In IAMs, the climate component tends to be very simple relative to most physical climate models. van Vuuren et al. (2011), Marten (2011) and others have evaluated the predictive performance of the simple climate modules in IAMs. In order to study the economic consequences of uncertainty about extreme warming in IAMs, it is helpful to first get a conceptual handle on the underlying physical uncertainties that drive the economic analysis. In this paper we employ the DICE model (Nordhaus 2008), so our discussion seeks to link its climate module to fundamental physical concepts, something that the bulk of the IAM literature neglects. DICE is one of the the most widely studied IAMs. In it economic damages from climate change depend exclusively on the global mean temperature change. In DICE, this quantity is calculated using Eq. 1.
$$ T_t = T_{t-1} + \xi_1 \left[ F_t - \frac{F_{2 \times CO_2}}{S}\left(T_{t-1}\right) - \xi_3\left(T_{t-1}-T^{LO}_{t-1}\right)\right] $$
(1)
-
T
t
::
-
Global mean surface temperature change at time t with respect to 1900
-
F
t
::
-
Radiative forcing at time t
-
\(F_{2 \times CO_2}\)::
-
Radiative forcing for a doubling of atmospheric CO
2
-
S::
-
Climate sensitivity
-
\(T^{LO}_t\)::
-
Temperature of the lower oceans at time t with respect to 1900
-
ξ
1::
-
‘speed of adjustment parameter’
-
ξ
3::
-
‘coefficient of heat loss from the atmosphere to oceans’
To understand the physical basis of this equation, start by considering the planet’s surface, lower atmosphere and oceans as “the system”—a box into which energy flows in and out. In equilibrium the rate of energy input to the system (from the sun) equals the rate of energy lost (through radiation to space) so the energy content remains constant. Increasing atmospheric greenhouse gas concentrations represent a forcing which decreases the rate of energy loss, leading to an increase in energy content until feedbacks, including rising temperatures, increase the rate of energy loss again bringing the system back into balance. Since we are principally interested in changes from an assumed equilibrium state, the forcings, which decrease the rate of energy loss, can be thought of as increasing the rate of energy input; the feedbacks being a consequential increase in the rate of energy output. This is encapsulated by Eq. 2 (Andrews and Allen 2008; Senior and Mitchell 2000). The right hand side represents the overall rate of energy input to the system: radiative forcing F, reduced by the increase in the rate of energy output to space, the feedbacks, which are taken to be proportional to the surface temperature change T. The left hand side represents the rate of change of the system’s energy content, measured in terms of the change in surface temperature multiplied by an effective heat capacity for the system as a whole.
$$ C_{\rm eff}\frac{dT}{dt} = F - \lambda T $$
(2)
-
C
eff::
-
Effective heat capacity of the climate system
-
T::
-
Surface temperature change from some equilibrium state
-
F::
-
Radiative forcing
-
t::
-
time
-
λ::
-
a feedback parameter
Over short timeframes, decades not millennia, much of the excess energy input leads to warming of the upper oceans (Levitus et al. 2000; Lyman et al. 2010). More slowly the energy penetrates to the deep, or lower, oceans. An extension of the above model is therefore to consider that our system includes not the whole ocean but only the upper ocean, which is taken to be a well-mixed layer (and therefore warms uniformly), coupled to a second box which represents the deep ocean and into which heat diffuses. The deep ocean is usually not taken to be well-mixed but rather to have temperatures which decrease with depth (Hansen et al. 1985; Frame et al. 2005). A simpler form of this extension, though, would assume that the deep ocean too is well-mixed and can be represented by a single temperature, call it T
LO. The flow of energy from the surface to the lower oceans is then taken to be proportional to their temperature difference. With this extension, we get Eq. 3. Note that T is still the surface temperature so the heat capacity now relates only to the upper box.
$$ \begin{array}{rll} C_{\rm up}\frac{dT}{dt} &=& F - \lambda T - \beta (T-T^{LO}) \label{Energy2box} \end{array}$$
(3)
-
C
up::
-
Effective heat capacity of the upper oceans, land surface and atmosphere
Equations 2 and 3 both represent energy conservation, but in one- and two-box systems respectively. Applying Euler’s method to discretize Eq. 3, and re-arranging we obtain Eq.4.
$$ \begin{array}{rll} T_t &=& T_{t-1} + \frac{\Delta t}{C_{\rm up}} \left[ F_{t-1} - \lambda T_{t-1} - \beta (T_{t-1}-T^{LO}_{t-1}) \right] \label{E2boxdiscT} \end{array} $$
(4)
-
t::
-
Now the number of the time-step not continuous time
- Δt::
-
length of the time-step
Consideration of Eq. 2 for the equilibrium response to doubling the atmospheric CO
2 concentrations shows that λ in Eq. 4 can be equated with \(\frac{F_{2\times CO_2}}{S}\) in Eq. 1. We have thus arrived at a formulation that is almost identical to the temperature equation in the DICE model (Eq. 1).Footnote 1
The physical basis of Eq. 4 makes it easier to relate different sources of uncertainty in Eq. 1 to the various sources of uncertainty discussed in the physical science literature. We focus here on uncertainty about the climate sensitivity, S, and the heat capacities, C
eff and C
up.
Climate sensitivity
The value of S is not known with certainty. Instead, over the last two decades a large literature has emerged that seeks to quantify uncertainty about S via probability distributions. Many distributions have been published (see Fig. 2), along with a number of review and meta-analysis papers utilizing collections of distributions (Meinshausen et al. 2009). From this literature three stylized facts emerge. First, there are differences—at times large—between the various estimates. Second, all have a large positive skew and in most cases it satisfies the definition of a fat tail. Third, there are large differences between the various estimates of the upper tail.
The pdfs generated represent different assessments of epistemic uncertainty, each conditioned on a different set of assumptions (only some of which are usually made explicit) and founded on different underlying observational and/or model data. It is important to note, however, that S is being used as a proxy for λ which represents the feedbacks relevant at some point in time and is state-, and therefore time-dependent; as is S. The relevant distribution of S to use in an IAM will change over time within the simulation as the strength of different feedback processes vary. (Consider, for instance, the role of sea ice in the albedo feedback—this may be small for small increases in temperature, large for temperatures when the sea ice rapidly declines and smaller again when the area of remaining sea ice is small.) The foundation of some of the pdfs may make them them more relevant in the short term, others in the longer term and still others of limited relevance over the next 400 years or so; a time period typical of IAM simulations. On top of this, each of the methods has methodological advantages and disadvantages. Thus it is not possible to identify from the literature a single distribution which is most suitable for use in an IAM.
Setting aside the issue of time dependence, it is tempting to combine the various estimates, but this too is problematic. The distributions are not independent in terms of either methodology or data constraints, yet their degree of dependence is unclear. Thus neither naive combination nor more complicated weightings can be relied upon to give “the right” distribution. For the time being the upshot is to accept that the science has produced many different distributions and the economics must accept relatively large uncertainty, not just in the value of S, but in the uncertainty in the value of S—particularly in the tails of the distribution. There are opportunities to narrow our uncertainty for economic applications, but the first step must be to try to understand what uncertainty about the tail shape implies for the robustness of the economic analysis.
Effective heat capacity
Examination of Eq. 1 suggests that uncertainty in transient temperature change is dependent not just on climate sensitivity but also on uncertainty in the parameters ξ
1 and ξ
3 (and also in \(F_{2\times CO_2}\), although \(F_{2\times CO_2}\) is considered well known). Historically the majority of the heat has remained in the upper oceans (Levitus et al. 2000; Lyman et al. 2010), so we focus our attention on ξ
1; ξ
3 only being important for the transfer of heat to the deep oceans.
Comparing Eqs. 1 and 4 shows that \(\xi_1 = \frac{\Delta t}{C_{\rm up}}\). The scientific literature does not provide constraints on C
up directly, but Frame et al. (2005) present uncertainty estimates for effective heat capacity, C
eff, giving 95 % confidence intervals of \((< 0.2~\textrm{GJm}^{-2}\textrm{K}^{-1}, > 1.7~\textrm{GJm}^{-2}\textrm{K}^{-1})\) for the latter half of the 20th century.Footnote 2 A first approximation of ξ
1 from the observational data would simply be \(\xi_1 = \frac{\Delta t}{C_{\rm eff}}\), but this can be refined using values from the first period of the DICE model to give the implied ratio of C
up to C
eff at the beginning of the 21st century; a value unlikely to be substantially different to that in the latter half of the 20th century and therefore comparable with observations. Equation 5, based on Eqs. 2 and 4, shows this relationship.
$$ \begin{array}{rll} \xi_1 &=& \frac{\Delta t}{C_{\rm up}} \approx \frac{\Delta t \left[ F_{1} - \frac{F_{2 \times CO_2}}{S} T_{1} \right]}{C_{\rm eff} \left[ F_{1} - \frac{F_{2 \times CO_2}}{S} T_{1} - \xi_3 (T_{1}-T^{LO}_{1}) \right]} \label{Auncert} \end{array} $$
(5)
The default DICE value for ξ
1 is 0.208, which translates into a value for C
eff of \(1.8~ \textrm{GJm}^{-2}\textrm{K}^{-1}\), on the high side of what the observations suggest is likely.Footnote 3 A natural question to ask, then, is what the consequences would be of assuming a lower heat capacity.
With a lower heat capacity, a given energy input produces more rapid warming, while the equilibrium temperature of the system is not affected. The main consequence, then, is to ‘front-load’ warming. As Fig. 1 illustrates, this front-loading can more than compensate for a lower climate sensitivity in the short term, producing higher temperatures even with a lower climate sensitivity. For a given S, a lower effective heat capacity will tend to lead to faster warming, causing greater damages to occur in the nearer term, when the relieving effect of discounting is least felt. Uncertainty about the value of the effective heat capacity could therefore have important implications for the economic analysis of climate change.