One of my more recent posts on the two-box model explored the space of possible underlying models for a given empirical fit by fixing heat capacities of the two boxes and varying the heat transfer rate. Keeping the time constants positive restricts the range of allowed heat capacities considerably, while forcing fraction (x) and temperature measurement fraction (y) also provide some constraints given the expectation they must lie between 0 and 1 (and must have actual solutions). Even among solutions satisfying those constraints, there is a further condition that the results look reasonable - as pointed out there and by Lucia here, some of the solutions produce wildly different response levels for the two boxes, which seems unrealistic for systems that should roughly correspond to sub-components of Earth's climate.
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.
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:
Continuing from the previous post, suppose we want to look at solutions where the (s) box is close to the short-time-constant (τ-) solution, and the (o) box then close to the long-time-constant (τ+) solution. That suggests setting the corresponding inverse time constants perturbatively close to τ- and τ+, respectively. Define dimensionless small numbers εs and εo as follows:
Eq. 27: αs = (1 - εs)/τ-
Eq. 28: αo = (1 - εo)/τ+
Then from the definitions of ν+ and ν- (unnumbered equation between 22 and 23 in the previous post) we find:
This is essentially a continuation of the math in the previous post. The same warnings apply!
The previous analysis indicated we have 3 free variables to play with. Let them be αs, αo and Cs (the initial inverse time constants of the two boxes, and the heat capacity of the "surface" box).
Equations 9 and 12 of the previous post show a relationship between w+s and w+o depending on the α's, γ's and τ+, and equations 10 and 13 show the same relationship for w-s and w-o with τ- instead of τ+.
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...
Ok, this time I'm going to start with the graph, and explain what's going on after. Seems to work for other folks... :