This will probably be painful for anybody who hasn't already been following this (which I'm sure is all but 1 or 2 readers, if that many). So skip this post unless you're really into solving systems of equations...
The following stems from continued discussion of the two-box model of Earth's climate at Lucia's blog. Use of a two-box model to try to match temperatures based on forcings with two time-scales for response implies a solution that specifically consists of a linear combination of exponential smoothings of the forcing, according to those time-scales. Lucia has been insisting that any such solution then implies a specific underlying physical model, where one can look at the parameters of interest and determine whether they make sense. In my view fitting observed temperature does not provide enough information to pick out a particular two-box model, and in fact I'll show below that you end up with a 3-dimensional space of possible two-box models to choose from. Whether that space contains solutions that physically make sense or not may be another issue.
In the following I will try to follow a consistent, symmetrical notation. Superscripted 's' or 'o' in a variable name implies it is associated with one of the two boxes ('surface' or 'ocean', nominally), while subscripted '+' or '-' in a variable name implies an association with one of the two ('+' = slow, '-' = fast) time-constants.
First of all, start by setting values for τ+ and τ- which satisfy eq. 4 from Lucia's previous post:
Eq. 1: -1/τ(+,-) = (1/2) (-b ± sqrt(b^2 - 4c))
b = (αo + αs) + (γo + γs)
c= (αo αs) + (αs γo + αo γs)
(substitute those b and c here, we'll re-use c).
So Eq. 1 represents 2 equations relating the 4 parameters - 2 αs and 2 γs - to our predetermined τs.
F+(t) = ∫-∞t exp[-(t-s)/τ+] F(s)/τ+ ds
and F-(t) similarly (F(s) is total forcing, the same as FT in Lucia’s notation).
Our input data consists of the time-dependent forcing F(s), two time constants τ+ and τ-, along with parameters a1, a2, and a3 from a numerical fit using those time constants and the forcing to observed temperature. The following argument holds as long as F(s), F+(s) and F-(s) can be considered independently varying (so that their respective coefficients in any time-dependent equations must match).
Let T(t) be that fit to the observational temperature:
Eq. 2: T(t) = a1 + a2 F+(t) + a3 F-(t)
Our question here is to define the class of underlying physical two-box models that could be compatible with these fitting coefficients. Denote the two components by (s) and (o) (tentatively "surface" vs "ocean") that respectively have temperatures (Ts and To), heat capacities (Cs and Co) and a heat exchange term (β). Note that the two γ terms mentioned in the two-box solution for τ(+,-) above are determined by β, Cs and Co. Then
Eq. 3: T(t) = c + y Ts(t) + (1-y) To(t)
introducing two new parameters, c and y. Ts(t) and To(t) represent the underlying physical system temperatures that have the same form of solution as T - i.e. they are similarly linear combinations of F+ and F-, but with a zero constant term:
Ts(t) = w+s F+(t) + w-s F-(t)
To(t) = w+o F+(t) + w-o F-(t)
which defines 4 more parameters in the respective weights.
From these definitions and Eq. 2 and 3 we get
Eq. 4: a1 + a2 F+(t) + a3 F-(t) = c + (y w+s + (1-y) w+o) F+(t) + (y w-s + (1-y) w-o) F-(t)
or, equating the constant term and the terms proportional to F+ and F- respectively:
Eq. 5: a1 = c
(1 equation fixing 1 parameter)
Eq. 6: a2 = y w+s + (1-y) w+o
Eq. 7: a3 = y w-s + (1-y) w-o
giving us 2 more equations relating the 5 parameters (y and the 4 w's) to the fitting coefficients.
Now we have the differential equations that Ts and To satisfy, dependent on an assumed constant split (x, 1-x) in forcing between the two boxes. First note the derivative of F(+,-) above is simple:
dF+(t)/dt = (F(t) - F+(t))/τ+
UPDATE: the factor of 1/τ+ was missing previously from the F(t) term - thanks to Lucia for noticing the problem in the comments here. This and subsequent instances have been corrected.
and similarly for F-(t). Then
Eq. 8: dTs/dt = -w+s/τ+ F+(t) - w-s/τ- F-(t) + (w+s/τ+ + w-s/τ-) F(t)
and similarly for To. The original two-box ODE's were:
dTs/dt = -(αs + γs) Ts(t) + γs To(t) + x F(t)/Cs
dTo/dt = γo Ts(t) - (αo + γo) To(t) + (1-x) F(t)/Co
so substituting Eq. 8 and the original equations for Ts and To and equating the terms mutliplying F+(t), F-(t) and F(t), respectively, we find:
Eq. 9: -w+s/τ+ = - (αs + γs) w+s + γs w+o
Eq. 10: -w-s/τ- = -(αs + γs) w-s + γs w-o
Eq. 11: x = Cs (w+s/τ+ + w-s/τ-)
(note Eq. 11 is the same as Lucia's Eq. 7 and 8, with the reference-points for Ts and To set to zero as they must be)
Eq. 12: -w+o/τ+ = γo w+s - (αo + γo) w+o
Eq. 13: -w-o/τ- = γo w-s - (αo + γo) w-o
Eq. 14: 1-x = Co (w+o/τ+ + w-o/τ-)
which gives us 6 equations relating the 4 w's, 2 α's, 2 γ's Cs, Co, and x (11 parameters) and our pre-determined τ's. Note that Eq. 9 combined with Eq. 12 gives the pair of eigenvalue equations for τ+, while Eq. 10 and Eq. 13 give the same pair for τ-, so there are only 4 independent equations between these 4 and the 2 eigenvalue solution equations of Eq. 1.
The γ's are also related to one another through:
Eq. 15: Cs γs = Co γo
(from the definition of γ in terms of the heat exchange rate β and heat capacity) and note the constraint that each γ >= 0 - also note that αs > 0 and αo > 0 must also be satisfied, as these are the inverse uncoupled time constants of the two boxes.
Additional constraints are 0 < x < 1 and 0 < y < 1, and that the Cs and Co should be reasonable for some physical partitioning of Earth’s climate system.
So by my count here we have 13 parameters (4 w's, 2 α's, 2 γ's, c, x, y, Cs, Co) and 10 independent equations (Eq. 5-7 and Eq. 9-15) along with at least 5 independent inequalities (for x, y, α's and γ - in principle the constraints on Cs and Co could also be expressed as inequalities).
Other questions are:
* does the solution space for any input parameters τ(+,-), a1, a2, a3 include a nonempty subspace satisfying the first 5 inequalities?
* do the solutions from fitting the observed temperature record include a nonempty subspace where Cs and Co are reasonable physical values for the Earth?
If I get around to it I'll devote another post to exploring the space of solutions a bit - I think the most interesting approach would be to be to allow αs, αo and Cs to vary freely and see what everything else looks like.
UPDATE (Sep 8, 2009) A minor typo in Eq. 8 (missing slash in the second term) has been corrected. No other results were affected by this change.