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Empirical evidence that humans are causing global warming

What the science says...

Select a level... Basic Intermediate

Less energy is escaping to space: Carbon dioxide (CO2) acts like a blanket; adding more CO2 makes the 'blanket' thicker, and humans are adding more CO2 all the time.

Climate Myth...

There's no empirical evidence

"There is no actual evidence that carbon dioxide emissions are causing global warming. Note that computer models are just concatenations of calculations you could do on a hand-held calculator, so they are theoretical and cannot be part of any evidence." (David Evans)

The proof that man-made CO2 is causing global warming is like the chain of evidence in a court case. CO2 keeps the Earth warmer than it would be without it. Humans are adding CO2 to the atmosphere, mainly by burning fossil fuels. And there is empirical evidence that the rising temperatures are being caused by the increased CO2.

The Earth is wrapped in an invisible blanket

It is the Earth’s atmosphere that makes most life possible. To understand this, we can look at the moon. On the surface, the moon’s temperature during daytime can reach 100°C (212°F). At night, it can plunge to minus 173°C, or -279.4°F. In comparison, the coldest temperature on Earth was recorded in Antarctica: −89.2°C (−128.6°F). According to the WMO, the hottest was 56.7°C (134°F), measured on 10 July 1913 at Greenland Ranch (Death Valley).

Man could not survive in the temperatures on the moon, even if there was air to breathe. Humans, plants and animals can’t tolerate the extremes of temperature on Earth unless they evolve special ways to deal with the heat or the cold. Nearly all life on Earth lives in areas that are more hospitable, where temperatures are far less extreme.

Yet the Earth and the moon are virtually the same distance from the sun, so why do we experience much less heat and cold than the moon? The answer is because of our atmosphere. The moon doesn’t have one, so it is exposed to the full strength of energy coming from the sun. At night, temperatures plunge because there is no atmosphere to keep the heat in, as there is on Earth.

The laws of physics tell us that without the atmosphere, the Earth would be approximately 33°C (59.4°F) cooler than it actually is.

This would make most of the surface uninhabitable for humans. Agriculture as we know it would be more or less impossible if the average temperature was −18 °C. In other words, it would be freezing cold even at the height of summer.

The reason that the Earth is warm enough to sustain life is because of greenhouse gases in the atmosphere. These gases act like a blanket, keeping the Earth warm by preventing some of the sun’s energy being re-radiated into space. The effect is exactly the same as wrapping yourself in a blanket – it reduces heat loss from your body and keeps you warm.

If we add more greenhouse gases to the atmosphere, the effect is like wrapping yourself in a thicker blanket: even less heat is lost. So how can we tell what effect CO2 is having on temperatures, and if the increase in atmospheric CO2 is really making the planet warmer?

One way of measuring the effect of CO2 is by using satellites to compare how much energy is arriving from the sun, and how much is leaving the Earth. What scientists have seen over the last few decades is a gradual decrease in the amount of energy being re-radiated back into space. In the same period, the amount of energy arriving from the sun has not changed very much at all. This is the first piece of evidence: more energy is remaining in the atmosphere.

 

Total Earth Heat Content from Church et al. (2011)

What can keep the energy in the atmosphere? The answer is greenhouse gases. Science has known about the effect of certain gases for over a century. They ‘capture’ energy, and then emit it in random directions. The primary greenhouse gases – carbon dioxide (CO2), methane (CH4), water vapour, nitrous oxide and ozone – comprise around 1% of the air.

This tiny amount has a very powerful effect, keeping the planet 33°C (59.4°F) warmer than it would be without them. (The main components of the atmosphere – nitrogen and oxygen – are not greenhouse gases, because they are virtually unaffected by long-wave, or infrared, radiation). This is the second piece of evidence: a provable mechanism by which energy can be trapped in the atmosphere.

For our next piece of evidence, we must look at the amount of CO2 in the air. We know from bubbles of air trapped in ice cores that before the industrial revolution, the amount of CO2 in the air was approximately 280 parts per million (ppm). In June 2013, the NOAA Earth System Research Laboratory in Hawaii announced that, for the first time in thousands of years, the amount of CO2 in the air had gone up to 400ppm. That information gives us the next piece of evidence; CO2 has increased by nearly 43% in the last 150 years.

 

Atmospheric CO2 levels (Green is Law Dome ice core, Blue is Mauna Loa, Hawaii) and Cumulative CO2 emissions (DOE Data Explorer). While atmospheric CO2 levels are usually expressed in parts per million, here they are displayed as the amount of CO2 residing in the atmosphere in gigatonnes. CO2 emissions includes fossil fuel emissions, cement production and emissions from gas flaring.

The Smoking Gun

The final piece of evidence is ‘the smoking gun’, the proof that CO2 is causing the increases in temperature. CO2 traps energy at very specific wavelengths, while other greenhouse gases trap different wavelengths.  In physics, these wavelengths can be measured using a technique called spectroscopy. Here’s an example:

Spectrum of the greenhouse radiation measured at the surface. Greenhouse effect from water vapor is filtered out, showing the contributions of other greenhouse gases (Evans 2006).

The graph shows different wavelengths of energy, measured at the Earth’s surface. Among the spikes you can see energy being radiated back to Earth by ozone (O3), methane (CH4), and nitrous oxide (N20). But the spike for CO2 on the left dwarfs all the other greenhouse gases, and tells us something very important: most of the energy being trapped in the atmosphere corresponds exactly to the wavelength of energy captured by CO2.

Summing Up

Like a detective story, first you need a victim, in this case the planet Earth: more energy is remaining in the atmosphere.

Then you need a method, and ask how the energy could be made to remain. For that, you need a provable mechanism by which energy can be trapped in the atmosphere, and greenhouse gases provide that mechanism.

Next, you need a ‘motive’. Why has this happened? Because CO2 has increased by nearly 50% in the last 150 years and the increase is from burning fossil fuels.

And finally, the smoking gun, the evidence that proves ‘whodunit’: energy being trapped in the atmosphere corresponds exactly to the wavelengths of energy captured by CO2.

The last point is what places CO2 at the scene of the crime. The investigation by science builds up empirical evidence that proves, step by step, that man-made carbon dioxide is causing the Earth to warm up.

Basic rebuttal written by GPWayne

Addendum: the opening paragraph was added on 24th October 2013 in response to a criticism by Graeme, a participant on the Coursera Climate Literacy course. He pointed out that the rebuttal did not make explicit that it was man-made CO2 causing the warming, which the new paragraph makes clear. The statement "...and humans are adding more CO2 all the time" was also added to the 'what the science says section. 


Update July 2015:

Here is a related lecture-video from Denial101x - Making Sense of Climate Science Denial

Last updated on 12 July 2015 by MichaelK. View Archives

Printable Version  |  Offline PDF Version  |  Link to this page

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Comments 76 to 100 out of 417:

  1. e, a constant forcing held forever will indeed results in a constat rise in temperature. When the forcing is applied and the temperature starts to increase the earth wiil progressively increase its thermal emission untill balance is reached again and the temperature stops increasing. Mathematically it is expressed by the term λΔT in the heat balance equation: C dΔT/dt = F−λΔT sooner or later this term will balance the forcing. Never overlook the stronger and faster negative feedback! :)
  2. Doug_Bostrom, I have now discovered that "Tamino" is the blog pseudonym of the author of the site to which Riccardo referred me. You wrote: "I may be wrong but as far as I know Tamino has never been found wrong w/regard to posts he's made on his site." I am then especially honoured to have been the very very first to have done so. Thank you. Riccardo, You introduced the reference to Schwartz to counter my arguments about the difficulty of reconciling various assumptions to an observed increase in OLR. Can we now agree that one cannot derive an expression for OLR from the Schwartz model in the way you attempted and return to the main thrust of the conversation? This is a serious, not a provocative, question. The impatience I expressed in #68 was because I suspected that you already understood the implications of my comments on interpretation of the F term in Schwartz, but you did not wish to acknowledge this. However, this may not be true, and hence, my question. Do I need to expand further on this subject? Do you question my interpretation of the Schwartz model? Or can we agree and move on?
  3. What, I'm supposed to take your word for it, PaulK? Sorry, not until you work it out w/Tamino directly. If he says you're correct then congratulations. But based on your waffling w/Riccardo I doubt you'll end up w/anything to celebrate.
  4. e, To expand a little on Riccardo's comment. The physics says that if one applies a positive impulse forcing to a system in steady state (input power equals output power), then the system will heat up. As the system heats up, it will increase its power output until the temperature restabilises at a new constant value. The small issue I have with Riccardo is definitional. If the input power is I(t) and the output power is O(t), then at time t0, in steady-state, we have I(t0)-O(t0) = 0. If one defines the net forcing over time, F(t), as the difference between input and output power, i.e. F(t) = I(t)-O(t), and one applies an impulse forcing F(t0) to the system, then to restore steady-state, F(t) must then decay to zero after a period of time as temperature restabilises. However, the expression used for F(t) in Riccardo's heat balance equation (C dΔT/dt = F−λΔT) is not a net forcing over time. Instead, it is equal to the difference between the input power at time t and the output power at time, t0. For this expression to make sense, mathematically, F(t) = I(t) - O(t0) - equivalent to a series of stacked impulse forcings applied to the input side of the power equation.
  5. PaulK, indeed F(t) is not equal to I(t)−O(t). No one ever said it is, Tamino even wrote it explicitly. Let's see if you like a different wording more: I(t) = Ie+F(t) O(t) = Oe+λΔT(t) (to first order) I(t) − O(t) = Ie + F(t) − Oe − λΔT(t) = F(t) − λΔT(t) with Ie and Oe equilibrium values. Straightforward, I'd say.
  6. PaulK, Riccardo, I was attempting to address this comment from PaulK > "Quote Now suppose that prior to our starting time, climate forcing was constant and equal to zero, and temperature departure was constant and equal to zero. After time t0, climate forcing increased to 1 W/m^2 and stayed there. Then the solution turns out to be: Theta(t) = (1 – exp(-lamda*t/C))/lamda Endquote It is hopefully evident to you that Theta (t), the temperature change from the forcing, must asymptote mathematically from this expression to a constant 1/lamda at large values of t. So now ask yourself the question whether it is possible in terms of first law of thermodynamics to have an imbalance of TOA radiative energy for an infinite time which results in a finite (constant) change in planetary temperature. If you can truly answer yes to this question , then I think that I am going to sign off, since I am wasting my time here. " I'll admit I probably misunderstood what you were trying to say Paul. I don't see what is wrong with Tamino's solution for Θ(t) in his scenario. To repeat what you said in your recent post: "As the system heats up, it will increase its power output until the temperature restabilises at a new constant value." Isn't this exactly what Tamino was showing in his solution for temperature departure Θ(t) in his scenario? How does this violate the first law of thermodynamics?
  7. e, You are the only person here who is NOT confused. There is nothing wrong with Tamino's solution for Θ(t) given his assumptions and approximations. My question was directed at Riccardo, who I believed had a conceptual misunderstanding of the F-term in Schwartz, based on his attempts to derive an expression for OLR from this model. (See my post #65 for context.) I'll address the reason for why I believed this in a separate post for Riccardo.
  8. Riccardo, The solution I proposed for OLR in posts #52 and #55 is mathematically exact, and makes very few assumptions. The solution is based directly on a flux response function which varies as a function of time, and therefore represents a generalized solution to a wide variety of energy balance models. Because it makes very few assumptions, the conclusions about the shape of the OLR response are mathematically robust but have low information content. I would prefer to use this model directly as a basis to discuss observed OLR response, rather than a more informative model, but with questionable assumptions. However, in response to the points I made about the geometry of the OLR response, you referred me to the Schwartz paper and you attempted to abstract a solution for OLR from that model (your posts #56 and #60). I did the same thing, initially mistakenly assuming that F was a net forcing in time (Q(t) – E(t)), since Schwartz rather loosely describes F as a Delta(Q-E). When I reexamined the solution form, however, it was apparent that F has to be an impulse forcing in Schwartz, as more conventionally applied. So I again recalculated OLR and found that I had a solution that was different from (both your original and) your corrected OLR solution. I concluded, since we had been talking originally about net flux response functions, that you were misunderstanding the F term in Schwartz. It is apparent from your last post that this is not the case. Therefore, I now believe that it is either your maths or mine which are wanting. For the case F=bt, I obtain the following result for OLR, using the Schwartz model and Schwartz nomenclature:- C * d(Deltat)/dt = CdT/dt = dH/dt =Q(t) – E(t) = net flux in time For constant input Q(t) = Q(0) for all t. We “transfer” the forcing to the output side as follows: Net flux in time = Q(0) –(E(t)-F(t)) = Q(0) – OLR(t) = C*d(Deltat)/dt Hence OLR(t) = Q(0) - C* d(Deltat)/dt But Deltat = b((t-tau) + tau*exp(-t/tau))/lamda and Therefore d(deltat/dt) = b(1-exp(-t/tau))/lamda Substituting into the solution for OLR we obtain : OLR(t) = Q(0) – Cb(1-exp(-t/tau))/lamda Note that at time t=0, we obtain OLR(0) = Q(0), and for t>>tau, OLR tends to a constant = Q(0)-Cb/lamda = Q(0) –b*tau. If we note that b can be written as F(tau)/tau, then this asymptote is equal to Q(0) – F(tau). All of this seems reasonable to me within the context of the assumptions made. The above solution may be compared with your corrected solution (post#60) : OLR(t) = β *((t-τ)+τ*exp(-t/τ))-β*t Clearly, the structural forms are very different. I emphasise that I am not wedded to the Schwartz model, but I believe that we need to get this out of the way if there is any hope of having a sensible conversation on the subject of whether an observed rise in OLR can be rendered compatible with common assumptions.
  9. Holy Cow, this is fun to watch.
  10. PaulK, "Clearly, the structural forms are very different." They are not. Indeed they're identical apart from the term Q(0) which comes from working with T instead of ΔT and the use of C/λ which is τ. It's just simple math. @Doug sorry to disappoint you. It was not much fun, just trivial math. :)
  11. Riccardo, Well, erm, OK. I guess if my bank manager were to say he were going to repay only the interest instead of capital and interest, you could describe that as structurally identical. I think what you are saying is that you wrote "OLR" when you intended to write "Delta OLR", where you would/could define the latter as OLR(t)minus the radiative input or output prior to the forcing being imposed. I think you are also suggesting that in context, I should have been smart enough to figure out what you meant rather than what you wrote. In this, I think you are right. I should have spotted what you intended to say. However, by a remarkable non-coincidental coincidence, the expression you wrote for OLR actually corresponds to the radiative imbalance perturbation function for the boundary condition of constant TSI - the thing I was focused on in the first instance, and which I would like to return to. This really did throw me off. Given that we do now (I believe) have a common understanding of Schwartz, what I would like to do is to take the general solution I offered in #55 and demonstrate that (a)it works perfectly when applied to Schwartz if one accepts the same (restrictive) assumptions as Schwartz and (b) that it is a lot more versatile in its ability to accept realworld data in order to assess how OLR should be moving. Unfortunately, I don't have time immediately, but I will post on (a) as soon as I do.
  12. Riccardo, I promised a post proving that the general solution I offered in #55 is easily reconciled with Schwartz if one accepts his assumptions. Generalised heating model: dH/dt = Q(t) – E(t) = absorbed SW (flux) – Outgoing LW(flux) at TOA Assume that at time t = 0, the system is in steady-state equilibrium: Q(0) = E(0). Now, keeping everything else unchanged, consider a positive impulse forcing F1 = constant which results in a perturbation, f(t), of the OLR. We can write: OLR(t) = Q(0) + f(t) ; Q(t) = Q(0) = constant ; dH/dt = -f(t) At this stage, we don’t know what f(t) looks like, but we do know some things about it: • Minus f(t) is positive definite on the open interval (0,te), where te is the equilibrium time. (Otherwise the (constant impulse ) forcing would have to cause a net cooling at some stage in its effect). • As the system restabilises at the equilibrium time, te, f(t) must go to zero. • It is both closed and integrable, since the area bounded by the curve represents the finite energy commitment associated with the impulse forcing, F1. This system can be solved for the total perturbation (and hence for OLR) by superposition. This permits one to model combinations of input and output forcings over time. But let us consider the simple case first of where we have an exponential or annually geometric growth in CO2, translated into a forcing which is linear with time: F(t) = bt. The solution is analytic for this condition, but for convenience later, we will choose a superposition timestep of 1 year. We set the first year forcing F1 = b. All subsequent years are then also equal to F1 to satisfy F=bt. This is equivalent to superposing each year the same perturbation function f(t) to obtain the total perturbation to the system. The solution is then OLR(t) = Q(0) + integral of f(t) from 0 to t for all t< te OLR(t) = Q(0) + integral of f(t) from 0 to te (i.e. a constant) for all t>=te Note then that independently of the choice/calculation of the perturbation function (f(t)), OLR is monotonically decreasing until the equilibration time, and stays constant thereafter for this case of F(t) = bt which is proxy for a geometrically increasing CO2 concentration. This is an analytic result. So, does the above solution work for the Schwartz model? In Schwartz, the equilibration time strictly speaking is infinity (NOT tau). The perturbation function for a constant impulse forcing F1 in Schwartz is given by f(t) = -F1*exp(-t/tau) = -b*exp(-t/tau) Substituting into the generalised solution for OLR above, we obtain: OLR (t) = Q(0) + integral of f(t) from 0 to t = Q(0) + b*tau*(exp(-t/tau) – 1) for all t less than te = infinity. You should then find that this is compatible with the solution we obtained directly from manipulation of Schwartz. Next stage is to better understand the difficulties of reconciling the increasing OLR with IPCC assumptions.
  13. Apart from your semi-deplorable fling about "junk science" PaulK, may I just say how refreshing it is to see people such as you and Riccardo, "e" roll up their sleeves and invest some serious mental elbow grease here? There are only a few folks hanging out on the site who can have a conversation of this nature and produce reasonable repartee, particularly when it comes to doing maths; Riccardo, BP and a very few others come to mind. I'm not sure about the importance of the discrepancy you mention as a dilemma, I'm off to see what's up in the recent observational department on that but it's certainly pleasant to see such detailed treatment.
  14. I'm not sure I see where the problem is regarding TOA imbalance, model versus observations (observed OLR vs. IPCC assumptions?). After cleaning up some obvious problems (>6W/m2, we'd cook more rapidly) with satellite measurements, we're left with direct imbalance observations of ~0.95W/m2 versus model predictions of ~0.85W/m2*. These rates are in reasonable agreement. I see PaulK's point w/regard to his formal look at the situation but something's not quite closing the circle; we can quibble about splicing etc. but it's pretty hard to simply say -all- OHC and atmospheric temperature measurements for the past 40 years are wrong and that temperature has not changed in that time, meanwhile both models and observations indicate an imbalance. Hmmm. A puzzle. *EARTH’S GLOBAL ENERGY BUDGET, K. E. Trenberth, J. T. Fasullo, J. Kiehl, Bull. Am. Meteorol. Soc. 90, 311 (2009) Full text here.
  15. Doug, Looking back at my posts, my attack on the article in the Open Mind site was completely ill-founded, and I retract my comments unreservedly. I claim temporary insanity since I was in the grip of an obsession that led me to believe incorrectly that Riccardo was using the Open Mind article to support an invalid definition of the forcing term as used in the Schwartz model. The author's reputation for infallibility remains untarnished - at least by me.
  16. PaulK, the process you describe is just an integration, which we already know. In some cases, like for example the linear forcing, we have the analytic solution. For a more general forcing we need to do it numerically. And we agree on this. But then you confuse the forcing with the OLR and never in you analysis does the net balance appear. Indeed you write dH/dt=-f(t); here f(t) should be the net energy (im)balance but then it cannot be equal to the OLR. You need to have both the forcing and the thermal radiation. I'd suggest to first write and solve the heat balance equation for ΔT (sorry if i keep using variations, why bother with the equilibrium values?). After that we can try to see who's that guy we call OLR.
  17. PaulK it's even more rare to encounter online folks capable of such contrite speech. Bravo. Not to sink into an opera of tearful counter-apologies but for my part my knee jerk estimation after reading your remark on Tamino was that I was seeing the emergence of another unreasonably intractable person. Sorry about that!
  18. Hi Doug, Thanks for the kind words. Re your post #89, I am certainly not saying that the temperature and OHC measurements for the past 40 years are "wrong", although they do carry some hefty measurement uncertainties. We have undoubtedly seen planetary heating over a long timeframe. The key issue here, for me at least, is climate sensitivity and, ultimately, the cause and attribution of the heating. A careful reading of the Trenberth and Fusillo paper reveals that they do not claim anywhere that the satellite measurements of radiative imbalance match the climate models. On the contrary, they state that they do NOT, and then use the error statistics on the satellite measurements to show that the satellite data can be adjusted within the error bars so that it is “not incompatible with” the residual imbalance inferred from climate models. This however then leaves Trenberth’s question of where the missing heat energy was going in the period from around 2002 to 2008, when OHC showed a flat/cooling trend (Willis, Levitus, Cazenave). For me the jury is still out on OHC from the Schuckman paper, which is not only an outlier relative to the three papers I mention, and has not been reconciled to the shallower data, but would also mean that all previous reconciliations of energy balance (which did not account for Schuckman’s variation in deep ocean heat content) were fundamentally flawed. But back on topic, in practical terms the relative error statistics on the radiative imbalance from satellite measurements are very large (error analysis on the difference between two large numbers always reveals poor statistics); the measurement noise on the difference turns out to be almost an order of magnitude greater than the signal we are interested in! On the other hand satellite measurements for OLR can be quite PRECISE, but INACCURATE in absolute terms. This implies that the relative errors on trends in the measurements should be smaller than the error in absolute magnitude of the measurement, and very much smaller than the (even larger) relative error in the difference between the measurements. It is quite possible - likely even - that we can then deduce more from the trends in individual measurements than we can from the absolute differences between those measurements. So, is it important if a climate model (from a simple analytic model to a CGCM) doesn’t match the observed trends in OLR? Well, it obviously depends on what information one is trying to abstract from the model. But if one is talking about attribution studies, I happen to believe that it is crucially important.
  19. Hi Riccardo, Your first paragraph (#91) raises a profound question, which I believe requires a separate post to deal with. Before I can get to it, however, your second paragraph suggests to me that we still have a gulf of understanding to bridge. You wrote: “But then you confuse the forcing with the OLR and never in you analysis does the net balance appear. Indeed you write dH/dt=-f(t); here f(t) should be the net energy (im)balance but then it cannot be equal to the OLR. You need to have both the forcing and the thermal radiation. I'd suggest to first write and solve the heat balance equation for ΔT (sorry if i keep using variations, why bother with the equilibrium values?). After that we can try to see who's that guy we call OLR.” Let me deal with this sentence by sentence to see if we can bridge the gap:- “You confuse forcing with OLR”. I don’t believe that I do anywhere. Can you be more specific about where this confusion occurs and I will try to address it. “...never in your analysis does the net imbalance occur. Indeed you write dH/dt= -f(t);...” The net imbalance IN FLUX is the basis for the analysis. I wrote dH/dt = Q(t) – E(t) = absorbed SW (flux) – Outgoing LW(flux) at TOA This IS the net imbalance at TOA. I also wrote:- Assume that at time t = 0, the system is in steady-state equilibrium: Q(0) = E(0). Now, keeping everything else unchanged, consider a positive impulse forcing F1 = constant which results in a perturbation, f(t), of the OLR. We can write: OLR(t) = Q(0) + f(t) ; Q(t) = Q(0) = constant ; dH/dt = -f(t) The net imbalance here is Q(t)-E(t), but Q(t) = Q(0) and E(t) after the forcing is equal to Q(0) + f(t). Hence the net imbalance is equal to Q(t)-E(t) = Q(0) – (Q(0)+f(t)) = -f(t). This is also by definition equal to the rate of change of energy entering or leaving the system, so we also have:- dH/dt = -f(t). “Here f(t) should be equal to the net energy imbalance.” No. It should not. The perturbation f(t) has the dimensions of FLUX. The negative form –f(t) is equal to the net FLUX imbalance for this boundary condition of constant input flux. This function represents a perturbation of OLR for a single pulse of CO2. It would have to be integrated w.r.t. time to give a net energy imbalance. “...but then it [f(t)] cannot be equal to the OLR. “ You are right. It is not equal to the OLR. It represents a perturbation of the OLR for a single impulse forcing . The OLR for this forcing (and a boundary condition of constant absorbed SW) = Q(0) +f(t) as stated. “You need to have both the forcing and the thermal radiation.” Agreed. They are both built into the perturbation. “I'd suggest to first write and solve the heat balance equation for ΔT.” Well as perhaps we will eventually get to, I am not sure what this should be in the real world. On the other hand if you want a solution for ΔT based on Schwartz-like assumptions, then you immediately have one from the solution I proposed by writing CdT/dt = C dΔT/dt = dH/dt. Since we know dH/dt, we can trivially calculate ΔT. This post is already too long, so I will answer the more difficult question that you posed in your first paragraph (Why use this new numerical solution when there is one already available?) in a separate post, when I get a little more time. But one thing which you said did strike me. I may be wrong, but I get the impression that you think I backed out the solution for OLR from Schwartz. I did not. I solved the superposition equation from the generalised definition, and then set all of the perturbations equal to each other for equal superposition timesteps. A “Chinese box” proof then demonstrates that the superposition solution is analytic for this boundary condition of F=bt. I then applied this solution to the more restrictive assumptions of Schwartz. It is possible that you are getting confused over the dimensionality of the solution I offered. Because the term for OLR involves an integral of flux, it may appear like there is confusion between energy and flux. There is no such confusion. The integral term is here effectively divided by time, but the superposition timestep equals 1 year. Hence the integral term here has the dimensionality of a flux. More later when I have a minute.
  20. PaulK, maybe I didn't understand your notation. Is your f(t) the same as what people usually call F(t)−λΔT? In other words, did you include both the forcing and the response to the forcing into f(t) so that it's not not just f(t) but f(t,ΔT(t))?
  21. Riccardo #95, Effectively, yes. Or at least it would be the same if one were to accept the assumptions in your version of the heat balance equation (and change the sign convention on the perturbation). With these assumptions, it is ALSO equal to -F*exp(t/tau) - the perturbation from a single impulse forcing.
  22. Riccardo #95, Erratum. I should have written ALSO equal to -F*exp(-t/tau). (A Taylor expansion of this form and then curtailment of the higher order terms yields your form of the heat equation.)
  23. PaulK, it's not clear to me what you mean by "the assumptions in your version of the heat balance equation". Which assumptions did "I" make? Did you find something wrong? I understand we're back to the beginning but I'm a bit lost with this discussion.
  24. Thank you for the detailed article. I have a question? In the section, co2 traps heat, you quote Evans 2006: The results lead the authors to conclude that "this experimental data should effectively end the argument by skeptics that no experimental evidence exists for the connection between greenhouse gas increases in the atmosphere and global warming." Yet the opening sentence in the link you provide, Evans states: "The earth's climate system is warmed by 35 C due to the emission of downward infrared radiation by greenhouse gases in the atmosphere (surface radiative forcing) or by the absorption of upward infrared radiation (radiative trapping)." It is the use of the conjunction "or" that I am querying. Which is it? Is it surface radiative forcing, or is it radiative trapping? I am *not* suggesting the above statement in some way invalidates Evans' proof, but it does cast doubt on the mechanism responsible for the warming. For completeness, I must point out that this sentence does not appear in the linked article - only the abstract.
    Response: "Which is it? Is it surface radiative forcing, or is it radiative trapping?"

    A little from column A, a little from column B. Greenhouse gases both absorb upward infrared radiation which warms the atmosphere, and also scatter or reemit infrared radiation in all directions, some of which heads back to Earth.
  25. It can't be a bit of both as you suggest. Warming the atmosphere can only lead to warming the surface if at least some of the warming is directed downwards. Specifically, radiative trapping produces (some) surface radiative forcing. But that is not what Evans said. He said it is one *or* the other. He didn't say it's a bit of both. Also, the sentence makes no sense if we substitute "or" for "and", so it is unlikely a typo. However, "caused by" would fit.

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