By nature of its derivation, the Eocene relationship reflects equilibrium climate states. Its usage is a consideration ‘from the end’: The eventually established temperature is pre-known from the CO2 concentration. Also in the industrial-age calculation above, the water vapor concentration is determined from the pre-known temperature. Now, the consideration will be ‘bottom up’, from the beginning of a disturbance to a given equilibrium state. Concentration and absorption are to determine the temperature from the disturbance itself without pre-knowledge of a ‘to-be’ temperature.
Model description
The above density proportionality scheme is adapted to the case of a gradual disturbance acting for some decades and then switched off. Two disturbance cases will be regarded: increase of radiation from the surface for 70 years and doubling/quadrupling of the CO2 concentration in 70/140 years. To briefly summarize the adapted density scheme: With a year’s disturbance, the longwave absorption in the atmosphere changes (effective after ocean heat uptake), in tail temperature changes (according to ideal blackbody), which in turn alters densities and thus absorption.
Model details: Starting point at year 0 is the same value set as in the previous paragraph: p0H2O = 0.4 %, p0CO2 = 280.3 ppmv (p for volume mixing ratios, typically termed concentration within the present work), absorption QA0=57 W/m2 at these concentrations, temperature 14°C. Radiation contributions are translated into temperature contributions via the blackbody assumption. The absorption change (ΔQA) is taken proportional to the concentration change. The radiation at the end of a given year (Q) and the corresponding temperature (T) are
T = (Q/σ)1/4, Q(year) = Q(year-1) + δQ(year), δQ = (1 - u) ∙ δQA, δQA = ΔQA(year) – ΔQA(year-1), ΔQA = (p - p0) / p0 ∙ QA0, pH2O(year) = pH2O(T(year-1)), pCO2: see text, p = pH2O + pCO2, | (1) |
in partly generic variable definitions, with u the ocean heat uptake ratio, σ the Stefan-Boltzmann constant, and starting at year = 1 with Q(0) = 385.5 W/m2 and ΔQA(0) = 0. (If δQA was subjected to the atmospheric window, all results were retained with QA0=63.4 W/m2, compare remark in the previous paragraph).
Radiation increase
This density proportionality scheme (Eq. 1) is applied to the scenario where the longwave radiation from the ground is steadily increasing within 70 years to an exemplary 3.2 W/m2 stronger radiation, constant from year 71 onwards. The CO2 concentration is set dependent on temperature via 20 ppmv/°C (from the Late Quaternary, cf. [1]) and the H2O concentration as in the previous paragraph.
With an uptake ratio u = 0.21 on the yearly absorption change (δQA), the result for the atmospheric radiation change is obtained as 2.35 W/m2 after the disturbance period, thus indicating the new equilibrium case.
This compares well with the 2.4 W/m2 obtained for the same variability scenario in the earlier energy budget studies [9], there also reflecting equilibrium conditions.
CO 2 increase
Next, the density scheme (Eq. 1) is applied to doubling and quadrupling of the CO2 concentration within 70 and 140 years, respectively, and the concentration kept constant afterwards.
It turns out that the equilibrium temperatures as given by the Eocene relationship are only reached to 39 % and 25 % for the two scenarios. Something is different to the former disturbance of (direct) radiation change from the ground. Also, gradual dilution of ocean heat uptake alone does not lead to the equilibrium temperatures. Thus, absorption increase from increased CO2 concentration appears too weak to drive temperatures to equilibrium (in the present proportionality scheme). There seems to be an additional process raising water vapor concentrations to the required equilibrium level. From overall impression, such process is indicated delayed to the original disturbance and vanishing with time (details not shown). The present scheme asymptotically delivers the Eocene equilibrium temperatures, if an additional radiation (to δQ) is introduced at ‘ramp-up’ termination (e.g. of 0.6 % in the doubling scenario) and afterwards, this additional radiation and the ocean heat uptake ratio are steadily decreasing by 6 %/100 years.
The results are summarized in Fig. 3: the temperature change versus time in years from the starting point of quadrupling (upper red data set) and doubling (lower blue data set) of the CO2 concentration, with quadrupling through the first 140 years and doubling through the first 70 years, both at constant annual rate, and the concentration kept constant afterwards. Results from previous sophisticated simulations [11] are depicted by dashed lines, with the equilibrium temperatures – as established after 2000–3000 years – indicated on the right side by horizontal lines. The single dots mark the corresponding temperatures according to the Eocene relationship. The dotted red and blue lines show the temperature results from the present model (Eq. 1 with precedingly described adaptation).
The temperatures of the Eocene relationship and the equilibrium simulation results are in good agreement. In the ‘ramp-up’ period, the density proportionality scheme of Eq. (1) is in good or fair agreement with the sophisticated simulations [11] without the extra radiation increase at ‘ramp-up’ termination (see above).
From the dotted blue line in Fig. 3, i.e. for doubled CO2 concentration, a rule of thumb may be extracted. Roughly ⅓ of the equilibrium temperature is rather contemporarily expected once the concentration limit is reached; ½, ⅔ and 80 % of the equilibrium temperature are reached about 70, 460 and 830 years from the beginning of the constant-concentration period.
Conclusion
The longwave absorption-to-density proportionality – as discovered for equilibrium states from the Eocene relationship – also delivers convincing results in the transient climate regime. The results are consistent with energy budget considerations (scenario of surface radiation change) and with sophisticated simulations in the ‘ramp-up’ period of a continuous CO2 change. In the latter case after disturbance switch-off, asymptotic approach towards equilibrium can be reproduced by a small extra-rise of the H2O concentration via radiation increase in the order of 0.6 % upon disturbance switch-off, and by diluting of this increase and the ocean heat uptake ratio with 6 % / 100 years. As a rule of thumb, final equilibrium temperatures are anticipated on the millennium time scale about 3 times the contemporarily emerging temperatures at termination of the CO2 concentration change.