A simple carbon cycle representation for economic and policy analyses

Abstract

Integrated Assessment Models (IAMs) that couple the climate system and the economy require a representation of ocean CO₂ uptake to translate human-produced emissions to atmospheric concentrations and in turn to climate change. The simple linear carbon cycle representations in most IAMs are not however physical at long timescales, since ocean carbonate chemistry makes CO₂ uptake highly nonlinear. No linearized representation can capture the ocean’s dual-mode behavior, with initial rapid uptake and then slow equilibration over ∽10,000 years. In a business-as-usual scenario followed by cessation of emissions, the carbon cycle in the 2007 version of the most widely used IAM, DICE (Dynamic Integrated model of Climate and the Economy), produces errors of ∽2^∘C by the year 2300 and ∽6^∘C by the year 3500. We suggest here a simple alternative representation that captures the relevant physics and show that it reproduces carbon uptake in several more complex models to within the inter-model spread. The scheme involves little additional complexity over the DICE model, making it a useful tool for economic and policy analyses.

Appendices

Appendix A: BEAM model equations, parameter values and initial conditions

1.1 A.1 Equations

BEAM consists of four independent equations: three that track total carbon in each layer (atmosphere, upper, and lower ocean- Eq. 10), and one that tracks acidity (Eq. 13). Carbon transfers in BEAM are described by:

$$ \frac{d}{dt} \left (\begin{array}{ccc} & M_{AT} \\ & M_{UP} \\ & M_{LO} \\ \end{array} \right) = \left (\begin{array}{rccccr} -k_{a} & & & k_{a} \cdot A \cdot B & & 0~ \\ k_{a} & & & -(k_{a} \cdot A \cdot B) - k_{d} & & \frac{k_{d}}{\delta} \\ 0~ & & & k_{d} & & -\frac{k_{d}}{\delta} \\ \end{array} \right) \left (\begin{array}{ccc} & M_{AT} \\ & M_{UP} \\ & M_{LO} \\ \end{array} \right) + E(t) $$

(10)

where the M _i s represent the mass of carbon (in CO₂ or dissolved inorganic carbon) in the atmosphere ( _{A T} ), upper ocean ( _{U P} ), and lower ocean ( _{L O} ); and E(t) is rate of anthropogenic CO₂ emissions. (Emissions units must match those of concentrations and are therefore specified in mass of carbon in CO₂.)

The parameter A is the ratio of mass of CO₂ in atmospheric to upper ocean dissolved CO₂, i.e. A is inversely proportional to CO₂ solubility. Solubility is set by ‘Henry’s law’, which prescribes that in equilibrium, the concentrations of CO₂ in the atmosphere and ocean are related by a coefficient dependent only on temperature. Henry’s Law may be written in various forms; for convenience we define the coefficient k _H as a dimensionless ratio of the molar concentrations of CO₂ in atmosphere and ocean. The parameter A is then

$$\begin{array}{*{20}l} A= k_{H} \cdot \frac{AM}{OM/(\delta+1)} \end{array} $$

(11)

where AM are OM are the number of moles in the atmosphere and ocean, respectively, and O M/(δ+1) signifies the upper ocean only. B is the ratio of dissolved CO₂ to total oceanic carbon, a function of acidity:

$$\begin{array}{*{20}l} & B= \frac{1}{1 + \frac{k_{1}}{[\mathrm{H^{+}}]} + \frac{k_{1}k_{2}}{[\mathrm{H^{+}}]^{2}}} \end{array} $$

(12)

where k ₁ and k ₂ are dissociation constants. Alkalinity, A l k, is used to determine [H ⁺] by solving the quadratic:

$$\begin{array}{*{20}l} & \frac{M_{UP}}{Alk} = \frac{1 + \frac{k_{1}}{[\mathrm{H^{+}}]} + \frac{k_{1}k_{2}}{[\mathrm{H^{+}}]^{2}}}{\frac{k_{1}}{[\mathrm{H^{+}}]} + \frac{2k_{1}k_{2}}{[\mathrm{H^{+}}]^{2}}} \end{array} $$

(13)

Three parameters are temperature-dependent (k ₁, k ₂, and k _H , Eqs. 14–16). Because the temperature dependence has a relatively minor aggregate effect, however, some users may wish to omit it.

1.2 A.2 Parameter values

Recommended BEAM parameter values are given in Table 3. Most are well established. Time constants k _a , k _d , and the ratio of upper to lower ocean (δ) are not well constrained; we use reasonable values from Bolin and Eriksson (1959). We determine alkalinity by assuming equilibrium in the pre-industrial ocean at pH=8.29 (see Appendix A.3). It is also possible to specify A l k and adjust pre-industrial pH, k _a , k _d , and δ for best fit to more complex models.

Table 3 Recommended BEAM parameters

To include temperature-dependent effects, replace k ₁, k ₂, and k _H with their temperature-dependent forms of Eqs. 14–16. Following Archer et al. (2004), we assume the global ocean temperature anomaly is equal to anomaly in mean surface temperature, so this anomaly is added to the baseline ocean temperature (i.e. T= 283.15 K+ ΔT). This assumption is likely an upper limit for ocean temperature change.

CO ₂ solubility (Henry’s law) (Weiss 1974)

$$\begin{array}{@{}rcl@{}} \mathbf{k_{H}}&=&\frac{1}{k_{0}} \cdot {\frac{\textit{liter seawater}}{1.027~kg}} \cdot \left(\frac{55.57~mol}{liter}\right) \ \ \ \ \ \ \text{with} \\ \ k_{0}&=&exp\left[\frac{9345.17}{T} - 60.2409 + 23.3585\cdot ln\left(\frac{T}{100}\right)\right. \\ &&{\kern2pc}+ \left.S\cdot\left(.023517 - .00023656\cdot T + .0047036\cdot \left(\frac{T}{100}\right)^{2}\right)\right] \end{array} $$

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First and second dissociation constants (Mehrbach et al. 1973)

$$\begin{array}{*{20}l} &\mathbf{k_{1}}=10^{-pK_{1}} \ \ \ \text{and}\ \ \ \mathbf{k_{2}}=10^{-pK_{2}} & \\ & \text{with} \quad pK_{1}\,=\,-13.721 + (0.031334 \cdot T) + \frac{3235.76}{T} + 1.3\cdot 10^{-5} \cdot S \cdot T \,-\, \left(0.1031\cdot S^{0.5}\right) & \end{array} $$

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$$\begin{array}{*{20}l} & \text{and} \quad pK_{2}\,=\,5371.96+(1.671221\cdot T) + (0.22913\cdot S) + (18.3802\cdot log(S)) \,-\,\frac{128375.28}{T} \end{array} $$

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$$\begin{array}{*{20}l} &\quad\quad\quad\quad\quad- (2194.30\cdot log(T)) \,-\, (8.0944\cdot 10^{-4}\cdot S\cdot T) \,-\,\left(5617.11\cdot \frac{log(S)}{T}\right) \\ &\quad\quad\quad\quad\quad+ 2.136\cdot \frac{S}{T} & \end{array} $$

where k ₀ has units $\frac {\textit {mol C}}{\textit {kg seawater } \cdot \textit {atm}}$, k _H is dimensionless, k ₁ and k ₂ have units m o l/k g seawater, T is temperature (K), and S is salinity (∼35g/kg seawater).

1.3 A.3 Initial conditions and model implementation

BEAM initial conditions are listed in Table 4. We set pre-industrial ocean carbon content by assuming equilibrium with atmospheric CO₂ at 280 ppm (IPCC 2007) and pH of 8.29. Pre-industrial pH is chosen so that after running forward with historical emissions (Boden et al. 2010), BEAM present-day pH matches that of UVic and CLIMBER-2 (∼8.16, Montenegro et al. (2007)). Matching pH is needed for a valid comparison because pH values markedly affect CO₂ uptake. Both pre-industrial and present-day pH levels are uncertain by ±.05 (Feely et al. 2009). We define ‘present-day’ as that point where atmospheric CO₂ matches 380 ppm, the 2007 DICE initial condition, which occurs near year 1994 for BEAM. The exact timing is not significant for subsequent CO₂ anomaly evolution.

Table 4 BEAM, DICE initial conditions

Because the atmosphere/upper ocean exchange timescale is short in BEAM, the present-day upper ocean is nearly in equilibrium with the atmosphere (M _{A T} /M _{U P} ∼A⋅B∼1.1), though the lower ocean is slightly out of equilibrium (M _{L O} /M _{U P} ∼49 while δ=50). Both DICE versions begin with excess atmospheric CO₂ relative to the upper ocean. (In 2007 DICE, M _{A T} /M _{U P} ∼ 0.6 but A⋅B ∼0.5.) The larger upper ocean carbon reservoir in DICE (2007 and 2010) than in BEAM reflects a larger equilibrium upper ocean/atmosphere carbon ratio and can be thought of as a deeper mixed ocean layer. DICE total ocean carbon is ∼1/2 the real-world value; this difference can be thought of as a smaller total ocean volume.

The more realistic BEAM representation does have one drawback, that the sensitivity of coefficients mandates fine timesteps to avoid instability during numerical integration. The figures shown here were generated using 0.01 year timesteps. Timesteps as coarse as 1/10 ^th year can produce oscillation in pH and uptake/release of oceanic CO₂ when emissions change rapidly (e.g. Oeschger et al. 1975). Resulting error in atmospheric CO₂ anomaly for the emissions scenario used here would reach several percent. Note that this instability means that BEAM cannot be used to simulate the response to an abrupt addition of CO₂. For code of the full BEAM representation, see www.rdcep.org/carbon-cycle-model.

B Temperature model

For completeness, we describe the 2007 DICE temperature model, which appears to adequately capture temperature evolution (Fig. 4). Just as the ocean takes up CO₂ in response to atmospheric CO₂ perturbations, it also takes up heat in response to surface warming, with a long equilibration time because of the large thermal inertia of the ocean. Heat uptake in DICE is represented by a linear model similar to that used for carbon uptake. Because heat uptake is in reality largely linear, this representation adequately reproduces climate behavior. As with the carbon cycle, many of the coefficients in the DICE 2007 temperature model given below are calibrated to the MAGICC model (Wigley et al. 2007) or taken from the IPCC (2001) and IPCC (2007).

Fig. 4

The two-box DICE temperature model appears to adequately capture temperature evolution in more complex models. We drive the 2007 DICE temperature model with atmospheric CO₂ anomalies from UVic (green) and CLIMBER-2 (maroon) from the A2 ⁺ emissions scenario and compare atmospheric temperatures from DICE (dashed) to those models’ own temperature representations (solid). DICE temperature evolutions differs somewhat in behavior but lies within the range of uncertainty

The DICE temperature model uses only two layers, the atmosphere and lower ocean; the upper ocean is assumed to follow atmospheric temperature (Eqs. 17–18, but note that we have re-organized parameters to be more intuitive). Radiative forcing F due to increased atmospheric CO₂ warms the atmosphere (and upper ocean), producing a disequilibrium with the lower ocean that is eroded with timescales 1/μ:

$$\begin{array}{*{20}l} T_{AT}(t)=&T_{AT}(t-1) \,+\, \mu_{AT}\cdot \left[{\Lambda}\cdot \left(T_{eq}(t) - T_{AT}(t\,-\,1)\right) \,-\, \gamma\cdot \left(T_{AT}(t\,-\,1) - T_{LO}(t-1)\right)\right] \end{array} $$

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$$\begin{array}{*{20}l} T_{LO}(t)=&T_{LO}(t-1) + \mu_{LO}\cdot \gamma\cdot \left(T_{AT}(t-1) - T_{LO}(t-1)\right) \end{array} $$

(18)

where the T _i s are atmospheric and lower ocean temperature changes (in ^∘C) since pre-industrial times; γ relates atmosphere-ocean heat transfer to temperature anomaly (γ=0.3 W/m ²/^∘C); Λ is the general climate sensitivity (1.3 W/m ²/^∘C, derived by dividing DICE assumptions of the forcing per doubling of CO₂ (α=3.8 W/m ²) by the assumed equilibrium warming after doubling of CO₂ (β=3.0 ^∘C / doubling)); and T _{e q} (t) is the equilibrium temperature that would be produced by the imposed forcing: T _{e q} (t)=F(t)/Λ. Forcing F(t) is assumed to be linear with the logarithm of the fractional change in CO₂ since pre-industrial times, a standard assumption in climate science⁹:

$$\begin{array}{*{20}l} F(t)=\,\,&\alpha\cdot log_{2}\left(M_{AT}(t)/M_{AT}(PI)\right) \end{array} $$

(20)

where M _{A T} (P I) is the mass of pre-industrial atmospheric carbon (596.4 Gt C, equivalent to ∼280 ppm CO₂).

Note that equilibration timescales for the atmosphere and lower ocean need not be equal since temperature is not a conserved quantity: μ _{A T} =0.22/10 years so τ _{A T} ∼45 years while μ _{L O} =(1/6)/10 years so τ _{L O} ∼60 years. The 2010 DICE temperature model uses the same equations with small adjustments to three coefficients: γ = 0.31 W/m ²/^∘C, μ _{A T} =0.208 (τ _{A T} ∼48 years), and β=3.2 ^∘C / doubling. Because DICE 10-year timesteps are long relative to these timescales, DICE coefficient values differ from those of a continuum representation. If the model is rewritten to use 1-year timesteps, Marten and Newbold (2013) recommend using γ=0.5072 W/m ²/^∘C, μ _{A T} = 0.0586, and μ _{A T} = 0.018336 to replicate 2007 DICE.