FIGURE 1

Schematic of the geometry of the box-diffusion-upwelling model used in these remarks.

FIGURE 2

Temperature versus time for the impulsive doubling of CO2 at t = 0 in a model with = 3.6°C.

a = ao/g, where g = system gain, and the relation between g and total feedback, f, is g = 1/(1 − f). Negative f's lead to g < 1, while f > 1 implies an unstable system. If we associate a doubling of CO2 with a ΔS (at the surface as opposed to the ''top of the atmosphere") of 1.8 W m2, then, in the absence of feedbacks, ΔT = 1.2°C. Models that yield ΔT2×CO2 = 4°C involve a gain of 3.3 (or f = 0.7). An idea of the nature of ocean delay can be seen in Figure 2, where we consider the transient response to an impulsive doubling of CO2 in a system where ΔT = 3.6°C. Clearly, the approach to equilibrium is not simply exponential. However, for simplicity of discussion, we will identify a response with a time scale τ, corresponding to the time it takes ΔT to reach to within (1 − 1/e) of its equilibrium value when the system is impulsively forced. Figure 3 shows how τ varies with g. Over the range of interest, the dependence is almost linear. (As Hansen et al. (1985) note, the relation becomes quadratic as H → ∞.)

A main point of these remarks is that in dealing with these simple models, it is occasionally useful to consider τ rather than g, and ΔT2×CO2 rather than ΔS2×CO2 as variables to focus on. Doing so will suggest constraints on both τ and ΔT2×CO2, which can be determined from considering responses to both volcanos and increasing greenhouse gases. It must be emphasized that it is just as useful to use data to constrain the likely response to doubled CO2 as it is to detect the actual response (which might prove small).

Figure 4 shows T vs. t for the IPCC "business as usual" (BAU) emissions scenario (which leads to a quadrupling of "effective" CO2 by 2100; viz., Houghton et al., 1990). The different curves correspond to various choices of equilibrium . We see (consistent with IPCC results) that expectations for current warming range from 0.15°C for = 0.24°C to 0.8°C for ΔT2×CO2 = 4.8°C. The observed warming over the past century of 0.45°C ± 0.15°C (viz., Figure 5) corresponds to about = 1.2°C,2 but given the natural variability in T, it is difficult to rule out any of the choices on the basis of the observed global

FIGURE 3

Characteristic ocean delay (τ) versus gain (g). See text for discussion.

FIGURE 4

Temperature change since 1880 for IPCC "business as usual" emissions scenario and various model sensitivities.

2  

Wigley and Raper consider the impact of sulfate cooling; however, according to recent calculations of Kiehl and Briegleb (1993), this has been reduced to 0.3 W m2 and is no longer of such great consequence.



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