Impulse Response Climate Models

For any complex, nonlinear system such as climate, it is permissible, for sufficiently small perturbations, to describe the response of the system to external forcing in terms of a linearized response model. In the case of the climate system, the linearization condition is approximately satisfied for the external forcing due to anthropogenic greenhouse gas emissions if the temperature change remains below about 2-3°C. Measured in Kelvin relative to absolute zero temperature at — 273°C, the global mean temperature of the earth (15°C) is 288 K. Thus a 3°C temperature change represents a perturbation of only 3/288 ~ 1% in the absolute temperature scale relevant for infrared greenhouse radiation effects. The climate change effects in this range can normally be computed to adequate approximation as a linear response.

An important caveat in the application of the linear response approximation, however, is that the reference state on which the linear perturbation is superimposed is not close to an unstable bifurcation point. In this case, the response of the climate system to even relatively small external forcing can differ significantly from the climate change computed with a linear response model and is basically unpredictable. A number of such potential instabilities have been discussed in the literature. One of the more serious possibilities, which has been observed in paleoclimatic records and simulated in models, is a breakdown of the North Atlantic ocean circulation (Maier-Reimer and Mikolajewicz, 1989; Rahmstorf and Willebrand, 1995; Rahmstorf, 1995; Schiller et al., 1997). This can be triggered through a warming and/or freshening of North Atlantic surface waters. The northward-traveling surface water of the Gulf Stream then no longer becomes dense enough when cooled at higher latitudes to sink to sufficient depth to drive the deep ocean circulation, which is the source of the balancing northward current. A breakdown of the Gulf Stream, which produces a 6°C warmer climate in Europe relative to the latitudinal mean, would clearly have dramatic consequences for the climate of Europe. Other potentially catastrophic instabilities are a collapse of the West Antarctic ice sheet, which would result in a global sea-level rise of 6 m, or a runaway greenhouse warming through the release of large quantities of methane (a very effective greenhouse gas) that are currently trapped in the permafrost regions of Siberia and in hydrates in the deep ocean. Current predictions of anthropogenic greenhouse warming suggest that such

Impulse Response Climate Model scalar 0=

high-dimensional vector, j = iT(x), p (x) , z (x) , ...y

FIGURE 3 Impulse-response representation of the response of the climate system to a 5-function C02 input. The net response is given by a convolution of the response of the atmospheric C02 concentration to a ¿¡-function C02 input (calibrated against a carbon cycle model) and the response of the physical ocean-atmosphere climate system to a step-function increase in the C02 concentration (calibrated against a coupled ocean-atmosphere GCM). For details, see Hasselmann etal., 1997.

instabilities are unlikely to occur if the global warming remains below 2-3°C, and we will ignore them in the following. However, the implications of low-probability, high-impact climate change instabilities should be kept in mind in integrated assessment studies.

As an example of an impulse-response climate model, we consider in the following the response of the climate system to C02 emissions e(t) (cf. Fig. 3), which represent about 60% of the total greenhouse warming today and are expected to contribute a still larger fraction in the future. For small perturbations, the relation between the C02 forcing and the climate-change response </i(x, t) can be expressed as a linear response integral t]4X. t) = R(X, t — t')e(t')dt', Jo where the impulse-response function R(x, t) represents the climate response to a S-function C02 input at time t = 0. The function R(x, t) can be calibrated against the response of the climate system computed with a fully nonlinear, state-of-the-art climate model for some given greenhouse gas emission scenario. Once calibrated, the climate response can be computed for arbitrary emission curves by superposition. Coupled with a similarly efficient socioeconomic model, the impulse-response climate model thus enables one to efficiently perform a large number of simulations, as required, for example, for optimal emission path computations or sensitivity studies.

An important feature of impulse-response models is that they entail no loss of information compared with the complete model: the function R(x, t) can contain the same number of degrees of freedom in the description of the climate change signal i/i(x, t) as the complete climate model against which it is calibrated.

Examples of typical impulse-response functions are shown in Figs. 4 and 5 (adopted from Hooss et al., 2001). The net impulse-response function of the coupled carbon-cycle/physical ocean-atmosphere system is given by a convolution (cf. Hasselmann et al., 1997) of the linear impulse-response function for the carbon cycle alone (representing the atmospheric C02 response to a S-function C02 input) and the response function of the physical ocean-atmosphere system (representing the response of the physical system variables to a step-function increase in the C02 concentration). Figure 4 shows the response function Rt. for the carbon cycle and the response functions RT and Rs for the global mean temperature and mean sea-level rise, respectively, for the physical climate system. The resulting net response functions for Rc, R(n, and R(s) for the coupled carbon-cycle/physical ocean-atmosphere system are shown in Fig. 5.

The linear impulse-response model has recently been generalized by Hooss et al. (2001) to include some of the dominant non-linearities of the climate system. The net response curves shown in Fig. 5 were computed using this generalized model, showing the impact of nonlinearities in the lower two-panel rows. The princi

FIGURE 4 Individual impulse-response functions for the carbon cycle (response Rc of the atmospheric C02 concentration to a 5-function C02 input at time t = 0, left) and the variables global mean near-surface temperature (center) and mean sea-level rise (right) for the physical ocean-atmosphere system (response to a step-function increase in the C02 concentration to a constant level at time t = 0). The units for temperature response RT and sea-level response Rs refer to the amplitudes of the first empirical orthogonal functions (EOFs) of the response patterns of the respective variables and are essentially arbitrary (see text). Adopted from Hooss et al., 2001.

FIGURE 4 Individual impulse-response functions for the carbon cycle (response Rc of the atmospheric C02 concentration to a 5-function C02 input at time t = 0, left) and the variables global mean near-surface temperature (center) and mean sea-level rise (right) for the physical ocean-atmosphere system (response to a step-function increase in the C02 concentration to a constant level at time t = 0). The units for temperature response RT and sea-level response Rs refer to the amplitudes of the first empirical orthogonal functions (EOFs) of the response patterns of the respective variables and are essentially arbitrary (see text). Adopted from Hooss et al., 2001.

pal effect is a lowering of the rate of decrease of the atmospheric C02 concentration at higher CO; concentrations. This is due to the slower uptake of CO, by the ocean in which solubility of C02 decreases with increasing C02 concentration.

Hooss et al. (2001) considered not only the responses of the global mean temperature and sea level, but also the changes in other climate variables, such as precipitation and cloud cover, and in addi tion the spatial dependence of these variables. They found that the latter could be well described by a single spatial response pattern for each variable considered (cf. Figs. 6, 7, 8; see also color insert). Thus, the spatiotemporal response properties of each climate variable can be represented as the product of a normalized pattern (the empirical orthogonal function, on EOF palten), and an associated pattern coefficient, whose time evolution must necessarily be the same as the

FIGURE 5 Net impulse-response functions R.. for the atmospheric CO, concentration (left column), the global mean temperature Rrh (center column), and the amplitude Rm of the first EOF of the sea-level rise (right column, proportional to the mean sea-level rise) for different magnitudes of a S-function CO, input at time t = 0. The differences in the responses in the three cases (from top to bottom, 15, 100 and 30% increase in initial CO, concentration, respectively) arise from nonlineari-ties in the generalized nonlinear impulse-response model of Hooss et al., 2001. (Adopted from this source.)

FIGURE 5 Net impulse-response functions R.. for the atmospheric CO, concentration (left column), the global mean temperature Rrh (center column), and the amplitude Rm of the first EOF of the sea-level rise (right column, proportional to the mean sea-level rise) for different magnitudes of a S-function CO, input at time t = 0. The differences in the responses in the three cases (from top to bottom, 15, 100 and 30% increase in initial CO, concentration, respectively) arise from nonlineari-ties in the generalized nonlinear impulse-response model of Hooss et al., 2001. (Adopted from this source.)

Near-surface air temperature 4xC02 - Ctrl

Klaus Hasselmann 1 .EOF 96.7%

FIGURE 6 Dominant spatial-response pattern (first EOF) of near-surface temperature change to increased atmospheric C02 concentration (from Hooss et al., 2001). See also color insert.

Precipitation 4xC02-ctrl 1.EOF 30.6%

180w 150w 120w 90w 60w 30w 0 30e 60e 90e 120e 150e 180e

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

FIGURE 7 Dominant spatial-response pattern (first EOF) of change in precipitation caused by an increase in atmospheric CO, concentration (from Hooss et al., 2001). See also color insert.

Sealevel 1. EOF 99.9%

Sealevel 1. EOF 99.9%

85E 145E 155W 95W 35W 25E

FIGURE 8 Dominant spatial-response pattern (first EOF) of change in sea level caused by an increase in atmospheric C02 concentration (from Hooss et al„ 2001). See also color insert.

85E 145E 155W 95W 35W 25E

FIGURE 8 Dominant spatial-response pattern (first EOF) of change in sea level caused by an increase in atmospheric C02 concentration (from Hooss et al„ 2001). See also color insert.

time evolution of the associated global mean variable (which is given by the spatial mean of the response pattern).

An important common characteristic of the response functions is their exceedingly long memory. The decrease in atmospheric C02 concentration (due to the gradual uptake of C02 by the oceans and the terrestrial biosphere) is an extremely slow process extending over several hundred years. Combined with the delayed temperature response RT of the coupled ocean-atmosphere system to a CO, increase induced by the large thermal inertia of the oceans, the net temperature response RT of the climate system to C02 input persists over several centuries. Thus, in assessing the climate-change impacts of human activities, one must consider time horizons far beyond the normal planning horizons of decision makers.

The impact of the long memory of the climate system on the climate response to anthropogenic C02 emissions is illustrated in Figure 9, which shows the C02 emissions, C02 concentrations, and temperature change computed for a "business-as-usual" (BAU) scenario and an alternative frozen-emissions scenario. The upper panels show the evolution over the next 100 years, the lower panels the evolution over the next 1000 years. In the case of the BAU scenario, all fossil fuel resources, estimated at 10,000 GtC, are assumed to be exploited within the next 500-700 years. The long-term impacts in the lower panel are seen to greatly exceed the climate change over the next 100 years, even for the frozen emissions case. Although the impulse-response computations are clearly unreliable for such large climate changes, the orders of magnitude of the computed warming, in the range of 10°C for the BAU case—exceeding the climate changes of the ice ages and beyond the range in which even fully nonlinear state-of-the-art models can be credibly applied—clearly demonstrate the danger of underestimating future climate-change impacts by limiting considerations of climate policy to only a few decades (see also Cline, 1992).

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