## Climate sensitivity radiative forcing and feedback

The simple model we have been studying affords us the opportunity to introduce the concepts of radiative forcing, sensitivity coefficient and feedback factor. These diagnostics can be applied across the whole spectrum of climate models, from the simplest to the most comprehensive.

Suppose that the mean surface temperature depends on some parameter A, and we wish to know how sensitive T is to changes in that parameter. For example, this parameter might be the Solar constant, or the radiating pressure. It could be some other parameter controlling the strength of the greenhouse effect, such as CO2 concentration. Near a given A, the sensitivity is characterized by dT/dA.

Let G be the net top-of-atmosphere flux, such as used in Eq. 3.11. To allow for the fact that the terms making up the net flux depend on the parameter A, we write G = G(T, A). If we take the derivative of the the energy balance requirement G = 0 with respect to A, we find dG dT dG ,

so that jrp dG

dA dT

The numerator in this expression is a measure of the radiative forcing associated with changes in . Specifically, changing A by an amount 5 A will perturb the top-of-atmosphere radiative budget by

dG;5A, requiring that the temperature change so as to bring the energy budget back into balance.

For example, if A is the Solar constant L, then dA = 1 (1 — a). If A is the radiating pressure prad, then dGA = — ddpLR. Since OLR goes down as prad is reduced, a reduction in prad yields a positive radiative forcing. This is a warming influence.

Radiative forcing is often quoted in terms of the change in flux caused by a standard change in the parameter, in place of the slope ^A itself. For example, the radiative forcing due to CO2 is typically described by the change in flux caused by doubling CO2 from its pre-industrial value, with temperature and everything else is held fixed. This is practically the same thing as dGA if we take A = log2pCO2, where pCO2 is the partial pressure of CO2. Similarly, the climate sensitivity is often described in terms of the temperature change caused by the standard forcing change, rather than the slope dTA. For example, the notation AT2x would refer to the amount by which temperature changes when CO2 is doubled.

The denominator of Eq. 3.14 determines how much the equilibrium temperature changes in response to a given radiative forcing. For any given magnitude of the forcing, the response will be greater if the denominator is smaller. Thus, the denominator measures the climate sensitivity. An analysis of ice-albedo feedback illustrates how a feedback process affects the climate sensitivity. If we assume that albedo is a function of temperature, as in Eq. 3.9, then