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 CO_{2}, and a transient response of 1.0 to 3.0 C for doubling under a 1% per year CO_{2} 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 α^{s} and α^{o} in the notation used to this point), the heat capacities of the two boxes (C^{s} and C^{o}), 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:

C^{s} dT^{s}/dt = F^{s}(t) - α^{s} C^{s} T^{s} + β (T^{o} - T^{s})

C^{o} dT^{o}/dt = F^{o}(t) - α^{o} C^{o} T^{o} + β (T^{s} - T^{o})

the long-term response can be found by setting F^{s}(t) = x F_{0}, F^{o}(t) = (1-x) F_{0}, 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 T^{s} and T^{o}:

Eq. 1: T^{s} = (α^{o}C^{o}x + β) F_{0}/(α^{o}C^{o}α^{s}C^{s} + β(α^{o}C^{o} + α^{s}C^{s}))

Eq. 2: T^{o} = (α^{s}C^{s}(1 - x) + β) F_{0}/(α^{o}C^{o}α^{s}C^{s} + β(α^{o}C^{o} + α^{s}C^{s}))

Note first several limits here: if F_{0} 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 T^{s} and T^{o} - 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 α^{s}C^{s} and α^{o}C^{o}, both T^{s} and T^{o} reduce to a single temperature with a response ratio equal to 1/(α^{s}C^{s} + α^{o}C^{o}).

The value of α^{s} 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_{+} ≈ (α^{s}C^{s} + α^{o}C^{o})/(C^{s} + C^{o})

Eq. 4 1/τ_{-} ≈ β(1/C^{s} + 1/C^{o})

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 γ^{s} must be smaller than or roughly of the same order as α^{s} and α^{o}, 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 - T_{m} = y T^{s} + (1-y)T^{o} (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 (C^{s} and C^{o}) and the normalized heat transfer rate γ^{s}.

**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 γ^{s} parameter is small - in particular there are no two-box solutions with small γ^{s} if the fast-box heat capacity C^{s} 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 γ^{s}.

One important constraint is that the two-box slow and fast time constants (and their inverses α^{s} and α^{o}) 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: φ^{s} = (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 γ^{s} 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)γ^{s}^2 + γ^{s}(1/τ_{+} + 1/τ_{-}) - 1/(τ_{+}τ_{-}) > 0

Note that the only time this is a concern is if ξ_{+} + ξ_{-} < 2, i.e. γ^{s} > (1/τ_{+} + 1/τ_{-})/2. The equality version of Eq. 7 is a quadratic in γ^{s}, and only the larger solution of that quadratic is larger than that limit. So we find the constraint on γ^{s} to prevent α^{s} from going negative in this solution space is:

Eq. 8: γ^{s} < (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 γ^{s} 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 α^{o}, i.e. φ^{o} obtained from Eq. 42 in the previous post. The denominator in Eq. 42 can be shown to always be positive (φ^{s} > ξ_{+} - 1) so the concern is with the numerator term. Again this can shown to not be negative unless ξ_{+} < λ (γ^{s} > 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 γ^{s}, and α^{o} becomes negative for values of γ^{s} between the two solutions, if there are two. The two end-points of this disallowed interval for γ^{s} are given by the two solutions to:

Eq. 10: γ^{s}_{l,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 α^{o} 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 γ^{s} values with negative α^{o}.

Note that this implies that, while α^{s} decreases monotonically as γ^{s} increases, α^{o} also first decreases, but then turns around and becomes larger for larger γ^{s}. That means there is a point where α^{s} and α^{o} 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 γ^{s} 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 γ^{s} 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 γ^{s}, but γ^{s} cannot be too large due to the above limits in order to keep α^{s} and α^{o} positive.

There is one more relationship to look at before taking a graphical look. Combining Eq. 11 and 14 from the original post and Eq. 24 and 25 from its continuation, we find:

Eq. 12A: x + (1-x) r_{+} = b_{2}(1 + r_{+}^2/λ)/(y + (1 - y)r_{+}) + b_{3}(1 + r_{-}r_{+}/λ)/(y + (1 - y) r_{-})

Eq. 12B: x + (1-x) r_{-} = b_{2}(1 + r_{+}r_{-}/λ))/(y + (1 - y)r_{+}) + b_{3}(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 b_{3} term in Eq. 12A and the b_{2} 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_{+}) = b_{2}(1 + r_{+}^2/λ)

and similarly with r_{-} and b_{3}. 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 (C^{s} = heat capacity of air, C^{o} = 1% of heat capacity of ocean), but allowing γ^{s} 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 α^{s} and α^{o} curves in violet and orange (these have been multiplied by τ_{-} for these graphs). Both decline for small γ^{s}. α^{o} stays barely positive and then turns up, and the two curves cross at a singular point with γ^{s} around 2.7e-08 s^-1, shortly after which α^{s} becomes negative and we're out of the realm of physically allowed γ^{s} 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 γ^{s} 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 α^{s} 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 γ^{s} 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 γ^{s} values disallowed because they result in a negative α^{o}. There are still valid ranges of solutions for x and y for small γ^{s}, but here the α^{o} 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 C^{s}.

**Figure 4**: Tamino case with slow-box heat capacity at 5% of ocean

The region of allowed γ^{s} 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 C^{o} value - here the heat capacity ratio is large enough that C^{s} violates the constraints of Eq. 36A and 36 B, so no small values for γ^{s} are allowed, but γ^{s} 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 γ^{s} are disallowed due to violation of Eq. 36A-B; the allowed γ^{s} 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 γ^{s} 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 C^{o} only 3% of the full ocean.

Allowed γ^{s} 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 γ^{s} 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 C^{o}, 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).

## Comments

## "Eq. 3 1/τ+ ≈ (αsCs +

"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."

I think this is true because with large Beta, the ocean will quickly heat or cool the atmosphere but Lucia mentioned overlapping boxes in which case the effective beta is reduced and the effective heat capacity of the top box is increased. This can effectively narrow the temperature difference while still allowing for a top box time constant that isn't too quick.

For more information on overlapping boxes, see my post at the climate audit message boards:

Overlapping Boxes

http://www.climateaudit.org/phpBB3/viewtopic.php?f=4&t=774

With regards to beta, I think the inequalities are important. If any tuning is done maybe treat it as a constrained optimization problem (like linear programming) where the parameters are only tuned within the limits of their uncertainty. I think there would be more uncertainty in the forgings then the heat transfer coefficients, therefore I think more tuning could be done with regards to the forgings then the heat transfer coeficients.