
\documentclass[galley,gbc]{agu2001}
\usepackage{lineno}

\usepackage{epsfig}
\authorrunninghead{MARINOV ET AL.}
\titlerunninghead{IMPACT OF OCEANIC CIRCULATION}
\authoraddr{I. Marinov, Dept. of Marine Chemistry and Geochemistry,
Woods Hole Oceanographic Institution, Woods Hole, MA, USA (imarinov@whoi.edu)}

\begin{document}

\title{\bf{Supplementary material: The relationship between atmospheric $p$CO$_2$, surface nutrients and globally averaged preformed PO$_4$}}


\par
Here we confirm through a set of nutrient depletion experiments the existence of a robust and predictive functional relationship between globally averaged preformed PO$_4$ and atmospheric $p$CO$_2$ as shown
in Fig 3c and described by Eq. (8) of the paper:
\[
pCO_{2\,a}\simeq c_1\cdot e^{-\frac{OCS_{soft}}{C_{buffered}}}=c_1\cdot e^{\frac{ r_{C:P}\cdot (\overline{PO_{4\,pref}} - \overline{PO_{4}}) \cdot V_{oc} }{C_{buffered}}}
\]

\par
Fig~S1 shows results of Southern Ocean nutrient depletion experiments, where we restore nutrients to zero everywhere at the ocean surface south
of 30$^o$S. The total amount of PO$_4$ in the ocean is kept constant at all times. The total amount of inorganic carbon in the ocean-atmosphere
system (total $DIC$ in the ocean plus total CO$_2$ in the atmosphere) is kept constant at all times and identical between all models, while
atmospheric $p$CO$_2$ is allowed to vary. The nutrient depletion experiments are carried out separately in all nine standard models (with
biological time scale $\tau$= 30days) described in Table 1 of the paper. Each model is run for more than 5000 years to a new equilibrium state
while continuously restoring nutrients to zero.

Surface nutrient depletion (equivalent in our models to a strong increase in surface production) results in a strong decrease in surface nutrients and a decrease in the global preformed nutrient inventory, followed by a strong decrease in atmospheric $p$CO$_2$. The extent to which atmospheric $p$CO$_2$ decreases depends on the decrease in global preformed PO$_4$, which is a function of both surface PO$_4$ and the net deep ocean ventilation in the model.

We also performed global depletion simulations in our standard models (with biological time scale $\tau$= 30days)  by continuously restoring
nutrients to zero everywhere at the ocean surface. In Fig~S2 we compile equilibrium results from all our sets of simulations: standard
($\tau$=30days) simulations, Southern ocean depletion simulations, global depletion simulations, as well as simulations with $\tau$=1 year and
$\tau$=3 days.

Our paper shows through both theory and numerical models that globally averaged preformed PO$_4$ (or equivalently $OCS_{soft}$) cleanly predicts
atmospheric $p$CO$_2$ in the limit of fast gas exchange. Figure~S2d above confirms that, for all cases explored, there is a clean functional
relationship between globally averaged preformed PO$_4$ and atmospheric $p$CO$_2$. This is the same exponential relationship shown in Fig. 3c of
the main text and described by Eq. 8.

By contrast, figure~S2a-c proves that there is no unique functional relationship between surface PO$_4$ and atmospheric $p$CO$_2$, regardless of
how we define our surface PO$_4$ average. This is the case because global preformed PO$_4$ depends both on the preformed surface PO$_4$ in the
deep water formation regions and on the change in the strength of AABW relative to NADW.

The "slope" of the $p$CO$_{2\,a}$ - surface PO$_4$ is not unique, and depends on how we define surface PO$_4$. Since deep water formation
happens in convective centers south of 60$^o$S, surface PO$_4$ south of 60$^o$S is perhaps the most relevant surface end-member. However, points
in Fig~S2c appear clustered and surface PO$_4$ south of 60$^o$S is clearly not a robust predictor of atmospheric $p$CO$_2$.

\par
To further illustrate the decoupling between the surface PO$_4$ and $p$CO$_{2\,a}$ we also note that
it is possible for two different models to have widely different surface PO$_4$ values but identical atmospheric $p$CO$_2$ values.
For example, the Southern Ocean surface PO$_4$ is lower by about 0.7 umol/kg in the nutrient-depleted windx2 model compared to the control LL model.
However, the global preformed PO$_4$ is nearly identical in these two models, because the stronger deep ocean ventilation in the former model compensates for its lower surface preformed PO$_4$.
As a consequence, atmospheric $p$CO$_2$ is nearly identical in these two cases.

\begin{figure} [ht]

\caption{
Atmospheric $p$CO$_2$ (ppm) versus (a) surface PO$_4^{3-}$ averaged over the entire ocean ($\mu$mol/kg), (b) average surface  PO$_4^{3-}$ south of 30$^o$S ($\mu$mol/kg),
(c) globally averaged preformed PO$_4^{3-}$.
Results shown for both control simulations (open symbols) and Southern Ocean surface nutrient depletion simulations (full symbols) in the nine models described in Table 1 of the paper with biological time scale $\tau$=30 days. Fast gas exchange assumed. Circles denote lower Southern Ocean ventilation models, triangles higher ventilation models.}
\label{fig:Fig_Suppl1}
\end{figure}

\begin{figure} [ht]

\caption{
Atmospheric $p$CO$_2$ (ppm) versus (a) surface PO$_4^{3-}$ averaged over the entire ocean, (b) average surface  PO$_4^{3-}$ south of 30$^o$S,
(c) average surface  PO$_4^{3-}$ between 60$^o$S and 80$^o$S,
(d) globally averaged preformed PO$_4^{3-}$. All PO$_4$ values in units of $\mu$mol/kg.
Results shown for control simulations (open circles and full black triangles), after Southern Ocean surface nutrient depletion (full blue triangles), or after global surface nutrient depletion (full red triangles) in the nine circulation models described in Table 1 of the paper. Fast gas exchange assumed. Globally integrated preformed PO$_4$ is clearly the best indicator for
atmospheric $p$CO$_2$.
}
\label{fig:Fig_Suppl2}
\end{figure}

\end{document}
