With this expression, Eq. 3.14 can be rewritten dT 1 r dG

1 da

In writing this equation we primarily have ice-albedo feedback in mind, but the equation is valid for arbitrary a(T) so it could as well describe a variety of other processes. The factor in square brackets in Eqn. 3.16 is the sensitivity the system would have if the response were unmodified by the change of albedo with temperature. The first factor determines how the sensitivity is increased or decreased by the feedback of temperature on albedo. If —1 < $ < 0 then the feedback increases the sensitivity - the same radiative forcing produces a bigger temperature change than it would in the absence of the feedback. When $ = — 1, for example, the response to the forcing is twice what it would have been in the absence of the feedback. The sensitivity becomes infinite as $ ^ —1, and for —2 < $ < —1 the feedback is so strong that it actually reverses the sign of the response as well as increasing its magnitude. On the other hand, if $ > 0, the feedback reduces the sensitivity. In this case it is a stabilizing feedback. The larger $ gets, the more the response is reduced. For example, when $ = 1 the response is half what it would have been in the absence of feedback. Note that the feedback term is the same regardless of whether the radiative forcing is due to changing L, prad or anything else.

As an example, let's compute the feedback parameter $ for the albedo-temperature relation given by Eq. 3.9, under the conditions shown in Fig. 3.10. Consider in particular the upper solution branch, which represents a stable partially ice-covered climate like that of the present Earth. At the point L = 1400W/m2, T = 288K on this branch, we find $ = —.333. Thus, at this point the ice-albedo feedback increases the sensitivity of the climate by a factor of about 1.5. At the bifurcation point L « 1330W/m2,T « 277K, $ ^ — 1 and the sensitivity becomes infinite. This divergence merely reflects the fact that the temperature curve is vertical at the bifurcation point. Near such points, the temperature change is no longer linear in radiative forcing. It can easily be shown that the temperature varies as the square root of radiative forcing near a bifurcation point, as suggested by the plot.

The ice-albedo feedback increases the climate sensitivity, but other feedbacks could be stabilizing. In fact Eq. 3.17 is valid whatever the form of a(T), and shows that the albedo feedback becomes a stabilizing influence if albedo increases with temperature. This could conceivably happen as a result of vegetation feedback, or perhaps dissipation of low clouds. The somewhat fanciful Daisyworld example in the Workbook section at the end of this chapter provides an example of such a stabilizing feedback.

The definition of the feedback parameter can be generalized as follows. Suppose that the energy balance function G depends not only on the control parameter A, but also on some other parameter R which varies systematically with temperature. In the previous example, R(T) is the temperature-dependent albedo. We write G = G(T, R(T), A). Following the same line of reasoning as we did for the analysis of ice-albedo feedback, we find

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