One more reasonable constraint on two-box empirical models of Earth's climate

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 long-term response of the two boxes to forcing in the model treated thus far is given by just setting the time derivatives to zero and fixing the forcing F at a constant value F0. Solving the resulting pair of equations gives:

(1) Ts/F0 = (αoCox + β)/(αoCoαsCs + β(αoCo + αsCs))

(2) To/F0 = (αsCs(1-x) + β)/(αoCoαsCs + β(αoCo + αsCs))

While we don't have to require these long-term linearized responses of the two boxes to exactly match (this is a steady-state, not equilibrium, problem, and Earth in steady state has many regions with widely differing temperatures), it seems unlikely that one box would absorb most of the temperature increase while the other remains substantially unchanged. So why not add a constraint that the long-term ratio of these two responses must be limited to a certain range - tmin to tmax:

(3) tmin < (αoCo x + β)/(αsCs (1-x) + β) < tmax

which can then be rearranged into constraints on the value of the forcing fraction x:

(4) (αstmin - γs(1 - tmin))/(αo/λ + αstmin) < x < (αstmax + γs(tmax - 1))/(αo/λ + αstmax)

The following figures map out these limits setting tmin to 0.8 and tmax to 1.2, i.e. allowing a 20% variation in long-term response temperature between the fast and slow boxes. The value of x in the following, to meet this constraint, must lie between the red and green curves.

tamino_x_gamma1.png
Figure 1: The fast box with heat capacity of Earth's atmosphere, and the slow box with 12% of Earth's full ocean heat capacity. There is a wide range of heat transfer rates giving solutions along the blue curve (+ choice for x,y) - essentially every value for which y can be found and is less than 1.

tamino_x_gamma5.png
Figure 2: The fast box with twice the heat capacity of Earth's atmosphere, and the slow box with 10% of Earth's full ocean. Here only a limited range of the (-) solution (black curve) is allowed, for γs values a bit above 1.0e-8 s^-1.

tamino_x_gamma4.png
Figure 3: Fast box with twice the atmosphere, slow box with just 5% of the ocean. Again a very limited range of solutions along the black curve, this time requiring the heat transfer rate γs to be near zero.

tamino_x_gamma3.png
Figure 4: Fast box with five times the atmosphere, slow box with 5% of the ocean. This time the limited range, again along the black curve, is near γs values of 1.0e-8 s^-1 again.

tamino_x_gamma2.png
Figure 5: Fast box with five times the atmosphere, slow box with 12% of the ocean. This time the allowed solution range is essentially all of the (short) blue curve with γs around 2.8e-8 s^-1.

This additional condition clearly limits the space of allowed two-box underlying models, but there are still a wide variety to choose from even meeting the original "Tamino" choices of 1-year and 30-year time constants.