By Valerio Lucarini
Climate change arguably presents one of society’s greatest universal challenges and opportunities for scientific enterprise. Earth System Models (ESMs) are currently the most advanced tools for climate change prediction; their future projections serve as key ingredients for the reports of the Intergovernmental Panel on Climate Change (IPCC) and are crucial for climate negotiations. In fact, Syukuro Manabe and Klaus Hasselmann recently received the 2021 Nobel Prize in Physics for their contributions to “the physical modeling of Earth’s climate, quantifying variability and reliably predicting global warming” (an explanation of the scientific motivations behind the prize is available online). In the case of IPCC-class ESMs, researchers usually construct future climate projections by defining various climate forcing scenarios that differ in terms of the way that atmospheric composition and land use change with time. They typically perform an ensemble of simulations—all with different initial conditions—for each scenario [6].
This approach incurs a bottleneck, as each additional forcing scenario requires a new ensemble of simulations. Additionally, disentangling the impact of the various forcing components is nontrivial. Finally, if one wants to consider different time modulations of a given forcing—e.g., a faster or slower carbon dioxide (CO2) increase—then morphing climate change projections is also nontrivial.
Response Operators
A possible strategy for modeling climate change relies upon the construction of response operators that can transform inputs from forcing scenarios into outputs in the form of climate change signals. Such signals are intended as ensemble averages. One may then be tempted to utilize the fluctuation-dissipation theorem, which establishes a relationship between the statistics of a system’s free fluctuations and its response to external forcings [10]. Unfortunately, forced fluctuations for nonequilibrium systems contain features that are absent from the free fluctuations of the unperturbed system [15]. This means that climate change will not merely project onto the natural modes of climate variability. Instead, it will lead to unprecedented events (climatic surprises) with statistical signatures that are not compatible with past statistics [7].
Nonetheless, it is possible to define response operators for general non-equilibrium chaotic systems using the Ruelle response theory [15]. However, it is also possible—and somewhat easier—to rigorously define a response theory for stochastic systems. A modern reformulation of response theory relies upon the formalism of the transfer operator [3].
Figure 1. Response of the globally averaged surface temperature (ST) to a one percent annual increase of CO2, up to doubling. The red line represents the ensemble average of the direct numerical simulations, the yellow area represents the range of response of the various ensemble members, and the blue line represents prediction via response theory. Figure courtesy of [11].
The direct implementation of the Ruelle response formulas is challenging because of the different properties of the contributions that stem from the unstable and stable directions of the flow [1]. A low-tech workaround that allows one to derive the crucial ingredient—the Green function—depends on the use of a simple yet rigorous experimental protocol; here I describe the protocol in the context of climate response.
Climate Change Projections: Surface Temperature
A critical quantity in climate science is the equilibrium climate sensitivity (ECS), which measures the change in the long-term globally averaged surface temperature (ST) due to doubling of CO2 with respect to pre-industrial levels. Because of the various feedbacks that act in the climate systems, relating the ECS to the transient climate response for a given pattern of CO2 forcing is far from obvious [14]. Additionally, state-of-the-art ESMs disagree on their simulated ECS [6].
Response theory allows one to write the ECS as proportional to the integral over time of the Green function of the globally averaged ST. It can also provide an explicit expression for the strength and characteristic timescales of the feedbacks, in terms of the spectrum of the unperturbed system’s transfer operator [12]. Note that Hasselmann and his collaborators provided an early heuristic but very insightful discussion of this point [8]. Researchers have successfully adopted this methodology to perform climate change projections in a hierarchy of climate models: low-dimensional, conceptual models; intermediate complexity climate models; and even state-of-the-art fully coupled ESMs [6].
Figure 2. Climate response at different time horizons of the surface temperature (ST) spatial field for a one percent CO2 increase until doubling. 2a. Projection obtained with response theory for a time horizon of 20 years. 2b. Projection obtained with response theory for a time horizon of 60 years. 2c. Difference between the ensemble average of the direct numerical simulations and the response theory prediction for a time horizon of 20 years. 2d. Difference between the ensemble average of the direct numerical simulations and the response theory prediction for a time horizon of 60 years. Figure courtesy of [13].
The key step for performing projections via response formulas is to compute the Green function of interest. The idea is to look at the ensemble average of the model response to an instantaneous doubling of the CO2 concentration and identify the Green function with the time derivative of such a signal. One can then perform linear climate change projections by convolving the Green function with a time pattern that is associated with the prescribed path of CO2 in order to construct a continuum of climate change scenarios. After performing the necessary ensemble of simulations, one can obtain response operators for any observable of interest [11, 13]. Figure 1 displays the adeptness of this type of response operator when projecting the change in the globally averaged ST to a one percent annual increase of CO2 up to doubling for a state-of-the-art ESM [11].
The procedure works when predicting global quantities and constructing maps of climate change, e.g., generating spatially relevant information. Figure 2 demonstrates the capability of linear response operators to perform convincing climate projections of the ST field on a multidecadal timescale [13].
Climate Change Projections: Ocean Circulation
When employing a comprehensive ESM, one can also treat slow climatic variables — such as those that are associated with the global ocean circulation. Indeed, the response operators perform very accurate projections of future changes in the strength of the Atlantic meridional overturning circulation (AMOC) and the Antarctic circumpolar current, which are among the most relevant features of global ocean circulation. Figure 3 illustrates that CO2 increases are expected to lead to a substantial temporary decrease in the strength of the two oceanic currents, which eventually recover [11]. It is possible to link the substantial decrease in AMOC strength with the current climate state’s proximity to the tipping point that is associated with its complete shutdown [4]. In fact, the transfer operator theory provides the following clear link: when the spectral gap shrinks to zero, the system is at a critical transition and the sensitivity to perturbations diverges [6, 12].
Figure 3. Same color scheme as Figure 1, but for (3a) Atlantic meridional overturning circulation (AMOC) strength and (3b) Antarctic circumpolar current (ACC) strength. Figure courtesy of [11].
Climate Change Projections: Warming Hole in the North Atlantic
A somewhat quirky feature of ongoing regional climate change is the presence of a warming hole in the North Atlantic. East of Labrador (in Canada) and south of Greenland is a region where surface temperatures are currently following a decreasing trend, which is likely to continue for several decades. This “cold blob” is caused by an interplay of local ocean and atmospheric processes that feature different characteristic timescales superimposed on the overall average warming tendency [9]. An important contribution in the cooling trend comes from the decrease in the advection of warm water due to the weakening AMOC. The cold blob has a multiscale and multiphysics nature, and one might reasonably believe that it would be difficult to capture using linear methods. Instead, Figure 4 proves that our proposed methodology performs accurate projections of the blob’s time evolution. The cold blob is mainly associated with the decades wherein the AMOC is the weakest [9].
Figure 4. “Cold blob” in the North Atlantic. Same color scheme as Figure 1, but for the average of the surface temperature in the box [53°W,26°W] x [53°N,69°N]. Figure courtesy of [11].
Conclusions
We have attempted to demonstrate how statistical mechanics—specifically response theory—can support the provision of flexible, accurate, and computationally parsimonious climate change projections via climate models with different complexity levels. A growing body of literature has been revealing applications of linear response theory to relevant climate problems like geoengineering [5] and the definition of safe operating spaces for the planet [2]. Progress in the development of statistical methods might be able to relax the need for ensembles of simulations to construct response operators [16]. This development would greatly increase response theory’s applicability to climate modeling, for which the construction of large ensembles is very expensive. Other developments suggest the use of certain observables as surrogate predictors of other observables [12].
Linear response’s application to climate problems is promising, but many challenges clearly lie ahead. One such challenge involves systematically understanding how to treat nonlinear effects, including those that stem from the interaction of multiple simultaneous forcings. Nevertheless, Ruelle’s theory does extend to higher orders of perturbations, meaning that the necessary mathematics—at least in principle—is already there.
Valerio Lucarini presented this research during a minisymposium at the 2021 SIAM Conference on Computational Science and Engineering, which took place virtually earlier this year.
References
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Valerio Lucarini ([email protected]) is a professor of statistical mechanics at the University of Reading, where he directs the Centre for Mathematics of Planet Earth. He received a 2018 Whitehead Prize of the London Mathematical Society and the 2020 Lewis Fry Richardson Medal of the European Geosciences Union. He will also deliver the American Geophysical Union’s 2021 Ed Lorenz Lecture in December 2021. |