### 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 eect and, therefore, over long periods of time, treats the ocean as absorbing more
CO_{2} than it really will. The BEAM model includes a simpli
ed version of ocean chemistry to account
for this eect 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
CO_{2} and $B$ represents the partitioning of upper ocean dissolved
CO_{2} 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 CO_{2} must balance the concentration
of CO_{2} 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 CO_{2} 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.