 3.2 W/m2/K is Planck feedback. If we add Planck feedback to the sum of all LWR ((de/dT)*oT^4) and SWR (da/dT*S/4) feedbacks, we have a single number that tell us how our planet’s radiative balance changes with temperature – the climate feedback parameter. (I haven’t discussed the sign of these terms, but incoming is positive and outgoing is negative. Planck feedback is -3.2 W/m2/K)

If you look at the heat capacity of the mixed layer of the ocean (and surface and atmosphere), a 3.7 W/m2 forcing is capable of producing an INITIAL warming rate of 0.74 K/year. Although radiative cooling to space increases as the planet warms, within about a decade most of the change to a new equilibrium temperature would be realized. Water vapor remains in the atmosphere an average of 9 days, clouds exist for shorter periods, trade winds circle the Earth in less than a month, the jet stream in a few days, and even 3/4 of sea ice melts every summer. The only thing that takes longer than a decade is changes in ice caps and vegetation.

Now let’s put a large well-insulated tank of water in our spacecraft – the deep ocean. In the long run, this doesn’t change the equilibrium temperature change we calculate above. ECS is depends only on a, S, e, de/dT, da/dT and F (=3.7 W/m2 by definition). However, it takes longer to reach a new equilibrium after a change in F. If Q is the amount of heat per unit area flowing into this tank of water on our spacecraft, we can write:

(1-a)*S/4 = eoT^4 + F + Q

And we can now calculate the a new transient temperature change (TCR) for any time when ocean heat uptake is Q. ARGO is measuring Q, so now we have a better idea of how to convert the transient warming we have already experienced into the future warming at equilibrium.

In climate science, we call this an energy balance model

]]>Can the arctic ice increase for a decade?

Antarctic Sea ice extent increased for some thirty years in the face of rising temperatures. So it’s possible that Arctic Sea ice could also start to increase for a while, especially since Antarctic Sea ice looks to be decreasing. The downward trend in average global sea ice extent has been fairly steady over the same period.

]]>You wrote: “And I am not convinced of the linearity of change we see today.”

For a simple mathematical function, as long as the response is continuous, it will be linear for sufficiently small changes. That is a consequence of Taylor’s theorem. There are three big ifs in that. (1) Is there a discontinuity (tipping point) in the system? (2) How large can the change be and still be small? Invoking either or both is a favorite ploy of alarmists. A tipping point will only amplify the response. Non-linearity could either amplify or dampen large changes. So far as I am aware, the only thing in the climate record that implies non-linearity is glacial termination. I don’t think any AOGCM’s show a tipping point. Almost all of them show nearly linear response up to four times CO2. Linearity is not assumed in the models, but the models could be missing something that creates a tipping point or strong non-linearity.

(3) Does the overall average response of a chaotic system behave as a simple mathematical function? I am not really up on the math of chaotic systems, but I think the answer is “yes” when the system shows predictability of the second kind. The usual concept of ‘climate’ assumes that to be so, and the models show such behavior. I am not convinced that the real Earth shows such behavior and suspect that the models have inadvertently baked it in.

You wrote: “Can the arctic ice increase for a decade?”

Well, it has increased for ballpark 80K years in the past.

I don’t think that the permanent ice is modeled. Surface albedo feedbacks are due to seasonal snow and ice, chiefly stuff that is present in spring and gone by summer.

]]>I can admit that the difference between feedbacks and forcing can be a bit challenging. Perhaps there are great timelags in the redistribution of heat that can mask something. The unforced change in temperature may have a greater and longer variability. So I don`t pretend to say anything about equilibrium.

It is about understanding the climate change we see today, and the mantra of TCR and ECS doesn`t help us very much. The other way around, I think the investigation of the earths energy budget can tell us a little about the realistic values of sensitivity. ]]>

This is a complicated question and you are confusing/intermixing a variety of concepts:

Are you talking about equilibrium or transient warming?

You appear to be confusing forcing (measured in W/m2) and feedbacks (measured in W/m2/K). Positive feedbacks in SWR (surface and cloud albedo) are important (to “high” climate sensitivity), but all feedbacks are equally important.

Is the change in GHGs the only forcing involved? What about aerosols?

DLR is a internal transfer of heat between the surface and the atmosphere, so changes in DLR do not warm or cool the planet. DLR just redistributes heat that is already in the planet and that redistribution is relatively fast – a few months at most to reach equilibrium.

Ocean heat uptake is also an internal redistribution of heat between the surface (including a mixed ocean layer that responds mostly within a decade) and the deep ocean (which requires centuries to approach equilibrium). This slow process creates the difference between transient and equilibrium warming.

Changes in the rate of heat transfer between the surface and deep ocean create “unforced” changes in temperature such as El Ninos.

The simplest answer is that TCR and ECS let us how much warming is caused by a change in GHGs.

]]>I think the question is actualized by the 2014 Donohoe et al paper, as they show that it is the absorbed solar radiation (ASR) that makes the earth warmer, perhaps together with IR downwelling radiation. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4250165/

«In computer modeling of Earth’s climate under elevating CO2 concentrations, the greenhouse gas effect does indeed lead to global warming. Yet something puzzling happens: While one would expect the longwave radiation that escapes into space to decline with increasing CO2, the amount actually begins to rise. At the same time, the atmosphere absorbs more and more incoming solar radiation; it’s this enhanced shortwave absorption that ultimately sustains global warming.»

« As longwave radiation gets trapped by CO2, the Earth starts to warm, impacting various parts of the climate system. Sea ice and snow cover melt, turning brilliant white reflectors of sunlight into darker spots. The atmosphere grows moister because warmer air can hold more water vapor, which absorbs more shortwave radiation. Both of these feedbacks lessen the amount of shortwave radiation that bounces back into space, and the planet warms rapidly at the surface.

Meanwhile, like any physical body experiencing warming, Earth sheds longwave radiation more effectively, canceling out the longwave-trapping effects of CO2. However, a darker Earth now absorbs more sunlight, tipping the scales to net warming from shortwave radiation.»

From:The missing piece of the climate puzzle. Researchers show that a canonical view of global warming tells only half the story. By Genevieve Wanucha, 2014.

Both measurements and models show that there is no reduction in IR radiation out at Top-of-atmosphere. Instead it has been a slight increase the last 40 years. https://www.ncdc.noaa.gov/teleconnections/enso/indicators/olr/

So the increased heat uptake is coming from increased absorbed solar radiation and from increased downwelling longwave radiation.

As I undersand SoD, his model shows that almost all increase of downwelling IR energy goes into the warming of ocean (by less cooling over the ocean skin). So, how much energy is this? Most studies operate with DLR increase of about 2 W m2 pr decade. Wang and Liang, 2009, have the value of 2,2 W, and is perhaps the most thorough study. The OHC has increased by 7,5W m2 pr decade between 1992 and 2015, according to Lijing Cheng, Kevin E. Trenberth, John Fasullo, Tim Boyer, John Abraham, and Jiang Zhu (2017): Improved estimates of ocean heat content from 1960 to 2015.

The conclusion of this should be that only one third of global warming can be attributed to increased Green House Gases. And the atmospheric part of heat uptake attributed to GHG should then only be about 0,5% of the total global heat optake. And I wonder if it also means that natural variations play a greater role than revised canonical view will admit. There are variations in wind pattern, ocean currents and arctic melting. ]]>

By the way, your comment got trapped in moderation because of the d word. All explained in The Etiquette.

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