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Get Real!

March 29, 2008 · 57 Comments

The last post witnessed the utterly ridiculous (by which I mean, worthy of ridicule) attempt to justify the garbage analysis by Jim Goodridge. Much of the vain attempt focused on trying to say that the accumulated departure from average of sunspot counts, as a proxy for accumulated departure from average of solar irradiance, was somehow a meaningful indicator of heat content of the climate system — as though heat loss from the climate system were somehow constant. Of course, under this model if solar irradiance remains above average, even by a little, heat content will increase without bound, as will temperature. If you actually believe that under those conditions, temperature will continue to rise at a roughly constant rate forever, then you’re in need of remedial critical-thinking classes. Sit in the front so you can pay better attention, and try to help Anthony Watts — he needs it. And by the way, Goodridge doesn’t suggest that model; he doesn’t suggest any model, instead he suggests that his analysis is a valid way to characterize trends in solar output. That’s even more ridiculous.


But since the subject came up, I thought I’d give a more realistic picture of things. Oops! I shouldn’t say “more realistic,” because that picture of things isn’t realistic at all. I’ll show a model of actual behavior which doesn’t reflect reality, but at least is a plausible crude first approximation to it. As far as I can tell, it’s the crudest model which can be posited as having any relation whatsoever to reality.

Let’s begin by defining some variables. Let H be the total heat content of the global climate system. Let T be the mean temperature of the global climate system. Then those two variables are approximately proportional to each other, with a constant of proportionality C which is the specific heat of the global climate system.

H = C T.

Of course, over time the heat content of the climate system doesn’t remain constant. Energy comes in, at a rate (i.e., power level) which I’ll call I, and energy goes out, at a rate which I’ll call O:

dH/dt = I - O.

The input power I isn’t constant, it shows changes due to changing solar irradiance, greenhouse gas concentration, reflective aerosols, albedo change, etc. etc. etc. These factors change over time, so instead of just writing I, I’ll write I(t) to indicate that input is a function of time.

The output power isn’t constant either, it changes primarily due to changing temperature. One approximation of the output power is the Stefan-Boltzmann equation:

O = \sigma T^4,

where \sigma is a constant. Note that output power goes as the fourth power of temperature — it’s certainly not constant as temperature changes. That fourth-power factor makes the ensuing equations more difficult to solve mathematically, so I’ll use an approximation, that the temperature doesn’t change much from some “reference value” T_0. Let \theta be the temperature difference from reference (we can think of it as temperature anomaly), so

T = T_0 + \theta.

Then the output power is

O = \sigma (T_0 + \theta)^4 = \sigma [ T_0^4 + 4 T_0^3 \theta + 6 T_0^2 \theta^2 + 4 T_0 \theta^3 + \theta^4 ].

We can write this as

O = \sigma (T_0 + \theta)^4 = \sigma T_0^4 [1 + 4 \theta/T_0 + 6 (\theta/T_0)^2 + 4 (\theta/T_0)^3 + (\theta/T_0)^4 ].

Now \theta/T_0 is quite small, compared to 1. Therefore (\theta/T_0)^2 is much smaller still, (\theta/T_0)^3 is much smaller even than that, and (\theta/T_0)^4 is super-tiny. So as our approximation, we’ll ignore the tiny terms of 2nd, 3rd, and 4th power, leaving

O \approx O_0 + \lambda \theta,

where O_0 is the power output at “reference” temperature T_0, and \lambda is a constant. In fact this equation applies even if the output power doesn’t follow the Stefan-Boltzmann equation, it just requires that output power is a smooth function of temperature, which it is. It’s a general property of smooth functions that within a small range of variation of the independent variable, the dependent variable is approximately a linear function of that independent variable.

We can now write our heat balance equation as

dH/dt = I(t) - O_0 - \lambda \theta.

But H=CT and C is a constant, so

C dT/dt = I(t) - O_0 - \lambda \theta.

Of course T = T_0 + \theta and T_0 is a constant, so

C d\theta/dt = I(t) - O_0 - \lambda \theta.

We’re almost there. Let’s take the term I(t) - O_0 and call it the climate forcing; it too is a function of time:

F(t) = I(t) - O_0.

We now have our basic dynamical equation, the crude model of global temperature:

C d\theta/dt = F(t) - \lambda \theta.

Although I haven’t bothered to write \theta(t), it’s still the case that \theta, the temperature departure from its reference value, is a function of time, just like the climate forcing F(t). The parameters C and \lambda are constants.

This is the crude model used by S. Schwartz to estimate climate sensitivity to climate forcing. I (and others) happen to believe that his result is incorrect, not just because the approximate model isn’t close enough, but mainly because his estimated values of the various constants (C and \lambda) are incorrect; you can read about that here. But the point is that I’m not the originator of this model; as crude as it is, it has actually been applied in peer-reviewed publications by legitimate scientists to estimate relevant climate parameters.

For temperature to remain constant, the input and output must balance, i.e.,

F(t) = \lambda \theta.

Hence for temperature to remain constant, climate forcing must also remain constant. If climate forcing is not constant, we have to apply our dynamical model of the global climate system’s response to changing input (climate forcing), in other words we have to solve that equation to determine temperature response. It turns out that the equation can be solved exactly for any form of the climate forcing F as a function of time:

\theta(t) = \theta(t_0) + e^{-\lambda t / C} \int_{t_0}^t e^{+\lambda s /C} F(s)/C ~ds,

where t_0 is the “starting time.”

I’m fully aware that there’s an integral in that equation, and that the climate forcing F is inside that integral. But note the exponential factor also inside that integral, and the exponential factor outside it.

Now suppose that prior to our starting time, climate forcing was constant and equal to zero, and temperature departure \theta was constant and equal to zero. After time t_0, climate forcing increased to 1 W/m^2 and stayed there. Then the solution turns out to be:

\theta(t) = (1/\lambda) [ 1 - e^{-\lambda t / C}].

In response to the sudden increase of forcing, temperature rises. But the rate of increase doesn’t remain constant, because as temperature rises so does the output energy flow from the system. Temperature approaches a new equilibrium value of 1/\lambda, and approaches that value at exponential decay. The ultimate equilibrium value due to increased forcing of 1 W/m^2 is the climate sensitivity, which for this model is 1/\lambda. The characteristic time scale for its exponential-decay approach to new equilibrium is \tau = C/\lambda.

After the last post, I witnessed an orgy of idiocy in reader comments. I suppose I should expect the same thing here. Let the games begin.

Categories: Global Warming · climate change

57 responses so far ↓

  • TCO // March 29, 2008 at 4:07 pm

    You’re getting kind of touchy. P.s. There are are still unresolved issues with the PCA. Will you dig into this more? Do you think that Mann should have noted his acentric correction within the text of his paper?

    [Response: Don't change the subject. There are several threads where that topic is relevant, this isn't one of them. That's also a subject which is genuinely in dispute. This isn't.]

  • Aaron Lewis // March 29, 2008 at 4:17 pm

    In a system containing substantial amounts of water, “H=CT” is not linear in the neighborhoods of T=0C and T=100C. For the subsequent math to be valid at any useful approximation, the domain must be defined to exclude these neighborhoods.

    Since freshwater freezes (and melts at 0C and sea water freezes (and melts) at -1.8C, this neighborhood is larger than those found on math department blackboards.

    Ice makes life more interesting.

    It allows a body (GIS, WAIS, EAIS) to absorb heat while its temperature and emmistivity remain relatively constant.

    Certainly, ice volumes are a tiny part of the Earth’s climate system, but they are large enough to affect sea level. And, they are large enough the change the heat flow and expected temperature caculations.

  • TCO // March 29, 2008 at 4:20 pm

    I agree. I was just getting lonely for you on those other threads. Thought maybe you had found a new love.

    Plus, what is the point of digging into this stuff which is not in dispute, when there’s interesting stuff that is in dispute. I mean are you going to refute even the idiocy from people who babble about the low concentration of CO2 or other silliness? Watts is a hack. Let’s talk some interesting PCA shiznet…

    [Response: I know he's a hack, you know he's a hack, but he's still getting 300,000 hits a month on his blog, he was an invited speaker at the recent climate-nonsense conference in NYC, and I'm constantly seeing other bloggers claim AGW is false and pointing to Watts as a key resource.]

  • Hansen's Bulldog // March 29, 2008 at 4:20 pm

    Note: a reader inquired privately about the change from t to s inside the integral, in the equation giving the solution of the dynamical equation. That change is correct; t is the time, and it appears as the upper limit of the integration range, but inside the integral the dependent variable is an integration variable, also sometimes called a “dummy variable.”

  • Armagh Geddon // March 29, 2008 at 5:17 pm

    While you guys are busy answering questions, here are some more from Roy Spencer that seem to me to deserve a thoughtful response.

    “Hey, Nobel Prize Winners, Answer Me This
    by Roy W. Spencer
    University of Alabama in Huntsville
    15 March, 2008

    As a climate scientist, I would like to see some answers to a few basic global warming science questions which I’m sure the U.N.’s Ministry of global Warming Truth (also known as the Intergovernmental Panel on Climate Change, IPCC) can handle. After all, since they are 90% confident that recent global warming is manmade, they surely must have already addressed these issues:

    1) Why are ALL of the 20+ IPCC climate models more sensitive in their total cloud feedback than published estimates of cloud feedbacks in the real climate system (Forster and Gregory, J. Climate, 2006)? If the answer is that “there are huge error bars on our observational estimates of feedback”, then doesn’t that mean that it is just as likely that the real climate system is very insensitive (making manmade global warming a non-problem) as it is to be as sensitive as the IPCC models claim it is?

    2) And regarding those observational estimates of (somewhat) positive cloud feedbacks: How do you know that the cloud changes that have been observed during temperature changes really are “feedbacks”? In other words, how do you know that the temperature changes caused the cloud changes, rather than the other way around? This basic distinction between cause and effect is critical because such a misinterpretation will ALWAYS make the climate system look more sensitive than it really is (e.g., it is energetically impossible for more low clouds to cause a warming). Doesn’t it seem like a coincidence that the ONE case were we know that there was a huge non-cloud forcing (the 1991 eruption of Mt. Pinatubo) resulted in a negative solar shortwave cloud feedback, whereas all other periods showed supposedly positive shortwave cloud “feedback”?

    3) As a follow on to question #2, we all agree that there has been strong global-average warming since the 1970’s. Well, how do you know this wasn’t the result of a small, natural change in cloud cover? Doesn’t it seem like (another) coincidence that the Pacific Decadal Oscillation (PDO) just happened to shift to a different mode in 1977, about the time that the warming started? (Please don’t say that the greater warming over land versus ocean is consistent with manmade greenhouse gas forcing…because it is also consistent with ANY kind of change in the Earth’s radiant energy budget, whether natural or manmade.)

    The fact is, we DON’T know how much of recent warming is natural, simply because we don’t have good enough global cloud observations back to the 1970’s (and earlier) to measure any long-term changes in cloudiness to the required accuracy – 1% or less.

    The same cause-versus-effect uncertainty is true of any other climate variable as well, for instance water vapor, our main greenhouse gas. A small change in precipitation efficiency (the main process which ultimately limits the strength of the natural greenhouse effect) could cause a change in average water vapor content, which then would change the average temperature. In other words, increased water vapor doesn’t have to only result from warming…warming can also result from increased water vapor.

    The fact that we don’t have a good enough understanding (or observations) of cloud changes, or precipitation efficiency changes, on decadal time scales to document such potential mechanisms seems like pretty weak justification for blaming all of our recent warming on mankind. And if you say, “well, the IPCC doesn’t claim that ALL of the warming is manmade…”, then tell me: About what percentage of the warming IS natural, and how did you come up with that quantitative estimate?

    I fear that the sloppy science that too many climate researchers have lapsed into could, in the end, hurt our scientific discipline beyond repair. The very high level of certainty (90%) claimed by the IPCC for their manmade explanation for warming can not be justified based upon the scientific evidence, and is little more than an expression of their faith that they understand the causes of climate variability – which they clearly don’t.

    For those scientists who value their scientific reputations, I would advise that they distance themselves from politically-motivated claims of a “scientific consensus” on the causes of global warming — before it is too late. Don’t let five Norwegians on the Nobel Prize committee be the arbiters of what is good science.”

  • dhogaza // March 29, 2008 at 5:43 pm

    For those scientists who value their scientific reputations, I would advise that they distance themselves from politically-motivated claims of a “scientific consensus” on the causes of global warming — before it is too late.

    I’m sure he has the same advice for evolutionary biologists, seeing as he’s a creationist.

  • kim // March 29, 2008 at 5:50 pm

    I think I’ve never heard so loud
    The quiet message in a cloud.
    =====================

  • winnebago // March 29, 2008 at 5:56 pm

    Roy Spencer, the creationist, worrying about other’s scientific reputations. Now that’s comedy gold.

  • TCO // March 29, 2008 at 6:08 pm

    Armagh Geddon: Your comment is way off topic. I’m serious.

  • cthulhu // March 29, 2008 at 6:17 pm

    The “difference from mean sunspot model of solar forcing” was particularly ridiculous. Saw it posted on some blog as evidence that the recent warming was solar caused. That anyone is defending it is ridiculous.

    Looks like the usual suspects are frantically trying to derail the subject now though. I recommend simply removing their posts if they are incapable of finding the “open thread”

  • David B. Benson // March 29, 2008 at 6:19 pm

    Armagh Geddon — After checking

    http://en.wikipedia.org/wiki/Roy_Spencer

    I’ll opine that his priors, which include ‘intelligent design’, preclude him from correctly understanding the role of carbon dioxide as a global warming (so-called greenhouse) gas.

  • Bill Illis // March 29, 2008 at 6:30 pm

    How does the model apply to a particular location on the planet; for example a location at 45 degrees north where the peak temperature of the summer occurs 40 days (July 30th) after the summer solstice on June 20th?

    [Response: It doesn't. As I said, it's the *most crude* model that bears any relation to reality; it only considers a single global "box," has no position dependence (which makes it a zero-dimensional model).]

  • steven mosher // March 29, 2008 at 7:06 pm

    I can get you 300K hits months, but you have to do exactly as I say.

  • Brian D // March 29, 2008 at 7:20 pm

    I think Spencer has more to answer for first, such as why he himself endorsed a purely faith-based argument for inaction on climate change, based on the absolute facts that the seas can’t rise, that natural cycles never change, and that the Earth’s biosphere is invincible simply because the Bible says so.

    (Google “Cornwall Alliance”; Spencer signed the open letter I’m getting this from, along with several other “usual suspects” (even those who aren’t evangelical Christians). It’s particularly amusing in the light of the recent Catholic church addition of environmental damage to the list of deadly sins.)

    Now let’s get back on topic.

    I don’t have much to add that isn’t already up there; Tamino’s exposition is clear to me. However, I have experience with thermal physics (not very much, I admit — it was years ago, and the professor teaching those courses could put a charging rhino to sleep, so I’m hardly an expert). I’d be interested in seeing what those without such a background would say on this — time to hit up Watts’ blog, I suppose.

  • fred // March 29, 2008 at 7:48 pm

    The tone is very regrettable; the questions however are real. If only we could keep to real questions and restrained tone, how much quicker we’d get to the bottom of this.

  • David B. Benson // March 29, 2008 at 8:17 pm

    Brian D — My thermal physics was of the Physics 101 sort. I found Tamino’s expostition, as always, admirably clear.

  • Timothy Chase // March 29, 2008 at 8:18 pm

    steven mosher wrote:

    I can get you 300K hits months, but you have to do exactly as I say.

    He can, too!

    Steven McIntyre and Anthony Watts owe all their success to him. But the price, the price…

  • Steve Bloom // March 29, 2008 at 8:20 pm

    Question for “Armagh Geddon”: How do we tell that the error-prone Spencer isn’t just an embittered crackpot who’s not worth responding to?

    Underlying the material you quoted is the idea that climate is more stable than the IPCC thinks. What I have never seen from Spencer is a discussion of how that concept jibes with paleoclimate data. But maybe you’ve seen something on that subject that you could post?

    FWIW, I think he avoids the subject because it conflicts with his fundamentalist Christian beliefs. Events in Revelations don’t much resemble anthropogenic climate change. Beyond that (and actually I don’t know the answer to this), is it possible that Spencer is a young earth creationist?

    This article by Spencer is perhaps illustrative. Take-home quote: “I was convinced of the intelligent design arguments based upon the science alone.”

  • kim // March 29, 2008 at 8:26 pm

    The creationist criticism of the evolutionists is that they faithfully believe in a naturalistic development of the so-called ‘irreducible complexities’, such as flagellae and the clotting cascade, despite the fact that the precise mechanism of development is not elucidated.

    The alarmists criticism of the skeptics is that they faithfully believe in sun driven climate despite the fact that the precise mechanism by which that is accomplished is unknown.

    Compare and contrast. At your peril.
    ====================

  • DocMartyn // March 29, 2008 at 8:41 pm

    May I just ask a really simple question, vis:-

    “Stefan-Boltzmann equation”

    I will just ask a couple of very simple questions about the Stefan-Boltzmann equation.

    1) Does the Stefan-Boltzmann equation apply to all materials, or does it only apply to materials in the solid state and when such solids have high rates of heat transference (as claimed by Boltzmann)

    2) Does the Stefan-Boltzmann equation only apply to objects which have a uniform temperature (as claimed by Boltzmann).

    3) Why does the Stefan-Boltzmann equation fail in the description of the Earth? The mantle/crust upper boundary is at a temperature of over 4,000°C. Why is that not radiated into our feet? Why does the crust act as a better insulator than does vacuum?

    4) Why should a solid, with an atmosphere, follow Stefan-Boltzmann equation at all?

    5) Why do you use equilibrium thermodynamics to describe and non-equilibrium system? Why not use steady state thermodynamics?

  • cthulhu // March 29, 2008 at 9:17 pm

    Well a bit hypocritical of me seeing as I complained about going off topic, but

    “Twenty years ago, as a PhD scientist, I intensely studied the evolution versus intelligent design controversy for about two years. And finally, despite my previous acceptance of evolutionary theory as “fact,” I came to the realization that intelligent design, as a theory of origins, is no more religious, and no less scientific, than evolutionism.”
    http://www.tcsdaily.com/article.aspx?id=080805I

    It is troubling he could come to such a conclusion after two years of “intensive” study on the subject. Even more troubling are the arguments he uses which a cursory google search would find lacking.

    Here’s a site much like skepticalscience but answering common arguments of evolution skeptics, he uses a fair few of these arguments:
    http://www.talkorigins.org/indexcc/list.html

    Some people would say it is wrong to write off his analysis of the field of climate science just because his 2 year intensive analysis of another field was totally wrong, but I am not one of these people.

  • cthulhu // March 29, 2008 at 9:24 pm

    [QUOTE]The creationist criticism of the evolutionists is that they faithfully believe in a naturalistic development of the so-called ‘irreducible complexities’, such as flagellae and the clotting cascade, despite the fact that the precise mechanism of development is not elucidated.[/QUOTE]

    The mechanism certainly is, it’s the precise pathway that isn’t known.

    “The alarmists criticism of the skeptics is that they faithfully believe in sun driven climate despite the fact that the precise mechanism by which that is accomplished is unknown.”

    That one is more correct. They propose some unknown solar mechanism caused the warming - see the very topic of this blog entry for example. They provide correlation but no mechanism rooted in physics.

  • David B. Benson // March 29, 2008 at 9:34 pm

    It would make for a cleaner organization to take the deconstruction of Roy Spencer to the current Open Thread.

    Please.

    [Response: Agreed. Those who wish to discuss Spencer or creationism, take it to the open thread.]

  • kim // March 29, 2008 at 9:50 pm

    Right cthulhu, the correlation is there, the causation is probably there; the precise mechanism will earn a Nobel, surely.
    ==============================

  • David B. Benson // March 29, 2008 at 10:19 pm

    DocMartyn stated “I will just ask a couple of very simple questions about the Stefan-Boltzmann equation.” He then proceeded to ask five.

    You know, you could go read somethiing first, and then maybe have some questions which are not immediately answerable by anybody who has:

    http://en.wikipedia.org/wiki/Stefan-Boltzmann_law

  • Hank Roberts // March 29, 2008 at 11:14 pm

    Or docmartyn could read his own previous postings, in which he lays out his Stefan-Boltzmann interpretation in detail. Elsewhere here, and at CA, and at other blogs.

    Google — find yourself.

  • Hans Erren // March 29, 2008 at 11:30 pm

    The variations in the oceanic heat content dwarf that of the atmosphere.
    http://climatesci.org/2008/03/28/a-short-tutorial-on-global-warming/

    “As repeatedly emphasized on Climate Science, the use of surface air temperature is not a measure of climate system heat since it has almost no mass associated with it. Heat of the climate system requires that temperature change (and heat associated with the phase of water) ocurr over mass. “

  • Erik Hammerstad // March 30, 2008 at 12:14 am

    Jason measures ocean altitude, Grace measures ocean mass, and Argo measures ocean heat content, and they should agree when combined. Currently they do not, see for example http://www.dgfi.badw.de/typo3_mt/fileadmin/20071015-17-Potsdam/mo_1410_03_chambers.pdf
    so Hans Erren’s quote from RP’sr is of no help, allthough in principle it is of course correct.

  • cthulhu // March 30, 2008 at 12:29 am

    I would expect surface air temperature is at least proportionally related to heat of the climate system over long enough periods. Example: surface air temperature in the past decade has been higher than it was in the 80s, and that should be a strong indication that there is more heat in the climate system today.

  • kim // March 30, 2008 at 12:29 am

    As Arnt Bernerts said “Climate is the continuation of the oceans by other means”.
    ==========================

  • Gavin's Pussycat // March 30, 2008 at 3:14 am

    > They provide correlation but no mechanism rooted in physics.

    Which correlation? There isn’t any. If either solar activity or solar brightness– or any proxy of those, like cosmic-ray intensity — had been increasing over the past three decades, it would perhaps be worthwhile to start looking for a mechanism. As it happens, they haven’t.

  • kim // March 30, 2008 at 4:13 am

    GP, sunspots are sparse or absent during the Grand Minima. I agree, the mechanism by which the sun evokes changing climate is unknown. Aren’t you curious?
    ========================

  • Cthulhu // March 30, 2008 at 5:09 am

    “Which correlation? There isn’t any. If either solar activity or solar brightness– or any proxy of those, like cosmic-ray intensity — had been increasing over the past three decades, it would perhaps be worthwhile to start looking for a mechanism. As it happens, they haven’t.”

    I was talking about the correlation with temperature if you graph difference of sunspot number from the mean. But there’s no proposed physical mechanism underlying doing that, which coincidentally is the subject of the last two posts here (im on topic for a change)

  • Hank Roberts // March 30, 2008 at 6:04 am

    Chuckle. A checkable source, albeit misspelled, and it’s the one with all the duplicate websites so it appears many times. It must be realer!
    http://www.google.com/search?num=100&hl=en&newwindow=1&safe=off&client=firefox-a&rls=org.mozilla:en-US:official&hs=F0V&lr=lang_en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=Bernaerts+said+%E2%80%9CClimate+is+the+continuation+of+the+oceans+by+other+means%E2%80%9D.&spell=1

  • Gavin's Pussycat // March 30, 2008 at 7:05 am

    kim, I meant last 30 years.
    Cthulhu, OK, I see what you’re getting at. But didn’t HB dispatch that one adequately?

  • Gavin's Pussycat // March 30, 2008 at 8:21 am

    The analysis presented is one which I was thinking of writing up myself, relating to the Schwartz analysis. Now somebody did it. Thanks HB!

    We can extend it a bit still. I have bad experiences entering formulas here, but here it comes.

    The differential equation found is called linear non-homogeneous, and we know that once we have found one solution, we can obtain all the solutions by adding any solution of the corresponding homogeneous equation:

    C d\theta/dt = -\lambda\theta

    The general solution of this is

    \theta(t) = A exp(-\lambda t / C)

    with A an arbitrary constant. This describes the exponential decay, from whatever initial state, to the equilibrium state — whatever that is, based on the solution to the non-homogeneous equation (so you should add this \theta to that earlier obtained \theta.)

    The time scale of the decay is \tau = C/\lambda: then you get for the solution

    \theta(t) = A exp(-t/\tau)

    as HB argued, but didn’t quite show.

    It gets interesting when you add a second heat reservoir (say, deep ocean) to the first. Then you get two equations:

    C1 d\theta1/dt = X(\theta2 - \theta1) - \lambda\theta1
    C2 d\theta2/dt = X(\theta1 - \theta2)

    where X describes energy transport between the two reservoirs, depending on temperature difference.

    As a matric equation this becomes (note that \theta has become a two-element vector, here written in Matlab notation):

    d/dt [\theta1; \theta2] = -M [\theta1; \theta2]

    where

    M(1,1)= (X+\lambda)/C1
    M(1,2)=-X/C1
    M(2,1)=-X/C2
    M(2,2)=X/C2

    This matrix is positive definite having a determinant of (\lambda X) / (C1 C2), where all symbols are known to be positive for physical reasons. It will have two positive eigenvalues, corresponding to two different time scales, both describing an exponential decay process. The solutions \theta1(t) and \theta2(t) will be (different) linear combinations of these two exponentials.

    So here we have the example of another thing lacking with the simplistic Schwartz approach: all you need is the vectorial AR(1) representation above to get more than one characteristic time scale, adding some sorely missing physical realism.

  • Gavin's Pussycat // March 30, 2008 at 9:54 am

    Chtulhu:

    But there’s no proposed physical mechanism
    underlying doing that, which coincidentally is the subject of the last two posts here (im on topic for a change)

    To expand on that, the problem isn’t so much that there is no proposed physical mechanism, as that, even with a working mechanism, the math doesn’t work out, as HB tried to show above: you would get a decaying exponential, not a growing one as we see in reality.

    This is perhaps clearest seen in a picture

    http://www.realclimate.org/BD3.jpg

    which is part of a December 2007 article by Pierrehumbert. You see the mean global temperature go up in an exponential fashion (red curve), but none of the measures of solar activity (solar irradiance, blue, cosmic ray flux, green, geomagnetic index, black) do. They just oscillate on the 11-year solar cycle period. And that does indeed affect the temperature record on the +/- 0.2 degree level.

    So, it’s just not true that climatologists deny the influence of the Sun on global temperature variation: that influence is well known and documented. It’s not even true that there’s no proposed physical mechanism — there are several tentative ones, but their effects just haven’t been quantified at this point. The point is that they don’t explain what happened over the last three decades.

    Ray Pierrehumbert’s full article is here:

    http://www.realclimate.org/index.php/archives/2007/12/les-chevaliers-de-l%e2%80%99ordre-de-la-terre-plate-part-ii-courtillots-geomagnetic-excursion/#more-504

  • Johan i Kanada // March 30, 2008 at 2:12 pm

    Tamino,
    There is no need to misrepresent what I and others argued re: your previous post. One does not have to be stupid to disagree.

    With this post you show very clearly (thank you!) that, if lambda is relatively small and with small temperature devations, O is close to constant, and dT/dt essentially becomes proportional to the I(t) anomaly (i.e. I(t) less O0) . Which then is what would be the underlying justification for comparing integrated energy input with the temperature trend. Which is what Goodridge attempted to do (albeait in not the most scientific manner).
    Of course, this is a simplified (special case) version of the most rudimentary model (and as such it has severe limitations), but does not seem to be a fundamentally un-sound or ill-formed model.

    Tamino & GP, regarding the oceans, what happens with the step response in a model where oceans (presumably a much slower and larger energy reservoir than the atmosphere) is included in the model?

    [Response: Sigh... it's nothing short of astounding how far you're willing to go, rather than accept the fact that Goodridge's analysis is bull.

    Even if we apply this zero-dimensional model with a very long time constant (and Goodridge doesn't give any model at all), do you have *any* idea how long the time constant would have to be for that model to be even approximate?

    Did you pay attention to the zero point which relates input to climate forcing? Think about that -- and how it relates to the "declining trend for the 1700 to 1935 period" Goodridge finds.]

  • Johan i Kanada // March 30, 2008 at 4:34 pm

    Tamino,

    Sigh…

    Nodody (at least not me) have argued that the Goodride essay is the height of scienctific writing. But you and others attacked the principle of integrating energy input over time, saying that it was invalid and one could only look at the “data” and not the “accumulation”. But integration (and differentiation) is of course completely valid and, I would assume, present in all existing climate models.

    Btw, how long is the time constant?

    [Response: Not only do you not understand what's going on, you don't understand what I've said.

    I have not "attacked the principle of integrating energy input over time, saying that it was invalid..." I pointed out, quite correctly, that using such a procedure to characterize *trends* is nonsense. And that's precisely what Goodridge did, but you are STILL trying to suggest that Goodridge's analysis is some sort of integration of solar output. Goodridge doesn't even *have* a model.

    As this post demostrates, integrals which include climate forcing can be part of realistic models. But integrals which include ONLY climate forcing, with no time-decay factor, are NOT.

    Have you even *thought* about the proper zero point? About the "declining trend from 1700 to 1935"?

    As for the time constant -- figure it out for yourself. I'm sick and tired of being your remedial-critical-thinking instructor.]

  • Johan i Kanada // March 30, 2008 at 7:11 pm

    Tamino,

    Chill, please. You’re a good instructor, and a good writer, but you don’t have to teach me critical thinking.

    For sure you can help me (and others) to understand the modeling and methods that you guys employ. I am, in all sincerity, interested in what the impact oceans might be, what the actual time constant could be, etc etc.

    [Response: I believe that you're sincerely interested. But you have to stop making *statements* about the validity of Goodridge's analysis, and just ask questions instead. It would greatly help if you'd invest more effort working things out, before aksing questions. And you really need to face the fact that Goodridge's work is bs. Say it out loud.]

  • L Miller // March 31, 2008 at 5:46 pm

    Johan

    One essential point you are missing is that there is no storage mechanism between surface temperature and outgoing IR ( Power out). The oceans do store energy and therefore take time to warm up or cool but that occurs before/as surface temperatures rise. All you need to disprove the assumption of constant power out, therefore, is to show a temperature trend . This negates any possible use of this particular type of accumulation in looking at temperature trends.

  • David B. Benson // March 31, 2008 at 7:16 pm

    Here is a similar question (for which much of the math is already done, but appears to need a computer to properly draw the graph of the answer):

    A cylindrical block of ice at a uniform temperature of 270 K is sitting in a perfectly insulating container. It is suddenly subjected to a constant source of heat, uniformly over the entire top. The melt water is assumed to be instantly removed. Describe the position of the top of the ice as a function of time.

    You’ll need the thermal conductivity of pure water ice and the latent heat of fusion.

    I’m interested in the answer, but don’t have time to work it out in detail. Anyone?

  • TCO // March 31, 2008 at 7:59 pm

    I’m just a high school algebra type, not a physicist or heat transfer jock.

    The heat will go into two things: melting of water and heating of the -3C block of ice. Given immediate removal of melt, you can treat the heating as coming from a OdegC .

    Some equations to describe the problem:

    A. Total energy:
    Total heat = heat rate(t)*total time= Vice*Cpice*3deg+Vice*dHfusion. (from this “end time” can be derived).

    Vice=piRsq*h

    B. Rates
    Heat rate (t) = rate of heat going into melting(t) + rate of heat going into ice temp changing (t)=constant.

    Heat transfer into the block of ice will be a function of the temperature delta (really the gradient as a function of time, this is not an equilibrium situation) and the thermal conductivity of ice. The temperature gradient in the block is a function of time and thus the changing geometry (shrinking), the heating up over time of different spots (along the profile). The Cpice will come back into play as will summation of warming of each spot.

    This should be solveable with some diffeqs or integration or whatever. I’m not going to derive that from first principles but looking up some stuff on plate heat transfer in a basic text would probably help here.

    Then the amount of heat going into melting as a function of time can be recorded as just the difference of constant overall heat rate and the amount going into block heating (partition will vary over time, I think).

    The height of the block as a function of time can just be calced based on Cp, geometry, and the time varying heat of melting rate.

  • David B. Benson // March 31, 2008 at 9:43 pm

    TCO — Thanks for the attempt. It led me to find

    http://www.britannica.com/eb/article-65644/ice-in-lakes-and-rivers#534086.hook

    which is about ice growth, but close enough. Since the thermal conductance of ice is high, first the ice (of moderate thickness) all gains 273 K and then the melting proceeds according at a rate proportional to sqrt(k0(F-t)+k1), where F is the final time at which there is no ice, t=0 is taken as the time the ice begins to melt and k0, k1 are constants.

    Hope I did this right, because it certainly is not what I had expected!

  • TCO // March 31, 2008 at 9:50 pm

    If the whole block becomes 273, than the rate of melt advance at that should be constant after ice is all ready to melt (constant heat input, constant rate of melt advance).

  • David B. Benson // March 31, 2008 at 10:45 pm

    TCO — That’s not what I obtain by running the ice in lakes equation backwards in time. Maybe that is a different problem. I’ll need to ponder it later.

    By the way, I didn’t write down correctly in my previous post. I won’t try again, since the equation is in the link.

  • TCO // April 1, 2008 at 12:08 am

    If the entire block is AT the melting temperature, and then you hit it with heat at a constant rate, then you will melt ice at a constant rate. That is just a basic property of matter.

    If you start the problem with the ice at some temp below freezing, than there is some partition of energy into heating up the block and into melting. I would imagine the total energy raise the entire block 3 deg is much less than that to melt it, therefore as a first approximation, you could just use constant rate of melting. However, you wanted the real functional form and I told you how to derive it.

    I have no idea what your ice in lakes equations shows, but it is very likely that it is a different situation than the example you gave in terms of boundary conditions (insulation, isostarting temp , removal of melt, etc.) so why are you surprised that there is a different answer?

  • David B. Benson // April 1, 2008 at 1:04 am

    TCO — The equation is not mine. I found it in the link to an Online Britanica page.

    I’m surprised that it is so completely different. As I said, I’ll need to ponder the situation some more…

  • David B. Benson // April 1, 2008 at 9:16 pm

    Here is a derivation of the time evolution equation for the height of the ice, as in Online Britiannica, except for the part about the bulck transfer coefficient:

    The ice is warmed to its freezing temperature, 273 K.

    Let the surface area be A, the depth h = h(t) and the density D. The mass is M = M(t) = DAh(t). It is necessary to add Q = ML joules, where L is the latent heat of fusion, to melt the ice.

    The ice is a heat pipe. Let k be the thermal conductivity of ice and dT the difference between the heat source and 273 K. The heat pipe transfer rate is
    H(t) = kAdT/h(t) with h(0) = h0 the intial condition.
    Acting for s seconds to transfer Q joules, we have

    DLAh = Q = Hs = kAdTs/h

    Multiplying on right and left by h and dividing by DLA, we obtain

    h^2 = (k/DL)dTs

    from which we obtain

    h(t) = sqrt((k/DL)dTt)

    describing the dynamics to be subtracted from the initial condition, h(0) = h0. About 70% of the ice is melted in the first s/2 seconds.

  • David B. Benson // April 1, 2008 at 10:53 pm

    Oops. I didn’t do the intial conditions properly, although the dynamics for h(t) is correct (under the usual Physics 101 simplifying assumptions). Rather than write down the correct, but messay looking, equation, just note that in the first s/2 seconds, Q/2 joules have been delivered into the system. This goes into

    (1) completely melting about 30% of the ice volume, and

    (2) adding 0.2Q to the remaining 70% of the ice volume (which needs 0.7Q to actually melt).

    There remaining 0.5Q is then added in the last s/2 seconds, completely melting the (perfect) ice.

    The lesson to be learned, for me, is that ice sheet melt, even assuming a constant temperature bath (instead of an, on average, increasing one), proceeds ever faster as the melt proceeds. While surely the details of an ice sheet melt are much more complex than this simplified model, it is nontheless approximately correct.

    So how much of Greenland’s ice cap will melt this melt season upon us? And the next? The next after that? …

  • null{} // April 2, 2008 at 10:05 am

    David B. Benson, the units in your equations are incorrect. A level of fundamental error that is seldom attained by even freshmen in high school. I’m surprised that tamino allows such errors to appear here. I suggest you check into the dimensional homogeneity of the fundamental laws of nature.

  • null{} // April 2, 2008 at 12:13 pm

    David B. Benson, see:

    http://en.wikipedia.org/wiki/Stefan_problem

    http://web.mit.edu/lienhard/www/ahtt.html

  • Dano // April 2, 2008 at 12:57 pm

    Someone really should tell The Google it needs a ‘Wisdom’ button.

    Best,

    D

  • null{} // April 2, 2008 at 1:54 pm

    Dano, another of you seemingly endless most excellent contributions to the technical issues.

    So long as you continue to fail to note incorrect technical issues, those issues will be taken to be correct as stated.

    The second reference above, by the way, contains the correct analytical solution for the problem David B. Benson is messing up. It’s a sophomore-level heat transfer problem.

  • Hank Roberts // April 2, 2008 at 2:22 pm

    David, you might ask our host to retroactively either fix (block-quote with a comment, or strikeout if the software will do that) errors in your math/theory posts as you later discover them –to simplify the thread for later readers.

    Else it’s just confusing, I think.

    Most blogs just leave errors up and comment on them later, but that leaves the errors where people will find them with search tools, not good.

  • David B. Benson // April 2, 2008 at 5:40 pm

    null{} // April 2, 2008 at 10:05 am wrote “David B. Benson, the units in your equations are incorrect.” I suppose this is possible, but I did check the units. What I derived agrees with the solution given in the Online Britannica link given in a prior post. [In your next message you provided two links. Thank you.]

    Hank Roberts // April 2, 2008 at 2:22 pm — Thank you for the suggestion, but I suspect that Tamino, aka “Hansen’s Bulldog”, has quite enough to do.

  • David B. Benson // April 2, 2008 at 9:27 pm

    This thread is mostly about Stefen’s Law, aka the Stefen-Botzman equation. The appendix, for the last several posts, considers another approximation having to do with ice melt. This is a difficult question, since geologists still don’t seem able to accurately model ice collapse, as the news of the last several years suggests. Indeed, here is a link to an attempt to model the southern Laurentide ice sheet

    http://www.geology.wisc.edu/~davem/abstracts/05-11.pdf

    which offers its own self-criticism in the conclusion.

    Rather than start with such complex models, which try to consider all aspects of ice growth/melt and flow, it is instructive to begin with an extremely simple, Physics 101 style, equation for ice growth and then melt. An approximate equation for growth, from

    http://www.britannica.com/eb/article-65644/ice-in-lakes-and-rivers

    is (with a change of notation to avoid subscripts)

    h(t) = sqrt((2k/DL)dTt + (k/B)^2) - (k/B)

    where h is the height, in meters, k is the thermal conductivity in (W/(m.K)), D is the density in (kg/m^2), L is the latent heat of fusion in (J/kg), dT is the temperature difference in K, t is tiem elapsed time in seconds, and B is the bulk
    transfer coefficient in W/(m^2.K). One readily determines that the units inside the square root are square meters, so the equation is dimensionally correct. Since (k/B) is about 0.1, so long as the ice is a meter or more thick, the terms involving (k/B) can be neglected to obtain an even more approximate solution. I did this in a previous post on this thread, and for the record, the determination is dimensionally correct. Let us now ‘run time in reverse’ in that equation to even more approximately see the time evolution of melting ice. We start time at s seconds with the ice depth being

    h(s) = sqrt((k/DL)dTs)

    via the more approximate equation and run time towards zero as the melt progresses. The point of the exercise is to discover the (now unsurprising) fact that the melt proceeds at an every increasing rate.

    null{} wrote “David B. Benson, the units in your equations are incorrect.” I will certainly acccept an apology, for while my Physics-101-concepts-only solution is certainly too simple, it is dimensionally correct. He further suggested looking for the solution in a heat transfer online text’book’. I went through Chapter 5 without finding the solution. Perhaps null{} can point out a particular page or pages.

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