## Abstract

We consider the dynamical evolution of a simple climate system that describes the average temperature of the Earth’s atmosphere owing to radiative forcing and coupling to a positive feedback variable such as the concentration of greenhouse gases in the presence of fluctuations. Analysing the resulting stochastic dynamical system shows that, if the temperature relaxes rapidly relative to the concentration, the time-dependent and stationary probability density functions (pdfs) for the temperature rise possess a fat tail. In contrast, if the feedback variable relaxes rapidly relative to the temperature, the pdf has no fat tail, and, instead, the system shows critical slowing down as the singular limit of positive feedback is approached. However, if there is uncertainty in the feedback variable itself, a fat tail can reappear. Our analysis may be generalized to more complex models with similar qualitative results. Our results have policy implications: although fat tails imply that the expectation of plausible damage functions is infinite, the pdfs permit an examination of the trade-off between reducing emissions and reducing the positive feedback gain.

## 1. Introduction and model

Global warming couples a variety of effects on multiple space and time scales, from fast atmospheric circulation on a daily or weekly time scale to slow, large-scale ocean circulation that varies on time scales of centuries to millennia. It is also greatly influenced by feedback mechanisms that include variations in polar ice-cap extent, desertification, water vapour concentration and cloud feedback, to name just a few causes (Graves *et al.* 1993; Scheffer *et al.* 2006; Torn & Hart 2006). The combination of these multi-scale effects and feedback leads to large uncertainties in the outcome of climate models, so that the natural language for climate variables, such as the global mean temperature, is its probability density function (pdf), rather than its deterministic value.

Understanding the effects of uncertainty from the properties of the pdf and its temporal evolution is thus of crucial importance. One approach to obtaining the pdf is via global climate models (GCMs) that are capable of simulating some of the highly complex radiative, chemical, thermodynamic and hydrodynamic processes. However, sheer practicality means that GCMs leave out effects associated with fluctuations that occur on fast time scales. Here, we analyse the effect of a single feedback mechanism on the dynamical evolution of the temperature, in a simple stochastic climate model, in order to sharpen the understanding of the role that these fluctuating effects play.

The simple model we adopt consists of a pair of coupled stochastic dynamical equations for the time evolution of the temperature *T*(*t*) (Tung 2007) and the concentration of greenhouse gases (GHGs), *c*(*t*):
1.1
and
1.2
Here, equation (1.1) is the radiative energy balance equation averaged over the entire Earth with an atmosphere that has density *ρ*, specific heat *C*_{p} and depth *H*. *σ* is the Stefan–Boltzmann constant and *g*(*c*) is a feedback parameter that modifies the Stefan–Boltzmann law in a concentration-dependent manner. The atmosphere heats or cools because of an imbalance between (i) the net solar influx *R*(*c*), which accounts for the incoming solar radiation and the effects of albedo and atmosphere in a composition-dependent manner, characterized in terms of the average concentration *c*(*t*) of GHGs, and (ii) the radiation from Earth, accounted for by a modification of the Stefan–Boltzmann law that introduces feedback in the presence of GHGs. Equation (1.2) is the scaled equation for the evolution of the concentration of GHGs, with being the addition rate of GHG per unit volume to the atmosphere. The terms and *f*_{c}(*t*) are independent random forcing terms, assumed to be Gaussian, that represent fast, uncorrelated, changes in temperature and GHG emission rate, respectively. These coupled stochastic dynamical equations are ubiquitous in modelling physical phenomena, and it may be pertinent to point to a formal resemblance of the current system to intermittency in single-molecule spectroscopy (Wang & Wolynes 1995, 1999; Cao & Silbey 2008).

To understand the effects of climate change driven by changes in concentration of GHGs, we consider small deviations from the equilibrium temperature *T*_{0} consistent with a pre-industrial equilibrium concentration of GHGs, *c*_{0}. Expanding about this state, i.e. substituting Δ*T*(*t*)=*T*(*t*)−*T*_{0} and Δ*c*(*t*)=*c*(*t*)−*c*_{0}, and keeping only leading order terms leads to a scaled form of the coupled climate model, equations (1.1) and (1.2) (see appendix A),
1.3
and
1.4
where *g*=*g*_{0}+*g*′_{0}Δ*c*, *γ*=*g*′_{0}/(1−*g*_{0}) with *g*_{0}=*g*(*c*_{0}), and the incremental radiative forcing, *λ*_{0}Δ*R*_{p}, to the lowest order in the concentration deviation, is *λ*_{0}Δ*R*_{p}=*λ*_{0}*R*′_{p}(*c*)Δ*c*, with *λ*_{0} a constant. Equation (1.3) couples the change in the equilibrium temperature via feedback to the change in GHG concentration only, although, in general, feedback may also depend on water vapour concentration, clouds, soil conditions, etc., each of which may evolve separately. Here, we focus on the influence of GHG concentration on the feedback and so assume *g*_{0}<1. Then, the feedback parameter *γ* couples the GHG concentration to the temperature, with *γ*>0 (*γ*<0) corresponding to positive (negative) feedback.

## 2. Analysis of model

There are two time scales in the problem defined by equations (1.3) and (1.4), set by the relaxation times, *τ*_{T} and *τ*_{c}, for the equilibration of temperature and concentration. Both relaxation times are determined by loss factors, such as transfer of heat or mass to the ocean, and by dissipation arising from the fluctuations. The potential for vastly different time scales for the relaxation of temperature and GHG concentration, both of which can themselves vary over many orders of magnitude, suggests that the dynamics of the simple climate system, equations (1.3) and (1.4), is best treated via an analysis of various limiting cases.

We first consider the case of fast temperature relaxation, i.e. *τ*_{T}/*τ*_{c}≪1. On the time scale *τ*_{c}∼*t*≫*τ*_{T}, radiative equilibrium is established. Incremental radiative forcing, Δ*R*_{p}(*c*), causes a temperature change Δ*T*=*λ*_{0}Δ*R*_{p}(*c*)/(1−*γ*Δ*c*). For simplicity, we consider the case of linear forcing, Δ*R*_{p}(*c*)=*α*Δ*c*, so that
2.1
It is useful to rewrite the above relations in terms of the equilibrium climate sensitivity parameter using the logarithmic derivative or elasticity, , following Torn & Hart (2006), and the sensitivity of the fluctuations, .^{1} When the radiative forcing is linear, *S*_{0}=(1+*γ*)/4, indicating greater sensitivity for larger positive feedback values, but the sensitivity to fluctuations is still greater, since *S*_{Δ}=*γ*Δ*c*(1−*γ*Δ*c*)^{−1}. In the absence of feedback, *S*_{0}=1/4, consistent with the Stefan–Boltzmann law, whereas *S*_{Δ}=0 since there is no incremental change in the temperature in the absence of GHG fluctuations.

For positive feedback, *γ*>0, we may write the above relation in two equivalent forms: Δ*T*/Δ*T*_{0}=*γ*Δ*c*/(1−*γ*Δ*c*) or *γ*Δ*c*=Δ*T*/(Δ*T*_{0}+Δ*T*). It is evident that these relations are valid only for ; as this point is approached, the average atmospheric temperature diverges since heat loss vanishes so that as .

The concentration, *c*(*t*), evolves on the slow time scale, i.e. *t*∼*τ*_{c}≫*τ*_{T}, according to equation (1.4). If the fluctuation *f*_{c}(*t*) is Gaussian, then the associated Ornstein–Uhlenbeck process can be described using the Ito procedure by a Fokker–Planck (FP) equation,
for the time-dependent pdf, *p*(Δ*c*,*t*) (Cao & Silbey 2008), where 〈*f*_{c}(*t*_{1})*f*_{c}(*t*_{2})〉=*Γ**δ*(*t*_{1}−*t*_{2}). For the case of constant emission rate, *q*(*t*)=*c**o**n**s**t*., the solution of the above FP equation is (Gardiner 2004)
2.2
following the initial increment in the concentration of GHGs and *β*(*t*)=2*τ*_{c}(1−*e*^{−2t/τc}). The appropriate boundary conditions for this FP equation are . The above expression is of the form and vanishes when and may thus be used to write the correctly normalized anti-symmetric combination valid over the allowed range.

The time-dependent pdf for the temperature is found by the change of variables *γ*Δ*c*=Δ*T*/(Δ*T*_{0}+Δ*T*), with *γ*≠0, which is restricted to the range, 0≤Δ*c*≤*γ*^{−1} (or equivalently, ), and transforms *p*^{c}(Δ*c*,*t*|Δ*c*(0)) according to *p*(Δ*T*,*t*|Δ*T*(0))=*p*^{c}(Δ*c*,*t*|Δ*c*(0))(*d*Δ*c*/*d*Δ*T*) so that
2.3
where
In figure 1, we show the temporal evolution associated with this function for parameters chosen to be consistent with current global warming scenarios. At very long times, the stationary pdf for the temperature difference,
exhibits a fat tail in agreement with the results of and implies that the expected value for all times. We note that, here the fat tail in the pdf does not reflect a mechanism giving rise to Levy flights, but is simply the consequence of the nonlinear change of variables in the pdf from concentration to temperature, with an underlying conventional Gaussian random process. For the parameters based on IPCC data and projections (Intergovernmental Panel on Climate Change 2007) (see figure 1 caption for details), the maximum in the pdf in 100 years occurs at Δ*T*=2.66^{°}C as the concentration increases from 350 to 550 ppm. However, the probability is 0.24 that Δ*T*≥4^{°}C and 0.10 that Δ*T*≥5^{°}C.

For negative feedback, *γ*<0,^{2} the steady-state condition now covers the entire concentration range, , but the temperature is restricted to the range 0≤Δ*T*≤Δ*T*_{0}. The expression for the time-dependent pdf, equation (2.3), remains valid with the modifications Δ*T*/Δ*T*_{0}=|*γ*|Δ*c*/(1+|*γ*|Δ*c*) and *Q*(Δ*T*)=[Δ*T*/(Δ*T*_{0}−Δ*T*)−1−|*γ*|*q**τ*_{c}]. Beyond the range Δ*T*>Δ*T*_{0}, *p*(Δ*T*,*t*|Δ*T*_{0}) is zero and there is no fat-tailed distribution.

We next consider the case of slow temperature relaxation, i.e. *τ*_{T}/*τ*_{c}≫1. This situation arises, for example, when considering the long time scale for energy transfer from the atmosphere to the ocean (Baker & Roe 2009). On the fast time scale, the concentration will adopt its steady-state value according to equation (1.4) and *c*_{s}(*t*)=*q*(*t*)*τ*_{c}. On this time scale, there is no dynamic feedback and the linearized stochastic equation for the temperature, equation (1.3), is
2.4
For the case of linear radiative forcing and constant *c*_{s}, the stationary temperature, Δ*T*_{e}, is shifted higher, according to the relation Δ*T*_{e}=Δ*T*_{0}*γ*Δ*c*_{s}/(1−*γ*Δ*c*_{s}). This stochastic equation can again be expressed as an FP equation with a solution given in terms of the temperature difference *δ**T*(*t*)=Δ*T*(*t*)−Δ*T*_{e} as
2.5
with and .

Although this pdf does not show a fat tail, the system now exhibits critical slowing down since and as (*γ*Δ*c*_{s})→1. This corresponds to either a relatively large value of the feedback factor *γ* or a large concentration of GHG in the atmosphere.

For the case of slow temperature relaxation, it is possible for *p*(*δ**T*,*t*|*δ**T*(0)) to exhibit a fat tail if there is a subsidiary pdf for the feedback parameter *γ* or equivalently *g*_{0}. For example, if the gain *g*=(1−*g*_{0})(1−*γ*Δ*c*_{s}) is distributed according to a pdf *h*(*g*), which is finite at *h*(1), then, as shown in appendix B, *p*(*δ**T*,*t*|*δ**T*(0))→*h*(1)*δ**T*^{−2} as .

## 3. Discussion

In summary, we find that, if the temperature relaxes on a fast time scale compared with that for concentration equilibration, the temperature is slaved to the variability in the concentration. For positive feedback, *γ*>0, both stationary and time-dependent pdfs for the temperature change have fat tails owing to the stochastic dynamics of the concentration field. In contrast, if the temperature relaxes on a slow time scale compared with the time scale for concentration equilibration, the stationary and time-dependent pdfs do not possess fat tails, and the stochastic dynamics of the temperature determines the form of the time-dependent pdf. In particular, the characteristic relaxation time exhibits critical slowing down, as the singular point of the positive feedback is approached. However, if uncertainty is introduced in the feedback variable, *γ*, a fat tail may reappear.

We conclude with an application of our results to a policy issue. Modellers of global climate know that feedback effects are crucial. They observe simulation outcomes that are skewed to high temperatures (Webster *et al.*2001, 2003; Forest *et al.* 2002). The existence of a fat tail is not an artefact of computational GCM simulation that will disappear with repeated Monte-Carlo trials, but rather an inherent consequence of the presence of positive feedback, as our analysis of the simple model examined here shows. pdfs with fat tails present formidable problems for conventional expected value cost–benefit analysis because of the relatively higher probability of high cost outcomes (in this case elevated global temperature) (Roe 2009; Weitzman 2009); for example, a simple consequence is that damage functions of the form with *α*≥1 will have unbounded expected value at all times. Any practical policy designed to reduce the likelihood or consequence of loss probability outcomes must address not only the shape of the stationary pdf, but also its time evolution. Our model analysis offers an insight into the relaxation times that may be encountered, and the explicit form for the pdf permits an examination of the trade-offs between parameters that might be achieved by policy action. For example, it is possible to determine all combinations of effluent flux, *q*, and positive feedback, *γ*, that give the probability, , of a temperature increase below a set amount, . If the cost is known of policy options that are available to lower the emission rate *q* or reduce the feedback parameter *γ* (by geo-engineering), then an optimal economic policy can be determined. Figure 2 presents an example of such a trade-off that exhibits a striking and somewhat unexpected linear relation between the combinations of *q* and *γ* that results in a given .

Our analysis of the simple climate model can be easily extended to the case in which there are several feedback mechanisms, both positive and negative, acting simultaneously (Roe 2009; Weitzman 2009). This approach can also be applied to a climate that shifts between positive and negative feedbacks by matching solutions for the time-dependent pdfs. In the case of fast temperature relaxation, if the flux *q* varies with time on the slow time scale, the resulting FP equations can be solved numerically. Extending the treatment to include the spatial variation of the concentration *c*(** r**,

*t*) and temperature

*T*(

**,**

*r**t*) requires including the effects of convection and/or diffusion in the transport equations. As the resulting equations are nonlinear, mode coupling will almost certainly result in a variety of new behaviours. However, the zero-mode behaviour reported here should persist.

## Appendix A. Derivation of stochastic dynamical system for climate

We begin with the deterministic energy balance equation for the atmosphere, obtained by integrating across its depth and over all latitudes, equation (1.1),
In the context of climate change driven by changes in the concentration of GHGs, we are interested in small deviations from the equilibrium temperature *T*_{0} consistent with a (pre-industrial) equilibrium concentration of GHGs, *c*_{0}, defined by
A1
Small deviations from this state, Δ*T*(*t*)=*T*(*t*)−*T*_{0}, lead to
where , with Δ*g*=*g*(*c*)−*g*_{0} and *g*_{0}=*g*(*c*_{0}). We now expand with respect to Δ*c*(*t*)=*c*(*t*)−*c*_{0} in two steps. First, expand *g*(*c*) to first order and keep the leading coupling term Δ*T*Δ*c*,
A2
where and *γ*=*g*′_{0}/(1−*g*_{0}), with (⋅)′≡*d*(⋅)/*d**c*. The radiative forcing term Δ*R*_{p}(*c*) can be nonlinear in *c*. Dividing both sides of equation (A2) by and defining and *τ*_{T}=*λ*_{0}*ρ**C*_{p}*H* yields the deterministic form of the incremental energy balance equation,
A3
If we now expand the radiative forcing to the lowest order in Δ*c*, this yields .

This derivation is consistent with the traditional approach that starts with a linearized energy balance equation given by (Tung 2007)
A4
where the right-hand side is the difference between the incoming solar flux *Q* on Earth with albedo *a*, *R*(*c*)=(1−*a*)*Q*, and the outgoing radiation is given by approximating the Stefan–Boltzmann black-body radiation law via a linearized approximation determined by fitting the form of infrared emission from the Earth to observational data on outgoing long-wave radiation (Torn & Hart 2006), with *A*=(*σ**T*^{4})_{0} and *B*=(*d*(*σ**T*^{4})/*d**T*), where the subscript 0 corresponds to the steady state. To make the connection, we again consider small perturbations in the temperature and the GHG concentration from the pre-industrial equilibrium, introducing Δ*T*(*t*)=*T*(*t*)−*T*_{0} and Δ*c*(*t*)=*c*(*t*)−*c*_{0}, in equation (A4), and obtain
A5
which is of the same form as equation (A2). Dividing both sides by *B*(*c*_{0}) and letting the scaled incremental radiative forcing *λ*_{0}Δ*R*_{p}=(*R*′(*c*_{0})−*B*′(*c*_{0})*T*_{0})/*B*(*c*_{0}), the characteristic relaxation time scale for temperature variations *τ*_{T}=*ρ**C*_{p}*H*/*B*(*c*_{0}) and *γ*=−*B*′(*c*_{0})/*B*(*c*_{0}) yields exactly the same deterministic version of the energy balance equation (A3). Adding the scaled form of a fluctuating term that characterizes rapid uncorrelated fluctuations in the radiative forcing yields equation (1.3). The equation for the evolution of GHGs, equation (1.2), is a first-order process in the presence of source and fluctuations.

## Appendix B. The role of feedback uncertainty in the case of fast relaxation of GHGs

For fast concentration relaxation, the general form for the average temperature change as a function of time is
B1
where . This can be formally integrated to yield
B2
The relaxation of 〈Δ*T*(*t*)〉 slows down as *g*(*c*_{s}(*t*))→1. If we assume *c*_{s}(*t*) is a constant, equation (B2) simplifies to
B3
We recall that, to first order [1−*g*(*c*_{s})]=(1−*g*_{0})[1−*γ*Δ*c*_{s}], so that the slowing down can occur either for *g*_{0}→1 or for *γ*→1. Different values of the gain, *g*=*g*(*c*_{s}), will lead to different average temperature changes; thus in equations (B2) or (B3), the temperature increase should be denoted by 〈Δ*T*(*t*)〉(*g*), indicating that the average temperature change is a function of *g*. Different values of {*g*_{i}} will also produce different trajectories for 〈Δ*T*(*t*)〉(*g*). If one assumes that the different values for *g* come from a pdf *h*(*g*) supported on 0≤*g*≤1, this produces a pdf for of the form
B4
with . As long as *h*(*g*) is finite at *g*=1, the pdf will be asymmetric and exhibit a fat tail. As an example, consider equation (B3), in the limit , *g*(*c*_{s})→1, with fixed. In this limit,
with Δ*T*_{s}=*λ*_{0}Δ*R*_{p}(*c*_{s}). This gives
and the pdf for the temperature increase,
over the range , which shows a fat tail.

## Footnotes

↵1 This abuse of notation making the temperature a function of the GHG concentration corresponds only to the case when the temperature is slaved to the concentration, as given by equation (2.1). Our definition of the equilibrium sensitivity is different from that of the Intergovernmental Panel on Climate Change (IPCC) and is given by the change in steady-state temperature that occurs with a doubling of atmospheric CO

_{2}, , so Δ*T*(2*c*_{0})=*S*_{eq}.↵2 In the IPCC Synthesis Report 2007, fig. 5.1 implies that climate action arises with negative feedback (Intergovernmental Panel on Climate Change 2007).

- Received August 23, 2009.
- Accepted November 11, 2009.

- © 2009 The Royal Society