Nonlin. Processes Geophys., 17, 113–122, 2010 www.nonlin-processes-geophys.net/17/113/2010/ Nonlinear Processes © Author(s) 2010. This work is distributed under in Geophysics the Creative Commons Attribution 3.0 License.

Another look at sensitivity

I. Zaliapin1 and M. Ghil2,3 1Department of Mathematics and Statistics, University of Nevada, Reno, USA 2Geosciences Department and Laboratoire de Met´ eorologie´ Dynamique (CNRS and IPSL), Ecole Normale Superieure,´ Paris, France 3Department of Atmospheric & Oceanic Sciences and Institute of Geophysics & Planetary Physics, University of California, Los Angeles, USA Received: 11 August 2009 – Revised: 15 January 2010 – Accepted: 24 February 2010 – Published: 17 March 2010

Abstract. We revisit a recent claim that the Earth’s climate 1 Introduction and motivation system is characterized by sensitive dependence to param- eters; in particular, that the system exhibits an asymmetric, 1.1 and its implications large-amplitude response to normally distributed feedback forcing. Such a response would imply irreducible uncer- Systems with feedbacks are an efficient mathematical tool for tainty in predictions and thus have notable modeling a wide range of natural phenomena; Earth’s cli- implications for climate science and climate-related policy mate is one of the most prominent examples. Stability and making. We show that equilibrium climate sensitivity in sensitivity of feedback models is, accordingly, a traditional all generality does not support such an intrinsic indetermi- topic of theoretical climate studies (Cess, 1976; Ghil, 1976; nacy; the latter appears only in essentially linear systems. Crafoord and Kall¨ en´ , 1978; Schlesinger, 1985, 1986; Cess et The main flaw in the analysis that led to this claim is in- al., 1989). Roe and Baker (2007) (RB07 hereafter) have re- appropriate linearization of an intrinsically nonlinear model; cently advocated existence of intrinsically large sensitivities there is no room for physical interpretations or policy con- in an equilibrium model with multiple feedbacks. Specifi- clusions based on this mathematical error. Sensitive depen- cally, they argued that a small, normally distributed feedback dence nonetheless does exist in the , as well may lead to large-magnitude, asymmetrically distributed val- as in climate models – albeit in a very different sense from ues of the system’s response. the one claimed in the linear work under scrutiny – and we Such a property, if valid, would have serious implications illustrate it using a classical energy balance model (EBM) for climate dynamics (Allen and Frame, 2007) and for mod- with nonlinear feedbacks. EBMs exhibit two saddle-node bi- eling of complex systems in general (Watkins and Freeman, furcations, more recently called “tipping points,” which give 2008). In this paper, we revisit the dynamical behavior of rise to three distinct steady-state , two of which are a general, equilibrium with genuinely nonlin- stable. Such bistable behavior is, furthermore, supported by ear feedbacks, and focus subsequently on a simple energy- results from more realistic, nonequilibrium climate models. balance model (EBM). We notice that the main, technical In a truly nonlinear setting, indeterminacy in the size of the part of RB07’s argument is well-known in the climate liter- response is observed only in the vicinity of tipping points. ature, cf. Schlesinger (1985, 1986), and thus it seems useful We show, in fact, that small disturbances cannot result in to review the associated assumptions and possible interpreta- a large-amplitude response, unless the system is at or near tions of this result. such a point. We discuss briefly how the distance to the bi- We rederive below in Sect. 1.2 the key equation of furcation may be related to the strength of Earth’s ice- RB07 and comment on their purportedly nonlinear analy- feedback. sis in Sect. 1.3. We then proceed in Sect. 2 with a more self-consistent version of sensitivity analysis for a nonlin- ear model. This analysis is applied in Sect. 3 to a zero- dimensional EBM. Concluding remarks follow in Sect. 4. Correspondence to: I. Zaliapin ([email protected])

Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union. 114 I. Zaliapin and M. Ghil: Climate sensitivity

which readily leads to 14

λ0  2 12 ∆T = ∆T0/(1-f) 1T = 1R +O (1T ) , (3) 1−f 10 6= Asymmetric as long as f 1.

C) response distribution o 8 RB07 drop the higher-order terms in Eq. (3) to obtain T ( T ∆ λ 6 1T = 0 1R, (4) 1−f 4 Symmetric feedback distribution which is their Eq. (S4). This equation leads directly to their 2 main conclusion, namely that a normally distributed feed- back factor f results in an asymmetric system response 1T 0 0 0.2 0.4 0.6 0.8 1 to a fixed forcing 1R. The purported sensitivity is due to Feedback factor, f the divergence of the right-hand side (rhs) of Eq. (4) (their Fig. 1. Temperature response 1T as a function of the feedback Eq. S4) as f approaches unity. Figure 1, which is analogous factor f in a linear climate model; modified after Roe and Baker to Fig. 1 of RB07, illustrates this effect. (2007). In the absence of feedbacks (f = 0), the response is given Roughly speaking, RB07 use the following argument: If o by 1T = 1T0 = 1.2 C. The response is amplified by feedbacks; the derivative of R(T ) with respect to T is close to 0, then it diverges (1T → ∞) as f → 1. The shaded areas illustrate the the derivative of T with respect to R is very large, and a small hypothetical symmetric distribution of the feedback factor f and change in the radiation R corresponds to a large change in the the corresponding asymmetric distribution of the system response temperature T . Such an argument, though, is only valid for 1T . Our study shows that this prominently asymmetric response an essentially linear dependence R ∝ T . Our straightforward is only seen in a linear model and is absent in a general nonlinear analysis in Sect. 2 below shows that the sensitivity effect of model (see Fig. 2 below). Fig. 1 is absent in climate models in which genuinely nonlin- ear feedbacks R = R(T ,α(T )) are present. It is worth noticing that, since one seeks the tempera- 1.2 Roe and Baker’s (2007) linear analysis ture change 1T that results from a change 1R in the forc- ing, it might be preferable to consider the inverse function We follow here RB07 and assume the following general T = T (R) or, more precisely, T = T (R,α(R)) and the corre- setup. Let the net radiation R at the top of the atmosphere sponding Taylor expansion be related to the corresponding average temperature T at the Earth’s surface by R = R(T ). Assume, furthermore, that ∂ T ∂ T ∂ α   = 1T = 1R + 1R +O (1R)2 . there exists a feedback α α(T ), which is affected by the ∂ R ∂ α ∂ R temperature change and which can, in turn, affect the radia- tive balance. Hence, one can write R = R(T ,α(T )). The main conclusions of our analysis will not be affected by To study how a small change 1R in the radiation is related the particular choice of direct or inverse expansion, provided to the corresponding temperature change 1T , one can use the nonlinearities are correctly taken into account. the Taylor expansion (Arfken, 1985) to obtain, as 1T tends to zero, 1.3 Roe and Baker’s “nonlinear” analysis

∂ R ∂ R ∂ α   In this section we address the analysis carried out by RB07 1R = 1T + 1T +O (1T )2 . (1) ∂ T ∂ α ∂ T in the supplemental on-line materials, pp. 4–5, Sect. “Non- linear feedbacks.” The main conclusion of that analysis was Here, O(x) is a function such that |O(x)| ≤ Cx as soon as that, given realistic parameter values for the climate system, 0 < x <  for some positive constants C and . the effects of possible nonlinearities in the behavior of the Introducing the notations function R = R(T ) are negligible and do not affect the sys- tem’s sensitivity. We point out here two serious flaws in their 1 ∂ R ∂ R ∂ α mathematical reasoning that, each separately and the two to- = , and f = −λ0 , λ0 ∂ T ∂ α ∂ T gether, invalidate such a conclusion. First, and most importantly, despite their section’s title, the for the “reference sensitivity” λ0 and the “feedback factor” analysis carried out by RB07 is still linear. Indeed, the Tay- f , we obtain lor expansion in their Eq. (S7) is given by

1−f  2 1 1R = 1T +O (1T ) , (2) 1R ≈ R01T + R00 1T 2, (5) λ0 2

Nonlin. Processes Geophys., 17, 113–122, 2010 www.nonlin-processes-geophys.net/17/113/2010/ I. Zaliapin and M. Ghil: Climate sensitivity 115 where (·)0 stands for differentiation with respect to T . But higher-order terms in the expansion of 1T are vanishingly RB07 immediately invert this equation for 1T subject to the small: assumption   1−f O (1T )2  1T ; 2 λ0 1T = 1T 1T0, and (b) the quantity in the rhs of this inequality is itself where 1T0 is a constant. Hence, instead of solving the nonzero. quadratic Eq. (5), Roe and Baker solve the following linear If one assumes, for instance, that O(1T )2 ∼ C(1T )2, approximation: where the precise meaning of g(x) ∼ f (x) is given by limx→0 g(x)/f (x) = 1, then the assumptions (a, b) above  1  1R = R0 + R00 1T 1T , hold for 1T that satisfy both of the following conditions 2 0 0 < 1T  (1−f )/(λ0 C) and 0 < 1T < , (7) and thus obtain the key formula (their Eq. S8) where C and  are defined after Eq. (1). The first of these −λ 1R conditions implies that the range of temperatures within 1T ≈ 0 ; (6) 1 00 which the approximation (4) works vanishes as the feedback 1−f − λ0 R 1T0 2 factor f approaches unity. Hence, all the results based on this 0 the last step uses the, correct, fact that R = (1 − f )/λ0. approximation – including precisely the main conclusions of This equation artificially introduces a divergence point for RB07 – no longer apply outside a vanishingly small neigh- 00 the temperature at f = 1−(λ0/2)R 1T0, which clearly can- borhood of f = 1.0. not exist in a quadratic equation. Equation (6) is thus a very The asymptotic behavior we assumed above for  crude approximation that significantly deviates from the true O (1T )2 is not exotic. Consider for instance the solution to the full quadratic Eq. (5) – which we discuss be- function R = T 2 in the neighborhood of R = 0. Its Taylor low in Sect. 2 – and thus cannot be used to justify general expansion statements about climate models.  2 The second flaw in the Roe and Baker (2007) reasoning is 1R = 2T 1T +O (1T ) that, using the model and parameter values they suggest, one can be used to obtain, ignoring the second-order term, readily finds that: 1T ≈ 1R/(2T ). (8) – R0 = −2, i.e., global temperature and radiation are neg- atively correlated, which is hardly the case for the cur- The last equation would seem to imply that the growth of rent climate (e.g., Held and Soden, 2000). We notice 1T is inversely proportional to T itself, so the change in T that the negative sign of the correlation follows directly should increase infinitely fast as T goes to 0, a rather annoy- from their Eq. (S10) and is not affected by particular ing contradiction. values of the model parameters. Furthermore, The way out of this conundrum is to realize that the change 1T given by Eq. (8) is only valid in a small vicinity of T = 0 – R00 = −0.03, which means that the model they consider and cannot be extrapolated to larger values. Of course, we all is, indeed, essentially linear, and thus not very realistic. know that the function R = T 2 is nicely bounded and smooth in the vicinity of 0, but it is essential to take into account Although, in this part of their analysis, Roe and Baker as- the second term in its Taylor expansion in this vicinity to sumed that f = 0.4, it is easy to check that, for all f < 0.95, obtain correct results. We show in Sect. 3 below that this their model satisfies |R00/R0| < 0.1 and is therewith very simple example depicts the essential dependence of the Earth close to being linear. To conclude, the effects on nonlinear- surface temperature on the global solar radiative input, for ities are indeed negligible in the particular model studied in conditions close to those of the current Earth system. this part of the RB07 paper, since the model is very close to In summary, the linear approximation of the function being linear; one cannot extrapolate, therefore, their conclu- R(T ) derived by RB07 from its Taylor expansion is not valid sions to climate models with significant nonlinearities. when f approaches unity. In this case – which is precisely We next proceed with a mathematically correct sensitivity the situation emphasized by these authors – the higher-order analysis of a general climate model in the presence of truly terms “hidden” inside O(1T )2, which they neglected, are nonlinear feedbacks. indispensable for a correct, self-consistent climate sensitivity analysis. 2 A self-consistent sensitivity analysis A correct analysis of the case when f approaches unity needs to start with a Taylor expansion that keeps the second- It is easily seen from the discussion in Sect. 1.2, especially order term from Eq. (2), that the relationship (4) is a crude approxima- 1−f 2  3 tion: it is valid only subject to the assumptions that (a) the 1R = 1T +a(1T ) +O (1T ) , λ0 www.nonlin-processes-geophys.net/17/113/2010/ Nonlin. Processes Geophys., 17, 113–122, 2010 10 I. Zaliapin and M. Ghil: Climate Sensitivity

opment: Past, Present and Future, Academic Press, San Diego, 285-325, 2000. Ghil, M., Chekroun, M. D., and Simonnet, E.: Cli- mate dynamics and fluid mechanics: Natural variabil- ity and related uncertainties, Physica D, 237, 2111–2126, doi:10.1016/j.physd.2008.03.036, 2008. Gladwell, M.: The Tipping Point: How Little Things Can Make a Big Difference, Little Brown, 2000. Hannart, A., Dufresne, J.-L., and Naveau, P.: Why climate sen- sitivity may not be so unpredictable, Geophys. Res. Lett., 36, L16707, doi:10.1029/2009GL039640, 2009. Held, I.M.: The gap between simulation and understanding in cli- mate modeling, Bull. Amer. Meteorol. Soc., 86, 1609–1614, 2005. Fig. 1. Temperature response Δ 𝑇 as a function of the feedback Held, I. M., and Suarez, M. J.: Simple albedo-feedback models of factor 𝑓 in a linear climate model; modified after Roe and Baker ice-caps. Tellus, 26, 613–629, 1974. (2007). In the absence of feedbacks (𝑓 =0), the response is given Held, I. M. and Soden, B. J.: and global warming, 𝑜 by Δ 𝑇 =Δ𝑇0 =1.2 C. The response is amplified by feedbacks; Annu. Rev. Energy Environ., 25, 441–475, 2000. it diverges (Δ 𝑇 →∞)as𝑓 → 1. The shaded areas illustrate the Hillerbrand, R. and Ghil, M.: Anthropogenic climate change: Sci- hypothetical symmetric distribution of the feedback factor 𝑓 and the entific uncertainties and moral dilemmas, Physica D, 237(14-17), corresponding asymmetric distribution of the system response Δ 𝑇 . 2132–2138, 2008. Our study shows that this prominently asymmetric response is only Hoffman, P.F., Kaufman, A.J., Halverson, G.P. and Schrag, D.P.: A seen in a linear model and is absent in a general nonlinear model Neoproterozoic snowball earth, Science, 281, 1342–1346, 1998. Hyde, W.T., Crowley, T.J., Baum, S.K. and Peltier, W.R.: (see116 Fig. 2 below). I. Zaliapin and M. Ghil: Climate sensitivity Neoproterozoic “snowball Earth” simulations with a coupled

climate/ice-sheet model, Nature, 405, 425-429, 2000. a=0 In general, one can consider an arbitrary number of terms 14 Lenton, T. M, Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahm- a=0.1 in the Taylor expansion of R(T ). The very fact that one re- a=1 liesstorf, on the S. validity and Schellnhuber, of the Taylor H. J.: expansion Tipping elements implies in that theR(T Earth’s ) a=10 12 is boundedclimate system, and sufficiently Proc. Natl. smooth; Acad. Sci. in other USA, words, 105, 1786–1793, a diver- gence2008. of the equilibrium temperature due to a small change in 10 theLorenz, forcing E.N.: contradicts Deterministic the very nonperiodic assumptions flow, on J. Atmos. which RB07 Sci., 20, based their sensitivity analysis. C) 8 130–141, 1963. o

T ( MacCracken, M. C.: Geoengineering: Worthy of Cautious Evalua- Δ 6 3tion? Sensitivity Climatic for Change, energy 77, balance 235–243, models 2006. (EBMs) McWilliams, J. C.: Irreducible imprecision in atmospheric and 4 Weoceanic consider simulations, here a classical Proc. Natl. climate Acad. model Sci. withUSA, nonlinear 104, 8709– feedbacks to illustrate that, in such a model: (i) the type of 8713, 2007. 2 sensitivity claimed by RB07 does not exist; and (ii) sensitive North, G. R.: Theory of energy-balance climate models, J. Atmos. dependence may exist, in a very different sense, namely in 0 0 0.2 0.4 0.6 0.8 1 theSci., neighborhood 32, 2033–2043, of bifurcation 1975. points, as explained below. Feedback factor, f North, G. R., Cahalan, R. F., and Coakley, J. A.: Energy-balance 3.1climate Model models, formulation Rev. Geophys., 19, 91–121, 1981. Fig.Fig. 2. 2.TemperatureTemperature response response Δ1T𝑇 as a a function function of of feedback feedback factor factor f in a nonlinear, quadratic climate model, governed by Eq. (9). The Pierrehumbert, R. T.: High levels of atmospheric 𝑓 in a nonlinear, quadratic climate model, governed by Eq. (9). The We consider here a highly idealized type of model that con- curves correspond (from top to bottom) to increasing values of the necessary for the termination of global glaciation, Nature, 429, curves correspond00 (from top to bottom) to increasing values of the nects the Earth’s temperature field to the solar radiative flux. magnitude a = R /2 of the quadratic term; the values of a for each 646–649, 2004. ′′ The key idea on which these models are built is due to magnitudecurve are given𝑎 = 𝑅 in the/2 of figure’s the quadratic legend, term;in the the upper-left values of corner.𝑎 for Theeach BudykoPoulsen,(1969 C.J.) and and Sellers Jacob,( R.L:1969). Factors They that have inhibit been sub- snow- curveuppermost are given curve in (red) the figure’s corresponds legend, to a in= the0 and upper-left is the same corner. as Thethe one shown in Fig. 1 above. Extreme sensitivity, expressed in the di- sequentlyball Earth generalized simulation, and used Paleoceanography, for many studies 19, of climate PA4021, uppermost curve (red) corresponds to 𝑎 =0and is the same as the vergence of 1T , is only seen in the linear model; it rapidly vanishes stabilitydoi:10.1029/2004PA001056, and sensitivity; see Held 2004. and Suarez (1974); North one shown in Fig. 1 above. Extreme sensitivity, expressed in the in the nonlinear model, as the nonlinearity factor a increases. (Quon,1975) C.,and andGhil Ghil,(1976 M.:), Multipleamong others. equilibria in thermosolutal con- divergence of Δ 𝑇 , is only seen in the linear model; it rapidly van- Thevection interest due to and salt-flux usefulness boundary of these conditions, “toy” J. models Fluid Mech., resides 245, ishes in the nonlinear model, as the nonlinearity factor 𝑎 increases. in two complementary features: (i) their simplicity, which 00  449–483, 1992. where a = R /2. If O (1T )3 is much smaller than the allows a complete and thorough understanding of the key other two terms on the rhs, then the temperature change can mechanisms involved; and (ii) the fact that their conclusions be approximated by a solution of the quadratic equation have been extensively confirmed by studies using much more detailed and presumably realistic models, including general 1−f 1T +a(1T )2 = 1R. (9) circulation models (GCMs); see, for instance, the reviews of λ0 North et al. (1981) and Ghil and Childress (1987). The main assumption of EBMs is that the rate of change of The real-valued solutions to the latter equation, if they exist, the global average temperature T is determined only by the are given by net balance between the absorbed radiation Ri and emitted  s  radiation Ro: 1 f −1 1−f 2 41R 1T1,2 =  ± + . dT 2 aλ aλ a c = R (T )−R (T ). (11) 0 0 dt i o For simplicity, we follow here the zero-dimensional (0- In particular, when f is close to 1.0, then D) EBM version of Crafoord and Kall¨ en´ (1978) and Ghil r1R and Childress (1987), in which only global, coordinate- 1T ≈ ± . (10) independent quantities enter; thus 1,2 a R = µQ (1−α(T )),R = σ g(T )T 4. (12) One can see from Eq. (10) that the proximity of the feed- i 0 o back factor f to unity no longer plays an important role in In the present formulation, the planetary ice-albedo feed- the qualitative behavior of the equilibrium temperature. This back α decreases in an approximately linear fashion with T , point is further illustrated in Fig. 2 that shows the climate sys- within an intermediate range of temperatures, and is nearly tem’s response 1T as a function of the feedback factor f for constant for large and small T . Here Q0 is the reference different values of the nonlinearity parameter a. The most value of the global mean solar radiative input, σ is the Stefan- important observation is that the climate response does not Boltzmann constant, and g(T ) is the grayness of the Earth, diverge at f = 1; moreover, the asymmetry of the response i.e. its deviation from black-body radiation σ T 4. The pa- due to the changes in feedback factor f rapidly vanishes as rameter µ ≈ 1.0 multiplying Q0 indicates by how much the soon as the dependence of 1R on 1T becomes nonlinear. global insolation deviates from its reference value.

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Ramanathan, V., Crutzen, P. J., Kiehl, J. T. and Rosenfeld, D.: Aerosols, climate, and the hydrological cycle, Science, 294(5549), 2119 – 2124, 2001. Roe, G. H. and Baker, M. B.: Why is climate sensitivity so unpre- dictable? Science, 318, Issue: 5850, 629–632, 2007. Ruelle, D., and Takens, F.: On the nature of turbulence, Commun. I. Zaliapin and M. Ghil: Climate sensitivity 117 Math. Phys., 20, 167–192, 1971. Schlesinger, M. E.: Feedback analysis of results from energy bal- (a) shown in the figure for smaller values of κ, correspond to ance and radiative-convective models. In MacCraken, M. C. and 0.9 Sellers-type models, in which there exists a transition ramp Luther, F. J. (eds.) Projecting the Climatic Effects on Increas- κ=0.05 between the high and low albedo values. Figure 3b shows 0.8 κ=0.1 ing Carbon Dioxide, DOE/ER-0237 US Department of Energy, the corresponding shapes of the radiative input Ri = Ri(T ). κ=0.2 Washington DC, 280–319, 1985. 0.7 The is parametrized, as in Crafoord and κ=1 Schlesinger, M. E.: Equilibrium and transient climatic warming in- α Kall¨ en´ (1978) and Ghil and Childress (1987), by letting 0.6 duced by increased atmospheric CO2, Climate Dyn., 1, 35–51,  6 g(T ) = 1−mtanh (T /T0) . (14) 1986. 0.5 Schneider, S.H. and Dickinson, R.E.: Climate modeling. Rev. Geo- Substituting this greenhouse effect parametrization and the phys. Space Phys., 25, 447, 1974. 0.4 one for the albedo into Eq. (11) leads to the following EBM: Sellers, W. D.: A climate model based on the energy balance of the Ice−albedo feedback, 0.3 ˙ earth-atmosphere systems, J. Appl. Meteorol., 8, 392–400, 1969. cT = µ Q0 (1−α(T )) 0.2 4 h  6i Simonnet, E., Dijkstra, H. A., and Ghil, M.: Bifurcation analysis of − σ T 1−mtanh (T /T0) , (15) ocean, atmosphere and climate models, in Computational Meth- 0.1 ods for the Ocean and the Atmosphere, R. Temam and J. J. Trib- 250 260 270 280 290 300 where T˙ = dT/dt denotes the derivative of global tempera- Temperature, T bia (eds.), 2009. ture T with respect to time t. (b) Smale, S.: Differentiable dynamical systems, Bull. Amer. Math. It is important to note that current concern, both scien- tific and public, is mostly with the greenhouse effect, rather Soc., 73, 199–206, 1967. than with actual changes in insolation. But in a simple Study of Man’s Impact on Climate (SMIC): Inadvertent Climate 250 EBM model – whether globally averaged, like in Crafo-

Modification. MIT Press, Cambridge (Mass.), 1971. i ord and Kall¨ en´ (1978) and here, or coordinate-dependent, as Stommel, H.: Thermohaline convection with two stable regimes of in Budyko (1969); Sellers (1969); Held and Suarez (1974); flow. Tellus, 13, 224–230, 1961. 200 North (1975) or Ghil (1976) – increasing µ always results in Temam R., 1997: Infinite-Dimensional Dynamical Systems in Me- a net increase in the radiation balance. It is thus convenient, κ=0.05 and quite sufficient for the purpose at hand, to vary µ in the chanics and Physics, Springer-Verlag, New York, 2nd Ed., 648 150 κ pp, 1997. =0.1 incoming radiation Ri, rather than some other parameter in κ=0.2 the outgoing radiation Ro. We shall return to this point in Thual, O. and McWilliams, J. C.: The catastrophe structure of Absorbed solar radiation, R κ =1 Sect. 4. thermohaline convection in a two-dimensional fluid model and 100 a comparison with low-order box models, Geophys. Astrophys. 3.2 Model parameters Fluid Dyn., 64, 67–95, 1992. 50 Watkins, N. W., and Freeman, M. P.: Geoscience - Natural com- 255 260 265 270 275 280 285 290 The value S of the solar constant, which is the value of Temperature, T plexity, Science, 320, 323–324, 2008. the solar flux normally incident at the top of the atmosphere along a straight line connecting the Earth and the Sun, is as- Wetherald, R. T. and Manabe, S.: The effect of changing the solar 𝑅 Fig. 3. Dependence of the absorbed incoming radiation i on the sumed here to be S = 1370 W m−1. The reference value of constant on the climate of a general circulation model, J. Atmos. Fig. 3. Dependence𝜅 of the absorbed incoming radiation𝛼 = 𝛼(𝑇R;i𝜅on) the steepness parameter : (a) ice-albedo feedback = ; , and the global mean solar radiative input is Q = S/4 = 342.5, Sci., 32, 2044–2059, 1975. steepness parameter κ: (a) ice-albedo feedback α α(T κ), and 0 (b) absorbed radiation 𝑅i = 𝑅i(𝑇 ; 𝜅), for different values of 𝜅; see (b) absorbed radiation Ri = Ri(T ;κ), for different values of κ; see with the factor 1/4 due to Earth’s sphericity. Eqs.Eqs. (12)(12) and (13).(13). The parameterization of the ice-albedo feedback in Eq. (13) assumes Tc = 273 K and c1 = 0.15, c2 = 0.7, which ensures that α(T ) is bounded between 0.15 and 0.85, as in We model the ice-albedo feedback by Ghil (1976); see Fig. 3a. The greenhouse effect parametriza- 1−tanh(κ (T −Tc)) tion in Eq. (14) uses m = 0.4, which corresponds to 40% α(T ;κ) = c1 +c2 . (13) −6 = × −15 −6 2 cover, and T0 1.9 10 K (Sellers, 1969; Ghil, 1976). The Stefan-Boltzmann constant is σ ≈ 5.6697 × This parametrization represents a smooth interpolation be- 10−8 W m−2 K−4. tween the piecewise-linear formula of Sellers-type models, like those of Ghil (1976) or Crafoord and Kall¨ en´ (1978), and 3.3 Sensitivity and bifurcation analysis the piecewise-constant formula of Budyko-type models, like those of Held and Suarez (1974) or North (1975). 3.3.1 Two types of sensitivity analysis Figure 3a shows four profiles of our ice-albedo feedback α(T ) = α(T ) as a function of T , depending on the value We distinguish here between two types of sensitivity analy- of the steepness parameter κ. The profile for κ  1 would sis for the 0-D EBM (11). In the first type, we assume that correspond roughly to a Budyko-type model, in which the the system is driven out of an equilibrium state T = T0, for albedo α takes only two constant values, high and low, de- which Ri(T0)−Ro(T0) = 0, by an external force, and want to pending on whether T < Tc or T > Tc. The other profiles see whether and how it will return to a new equilibrium state,

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600 μ=2.0

500

400

μ=1.0 300 Radiation, R

200 μ=0.5

100

0 150 200 250 300 350 400 Temperature, T 400

Fig. 4. Outgoing radiation 𝑅o (red, dashed line) and absorbed in- 350 κ coming solar radiation 𝑅i (blue, solid lines) for our 0-D energy- =0.01 balance model (EBM), governed by Eq. (15). The absorbed radia- 𝜇 =0.5, 1 300 tion is shown for and 2.0 (from bottom to top), while κ=1 𝜅 =0.05. Notice the existence of one or three intersection points κ 250 κ=0.05 =0.1 between the 𝑅o curve and one of the 𝑅i curves, depending on the 𝜇

value of ; these points correspond to the equilibrium solutions of Temperature, T (15), i.e. to steady-state climates. 200 12 I. Zaliapin and M. Ghil: Climate Sensitivity 118 I. Zaliapin and M. Ghil: Climate sensitivity 150

400 100 0 0.5 1 1.5 2 2.5 600 μ=2.0 μ

350 500 current Earth state Fig. 6. Equilibrium solutions of the EBM (15) for ice-albedo feed- 𝛼 = 𝛼(𝑇 ; 𝜅) 300 back functions given by Eq. (13) and corresponding 400 to the steepness parameter values 𝜅 =0.01, 0.05, 0.1 and 1. Notice

μ=1.0 250 the deformation of the stable (blue, solid lines) and unstable (red, 300 dashed lines) solution branches: the “cold” tipping point moves to Radiation, R Temperature, T 𝜇 =1.0 𝜅 200 the left, away from ,as increases; compare Fig. 3a. 200 μ=0.5

100 150

0 100 150 200 250 300 350 400 0 0.5 1 1.5 2 2.5 Temperature, T Fractional insolation change, μ 400 Fig. 4. Outgoing radiation R (red, dashed line) and absorbed in- Fig. 4. Outgoing radiation 𝑅o (red, dashed line) and absorbed in- Fig.Fig. 5. 5. EquilibriumEquilibrium solutions solutions of the EBM ((15)15) depending on on the the 350 coming solar radiation Ri (blue, solid lines) for our 0-D energy- fractional change µ in insolation. Notice the existenceκ of two sta- coming solar radiation 𝑅i (blue, solid lines) for our 0-D energy- fractional change 𝜇 in insolation. Notice the existence=0.01 of two sta- balance model (EBM), governed by Eq. (15). The absorbed radi- ble (blue, solid lines) and one unstable (red, dashed line) solution balance model (EBM), governed by Eq. (15). The absorbed radia- ation is shown for µ = 0.5,1 and 2.0 (from bottom to top), while blebranches. (blue, solid The arrows lines) and show one the unstable direction (red, in which dashed the line) global solution tem- = 𝜇 =0.5, 1 300 tionκ is0.05. shown Notice for the existenceand of 2.0 one (from or three bottom intersection to top), points while branches.perature will The change arrows after show being the direction displaced in from which a nearby the globalκ=1 equilib- tem- 𝜅between=0.05 the. NoticeRo curve the existence and one of of the oneRi orcurves, three intersection depending on points the rium state by external forces. The current Earth state corresponds perature will change after being displaced from a nearbyκ equilib- value of µ; these points correspond to the equilibrium solutions of 250 κ=0.05 =0.1 between the 𝑅o curve and one of the 𝑅i curves, depending on the to the upper stable solution at µ = 1; in this figure κ = 0.05. Eq. (15), i.e. to steady-state climates. rium state by external forces. The current Earth state corresponds 𝜇 value of ; these points correspond to the equilibrium solutions of to theTemperature, T upper stable solution at 𝜇 =1; in this figure 𝜅 =0.05. (15), i.e. to steady-state climates. 200 which may be different from the original one. This analysis The existence of the three equilibria – two stable and one refers to the “fast” dynamics of the system, and assumes that unstable – has been confirmed by such results being obtained 150 Ri(T )−Ro(T ) 6= 0 for T 6= T0; it is often referred to as linear by several distinct EBMs, of either Budyko- or Sellers-type stability analysis, since it considers mainly small displace- (North et al., 1981; Ghil, 1994). Nonlinear stability, to large 400 100 ments from equilibrium at t = 0, T(0) = T0 + θ(0), where perturbations0 in the0.5 initial state,1 has been1.5 investigated2 by in-2.5 θ(0) is of order , with 0 <   1, as defined in Sect. 2. troducing a variational principleμ for the latitude-dependent The350 second type of analysis refers to the system’s “slow” EBMs of Sellers (Ghil, 1976) and of Budyko (North et al., dynamics. Wecurrent are Earth interested state in how the system evolves 1981Fig. 6.) type,Equilibrium and it confirms solutions the of the linear EBM stability (15) for results. ice-albedo feed- 𝛼 = 𝛼(𝑇 ; 𝜅) along300 a branch of equilibrium solutions as the external force back functions given by Eq. (13) and corresponding changes sufficiently slowly for the system to track an equi- 3.3.2to the steepness Sensitivity parameter analysis values for𝜅 a=0 0-D.01 EBM, 0.05, 0.1 and 1. Notice librium state; hence, this second type of analysis always as- 250 the deformation of the stable (blue, solid lines) and unstable (red, sumes that the solution is in equilibrium with the forcing: We analyze here the stability of the “slow,” quasi-adiabatic dashed lines) solution branches: the “cold” tipping point moves to − = (in the statistical-physics sense) dynamics of model (15). RiTemperature, T (T ) Ro(T ) 0 for all T of interest. Typically, we want the left, away from 𝜇 =1.0,as𝜅 increases; compare Fig. 3a. to know200 how sensitive model solutions are to such a slow The energy-balance condition for steady-state solutions Ri = change in a given parameter, and so this type of analysis is Ro takes the form called150sensitivity analysis. In the problem at hand, we will 4 h  6i study – again following Crafoord and Kall¨ en´ (1978) and Ghil µQ0 (1−α(T )) = σ T 1−mtanh (T /T0) . (16) and100 Childress (1987) – how changes in µ, and hence in the global insolation,0 0.5 affect the1 model’s equilibria.1.5 2 2.5 We assume here, following the previously cited EBM work, Fractional insolation change, μ A remarkable property of the EBM governed by Eq. (11) that the main bifurcation parameter is µ; this happens to is the existence of several stationary solutions that describe agree with the emphasis of Roe and Baker (2007) on cli- Fig.equilibrium 5. Equilibrium climates solutions of the Earth of the (Ghil EBM and (15) Childress depending, 1987 on the). mate sensitivity as the dependence of mean temperature T 𝜇 fractionalThe existence change andin linear insolation. stability Notice of the these existence solutions of two result sta- on global solar radiative input, denoted here by Q = µQ0. blefrom (blue, a straightforward solid lines) andbifurcation one unstableanalysis (red, dashed of the line) 0-D solution EBM Figure 4 shows the absorbed and emitted radiative fluxes, branches.(11), as well The arrows as of its show one-dimensional, the direction in which latutude-dependent the global tem- Ri and Ro, as functions of temperature T for µ = 0.5,1 and peraturecounterparts will change (Ghil, after1976 being, 1994 displaced): there fromare two a nearby linearly equilib- sta- 2.0. One can see that Eq. (16) may have one or three solu- riumble solutions state by external – one forces.that corresponds The current to Earth the presentstate corresponds climate tions depending on the value of µ: only the present, relatively and one that corresponds to a much colder, “snowball Earth” warm climate for µ = 2.0, only the “deep-freeze” climate to the upper stable solution at 𝜇 =1; in this figure 𝜅 =0.05. (Hoffman et al., 1998) – separated by an unstable one, which for µ = 0.5, and all three, including the intermediate, un- lies about 10 K below the present climate. stable one for present-day insolation values, µ = 1.0. These

Nonlin. Processes Geophys., 17, 113–122, 2010 www.nonlin-processes-geophys.net/17/113/2010/ I. Zaliapin and M. Ghil: Climate sensitivity 119 steady-state climate values are shown as a function of the in- of their five model levels, exhibits such a parabolic depen- solation parameter µ in the bifurcation diagram of Fig. 5. dence on fractional radiative input; see Fig. 5 in their paper. The “fast” stability analysis (not presented here) shows Moreover, these authors emphasize that “As stated in the In- that small deviations θ(0) from an equilibrium solution, troduction, it is not, however, reasonable to conclude that the while all parameter values are kept fixed, may result in two present results are more reliable than the results from the types of dynamics, depending on the initial equilibrium T0: one-dimensional studies mentioned above simply because fast increase or fast decrease of the initial deviation (Ghil and our model treats the effect of transport explicitly rather than Childress, 1987). The fast increase characterizes unstable by parameterization. [...] Nevertheless, it seems to be sig- equilibria: a small deviation θ(0) from such an equilibrium nificant that both the one-dimensional and three-dimensional T0 forces the solution to go further and further away from models yield qualitatively similar results in many respects.” the equilibrium. In practice, such equilibria will not be ob- In fact, rigorous mathematical results demonstrate that served, since there are always small, random perturbations the saddle-node bifurcation whose normal form is given by of the climate present in the system: just think of weather as Eq. (17) occurs in several systems of nonlinear partial differ- representing such perturbations. ential equations, such as the Navier-Stokes equations (Con- The fast decrease of the initial deviation θ(0) character- stantin et al., 1989; Temam, 1997), and not only in ordinary izes stable solutions; only such equilibria can be observed in differential equations, like Eqs. (11) and (15) above. We practice. The two stable solution branches of Eq. (15) are emphasize, though, that this does not cause the temperature shown by solid lines in Fig. 5, while the unstable branch is to increase rapidly due to small changes in insolation: the shown by the dashed line. The arrows show the direction in presence of the bifurcation point will result in small, positive which the temperature will change when drawn away from an changes of global temperature for slow, positive changes in equilibrium by external forces. This change, whether away µ, while it may throw the climate system into the deep-freeze from or towards the nearest equilibrium, is fast compared to state for slow, negative changes in µ. the one that occurs along either solution branch (Ghil, 1976, 1994). 4 Discussion 3.3.3 Bifurcation analysis 4.1 How sensitive is climate? Given the choice of model parameters, the present climate state corresponds to the upper stable solution of Eq. (15), Making projections of climate change for the next decades at µ = 1 (see Fig. 5). It lies quite close to the bifurcation and centuries, evaluating the human influence on future Earth point (µ,T ) ≈ (0.9,280 K), where the stable and unstable so- temperatures, and making normative decisions about current lutions merge. and future anthropogenic impacts on climate are enormous The so-called normal form of this bifurcation is given by tasks that require solid scientific expertise, as well as respon- the equation sible moral reasoning. Well-founded approaches to handle X˙ =µ ¯ −X2, (17) the moral aspects of the problem are still being debated (e.g., Hillerbrand and Ghil, 2008). It is that much more important where X is a suitably normalized form of T , and µ¯ is a nor- to master existing tools for acquiring accurate and reliable malized form of µ. Equation (17) describes the dependence scientific evidence from the available data and models. Sev- between T and µ in a small neighborhood of the bifurca- eral of these tools come from the realm of nonlinear and com- tion point. In particular, the stable equilibrium branch is de- plex dynamical systems (Lorenz, 1963; Smale, 1967; Ruelle scribed by and Takens, 1971; Ghil and Childress, 1987; Ghil, 1994; Ghil et al., 2008). X = +pµ¯ ; A straightforward analysis, carried out in Sect. 2 of this pa- this result has exactly the same form as the positive solu- per, shows that a proper treatment of the higher-order terms tion of Eq. (10), given by our self-consistent analysis of cli- in a climate model with nonlinear feedbacks does not reveal mate sensitivity in the presence of genuine nonlinearities, cf. the exaggerated sensitivity to forcing that was used in RB07 Sect. 2. Hence, the derivative dX/dµ¯ , and thus dT/dµ, goes to advocate intrinsic unpredictability of climate projections. to infinity as the model approaches the bifurcation point; this We emphasize that the error in Roe and Baker’s analysis is is exactly the situation discussed earlier in Sect. 2. not related to their choice of model formulation or of the It is important to realize that the parabolic form of tem- model parameters nor to their interpretation of model results. perature dependence on insolation change is not an accident The problem is purely a matter of elementary calculus, and due to the particularly simple form of EBMs. Wetherald and is due to inappropriate, and unnecessary, linearization of a Manabe (1975) clearly showed, in a slightly simplified GCM, nonlinear model. that not only the mass-weighted temperature of their total at- Our analysis complements, reinforces and goes beyond mosphere, but also the area-weighted temperatures of each that of Hannart et al. (2009), who also showed that the claim www.nonlin-processes-geophys.net/17/113/2010/ Nonlin. Processes Geophys., 17, 113–122, 2010 120 I. Zaliapin and M. Ghil: Climate sensitivity of RB07 “results from a mathematical artifact.” We notice It has become common in recent discourse about po- simply that Hannart and colleagues did not even question the tentially irreversible climate change to talk about “tipping linear approximation framework of RB07 and still concluded points”; e.g., Lenton et al. (2008). The term was originally that the claims of irreducibility of the spread in the envelope introduced into the social sciences by Gladwell (2000) to de- of climate sensitivity are not supported by the RB07 analysis. note a point at which a previously rare phenomenon becomes To summarize, while the general human concern about cli- dramatically more common. In the physical sciences, it has mate sensitivity expressed by RB07 should be reasonably been identified with a shift from one stable equilibrium to an- shared by many, their scientific conclusions do not follow other one, i.e., to a saddle-node bifurcation, as seen in Fig. 5 from their model and its results, when correctly analyzed, as here and explained in Sect. 3.3.3 above. done here in Sect. 2. Nor are these conclusions supported In the EBM context of Fig. 5, it would require an enor- by other models of greater detail and realism, when properly mous, almost twofold increase in the insolation in order for a investigated. Accordingly, conclusions about the likelihood deep-freeze-type equilibrium to reach the bifurcation point at of extreme warming resulting from small changes in anthro- µ ≈ 1.85 and jump from there to T ≈ 350 K, a temperature pogenic forcing can hardly be used to support political pro- that sounds equally unpleasant. Within the broader context posals (e.g., Allen and Frame, 2007) that claim to provide of the recent debates on how to exit a snowball-Earth state, future directions for the climate-related sciences. very large, and possibly implausible increases in CO2 levels Still, this paper’s analysis does not preclude in any sense would be required (Pierrehumbert, 2004). the Earth’s temperature from rising significantly in coming Indeed, the likelihood to actually reach the tipping point years. The methods illustrated here can only be used to study to the left of the current climate in Fig. 5 seems to be quite climate sensitivity in the vicinity of a given state; they can- small. Mechanisms for entering a snowball-Earth climate not be applied to investigate climate evolution over tens of have been recently studied with a number of fairly realistic years, for example in response to large increases in green- climate models (Hyde et al., 2000; Donnadieu et al., 2004; house gases or to other major changes in the forcing, whether Poulsen and Jacob, 2004). Both modeling and independent natural or anthropogenic. This latter problem requires global geological evidence suggest that Earth’s climate can sustain interdisciplinary efforts and, in particular, the analysis of the significant fluctuations of the solar radiative input, and hence entire hierarchy of climate models (Schneider and Dickin- of global temperature, without entering the snowball Earth, son, 1974), from conceptual to intermediate to fully coupled and evidence for Earth ever having been in such a state is still GCMs (Ghil and Robertson, 2000). It also requires a much controversial. more careful study of random effects than has been done Nevertheless, the existence of the upper-left tipping point heretofore (Ghil et al., 2008). shown in Fig. 5 is confirmed by numerous model studies, It seems to us that Roe and Baker’s title question “Why Is including GCMs, and we have already cited some evidence Climate Sensitivity So Unpredictable?” still remains open. also for the lower-right tipping point in the figure. Several hypothetical tipping points on the “warm” side have been 4.2 Where are the “tipping points”? identified by Lenton et al. (2008) and references therein, among many others. But only few of these have been stud- The S-shaped diagram of Fig. 5 – see also Fig. 10.6 in Ghil ied with the same degree of mathematical and physical detail and Childress (1987) and Fig. 4 in Ghil (1994) – was used as the ones of Fig. 5 here. One worthwhile example is that here to show the smoothness and boundedness of tempera- of the oceans’ buoyancy-driven, or thermohaline, circulation ture changes as a function of insolation changes, away from (Stommel, 1961; Bryan, 1986; Quon and Ghil, 1992; Thual a saddle-node bifurcation, like that of Eq. (10) in Sect. 2 or and McWilliams, 1992; Dijkstra and Ghil, 2005). of Eq. (17) in Sect. 3.3.3. Accordingly, humankind must be careful – in pursuing This S-shaped curve nevertheless reveals the existence of its recent interest in geoengineering (Crutzen, 2006; Mac- sensitive dependence of Earth’s temperature on insolation Cracken, 2006) – to stay a course that runs between tipping changes, or on other changes in Earth’s net radiation budget, points on the warm, as well as on the “cold” side of our cur- such as may be caused by increasing levels of greenhouse rent climate. In any case, the existence, position and prop- gases, on the one hand, or of aerosols, on the other. This sen- erties of such tipping points need to be established by physi- sitive dependence is quite different from the one advocated cally careful and mathematically rigorous studies. The “mar- by RB07. Namely, if the parameter µ were to slightly de- gins of maneuver” seem reasonably wide, at least on the time crease – rather than increase, as it seems to have done since scale of tens to hundreds of years, but this does not eliminate the mid-1970s, in the sense described in the last paragraph the possibility to eventually reach one such tipping point, and of Sect. 3.1 – then the climate system would be pushed past thus we are led directly to the next question. the bifurcation point at µ ≈ 0.9. The only way for the global temperature to go would be down, all the way to a deep- freeze Earth, with much lower temperatures than those of re- cent, Quaternary ice ages.

Nonlin. Processes Geophys., 17, 113–122, 2010 www.nonlin-processes-geophys.net/17/113/2010/ 12 I. Zaliapin and M. Ghil: Climate Sensitivity

600 μ=2.0

500

400

μ=1.0 300 Radiation, R

200 μ=0.5

100 I. Zaliapin and M. Ghil: Climate sensitivity 121 0 150 200 250 300 350 400 Temperature, T 400 current computing capabilities, along with well-developed methods of numerical bifurcation theory (Dijkstra and Ghil, 𝑅 Fig. 4. Outgoing radiation o (red, dashed line) and absorbed in- 2005; Simonnet et al., 2009). This approach holds some 350 κ coming solar radiation 𝑅i (blue, solid lines) for our 0-D energy- =0.01 promise in evaluating the distance of the current climate state balance model (EBM), governed by Eq. (15). The absorbed radia- from either a catastrophic warming or a catastrophic cooling. 300 tion is shown for 𝜇 =0.5, 1 and 2.0 (from bottom to top), while κ =1 Acknowledgements. We thank two anonymous referees for their 𝜅 =0.05. Notice the existence of one or three intersection points κ 250 κ=0.05 =0.1 constructive comments that helped improve the paper. This study between the 𝑅o curve and one of the 𝑅i curves, depending on the was supported by US Department of Energy grants DE-FG02- 𝜇

value of ; these points correspond to the equilibrium solutions of Temperature, T 07ER64439 and DE-FG02-07ER64440 from its Climate Modeling (15), i.e. to steady-state climates. 200 Programs.

Edited by: H. A. Dijkstra 150 Reviewed by: P. Ditlevsen and another anonymous referee

400 100 0 0.5 1 1.5 2 2.5 μ References

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