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### BEAM simplified model

The default carbon model in webDICE runs quickly and is reasonable accurate over short periods of time. Over longer periods, however, it is less accurate. The reason is that as carbon concentrations in the ocean go up, its ability to absorb additional carbon goes down. The default model does not account for this eect and, therefore, over long periods of time, treats the ocean as absorbing more CO2 than it really will. The BEAM model includes a simpli ed version of ocean chemistry to account for this eect and is more accurate over longer time periods. The cost of choosing BEAM is that webDICE will run more slowly, particularly in optimization mode, where the model may time out. more

Glotter et al. demonstrate that the default DICE carbon cycle representation fails to accurately model oceanic carbon uptake [1]. In webDICE, the atmospheric temperature anomaly is a function of the concentration of carbon in the atmosphere. As was discussed in section 5.1, the atmospheric concentration of carbon in each time period in default DICE is determined by the concentration from the previous time period augmented by emissions and reduced by a constant fraction which is absorbed by the ocean. In the default DICE carbon cycle, the atmospheric concentration of carbon in each time period is determined by the concentration from the prior time period augmented by emissions and reduced by a constant fraction which is absorbed by the ocean. It is this constant fraction $(1-\phi_{1,1})$, which accounts for the unphysical linear absorption of carbon by the oceans in the default DICE model.

Actual carbon uptake by the ocean is highly non-linear, characterized by a rapid initial uptake period followed by a long-tail equilibrium stage. The following graph compares the pathway of carbon mass in the atmosphere as prescribed by DICE verses BEAM (the Bolin and Eriksson Adjusted Model — based on an established model first published in 1958) [2] given the same emission trajectory (IPCC Scenario A2+) [3]

#### Figure 1: Comparison of DICE carbon cycle with BEAM and two other physical models

The BEAM model does a much more complete job than DICE at capturing the relevant physics of oceanic carbon uptake. webDICE uses the simplified version of BEAM presented in Glotter et al. The authors demonstrate that their abbreviated version of the model, which excludes temperature-dependent coefficients, offers very similar results to the full BEAM model. webDICE utilizes simplified BEAM for ease of computation. Glotter et al. construct a similar three reservoir model to default DICE, $\frac{d}{dt}\left[\begin{array}{c} M_{AT}(t)\\ M_{UP}(t)\\ M_{LO}(t) \end{array}\right]=\left[\begin{array}{ccc} -k_{a} & k_{a}\cdot A\cdot B & 0\\ k_{a} & -(k_{a}\cdot A\cdot B)-k_{d} & \frac{k_{d}}{\delta}\\ 0 & k_{d} & -\frac{k_{d}}{\delta} \end{array}\right]\left[\begin{array}{c} M_{AT}\\ M_{UP}\\ M_{LO} \end{array}\right]+\frac{dE(t)}{dt},$ where $A$ is the ratio of atmospheric carbon to upper oceanic dissolved CO2 and $B$ represents the partitioning of upper ocean dissolved CO2 to total inorganic carbon and $\frac{dE(t)}{dt}$ is the emissions rate.

BEAM will estimate that carbon stays in the atmosphere longer than with the default carbon cycle. As a result, damages from climate change will persist for a longer period of time and be higher overall.

Additional technical details According to Henry’s Law the partial pressure of atmospheric CO2 must balance the concentration of CO2 in the upper ocean: $A=\frac{AM}{OM/(\delta+1)}$, where $AM$ is the number of moles in the atmosphere, $OM/(\delta+1)$ is the number of moles in the upper ocean, $k_{H}$ represents the solubility of CO2 in seawater.

$B=1/(1+\frac{k_{1}}{[H^{+}]}+\frac{k_{1}\cdot k_{2}}{[H^{+}]^{2}})$ where $[H^{+}]$ can be solved for by solving the following:

$\left\{ 1+\frac{k_{1}}{[H^{+}]}+\frac{k_{1}\cdot k_{2}}{[H^{+}]^{2}}\right\} /\left\{ \frac{k_{1}}{[H^{+}]}+\frac{2k_{1}\cdot k_{2}}{[H^{+}\}^{2}}\right\} =\frac{M_{UP}}{Alk}$ where $Alk=662.7\: Gt\: C$ the alkilinity.

$\frac{dE(t)}{dt}=E(t)/10$ in webDICE to adjust for different timescales.

[1] Glotter, Michael and Pierrehumbert, Raymond T. and Elliott, Joshua and Moyer, Elisabeth J., A Simple Carbon Cycle Representation for Economic and Policy Analyses (September 1, 2013). RDCEP Working Paper No. 13–04. Available at SSRN: http://ssrn.com/abstract=2331074 or http://dx.doi.org/10.2139/ssrn.2331074

[2] Bolin, Bert and Erik Eriksson, 1958. Changes in the Carbon Dioxide Content of the Atmosphere and Sea due to Fossil Fuel Combustion. In The Atmosphere and the Sea in Motion: Scienti c Contributions to the Rossby Memorial Volume. Bert Bolin, ed. New York, Rockefeller Institute Press, 130142.

[3] Montenegro, A., V. Brovkin, M. Eby, D. Archer, and A. J. Weaver (2007), Long term fate of anthropogenic carbon.