The following proved a little long to be just an update to the previous post; I guess one should never say never. Nevertheless I don't anticipate a need for anything more on this model.
First, let's go back to why I've been looking at this at all. Modeling climate is typically done using sophisticated and detailed models that actually simulate the behavior of the ocean and atmosphere as best we can using the appropriate physical parameters and equations, for example the NCAR Community System Model, which is freely available for download with extensive documentation. The output of such models can tell us a lot about natural variations in Earth's systems, the response to changes in radiation input from the Sun, to changes in greenhouse gas concentrations, and to other factors that alter the energy balance of the planet. Spencer Weart's website The Discovery of Global Warming is an excellent review of the lines of reasoning that went into developing the sophisticated models used today.
The most basic response of Earth's climate to changes in "forcings", (factors that modify the energy balance) is a change in overall temperature. Additional heating should cause Earth's temperature to rise until output energy matches the increased input; similarly a negative forcing should cause temperatures to drop. This response should be roughly linear in the forcing, but it may take some time for the balance to be restored. This temperature response to forcing is known as the "sensitivity" of the climate system. A larger sensitivity means the temperature increase will be larger for a given change (for example in CO2 concentrations) while a smaller sensitivity means a smaller increase. Estimates can be made of this sensitivity based on past climate or through the extensive simulation processes of the climate models. The 2007 IPCC report summarized the evidence for sensitivity to get a long-term response of 2.0 to 4.5 degrees C with a best estimate of 3 C for doubling CO2, and a transient response of 1.0 to 3.0 C for doubling under a 1% per year CO2 increase (see Chapter 10, p. 749 of the WG1 IPCC report for the discussion of transient climate response).
The difference between the IPCC"s transient and equilibrium response values is directly related to the speed with which the climate system can respond to changes in forcings. The atmosphere responds quickly, while the land more slowly and the ocean even more slowly, and full response may be arbitrarily long thanks to the diffusive nature of heat transport into the bulk below surface. Looking just at the instantaneous relationship between forcings and temperature in the historical record will give you only the very fastest components of the response. To estimate the full response from historical records requires some accounting for the slow components, and there was some interest a couple of years ago in a simple model with a single time-constant to account for that; Lucia Liljegren has also illustrated such a "lumped parameter" model.
But the Earth is clearly not a single-time-constant planet. Tamino's original two-box model post gave a general picture of how to model the planet with two components with different time scales, as a generalization of the single-time-constant approach. A better generalization might be to look at diffusive responses - Alexander Harvey in Comment #18475 here suggested making the ocean box diffusive, and that approach does seem worth exploring. Another relatively simple possibility might be to approximate the system by three boxes where the short time-constant is 0 (instantaneous response), the long-time constant is infinite (deep ocean or ice-sheet heat sink that doesn't change temperature at all) and there's really only one middle-scale time constant to discuss.
First, though, let's finish the exploration of the two-box model. The defining parameters of the model are the two time constants (the inverses of αs and αo in the notation used to this point), the heat capacities of the two boxes (Cs and Co), and the heat transfer rate between them (proportional to their temperature difference with a coefficient β). If all those parameters are given, the magnitude and time-dependence of the response to given forcings is determined. In particular, from the basic equations of the system:
Cs dTs/dt = Fs(t) - αs Cs Ts + β (To - Ts)
Co dTo/dt = Fo(t) - αo Co To + β (Ts - To)
the long-term response can be found by setting Fs(t) = x F0, Fo(t) = (1-x) F0, i.e. to time-independent values. After some transient, the temperatures will stop changing and we can set the left-hand sides to zero and solve for Ts and To:
Eq. 1: Ts = (αoCox + β) F0/(αoCoαsCs + β(αoCo + αsCs))
Eq. 2: To = (αsCs(1 - x) + β) F0/(αoCoαsCs + β(αoCo + αsCs))
Note first several limits here: if F0 is zero (no forcing), the temperatures become zero - the temperatures in this model are measures of change in temperature, not absolute temperatures, and their baseline is zero. If β is zero, the two boxes become uncoupled, and the response ratios are simply the inverse of the product of heat capacity C and inverse time constant α.
Even with positive β values, this long-term response will generally have different temperatures for Ts and To - this system represents a steady-state system with energy flowing through it, not a thermodynamic equilibrium where all temperatures must become the same in the end. However, the larger β is, the more the two temperatures are forced to be the same. For very large values of β relative to αsCs and αoCo, both Ts and To reduce to a single temperature with a response ratio equal to 1/(αsCs + αoCo).
The value of αs is the inverse time constant of the "fast" box in isolation, and similarly for the "slow" box, but when the two are coupled together the effective time constants of the system shift - Lucia went through the algebra here (see her Eq. 4) and Tamino agreed. In the limit of very large β the two inverse time constants 1/τ+ and 1/τ- become
Eq. 3 1/τ+ ≈ (αsCs + αoCo)/(Cs + Co)
Eq. 4 1/τ- ≈ β(1/Cs + 1/Co)
so τ- becomes very short, and the model reduces essentially to a one-box model with time-constant τ+ and response as noted above.
So a meaningful two-box model needs a value of β that is not too large - in the notation of the previous post in this series, the value for γs must be smaller than or roughly of the same order as αs and αo, or the two-box assumption fundamentally breaks down. The existence of large regions of Earth's surface and near-surface bulk with considerably different temperatures from one another suggests that in the physical Earth system there should be many partitionings that allow for two "boxes" of differing temperature histories with a heat transfer rate that does not overwhelm the response.
Now the arguments in my recent series of posts culminating in the most recent discussion of possible parameter values have come from a different angle. Rather than assuming we have a given two-box system, start from the point of a temperature fit to two given time-constants τ+ and τ- and find the parameters for all possible two-box systems that could give such a result. The temperature fit introduces another parameter 'y' that describes the relationship between the measured temperatures and the temperatures of the two boxes - Tm = y Ts + (1-y)To (plus an arbitrary baseline determined by the baseline of measured temperatures and forcings). The resulting space of solutions has three free parameters which in the most recent post I took as the two heat capacities (Cs and Co) and the normalized heat transfer rate γs.
CORRECTION: the following text and figures through to the end of this post have been modified from the original to reflect the corrections in previous posts due to an original error in Eq. 26
The previous post on a perturbative approach examined the constraints when the γs parameter is small - in particular there are no two-box solutions with small γs if the fast-box heat capacity Cs is within a range determined by the fit parameters and the heat capacity ratio λ (Eq. 36A and 36B). Now let's look more directly at the constraints associated with larger values of γs.
One important constraint is that the two-box slow and fast time constants (and their inverses αs and αo) must be positive. From the notation of Eq. 44 of the previous post and surrounding results, that means for the fast box time constant:
Eq. 5: φs = (ξ+ + ξ- - 2 + sqrt((ξ- - ξ+)^2 - 4λ))/2 > 0
(recall we chose the (+) sign in Eq. 44; the constraint would be worse for the (-) sign).
If ξ+ + ξ- > 2 then we don't have a problem. However, from their definitions, as γs increases the ξ's get smaller, and it is that limit that gives us a real constraint. So in the case that ξ+ + ξ- < 2, we have:
Eq. 6: (2 - ξ+ - ξ-)^2 < ((ξ- - ξ+)^2 - 4λ)
or, multiplying through, canceling terms, dividing by 4 and rearranging:
ξ+ + ξ- - ξ+ξ- - 1 - λ > 0
and returning to the definitions of the ξ's this gives:
Eq. 7: -(1+λ)γs^2 + γs(1/τ+ + 1/τ-) - 1/(τ+τ-) > 0
Note that the only time this is a concern is if ξ+ + ξ- < 2, i.e. γs > (1/τ+ + 1/τ-)/2. The equality version of Eq. 7 is a quadratic in γs, and only the larger solution of that quadratic is larger than that limit. So we find the constraint on γs to prevent αs from going negative in this solution space is:
Eq. 8: γs < (1/τ+ + 1/τ- + sqrt((1/τ+ - 1/τ-)^2 - 4λ/τ+τ-))/(2(1 + λ))
in terms of the fitted time constants and heat capacity ratio λ. If τ- is very small the limit on γs becomes 1/(2(1 + λ) τ-), i.e. this limit is less constraining the shorter the short fitted time constant is. So large heat transfer rates only work if the short time constant is very short.
There is a similar constraint for αo, i.e. φo obtained from Eq. 42 in the previous post. The denominator in Eq. 42 can be shown to always be positive (φs > ξ+ - 1) so the concern is with the numerator term. Again this can shown to not be negative unless ξ+ < λ (γs > 1/(λτ+)) and rearranging the terms resolves to the condition:
Eq. 9: λ^2 + λ - λ(ξ+ + ξ-) + ξ+ξ- > 0
The equality turns into a quadratic in γs, and αo becomes negative for values of γs between the two solutions, if there are two. The two end-points of this disallowed interval for γs are given by the two solutions to:
Eq. 10: γsl,r = (λ (1/τ- + 1/τ+) ± sqrt(λ^2 (1/τ- - 1/τ+)^2 - 4 λ/(τ+τ-)))/2(λ^2 + λ)
If the term under the square root is negative, then there are no such solutions and αo is always positive - this holds when λ is small enough relative to the fitted time constants:
Eq. 11: λ < 4 τ+τ-/(τ+ - τ-)^2 ≈ 4 τ-/τ+
If λ is large (the heat capacities of the two boxes are close) while the time constant ratio is small the constraint of Eq. 11 would be violated and there would be an interval of γs values with negative αo.
Note that this implies that, while αs decreases monotonically as γs increases, αo also first decreases, but then turns around and becomes larger for larger γs. That means there is a point where αs and αo cross - of course this will be a singular state for the two-box model, with both of the original time constants equal to one another. For γ values larger than this crossover point the "slow box" actually has a shorter bare time constant than the "fast" one under the given fitted conditions; this gives a perhaps valid but unusual solution to the problem.
The above constraints (Eq. 8 and the excluded bounds given by Eq. 10 here) limit the allowed values of γs for physical solutions, but quite large heat transfer coefficients are allowed if the fast time constant is short enough. The previous constraints on heat capacity values (Eq. 36A, B) were determined in the perturbative limit when γs is small, and the actual constraint provided by the requirement of a positive square-root term in the quadratic for Eq. 26 is slightly less limiting for larger values of γs, but γs cannot be too large due to the above limits in order to keep αs and αo positive.
Eq. 12A: x + (1-x) r+ = b2(1 + r+^2/λ)/(y + (1 - y)r+) + b3(1 + r-r+/λ)/(y + (1 - y) r-)
Eq. 12B: x + (1-x) r- = b2(1 + r+r-/λ))/(y + (1 - y)r+) + b3(1 + r-^2/λ)/(y + (1 - y) r-)
The earlier relationships for r+ and r- turn out to force the product r+r- = -λ which zeros out the b3 term in Eq. 12A and the b2 term in Eq. 12B, and leaves us with a symmetrical relationship between x and y:
Eq. 13: (x + (1-x) r+)(y + (1-y)r+) = b2(1 + r+^2/λ)
and similarly with r- and b3. What that means is that whenever we have one solution (x = x0, y = y0), then the reverse (x = y0, y = x0) is also a solution here. Since that gives two different values of y satisfying the original Eq. 26, those must correspond to the (+) and (-) solutions for y - i.e. we obtain only 1 (x,y) pair from the (+, -) solutions of Eq. 26 (however the consequent values of w+s and w-s and the resulting individual box temperature curves are themselves are distinct for the (+) and (-) cases).
Let's turn from the algebra to some graphical representations of these solutions. First let's look at the "Tamino" conditions from the previous post, with 1-year and 30-year time constants and the resulting fit parameters, the heat capacities I originally chose (Cs = heat capacity of air, Co = 1% of heat capacity of ocean), but allowing γs to vary:
NOTE: figures and text below were inconsistent - revisions "under construction" I guess I should have posted - for a few hours on 9 Sep 2009 - all should be consistent as of this afternoon (3:50 pm)
Figure 1: Tamino case with original heat capacity choice
First note the heat capacity ratio λ has been drawn in yellow, as a reference straight line (it appears to constrain the value of 'x', for instance, as Nick Stokes has pointed out in some of his analysis on the ClimateAudit BB). Next note the αs and αo curves in violet and orange (these have been multiplied by τ- for these graphs). Both decline for small γs. αo stays barely positive and then turns up, and the two curves cross at a singular point with γs around 2.7e-08 s^-1, shortly after which αs becomes negative and we're out of the realm of physically allowed γs values.
The black and blue curves represent the (+) solution for y, the fraction of measured temperature in the fast box, and the corresponding values for x (fraction of forcing in the fast box), or correspondingly the (-) solution for x and y, respectively. Only a small interval of γs values near 0 is allowed - up to a little under 3x10^-9 s^-1. So the limit on allowed values for x and y is more stringent than the αs positivity constraint in this case. This is the region from which I picked the "Case 1" (and reversed "Case 2") solutions in the previous post.
Figure 2: Tamino case with slow-box heat capacity at 3% of ocean
Increasing the slow-box heat capacity results in a greatly extended valid region allowing γs up to almost 10^-8 s^-1.
Figure 3: Tamino case with slow-box heat capacity at 3% of ocean and fast-box 5x atmosphere
Here we have a case where the Eq. 11 constraint is violated, and there is a broad region of γs values disallowed because they result in a negative αo. There are still valid ranges of solutions for x and y for small γs, but here the αo constraint limits the maximum value of y (or x) to about 0.9, instead of 1.0. Note the absolute heat transfer rate β is higher here because of the larger value for Cs.
Figure 4: Tamino case with slow-box heat capacity at 5% of ocean
The region of allowed γs values expands further - up to about 1.4x10^-8 s^-1.
Figure 5: Tamino case with slow-box heat capacity at 10% of ocean
An even larger Co value - here the heat capacity ratio is large enough that Cs violates the constraints of Eq. 36A and 36 B, so no small values for γs are allowed, but γs values in the range 1.0 to 2.5x10-8 s^-1 have valid x,y solutions.
Figure 6: A case I looked at in my original post on this, with slow-box heat capacity at 5% of ocean.
Again small values of γs are disallowed due to violation of Eq. 36A-B; the allowed γs range here is about 0.8 to 1.3x10^-7 s^-1. Because of the shorter τ- here (0.12 years), the allowed range of γs goes to much higher values than in the "Tamino" cases with a 1-year short time constant, so these are larger absolute heat transfer rates than in the previous examples.
Figure 7: The same case as Fig 6, but with a Co only 3% of the full ocean.
Allowed γs values in this case are from 0 to about 8x10^-8 s^-1.
Figure 8: Another example looked at in the early post, with slow heat capacity 3% of the full ocean.
In this case the short time constant is considerably longer, at 2 years - this was the optimal fitted case to 20th century temperatures (though there was little dependence of the fit on the short time constant). Due to the longer τ- here, the range of allowed γs values is lower than in the other cases looked at, leading to a lower maximum heat transfer rate needed for this solution.
The main point of Lucia's recent post was to show the temperature curves for the slow and fast boxes - she picked my "case 1" which turns out to have a very hot slow box, probably not realistic. "case 2" (as she posted on here, later, leading to all these revisions!) has a hot fast box, again not very realistic:
Figure 9: Case 1 "Tamino" temperature fit with low Co and very low heat transfer (+ solution)
Figure 10: Case 2 "Tamino" temperature fit with low Co and very low heat transfer (- solution)
After revision I added new Case 3 and 4 examples, which look a lot better with both temperatures keeping pace relatively nicely:
Figure 11: Case 3 "Tamino" temperature fit with higher Co and heat transfer (+ solution)
Figure 12: Case 4 "Tamino" temperature fit with higher Co and heat transfer (- solution)
The same can be done for the many other possible solutions - the following two are for other fits from the earlier post:
Figure 13 Temperature curves for 0.12-year short time constant, 20-year long time constant, (+ solution)
Figure 14 Temperature curves for 0.12-year short time constant, 20-year long time constant, (- solution)
Figure 15 Temperature curves for 2-year short time constant, 20-year long time constant, (+ solution)
Figure 16 Temperature curves for 2-year short time constant, 20-year long time constant, (- solution)
Some of these look very reasonable (Tamino case 3 and 4, and Figure 13 and 14) and some are almost certainly not realistic for a partitioning of Earth's climate system. One thing to notice here is the contrast between the (+) and (-) solutions for y - for (+) y, the measured temperatures should hue more closely to the fast box (y closer to 1), which leaves the slow box more unconstrained; the reverse is true for the (-) solution. This is consistent in the unrealistic-looking temperature graphs (Tamino case 1 and 2, and Figure 15-16) with the (+) solutions having excessively sensitive "slow" boxes, and the (-) solutions with excessively sensitive "fast" boxes.
This suggests that a real physical constraint is associated with requiring the long-term (or steady-state) sensitivities for the "fast" and "slow" boxes to be comparable. I don't think they can be constrained to be exactly equal - there's no particular physical reason one part of Earth has to change temperature in lock-step with another - but they shouldn't be too far apart. More on this constraint in what looks like will need to be yet another follow-up post, later!
UPDATE 9 Sep 2009: Since the original Eq. 26 for y was faulty, as was the constraint of the old Eq. 36 on Co, all the figures and some of the discussion here has been updated to reflect the corrected relationships. Also an extra slash in Eq. 3 was removed (this had no effect on anything else).