I was wanted to show graphs illustrating the temperature series for the “atmosphere” and “ocean” for
Arthur’s “Case 1” & “Case 2” solutions, which he says correspond to a 2 box model that he deems a reasonable representation of the earth’s atmosphere. Round off error was causing me some difficulties; but Arthur kindly provided a larger number of significant figure in comments. I can now show a graph of the temperature for “Case 1”.
Data in Arthur’s post
The narrative in Arthur’s post provides some data for a solution he calls “Case 1”. He tells us the time constants and heat capacities as:
Set Ï„+ to 30 years, Ï„- to 1 year. The fitted parameters are a2SI = 0.739 and a3SI = 0.038. Assume the short-time-constant-box heat capacity Cs is just that of the atmosphere, 1.34×10^7 J/Km^2. Give the long-time-constant-box Co a heat capacity of 1% of the full ocean, so 1.06×10^8 J/Km^2.
Whether or not one one may simply ignore 99% of the climate system when fitting temperature to a two box model, and never give the heat transfer out of the lower box a second thought is debatable (and is being debated in comments.) For now, I note Arthur used:
- Cs=1.34×10^7 J/Km^2 which corresponds to the full atmosphere
- Co = 1.06×10^8 J/Km^2 which would correspond to approximately 20 meters of ocean.
- Ï„+ = 30 years and
- Ï„– = 1 year
Arthur then says:
This leaves the heat transfer rate γs as a free parameter. We’ll start with γs = 3.17×10^-11 s^-1 which is roughly a factor of 30 smaller than 1/Ï„+ = 1.06×10^-9 s^-1.
So, Arthur selected γs = 3.17×10^-11 s^-1=3.16E+07 s/year or 1.00E-03 years-1.
For the engineers who might want to compare this value to typical values for air/solid interfaces , this corresponds to a heat transfer coefficient of (3.17×10^-11 1/s ) * ( 1.34×10^7 J/Km^2 ) = 4.25E-04 Watts/m^2 K. While one generally does not consult Wikipedia when doing heat transfer calculations, it is nevertheless useful to not that Wikipedia reports typical values of heat transfer coefficient in air of 10 to 100 W/(m2K). So, some might consider Arthur’s choice a smidge low. (Arthur does later increase this value by a factor of 100 resulting in heat transfer coefficients only 4 orders of magnitude lower than the lowest conceivable values for realistic heat transfer coefficients.)
But the question of whether Arthur’s choice for γs is realistic does not prevent us from reconstituting the temperature series for his box. Returning to his computation, Arthur chose γs = 1.00E-03 years-1, then for his choices of heat capacities results in γo =γs * (Cs/Co) = 1.26E-04 years-1.
For case 1 in Arthur’s narrative, we learn he obtained:
αs = 3.1656396 x10^-8 s^-1 = 1.00E+00 years1
αo = 1.052268 x10^-9 s^-1 = 3.31E-02 years1
(The rounding of αs in the first post was resulting in an difficulty that potentialy “mattered” vs. a vs. the 2nd law. But that’s ok now.)
The assessment of whether or not αo is reasonable for the earth depends, to some extent, on what use for γo. After all, the top surface of the ocean does radiate energy with the amount depending on the surface temperature and the emissivity of the surface. Those photons either escape to the universe, and contribute to the magnitude of αo or they are intercepted by the atmosphere which is reflected in γo. A further complication arises if we recognize the remaining 99% of the ocean acts a an infinite heat sink with a constant tempeature, then the magnitude of αo must rise to reflect this heat loss form the ocean box to the lower ocean.
Since that evaluation is complicated and you all want to see graphs, I defer discussion of whether αo is realistic when used to describe heat loss from this thin layer of ocean to the regions outside both boxes.
Now, for temperature graphs!
I created plots based on these additional values listed for Case 1 in Arthur’s post:
- x=0.0179 (meaning 1.79% of the forcing goes in the atmosphere box and the rest in the ocean box; that is 50 times as much energy goes in the ocean box.)
- y=0.91 which means the thermometers are mostly in the atmosphere box.
- I don’t need to use the weights as I can integrate the temperatures based on the box parameters.
As you can see by comparing the green and yellow curves, the mixed temperature computed by adding 91% of the atmosphere temperature and 9% ocean temperature does fit the data rather well.
In contrast, the temperature of the ‘ocean’ does not match the data. In this box model, the ocean has warmed much more rapidly than the atmosphere and also more rapidly than the average thermometer measuring global surface temperature.
I don’t have ocean heat content data for the top 1% of the ocean handy; if anyone has average temperature for that reason, I would be happy to add that to the graph.
The phenomenological explanation for the rather large increase in temperature for the ocean box, is this: In Arthur’s box model, the ocean box absorbs 50 times as much of the forcing as the atmosphere box. Meanwhile, the heat capacity of Arthur’s ocean is only 10 times as large as the atmosphere, so one might expect the ocean to warm up more readily than the atmosphere.
In addition, Arthur chose a very,very low heat transfer coefficient between the lower box and the atmosphere. The solution also resulted in a rather low effective emissivity for the surface of the ocean; so heat can’t radiate away very well.
So, lots of heat enters his very lightweight ocean, it can’t escape to the atmosphere, it can’t escape to the lower ocean, it heats up. I can say this about Case 1: if I correctly diagnosed the typo, it appears that these choices do create a box model that does not violate the 2nd law of thermo.
I think it is up to each reader to decide whether the parameters of this box model correspond to any sort of “physically realistic” representation of the earth. After all: Who am I to tell people what to think?
Lucia, I don’t believe you can just “integrate the temperatures based on the box parameters” because of the heat exchange term (although in this particular case I agree I probably did choose it somewhat low, for illustrative purposes).
I’m glad we’re agreed now that there’s no need for a second-law violation.
The fitted parameters for Tamino’s 30-year time constant allow at most about 80 meters of ocean in the slow-box heat capacity. My choice of 20 was perhaps too small, but it can’t be hugely bigger without the two-box model completely failing. Your analysis here, if correct, merely suggests more constraints on physically reasonable results. Write those constraints down and we’ll see what comes out.
Analysis is always fun, isn’t it?
Lucia,
I think something’s wrong with the scale there. 6C temperature anomaly?
Lucia, I do not appreciate your mocking tone here. If you have more constraints you want to apply for physical realism, write them down and we can explore whether or not they can be satisfied. So far I have found an infinity of solutions that satisfy every constraint you have imposed to this point, except the clearly unphysical claim that the total heat capacity has to be that of the entire ocean. Perhaps out of that infinity I picked a couple that have other problems – well, let’s add in the additional constraints so we can narrow down the solution space further if necessary.
I have in fact shown several of your claims to this point to be wrong. There is no second-law violation inherent in Tamino’s example. There is a 3-dimensional infinity of possible two-box models that match it, not just the 1 you claimed. The specific value of the short time constant makes essentially no difference to the two-box fit or resultant climate sensitivity, nor does it significantly change the corresponding two-box solutions. And so on.
I don’t recall a single time you’ve actually admitted to being wrong on any of these things though, you just move on to the next opportunity to ridicule.
I’m sorry to sound angry, but this is really incredibly frustrating.
Will you admit you were wrong about your initial claims that this has anything to do with the second law?
Nick–
Those are the values for Arthur’s result.
Arthur–
When you find any solutions that are physically realistic, let us know.
I understand you can be a little emotional at times. But you might want to reflect and recognize that you are claiming to have corrected me on claims I did not make. I suspect you are “correcting” me for claims Tamino accused me of making.
You might want go go back to my original comment that pissed Tamino off Taminos to review what I was asking from the start. I will hightlight the bits that don’t seem to register with you.
Not withstanding your interpretation that my concerns could be addressed by finding solutions with unrealistic parameters that don’t violate the 2LOT, -it should be quite clear that I from the start was always considering other issues than merely violating the second law.
In comments I have consistently repeated this notion in response to your various suggestions of how we can constitute a two-box model that doesn’t violate the 2LOT. I have consistently made the point that the parameters should correspond to two-box earths’ that describe Krypton rather than earth.
I recognize that you have not understood the notion that I am concerned about whether or not the box model is physically realistic and that not violating the 2nd law is one of the criteria but not the only one.
Others in comments have recognized that you have not understood the notion I was communicating. I and others have tried to explain it to you in many ways. I even organized my post to emphasize this.
For example my post on August 31, specifically mentioned the cascade of tests. Specifically
Why you think merely finding some cases that don’t violate the 2nd law with a two box model that has an atmosphere with a realistic heat capacity but in which the ocean and atmosphere are separated by a thick layer of Owens-Corning pink insulation is proving me “wrong”, I cannot imagine. I have always said “with physically realistic parameters”. I’m willing to give a lot of slop to those physically realistic parameters.
Again: When you find any solutions with parameters that are remotely physically realistic that don’t violate the 2nd or first law, let us know.
Nick–
I think in Arthur’s notational convention, the sensitivity for the ocean is
(w+s * r+) + (w-s + r-)
His case 1 values are:
w+s = 8.40×10^-3 K/(W/m^2)
w-s = 4.18×10^-2 K/(W/m^2)
and r+ = 967
r- = -1.31×10^-4
So, I think doubling of CO2 will make his ocean more than boil away.
lucia,
You clearly stated many times that you thought there would be a unique 2-box solution that matches Tamino’s analysis. You mocked Arthur Smith for suggesting that there would be an infinite number of solutions. I believe you said math wasn’t his strong point.
He has proven that there is an infinite number of solutions in 3-dimensional space. Are you willing to at least concede you were wrong on that point?
JohnV–
What haven’t I conceded? At one point, I thought there might be enough equations to map into one solutions– but I also said I wasn’t sure that the equations I counted up might be linearly independent.
They weren’t– and I admitted that long ago. But we could still solve the set of equations contingent on choices– which is what I always believe to be true. If we specify some, we can solve for the rest of the box model. That is precisely what Arthur was trying to do.
In fact, this was determined that we could compute box parameters based based on three arbitrary choices, before my August 31 posts where I discussed that we do, indeed, need to pick three free parameters.
So, once gain, what haven’t I conceded?
My point has always been that I said that we need to figure out if any of that infinity of cases overlap physically realistic systems for the earth. That’s the point of the venn diagrams, the various discussion about unrealistic treatments of the ocean (in the context of two box models) etc.
I still don’t know they don’t. It may be possible that some of that infinity of solutions map into solutions that are realistic for the earth. What I do know is, not withstanding Arthur’s tendency to resort to rather high toned language, (alternating with acting injured) he has not shown this:
He has shown something barely disputed: that there are an infinite number of two-box models corresponding to the two time constant. What he has not showns is that any of two box models “resemble” components of the earth’s climate system in any meaningful way.
Arthur
Why do you think that’s a problem? I don’t see any.
On the “for illustrative purposes”, it might be wiser to pick something within 4 orders of magnitude of what is possible. It also might be wise to actually consider the problems I suggested when you first suggested you could just throw away portions of the ocean.
Should you make another attempt, I suggest considering the fact that heat transfer both by natural convection but also evaporation, and then rain falls back down.
Why do you think it’s my job to specify every constraint?
This all started with my asking a question of Tamino. I didn’t know the answer then and I still don’t know it. He claimed he’s already done all this. Ask him.
There’s some merit at this stage in revisiting Steve Schwartz analysis. He inferred a total effective heat capacity for time puposes, air+ocean, of 0.44 GJ/m2/K, which is about 4x the figure quoted here as that which Arthur used – not so different. On that basis, Arthur’s figure has respectable ancestry.
“Where climate talk gets HOT!”
Nick–
And yet, Steve Schwartz analysis of a one-box model is precisely the analysis that Tamino trashed. 🙂
As I have said: It’s no surprise the two-box model results in a regression that fits data. After all, the one box model does.
What’s not clear is whether the two box model teaches us anything we didn’t already know with a 1 box model, and it’s not clear either can be relied on to provide good empirical estimates of the climate sensitivity. They may. They may not.
The 1 box model gives a value on the low range. It gives us short time constants. What that means…. dunno.
Crickets…
I was also upset in my second comment here because the first post looked like it had failed (got a server error from your site) and I hadn’t kept a copy of what I wrote. Comment #19286 was my first reaction to your post here – you are adding what seem to be many additional, unspecified, constraints on what you originally meant by “physically realistic space”, and as far as I can tell it’s a constantly moving target. We knew from the start that the two-box model was a great simplification, so for example the temperatures are assumed to vary uniformly within each box, when they clearly do not. Ok, it’s unphysical. Does that mean this was all pointless?
If you have any desire to pursue this further with me, please describe what you would actually consider “physically realistic space”, preferably with a set of numerical constraints in the notation we’ve developed to this point and a physical argument for why you think such constraints are actually necessary.
If you demand Cs + Co = CF + Ca, then we’re done, I’ve already shown that’s impossible to satisfy.
Specify something else for what exactly would satisfy your concerns, and we’ll see if it’s possible or not. Without some details I just have no further desire to discuss it here. I may try a couple more examples over on my blog for the curious.
Arthur–
Yes. Something is weird with comments right now. I don’t know what it is. I’m visiting spam to free things.
No. But those are the assumptions of any two box model. So, we concede that these are being made. But given that these are made, you still make other assumptions that made sense given this content.
Arthur: I think this is your job.
I thinks inefficient to try to do much before the first cut to eliminate boxes that won’t work for any choice of heat transfer coefficient etc. The reason it’s inefficient is you will end up trying to estimate heat transfer coefficients for geometric configurations that can’t work at all. (For example: Too deep of oceans.)
That’s why my list of steps has it after the first cut.
But even if I thought it efficient to estimate these first, how could I begin to tell you what the constraints are when you don’t even accept the notion that I think the bottom box needs to contain all the notion? You want to retain the flexibility of deciding what’s in each box, you have to assign yourself the task of figuring out how these geometric choices affect the proper magnitude of the parameters for the box model.
I can only provide rough guidance as follows:, any two box model has parameters αs, αo, γo, γs, Cs and Co and you apply forcings which are distributed using ‘x’.
Each of these is meant to represent something physical (as discussed in my first blog post.) Also, you have introduced “y” some distribution of thermometers (which I think is odd. But if you can make sense of it, I’m willing to keep that in the equations and wait to see what you get.)
You should be able to look at what you think belongs in each box, (air/deep ocean/ shallow ocean etc.) and based on those contents crack out some books to estimate what makes sense for the magnitude of αs and γs based on what they represent. It’s a bit tedious.
Everyone is going to give you some slop (because I’m sure we’ll discover no-one knows the heat transfer at the air water interface to better than within an order of magnitude!) But you need to slog through this tedium before you decree you’ve proven that Tamino’s method is simultaneously “simple and elegant” and “produces numbers that we can believe more than a 1 box model or even just some econometrics curve fit”. (By the time you slog through, you may decide that it’s not so “simple”! )
Ok, it’s unphysical. Does that mean this was all pointless? I could see that coming.
“Does that mean this was all pointless?”
YES YES YES
Andrew
It is only pointless if NO ONE LEARNED ANYTHING in the exchange.
As many readers don’t post, that may be hard to tell.
It ain’t over till the mass limited person vocalises.
Pointless no.
That it is meaningless I am certain.
But one can learn from such meaningless excercises what is pointless: and possibly better still what is useful.
Kindest Regards.
This is probably a stupid question, given that I do not understand much of this.
Under what circumstances would energy move from box 2 to box 1?
In a warming phase, it would seem to me that box 1 is always going to be warmer than box 2, and thus energy transfer should only go from 1 to 2. (This is of course assuming well mixed boxes).
Is this correct?
This series of threads is certainly not pointless. At a minimum it shows:
1) The potential problems/pitfalls of creating a greatly simplified model to represent a very complicated system.
2) That a curve fit can often be constructed which tracks data very well, but which may or may not reveal anything meaningful about the system from which that data comes.
3) There are a lot of smart, technically competent, and sincere people who care enough about climate projections to spend valuable time and effort to better understand them.
4) Lucia maintains an open forum for (sometimes heated) technical exchanges about climate science, which contrasts starkly with certain other well known sites about climate.
I ask all to consider if these threads could (or more correctly, would ever be allowed) to take place at RC or Tamino’s. I think the honest answer is no, and I think this speaks volumes about the attitudes of the folks who maintain those blogs.
Steve F,
The threads aren’t pointless. You are correct.
The Climate Models themselves… (wince)
Andrew
“4) Lucia maintains an open forum for (sometimes heated) technical exchanges about climate science, which contrasts starkly with certain other well known sites about climate.
I ask all to consider if these threads could (or more correctly, would ever be allowed) to take place at RC or Tamino’s. I think the honest answer is no, and I think this speaks volumes about the attitudes of the folks who maintain those blogs.”
And don’t forget that the people who run those blogs are prominent climate scientists. They prefer their science settled, not stirred.
Lucia
I think Dale Carnegie put it best in his book “How to Win Friends and Influence People”, ” a man convinced against his will is of the same opinion still”.
Although it’s difficult for me to understand all of the mathematics involved, I do not think there is an argument that can made here that will convince Arthur that Tamino’s method is anything but “simple and elegant” short of a confession by Tamino that his two box model was a “dream sequence” and that we can all go back to our normal lives in a physical world where the 2nd law applies.
Thanks
Edward
It does illustrate how difficult it is to contruct a climate model that accurately covers all the interactions. At numerous points, one would have to take shortcuts, build-in parameterized assumptions etc.
I still think it is worth pursuing.
Temperatures are not keeping up with the original trendlines predicted by the theory and the rationale for that is that the ocean boxes are absorbing the forcing. Current theory puts that lag at 30 years but it could be considerably longer or it could be significantly overstated. Those participating should continue exploring the question.
David Gould–
Are you asking for the real world? Or Arthur’s box model?
I’ll assume you are asking for the two0box model which is easier to answer.
If the two-model is constituted such that there is no violation of the 2nd law then energy moves from box 2 to box 1 when the temperature of box 2 is higher.
Let’s give a hypothetical example with using “units” for energy, thermal mass, time and temperature. (So, none are real. For a real problem, you use real values and real units.)
In Arthur’s case 1, box 2 has about 10 times as much “thermal mass” as box 1. At time zero, both boxes are at temperature ‘0’.
Suppose heat were added. For each H=50 units of heat added to the lower box with thermal mass “C=10”, So, during a small amount of time “dt = 0.01”, the lower box temperature rises approximately (H/C)* dt = (50/10)* 0.01= 0.5 “units” of temperature.
Meahwhile, 1 unit of heat is added to the upper box (with thermal mass 1). So it’s temperature rises (H/C) * dt = (10.1)* 0.01 = 0.1 “unit”.
So, not the temperature of the lower box is 0.1 and the temperature of the upper box is 0.5. So, at this point, some temperature will flow from the warmer box (i.e. 2) to the colder box (i.e. 1)
For a two-box model, if all the heat were added to box 1, yes. Even if the majority of the heat was added to box 1. In Arthur’s box case, 1 will first initially warm faster if the lower box were made larger by a factor of 5.
So, if what you are saying is it seems to you that on the real earth the atmosphere would warm before the ocean, then you would screen out box models that did not do this as “unrealistic”. You would then reject the Arthur’s case 1 box model as “unrealistic” based on it’s behavior. (And likely based on the observed fact that the ocean has not warmed like that.)
You would then suggest at least some of the examples in the collection of samples he says have realistic properties for earth do not have realistic properties for earth, and he needs to screen the group further.
He needs to keep screening until he finds a subset that do have realistic properties for two-box models of earth and map into the regression parameters τ=1 year and 30 years.
He hasn’t done that yet. If Tamino has, he hasn’t revealed his work. (It may be possible to show to box models exist hat map into those parameters. I don’t know. But I think it’s more work than Arthur anticipated.)
“Those participating should continue exploring the question.”
They should be studying the actual climates and recording their findings. You know… building an ever expanding knowledgebase instead of pretending they already know stuff they clearly don’t.
Andrew
Lucia — given bugs, a jones and Arthur Smith’s “attitude” to your analysis you should have call this thread the graphs of wrath
Lucia, Aurthur, Nick, Carrick and others, thank you for this excercise in scientific triangulation. Science is bigger than one person. It is usually through an open, back and forth exchange such as this that paves the way to understanding by pushing to the dead ends.
Aurthur, I appreciate your participation, but I think you are confusing firm “claims” with soft “hunches” or “assumptions for the sake of argument”. I think Lucia has done a pretty good job of qualifying her statements with the appropriate caveats. However, a real time discussion on a blog leaves a lot of comments that can be misinterpreted.
Lucia,
You claim that “Wikipedia reports typical values of heat transfer coefficient in air of 10 to 100 W/(m2K)”. Indeed it does, but such a claim makes no sense. Heat transfer coefficients refer to a boundary between two systems of a specific geometry, and the quoted wikipedia statement gives no reference geometry, although the surrounding context suggests the author of the article was focused on fluid flow through pipes, i.e. systems a few centimeters in size.
Size matters critically to heat transfer because with both convection and conduction the rate of heat transfer is proportional to the gradient of temperature, i.e. temperature difference divided by distance, rather than proportional simply to the temperature difference in itself.
Here we are talking about two bodies that have dimensions on the order of 100’s of meters to km, so the distance in question is on the order of 10^3 to 10^4 times larger than the reference system in the wikipedia quote. Hence one would naturally expect heat transfer coefficients between systems of that size to be several orders of magnitude smaller than the wikipedia reference.
Also note that practical insulation products work mainly by preventing convection; the bulk of their volume is still air; air’s thermal conductivity is very low.
Dr. Spencer:
August 2009 Global Temperature Update: +0.23 deg. C
September 4th, 2009
It does illustrate how difficult it is to contruct a climate model that accurately covers all the interactions.
Which, if you were a cynic, would be what you would say Lucia tried to do all along, make it all sound too hard and pointless. Tamino’s point all along was that you could take a very simple model that didn’t try do anything more than cover the most basic interactions, and still come up with a reasonable answer.
Arthur–
a) I know how insulation works. I also know it is used because free convection heat transfer occurs at the surface of roofs. It also occurs at the surface of lawns etc.
b) The ocean surface is rippled and wavy. The air is no motionless as in the cells of insulation.
c) If you need an estimate of the heat transfer coefficient, I might suggest you consult the literature instead of trying to do the scaling analysis, which is rather difficult at the ocean/air interface, which is pretty dynamic. I googled and found a woodshole report where heat transfer fluxes were measured.
Lucia,
Ok, I have just one more question for you. I’ve done the rest of the work on my end and I’m satisfied with several things that were still slightly bugging me – I’ll post on my blog when I get a chance this weekend.
Here’s the question. You have just devoted an entire post to my “case 1” example, which occupied all of 5 lines in my last long post, exploring the “+” solution for y. Will you devote a second entire post to my “case 2” example, for the “-” solution? If not, why did you choose not to look at it in this post?
The point being, “case 2” is yet another of the infinitely many two-box solutions for Tamino’s case, and aside from the possibly low heat transfer rate (a new criterion you introduced with this post) it looks to me like it satisfies all the other criteria you have specified. Why didn’t you try posting “graphs of Arthur’s Case 2”? I certainly hope this isn’t just another case of cherry-picking, is it?
Hey Lucia. I’ve got some results up for CCCMA CGCM 3.1 T47. I calculated the TLT anomalies. This model runs pretty hot!
Arthur–
I didn’t post case 2 because it looks wrong too and I didn’t want to pile on. Why do you think it looks ok?
I think it looks ok because I’ve plotted the temperature curves myself.
You didn’t want to “pile on”, huh? Am I Charlie Brown?
Arthur–
I plotted them too. From a computational point of view, the only difference between case 1 and case two is x = 0.177. Other than that, the values of αs, αo, γs, γs and τ+ and τ- are as discussed in my current post.
With x = 0.177, the ocean temperature still soar. The air temperatures are “wild” to say the least. Both behaviors are consistent with your values of W+r+ ~ 0.8 K and w-s = 0.417 K/(W/m^2).
Are there typos in your listing of case 2? Do you mean some other case 2?
After checking 2 cases, I have to admit I figure it makes sense to just explain the problem with your screening and afterwards you can start doing the screenings yourself.
Plus Arthur–
Wouldn’t it make more sense to let you rerun these with reasonable values of heat transfer between the layers? And let you explain why you think they are even worth considering based on the values of αs, αo, γs, γo relative to what might be remotely reasonable for the earth? None of your cases use reasonable values.
“Arthur Smith (Comment#19358) September 4th, 2009 at 1:44 pm
Lucia,
You claim that “Wikipedia reports typical values of heat transfer coefficient in air of 10 to 100 W/(m2K)â€. Indeed it does, but such a claim makes no sense. Heat transfer coefficients refer to a boundary between two systems of a specific geometry,”
Isn’t Geometry here irrelevant to the coefficient because it says per m^2 per Kalvin. In other words it is a function of the surface area. Clearly if one medium is much more conductive then air, then the geometry should have little relevance. I guess in the case of the ocean, waves would effectively increase the surface area for heat dissipation. Of course the above constant doesn’t take into account wind but it gives a wide range, so maybe this covers a wide range of wind velocities.
Lucia, if you find the ocean (slow box) temperatures still soar in case 2, you are doing something wrong. Post your version of the curves and perhaps we can see where the problem is. It’s simple enough to just multiply the w+ and w- coefficients to the exponentially smoothed curves and add them together.
John–
Geometry is not irrelevant. But 2 minutes of googling for literature confirms that the range at Wikipedia listed for ‘air’ does include the ocean/atmosphere interface.
This should be no big surprise to people who were discussing the magnitude of these parameter in comments. For example DeWitt Payne was discussing them at some length.
Arthur–
The method I am using is simple. It uses α γ and x. You derived your w’s from those. The two methods are the same. Of they aren’t then either
a) there are typos in your equations or
b) you miscalculated your w’s or
c) you miscalculated your x’s.
Are there any typos in your equations?
(Comment#19391) Lucia I meant irrelevant as a first order approximation (Especially given there were such a wide range of values). Anyway, moving on, I noticed in some of the above posts, we were accessing how realistic a model was based on the transfer coefficients.
If the top box includes some of the ocean, then the transfer coefficients must have a smaller transfer coefficient then if it doesn’t because because the heat transferred is now only the fraction of that heat that is not already in the top box. That is some of the heat in the top box, includes some of the ocean heat and the transfer coefficients are based on the temperature difference between the surface box and the ocean.
The new transfer coefficient is given by:
\beta’=beta(1-\alpha) where:
\alpha is the fraction of the ocean which is included in the top box.
I discuss this a bit in my post:
Overlapping boxes:
http://www.climateaudit.org/phpBB3/viewtopic.php?f=4&t=774
Lucia,
I ran into a puzzle with your results
I assume that the bottom row is for top box being air only (Z=0)? But in any case, from your equations (1d) and (2d), the eigenvalues should be 1 and 1/30, as I understand. But for the second solution, I got (1.2296 and -0.1384) and for the first, (0.7346 and 0.2234). The results you gave are only to 2 sig figs, but I don’t think rounding is the explanation.
I was trying to reconcile your numbers with mine. A big difference is that you are putting only 2.7% of the forcing into box 1, whereas I am putting in the lot there. I’ve extended the R code to allow for split forcing, and so get for (x, αs,αo,γs,γ0):
(2.7%, -2.951, 0.04900, -3.9300, -0.00497)
The negative αs seems unphysical. Again, I think the underlying reason is the assumption that the whole ocean responds as one (well-stirred) box. If I reduce the heat capacity ratio from 791 to 150 I get (2.7%, 0.1833, 0.0325, -0.8121, -0.00541), which is OK.
I’ll put the new R code up soon (with revised text) at CA.
I’ve found approximations for the split-forcing case. I’ll write them up at CA. But the bottom line is that:
αo,γs,γo will always have the physically correct sign.
But for αs to be +ve, the criterion is (to a good approx)
x < (C_s/C_o)(1+(λ_1*a_1)/(λ_2*a_2))
where a_1, a_2 are the regression coefficients.
That’s a pretty strong upper limit. For C_o/C_s=791 (box=full ocean), x has to be less than 0.6% in the Tamino case (λ=1,1/30). Most of the forcing has to be on the ocean, to meet this physicality criterion.
Nick–
Thanks.
I’m looking that to reconcile with yours. Yes– if there aren’t typos, my intention for the case with number listed has all air in the top box. (Box 1)
I’ve been looking at a “merely looks like the data” that Dave Gould suggested in comment yesterday. (He may not know he suggested it.) that one give another requirement that to make the ocean warm more slowly than the air in under impulse response, the ocean has to be big.
So, I’m not sure Tamino’s values can work.
Nick–
Are you exercising Arthur’s linear combination of temperature “y” yet? (I didn’t in any equations I’ve posted. I don’t need to derive any equations for mapping the fits into these parameters because Arthur starts with the box parameters, and then supplies the “x” and “y”.)
Lucia,
No, I haven’t implemented y – I concentrated on x because it gives a physicality test.
I’ve put up the new theory and spreadsheet at CA BB.
I think now that one can’t say that the box model is invalid. On this theory, you can always find a range of forcing split factor x that gives a valid matrix.Whether you think that split is reasonable is another matter, but it doesn’t violate any laws, and doesn’t affect the regression fit and sensitivity.
btw, I earlier gave some results with negative gamma’s, and said they were OK. I think in your notation, they should have been positive (and OK). I think you may have also changed the sign in the results you gave.
Nick–
I’ll look at what you did later because I am still of the opposite conclusion. But of course, it may be that I am wrong or I am considering constraints you have not imposed and so our basis for conclusions differ.
I’ve got a busy weekend with company.
Nick–
Ok. I looked here.
I think that is a great start. It provides bounds for those cases that don’t volate the 2nd law and should be discarded for that reason, But now we need to look at the other requirements which are: the values for αs and γs must result in a box model that corresponds to earth and not, for example, Krypton or an aging resin filter module one used at K-basin. (I once did a two box model for mass transfer in one of those.)
However, in my initial comment at Tamino’s and my first post here, it’s clear that I don’t consider merely not violating the second law with arbitrarily selected parameters that might be utterly unrealistic for the earth is sufficient to make the two-box model physically realistic.
I think it’s important that we be able to find a two-box model with parameter α γ etc. that are remotely realistic for the earth. This idea has been reflected in my comments from the beginning– including the comment that make Tamino flip out over at his blog, and in my first post where I specified “realistic” values for the heat capacity of the earth and also the plausible coefficient for a linearized expression for radiative heat transfer.
(I’ve always been perfectly willing to expand those and it appears Dewitt Payne may have notions about a better value for the linearized expression for radiative heat transfer. But we know there is some plausible value that prevents the earth from having a venus like or mars like temperature.)
New post with more algebra, discussion of heat transfer rate constraints, and lots and lots of pretty pictures here…
In particular here’s the “Case 2” plot that Lucia didn’t choose to show above so as not to “pile on”:
http://arthur.shumwaysmith.com/life/sites/default/files/two_box/check_temps_gamma3.17e-11.png
R source code for this plot is here – just change the ‘.png’ extension to ‘.R’ to get the R code for any of these plots.
The wsplus, wsminus, rplus, rminus values are exactly as I posted under the case 2 line. The two temperature curves certainly look reasonable. Note that the baseline on all three curves is essentially arbitrary in this plot (set to zero in 1880 for Ts and To, while Tm has the GISS baseline) so the important thing is reasonable trends and responses rather than relative temperature anomaly values at a given point in time.
The additional case with 100x greater heat transfer rate is also posted on the page I just put up, as is the one for a heat transfer rate about as high as is possible with these parameters, another 6-fold higher. It makes little difference to the fitted temperature curves. Additional cases with shorter and longer short-time constants are also posted. Going to much shorter “fast” time constants allows higher heat transfer rates in the solution, and I looked at numbers as high as γs = 1.5e-7 s^-1 which, by Lucia’s math above, amounts to a heat transfer rate of 2 W/m^2 K. I’m not sure such high heat transfer rates actually make sense for this “box” version of Earth’s climate, but they’re within the solution space for short time constants, at least.
For a time constant of 1 year, the maximum possible γs value is about 3e-8s^-1, or about 0.4 W/m^2 K if Cs = Ca. Allowing the fast heat capacity to be larger for this longer time constant also allows for larger heat transfer rates – Cs = 5 Ca brings this up to 2 W/m^2 K again.
In any case, there is a somewhat limited, but still very broad range of heat capacities and heat transfer rates for the two boxes in Tamino’s fit and in fits with shorter time constants that satisfy what look like all the basic physical constraints, and appear to allow values that are reasonable for components of the Earth. The only conditions that are not really met are being able to include the full ocean in the slow box (at most about 10% worked in the examples I looked at) and reaching the very high heat transfer rates Lucia seems to think are needed here.
Arthur–
I will be posting on Tuesday, after the holiday. Have a nice labor day.
I don’t think trashed is right – T’s main issue was what he echoed more recently, that one time constant wasn’t enough. He wrote a comment to JGR, and SS conceded the point, and actually did a two-time analysis, which sounds fairly like what Tamino has done now. It raised the sensitivity. In fact, Schwartz got a long time constant, optimised, of 8.5 years, and sensitivity 0.51 ± 0.26 K/(W m2). My spreadsheet with time constants of 1 and 8.5 years gives sensitivity 0.453.
I did an optimisation of the fit (Sum squares of residuals, without SOI) over 2 time constants, and got (1.058, 21.50) years, with sensitivity 0.607. Including SOI changed the optimum slightly to (0.985, 20.18)years. with sensitivity 0.569.
At one time, lucia (I believe) calculated that the short time constant should be 0.09 years.
In terms of physicality, this is likely the time constant for the atmosphere.
The earth surface and the atmosphere lag behind the solar energy cycle by 0.09 years = 34 days. The summer solstice and the maximum solar forcing is on June 21 while atmospheric temperatures peak about 34 days later, around July 25. They continue to lag behind the solar energy cycle by the same 34 days throughout the year.
The sea surface has a slightly longer lag, about 83 days – sea surface temperatures peak about September 12th each year or about 0.23 years behind the annual solar energy cycle (and this also continues throughout the year with minimum sea surface temperatures occuring 83 days after the winter solstice).
I don’t know about ocean temperatures as you go lower than the surface but the actual physical time constants should be 0.09 years for the atmosphere and 0.23 years for the immediate 3M ocean surface.
Nick– The original paper in JGR said the one point model was big simplification, and the blog articles in response were quite a bit less measured than the Foster JGR comment. “Trashed” is the correct word.
It is true that Schwartz added another time constant, which makes the fit better as did Scaffetta. So, the notion of using two time constants is hardly new.
When Schwartz added the time constant, he got the result for a time constant I got in my November blog post reaction to his paper which shows that if we assume the signal is from a 1 lump parameter + white measurement noise. This is for a sufficiently short time constant, you can’t distinguish physics from measurement noise with monthly data. It may be one; it may be the other, you can’t tell.
I’ve put up a revised post at the CA BB. I had made a mistake with the calculation of the inferred ocean temp T_o. With that corrected, then in the case where x is restricted to “physical” values, T_o now has more interesting behaviour. A plot is shown on the revised spreadsheet.
Lucia, in my #19407 I said I thought you might have changed the sign of the gamma’s you showed. That was based on a misreading of your post.
I now see another difficulty with achieving “physicality”. The solutions I have been getting choose a very low x to satisfy 2LOT requirements. But I have been overlooking a 1LOT issue. I’ve introduced a factor b, which scales the flux to match the observed temperatures. This has conservation of energy issues, in that there should be a constraint to prevent the total heat rise exceeding the forcing energy supplied. There isn’t, and in this low x regime, that clearly doesn’t happen. The forcing causes the whole ocean to heat substantially, which it can’t.
So more thinking required, and it’s very late here, so that’s for tomorrow. The earlier curve fittings, and the associated sensitivity calculations, aren’t affected.
The equations for the boxes are models of energy content and changes in the content by means of heat transfer between the boxes plus energy addition to the boxes by means of an external supply. The energy content represents the state of the average for all the mass in the boxes.
The temperature measurements that are used to determine the numerical values of the parameters in the model equations, on the other hand, are measurements of the effects of the redistribution of the energy content of the physical system. Redistribution both within a subsystem and between subsystems.
Those measurements are not sufficient to determine that the energy content of either a subsystem or of the total system has either increased or decreased.
Attempts to remove the redistribution effects from the actual energy-content changes have not yet accounted for the possibility that the redistribution effects alone can exhibit responses and behaviors that would be the same as those for an actual change in the energy content.
The physical phenomena and processes responsible for the energy redistribution are not, and cannot, be captured into the framework of simple box models. These important phenomena and processes are the energy transported with large-scale motions of a part of the matter that makes up a subsystem. The bulk-average energy content is not a part of this transport and the subsystem-to-subsystem energy exchange does not occur at the bulk-average temperature of a subsystem. Instead, the energy exchange occurs at the interfaces between subsystems and the driving potential for heat transfer ( energy exchange ) at these spatial locations is generally much different from the potential represented by bulk-average temperatures.
That the resulting equations from the box approach can be made to kind-of ‘look like’ the measured values is only a consequence of curve-fitting. Especially, curve-fitting a small segment ( short time series ) of the physical world for which almost any old curve fit can be made to ‘work’. These exercises are nothing more than fitting curves to the observed response of the material; in contrast to mechanistic modeling of the material, its physical phenomena and processes, and its interactions with other materials. In other words, these are empirical correlations of observed response and additionally the observed response does not necessarily correspond to the intent of the basic model equations.
Extrapolation of such empirical curve-fitting results is always strongly discouraged. In the present case, consider the results the curve fits give for very long-term time scales. They give states of the subsystems and the total system that have never occurred, and will never occur, in the physical system.
The box-model approach might possibly be more successful if distributions of the energy content within the subsystems is taken into account. And the interactions between subsystems is based on the state of the systems at the interfaces between the subsystems and not the bulk-average states. Additionally, accounting for the important physical phenomena and processes of redistribution of the energy content by motions of parts of the subsystems is also necessary. And, finally, accounting for the changes in the external energy addition and its interaction with the subsystems due to changes in the subsystems is necessary.
It is also important to note that the numerical values for the forcings are, in many cases, extremely fuzzy. And that the more important forcings are generally the most fuzzy. A more nearly complete analysis would propagate the effects of the uncertainties of these fuzzy numbers through the analysis and determine the range of the response function(s) that the uncertainties introduce. These ranges will be huge.
Box-model results for the sensitivity of the Earth to changes in the composition of the atmosphere will not offer improvement over the well-established, for over four decades now, range.
I’ve figured out the problem that I perceived last night. It’s somewhat in line with Dan’s comment above, and means I have to do some of the arithmetic of absolute heat contents that Lucia and Arthur have been doing, and I’ve been trying to avoid.
I’ve introduced a multiplier b=w_1 λ_1 + w_2 λ_2, where w_i are the fitted coefs and λ_i are the inverse assumed time constants. But b is also 1/C_s in the model where x=1 – ie all forcing is applied to box 1. So the fitting actually determines C_s, which is the heat content of box 1. It therefore fixes how box 1 can be interpreted.
To get a SI value, the time constants have to be expressed in seconds. Then from the coefs 0.0798 and 0.6283 with times of 1 and 30 years, C_s=3.177 MJ/m2/K. Arthur quotes the atmosphere as 13.4 MJ/m2/K. So the fitted figure is a bit small, but in the ballpark for the atmosphere.
Now as I said above, there is a criterion for physicality that xR < 1 + w_1 λ_1/(w_2 λ_2), R=C_o/C_s. To meet it, I had been drastically reducing the forcing partition factor x, as Lucia and Arthur did. But that has the effect of proportionally reducing the inferred C_s, which was already on the small side. If you fix a ratio R, then C_o is also reduced.
So that is the solution to the apparent failure of conservation of energy when x is small, and T_o shows variations which the forcing flux could not produce on the full ocean. In this case, C_o does not represent the full ocean, but only a tiny fraction of it, and the inferred T_o is reasonable. So is the outcome – a fully stirred ocean was never realistic.
However, I think now that it is unwise to meet the physicality criterion by reducing x. It is better to reduce R, keeping C_s realistic, but recognising that C_o represents only a top layer of ocean. For example, Schwartz, in the paper I cited above, had C_o=440 MJ/m2/K, which corresponds to about the top 120m (down to thermocline), and a R of about 30. Even then, the forcing fraction x has to be reduced to about 0.16.
So, bottom line, I don’t need to change my previous posts or spreadsheet. Users need to be aware that a choice of x and R, which are input data, implies values for C_o and C_s. For these implied values, 1LOT is satisfied, and the inferred T_o is indeed realistic. But C_o represents only a top layer of the ocean. It can’t really be otherwise – the time scale for heating the whole ocean is millenial, and we can’t fit that with a century of observations.
Hi Lucia – could you kindly inline the figure, first URL, in my comment #19421 above? Thanks.
Dan –
I don’t think anybody’s suggesting they improve on the established sensitivity range! The original point of doing it was they provide some justification for the established range using purely historical data and simple analysis, rather than the complex modeling exercise that some people find difficult to trust. I do agree that expecting the boxes to correlate closely with any particular physical subsystems for our planet is not going to be fruitful; on the other hand there are likely to be approximate eigenvector-like breakdowns of Earth’s many components that this sort of approach is capturing in some fashion with “lumped” parameters. And I agree getting error bars would be good too.
I made a calculation error above #19436 – the fitted coefficients imply a C_s of 313.7 MJ/m2/K, not 3.177. So it’s actually much larger than the atmosphere (about 25x) and suggests a top box which includes a lot of ocean. It also means that it’s now quite realistic to choose a reasonably small partition fraction x. For example, if I choose R=30, x=0.1, then C_s is about 2.34x atmosphere, and C_o corresponds to the top 178 m of ocean (and the results are “physical”).
I’ll amend the spreadsheet to show these calculations.
Arthur Smith (Comment#19438) September 6th, 2009 at 4:42 pm
I think the “likelihood” of useful eigenvalue decompositions of Earth’s climate is something which needs to be proved, not asserted!
Barring that, what sort of useful error bounds could one hope to establish? Limits on the regression-derived time constants?
What’s been neglected here so far is that water has a (large) heat of vaporization and a (highly) temperature dependent vapor pressure. There is no way you can have an air/water interface without latent heat transfer back and forth and still claim you are simulating an earthlike planet. A first approximation would be to assume a separate water vapor only exponential atmosphere with the surface pressure equal to the vapor pressure of water at the ocean temperature, assuming a well mixed ocean. Then the mass of water in the atmosphere per unit area would be equal to the surface pressure divided by the gravitational acceleration. That won’t change the heat capacity of the atmosphere box considered by itself because at normal temperatures the water vapor content is small and the heat capacity of water vapor isn’t all that different from dry air. But it does mean that you can’t heat the ocean without putting some of that heat into the atmosphere in the form of increased water vapor mass as well as sensible heat to keep the relative humidity less than or equal to 100%. And you probably can’t heat the atmosphere without evaporating more water either. Constant relative humidity is one possible assumption. Then the effective heat capacity of the atmosphere becomes a strong function of temperature and can easily be a factor of two to three larger than dry air at temperatures on the order of 298 K. The reason for an exponential atmosphere is that a one box constant temperature atmosphere would contain the same amount of water vapor per cubic meter everywhere and require a far larger heat input to change the temperature and stay in equilibrium. I think. Maybe.
Yet another 2 box model:
I made a simple 2 box model in excel that just integrates the heat balance with one month timesteps. It seems to fit the GISS data pretty well with fairly realistic parameters.
What is the best way to make this model available? The ClimateAudit site where Nick put his model does not seem to be taking new accounts.
DeWitt Payne (Comment#19465) September 7th, 2009 at 9:41 pm
The diffusive model for heat transfer doesn’t distinguish causes. All it says is that heat flow is downgradient (from the hotter box to the cooler box) with a rate proportional to the temperature differential.
(Comment#19472)
“What is the best way to make this model available? The ClimateAudit site where Nick put his model does not seem to be taking new accounts.”
The climate Audit forums will take new accounts but you have to contact John A. See Post:
http://www.climateaudit.org/phpBB3/viewtopic.php?f=9&t=5
DeWitt Payne (Comment#19465) September 7th, 2009 at 9:41 pm
Oliver is right that the model does not distinguish causes. And it’s true that latent heat is an important contributor to heat transport. But its contribution to heat storage is negligible. There’s about 40mm liquid equivalent of water vapor in the air. LH is about 540x SH – ie if all the air’s WV condensed into the ocean, it could only heat the top 216 m about 0.1 degree C.
Thanks, John. I will do that.
In the mean time, this is the graph I get:
http://img5.imageshack.us/img5/4442/2boxk.png
Nick Stokes (Comment#19497) September 8th, 2009 at 4:16 am,
The heat content of precipitable water vapor in the air would have little effect on the ocean. But we already know that the heat capacity of the ocean is three orders of magnitude larger than the atmosphere so that’s not news. But latent heat transfer will affect the atmospheric temperature compared to no latent heat transfer. The atmospheric temperature will be lower than it would otherwise be in the absence of water vapor for a given heat content. And what we’re fitting is temperature data not heat content data. And it’s temperature data from the surface where humidity will have the greatest effect. Moist adiabats curve downward as they approach the surface so the average temperature of the whole atmosphere is indeed lower than it would be in the absence of water vapor.
This is what the post says, which you have to be registered to read:
DeWitt Payne:
So it looks to me like we need to be fitting to virtual temperature in the atmosphere. As I recall this doesn’t shift things by too much near the surface (maybe 1°C).
So I’d say quantitatively it’s not that important for this particular application.
Carrick (Comment#19503) September 8th, 2009 at 8:04 am ,
Actually, I was wrong again. The curvature of the moist adiabat means the average temperature of the atmosphere as a whole is significantly higher than otherwise, i.e. declining with altitude at less than the 10 K/km of the dry adiabat for exactly the same surface temperature. Which, now that I think about it, makes much more sense. It still means that more heat is required to raise the temperature of the atmosphere by one degree than for a dry atmosphere not in contact with water. The slope of the saturated moist adiabat decreases with temperature as well.
This is, IMO, also the principal mechanism of polar amplification. An increase in forcing in the tropics increases heat transfer to the poles because the heat content of the air in the Hadley and Ferrel cells increases faster than temperature. So temperature in the tropics increases less rapidly and polar temperatures increase more rapidly than if there were no change in meridional heat transfer.
Nick Stokes (Comment#19497) September 8th, 2009 at 4:16 am ,
To put things further into perspective, if you condensed that 40mm of precipitable water/m2 and put the heat (1E8 J/m2) in the atmosphere rather than the ocean, it would raise the temperature of the entire atmosphere by 6.4 degrees, assuming that the heat capacity of the atmosphere is equivalent to 3.2 m of water/m2.
I suggest interested parties peruse Roger Pielke, Sr.’s web site and see what he thinks about the use of atmospheric temperature alone as a proxy for planetary heat content.
DeWitt–
The equations we are using are anomaly equations. So the only thing that “matters” to the two box model in the incremental increase in heat content and the constant of proportionality between the heat content and the temperature.
The equations are already linearized about the baseline, so if the issue of the additional water vapor evaporated as a function of both the air and water temperature is important, that could be incorporated into the assessement of whether or not the parameters corresponding to Tamino’s solution are “realistic”.
My current sense is that that issue “won’t matter” to the evaluation of whether or not the two-box model maps into physically realistic space. The reason I think this is that I think we will get the same result whether or not we account for this effect. In both cases, the two-box model is going to have a difficult time mapping into physically realistic space.
Carrick (Comment#19503) virtual temperature should be relevant to the energy content of the air but not relevant to the heat exchange.
Evaporation is certainly relevant to the model. It is a method of the ocean giving off heat without significantly effecting surface temperatures. In terms of the two box model this should mean a smaller ocean time constant \Tau_s.
lucia,
I agree that even with water vapor, a two box model will almost certainly fail to provide a physically realistic solution. The problem with linearization, though, is that the effect of water vapor is highly non-linear so it depends strongly on the value of the absolute temperature. Which makes me wonder yet again how anyone can think that GCM’s are anything like valid when the global average absolute temperature can vary by degrees from run to run or maybe it’s model to model.
DeWitt Payne (Comment#19518) September 8th, 2009 at 12:35 pm
Linearizing around the current state removes the absolute temperature dependency (assuming we will not deviate more than a couple degrees in any case).
DeWitt–
Equations are linearized when effects are nonlinear. Then, people claim they should be valid over a small range of change in temperature anomaly. A 5K change relative to the base of 273K might, plausibliy, be a small change relative to the baseline.
If the effect is excessively no-linearly, the simplified model will break down. In that case, what we should find is that the Tamino’s fit would not map into the simplified model.
In other words: We would find the data show the two-box model idea might make a nice math problem but doesn’t really correspond to realistic physics.
lucia,
If the base temperature is 273 K, then water vapor can be ignored because it’s very low and doesn’t change very fast. At that temperature, though, (sarcasm) you might have to worry about the heat of fusion (/sarcasm). That won’t be true at 300K, corresponding to the approximate surface temperature in the tropics or something like 50% of the surface of the planet, where even a range of +/-1 K is going to show significant deviation from linearity if you include water vapor. 288 K, the surface temperature for the 1976 standard atmosphere, would be a more appropriate choice of a base temperature, at least for the surface. I haven’t done any calculations at that temperature to see what the effective heat capacity might be if you included water vapor.
Sorry Dewitt–
Yes. I typed freezing. But 5 K is still a small change relative to 300K.
When linearizing, you linearize about whatever you think is the base case. That’s the way it works.
We don’t care about the effetctive heat capacity. We care about the rate of change “d(H)/dT = mH “near the baseline temperature where H is the heat content.
dH/dt = mH T. We neglect the higher order terms in T as small.
Of course this only works if the linear term is large compared to the higher order terms. We don’t need to worry about these details unless Tamino’s model shows even the slightest risk of mapping into anything anywhere near realistic.
It doesn’t.
lucia,
dH/dT is what I meant by effective heat capacity.
Can you take the Clausius-Clapeyron equation for water to water vapor and rearrange it to get dH/dT? I’m way too rusty to even begin to try. Probably somebody has already done it, but it doesn’t look like something easy to google.
I’m just curious. Like you I think there is no chance of a physically realistic two box solution. Tom Vonk may well be correct that the climate sensitivity isn’t a constant either.
DeWitt–
It could be done. But right now, it’s not necessary because the two-box model looks like it would be wrong period.
I try to avoid algebra that is not necessary to the particular claim being tested. And right now, even though you have a point that water could affect the effective heat capacity, it doesn’t look like it would matter.
Out of curiosity, does anyone know if climate modelers predict the upper ocean will warm more than the atmosphere at equilibrium?