The concept of feedback is related to the climate sensitivity or climate stability. It is useful to have a measure of the strength of various feedback processes which determine the ultimate response of the climate system to any change in radiative forcing. In general terms, an initial change in temperature due to a change in radiative forcing, T_{forcing}, is modified by the complex combination of feedback processes such that:

T_{final} = T_{forcing} + T_{feedback}[Equation 5]

where T_{feedback} is the temperature change resulting from feedback and T_{final} is the overall change in temperature between the initial and final equilibrium states. The degree to which feedback processes influence the final climatic response is a measure of the sensitivity of the climate system.

Equation 5 can be rewritten as follows:

T_{final} = fT_{forcing}[Equation 6]

where f is called the feedback factor (Henderson-Sellers & Robinson, 1986). When only one feedback mechanism is operative the solution to equation 6 is simple, assuming both f and T_{forcing} are known. When more than one feedback is operative, matters become more complicated. For two feedbacks, the net effect is given by:

f = f_{1}f2 / (f_{1 }+ f2 – f_{1}f2)[Equation 7]

where f_{1} and f2 are the feedback factors of the two feedback processes (Henderson-Sellers & Robinson, 1986). Clearly, the feedback factors are neither additive nor multiplicative. A feedback operating alone with a factor of 2 would double the initial climatic response to forcing. If a second feedback with factor 1.5 acted with it, the overall feedback would be enhanced by a factor of 6. It can be seen, then, that a combination of feedback processes could dramatically affect the climate as it responds to only a small change in radiative forcing.

The climate’s sensitivity can be mathematically determined in another way. From satellite observations, it has been shown that changes in global temperature are approximately proportional to changes in radiative forcing. If we assume an instantaneous change in climate, from one equilibrium state to another, then:

Q = T[Equation 8]

where Q is change in radiative forcing (expressed in terms of the net downward radiative flux at the top of the troposphere), T is the change in global temperature, and is a measure of the climate sensitivity.

Based on equation 8, the climate sensitivity is usually expressed in terms of the temperature change associated with a specified change in radiative forcing, usually a doubling of the atmospheric carbon dioxide content. Thus, the carbon dioxide doubling temperature, T_{2x}, is given by

T_{2x} = Q_{2x} / [Equation 9]

where Q_{2x} is 4.2 Wm^{-2}. The magnitude (and sign) of T_{2x} will depend on , the climate sensitivity, which is determined by the net affect of the climate feedback processes. Despite extensive climate modelling over the last 2 decades to understand the problem of contemporary **global warming** (see chapter 6), it is this parameter that is proving hard to define numerically.

As explained earlier, the idea of static equilibrium and instantaneous climatic response represents an unrealistic situation. To take into account the dynamic and transient nature of the climate’s response to forcing, a more complicated equation linking T and Q is required if the evolution of the response with time is to be determined. In the case:

Q = T + C T/ t[Equation 10]

Here, the change in radiative forcing, Q, is balanced by:

1) the change in the outgoing radiative flux at the tropopause caused by the response of the climate system including feedback; and

2) the energy stored in the system, C T/ t, where C is the system’s heat capacity and t is time.

The latter term in equation 10 simulates the time-dependent nature of the climate system’s response. The main contributor to the system’s heat capacity is the world’s oceans. The heat capacity of water is large compared to that of air and is therefore able to store much more energy (see section 1.3.1). Additionally, the high heat capacity means that the oceans take time to heat up (or cool down) and hence slow the surface temperature response to any change in radiative forcing: the transient response will always be less than the equilibrium response.

Solution of Equation 10 leads to the definition of the response time of the climate system, , such that:

= C / [Equation 11]

If the heat capacity of the climate system is large, the response time is large. Equally, if the climate sensitivity is small the response time is large. is also known as the radiative damping coefficient. Here, analogy is made between the response of the climate system and an oscillating spring. If a spring has a high damping coefficient, it will stop oscillating soon after it has been set in motion. Similarly, if the radiative damping coefficient is large, the climate will respond quickly and will be small.

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