This will probably be my final post on this question - however I may append updates if other issues come up. In particular this post will look first at whether the fitted parameter values for two-time-constant fits to temperature vs forcing data for Earth's climate system have a collection of underlying physical two-box models that satisfy the basic physical constraints on such systems, and then whether the range of physical parameters in these matching models appears to correspond roughly with appropriate associated physical properties of the real Earth climate system.

**CORRECTION: The following text has been modified significantly due to errors in the preceding post that nullify most of the original discussion here**:

The most recent post using a perturbative approach provides some initial guidance on these issues, but we'll return to the full model shortly since it's not really much more complicated to solve. From the perturbative solution, for small values of the thermal coupling (β, i.e. γ's small compared to 1/τ_{+}), the constraints of Eq. 36A and 36B of the previous post limit the allowed values for the fast-box areal heat capacity C^{s} given a heat capacity ratio λ (and the fitted parameters for which we're trying to develop a solution). For the "Tamino" case, if λ is in the range 1 to 100, the excluded range for C^{s} can range from around 10^7 to over 10^10 J/Km^2, which includes the physical range of the atmosphere plus ocean system. That means there is a large subspace of choices in the physically realistic region for C^{s} and C^{o} that are excluded (at least for small values of the γ^{s} heat transfer rate) because the fit cannot lead to a real value for the y parameter. Even for those with real y, we may not get a result within the allowed range 0 to 1. So for any given case we'll need to look at the particular constraints in some detail to find solutions that are allowed by the math of the problem.

The upper bound of Eq. 40 provides a simpler constraint on the fast-time-constant heat capacity C^{s}:

**END OF CORRECTION**

C^{s} < τ_{-}/a_{3}

since y must be at most 1. From the earlier analysis, smaller assumed τ_{-} values led to smaller fitted coefficients a_{3}, so the fit is probably associating a physical heat-capacity-related constraint of this sort on the resulting a_{3} values. For the sample case of Tamino's τ_{-} = 1 year and translating to SI units as above we get

C^{s} < 3.16x10^7 J/Km^2 /a_{3}^{SI}

On the other hand, C^{s} should be at least as large as the heat capacity of the well-mixed atmosphere (the troposphere which contains the bulk of the atmosphere's mass, and heat capacity). Lucia's value of about 1.34x10^7 J/Km^2 seems reasonable, so from both these constraints we should expect a reasonable short-time-constant heat capacity to be on the order of a few times 10^7 J /K m^2, certainly not much less, and not a lot more than that.

Now, returning to the solution space, as noted at the end of the perturbative discussion, the choice of α^{s} and α^{o} as two of the three free parameters was perhaps not so wise. In the following the three free parameters in our two-box system will be the two heat capacities C^{s} and C^{o} (therefore determining their ratio λ = C^{s}/C^{o}) whose physical values we have just considered and somewhat constrained, and the heat transfer coefficient γ^{s} (which then determines γ^{o} = λ γ^{s} - and of course the original β = C^{s} γ^{s}). The perturbative approach works when γ^{s} is small compared to 1/τ_{+} (and γ^{s} must be positive) so the range for γ^{s} to look at for the full solution is from close to zero to on the order of 1/τ_{+} or perhaps higher. Higher values for the heat transfer physically imply that the two boxes are not really so isolated from one another.

The solution process of the earlier post can still be followed, but we need to rearrange Eq. 22 and 23 to solve for α^{o} and α^{s} in terms of γ^{s} and λ now, instead of the other way around. The best starting point is actually the pair of equations (with (+) or (-)) represented by the unnumbered expression just before Eq. 22 in that post. Divide that expression by γ^{s} and simplify to dimensionless variables by defining:

φ^{s} = α^{s}/γ^{s}

φ^{o} = α^{o}/γ^{s}

ξ_{+} = 1/(γ^{s}τ_{+})

ξ_{-} = 1/(γ^{s}τ_{-})

(note that all of these become large in the perturbative limit γ^{s} close to 0) we then get:

Eq. 41: ξ_{+} - φ^{o} + ξ_{+} φ^{s} - φ^{o}φ^{s} - λ φ^{s} - ξ_{+}^{2} + ξ_{+}φ^{o} + λ ξ_{+} = 0

together with the identical equation replacing ξ_{+} with ξ_{-}.

Rearranging Eq. 41 to solve for φ^{o} in terms of the others, we find:

φ^{o}(1 + φ^{s} - ξ_{+}) = ξ_{+} + ξ_{+} φ^{s} - λ φ^{s} - ξ_{+}^{2} + λ ξ_{+}

or

Eq. 42: φ^{o} = (ξ_{+} + ξ_{+} φ^{s} - λ φ^{s} - ξ_{+}^{2} + λ ξ_{+})/(1 + φ^{s} - ξ_{+})

and the same equation should hold for the (-) version, which we can combine on the way to finding an expression for φ^{s}:

Eq. 43: (1 + φ^{s} - ξ_{+})(ξ_{-} + ξ_{-} φ^{s} - λ φ^{s} - ξ_{-}^{2} + λ ξ_{-}) = (1 + φ^{s} - ξ_{-})(ξ_{+} + ξ_{+} φ^{s} - λ φ^{s} - ξ_{+}^{2} + λ ξ_{+})

Bringing the right-hand expression in Eq. 43 to the left, canceling out the terms symmetric in ξ_{+} and ξ_{-}, organizing in powers of φ^{s} and dividing by (ξ_{-} - ξ_{+}) we are left with:

Eq. 44: φ^{s}^2 + (2 - ξ_{-} - ξ_{+})φ^{s} + 1 + λ - ξ_{-} - ξ_{+} + ξ_{+}ξ_{-} = 0

which is a standard quadratic equation with two solutions for φ^{s}:

φ^{s} = (-b ± sqrt(b^2 - 4c))/2

where -b = ξ_{+} + ξ_{-} - 2 and c = 1 + λ - ξ_{-} - ξ_{+} + ξ_{+}ξ_{-} so the condition b^2 - 4c > 0 becomes:

Eq. 45: (ξ_{-} - ξ_{+})^2 > 4 λ

With τ_{-} the short time constant, ξ_{-} > ξ_{+}, so going back to our definitions for the ξ's, this constraint becomes:

Eq. 46: (1/τ_{-} - 1/τ_{+}) > 2 sqrt(λ) γ^{s}

which gives a limit on the magnitude of the heat transfer coefficient γ^{s}. If γ^{s} becomes too large relative to the τ's and λ as indicated here, then the corresponding time-constant becomes complex, and the parameter space is no longer suitable for the original physical interpretation of the two-box model.

Note that if we want α^{s} to be closer to the inverse of the short time constant τ_{-} then we need to take the (+) sign in Eq. 44, which will be assumed going forward.

Eq. 42 then gives φ^{o} and hence α^{o}, and we can continue to find the remaining parameters as in the earlier post.

Let's plug in values appropriate to Tamino's time-constants and see how they do. Set τ_{+} to 30 years, τ_{-} to 1 year. The fitted parameters are a_{2}^{SI} = 0.739 and a_{3}^{SI} = 0.038. Assume the short-time-constant-box heat capacity C^{s} is just that of the atmosphere, 1.34x10^7 J/Km^2. Give the long-time-constant-box C^{o} a heat capacity of 1% of the full ocean, so 1.06x10^8 J/Km^2. For these time-constants and heat capacities the associated constraints listed earlier (associated with Eq. 36A, B and 40) are all satisfied.

This leaves the heat transfer rate γ^{s} as a free parameter. We'll start with γ^{s} = 3.17x10^-11 s^-1 which is roughly a factor of 30 smaller than 1/τ_{+} = 1.06x10^-9 s^-1.

With the input time constants and coefficients set, and the free heat capacities and transfer rate given these physically reasonable values, we find for this "Tamino" set of parameters:

α^{s} = 3.17x10^-8 s^-1 (very close to 1/τ_{-})

α^{o} = 1.05x10^-9 s^-1 (very close to 1/τ_{+})

r_{+} = 967

r_{-} = -1.31x10^-4

(a quick check shows eq. 17 is satisfied by these when the precise values for the α's are used).

Then Eq. 26 allows for two complementary values for y, one in which the measured temperatures are close to those of the fast box, and one where they are close to the slow box:

**CORRECTED:**

Case 1 (+ sign in solution from Eq. 26):

y = 0.917

w_{+}^{s} = 9.08x10^-3 K/(W/m^2)

w_{-}^{s} = 4.15x10^-2 K/(W/m^2)

x = 0.0177

Case 2 (- sign in solution from Eq. 26):

y = 0.0177

w_{+}^{s} = 7.79x10^-4 K/(W/m^2)

w_{-}^{s} = 2.16 K/(W/m^2)

x = 0.917

Neither of these are actually very good solutions because the split of forcings that results (x) is far too asymmetrical - in case 1 the "slow" box heats up significantly while the fast box does little, but most of the weight in measured temperature comes from the fast box, so neither one makes sense as a model of a part of Earth's climate. In case 2 the reverse happens with the same result.

So, let's try a few more examples for the "Tamino" parametrization. Set C^{o} 10 times larger, to 1.06x10^9 J/Km^2 (about 10% of the ocean), and increase γ^{s} to 1.0x10^-8 s^-1 (300 times larger than before, so about 10 times bigger than 1/τ_{+}) with the remaining input parameters the same. That gives us

α^{s} = 2.16x10^-8 s^-1

α^{o} = 9.71x10^-10 s^-1

r_{+} = 3.06

r_{-} = -4.13x10^-3

and two more cases to look at -

Case 3 (+ solution for y):

y = 0.156

w_{+}^{s} = 0.270 K/(W/m^2)

w_{-}^{s} = 0.249 K/(W/m^2)

x = 0.110

Case 4 (- solution for y):

y = 0.110

w_{+}^{s} = 0.261 K/(W/m^2)

w_{-}^{s} = 0.358 K/(W/m^2)

x = 0.156

The x values and response numbers now do look somewhat sensible - perhaps a little too sensitive to the short time constant in the second case, but not far off.

**Note:** the following text is obsolete, and has been replaced by the above discussion.

Case 1 (+ sign in solution from Eq. 26):

y = 0.910

w_{+}^{s} = 8.40x10^-3 K/(W/m^2)

w_{-}^{s} = 4.18x10^-2 K/(W/m^2)

x = 0.0179

Case 2 (- sign in solution from Eq. 26):

y = 0.0912

w_{+}^{s} = 8.42x10^-4 K/(W/m^2)

w_{-}^{s} = 0.417 K/(W/m^2)

x = 0.177

As far as I can tell, both the above sets of parameters constitute a physically reasonable realization of a two-box model for Tamino's original fit. And these are just two of an infinite collection that are both physical in principle and in practice have values (for heat capacity, time constants, etc.) that are in the reasonable range for our planet.

Increasing the value of γ^{s} by a factor of 100, so that it is actually larger than 1/τ_{+}, changes these values enough that the (+) solution for y is larger than 1, and so probably implausible. However, the (-) solution is still viable, and has:

α^{s} = 2.85x10^-8 s^-1

α^{o} = 6.97x10^-10 s^-1

r_{+} = 9.65

r_{-} = -1.31x10^-2

y = 0.106

w_{+}^{s} = 8.46x10^-2 K/(W/m^2)

w_{-}^{s} = 0.403 K/(W/m^2)

x = 0.172

Quintupling C^{s} to 6.7x10^7 J/K m^2 changed almost none of these parameters - the only ones substantially affected were r_{-} and x, which were both made larger by roughly a factor of 5 (so unphysical values of x, larger than 1, would obtain if C^{s} was much larger than this).

Quintupling C^{o} however was sufficient to violate the Eq. 36 constraint, so that the fraction y became a complex number and the remainder of the solution made no sense. So the results here make plenty of sense, but only if you choose reasonable values (that can be justified relative to Earth's climate system) for the heat capacities of the two boxes in question.

In short, there certainly do exist (an infinite number of) two-box models with physical parameters that resemble components of Earth's climate system, which correspond to the fitting parameters found in Tamino's elegant demonstration of simple ways to estimate climate sensitivity from the measured data.

**UPDATE**: I've posted a Google Docs Spreadsheet that does the math on these and other examples. The above-mentioned results are from rows 14 through 17.

**UPDATE 2 - Sep 8, 2009**: It looks like I made a boo-boo somewhere. Lucia pointed out a problem here (her Eq 1N, 2N) which indicate my solutions don't satisfy my own Eq. 11 and 14. I've added checks to the above spreadsheet (columns Y through AF) to compare the solutions to the various equations stated so far, and indeed there are some problems. Well, better to have algebra checked by blog post than to go too far with wrong-headed results!

**UPDATE 3 - Sep 8, 2009**: See corrections to the previous posts - the problem was in the original Eq. 26, which led to the solutions for y; the original post here had extensive discussion on plausible values for C^{o} based on an associated relationship, but that turns out to be wrong - the constraint is instead on the combination of C^{s} and the heat capacity ratio λ, and isn't nearly as simple as I thought. I have simply removed most of the original discussion of this since in the correct context it is unjustified. Thanks to Lucia for suggesting some likely sources of problems.

## Comments

## Hi Arthur: I want to plot

Hi Arthur:

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

τ+ = 30 years, τ- =1 year

αs = 3.17x10^-8 s^-1 = 1.0004E+00 /years

αo = 1.05x10^-9 s^-1 (very close to 1/τ+) = 3.31E-02

(I used 31556926 sec/year)

So,

(1/ τ+ + 1/τ-) - (αs + αo) = -1.28E-03

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

So

(1/ τ+ + 1/τ-) - (αs + αo)= + (γs+ γo)

But you are reporting positive values of γs & γo?

Is this a rounding issue?

## Yes, it is a rounding issue -

Yes, it is a rounding issue - I only posted the values to 3 significant figures. Also, I used the 31557600 seconds in a Julian year, sorry should have been clearer on that, though the small difference doesn't amount to much. The γ values are forced to be positive because they are free parameters whose value I set. Here are more complete numbers for the first case:

α

^{s}= 3.1656396 x10^-8 s^-1 = 0.9989999 year^-1α

^{o}= 1.052268 x 10^-9 s^-1 = 0.0332071 year^-1and the difference you are looking at is +0.001 year^-1

## Oh-- On my comment, I can

Oh-- On my comment, I can read it at my blog. I typed the negative value that also subtracts off the gammas. But I am puzzled.

## Arthur-- Yes. I knew the

Arthur-- Yes. I knew the difference is small compared to the leading order of alpha_o. But it matters. :)

## Arthur, In my latest version,

Arthur,

In my latest version, which implements a split of forcing but not your mixed temperature, I've identified a criterion for physicality. It appears as a (low) upper limit on the forcing split x. There's a spreadsheet that implements it, at the CA site.

## Hi Nick - I'm working on some

Hi Nick - I'm working on some charts, but basically everything seems to agree with that constraint - x is on the order of λ, slightly less or slightly more, in pretty much all cases. I'm also working on some more constraint algebra which may agree with that, though I think we're starting from different free parameters which would make my 'x' constraint more a consequence than the constraint you found.

## Hi-- I want to shift my

Hi--

I want to shift my graphs so I can better compare to sattellite and ocean temperatures. Do you have the magnitude of the constant parameter somewhere?

## Good question. I thought I

Good question. I thought I was keeping good notes on this, but now I have no idea where I got those values of 0.739 and 0.038 for the 1-year + 30-year time-constant fit. When I run Nick's script for that now I get:

a1 = -0.06507 (intercept)

but also:

a2 = 0.61333

a3 = 0.07981

0.739 is more like what you get at 40 years for τ

_{+}. Hmmm. That doesn't affect the α's or r's for given heat capacity and transfer rates, but it does affect the final w's and x,y values. For these fit coefficients, then, the corrected case 3 and 4 numbers are:Case 3:

y = 0.399

w

_{+}^{s}= 0.274w

_{-}^{s}= 0.201x = 0.0894

Case 4:

y = 0.0894

w

_{+}^{s}= 0.213w

_{-}^{s}= 0.932x = 0.399

- case 3 still looks very nice, but the fast-box temperatures are pretty wild in case 4 now (x too large).

## Arthur-- On the notes... heh.

Arthur-- On the notes... heh. Yours must be as scattered over your desk as mine. Thanks.

It might actually be a good thing to work out the algebra on "mistakes". That way we can all say how reveal what we think looks "good" or "bad" with wrong values. Then, when the right values magically appear.... well... let's just say that having been forced to make judgment with the "wrong" values can sometimes avoid confirmation bias.

(Also for some reason, WP is echoing my first few words. I'm really not typing the first few words in twice.)

## The repeat is because it's

The repeat is because it's treating the first few words of your comment as the "title" of the comment. This site is built with drupal, not WordPress - I was trying to learn how it worked, but it's pretty complex and there are some big gaps still in the functionality I've been able to put together so far...

## I agree with you that, one of

I agree with you that, one of the graphs passes the first cut eyeball test. (Whether it would pass after comparing to ocean heat content data, I can't say. But then... so far I don't have that.)

I agree the other one is a big "jumpy"! I think we can all be glad we don't live on that two-box model planet.

My issue right now is that if I try to relate this to the associated "baseline" the baseline planet seems too cold. (I have not explained this at the blog, and I don't have time to do it today.)

Also, when Steve Reynolds posted, something DeWitt said a while back about latent heat suddenly registered. We might need to discuss the heat capacity of the atmosphere. (This is a D'oh moment for me because I thought DeWitt was discussing heat transfer.)