125 thoughts on “Two Box Model: A few results”

1. Lucia,
   Here are my results for that case (1,30). With the format
   T’ = -AT + bF
   with forcing applied to box 1 only, I get
   > A
   [,1] [,2]
   [1,] 0.14305119 8.6239125
   [2,] 0.01090254 0.8902821
   and after multiplying the bottom row by 791, the heat capacity ratio:
   CA
   [,1] [,2]
   [1,] 0.1430512 8.623913
   [2,] 8.6239125 704.213173
   And yes, the off diagonal terms are positive, which since -A is the multiplier, is unphysical. I don’t currently know why that is. It may be because of applying the forcing only to box 1. There’s also an issue with offsets. I’m currently not taking account of them; basically set to 0 for lack of info to the contrary.

2. Off sets meaning the constants term in the fit? I just used weighting factors supplied to me. Yours look slightly different.

   My sense of what goes wrong with the regression is this:

   We know there is noise in the temperature fits and the forcings are estimated. Also, the two box model (no matter how you shake it) is approximate no matter what– plus it only even attempts to describe the “expected value” of the temperature. You would have “weather noise” no matter what.

   When you fit the regression, only one temperature series is constrained to reality. The fit coefficients adapt to give the best fit for that time series, but there is bupkis to constrain the other box. Any errors do to anything can really screw up the fit.

   I suspect if made a synthetic two box system and super imposed noise, plus errors and drove it, we would find Tamino’s method of teasing out the sensitivity from one temperature profile would have huge uncertainty intervals. Huge.

   It might be fun to do if we are really interested in that as a method. It’s pedagogically interesting– as an example of something that should be done for any proposed method of determining sensitivity from a model. But I doubt it would be a paper unless someone wrote up the first fit and got it past some reviewers who didn’t ask the right questions!

3. Lucia, The weighting factors should be the same – they do not appear directly in A. And yes, offset means the constant term in the regression, and also in the ode system, where it acts as a centering on the forcing.

   In my formulation, the coupling between the box 1 regression and box 2 comes from eq (6), which basically says that the non-zero transfer coefficient means that A is not diagonal – so the eigenvalue matrix P is nontrivial, and so there is one equation expressing a linkage. The other linkage equation comes from the requirement that the heat transfer matrix is symmetric (energy cons).

   As I mentioned in the last thread, my problem with the inferred ocean temp is the opposite of yours – it’s just too immobile, which results from the huge heat capacity and the implied assumption of good mixing.

4. Nick–
   My weighting factors for solutions differed from Arthur’s by 1/tau because of the way we define our equations. It shouldn’t make a difference to the final temperature reconstructions as long as you don’t switch from one to the other.

   The temperature of the real ocean is too immobile. I admit this is a problem for fitting. But the problem with not using the ocean temperature is that the regression permits solutions that have temperature series that let the ocean do anything. If you reconstruct the ocean corresponding to the fit, you’ll see it does do wild things.

   So, my sense is that using some ocean data would be better than none. (Still, the bigger problem is that the two-well-mixed boxes model may simply not be useful, particularly if one of the other box or a linear combination is supposed to be represented by GMST.)

5. Lucia,
   If you are getting big ocean temperature changes, that implies huge heat fluxes in and out. Since the efflux to space is on a 30 yr timescale, that means the heat must be going into box 1, which would produce even wilder swings. This heat flux must show up in big numbers moving through your system?

6. Nick–
   In some of the solutions corresponding to the weights from the regression, the overwhelming majority of the heat goes in the ocean first. That applies to the illustration with the wild temperature swings above. This is made even wilder by the negative heat transfer coefficients!

   I didn’t discuss the physics of those “swings” for that reason.

7. Nick– I could show the response to a step impulse for some of those. They are weird due to the 2nd law violations and also to the way heat is apportioned to the boxes.

8. Nick–
   I did find an error in my solution for the quadratic. I had a * where I should have a +. I’ll revise this and repost.

   Heck, it could change my conclusions.

9. lucia,

   I’m not surprised that these examples are unphysical because they bear no relation to how the atmosphere/ocean system works on Earth, or at least my understanding of how it works, and what the temperature series actually measures. Your choice of heat capacities for the fast box are wildly unrealistic. The low is orders of magnitude too low and the high is about a factor of 5 too high. For the low choice of atmosphere only, I’ve calculated that the temperature over a 24 hour period would look much like the moon over a full lunar day with a diurnal temperature range at the equator of nearly 172 K (I was using lunar albedo of 0.11 rather than 0.30 so the delta T is now lower than the 197 K calculated before) if it were isolated from the ocean box. But of course it isn’t, so most of the heat will end up in the ocean, which, because that heat capacity is very high, wouldn’t change temperature at all to a first approximation. But of course we do have a temperature series to fit to, so the coefficients will be forced to non-physical values and voila, a second law violation.

   For the 5% ocean included in the ‘fast’ box, or a water column depth of 130m, the diurnal temperature change is only 7.2 degrees. So now the temperature change with forcing is too small, and once again the calculated coefficients will have to assume unphysical values to agree with the fitted model and there is a second law violation.

   Try 1.34E8 J/(Km2). for the fast box. It probably still won’t work because the thermometer placement assumption is wrong, but it might look somewhat better. Even better would be to also make y=0.2 or whatever the value is to put 80% of the thermometers in the ocean box.

10. The best way to get physical solutions is to write the states as follows:
    (one state is the difference of the temperatures)
    x_1=t_2-t_1
    And the other state is the total energy
    x_2=c_1t_1+c_2t_2

    The first state should be associated with the diffusion of heat between the two boxes and the second state is associated with the total energy.

    The differential equations are written as follows:
    \dot x_1=k x_1+Fd=(t_2-t_1)
    \dot x_2=c_1t_1+c_2t_2+Fs

11. Lucia

    I did find an error in my solution for the quadratic.

    I have a similar problem. My system of equations (10) and (11) has two solutions, where c1 and c2 swap signs. The only change it makes is that the off diagonal component A_12 changes sign. That goes from “unphysical” to “physical”. And from the algebra, there’s no reason to say which solution is correct. More thinking needed.

12. Nick–
    I still get a quadratic for the solution to r₊. I think that’s actually ok. If you go from the box parameters to the time constants, the you get one set of parameter. But in the other direction, you don’t. ( I’m pretty sure when I do the solution for y=0, which “swaps” the boxes, we’ll see the two eignen value solutions are the same. So, the “mystery” is the regression doesn’t “know” which box is the “top” box or which box is the “lower” box.

13. lucia,

    Diurnal temperature range in the Western Pacific is less than 1.5 C according to this publication by Webster, et.al. For clear sky conditions you’re looking at up to 958 W/m2 peak insolation. As I said, it dwarfs the greenhouse forcings. For land, the local humidity plays a role, higher humidity means lower diurnal range. But the average diurnal range over land is still on the order of tens of degrees. That constrains the land only heat capacity to a fairly narrow range with the lower limit at least an order of magnitude higher than atmosphere only. Obviously, the diurnal range far inland is not going to be affected much by the ocean.

14. Dewitt–
    It’s not isolated from the ocean box. The time constants for the solution and the time constants if you assume the two boxes are isolated differ dramatically. The weights for the solution matter.

    My plan after I look at selected case is to run a program that steps through a full range to let the program “discover” whether there are any choices of time constants for a two box model that do not violate the 2nd law. Once we find those, we can report what other properties it gives for the two boxes.

    Right now, the cases that correspond to regression variables look… odd… In oh so many ways. Some have heat transfer from a cold ocean to a hot atmosphere. Some have radiation heat transfer work backwards. Some correspond to a stefan-boltzman constant for the ocean exceeding that for the atmosphere by … oh.. thousands.

    But, all this is provisional on my not finding another algebra error like the one I found based on Nick’s comment.

15. Nick–
    My algebra looks ok. It’s just that I had a (Cs a+) in the numerator for “B” in a quadratic. I needed that in the *denominator.*

    I’m paranoid about going back and checking all my equations. For example, if I solve a 2×2, afterwards, I enter a cell for the two original equations and check.

    However… I didn’t enter the original for the quadratic to solve for r+, and the double checks for the eignenvalues assumed the r+ and r- were right!

    I guess I’ll do more checking tomorrow. But I think I have these right now.

16. Dewitt–
    I don’t think the diurnal problem information helps here. These have been defined as anomalies and the daily fluctuations are out.

    If your notion is the two box model is unrealistic: Quite possibly so. But the atmosphere does not include the dirt and the thermometers are not in the dirt. If we include the dirt, we can’t use a two box model.

17. lucia,

    It certainly is too isolated ,on a diurnal basis, or at least the temperature measurement is. There just isn’t time for the heat to move from one box to another over hundreds or thousands of miles. How far inland do you have to go before the moderating influence of the ocean on daily temperatures isn’t felt?

    Obviously the coupling is there over years and decades. I just think you have two independent measures of the local heat capacity for land and ocean that you could be using and aren’t.

18. Dewitt–

    But we know that either the surface is coupled with a deep water column and/or that evaporation/condensation of water contributes substantially to the effective heat capacity because we know the diurnal temperature variation of SST, and it’s small, on the order of 1 K or less.

    This is in the heat transfer term. That’s the coupling. When large, it increases the faster of the two time constant in the eigenvalue solution– which reflects the physical linkage.

    You don’t need to artificially call this a higher heat capacity to get the slower response time. You call actual heat capacity heat capacity, and put the heat transfer in the heat transfer term where it belong. This is not a problem.

19. lucia,

    The land thermometers aren’t in the dirt, but the ocean thermometers are certainly in the water. Also in most places, the dirt has water in it, which means good latent heat transfer to the boundary layer where the temperature is being measured.

20. Dewitt–
    I think you are confusing heat capacity with something you consider to be “effective heat capacity”. The time constant for the atmosphere is lower than you would think based on the radiative heat transfer and the mass of the atmosphere. This is due to the coupling to the ocean. The physics is captured in the heat transfer term, not by pretending there is more mass in the atmosphere than exists.

    So, your method is useful for estimating the the time constants in in the eigenvalue solutions but not getting the heat capacity for the atmosphere. We can estimate that from the actual mass of the atmosphere.

21. Dewitt–
    The method is already sufficiently flexible to deal with ocean thermometers in the ocean.

    If we are going to treat the temperature of the dirt is not equal to the temperature of the air around here. The ag guys monitor it and have a web page to inform the farmers. If you want to include the dirt in a two-box model as anything other than an infinite sink, you need a third box.

    I don’t think the diurnal problem tells us much about the proper constitution of a two box model anyway. The diurnal motion constitutes the “internal mixing” that might, possibly, make the boxes “well mixed” from the point of view of a two box model. But it’s not described by this two box model. (At least I don’t think so.)

22. lucia,

    So that means the diurnal temperature range of sea surface temperature is low because the heat transfer is slow? That makes no sense to me. Somehow I think we’re talking past each other.

    I still think you should give up on the atmosphere being a separate box from the surface. I don’t see how that will ever provide a physically valid solution from the 1 and 30 year time constant fit and I don’t believe that is what Tamino had in mind. A land box(including atmosphere above the land) and an ocean box (plus the rest of the atmosphere) seems much more realistic to me. Then you have values for y and z(?) that fall between 0 and 1 rather than being identically 0 or 1. You still have heat flow from one box to the other. My math fu is weak, but I don’t see how that changes the equations and their solutions.

23. lucia,

    We only have data for thermometers in the air above the dirt so we have no idea about the dirt temperature. However, the forcing is deposited in the dirt first and only then transferred to the air close to the dirt. Over the ocean, the energy is transferred to the ocean, which is also where the ocean thermometers are. If you then treat the atmosphere as a single unit, you have a problem because the atmosphere temperature is going to vary a lot more and more quickly over dirt than over water.

24. Dewitt–
    No. You were arguing the heat capacity of the atmosphere exceeds that associated with the mass of the atmopshere. You were suggesting this arose as a result of latent heat transfer between the ocean and the atmosphere (and to some extent the dirt under the atmosphere on land).

    In the two box model, the latent heat transfer is in the heat transfer term that couples it to the atmosphere to the ocean. The more effective the heat transfer, the “slower” the atmosphere (which is lower heat capacity than the ocean) will seem to be.

    So, yes, depending on how you view the problem, the “effective” heat capacity of the atmosphere may seem high compared to the heat capacity of the mass of air in the atmosphere. But that’s not a problem for the two box model. It is no reason to “add” mass to the atmosphere. In a two-box problem you keep the heat capacity of the air equal to the heat capacity of the air and recognize the heat transfer with the ocean is large. This automatically adjusts the time constant of the system so that the atmosphere appears to respond on the time scale associated with the eigenvalue not the time constants you are getting by using the literal value of the heat capacity of the air.

    The ocean, in contrast has a slow response because it is massive.

    While you may be right that we can’t get the right answer without accounting for dirt.

    However, you can’t add a “dirt” to the atmosphere without a third box.

25. Dewitt–

    However, the forcing is deposited in the dirt first and only then transferred to the air close to the dirt.

    Oddly… no. This is anomalous forcing. The ghg part is mostly a reduction in upwelling radiation! The other bit is in the non-anomaly part of the problem which was subtracted out. Bizzarre… yes..

    This is a very confusing thing about anomaly problems.

    If you then treat the atmosphere as a single unit, you have a problem because the atmosphere temperature is going to vary a lot more and more quickly over dirt than over water.

    This is an argument against using the two-box model at all. You may very well be right that it will fail. But that argument can’t suggest how to consider all the possible partitioning of the climate system into two boxes.

    You also seem to be trying to develop your own model in your head and considering how you want to partition stuff. You are explaining details you think are important which would preclude a two box model.

    That a perfectly reasonable thing to think about but….it’s not what we are doing here. We are trying to discover whether, given the properties of the earth, the regression coefficients from a curve fit based on the notion of “two boxes” can correspond to any two-box model.

    The reason we are doing this is that Tamino and Arthur seem to believe we should believe we can get meaningful sensitivity coefficients out of this two-box model.

26. lucia,

    I ran the numbers through MODTRAN. Doubling CO2 does indeed reduce IR radiation to space by about 2.8 W/m2 looking down from 100 km, 1976 standard atmosphere. But you also need to go to zero altitude and look up. When you do that, you find that downwelling IR increases by 3.2 W/m2. So the energy balance of the atmosphere as a whole doesn’t change much (in fact it cools slightly), but the surface temperature will change.

27. Dewitt—
    Interesting.

    But that’s still ok. The two box model doesn’t care whether the heat comes in through the lower surface, upper surface volumetrically or whatever. It’s just not that detailed. It only knows “heat dumped in here”. That’s it.

    That forcing is an estimate of the amount of heat added to the two boxes. It goes in one or the other.

28. lucia,

    I went back and read Tamino again. He does say atmosphere and ocean. I still think a two box model of land and ocean can be formulated that will be physical and have time constants close to what he used, but not an ocean and atmosphere only two box model. That’s hopeless.

    Of course, what he’ll say in the end is that he didn’t really mean atmosphere and ocean specifically, just prompt and slow.

29. Lucia,

    I’ve been trying to run the numbers too, particularly using the perturbative approach I most recently looked at and letting Cs, Co and γ_s be the free parameters. I’ll have to do some double-checking and perhaps look at the full solution too, but the only time I find a constraint violation is when I let Cs get way too large (my Eq. 36 constraint gets worse as the inverse square of Cs, if Co is held fixed – but actually Eq. 40 is a tighter constraint that eliminates the smaller y solution in most cases and even for y = 1 still fails too soon if you increase Cs). Of course I can also let the thermal coupling γ_s get too large (or be negative) which leads to negative α or γ values, but I think it’s supposed to be small (and positive) – not sure I’ve seen anybody put constraints on that yet, so the small values I’ve looked at seem not to be unphysical.

    The first interesting thing is that in almost all cases my Eq. 36 is very well satisfied by the fitted solution (a2 is much less than the limit) and so the value for y has to be very close to 1. Which is what has been argued all along here anyway – the measured temperature we’re looking at is essentially the “surface” box. That’s nice confirmation.

    The second interesting thing I’ve found is the value for ‘x’ – the split of the forcing between “surface” and “ocean” boxes – in almost all solutions (including the one for Tamino’s time constants and your values for Cs and Co) has to be quite small – from a few percent up to 20% or so. So most of the forcing is going into the slow box, not the fast one, according to this analysis. Except when you let Cs get large and Co get smaller, then the split heads the other way.

    Anyway, either I’ve messed up the math there or there’s something funny about your solutions. As far as typos, the line:

    The solutions for (2.7%, &alphas,&alphao,&gammas,&gamma0) …

    is missing semicolons on the entity names, I suspect – but also, what’s that “2.7%” term?

    And what are you seeing for the value of the forcing split, ‘x’?

30. Arthur

    The first interesting thing is that in almost all cases my Eq. 36 is very well satisfied by the fitted solution (a2 is much less than the limit) and so the value for y has to be very close to 1. Which is what has been argued all along here anyway

    Clearly, you and I have been involved in entirely different arguments. There has never been any argument about this point. Ever. It is obvious that if one parameter models fit this well, fits with two parameters will fit this well. No one has ever claimed otherwise.

    The 2.7% is the fraction of ocean heat going into the top box for that solution.

31. Hmm, something else that looks wrong – in your stated solutions for γ_s and γ_o the ratio of the two is about 81. It should be the same as the heat capacity ratio, about 10 times larger – what happened there? Looking at my notes I may have an issue with this too, some checking needed…

32. Lucia (#19108):

    Arthur

    The first interesting thing is that in almost all cases my Eq. 36 is very well satisfied by the fitted solution (a2 is much less than the limit) and so the value for y has to be very close to 1. Which is what has been argued all along here anyway

    Clearly, you and I have been involved in entirely different arguments. There has never been any argument about this point. Ever. It is obvious that if one parameter models fit this well, fits with two parameters will fit this well. No one has ever claimed otherwise.

    Huh? I guess we must really be talking past one another, because I don’t see how your response has anything to do with the fact that I’m finding ‘y’, the proportion of the measured temperature in the surface box, is for whatever the assumed input parameters (with reasonable values, including your assumptions, for Cs and Co) coming out to 100%. Goodness of fit or two-parameter vs. 1-parameter is a completely unrelated issue, I think!

    And what are you finding for ‘x’, the forcing ratio?

33. Arthur–
    The current tests have the top box include some ocean in the atmosphere. I discussed the three branches of testing things.

    But yes.. The one set I pasted is for all atmosphere in the top box. It looks like I cut and pasted from the wrong case when I updated. I’ll fix that.

34. Oh– Well then I have no idea what your equation 36 is.

    The ‘x’ is the forcing ratio. Since under my current solution method, I permit some ocean in the top box, I get different values for ‘x’ and it’s listed in the figures. In the firt example, it’s 2.7%.

    However, I also show cases where it is 100%– because I draw a “magic line” permitting the top box to contain some ocean. But we can never get much ocean into the top box without suddenly having global warming suck heat out of the ocean and dump extra in the atmosphere. (That is, make x exceed 1.)

    I orgainzed it this way to permit the possibility that Tamino’s initial time constants of 30 years and 1 would work out. But… they don’t. Not even with letting the “top” box become more massive and having a longer time constant.

35. Ok– went to look. I admit I didn’t pay much attention to your perturbation solution because I don’t see the computational advantage. We can just pick three parameters and solve for the others without taking any limits. Are you actually using the low heat transfer limit as your solution method?

36. Of course, what he’ll say in the end is that he didn’t really mean atmosphere and ocean specifically, just prompt and slow.

    Maybe. But you still need to associate things with two boxes and there needs to be some connections with the earth’s properties.

37. The reason we are doing this is that Tamino and Arthur seem to believe we should believe we can get meaningful sensitivity coefficients out of this two-box model.

    I thought Tamino was already satisfied that he had done just that.

38. lucia,

    In the legend of the graph where you have non-zero Z, it says Z=45%. I’m assuming that’s a typo as in your discussion you say it’s 5%. Also, 1060 for CsubF should be 10600 if it’s MJ/(m2K).

39. Bugs, – Tamino may do more than “seem to believe”. He may indeed believe it. However, it appears his belief may well be misplaced since additional fiddling is pointing toward continued violations of the 2nd law even with the reduction of the short time constant. Nick and I seem to be agreeing on things looking bad for Tamino; Arthur seems to think otherwise.

    Since we don’t all agree, it’s likely someone may have committed algebra or coding errors.

    I think we’ll sort it all out, but it’s actually faster if we show results because the others point out things that don’t make sense in the answer. (Nick found a bug in mine. Looking at mine, he thinks he found a bug in his. Arthur just posted a “hmmm… and seems to be looking for some in his. Well, this blog is called “The Blackboard” for a reason. We are all putting our notes on the board in real time.)

40. Dewitt– The figures are right. After some discussion with Nick I found a typo in my spread sheet. The figure captions auto-fix, but the text in the post doesn’t!

41. lucia,

    Only the caption for figure 2 has updated. Figures 1, 3 and 4 still have 1060 for CsubF.

    5% still seems like an order of magnitude too much for Z. That’s a 40 times larger value for Cs than for Z=0. According to Arthur, things fall apart if Cs gets too large, which seems quite reasonable to me.

43. lucia (Comment#19113) September 1st, 2009 at 9:59 pm

    I admit I didn’t pay much attention to your perturbation solution because I don’t see the computational advantage…Are you actually using the low heat transfer limit as your solution method?

    The immediate question that comes to mind: why this assumption?

44. Perhaps a three box model should be considered.

    http://explainthisimage.com/unexplainable-photo/4936-schrodinger-improves-acurac

    As you can see, all boxes start with the same state, but controlling the entropy may prove difficult.

45. Ah!!! In my perturbative analysis I’d originally transcribed the value of λ (the heat capacity) upside down – it’s Cs/Co, not Co/Cs, and therefore small, not large. That makes quite a difference when you plug in the numbers!

    In particular, my eq. 36 inequality becomes:

    0 < a2 < τ+/4Co

    and doesn’t depend on Cs at all. So my conclusions about changing Cs and Co above were completely wrong – it is Co that provides the strongest constraint on the fit coefficient (a2) to get a physical result, and Co cannot be too large. Rearranging the inequality, in fact, we have:

    Co < τ+/4a2

    If τ+ is 30 years and a2 (slow box sensitivity) is about 0.5 K/(W/m^2) that constrains Co to be:

    Co < 4.7×10^8 J/(K m^2)

    So the “slow” heat capacity here has to be less than 4.4% of Lucia’s total ocean heat capacity CF, if we are to have a physically reasonable result with a slow time constant of 30 years.

    Alternatively one can figure out what the time constant τ+ has to be:

    τ+ > 4 a2 Co

    so for Co = CF and if a2 is still around 0.5 that gives τ+ > 670 years.

    In other words, Lucia’s results posted so far are guaranteed to be unphysical because they assume the slow box is the full ocean, and yet the time constant is unphysically short.

46. Big new development. I’ve fixed an error regarding normalisation, and Tammy is looking good again. The sign ambiguity I feared was not there, and now the heat trabsfer has the right sign – the solutions are “physical”.

    But more importantly, there is a very simple approximate solution, which is very good indeed. You can directly proceed from the given numbers (after regression) to compute with simple steps the full transfer array, and then the ocean temperature. No complicated algebra or numerics. It’s all at the updated CA site (scroll down).

    The new numerical results – for 1yr and 30 yrs:
    > A
    [1,] 0.80031652 -11.0065759
    [2,] -0.01391476 0.2330168
    Note the correct negative off-diag coeffs. The explicit approx says this sign should always apply.

47. Arthur–

    “In other words, Lucia’s results posted so far are guaranteed to be unphysical because they assume the slow box is the full ocean, and yet the time constant is unphysically short.”

    1)I’m plannning to do three branches as discussed.
    2) In this analysis, the ocean does not always have Co=Cf. Some of the ocean is in the top box. But I limit it to an amount that ensure we aren’t saying GHG’s cause negative forcing in the lower box.

48. Dewitt

    5% still seems like an order of magnitude too much for testing purposes. That’s 40 times larger total heat capacity.

    That the maximum that avoids looking at solutions with negative forcing in the ocean while there is positive forcing in the top box.

    If you are suggesting that if the only non-2nd law violating solutions say the surface thermometers represent the well mixed temperature of the atmosphere and 5% of the ocean does not make any sense, I would suggest you are correct. That’s why I discussed layers of screening.

    On the first screen, I’m just applying to totally obvious non-contentious screens: 1) Don’t violate send law. 2) Don’t have forcings in one box positive while the forcing in the other box is negative.

    If a solution corresponding to a regression passes this, then we look at other issues:
    1) Does the magnitude and/or the ratio of the “alphas” make sense. (Both represent radiation to the universe. Does it “make sense” for the stefan-Boltzman constant to differ by a factor of 1000 for the ocean and atmosphere? I would suggest not!)
    2) The previous screening only required γ to be positive. But if it passes that, we could look up the literature to see whether some positive magnitude makes any sense at all.

    Right now, that 5% solution you think doesn’t make sense for matching the thermometers also violates the 2nd law. So, the issud of “is 5% a realistic result” is moot.

49. Nick–

    [1,] 0.80031652 -11.0065759
    [2,] -0.01391476 0.2330168

    Is this the A matrix?
    Equation (7) in the climate audit forum requires A_12/A_21 = C_o/C_s. You have 0.80/0.23 ~4, but the correct ratio is closer to 1000. Is this something else?

50. By the way, I like the method of hunting for solutions with A_12/A21 = C_o/C_s.

51. Lucia,
    Yes, it’s A. The ratio is A_12/A_21=-11/-0.0139 = C_o/C_s=791
    If you multiply the bottom row by 791, it’s symmetric.

52. Oh– Shoot. Ok. Yes. I get it. I need to drink my coffee.

    I was thinking you’d added the addtional constraint of A11~Co/CsA22. (Why I thought this I don’t know. I suspect it’s because I went there after posting a comment about magnitudes of alpha that “make sense”.

    But I’m puzzled you are finding the Tamino original result “makes sense” because, at least with Cs with a realistic magnitude, it can’t. We don’t even need to solve the regression to find it can’t.

53. By “looking good” I didn’t mean “makes sense” – OK, sounds odd, but I meant only that the sign of A_12 was not unphysical. Yes, the discrepancy in heat capacity is a big problem.

54. Nick–
    Ok, I’m trying to sort out notation and translate. Your A matrix is

    A11 A12
    A21 A22

    Relative to my αs and γs this is
    A12 = -γ_s
    A11 = α_s+γ_s
    (If there was no radiative heat transfer to the universe, A11 = -A12. &gamma must be positive. α must be positive or radiative heat transfer works backwards.)

    A21 = γ_o
    A22 = α_o-γ_o

    Is this right? Or am I missing something?

55. Nick– I think I’m sorting out what you are doing. It’s a bit odd you, me and Arthur all doing things differently with different notation, and all “live” on the web. (I like it, but it is weird drinking coffee and trying sort out the notation.)

    If I understand what you did, the math is probably ok… but… the result is unphysical! (Insert numbers into comment above…. I didn’t want to do it because I’m afraid I’m totally misunderstanding.)

56. Lucia – putting “some of the ocean” in the top box doesn’t help – it diminishes Co slightly, but there’s no way you can get Co down to 4.4% or less of CF unless you swap the meanings of Co and Cs, and that leaves things just as bad on the Cs side.

    The problem here is that you are being far too rigid in your interpretation of what a “two-box model” can physically mean. It is an extremely simplified model of the Earth, obviously. There should be some correspondence with components of the Earth system in that the physical parameters should be reasonably interpretable as corresponding in magnitude to the appropriate portions of the planet.

    But nobody has ever asserted that the full ocean can respond to the GHG radiative energy flow changes we’re experiencing on a time-scale of just a few decades, it’s always been more like 1000 years or so. So plugging in the full ocean heat capacity here doesn’t make sense from the start.

    Anyway, for a 30-year time constant and a reasonable fitted sensitivity value, the real constraint on heat capacities is even stronger than I asserted in comment #18939 and #18947 on the previous thread here the other day – it should be:

    Ca <≈ Cs < Co < 0.1 CF

    with that upper bound even less if a2 is larger (0.044 CF for a2 = 0.5 K/(W/m^2), 0.022 CF if a2 = 1.0 K/(W/m^2).

    Plugging in larger values for Co is guaranteed to give nonsense for this sort of two-box model with a decadal long time constant.

57. Arthur–

    unless you swap the meanings of Co and Cs, and that leaves things just as bad on the Cs side.

    Yes. That’s the next step.

    But nobody has ever asserted that the full ocean can respond to the GHG radiative energy flow changes we’re experiencing on a time-scale of just a few decades, it’s always been more like 1000 years or so. So plugging in the full ocean heat capacity here doesn’t make sense from the start.

    If you want to separate out the rest of the deep ocean from the earth’s climate system, you need a three box model with three time constants. You can’t just ignore it, set the heat transfer from the second box to the deep ocean to zero and then pretend you have two time constants for some earth with no deep ocean.

58. lucia,

    You can’t just ignore it, set the heat transfer from the second box to the deep ocean to zero and then pretend you have two time constants for some earth with no deep ocean.

    Sure you can. It makes a lot more physical sense than your assumption that the entire ocean is well mixed when we know for a fact that it isn’t. Then there’s the fact that you have no provision for the heat capacity of the land surface, which is apparently assumed to be zero, that the atmosphere above land behaves very differently than the atmosphere above the ocean and that the thermometers cannot all be in one box. If you were modeling over a time scale of millenia, I would say you would need a third box with heat transfer or a fully diffusive ocean. But we’re not. On a time scale of a century, heat transfer to the deep ocean is negligible for the magnitude of the temperature changes being modeled.

    Your choice of boxes as just the atmosphere and just the entire ocean make the whole exercise moot. It’s equivalent to a straw man argument and you won’t have proved Tamino’s model is unphysical since your assumptions are unphysical to start with.

59. DeWitt:

    It’s equivalent to a straw man argument and you won’t have proved Tamino’s model is unphysical since your assumptions are unphysical to start with.

    Have to disagree with you here.

    Anytime you simplify a system with e.g. a two- three- or even five-box model you are going to leave physics out.

    Lucia, it appears to me, is making a good-faith attempt at setting down Tamino’s assumptions about his two-box model. Unless she assumes something like e.g., F=mv or E=1/2 m v^4 or dS/dt < 0, the charge you make here is entirely false.

    It may be—due to criticisms such as the ones you raise—that the models does a very poor job of describing the actual processes of climate but that's a very different thing than saying it's "unphysical".

    Assuming a "well-mixed" ocean when it isn't doesn't make the simplified system unphysical. It is just one of the many ways in which the two-box model represents a simplification of the underlying complex system it's meant to (in some sense) represent.

    It may well be that there are unphysical assumptions made by Tamino that are leading to unphysical results, but I don't see how assuming well-mixed versus stratified oceans is one of those.

60. Arthur–
    I should note that whether explicitly asserted or not, what is being done is expanding on only two time scales which have been selected for heaven knows what reason. And then what is being claimed is that one can pull the sensitivity from those fitting parameters.

    You argument currently amounts to “there is really a secret other time scale and it’s important. We’ll do the fitting pretending the other– quite important timescale–should be ignored. We’ll pretend this is still a “physics based fit”. Then, if anyone challenges this 2 parameter fit as “not physics based”, we’ll just say, “Well… yeah. We know there is another time scale. We know it’s important. If we included it, our system would not violate the 2nd law. But, we don’t want to include it, so we won’t. And we’ll still claim our curve fit is supported by valid physics. Even though it’s not, and we know it. But we get to say it is. Because we know it’s not and periodically admit it. But that doesn’t mean we can’t turn around and insist it is physics based– provide we point out that we know it’s not.”

61. Carrick,

    Unphysical is too strong a word. But I don’t see how a simplified assumption of a two layer ocean with negligible heat transfer from the top to the bottom layer is any less valid than the assumption that the the ocean is well mixed and there either there is no land surface at all or the heat transfer to the land can be neglected. Neither one is accurate but the stratified ocean model in one box and the land in the other seems to me to be a lot more like the real world even if it isn’t exactly what Tamino said his boxes were.

62. Dewitt–

    It’s equivalent to a straw man argument and you won’t have proved Tamino’s model is unphysical since your assumptions are unphysical to start with

    The definition of arguing a strawman is to address a claim that was never made. Strawmen are not flimsy arguments, they are arguments that were never made. Tamino presented a fit and appears to claim it is based on a physical model which is a two box model. I am addressing that claim. It is not a “strawman”.

    You seem to be suggesting that his box model is obviously unphysical, so proving I don’t have to prove it. Clearly, I don’t need to prove it’s unphysical to people, like you,who don’t think it’s physical in the first place. They already think the model is flawed. Mathematical-physical proofs are irrelevant to them.

    Proof is required for those who either insist or which to believe is physical. Like Arthur.

    In that context, we try to find the best possible two-box model based on the properties of the earth, and compare that to the fit. Arthur wants to sometimes allude to a third box, and suggest that somehow because we can imagine material in the third box, we can then do a fit with two boxes ignoring the existence of the third box.

    Either the third box is there — in which case, Arthur must account for it with an extra time constant in the expansion — or the third box is no there– in which case, the full climate system must be partitioned in the two boxes. There is no alternative.

    Mind you, Arthur may be able to do the problem for three boxes, and then do a perturbation to deal with the third time constant, prove it somehow ‘doesn’t matter’ and etc. But that’s a lot of work and it’s not at all clear that if the first two time constants are 0.09 years and 30 years that we can utterly neglect any time constant of 300 years or even 1000 years.

    (Plus, really, Arthur needs to be careful about suggesting time constants of 1000 years. If he does the math, some denialist will come along, point out that something with a time constant of 1000 could easily not have been in anything near equilibrium, add the possibility that the earth was not in equilibrium in 1880 to the problem and show possibilities Arthur is not going to “like”.)

63. DeWitt–

    But I don’t see how a simplified assumption of a two layer ocean with negligible heat transfer from the top to the bottom layer is any less valid than the assumption that the the ocean is well mixed and there either there is no land surface at all or the heat transfer to the land can be neglected.

    If you are going to neglect heat transfer at the top of the lower layer, you need to justify this assumption over the time scales of interest.

    To justify this, you have to write down the equations for the 3 box model, do the math and show under what circumstances it makes sense to treat the lower box as permitting zero heat transfer over the relevant time scales. It can’t just be small– it has to be effectively zero over the relevant time scales.

    This is pretty much the same answer I have been giving Arthur. If you want to throw away part of the mass of the climate system, you have to start with the three box system, show the circumstances in which you can neglect the lower layer and then show those are realistic over the time scales of interest.

    Lots of engineered systems would treat the lower ocean as an infinite sink for the purpose of sizing cooling coils for a submarine or something. They do this because the time scale of interest is seconds, minutes or days. But this is not the case for the current model where we are fitting data collected over a century to a series of temperature measurements to estimate parameters that describe the sensitivity of the climate system at equilibrium– that is after an infinite amount of time has passed.

64. Re Comment #19174

    “some denialist will come along, point out that something with a time constant of 1000 could easily not have been in anything near equilibrium, add the possibility that the earth was not in equilibrium in 1880 to the problem and show possibilities Arthur is not going to “like”.”

    That should be easy enough.
    First, at equilibrum heat flow must always be from the deep ocean to the shallow ocean. All heat that goes down into the deep ocean (mostly through the thermohaline circulation) must ultimately come back up (there is nowhere else for it to go). Further the geotermic heat flow must also come up the same way. This is usually said to be about 0.1 Wm-2 but it might actually be rather higher. Undoubtedly the heat flow between shallow and deep ocean fluctuates over time but in the long run the net flow must be upwards.
    Second that 1000 year time constant is probably reasonable. That is on the order of the turnaround time for the thermohaline circulation, which of course opens up the possibility that this warming trend we have had since 1880 is just the warmer NADW (North Atlantic Deep Water) of the MWP coming back to the surface…..

65. lucia,

    Writing equations is not my strong suit. Besides, I haven’t thought this through yet.

    Thinking out loud, as it were: In the real ocean, there is a temperature difference between the top layer and the bottom layer. So heat must transfer from top to bottom by some sort of diffusion. If that were all there was, then the bottom layer would gradually warm and the temperature gradient would flatten and move lower. But we know that doesn’t happen. I’ve linked elsewhere to the data that shows that. So, the bottom layer must be coupled to space to radiate away the energy transferred and there must be upward flow everywhere to match the downward heat transfer. The coupling to space is most likely to occur at the poles where sea water cools and sinks. But that’s equilibrium.

    So what happens when we increase the temperature of the top layer slightly? Downward heat transfer would increase, but by how much? The temperature differential is only about 6 C max (10 to 4). The rate of heat transfer can be calculated by the rate of upwelling. I should go back and look it up, but IIRC someone calculated it as about 9 W/m2. Then heat transfer down would increase in proportion to the change in temperature difference. According to the ocean only anomalies that would be ~0.6 C max or about 0.9 W/m2 over the 120 years of interest. But I’m going to assume that upwelling flow will also increase to match the increase in heat transfer down to keep the interface stable. I see this as no different in principle than the assumption that the full ocean is well mixed. We also know from the 1966 and 1985 measurements that showed no change in the temperature profile that this is not an unreasonable assumption. That would mean that radiation to space from the bottom box would have to increase by the same amount. But that means that heat transfer to space from the ocean as a whole looks exactly as if all the heat were radiating from the surface directly rather than some of it going through the bottom layer. And that’s the same assumption as for a full ocean well mixed model except the heat capacity of the top layer is smaller than for the full ocean.

    The problem I see with this is that there may be an additional time constant involved. But it’s not the thousand years it takes an individual packet of water to travel the full circuit no more than signal transmission through a copper wire is dependent on the speed the actual electrons move through the wire. So I postulate it’s negligible too. That may lead to momentum change issues, but the flow per unit area is low.

    And yet, this is still not the Earth, but a planet entirely covered with water like the Venus of 1940’s science fiction. In which case perhaps we should be fitting the ocean only temperature anomalies rather than the mixed land and ocean anomalies. I’ve done that and the short time constant fit is no longer significant.

66. Dewitt–

    In which case perhaps we should be fitting the ocean only temperature anomalies rather than the mixed land and ocean anomalies.

    It may well be the case that you can develop a superior two-box or not-two box model that needs to be regressed on ocean temperature anomalies. (You’ve got a lot of physics in those paragraphs, and if you drew control volumes and various diagrams, you’d see that what you are describing is not a two-box model or any sort)

    While interesting, your discussion has nothing to do with the question of whether or not a fit based on a two-box model using surface temperatures corresponds to a two-box model that could, remotely, approximate the earth. I’m focusing on the second. I’m not seeking to develop a “better more physical simple model”.

    The better, more physical models already exist. One group is called the “upwelling diffusion models”. These were tuned and used in the TAR and SAR. Tamino did not use these to form the basis of his regression. So, while it is quite clear that climatologist see the value of “upwelling diffusion models”, it is equally clear they have nothing to do with the method Tamino proposed and whose adequacy I am trying to think about.

    If you just want to discuss “How would I develop a more ideal simple model”, I would be glad to open a thread. Alexander Harvey was discussing one; oliver was discussing one. But those discussions are separate from the issue of testing the regression coefficients for the two-box model.

67. I’m not sure if the question of how to find the time constants was answered but here is an observation:

    There is no need to assume anything about the time constants. Here equations are:

    Code: Select all
    CsdTs/dt = -(Cs/ τs) Ts + β (To-Ts) + Fs (1a)

    CodTo/dt =- (Co/ Ï„o) To – β (To-Ts) + Fo (2a)
    http://rankexploits.com/musings/2009/two-box-models-the-2nd-law-of-thermodynamics/

    Let’s transform these equations. Take equation one and subtract equation 2:

    CsdTs/dt-CodTo/dt=2 β (To-Ts)+F_s+F_o

    Take To*(1a)+Ts*(2a)

    CsCo(Ts/dt+To/dt)=-TsTo((Cs/ Ï„s)+(Co/ Ï„o))(Ts+To)+FsTo+FoTs

    Now take as your new variables y1=(To-Ts) and y2=(To+Ts)

    These equations can be estimated independently of each other.

68. lucia,

    From your original post:

    What is the point of fitting temperatures measured 2m above the earth’s surface to a statistical to a model with parameters that apply to Krypton or Jupiter? Note the number of times I mention use the word earth in the post above.

    His model may be fine for Jupiter or Krypton. I have no idea.

    My question is, what is the point of creating a physical model that does not in fact describe an earthlike planet or how the temperature anomalies were actually measured. While an interesting exercise, it bears no relation to why the land plus ocean temperature anomalies can be fit with a two time constant model and whether the fitting constants are equivalent to the climate sensitivity, much less whether there is a second law violation. The two time constants happen because the Earth has land surface which responds quickly to changes in forcing and ocean surface which responds more slowly and the temperature anomalies are a linear combination of temperatures over land and in the ocean.

    Your model may be fine for a fictional Venus. But we don’t have temperatures and forcings for that planet.

69. Lucia #19142
    Yes, I think that correspondence is correct. The “physical” condition is that A_12<0 (and A_21<0). This is because I've written the coefficient as -A. The reason for that is that the eigenvalues of A are the original assigned values (eg 1, 1/30), and if they are to be positive, the coefficient is -A. That eigenvalue correspondence is why I used A as the primary coefficient.

    You can back calculate to get the αs and γs from A.

70. lucia,

    I thought of something else and I can’t wade through all the algebra to see if it’s included. For the atmosphere box there are two temperatures or maybe three not one. There’s the surface brightness temperature and the TOA brightness temperature which controls radiation to the surface and to space and the thermodynamic temperature. The surface temperature always being higher than the TOA temperature by the amount of the greenhouse effect and the thermodynamic temperature somewhere in between. When an internal forcing, as from a change in ghg concentration, is applied they move in different directions. The TOA brightness temperature goes down and the surface brightness temperature goes up, but the thermodynamic temperature doesn’t change, initially anyway. An external forcing like solar or albedo change due to snow or aerosol should not change the difference between the surface and TOA but will change the thermodynamic temperature. I’m not sure about black carbon or ozone. It depends on where both are. The ozone may be stratospheric, tropospheric or both. Black carbon may or may not be suspended in the atmosphere. Presumable it is in the atmosphere so it could directly increase the thermodynamic temperature without lowering the TOA temperature by absorption of sunlight or it could act like a ghg by absorbing surface IR and emitting at a lower brightness temperature than the surface or both.

    I think that means you have to have an additional parameter for the effective thermal conductivity of the atmosphere box and that all forcings are not equal. Of course it may already be there and I missed it. Or maybe it all comes out in the wash and can be neglected. But I didn’t see anything related in the discussion.

71. Dewitt–
    The goal is to determine what two box models correspond to the regression and see if those parameters could map into a two-box model with parameters that match the earth. If it does, one could suggest that model is useful for the purpose of determining sensitivity, even if it wasn’t useful for anything else. But if the only two box models that correspond to the regression can’t possibley be two-box models of the earth, then we would conclude that the test itself shows the two-box model is not useful for this purpose.

    In this context, that fact that full blown GCMS capture more details than upwelling diffusion models which capture more details than 3 box models, which capture more details than 2 box models (which themselves capture more details than 1 box models) is a true fact, but not relevant to testing whether a two box model regressed on atmosphere only data is useful for determining the climate sensitivity.

72. Nick…

    You can back calculate to get the αs and γs from A.

    I have…. If I didn’t subtract wrong, the results aren’t physical! I’m cooking dinner. I’ll create a graph and discuss this more later.

73. Dewitt

    For the atmosphere box there are two temperatures or maybe three not one. There’s the surface brightness temperature and the TOA brightness temperature which controls radiation to the surface and to space and the thermodynamic temperature.

    Two box models cannot capture this level of detail. There are two temperature. Period.

74. Then I guess you don’t care that because the atmosphere isn’t gray, the rate of change of radiant power to space with temperature isn’t 4*sigma*Ts,o**3 either, it’s quite a bit less, not to mention it isn’t proportional T**3 even if you did have the temperature that gave the correct rate of change, which, btw, isn’t equal to the TOA brightness temperature either.

    Final thoughts and then I give up on this. Tamino’s model proves nothing about climate sensitivity. The response surface of the two time constant fit is so flat you can get almost any climate sensitivity you want by the choice of the slow time constant and still explain about 80% of the variation in temperature anomaly. OTOH, your two box physical model is so unrelated to the real Earth that the results of its analysis won’t prove anything either.

75. Dewitt–
    1) I do care if the constant of proportionality for the radiative heat loss is lower. That appears in the two box model. I don’t care about zillions of other details that do not appear in any two-box model of any sort.

    2) If your view is that no two box model can be useful, obviously, showing a particular one is not useful would seem pointless to you. I nevertheless think it’s a worthwhile exercise because others have suggested the two-box model is useful for the specific purpose of getting an empirical fit.

76. I should modify what I said above:
    When evaluating a two box model, I don’t care about details that do not appear in any two-box model of any sort.

    If we were trying to come up with our own model for some other purpose, I would care about those details.

77. DeWitt Payne (Comment#19205) September 2nd, 2009 at 5:40 pm

    …your two box physical model is so unrelated to the real Earth that the results of its analysis won’t prove anything either.

    The earth and moon aren’t point masses either, but we often start with such an oversimplified model when we begin exploring orbital parameters; then we fix it up when it (unsurprisingly) fails to be perfect.

    It isn’t SOOOO blasphemous, is it? 😛

    There are lots of things we just don’t know observationally about the climate system, so it helps to have something simple that can help us gain some intuition. But of course we want that something to make internal sense, which I think is how this whole project got started.

78. lucia,

    I’ve wondered into this large discussion pretty late. I have a couple of observations that may or may not have been covered already.

    First, when Arthur Smith said “Which is what has been argued all along here anyway” he was using the “to maintain in reasoning” definition of “argued” — not “argued about”. It was more like “Which is what has been *stated*…”. Not “Which is what has been argued *about*…”.

    Second, you can’t insist that the entire ocean be included in Co. It’s pretty clear that Co is for the well-mixed upper ocean. Tamino said as much in his reply to your first comment.

    You *could* build a two-box model for longer time scales of 1000+ years. If you did, it would make sense for the slow time constant to include the lower ocean. For this model with a slow time scale of ~30 years, it’s clear that the slow time scale does not include the entire ocean.

    If you insist that the entire ocean needs to be included in Co, then what is your reasoning for the entire ocean not being included in Lumpy?

79. There’s been talk of a three box model. One of the problems is that the range of time constants that we can get information about from present data is very constrained. Less than 1 year runs up against the yearly frequency of forcing estimates (and interpolation is not a solution). And more than 30 years runs up against our 124 years of data.

    However, in a sense we already have a 3 box model. The time constants are applied via exponential smoothing, and there 1000 years might as well be forever, relative to our 124 years of data – it returns a constant. And indeed, in the original regression, we do have a constant (“intercept”).

80. If we were trying to come up with our own model for some other purpose, I would care about those details.

    I think you have more or less come to Tamino’s position.

81. JohnV–

    Second, you can’t insist that the entire ocean be included in Co. It’s pretty clear that Co is for the well-mixed upper ocean. Tamino said as much in his reply to your first comment.

    If you leave part of the climate system out, you have to account for the heat transfer to that part. So, either a) you leave it out or b) you account for that heat transfer.

    If you insist that the entire ocean needs to be included in Co, then what is your reasoning for the entire ocean not being included in Lumpy?

    Who says Lumpy is physically realistic?

82. Nick–
    It’s perfectly possible for someone to formally specify a three box model, find the best fit time constants, and then discuss whether or not that model can gracefully degrade to three boxes. We can then test the two box model, and see whether the parameters picked make sense. But if you are going to throw away part of the ocean, we need to test whether the amount thrown away remains consistent with a two box model fit.

    You can no more just throw away whatever amount of ocean you want to throw away without testing than you can pick two parameters out of the blue and assume they will be ok either.

    bugs– I don’t know what Tamino’s position is. I doubt you do. But, in the context of picking a two-box model, we ignore the features that are not captured by a two box model. This does not mean pick any time constants we like, and attribute physical meaning to the result. It also doesn’t mean we can use any short time constant we like. So, I doubt if I’ve come to the same position as Tamino, because impression when last he blogged was he things any short time constant is fine.

83. Lucia: “If you leave part of the climate system out, you have to account for the heat transfer to that part. So, either a) you leave it out or b) you account for that heat transfer.”

    Wouldn’t it be fairly straightforward to allow for (likely a small amount of) heat transfer from the well mixed upper ocean (with moderate heat capacity and ~30 year time constant) to a heat sink (fixed temperature) lower ocean (in effect infinite or greater than 500 years time constant)?

    Then at least you still only have 2 boxes.

84. SteveR–
    Either the part below the lower box is
    a) a heat sink because it’s humogonourmous and well mixed
    b) has finite time constant (like say 500 years.) or
    c) it somehow seems to behave as some sort of barrier with zero heat transfer over a very long time scale.

    If it’s the 2nd, then you include that as a third time constant, and do the test for consistency on that basis. If it’s either of the first two, you have 2 time constants, but when checking for consistency.

    If you think the third lower box is a heat sink with constant temperature, you do only have two time constants. However, after doing the fit, you evaluate the α coefficient for the 2nd box to see how much heat flux it suggest is leaving out the bottom of the box, figure out if that makes sense for the amount of ocean you treated as a heat sink, and then test whether it makes sense that the part of the ocean you treated as a heat sink could, plausibly act as a heat sink.

    If you think the lower box is an impenetrable barrier with zero heat transfer, you also have a two box problem. But once again, you have to figure out whether your two-box solution makes sense for the properties of the earth. Basically, you ask whether it makes sense no heat transfers out that part.

    So, I return to what I said: If you can’t just throw away heat capacity wily-nily. You may be able to partition bits off from the two boxes and treat them as “outside” the two boxes. But after you do that, you have to check whether the part you partitioned off really can be treated that way. If you check, you will find that your initial assumption about being able to partition off parts of the lower ocean was not consistent with the result fro the regression.

85. Maybe Tamino’s Daemon is taking care of the 2nd Law problems. Oops, sorry Lucia, that would be you.

86. To warm up the deep oceans, you really need to warm up their source – the polar oceans. The poles have to warm up so there is no sea ice and average polar sea surface temperatures need to rise above 3.0C, 4.0C and so on. After that, it will still take 500 to 1,500 years before the deep oceans as a whole warm up. There is, of course, still some downward mixing of temperatures from the topical ocean surface but the poles also need to warm up.

    I’ve seen a much better illustration of this concept than the following image, but this is the only one one I can find right now.

    http://oceanography.earthednet.org/Mini_Studies/Deep_Ocean_Circulation/Deep_Ocean_Circulation_files/image003.jpg

87. I think we must start calling our Hostess St. Lucia of Rank Exploits!!!! ;>)

88. Two-box simply gives you nonsense if Co is too large. CF is too large, so you’re getting nonsense.

    I’ve put together a final post on the full solution with all constraints. As long as you use reasonable Earth-system-sensible values for Cs and Co (even quite large values for Cs, relative to Ca) you get reasonable numbers with no constraints violated. In particular, Lucia’s original demonstration claiming “second law violation” is actually wrong, I believe because it neglected the role of the splitting factors ‘x’ and ‘y’ in the real solution. Even with a short time constant as long as 1 year and a heat capacity Cs = Ca, there’s a perfectly reasonable (infinitely many, actually) two-box model that works with those time constants and fitting coefficients.

89. I have posted it at CA so I might as well post it here .
    .
    The 2 box “model” which is supposed to describe (approximately) the climate dynamics is 2 dimensional – the 2 time constants define a phase space of dimension 2 .
    Yet the dimension of the climate attractor has been estimated in many papers (f.ex http://ams.allenpress.com/perlserv/?request=get-abstract&issn=1520-0469&volume=43&page=419) .
    This dimension is around 5 .
    That means that at least 5 independent parameters are necessary to describe the climate attractor . Of course as we are dealing with a chaotic system , this doesn’t describe completely the dynamics but it gives at least the invariant set .
    .
    Now 5 is between 2 and 3 times 2 .
    So clearly a phase space of dimension 2 even if it happened by some chance to be a subspace of the attractor of dimension 5 , can’t describe the real Earth .
    In the best case where the 2 dimensionnal space has a non empty intersection with the 5 dimensional attractor , there would be SOME properties of the climate that would seem to be captured with SOME choices of the 2 parameters but other properties would be hopelessly wrong and one could of course not predict which would be right (during a limited time) and which would be wrong .
    In the worst case there would be an empty intersection and the “model” would just be garbage that could be proven contradicting all kind of physical laws .
    .
    The reason for that is simple – if mother Nature saw fit to comply with all her constraints that we call physical laws via a 5 dimensional space then it is that she needed 5 dimensions and not less .
    Any attempt to do as well (even approximately) with less dimensions via some “model” is a waste of time bound to fail .
    If we were not dealing with a chaotic system we could capture some crude low dimensionnal features but it happens that we deal with a chaotic system .

90. Lucia for some reason the link doesn’t want to get in the post . I retry here : http://ams.allenpress.com/archive/1520-0469/43/5/pdf/i1520-0469-43-5-419.pdf

91. All this talk of models has made me wish I hadn’t scraped my climate model project. It had a 5° x 5° grid with a mixed layer depth of 70 m and atmospheric layer of about 1 km. I downloaded land/ocean emissivity data to help with calculating the radiative heating/cooling of the surface. I also wrote some code to calculate the emissivity of the atmosphere as a function of CO2 and H2O concentration. I had to use time steps of about 15 minutes, otherwise, the simulation would blow up. Unfortunately, the ocean would freeze over after a few model days. But before it began freezing, the temperature distribution over the globe became realistic after one or two model days. After the sun would set in any portion of the globe afterwards, the ocean started losing heat too fast by radiation, and the radiation from the atmosphere wasn’t putting a break on it. I tried using actual temperature data to estimate a heat transport coefficient (I previously tried guessing it by trial-and-error) for each grid box to force the simulation to at least be relatively stable. I was basically using the flux adjustment method. I gave up after the coding and calculations got too complex and too long to complete in a reasonable amount of time.

92. Lucia
    bugs– I don’t know what Tamino’s position is. I doubt you do. But, in the context of picking a two-box model, we ignore the features that are not captured by a two box model. This does not mean pick any time constants we like, and attribute physical meaning to the result. It also doesn’t mean we can use any short time constant we like. So, I doubt if I’ve come to the same position as Tamino, because impression when last he blogged was he things any short time constant is fine.

    I quoted what he said. Just as Tamino did, you are drawing a line in the sand on just what your model does and does not do. Tamino’s had shortcomings, but gave a good representation of the forcings on the climate. You are doing a lot more work, but still not enough to satisfy the next person who is finding your model ‘unphysical’.

93. I have now posted at the CA BB a XL spreadsheet, which has the functionality of previous R programs. You can just enter the key numbers (red, top left) and it does the regressions, figures the box parameters, and plots the inferred ocean temperatures. It’s highly experimental and may change – please regard with suspicion and check. It implements the theory on that CA BB page.

94. Arthur–

    a) You think my solution using the “atmosphere” and “ocean” as “atmosphere” and “ocean” was “just wrong” when that’s what Tamino described?

    b) Your solution uses 2% of the water in the ocean and treats the rest as an either an infinite heat sink? Do you seriously think this is a remotely reasonable “earth value”?! That’s utterly ridiculous.

    c) Can you actually claim, with a straight face, that the notion 90% of the thermometers, all located near in the atmosphere or above the surface) are measuring the “slow box” (i.e. ocean) is reasonable? (Wow.) Meanwhile the alternate of 17% in the slow box is also reasonable? (Double wow for not even trying to decide what types of combinations might be reasonable given the actual location of the thermometers or possible ideas of boxes! )

    d) Other than being merely positive, you make no checks what-so-ever to see whether the heat transfer rates between your boxes are physically realistic? What do these correspond to?

    and finally on top of all this:
    e) You find if the “ocean box” contain even on iota more than 10% of the ocean, Tamino’s values failed even by your very liberal standards of considering a solution valid. (And you do nothing to account for heat flow out of the bottom 10%?!)

    Let’s just say I don’t think you showed Tamino’s method makes physical sense. I’d say, assuming this is your last post, you have convincingly shown
    a) Tamino’s post does not make physical sense and
    b) You want so much to believe it does you will throw away 98% of the ocean without the tiniest back of the envelope computation to examine the effect of adding the terms to estimate the heat flux out of the bottom of that ocean box!
    c) You forgo other checks to see if your results are reasonable.

95. Arthur–
    I think there may be something wrong with your parameters.

    For case1, could you tell me what you get for your betas in whatever units you prefer?

96. Arthur–
    Are comments closed on that post?
    Could you check whether your values for alpha and beta result in

    (1/τ_– + 1/τ₊) gives the proper balance for your ratio of specific heats in case 1? (And presumably all others?) There may be a typo in your post.

97. lucia,

    Tamino’s model did not *not* assume Co was for the entire ocean. It is commonly known that the deep ocean is weakly coupled to the upper ocean. In Tamino’s reply to your first question he said this:

    “It seems to me that a reasonable interpretation is that the fast box is the atmosphere and the slow box is the upper ocean. I suspect the time scale for the deep ocean is much longer than 30 years.”

    In any parameterization, you must simplify. In the case of a two-box model with sub-century time scales you clearly can not explicitly include the deep ocean.

    In your early posts on Lumpy you talked about it having a heat capacity equivalent to 150m of ocean:

    ” I obtained Ï„= 14.5 years and α-1=1000 MJ/m^2 (This corresponds to a body of water with a depth of 150 m; Hansen et al. 1988 report the global averaged value of the mixed layer depth of the ocean is 240 m.)”
    http://rankexploits.com/musings/2008/how-large-is-global-climate-sensitivity-to-doubled-co2-this-model-says-17-c/

    You clearly understand the concept of the well-mixed upper ocean being de-coupled from the deep ocean. You clearly spoke of Lumpy being physically reasonable.

    Conceptually, think of Tamino’s two-box model as Lumpy with an independent atmosphere — not Lumpy plus the deep ocean.

98. JohnV–
    How weak? This has to be justified in a two box model? You need a third box for the deep ocean.

    With Lumpy–

    1) Have advocated those value for sensitivity as giving correct answers? Or even good estimates? I have not. That’s what that would correspond to. Lumpy can be (and has been) criticized for lacking the flexibility to provide decent values for sensitivities and time constants. I conceded this when I was blogging about Lumpy.

    Lumpy is a curve fit that corresponds to some physics. It explains just as well as the two body curve fit.

    2) Did I just pick Lumpy’s values to be whatever I liked? As Tamino did? No.

    3) If Tamino’s model was Lumpy with an independent atmosphere, why did the initial choice of time constant for the atmosphere have nothing to do with the atmosphere but would, if one thought of his model as ocean + atmosphere, violat the 2nd. law? And failing that, why not find what time constants work best rather than just picking them?

99. Just a few facts about the ocean:
    – Average depth: 3790m (Wikipedia’s ocean page)
    – Well-mixed depth: 240m (Lucia’s reference to Hansen above)
    – Other references online: 20-100m
    – Well-mixed fraction: < 6.3%

    Also from Wikipedia's thermocline page:

    "In the ocean, the thermocline may be thought of as an invisible blanket which separates the upper mixed layer from the calm deep water below.“

100. lucia,

     I’m not going back into the whole argument about what Tamino meant.

     (I will say that since he was using annual data his choices for the atmospheric time constant were 0 or 1 years. A time constant of 0.09 years or 0.5 years is effectively the same as 0.

     I’m just talking about whether Co should include the deep ocean. If I understand correctly what’s been happening here, the supposed 2nd law violation only holds up under your assumptions of Co including the entire ocean. What happens to your results if Co only includes the well-mixed layer?

101. JohnV–
     1) It is untrue that using annual average value restricts on to decribing the atmosphere with 0 or 1 year. One can select from a range of analytical methods and some permit you to chose reasonalbe time constants. (In anycase, zero is closer, is it not.)

     2) If one discovers annual average data are not suitable for a proposed analysis, one can use monthly data.

     3) If Co does not include the deep ocean, one must extend a real box model to include that before over interpreting the meaning of the results. This is why I do not over interpret the results of Lumpy, and in fact, have rarely discussed as anything other than a heuristic device.

     4) I don’t know what my results look like if Co is only the well-mixed layer. For the partitioning I discussed, I have been getting violations of the 2nd law. This contradicts what Arthur is claiming. However, a) I haven’t varied ‘y’ the linear proportionality constant Arthur used instead I b) varied the partitioning of the boxes. (This is a way above and beyond what Arthur used. I discussed that I was going to look at all of them. And FWIW, I was testing closer to the order Nick was rather than arthur. )

     Finally, I’ve been doing a quick check, and I can’t square Arthur’s values for various various quantities for Case 1 or 2 with the simplest requirements for eigenvalues. This may be me, or it may be him, but comments appear closed over there. I’m double checking to see if I can figure out how to get his combination of results.)

102. Oliver,

     Your example is flawed. What is being done here is more like calculating orbital parameters with a solar mass ten times its actual value and being surprised that the calculated parameters do not match observations.

     As usual, Tom Vonk makes my point in his post above far better than I ever could.

103. Sorry, comments were accidentally closed on my post last night (it was late, I must have clicked a wrong button somewhere) – come on over if you have questions.

     Lucia, in what way are you doing anything more general than I am? You’re just setting Cs and Co to different values – well they’re free parameters in my case too. You are setting y = 1, losing one free parameter there, while I am using the heat transfer value γ as my additional free parameter, and figuring out what y should be from that.

     And as I showed, no matter what you pick for γ, as soon as Co is assumed to be too large for the sort of fitting coefficients we’ve been finding with a decadal long-time-constant, the solution for y disappears, and there is no corresponding physical two-box model.

     No matter what exactly Tamino said or meant, the fact is that a too-large value for the slow-box heat capacity breaks the two-box model, and anything you talk about beyond that is nonsense. If Tamino insisted that his slow box was the full ocean, he would have made a serious mistake there. But it doesn’t appear he did.

     Since the original fit was to the GISS global land-ocean temperature index and 70% of Earth’s surface is ocean, I wouldn’t be particularly surprised to find that a reasonable two-box model fit finds up to 70% of measured temperatures in the “slow”, “ocean” box. 90% is a bit high, I agree, but I can’t rule it out as ridiculous, not sure why you’re so certain on that.

104. lucia,

     I will concede your points 1 and 2 because they are distracting me from my main point and I never should have responded in the first place.

     Getting back to what should be included in Co:
     In my opinion, any two-box model is a simplification. Since there is a well-known layer with minimal heat transport between the well-mixed and deep ocean layers, that’s an obvious place to cut off the model. (I haven’t been able to find a numerical value for “minimal heat transport” but it’s easy to find references that discuss the thermocline and halocline between the upper and deep oceans).

     If I understand correctly, you are saying that any n-box model (with n>1) must include the deep ocean. That you can not cut off the model at a thermal boundary layer. By that reasoning, must you not also include the mantle of the earth since there is heat transport from the air and ocean into the ground underneath? Do you see how it starts to get silly after a while? Models involve simplifications and partitioning the well-mixed layer from the deep ocean is an obvious simplification to make.

     Also remember that there are an infinite number of physical two-box models (with heat transfer between the boxes, etc) that can satisfy Tamino’s two-box analysis. If you do the math (which would be very difficult) there is also an even more infinite 🙂 number of three-box models (third box is deep ocean) that satisfy the constraints of Tamino’s two-box analysis.

     Finally, don’t be upset that Tamino may have gone too far in his small post a couple of weeks ago. The issue here, which you created, is whether his analysis violates the 2nd Law. Does it violate the 2nd Law if Co is the heat capacity of the upper well-mixed ocean (as Tamino suggested in his very first response to your questions)?

105. On “3-box” models: in effect a very long time-constant gives not a constant term in the temperature response, but a term proportional to the cumulative forcing (integrated over time, divided by the very long time constant).

     However, I think it would probably be more fruitful to look at diffusive models as Alex Harvey suggested – I’ll probably take a look at that if I get a chance to get back to this in the next few weeks.

106. The thickness of the well-mixed layer depends on latitude.

     Discussion here.

     However, I’m not sure the mixed layer is what the “ocean” time constant is measuring. I think that mixing time constant is pretty short…. A 30 year time constant would probably have more to do with upwelling and downwelling currents and the vertical fluid transport associated with them.

     Also you aren’t actually ignoring the land masses so much as averaging in the temperature response of the land with the oceans. This is similar to what happens in gravity models where you average land and ocean and end up with a 1-km thick layer of water on the surface…this is DeWitt’s “Earth=Venus” model.

     Whether that leads to unphysical values of the physical parameters is something that could be estimated without having to go to a full 3-box model fit.

107. I meant to link this figure of the ocean layers.

     It’s from this link

108. bugs:

     I quoted what he said. Just as Tamino did, you are drawing a line in the sand on just what your model does and does not do. Tamino’s had shortcomings, but gave a good representation of the forcings on the climate. You are doing a lot more work, but still not enough to satisfy the next person who is finding your model ‘unphysical’.

     I’ve done a fair amount of simplified models or more complex physical systems, usually in conjunction with modeling cochlear physics. My group has a single limit-cycle oscillator model that we use to model spontaneous otoacoustic emissions, which are tonal emissions generated by the cochlea in the absence of external stimuli. The actual cochlear system is much more complex than this of course…

     The full model is a coupled 3-dimensional system with an active, nonlinear partition separating two volumes. You can produce a bulk model version of this where you average over a cross-section (and end up with net fluid flows in each volume for example).

     The two approaches are very different. In one case you’re writing down a simplified model that captures some of the relevant physics, in the second you are actually starting from first principles and developing a lower dimensional version of the original model.

     Anyway this doesn’t directly relate to what bugs said, but my limit-cycle oscillator models can’t be directly related to the physical constants of the underlying cochlea, but it still captures much of the same dynamics shared by a much more complex physical model that is very difficult to analyze analytically or even solve efficiently numerically.

     I would consider things like the limit-cycle oscillator model or Tamino’s two box models to be “toy models”. Even these models eventually have to match relevant physical constraints like causality and such. Tamino’s model would certainly be wrong is it violated the 2nd Law of Thermodynamics.

     You’d have to do some sort of bulk parameter analysis to show how the more complex climate system reduces in some approximation to a two box model before you could definitively say “this parameter is too large or too small”.

     When I use words like “unphysical” I mean things like Cs < 0, not Cs isn't in the range we expect for air.

109. Arthur–

     Lucia, in what way are you doing anything more general than I am?

     I am considering
     a) the “y” you introduced which you use for your linear combination of value and
     b) considering the possibility that the we may draw the line for the boxes somewhere other than the air/ocean interface. (I introduced a “z”.

     You only consider the “y”.

     And as I showed, no matter what you pick for γ,

     Oh? Did you consider any actually realistic value for γ? Or only values that are more than 4 orders of magnitude lower than one would expect for heat transfer coefficients on the air side of an air/solid interface?

110. Arthur–
     On the numbers, I think part of my trouble in reproducing your values may be rounding error in the value of α_o relative to α_o. Did you use exactly

     α_s = 3.17×10^-8 s^-1
     α_o = 1.05×10^-9 s^-1
     and
     γs = 3.17×10^-11 s^-1 ?

     Did you convert from years to seconds using 31556926 seconds/ year?

111. DeWitt Payne (Comment#19248) September 3rd, 2009 at 9:24 am

     Oliver,
     Your example is flawed. What is being done here is more like calculating orbital parameters with a solar mass ten times its actual value and being surprised that the calculated parameters do not match observations.

     This is a silly interpretation of what I said. You have brought up vertical heat differential and latitudinal variation within boxes, advective balances between boxes, and not enough boxes. All of these are valid criticisms of the physics of a two-box model, but Lucia has maintained her focus on the original question: is there a meaningful way to construct a two-box model from the forcing and surface temperature records.

     To accept your analogy: this model makes no assumption of the solar mass; however, if of our simplified orbital mechanics fit indicated a solar mass 10x the known number, this would serve as a reality check for the model would it not?

     As usual, Tom Vonk makes my point in his post above far better than I ever could.

     Tom Vonk believes that the degrees of freedom in the system make such a fit impossible, but I don’t see why this makes the exercise futile. If the two-box dynamics can be described, we might be able to gain some intuition about how a very simplified case involving only two time constants and a very simple heat transfer can behave. Meanwhile, if the fit fails to be meaningful, we may be able to understand why this is so by examining the system of ODEs and some of the behavior graphically.

     Oliver

     P.S.: It seems I am repeating some of what Carrick just posted.

112. Hi Arthur–
     Here’s what I posted:

     I want to plot things out, but I’m having trouble with your values.

     Ï„+ = 30 years, Ï„- =1 year
     αs = 3.17×10^-8 s^-1 = 1.0004E+00 /years
     αo = 1.05×10^-9 s^-1 (very close to 1/Ï„+) = 3.31E-02

     (I used 31556926 sec/year)

     So,
     (1/ Ï„+ + 1/Ï„-) – (αs + αo) = -1.28E-03

     But we know from the solution for the eigenvalues (αs + αo) + (γs+ γo) = (1/ τ+ + 1/τ-).
     So
     (1/ Ï„+ + 1/Ï„-) – (αs + αo)= + (γs+ γo)

     But you are reporting positive values of γs & γo?
     Is this a rounding issue?

113. JohnV

     Also remember that there are an infinite number of physical two-box models (with heat transfer between the boxes, etc) that can satisfy Tamino’s two-box analysis.

     It is true that there are an infinite number of models that correspond to tamino’s regression analysis. However, as far as I can see, none happen to make realistic box models for the earth.

     To provide a stooopid analogy, the range from [-infinity ,0] is infinite, but if you need that infinite group of numbers to match a value from [5,10], the fact that you can pick a number from [-infinity ,0] is inadequate to your needs.

114. lucia,

     You did a check using only 10% of the ocean’s mass in the ocean box. Would it take much effort to do a check using only the well-mixed upper layer? Remember that the well-mixed upper layer is at most 6% of the total ocean, and probably more like 2.5% (based on well-mixed depths between 100m and 240m).

     Does your 2nd Law violation hold up using only the well-mixed layer as Tamino described in his first response to your questions?

115. JohnV–
     I get negative heat transfer coefficients across the air/water interface for oceans 2.5% as large as the full ocean and the top box air only or air and the maximum amount that water that means we ghg’s don’t air condition the ocean.

     This differs from Arthur, so I’m looking for computational errors on my part. I’m also waiting to learn why the combination of α and τ’s in Arthur’s post don’t seem to add up. (The issue will probably be a typo or my misreading, but I’m currently puzzled.)

116. JohnV (Comment#19266),

     According to Josh Willis (NASA JPL), who does ocean heat analysis from Argo data, the average thickness of the well mixed layer is about 50-60 meters, not 100 to 240.

117. lucia:
     Thanks for checking your results using 2.5% of the total ocean. I will wait for you and Arthur Smith to reconcile the differences in your results.

     SteveF:
     Thanks. I’ve seen numbers from 20m up to 240m for the well-mixed layer. From what I understand, the depth varies by season and by latitude. It may also depend on the context in which the term is used — perhaps the thermocline is shallow but the effective heat capacity includes a small fraction of the total ocean (60m + 2% of 3790m = 140m). I’m just guessing here.

118. JohnV :
     It does indeed vary a lot, from >250 meters in quieter tropical areas to zero at high elevations. Varies with season a lot as well, at least outside the tropics. At high enough latitude to have substantial summer/winter swings, and/or a thin or non-existent WML, I’m not sure what the contribution should be to a box model. The complexity of the ocean suggests that any simple thermal model will be far from an accurate representation.

     Regardless of the ocean structure, we can be pretty sure that the majority of solar energy is converted to heat in the top 120 meters of ocean. Whether this layer is considered to be part of the “top box” or the “bottom box”, it is where most of the thermal energy is deposited; the ocean heats the air.

119. Lucia (#19260) – γs for me is a free parameter, which I forced to be positive, so there’s no way it can be negative in my (or any physical) solution.

     I only posted numbers to 3 significant figures on my blog – so yes, you are running into a rounding error because αs and αo are very close to the τ’s. Also I used a Julian year (31557600 seconds). More digits from the basic “Tamino” result gives:

     αs = 3.1656396 x10^-8 s^-1 = 0.9989999 year^-1
     αo = 1.052268 x 10^-9 s^-1 = 0.0332071 year^-1

     and the difference you are looking at is +0.001 year^-1 (about the same size as γs).

     Note that I also posted a second set of results with γs 100 times larger, with nothing particularly unphysical about the resulting numbers. And for a slightly smaller value of γs you would still have two acceptable values for the y parameter also, one of them very close to 1. I haven’t seen you post an estimate of what γs magnitude should be, but if it works when it’s very small and works when it’s bigger than αo, it should work whatever a reasonable physical number is.

     As to “generality” (your comment #19257), I haven’t seen you write down what ‘z’ means, but I’ve been assuming it’s purpose is to set the two heat capacities Cs and Co. I.e. something like:

     Cs = Ca + z CF
     Co = (1-z) CF

     Is that what you’re doing? If so, you have reduced my two free heat-capacity parameters (Cs and Co) down to just one (z), and my results are more general than yours.

     But maybe z means something else? What exactly?

120. I’ve posted a spreadsheet that does my version of the math here on Google Docs. Lines 14-17 are the “Tamino” numbers I mentioned, first with small γs and then with the 100-times larger value. Line 18 is with a 5-times larger Cs, and line 19 with a 5-times larger Co. The earlier lines are various other cases I was playing with to make sure I got the math right and understood what was happening.

121. Arthur–

     γs 100 times larger

     Yes. I noticed. So did you use any realistic values?

122. By realistic, I mean: Have you considered the lower range of possible heat transfer coefficients between an atmosphere and ocean which are not separated by a layer of Owings/Corning pink insulation?

123. I can’t seem to let this go.

     The history as I see it:

     Tamino says he can calculate climate sensitivity using a mathematical model with two time constants and a linear combination of temperature anomalies from each time constant to fit a particular set of forcings and the annual global temperature anomalies from 1880 to 2003. Lucia asks Tamino if his model violates the second law. This question only makes sense if Tamino’s mathematical model is derived from a physical model containing only two boxes with earthlike physical properties, which it clearly wasn’t. Tamino thinks the question is irrelevant, which from his point of view it was. Mayhem ensues.

     Nick Stokes produces R code that looks a lot like and may well be exactly equivalent to what Tamino did so now everybody can play around with time constants. and generate linear coefficients.

     lucia and Arthur try to construct a physical model containing only two boxes with earthlike physical properties to see if the solutions generated by Tamino’s mathematical model map into physical variables that make sense. There is some disagreement about whether there are an infinite number of solutions or not, but the key point is that a two box only model consisting of a well mixed ocean of the mass and surface area of the earth’s ocean and an atmosphere with the mass of the earth’s atmosphere has time constants different from Tamino’s 1 and 30 years (which appear to be purely arbitrary choices as the best fit is more like 1 and 20 years). The atmosphere box is on the order of 0.1 year and the well mixed ocean box is on the order of one hundred years. This is probably sufficient information to say that Tamino’s two box math model cannot be mapped into a two box only physical model with physical properties that do not violate thermodynamics and be earthlike. But we continue nevertheless almost ad nauseum. After several false starts it is now crystal clear that a two box only thermodynamically and physically valid physical model of the earth as we know it cannot be constructed.

     Does this mean that Tamino’s calculation of the climate sensitivity is wrong? Well, not necessarily. The validity of Tamino’s calculation depends only on whether the forcings are correctly calculated, that the temperature anomalies are unbiased and that significant quantities of heat aren’t being transferred to and stored in some hidden box. If so, the measured temperature anomalies should be in some way proportional to changes in global heat content.

     The reason is that the temperature measurements come from the real world that is not allowed to violate thermodynamics. This world may need a very large number of coupled boxes in its physical model. But we don’t know (well, we have some idea about some of the other boxes like the deep ocean, which is the only reasonable candidate for the hidden box) or need to know, for the purposes of calculating climate sensitivity, what happens in those other boxes because we know that the system as a whole obeys the laws of thermodynamics by definition.

     What would be an interesting exercise now would be to take the IPCC ranges of the various forcing and do a Monte Carlo analysis of the climate sensitivity calculated by Tamino’s method. The tropospheric aerosol forcings (the last two columns in gisforc.txt) come to mind immediately. If you eliminate them entirely, the calculated climate sensitivity is much lower, ~0.25 Km2/W. But you also don’t see much effect from the stratospheric volcanic aerosol forcing. That’s suspicious. So some reduction in total forcing does seem warranted.

124. Does this mean that Tamino’s calculation of the climate sensitivity is wrong? Well, not necessarily.

     Correct. It could be right.

     After all, we have temperature changes. We have forcing estimates. Lots of people have come up with lots of way to estimate climate senstivity based on temperature changes and forcings.

     With regard to the two box model, the issues are something like “Is there any reason to believe this method of estimating is any better than any other method? For example Schwartz 1 box method? Or hansens using longer term data? Might it be worse? Have we learned anything new? Is the new information independent form other information? (After all the 30 year time constant was supposedly picked based on a climate model driven by the same forcings.)”

125. Carrick
     “You’d have to do some sort of bulk parameter analysis to show how the more complex climate system reduces in some approximation to a two box model before you could definitively say “this parameter is too large or too small”. ”
     .
     Yes this basically what I have been saying in other words .
     When one deals with chaotic systems , phase space is always much more relevant to describe and understand processes .
     What you call “bulk parameter analysis” , I’d call the proof that a 2 dimensionnal arbitrary surface has a non empty intersection with a 5 dimensionnal finite volume in the phase space .
     .
     Now even supposing that there is a non empty intersection , the second question would come if the intersection surface says anything relevant about the 5 dimensionnal volume .
     In the general case it says nothing because it is just again a surface with exactly the same properties as the original surface and a more or less complicated boundary which depends on how the 5 dimensionnal volume has been cut .
     Now if the 5 dimensionnal volume was HIGHLY regular (f.ex a sphere) then the shape of the intersection boundary would tell you a very small something about “the circularity” .
     Now I suppose that nobody believes that the climate attractor is highly regular and that an intersection with an arbitrary surface says anything interesting .
     That is for me largely enough to say that the 2 box “model” has either no intersection with the climate attractor (what I believe because if Lucia showed that it violates 2 LOT even for a “reasonable” choice of parameters , then it is surely worse for the infinity of other choices) or if it has an intersection then it says nothing relevant .
     I also find it strange to choose precisely time constants to define a surface in the phase space because it is equivalent to restrict the frequency domain to 2 dimensions . But the frequency domain of chaotic systems is always infinite dimensionnal so this particular reduction can’t describe the real Earth .
     .
     de Witt
     I basically agree with what you wrote .
     But I don’t agree with “Does this mean that Tamino’s calculation of the climate sensitivity is wrong? Well, not necessarily. ” .
     This statement depends again on the hypothesis that the climate sensitivity is a constant .
     I don’t see any reason why it should be constant .
     But if it is not , then the 2 box “model” can’t obviously calculate it because it misses at least 3 dimensions .

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