In using the highly-idealised abruptCO2 experiments, it is essential that their physical relevance (traceability) to more realistic gradual forcing experiments is determined. Some GCMs could respond unrealistically to the abrupt forcing change. A key tool here is the step-response model (described below). This response-function method aims to predict the GCM response to any given transient-forcing experiment, using the GCM response to an abruptCO2 experiment. Such a prediction may be compared with the GCM transient-forcing simulation, as part of a traceability assessment (discussed in detail in section 5).
Once some confidence is established in traceability of the abruptCO2 experiments to transient-forcing scenarios, the step-response model has other roles: to explore the implications, for different forcing scenarios, of physical understanding gleaned from abruptCO2 experiments; to help separate linear and nonlinear mechanisms (section 5); and potentially as a basis for GCM emulation. The method description below also serves to illustrate the assumptions of linear system theory.
The step-response model represents the evolution of radiative forcing in a scenario experiment by a series of step changes in radiative forcing (with one step taken at the beginning of each year). The method makes two linear assumptions. First, the response to each annual forcing step is estimated by linearly scaling the response in a CO2 step experiment according to the magnitude of radiative forcing change. Second, the response yi at year i of a scenario experiment is estimated as a sum of responses to all previous annual forcing changes (see Figure 1 of Good et al., 2013 for an illustration):
where xj is the response of the same variable in year j of the CO2 step experiment. scales down the response from the step experiment (xj) to match the annual step change in radiative forcing from year i to year j of the scenario (denoted ):
where is the radiative forcing change in the CO2 step experiment. All quantities are expressed as anomalies with respect to a constant-forcing control experiment.
This approach can in principle be applied at any spatial scale for any variable for which the assumptions are plausible (e.g. Chadwick et al., 2013).
Even in a linear system, regional climate change per K of global warming will evolve during a scenario simulation. This happens because different parts of the climate system have different timescales of response to forcing change.
This may be due to different effective heat capacities. For example, the ocean mixed layer responds much faster than the deeper ocean, simply due to a thinner column of water (Li and Jarvis, 2009). However, some areas of the ocean surface (e.g. the Southern Ocean and south-east subtropical Pacific) show lagged warming, due to a greater connection (via upwelling or mixing) with the deeper ocean (e.g. Manabe et al., 1990;Williams et al., 2008). The dynamics of the ocean circulation and vegetation may also have their own inherent timescales (e.g. vegetation change may lag global warming by years to hundreds of years, Jones et al., 2009). At the other extreme, some responses to CO2 forcing are much faster than global warming: such as the direct response of global mean precipitation to forcings (Allen and Ingram, 2002;e.g. Andrews et al., 2010;Mitchell et al., 1987) and the physiological response of vegetation to CO2 (Field et al., 1995).
In a linear system, patterns of change per K of global warming are sensitive to the forcing history. For example in Figure 1, a scenario is illustrated where forcing is ramped up, then stabilized. Three periods are highlighted, which may have different patterns of change per K of global warming, due to different forcing histories: at the leftmost point, faster responses will be relatively more important, whereas at the right, the slower responses have had some time to catch up. This is illustrated in Figure 2 for sea-level rise. The blue curves show that for RCP2.6, global-mean warming ceases after 2050, while sea-level rise continues at roughly the same rate throughout the century. This is largely because deep ocean heat uptake is much slower than ocean mixed-layer warming.
By design, abruptCO2 experiments separate different timescales of GCM response to forcing change. This is used, for example, (Gregory et al., 2004) to estimate radiative forcing and feedback parameters for GCMs: plotting radiative flux anomalies against global mean warming can separate 'fast' and 'slow' responses (see e.g. Figure 3).
Nonlinear mechanisms arise for a variety of reasons. Often, however, it is useful to describe them as state-dependent feedbacks. For example, the snow-albedo feedback becomes small at high or low snow depth. Sometimes, nonlinear mechanisms may be better viewed as simultaneous changes in pairs of properties. For example, convective precipitation is broadly a product of moisture content and dynamics (Chadwick and Good, 2013;Chadwick et al., 2012). Both moisture content and atmospheric dynamics respond to CO2 forcing, so in general we might expect convective precipitation to have a nonlinear response to CO2 forcing. Of course, more complex nonlinear responses exist, such as for the Atlantic Meridional Overturning Circulation.
In contrast to linear mechanisms, nonlinear mechanisms are sensitive to the magnitude of forcing. For example, the two points highlighted in Figure 4 may have different patterns of change per K of global warming, due to nonlinear mechanisms.
An example is given in Figure 5, which shows the albedo feedback declining with increased global temperature, due to declining snow and ice cover, and the remaining snow and ice being in areas of lower solar insolation (Colman and McAvaney, 2009).
AbruptCO2 experiments may be used to separate nonlinear from linear mechanisms. This can be done by comparing the responses at the same timescale in different different abruptCO2 experiments. Figure 6 compares abrupt2xCO2 and abrupt4xCO2 experiments over years 50-149. A 'doubling difference' is defined, measuring the difference in response to the first and second CO2 doublings. In most current simple climate models (e.g. Meinshausen et al., 2011), the radiative forcing from each successive CO2 doubling is assumed identical (because forcing is approximately linear in log[CO2], Myhre et al., 1998). With this assumption, a linear system would have zero doubling difference everywhere. Therefore, the doubling difference is used as a measure of nonlinearity. The question of which abruptCO2 experiments to compare, and over which timescale, is discussed in section 5.
In some GCMs, the forcing per CO2 doubling has been shown to vary with CO2 (Colman and McAvaney, 2009;Jonko et al., 2013). However, this variation depends on the specific definition of forcing used (Jonko et al., 2013). Currently this is folded into our definition of nonlinearity. If a robust definition of this forcing variation becomes available in future, it could be used to scale out any difference in forcing between pairs of abruptCO2 experiments, to calculate an 'adjusted doubling difference'.
As an example, Figure 7 maps the response to abrupt2xCO2 and abrupt4xCO2, and the doubling difference, for precipitation in HadGEM2-ES over the ocean (taken from Chadwick and Good). The nonlinearities are large - comparable in magnitude to the responses to abrupt2xCO2, albeit with a different spatial pattern.