How Does Climate Sensitivity Relate To Radiation Flux?

Let’s first remember that, as we previously talked about, Climate sensitivity corresponds to the amount of warming that will occur from a certain amount of “forcing”-change in the Earth’s radiative cooling that arises form something other than temperature changes themselves. In order for the Earth to maintain a constant temperature, the amount of radiation it receives from the sun must equal the amount that it reflects or re-emits back to space. Otherwise, because it is gaining or losing energy, it will warm or cool. Of course, warmer bodies emit more infrared radiation, so the Earth just needs to warm or cool enough to make the net radiation flux equal to zero again. The precise amount of temperature change necessary corresponds to the climate sensitivity. Of course, once again, there are caveats, which related to my previous post. But let’s forget those for a moment. Is it possible, in principle, to determine the climate sensitivity from looking at the Earth’s radiation budget and it’s response to temperature changes (feedback)? In fact, it is, in principle, possible, and one can even quantitatively describe how this can be done. However, keep in mind that it will only be possible to do so if the changes in the Earth’s radiation budget can be explicitly isolated as changes due to temperature, and changes due to other things, like CO2 (“forcing”).

This point is greatly emphasized by Roy Spencer, but our interest here is only in the Mathematics so this can be better described to people. I may not be the best person to do this, but it interests me and I’m going to try. The sensitivity of the climate relates to the theoretical response with no feedback as:

ΔT=ΔT0/(1-f)

Where delta T corresponds to the response with feedback, delta T naught to the zero feedback response, and f to the feedback factor. The feedback factor in turn depends on the radiation flux caused by temperature change (the total flux minus the forcing flux) as:

ΔF/K=-3.3(f)+3.3

Where ΔF/K is radiation flux per degree Kelvin, and 3.3 corresponds to the zero feedback flux per degree Kelvin, so for a feedback factor of 0, the ratio would be 3.3 W/m2 and if the feedback factor is one (corresponding to division by zero in our sensitivity formula), then no amount of change in K will create any change in ΔF, corresponding to infinite sensitivity. A feedback factor of -1, corresponding to half the zero feedback sensitivity, requires that the flux per Kelvin be double the no feedback value. Roy has been finding values of flux per Kelvin of ~6 W/m2, which corresponds to a feedback of ~-0.82, almost cutting the zero feedback sensitivity in half. The amount of temperature change to eliminate the flux from doubling CO2 (3.7 W/m2) would be a little over .6 degrees Kelvin, which is a very low sensitivity. Of course, determining the value of ΔF/K is difficult, as Spencer has himself pointed out. But the values found in climate models appear to be distinctly different from the observed fluxes.

Anyway, the main point of this post was to explain the math involved. I may revisit this later to investigate the feedbacks myself. It’s perhaps a topic for future investigation.

I would also like thank Roy for confirming my calculations.

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11 Responses to “How Does Climate Sensitivity Relate To Radiation Flux?”

  1. Steve Fitzpatrick Says:

    Hi Andrew,

    Your analysis seems correct, as far as it goes. The problems with determining climate sensitivity from the balance of radiative fluxes is that you have to quantify and analyze the difference between the incoming solar energy (fairly constant) and the sum of two very noisy measurements: the outgoing short wave radiation and the outgoing long wave radiation. Since the energy imbalance postulated for GHG forcing plus water vapor amplification is only on the order of 1%-2% of the time average solar energy flux, you need to have extremely good measurements of the total outgoing fluxes to determine the sensitivity with any degree of certainty. Most likely you will become very frustrated by the uncertainties involved (a la Trenberth… “Where is the missing heat!?!”).

    It seems to me that the most reasonable way to examine radiative imbalance (and constrain climate sensitivity) is through accurate measurments of ocean heat, since accumulation of heat in the ocean is almost identical to the true radiative imbalance. Since the ocean heat is a direct measure of the imbalance rather than the mathematical difference between measurements of incoming and (noisy) outgoing fluxes, it provides a much better signal to noise ratio measure of the net imbalance.

  2. Andrew Says:

    Steve, can’t respond fully at this time, since I’m at the University library, but I’m not sure if the accuracy of the measurements in absolute terms is what’s important. If that were the case, then the sensitivity would just be the solar flux incoming divided by the surface temperature in Kelvin:

    https://devoidofnulls.wordpress.com/2010/05/26/timescales-special-cases-limitations-etc/

    However, as I note, this is not really analogous to the case of doubling CO2, since the physics that matters at very cold temperatures is completely different.

    What should matter are the “relative” uncertainties in the data as they change over time. That will probably be different. I realize it might not be entirely clear what I’m saying, but I can’t elaborate much more now. As far as OHC goes, just as with the satellite data, you actually have combined forced and feedback flux, and I am unaware of OHC data available with the fine resolution I want, or how to convert the OHC to information on Radiation, which is more intuitively related to sensitivity and feedback.

    The issues here are at the heart of the recent analyses by Spencer, as well as Lindzen and Choi (2009) and Lindzen and Choi (2010, submitted), which attempt to use the satellite data to determine the sensitivity. It may be useful to look at OHC to further constrain analyses, and provides an independent line of evidence/assessment, which may help confirm the radiation analyses (one would hope, but they might also contradict them, in which case I’d be pulling my hair out just like Kevin). Of course OHC has it’s problems as well-who’s data do you go with? For the feedback analysis, it may matter a great deal, except perhaps recently (Josh Willis says that the groups pretty much agree on most of the ARGO era).

  3. Steve Fitzpatrick Says:

    Andrew,

    Any change in the OHC over a short period (dH/dt if you will) seems to me to be essentially identical to the net radiative imbalance for that period. That is, the OHC change essentially identical to the difference between incoming and outgoing radiation…. there just is no place except the ocean where much heat can be accumulated. Many people object to the Lindzen and Choi papers at least in part because (it is said) that the analysis convolutes cause and effect. OHC does not suffer form this complication; an accurately measured change in ocean heat is a near certain indication of net energy imbalance (positive or negative!).

    There is of course some uncertainty in the ocean heat content data, but from 2004 forward the ARGO system has dramatically reduced that uncertainty. The email exchange between Pielke Sr, Josh Willis and Kevin Trenberth on ocean heat content makes it clear that Trenberth does not believe the ARGO data, but that Josh Willis believes it is pretty solid. There is no surprise in this disagreement, since the best available OHC analyses clearly show a very relatively slow accumulation of heat since 2004, casting doubt on both the accuracy of the satellite based measurements of radiation balance and on postulated very long ocean lags in responding to radiation imbalances… leading to more of Kevin Trenberths hair being torn out.

    I note that before the ARGO system was completed, it got a whole lot more attention and discussion about it’s importance from the ‘climate science community’ than it gets today. I don’t think this is a mere coincidence; much of the community just doesn’t like what the ARGO data say.

  4. timetochooseagain Says:

    “Many people object to the Lindzen and Choi papers at least in part because (it is said) that the analysis convolutes cause and effect. OHC does not suffer form this complication; an accurately measured change in ocean heat is a near certain indication of net energy imbalance”

    The convolution of cause and effect was mentioned in my post:

    “However, keep in mind that it will only be possible to do so if the changes in the Earth’s radiation budget can be explicitly isolated as changes due to temperature, and changes due to other things, like CO2 (“forcing”).”

    However, although OHC may measure the radiative imbalance, the radiative imbalance will always convolute cause and effect! This is because radiative imbalances can occur because of temperature changes (feedback) and because of forcings, and this is what the satellites are supposed to measure. These two things are also present in the OHC, in principle, and are again, difficult to seperate from one-another.

    I also believe I noted above that if you do not remove the forcing component of the radiative fluxes, this biases towards no change in the radiation with temperature, since the flux response to temperature is meant to cancel the radiative imbalance due to forcing, at least in equilibrium. This biases the analyses towards high sensitivity whenever a forcing component remains in the data, since a smaller slope corresponds to more positive or less negative feedback.

  5. Steve Fitzpatrick Says:

    Andrew,
    “However, although OHC may measure the radiative imbalance, the radiative imbalance will always convolute cause and effect! ”

    Sure, but mainstream climate science says that radiative forcing from GHG’s must be amplified, not reduced by feedbacks. The OHC data removes a ‘degree of freedom’ from the analysis. Nobody can just say that most of the added heat calculated from satellite radiation measurements must be accumulating in the ocean, like you could if OHC data were much less reliable (eg. no ARGO data). Trenberth tears out his hair specifically because the satellite measurements are in conflict with what appear to be more reliable OHC data from ARGO. Even the steric contribution to sea level rise over the past several years is consistent with quite low heat accumulation.

    My only point is that efforts to determine climate sensitivity from (very noisy) measurements of radiative fluxes seems to me likely a futile exercise.

  6. timetochooseagain Says:

    Steve-OHC wouldn’t be any easier to use to determine the sensitivity, I think, because my point is that as long as the OHC is confounding the forcing and feedback, the equations above are essentially inapplicable. The apparent “noise” in the data may well have a physical interpretation-radiation changes that are not feedback, but forcing.

    If you look at this post by Roy:

    http://www.drroyspencer.com/2010/04/a-response-to-kevin-trenberth/

    the LW plot in particular looks similar to the temperature plot. Part of this is the “planck response” mentioned in my post, but there may be some feedback, and forcing, there, too.

  7. Steve Fitzpatrick Says:

    Andrew,

    I had read Spencer’s response some time back. And regrdless of if it is valid or not, it looks like nobody in climate science accepts Spencer’s work. There is too much wiggle room in any analysis of the radiation balance data to allow a meaningful determination. And besides, the OHC data fairly well screams that the radiative balance data is just plain wrong. How can you try to generate a reasonable estimate of climate sensitivity from radiation fluxes when the OHC says the basic data are not of a quality that allow the sensitivity to be determined?

    I will likely be dead before this is completely sorted out, but you will likely be around. I believe that the twin realities of less than expected OHC increases and significantly less rise in temperatures than predicted will ultimately force a substantial downward revision in ‘best estimates’ of climate sensitivity, to somewhere under 1.5C per doubling.

  8. timetochooseagain Says:

    The point of the post was to address the issue of how the fluxes should theoretically translate to feedback. I am not familiar enough with the OHC data or the radiation data to have an opinion as to whether they are consistent with one another, and which is more accurate. But I also suspect that the sensitivities will ultimately be found to have been overestimated, perhaps by quite a large factor.

    Ultimately, I also think we’ll be able to sort out discrepancies between the radiation and OHC data. What will be the important insight that allows for this? Dunno. Examining both datasets in detail might help. I hadn’t planned on looking at OHC but am now intrigued by the possibilities it may have.

  9. Radiation Redux « Hypothesis Testing Says:

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