Predicting Climate Change using Response Theory: Global Averages and Spatial Patterns
PPredicting Climate Change using Response Theory: GlobalAverages and Spatial Patterns ∗ Valerio Lucarini , [ [email protected] ] Meteorologisches Institut, CEN, University of HamburgHamburg, Germany Department of Mathematics and StatisticsUniversity of Reading, Reading, UKFrancesco Ragone Laboratoire de Physique de l’ ´EEcole NormaleSup´eerieure de Lyon Lyon, FranceFrank Lunkeit Meteorologisches Institut, CEN, University of HamburgHamburg, GermanyMarch 17, 2016
Abstract
The provision of accurate methods for predicting the climate response to anthropogenicand natural forcings is a key contemporary scientific challenge. Using a simplified and efficient ∗ Paper prepared for the special issue of the Journal of Statistical Physics dedicated to the 80 th birthday of Y.Sinai and D. Ruelle. a r X i v : . [ phy s i c s . a o - ph ] M a r pen-source general circulation model of the atmosphere featuring O(10 ) degrees of freedom,we show how it is possible to approach such a problem using nonequilibrium statistical mechan-ics. Response theory allows one to practically compute the time-dependent measure supportedon the pullback attractor of the climate system, whose dynamics is non-autonomous as a resultof time-dependent forcings. We propose a simple yet efficient method for predicting - at anylead time and in an ensemble sense - the change in climate properties resulting from increasein the concentration of CO using test perturbation model runs. We assess strengths andlimitations of the response theory in predicting the changes in the globally averaged values ofsurface temperature and of the yearly total precipitation, as well as in their spatial patterns.The quality of the predictions obtained for the surface temperature fields is rather good, whilein the case of precipitation a good skill is observed only for the global average. We also showhow it is possible to define accurately concepts like the the inertia of the climate system orto predict when climate change is detectable given a scenario of forcing. Our analysis can beextended for dealing with more complex portfolios of forcings and can be adapted to treat, inprinciple, any climate observable. Our conclusion is that climate change is indeed a problemthat can be effectively seen through a statistical mechanical lens, and that there is great po-tential for optimizing the current coordinated modelling exercises run for the preparation ofthe subsequent reports of the Intergovernmental Panel for Climate Change. The climate is a forced and dissipative nonequilibrium chaotic system with a complex naturalvariability resulting from the interplay of instabilities and re-equilibrating mechanisms, nega-tive and positive feedbacks, all covering a very large range of spatial and temporal scales. Oneof the outstanding scientific challenges of the last decades has been the attempt to provide acomprehensive theory of climate, able to explain the main features of its dynamics, describeits variability, and predict its response to a variety of forcings, both anthropogenic and natural[1, 2, 3]. The study of the phenomenology of the climate system is commonly approached byfocusing on distinct aspects like: • wave-like features such Rossby or equatorial waves, which have enormous importance interms of predictability and transport of, e.g. , energy, momentum, and water vapour; • particle-like features such as hurricanes, extratropical cyclones, or oceanic vortices, which re of great relevance for the local properties of the climate system and its subdomains; • turbulent cascades, which determine, e.g. dissipation in the boundary layer and devel-opment of large eddies through the mechanism of geostrophic turbulence.While each of these points of view is useful and necessary, none is able to provide alone acomprehensive understanding of the properties of the climate system; see also discussion in[4]. On a macroscopic level, one can say that at zero order the climate is driven by differencesin the absorption of solar radiation across its domain. The prevalence of absorption at sur-face and at the lower levels of the atmosphere leads, through a rich portfolio of processes, tocompensating vertical energy fluxes (most notably, convective motions in the atmosphere andexchanges of infrared radiation), while the prevalence of absorption of solar radiation in thelow latitudes regions leads to the set up of the large scale circulation of the atmosphere (withthe hydrological cycle playing a key role), which allows for reducing the temperature differ-ences between tropics and polar regions with respect to what would be the case in absence ofhorizontal energy transfers [5, 6].It is important to stress that such organized motions of the geophysical fluids, which act asnegative feedbacks but cannot be treated as diffusive Onsager-like processes, typically resultfrom the transformation of some sort of available potential into kinetic energy, which contraststhe damping due to a variety of dissipative processes. See [7] for a detailed analysis of therelationship between response, fluctuations, and dissipation at different scales. Altogether, theclimate can be seen as a thermal engine able to transform heat into mechanical energy witha given efficiency, and featuring many different irreversible processes that make it non-ideal[8, 9, 2].Besides the strictly scientific aspect, much of the interest on climate research has beendriven in the past decades by the accumulated observational evidence of the human influenceon the climate system. In order to summarize and coordinate the research activities carriedon by the scientific community, the United Nations Environment Programme (UNEP) andthe World Meteorological Organization (WMO) established in 1988 the International Panelon Climate Change program (IPCC). The IPCC reports, issued periodically about every 4-5years, provide systematic reviews of the scientific literature on the topic of climate dynamics,with special focus on global warming and on the socio-economic impacts of anthropogenicclimate change [10, 11, 12]. Along with such a review effort, the IPCC defines standards for he modellistic exercises to be performed by research groups in order to provide projectionsof future climate change with numerical models of the climate system. A typical IPCC-likeclimate change experiment consists in simulating the system in a reference state (a stationarypreindustrial state with fixed parameters, or a realistic simulation of the present-day climate),raising the CO concentration (as well as, in general, the concentration of other greenhousegases such as methane) in the atmosphere following a certain time modulation in a certain timewindow, and then fixing the CO concentration to a certain value to observe the relaxationof the system to a new stationary state. Each time-modulation of the CO forcing defines a scenario , and it is a representation of the expected CO increase resulting from a specific pathof industrialization and change in land use. Note that the attribution of unusual climaticconditions to specific climate forcing is far from being a trivial matter [13, 14].While much progress has been achieved, we are still far from having a conclusive frameworkof climate dynamics. One needs to consider that the study of climate faces, on top of all thedifficulties that are intrinsic to any nonequilibrium system, the following additional aspectsthat make it especially hard to advance its understanding: • the presence of well-defined subdomains - the atmosphere, the ocean, etc. - featuringextremely different physical and chemical properties, dominating dynamical processes,and characteristic time-scales; • the complex processes coupling such subdomains; • the presence of a continuously varying set of forcings resulting from, e.g. , the fluctuationsin the incoming solar radiation and the processes - natural and anthropogenic - alteringthe atmospheric composition; • the lack of scale separation between different processes, which requires a profound revisionof the standard methods for model reduction/projection to the slow manifold, and calls forthe unavoidable need of complex parametrization of subgrid scale processes in numericalmodels; • the impossibility to have detailed and homogeneous observations of the climatic fieldswith extremely high-resolution in time and in space, and the need to integrate directand indirect measurements when trying to reconstruct the past climate state beyond theindustrial era; • the fact that we can observe only one realization of the process. ince the climate is a nonequilibrium system, it is far from trivial to derive its responseto forcings from the natural variability realized when no time-dependent forcings are applied.This is the fundalmental reason why the construction of a theory of the climate responseis so challenging [3]. As already noted by Lorenz [15], it is hard to construct a one-to-onecorrespondence between forced and free fluctuations in a climatic context. Following thepioneering contribution by Leith [16], previous attempts on predicting climate response basedbroadly on applications of the fluctuation-dissipation theorem have had some degree of success[17, 18, 19, 20, 21], but, in the deterministic case, the presence of correction terms due to thesingular nature of the invariant measure make such an approach potentially prone to errors[22, 23]. Adding noise in the equations in the form of stochastic forcing - as in the case of usingstochastic parametrizations [24] in multiscale system - provides a way to regularize the problem,but it is not entirely clear how properties convergence in the zero noise. Additionally, oneshould provide a robust and meaningful construction of the model to be used for constructingthe noise: a proposal in this direction is given in [25, 26, 2].In this paper we want to show how climate change is indeed a well-posed problem atmathematical and physical level by presenting a theoretical analysis of how PLASIM [27], ageneral circulation model of intermediate complexity responds to simplified yet representativechanges in the atmospheric composition mimicking increasing concentrations of greenhousegases. We will frame the problem of studying the statistical properties of a non-autonomous,forced and dissipative complex system using the mathematical construction of the pullbackattractor [28, 29, 30] - see also the closely related concept of snapshot attractor [31, 32, 33, 34]- and will use as theoretical framework the Ruelle response theory [35, 22] to compute theeffect of small time-dependent perturbations on the background state. We will stick to thelinear approximation, which has proved its effectiveness in various examples of geophysicalinterest [23, 36]. The basic idea is to use a set of probe perturbations to derive the Greenfunction of the system, and be then able to predict the response of the system to a large classof forcings having the same spatial structure. In this way, the uncertainties associated to theapplication of the fluctuation-dissipation theorem are absent and the theoretical framework ismore robust. Note that, as shown in [37], one can practically implement the response theoryalso to treat the nonlinear effects of the forcing.PLASIM has O(10 ) degrees of freedom and provides a flexible tool for performing theo-retical studies in climate dynamics. PLASIM is much faster (and indeed simpler) than thestate of the art climate models used in the IPCC reports, but provides a reasonably realistic epresentation of atmospheric dynamics and of its interactions with the land surface and withthe mixed layer of the ocean. The model includes a suite of efficient parametrization of smallscale processes such as those relevant for describing radiative transfer, clouds formation, andturbulent transport across the boundary layer. It is important to recall that in climate scienceit is practically necessary and conceptually very sound to choose models belonging to differentlevels of a hyeararchy of complexity [38] depending on the specific problem to be studied: themost physically comprehensive and computationally expensive model is not the best model tobe used for all purposes; see discussion in [39].In a previous work [36] we have considered a somewhat unrealistic set-up for the model,where the meridional oceanic heat transport was set to zero, with no feedback from the climatestate. Such a limitation resulted in an extremely high increase of the globally averaged surfacetemperature resulting from higher CO concentrations. In this paper we extend the previousanalysis by using a better model and by considering a wider range of climate observables able toprovide a more complete picture of the climate response to CO concentration. In particular,we wish to show to what extent response theory is suited for performing projections of thespatial pattern of climate change.The paper is organized as follows. In Section 2 we introduce the main concepts behindthe theoretical framework of our analysis. We briefly describe the basic properties of thepullback attractor and explain its relevance in the context of climate dynamics. We thendiscuss the relevance of response theory for studying situations where the non-autonomousdynamics can be decomposed into a dominating autonomous component plus a small non-autonomous correction. In Section 3 we introduce the climate model used in this study,discuss the various numerical experiments and the climatic observables of our interest, andpresent the data processing methods used for predicting the climate response to forcings. InSection 4 we present the main results of our work. We focus on two observables of greatrelevance, namely the surface temperature and the yearly total precipitation, and investigatethe skill of response theory in predicting the change in their statistical properties, exploringboth changes in global quantities and spatial patterns of changes. We will also show how toflexibly use response theory for predicting when climate change becomes statistically significantin a variety of scenarios. In Section 5 we summarize and discuss the main findings of this work.In Section 6 we propose some ideas for potentially exciting future investigations. Pullback Attractor and Climate Response
Since the climate system experiences forcings whose variations take place on many differenttime scales [40], defining rigorously what climate response actually is requires some care. Itseems relevant to take first a step in the direction of considering the rather natural situationwhere we want to estimate the statistical properties of complex non-autonomous dynamicalsystems.Let us then consider a continuous-time dynamical system˙ x = F ( x, t ) (1)on a compact manifold Y ⊂ R d , where x ( t ) = φ ( t, t ) x ( t ), with x ( t = t ) = x in ∈ Y initialcondition and φ ( t, t ) is defined for all t ≥ t with φ ( s, s ) = . The two-time evolutionoperator φ generates a two-parameter semi-group. In the autonomous case, the evolutionoperator generates a one-parameter semigroup, because of time translational invariance, sothat φ ( t, s ) = φ ( t − s ) ∀ t ≥ s . In the non-autonomous case, in other terms, there is an absoluteclock . We want to consider forced and dissipative systems such that with probability one initialconditions in the infinite past are attracted at time t towards A ( t ), a time-dependent familyof geometrical sets. In more formal terms, we say a family of objects ∪ t ∈ R A ( t ) in the finite-dimensional, complete metric phase space Y is a pullback attractor for the system ˙ x = F ( x, t )if the following conditions are obeyed: • ∀ t , A ( t ) is a compact subset of Y which is covariant with the dynamics, i.e. φ ( s, t ) A ( t ) = A ( s ), s ≥ t . • ∀ t lim t →−∞ d Y ( φ ( t, t ) B, A ( t )) = 0 for a.e. measurable set B ⊂ Y .where d Y ( P, Q ) is the Hausdorff semi-distance between the P ⊂ Y and Q ⊂ Y . We havethat d Y ( P, Q ) = sup x ∈ P d Y ( x, Q ), with d Y ( x, Q ) = inf y ∈ Q d Y ( x, y ). We have that, in general, d Y ( P, Q ) (cid:54) = d Y ( Q, P ) and d Y ( P, Q ) = 0 ⇒ P ⊂ Q . See a detailed discussion of these conceptsin, e.g. , [28, 29, 30]. Note that a substantially similar construction, the snapshot attractor , hasbeen proposed and fruitfully used to address a variety of time-dependent problems, includingsome of climatic relevance [31, 32, 33, 34].In some cases, the geometrical set A ( t ) supports useful measures µ t (d x ). These can be ob-tained as evolution at time t through the Ruelle-Perron-Frobenius operator [41] of the Lebesguemeasure supported on B in the infinite past, as from the conditions above. Proposing a mi-nor generalization of the chaotic hypothesis [42], we assume that when considering sufficiently igh-dimensional, chaotic and non-autonomous dissipative systems, at all practical levels - i.e. when one considers macroscopic observables - the corresponding measure µ t (d x ) constructedas above is of the SRB type. This amounts to the fact that we can construct at all times t ameaningful (time-dependent) physics for the system. Obviously, in the autonomous case, andunder suitable conditions - e.g. in the case of of Axiom A system or taking the point of viewof the chaotic hypothesis - A ( t ) = Ω is the attractor of the system (where the t − dependenceis dropped), which supports the SRB invariant measure µ (d x ).Note that when we analyze the statistical properties of a numerical model describing anon-autonomous forced and dissipative system, we often follow - sometimes inadvertently -a protocol that mirrors precisely the definitions given above. We start many simulations inthe distant past with initial conditions chosen according to an a-priori distribution. Aftera sufficiently long time, related to the slowest time scale of the system, at each instant thestatistical properties of the ensemble of simulations do not depend anymore on the choice ofthe initial conditions. A prominent example of this procedure is given by how simulations ofpast and historical climate conditions are performed in the modeling exercises such as thosedemanded by the IPCC [10, 11, 12], where time-dependent climate forcings due to changes ingreenhouse gases, volcanic eruptions, changes in the solar irradiance, and other astronomicaleffects are taken into account for defining the radiative forcing to the system. Note thatfuture climate projections are always performed using as initial conditions the final states ofsimulations of historical climate conditions, with the result that the covariance properties ofthe A ( t ) set are maintained.Computing the expectation value of measurable observables for the time dependent measure µ t (d x ) resulting from the evolution of the dynamical system given in Eq. 1 is in general far fromtrivial and requires constructing a very large ensemble of initial conditions in the Lebesguemeasurable set B mentioned before. Moreover, from the theory of pullback attractors we haveno real way to predict the sensitivity of the system to small changes in the dynamics.The response theory introduced by Ruelle [35, 22] (see also extensions and a different pointof view summarized in, e.g. [43]) allows for computing the change in the measure of an AxiomA system resulting from weak perturbations of intensity (cid:15) applied to the dynamics in terms ofthe properties of the unperturbed system. The basic concept behind the Ruelle response theoryis that the invariant measure of the system, despite being supported on a strange geometricalset, is differentiable with respect to (cid:15) . See [44] for a discussion on the radius of convergence(in terms of (cid:15) ) of the response theory. n this case, instead, our focus is on saying that the Ruelle response theory allows forconstructing the time-dependent measure of the pullback attractor µ t (d x ) by computing thetime-dependent corrections of the measure with respect to a reference state. In particular, letus assume that we can write ˙ x = F ( x, t ) = F ( x ) + (cid:15)X ( x, t ) (2)where | (cid:15)X ( x, t ) | (cid:28) | F ( x ) | ∀ t ∈ R and ∀ x ∈ Y , so that we can treat F ( x ) as the backgrounddynamics and (cid:15)X ( x, t ) as a perturbation. Under appropriate mild regularity conditions, it ispossible to perform a Schauder decomposition [45] of the forcing, so that we express X ( x, t ) = (cid:80) ∞ k =1 X k ( x ) T k ( t ). Therefore, we restrict our analysis without loss of generality to the casewhere F ( x, t ) = F ( x ) + (cid:15)X ( x ) T ( t ).One can evaluate the expectation value of a measurable observable Ψ( x ) on the time de-pendent measure µ t (d x ) of the system given in Eq. 1 as follows: (cid:90) µ t (d x )Ψ( x ) = (cid:104) Ψ (cid:105) (cid:15) ( t ) = (cid:104) Ψ (cid:105) + ∞ (cid:88) j =1 (cid:15) j (cid:104) Ψ (cid:105) ( j )0 ( t ) , (3)where (cid:104) Ψ (cid:105) = (cid:82) ¯ µ (d x )Ψ( x ) is the expectation value of Ψ on the SRB invariant measure ¯ µ (d x )of the dynamical system ˙ x = F ( x ). Each term (cid:104) Ψ (cid:105) ( j )0 ( t ) can be expressed as time-convolutionof the j th order Green function G ( j )Ψ with the time modulation T ( t ): (cid:104) Ψ (cid:105) ( j )0 ( t ) = (cid:90) ∞−∞ dτ . . . (cid:90) ∞−∞ dτ n G ( j )Ψ ( τ , . . . , τ j ) T ( t − τ ) . . . T ( t − τ j ) . (4)At each order, the Green function can be written as: G ( j )Ψ ( τ , . . . , τ j ) = (cid:90) ¯ µ (d x )Θ( τ j − τ j − ) . . . Θ( τ )Λ S τ . . . S τ j − Λ S τ j Ψ( x ) , (5)where Λ( • ) = X · ∇ ( • ) and S t ( • ) = exp( tF · ∇ )( • ) is the unperturbed evolution operatorwhile the Heaviside Θ terms enforce causality. In particular, the linear correction term can bewritten as: (cid:104) Ψ (cid:105) (1)0 ( t ) , = (cid:90) ¯ µ (d x ) (cid:90) ∞ d τ Λ S τ Ψ( x ) T ( t − τ ) = (cid:90) ∞−∞ d τ G (1)Ψ ( τ ) T ( t − τ ) , (6)while, considering the Fourier transform of Eq. 6, we have: (cid:104) Ψ (cid:105) (1)0 ( ω ) = χ (1)Ψ ( ω ) T ( ω ) , (7)where we have introduced the susceptibility χ (1)Ψ ( ω ) = F [ G (1)Ψ ], defined as the Fourier transformof the Green function G (1)Ψ ( t ). Under suitable integrability conditions, the fact that the Green unction G ( t ) is causal is equivalent to saying that its susceptibility obeys the so-called Kramers-Kronig relations [46, 23], which provide integral constraints linking its real and imaginary part,so that χ (1) ( ω ) = i P (1 /ω ) (cid:63) χ (1) ( ω ), where i = √− (cid:63) indicates the convolution product, and P indicates that integration by parts is considered. See also extensions to the case of higherorder susceptibilities in [47, 48, 37, 49].As discussed in [23, 2, 36], the Ruelle response theory provides a powerful language forframing the problem of the response of the climate system to perturbations. Clearly, givena vector flow F ( x, t ), it is possible to define different background states, corresponding todifferent reference climate conditions, depending on how we break up F ( x, t ) into the twocontributions F ( x ) and (cid:15)X ( x, t ) in the right hand side of Eq. 2. Nonetheless, as long as theexpansion is well defined, the sum given in Eq. 3 does not depend on the reference state. Ofcourse, a wise choice of the reference dynamics leads to faster convergence.Note that once we define a background vector flow F ( x ) and approximate its invariantmeasure ¯ µ (d x ) by performing an ensemble of simulations, by using Eqs. 3-6 we can constructthe time dependent measure µ t (d x ) for many different choices of the perturbation field (cid:15)X ( x, t ),as long as we are within the radius of convergence of the response theory. Instead, in orderto construct the time dependent measure following directly the definition of the pullbackattractor, we need to construct a different ensemble of simulations for each choice of F ( x, t ).One needs to note that constructing directly the response operator using the Ruelle formulagiven in Eq. 6 is indeed challenging, because of the different difficulties associated to the con-tribution coming from the unstable and stable directions [50]; nonetheless, recent applicationsof adjoint approaches [51] seem quite promising [52].Instead, starting from Eqs. 6-7, it is possible provide a simple yet general method forpredicting the response the system for any observable Ψ at any finite or infinite time horizon t for any time modulation T ( t ) of the vector field X ( x ), if the corresponding Green functionor, equivalently, the susceptibility, is known. Moreover, given a specific choice of T ( t ) andmeasuring the (cid:104) Ψ (cid:105) (1)0 ( t ) from a set of experiments, the same equations allow one to derivethe Green function. Therefore, using the output of a specific set of experiments, we achievepredictive power for any temporal pattern of the forcing X ( x ). In other term, from theknowledge of the time dependent measure of one specific pullback attractor, we can derive thetime dependent measures of a family pullback attractors. We will follow this approach in theanalysis detailed below. While the methodology is almost trivial in the linear case, it is inprinciple feasible also when higher order corrections are considered, as long as the response heory is applicable [48, 37, 49].We also wish to remark that in some cases divergence in the response of a chaotic systemcan be associated to the presence of slow decorrelation for the measurable observable in thebackground state, which, as discussed in e.g. [53], can be related to the presence of nearbycritical transitions. Indeed, we have recently investigated such issue in [54], thus providing thestatistical mechanical analysis of the classical problem of multistability of the Earth’s systempreviously studied using macroscopic thermodynamics in [55, 56, 57]. The Planet Simulator (PLASIM) is a climate model of intermediate complexity, freely avail-able upon request to the group of Theoretical Meteorology at the University of Hamburg( )and includes a graphical user interface facilitating its use. By intermediate complexity we meanthat the model is gauged in such a way to be parsimonious in terms of computational cost andflexible in terms of possibility to explore widely differing climatic regimes [39]. Therefore, themost important climatic processes are indeed represented, and the model is complex enoughto feature essential characteristics of high-dimensional, dissipative, and chaotic systems, as theexistence of a limited horizon of predictability due to the presence of instabilities in the flow.Nonetheless, one has to sacrifice the possibility of using the most advanced parametrizationsfor subscale processes and cannot use high resolutions for the vertical and horizontal direc-tions in the representation of the geophysical fluids. Therefore, we are talking of a modellingstrategy that differs from the conventional approach aiming at achieving the highest possibleresolution in the fluid fields and the highest precision in the parametrization of the highestpossible variety of subgrid scale processes [12], but rather focuses on trying to reduce the gapbetween the modelling and the understanding of the dynamics of the geophysical flow [58].The dynamical core of PLASIM is based on the Portable University Model of the At-mosphere PUMA [59]. The atmospheric dynamics is modelled using the primitive equationsformulated for vorticity, divergence, temperature and the logarithm of surface pressure. Mois-ture is included by transport of water vapour (specific humidity). The governing equationsare solved using the spectral transform method [60, 61]. In the vertical, non-equally spaced igma (pressure divided by surface pressure) levels are used. The parametrization of unresolvedprocesses consists of long- [62] and short- [63] wave radiation, interactive clouds [64, 65, 66],moist [67, 68] and dry convection, large-scale precipitation, boundary layer fluxes of latent andsensible heat and vertical and horizontal diffusion [69, 70, 71, 72]. The land surface schemeuses five diffusive layers for the temperature and a bucket model for the soil hydrology. Theoceanic part is a 50 m mixed-layer (swamp) ocean, which includes a thermodynamic sea icemodel [73].The horizontal transport of heat in the ocean can either be prescribed or parametrized byhorizontal diffusion. In this case, we consider the second setting, as opposed to what exploredin [36], because it is well known that having even a severely simplified representation of thelarge scale heat transport performed by the ocean improves substantially the realism of theresulting climate. We remind that the ocean contributes to about 30% of the total meridionalheat transport in the present climate [6, 74, 75]. A detailed study of the impact of changingoceanic heat transports on the dynamics and thermodynamics of the atmosphere can be foundin [76].The model is run at T21 resolution (approximately 5 . o × . o ) with 10 vertical levels.While this resolution is relatively low, it is expected to be sufficient for obtaining a reasonabledescription of the large scale properties of the atmospheric dynamics, which are most relevantfor the global features we are interested in. We remark that previous analyses have shown thatusing a spatial resolution approximately equivalent to T21 allows for obtaining an accuraterepresentation of the major large scale features of the climate system. PLASIM features O (10 )degrees of freedom, while state-of-the-art Earth System Models boast easily over 10 degreesof freedom.While missing a dynamical ocean hinders the possibility of having a good representation ofthe climate variability on multidecadal or longer timescales, the climate simulated by PLASIMis definitely Earth-like, featuring qualitatively correct large scale features and turbulent atmo-spheric dynamics. Figure 1 provides an outlook of the climatology of the model run withconstant CO concentration of 360 ppm and solar constant set to S = 1365 W m − . Weshow the long-term averages of the surface temperature T S (panel a) and of the yearly totalprecipitation P y (panel b) fields, plus their zonal averages [ T S ] and [ P y ] . Despite the simplifi-cations of the model, one finds substantial agreement with the main features of the climatology We indicate with [ A ] the zonally averaged surface value of the quantity [ A ]. btained from observations and state-of-the-art model runs: the average temperature mono-tonically decreases as we move poleward, while precipitation peaks at the equator, as a resultof the large scale convection corresponding to the intertropical convergence zone (ICTZ), andfeatures two secondary maxima at the mid latitudes of the two hemispheres, corresponding tothe areas of the so-called storm tracks [6]. As a result of the lack of a realistic oceanic heattransport and of too low resolution in the model, the position to the ICTZ is a bit unrealisticas it is shifted southwards compared to the real world, with the precipitation peaking justsouth of the equator instead of few degrees north of it.Beside standard output, PLASIM provides comprehensive diagnostics for the nonequilib-rium thermodynamical properties of the climate system and in particular for local and globalenergy and entropy budgets. PUMA and PLASIM have already been used for several theoret-ical climate studies, including a variety of problems in climate response theory [36, 2], climatethermodynamics [77, 78], analysis of climatic tipping points [55, 54], and in the dynamics ofexoplanets [56, 57]. We want to perform predictions on the climatic impact of different scenarios of increase inthe CO concentration with respect to a baseline value of 360 ppm , focusing on observablesΨ of obvious climatic interest such as, e.g. the globally averaged surface temperature T S . Wewish to emphasise that most state-of-the-art general circulation models feature an imperfectclosure of the global energy budget of the order of 1 W m − for standard climate conditions,due to inaccuracies in the treatment of the dissipation of kinetic energy and the hydrologicalcycle [75, 2, 79, 80]. Instead, PLASIM has been modified in such a way that a more accuraterepresentation of the energy budget is obtained, even in rather extreme climatic conditions[55, 56, 57]. Therefore, we are confident of the thermodynamic consistency of our model,which is crucial for evaluating correctly the climate response to radiative forcing resultingfrom changes in the opacity of the atmosphere.We proceed step-by-step as follows: • We take as dynamical system ˙ x = F ( x ) the spatially discretized version of the partialdifferential equations describing the evolution of the climate variables in a baseline sce-nario with set boundary conditions and set values for, e.g. , the CO = 360 ppm baselineconcentration and the solar constant S = 1365 W m − . We assume, for simplicity, that
13) b)c)Figure 1: Long-term climatology of the PLASIM model. Control run performed with backgroundvalue of CO concentration set to 360 ppm and solar constant defined as S = 1365 W m − : a)Surface temperature (cid:104) T S (cid:105) (in K ); b) Yearly total precipitation (cid:104) P y (cid:105) (in mm ); c) Zonally averagedvalues (cid:104) [ T S ] (cid:105) (red line and red y − axis) and (cid:104) [ P y ] (cid:105) (blue line and blue y − axis).14 ystem does not feature daily or seasonal variations in the radiative input at the top ofthe atmosphere. We run the model for 2400 years in order to construct a long controlrun. Note that the model relaxes to its attractor with an approximate time scale of 20-30years. • We study the impact of perturbations using a specific test case. We run a first set of N = 200 perturbed simulations, each lasting 200 years, and each initialized with thestate of the model at year 200 + 10 k , k = 1 , . . . , X ( x ) the additional convergence of radiative fluxes due to changes in the atmospheric CO concentration. Therefore, such a perturbation field has non-zero components onlyfor the variables directly affected by such forcings, i.e. the values of the temperatures atthe resolved grid points of the atmosphere and at the surface. In each of these simulationwe perturbed the vector flow by doubling instantaneously the CO concentration. Thiscorresponds to having ˙ x = F ( x ) → ˙ x = F ( x ) + (cid:15) Θ( t ) X ( x ). Note that the forcing is wellknown to scale proportionally with to the logarithm of the CO concentration [6]. • By plugging T ( t ) = T a ( t ) = Θ( t ) into Eqs. 6, we have that :dd t (cid:104) Ψ (cid:105) (1)0 ( t ) = (cid:15)G (1)Ψ ( t ) (8)We estimate (cid:104) Ψ (cid:105) (1)0 ( t ) by taking the average of response of the system over a possibly largenumber of ensemble members, and use the previous equation to derive our estimate of G (1)Ψ ( t ), by assuming linearity in the response of the system. In what follows, we presentthe results obtained using all the available N = 200 ensemble members, plus, in someselected cases, showing the impact of having a smaller number ( N = 20) of ensemblemembers.It is important to emphasize that framing the problem of climate change using the formal-ism of response theory gives us ways for providing simple yet useful formulas for definingprecisely the climate sensitivity ∆ Ψ for a general observable Ψ, as ∆ Ψ = (cid:60){ χ (1)Ψ (0) } . Fur-thermore, if we consider perturbation modulated by a Heaviside distribution, we havethe additional simple and useful relation:∆ Ψ = 2 π(cid:15) (cid:90) dω (cid:48) (cid:60){(cid:104) Ψ (cid:105) (1)0 ( ω (cid:48) ) } , (9)which relates climate response at all frequencies to its sensitivity, as resulting from thevalidity of the Kramers-Kronig relations. e remark that using a given set of forced experiments it is possible to derive informationon the climate response to the given forcing for as many climatic observables as desired.It is important to note that, for a given finite intensity (cid:15) of the forcing, the accuracyof the linear theory in describing the full response depends also on the observable ofinterest. Moreover, the signal to noise ratio and, consequently, the time scales overwhich predictive skill is good may change a lot from variable to variable. • We want to be able to predict at finite and infinite time the response of the system to someother pattern of CO forcing. Following [36], we choose as a pattern of forcing one of theclassic IPCC scenarios, namely a 1% yearly increase of the CO concentration up to itsdoubling, and we perform a set of additional N = 200 perturbed simulations performedaccording to such a protocol. This corresponds to choosing a new time modulation thatcan be approximated as a linear ramp of the form T τb ( t ) = (cid:15)t/τ, ≤ t ≤ τ, T τb ( t ) = 1 , t > τ, (10)where τ = 100 log 2 years ∼
70 years is the doubling time. Therefore, for each observableΨ we compare the result of convoluting the estimate of the Green function obtained inthe previous step with time modulation T b ( t ) with the ensemble average obtained fromthe new set of simulations. The response theory sketched above allows us in principle to study the change in the statisticalproperties of any well-behaved, smooth enough observable. Nonetheless, problems naturallyemerge when we consider finite time statistics, finite number of ensemble members, and finiteprecision approximations of the response operators. The Green functions of interest arederived using Eq. 8 as time derivative of the ensemble averaged time series of the observedresponse to the probe forcing whose time modulation is given by the Heaviside distribution.Clearly, the response is not smooth unless the ensemble size N → ∞ . Therefore, taking nu-merically the time derivative leads to having a very noisy estimate of the Green function,which might also depend heavily on the specific procedure used for computing the discretederivative. This might suggest that the procedure is not robust. Instead, we need to keep inmind that we aim at using the Green function exclusively as a tool for predicting the climate esponse . Therefore, if we convolute with the Green function with a sufficiently smooth mod-ulating factor T ( t ) as in Eq. 6, the small time scales fluctuations of the Green function, albeitlarge in size, become of no relevance, because they are averaged out. This is further easedif, instead of looking for predictions valid for observables defined at the highest possible timeresolution of ur model, we concentrate of suitably time averaged quantities. Clearly, while itis in principle possible to define mathematically the climate response on the time scale of, e.g. ,one second, this has no real physical relevance. Looking at the asymptotic behaviour of thesusceptibility it is possible to derive what is, depending on the signal-to-noise ratio, the timescale over which we can expect to be able to perform meaningful predictions; see discussion in[36]. a) b)Figure 2: Climate response to instantaneous [ CO ] doubling for (a) globally averaged mean annualsurface temperature (cid:104){ T S }(cid:105) ; and (b) the globally averaged annual total precipitation (cid:104){ P y }(cid:105) (b).The black line shows the N = 200 ensemble averaged properties of the doubling CO experiments.The light red shading indicates the variability as given by the 2 σ of the ensemble members. In eachpanel, the inset portrays the corresponding Green function. In order to provide an overlook of the practical potential of the response theory in addressingthe problem of climate change, we have decided to focus on two climatological quantities ofgeneral interest, namely the yearly averaged surface temperature T S and the yearly totalprecipitation P y . Such quantities have obvious relevance for basically any possible impactstudy of climate change, and, while there is much more in climate change than studying thechange in T S and P y , these are indeed the first quantities one considers when benchmarking he performance of a climate model and when assessing whether climate change signals can bedetected.Another issue one needs to address is the role of the spatial patterns of change in theconsidered quantities. The change in the globally average surface temperature { T S } hasundoubtedly gained prominence in the climate change debate and in the IPCC negotiationstargets are tailored according to such an indicators. Nonetheless, the impacts of climate changeare in fact local and one needs to investigate the geography of the change signals [12]. Evidently,one expects that coarse grained (in space) quantities will have a better signal-to-noise ratioand will allow for performing higher precision climate projection using response theory. Onthe other side, the performance of linear response theory at local scale might be hinderedby the presence of local strongly nonlinear feedbacks, such as the ice-albedo feedback, whichhave less relevance when spatial averaging is performed. In what follows, we will considerobservables constructed from the spatial fields of T S and P y by performing different levels ofcoarse graining. We will begin by looking into globally averaged quantities, and then addressthe problem of predicting spatial pattens of climate change. We begin our investigation by focusing on the globally averaged surface temperature { T S } andthe globally averaged yearly total precipitation { P y } . In what follows, we perform the analysisusing the model output at the highest available resolution (1 day) but present, for sake ofconvenience and since we indeed focus on yearly quantities, data where a 1 − year band passfiltering is performed. Figure 2 shows the ensemble average performed over N = 200 members of the change of (cid:104){ T S }(cid:105) (panel a) and (cid:104){ P y }(cid:105) (panel b) as a result of the instantaneous doubling of the CO concentration described in the previous section. We find that the asymptotic change in thesurface temperature is given by the equilibrium climate sensitivity ECS = ∆ { T S } ∼ . K ,which is just outside the likely range of values for the ECS elicited in [12]. Note that the modelsdiscussed in [12] include more complex physical and chemical processes and most notably acomprehensive representation of the dynamics of the ocean, plus featuring a seasonal and daily We indicate with { A } the globally averaged surface value of the quantity A . ycle of radiation, so that the comparison in a bit unfair. Nonetheless, we get the sense thatPLASIM features an overall reasonable response to changes in the CO . This is confirmed bylooking at the long terms response of (cid:104){ P y }(cid:105) to [ CO ] doubling, for which we find ∆ { P y } ∼ mm , which corresponds to about 11 .
6% of the initial value. These figures are also in goodagreement with what reported in [12]. We will comment below on the relationship betweenthe climate change signal for { T S } and for { P y } .In each panel of Fig. 2 we show as inset the corresponding Green function computed accord-ing to Eq. 8. We find that both Green functions have to first approximation an exponentialbehaviour, even if one can expect also important deviations, as discussed in [36]. We will notelaborate on this. Instead we note that G (1) { P y } is more noisy that G (1) { T S } , as a result of the factthat (cid:104){ P y }(cid:105) (1) ( t ) has stronger high frequency contribution to its variability than (cid:104){ T S }(cid:105) (1) ( t ), i.e. (cid:104){ P y }(cid:105) (1) ( t ) has, unsurprisingly, has a much shorter decorrelation time, because it refersto the much faster hydrological cycle related processes.Figure 3 provides a comparison between the statistics of (cid:104){ T S }(cid:105) ( t ) and (cid:104){ P y }(cid:105) ( t ) obtainedby performing N = 200 simulations where we increase the CO concentration by 1% per yearuntil doubling, and the prediction of the response theory derived by performing the convolutionof the Green functions shown in Fig. 2 with the ramp function given in Eq. 10. We have thatthe prediction of the ensemble average δ (cid:104){ T S }(cid:105) (1)0 and of the ensemble average δ (cid:104){ P y }(cid:105) (1)0 (bluethick lines) obtained using N = 200 ensemble members for the doubling CO experiments isin good agreement for both observables with the results of the direct numerical integrations.More precisely, we have that the prediction given by the climate response lies within thevariability of the N = 200 direct simulations for basically all time horizons. The range ofvariability is depicted with a light red shade, centered on the ensemble mean represented bythe black line. Instead, within a time window of about 40 to 60 years, the response theoryslightly underestimates the true change in both { T S } and { P Y } : we will investigate below thereasons for this mismatch.It is well known that it is hard to construct a very large set of ensemble members in thecase of state-of-the-art climate models, due to the exorbitant computing costs associated witheach individual run. Additionally, in general, in the modelling practise the continuous growthin computing and storing resources is typically invested in increasing the resolution and thecomplexity of climate models, rather than populating more attentively the statistics of the runof a given version of a model. Therefore, even in coordinated modelling exercises contributingto the latest IPCC report, the various modelling groups are requested to deliver a number of
19) b)Figure 3: Climate change projections for the globally averaged mean annual surface temperature (cid:104){ T S }(cid:105) ; and (b) the globally averaged annual total precipitation (cid:104){ P y }(cid:105) . The black line shows the N = 200 ensemble averaged properties of the experiments where we have a 1% per year [ CO ]increase up to [ CO ] doubling. The light red shading indicates the variability as given by the 2 σ ofthe ensemble members. The blue shading indicates the interannual variability of the control run.The thick blue line is the projection obtained using the Green functions derived using N = 200ensemble members of the instantaneous doubling [ CO ] experiments. The dashed lines correspondto ten projections each obtained using Green functions derived from N = 20 ensemble members ofthe instantaneous doubling [ CO ] experiments. ensemble member of the order of 10 [12].In order to partially address the problem of assessing how the number of ensemble memberscan affect our prediction, we present in Fig. 3 the result of the prediction of climate changesignal for (cid:104){ T S }(cid:105) and (cid:104){ P y }(cid:105) obtained by constructing the Green function using only N = 20members of the ensemble of simulations of instantaneous CO concentration doubling. Tenthin dashed blue lines represent in each panel the result of such predictions. Interestingly,each of the prediction obtained with a reduced number of ensemble members also agrees withthe direct numerical simulations when we consider (cid:104){ T S }(cid:105) , because the spread around our bestestimates obtained using the full set of ensemble members is minimal.Instead, when looking at (cid:104){ P y }(cid:105) , we have that only some of the predictions derived usingreduced ensemble sets lie within the variability of the direct numerical simulations, with a uch larger spread around the prediction obtained with the full ensemble set. This fact isclosely related to the fact that the corresponding Green function is noisier, see Fig. 2, andsuggests that in order to have good convergence of the statistical properties of the responseoperator a better sampling of the attractor is needed.We conclude that the computational requirements for having good skill in predicting thechanges in { P y } are harder than in the case of { T S } . Indeed, the surface temperature is a good quantity in terms of our ability to predict it, and, in terms of being a good indicator ofclimate change, as it allows one to find clear evidence of the departure of the statistics from theunperturbed climate conditions. This is in agreement with the actual practice of the climatecommunity [12]. We dedicate some additional care in studying the climate response in terms of changes in theglobally averaged surface temperature. We wish to use the information gathered so far forassessing some features of climate change in different scenarios of modulation of the forcing.In particular, we focus on studying the properties of the following expression: δ (cid:104){ T S }(cid:105) (1)0 ( t, τ ) = (cid:90) ∞ d sG (1) { T S } ( t − s ) T τb ( s ) (11)when different values of the CO concentration doubling time τ are considered. This amountsto performing a family of climate projections where the rate of increase of the CO concen-tration is r τ = 100(2 (1 /τ ) −
1) % per year (where τ is expressed in years). As limiting cases,we have τ =0 - instantaneous doubling, as in fact described by the probe scenario T a ( t ), and τ → ∞ , which provides the adiabatic limit of infinitesimally slow changes.We want to show how response theory - and, in particular, Eq. 11 - can be used forproviding a flexible tool in the problem of climate change detection. The definition of a suitableprobabilistic framework for assessing whether an observed climate fluctuations is caused bya specific forcing is extremely relevant (including for legal and political reasons) and is sincethe early 2000s the subject of an intense debate [13, 14]. In this case, given our overall goals,we provide a rather unsophisticated treatment of the problem. In Figure 4 a) we present ourresults, where different scenarios of forcings are considered. The black line tells us what is thetime it takes for the projected change in the ensemble average to lie outside the 95% confidenceinterval of the statistics of the unperturbed control run, i.e. practically being outside its range f interannual variability. More precisely, the black line portrays the following escape time: t τmin, = min t | δ (cid:104){ T S }(cid:105) (1)0 ( t, τ ) ≥ σ ( { T S } ) , (12)where σ ( { T S } ) ∼ . K is the standard deviation of the yearly averaged time series of { T S } in the control run. a) b)Figure 4: Climate response at different time horizons. a) Time needed for detecting climate climatechange: ensemble average of the response is outside the interannual variability with 95% statisticalsignificance (black line); no overlap between the 95% confidence intervals representing the inter-annual variability of the control run and the ensemble variability of the projection (red line). b)Transient climate sensitivity - TCS - as measured by the ensemble average of the (cid:104) T S (cid:105) at time τ where doubling of [ CO ] is realized following an exponential increase at rate of r τ = 100(2 (1 /τ ) − τ is expressed in years. The equilibrium climate sensitivity (ECS) is indicated. Nonetheless, we would like to be able to assess when not only the projected change in theensemble average is distinguishable from the statistics of the control run, but, rather, when aan actual individual simulation is incompatible with the statistics of the unperturbed climate,because we live in one of such realizations, and not on any averaged quantity. Obviously,in order to assess this, one would require performing an ensemble of direct simulations, thusgiving no scope to any application of the response theory. We can observe, though, from Fig.3a), that the interannual variability of the control run and the ensemble variability of theperturbed run are rather similar (being the same if no perturbation is applied). Therefore,we heuristically assume that the two confidence intervals have the same width. The red line ortrays the second escape time t τmin, = min t | δ (cid:104){ T S }(cid:105) (1)0 ( t, τ ) ≥ σ ( { T S } ) , (13)such that for t ≥ t τmin, it is extremely unlikely that any realization of the climate changescenario due to a forcing of the form T τb perturbed run has statistics compatible with that ofthe control run. In other terms, t τmin, provides a robust estimate of when detection of climatechange in virtually unavoidable from a single run, while t τmin, gives an estimate of the timehorizon after which it makes sense to talk about climate change. We would like to remark thatusing the Green funtcions reconstructed from the reduced ensemble sets as shown in Fig. 3,one obtains virtually indistinguishable estimates for t ≥ t τmin, and t ≥ t τmin, for all values of τ . This suggests that these quantities are rather robust.We can detect two approximate scaling regimes, with changeover taking place for r τ ∼ • for large values of r τ ( ≥
1% per year), we have that t τmin, , t τmin, ∝ r − . τ • for moderate values of r τ ( ≤
1% per year), we have that t τmin, , t τmin, ∝ r − τ . Thelatter corresponds to the quasi-adiabatic regime and one finds that, correspondingly,that t τmin, , t τmin, ∝ τ .A quantity that has attracted considerable interest in the climate community is the so-calledtransient climate sensitivity (TCS), which, as opposed to the ECS, which looks at asymptotictemperature changes, describes the change of { T S } at the moment of [ CO ] doubling followinga 1% per year increase [81]. The difference between ECS and TSC gives a measure of the inertiaof the climate system in reaching the asymptotic increase of { T S } realized with doubled CO concentration. Using response theory, we can extend the concept of transient climate sensitivityby considering any rate of exponential increse of the CO concentration. Using Eq. 11, wehave that: T CS ( τ ) = δ (cid:104){ T S }(cid:105) (1)0 ( τ, τ ) (14)describes the change in the expectation value of { T S } at the end of the ramp of increase of CO concentration. As suggested by the argument proposed in [81], one expects that the TCSis a monotonically increasing function of τ (see Fig. 4 b), and the difference between the ECSand TCS becomes very small for large values of τ , because we enter the quasi-adiabatic regimewhere the change in the CO is slower than the slowest internal time scale of the system. .1.3 A Final Remark We would like to make a final remark of the properties of the response of the global observables { T S } and { P y } . Looking at Figs. 2 and 3, one is unavoidably bound to observe that thetemporal pattern of response of { T S } and { P y } are extremely similar. In agreement with [81](see also [12]), we find that to a very good approximation the following scaling applies for allsimulations: δ (cid:104){ P y }(cid:105) (1)0 ( t ) (cid:104){ P y }(cid:105) ∼ . δ (cid:104){ T S }(cid:105) (1)0 ( t ) K , where K is one degree Kelvin. In other terms, the two Green functions G (1) { T S } and G (1) { P y } are,to a very good approximation, proportional to each other when yearly averages are considered .Note that this scaling relations does not agree with the naive scaling imposed by theClausius-Clapeyron relation controlling the partial pressure of saturated water vapour. Infact, were the Clausius-Clapeyron scaling correct, one would have δ (cid:104){ P y }(cid:105) (1)0 ( t ) (cid:104){ P y }(cid:105) ∼ . δ (cid:104){ T S }(cid:105) (1)0 ( t ) K .
The reasons why a scaling between changes in { T S } and { P y } exists at all, and why it lookslike a modified version of a Clausius-Clapeyron-like law, are hotly debated in the literature[82, 83, 84]. While there is a very strong link between the change in the globally averaged precipitationand of the globally averaged surface temperature, important differences emerge when lookingat the spatial patterns of change of the two fields [83]. We will investigate the spatial featuresof climate response in the next subsection.The methods of response theory allow us to treat seamlessly also the problem of predictingthe climate response for (spatially) local observables. It is enough to define appropriatelythe observable Ψ and repeat the procedure described in Section 3.1. As a first step in thedirection of assessing our ability to predict climate change at local scale, we mostly concentratethe zonally (longitudinally) averaged fields [ T S ]( λ ) and [ P y ]( λ ), where we have made explicitreference to to the dependence on the latitude λ . Studying these fields is extremely relevantbecause it allows us to look at the difference of the climate response at different latitudinalbelts, which are well known to have entirely different dynamical properties, and, in particular,to look at equatorial-polar contrasts. We show in Fig. 5 the long-term change in the climatology of the [ T S ], i.e. , the climatesensitivity for each latitudinal band. We have confirmation of well-established findings: theresponse of the surface temperature is much stronger in the higher latitudes than in the tropicalregions, as a result of the local ice-albedo feedback and, secondarily, of the increased transportresulting from changes in the circulation. Additionally, there is a clear asymmetry betweenthe two hemispheres, with the response in the northern hemisphere being notably larger, as aresult of the larger land masses [12]In this case, we need first to construct a different Green function for each latitude fromthe instantaneous CO doubling experiments. Then, we perform the convolution of the Greenfunctions with the same ramp function and obtaining the prediction of the response to the 1%per year increase in the CO concentration for each latitude.Figure 6 shows the result of our application of the response theory for predicting theresponse of the zonally averaged surface temperature to the considered forcing scenario: panela) displays the projection performed using response theory, and panel b) shows the differencebetween the results obtained from the actual direct numerical simulations. We first observethat the agreement is extremely good except for the time window 20 - 60 y in the high latitudes T S . a) Projection of the change of[ T S ]. The black and red lines indicates the escape times as presented in Fig 4 for τ = 70 y. b)Difference between the ensemble average of the direct numerical simulations and the predictionsobtained using the response theory. of the Southern Hemisphere and 40 - 60 y in the high latitudes of the Northern Hemisphere,where the response theory underestimates the true amount of surface temperature increase.Something interesting happens when looking at the latitudinal profile of the time horizonsof escape from the statistics of the control run presented in Fig. 4a given by t τmin, and t τmin, , where τ is 70 y . Interestingly, we find that, while the climate response is weaker inthe tropics, one is able to detect climate change earlier than in the high latitude regions,the reason being that the interannual variability of the tropical temperature is much lower.Therefore, the signal-to-noise ratio is more advantageous. We need to note that our modeldoes not feature processes responsible for important tropical variability like El-Ni˜no-SouthernOscillation (ENSO), so this result might be not so realistic, yet it seems to have some conceptual
26) b)c) d)Figure 7: Climate response at different time horizons for the T S spatial field. a) Projection obtainedusing response theory for a time horizon of 20 y . b) Same as in a), but for 60 y . c) Difference betweenthe ensemble average of the direct numerical simulations and the prediction of the response theoryfor a time horizon of 20 y . d) Same as in c), but for 60 y . merit.It is also rather attractive the fact that we are now able to relate to specific region thecold bias of the prediction for { T S } already seen in Fig. 3. The reason for the presence ofsuch discrepancies concentrated in the high latitude regions is relatively easy to ascertain. Wecan in fact attribute this exactly to the inability of a linear method like the one used hereto represent accurately the strong nonlinear ice-albedo feedback, which dominates the climateresponse of the polar regions, and especially over the sea areas.This can be made even more clear by looking at the performance of the response theory inpredicting the 2D patterns of change of T S . This is shown in Fig. 7. where we see more clearly P y . a) Projection of the changeof the zonal average of P y for different time horizons. The black and red lines indicates the escapetimes as presented in Fig 4 for τ = 70 b) Difference between the ensemble average of the directnumerical simulations and the predictions obtained using the response theory. the geographical features of the changes in T S described above, and find confirmation thatthe sources of biases come exactly from the high-latitude sea-land margins, where sea ice ispresent. We also note that while for the time horizon of 20 y the bias between the simulationsand the predictions of the response theory is comparable to the actual signal of response, thesituation greatly improves for the time horizon of 60 y . As we see here, there is good hope inbeing able to predict quite accurately the climate response also at local scale, with no coarsegraining involved, at least in the case of the T S field.As a final step, we wish to discuss climate projections for the quantity [ P y ]. Figure 5 showsthat our model give a picture of long-term climate change that is overall compatible with the ndings of more complex and modern models: we foresee an increase of the precipitation in thetropical belt and in the regions of the storm tracks in the mid-latitude of the two hemisphere,whereas the change in small or negative in the remaining parts of the world. In other termsthe regions that get a lot of rain are going to get even more, while drier regions do not benefitfrom the overall increase in the globally averaged precipitation. As we see, there is basicallyno correspondence between the patterns of change of [ P y ] and [ T S ] (despite the strong linkbetween the response of the two globally averaged quantities) because precipitation patternschange as a result of a complex interplay of small and large scale processes, involving localthermodynamic exchanges, evaporation, as well as purely dynamical processes [83].Panel a) of Figure 8 shows how response theory predicts the change in [ P y ] at different timehorizons. We note that climate change is basically detectable only in the regions where strongincreases of precipitation takes place, and the horizon of escape from the control is much laterin time compared to the case of [ T S ], the basic reason being that the variability of precipitationis much higher.Panel b) of Fig. 8 shows the bias between the ensemble average of the numerical simulationsand the prediction of the response theory. We notice that such biases are much stronger thanin the case of [ T S ] (Fig. 6b). In particular, we find rather interesting features in the timehorizon of 20 − y for basically all latitudes. As we know from Fig. 3b), the global average ofsuch biases is rather small, but they are quite large locally. The projections performed usingresponse theory underestimate the effects of some (relevant) nonlinear phenomena that impactthe latitudinal distribution of precipitation, such as: • Change in the relative size and symmetry of the ascending and descending branches ofthe Hadley cell, and in the position of its poleward extension: it is well known that strongwarming can lead to a shift in the ITCZ, where where strong convective rain occurs, andto an extension of the dry areas of descending [85]. Correspondingly, the projectionsperformed using the response theory are biased dry in the equatorial belt and biased wetin the subtropical band. • Impact on the water budget of the mid-latitudes of the increased water transport fromthe tropical regions taking place near the poleward extension of the Hadley cell, plus thechange in the position of the stork track [86]. As a result, the projection performed usingthe response theory is biased dry compared to the actual simulations in the mid-latitudes.Comparing Figures 3a), 3b), 6b), and 8b) makes it clear that the performance of methods ased on linear response theory depends strongly on the specific choice of the observable. Infact, when we choose an observable whose properties are determined by processes that arerather sensitive to our forcing, higher order corrections will be necessary to achieve a goodprecision. This paper has been devoted to providing a statistical mechanical conceptual framework forstudying the problem of climate change. We find it useful to construct the statistical propertiesof an unavoidably non-autonomous system like the climate using the idea of the pullbackattractor and of the time-dependent measure supported on it. Response theory allows topractically compute such a time-dependent measure starting from the invariant measure of asuitably chosen reference autonomous dynamics.Using a the general circulation model PLASIM, we have developed response operators forpredicting climate change resulting from an increase in the concentration of CO . The modelfeatures only O (10 ) degrees of freedom as opposed to O (10 ) or more of state-of-the-artclimate models [12], but, despite its simplicity, delivers a pretty good representation of Earth’sclimate and of its long-term response to CO increase. PLASIM provides a low-resolution yetaccurate representation of the dynamics of the atmosphere and of its coupling with the landsurface, the ocean and the sea ice; it uses simplified but effective parametrizations for subscaleprocesses including radiation, diffusion, dissipation, convection, clouds formation, evaporation,and precipitation in liquid and solid form. The main advantage of PLASIM is its flexibility andthe relative low computer cost of launching a large ensemble of climate simulations. The maindisadvantage is the lack of a representation of a dynamic ocean, which implies that we havea cut-off at the low frequencies, because we miss the multidecadal and centennial variabilitydue to the ocean dynamics. We have decided to consider classic IPCC scenarios of greenhouseforcing in order to make our results as relevant as possible in terms of practical implications.The construction of the time dependent measure resulting from varying concentrations of CO has been achieved by first performing a first set of simulations where N = 200 ensemblemembers sampled from a long control run undergo an instantaneous doubling of the initial CO concentration. Through simple numerical manipulations, we have been able to derivethe linear Green function for any observable of interest, which makes it possible to performpredictions of climate change to an arbitrary pattern of change of the CO concentration sing simple convolutions, under the hypothesis that linearity is obeyed to a good degree ofapproximation.We have studied the skill of the response theory in predicting the change in the globallyaveraged quantities as well as the spatial patterns of change to a forcing scenario of 1% peryear increase of CO concentration up to doubling. We have focused on observables describingthe properties of two climatic quantities of great geophysical interest, namely the surfacetemperature and the yearly total precipitation. The predictions of the response theory havebeen compared to the results of additional N = 200 direct numerical simulations performedaccording this second scenario of forcing.The performance of response theory in predicting the change in the globally averagedsurface temperature and precipitation is rather good at all time horizons, with the predictedresponse falling within the ensemble variability of the direct simulations for all time horizonsexcept for a minor discrepancy in the time window 40 −
60 y. Additionally, our results confirmthe presence of a strong linear link in the form of modified Clausius-Clapeyron relation betweenchanges in such quantities, as already discussed in the literature.We have also studied how sensitive is the climate projection obtained using response the-ory to the size of the ensemble used for constructing the Green function. This is a matter ofgreat practical relevance because it is extremely challenging to run a large number of ensem-ble members for specific scenarios using state-of-the-art climate models, given their extremecomputational cost [12]. We have then tested the skill of projections of globally averagedsurface temperature and of globally averaged yearly total precipitation performed using Greenfunctions constructed using N = 20 ensemble members. We obtain that the quality of theprojection is only moderately affected, and especially so in the case of the temperature ob-servable.By performing convolution of the Green function with various scenarios where the CO increases at different rates, we are able to study the problem of climate change detection,associating to each rate of increase a time frame when climate change becomes statisticallysignificant. Another new aspect of climate response we are able to investigate thanks tothe methods developed here is the rigorous definition of transient climate sensitivity, whichbasically measures how different is the actual climate response with respect to the case ofquasi-adiabatic forcings, and defines the thermal inertia of climate.We have shown that response theory allows to put in a broader and well defined contextconcepts like climate sensitivity: we understand that the equilibrium climate sensitivity relates to the zero-frequency re-sponse of the system to doubling of the CO concentration: it is then clear that if we arenot able to resolve the slowest time scales of the climate system, we will find so-calledstate-dependency [87, 88] when estimating equilibrium climate sensitivity on slow (butnot ultraslow) time scales, because we sample different regions of the climatic attractor; • we have that concepts like time-dependency [89] of the equilibrium climate sensitivityare in fact related to the more concept of inertia of the climate response, which can beexplored by generalizing the idea of transient climate sensitivity [81] to all time scales ofperturbations.The analysis of the spatial pattern of climate change signal using response theory is entirelynew and never attempted before. Clearly, when going from the global to local scale we haveto expect lower signal-to-noise ration, as the variability is enhanced, and the possibility thatlinearity is a worse approximation as a result of powerful local nonlinear effects. Responsetheory provides an excellent tool also for predicting the change in the zonal mean of the surfacetemperature, except for an underestimation of the warming in the very high latitude regionsin the time horizons of 40 −
60 y. This is, in fact, the reason for the small bias found alreadywhen looking at the prediction of the globally averaged surface temperature. By looking atthe 2D spatial patterns, we can associate such bias to a misrepresentation of the warming inthe region where the presence of sea-ice is most sensitive to changing temperature patterns.The fact that linear response theory has problems in capturing the local features of a stronglynonlinear phenomenon like ice-albedo feedback makes perfect sense. In is remarkable that,instead, response theory is able to predict accurately the change in the surface temperaturefields in most regions of the planet.The prediction of the spatial patterns of change in the precipitation is much less successful,as a result of the complex nonlinear processes controlling the structure of the precipitativefield. In particular, response theory is not able to deal effectively with describing the polewardshift of the storm tracks, in the widening of the Hadley cell, and in the change of the ICTZ.This paper provides a possibly convincing case for constructing climate change predictionsin comprehensive climate models using concepts and methods of nonequilibrium statisticalmechanics. The use of response theory potentially allows to reduce the need for runningmany different scenarios of climate forcings as in [12], and to derive, instead, general tools forcomputing climate change to any scenario of forcings from few, selected runs of a climate model. dditionally, it is possible to deconstruct climate response to different sources of forcings apartfrom increases in CO concentration, e.g. changes in CH and aerosols concentration, in landsurface cover, in solar irradiance - and recombine it to construct very general climate changescenarios. While this operation is easier in a linear regime, it is potentially doable also in thenonlinear case, see [49] for details. The limitations of this paper point at some potentially fascinating scientific challenges to beundertaken. Let’s list some of them: • A fundamental limitation of this study is the impossibility to resolve the centennialoceanic time scales. It is of crucial importance to test whether response theory is ableto deal with prediction on a wider range of temporal scales, as required when oceandynamics is included. We foresee future applications using a fully coupled yet efficientmodel like FAMOUS [90]. • One should perform a systematic investigation of how appropriate linear approximation isin describing climate response to forcings, by computing estimates of the Green functionusing different level of CO increases and testing them against a wide range of timemodulating functions describing different scenarios of forcings. • It is necessary to study in greater detail what is the minimum size of the ensemble neededfor achieving a good precision in the construction of the Green function; the requirementdepends on the specific choice of the observable, including how it is constructed in termsof spatial and temporal averages of the actual climatic fields. • It is crucial to look at the effect of considering multiple classes of forcings in the climatechange scenarios and test how suitable combination of the individual Green functionsassociated to each separate forcing are able to predict climate response in general.Different points of view on the problem of predicting climate response should as well befollowed. The ab initio construction of the linear response operator has proved elusive be-cause of the difficulties associated with dealing effectively with both the unstable and stabledirections in the tangent space. It is promising to try to approach the problem by using theformalism of covariant Lyapunov vectors (CLVs) [91, 92, 93, 94] for having a convincing rep-resentation of the tangent space able to separate effectively and in an ordered manner the ynamics on the unstable and stable directions. CLVs have been recently shown to have greatpotential for studying instabilities and fluctuations in simple yet relevant geophysical systems[95]. By focusing on the contributions coming form the stable directions, one can also expectthat such an approach might allow for estimating the - otherwise hard to control - error inthe evaluation of the response operator introduced when applying the standard form of thefluctuation-dissipation theorem in the context of nonequilibrium systems possessing singularinvariant measure. This would help understanding under which conditions climate-relatedapplications of the fluctuation-dissipation theorem [17, 18, 19, 20, 21] have hope of beingsuccessful.One of the disadvantages of the CLVs is that constructing them is rather demanding interms of computing power and requires a global (in time) analysis of the dynamics of thesystem, in order to ensure covariance, thus requiring relevant resources in terms of memory.Additionally, one expects that all the CLVs of index up to approximately the Kaplan-Yorkedimension of the attractor of the system are relevant for computing the response. Such anumber, albeit typically much lower than the number of degrees of freedom, can still beextremely large for an intermediate complexity or, a fortiori, in a comprehensive climate model.At a more empirical level, a cheaper and effective alternative may be provided by the useof Bred vectors (BVs). See comprehensive presentations in [96, 97]. BVs are finite-amplitude,finite time vectors constructed as the difference between a background trajectory and a set ofperturbed trajectories, where the perturbations (initially chosen at random having a small, yetfinite norm) change following the nonlinear evolution of the trajectories, and are periodicallyrescaled to the prescribed initial norm. BVs provide a very efficient method for describing themain instabilities of the flow, taking into account nonlinear effects. In fact, what is extremelyinteresting about BVs is that a) their growth factors are strongly dependent on the region ofthe phase space where the system is; and b) the choice of the reference norm of the perturbationand of the time interval between two successive renormalization procedures (breeding period)effects strongly the properties of the dominant instabilities specifically active on the chosentime scales. Clearly, in the limit of infinitesimally small reference norm and infinitely longbreeding time, all BVs converge to the first CLV. This is not the case when finite size effectsbecome relevant. Instead, by considering not too small perturbations and a long enough timeinterval, the very fast instabilitities associated to the first CLVs are washed away by the lossof correlation due to nonlinear effects. In many applications of meteo-climatic relevance ithas been shown that a relatively low number of BVs is extremely effective for reconstructiong the properties of the unstable space, and that BVs contain useful information on spatiallylocalized features, so that it may be worth trying to construct an approximation to the Ruelleresponse operator using the BVs. One may note that considering different constructions ofthe BVs as discussed above might lead to the useful result of underlining different processescontributing to the response of the system.Another promising approach for studying climate response relies on reconstructing theinvariant measure of the unperturbed system using its unstable periodic orbits (UPOs) [98].Unstable periodic orbits has been shown to be a useful tool for studying persistent patternsand transitions in the context of simple atmospheric models [99, 100, 101]. Since resonancesin the susceptibility describing the frequency-dependent response of specific observables canbe associated to dominating UPOs (compare, e.g. , [37] and [102]), one can hope to be able toconstruct hierarchical approximations of the response operators by summing over larger andlarger set of UPOs.Finally, it is worth mentioning that response theory can be approached in terms of studyingthe properties of the Perron-Frobenius transfer operator and of its generator [103] of theunperturbed and of perturbed system. In other terms, the focus is on studying directly how theinvariant measure changes as a result of the applied perturbation and the challenge is in findingappropriate mathematical embedding in terms of suitable functional spaces [104, 105, 43, 106].See [3] for a proposal going in the direction of studying climate change by looking directly atmeasures rather than at observables, as instead done here.While the practical application of such an approach in a very high-dimensional problem likein the case of climate might in principle faces problems related to the curse of dimensionality,it has been advocated that it could provide an excellent framework for studying vicinity of theclimate to critical transitions [53], i.e. anticipating where there is no smoothness of the invari-ant measure with respect to perturbations. Such transitions are flagged by presence of roughdependence of the system properties on the perturbation due to presence of Ruelle-Pollicottresonances. This idea has been recently confirmed also analyzing long simulations performedwith PLASIM and constructing a reduced space from two carefully selected observables [54].Recently, a comprehensive response theory for Markov processes in a finite state spacehas been presented in the literature. Such a theory provides explicit matricial expressions ofstraightforward numerical implementation for constructing the linear and nonlinear responseoperators, including estimates of the radius of convergence [107]. Using such results in areduced state space might provide a novel and effective method for approaching the problem f climate response. Acknowledgements
The authors wish to thank M. Chekroun, H. Dijkstra, M. Ghil, A. Gritsun, A. von der Heydt,and A. Tantet for sharing many ideas on climate sensitivity and climate response. Withoutsuch stimulations this work would have hardly been possible. The authors want to thank M.Ghil for providing a constructive and insightful review of this paper. VL wishes to thank D.Ruelle for sharing questions, answers, ideas, and doubts with such a special intellectual depthand generosity.
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