Introduction

The global atmospheric concentration of CO₂ has increased dramatically as a result of human activities since 1750 and now far exceeds pre-industrial values. Indeed, present-day CO₂ are higher now than at any time in the past 650,000 years.

The reason we care about this increase of course is because of concerns over climate change. CO₂ is a greenhouse gas (GHG). That is, it absorbs outgoing longwave radiation and thus warms the atmosphere. CO₂ is not the only GHG that is increasing due to human activity, but as this table from the IPCC report shows, its radiative effect is greater than that of all all other anthropogenic GH gases.

Given the importance of CO₂ for climate, a key question is where does this CO₂ come from and where does it go. The cartoon below attempts to summarize our current knowledge of the sources and sinks of anthro CO₂. There are two principal sources. The largest is the burning of fossil fuels which emits on the order of 5.4 PgC/y. The second largest is changes in land use, primarily deforestation in the tropics to make way for agriculture. Estimates for this source are highly uncertain however, ranging from 0.6 to 2.5 PgC/y. Together, these two sources contributed somewhere between 340 to 420 PgC between 1800 and 1994. So what happened to this CO₂? Well, less than 50% of it or 165 PgC currently resides in the atmosphere so the balance must have been taken up by the ocean or the terrestrial biosphere. The net ocean uptake appears relatively well constrained at about 1.9 PgC/y, but because of the uncertainty in the land use source, the terrestrial biosphere could either be a net source or sink of CO₂. If it turns out to be the former, then the ocean could have an even bigger role in mitigating the impact of anthropogenic CO2 emissions. Indeed, some 85% of all the anthropogenic carbon will eventually dissolve in the ocean, "eventually" being several hundred years hence. Thus, it is important to better quantify and understand the ocean's role in the perturbed carbon cycle, which is the primary aim of our research.

Specifically, our goal is to:

Reconstruct the history of anthropogenic carbon in the ocean over the industrial period

An historical resconstruction of the spatial distribution of CO2 in the ocean provides us with insights into the physical mechanisms driving ocean uptake, and allow us to rigorously evaluate and improve forward biogeochemical models to more accurately predict future uptake.

What do we know about anthropogenic CO₂ in the ocean?

To appreciate why is it so difficult to estimate anthropogenic carbon in the ocean, it is useful to review some basic aspects of the problem. There are 4 key points to keep in mind:

• Anthropogenic carbon is not a directly measurable quantity. It has to be estimated using indirect means.
  
• The anthropogenic signal in the ocean (ΔDIC) is only a few percent of the (unknown) natural background of dissolved inorganic carbon (DIC).
• Carbon in the ocean participates in rather complex in situ biogeochemisry.
• Due to long transport time scales, the ΔDIC distribution in the ocean is highly heterogenous.

Thus, unlike the atmosphere, which is relatively well mixed, and where we have ice core and instrumental data going back many thousands of years, the ocean is much more challenging in this regard. What we do know about ΔDIC in the ocean is based on so called "back calculation" methods which attempt to separate the small anthro perturbation from the large background. The essential idea - which goes back to work by Brewer, Chen, and Millero - is that we can estimate anthropogenic carbon by correcting the measured total DIC for changes due to biological activity. The basic equation looks  something this, where the first term is the measured DIC, the second is the change in DIC due to soft tissue remineralization and carbonate dissolution, and the 3d term is the air sea CO2 disequilibrium when the water sample was last in contact with the surface.

∆DIC = DIC_meas - ∆DIC_bio - DIC_diseq + …

Estimating these terms is a messy business to say the least, and it requires  making several critical assumptions, among them:

• The stociometric or so called Redfield ratios necessary to account for the biology are known and constant,
• Mixing in the ocean is a negligble component of tracer transport compared with advection (the so-called "weak mixing" assumption), and
• The air-sea disequilibrium has remained constant over the industial era.

How valid are these assumptions? As we see below, not very. And just to illustrate the potential difficulties, the figure below compare the results of applying three different back calculation methods to estimate anthropogenic carbon in the Indian Ocean. Clearly, while there is some qualitative agreement between them, there are large quantitative differences as well. In fact, integrated inventories differ on average by 20% between the different methods. And many of these methods, including the widely applied "ΔC*" method, give negative values of anthropogenic carbon, which points to serious problems. Finally, back calculation methods can only provide us with a single snapshot of the distribution of anthropogenic carbon in the ocean.

Assumption I: How well do we know Redfield ratios?

A key assumption made by back calculation methods is that we know what Redfield ratios to use to correct for the biology. The difficulty with this is that there are in fact large uncertainties in our knowledge of the Redfield ratios and this translates into a correspondingly large error in the inferred DDIC. As an example, the plot below, from Wanninkhof et al. (1999), shows the fractional uncertainty in the inferred ΔDIC using the Gruber ΔC* method by propagating a typical (12 %) uncertainty in the C:O remineralization ratio. The error is not small even for large values of ΔDIC. In the upper ocean, this can easily translate into a 30-50% uncertainty in the inferred ΔDIC.

Assumption II: Is ocean transport dominated by advection?

A second implicit but crucial assumption is that ocean transport is largely advective in nature. This is an implicit assumption because back calculation techniques use transient tracers such as CFCs to infer the time it takes for a fluid parcel to go from the surface to the interior. This works as follows. If you assume there is no mixing, then the measured interior tracer concentration is related to the surface history of the tracer through a simple time lag. If you know the surface history you can calculate the time lag. This is shown schematically in the figure on the left below.

Mathematically, we may write this as:

However, even though it underlies much of chemical oceanography, this simple picture of the ocean is fundamentally incorrect. The ocean is turbulent and diffusive, and in the presence of mixing there is no unique time scale or pathway that connects the ocean surface to an interior point. So instead of a single transit time, there is a probability distribution of transit times, or "transit time distribution" (TTD). Consequently, the measured tracer concentration is a weighted average of the surface history of the tracer, the appropriate weight being given by the TTD and expressed mathematically as a convolution integral:

You can think of G as a way to partition each water parcel according to when and where it was last in contact with the surface.

There is plenty of evidence for this messier view of the ocean. As an example, the figure on the left below shows the observed relationship between two different tracers (CFC-11 and CFC-12) in the Indian Ocean. The gray dots represent data while the various curves are attempts at modeling the observed tracer distributions with TTDs of different widths. Evidently, the data are best explained by broad TTDs implying strong mixing. The purely advective case is completely inconsistent with the data, something we find in other ocean basins as well.



The plot on the right shows simulations in a 1 deg data assimilated ocean model.  (The simulations were performed using the Transport Matrix Method (TMM) developed by us, an extremely efficient new technique for performing biogeochemical tracer simulations.) Here too, we see broad TTDs, a feature that seems independent of model resolution.

The weak mixing bias:

So how does the neglect of mixing impact estimates of anthropogenic carbon? Th e figure shows the inferred DDIC as a function of measured CFC-12 concentration for two scenarios. The first, shown in blue assumes perfect advection. The second shown in red assumes strong mixing. You can see that in the upper ocean (higher CFC concentrations) the difference between the two is small, i.e., CFC is a good proxy for DDIC. But at intermediate depths the no mixing assumption predicts substantially higher values of DDIC. This is because the no mixing estimate is based on tracer ages which are biased toward younger values. Since anthropogenic carbon in the surface ocean is increasing over time, this results in a higher estimate of anthropogenic carbon in the interior. In the deep ocean, the bias is the opposite. If there is no detectable CFC, we assume that there is no anthropogenic carbon, which really cannot be true in the presence of finite mixing and the fact that anthropogenic carbon has been around for a lot longer than CFCs. Hence, this leads to an underestimate.

Assumption III: Constant air-sea disequilibrium

The third assumption is that the ocean surface has kept up with increasing levels of atmospheric CO2. This is known as the constant disequilibrium assumption. (Incidentally, there are two main reasons for why the ocean is not in equilibrium with the atmosphere. The first is the fact that it takes a finite amount of time for air-sea exchange of CO2. This is about a month for most gases, but for CO2 it is about a year because of it s buffer chemistry in seawater. The second reason is ocean circulation which is continuously pumping away CO2 depleted waters away from the surface at high latitude and bringing up CO2 enriched water in the tropics. This is why the subpolar ocean surface is highly undersaturated while the tropics are oversaturated.) To evaluate how well this assumption holds, we have performed explicit simulations of anthropogenic carbon in an ocean biogeochemical model. The top right panel in the figure below shows the preindustrial disequilibrium (surface ocean pCO2 minus atmospheric pCO2) in the model. The bottom panel shows the change in disequilibrium between the preindustrial and 2005. It is evident that the changes are quite substantial. In fact, in the Labrador Sea the disequilibrium changes by ~80%, while in the tropical pacific it changes by ~40%. Since much of the anthropogenic carbon enters the ocean at high latitude regions such as the Labrador sea, the constant disequilibrium assumption leads to an overestimate (by almost 20%) in the estimated anthropogenic CO2.

The Transit-time Distribution (TTD) Method

To overcome the difficulties associated with traditional back calculation methods, we have developed an alternative approach. In many ways it is a much simpler and cleaner approach than the back-calculation methods because it makes no attempt to separate the small anthropogenic component from the large background. And it not only allows us to relax many of the assumptions traditionally made, it uniquely provides us with the time-evolving 3-d distribution of anthropogenic carbon over the entire industrial era. The basic ideas and assumptions are as follows:

• The anthropogenic perturbation is sufficently small for us to treat ∆DIC as a conservative tracer that is transported by the circulation from the surface to the interior,
• Ocean transport can be characterized by a transit time distribution (TTD), and
• Ocean circulation is in steady state.
  

Given these assumptions, we can write the interior anthropogenic carbon concentration as a convolution of the surface history of ∆DIC with the TTD:

The summation here is over multiple source regions each of which will, in general, have a different time history. To apply this equation, we need two pieces of information: The transit time distribution G, and the surface history of anthropogenic carbon.

Estimating the TTD from tracer observations

To estimate G, we use measurements of transient and steady state tracers along with their known surface history. Each passive tracer satisfies a  convolution integral similar to that for DDIC:


Our goal is to use discrete measurements of various tracers to estimate G. However, since at any given location there are only a handful of observations, this is a highly  underdetermined deconvolution problem! We regularize it using a Bayesian method known as the "maximum entropy" method (MEM). The MEM solution to this problem can be written as:


where H is a prior solution, and the α's are Lagrange multipliers that satisfy the observational constraints. Substituting the above solution into the convolution integral for each tracer results in a small nonlinear problem whose solution is the α's.

The figure belows shows a particularly successful example of the inversion. The blue line shows the directly simulated TTD in an ocean model. Several tracers were also simulated in the same model and the MEM was applied to these synthetic data.  The red line shows the resulting maximum entropy inverse solution. The prior solution used is shown by the green line. Here, for illustrative purposes we have used a uniform prior. In practice, we use analytical solutions to the 1-d advection-diffusion equation as priors.

Estimating the surface history of ∆DIC

The second piece of information we need is the surface ∆DIC history. To estimate it, we assume that transport in the ocean is to leading order isopycnal (see schematic below).



We then equate the instantaneous air-sea flux of anthropogenic carbon into the layer to the instantaneous rate of change of inventory:


The air-sea flux of anthropogenic CO2 can be written as:


The various variables are:

• ∆CO2(t) = CO2(t) - CO2(1780) [1780 = preindustrial]
• CO2 = g(DIC) = nonlinear carbonate chemistry
• α = solubility of CO2
• k = gas exchange coefficient
• DIC(1780) = DIC_meas(t_o) - ∆DIC(t_o)

Finally, we can write the inventory in terms of the TTD:



These expressions must hold for all time t, which once again gives us a nonlinear system of equations whose solution is the required surface history of anthropogenic carbon.

Verification of TTD method in an Ocean Model

As a first step, we have applied the TTD method to synthetic tracer "observations" simulated in a global ocean model. Tracers simulated include, CFCs, ¹⁴C, nutrients, and O₂. As "truth", we also simulate anthropogenic carbon.

The figure below compares the column inventory of DDIC simulated in the model (left) with that obtained using the maximum entropy method.



The agreement between the two is remarkably good, with maximum differences of O(5 mol/m²) in Southern Ocean. The total inventory simulated in the model is 123.7 PgC, while the inverse solution gives 127.6 PgC. The figure below compares the simulated preindustrial pCO2 on each of the 21 surface patches used in the inversion, with the inverse solution. Again, the agreement is quite good, with a maximum error of roughly 5 ppm.

Application to Ocean Observations

Having gained confidence in the TTD method from the synthetic inversion, we next applied it to the data from the Global Ocean Data Analysis Project (GLODAP) database. GLODAP is a database of tracer measurements made during the WOCE period (1990’s). It includes CFCs, carbon, ¹⁴C, O₂, and nutrients. Additionally, temperature and salinity are available from the World Ocean Atlas.

The animation below shows the column inventory of anthropogenic carbon over the industrial period. The movie runs from 1775 to 2007. The total inventory is shown in the upper left corner.


The figure below shows the anthropogenic carbon uptake history from 1765-2005. The TTD-based global uptake (1980’s-1990’s) is  1.7-2.1 PgC/y, which compares favorably with several independent estimates such those based on the atmospheric O₂/N₂ ratio (1.9 ± 0.6; Manning and Keeling, 2006), the air-sea pCO₂ difference (2.0 ± 60%; Takahashi et al, 2002), and the air-sea ¹³C disequilibrium (1.5 ± 0.9; Gruber and Keeling, 2001).



Finally, we estimate a total inventory (in the mid-1990's) of 126.2 ± 14 PgC. This is substantially higher than the 106 ± 21 PgC value given by the ∆C* method (Sabine et al, 2004). The main reason for the discrepancy is the constant disequilibrium assumption made by the latter which leads to at least a 20% overestimate.

Summary and Conclusions

• Tracer-constrained TTDs have been used to reconstruct the history of anthropogenic carbon in the ocean.
• The TTD-derived inventory of anthropogenic carbon in the ocean (in the mid-1990s) is 126.2 ± 14 PgC.
• In contrast, the ∆C* method gives an inventory of 106 ± 21 PgC.
• There are substantial differences in the spatial distribution of ∆DIC between TTD and ∆C* estimates which can be explained by the assumptions of weak mixing and constant disequilibrium made by the latter.
• The TTD method gives a global ocean uptake of 1.7-2.1 PgC/y during 1980’s-1990’s.