The Azimuth Project
Experiments with ocean carbon box model



In this experiment I want to zoom in on the carbon cycle and give detailed description of the ocean modeled with box models. In the first experiment,Experiments with carbon cycle box model I intentionally focused on the global view from the atmosphere, as I also introduced a lot of new terminology.

But here we take those for granted and look at what processes that are involved in the marine part of the carbon cycle. According to Joel Thornton’s slides - see the reference section - the volume is about 1.3 10 1810^18 m^3 and 50% of the photosynthesis on Earth takes place in the oceans. So the ocean area is 349 10 12m 210^12 m^2. Which we will use when calculating the box volumes.


Geochemical processes

Due to the fact that CO 2CO_2 is highly soluble some of the carbon dioxide CO 2CO_2 in the atmosphere reacts when it reaches the water in the ocean and forms dissolved inorganic carbonates (DIC), like liquid CO 2CO_2, carbonic acid, carbonate and bicarbonate. I will refer to the sum of all of these as DIC. Then depending on the alkalinity (the ability of the ocean to neutralize acid, not to be confused with basicity) of the ocean, a couple of chemical process can occur which is illustrated here:

ocean biopump

One third of the DIC is transported down into the deep ocean due to the solubility pump, that colder water has higher density than warm and also that more salty water has higher density than less salty water. If you’re interested check out our Thermohaline circulation ( abbreviated THC below) page. I might take this up again in the carbon or water cycle, as I am curious myself to see what has been written and modeled around them together). Here is a diagram showing how the DIC changes going towards the ocean bottom:

DIC profile

So the remaining two thirds of the DIC is processed in the upper layer of the ocean less than 1000 m, by the biological pump - producing both hard and soft biological tissues. So it makes sense to use 2 boxes or more to model the ocean.

The ocean also has the ability to buffer the DIC which is caused by Ph-value and the capacity to buffer is measured with the Revelle factor. Here is an image showing how it varies (as described by Sabine et. al 2006):

revelle factor in oceans

The pre-industrial amount of carbon in the oceans was 38 000 Gt of dissolved inorganic carbon. The pre-industrial flux of carbon was almost in balance or was a slight source of carbon. And as I showed in the first experiment that the current ocean acts as an increasing carbon sink, today about 2 Gt annually. The amazing thing is that despite these complicated interacting pumps and buffering - scientists have been able to measure and estimate the amount of carbon corresponding to human emissions, at 180 GtC in the mid 1990’s according to David Archer- in The Global Carbon cycle. So this is a third of the total human emission.

So what do this imply for the future behavior of the carbon cycle? In short David states that that more and accelerating CO 2CO_2 in the air will make the ocean buffering weaker.

Modelling the ocean

I decided to follow the 3-box model presented in Marinov and Sarmiento 2004 as it also models the Thermo-Haline Circulation (see the references). This was also proposed by Oeshger writes Jorge Sarmiento and J Toggweiler in their original paper from 1984, and Knox and McElroy. One can model the ocean with 2 boxes only with one box for the ocean top and one for the ocean bottom, but it has limitations according to Marinov and Sarmiento:

The limitation of the two box model is related to the fact that most water that participates in the formation of deep water originates in small surface regions of the high latitudes and therefore has properties different from the global surface mean.

The equation numbers are from the original 1984 research paper.

2-box model

We look at this as it explains how the ocean pumps work. So when air meets the ocean it is dissolved into water and also reacts by forming bicarbonate and carbonate ions:

DIC=|CO 2|+|HCO 3 |+|CO 3 2|(1) DIC = |CO_2| + |HCO_3^-| + |CO_3^2-| (1)

The alkalinity is approximated by Sarmiento and Marinov as:

ALK=|HCO 3 |+|2CO 3 2|+|B(OH 4 )|+|OH ||H +|+others(2) ALK = |HCO_3^-| + |2 CO_3^2-| + |B(OH_4^-)| + |OH^-| - |H^+| + others (2)

Photosynthesis uses DIC and PO 4PO_4 at the surface and then it sinks to the bottom as DIC or particles to the bottom and any organic matter that reaches the bottom is transformed back to DIC by re-mineralization. So this is the soft tissue part of the biological pump, that gives a net transport of DIC to the ocean bottom.

The other important part of the biological pump is the formation and dissolution of calcite and aragonite done by single cell algae and marine plankton:

Ca 2++CO 3 2=Ca 2CO 3(3) Ca^{2+} + CO_3^{2-} = Ca_2CO_3 (3)

Then I can solve the mass balance equations for PO 4PO_4 and DICDIC and solve them.

3-box model

Simple 3-box model

Let me first use the same approach as in the first experiments with carbon box model and use current and preindustrial values and fluxes and use the slides that follows Daniel Jacob’s book on geochemistry.

More advanced box models

The way that the THC is modeled is by a constant circular flux between the three boxes and that got me interested to see if I can use the THC as a parameter in the modeling. Here is an image from the article by Archer and Broecker, of that model:

ocean 3-box model

In order to make the 3-box model we divide the sea surface box in two in order to model the solubility pump. The flows are given in Sverdrup (1 Sv is 10 6m 3s 110^6m^3 s^-1). So the circular flux of 19 Sv represents the thermohaline circulation in this model. Archer and Broecker claims that this simple box models are more latitude-sensitive than several other General circulation models. So let me use their parameter values as reference in my modelling. They use a model that have one box for the high latitudes - 2.5 degrees Celsius for both poles and low latitudes - 15 degrees Celsius which have depths of 100-250 m.

First in order to get started and use Sage I’ll model the ocean as a 3-box and the atmosphere as one additional box and I will use pre-industrial values for flows and box mass.

The idea is to code this in Sage and I am in the process of doing that now and try to make an interactive version that allows you to control the number of boxes and their respective quantative values, e.g. the carbon masses or concentrations and flows in the model. I calculated the box volumes and the low-latidude box is 2.97x10 16m 32.97x10^16 m^3, the high-latitude box volume is 1.31x10 16m 31.31x10^16 m^3 and the deep box volume is 1.25x10 18m 31.25x10^18 m^3.


# Initialize the ocean box model

ocean_volume = 1.292e18 # m^3
ocean_area = 349e12 # m^2
atm_volume = 1.773e20 # moles
depth_low = 100. # m
depth_high = 250. # m
part_high = 0.15
low_volume = ocean_area*(1-part_high)*depth_low
high_volume = ocean_area * part_high *depth_high
low_temp = 21.5 # C
high_temp = 2. # C
ocean_salinity = 34.7 # ppm remember 
ratio_rcorgp = 130 # redfield ratio
ratio_fca = 0.20
ratio_rcp = ratio_rcorgp / (1.-ratio_fca)
ratio_alk_p = 2*ratio_fca*ratio_rcp - 15.
ratio_2_p = 169.
delta13_c = -0.23

# flows assumed fixed
pvl,pvh = 3., 3. # m/day
p_conveyour = 20 # Sv = 10^ 6 m^3 / s

# variables
# The  formation  of bottom  water  around  Antarctica
f_hd = randrange(3,300,1) # 3-300 Sv

f_h = 0.075 + random()*7.5 # fixed. .075-7.5 sinking flux moles C m^-2year^-1


Data sources

Abstract: Net oceanic uptake of the greenhouse gas carbon dioxide (CO2) reduces global warming but also leads to ocean acidification [Intergovernmental Panel on Climate Change (IPCC), 2007]. Understanding and predicting changes in the ocean carbon sink are critical to assessments of future climate change.

Surface water CO2 measurements suggest large year-to-year variations in oceanic CO2 uptake for several regions [Doney et al., 2009]. However, there is much debate on whether these changes are cyclical or indicative of long-term trends. Sustained, globally coordinated observations of the surface ocean carbon cycle and systematic handling of such data are essential for assessing variation and trends in regional and global ocean carbon uptake, information necessary for accurate estimates of global and national carbon budgets.

Slides from courses

Open access, Creative commons (CC)

Free books to download for students, but not CC

Not open

Compares several box models which includes high latitude sensitivity with ocean GCM.